Precision Measurement and

Fundamental Constants II

Proceedings of the Second International Conference

held at the National Bureau of Standards

Gaithersburg, MD, June 8-12, 1981

Edited by:

B.N. Taylor and W.D. Phillips

Center for Basic Standards

National Measurement Laboratory

National Bureau of Standards

Gaithersburg, MD 20899

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued August 1 984
Library of Congress Catalog Card Number: 84-601083

National Bureau of Standards Special Publication 617

Natl. Bur. Stand. (U.S.), Spec. Publ. 617, 646 pages (Aug. 1984)

CODEN: XNBSAV

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1984

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402
Abstract

This volume presents the Proceedings of the Second International Conference on Precision Measurement and Fundamental Constants, held at the National Bureau of Standards in Gaithersburg,

MD, from June 8-12, 1981. Like its 1970 predecessor, the Conference provided an international

forum for theoretical, experimental, and applied scientists actively engaged in research on precision

measurements relating to the fundamental physical constants, and on the testing of related fundamental theory. More specifically, the purpose of the Conference was to assess the present state of the precision measurement-fundamental constants field, to examine basic limitations, and to explore the prospects for future significant advances. The principal subjects covered were: fre- quency, wavelength, spectroscopy, quantum electrodynamics, the gas constant, x-ray interferometry, nuclidic masses, uncertainty assignment, gravitational acceleration, mass, electrical quantities,

gravity, and relativity. These proceedings contain the vast majority of both the invited review

papers and the contributed current research papers presented at the Conference. The new results

reported at the Conference were considered for inclusion in the 1983 least-squares adjustment of

the constants carried out under the auspices of the Task Group on Fundamental Constants of the

Committee on Data for Science and Technology (CODATA).

Key words: data analysis; experimental tests of theory; fundamental constants; least-squares adjustments; precision measurements.

iii

Preface

Like its 1970 predecessor, the Second International Conference on Precision Measurement and

Fundamental Constants (PMFC-II), held at the National Bureau of Standards, Gaithersburg, MD,

June 8-12, 1981, provided an international forum for scientists actively engaged in experimental

and theoretical research on precision measurements relating to the fundamental physical constants,

and on the testing of related fundamental theory. More specifically, the Conference assessed the

present state of the precision measurement-fundamental constants (PMFC) field, examined its current limitations, and explored the prospects for future significant advances.

The Conference was unique in that it brought together to an even greater extent than did PMFCI scientists from fields as apparently diverse as precision mass metrology and quantum field theory.

But this of course was the main reason for holding the Conference; that is, its principal purpose was to foster the interchange of information and ideas among researchers who at first glance have

Httle in common but who upon closer scrutiny have much to gain from each other because of their

mutual interest in and need for precision measurements and accurate values of the fundamental

constants. The reasons for this interest and need were well described in the Preface to the pro- ceedings of PMFC-I by D. N. Langenberg and B. N. Taylor:

Precision measurement plays an essential and crucial role in the development of all science and tech- nology. Scientific theories cannot be tested, instruments and machines cannot be designed and built, and even the daily routine of the market place cannot proceed without precise quantitative measurement. The demand for precision and accuracy is greatest at the frontiers of science and technology. Here our increasing understanding of the fundamental nature of the universe and our increasing ability to put our knowledge to practical use depends on continuing refinement of our theories and of the tech- nology used in testing these theories against reality. In all of our fundamental theories there appear a certain few parameters which characterize the fun- damental particles and interactions we find in nature. These are the fundamental physical constants. The precision determination of the numerical values of these constants has long been and remains one

of the principal objectives of experimental science. This is not because there is any intrinsic virtue in accumulating lists of ever longer numbers, but because the fundamental constants are the quantitative

links between our most basic theories and the physical reality we wish them to describe. Our theories must stand or fall according to their ability to make quantitative predictions which agi-ee with experi- mental observations to the maximum accuracy possible. The progress of our understanding of the physical world is therefore very much intertwined with the advance of the art of precision measurement and

its application to the determination of the fundamental physical constants. To this we must add the idea that because of the interrelationships which exist among the fundamental constants and related quantities, for example, the proton gyromagnetic ratio, the Faraday, Avogadro, and fine-structure constants, 2e//? via the Josephson effect, the absolute ohm and

ampere, the quantized Hall resistance, the electron and muon anomalous magnetic moments, the

proton to electron mass ratio, etc., it is appropriate, indeed even necessary, for the solid state

theorist to be aware of the work of the electrical metrologist; the atomic mass spectroscopist of

the work of the precision electrochemist; and the quantum electrodynamics theorist of the work of

the experimental semiconductor physicist. The fundamental constants of nature and closely related

precision measurements are truly the common meeting ground for many of the disciplines of sci- ence and technology.

The technical program of the Conference was designed to give participants both a broad and in- depth view of the PMFC field as well as many opportunities to interact with one another. The

Keynote Address by Ian W. Roxburgh served to stimulate thinking about the fundamental constants and their relationship to the laws of nature; the invited review talks provided a broad over- view of the PMFC field including its current limitations and future possibilities; the orally

presented contributed papers gave a more detailed sampling of the current research underway in the field; the five poster sessions (one each day with about 20 papers per session) allowed every- one to hold in-depth discussions of their current research work with interested colleagues; and the

informal, off the record, evening session enabled those who were sufficiently brave to present

their most highly speculative or "far out" ideas. Each day of the Conference was devoted to a small number of broad topical areas and each of

the papers presented that day—invited review, oral contributed, and poster—more or less dealt

with the selected topics. This arrangement has been followed in this volume except for slight

modifications in order to improve overall coherence. Thus, included in these Proceedings are the

written versions of the Keynote Address, the 21 invited review papers (one in abstract form only),

the 16 oral contributed papers (one in abstract form only); and 93 of the 100 poster papers (one in abstract form only). Nothing has been included from the evening session because it was truly "off

the record."

V
A quick perusal of the contents of these Proceedings would convince even the most casual observer

that the precision measurement-fundamental constants field is full of vitality. Enormous advances

have been made over the last decade or so since PMFC-I. The uncertainty in c, the speed of light,

is now limited by the present *^^Kr definition of the meter; a definition of length based on c is destined

to replace the krypton definition within the next several years. The Rydberg constant is now

known to one part in 10^, the Avogadro constant to one part in 10^, the fine structure constant to one part in 10"^, the Faraday constant to better than two parts in 10^, the proton-to-electron mass

ratio and proton gyromagnetic ratio to two parts in 10'^, the anomalous moment of the electron to 3 parts in 10^, and the Newtonian gravitational constant to 6 parts in 10^. On the theoretical side,

significant advances have been made in the theory of the anomalous moment of the electron and

muon, Lamb-shift, and muonium hyperfine splitting. Moreover, completely new techniques and

phenomena which promise to have a significant impact on the future of the PMFC field have been

developed and identified. Especially noteworthy in this regard is the methodology for trapping a

single, charged particle pioneered by H. G. Dehmelt and colleagues, and the discovery of the

quantized Hall resistance by K. v. Klitzing. This latter phenomenon, which may eventually yield

the most accurate value of the fine-structure constant, has already attracted considerable attention

since it was first reported in June 1980 as evidenced by the eight papers on the subject presented

at the Conference. It could have the same impact on the PMFC field in the 1980's as did the

Josephson effect in the 1960's. The one to two orders of magnitude reduction in the uncertainties of many of the constants which

has taken place since PMFC-I continues the trend of the last 20 to 30 years and again raises the

question as to whether it can continue indefinitely. Will improvements in and the appHcation of

existing technologies such as high-speed computers, lasers, and cryogenics, and the discovery of new techniques and phenomena, allow the proceedings editors of a hypothetical PMFC-III to make

the same comment a decade or so hence? We are incHned to beUeve that those editors will be able

to do so, especially if the enthusiasm of the Conference participants was any indication of what hes

ahead. In this regard we were especially pleased to see so many in attendance from outside of the

United States (103 out of 257 or 40%) and to see the large number of graduate students who are being trained to become the next generation of "precision measurers."

We would like to thank personally the many organizations and individuals who helped to make

PMFC-II the highly worthwhile Conference it turned out to be. These include its sponsors, contributors, and the members of the organizing committees as listed on the following pages, as well

as many members of the staff of the National Bureau of Standards, especially Joanne Lorden,

Greta Pignone, and Kathy Stang.

B. N. Taylor

W. D. Phillips

vi
Conference Organizing Committees

Conference Committee

K. M. Baird (National Research Council, Canada)

E. R. Cohen (Science Center, Rockwell International, U.S.A.)

R. D. Deslattes (National Bureau of Standards, U.S.A.)

J. L. Hall (Joint Institute for Laboratory Astrophysics, U.S.A.)

K. G. Kessler (National Bureau of Standards, U.S.A.)

D. Kleppner (Massachusetts Institute of Technology, U.S.A.)

D. R. Lide (National Bureau of Standards, U.S.A.)

R. C. Ritter (University of Virginia, U.S.A.)

W. G. Schweitzer (National Bureau of Standards, U.S.A.)

B. N. Taylor (Chairman; National Bureau of Standards, U.S.A.)

International Advisory Committee

S. J. Brodsky (Stanford Linear Accelerator Center, U.S.A.) H. G. Dehmelt (University of Washington, U.S.A.)

V. I. Goldansky (Institute for Chemical Physics, U.S.S.R.)

T. W. Hansch (Stanford University, U.S.A.)

H. W. Hellwig (Frequency and Time Systems, Inc., U.S.A.)

T. Kinoshita (Cornell University, U.S.A.)

B. Kramer (Physikahsch-Technische Bundesanstalt, F.R.G.) M. Morimura (National Research Laboratory of Metrology, Japan)

B. W. Petley (National Physical Laboratory, U.K.)

T. G. Quinn (Bureau International des Poids et Mesures, France) N. F. Ramsey (Harvard University, U.S.A.)

A. Rich (University of Michigan, U.S.A.)

Honorary Committee

E. Ambler (National Bureau of Standards, U.S.A.)

W. R. Blevin (CSIRO Division of Applied Physics, Australia)

L. M. Branscomb (International Business Machines Corporation,

U.S.A.)

P. Dean (National Physical Laboratory, U.K.)

A. Ferro-Milone (Institute Elettrotechnico Nazionale "Galileo Ferraris," Italy) P. Giacomo (Bureau International des Poids et Mesures, France)

Kanjie Ju (National Institute of Metrology, P.R.C.) M. Kawata (National Institute of Metrology, Japan)

D. Kind (Physikahsch-Technische Bundesanstalt, F.R.G.)

H. Preston-Thomas (National Research Council, Canada)

Y. V. Tarbeyev (Mendelelyev Research Institute of Metrology,

U.S.S.R.)

J. Terrien (Bureau International des Poids et Mesures, France)

vii
Sponsors

International Union of Pure and Applied Physics

Committee on Data for Science and

Technology of the International

Council of Scientific Unions

U.S. National Academy of Sciences

National Research Council, Committee

on Fundamental Constants

Bureau International des

Poids et Mesures

National Bureau of Standards

U.S. Department of Commerce

Contributors

The Second International Conference on Precision Measurement and Fundamental Constants expresses its deep appreciation to the fol- lowing organizations for their generous financial support which contributed greatly to making this meeting possible.

International Union of Pure and Applied Physics

Leeds & Northrup Company

Matheson Gas Products

Mettler Instrument Corporation

The 3M Company

Mobil Research and Development Corporation

Monsanto Company

Motorola Inc. National Bureau of Standards

National Conference of Standards Laboratories

National Science Foundation

Rockwell International Sandia Laboratories

Shell Development Company

Tektronix, Inc. Texas Instruments, Inc. U.S. Air Force Geophysics Laboratory

U.S. Army Research Office U.S. Office of Naval Research

Westinghouse Electric Corporation

Xerox Corporation

Airco, Inc. Avco Corporation

Ball Technical Products Group

Beckman Instruments, Inc. Bendix Automation & Measurement Division COMSAT Laboratories Corning Glass Works

E.I. duPont de Nemours & Company, Inc. EG&G Princeton Applied Research Corporation

Electro Scientific Industries, Inc. Exxon Research and Engineering Company

Fisher Scientific Company

John Fluke Mfg. Co., Inc. The Foxboro Company

Frequency and Time Systems, Inc. General Atomic Company

General Electric Company

Guildline Instruments Inc. Gulf Research & Development Company

Hewlett-Packard Company

IEEE Instrumentation and Measurement Society

International Business Machines Corporation

viii
ix

Contents

Page

Preface v

Conference Organizing Committees vii

Conference Sponsors and Contributors viii

Frontispiece ix

Keynote Address: The Laws and Constants of Nature 1

I. W. Roxburgh

Frequency, Wavelength, and Stored Ions

Frequency Standards Based on Magnetic Hyperfine Structure Resonances (Review) 11 H. Hellwig

Performance of the Three NRC 1-Meter CsVI Primary Clocks 21 A. G. Mungall, H. Daams, and J.-S. Boulanger

Prospects for Cesium Primary Standards at the National Bureau of Standards 25

L. L. Lewis, F. L. Walls, and D. A. Howe

A Cesium Beam Atomic Clock with Laser Optical Pumping, as a Potential

Frequency Standard 29

M. Arditi

Frequency Measurement of Optical Radiation (Review) 35

K. M. Baird

Optical Frequency Standards: Progress and Applications (Review; abstract only) 43

J. L. Hall

Measurement of Frequency Differences of Up to 170 GHz Between Visible

Laser Lines Using Metal- Insulator-Metal Point Contact Diodes 45

H.-U. Daniel, M. Steiner, and H. Walther

Precision Frequency Metrology for Lasers in the Visible and Apphcation to Atomic

Hydrogen , 49

B. Burghardt, H. Hoeffgen, G. Meisel, W. Reinert, and B. Vowinkel

System for Light Velocity Measurement at NRLM 53

K. Tanaka, T. Sakurai, N. Ito, T. Kurosawa, A. Morinaga, and S. Iwasaki

'Laser Wavelength Measurements and Standards for the Determination of Length (Review) ... 57

W. R. C. Rowley

Double-Mode Method of Sub-Doppler Spectroscopy and Its Apphcation in Laser

Frequency Stabilization 65 N. G. Basov, M. A. Gubin, V. V. Nikitin, A. V. Nikulchin, D. A. Tyruikov, V. N. Petrovskiy, and E. D. Protscenko

He-Ne C'-n.,) Lasers at 0.633 ixm (and at 0.604 (xm) 69

F. Bertinetto, B. I. RebagHa, P. Cordiale, S. Fontana, and G. B. Picotto

Recent Work on 612 nm He-Ne Stabilized Lasers 73

A. Brillet, P. Cerez, and C. N. Man-Pichot

Iodine and Methane Stabilized He-Ne Lasers as Wavelength Standards 77

Shen Nai-cheng, Wu Yao-xiang, Sun Yi-min, Li Cheng-yang,

Zhang Xue-bin, and Wang Chu

xi
Page

Spatial Coherence and Optical Wavelength Metrology 81

P. Bouchareine

Spectroscopy of Stored Ions (Review) 83

D. J. Wineland

Progress Toward a Stored Ion Frequency Standard at the National Bureau of Standards 93

W. M. Itano, D. J. Wineland, J. C. Bergquist, and F. L. Walls

High Resolution Microwave Spectroscopy on Trapped Ba^ Ions 99

W. Becker, R. Blatt, and G. Werth

Observation of High Order Side Bands in the Spectrum of Stored ^He^ Ions 103 H. A. Schuessler and H. S. Lakkaraju

Quantum Limits in the Measurements of e.m. Fields and Frequency (Abstract only) 109 V. B. Braginsky

Spectroscopy, Quantum Electrodynamics, and Elementary Particles

Precision Laser Spectroscopy (Review) Ill

T. W. Hansch

Atomic Beam, Linear, Single-Photon Measurement of the Rydberg Constant 117

S. R. Amin, C. D. Caldwell, and W. Lichten

Current Work on Two Photon Excitation in a Hydrogen Beam for the

Measurement of the Rydberg Constant and mjmp 123 D. Shiner and C. Wieman

Measurement of the 2^P3/2- 2'^Sy2 Fine-Structure Interval in Atomic Hydrogen 127

K. A. Safinya, K. K. Chan, S. R. Lundeen, and F. M. Pipkin

Measurement of the Lamb Shift in Hydrogen, n = 2 131

S. R. Lundeen and F. M. Pipkin

Atomic Interferometer Method Measurement of the Lamb Shift in Hydrogen (n = 2) 135 Y. L. Sokolov

Measurement of the 4'^Sy2 - 4'^'Pv2 Lamb Shift in He+ 141

J. J. Bollinger, S. R. Lundeen, and F. M. Pipkin

Lamb Shift in the Hydrogenic Ion CU"^ 145

E. T. Nelson, 0. R. Wood II, C. K. N. Patel, M. Leventhal,

D. E. Murnick, H. W. Kugel, and Y. Niv

Helium Fine Structure and the Fine Structure Constant 149

W. Frieze, E. A. Hinds, A. Kponou, V. W. Hughes, and F. M. J. Pichanick

Preliminary Measurement of the J = 0 to / = 2 Fine Structure Interval

in the 3 ^P State of Helium 153

M. Feldman, T. Breeden, L. DiMauro, T. Dong, and H. Metcalf

Laser Microwave Precision Measurements of 2 -^Si and 2 '^P Term Splittings

in Helium-Like Li^ 159 U. Kotz, J. Kowalski, R. Neumann, S. Noehte, H. Suhr,

K. Winkler, and G. zu Putlitz An Optically Pumped Metastable Hydrogen Beam 163

K. C. Harvey

Precise Determination of the S and P Quantum Defects in Sodium and Cesium

by Millimeter and Submillimeter Spectroscopy Between Rydberg States 169

P. Goy, J. M. Raimond, G. Vitrant, C. Fabre, S. Haroche, and M. Gross

A New Method of Measuring the Fine-Structure Constant Using Stark Spectroscopy 173

M. G. Liftman and W. D. Phillips

Time Resolved Sub-Natural Width Spectroscopy 177

W. D. Phillips and H. J. Metcalf

xii
Page

The Spectroscopy of Atoms and Molecules in Gases: Corrections to the

Doppler-Recoil Shift 181

M. P. Haugan and F. V. Kowalski

The Implications of QED Theory for the Fundamental Constants (Review) 185

G. P. Lepage and D. R. Yennie

Uncertainties in QED Fine Structure Calculations 195

G. W. Erickson

Sixth Order Contributions to g-2 of the Electron 201

M. J. Levine and R. Z. Roskies

Calculation of the Eighth Order Anomalous Magnetic Moment of the Electron 203

T. Kinoshita and W. B. Lindquist

Experimental Determinations of the Anomalous Magnetic Moments of the Free

Leptons (Review) 207

R. Conti, D. Newman, A. Rich, and E. Sweetman

Preliminary Comparison of the Positron and Electron Spin Anomalies 215

P. B. Schwinberg, R. S. Van Dyck, Jr., and H. G. Dehmelt

Geonium Without a Magnetic Bottle—A New Generation 219

G. Gabrielse and H. Dehmelt

The Magnetic Moment of Positive Muons in Units of the Proton Magnetic Moment 223

E. Klempt, R. Schulze, H. Wolf, M. Camani, F. N. Gygax, W. Riiegg, A. Schenck, and H. Schilling

Measured Gy-Factor Ratio of "He^d ^Sy2) and ^He(2 ^Si) 229

H. G. Robinson and C. E. Johnson

Determination of the Neutron Magnetic Moment 233

G. L. Greene, N. F. Ramsey, W. Mampe, J. M. Pendlebury, K. Smith,

W. B. Dress, P. D. Miller, and P. Perrin

Precision Exotic Atom Spectroscopy (Review) 237

V. W. Hughes

Fundamental Tests and Measures of the Structure of Matter at Short Distances (Review) 249

S. J. Brodsky

Experimental Limit for the Charge of the Free Neutron 257

R. Gahler, J. Kalus, and W. Mampe

Gas Constant, X-Ray Interferometry, Nuclidic Masses, Other Constants,

and Uncertainty Assignment

Methods for the Determination of the Gas Constant (Review) 263

A. R. Colclough

An Ultrasonic Determination of the Gas Constant 277

W. C. Sauder

Spherical Acoustic Resonators: Promising Tools for Thermometry and

Measurement of the Gas Constant 281

M. R. Moldover and J. B. Mehl

Proposed Method for the Determination of the Molar Gas Constant, R 287

L. A. Guildner and M. L. Reilly

A Radiometric Determination of the Stefan-Boltzmann Constant 291

T. J. Quinn and J. E. Martin

On the Radiometric Measurement of the Stefan-Boltzmann Constant at NRLM 299

A. Ono

Applications of X-ray Interferometry (Review) 303

R. D. Deslattes

Work Related to the Determination of the Avogadro Constant in the PTB 313

P. Seyfried

xiii
Page

Absolute Determination of the (220)-Lattice Spacing in Silicon 317

P. Becker and H. Siegert

Systematic Uncertainties in the Determination of the Lattice Spacing d(220) in Silicon 321 H. Siegert and P. Becker

High Precision Studies of Pionic X Rays: Some Past Results and Future Prospects 325

G. Dugan, L. Delker, C. S. Wu, and D. C. Lu

Isotope Shifts of K X-Rays of Lead 331

G. L. Borchert, 0. W. B. Schult, J. Speth, P. G. Hansen, B. Jonson, H. Ravn, and J. B. McGrory

The Measurement of Atomic Masses by Mass Spectroscopic Methods and a Role for Atomic Masses in the Determination of the Fundamental

Constants (Review) 335

W. H. Johnson

Absolute Determination of the Threshold Energies of "Li(p,n), ^"B(p,n), and '^N(p,n) 345

P. H. Barker, M. J. Lovelock, H. Naylor, R. M. Smythe, and R. E. White

Preliminary Proton/Electron Mass Ratio Using a Precision Mass-Ratio Spectrometer 349

R. S. Van Dyck, Jr. and P. B. Schwinberg

A Direct Determination of the Proton-Electron Mass Ratio 353

G. Graff, H. Kalinowsky, and J. Traut

A New Determination of the Atomic Weight of Silver and an Improved Value

for the Faraday 357

L. J. Powelh T. J. Murphy, and J. W. Gramhch

High Resolution Magnetic Measurements on Rotating Superconductors to Determine hhyi^ 359

B. Cabrera, S. B. Felch, and J. T. Anderson

High Precision Measurement of the Electron Compton Wavelength ihlm^.) Using

Cryogenic Metrological Techniques 365

j. C. Gallop, B. W. Petley, and W. J. Radcliffe

Preliminary Determination of 369

E. Kriiger, W. Nistler, and W. Weirauch

The Assignment of Uncertainties to the Results of Experimental Measurements (Review) 375

J. W. Muller

On the Statement of a Total Confidence Interval Based on the Concept of ^ Randomization of Systematic Errors: Large and Small Sample Sizes 383

W. Woger

Measurement Assurance 385

A. F. Dunn

An Extended-Least -Squares Treatment of Discrepant Data 391 E. R. Cohen

Gravitational Acceleration, Mass, and Electrical Quantities

Present Status of the Absolute Measurement of Gravitational Acceleration (Review) 397

A. Sakuma

A New, Portable, Absolute Gravimeter 405

M. A. Zumberge, J. E. Faller, and R. L. Rinker

"Super Spring"—A Long Period Vibration Isolator 411

R. L. Rinker and J. E. Faller

Transportable Gravimeter for the Absolute Determination of Gravity 419

Guo You-guang, Huang Da-lun, Li De-xi, Zhang Guang-yuan, Gao Jing-long,

Fang Yong-yuan, and Huang Cheng-qing

New Techniques for Absolute Gravity Measurement 423

J.A.Hammond, R. L. liiff, and R. W. Sands

xiv
Page

The Mass Unit "Kilogram," Precision Measurement of Mass, Attainable

Uncertainties, and Possibilities of a New Definition (Review) 427

M. Kochsiek . Measurement of Air Density for High Accuracy Mass Determination 437

D. B. Prowse

On a More Precise Correction for Buoyancy and Gas Adsorption in Mass Measurement 441

Y. Kobayashi

Density Standards—The Density and Thermal Dilatation of Water 445

G. A. Bell and J. B. Patterson

Precision Measurements on Solid Artifacts for a Redetermination of the Density of Water — 449

A. Peuto, A. Sacconi, R. Panciera, W. Pasin, and M. Rasetti

The Influence of Dissolved Air on the Density of Water 453

G. Girard and M.-J. Coarasa

The Helium Melting Curve and the Linkage of Fundamental Constants,

Pressure, Density, and Mass , 457

C. T. Van Degi'ift

Reahzation of the Electrical SI Units (Review) 461

B. P. Kibble

An Absolute Determination of the Volt at LCIE 465

N. Elnekave and A. Fau

The CSIRO Absolute Volt Project 469

G. J. Sloggett, W. K. Clothier, D. J. Benjamin, M. F. Currey, and H. Bairnsfather

Status of the Measurement of the NBS Ampere in SI Units 475

P. T. Olsen, W. D. Phillips, and E. R. Williams

A Feasibihty Study of an Absolute Determination of the Magnetic Flux Quantum 479

K. Hara, F. Shiota, and T. Kubota

The Work Done at the Mendeleyev Research Institute of Metrology (VNIIM)

To Improve the Values of the Fundamental Constants 483

Y. V. Tarbeyev

Reahzation of a Josephson Potentiometer 489

M. Koyanagi, T. Endo, and A. Nakamura

A Transportable Josephson Voltage Standard 493

K. Lahdenpera, H. Seppa, and P. Wallin

The Proton Gyromagnetic Ratio in HoO—A Problem in Dimensional Metrology (Review) 497

E. R. Williams, P. T. Olsen, and W. D. Phillips

The Development of Precision Measurement and Fundamental Constants in China 505

Wang Zhu-xi

The 7p-Experiment at PTB 509

K. Weyand

A New Method for the Determination of the Proton Gyromagnetic Ratio 515

G. L. Greene

The Quantized Hall Resistance in Two-Dimensional Systems (Review) 519

K. von Klitzing, H. Obloh, G. Ebert, J. Knecht, and K. Ploog

Hall Effect in Silicon MOS Inversion Layers for hle^ Determination 529

C. Yamanouchi, K. Yoshihiro, J. Kinoshita, K. Inagaki, J. Moriyama,

S. Baba, S. Kawaji, K. Murakami, T. Igarashi, T. Endo, M. Koyanagi,

and A. Nakamura

Cryogenic Method for the Determination of the Fine-Structure Constant

by the Quantized Hall Resistance 535

E. Braun, P. Gutmann, G. Hein, F. Melchert, P. Warnecke, S. Q. Zue, and K. v. Klitzing

XV
Page

Status of the NBS-NRL Determination of the Fine-Structure Constant

Using the Quantized Hall Resistance Effect 539

M. E. Cage, R. F. Dziuba, B. F. Field, C. F. Lavine, and R..J. Wagner

Use of a Cryogenic Current Comparator to Determine the Quantized Hall

Resistance in a Silicon MOSFET 543

A. Hartland

A Resistance Standard Using the Quantized Hall Resistance of GaAs-Al,Gai„ ,.As Heterostructures 549

A. C. Gossard and D. C. Tsui

A Quantitative Theory for the Determination oihie- from the Hall Effect in Two-Dimensional Conductors 553

L. BHek and G. Hein

Quantum Hall Effect: Role of Inversion Layer Geometry and Random Impurity Potential 557

R. W. Rendell and S. M. Girvin

Gravity and Relativity

Experiments Relating to the Newtonian Gravitational Constant (Review) 561 H. de Boer

Redetermination of the Newtonian Gravitational Constant "G" 573

G. G. Luther and W. R. Towler

The Design of a Beam Balance for a Determination of G 577

C. C. Speake and A. J. F. Metherell

Optimizing the Shape of the Attracting Mass in Precision Measurements of G 581 A. J. F. Metherell, C. C. Speake, Y. T. Chen, and J. E. Faller Vacuum Polarization and Recent Measurements of the Gravitational

Constant as a Function of Mass Separation 587

D. R. Long

Tests of the Gravitational Inverse Square Law Using Torsion Balances 591

J. K. Hoskins, R. Newman, J. Schultz, and R. Spero

Measurement of Gravitational Forces at Separations Around 10 Meters 595

H. Yu, W. Ni, C. Hu, F. Liu, C. Yang, and W. Liu

Non-Newtonian Gravity: Geophysical Evidence 597

F. D. Stacey and G. J. Tuck

Experimental Test of a Spatial Variation of the Newtonian Gravitational

Constant at Large Distances 601 H. A. Chan and H. J. Paik

The Measurement of G for Small Inter-Mass Spacings 607

W. C. Oelfke

Tests of Gravitation and Relativity (Review) 611

R. F. C. Vessot

Is the Gravitational Constant Changing? 625

T. C. Van Flandern

Experiments on Variation of the Gravitational Constant Using Precision Rotations 629

G. T. Gillies and R. C. Ritter

Interpreting Dirac's Large Numbers Hypothesis 635

W. Davidson

New Laboratory Test of the Equivalence Principle 639

P. T. Keyser, J. E. Faller, and K. H. McLagan

What Test Masses Are Best for an Eotvos Experiment? 643

D. F. Bartlett, J. Shepard, and C. D. Zafiratos

xvi
Page

Equivalence Principles and Precision Measurements 647

W. Ni

An Inertial Clock to Test the Non-Metricity of Gravity 653

W. S. Cheung and R. C. Ritter

Assessment of the Prospects for a Measurement of Relativistic Frame Dragging by 1990 659

R. A. Van Patten The Status of the Velocity of Light in Special Relativity 667

E. Breitenberger

Relativistic Time Dilation: A Latter-Day Ives-Stillwell Experiment 671

P. Nachman, M. D. Rayman, and J. L. Hall

An Experiment to Measure Relative Variations in the One-Way Velocity of Light 675

D. G. Torr and P. Kolen

High-Energy Gamma Rays Might Be Faster than Visible Light 681

K. Fujiwara

Fiber Optic Ring as a Gravitational Wave Detector 685

C. L. Mehta, D. Ranganathan, and G. Bose

A Possible Laser Gravitational Wave Experiment in Space (Abstract only) 689

J. E. Faller and P. L. Bender

Author Index 691

Conference Registrants 695

Papers in this volume, except those by National Bureau of Standards authors, have not been

significantly altered by the National Bureau of Standards. Opinions expressed in non-NBS papers

are those of the authors, and not necessarily those of the National Bureau of Standards. Non-NBS

authors are solely responsible for the content and quality of their submissions.

The mention of trade names in the volume is in no sense an endorsement or recommendation by

the National Bureau of Standards.

xvii

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

KEYNOTE ADDRESS

The Laws and Constants of Nature

Ian W. Roxburgh

Department of Applied Mathematics, Queen Mary College, University of London

The paper concentrates on a few problems; geometry, mechanics, gravitation, and the large

numbers {10'^^) that relate microphysics to the large scale stracture of the universe. My purpose is not

so much to describe what is known, but to question how well anything is known and to provoke the reader into asking questions and proposing experiments that probe the foundations of our understanding. Is geometry locally Euclidean? Do different clocks keep the same time? Does our existence depend

on the exact form of the laws of nature and the exact values of the constants of nature? Do the con- stants of nature vary in time? Why are there laws at all? Readers can, and I hope will, add questions of their own. Key words: anthropic principle; constants of nature; fundamental natural laws; gra\itation; large

number hypothesis; time variation of constants.

1. Introduction

Science as we know it probably began with man's at- tempts to understand the regularity of nature as seen in the rising and setting of the sun, the phases of the moon,

the annual recurrence of the seasons, and the motion of the planets across the sky. Indeed, the Shu Ching, the

ancient Chinese Book of History, opens in the Cannon of Yao with the sovereign giving instructions [1]:

"He commanded the brothers Hsi and the brothers Ho, in reverent accord with their observations of the wide heavens, to calculate and delineate the move- ments of the sun, the moon, the stars and the zodiacal spaces and so to deliver respectfully the seasons

to be observed by the people."

Science in those days was a dangerous activity; we later learn that the astronomers Hsi and Ho were executed for

failing to predict a solar ecUpse. If the supporting details are correct (the moon was in Sieou Fang in the center of Tsing Lung on the first day of autumn) this was in the

year 2137 B.C.

Whilst we do not know how the astronomers could have

predicted eclipses at that time, we learn from the 'Chou

pi' of the 'Kai-T'ien' model of the motion of the sun [2]. The 'Chou pi' probably dates from about 1100 B.C., but

it is possible that the cosmological model was a later addition either indigenous to China or imported from India

where a very similar model is described in the

Suryapi-ajnapti (circa 500 B.C.) [3]. The model has the sun describing a set of seven circles about the celestial pole at a height of 80,000 li above a flat earth; the innermost circle has a diameter of 238,000 li, the outermost

circle a diameter of 476,000 li. The sun moves on the

inner circle at summer solstice and moves from circle to

circle during the course of six months until at winter solstice it is on the outermost circle. The sun's rays only

extend a distance of 167,000 li so explaining night and

day.

This model is quite good. Indeed, one can compute that

at a latitude of 36° the model correctly predicts the

elevation angles of the sun at noon on both midsummer's

and midwinter's day. But is it Science? Indeed, what is Science?

The ancient Chinese were clearly aware of the regularity of the motion of the sun across the sky and of the sea- sons. They constructed a model based on these observations, the model explained past observations (at least to some level of accuracy) and successfully predicted future

observations. What more do we require before giving

such a model the accolade of being scientific? Is not this model as scientific as Kepler's Laws?

The ancient Chinese scientists had made that first step

in science of beheving that the future could be predicted on the basis of past experience. This is the act of faith we

make in science although as pointed out by Hume, it is an

act of faith and we cannot on the basis of past experience

prove that our predictions will be true in the future [4]. The Chinese model was quantitative and not just qualitative. It was narrow, applying only to one set of phenomena, so it lacked the generality we associate \nth

scientific reasoning. But I nevertheless think we should

describe these Chinese atronomers as early scientists.

2. Geometry as an Example of Science

A better example of early science is geometry. The ori- gins of geometrical knowledge also go veiy far back in the history of mankind; some special laws such as Pythagoras' Theorem in the form 6- + 8- 10- were known to the Chinese. The fact that the circumference of a circle is a constant multiple of its diameter was also

1
known to several ancient civilizations, as v^^as the existence of similar triangles. However, the main body of geometrical knowledge was accumulated by the Greeks

and it is to this culture that I turn for my discussion on geometry as a science.

Let us take a few simple geometrical results and ask

what they tell us about the world. Three such examples

are:

(a) The sum of the angles of any triangle equals two

right angles.

(b) In a right angled triangle the sum of the squares

of the lengths of the two smallest sides equals the

square of the length of the hypotenuse.

(c) The circumference of any circle is a constant multiple of its diameter.

These results can properly be considered as laws of nature. They are found experimentally by drawing trian- gles and circles and measuring lengths and angles. They

tell us quite a lot about the world we live in: The results are the same no matter where we carry out the experi- ments; the laws are true for all orientations of triangles;

they were true in 400 B.C. and are true today; in order

to conduct such experiments we need to define standards

and measuring procedures These laws are

general, that is when stating them we are not only say- ing that these laws were satisfied for all triangles and

circles that have been measured, but that they are true

for all triangles and circles. We have made that inductive

leap from the particular to the general that is characteristic of science and are prepared to use these laws to predict the outcome of future experiments. In doing so the ancient Greeks were accepting that nature satisfies

' laws of nature' that can be uncovered by experiment.

But the contribution of the Greeks does not stop at the

discovery of such laws. Their main contribution to geometry was to recognize that all these empirical laws

could be deduced from a small set of fundamental laws: the axioms of Euclidean geometry together with defini- tions of the objects that enter those laws and the rules of

logic that we use in deductive reasoning. Examples of these latter two categories are [5]: A straight line is a Hne which lies evenly with the

points on itself. Things which are equal to the same thing are equal

to each other. The axioms or fundamental laws as defined by Euclid are

[5]:(1) It is possible to draw a straight line from any

point to any point.

(2) It is possible to extend a finite straight line con- tinuously in a straight line.

(3) It is possible to describe a circle with any center and any radms.

(4) All right angles are equal to one another.

(5) If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which

the angles are less than two right angles.

These fundamental laws clearly represent an extrapolation from experience. Our experience is finite. Within

this finite domain we find these laws to be satisfied (to some level of accuracy!) so we postulate that they can be

generalized to the above. Given these fundamental laws

all the other laws of geometry can be deduced by logical

(i.e., mathematical) reasoning. This is just what we try

to do in modern physics. We find laws applicable to different sets of phenomena and then seek more fundamental laws from which this variety of subsidiary laws can be

deduced.

3. The First Constant of Nature?

Let me return to one of the early geometrical laws:

"The circumference of any circle is a constant multiple

of the diameter."

What is this constant multiple, and how can it be

determined? The multiple is given the label -tt and we

determine it by measurement. If I take a circle and some

measuring device, for example my wife's tape measure

(borrowed from her sewdng box!) I can first of all test that it is a circle by measuring the diameter at different

orientations. I always find the same value, say 8 inches.

I then measure the circumference and find that it is say

25 inches. The ratio is -tt = 25/8 = 3.125. I repeat the ex- periment in London and in Washington, and I find more

or less the same answer. I do the same experiment with

different size circles and I find more or less the same

answer. I have therefore determined the first known

constant of nature as tt = 3.125. Such a simple experiment raises several interesting

questions: First one might ask about the measuring ap- paratus. A plastic tape measure is not perhaps the most

accurate of measuring rods and I doubt if it would compare very favorable with the standard definition of

length whether by reference to the old standard meter or

to the new standard using the velocity of light. Yet it has some advantages. I can fold it back on itself and compare

the one inch intervals on one part of the tape with those on another part; they seem to agree. Thus the subdivisions may not be equal to a standard inch but they are equal to each other. But since the value of tt is given by

the ratio of the circumference to the diameter it does not matter whether the scale on my measure agrees with the

standard or not; the ratio of lengths and therefore the

value I find for tt is only dependent on an accurate divi- sion of my measuring tape. This independence of the definition of a measuring standard is not confined to the

determination of tt; the fine structure constant, or the ratio of proton to electron masses are pure numbers and

therefore independent of the standard units of mass,

length, charge, or time. Units are only an intermediary

that enable us to attach labels of mass, length, etc., to other quantities, but the constants of nature are really

pure numbers, independent of units. This is a point I will return to later on in this article. Again this simple experiment can teach us something

about experimental procedures. If I repeat the experiment at different times on circles of different dimensions

and at different places I do not get exactly the same

result, only 'more or less' the same result. Thus a se- quence of five measurements may give the values

3.125 3.163 3.142 3.120 3.157

I now recognize that one measurement does not give an

exact result so I take an average and obtain tt = 3.1414,

and believe that this is a more accurate estimate of tt. Indeed, if I follow the standard practice of estimating the

uncertainty to be attached to this value I would then cal- culate the standard deviation to be 0.01695 and give the

experimental determination of tt as

TT - 3.141 ± 0.017.

It is important to realize that in following the above

procedure I have made an act of faith, namely that there

2
is a unique value of it that can be determined to some ac- curacy by taking many measurements. I have ruled out

the possibility that it is really a stochastic quantity and

that my measurements reflect such a fact about nature. On the basis of my experimental determination of I cannot justify such an assumption. The same problem

necessarily arises with all measurements, for example,

why should not the fine structure constant be a stochastic quantity? It is true that past experience shows that as we make more accurate experiments the latest determinations do not show the earlier fluctuations so we deduce

that these earlier fluctuations were experimental error. But if we then claim that this experience proves the

present set of values do not reflect the stochastic nature

of the world, we are again up against Hume's argument

that we cannot prove that our laws will hold true in the

future just because they held good in the past.

Of course if we now wish to evaluate -n- we would not do it by an experiment of the kind I described; we would

calculate it mathematically. For example we might use

the power series for tan'^x obtained from the integral

obtained by Newton, Leibniz, and Gregory [6]:

tan

•^0 1+2/2 S ^ 7

-ITaking x - 1 gives the Leibniz expression [6] - =1-1 + 1-1 + 1

4 3 5 7 9

This result, though elegant in relating tt to simple frac- tions, is of no practical use. A much better formula is that obtained by Machin [7] which is i=4ta„-'i-ta„-'^

= 4 1

5

1 239

1 J_

3 53

1 1 3 2393

1 J_

5 5^

+ .

1 J_

7 5^ 9 5^

Thus IT can be determined to any required accuracy; it is no longer a constant of physics but a constant of

mathematics! What does this mean? If a physical con- stant Hke the ratio of the circumference to the diameter

of a circle can be determined by mathematics, why not other physical constants like the fine structure constant,

the mass ratios of elementary particles .... Is physics

reducible to mathematics?

4. The Relationship Between IVIathematics

and Physics

The mathematical reasoning that led to the power

series for tan^a; and hence for tt is part of Euclidean

geometry; that is, provided the fundamental laws, or Euclid's axioms, are true, then the ratio of the circumfer- ence of a circle to its diameter is calculable from the

above series. But what does it mean to ask if the axioms

of a branch of mathematics are true? All that we ask in mathematics is that the axioms be consistent, that is they do not lead to contradictory conclusions. Mathematics is not true or false, it is simply a procedure for deducing the consequences of a particular set of axioms. What

we are doing is to ask whether the axiom set of Euclidean geometry is an appropriate mathematical model

for the physical world, so that the theorems of Euclidean

geometry can be taken over into statements about physical geometry. This clearly requires us to define the objects we call lines and angles in a physical way. Whether

these definitions then satisfy the axioms of Euclidean

geometry is an empirical question that can only be set- tled by experiment.

There are several definitions of a straight line that con- form to everyday concepts. One is the path of a light ray. Indeed, it is possible that this is what was behind

Euchd's statement that "a straight line is a line which

lies evenly with the points on itself" but it is not the

only definition. Another is the geodesic definition that a straight line is the shortest distance between two points;

that is, of all the possible paths joining two points the

straight line is that which requires the minimum number

of measuring rods laid end to end to cover the distance

from one point to another. A right angle can be defined

as one quarter of a revolution such that four such angles

which coincide when laid one on top of another constitute a complete revolution when laid side by side. An alternative definition is that it is the angle between two straight

lines such that the distance from a point on one line to a point on the other is a minimum. Wliether these alternative definitions are equivalent is again an empirical issue,

and having chosen one definition it is again an empirical

issue as to whether these physical definitions have the same properties as the lines and angles in Euclidean

geometry. Of course we can only verify that they do to a certain level of accuracy, so whilst our mathematical

deductions from the axioms are correct we cannot be

sure that they correspond to consequences for the physical behavior of objects.

There is a philosophical point here of some substance

as was emphasized by Poincare [8]. Suppose we found

that our physical geometry did not agree with Euclidean

geometry. How are we to interpret such a finding? One

possible route is to stick to our definition of a straight

line as, say, the path of a light ray, in which case the

geometry of light rays is non-Euclidean. On the other

hand we could say that light does not follow straight

lines since they do not satisfy Euclidean geometry. The

physicist would then seek to explain this departure from

Euclidean geometry by saying that light rays are bent by

the presence of other objects in the universe. Within

present day science the situation is somewhat confused.

On the one hand we talk about geodesies and curved

space—on the other hand we talk about the bending of

light, that is bending compared to a EucHdean straight

line. This illustrates Poincare's conventionality thesis,

subsequently elaborated by Reichenbach and other philosophers of science, that there are many ways of inter- preting a given physical situation depending on the definitions used. Contrary to oft repeated claims, there is no

information conveyed in the statement "space is non Euclidean." Our knowledge is of the behavior of objects, not space. It may be convenient to talk of light as defining straight hnes and that the light geometry is non Euclidean, but it is no more (or less) correct than to claim

the straight Hnes are those that satisfy Euclidean

geometry and that light rays are bent.

Yet another lesson to be learnt from geometry is that

there is no clear division between what we consider as axioms or fundamental laws, and what we consider as theorems or deductions from those laws. For example,

the axioms of Euclid can be used to prove Pythagoras'

theorem, or alternatively, Pythagoras' theorem can be

taken as the fifth axiom and then the statement that

straight lines meet if produced indefinitely becomes a theorem. Alternatively, the fifth axiom could be taken to be the statement that similar triangles exist, or the sum

of the angles of a triangle is half a revolution One

3
man's axiom is another man's theorem! So it is in the

other branches of science; we choose to take particular statements as fundamental laws and others as deductions

from those laws but laws and consequences can be inter- changed and we have no way of establishing one set of laws as 'the laws of nature'.

5. The Truth of Euclidean Geometry

Even in Greek times there was some unease about the

fifth axiom and attempts were made either to replace it by some more self-evidently true axiom or to deduce it from the other four. The best known attempt was by Pto- lomy who tried to deduce the parallel axiom from the

first four of Euclid's axioms together with the first twenty eight theorems derived by Euclid which are in- dependent of the fifth axiom, but in this as in other at- tempts some other assertion entered the argument and

effectively replaced the fifth axiom.

However, there does not seem to have been much

doubt that Euclidean geometry was indeed the physical

geometry of the world. Indeed, it was so firmly held to be true that attempts to demonstrate the validity of other branches of mathematics used the model of Euclidean geometry. Almost all philosophers, mathematicians, and scientists held Euclidean geometry to be

necessarily true. Isaac Barrow (Newton's predecessor at Cambridge) held that this truth was derived from innate

reason. Newton, Leibniz, Lock, and Hobbes held that

Euclidean geometry was inherent in the design of the

universe. One notable exception was David Hume, to whose ideas I have already drawn attention. Hume ar- gued both against the necessity of laws and contended

that no amount of past experience could demonstrate that

the future vdll obey such laws; knowledge is empirical

and the laws of Euclidean geometry, like other laws, are not necessary truths. On the other hand Immanuel Kant maintained that Euclidean geometry is an a priori synthetic truth, that is (roughly interpreted) that our minds compel us to organize our experiences around certain intuitions of space and

time and that Euclidean geometry was one such a priori

synthetic truth. So pervasive was Kantian philosophy

that even when Gauss had convinced himself that non- Euclidean geometry was possible he states in a letter to Bessel that he would not pubHsh his findings because he

feared ridicule. As with many of the discoveries of science and

mathematics there is no one person that can be called the

discoverer of non-Euclidean geometry. Lambert reached

the conclusion that there could be many geometries

derived from different axioms but it was Gauss who

recognized that the non-Euclidean geometry could apply

to the physical world; in a letter to Olbers written in 1817 he concluded [9] "... we must place geometry not

in the same class as arithmetic which is purely a priori

but with mechanics ..."

This empirical approach to geometry led to Gauss sur- veying a triangle formed by the mountain peaks Brocken,

Hohenhagen, and Inselsberg to determine the sum of the

angles of a triangle [10]. In fact the result obtained was

180° 14.85" but Gauss realized that the error exceeded

14.85" so that the correct result could have been 180° or

less, and that it was any way unlikely that we could find a positive result with such a small triangle (the sides were 69, 85, and 197 km). However, at least the distinction had been made between physical and mathematical

knowledge.^

6. Newtonian Gravitation and Mechanics

I now wish to examine the laws of mechanics and gravitation by which Newton and his successors were able to account for the motion of planets, the rate of pendula

clocks, the motion of projectiles . . . According to Newton's laws the motion of a body of mass mi under

the gravitational attraction of a second body of mass mz

is given by [11]

dhi2 _ -G(mi + mi)

~]72 3

where ri2 is the (vector) distance between the two masses, G

is the Newtonian constant of gravitation, a universal con- stant of nature to be determined by experiment [11]. This constant G is of a different kind to tt. It has di- mensions and therefore its value depends on the units in which it is measured. The quantity tt on the other hand is a pure number; its value is unit independent. This should

make us ask what is meant by calling G a constant of nature. Let us examine carefully the content of Newton's law.

It contains the symbols r for length, t for time, m for mass. What do these symbols mean? How are mass,

length, and time to be measured?

Firstly, the laws only apply in an 'inertial frame', that

is in a system of reference in which freely moving particles continue with constant velocity. Yet to know

whether particles have a constant velocity presupposes a measure of time and distance! Suppose we use a rigid rod

to measure distance. We can now define a measure of

inertial time such that a freely moving particle covers

equal distances in what we define as equal times. This

particle, of course, necessarily covers equal distances in equal times. It then becomes a matter of empirical investigation as to whether other particles also cover equal

distances in equal times as measured by that inertial clock; Newtonian mechanics is based on the hypothesis

that such frames of reference do exist in nature. In practice time was not measured using freely moving particles, but either by the rotation of the earth around its axis, or some gravitational clock such as a pendulum. Do

these clocks keep Newtonian inertial time?

It is an elementary deduction from Newton's laws that

the total angular momentum of an isolated system of in- teracting particles does not change in inertial time. Thus

the angular velocity of a rigid body is constant in inertial time so the rotation of the earth provides a tolerably

good clock. However, it is not self evident that Nevi1:on's

laws are true. We could imagine a world in which the an- gular momentum of an 'isolated' body changed (slowly) as measured in inertial time. Whether or not this is the case is an empiiical question.

A pendulum clock is a gravitational rather than an

inertial clock, as is the motion of the moon around the

earth, the earth around the sun, etc.; these depend on

the law of gravitation as well as the laws of motion.

Newton's law of gravity is not self evidently true and

As far as I am awai'e, no one has proposed an improved version of Gauss' experiment; for 3 satellites in orbit around the earth the sum of the

angles according to general relativity would be 180° plus an excess of about 0.005".

4
needs to be tested: is the symbol t that enters the law of

gravity the same as inertial time? Is the constant of gravity, G, really a constant in time and/or space? These

are empirical questions that can only be decided by ex- periment. Such conceptually simple experiments would

compare the period of a pendulum, or the length of the

year with the length of a day. Suppose Newton's law of gravity is true for some time scale tg. The period of a pendulum P^, and the length of the year would then

be

M^

1/2 1 G

y2 Pk =

27: Ll^

M 1/2 ' 1/2

where I is the length of the pendulum, Me the mass and

the radius of the earth, Le the radius of the earth's

orbit around the sun, and Mq the solar mass. If Pc and

Pe varied in time (as measured by an inertial clock) we

could interpret this as the constant of gravity G = G{t).

However, this would require P e /P c to be independent of time since they are both gravitational clocks. If P^/Pq

varied in time then we would require a spatial as well as temporal variation in G. So far I have paid little attention to the meaning of the

symbols r for length and m for mass. Within Newtonian

mechanics we measure length by two means, rigid rulers,

and triangulation. It is an empirical fact about the world

that objects that are the same length at one place remain

equal in length when transported to another place. We

can therefore usefully base our measurement of length on

such a basis. We cannot however test that bodies do not change in length when transported, so it is purely a con- vention to assume that they do not. The triangulation

method of length measurement is based on the assertion

that Euclidean geometry is valid both for rigid bodies

and for the path of light rays. As I pointed out earlier,

whether or not these assertions are true are empirical

questions.

The mass that enters Newton's laws is the inertial mass defined in terms of momentum conservation. Having set up a measure of length and time we can choose one standard mass mo. By measuring initial and final velocity in a collision experiment we determine the inertial mass M of another body through the relation

mo u\ + MU\ = mo U{ + MUj,

where {Ui, UO and {U{, Uf) are the initial and final velocities of mo and M. Any one such experiment gives a value for M. The empirical content in the law of momentum conservation is that any initial conditions (Ui, U\)

give the same value of M for a given object. Newton's

law of gravity states that the source of the gravitational

field is proportional to the inertial mass, or that the constant of gravity, G, is actually a constant independent of the nature of the bodies.

7. The Newtonian Constant of Gravity

Having laid down a procedure for measuring mass,

length, and time we can now ask what meaning can be

given to the constant of gravity within mechanics and

gravitation. The basic units are arbitrary; let us call them mo, Iq, to. We can now imagine an experiment to determine G by measuring the period of a pendulum of length L = Ni Iq, on the surface of a planet of mass M = Nz mo, whose radius is R - N'^ Iq; ii the period

P - Ni toVfe have

G = 4tt2 Nj_Nj_

mo N2 Ni

1 3

''0

Clearly A^i, N2, N3, are measurable (by counting)

so G can be determined in units of mo, lo, and to. Moreover we could imagine a world in which this experiment

would give a different value of AT 4 and hence of G

depending on position and time. It is therefore meaningful to talk about a variation of the constant of gravity

within the framework of Newtonian mechanics.

If we also introduce the velocity of light, c, we can compare a gravitational clock with a light clock. For ex- ample we could imagine an experiment in which two bodies were in orbit about each other and reflected a light

signal back and forth. The number of such round trip signals in one orbit (relative to an inertial frame) is a measurable; it could be constant in time or vary in time.

Similarly, given our choice of standards Iq, to, we can measure the velocity of light c. This could be constant or varying as measured in inertial time. Indeed, we could

imagine a world in which both c and G varied in inertial time and in which the number of round trip light signals

in a gravitational orbit also varied.

While we can imagine a world in which c - c{t),

G = Git), what would it mean to say that G were con- stant but different in value? How would this manifest it- self in measurable ways? To say that G ^ Git ) we mean

that having chosen our units mo, lo, to, we find a dif- ferent result of a given experiment when done at dif- ferent times but referred to the same units. But we can- not transport our arbitrary units mo, ^o. ^0 from this world to some other I might imagine. The problem is that

the world of Newtonian mechanics is continuous. There

are no intrinsic standards of mass, length, and time defined within the system so that measurement cannot be

entirely reduced to counting and I cannot compare my

two worlds. Within Newtonian mechanics the value of G

is meaningless; it just reflects the arbitrariness of our

choice of units. For illustration let us imagine that with

the standard pre-atomic definitions of kilogram, meter,

and second, we found that

G = 6.67 X 10" kg"

The relative behavior of the moon and planets would be

unchanged. The only change would be in the number of

days in a year, and the number of swings of a 1 meter

pendulum in one day; that is, comparisons of gravitational phenomena with our arbitrarily defined inertial clock. The observed phenomena could be equally ex- plained by stating that the earth was spinning ten times

faster so that there were only 8640 seconds in a day and

G had its usual value.

This should be contrasted with a world model with

discrete intrinsic structure. For example, if all bodies were made up of a basic unit m0 of radius / 0, by using the propagation of light 1 0 can also be a unit of time 1 0, and all measure- ments are now reducible to counting. In terms of these basic

units G is a number; if it had a different value physics would

be different. For example the period of rotation of the earth

is some pure number times the intrinsic unit <o so I can meaningfully state that the rotation rate is different and so separate a change in the earth' s rotation from a change in G. Since our present models of the world do have a discrete

quantized structure, an absolute determination of G be- comes meaningful.

8. Quantum Clocl(s

The previous discussion is somewhat removed from ac- tual practice. Time and distance standards are normally

based on quantum phenomena and the properties of light.

Electric charge apparently comes in identical 'lumps", in5
teger multiples of the charge on an electron e (or possibly

multiples of e/3?). Matter also comes in lumps, m^, m^,

etc., and angular momentum comes in multiples of n. This allows us to define an atomic unit of time, for example the period of revolution of an electron in the ground

state of a hydrogen atom as ^0 4 ,

and a unit of length, Yq, as the radius of this orbit:

1/2 1/2

Gnil Mo

We can now measure time in units of Pq, length in units of ro, and mass in units of m^. If Newton's laws are true the period Pe of the earth around the sun is given

by

u f o 1 1/2 f 1 1/2 f ^ -] 1/2 E Po

Each term in parentheses is a pure number and therefore

represents at least in principle an empirical measurement. Thus we can meaningfully ask if this relation is satisfied (i.e., is Newton's law true in atomic units) and

determine the value of G by determining the dimensionless number e^/Gmi. We can also meaningfully ask if this dimensionless number (=10^^) is constant in time, that is do we get the same value when we conduct an experiment at different times? We should, however, remember

that should we find a time variation we have a choice

between saying e and m^, are constant in time and G

varies, or taking G to be constant and letting e and/or

vary in time.

9. The Fine-Structure Constant and Other

Numbers

So far we have not considered the velocity of light.

Since we have a quantum definition of length and time it is now an empirical question as to whether the velocity of

light is constant as measured in these quantum units. For

example, the time required for a light signal to travel a

distance measured in units of Pq is 2-nf 0 cPi 1ic which is the fine-structure constant. Whether or not a is constant in time is therefore an empirical result about

the world, and this can only be settled by experiment.

Similarly, we may ask if the mass ratio of elementary

particles, i.e., m^/nig, varies in time. Indeed, any pure

number is independent of our choice of units and can be

considered as a possible candidate for time variation. In the end it is experiment that has to decide; all philosophy can do is to try and separate the meaningless from the

meaningful questions.

One might reasonably ask why we should expect any

time variation of numerical constants. To answer this we

must consider the large scale structure of the universe.

10. Cosmology

There are very few measured properties of the large

scale structure of the universe, and even fewer that have

any claim to accuracy. The best known is probably the

Hubble parameter Hq. This is often thought of as the constant of proportionality in the velocity-distance rela- tion for the expanding universe, V = H^r. However,

distances and velocities are inferred quantities; the ac- tual measurables are the energy per unit time per unit area F, and the red shift Z, of light from distant galaxies; Ho is then calculated from the relation

F = [1 + (go - 1)Z + ...]

where L is the intrinsic luminosity of such galaxies. Estimates of L and hence Ho vary by factors of one to four

(and maybe more). Consequently Ho is only known to a

factor of about 2 (or more!); to order of magnitude Hq^ —

10^^ years. The parameter go in the above expression is called the

deceleration parameter and is related to the rate of change of the Hubble constant Ho- One of the challenging

problems of observational cosmology is to determine the

actual value of go since (with the aid of some theory!) it" will tell us whether the universe will continue to expand

or contract. If the mean density of matter in the

universe, po, can also be determined, these parameters

will tell us whether our simple cosmological models are

valid. This mean density, po, is difficult to estimate. If we

just estimate the masses of galaxies then po — O.Olpc

where pc is called the critical density and is such that

1 = 1

S H^ 2-

But of course there may be mass that we have not yet

detected; halos of galaxies, matter in intergalactic space

(e.g., bricks!), dead stars, black holes, massive neutrinos, gravitational waves As I said earlier there

are few reliable measurements in cosmology!

11. Dimensionless Constants of Nature

The two cosmological parameters. Ho and po, can be

combined with other constants of nature, G, c, e, and m^,

to give the following pure numbers:

6 X 10

GfUf

,40 4 X 10'"';

= 4 X 1042

The first parameter is often referred to as the age of the

universe in atomic units; the second is the ratio of atomic

to cosmological density; and the third is the ratio of

electrical to gravitational forces. Other pure numbers we have considered are the fine-structure constant and

the mass ratio of the electron to proton:

a —

1ic

1 137'

1 1836

Why do these numbers have these values? Why is the

ratio of the circumference to radius of a circle equal to 3.1415926. . .? Can these numbers be deduced from a sin- gle theory such that no other values are possible? Are

these numbers just random initial conditions for the

universe? What would the universe be Hke if these

numbers were different?

I think most scientists would conjecture that one day we will be able to explain why the particle masses are as observed, and why the fine-structure constant has its particular value. At the moment we do not have a deep

enough understanding of particle physics to provide the

explanation. A similar view can be taken about the

cosmological numbers of the order of 10^". Such fundamental theories linking cosmology and microphysics have

6
been pursued by several people, but without apparent

success [12, 13]. The point of view of Dirac [14] is that

whatever the theory is, it will entail that where such

large numbers as 10^° occur, then parameters of this magnitude are proportional to each other, so that

Gm'i " e^Ho

where ki and k2 are of order unity. This statement is known as The Large Number Hypothesis.

I should point out that the standard, general relativistic cosmological model does not satisfy this hypothesis;

the first parameter is constant in time, the second

parameter varies linearly vdth time, and the third

parameter varies with the square of the time. (These

statements are only strictly true for the Einstein-de

Sitter model.) It is therefore pure accident that, these

three numbers happen to have approximately the same

value at the present time.

The large number hypothesis leads to the result that

either all three parameters are constant in time, or they

vary in unison. The first proposal leads directly to the

steady state cosmology in which G, Ho, and po are constant. This in turn implies the creation of matter [15]. The steady state theory has few advocates these days as

its very specific predictions are apparently contradicted

by observation, for example, the radio source counts. I must admit that I am not wholly convinced that some

form of the steady state theory cannot stand up to this data, for example, a statistically steady state with local fluctuations.

The second proposal advanced by Dirac is that as H

is a time, related to the age of the universe, then we

should find

H, 1 Po This is one origin of the idea that the Newtonian constant

of gravity varies in time, although this proposal was advanced somewhat earlier by E. A. Milne [16]. Such a

variation is incompatible with general relativity and some new theory needs to be advanced that successfully explains this time variation, and whose predictions can be

evaluated and subjected to empirical investigation. The

scalar-tensor theory of Brans and Dicke [17] was such an

attempt, but it cannot successfully explain the time vari- ation of G and po [18]. I have considered many other possible theories but I have not, as yet, been able to find one that explains all the large number relations and

agrees with solar system tests on the orbits of planets

[19]. Recently Dirac introduced a two metric framework. In the gravitational metric, dSg, Einstein's equations are

valid, and this is conformally related to an atomic metric,

dsA = dSg/p, where p is to be determined [20]. Without

any theory for the determination of p all that can be done

is to ask general questions such as what classes of theory

satisfy the large number hypothesis [21] and what conse- quences would such theories have in other branches of

physics [22]. My own view is that little progress can be

made without some more precise theory.

However, experimentally there is clearly a testable

prediction, namely that the constant of gravity G de- creases in atomic time, G/G — -10^^ years. This can be

tested either by radar ranging experiments or by laboratory tests. A second prediction is that there could be

creation of matter proportional to matter already

present, and this might be detectable in the laboratory

[23]. At the present time no experiments are sensitive

enough to detect these effects.

12. The Anthropic Principle

I have already pointed out that in standard theory (i.e. general relativity and quantum mechanics) the large

numbers vary in time, and it is just a chance event that

these three numbers happen to be about 10""^. These

numbers could be different, in which case the properties

of the universe would be different, and in particular we, man the observer, would not be here to ask why the

universe is as it is. This idea has been elevated to the

status of The Anthropic Principle [24, 25, 26]. Since man contains heavy elements there must have

been nucleosynthesis, therefore there must have beer- stars that have evolved to supernovae and ejected heavy

elements into the interstellar material. This material

then formed other stars with planets on which life as we

know it evolved.

A star is an object in which pressure balances gravity

and in which gas pressure dominates over radiation pressure; this is sufficient to show that stars have masses up

to a maximum Mq where

Mi _

nip Gm2

and an - ^

^ nc On the other hand the central temperature must be high

enough for nuclear reactions to take place before the gas

is degenerate. This gives a minimum mass of about 0.01 Mq. The luminosity of a star is essentially determined by

radiative energy transport. The principal opacity source

is scattering by electrons, and this permits the evaluation of the luminosity Lq. The evolution time for a star is essentially the time to convert a significant fraction of hydrogen to helium and so is proportional to Mq/Lq. The

actual result gives

evol 1

o-G mpC 2

•

The anthropic principle is now used to argue that if the

age of the universe Hq^ is less than ^evoi then no stars would have evolved. On the other hand, if the universe is very much older than t^voi there would be no stars left. Conditions are therefore ripe for Hfe to develop when the

universe is just a little older than ^evoi or

_1 J_ n rtipC^

Hq olq ninC^ e^Ho Gm^

1

go =

where a is the fine-structure constant. This 'explains'

why this large number coincidence is observed.

The second large number coincidence is 'explained' by

appealing to cosmological models. In such general relativistic models the deceleration parameter go is

4tt G Po S H§

In the Einstein de Sitter model qo ^ 1/2. If 90 were very

much less than 1/2 the universe would not be unstable to the formation of galaxies, and hence stars. On the other

hand, if go were much greater than 1/2, the age of the

universe, which for such models is approximately

1/Hoqo^, would be so short that there would not have

been time for stars to have been formed and to have

evolved. Hence we expect to find qo ~ 1/2. An alternative explanation is that general relativity is incomplete since it does not predict a unique cosmological

model. When we have found how to correct this defi- ciency we will find that the only possible cosmological

model is the Einstein-de Sitter model with qo = 1/2.

7
Either way the combination of go = 1/2 with the other

large number relation gives the results:

e^po Gnii e^Ho

'

This line of argument has been pushed further by Carter

[25] who argues that we need conditions which give some

stars with convective outer envelopes since these stars have planets, but we cannot have completely mixed con- vective stars since supernovae require unmixed stars. This argument is quantified by requiring that the effec- tive temperature of stars be approximately the ionization

temperature of hydrogen {OAa^rUgc'^/k). This condition

determines the mass of such stars as

The argument is now used that ac^'^a^*^ = 1 since if it were much greater all stars would be convective and

mixed, whereas if it were much less then there would be

no stars with outer convective zones, and so no planets.

With ac — a^" we have ac ~ 10"^^ and the actual magnitude of the large numbers is 'explained'. Similar argu- ments to those above can be used to explain other numerical constants of nature [26].

" 13. Critique of the Anthropic Program

My first objection to the anthropic program is that the

technical arguments have not been carried through; the

behavior of models with substantial different parameters

awaits evaluation. For example, could we not have a universe with qo = 10'^^? In such a model stars would

have evolved provided

1 !_ t

The arguments on galaxy and star formation are a little weak since we are still struggling to understand how

gravitational condensations occur. The argument ad- vanced by Carter to explain why ag ~ lO"*^ is very

weak. Firstly supernovae could occur in homogeneous

evolution after evolution through to carbon stars, and

secondly it is by no means obvious that convective zones

are a prerequisite for planetary formation. This criticism

however can and should be quantified by calculations on

the properties of a universe with different values of the

large numbers.

But if the program is successful, what does it tell us? There are laws of nature (L, ) containing arbitrary con- stants (a,) and a set of observations (Ot); and a subset of these observations is the existence of ourselves. If the

theories are 'correct' then with the observed values of the constants they must predict our existence. If they

did not predict our existence the theories would be

wrong—that is contradicted by observation. If we had

the same theories with different constants the predictions would necessarily be different, and there is no rea- son why those predictions should include the emergence

of mankind. The anthropic program is therefore bound to be successful. We exist in the universe and if we had a

full knowledge of the laws of physics, we would predict our own existence.

The more fundamental problem is why there are any

constants of nature at all, or indeed why there are any

laws of nature. Is there a Creator who set up the

universe with just those initial conditions, laws, and con- stants of nature that would lead to the emergence of mankind? Is it just chance? Is there an infinity of 'parallel' universes with different laws and different constants,

each with its own properties? Our universe which contains us then necessarily has those laws that give rise to mankind. Does the universe go through cycles in which

the laws and constants of nature change? Could it be

that there are really no laws or constants of nature, but

that we just live in a statistical fluctuation of order in a universe of chaos? Or are there unknown laws, constraints, consequences yet to be discovered that will ulti- mately lead us to the conclusion that there can only be

one universe, one set of laws, one set of constants, one

geometrical structure? Is there only one possible

universe with the properties we observe including our own existence?

14. Conclusions

I have tried to emphasize throughout this article that

experiments are the final arbiter; in Poincare's words

"Experiment is the sole source of truth" [27]. Yet

mathematical physics has a vital role to play in developing ideas through to predictions that can then be compared to experimental data; and philosophy too has a role

to play in challenging our firmly held views, in separating the meaningful from the meaningless questions.

I would therefore like to finish by encouraging the

reader to let his hair down occasionally and think of the

outlandish possibilities, which can in turn be developed

and tested. Is 'space' as determined by light signals lo- cally Euclidean? Is gravitational time the same as inertial time, spin time, electric time Are the laws of physics 'here and now' the same as they were 'there and

then'? Is matter being created or destroyed? At what

level of accuracy will we find general relativity to be in- correct? Does special relativity hold good as v cl Do

the fine-structure constant and the particle mass ratios

vary in time at some level? The reader can add his own

ideas to such a list!

References

[1] Shu Ching, The Book of History (circa 2200 B.C.). For translations, see J. Legge, The Shoo King (The Author's Hongkong, London, 1865); C. Waltham, Shu Chiyig (Allen and

Unwin, 1971). See also C. P. S. Menon, Early Astronomy

and Cosmology (London, 1932).

[2] Chou Pi (circa 1100 B.C.). For a discussion of this model, see Thaibut, J. Asiatic Soc. of Bengal (1880); and C. P. S. Menon, Early Astronorny and Cosmology (London, 1932).

[3] Suryaprajnapti, Book 1, chapter 8 (circa 500 B.C.); see also W. Brennand, Hiridu Astronomy (London, 1896).

[4] D. Hume, An Inquiry Concerni7ig Human Understanding

(1748) [see also Open Court Edition, Chicago (1927)].

[5] Euclid, Elements (circa 300 B.C.); see also T. L. Heath, The

Thirteen Books of Euclid's Elements (Dover, New York,

1956).

[6] G. Leibniz (1674), published in Math. Schrifters, vol. 5, Ed. by C. I. Gerhardt (Ascher Schmidt, 1849), p. 88. See also M. Kline, Mathematical Thought from. Ancient to Modem

Times (Oxford University Press, i972).

[7] J. Machin, in W. Jones (Synopsis, London, 1706); see also K. Kopp, Infinite Series (Hafner, New York, 1971), p. 253.

[8] H. Poincare, La Science et I'Hypothese (Villiers, Paris,

1902). [English transl.: G. B. Halsted (Dover, New York,

1952.]

[9] K. F. Gauss, Werke vol. 8 (B. G. Teubner, Leipzig, 1900), p. 177.

[10] K. F. Gauss, Werke, vol. 4 (B. G. Teubner, Leipzig, 1900),

p. 258.

['11] I. Newton, Philosphial Naturalis Principia Mathem/itica,

London (1687).

8
[12] A. S. Eddington, Fundamental Theory (Cambridge University Press, 1946); E. A. Milne, Kinematic Relativity (Oxford University Press, 1948).

[13] I. W. Roxburgh, in The Encyclopedia of Ignorance, Ed. by

R. Duncan and M. Weston-Smith (Pergamon Press, Oxford, 1977), p. 38.

[14] P. A. M. Dirac, Proc. R. Soc. London, Ser. A: 165, 199

(1938).

[15] H. Bondi and T. Gold, Mon. Not. R. Astron. Soc. 108, 372

(1948).

[16] E. A. Milne, Nature (London) 130, 508 (1932); and Relativity, Gravitation and World Stnicture (Oxford University Press, 1935).

[17] C. H. Brans and R. H. Dicke, Phys. Rev. 124, 925 (1961).

[18] S. Weinberg, Cosmology (Wiley, New York, 1972), p. 619.

[19] I. W. Roxburgh, Bicentennial Lectures on Cosmology,

University of Virginia, 1976 (unpublished).

[20] P. A. M. Dirac, Proc. R. Soc. London, Ser. A: 338, 439

(1974).

[21] I. W. Roxburgh, Nature (London) 268, 504 (1977).

[22] V. M. Cannuto, P. J. Adams, S. H. Hsieh, and E. Tsiang,

Astrophys J. Suppl. Ser. 41, 243 (1979).

[23] R. Ritter, these proceedings.

[24] R. H. Dicke, Nature (London) 192, 440 (1961).

[25] B. Carter, in Confrontation of Cosmological Theories and

Observation Data: Proceedings, Ed. by M. S. Longair

(Kluwer Boston, 1974).

[26] B. J. Carr and M. J. Rees, Nature (London) 278, 605 (1979).

[27] H. Poincare, La Science et I'Hypothese (Villiers, Paris,

1902). [English transl.: G. B. Halsted (Dover, New York,

1952), p. 140.]

9

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984). FREQUENCY, WAVELENGTH, AND STORED IONS

Frequency Standards Based on Magnetic Hyperfine Structure Resonances

H. Hellwig

Frequency and Time Systems, Inc., 34 Tozer Road, Beverly, MA 01915

Magnetic hyperfine resonances in alkali atoms were the physical basis of practical atomic frequency

standards at the beginning of the age of the realization of the unit of time via atomic phenomena. This

dates back more than 25 years; it appears to be no accident that today's time and frequency technology

still rests on the same physical basis: practical frequency standards and clocks use magnetic hyperfine

transitions in cesium, rubidium, and hydrogen, and the unit of time is defined—as well as practically realized—via the cesium resonance. This paper explores the basis for this phenomenon which is a result of a combination of mature

electronics and physics technologies with proven principles of experimental physics such as beam spec- troscopy and optical pumping. This paper will also address the limitations of these "traditional" or "microwave" frequency standards as well as opportunities still open for further improvements. These

limitations and opportunities center around the desirability to achieve a spectrally narrow line; i.e., a high line-Q within the microwave region. Options to realize improved Q-values will be discussed. Key words: atomic clocks; cesium beam; frequency standards; hydrogen maser; rubidium gas cell.

1. Introduction

The resonances of cesium, rubidium, and hydrogen in the microwave region have dominated the atomic time

and frequency standard field since the 1960's. In fact,

rubidium and cesium standards now have been commercially available for two decades [1, 2]. The 1960's saw re- finements in atomic clock design and their use in many

scientific applications and in metrology including the generation of time on a world-wdde basis. The 1970' s can be

characterized by the beginning of large-scale system-use

of atomic time and frequency standards reaching far beyond scientific studies, making possible today's communication and navigation systems.

An atomic time and frequency standard consists of three elements: the external interfaces, the electronic

systems, and the atomic resonator. The atomic resonator

is the heart of the atomic standard and serves the same

function as the pendulum in a mechanical clock; however,

with one most fundamental difference: The atomic reso- nance frequency itself is given by nature, it will not

drift, nor will it age according to all of our present ex- perimental and theoretical knowledge. Furthermore, all atoms are the same. A standard built independently from

another standard will show the same characteristics and

the same frequency as its predecessors or as standards in different locations. Atomic clocks feature an a priori ac- curacy; therefore, atomic clocks and frequency standards

are often called primary standards.

The frequency of the emitted or absorbed radiation is related to the energy difference between the two atomic

energy levels. In the three alkali atoms, hydrogen, ru- bidium, and cesium, which are most prominently in use

today, as they were for several decades, the magnetic

hyperfine transition in the ground state (^Si/2) is being

used. Table 1 shows a summary of these three standards

and their characteristic frequencies. The transitions are between energy levels which are created by the spin-spin

interaction between the atomic nucleus and the outer

electron in the ground state of the atom. As such, the

separation of the two energy levels, i.e., the transition

frequency, is a function of the magnetic field. This

dependency is described by the Breit-Rabi equation [3, 4] and depicted in Fig. 1. In order to define the precise resonance frequency, the

external magnetic field has to be well defined and kept

Table 1. Summary of rnost ividely used atoms.

Atomic Nuclear Atom Mass Spin vr(Hz)

Hydrogen (H) 1 1/2 1 420 405 752 Rubidium (Rb) 87 3/2 6 834 682 608 Cesium (Cs) 133 7/2 9 192 631 770

(F.mp)

0 12 3 X

Figure 1. Energy level diagram of ^'^^Cs in the ^Sj!^ groiaid

state as a function of the applied magnetic field.

11
constant. This necessitates magnetic shielding which is a

characteristic design feature of all presently used atomic

freqliency standards. The shielding can be quite ela- borate, and reduces the external magnetic fields,

foremost the earth's magnetic field, to one percent or

less of its normal value. It is obvious that only the

Amp = 0, the so-called clock transition, can provide

highly stable reference frequencies. The transition probability for this transition must be maximized and the

corresponding probabilities for all other magnetic-fielddependent transitions must be minimized. This is done by

applying a constant magnetic field, the so-called C-field. In order to force the Amp = 0 transition, the magnitude

of the C-field is typically at least an order of magnitude

above all anticipated residual field components, and its orientation is such that its vector is parallel to the vector

of the oscillating field w^hich is coupled to the resonance

of the atom. The resulting frequency shift can range up

to parts in 10^ as compared to the frequency of the atom

in a zero magnetic field. The magnitude of the field can be measured quite precisely, actually to an accuracy which is much better than

required in view of all other frequency stability limitations. Such a measurement is done by using the atom it- self as the magnetometer. In Fig. 2, which illustrates cesium for regions of relatively low magnetic field [4], the clock transition is indicated by the solid line. The

magnetic field H can then be measured precisely by

measuring the magnetic field dependent or Zeeman reso- nances at higher or lower frequencies as indicated by the

dotted lines between the energy' levels. We note that we

can measure a magnetic field H by measuring at mi- crowave frequencies or, alternately, by injecting a low

frequency.

Wobs = Wo + K V - --

—1 rr- ^

-H

1

1 H

1

1

t ^ " rfJ

i_L_

-Lt

i

MAGNETIC FIELD "o ENERGY STATES

AT H " Hn

Figure 2. Magnetic field dependence of the hyperfine energy

levels in the groundstate of the cesium- atom,.

2. Design Principles of an Atomic

Resonator

Atomic frequency standards, fundamentally, are devices which allow the measurement of a microwave fre- quency with very high resolution and accuracy. The

Doppler effect represents perhaps the most important

problem limiting the accuracy and resolution of any fre- quency standard. In the usual way we can say that if an

absorber of radiation moves relative to the source, the

observed resonance frequency is shifted to the value [5]

2M?

where the velocity v and wave-vector k are measured rel- ative to the source. The first order Doppler shift (k • v), the second order Doppler shift (v/c)^/2, and the recoil

shift (the last term) can be understood in terms of conser- vation of energy and momentum in the emission or absorption process. Specifically, the so-called second order

Doppler shift is merely the relativistic time dilation factor resulting from the movement of the atom relative to the

apparatus. Its effect is small but ultimately important.

The term "natural linevddth" normally refers to the

emission of visible light where the natural linewidth

describes the spectral linewidth associated with an optical transition. Lifetimes of optical transitions can be cal- culated from quantum mechanics using Planck's equation

for the black-body radiator. The lifetime in a given state

is proportional to the cube of the frequency; thus, at microwave frequencies we are faced with natural life- times of the states involved of hundreds of seconds. In other words, the natural linewidth is unmeasurably small

in the microwave region as compared to other limitations. As a consequence, one may interpret for micro- wave frequency standards (Fig. 3), that the lifetime is given by the observation time allowed by the frequency

standard apparatus or atomic resonator to coherently ob- serve the atomic radiation. In general, this lifetime is limited by either of two general effects: (1) the atom

enters and leaves the apparatus after a time At, or

(2) the atom stops oscillating due to a collision with other

atoms or the walls of the container after a time A t. Thus

atomic frequency standards are characterized by maximizing At and, vdth it, the line-Q.

A

Doppler / Width ^

\ Gaussian

B

Lorentzian

Natural _^

Linewidth

\^^^^^^aL|ss|an

FREQUENCY FREQUENCY

Figure 3. Spectrum of an atomic transition. On the left, part A

shows the situation when the atoms are unbound and the reso- nance feature has the full Doppler width Av~ (vie) vq. When

the atom is confined to dimensions less than the wavelength,

the Doppler profile is suppressed and the central feature has

the natural undth A v as shown in part B. (From Ref. [5].) Fundamentally related to this, again via Planck's equation, is the fact that the population difference between the

two levels separated by microwave frequencies is very

small. The population difference is a function of v. In the

low gigahertz region where h v « kT, the population

difference is only a very small fraction of one percent.

Therefore, in order to observe effectively the atomic reso- nance, it is a common design principle in microwave fre- quency standards to modify the population in the upper

and lower state. Either one has to be significantly depopulated. The result is an atomic ensemble which either has a

significant net emission or net absorption of energy. The

way in which this is accomplished is a significant determinant in the design of an atomic resonator. In summary, the design of an atomic resonator is characterized by the particular method of maximizing the

coherent observation time, minimizing first order Doppler

effects, and achieving a significant net population differ- ence between the two microwave energy levels.

12
3. Performance Principles

Time and frequency standards can be characterized in numerous ways. However, most frequently, a time

domain characterization is most lucid and useful. The

quantity measured [6] is the two sample or Allan vari- ance (Tyir) defined as

where y is the fractional frequency of adjacent measure- ments each with a sample time t, and the brackets denote an infinite time average.

An idealized presentation [1] for all precise time and

frequency standards is shown in Fig. 4. The first part, I, of the stability plot is determined by the fundamental

noise processes present in the standard and can be

characterized by the equation a,/T) = /cit ^ The second

part, II, is called the flicker or frequency floor, which

describes an independence of o-,/(t) from the averaging

time. We shall call this value dyp. Its level depends on

the particular frequency standard and is not fully understood in its physical basis, but relates to fluctuations in the value of critical, frequency-determining parameters

of the standard. The last section. III, can again be

characterized by a,y(T) ^ k-zT ~. The coefficients ko and

cannot usually be determined very accurately because of the long measurement times needed in order to obtain

statistical confidence. These coefficients are subject

largely to environmental effects and are thus not stable.

If linear frequency drift is present, az - +1. Of course,

drift is always caused by some systematic change in a frequency determining parameter with time. In contrast. Regions I and II have more quantifiable physical rela- tionships. The approximate relationship between fre- quency stability in Region I and the signal to noise avail- able in the device can be given by (JyiT) — (Q S/N)"\

S/N is the signal to noise ratio which, of course, is a

function of the sampling time t. The quality factor of the resonance line enters as a key determining factor: the

larger the Q, the better the stability. However, Q and

S/N can be traded against one another.

A device with a relatively low Q and a large S/N gives

the same short-term stability performance as a device

with a high Q and a low S/N. S/N can be expressed in terms of physical parameters of the atomic resonators which typically are shot-noise-limited: If n is the number

of signal events per unit time, S/N = VnV.

II

.

III.

WHITE OR FLICKER OF PHASE NOISE

WHITE FREQUENCY NOISE FLICKER "FLOOR" (FLICKER OF FREQUENCY

)

1 PURE FREQUENCY DRIFT

(AGING)

The flicker floor. Region II in Fig. 4 (cTyf), appears to be related to the Q-value of the resonance. Qualitatively,

this is not surprising. Atomic resonators with Q-values of

10* or higher may be expected to have accuracies of one

part in 10* or better because the linewidth is acting like an ultimate limit on the variability of the frequency of the resonance [7]. We can summarize: A high resonator Q

is essential to achieve high accuracy and ultimate stability. Thus, the time keeping potential of an atomic clock is governed by the Q-value and not by the signal-to-noise.

Signal-to-noise is only important to assure sufficient short-term frequency stability so that measurement

periods of acceptably short duration are needed to realize the available accuracy.

4. Electronic Systems

Figure 5 depicts the general concept of the overall design of an atomic standard where the atomic resonator serves as the frequency reference for a slave oscillator (usually a quartz crystal) in a servo loop configuration.

Figure 5. Block diagram of an atomic clock mth a frequency

lock servo.

Frequency-lock servos are used in passive atomic

clocks (e.g., cesium beam and rubidium gas cell) to steer the frequency of the reference oscillator so that it coin- cides with that of the atomic transition. The atomic reso- nator behaves similarly to a simple band-pass filter followed by a square-law detector. Therefore, in the fol- lowing, we will assume that the "line shape" of the

detected signal S (co) versus applied rf frequency is Lorentzian, i.e.,

S(co) = So (1 + (v - vo)2/Avf )-\

where vq is the atomic frequency, Av; is the line width,

and V is the frequency of the source (multiplied reference

oscillator). S (w) is plotted in Fig. 6. Although some de- vices do not exhibit such line shapes, the qualitative

results discussed below still apply. We can use the S (co) curve as a frequency discriminator; one common way is to use sinewave frequency modulation (FM) on the

source. In this case v — + A v», sin aj,„ t where A is the FM swing and co,,, = 2Tr v,,, the modulation frequency.

If we now mix this signal with a reference signal pro- portional to sin 0),,, t, as shown in Fig. 5, then the output

of the mixer has output near dc (i.e., neglecting 2w,„

terms) of V. - -F Av„i(vs - Vo)

(Av,)2 1 +

A V;

+ D ,

Figure 4. Typical frequency stability behavior of a frequency

standard. The two-sample variance is ^ (j). where F depends on the input levels to the mixer and

mixer efficiency and Z) is a possible mixer output offset.

13
Figure 6. Plot of atomic resonance features assuming a

Lorentzian line shape. S (w) is detected signal versus fre- quency. Vq(w) is the detected signal compoyieyit (at frequency

v„, ) wheyi the amplitude of the frequency modulation is small

(Av,„ « Av;).

This output voltage as a function of frequency is plotted

in Fig. 6 and can be used to servo the frequency of the oscillator (VCO) as shown in Fig. 5. For good long-term stability and accuracy, we want the gain G (co = 0) ^ ao; therefore this part of the servo is usually an integrator.

Depending on the particular design of the servo electronics, frequency offsets, frequency instabilities, and fre- quency drifts can be induced which usually relate to phase

and gain changes. However, proper design and alignment

can nearly arbitrarily reduce such phenomena.

It is possible to operate an atomic standard as an ac- tive oscillator or maser oscillator. It is necessary,

nevertheless, to include frequency synthesis and elec- tronic servo principles in such a device because the out- put signal is not a standard frequency and, typically, has

a very low power level (order of picowatts).

The electronic system of a maser oscillator frequency

standard is depicted in Fig. 7. The output of the maser

is amplified in a wideband amplifier, and then fed into a double heterodyne receiver which ultimately translates

the frequency down to dc. This dc voltage is used to control the frequency of the crystal oscillator which drives

the double heterodyne system. In other words, the crystal oscillator is phase-locked to the maser frequency.

Such frequency standards are called "active" devices (in contrast to the "passive" principles discussed above).

|SYNTHESIZE«] 1^^>—- [~HI X

| A

OUTPUT FREQUENCr

VARACTOB

TUNABLE

X-TAL OSCILLATOR

In both "passive" as well as "active" systems, the

slave oscillator is of special importance because its per- formance determines, in part, the frequency stability of the whole standard. For averaging times of less than

those corresponding to the unity gain of the servo loop,

the frequency stability is essentially that of the slave oscillator; for times larger than that, the atomic resonator dominates the stability performance. It must be noted

here that an inferior slave oscillator, i.e., an oscillator with a frequency stability worse than that of the atomic resonator at the unity gain point, will correspondingly

reduce the frequency stability of the system for some

range of averaging times. Thus, in the design of a complete frequency standard, the choice of both the stability

characteristics of the slave oscillator as well as the

atomic resonator in combination with the selection of the

time constant of the servo loop or bandwidth which

determines the unity gain point, is critical to achieve op- timum overall system performance.

5. The Cesium Beam Standard

The cesium standard was the first and still is the most

important of all atomic standards. Developed in the early

1950' s, its first operational use as a laboratory clock came about in 1955 at the National Physical Laboratory

in the United Kingdom, and its first commercial realiza- tion took place in 1958 at the National Company in Massachusetts, U.S.A. Today, a variety of laboratory-type

and commercial standards exist [1], and the world's

internationally coordinated time system is based on a

large number of cesium standards of worldwide distribution. The laboratory devices, or primary standards, are used to define the unit of time, the second [8]. This is done by constructing the apparatus in such a way that

most known perturbing parameters can be independently

and separately measured. Then, corrections can be applied to the output frequency in order to realize, as best

as possible, the unperturbed resonance frequency of the

cesium atom at rest in free space [9, 10, 11]. Figure 8 depicts the schematic design of a cesium beam

standard omitting the electronic systems. The oven is constructed in such a way that an ampule containing

cesium can be inserted and opened inside of the oven via a remote mechanism after the beam device has been

evacuated to very low pressures. Pressures of better

than 10^^ Torr are needed to allow sufficient mean free path from source to detector. Oven temperatures used

vary between 60 and 120 °C, providing vapor pressures

MAGNETIC SHIELD

Figure 7. Schematic of a maser oscillator frequency standard.

Figure 8. Schematic of a cesium atomic-beam resonator. The

frequency input is derived from a quartz-crystal oscillator (typically at 5 MHz) with a frequency multiplier and synthesizer to generate the atomic resonance frequency. Afeedback servo from

the detector output then controls this oscillator.

14
in the range from 10"'' to 10"^ Torr inside of the oven. The state selectors for a cesium beam are typically dipole

magnets, but multiple dipole, double-dipoles, and hexapole magnets have been used. Dual-beam devices have

been built which utilize simultaneously the upper and

lower state by generating two different beams deflected

in opposing directions [10]. Vacuum is typically maintained by ion pumps; these

feature rather sizeable pump capacities in the case of

laboratory-type standards, but are as small as 0.1 l/s internal pumps in the case of commercial cesium standards. These small pumps serve only to scavenge residual gases, most importantly noble gases. Since prior to per- manent seal-off a commercial tube is baked at high temperatures, there is very little evolution of gases inside of

the tube; furthermore the expended cesium acts as an ef- fective getter for many gases. The waste-cesium itself is gettered by strategically placed graphite in solid, as well

as in surface deposition form.

The detector in cesium standards is typically a hot

wire. The low ionization potential of the cesium atom

makes it possible , to surface-ionize cesium atoms on metals with a sufficiently large work function. Metals which

are being used include tungsten, niobium, and tantalum,

as well as platinum-iridium alloys. Since most of these

metals contain significant amounts of contaminants, in particular, potassium, the ions formed on the surface of the hot wire are extracted through a mass spectrometer.

The mass spectrometer is typically a magnetic field, bending the ions by 90 degrees out of the original path of the incoming neutral atoms. The ions are collected on an

electrode. In some high intensity beam laboratory devices the current is directly detected by an electronic

amplifier [9, 10, 11], typically with a field-effect transis- tor front end. In all commercial devices, however, a

tube-internal electron multiplier is used. Thus, the col- lector electrode becomes the first dynode of the multiplier. The electron multiplier provides sufficient gain to

raise the signal level by orders of magnitude above the

noise level of electronic amplifiers. The surface ionization

efficiency, together with the low noise of an electron

multiplier, allows an essentially noise-free detection of

single atoms in cesium beam devices.

The speed of the atoms and the length of the interaction region determine the Q of the cesium resonator. Atomic velocities at the most probable thermal velocity

are 200 to 250 m/s. In many practical cesium beam tubes,

a lower velocity is selected by the beam optics, typically

close to 100 m/s. Thus we find Q values ranging from a few 10^ to a few 10^ over the spectrum of commercial and

laboratory devices. Typical beam intensities at the detector are of the order of 10'' to 10^ atoms per second

corresponding to electronic currents of the order of picoamperes. The resulting signal-to-noise for averaging

times of one second is several thousand and allows fractional frequency stabilities at one second in the range

from 10"^" to 10"^*^. Cesium is preferred over other atoms

because of its convenient frequency, permitting the use

of readily available electronic techniques while featuring

a sufficiently high resonance frequency for achieving high

Q values; cesium has sufficient vapor pressure not far above room temperature, and the detectability of cesium

by surface ionization is nearly 100% efficient. The most significant limitation to frequency stability

and accuracy of cesium beam devices appears to be the

distributed cavity phase shift [12, 13]. It arises from the

fact that the cavity phase is distributed along the axis as

well as across both coordinates of the plane perpendicular to the beam. As a result, different trajectories sample

different cavity phases in each of the two interrogation

regions; and the cavity phase difference which affects the

shift of the Ramsey pattern becomes trajectory dependent. Since with spatial state selection the trajectories are velocity dependent, we have a two-parameter dependence of the effective cavity phase shift, and thus of the

apparent resonance frequency. Since the distributed cavity phase shift can be assumed to be of the order of 10

""^

to 10"^ radian, cesium tubes suffer from this effect,

depending on their resonance line Q, anywhere between

10"^"^ to 10"". Since optimum power is a function of the

velocity, we have a change of the average trajectory lo- cation with microwave power, and thus frequency shifts as a function of microwave power. Other related shifts [13] include temperature gradient related frequency

shifts, acceleration or gravitationally induced frequency

shifts, vibration induced perturbations, and voltage re- lated frequency shifts, e.g., those caused by change in the

collector voltage potential near the hot wire [14] (causing

a samphng of different regions of the hot wire

corresponding to different atomic trajectories). Thus,

state selection and detection techniques which are uni- form across the beam and retain spatial isotropy of the

velocity distribution would essentially remove all effects related to the distributed cavity phase shift. Optical

pumping appears to be an elegant means to this end, but

at this point in time insufficient experimental data are

available to support its practical feasibility. Optical

detection would be a combined function of the second

state selector which, if optically executed, would interrogate the population levels and act as a photon

transformer for the detection of microwave-induced

atomic transitions.

6. The Rubidium Gas Cell Standard

The traditional design of a rubidium standard features

a lamp filled with rubidium 87 isotope, a filter with ru- bidium 85 isotope, and the cell featuring again rubidium

87 isotope [15]. As illustrated in Fig. 9, it is a happy

coincidence that the optical absorption lines of rubidium

85 largely overlap with one of the hyperfine splitting

components of the rubidium 87 pumping light; thus the

insertion of the rubidium 85 filter effectively suppresses

the light which otherwise would couple to the upper

microwave level of rubidium 87. This results in effi- cient optical pumping, i.e., in population of the upper

-6.835 GHz

ENERGY = hi/

Figure 9. The solid curves represent optical emission

correspondim to the desired (F = 1) ayid the undesired (F = 2) states of Rb. The dashed curves represent absorptions

corresponding to the filter action of^^Rb.

15
microwave level, while the lower level essentially is fully

depopulated.

Figure 10 shows the schematic of a rubidium gas cell frequency standard. In order to achieve full efficiency of

the system, the vapor pressure in the gas cell has to be

adequately high; partial pressures of 10"^ to 10"^ Torr

are a reasonable compromise between achieving good sig- nals while avoiding significant numbers of spin exchange

collisions between rubidium atoms which would lead to frequency shifts [16] and also shorten the relaxation time

and thus the line Q. In order to obtain the needed light

intensity, the rubidium pressure in the lamp has to be at a higher level, and, typically, temperatures between

120 °C and 180 °C in the lamp are used. The filter temperature is adjusted to a temperature between those of the lamp and the gas cell. The gas cell features a buffer

gas mixture typically containing two components, one

with a positive pressure shift on the rubidium frequency,

the other with a negative shift, compensating each

other's temperature coefficients [17]. In practice the

residual temperature coefficient is not much less than 1 part in

MAGNETIC SHIELD

Rb LAMP

Rb-85 BUFFER

GAS J L

r.f

.

LAMP EXCITER

Rb-87

t BUFFER

GAS

POWER SUPPLIES

FOR LAMP, FILTER AND ABSORPTION

CELL THERMOSTATS

Figure 10. Schematic of a ruhiduyn gas cell frequency standard.

Usually, the rubidium is in a liquid/gas equilibrium, provided via a relatively cooler appendix in the

rubidium gas cell. The appendix is cooled via thermoelectric cooling or by providing a thermal gradient oven

with higher temperatures for most of the gas cell with

the exception of the appendix. Alkali-resistant glass

must be used to minimize reaction as well as absorption

and diffusion of rubidium into the glass. Some diffusion

of rubidium into the glass always exists; thus, overfilling

with rubidium is typically the procedure in manufacturing the lamp, filter, as well as the gas cell. Rubidium depletion into the glass is responsible for the fact that the

rubidium vapor pressure is not in true equilibrium with

the liquid phase as would be calculated from the temperature of the appendix containing the liquid phase. The

lamp is typically excited by an rf-discharge in the hundred megahertz range. The rubidium gas cell is housed in a cavity with low Q. The microwave signal can be generated externally or by harmonic generation internal to the cavity using a varactor diode. A separate filter cell is not absolutely necessary. If the rubidium gas cell is filled with an isotopic mixture of rubidium 85 and rubidium 87, the light of the lamp penetrating the first layers

of this isotopic mixture will shift in its spectral characteristics in such a way as to effectively pump the

remainder of the gas cell. In other words, this "in- tegrated" gas cell [18] acts hke a filter cell at the side

facing the lamp and, on the opposite side, like the gas

cell resonator. The detector is typically a photocell, i.e..

a photo-voltaic device, which has a surface area commensurate with the effectively used light beam (of the order

of 1 centimeter diameter). Unlike the case of the cesium

standard, there is no optimum power. The reason for this

is the fact that we have no transit-time phenomena as in a cesium beam, but rather the relaxation spectrum of the

rubidium atoms due to collisions or radiation phenomena.

Best conditions are obtained by selecting the microwave

power in such a way that the product of signal-to-noise

and line Q is optimized.

As discussed above, the atomic ensemble does not per- form spatial averaging but rather acts as a superposition

of the individual resonance frequencies of the spatially

fixed atoms leading to an inhomogeneous resonance line. As a result, dependencies occur against all parameters

which are spatially selective [19]. This includes light in- tensity, magnetic field gradients across the cell, and

microwave power due to the mode structure of the micro- wave cavity. In summary, the rubidium gas cell standard

is a less "primary" device than the cesium beam tube.

Basic frequency shifts due to buffer gas and light are of the order of parts in 10^ to parts in 10^. The rubidium

standard thus is sensitive to the stability in time of those

parameters which cause these shifts [20]. Therefore, it is no surprise that rubidium gas cell standards show sys- tematic frequency changes on the order of 10"^^ to 10"^^/day, corresponding to parameter changes as a function of time on the order of 10"^ to 10^^/day.

One of the primary limitations of the rubidium gas cell standard is the fact that it features an inhomogeneous

line. In order to transfer the rubidium gas cell into a standard with a homogeneous line, the buffer gas has to be essentially removed in order to realize effective spatial averaging of the individual atoms. As a result, wall

collisions dominate as was shown experimentally [21] us- ing high-polymer coatings such as paraffin and po- lyetheylene as wall materials. This approach allows the

removal of the buffer gas while retaining the optical

pumping principle with separate filter or isotope mixture

in the gas cell. Unresolved at this time is the stability in time of the coating materials when exposed to rubidium.

Another limitation of rubidium standards appears to be

the residual light shift. One portion of the light shift is due to the inhomogeneous line and would be cured with

the wall coating approach and spatial averaging. The

other remaining light shift is due to the insufficient overlap of the rubidium 85 absorption and the rubidium 87

emission (see Fig. 9). This leads to an asymmetric spectral filtering of the rubidium 87 line which pumps the

lower microwave" level. This light shift could be removed

if a light source is employed which is centered and symmetrically coupled to the upper microwave level. Obviously, this can only be a laser source. The limitation of a

practical standard would then be the stability of the laser frequency with temperature.

7. The Hydrogen Maser

An atomic hydrogen maser is depicted schematically in Fig. 11. It is shown in operation as an active maser oscil- lator. The beam source is a glass container in which

atomic hydrogen is created in a radio-frequency

discharge from molecular hydrogen. Molecular hydrogen

is supplied from a hydrogen source, such as a bottle, or

in some more recent experiments, from metal-hydrides at elevated temperatures. The hydrogen beam is formed by

a collimator which can be a single channel or a multichannel design. The hydrogen beam intensity is determined

by the power of the dissociating discharge, as well as the

16
"C-FIELO"-

VACUUM ENVELOPE -

KASER

FREQUENCY

OUTPUT 1.4?0.405,7S1 Hz

MAGNETIC SHIELD

. OlSSOCIATOB/BEAH SOURCE

VALVE POWER SUPPLY

Figure 11. Schernatic of an atomic hydrogen maser frequency

standard.

flux of molecular hydrogen. In order to achieve stable beam intensities, it is mandatory to regulate both the

power of the dissociating discharge as well as the supply

of molecular hydrogen. Palladium metal has the property

that, at elevated temperature, it has a very high diffu- sion coefficient for molecular hydrogen, while essentially

remaining impermeable to all other gases. Thus via temperature changes it can operate as a regulated leak while

at the same time purifying and rejecting impurities in the

hydrogen gas. For effective dissociation, the typical

pressure in the source which operates at around 100 MHz

is 10"^ Torr. The beam, W'hich is formed at the output of the source, must be state selected [22]. The state selector almost al- ways is a multipole, axially symmetric magnet, which

achieves high field strengths while providing an axially

symmetric state selected beam as well as some limited

spatial focusing. The most probable velocity is focused on the entrance aperture of the storage bulb. The storage

bulb is placed inside of a TEOll cavity. In order to retain a high cavity-Q, the storage bulb is typically made out of quartz which shows very low dielectric losses. The inside

of the storage bulb is coated with a high-polymer substance. Fluorocarbons such as Teflon have provided the

best surfaces, allowing highly elastic colhsions. Typically, many thousands of collisions can take place before

sufficient phase error is accumulated to de-correlate the

radiation process. If the cavity-Q is high enough, i.e., cavity losses are low and the supply of state selected hydrogen is sufficiently high, the energy transferred into

the cavity by the state selected hydrogen atoms exceeds

the energy losses of the cavity and its associated elec- tronics: the maser becomes an oscillator, providing a fre- quency close to the atomic resonance frequency of atomic

hydrogen at 1,420,405,752 Hz. It is desirable to achieve

oscillations at the lowest possible beam intensities to make the storage time in the storage bulb as long as possible. Since storage times in excess of one second are

possible, the escape relaxation time of the bulb, i.e., the

flow of gas in or out of the bulb, is arranged in such a way as to be just below the wall collision relaxation time

of the hydrogen atoms for optimum performance. Storage

times of about one second lead to line-Q values of above

10^. These are the highest Q values in all presently used

atomic standards, and are the basic reason for the exceptional performance of the atomic hydrogen maser. The

output of the device is a signal at the picowatt level. For an active maser oscillator, the crystal oscillator is

still needed in order to provide a standard frequency

output. A signal which is derived from the crystal oscillator is generated at the atomic hydrogen maser frequency.

This signal is compared to the output of the atomic

hydrogen maser using a superheterodyne, multistage

receiver, as was discussed in section 4. Ultimately, the

output of a phase detector is used to servo the crystal oscillator to be phase-coherent with the atomic hydrogen maser output signal. For this type of servo, which is phase sensitive, the frequency stability improves as the

inverse of the averaging time t, as compared to a t"^

dependency for frequency-lock servos such as those in the "passive" cesium beam and rubidium gas cell standards. In an active device, cavity pulling is more pro- nounced [23]. Cavity pulling can be approximated by the

following equation:

V - VR - -— (Vc - Vr) . We note that the pulling factor translating the cavity

offset into the frequency offset of the standard is the

simple ratio of the cavity-Q and the line-Q as contrasted

to the square of this ratio for passively operating devices

(far below oscillating threshold). The puUing factor in typical hydrogen masers is of the order of 10"^ for the

translation of cavity frequency offsets into output fre- quency offsets. Thus, active hydrogen masers are characterized by designs to stabilize the microwave cavity. This

stabilization can be done by using low thermal expansion

materials such as pure quartz as the cavity material

within an oven enclosure. Such ovens may yield temperature stabilities of better than one millidegree.

There is an alternate way to stabilize the microwave

cavity's resonant frequency [22]. By increasing the

amount of hydrogen supplied to the storage bulb, an in- creasing number of spin-exchange collisions between the

hydrogen atoms take place, shortening the radiation life- time of the hydrogen atoms. The result is a broadening of the resonance line, i.e., a decrease in the line-Q. If the

cavity is not tuned perfectly, the pulling equation leads

to a frequency change as a result of the change in Q due

to spin-exchange. Thus, a modulation of the hydrogen

beam intensity results in a modulation of Q which can be

used to determine the point where (v - vr) becomes zero and the cavity is properly tuned.

For hydrogen beam intensities insufficient to provide

self-sustained oscillations, microwave radiation, i.e., its amplification, can be detected by using a microwave re- ceiver, and a crystal oscillator can be servoed to the

atomic resonance signal [24]. In this process, frequency

or phase modulation in conjunction with either the ab- sorption feature or the dispersion feature of the atomic resonance can be used to generate the reference signal

for locking the crystal oscillator. This is the principle of the passive atomic hydrogen maser [25]. Since the detection of the atomic resonance still relies on the (phasesensitive) measurement of the atomic radiation, the

pulling equation for the active maser still applies. Therefore, cavity stabilization is as important for a passive maser as for the active one with the only difference that

lower cavity-Q values may be used. An elegant method

for cavity stabilization is the absolute probing of the cavity resonance by injecting sidebands into the \vings of the

cavity resonance and detecting any mistuning of the cavity. This is shown in Fig. 12 where two servos are employed [25], one at a high frequency such as 10 kHz to probe the cavity resonance and a much slower modulation, e.g., at 1 Hz, locking the crystal oscillator to the

atomic reference frequency.

It must be noted that one of the exquisite features of the atomic hydrogen maser in the active mode is that its

17
1420 MHi CAVITY

Figure 12. Block diagram of a passive hydrogen maser fre- quency standard.

stability improves as t"^ Stabilities of 10"^^ at 1000 sec- onds have been achieved and stabilities in the 10"^^ range

have been repeatedly reported. We are rejninded that

the main reason for this superb performance of the hydrogen maser is the line-Q which exceeds that of the

longest cesium beam tube by almost an order of magnitude. We remember that the concept of optimum power,

so convenient in the case of the cesium atomic resonator,

is absent in the case of the hydrogen storage principle, as

it was in the case of the rubidium gas cell. Again, we

have an exponential relaxation phenomenon in the

storage bulb, which is due to the escape of atomic hydrogen out of the storage bulb. Thus, increasing microwave

power leads to interrogation of atoms with lesser relaxation times and corresponding line broadening, much as in the case of the rubidium gas cell. Again, in the passive

hydrogen maser, the microwave power is set in such a way as to maximize the product of line-Q and signal-to- noise.

It is difficult to imagine a storage-type device with a performance exceeding that of the hydrogen maser. This

is based on the fact that the hydrogen atom is the smallest of all atoms and permits a maximum number of colli- sions with minimum relaxation.

As we discussed many times, the very high line-Q of the hydrogen storage principle leads to a very low flicker or frequency floor, as well as to excellent long term

time-keeping performance. The key to frequency fluctua- tions in medium and long term is cavity pulling. The

elimination of cavity pulling appears most elegant with

the passive maser principle. The passive principle also al- lows less costly, more practical and smaller devices

[26, 27, 28, 29]; however, the reduced cavity volume (e.g., of a dielectric cavity) leads to less available volume for the storage container, and thus increased wall-relaxation

and consequential reduction in the line-Q. Unfortunately,

for the purpose of this discussion, it appears that the re- duced sensitivity to cavity pulling due to the lower cav- ity-Q and the passive control principle is compensated by

the lower line-Q which leads to increased susceptibility to frequency perturbing parameter changes. Thus, it may

be concluded that for excellence in clock performance

(i.e., a very high line-Q), a so-called full-size passive hydrogen maser appears to be the most promising device

realization.

8. Fundamental Advances

As compared to today's microwave frequency standards, fundamental improvements in both Q and signalto-noise appear to be possible only by the use of higher

frequencies, including infrared and optical frequencies. If

it is assumed that these higher frequencies do not pose

additional technical problems, the advantages are obvious: For the same length of the device, the Q improves

with the resonance frequency. Thus a device with a reso- nance at 100 GHz and otherwise similar dimensions, has

a 10 times improved Q as compared to cesium, and in the near infrared and visible regions, an enhancement by a

factor of 10,000 is available. With such enhancement fac- tors, Q could be traded against signal-to-noise, by build- ing smaller devices making more efficient use of the

spatial acceptance limitations, or by multiple interrogation of the atoms or molecules, cycling each atom several

times through its two states. Another technically elegant

solution is optical pumping for state selection and possibly also detection. Optical pumping was tried on the ru- bidium beam and, more recently, on the cesium beam,

originally with rf-excited lamps, but more recently with

lasers, including diode lasers and dye-lasers. The principle advantage of this technique is that it removes the dis- tributed cavity phase shift limitation. Apparently, ion storage is the only known technique

offering line-Q' s well in excess of 10^*^ in the microwave

region. Although ions with resonance frequencies in the

infrared or visible radiation region are typically considered, very attractive resonances can be found in the

GHz region. Most importantly this includes mercury with

hyperfine resonances at about 26 and 40 GHz

[30,31,32,33]. The achievement of very high line-Q's

is a result of stable storage in electromagnetic fields [34]. The most attractive and elegant solution is the radiofrequency ion trap. An analysis of the frequency shifts due to this confinement method has shown their influence

to be less than 10"^^ The main drawback of this method

is the difficulty in determining the speed of the ions

which is needed to calculate the second order Doppler

effect. Both the first and second order Doppler shifts could be

reduced in a fundamental way if the atoms could be

slowed down or cooled. Ion traps offer the possibility of

using radiation pressure to cool. Recently tunable,

narrow-band optical sources have become available which

make detection of the small number of ions much easier

than in the past and effective cooling possible. Typically, not many more than 10"* ions can be stored and cooling of

electromagnetically confined atoms or ions has been

demonstrated [35, 36]. Optical pumping appears to be

necessary to effectively interrogate any stored ion- ensemble without perturbation. Other auxiliary tech- niques in need of refinement for practical uses include

the generation of ions and the interrogation by a micro- wave signal. J

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18
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(Electronics Ind. Assoc., Wash., D.C., 1977), p. 555.

[15] P. Davidovits and R. Novick, Proc. IEEE 54, 155 (1966).

[16] M. Tetu, R. Fortin, and J. Y. Savard, IEEE Trans. In- strum. Meas. IM-25, 477 (1976).

[17] G. Missout and J. Vanier, IEEE Trans. Instrum. Meas. IM24, 180 (1975).

[18] E. Jechart, Proc. 27th Annual Symp. on Freq. Control (Elec- tronics Ind. Assoc., Wash., D.C. 1973), p. 387.

[19] A. S. Risley and G. Busca, Proc. 32nd Annual Symp. on

Freq. Control (Electronics Ind. Assoc., Wash., D.C,

1978), p. 506.

[20] P. Cerez and F. Hartmann, IEEE J. Quantum Electron. QE-13, 344 (1977).

[21] G. Busca, R. Brousseau, and J. Vanier, IEEE Trans. In- strum. Meas. IM-24, 291 (1975).

[22] D. Kleppner, H. C. Berg, S. B. Crampton, N. F. Ramsey,

R. F. C. Vessot, H. E. Peters, and J. Vanier, Phys. Rev.

138, A972 (1965).

[23] J. Viennet, C. Audoin, and M. Desaintfucien, IEEE Trans. Instrum. Meas. IM-21, 204 (1972).

[24] H. Hellwig and H. E. Bell, Metrologia 8, 96 (1972).

[25] F. L. Walls and H. Hellwig, Proc. 30th Annual Symp. on

Freq. Control (Electronics Ind. Assoc., Wash., D.C,

1976), p. 473.

[26] H. E. Peters, Proc. 32nd Annual Symp. on Freq. Control

(Electronics Ind. Assoc., Wash., D.C, 1978), p. 469.

[27] H. T. M. Wang, Proc. 34th Annual Symp. on Freq. Control

(Electronics Ind. Assoc., Wash., D.C, 1980), p. 364.

[28] E. M. Mattison, E.L. Blomberg, G.U. Nystrom, and R. F.

C. Vessot, Proc. 33rd Annual Symp. on Freq. Control

(Electronics Ind. Assoc., Wash., D.C, 1979), p. 549.

[29] D. A. Howe, F. L. Walls, H. E. Bell, and H. Hellwig, Proc. 33rd Annual Symp. on Freq. Control (Electronics Ind. Assoc., Wash., D.C, 1979), p. 554.

[30] M. Jardino, M. Desaintfuscien, R. Barillet, J. Viennet, P. Petit, and C. Audoin, Proc. 34th Annual Symp. on Freq. Control (Electronics Ind. Assoc., Wash., D.C, 1980),

p. 353.

[31] H. A. Schuessler, Metrologia 7, 3 (1971).

[32] M. D. McGuire, Proc. 31st Annual Symp. on Freq. Control

(Electronics Ind. Assoc., Wash., D.C, 1977), p. 612.

[33] D. J. Wineland, W. M. Itano, J. C. Berquist, and F. L. Walls, Proc. 35th Annual Symp. on Freq. Control (Elec- tronics Ind. Assoc., Wash., D.C), to be published.

[34] H. G. Dehmelt, Adv. At. Mol. Phys. 3, 53 (1967).

[35] D. J. Wineland, R. E. Drullinger, and F. L. Walls, Phys. Rev. Lett. 40, 1639 (1978).

[36] W. Neuhauser, M. Hohenstatt, P. Toschek, and H. Dehmelt,

Phys. Rev. Lett. 41, 233 (1978).

19

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Performance of the Three NRC 1 -Meter CsVI Primary Clocks

A. G. Mungall, H. Daams, and J.-S. Boulanger

Division of Physics, National Research Council, Ottawa, Ontario, Canada, K1A 0R6

The performance of the three 1-m interaction length NRC primary cesium clocks, CsVIA, CsVIB,

and CsVIC, is outlined for their initial year and a half of operation as primary clocks, which com- menced in December, 1979. Measurements of their systematic corrections made then and again early in 1981 indicate only small changes, within the accuracy limit of ± 1.5 x lQ~^'-\ The frequencies of the three clocks compared with that of CsV remained within ± 2 x 10"''' despite certain problems concerning the electronics systems which became apparent during this test period and which limited their long- term frequency stability. These problems, which caused frequency shifts arising from spectral impurities in the microwave exciting signal, have been partially corrected, with consequent improvement of the agreement between all four NRC primary clocks to better than 5 x 10"'^. The potential long-term

frequency stability appears to be a few parts in 10'^. Key words: frequency stability; performance; primary cesium clocks; spectral impurities; systematic

corrections.

1. Introduction

When the CsVI clocks were designed early in 1976, the

intent was to achieve a stability as good as that of the

NRC long beam primary cesium clock, CsV, and an accu- racy only slightly less, in a unit much smaller in physical

dimensions at a significantly lower cost [1-4]. It was also hoped that clocks embodying such a design might prove

attractive from the point of view of later commercial exploitation, since their potential accuracy would be about

•50 times better than that of currently available commercial clocks. Three cesium clocks were built of this design,

as shown in Fig. 1, during 1977 and 1978, and began

operation as secondary clocks late in 1978. As mentioned

in an earlier publication [5], which outlined their salient

design characteristics, certain difficulties arising from

thermo-electric currents flowing in the magnetic shields

and the aluminum interior support cylinder, and from

mechanical distortions were encountered. The former were overcome by improved clock and room temperature

control, and the latter by spring-mounting the ion pumps

and the microwave excitation system.

Following an initial test, evaluation, and adjustment

period, a complete accuracy evaluation was made of each

Figure l. The three CsVI clocks.

21
clock late in 1979. This was repeated early in 1981, and

only minor alterations in the systematic corrections were

observed.

Since December, 1979, all three clocks have operated

completely independently as primary standards, producing the scales of proper time PT(NRC CsVIA), PT(NRCCsVIB), and PT(NRC CsVIC). The relations between

these three scales and UTC(NRC), which is based solely on CsV, have been reported at 10-day intervals to the

Bureau International de VHeure.

This paper will describe their performance during the

first year and a half of their operation as primary clocks,

based not only on their accuracy evaluations, but also on comparisons between their scales of proper time and

UTC(NRC), and between UTC(NRC) and TA(NRC), and

UTC(USNO), UTC(BIH), TA(NBS), and TA(PTB).

2. Results

2.1 Accuracy Evaluations

It is essential in primary clock operation that the vari- ous systematic corrections such as the cavity phase

difference, the second order Doppler shift, and the fre- quency offset dependent on the direction of the C field remain constant. They should also be as small as possible, with magnitudes similar to that of the total clock uncertainty. The two complete evaluations so far performed

on the CsVI clocks show that these two requirements

have been met. None of the corrections exceeds 2 or 3 x

10"^"^, and each remains constant within several parts in

lO^'* except when known changes have been made or have

occurred to some component of the clock. The latter proviso is particularly applicable in the case

of the C-field reversal effect, which appears sensitive

primarily to changes in the form of the C field in the vi- cinity of the two microwave interaction regions. Such a

sensitivity is to be expected on the basis of an analysis

presented in a previous publication [6]. Measurements of

the C-field uniformity, using the set of six axial coils which excite the (4, -4) - (4, -3) transition, are per- formed regularly to check the constancy of this frequency

shift. Measurements during the first year of operation, during 1980, disclosed unexpected instabilities in the mi- crowave excitation systems which originated in all the

component units, the 5-90 MHz frequency multipHer, the

12.6 MHz frequency synthesizer, the step recovery diode

matching network, and the Gunn diode phase-locked fre- quency source. These all affected the spectral purity of the beam excitation signal and led to time-dependent fre- quency shifts. Modifications made to all these units have

improved their performance, but further work is neces- sary to attain consistently the long-term frequency stability of which the clocks are capable.

2.2 Short-Term Frequency Stability

Measurements of the short-term frequency stability of

each of the CsVI clocks intercompared, or compared with

CsV, indicate that for values of t between 2 and 10^ s, the value of ct(2, t) for any combination of the clocks can be described by the expression 4.6 x 10~^^t The

value of ct(2, 10'^ s) is about 1.2 x 10 or 8 x 10 per

clock. The dependence of a on t is shown in Fig. 2.

10''^ '

I I 1

I I I I I mI I '

10 10^ 10^ lo'' 10* V, SECONDS

Figure 2. The short-term frequency stability of the CsVI docks

both intercompared and compared with CsV.

2.3 Long-Term Frequency Stability

The long-term frequency stability of the CsVI clocks is indicated in Fig. 3, which shows comparisons of their scales of proper time and UTC(NRC), based on CsV.

Also shown in the figure are comparisons of UTC(NRC)

and TA(NRC) with UTC(BIH), UTC(USNO), TA(PTB)

and TA(NBS). The complete accuracy evaluations for both CsV and the CsVI clocks are indicated in Fig. 3 as well as the dates for a number of partial evaluations of

the latter. Also shown are the times at which a variety of

clock adjustments or equipment modifications were

made. In a number of cases, slope changes, indicating

frequency shifts, are evident at these times. The figure

shows that throughout the measurement period the rela- tive frequencies of CsV and the CsVI clocks were generally within ±2 x 10"^'^ despite recognized difficulties with the electronics systems. Subsequent to the 1981

CsVI evaluations, and after partial correction of these

difficulties, this value decreased to about 5 x 10^^"*. During periods for which no adjustments or modifications were made the long-term relative frequency stability, a(2, t), drops below 1 x '10"", as indicated in Table 1. This table also shows that if the C-field fre- quency steering corrections required in primary clock

operation are taken into account then the value of a(2, t) decreases to a few parts in 10''^. Examination of the lower portion of Fig. 3 showing

comparisons between the NRC time scales based on CsV

and those of the BIH and other national laboratories indi- cates that the frequency variations between these scales are of the order of ±2 x 10"''^ with the sole exception of

TA(NRC) - TA(PTB), which exhibits maximum changes

of about ±1 X 10". These comparisons, based principally on LORAN C measurements, are subject to daily and

annual signal propagation delay fluctuations, but the

latter appear to be appreciably smaller than the variations shown in the figure. All these variations are similar

in magnitude to those measured for the CsVI clocks with

respect to CsV during a period when known causes existed to explain the CsVI frequency fluctuations. It should therefore be possible to reduce the limits of the

fluctuations between the NRC primary clocks to values

significantly lower than those at present occurring

between the scales of the various different laboratories.

3. Conclusions

The three 1-m NRC CsVI clocks, now in operation for about two and a half years, provide first order accuracy

22
COMPARISONS BETWEEN UTC(NRC) AND PKNRC CsVIA), PKNRC CsVIB), AND PKNRC CsVIC)

i CsVI CLOCK SYSTEMS FAILURES OR ADJUSTMENTS

I

CsVI PARTIAL RE-EVAUJATIONS

MJD

Figure 3. Time scale comparisons between UTC(NRC) and PT(NRC CsVIA), PT(NRC CsVIB), and

PT(NRC CsVIC) shown in the upper portion, and UTC(NRC) and TA(NRC) mth UTC(BIH),

UTC(USNO), TA(NBS), and TA(PTB) in the lower portion.

Table 1. Relative frequency stability for the NRC scales of proper time, unth and without corrections for C field steering of the clock frequencies

Time scales Meas. Period Total time, T a (2, t) X 10-15 compared MJD to MJD days days Uncorrected Corrected

CsVIA-CsV 44303 44354 51 17 6.2 2.2 CsVIB-CsV 44583 44658 75 25 1.2 1.7 CsVIC-CsV 44301 44430 129 43 6.3 3.6

and frequency stability in a clock appreciably smaller

than CsV or other comparable primary standards. It is likely that further improvements in the electronics sys- tems will lead to the reahzation of a long-term stability of a few parts in 10^^. References

[1] A. G. Mungall, R. Bailey, H. Daams, D. Morris, and C. C.

Costain, Metrologia 9, 113 (1973).

[2] A. G. Mungall, H. Daams, D. Morris, and C. C. Costain,

Metrologia 12, 129 (1976).

[3] A. G. Mungall and C. C. Costain, Metrologia 13, 105 (1977).

[4] A. G. Mungall, IEEE Trans. Instrum. Meas. IM-27, 330

(1978).

[5] A. G. Mungall, H. Daams, and J.-S. Boulanger, IEEE Trans. Instrum. Meas. IM-29, 291 (1980).

[6] A. G. Mungall, Metrologia 12, 151 (1976).

23

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Prospects for Cesium Primary Standards at the National Bureau of Standards

L. L. Lewis, F. L. Walls, and D. A. Howe

Time and Frequency Division, National Bureau of Standards, Boulder, CO 80303

An application of optical pumping, in conjunction with a number of design improvements, may per- mit the development of a cesium primary standard vnth an accuracy an order of magnitude better than

that of our present primary frequency standards, NBS-4 and NBS-6. Limitations to short-term stability, as well as possible errors in accuracy, are discussed. Key words: atomic clock; atomic frequency standards; cesium frequency standard; light shift; optical

pumping.

1. Introduction

The United States' primary frequency standard, NBS6, was placed into service in 1975. This cesium standard

has an accuracy of about 8 x 10"^'*, long-term stability

better than 1 x 10"^'*, and short-term stability of about

5 X 10"^^ T"^. Research in progress at NBS is directed

tovi^ards the design of a new cesium standard with an

order of magnitude improvement in accuracy and longterm stability and a factor of two or three improvement

in short-term stability. Much of this improvement may be

obtained through the use of optical pumping state preparation and fluorescence detection, and the rest of

the advancement vdll come from a number of technological innovations in the clock system.

The accuracy of NBS-6 is limited in large part by mi- crowave phase shifts associated with losses in the Ramsey cavity [1]. The results of the 1980 evaluation of

NBS-6 are given in Table 1, clearly illustrating this fact.

The size of the apparent shift of the microwave resonance frequency is approximately

Av4> (1)

where Acj) includes both end-to-end and distributed cavity phase shift, and A v^^ is the microwave linewidth of the

central Ramsey feature. Since Av^ ~ 30 Hz, one con- cludes from Table 1 that A(J) ~ 3 x 10"^ radian. This is a reasonable value for a cavity structure as large as the one used in NBS-6 [2]. Assuming a linear dependence of the distributed cavity phase shift upon position across the microwave cavity window of ~ 1 x 10"^^ rad/mm [1, 2], it would be necessary to obtain retrace of the cesium

beam to within ~ 100 n-xn in order to ensure a frequency

error associated with the cavity phase shift of less than

10"^^. We believe that the use of optical pumping state preparation and fluorescence detection techniques in a cesium atomic beam frequency standard should make

Table 1. NBS-6 uncertainties, 1980 evaluation.

Source of Uncertainty Bias {uy x 10^3) Uncertainty (x lO^^)

(a) Cavity Phase Shift (for one direction) (residual first-order Doppler Shift)

(b) Second-order Doppler

Shift

Pulling by neighboring transitions

Magnetic Field Effects

(a) Offset due to finite field

(b) Magnetic field inhomogeneity

(c) Majorana Transitions

Servo System Offset

(a) Amplifier offsets

(b) Second harmonic

distortion RF Spectrum

Cavity Pulling

3.3

(typical)

-2.8

(typical)

-hO.3

(typical) +0.02

0.80

0.10

0.20

0.02

0.02

?

0.1 0.2

0.1

0.01

RMS error due to systematic frequency biases 0.87

Random Uncertainty 0.15

25
such a precision retrace possible. In addition, different

microwave cavity structures than that used in NBS-6

may reduce the cavity phase shift dependence upon position, thereby relaxing the retrace requirement by as much as an order of magnitude.

The effect of Majorana transitions on the accuracy of

Cs standards is not clear. Various authors [3] have sug- gested that they may produce uncertainties as large as one part in 10^^. If optical pumping is used for state preparation and detection, such Majorana transitions

should not occur, since a uniform C-field strength can be

maintained throughout the clock. In addition, by using

laser techniques, it should be possible to measure Majorana transitions which might occur. Optical pumping may reduce other sources of uncertainty as well. If only a single magnetic sublevel is prepared, there will be no Am^ = 0 neighboring transitions which would shift the central resonance. This would

permit a much lower value of C-field, which would relax

restrictions on measurement of the finite field, as well as reduce the effect of magnetic field inhomogeneities and

magnetic field changes with time.

The other contributions to uncertainty listed in Table 1 are not considered to be serious limitations to accuracy

at the 10"'^ level, assuming some reasonable improvements in electronics and measurement techniques are made. However, as discussed below, the introduction of

optical pumping will create new sources of error which

must be considered.

2. Optical Pumping and Fluorescence

Detection

The development of high performance, cw, single-mode

diodes for the communications industry has fortunately

provided an optical pumping source appropriate for use

in atomic frequency standards. The linewidth, intensity

noise, frequency noise, and wavelength tuning characteristics of these devices make them very attractive for

this purpose [4]. At the present time, it is reasonable to expect laser diodes to operate tuned to an atomic transi- tion for months at a time.

The first use of laser diodes to optically pump a cesium

beam atomic clock was made by Arditi and Picque [5]. They used a single GaAlAs laser tuned to the D2 line of Cs (852 nm) in order to pump atoms into one of the

F =3, 4 ground-state hyperfine levels (Pig. 1). Detection

of a microwave transition within the Ramsey cavity was

accomplished by fluorescence detection using the same

laser. The work at the National Bureau of Standards (NBS)

has moved in a somewhat different direction, requiring more than one laser. It is possible to pump nearly every

Cs atom in an atomic beam into a single magnetic sublevel [6]. If one laser is tuned to • the 6 1/2 F=3 ^ 6 ^Pa^ F'-^ transition, and a second laser is tuned to the 6 1/2 ^ =4 6 F'=4 transition with

plane electric polarization parallel to a weak magnetic

field (-rr-polarization), only atoms in the F =4, m^-^O sublevel of the ground state will remain unaffected. This

selection rule is clear when one notes that the ClebschGordon coefficient <Fjm,Fmj\F'mf> - <F100| F0> = 0, where j is the angular momentum of the photon. Eventually, most of the atoms will be pumped into

this magnetic sublevel. Alternatively, a similar arrangement can pump atoms into the F =3, mp -0 state. Fluorescence detection can be performed with a third

laser which is tuned to either the F =4 F' =5 or the

Mp= -4-2 0 2

I I I I I I I I 253 MHz

I I I I I I I I 203 MHz

I I I I I I I 152 MHz

I I I I I

5-1

4

3

F'=2

6^P3/2

852 nm

M = -4-2024 I I I I I I I I I 9.19263177 GHz

II

F=3

6^Si/2

Figure l. Term diagram for Cs. -

F -S F' -2 transition. In these two cases (with

appropriate polarization of the F =B ^ F' =2 laser light to avoid pumping into the = ±3 sublevels), atoms in the excited state return to the original ground-state

hyperfine level. Thus, a large number of fluorescence

photons (limited by excited-state energy separations)

may be obtained from each atom in the atomic beam. This

technique should permit unity quantum detection effi- ciency, even if the light collection efficiency is considerably less than one. However, as discussed below, there

may be systematic effects which would make this detection method undesirable. In this case, it is still possible

to use a laser tuned to a pumping transition, and take

care to obtain a high collection efficiency at the detector. An additional advantage a pumping transition provides in detection is that every atom is weighted equally regardless of velocity, which may provide immunity from cer- tain systematics. On the other hand, detection by

fluorescence on a cycling transition produces a relatively

larger signal for slower atoms which remain in the laser beam longer, which would increase the effective

microwave Q. Still another advantage of optical pumping

is that it should be possible to operate atomic beams in opposite directions simultaneously without interference.

This would permit direct measurement of retrace, as well as very rapid modulation of beam direction (or even con- tinuous operation of two beams) for purposes of cancella- tion of cavity phase shift. A final consideration in the use

of fluorescence detection is that the laser beam can be

made to intersect the atomic beam at a slight angle,

thereby selecting low velocity atoms. This would cause an increase in the microwave transition Q, with some

accompanying loss of signal. This result is a consequence

of the Doppler shift, which changes the effective velocity

distribution of the cesium beam. As further experiments

are performed, the relative merits of these various detection methods will become clearer. Using values of atomic beam current Ib -1.0 xiA

(6 X 10^ atoms/s) and microwave transition Q ~ 10^, it should be possible to obtain stabilities of — 1 x 10~^^ T~^, even if a pumping transition is used for detection

purposes. The high beam current suggested here would come partially from the increased number of useful

cesium atoms (a factor of 8 more than that of NBS-6,

26
through optical pumping of all magnetic sublevels), and

partially from new oven design. A reflux-type oven is presently under consideration.

3. Additional Uncertainties and Biases

Perhaps the most serious source of frequency uncertainty introduced by optical pumping techniques is that

of near-resonant light shifts [7, 8]. Although it should be

possible to prevent stray laser light from entering the

microwave region, fluorescence hght from the atoms will pass through the cavity, and interact with atoms in the

"C" region, changing the ground-state hyperfine splitting shghtly. The form of this light shift is dispersive,

but averaging over excited-state hyperfine levels and in- cluding Doppler shifts associated with the MaxwelHan

velocity distribution in the atomic beam, gives a nonzero

value for the shift. Preliminary calculations have been

made at NBS [9] for the frequency shift from fluores- cence light for pumping by two lasers of -ir-polarization,

with the C-field transverse to the atomic beam direction,

and including the effect of the tensor light shift [8]. The computations, assuming lasers of equal power

driving the F =4^F' =4andF = 3^F' = 4 transitions

in the pumping region, give a light shift of approximately

4 Hz for a power flux of 1 ixW/cm- at the site of the

cesium atom in the C-field region. A simple computer cal- culation gives ten as the average number of photons emitted per atom in this pumping scheme. Referring to Fig. 2, reasonable values of the clock dimensions would

be li = I2 ^ h = 50 cm, and L = 200 cm. Assuming a beam flux of 6 x 10^ atoms/s at the detection region, and

a free aperture of 2 mm everywhere in the cesium

beam, the total fluorescent power arriving at the first window of the Ramsey cavity from the pumping region

would be about 2 x 10"^ jo-W/cm^. This gives a fractional

frequency shift of ~ 9 x 10"^^, which is comparable to the desired accuracy. Additional coUimation of the atomic

beam before the pumping region would reduce this predicted light shift by a factor of ten or more.

Nevertheless, a more careful calculation of the effect will be made, including the effective velocity distribution as- sociated with cycling fluorescence detection, and including the shift caused by light originating in the detection

region. In addition, versions of optically pumped fre- quency standards being considered contain provision for measurement of the light shift.

^3

>< ^ >

[Beam

A /

A

Cs

Oven

Pump

Laser

Ramsey

Cavity

\

Detection

Laser

Figure 2. Schematic of Cs atomic beam frequency standard. A second source of trouble associated with the laser

optical pumping is deflection of the atomic beam through

photon recoil. The deflection is about 1.6 x 10"^ rad/photon for an average atomic velocity of ~ 200 m/s.

and proportionately greater for slower velocities. If the

Cs beam is irradiated symmetrically on two sides, an ex- pansion of the atomic beam should occur, with magnitude ~ 5 X 10"^ rad. With the same assumptions used above

to estimate the limitations imposed by beam retrace, this should give an uncertainty in frequency considerably less than 10-1^

Still another correction to the Cs microwave frequency,

not unique to optical pumping, but which has not been

fully considered in the past, is the light shift caused by

blackbody radiation. The ac Zeeman shift of the groundstate hfs of Cs, due to the magnetic field of the blackbody radiation is only ~ 10"^^ [11]. However, the light

shift associated with the rms electric field of the blackbody radiation is considerably larger [10]. The approximate magnitude of the shift correction at T = 300 K is about

8Vss

Vhfs - 1.8 X 10 -14 (2)

The sensitivity of an anticipated new cesium primary

standard would be adequate to measure this effect in a

suitably designed apparatus.

4. Conclusion

Optical pumping techniques may improve both the ac- curacy and short-term stability of Cs primary frequency

standards. The greatest anticipated improvement comes

from better retrace upon beam reversal. New problems

associated with laser diode optical pumping will require

serious consideration, but are not seen as major obstacles

to the design of an improved standard.

The authors wish to thank their colleagues in the Time

and Frequency Division for many fruitful discussions.

The contributions of D. Wineland, E. Smith, and D. Glaze were especially helpful.

References

[1] D. J. Wineland, D. W. Allan, D. J. Glaze, H. W. Hellwig,

and S. Jarvis, Jr., IEEE Trans. Instrum. Meas. IM-25,

453 (1976).

[2] R. F. Lacey, Proc. 22nd Annual Symp. on Frequency Control, U.S. Army Electronics Command, Ft. Monmouth,

NJ, 545 (1968).

[3] G. Becker, IEEE Trans, Instrum, Meas. IM-27, 319 (1978); D. W. Allan, H. Hellwig, S. Jarvis, D. A. Howe, and R. M. Garvey, Proc. 31st Annual Symp. on Frequency Control, Ft. Monmouth, NJ, 555 (1977); S. Urabe, K. Nakafiri, Y. Ohta, M. Kabayashi, and Y. Saburi, IEEE Trans. In- strum. Meas. IM-29, 304 (1980).

[4] L. L. Lewis and M. Feldman, Proc. 35th Annual Symp. on Frequency Control, U.S. Army Electronics Command, Ft. Monmouth, NJ (1981) 612.

[5] M. Arditi and J.-L. Picque, J. Phys.-Lett. (Paris) 41, L-379

(1980); see also M. Arditi, these proceedings.

[6] This method was made known to us by L. Cutler. See also H. J. Gerritsen and G. Nienhuis, Appl. Phys. Lett. 26, 347

(1975).

[7] We are grateful to A. Brillet for emphasizing the seriousness

of this systematic to us.

[8] B. S. Mathur, H. Tang, and W. Happer, Phys. Rev. 171, 11

(1968); W. Happer and B. S. Mathui-, Phys. Rev. 163, 12

(1967).

[9] E. Smith, private communication.

[10] W. M. Itano, L. L. Lewis, and D. J. Wineland. Phys. Rev. A

25, 1233 (1982).

[11] T. F. Gallagher and W. E. Cooke, Phys. Rev. Lett. 42, 835

(1979).

27

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

A Cesium Beam Atomic Clock with Laser

Optical Pumping, as a Potential Frequency Standard

M. Arditi

Institut d'Electronique Fondamentale*, Universite Paris XI, Bat. 220, 91405 Orsay, France

A passive microwave cesium beam resonator using optical pumping and optical detection, with a cw

tunable GaAs diode laser, has been realized. The "0-0 clock transition" is detected through a change in the intensity of the fluorescence of the cesium beam. Experimental recordings of the Ramsey pattern agree with a Maxwellian distribution of atomic velocities. Results of prehminary tests, to an accuracy

of a few parts in 10", show good potential for a frequency standard of higher accuracy. Key words: atomic clock; cesium beam; frequency standard; laser diode; optical pumping.

1. Introduction

With a cesium beam frequency standard of the RabiRamsey type, an accuracy better than 10~^^ in the reali- zation of the second appears to be a realistic probability

[1]. This accuracy is limited mainly by second-order

Doppler-shift and cavity phase-shift w^hose estimation

depends on a precise measurement of the velocity distri- bution of the atoms in the beam [2]. The purpose of this paper is to describe a cesium beam

atomic frequency standard where optical pumping could,

perhaps, further extend the limits of accuracy due to these effects. As early as 1950, A. Kastler [3] had proposed to re- place the Stern-Gerlach magnets in a Rabi classical atomic

beam magnetic resonance apparatus by "optical pumping," to create and detect differences in the Zeeman sublevel populations of the ground state of the alkali atoms

[4]. A successful application of these principles was obtained in a **''Rb atomic beam to measure accurately the

hyperfine structure separation of the ground state [5]. In that experiment, optical pumping was achieved using the

isotopic shift of the '^^Rb and *^''Rb resonance lines emitted

by spectral lamps. However, limited by a weak signal to noise ratio of the detection, the device could not be used

conveniently as a practical frequency standard.

The advent of stable, monochromatic and tunable laser sources (diode lasers or dye lasers) in the near

infra-red, had raised the hope of increasing the signal to noise ratio in optically pumped atomic beam clocks. After

the successful experiments of J.-L. Picque on hyperfine

pumping of a cesium beam using a GaAs laser diode [6], an attempt was made to set up an optically pumped

cesium beam clock with optical detection. The preliminary results were described in a brief report [7]. In this paper more details are given on the experimental pro- cedure and on the potentialities for a fully developed fre- quency standard.

2. Optical Pumping of a Cesium Beam

The experimental set-up is shown schematically in Fig.

1. In region A, optical pumping increases the population difference between the F = 4 and F — S levels of the

ground state of cesium atoms. In region C, microwave in- teraction takes place, and in region B the microwave

resonance is detected with a photocell by a change in the

intensity of the beam's fluorescence. In this device, the atomic beam is axially symmetrical,

and the velocity distribution of the cesium atoms is the same across the beam since there is no spatial selection as with the A magnet, thus reducing considerably the

problem of the spatially distributed cavity phase-shift

[8]. Moreover, the device can be made completely reversible by placing a cesium oven at each end, thus simplifying the retrace measurement of the residual cavity

phase-shift. Also, as will be seen later, because of the

high spectral density of the laser light, the cesium atoms

wall all be pumped with the same efficiency, regardless of their velocity, so that the velocity distribution will be

broad, very close to a Maxwell-Boltzmann distribution

and thus the second-order Doppler-shift can be more

easily evaluated.

PHASE

DETECTOR MOD

2 KHz

FREQUENCE CONTROL

GaAs LASER

192 MHz

RECORDER

® [B. ^..s.°y

1.2,3 PHOTOCELLS

*Laboratoire associe au C.N.R.S. Figure l. Cesium beam resonator with optical pumping and

optical detection.

29
The time evolution for the population of the ground

state hyperfine Zeeman levels of cesium atoms has been

previously analyzed for the case where a cesium beam is optically pumped under different conditions of polarization, either with a weak monochromatic laser light or with the resonant light from a spectral lamp filtered by a cesium cell [9]. With cesium spectral lamps and cesium filter-cells, no good experimental results could be obtained in population

changes in a cesium atomic beam, although very good

signal to noise ratio resulted in the optical detection of

the 0-0 transition in a gas cell [10]. This failure could be

attributed perhaps to a lack of pumping light due to too much light absorption by the filter-cell. More successful results were achieved with a semiconductor laser diode (type LD-33, from Laser Diode

Lab. Inc.) originally intended for pulse-operation and

adapted by J.-L. Picque for cw, single mode operation in a temperature-stabilized cryostat at about 25 K. For an injection current of 200 mA this laser diode

delivers a few milliwatts of power, and the output

wavelength is coarsely adjusted in the vicinity of the

cesium D2 resonance jine (852.1 nm) by changing the temperature (about 2.4 A per °C) and finely tuned to one of

the hyperfine components by varying the injection current (about 200 MHz per mA). The phase-jitter of the

laser light can be reduced considerably by locking the

frequency of the laser to an external Fabry-Perot resonator. More details on the operation of the laser diode and

cryostat can be found in Ref. [11]. By sweeping the

piezo-crystal of the Fabry-Perot with a dc voltage ramp,

the laser frequency can be tuned over a wide range. A

photocell, under the cesium beam, is used to detect the

fluorescence of the beam.

The cesium atomic beam is produced in a stainless steel vacuum chamber about 1.5 m long, fitted with several

glass windows to admit the light of an external laser

diode. The collimator of the cesium oven is an array of about 185 nickel tubes (diam. = 0.25 mm, length = 10 mm) inside a 6 mm diam. cylinder. The directivity of the

beam being rather poor at the oven exit, the cesium

beam is further collimated by several apertures (diam. -

8 mm) distributed along the axis, and resulting in a beam

divergence from the axis of less than 1 degree. The aper- ture supports are coated with lamp-black to act as an ef- ficient getter for diverging cesium atoms, and a movable

stop is provided in front of the oven to check that the

detected signals are due to the atomic beam and not to some residual cesium vapor. At the end, an ion detector can be placed to give a measurement of the cesium

atomic density in the detection region, which typically

for an oven temperature of 96 °C is of the order of 2 x

10^ atoms per cm^. Right under the cesium beam, in the

regions of illumination, silicon photocells are placed in shallow cups (to protect them from cesium deposits) to detect the fluorescence of the beam.

Figure 2(b) shows a recording of the derivatives of the

absorption lines originating on the F = 3 level and

detected by the fluorescence of the beam as the laser diode was swept in frequency. The line-width of each

hyperfine component is about 50 MHz. Similar results were obtained for transitions originating on the F = 4

level. Due to selection rules which allow only AF = 1, ^nif = 0, ± 1 transitions to take place, atoms in the

ground state undergoing 3 —> 2 or 4 ^ 5 transitions can

only return to their original level and the total population

of these ground levels should not change due to these

155 153 I ma

Figure 2. a) Energy levels for cesium D2 transitions;

h) Derivatives of D2 absorption lines originating on F = 3

level, recorded by fluorescence of a cesium, beam.

two transitions. However, as was shown in Ref. [9] when detailed balancing of the populations of all the Zee- man sublevels is carried out, a population difference can appear between the F = 3, mp = 0 and F = 4, mp = 0

levels, even though the overall population of the ground

level should not change.

To study in detail these effects, the set-up of Fig. 1 was completed by placing a rotating light-chopper in front of the pumping window. Without microwave excitation, by alternatively pumping the cesium beam on and

off, the populations are redistributed in the various Zee- man sublevels, and an unmodulated probe beam from the

laser can be used to detect the total population changes

by monitoring the modulation of the beam fluorescence.

Figure 3(a) shows such a recording obtained by sweeping the laser frequency. It can be seen that the largest

signal is obtained with the laser diode tuned to the 3^3

transition frequency. It is interesting to note also that a

distorted signal is detected at the 3^2 transition

although, according to the selection rules, the total population of the F = 3 level should not change. Similar

recordings were obtained with the laser tuned to the fre- quencies of the transitions originating on the F = 4 level. Changing the polarization of the laser light from linearly to circularly polarized, in all possible combinations for pumping or detecting, did not change significantly the

signal shapes or magnitudes and it is finally simpler to use the light as it comes directly from the laser diode.

Since there is no saturation with the laser power

involved in these experiments, it is easy to compute the

relative change of fluorescence in the detection region B

produced by optical pumping of the cesium beam in region A: here, it is about 7 percent, for the value of Igt

corresponding to experimental conditions {h = pumping

laser intensity: 6 x 10^^ photons /sec, and t = pumping

time illumination: about 50 microseconds). A dual fre- quency optical pumping scheme, with two lasers, has also

30
3-3

Figure 3. a) Optical detection of population change in F = 3

level produced by optical pumping of cesium beam, b) Optical

detection of 0-0 microwave transitioyi.

been suggested, by which all of the atoms could be

pumped into the F = Z, nip = 0 (or = 4, rrip = 0) level

[12]. With the same experimental set-up, if in addition the

microwave cavity is excited at the 0-0 frequency, i.e., 9192 ,

. . . MHz, it is found that the signals previously

recorded in Fig. 3(a) are reduced by about 1/7 of their

value, indicating that the change of fluorescence pro- duced by the microwave interaction is about 1 percent of the total fluorescence of the cesium beam in B. This is also in agreement with the value given by a simple linear

theory.

To increase the signal to noise ratio of the optical

detection of the 0-0 transition, of interest in atomic

clocks, the experimental set-up was modified as follows: The light-chopper is removed and a low frequency

modulation is applied to the frequency of the microwave

oscillation. A maximum signal is obtained in photocell B

when the microwave frequency is set to the side of max- imum slope of the 0-0 resonance curve. With the

microwave frequency fixed at this value, the laser diode

is swept in frequency and the signal out of photocell B is recorded with a synchronous detector. Figure 3(b) shows

such a recording (similar results were obtained for laser frequencies corresponding to transitions originating on

the F = 4 level). Here again there is a predominance of

signal at the 3^3 transition frequency and also an

unstable and peculiar shape signal from the 3 —» 2 transi- tion frequency. In this 0-0 detection, the width of the

laser frequency range corresponds to the width of the

absorption lines in the cesium beam, about 50 MHz.

In conclusion: from these experiments it appears that

the laser frequency giving the largest signal in the optical detection of the 0-0 transition is the frequency of the

3^3 transition, although, as we shall see later, the 3

4 transition frequency gives a better signal to noise ratio

of the detection (in most of these experiments, the photon flux collected by fluorescence by the photocell in B

was of the order of 5 x 10^*^ photons/sec).

For applications to atomic frequency standards, it is much better to lock the laser frequency directly to one

of the absorption lines of the cesium beam itself [11]. As

shown in Fig. 2(b), the derivative of these absorption

Hnes are narrow-width S curves with a well defined cross-over which can be used for locking the laser fre- quency, and this was done in the following experiments.

3. Optical Detection of the 0-0 "Clock

Transition"

The experimental set-up of Fig. 1 was modified as fol- lows: the laser frequency was set to correspond to one of the absorption lines of the cesium beam and was slightly

frequency modulated (at a 3 kHz rate, for example). The

modulated fluorescence of the cesium beam was detected

by a photocell in A feeding a phase detector. The output

of this phase detector controls the current of the laser diode, thus locking the laser to the absorption frequency

of the cesium beam (see also Fig. 5). The microwave cavity was of the Ramsey-type with oscillating fields separated by a distance L. Circular holes

(8 mm diam.) were provided at cavity ends for the pas- sage of the atomic beam. This cavity was made of a rec- tangular waveguide operating in a TEoi^v mode and is resonant for a length equal to N\J2. Experimentally

two different cavities have been used: one with L = 21.5 cm (A^ = 14) and the other with L = 36.5 cm (A^ = 20), in order to compare the experimental resonant curves with

the theory. The cavity was finely tuned to resonance

with a E - H waveguide tuner in the cavity feed. The uniform magnetic field in region C was produced

by a U shaped electromagnet of Conetic metal. The

whole assembly, cavity and electromagnet, was placed in- side a cylindrical magnetic shield of |jL-metal which was

demagnetized by means of a 50 Hz ac current. The distri- bution of the residual magnetic field inside the cavity was measured with a Hall effect magnetometer probe.

The mean value of the field could be obtained also, in operation, by inducing Zeeman low frequency transitions

with two coils placed at the cavity ends, which at reso- nance perturbed the 0-0 detected signal.

A photocell in B detected the change of fluorescence of the cesium beam when the 0-0 transition was induced by

microwave resonance. To study the shape of the 0-0 reso- nance curve, the microwave frequency was swept slowly

around the resonance frequency. By modulating, at low

frequency, the frequency of the microwave excitation,

the output of a phase detector fed by photocell B gave a dc signal, the polaiity of which depended on the relative

position of the microwave excitation and the 0-0 reso- nance frequencies. By sweeping the exciting frequency

this signal gave the derivative of the resonance curve. An experimental curve is shown in Fig. 4 for a cavity

with 1' = 1 cm and L = 21.5 cm (f = length of the oscillat- ing field region, and L = distance between the two

separated oscillating fields), and an oven temperature of about 85 °C. According to Ramsey [13] the separated oscillating

fields transition probability for a cesium atom in the

beam, with velocity v, is: P = sin2(25T)cos2[(coo - a))L/2i'] , (1) where b the Rabi nutation frequency, is proportional to the square-root of the microwave power in the cavity, (ai the Bohr resonant frequency, co the excitation frequency,

31
Figure 4. Experimental derivative of Ramsey pattern of

cesium, beam, resonator tvith optical pumping by laser (solid

curve) and theoretical values (dots) for Maxwellian distribution of atomic velocities (sweep speed: 16 Hz Is; recorder time

constant: 3 s).

and T the time the atom spends in each end of the cavity.

Assuming the usual Maxwellian velocity distribution

for effusive beams:

/(?;)= 2( V 3/a4) exp ( -v Va^) , (2) where a is the most probable velocity in the atomic

beam, a = {2kT/m)^ {ni, mass of the cesium atom; T, temperature of the cesium oven), the transition probability averaged over this velocity distribution is given by:

<P >

with

" y''sm-\ —an

COS" 2ay

y = via

dy (3)

(4) Near resonance, the derivative of <P> versus frequency

reduces to:

dj a

(5)

+ mw (x -

-il/2)K'{x),

with X — ~ (oo)L/a and K'ix)

e~"^ y-sinixly) dy.

a

(6)

On Fig. 4 such a theoretical curve has been plotted for the following values of the parameters: f = 1 cm, L -

21.5 cm, a = 213 m/sec and 2b ( /a = 2.0 (corresponding

to a microwave signal power slightly over that for op- timum transition probability) and vdth the ordinate of

the curve normalized to coincide with the maximum of

the experimental curve. The agreement between the ex- perimental curve and the computed points is quite good,

and gives evidence of a broad Maxwellian distribution of

velocities in the beam. The frequency width between the

central peaks of the derivative curve is about 500 Hz and

320 Hz for the microwave cavities with lengths L =

21.5 cm and 36.5 cm, respectively.

The shape of the resonance curve in this experiment is different from the one obtained in the optical pumping of a rubidium beam with spectral lamps [5] where there was a line-narrowing due to an enhanced contribution of the

slower atoms to the overall signal [4]. This is probably due

to the fact that the spectral density of the monochromatic

laser radiation is about 10^ times higher than for a spectral

lamp, so that the efficiency of optical pumping is about the same for all the atoms, regardless of their transit time

across the light beam. However, the lack of this resonance narrowing with laser pumping is more than compensated

by the large increase in the signal to noise ratio (SIN) of

the detection in this case and this makes the device attrac- tive for atomic clock applications. In the present experiments, the SIN is limited by the

noise due to residual phase-jitter of the laser diode. In

this respect, it was observed that the SIN was very bad

when locking the laser frequency to the 3 2 absorption

line of the cesium beam, and better with the 3^4 than

with the 3^3 transition. This effect may have been in- strumental, the slope of the S curve (Fig. 2(b)) for the

3-^4 transition being smaller than for the 3 ^ 2 or 3 ^

3 transitions. Improvement in the SIN could be obtained

by increasing the laser intensity at the pumping region,

and by increasing the bandwidth of the amplifier and the

loop gain of the laser frequency servo. Also electrooptical devices could be used to reduce amplitude noise of

the diode laser. Further work is contemplated in that

direction.

4. Operation as an Atomic Clock

4.1 Experimental Set-Up

Referring to Fig. 5, the microwave excitation signal is synthesized from a stable 5 MHz voltage-controlled

quartz oscillator (H.P. 107 BR). The error signal from

the phase detector acts on a voltage-variable capacitor to lock the frequency of this oscillator to the atomic transition. The output of the 5MHz oscillator is compared with

the frequency of a commercial cesium-beam frequency

standard (H.P. 5061 A) either by frequency multiplication and counting, or in a phase comparator. In this way

the precision resetability of the clock and its accuracy

could be checked versus various parameters such as: magnetic field, rf power, cavity tuning, rf modulation,

laser light intensity or polarization, oven temperature,

servo-gain, etc. However, in these preliminary measurements, the elec- tronics and the mechanical assembly were not sufficiently

developed for measurements of 2nd-order Doppler shift or cavity phase-shift of the order of 10"^^ to 10"^"^. Since

the commercial standard for frequency comparison has a

limited accuracy of ±2 x 10"^\ and with the inclusion of various biases in the electronics, it is believed that the

absolute accuracy of the measurements in these first tests was perhaps no better than ±3 x 10"^^ Also, since

the SIN could probably be improved considerably with

further work on the laser servo, the emphasis of the

measurements was mostly on the accuracy of the clock

rather than on short-term stability. Finally it should be

32
OUTPUT 5MHi

CRYSTAL

FC, OSCILL 5MHz

FREQUENCY SYNTHESIS

MOO.

30Hz, FREQUENCY

MULTIPLIER

AMPLI 3KH2

FC

9 192^1

AMPLI

30Hz.

MOD

SKHz

GaAs

® L° {D--°' -.T---I

LASER

(P PHASE DETECTION FC FREQUENCY CONTROL

Figure 5. Overall experimental set-up for optically pumped

cesium beam frequency standard.

the theoretical value: _

/hz =_9 192 631 770.05 + 426.7fl^ ^ (7) (H in gauss = 10 tesla), with a regression coefficient = 0.99978.

5. Conclusions

Although these preliminary measurements are too coarse to establish claims of high accuracy regarding

2nd-order Doppler shift or cavity phase-shift, the results

obtained, are encouraging and the experiments will be

continued with improved electronics, especially concerning the stabilization of the laser frequency (which in- cidentally, could be used also, in the same apparatus, as a secondary length standard).

The potentiality of this frequency standard for accu- racy of the order of 10"^'* will also depend on the absence

of an eventual "light-shift" produced by stray laser light or fluorescence of the beam and precautions should be

taken against this effect. Also, the need for liquid

helium for the laser cryostat is a drawback for operation

of the clock over a long period of time. However, laser diodes have already been developed for operation at room

temperature, with lifetimes in excess of several

thousands of hours of operation, and this should allow the

realization of more practical devices than the laboratory

apparatus just described.

noted that the laser diode used could lase on the same

frequency for slightly different conditions of current and

temperature, corresponding to different modes of oscilla- tion, and since some modes were less noisy than others,

the values of stability or precision quoted here may vary

slightly from one experiment to the other.

4.2 Experimental Results

Measurements of the locked oscillator frequency every

10 seconds, for a period of one hour, show that the distri- bution of the data is Gaussian (x^ = 3.6) with a standard

deviation of about -S = 5 x 10"^^ and an Allan variance

a^(lO) = 3 X 10-11. As said previously, the short-term stability is better

with the laser frequency locked to the 3^4 than to the

3^3 transition. For example, in one experiment, with

the 3 -» 4 transition: S = 7.8 x lO'^^ and UyilO) - 3 x 10"" whereas with the 3 3 transition: S = 1.4 x 10"!° and (Ty(lO) = 4.6 x The precision resetability from an offset resonance fre- quency was on the average ±3 x lO'^^. The effect of rf power variations on the precision of the

clock depended very much on cavity tuning. For example,

for a slight frequency detuning of the microwave cavity

the frequency shift could be as large as 1.3 x 10"" for a

3 dB variation in rf power from optimum power. However,

with fine adjustment of the cavity tuning, this shift could

be made smaller than 2 x 10 ~ " and probably much smaller

with improvements in the measuring equipment.

Variation of the laser light intensity was also studied

for possible frequency shifts due to the fluorescence of the

cesium beam. Within the accuracy of the measuring equipment (±2 x 10 ") no light-shift of the microwave reso- nance could be detected when the laser light was reduced

by as much as 50 percent of its maximum intensity.

The absolute accuracy of the clock was observed by

changing the static magnetic field H in the microwave

cavity and measuring the corresponding frequency shifts. With a large number of observations, a least-squares fit of the results gave the following relation, very close to

The author wishes to thank Professor A. Kastler, who

initiated the experiment, for his continuous encouragement. The work was done in collaboration with Dr. J.-L. Picque of Laboratoire Aime Cotton, Orsay, and with the

helpful advice of S. Roizen. Appreciation is also ex- pressed to Dr. T. Yabuzaki' and I. Hirano- for temporary

help in the early phase of the work, and to G. Faucheron,

D. Guitard, and P. Pages, for contributing technical help.

References

[I] D. J. Glaze, H. Hellwig, D. W. Allan, and S. Jarvis, .Jr., Metrologia 1.3, 17 (1977).

[2] H. Hellwig, S. Jarvis, D. Halford, and H. E. Bell, Metrologia 9, 107 (1975).

[3] A. Kastler, J. Phys. Radium 11, 155 (1950); J. Opt. Soc. Am. 47, 460 (1957).

[4] J. Brossel, B. Cagnac, and A. Kastler, J. Phys. Radium 15, 6

(1954).

[5] P. Cerez, M. Arditi, and A. Kastler, C. R. Acad. Sci. Ser. B: 267, 282 (1968). M. Arditi and P. Cerez, IEEE Trans. Instrum. Meas. IM-21,

391 (1972). P. Cerez and F. Hartmann, IEEE Trans. Quantum Electron. QE-13, 344 (1977).

[6] J.-L. Picque, Metrologia 13, 115 (1977).

[7] M. Arditi and J.-L. Picque, C. R. Acad. Sci. Ser. B: 290, 461

(1980), also in English: J. Phys. (Paris) Lett. 41, L379

(1980).

[8] S. Jarvis, Jr., Metrologia 10, 87 (1974).

[9] M. Arditi, I. Hirano, and P. Tougne, J. Phys. D. 11, 2465

(1978).

[10] E. Bernabeu, P. Tougne, and M. Ai-diti, C. R. Acad. Sci. Ser. B: 268, 321 (1969). M. Arditi and P. Tougne, C. R. Acad. Sci. Ser. B: 280, 405

(1975); Rev. Phys. Appl. 11, 665 (1976).

[II] J.-L. Picque, S. Roizen, H. H. Stroke, and 0. Testard, Appl.

Phys. 6, 373 (1976).

^Ionosphere Research Laboratory, Kyoto, Japan

-National Research Laboratory of Metrologj-, Ibai-aki, Japan

33
J.-L. Picque, and S. Roizen, Appl. Phys. Lett. 27, 340

(1975). L. S. Cutler and L. Lewis, National Bureau of Standards,

Boulder, CO (private communication).

[13] N. F. Ramsey, Molecular Beams, (Oxford University Press, London, 1963), p. 129.

[14] P. Cerez, C. R. Acad. Sci. Ser. B: 272, 897 (1971).

34
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Frequency Measurement of Optical Radiation

K. M. Baird

National Research Council of Canada, Ottawa, Canada K1A 0R6

The feasibility of directly relating the frequency of visible radiation to microwave standards has been demonstrated and a number of frequency comparison systems linking infrared frequencies to the cesium primary standard have already been operated. These have yielded sufficient accuracy that to- gether with wavelength measurement based on the ^^Kr line used to define the Meter, the standard of length can now be based without fear of a significant discontinuity, on a conventional value for the speed of light and the Cs standard for time. This paper reviews present and proposed frequency comparison chains and discusses their possibilities. Limitations for the general use of frequency comparison

methods in the optical region are described. Key words: frequency chains; laser frequency phase locking; optical frequency measurement; standard

of length.

1. Introduction

Among the remarkable facts of modern technology, I think one of the most striking is that the possibility is at hand of making an exact count of events that occur at a

rate of over 500 million in a microsecond. As you will hear in this and a later paper of this conference, the re- cent work leading to a redefinition of the Meter has in- volved the measurement of optical frequencies. Although

the actual counting of optical frequencies or phase lock- ing them to the cesium frequency standard has not yet

been achieved, the feasibility of doing this has been

demonstrated and a number of laboratories are now en- gaged in setting up equipment for its realization. Before

discussing the techniques that are used to accomplish the

remarkable feat mentioned above, let us look briefly at the basic principles used to measure a high frequency in terms of a lower one. Figure 1 illustrates how a device that has a non-linear

response will convert a sinusoidal signal into an output

that is distorted, i.e., it contains harmonics of the origi- nal signal. Depending on the response characteristic, the

output may contain harmonics of significant amplitude up

to very high orders (a hundred or more). Similarly, as shown in Fig. 2, such a device will convert a mixture of

=)o

3

3

O

INPUT

Figure 1. A purely sinusoidal input is converted by a device whose 7'esponse is 7ion-linear, as shoivn at the upper left, into a

distorted sinusoidal output, i.e., one that contains one or more

harmonics as shown in the lower right.

INPUT

Figure 2. A signal consisting of the superposition oj two purely sinusoidal waves of slightly different frequencies is con- verted by a device whose response is non-linear into an output

that contains a component whose frequency is equal to the difference or beat between the original signals.

two sinusoidal signals of slightly different frequencies,

into a distorted output that contains a signal corresponding to the difference frequency or beat between the two

original signals. A count of the beats over a period of time yields exactly the difference in the number of cycles

of the original signals. It can be seen that if two signals, one of which is nearly equal to a harmonic of the other, are impressed upon such a device having a suitable nonlinear response, the rather complicated output will contain a low frequency signal of frequency equal to the

difference between the one and the harmonic of the

other. Thus a measurement of the frequency of the beat,

/b, will yield the value of the higher frequency /h in terms of the lower /i, /h = Nfi + /b. The frequency of the beat can be low enough to handle in convenient electric circuits and be measured or counted. The only re- quirement for high speed is in the device for harmonic

generation and mixing. Of course the oscillations must be

sufficiently coherent (i.e., they must suffer frequency

changes sufficiently slowly) to allow observation of enough beat oscillations to make a significant measurement. When suitably applied, the method makes it possi- ble to measure the number of oscillations of the higher

frequency signal during a given number of periods of the

lower, without missing one.

35
2. Techniques for Measurement of Optical

Frequencies

Two major developments have made it possible to ap- ply the above methods to extend frequency measurement

to the optical region of the spectrum: the first was the in- vention of the laser, which satisfied the coherence re- quirement, and with the associated development of Doppler-free spectroscopy and techniques for stabilization, provided very precise reference standards; the

second was the development of very high speed nonlinear devices of which the most important for the

present discussion is the point contact metal-oxide-metal

(MOM) diode. It can be used for harmonic generation and

mixing to produce an electrical output from signals in the

range from dc to the infrared. Of nearly equal importance was the production of optical non-linear crystals which

can be used to generate second harmonics and for the addition or subtraction of infrared and optical frequencies.

They require a detector for conversion into an electrical

signal.

The development of the MOM diode was an extension

of the well known point contact technique (such as used

in old crystal radio receivers). This has been adapted to achieve very high response speed by the use of small,

low resistance, low capacitance junctions that result when the very fine tip (Fig. 3) of an etched tungsten

wire is brought into contact with an oxidized nickel post.

The contact area can be of the order of 10"^^ cm^ so that

the very low capacitance and the high speed of the electrical response, thought to be due to electron tunneling

through the 8-10 A nickel oxide layer, allows its use as an electrical device at frequencies up to about 200 THz

(X = 1.5 |xm).

Figure 3. An electroyi photomicrograph of the etched tip of a

fine tungsten ivire showing the very small radius that can be

2ised to create junction diodes of extremely small area. Overlapping this region, from about 30 THz (10 ixm) up

to frequencies of radiation in the visible and ultraviolet,

non-linear optical crystals are used for harmonic generation and signal mixing, giving an output that with suit- able sensitive detectors yields beat frequencies in the

form of usable electrical signals. There are, however,

limitations to the use of crystals: firstly, the crystals

have limited ranges of transparency, and secondly, there

is the problem of phase matching. This problem arises

because the harmonic signal generated at a given place in the crystal is extremely small. In order to get usable signals, the contributions along a considerable length of path must add up in phase, i.e., the harmonic generated

"downstream" must be in phase with the harmonic that

has come from "upstream." Because of dispersion the

phase velocity of the generating beam and the harmonic

will normally differ and this adding up will not occur, but

it can often be effected by compensating for dispersion

by making use of double refraction, a trick that requires

precise adjustment of the indexes of refraction at the two

frequencies, either by temperature or by the direction of propagation with respect to the crystal axes. Although the above developments have opened up the

possibility of direct frequency comparison of optical with

microwave frequencies, its realization is far from simple

and a great deal remains to be done before the frequency

of visible radiation can be correlated directly with mi- crowave standards. Problems arise from the fact that the

frequencies involved are so very high, so that even very

small relative differences result in beat frequencies that are high compared to those that can be handled in con- venient state-of-the-art circuitry. This makes trouble in two ways: small frequency instabilities in the lasers cause very large excursions in the beat . . . e.g., a jitter

of 10- J at 100 THz (\ = 3 ^JL m) is 100 MHz, and if the

sought for beat signal is of this order, measurement be- comes very difficult to say the least. Secondly, matching

the harmonic of one source to the frequency of another

source to produce low frequency beats is not easy be- cause of the limited choice of reference lasers and of their limited tuning range. This situation is illustrated in Fig. 4; the top half shows on a log scale the five decades

in frequency from the cesium standard at 9 GHz to the

visible. The ranges of klystrons and the range of operation of the W-Ni MOM diode are indicated, as well as important bench mark frequencies of Cs, UCN, H2O, CO2,

etc., up to the L lines on which HeNe and argon lasers can be stabilized. Since our frequency comparison

method depends to a considerable extent on the simple

addition of frequencies, particularly at the higher ranges

and since it yields output signals that correspond to differences, the difficulties are more appropriately illus- trated by the use of a linear scale, as done in the bottom

half of the figure. On this scale the total range of fre- quency comparison and measurement from sub-kHz

through the GHz microwave bands that have been made

possible with commercial equipment development and by

the pioneering experiments in the infrared done in the

sixties at MIT covers only a very small part at the left of

the scale. The enormous range yet to be covered to reach

the visible as well as the large size of the gaps between

available bench mark lasers is evident. This can be emphasized further by considering an appHcation of the rich 10 |JL m bands of CO2 laser lines indicated on the figure.

LOG FREQUENCY (Hz)

9'''III, '0 II 12 13 14 15

Cs HCN HjO COg CH^ Ne

^ \ \ 1

H-ii\

KLYSTRONS

VISIBLE W-Ni DIODE

FREQUENCY (THz)

200 300 400 500 600

1

\

1 Ne

1

Ne —1

1

1 1

1 1-9 3.39 1.52 1.15 X (^m) H

\ \ .633.612.576 .515 W-Ni DIODE VISIBLE

Figure 4. The spectrum covered by optical frequency measure- ment on a log scale (top) and a linear scale (bottom) of fre- queyicy showing the important "bench mark" lines.

36
One of these bands is shown in enlarged detail in Fig. 5. These very conveniently produced lines are spaced about

40 GHz apart covering a range of about two terahertz

and a number of such bands can be produced using vari- ous isotopes. However the gap between the lines is usually more than 100 times the gain width (i.e., tuning

range) of the lines; in other words the band is more than

99% empty space. Thus, for example it might be thought

straightforward to measure a CO laser line (at about

5 (xm) by comparison with the second harmonic of one of the CO2 lines; in fact the chance of finding a coincidence

closer than several GHz is very slight. In such a case the

gap usually has to be covered by the addition of a klys- tron generated frequency as will be seen in several examples below.

27.5

Frequency (THz) 28.0 28.5 29.0 29.5

Figure 5. The 10 [xm band of the CO, laser.

The CO2 band shown and similar overlapping bands

produced by the use of CO2 sequence hnes and CO-, of dif- ferent isotopic composition provide a very important set having frequencies very accurately known relative to one

another and covering the range from about 26 to 33 THz

(9 to 11 |ji,m). These radiations and their harmonics as well as radiations generated by the sums and differences

of pairs, and harmonics of the differences, can be added

to and subtracted from other laser radiations by the use

of W-Ni diodes up to about 1.5 ixm; above this point the

diodes appear not to respond electrically. The CO2 laser

is thus of great importance in frequency comparison

chains. In certain cases CO2 laser frequencies can be added to radiations above this limit, thus making it possible

in effect to transfer the band of frequencies to another

region of the spectrum, (e.g., at 0.633 (xm) by the use of

non-linear crystals. However the requirement that the

crystal be transparent to all radiations involved, the re- quirement of phase matching, and the very small non- hnear coefficients very much limit the possibilities, as will be seen. Referring again to Fig. 4 the scarcity of bench mark

laser lines can be somewhat alleviated by the use of tunable dye lasers in the visible, and, recently, of color center lasers in the range from —2.5 [xm to 1 \xm as will be discussed later. From the foregoing discussion it is evident that,

despite the simplicity of the basic principle, extension to the optical region of frequency comparison with respect

to microwave standards is far from easy. Nevertheless

the first measurement of a visible frequency has already

been demonstrated [1], and several measurements of

infrared frequencies have been made with very high ac- curacy [2-4]. Taken in conjunction with wavelength

measurement with respect to the **Kr length standard,

they make possible a new definition of the Meter based

on the standard of time and the adoption of a conventional value for the speed of light; this will be discussed

in another paper at this conference.

3. Chains to Compare Infrared and

Microwave Frequencies

The frequency chains to the infrared will be described

with reference to Fig. 6 where they are shown, greatly

simplified, in roughly the chronological order in which

they were successfully operated (or are expected to operate). The first measurement of the frequency of a CH4 stabihzed He-Ne laser was made at NBS (Boulder)

by Evenson and his colleagues [5]. Following up the

pioneering work at MIT on W-Ni diodes in the far in- frared, they gradually extended the chain, and succeeded

in measuring the CHj line at 88 THz (3.39 ixm), in 1972. Almost simultaneously the group at NPL in London set up a very similar chain [6] and obtained results in very

good agreement with NBS. In these chains as shown, the

output of a klystron of frequency 74 GHz, was compared

with an HCN laser by generation of the 12^^ harmonic

and mixing in a conventional tungsten silicon microwave

diode. The HCN laser output was in turn compared to the H2O laser line at about 10 THz by generation of its

12'*' harmonic and mixing in a W-Ni diode. The H2O line was multiplied by three and compared with CO2 laser ra- diation at 30 THz, again in a W-Ni diode, and finally, the

30 THz radiation was multiplied by three and compared

with a CH4 stabilized HeNe laser at 88 THz. As mentioned, this description is very much simplified: for example, at nearly all stages above the first klystron stage,

additional klystron frequencies had to be added or subtracted to reduce the output beat signal frequency to the

MHz range so as to be convenient for measuring; also the

fjiVi-lR Frequency Chains

0.1 I 10 100 THz

T NBS\

NPLi

NRC

NPL

IS

LPTF -+-

Kl

—I—

HCN

W W

' 3 ' 3~

H,0

I I I I I

w

W-Si

Ale W , W

HCN ^ D2O

W-Si

I W I

NRLM

PTB

NBS

NBS

+4h

W-Si W-GaAs

Care

Ale C.C.

Optical Synchrotron ^ ?

Figure 6. Simplified schematic diagram offrequency compari- son chains at the National Laboratories indicated on the left, covering the range from the microivave region to the 3 \im

wavelength region, the numbers indicate the harmonics used,

Kl = klystron; W = tungsten-nickel point contact diode; W - Si = tungsten-silicon diode, etc.; A = difference frequency generated by two CO2 lasers.

37
CO2 line used for comparison to the H2O line was not the same line that was used in the comparison with CH4 and

the difference had to be measured against a klystron fre- quency. At NPL a different pair of CO2 hues was used in this stage. These early experiments suffered loss of ac- curacy in the transfer from one stage to the next but in more refined repetitions of the experiments produced

frequency values for the CH4 and CO2 lines of accuracy

approaching better than 10"^. It was these values that,

taken together with wavelength measurements at NBS,

NPL, and several other national laboratories, led to the

acceptance of 299792458 m/s as the best value for the

speed of light [7]. More recently, at NEC (Ottawa) [8] and at IS (Moscow)

[2] the frequency chains shown were put into operation.

At NRC the HCN and H2O laser stages were replaced by

the use of difference frequencies generated in W-Ni

diodes by simultaneous input from two CO2 lasers operating on appropriately chosen transitions. A considerable

advantage in the simplicity of the lasers is partially

offset by the low signal strengths of the difference fre- quencies. This required the addition of an extra stage to reduce the harmonic numbers to 3, 3, and 4. The accu- racy of the first measurement at NRC was limited by the

fact that each stage was measured separately, depending

on saturated fluorescence stabilization of the CO2 lasers

for the transfer accuracy. This defect is being corrected

by the use of phase-locking in a new system now being

set up; phase-locking of the beat frequency of two lasers,

operating on lines of different CO2 isotopes, to a rubidium standard has already been performed successfully.

At IS, a D2O laser and harmonic factors of 8 and 3 were

used instead of the H2O laser and factors of 12 and 3 used

at NBS; an OSO4 stabilized CO2 laser was used as the

30 THz transfer. In a recent revised version of the NPL chain [3] the

number of stages was reduced as shown by the use of 43"' harmonic generation in a Josephson junction to go from the

klystron stage to a CO2 pumped alcohol laser at 4.25 THz;

from there two stages of 7x and 3x were used to go via a CO2 laser to the CH4 stabilized HeNe laser at 3.39 [xm. In

this measurement the stages were operated simultaneously and phase locking or continuous beat frequency

counting was used at all points to give a much higher accu- racy (3 X 10"^^) than in the previous experiment.

The last chain that, to date, has been reported as having operated is that at LPTF (Paris) [4]. It makes use of HCN and an alcohol laser in the lower stages. The

NRLM (Tokyo) chain, which is to be described at this conference, makes use of alcohol and HoO lasers to reach

the CO2 lines at 30 THz. The PTE (Braunschweig) chain

is in the stage that all the parts are operational but final stabilization and locking to make a measurement of the

CH4 line has yet to be done.

The newest NBS chain, shown near the bottom of

Fig. 5, is part of a continuous system designed to go to the visible. It will use harmonic numbers of 7, 7, 7, and 5

to go from the klystron region, via alcohol and CO2

lasers, to a color center laser at 2.52 [xm (130 THz); more

wiW be said later about the upper stages to the visible. The last system shown in the figure, also being set up

at NBS (Boulder), is quite different from all the others

and is intended to go directly from the rf region to at least the infrared by using the principle of the synchrotron. An electron orbiting in synchronism with an rf field will pass through a focussed laser beam in such a manner

that, if the orbital period, i.e., the rf, is an exact submultiple of the laser frequency, the electron will get a

'

' kick' ' from the electric field of the laser on each revolution and a resonance will be observed. The method has an

advantage, in addition to that of having only one stage,

in that it is a frequency division, rather than frequency

multiplication method, and as such will not suffer from

the problem of phase jitter amplification.

Before proceeding to a description of techniques to extend frequency measurements into the visible, it is ap- propriate to consider the accuracy demonstrated in meas- urements of the CH4 line frequency reported up to the

present. Figure 7 shows the latest values reported by

each laboratory with standard deviation error bars. It is seen that these agree to within one in 10'', i.e., well

within the precision of the present meter definition. There is only one case (IS) where the error bars do not

overlap.^ The uncertainties are still far from the limit imposed in principle by proper frequency comparison with

the Cs standard (lO'^^) but the NPL value (3 x lO'^^) is within the uncertainty known to be attributable to lack of reproducibility of the CH4 lasers as used. Evidence for

this uncertainty comes from two international direct comparisons of CH4 lasers, one amongst NPL, PTB, and

BIPM [9] and the other between BIPM and VNIIFTRI

(Moscow) [10].

F = 88 376 181 000 kHz +

400 500 600 700 800

. NBS I • 1 NPL I ' 1

1--O--1

NRC

IS

NPL

LPTF H.H ^

9

Figure 7. Results obtained at various laboratories for the fre- quency of the He-Ne laser line at 3.39 p^m stabilized to the P(7)

absorption line in methane.

4. Extension of Frequency Measurement to

the Visible

Systems designed to extend direct frequency comparison to the visible part of the spectrum are shown in very

simplified schematic form in Fig. 8. Only the first of these, the NBS-NRC chain [11, 1], has been demonstrated to completion, yielding a frequency for an I2 absorption line in the visilDle. The ISP (Novosibirsk) system

has been demonstrated to function but the component

frequencies have not been measured [12]. The other

three systems are under active development and many

parts of them are already operational. In the NBS-NRC experiments the frequency of a Xe

laser at 150 THz (2.02 |xm) was first measured by comparison in a W-Ni diode with the sum frequency of a HeNe laser at 3.39 fxm (88 THz) and two 10 |xm CO2 lines

Recent measurements have revised the IS value upwards by 17.4 kHz

[20].

38
IR-Visible Frequency Chains

0 100 200 300 400 500 THz

Figure 8. Simplified schematics of frequency comparison

chains at the National Laboratories indicated on the left cover- ing the range from 10 waveleyigth (30 THz) to the visible (600 THz). C = CO., laser; CC = color center laser; X = Xe

laser; N = He-Ne or pure Ne laser; LN = frequency generated

in a LiNbO^ crystal.

(-30 THz). The sum of the Xe Une and a CO Une at 50 THz was used to measure the 1.5 |xm (196 THz) HeNe laser line, again in a W-Ni diode. In the final part of the experiment done at NBS, the He-Ne 1.5 (xm Hne was

added in a proustite crystal to the sum of two COo lines (produced by addition in a CdGeAs crystal) resulting in a sum frequency very near 1.15 jjim. A pure Ne 1.15 jjtm

(260 THz) laser output was compared by the use of a Schottky diode. The Ne laser, stabilized on its Lamb dip, was taken to NRC and compared to a 1.15 |xm He-Ne

laser whose radiation, after doubling, was locked to an I2 line by saturated absorption. This experiment, done in 1979, thus demonstrated for the first time the direct

measurement of a frequency in the visible. It ought to be

mentioned however that the CO frequency was not

directly measured but was inferred from molecular constants based on frequency measurements. In fact, to date, the best knowledge of the 520 THz frequency is derived from its measured wavelength. The weak CO link could be corrected, but present efforts are directed to what appear to be more attractive systems as described

below.

In the ISP system three Ne transitions at 3.39, 2.39,

and 1.15 ixm were excited simultaneously in a He-Ne

plasma; non-Hnear interaction in the plasma itself pro- duced a sum frequency at 0.633 |xm (474 THz). Although

very elegant in principle the scheme has not been widely

adopted because of apparent difficulties in measurement

and control of the component frequencies.

The third chain shown, under development at NBS

(Boulder), starts with a CO2 Hne (shown at the end of

their chain in Fig. 6) whose fifth harmonic generated in a W-Ni diode is used to control the frequency of a color center laser at 130 THz. The frequency of the color center laser is doubled to 260 THz in a LiNbOs crystal,

and used to control a He-Ne 1.15 laser. The fre- quency of the latter is doubled in a second LiNbO^ cry.s- tal for reference to the I2 line at 520 THz as in the NBSNRC experiment. An additional stage has been proposed

that would use the difference frequency of the 520 THz

line and a carbon monoxide laser line to measure the

widely used He-Ne, I2 stabilized 0.633 \xm line. However

a suitable crystal was not identified. The fourth system shown in the figure, being

developed at NRC, makes use of two Xe laser Hnes excited simultaneously in the same laser and measured by

reference to CO2 laser lines in W-Ni diodes. The Xe lines are added in LiNbOs and compared with the difference

frequency generated by mixing a 1.15 |xm He-Ne laser and a CO2 line in a proustite crystal, thus giving the fre- quency of the He-Ne line. The latter can be compared

with the I2 line at 520 THz as described above. Some dif- ficulties in this system are associated with the proustite

difference frequency generation and low S/N but the

problem could be very much alleviated by the use of a AgGaS2 crystal, expected to be available in the near

future. The last system shown, nearing completion at NPL, is similar to the NBS system in the use of a color center

laser whose frequency is doubled in a LiNbOs crystal but

will make use of parametric conversion in LiNbOs to detect the difference between an I2 stabilized He-Ne

laser (0.633 jjim) and four times the frequency of the

color center laser. Also it uses only the fourth harmonic

of a different starting CO2 line and a lower frequency

color center laser, than in the NBS case, to reach the

lower frequency (474 THz) visible line. In concluding this description of the schemes for reaching visible frequencies it is emphasized that, as in the

case of the infrared frequency measurement, the experiments are much more complex than one might judge.

This can be illustrated by describing in more detail the

apparently simple stage of stabilizing doubled Ne

1.15 |jLm radiation with reference to an lo line as used in the NRC-NBS experiment.

This elegant experiment, performed by Hanes at NRC

[13] is shown schematically in Fig. 9. It employs a double resonant cavity seen at ABCBE and EBFG. The 1.15 ixm

radiation produced by the He-Ne plasma tube is focussed

into the LiNbOa crystal which is accurately temperature

controlled for phase matching so as to produce doubled

frequency at 520 THz (0.576 |xm). An additional phase

matching requirement is satisfied by a special dispersive

reflector at E to ensure that the reflected second harmonic is in phase with the second harmonic generated

from the reflected fundamental radiation (1.15 iJim). An

Figure 9. Schematic diagram of apparatus for stabilization of doubled He-Ne 1.15 fjL>?z laser on an line at 576 nm

(520 THz).

39
I2 cell in the second cavity, resonant at the doubled fre- quency, produces saturated absorption features that can be used for stabilization. Scanning and servo control of the cavity arms AC, CE, and EG are suitably coordinated so as to be resonant simultaneously at the required

frequencies. About 100 |jlW of 1.15 \xm radiation is emitted at A and about 20 |jlW of 0.576 |xm radiation at G. The fortunate I2 hyperfine spectrum at 520 THz is shown

in Fig. 10. Note the strong component at the left where

the gain has been reduced by a factor of 10. At present it looks as if the optical frequency measure- ment systems described above stand an excellent chance

of working and could, at least in principle, result in phase correlation of visible laser radiation with the

standard Cs frequency. This might be done by a completely phase locked system or by a simultaneous count

of beats at some of the stages. In any case the frequency

of good reference lines in the visible will be measured,

but they may amount to only a few bench marks at 0.576 |xm, 0.633 ixm, probably at 0.612 jjim, and possibly an I2 line near the Ha line at 0.656 fxm. Let us next con- sider the problems and some methods for establishing

and of making use of such bench marks for measurement

of other lines in the visible and infrared.

200 400 600 800 FREQUENCY CMHz)

o nm 1 k J

I hg f edcb a

Figure 10. I , Hyperfine spectrum at 520 THz obtained with the apparatus of Fig. 9.

5. The Use of Optical Frequency Standards

Once a bench mark is established in a given part of the

spedtrum, either in the form of a precisely reproducible

absorption line or a laser locked to a frequency chain,

other lines or bench marks can be measured by the use of methods like those described; this may require one or a number of steps. The process has become relatively

straightforward in the region covered by the point contact MOM diodes, i.e., up to about 1.5 |xm (200 THz).

Differences of up to a few tens of GHz can be measured

directly in the diode output; for greater separations, up

to about 100 GHz, klystron radiation can be mixed to measure the differences. For yet greater separations, up

to about seven terahertz, two CO2 lasers having the ap- propriate difference in frequency can be used; and

finally, CO2 or other appropriate laser emission can be

used to measure separations upwards of 25 THz.

One example of such a process is the work of Clairon et

al. [14], who have established a grid of very precise

standards, separated by about 50 GHz, covering a range

of 265 GHz in the 10 ixm region. This was done by stepping off from a well measured OSO4 standard [2] by the use of a klystron, to OSO4 and SFe absorption lines that

coincide with CO2 laser lines. Saturated absorption in the

very narrow OSO4 and SFe lines made possible such precise setting (±1 kHz and ±3 kHz respectively) that they

contributed practically no significant error, even allowing

for accumulation in a number of steps. Another example

of technique is the method used by Siemsen at NRC [15] to measure the frequency of a laser that is well outside a grid of reference lines. As shown in Fig. 11, when two

appropriate known frequencies / 1 and /2 are mixed with

the unknown frequency /3, the latter can be deduced

from the beat frequency / b = (/i - f2) - if 2 - fz)-

(f2- fa) beat frequency

—^— fa lOOmW

=^—

^— f3 > 10mW

^ -80dBm

/

-40dBm

Volt Watt Figure 11. The use of three wave mixing to measure the fre- quency of a laser line that lies outside a grid of known lines. The measurement of large frequency differences gets

considerably more difficult above the electrical response

limit of the MOM diodes (—1.5 iJirn). Photo electric detectors can be used as square law devices to measure

directly differences up to several GHz in the region from

3 |jLm through the visible. Schottky diodes have also been

used to measure differences up to 122 GHz [16] by mixing

with klystron generated frequencies. In order to measure

greater differences or to generate harmonics, non-linear

crystals must be used.

The use of non-linear crystals for frequency comparison

involves far more restrictions than does the application of

MOM diodes because of their limited range of transpar- ency, the problems of phase matching, and the small

non-linear coefficients; they cannot be used to mix klys- tron frequencies. We have seen examples of their application in the frequency chains described but these tend

to be special cases and general applicability cannot be as- sumed. For example AgGaS^ is a rather remarkable crystal that is transparent from 13 (xm to well into the visible and has a good conversion efficiency: one would expect it to be ideal for the last stage in the NBS chain going from

520 THz to the 0.633 laser line. Unfortunately however,

according to published data [17], it is not possible to realize phase matching for this case. The number of "special

cases," with the right transparency, phase matching,

and suitable laser lines may well become quite large,

however, with the further development of crystals and

tunable dye and color center lasers. There are cases al- ready where one can, in effect, transfer the 7 THz-wide

grid of COo laser frequencies into part of the visible spec- trum by mixing in proustite, and perhaps it is not unrea- sonable to hope that more possibilities like this will turn

up.Special tricks can be used to increase the measurement

separation limit beyond that imposed by the photo electric detector. One such is that suggested by Hansch and

Wong [18] whereby a widely tunable laser is frequency

modulated so as to match the side band separation to the

40
intermode spacing; a comb of frequencies covering about

0.5 THz in the visible may be generated and locked to an

absorption reference by two-photon absorption. Another

example is the stepping procedure proposed by Meisel

and his colleagues [19]. Using two stabilized dye lasers,

they plan to make two hundred steps of 80 GHz each in order to measure the He, line with respect to the 0.633 I2 laser line.

6. Conclusion

It is clear fi'om the foregoing review that, although the

direct measurement of frequencies in the optical region is now possible, it still often involves the use of individual

ingenious and difficult experiments, particularly in the

visible spectrum. For some time to come the best means

for interpolation or extrapolation with respect to the few

very accurate bench marks will be by the use of

wavelength interferometry; certainly it is the most con- venient for accuracies of 10"*^ or less. On the other hand

perhaps it is not an unreasonable hope that development

of new devices, such as possibly a broad band non-linear

reflector, will make possible "day-to-day" use in the optical region of the great accuracy and convenience in- herent in the methods of frequency comparison.

The author acknowledges with thanks the help of his colleagues and especially the major contribution of G. R. Hanes to the preparation of this paper.

References

[1] K. M. Baird, K. M. Evenson, G. R. Hanes, D. A. Jennings,

and F. R. Petersen, Opt. Lett. 4, 263 (1979).

[2] Y. S. Domnin, N. B. Koshelyaevskii, V. M. Tatarenkov, and

S. Shumyatskii, Pis' ma Zh. Eksp. Teor. Fiz. 30, 273 (1979) [JETP Lett. 30, 253 (1979)].

[3] D. J. E. Knight, G. J. Edwards, P. R. Pearce, and N. R.

Cross, Nature 285, 388 (1980).

[4] A. Clairon, B. Dahmani, and .J. Rutman, IEEE Trans. In- strum. Meas. IM-29, 268 (1980).

[5] K. M. Evenson, J. S. Wells, F. R. Petersen, B. L. Danielson, and G. W. Day, Appl. Phys. Lett. 22, 192 (1972).

[6] C. C. Bradley, G. Edwards, and D. J. E. Knight, Radio

Electron. Eng. 42, 321 (1972).

[7] Comptes Rendus des Seances de la Conf. Gen. des Poids et Mesures, 15th, 103 (1975).

[8] B. G. Whitford, Opt. Commun. 31, 363 (1979).

[9] B. W. Jolliffe, G. Ki-amer, and J.-M. Chartier, IEEE Trans. Instrum. Meas. IM-25, 447 (1976).

[10] N. B. Koshelyaevskii, A. Obukhov, V. M. Tatarenkov, A. N.

Titov, J.-M. Chartier, and R. Felder, Metrologia 17, 3

(1981).

[11] D. A. Jennings, F. R. Petersen, and K. M. Evensen, Appl.

Phys. Lett. 26, 510 (1975); Opt. Lett. 4, 129 (1979).

[12] V. P. Chebotayev, V. M. Klementyev, and Y. A. Matyugin,

Appl. Phys. 11, 163 (1976).

[13] G. R. Hanes, Appl. Opt. 18, 3970 (1979).

[14] A. Clairon, A. VanLerberghe, C. Salomon, M. Ouhayoun,

and C. J. Borde, Opt. Commun. .35, 368 (1980).

[15] K. J. Siemsen, Opt. Lett. 6, 114 (1981).

[16] H. -U. Daniel, M. Steiner, and H. Walther, Appl. Phys. 25, 7 (1981).

[17] G. D. Boyd, H. Kasper, and J. H. McFee, IEEE J. Quantum

Electron. QE-7, 563 (1971).

[18] T. W. Hansch and N. C. Wong, Metrologia 16, 101 (1980).

[19] B. Bukhard, H. J. Hoeffgen, G. Meisel, W. Reinert, and B. Vowinkel, these proceedings.

[201 U.S.S.R. State Committee of Standards Bulletin 16, Domnin, ei al., (1981) [ISSN 0135-2415].

41

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Optical Frequency Standards: Progress and Applications

J. L. Hall

Joint Institute for Laboratory Astrophysics, National Bureau of Standards and University of Colorado,

Boulder, CO 80309

The decade since the First International Conference on Precision Measurement and Fundamental

Constants has witnessed dramatic progress in stable laser technology. For example, frequency stability

[1], linewidth [2], and reproducibility [3] of ~3 x 10"''* have been reported for the methane-stabilized HeNe laser at 3.39 |xm. Impressive performance has also been obtained wdth CO2 lasers stabilized with CO2 [4] and other molecular resonances [5]. In the visible, argon ion lasers with an I2 molecular

reference have given 10"'^ reproducibility [6]. Recently, a frequency reproducibility of 6 x 10"'^ has been reported for the orange HeNe laser (612 nm) stabilized to an external I2 cell [7]. HeNe lasers operating on the usual red line can give reproducibility in the 10"'' domain using intracavity absorption in I2 [8]. However, to reach spectral transitions of particular physical interest (e.g., H [9]) or of special promise as standards (e.g., Ca at 657 nm [10]) in general will require use of a broadly tuneable laser, typically using color center crystals or a flovdng dye solution as the active medium. Comparable stabiliza- tion results with such tuneable lasers—especially dye lasers—is vastly more difficult than with gas

lasers, although kilohertz linevndth dye lasers have just been reported [11]. A technique suggested by Drever [12] recently allowed achievement of sub-100 Hz dye laser linewidth [13]. High resolution optical interaction techniques based on Ramsey's method of "separated oscillating

fields" have been studied theoretically [14] and demonstrated experimentally using two-photon [15] and/or multizone saturated absorption techniques [16] in atomic beams. As in the high resolution methane cell work [17], uncertainty in the second order Doppler shift provides the principal limitation

to the accuracy of optical frequency standards [18], even using atomic beam/Ramsey resonance tech- niques [19]. Measurement of atomic beam velocity distributions may be accomplished with gated excitation of the Ramsey zones, analogous to the techniques employed vdth cesium microwave standards [20]. Alternatively, the interacting molecular beam velocity may be precisely defined with dual frequency,

longitudinal saturation spectroscopy [21]. Useful signal/noise ratios may be feasible using sensitive cry- ogenic bolometeric detection of the excited beam [22] or using the recently-introduced optical hetero- dyne technique [23, 13] which appears to approach closely the fundamental quantum detection limit

[24]. The ultimate solution to the Doppler shift problem is surely to slow [25] and /or deflect [26] the atomic beam or to cool [27] electromagnetically-trapped ions using radiative processes. The only meaningful limitation to the latter technique may be inferred from the following: we provide an expanded

list of interesting candidate transitions for atomic beam frequency standard investigations; however

there are only three entries in a comparable list of suitable ions, filtered for compatibility with existing

potentially-stable laser sources. Although the ultrastable laser technology is still immature and of imperfect reliability—especially

for cw dye lasers—there is a certain temptation to begin applying these optical frequency standard

techniques to physical measurement problems of outstanding fundamental interest. One such measurement, a HeNe laser version of the Michelson-Morley isotropy of space experiment, has already

appeared [28]. Other precision experiments underway include: determination of the ground state Lamb-shift using two-photon spectroscopy [9] or construction of a fundamental frequency standard

using the same 1S-2S transition [29]; remeasurement of the Rydberg constant using precision atomic

beam spectroscopy [30]; precise measurement of metastable and Rydberg energy levels in helium to test the quantum-defect formalism; remeasurement of the relativistic time dilation with vastly higher

accuracy; and measurements to set a Hmit on any possible vector anisotropy in the speed of light. These and related stable laser techniques also appear relevant to interferometric gravity wave de- tectors [31] and some methods of detecting the influence of parity-nonconservation effects [32].

References

[1] J. L. Hall, in Atomic Physics 3, Ed. by S. J. Smith and G. K. Walters (Plenum, New York, 1973), p. 615.

[2] 7 X lO'l S. N. Bagev, L. S. Vasilenko, V. G. Gol'dort, A.

K. Dmitriev and A. S. Dychkov, Sov. J. Quantum Electron. 7, 665 (1977).

[3] S. N. Bagev and V. P. Chebotayev, Appl. Phys. 7, 71 (1975).

[4] C. Freed and R. G. O'Donnell, Metrologia 13, 151 (1977).

[5] C. J. Borde, M. Ouhayoun, A. vanLerberghe, C. Salamon, S. Avrillier, C. D. Cantrell, and J. Borde, in Laser Spectros- copy IV, Ed. by H. Walther and K. W. Rothe (SpringerVerlag, Heidelberg, 1979), p. 142.

[6] L. A. Hackel, R. P. Hackel, and S. Ezekiel, Metrologia 13. 171 (1977).

[7] P. Cerez, A. Brillet, C. N. Man-Pichot, and R. Felder,

IEEE Trans. Instrum. Meas. IM-29, 352 (1980).

[8] G. R. Hanes and K. M. Baird, Metrologia 5, 32 (1969). W. G. Schwitzer, E. G. Kessler, R. D. Deslattes, H. P. Laver, and J. R. Whetstone, Appl. Opt. 12, 2927 (1973). J. M.

Chartier, J. Helmcke, and J. A. Wallai-d, IEEE Ti-ans. In- strum. Meas. IM-25, 450 (1976).

[9] See for example A. I. Ferguson, J. E. M. Goldsmith, T. W.

Hansch, and E. W. Weber, in Laser Spectroscopy FV, op.

cit., p. 31.

[10] R. L. Barger, T. C. English, and J. B. West. Opt. Commun.

18, 58 (1976).

43
[11] J. Helmcke, S. A. Lee, and J. L. Hall, Appl. Opt. 21, 1686

(1982).

[12] R. W. P. Drever, private communication.

[13] R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, and A. J. Munley, in preparation.

[14] Y. V. Baklanov, B. Y. Dubetsky, and V. P. Chebotayev,

Appl. Phys. 9, 171 (1976).

[15] S. A. Lee, J. Helmcke, and J. L. Hall, in Laser Spectroscopy

IV, op. cit, p. 130.

[16] J. C. Berquist, S. A. Lee, and J. L. Hall, Phys. Rev. Lett.

38, 159 (1977).

[17] J. L. Hall, C. J. Borde, and K. Uehara, Phys. Rev. Lett. 37, 1339 (1976).

[18] J. L. Hall, C. Borde, and C. V. Kunasz, Bull. Am. Phys.

Soc. 19, 448 (1974).

[19] R. L. Barger, Opt. Lett. 6, 145 (1980).

[20] D. Halford, H. Hellwig, and D. Glaze, private communication.

[21] J. L. Hall, Opt. Commun. 18, 62 (1976).

[22] T. E. Gough, R. E. Miller, and G. Scoles, Appl. Phys. Lett.

30, 338 (1977).

[23] G. C. Bjorklund, Opt. Lett. 5, 15 (1979).

[24] L. Hollberg, T. Baer, H. Robinson, and J. L. Hall, Appl.

Phys. Lett. 39, 680 (1981).

[25] T. W. Hansch and A. Schawlow, Opt. Commun. 13, 68 (1975).

[26] J. E. Bjorkholm, R. R. Freeman, and D. B. Pearson, Phys. Rev. 23, 491 (1981).

[27] D. Wineland and H. Dehmelt, Bull. Am. Phys. Soc. 20, 637

(1975).

[28] A. Brillet and J. L. Hall. Phys. Rev. Lett. 42, 549 (1979).

[29] E. V. Baklanov and V. P. Chebotalyev, Opt. Commun. 12, 312 (1974).

[30] C. Weiman, private communication; W. L. Lichten, private

communication. (See also these proceedings.)

[31] R. W. P. Drever and K. S. Thorne, private communication:

P. L. Bender and J. E. Faller, private communication.

[32] I. B. Khriplovich, JETP Lett. 20 315 (1974).

44
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Measurement of Frequency Differences of Up to 170 GHz Between Visible

Laser Lines Using Metal-lnsulator-Metal Point Contact Diodes*

H.-U. Daniel^ M. Steiner^^ and H. Walther^'^^

Frequency differences of up to 170 GHz between the lines of a cw dye laser and a krypton laser at 568 nm were measured by mixing laser and microwave radiation in a metal-insulator-metal point contact diode. The beat signals exhibit good signal-to-noise ratio and no frequency "roll-off is observed

when increasing the laser frequency difference from a few hundred MHz to 170 GHz. It follows that the point contact diode could be used at still much higher difference frequencies. Furthermore, these

investigations show a diode response which is different at microwave and visible laser frequencies.

Video detection experiments performed in the visible show the influence of thermal phenomena in the diode junction having a roll-off frequency of a few megahertz.

Key words: frequency measurements; heterodyne spectroscopy; metal-insulator-metal diodes.

1 . Introduction

Metal-insulator-metal point contact diodes (MIM di- odes) have already been weW knovi^n for about 15 years as

effective nonlinear mixers and harmonic generators for infrared and microwave radiation [1]. On these grounds

they have been extensively used for absolute measure- ments of infrared laser frequencies [2] and for precision

infrared heterodyne spectroscopy (see, for example, Ref.

[3]). HoM^ever, until recently all efforts to use MIM

diodes in the visible spectrum for these purposes have

failed [4], thus causing a rather extensive discussion on

the diode action in general. Free electron tunnelling [5], thermally enhanced field emission [6], and photoexcitation of tunnel electrons [7] are but a few of the possible

mechanisms proposed.

This contribution reports on recent efforts made by the

authors [8, 9] to extend the range of application of MIM

diodes into the visible spectrum. By mixing the radiation

of a cw ring dye laser and a krypton laser at 568 nm with

an appropriate microwave frequency it could be demonstrated that MIM diodes are valuable tools for difference

frequency measurements even in the visible: The

170 GHz we report is the largest frequency difference

yet measured between two visible laser lines. Further- more, our results strongly indicate different mechanisms

to be effective in the microwave and visible frequency

ranges. These different mechanisms are found to cause

serious signal losses when mixing microwaves and visible laser radiation and, therefore, make it difficult to use microwave harmonics simultaneously generated in the

diode.

2. Experimental

Video detection experiments could be done simply by

focusing the multi-mode radiation of an argon ion laser

(514.6 nm) onto the point contact diode and measuring

the laser-induced diode currents at the kilohertz chopping frequencies and/or at the intermode beat frequen-

*Work supported in part by the Deutsche Forschungsgemeinschaft.

tMax-Planck-Institut f. Quantenoptik, D-8046 Garching, Fed. Rep. Germany.

ttSektion Physik, Universitat Miinchen, D-8046 Garching, Fed. Rep. Germany.

cies. Mixing experiments, however, were partly per- formed with an optically pumped sodium dimer ring laser emitting two or more lines simultaneously and in the

single-mode regime [8]. As this set-up showed certain

drawbacks, the main part of the mixing experiments was

done with the apparatus schematically shown in Fig. 1

OSA

Figure 1. Block diagram of the experimenial set-up.

[9]. The dye laser used was an actively stabilized ring

dye laser (Coherent 699) providing a single-mode output

of up to 200 mW at 568 nm (Rhodamine 6G). A singlemode krypton ion laser (Spectra Physics 171 with intra- cavity etalon) which was locked to a temperature controlled confocal Invar Fabry-Perot cavity, served as a second laser source. Additionally its intensity was stabil- ized using an external ADP crystal device. The detuning

of the dye laser emission relative to the fixed kiypton

laser frequency could be monitored by measuring an lo absorption spectrum (see Fig. 1) and comparing it with

the iodine line atlas of Gerstenkorn and Luc [10]. Both laser beams were adjusted for optimal collinearity, expanded in a telescope and finally focused onto the

diode by means of a microscope objective. The diode it- self consisted of a 25 ixm thick tungsten wire with an

electrolytically etched tip (radius of curvature 50-80 nm)

in mechanically adjustable contact with a polished cobalt

45
platelet. This electrode material seemed to offer a some- what superior stabihty compared with previously used

metals.

Microwave frequencies tunable from 60 to 90 GHz

were produced by a wobble generator (Marconi Model

6600/1 with a Siemens backward wave oscillator BWO

6655, peak output power 50 mW) and could be controlled

by a three-stage frequency locking stabilization scheme

which has been described elsewhere [8]. In the G-band a

reflex klystron (Varian VRT 2122 A) provided an ap- proximate power of 10 mW at 170 ± 1 GHz; however, the

spectral density was rather low since, owing to a power

supply failure, the line width of the klystron emission was 10 MHz. The microwave frequencies were coupled

into the MIM diode while it stood in a small groove in the

front end of a suitably twisted piece of waveguide. In the

E-band a small gold reflector at an appropriate distance

from the diode and the waveguide helped to form a fa- vorable cavity-like field distribution.

Beat signals detected by the MIM diode were amplified

(Avantek AMT 2006 M, 0.1-2 GHz, 49.5 dB gain, 4 dB

noise figure) and then measured in a microwave spectrum

analyzer. Ovdng to mechanical chopping of the laser beams the video output of each spectrum analyzer scan

could be detected by lock-in techniques. When the laser difference frequency and a microwave harmonic were

mixed, a digital signal averager following the lock-in amplifier had to be applied to recover the beat signals from

noise.

3. Frequency Difference Measurements

For reasons yet to be explained we always found the

optimal detection of visible laser light in low ohmic point contact diodes. Evidently, this caused signal losses when

mixing laser radiation and microwaves because the latter are best detected by highly nonlinear, i.e., high-resistance, diode junctions. Consequently, mixing experiments always had to begin with the observation of a low

frequency beat note (about 40 dB above noise) between

the two laser lines with a low resistance (10-20 O) MIM

diode. Increasing the diode resistance subsequently led to improved microwave detection and finally to reproducible observation of laser-microwave beat signals. These

beats were obtained within a wide range of diode resistances, but for stability reasons the best results fell into

the 30-50 fl range. Furthermore, most effective mixing was achieved when the detected dc signals of laser and

microwave intensity showed positive polarity.

Figure 2 shows mixing signals between the two dif- ferent laser hnes and the 170 GHz klystron emission obtained with a 30 ohm diode. The best signal-to-noise

ratio observed was 10 dB. We found similar mixing signals at a few hundred MHz as well as in the whole band

from 60 to 90 GHz with amplitudes up to 14 dB above

noise. This, however, certainly does not indicate a fre- ^ quency roll-off, but has to be attributed to the mentioned

fact that the klystron emission hne width at 170 GHz

was nearly a hundred times as broad as the BWO emission Hne width.

All beat signals between laser lines and fundamental

microwave frequencies were observed by real-time spec- trum analysis and without the need of signal averaging.

Averaging was found necessary, however, in the case of

mixing the laser light with a second harmonic of a micro- wave frequency generated simultaneously on the diode.

The beat signals achieved between two green Na2 dimer

laser lines (frequency difference 122 GHz) and the

second harmonic of a 61 GHz microwave frequency were

heavily buried in noise (about -20 dB). This conversion

loss of more than 30 dB is considerably higher than the

loss usually observed when generating microwave harmonics in a MIM diode. As mentioned above it has to be

ascribed to the low-resistance diode characteristic neces- sary for the light mixing experiment. Consequently, the

latter still turns out to be a major obstacle in mixing the

laser light with microwave harmonics; solving this problem would open an even vdder range of application to MIM point contact structures.

4. Detection IVIechanisms

While elastic electron tunnelling has finally been established as the diode mechanism effective in the infrared

and millimeter ranges [11], the response of the diode to

visible laser hght is still under discussion. Here thermal,

Figure 2. Beat signal obtained by mixing the light of the dye

laser and the krypton laser with a microwave frequency of 170 GHz. a: photograph of the beat signal (central peak

between two strong noise bursts) taken from spectrum analyzer

(vertical scale 10 dB/div, intermediate frequency 409 MHz, IF

bandwidth 1 MHz, scan width 5 MHz/div, scan time 5 msl

MHz), b: the same signal after phase-sensitive detection.

OdB

-20

-AO

-60 -

n I I I >iM| I—I

I 1 1 iM| r "ni 1—I 1 1 1 III

-J I I 1 1 1 III I I 0,1MHz 1MHz 10MHz 100MHz 1GHz

FREQUENCY DIFFERENCE

Figure 3. Laser induced diode current increase versus fre- queticy of the laser intensity modulation (diode resistance 50 ohm).

46
photoemissive, and geometrical phenomena are involved,

thus leading to rather complex diode behaviour.

Two especially interesting findings which have been

discussed in detail elsewhere [8] should still be mentioned here: In video detection experiments there is fre- quency roll-off observed in the detection characteristic of

MIM diodes (Fig. 3), and there can be laser driven

currents of different polarity depending on their modulation frequency [8]. Both indicate that there is a substantial change in the diode action when going from near dc

detection, where most of the investigations were per- formed (see, for example, Ref. [5]), to true mixing

conditions.

The results given in Fig. 3 were obtained by detecting

the diode current induced by multimode argon ion laser radiation (514.6 nm) in a 50 ohm junction. Between

50 kHz and 80 MHz the beam was modulated in an

acousto-optical device; higher intermediate frequencies were produced by intermode beat signals. While the high

frequency roll-off is due to diode mismatch, the low fre- quency decrease of 30 dB is caused by the time characteristic of the thermal diode response. The solid line drawn in this frequency range (Fig. 3) displays the

theoretical tunnelling current driven by thermal modulations of a whisker tip with a 1.7 MHz thermal cut-off frequency [6, 8]. In addition, noise temperature measure- ments give tip temperatures between 500 and 1000 K. It

is therefore concluded that thermal heating of the junction provides the main contribution to the diode response

below 100 MHz. At higher frequency differences it probably assists the field emission of electrons at both sides

of the contact [8, 12], thus causing the polarity changes

mentioned above. So the main difference between the ob- served diode response to infrared and microwave fre- quencies and to visible laser light seems to consist of thermal influences in the latter case.

5. Summary

It has been shown that MIM point contact diodes can be used as effective nonlinear mixer elements for visible laser light and microwave frequencies. The signal-tonoise ratio can be further improved by using more

powerful and frequency stabilized microwave sources and

a better mechanical adjustment for the diode. There was

no frequency roll-off observed in the mixing experiments;

this suggests that for the measurement of larger fre- quency differences the microwaves can be replaced by

infrared or far-infrared laser frequencies, which are known to couple even better to the diode whisker. Such a

link between the visible and infrared regions is of considerable importance for metrology. In the visible spectrum

the range of frequency differences now obtainable with

MIM diode mixers is far wider than before, which will make it possible to exploit the obvious advantages of fre- quency measurements as compared to wavelength measurements. The simphcity of the MIM diode technology

and already widespread knowledge of it should be of particular advantage compared with, for example, the use of integrated Schottky mixers.

References

[1] L. 0. Hocker and A. Javan, Phys. Lett. 26A, 255 (1968).

[2] K. M. Evenson, D. A. Jennings, F. R. Petersen, and .J. S. Wells, in Laser Spectroscopy III, Ed. by J. Hall and J. L. Carlsten (Springer, Berlin, New York, 1977) p. 56.

[3] J. S. Wells, G. E. Streit, and F. R. Petersen, Opt. Commun.

19, 248 (1976).

[4] D. A. Jennings, F. R. Petersen, and K. M. Evenson, in Laser

Spectroscopy IV, Ed. by H. Walther and K. W. Rothe

(Springer, Berlin, 1979) p. 39.

[5] S. M. Faris, T. K. Gustafson, and J. C. Wiesner, IEEE J. (Quantum Electron. QE-9, 737 (1973).

[6] A. A. Lucas and P. H. Cutler, Solid State Commun. 13, 361

(1973).

[7] G. M. Elchinger, A. Sanchez, C. F. Davis, Jr., and A. Javan,

J. Appl. Phys. 47, 591 (1976).

[8] H.-U. Daniel, M. Steiner, and H. Walther, Appl. Phys. 25, 7

(1981).

[9] H.-U. Daniel, M. Steiner, and H. Walther, Appl. Phys. 26, 19 (1981).

[10] S. Gerstenkorn and P. Luc, Atlas du spectre d'absorption de

la molecule de I'iode, U800-20000 cw"', (Editions du

CNRS, Paris 1978).

[11] A. Sanchez, C. F. Davis, Jr., K. C. Liu, and A. Javan, J. Appl. Phys. 49, 5270 (1978).

[12] M. J. G. Lee, R. Reifenberger, E. S. Robbins, and H. G. Lin- denmayr, J. Appl. Phys. 51, 4996 (1980).

47

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Precision Frequency Metrology for Lasers in the

Visible and Application to Atomic Hydrogen

B. Burghardt, H. Hoeffgen, G. Meisel, W. Reinert, and B. Vowinkel

Institut fur Angewandte Physik and Radioastronomisches

Institut, Oniversitat Bonn, D-5300 Bonn, F.R.G.

A multi-step method is discussed that permits the determination of frequency differences between

lasers in the visible in cases where the beat frequency is too large for direct detection. A step width of 80 GHz is used; the beat signal is picked up with millimeter-wave GaAs photodiodes. The resulting beat signals can be measured without further smoothing using a frequency counter. We report on ex- periments with atomic hydrogen, applying the method to measure transition frequencies aiming to determine the Rydberg frequency and the electron/proton mass ratio with increased precision.

Key words: atomic hydrogen transitions; electron-proton mass ratio; frequency measurement for visible laser radiation; Rydberg frequency.

1. Introduction

Heterodyne techniques have been proven to be ex- tremely powerful in all fields of frequency metrology.

For optical laser frequencies this principle is applied by

superimposing the parallel beams of the lasers so that

the beat frequency can be detected via a photodiode. The

result is the difference betw^een the frequencies of the

two lasers. This method is widely used to determine the

spacings, e.g., between hyperfine components and other

details in optical spectra [1]. An upper limit for the size of the measurable spacings is set by the reaction speed of

the diode. Commercial fast photodiodes have cut-off fre- quencies ranging from 5 to about 10 GHz with a fast fall- off if they are used at higher frequencies.

An alternative task is to determine the absolute fre- quency of laser radiation. The laser can be one that has

been tuned to some atomic or molecular transition of in- terest or it may be a new reference laser with a stable

but not yet accurately known frequency. There are three

ways to determine the frequency of such a laser: First,

the laser can be compared with the frequency of one or

several other lasers through a frequency chain that

reaches into the visible [2]. Nonlinear crystals and MIM

diodes can be used for this purpose though the method is difficult to apply routinely since it requires that phase

matching conditions for laser radiation over a very wide

range be met. The nonlinear efficiencies of crystals that

meet such conditions are small, so that in many cases the

signals may be too small. Even if this method can be applied to selected cases only, its great value lies in the fact that it can be used to establish one or several reference

laser frequencies in the visible with high accuracy. The second method is to compare the wavelength of the

laser v\dth unknown frequency vdth that of a laser of

known frequency [3]. The main problems arise from the

fact that in an interferometric wavelength comparison

geometric properties of the two laser fields such as their

parallelism and wavefront shape influence the result systematically. Some progress has been achieved in this

field by carefully controlling the laser beam parameters

[4].

2. Optical Laser Frequency Measurement

A third method by which to determine the absolute fre- quency of visible laser radiation is discussed in this paper. It makes use of a reference laser with well-known

frequency and a beat frequency determination. Thus it avoids the dangers of interferometric methods and gains

from the fact that beat frequencies are not shifted if the

two superimposed beams are out of parallelism. The ob- vious principle of the method is to determine the fre- quency difference between the reference laser and the

laser whose frequency is to be determined. The differ- ence is added to or subtracted from the reference fre- quency for the final result. It is the scope of this paper to discuss how this method can be applied in practice.

Clearly, the reference frequency should be close to the

unknown frequency so that the beat frequency is low and

thus easily measurable. The octave of visible light, however, spans about 375 THz. If a set of, e.g., 10 reference

lasers with almost equally spaced frequencies is established vdthin this band, the difference vdth respect to an

arbitrarily chosen unknown frequency can be as large as 20 THz approximately. At present the only well-known

reference lasers are stabilized to iodine transitions at 633

nm (HeNe, [5]) and 514 nm (Ar, [6]). There are no photodiodes that are fast enough to follow a 10 THz beat oscil- lation. A solution to the problem of measuring such a high difference frequency is to cut it into many smaller

differences that can be measured separately. This is achieved by using two cw dye lasers as interpolating oscillators. The principle is schematically presented in Fig.

1: Laser 1 has the unknown (or only approximately

knovra) frequency Vj. . Dye laser 2 is tuned to a frequency

V2 so that (v2 - Vj:) is measurable. Dye laser 3 in turn is tuned to V4, etc., until Vref is reached. The sum of all partial differences is the final large difference.

It is clear that the practical application of this scheme

requires that the step width be as high as possible in order to avoid excessively large step numbers and cumulative errors. We found a step width of 80 GHz (approximately 1 A at 6OOOA) a reasonable compromise between

costs and efficiency of the method [7]. A special GaAs

49
Figure 3. Schematic diagram of the E-band heat frequency

Figure 1. Schematic diagram to illustrate the procedure using detector, two interpolating oscillators (lasers 2 and 3) to close the gap

between laser 1 of unknow7i frequency ( vj and a reference

laser ( Vy^f).

2 \im

Figure 2. Cross section of the GaAs Schottky photodiode used

to detect beat signals of 80 GHz and over.

Schottky diode was used to pick up the beat oscillation

[8]. The active area of the diode has been enlarged to in- crease the beat signal. The resulting signal/background

level was high, namely 40 db for two 2 mW laser beams.

Figure 2 is a cutaway view of a single diode. Several

thousand are manufactured on a diode chip which is mounted inside an E-band wave guide light detector

(Fig. 3). The diodes were used up to a 70 mW cw light level focused to a spot size of 10 ixm vdthout damage to the diode.

3. Application to Hydrogen Transitions

An experiment is being prepared to measure the fre- quency of the hydrogen H„ transition and of other transitions in this way. For He, the HeNe reference laser will be used, which requires about 210 steps of 80 GHz each. The expected uncertainty is 0. 1 to 1 kHz per step result- ing in an overall error of 20 to 200 kHz for the total difference with respect to the HeNe laser. The other

main contribution to the error arises from the uncertainty to which laser 1 (which in this case is identical to laser 3) can be tuned to the center of the 30 MHz wide H„

transition. A conservative estimate of this uncertainty is 1% of the line width or 300 kHz [9]; with careful control

of the atomic lineshape and the laser light distribution

the uncertainty might be as low as 10"^ of the linewidth

or 30 kHz. Altogether it is planned to determine the He

frequency to within 50 to 500 kHz or with a relative uncertainty of 10^ to 10^^°. The result will be an improved

value of the Rydberg frequency, R^c with an accuracy

approaching the 10" level. The experiment is performed with the "free" atoms of an atomic beam in order to reduce any perturbations to the lowest possible level. The Hght interaction region is designed to avoid Doppler effects, making use of two spatially separated light fields from two exactly counter- running laser beams [10]. This method ensures that the

resulting Lamb dip cannot be shifted by non-perfect

alignment of the atomic beam with respect to the laser. Other atomic hydrogen transitions that are within the

range of available cw lasers are, e.g.. Ha (4860 A) and H^

(4341 A). Since the 4p and 5p states involved have considerably longer natural lifetimes than the 3p state associ- ated with the Ha transition, the accuracy might even be

higher. If one accurately measured hydrogen frequency

is divided by another, the Rydberg frequency as a lead- ing factor is eliminated, which allows interesting tests

for the remaining calculated factors [11]. In another experiment we are trying to observe the

two-photon transition from 2s to 7s, 8s or 9s of hydrogen

which requires readily available laser radiation between

750 and 800 nm. Besides an absolute frequency determination, the experiment aims to measure the isotope shift in order to improve the accuracy of the proton/electron mass ratio.

50
References

[1] G. Nowicki, H. Bekk, S. Goring, A. Hauser, H. Rebel, and

G. Schatz, Phys. Rev. C 18, 2369 (1978).

[2] K. M. Baird, K. M. Evenson, G. R. Hanes, D. A. Jennings, and F. R. Petersen, Opt. Lett. 4, 263 (1979).

[3] H. P. Layer, R. D. Deslattes, and W. G. Schweitzer, Jr., Appl. Opt. 15, 734 (1976).

[4] J. P. Monchalin, M. J. Kelly, J. E. Thomas, N. A. Kurnit, A.

Szoke, F. Zernike, P. H. Lee, and A. Javan, Appl. Opt.

20, 736 (1981), and references therein.

[5] F. Bayer-Helms, Ed., PTB-Bericht ME-17 (PhysikalischTechnische Bundesanstalt, 1977).

[6] F. Spieweck, IEEE Trans. Instrum. Meas. IM-29, 361

(1980).

[7] B. Burghardt, H. Hoeffgen, G. Meisel, W. Reinert, and B. Vowinkel, to be published

[8] B. Burghardt, H. Hoeffgen, G. Meisel, W. Reinert, and B. Vowinkel, Appl. Phys. Lett. 35, 498 (1979).

[9] B. Burghardt, M. Dubke, W. Jitschin, and G. Meisel, Phys.

Lett. 69A, 93 (1978).

[10] B. Burghardt, H. Hoeffgen, H. Kritz, and G. Meisel, PTBBericht E-18 (Physikalisch-Technische Bundesanstalt,

1981).

[11] G. W. Erickson, J. Phys. Chem. Ref. Data 6, 831 (1977).

51

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

System for Light Velocity IVIeasurement at NRLIVI

K. Tanaka, T. Sakurai, N. Ito, T. Kurosawa, A. Morinaga, and S. iwasaki

National Research Laboratory of Metrology 1-4, 1-chome, Umezono, Sakura-Mura, Niihari-Gun, IbarakI 305,

Japan

A system for making an absolute measurement of the wavelength and frequency of a stabilized carbon-dioxide laser is under construction. The wavelength has been measured by an up-conversion tech- nique using Proustite vwth reference to an iodine stabilized laser. The determined value is 9.31724631 (im

with a standard error of the mean of 1.4 x 10^* of the wavelength and the systematic uncertainty is roughly estimated to be 3 x 10"^ of the wavelength. For the frequency measurement, a water vapor laser and an optically pumped alcohol laser have been constructed. Tungsten-nickel and tungsten-cobalt point

contact diodes with precision mounts as harmonic generators and mixers have been developed and used

for evaluating the stability of the carbon-dioxide laser by beat frequency counting.

Key words: CO2 laser; light velocity; optical frequency difference; wavelength.

1 . Introduction

Since a precision value of the velocity of light was re- ported by Evenson et al. [1] at the fifth meeting of the

CCDM in 1973, the value was confirmed by Blaney et al.

[2] and Baird et al. [3]. The present authors, considering

the likelihood that a definition of the meter based on the

velocity of light will be adopted in the very near future,

have been developing a system, which is shown in Fig. 1, for measuring the absolute wavelength and frequency of

a carbon-dioxide laser to reconfirm the light velocity and

to provide for the establishment of a future wavelength

and optical frequency standard.

X-BAND KLYSTRON (lOGHz)

E-BAND KLYSTRON tSSGHz)

BEAT

(O.93GH2)

ICH3OH LASER (ll94GKzl|

'

^ (-0.05 BEAT

GHz)

^

J H—

1

K - BAND KLYSTRON

1 25 GHz)

|H;0 LASER (10718 GHz)

I

' BEAT > V (~ 0.05 GHz) y

1 ® K-BAND KLYSTRON

122 GHz)

I |C02 LASER (32176 GHz

WAVELENGTH )

>

9.3

HIGH POWER He-Ne LASER

LOCKED TO I2 STABILIZED LASER (633 nm)

FABRY -PEROT

INTERFEROMETER

^

66^^ (606nm) ^

2. Wavelength Measurement

In order to measure the absolute wavelength of a 9.3 |jim stabilized CO2 laser, 9.3 |xm radiation is up- converted to deep red 0.679 p.m radiation generated by

difference frequency mixing vdth the radiation from a 0.633 |xm stabilized He-Ne laser in a nonlinear crystal,

Proustite (AgsAsSs) [4, 5]. As the speed of light is independent of wavelength in vacuum, the relation I/X9.3 = l/Xo.ess ~ 1/^.679 holds. The

wavelength of 9.3 ixm can be calculated from the two

visible wavelengths, if they are knowni. The wavelength

of 0.679 |xm is measured with a pressure scanned FabryPerot interferometer vdth reference to a 0.633 ixm offset lock He-Ne laser whose wavelength is determined from

the ^^l2 stabilized He-Ne laser. The experimental system is schematically shovra in Fig. 2. The 1.6 m long CO2 laser having an intracavity

CO2 cell is stabilized to the R (12) line of the 9.4 |xm

band by the Javan-Freed method, locking to the zero

crossing point of the first derivative signal of 4.3 ixm sa- turated fluoresence. TEMoo output power of this laser is typically 0.3 W (after a chopper), and its frequency stability is 3 X 10-1^.

stab. C02 loser

Figure 1. System for light velocity measurement. Figure 2. Experimental systemfor tvavelength measurement of

carbon-dioxide laser.

53
The 2 m long 0.633 |xm He-Ne laser having a FoxSmith type mode selector is operated at a single fre- quency [6]. Its frequency is + 4.8 MHz offset locked,

with a relative stability of 1 x 10"^\ to the i-component

of the R(127) line of a ^^'^h stabilized He-Ne laser, which

has a reproducibility of 6 x 10"^^ [7]. The absolute

wavelength of this laser locked to the i-component of ^^^la has been evaluated to be 0.632 991 400 0 jjim with a standard error of the mean of 3 x 10"^ of the wavelength

[8]. As we have confirmed that the measured value

agrees with the recommended value of the CCDM in 1973

within the uncertainty of the measurement, the absolute vacuum wavelength of the offset locked laser is determined as 0.632 991 392 6 jxm based on the recommended

value of the iodine stabilized laser. The usual output

power of the TEMqo mode of this laser is 5 mW whereas

that of the iodine stabilized laser is 30 |jlW.

The beams of both the CO2 laser and the He-Ne laser are weakly focused into the center of a 6 mm long Proustite crystal so that the same confocal parameter of 0.2 m

can be obtained. The Proustite is cut and polished so that Type II phase matching is realized by angle tuning,

and it is positioned in a liquid nitrogen cryostat cooled to 77 K. The 0.679 ixm radiation of about 10"^ W is introduced to the Fabry-Perot interferometer along with that from the

offset locked He-Ne laser. A Glan-Thompson prism and a

grating monochrometer are used for suppressing the in- tensity of background 0.633 |xm radiation. A plastic

diffuser and a set of lenses are also used for achieving uni- form illumination onto the Fabry-Perot interferometer.

Scanning of the optical path length of the interferometer is made in a nearly linear fashion by introducing dry

nitrogen gas as a supersonic flow into the Fabry-Perot

chamber through a needle valve. Simultaneous scanning

for both radiations of 0.679 |xm and 0.633 jxm eliminates

the systematic error caused by thermal drift and

misalignment of the interferometer, or by the difficulty

in reaUzing the same reference pressure, 4 Pa, when the

scanning is started. The combined light of the two lasers

is detected by a photomultiplier, and its output signal is suppUed to two lock-in amplifiers, which are synchronized with two light choppers at two different chopping

frequencies of 400 Hz and 1 kHz, respectively. The out- puts from the two lock-in amplifiers are fed into a 2-pen

recorder, and a pair of interferograms are obtained on

the same chart. Dispersion of phase change on reflection is eliminated

by using a pair of data of fractional orders which are ob- tained by using two spacers of the Fabry-Perot inter- ferometers with the lengths of 50 mm and 170 mm,

respectively.

Figure 3 shows typical data of Fabry-Perot fringes of

0.679 (xm and 0.633 \xm. The centers of the interference

fringes are determined from abscissa readings on both

sides of each peak at six intensity levels between 30 and

80% of the peak height. Fractional order number is de- termined with a standard deviation of 0.001 fringe by ex- trapolating from four or five peaks by means of a least- squares method.

Thirty-three independent measurements were made for each of the short and long interferometers respectively.

The absolute wavelength of the 0.679 ixm difference fre- quency radiation was calculated using the method of exact

fractions with respect to the "virtual spacer," whose length

is the difference between the lengths of the long and short

interferometers. On the basis of the wavelength of the

0.633 |jLm offset locked laser, the vacuum wavelength of

-— pressure increase

Spacer Length 50nnm

Figure 3. Interferograms of 0.679 \im up-converted light and

0.633 |xTO helium-neon laser offset-locked to iodine stabilized

laser.

the 0.679 jxm radiation was determined to be 0.679 129

847 4 li-rn with a standard error of the mean of 1 x 10"^ of the wavelength. Thus, the infrared wavelength is calcu- lated to be

X9,3 = 9.317 246 31 (jLm

.

It should be noted that the uncertainty in X.93 is in- creased by a factor of 14. Therefore, the statistical uncertainty is estimated to be 1.4 x 10"^ of the wavelength

as a standard error of the mean. Although the systematic

uncertainties are not fully investigated, the uncertainty

due to interferometric measurement is roughly estimated

to be 3 X 10"^ of the wavelength.

3. Optically Pumped FIR Laser

An optically pumped far-infrared laser shown in Fig. 4

has been studied. The 2 m long carbon-dioxide laser for the pumping is stabilized by an opto-acoustic method.

The laser has a 1.5 m long, 8 mm bore discharge tube. A

25% CO2, 20% N2 and 55% He gas mixture flows at a pressure of 2.0 kPa at the entrance port. A maximum

output power of 31 W has been obtained under fundamental mode operation. The FIR laser has a 2.1 m long,

38.5 mm bore Pyrex waveguide. The resonator consists

of a gold coated flat mirror of fused silica with a 2 mm

entrance hole, and a flat aluminium mirror with a 5 mm

CH3OH

M2

FIR LASER

—2100mm

I OUT

M)

POWER

METER

PYREX GLASS TUBE

CO2

SPE.ANA.

XQ

PYRO- DET. •o

PARAL.

SPR.

i V - GRATING

I WATER jW.OUT

| ,|oo L/mm)

with 3 micro-m. 2000mm \ CO2 LASER

Figure 4. Schematic diagram of optically pumped FIR laser.

54
FIR coupling hole. Frequency tuning of the resonator is made by a parallel spring mechanism and a micrometer

driven by a small motor. We have made studies

concerned with increasing its output power and also sur- veyed the most suitable oscillation line for frequency synthesis vdth a water vapor laser and an E-band klystron.

4. Water Vapor Laser

Two internal-mirror-type water vapor lasers have been

constructed to examine the optimum discharge conditions, the characteristics of the output beam and the stabilities of output power and frequency. One has a 2.4 m

long resonator and a 25 mm bore discharge tube, and the

other has a 5.4 m long resonator and a 40 mm bore

discharge tube. Both lasers have similar designs. The

laser tube has a water cooling jacket, a copper anode and

a water cooled copper hollow cathode in the side arms. The laser resonator is composed of two gold coated Pyrex

mirrors which are connected rigidly by four invar rods. Frequency tuning of the resonator can be done by the same mechanism as that of the FIR laser. The water vapor is generated from distilled water in a

reservoir and gas additives are supplied to the laser tube

from the anode side; they are pumped out from the

cathode side by using a continuous gas flow system. The

mass flow rate of each gas is regulated by stainless steel needle valves and the gas pressure in the laser tube is adjusted by an exhaust valve. The short resonator laser was used to study the

operating characteristics and discharge conditions of the

28 |jLm output. It was found that the peak output power

was obtained at a discharge current which dissociated

the H2O rapidly [9]. An output power of about 30 mW is obtained under conditions of a H2O pressure of 80 Pa and

a H2 pressure of 120 Pa. The long resonator laser oscil- lated nearly at the condition which was estimated using a scaling law and the lasing condition in the short resonator laser. The output power was more than 150 mW multimode, but it was found that the optimum output power

was obtained at a different condition due to different sys- tems of gas flow in the two lasers. For extracting the 28 \xm output from the laser resonator, there are three methods where a coupling hole, an

intracavity beam divider and a Michelson-type coupler

are used. In the coupling hole method, higher order

transverse modes oscillated easily and the polarization

azimuth of the output beam varied with cavity-scanning

[10]. On the other hand, the alignment of the Michelsontype coupler was not easy. Therefore, the output beam

was coupled out using a 45° polyethylene intracavity

beam divider. The output beam was linearly polarized.

The TEMoo mode was obtained easily by inserting an

aperture in the cavity.

At present, we are aiming to stabilize the frequency of the laser to the Lamb dip with an FWHM of about

5 MHz. The frequency of the laser is stabilized by using a

piezoelectric translator. To obtain a good reproducibility,

the shape of the Lamb dip is being investigated. By

analyzing the beat frequency between two water vapor

lasers, origins of frequency fluctuations will be clarified.

5. M-l-M Point Contact Diode

For making precise frequency measurements, we have

made tungsten-nickel and tungsten-cobalt point contact

diodes [12, 13] of the metal-insulator-metal type [11]. A

short tungsten wire 25 |xm in diameter was spot welded

to the top of a 2 mm thick and 15 mm long brass rod. The

wire was bent and the top of the bent wire was etched by

conventional electrolytic polishing techniques. The flat, top surfaces of nickel or cobalt posts 30 mm in length and

3 mm in diameter were polished so that they had a flat- ness of better than one interference fringe with the

589 nm line of a sodium lamp. To obtain resettability and

stability of the point contact diode, a precise diode mount

which has fine mechanical adjustability and stability of contact pressure, was made. The tungsten-cobalt point contact diode had a S/N ratio 5-10 dB better than that

of the tungsten-nickel diode for detecting the rf beat note between harmonics of an X-band klystron and two CO2

lasers locked to 9R(24) and 9R(26) lines, respectively, as shown in Fig. 5. This diode was used for measuring the

stability of our CO2 laser, which was 3 x 10"^*^.

lOd

OdBm

70 MHz 0.5 MHz

Figure 5. Specty^um of heat signal detected using W-Co point

contact diode. Beat is obtained by mixing outputs of X-band

klystron and two CO2 lasers locked to 9R(24) and 9R(26) lines.

6. Conclusion

The absolute wavelength of a frequency stabilized carbon-dioxide laser has been evaluated to be

9.31724631 ixm vdth a statistical uncertainty of 1.4 x

10"^. A water vapor laser with a 28 fxm output of 150 mW and a tungsten-cobalt point contact diode with a S/N ratio improved by 5-10 dB have been developed.

We have been studying ways to increase the output of the FIR laser for Hnking the CO2 laser to the cesium fre- quency standard.

References

[1] K. M. Evenson, J. S. Wells, F. R. Petersen, B. L. Danielson, G. W. Day, R. L. Barger, and J. L. Hall, Phvs. Rev.

Lett. 29, 1346 (1972).

[2] T. G. Blaney, C. C. Bradley, G. J. Edwards, B. W. Jolliffe. D. J. E. Knight, W. R. C. Rowley, K. C. Shotton, and P. T. Woods, Nature 2.51, 46 (1974); Proc. Soc. London Ser. A

355, 89 (1977).

[3] K. M. Baird, D. S. Smith, and B. G. Whitford, Opt. Commun.

31, 367 (1979).

[4] K. M. Baird, H. D. Riccius, and K. J. Siemsen, Opt. Commun. 6, 91 (1972).

[5] B. W. Jolliffe, W. R. C. Rowley, K. C. Shotton, A. J. Wallard, and P. T. Woods, Nature 251, 46 (1974).

[61 A. Morinaga and K. Tanaka, Jpn. J. Appl. Phys. 17, 881 (1978).

[7] K. Tanaka, T. Sakurai, and T. Kurosawa, Jpn. J. Appl. Phys.

16, 2071 (1977).

55
[8] N. Ito and K. Tanaka, Metrologia 14, 47 (1978).

[9] A. Morinaga and K. Tanaka, IEEE J. Quantum Electron. QE-16, 406 (1980).

[10] A. Morinaga, Appl. Opt. 20, 2395 (1981).

[11] L. D. Hocker, D. R. Sokoloff, V. Daneu, A. Szoke, and A.

Javan, Appl. Phys. Lett. 12, 401 (1968).

[12] T. Kurosawa, T. Sakurai, and K. Tanaka, Appl. Phys. Lett.

36, 751 (1980).

[13] T. Kurosawa, A. Kuriyagawa, and K. Tanaka, Bull. Natl. Res. Lab. Metrology (Japan) 30, 8 (1981).

56
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Laser Wavelength Measurements and Standards

for the Determination of Length

W. R. C. Rowley

Division of IVIechanical and Optical Metroloqy

National Physical Laboratory, Teddington, Middlesex TW11 OLW, U.K.

The light emitted by portable stabilized lasers used as wavelength standards for length and spectroscopic measurements is reproducible to at least three parts in 10 , and different wavelengths can be

intercompared to this level of uncertainty by interferometry. Their absolute wavelength accuracy, limited at present to four parts in 10^ by the ^°Kr standard of the meter, wall be improved at least tenfold by a redefinition of the meter, based on the fixed value 299 792 458 m/s for the speed of light. Length

measurements, however, are seldom more accurate than one part in 10^, except in lunar and interplanetary ranging; although changes in length can be measured to better than one part in 10^'*. Key words: meter definition; precision length measurement; stabilized lasers.

1. Introduction

Developments in precision length and wavelength

measurement over the last decade have been dominated

to a large extent by laser techniques. For example, the

primary unit of length, the meter, was defined in 1960 by

the wavelength of a krypton-86 transition. By the mid1960' s, stabilized visible helium-neon lasers were becoming widely used for the practical measurement of length

by interferometry. The lasers available at that time were

stabilized to the Lamp dip in the center of the power tun- ing curve. The stability and reproducibility, at a few

parts in 10^, was significantly worse than that afforded

by the ^^Kr lamp. This accuracy was, and to a large extent still is, adequate for the majority of length measure- ment tasks involving material objects, such as length

bars and standard gauges. Gradually laser measurement

systems have become so widespread that, for example,

the majority of precision length measurement tasks at the National Physical Laboratory are now carried out vdth laser interferometer systems.

Lasers for interferometric measurement are important

because of three main characteristics:

(a) Temporal coherence—Their narrow monochromatic

bandwidth removes the former path-length restriction

of less than 1 m.

(b) Spatial coherence—The laser emits in a narrow beam of almost plane wavefront, so that the light can be easily and efficiently used in optical systems.

(c) Intensity—The amount of light and its efficiency of use makes photoelectric detection easy, leading to the

application of electronic methods and automated measurements. The length measurement technique that is normally

used with lasers is fringe-counting interferometry. Commercial systems are available for general measurements

of modest accuracy up to lengths of a few tens of meters. For the highest accuracy, however, purpose-built

mechanical systems are mandatory, with particular care

taken to avoid alignment, temperature, and refractive in- dex errors. The accuracy then attainable when using a well calibrated laser standard is as good as can be

achieved by using the ^^Kr standard lamp directly. For a 1-m length bar this accuracy is about ±1 x 10"^, or up to a factor of two better, at the 99% confidence level [1].

2. Saturated Absorption Stabilization

A great advance in the performance of lasers as refer- ence standards for length measurement was the introduction of the saturated absorption technique of stabilization

[2, 3]. It improved the reproducibility of laser standards

from being a factor of 10 worse than the '^^Kr lamp to a factor of 100 or more better than *^Kr. For several years, two such stabilized laser systems were widely studied

and developed. One of them, the 633-nm visible helium- neon laser, stabilized with iodine, is of particular importance because of its direct application to length measure- ment and its use as a standard for the calibration of other

lasers by beat frequency or interferometric wavelength

comparison. The other laser system is the methane-stabilized He-Ne laser at 3.39 ixm. Although not so suitable

for length measurement, this laser afforded the promise

of better stability and reproducibility. It has proved to be a particularly important link in the chain of frequency

measurements from the cesium frequency standard to- wards visible frequencies. The 3.39 p-m wavelength is also close enough to the visible for infrared /visible

wavelength measurements to be made. This has enabled

the ^^Kr and ^^^Cs standards of length and time to be in- terrelated, giving a value of the speed of light to the ac- curacy limitation imposed by the *"Kr standard [4]. It is usual and convenient for these stabilized lasers to have the saturable absorber within the laser cavity itself. This gives a neat and portable laser system. It is, however, not ideal. The interaction region is too small in di- ameter so that thermal movement takes the absorbing

molecules out of the beam, shortening the effective life- time; the length of the cell is restricted, so that the pres- sure of molecules must be raised to give sufficient signal;

57
and the high beam power density oversaturates the transitions, accentuating any asymmetry. Nevertheless, the

performance of these lasers is impressive enough. The

stability of a typical internal-cell 633-nm iodine-stabilized

He-Ne laser is 3 x 10"^^ for an averaging time of 10 s, and the corresponding figure is 1 x 10"^^ for the 3.39 |xm

methane-stabilized laser [5]. A more important measure of performance, however, is the reproducibility of the stabilized frequency (or

wavelength). As a result of many comparisons between

lasers of different origin, at various times and in various

laboratories, it seems that a reproducibility of ±2 x 10"^^ can normally be obtained with internal-cell 633-nm HeNe lasers, provided that the operating conditions are the same. The stabilized frequency is affected, for example,

by alignment errors, the iodine pressure, the amplitude

of frequency modulation, the iodine cell wall temperature, the internal power, etc. The iodine cell must also be

free from impurities, such as may result from outgassing

during or after sealing-off. From a joint study made re- cently between NPL and BIPM, it is clear that such con- tamination is not rare, and accounts for at least some of

the discrepancies of inter-laboratory reproducibility that are occasionally observed. It is clear that the quality of iodine cells needs strict monitoring [6, 7]. With internal-cell 3.39 iJim methane-stabilized lasers,

the methane hyperfine structure is not resolved, and an

inter-laboratory reproducibility of only ±3 x 10"'^ has

been reported in one study, although the reproducibility

figure of ±5 x 10"^^ has been obtained for lasers con- structed and compared within one laboratory [5]. By

putting the saturable absorber outside the laser cavity,

however, the conditions may be optimized, and much

better reproducibility achieved. The penalty is a nonportable system. Thus although a reproducibility of 1 x

10"^^ has been reported [5], it has not been possible to confirm, by direct beat-frequency comparison, that this can be achieved internationally with independently con- structed laser systems.

External-cell systems are also advantageous for iodine-stabilized lasers in the visible, and are probably 10 or 100 times better than the internal-cell systems. A

number of radiations of the ionized argon and krypton

lasers have been thus stabilized for use as wavelength

standards, particularly the argon 514-nm line [8]. The

orange 612-nm radiation and other transitions of the

helium-neon system also show great promise for excellent

stability and reproducibility [9]. Indeed, the number of

laser wavelength standards that could be realized, particularly by using dye lasers, is almost unlimited. The limit

will be set in practice by requirements and resources.

3. The Role of Wavelength Measurements

and Standards

Wavelength standards are the basis for interferometric

length measurement, but they serve a wider purpose. They are, for example, used in spectroscopy, where it is customary to specify absorption and emission lines by

their wavelength or wavenumber. The reason for this is historical and a matter of experimental convenience, as spectrographs allow such measurements to be made by

interpolation from a reference set of wavelength standards. It would be more fundamental, however, to specify

such radiations by the energy difference between the two

states involved in the transition. Alternatively, and more

conveniently, the frequency of the radiation, which is directly related to energy by the well known relation

E = /iv, is more fundamental than wavelength. I hope

and expect that there will be a gradual changeover from

the specification of wavelength or wavenumber to the

identification of spectral transitions by their frequency.

Thus one should perhaps talk about laser frequency

standards rather than laser wavelength standards. In comparing similar stabilized lasers to measure their

reproducibility, beat frequency techniques are used.

These are rapi'd, convenient and precise. Unfortunately it is not yet possible to use beat frequencies in the general

case to compare two lasers operating on completely dif- ferent spectral transitions. The frequency differences are

too great to be detected so easily. It is possible to work

out schemes to overcome this problem using non-linear

crystals as mixers and harmonic generators, synthesizing

the difference frequency approximately from infrared and

microwave oscillators. The final mixer/detector need

then only generate a low-frequency beat. At present,

however, it is not clear whether such schemes will be

viable for more than a few visible radiations of special in- terest. The alternative well-established technique is in- terferometric wavelength comparison. Although widely

applicable, it is unfortunately subject to a number of practical limitations that introduce errors and thus limit the accuracy. Nevertheless, accuracies of a few parts in

10^^ have been achieved [10]. Wavelength comparison is thus still a vital means for establishing the wavelength

ratios, and thus also the frequency ratios, of visible and

near infrared stabilized lasers. As such, it complements

the frequency-chain measurements that are now extending to visible frequencies. It will be interesting to see, however, to what extent frequency techniques may re- place the present role of wavelength measurement.

4. Interferometer Design

Wavelength measurements are carried out by inter- ferometric comparison, using the optical length of the

interferometer as a temporary or virtual intermediate

reference. The interferometers that are used may be di- vided into two main classes:

(a) Low-finesse, usually with two-beam interference,

such as the Michelson interferometer and its variants.

(b) High-finesse, with multiple-beam interference, of which the Fabry-Perot etalon is the normal form.

In their basic forms, both these systems form a Haidinger interference ring pattern in the image plane of an extended source, the transmission maxima for a radiation

of wavelength A. being given by:

m \ ^ 2t cos 8 , where m is the integer order of interference, t is the real or effective separation of the reflecting surfaces, and 6 is the angle of incidence.

As illustrated in Fig. 1, the main difference between

the two types of interferometer is the distribution of

light in the pattern between the maxima. The two-beam

system has a sinusoidal variation superposed on a con- stant background, whereas the multiple-beam system

gives rise to a sharply peaked intensity distribution. It would seem that the multiple-beam system defines the in- tensity maxima more precisely and should thus be advantageous. In practice, however, this advantage is offset by

other factors, such as transmission loss and asymmetry

of the peaks due to maladjustment or optical defects. Thus, both types of interferometer are roughly equally

utihzed in precision measurement.

58
Two - beam Curved mirrors are also used with Fabry-Perot

etalons. With laser sources these take the form of modematched cavities, similar to laser cavities. Such cavities can be made long without i'equiring tiny pinholes, and

they have a high finesse and transmission, giving excellent sensitivity.

A further subdivision of interferometer designs is ac- cording to the method of illumination:

(a) Spatially incoherent, or diffused laser light

(b) Coherent laser beam.

Multiple -beam

6t

^

W2 61

The forms of interferometer shown in Fig. 1 are designed

for an incoherent source that has a uniform intensity over the area corresponding to the exit pinhole, and that

also radiates uniformly over a sufficient solid angle to fill the entrance lens. Laser beams do not provide such il- lumination [12]. Partially coherent light, however, may

be formed with a diffuser. This gives rise to a speckle

field, and quasi-uniform illumination is often achieved by

moving or rotating this diffuser. Care must be taken, as

illustrated in Fig. 2, to avoid a Doppler shift due to deviation of the light at the diffuser. Reversing the direction

of rotation reverses the shift, so that it may be measured

or cancelled.

Figure 1. Interferometers used for wavelength measurement.

The two-beam (Michelson) form gives a sinusoidal intensity

variation, the multiple-beam (Fabry-Perot) gives sharp intensity maxima.

Interferometer designs may also be divided into vari- ous sub-classes. One such sub-class is according to the

reflector shape:

(a) Flat reflectors

(b) Curved (spherical) reflectors.

Flat reflectors are the classical form which, as just

described, give rise to a Haidinger ring pattern. Measurements can be made by measuring the diameters of

these rings, so that the fractional order at the centre may be determined by extrapolation. This used to be the common measurement technique, but it is now rarely employed. More usually, as illustrated in Fig. 1, the central

portion of the interference pattern is isolated by

transmission through a small hole, and measurements are made by varying the optical length in some linear or predictable manner. The angular size of the pinhole that

may be used depends inversely on the finesse and the re- flector separation t. Thus, for example, with the 1-m

Fabry-Perot instrument at NPL, the optimum size of the

pinhole is only 0.1 mm radius for a focal length of 1 m.

This makes the instrument sensitive to focus imperfections and aberrations of the imaging system.

Curved mirrors in the form of adjustable cat's-eye re- flectors are used in the two-beam interferometer at the

BNM in France [11]. This field-widened interferometer

enables a much larger detector pinhole to be used, which

is particularly advantageous when using relatively weak

incoherent light sources such as the ^^Kr lamp. With

laser sources, the field-vddening technique enables

greater reflector separations to be contemplated than

with flat mirrors, and it is planned to extend the BNM

instrument to t = 2 m.

Incident light

Frequency I'

Moving diffuser

velocity V

?^ (l + f

sin 0)

Eg

Disc at 10 rev/s, with beam 50mm from rotation axis. Doppler shift = 2x10""'° for 6 = 1 degree.

Figure 2. Doppler shift from a moving ( or rotating) diffuser.

Coherent laser beams, however, are essential for the

mode-matched form of interferometer. As showTi in Fig.

3, the equation specifying the exact length at which the

transmission maxima occur is slightly different, and in- corporates a phase shift term [13]. This term arises from

the diffraction limited propagation of a laser beam [14]

and may have a magnitude of a significant fraction of a wavelength. Undiffused laser beams may also be used in two-beam interferometers. As shown in Fig. 4, the two

wavefronts that interfere have different radii of curva- ture and different diameters. The combined effect causes

a slight reduction of visibility [15] together with a phase

shift, which is additional, and of comparable magnitude,

to the propagation phase shift [16, 17, 18, 19].

59
Plane parallel Fabry -Perot (Incoherent illumination)

Mode - matched cavity

(Laser illumination)

X = 2t Cos e

Radius R X = 2t - X 0

where © zCos-'(l -i- )'"

Figure 3. Resonance conditions (giving transmission maxima) for the plane-parallel

Fabry-Perot etalon, and mode-matched cavity interferometers.

I

In addition to the propagation phase shift, there is a phase shift due to the unequal wavefront curvatures

Combined effect: = D ~ arctan —^—7- 27r 2 TT Wo

Figure 4. Michelson interferometer with coherent laser illumination.

5. The Evolution of Wavelength

Measurement Technique

During the last decade, the form of interferometer most

used for precision wavelength measurement has been the

pressure-scanned plane-mirror Fabry-Perot. With dif- fused laser light, it is very suitable for measurements

directly with the lamp. The use of the primary standard radiation, however, restricts the precision because its bandwidth is wide and the etalon length must normally be

less than 20 cm. At the NPL [20, 21], the etalon was used

at near atmospheric pressure and scanned with a motorized piston. By scanning alternately with pressure in- creasing and decreasing, some effects of unequal elec- tronic time delays could be eliminated, but scanning at high pressure causes unfortunate temperature changes

and non-linearities. At the NBS (Gaithersburg) [22], the

NRC (Canada) [23], and NRML (Tokyo) [24] the etalon was placed in a vacuum chamber and scanned by leaking

dry nitrogen in through a supersonic nozzle. The interference patterns are usually detected photoelectrically using

chopped light, and digitized for computer analysis. At the

NRC, however, a moving photographic plate is used, and

this avoids the problems of unequal electronic delays. At

the IMGC (Italy) [25] the interferometer is kept evacu- ated and the Haidinger ring image is scanned across an

analysing slit in front of the photodector. The FabryPerot of the NSL (Austraha) [26,27] is not usually

scanned, but is adjusted onto the interference maxima

with a magnetic coil system that has a linear translation

characteristic and good reproducibility.

Michelson interferometer measurements of stabilized lasers relative to ^^Kr have also been carried out in several laboratories. In the interferometers at BIPM

[28] and PTB (West Germany) [17] the compensator plate

may be tilted slightly so that the phase of the interfer- ence pattern can be calculated by fitting a sinusoid to the

intensities at four equispaced values of path difference.

The instrument at NPRL (South Africa) [29] has one mir- ror on a parallel-spring translation member, driven by a

60
piezoelectric tube, with its position monitored by a ca- pacitance transducer. The linearity and calibration of

this transducer are checked every time measurements

are made. The observations are made by servolocking the

mirror to the positions corresponding to interference

maxima or minima and determining fringe fractions from

the capacitance transducer readings. With the field- widened instrument of the BNM (France) [11], one of the

cat's-eye reflectors may be moved on a parallel-spring

system by a differential screw together with a piezoelectric element. It may be scanned continuously over a few

fringes, or servolocked to successive fringes of the refer- ence laser. The measurement uncertainties in all these

measurements of stabilized lasers relative to ^^Kr is generally about 1 X 10~^ due to the practical limitations

imposed by the ^^Kr source, apart from the systematic

error in realizing the defined wavelength.

Significantly better accuracies are possible

in laser/laser comparisons. The path lengths may be

longer, and the source intensities are greater, so that more powerful observational methods may be used. In my opinion the most important development in this field has been the introduction of servolock methods, together

with tunable "slave" lasers. An early foretaste of this was

the work of Bay, Luther, and White [30] shown in Fig. 5. Sidebands separated by ± w from the laser carrier v pass

through an optical isolator and the mode-matched cavity,

and one servo-system locks the length of the cavity to the mean of the two fringe patterns, while the other modifies

the frequency co so that the two sidebands resonate at the same cavity length. As well as pointing the way to ser- volock methods, the modulation technique used in this ex- periment is a useful way of estabhshing the approximate

length, and hence the integer order number, of a long

interferometer [13, 31].

The use of a slave laser as the wavelength tunable ele- ment was introduced by Barger and Hall at the NBS,

Boulder [32]. The slave laser is servolocked so as to give

a maximum transmission of the interferometer, and

hence an integer number of wavelengths, and its detuning from the stabilized laser is measured by the beat fre- quency. Their particular measurement was relative to ^Kr so that the accuracy of the result was modest. Their

technique was, however followed at the NBS, Gaithersburg [13] in a measurement of the 3.39 |xm

methane-stabilized He-Ne laser relative to the 633-nm

iodine-stabilized laser. The reproducibility of these meas- urements was a few parts in 10^^, although the accuracy was limited to two parts in 10^'^ by systematic effects. The high precision of such measurements is due to the

advantageous use that is made of the observation time,

by comparison with scanning methods, and by the accu- racy of beat frequency counting instead of fringe interpolation as the measurement procedure. Another advantage

of the method is that because of the high precision, sys- tematic effects, due for example to adjustment errors, may be investigated quickly and without involving an un- due amount of data gathering. The method is also used at the PTB [34] and NPL [21, 33], and enables an accuracy

of a few parts in 10^^ to be achieved in visible laser comparisons [10]. Another wavelength measurement technique that has

been developed in the last decade is that of fringecounting, in which a moving reflecting element is traversed over a significant fraction of a meter, while

counting the interference fringes due to the unknown and

a reference laser. Its advantage is that no prior approximate knowledge is needed of the unknown wavelength,

or of a path difference. It is thus particularly suited to the swift determination of the wavelength output of tun- able dye lasers to an accuracy of 1 x 10"*' or 1 x 10"^. As

a precision measurement technique it is generally limited

by alignment errors due to the use of cube-corner reflectors; but when designed with accuracy in mind, the tech- nique may be as good as other two-beam interferometers.

Thus with the MIT instrument [19] an accuracy of a few

parts in 10^^ should be possible.

The main limitations to accuracy in wavelength measModulalor Isolator

10 GHz vco

Counter

Mode- matched

A+B —<

PSD

A-B

Figure 5. The apparatus of Bay, Luther, and White (1972), usnig a servo-locked modematched cavity interferometer, with feedback tuning of the source so that both sidebands

(v -\- u)) and (v — (ji) resonated simultaneously.

61
urement are:

(a) Flatness errors (or non-uniformity of curvature)

of surfaces,

(b) Non-uniformity of illumination (or mode impurity),

(c) Defects of imaging (or mode matching),

(d) Non-linearity of interpolation,

(e) Diffraction effects,

(f) Prismatic dispersion of windows,

(g) Parallelism and alignment errors,

(h) Electronic imperfections.

The flatness or errors of figure are the most serious

problem. A 1-m interferometer is roughly 3 x 10^ half- wavelengths in length, so that to achieve an accuracy of

3 X 10" each interference fringe must be subdivided to 10"^ of the fringe separation. The flatness of optical sur- faces is seldom better than X./100. The extra accuracy is achieved by averaging over the irregularities. The problem is to get the same average with both the radiations

being compared, and this is a particularly serious problem with interferometers that use spatially coherent

light, as the illuminated area then depends upon the

wavelength.

6. Wavelength Results

Many precision wavelength measurements have been

carried out on the methane- and iodine-stabilized He-Ne

laser radiations at 3.39 |xm and 633 nm, a few on the COo

laser system, and a few also on other iodine-stabilized

visible radiations. Measurements of the 633-nm laser, re- lative to ^''Kr have been reported by 10 laboratories, us- ing a variety of interferometer designs. As mentioned

above, the measurement uncertainties are generally in the region of 1 x 10"^, and their agreement well within

4 x 10"^. Measurements of the 3.39 [xm radiation directly

with respect to ^^Kr, reported by four laboratories, have

a slightly worse measurement uncertainty of about 4 x 10"^

The ratio of the two wavelengths 3.39 |jLm/0.633 fxm has

also been determined interferometrically. Direct meas- urements suffer the problem of the large diffraction ef- fect for the infrared radiation. In the NBS (Gaithersburg) measurement [13] this limited the accuracy to 2 x

10""^. A measurement at the PTB [34], however, was

limited to an uncertainty of 2 x 10 as also was a meas- urement at the NPL.

A measurement by the NRC [34] avoided this difficulty

by mixing the infrared radiation with that from a 633-nm

laser in a non-Hnear crystal, so as to give sum-frequency

radiation in the green. From a wavelength measurement

of this green light relative to a 633-nm iodine standard,

the required 3.39/0.633 ratio was determined. By this method the severe diffraction problem is avoided; but the

penalty is that the uncertainty of measurement is multiplied by the wavelength ratio in calculating the infrared

wavelength. Thus the uncertainty of the infrared /visible ratio was 7 x 10"^. This non-linear optical mixing method has also been

used to measure the wavelength ratio of a 9.7 |xm CO2

laser to the 633-nm laser, in this case by generating the

difference frequency in the deep red end of the visible spectrum. A measurement at the NPL by a pressure

scanning method [21] had an uncertainty of 5 x 10"^. Subsequently with a servolock flat-plate Fabry-Perot

system an uncertainty of 4 x 10"^" was achieved for the

infrared /visible ratio [10]. A direct measurement of the

CO2 wavelength is subject to serious limitation by

diffraction. Thus for the MIT measurement [19], the dif- fraction correction was -2 x 10"^ with a ±20% uncertainty, corresponding to an uncertainty in the

wavelength ratio of 4 x 10"^ and making the total wavelength ratio uncertainty 6 x 10~^. Only a few visible /visible laser wavelength ratios have

been measured precisely. The ratio of the iodine-stabilized Ar^ laser radiation to the 633-nm laser has been

measured with an uncertainty of 4 x 10"^" at the BIPM

[35] and by a servolock two-beam method at the PTB

[34], with an uncertainty of 5 x lO'^"^. The BNM have

also made a measurement [36], with an uncertainty 3 x

2Q-io_ rpj^g 612-nm iodine-stabilized He-Ne laser has been

measured with respect to the 633-nm laser by the BIPM

with an uncertainty of 3 x lO"^'' [35]. These results, however, do not represent the limit of what may be achieved by existing techniques. With the

MIT system it is suggested that with visible radiations,

where diffraction is less important, the measurement uncertainty would be reduced to a few parts in 10". The

BNM instrument is also believed to be capable of similar

accuracy [37], and a redesigned PTB instrument is being

considered that would also give performance to this

level. The C02/visible measurement at the NPL [10] in- volved a visible /visible wavelength ratio determination

vdth an uncertainty of 3 x 10"^ . It is thus realistic to quote 3 x 10"" as the level of performance that may be

achieved for visible laser radiations by current methods

of interferometric wavelength measurement.

The present wavelength standards shown in Table 1 have their uncertainties set by the 4 x 10"^ figure associ- ated with the ^^Kr standard. If it were not for this limit.

Table 1. Reference standards for length measurements.

^^Kr, transition 2p iq - 5^5, wavelength 605 780 211 fm

c, the speed of light 299 792 458 m/s

Laser Absorber Transition Component Wavelength, fm

He-Ne CH4 V3 ; P(^) FP 3 392 231 400 He-Ne 11-5 ; i2(127) ; i 632 991 399 He-Ne 9-2 ; i2(47) ; 0 611 970 771 Ar+ 127i, 43-0 ; P(13)' ; as 513 673 467

"All have identical uncertainties of 4 parts in 10^.

the uncertainties of the laser standards would now be a

factor of 10 better, with prospects for a further factor of 10 improvement. There is thus pressure to change the

definition of the meter so as to avoid this limitation to ac- curacy. The discussion on this has been active during the

past decade, and action is now imminent. The proposed

redefinition is such that the speed of light will become a

fixed constant with the value given in 1973 (299 792 458

m/s). This value was determined from frequency and

wavelength measurements of the 3.39 |xm methane-stabilized laser (c = / X), with an uncertainty that reflects essentially only the ^''Kr uncertainty of realization. A

number of different formulations have been suggested for the wording of the new meter definition [38, 39, 40]. After a joint CCDM/CCU discussion in April 1981, however, only two alternatives remain, with the bulk of opin62
ion being in favor of:

(a) "The meter is the length equal to the distance

travelled by plane electromagnetic waves in free space in a time interval of 1/299 792 458 of a second."

The alternative is:

(b) "The meter is the length equal to //299 792 458

wavelengths in free space of plane electromagnetic waves of which the frequency, expressed in hertz,

has the numerical value /."

The majority of people prefer wording (a) as it is easier

to understand, being simpler and more elegant in concept. A minority of metrologists prefer wording (b) be- cause it is closer to the means of realization of 1-m

lengths by means of laser wavelength standards. It is most hkely, however, that wording (a) will prevail.

A redefinition of the meter in this form may be recom- mended in 1982, for formal adoption in 1983 by the General Conference of Weights and Measures. When this takes place, the uncertainties of the stabilized laser wavelengths will have to be revised, and so will their numerical values. Present evidence suggests that the 3.39

|jLm wavelength will be reduced by 3 fm, and the three

visible laser values will each be reduced by about 1 fm,

with an extra digit being added after the decimal point,

as the uncertainties improve by a factor of 10.

7. Length Measurement

In interferometric wavelength comparison, the mechanical length of the interferometer is a kind of inter- mediate standard. It is effectively "measured" by the

standard radiation, although its length is not actually cal- culated. An analogous case is the Michelson-Morley type

of experiment carried out by Brillet and Hall [41] in which the length of a 30-cm mode-matched cavity was

monitored by reference to a 3.39 |xm methane-stabilized

laser. The cavity length drifted at a rate of about 5 x

10"^^ /s and distorted by ±1 x 10"^- by gravitational

stretching as it was rotated. By averaging over many

days, however, it was possible to show that its optical

length was independent of its orientation in inertial space

to an uncertainty of 2.5 x 10"^"^. This is metrology of ex- treme precision.

Even better precision is sought in interferometer sys- tems designed to detect gravitational waves. The first such system was built and operated at the Hughes

Research Laboratories in Malibu, California, in the early

1970' s [42]. This instrument had folded arms of effective

length 4.25 m and had a strain resolution of 10"^^ (Hz)"^, so that it would have been capable of detecting

gravitational waves that gave a total strain level of 10~ over the audio band (1 to 20 kHz). Now several gi'oups

around the world are developing interferometers of greatly improved sensitivity. At the California Institute

of Technology, for example, the aim is to build an inter- ferometer with a baseline of 40 m so as to detect strains

of about 3 X 10"^^ on miUisecond timescales. A long term

goal is to construct large scale systems, with baselines of about 1 km, in order to achieve a strain sensitivity of

10"^^^ for frequencies from about 30 Hz to 10 kHz. Such

sensitivity would match the estimated strength of gravitational waves that should occur reasonably often. Apart

from the mechanical problems, such measurements must

be limited by (a) the detection statistics (photon-counting

error), which means that high-power lasers should be

used, and (b) the disturbance to the momentum, and

hence the subsequent position of the reflecting mirrors,

by radiation pressure fluctuations. An interesting discussion has been taking place on this latter aspect [43], re- garding details of its applicability, and whether or not some arrangement may be devised to reduce its effect.

If interferometers are rigidly connected to the ground,

then they measure earth strain. A number of such instruments have been developed around the world for geophysical studies. The motion of the earth's surface behaves

like a stochastic process, with the notable exception of earthquakes and earth tides. This strain noise spectrum

[44] seems to be similar at different sites, and forms the

limitation to observations. Against this background, the

normal modes of oscillation of the earth may readily be

measured after an earthquake, and earth tides studied in detail and correlated with the local geology and topography [45]. Distances of kilometer dimensions measured in the

earth's surface by optical means are limited in accuracy

by the refractivity of the earth's atmosphere. This is of magnitude 3 x lO"'', so that to get an accuracy of 10"' an

accurate correction must be applied. The use of two optical wavelengths allows partial cancellation, but the water

vapor still presents some problem. The addition of a radio-frequency measurement should be a great improvement. A three-wavelength instrument of this kind being

developed in NBS Boulder [46] is aimed at a measure- ment accuracy of 5 x 10"* over a 50 km range. This is, incidentally, similar to the relative accuracy achieved for 1-m length bars measured by interferometry.

Over the larger distances in space beyond the earth's

atmosphere, refractive index is no longer a problem.

After a decade of observations at the McDonald Observatory, lunar ranging has provided a lot of information

about the orbital dynamics of the earth-moon system

[47, 48]. The fitting of this data to the various models of the orbits, rotations, etc. is a complex task. As a length

measurement project, the problems are formidable.

Nevertheless, a range precision of 4 x lO"-'"^ is achieved;

a remarkable achievement on a moving target at which

10^^ photons are transmitted, but only one comes back

2.5 seconds later. It should be possible to use lunar ranging results made at two different earth stations to measure the secular variation of the distance between them.

Data gathered over 10 years could determine this parameter to an accuracy of 1 cm/year.

Even higher relative accuracy has been achieved in the

radio ranging experiments made for the Viking relativity

experiment [49, 50]. Two Viking landers implanted on

the surface of Mars, and orbiting spacecraft, echoed

modulated microwave signals back to earth with a round

trip of 2500 seconds. The total time uncertainty of 10 ns, in favorable circumstances, corresponds to a fractional

precision of 4 x 10"^- in the distance to Mars. The proposed redefinition of the meter will allow such

precise measurements to be expressed absolutely in meters instead of only in time delay, and it will thus benefit

astronomers, as well as all who are concerned with precise laboratory measurements and stabilized lasers. References

[1] Report of the Advisory Committee for the Definition of the Meter, 1979, Appendix M 3, pp. M46-M55. (Bm-eau Inter- national des Poids et Mesures, F92310 Se\Tes, France,

1979).

[2] R. L. Bai-ger and J. L. Hall, Phvs. Rev. Lett. 22. 4 (1969).

[3] G. R. Hanes and K. M. Baird, Metrologia 5, 32 (1969).

63
[4] Report of the Advisory Committee for the Definition of the Meter, 1973. Recommendation M2. (Bureau International des Poids et Mesui-es, F92310 Sevres, France, 1979).

[5] Report of the Advisory Committee for the Definition of the Meter, 1979. Appendix M2-C, pp. M31-M45. (Bureau

International des Poids et Mesures, F92310 Sevres,

France, 1979).

[6] B. R. Marx and W. R. C. Rowlev, NPL Report MOM 51, April 1981.

[7] W. R. C. Rowley and B. R. Marx, Metrologia 17, 65 (1981).

[8] F. Spieweck, IEEE Ti-ans. Instrum. Meas. IM-29, 361

(1980).

[9] P. Cerez, A. Brillet, C. N. Man-Pichet, and R. Felder,

IEEE Trans. Instrum. Meas. IM-29, 352 (1980).

[10] P. T. Woods, K. C. Shotton, and W. R. C. Rowley, Appl.

Opt. 17, 1048 (1978).

[11] P. Bouchareine, Bulletin BNM, No. 24, 3, (1976).

[12] D. A. Solomakha, Meas. Tech. 16, 1167, (1973).

[13] H. P. Layer, R. D. Deslattes, and W. G. Schweitzer, Jr., Appl. Opt. 15, 734 (1976).

[14] H. Kogelnik and T. Li, Appl. Opt. 5, 1550 (1966).

[15] W. R. C. Rowley, Opt. Acta 16, 159 (1969).

[16] F. Bayer-Helms, PTB-Bericht Me-16 (PhysikalischTechnische Bundesanstalt, January 1977).

[17] G. Bbnsch, PTB-Bericht Me-17 (Physikalisch-Technische

Bundesanstalt, May 1977).

[18] K. Dorenwendt and G. Bbnsch, Metrologia 12, 57 (1976).

[19] J. -P. Monchalin, M. J. Kelly, J. E. Thomas, N. A. Kurnit, A. Szoke, F. Zernike, P. H. Lee, and A. Javan, Appl. Opt.

20, 736 (1981).

[20] W. R. C. Rowley and A. J. Wallard, J. Phys. E. 6, 647

(1973).

[21] T. G. Blaney, C. C. Bradley, G. J. Edwards, D. J. E. Knight, W. R. C. Rowley, K. C. Shotton, and P. T. Woods, Proc. R. Soc. London Ser. A: 355, 89 (1977).

[22] W. G. Schweitzer, Jr., E. G. Kessler, Jr., R. D. Deslattes, H. P. Layer, and J. R. Whetstone, Appl. Opt. 12, 2927

(1973).

[23] G. R. Hanes, K. M. Baird, and J. DeRemigis, Appl. Opt. 12, 1600 (1973).

[24] N. Ito and K. Tanaka, Metrologia 14, 47 (1978).

[25] F. Bertinetto and A. Sacconi, in Atomic Masses and Fundamental Constants 5, Ed. by J. H. Sanders and A. H. Wapstra (Plenum Press, New York, 1976), p. 357.

[26] J. B. Cole and C. F. Bruce, Appl. Opt. 14, 1303 (1975).

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(1975).

[28] A. J. Wallard, J. M. Chartier, and J. Hamon, Metrologia 11, 89 (1975).

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[31] F. Bien, M. Camac, H. J. Caulfield, and S. Ezekiel, Appl.

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(1973).

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1976), p. 410.

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Sevres, France, 1979).

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[36] P. Bouchareine and B. Rougie, Document CCDM/79-18, sub- mitted to the Advisory Committee for the Definition of the Meter, 1979. (Bureau International des Poids et Mesures,

F92310 Sevres, France, 1979).

[37] P. Bouchareine, Bulletin BNM, No. 43, 9, (1981).

[38] Document GT-M.U./N° 2. submitted to the CCDM/CCU

working group on the Definition of the Meter, April 1981. (Bureau International des Poids et Mesures, F92310

Sevres, France, 1981).

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[40] P. Bouchareine, Bulletin BNM, No. 43, 12, (1981).

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[42] R. L. Forward, Phys. Rev. D 17, 379 (1978).

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[46] S. E. Moody and J. Levine, Tectonophys. 52, 77 (1979).

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64
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Double-Mode Method of Sub-Doppler Spectroscopy and Its Application in

Laser Frequency Stabilization

N. G. Basov, M. A. Gubin, V. V. Nikitin, A. V. Nikulchin, and D. A. Tyruikov

P. N. Lebedev Physical Institute, U.S.S.R., Moscow, 117924

and

V. N. Petrovskiy and E. D. Protscenko

Institute of Physical Engineering, U.S.S.R., Moscow, 115409

We present some results of the investigation of the proposed high sensitivity method of sub- Doppler spectroscopy and laser frequency stabilization which is based on the parameters of a double- mode (DM) gas laser containing an internal absorption cell. A short DM He-Ne/CH4 laser was con- structed which has relative frequency stability better than 10"^'^ and radiation spectral width < 10 Hz. When a telescopic beam expander was used inside the cavity of the short DM He-Ne/CH4 laser, super- narrow reference spectral lines of about 3 kHz in width were obtained, and the magnetic hyperfine

structure (hfs) of the F,-' methane line was resolved. Key words: double-mode method; He-Ne laser; methane cell; sub-Doppler spectroscopy.

1. Introduction

The recent development of optical frecjuency standards

is based on obtaining super-narrow spectral lines with

nonlinear sub-Doppler laser spectroscopy methods. Extremely small values of collision, saturation, and transit

flight time broadening are required for obtaining spectral

lines of 10"- 10'^ Hz in width. In particular, if one uses the

method of inverted Lamb-dip with an internal or external

absorption cell, very low gas pressures and weak optical

fields are needed. An essential difficulty in experimentally obtaining such narrow resonances is the rapid de- crease of the signal-to-noise ratio when low absorption gas

pressures of 0. 1 to 1 mPa are used.

The present paper reports the results of studying the

double-mode method of sub-Doppler spectroscopy, which

has much higher sensitivity for obtaining narrow reso- nances in a laser with an internal absorption cell as compared to single mode laser operation. The investigations

have been carried out with a double-mode He-Ne /CH4

laser (at wavelength K = 3.39 [xm, line of methane),

which attracts attention as a possible high accuracy optical frequency standard [1]. Stable double-mode operation of the laser is realized

with the help of a Fabry-Perot cavity, containing two

quarter wave plates [2]. The laser provides stable operation near the gain curve center on the two adjacent axial modes possessing orthogonal linear polarizations. In contrast to a single-mode laser (SML) two types of narrow resonances, "amplitude" and "frequency"

(which possess a number of advantages) can be observed

in the double-mode laser (DML) radiation (Fig. 1).

2. Amplitude Resonances of the DML

In the observation of the output power of a separate

mode there appear "amplitude" resonances (AR) (Fig. 2) which are similar to the inverted Lam.b-dip, but may be

DOUBLE- MODE LASER

^^^^^^^^ polarizer

^// [h?n3 [ch] //

„5L0W" /4FC

SY5TEM

.FAST" AFC

SYSTEM

ER

REGISTRATION

AR

REGISTRATION

E*E2 cos tOfst

Figure 1. The scheme of the AR and FR observation in the DML. With respect to polarizer position, AR (1), FR (3), or

both resonances simultaneously (2) may be observed. In position (2) the resonances are used as discriminators for the twoloop AFC systems.

about two orders of magnitude higher than the SML

resonances. Such a sharp increase in the sensitivity is reached by creating a strong interaction regime between

orthogonally polarized modes. The ratio between reso- nance amplitude (a _) in SML and DML is as follows:

a -(DML)

a - (SML)

./ int (1)

where /int = 10 ^ - 10 - is the mode interaction parameter [2]. The physical mechanism of the sharp increase in the

AR in the DML consists in the following: when the fre- quencies of the two modes (wi, mo) are scanned their

intensities can be made approximately equal. If, for ex- ample, the first mode frequency wi coincides with the

65
Figure 2. The a mplitude of resonances of the DML. Ei is the in- tensity of one of the two }nodes; Ei the total DML intensity; p. = iOO mPa, 7_ = 150 kHz.

center of an inhomogeneously broadened absorption line (wi ^ (si-), the mode losses change slightly due to the for- mation of the Lamb-dip. For weakly absorbing or poorly

saturated transitions in molecular gases at low pressures, the change in losses is usually about 0.1% and may be an

order of magnitude smaller. Approximately the same contrast of narrow inverted Lamb-dip resonances is ob- served in SML operation.

But in DML operation the strong coupling between the

modes leads to a sharp intensity re-distribution among

the modes. It is possible to realize the situation when ap- proximately all the active atoms radiate in the first mode.

AR contrast may constitute 30-40% of the total laser power even in the case of a short laser. When the length

of the He-Ne cavity ('o is 60-80 cm, the transverse size of the mode distribution, f/_, in the cell about 2 mm, and

the methane pressure p _ within the range of (0.2-0.7) Pa, the AR width y- is equal to 100-150 kHz.

Under these conditions a large value of AR and high S/N

ratio allowed us to obtain a laser frequency stability of better than 10"^^ (Allan variance for t = 10 sec averaging

time). Increase of the AR by changing the mode interaction

(decreasing /int) is limited by two factors: the increase in natural laser intensity noise and instability of the

double-mode regime.

3. Frequency Resonances of the DML

The frequency resonances (FR) of the DML are ob- served as changes in the beat frequency between modes

and are caused by resonant changes in the Dopplerbroadened dispersion line of the absorption medium

under the influence of standing wave field saturation

(frequency pulling to the line center [3]). In the DML

with an internal absorption cell the frequency spacing

between modes coi2 =0)1-0)9 changes in the following

way (with |o)i - o)_ | ~ -y_; » 7-):

0)12 = cof. + Aa)io ~ o)ll + J-^^'

~

.^(2) lo 7- + (o)i - o)_)-

where a_ is the nonsaturated absorption coefficient per

unit length; c is the speed of light; and is the normalized intensity of the mode which coincides with o)„. In our experiments with the He-Ne /CH4 DML containing two \/4 plates the beat frequency 0)12 was determined

for the range (2-5) MHz, and its change (A 0)12) was ~ 10^ Hz, when p_ = (0.2 - 0.4) Pa and L'. ~ 0.5 m.

Reference [4] describes the first observation of FR in the DML with an absorption cell. FR in the SML has also been studied [5, 6] with the use of an additional heterodyne laser. In the DML the heterodyne function is per- formed by the second mode.

3.1 Increasing DML Short-Term Frequency Stability

One of the requirements for obtaining super-narrow

spectral lines with 7- ~ 10^ Hz is high short-term fre- quency stability of the laser. The laser spectral width

(Ao)s) must be considerably less than the resonance width

7- . It is well known that the frequency spectrum narrowing in any type of laser used in precision spectroscopy is accomplished by means of frequency stabilization with a broad-band automatic frequency control (AFC) system.

The best results in He-Ne /CH4 laser spectrum narrowing

have been achieved by Chebotayev et al. [7]. They obtained a spectral width A o)s ~ 1 Hz when using intense

(a _ ~ 1 mW) and narrow (7- ~ 50 kHz) resonances in a 5- meter length laser. In the present paper we suggest another method of laser spectrum narrowing. The main idea is that the laser frequency be stabilized to the reference line by a twoloop AFC system, using different molecular frequency

discriminators (Fig. 1). The first AFC reduces "slow"

frequency fluctuations in the range 0-10 Hz and uses as a reference point the top of the AR. This is the com- monly used AFC where laser frequency changes are detected as intensity changes. The second AFC reduces

"fast" fluctuations with a frequency of more than 10 Hz. The FR of the DML is used as a frequency discriminator

for this AFC.

Note the main advantages of such a combined system:

the "fast" feedback loop does not need laser cavity modulation and, consequently, the AFC band (A/afc) is limited

only by 7-, not by the modulation frequency, and may be,

therefore, 5-10 times wider than in ordinary AFC. As the

amplitude detection is replaced by frequency detection, so the demands on AR intensity are considerably less. At

the same time the value of FR(A 0)12) is proportional only

to a linear absorption and hence, weakly depends on AR

contrast. Therefore one can obtain for short lasers a FR of ~ 10'^ Hz in magnitude.

Figure 3b shows gain-frequency dependences of the

constructed ARC, which has frequency band width

A/afc = 20 kHz.

Figure 3a shows the mode beat signal, observed on a spectrum analyzer screen with the DML frequency stabil- ized by the double AFC. The full width at half maximum

(FWHM) of the observed signal corresponds to the laser frequency spectrum, multiplied by the derivative

d (x)i/d 0)12 in the center of the absorption lines. For our experimental conditions the value of d oji/d 0)12 varied

within the range 5-1.2. Taking into account the value

of d uii/d o)i2 the measurements gave the stabilized laser frequency spectrum A ojg'^' < 10 Hz. The passive laser frequency spectrum was about 8 kHz. The main contribution to the Ao)!*^' was made by the fluctuations of a radiogenerator used in the AFC frequency-to-voltage

converter. Other parameters of the DML are as follows:

t'o = 1.5 m, ('- = 0.8 m, d = 2.10-3m, p_ = 0.3 Pa. We

66
(S/N)kr

(S/N)AR add fm

E \ E 2 (3)

fASV'AfC

Figure 3. a) The mode beat frequency spectrum of the DML

stabilized with the two-loop AFC system, b) The gainfrequency dependence of the "slow" and 'fast" AFC parts. The

unity gain is at 20 kHz.

hope that further development of the method would allow one to obtain the natural laser spectral width.

3.2 Sensitivity of FR Registration in DI\/IL

The investigation of the FR shows that frequency detection of resonances in the DML may be more sensitive in comparison with AR and FR registration in the SML. This

feature of the DML FR is especially important for super- narrow reference line detection (7- < 1 kHz) with low

pressures of the absorption gas (p - =1-10 mPa).

Three noise sources must be taken into account for the

FR detection: additive noise, F/V transformation noise,

and natural frequency noise of the radiation. Using well known relations [8] we may show that for additive noise

the S/N ratio (by voltage) for the frequency detection is essentially more than for the amplitude detection:

where E i, E 2 are electric fields of the modes; / „, is the

modulation frequency (synchronous detection i.s supposed); 0 - is the value of AR in SML or in DML; and

A 0)12 is defined by Eq. (2). The ci-IEiEo ratio is equal approximately to the AR

contrast. Equation (3) shows that when /„, ~ 10" Hz,

a-/EiE2-~ 10"\ A 0)12 ~ 10'' Hz the gain in the S/N ratio may be a few orders of magnitude. In practice the fre- quency characteristics of the photodetector must be

taken into account in addition to (3). As to the F/V transformation noise the main contribution to its spectral density is made in the optical heterodyne process. In the DML the second mode is used as the

internal local oscillator and its frequency technical fluc- tuations are strongly correlated with the first mode's

fluctuations if the following condition is realized: -7- «

A 0)12 « 7+, (7+ is the gain line homogeneous width,

Ao)if is the Doppler width of the gain and absorption

lines). In the SML an external heterodyne laser is needed for the FR registration. If the heterodyne laser frequency is highly stabilized with the help of a frequency-feedback

system to the reference laser, the main contribution to the F/V transformation noise is made by the short-term

instability of the laser under investigation.

The spectral density of the technical frequency noise of gas lasers in the low frequency range (less than 10 kHz)

is about 10"* - lO'^ Hz^/Hz. When synchronous detection

is used the S/N ratio is determined by the spectral density at the modulation frequency. The problem can't be

solved by means of active laser stabilization since the

AFC system will decrease the frequency fluctuations

simultaneously with the signal (A 0)12) and hence the S/N

ratio will not be improved. It is possible in Refs. [5, 6] that poor passive short-term stability gave no opportunity to obtain any gain in the S/N ratio in case of the FR

registration in contrast to AR.

So, DML allows one to detect FR with the noise level equal to the natural frequency noises. On the other hand

the FR signal, given by Eq. (2), under conditions where

^1 ~ 1, is determined only by the linear absorption coeffi- cient, i.e., linearly depends on the pressure p_. (As is known, for the AR a_ ~ p'^ -.) We now estimate the limit of sensitivity of weak line detection by the proposed method. As was mentioned

above for the F^*"* methane line, which has an absorption

coefficient per unit length of 0.135 m"^ Pa"\ the FR is about 10^ Hz when p_ = 100-200 mPa, £ _ - 0.5 ^o- Assuming that the spectral density of the natural frequency

noises is ~ 10"^ Hz-/Hz one may find that the minimum

detectable methane pressure is (10"^ - 10 '') mPa. Therefore FR registration with S/N ~ 10''' is possible if ~

10'^ mPa, T = 1 s and short absorption cells are used.

3.3 Resolution of the F^"^^ Methane Line hfs in DML

For magnetic hfs resolution of the Fi"' line [9] a HeNe/CH4 DML with an internal telescope expander (TE) was constructed. The light beam diameter in the absorption cell was about 3 x 10"- m, while the spot size in the

active medium was 3 x 10"^ m; t'- = 0.8 m, I'o = 1.8 m. The whole experimental set-up also consisted of the

67
reference DML and heterodyne laser. The present

scheme is acceptable when precision spectroscopic investigations are made. The reference He-Ne/CH4 DML

(without TE) has a narrow spectrum (Awg'"' ~ 10 Hz) and was constructed according to the scheme described in section 3.1. DML/TE was studied in the pressure range

jo - ^ 300 mPa. Since there exists a time shift of the

beat frequency a){2, e.g., due to a slow change of the ac- tive medium parameters, synchronous detection of the

FR with /m = 400 Hz was used.

The first and second derivative signals of the FR are shown in Figs. 4 and 5 for different p_, /i values. Under

high methane pressures (Fig. 4a, p _ = 270 mPa) when

magnetic hfs isn't resolved the DML/TE frequency sta- bilization was made by means of the second derivative of the FR. The measurement of the relative frequency stability of the DML/TE and DML gave an Allan variance a = 0.5 X 10"" (t = 10 s) which is better by two orders

of magnitude than the one reached in Refs. [5, 6]. (In

Refs. [5, 6] a SML stabilized by the FR was used.) The

asymmetry of the FR and the "crossing" lines [9] are observed clearly at the p _ =40 mPa (Fig. 4b).

a)

Figure 4. The first and second derivative FR signals, a) p ^ =

270 mPa; the time constant t = 0.3 s; b) p _ = 40 mPa, j = 1 s; the "crossing" lines are observed at the red side of the main

components.

The spectrograms in Fig. 5a, b were obtained at equal

pressure (p_ = 3-4 mPa) but under two different intensities of the first mode; /i" = 0.2 and / ^ 1.0. Under

conditions corresponding to Fig. 5b, the half width at half maximum (HWHM) of the first derivative FR signal

is 7- = (2.7 ± 0.3) kHz. The transit flight time width of the FR is about 5 kHz. The last spectrograms show that

the spectral resolution and S/N ratio obtained in the

present work are equal to those of Ref. [9] where the hfs of the 7^2"' '^^6 had been resolved for the first time. However the parameters of the experimental set-ups differ in essential ways. Comparison of the absorption particle

number in the mode volume and the signal registration

time show that the detection sensitivity of the super- narrow resonances reached in the DML is about two or- ders higher than the one in Ref. [9]. In conclusion it may be noted that the DML with inter- nal absorption cell represents in itself an optical spectroscope with frequency detection of the signal which may

significantly increase the sensitivity and resolution of sub-Doppler spectroscopy and distinguish high quality

reference lines as well.

Figure 5. Magyietic hfs of the F'/' methane line. p_ =

3-lt mPa, T = 15 s, tight beam diameter = 30 mm, leyigth of

the cell = 0.8 m; a) /, = 0.2; b)/, = 1.0. References

[1] J. L. Hall, Science 202, 147 (1978).

[2] M. A. Gubin et al, Kvant. Elektron. (Moscow) 6, 63 (1979) [Sov. J. Quantum Electron. 9, 34 (1979)].

[3] V. S. Letokhov, Pis'ma Zh. Eksp. Teor. Fiz. 6, 597 (1967) [JETP Lett. 6, 101 (1967)].

[4] Yu. A. Vdovin et al., Kvant. Elekton. (Moscow) 2, 105 (1973). [Sov. J. Quantum Electron. 2, 565 (1973)].

[5] G. Kramer et al., Z. Naturforsch. 30a, 1128 (1975).

[6] S. N. Bagaev et al., Appl. Pliys. 10, 231 (1976).

[7] V. P. Chebotayev, Report at XIX General Assembly of URSI, Helsinki, Finland (1978).

[8] J. Klapper and J. Frankle, Phase-locked and FrequeyicyFeedback Sijstem.<i, (Academic Press, New York, 1972).

[9] J. L. Hall and C. J. Borde, Phys. Rev. Lett. .30, 1101 (1973).

68
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

He-Ne (^^^Ig) Lasers at 0.633 fjim (and at 0.604 ixm)

Fabrizio Bertinetto, Bruno I. Rebaglia, Paolo Cordiale,

Sergio Fontana, and Gian Bartolo Picotto

Istituto di Metrologia G. Colonnetti, Torino, Italy

Although iodine stabilized, 0.633 |xm lasers are used as practical wavelength standards, no com- mon agreement exists as to the operating conditions. It is shown that on the basis of such an agi-eement, reproducibility of ±20 kHz or ±4 x 10"^^v can be attained. This study proposes such conditions and shows that for reproducibility to exceed lO'^^v, certain cavity configurations must be discarded. Preliminary observations of strong absorption lines of iodine at the emission wavelength of 0.604 \i.m are also reported.

Key words: frequency stabilized lasers; frequency standards; wavelength standards.

1. He-Ne C^\) Lasers at 0.633 |xm

1.1 Stability and Reproducibility

Figure 1 is a plot of the Allan variance of two independently stabilized He-Ne (^^'^12) lasers with the temperature of the iodine cell walls as a parameter. The

operating conditions were: iodine pressure P = 17.4 Pa

and modulation depth A v„ -p =6 MHz. The improvement

in the short term stability due to enhanced signal to noise ratio [1] is evident and can be appreciated in the

practical utilization of the laser.

i=28X

i=mx

t=200°C

10-" r

10-'^

c^"

1-13 V. 10

10^

Figure 1. Allan variance plot for two 0.633 |xm He-Ne (^^^I^) lasers at different temperatures of the iodine cell.

In order to check the frequency reproducibility, three

lasers with identical cavity structures and servo-systems were used. Interchanging these servo-systems amongst

the lasers, all combinations of lasers and servos were obtained. For each of these combinations separately, fre- quency differences between the lasers were measured in such a way that in every two sets of measurements, i.e., two different combinations of lasers and servos, one laser with its servo-system was kept unaltered so as to be

used as a reference laser. It was further assumed that

the frequency of this reference laser remained unchanged

between two measurements. As a result the reproducibility was approximately ±5 kHz or 10 "v as shown in Fig. 2. Similar results can be obtained in any metrological la- boratory and it has already been shown that lasers con- structed in different laboratories, but under the same

operating conditions, show a repeatibility of ±20 kHz or ±4 X 10"^^ V [2]. However, for arbitrary operating conditions and for certain cavity geometries, the repeatibility

of the frequency output^ of He-Ne (^^^12) lasers can be no

better than ±5 x 10 "^°v, as will be shown below.

10 ,

A3 1 A3 1 CI 1 A2 1 C3 1 Al

Bl 1 CI 1 A2 1 Bl 1 Al 1 B3

C2 1 B2 ! B3 1 C3 1 B2 1 C2

I X—!—

X

N

0 ..

-10 i

9

Xo

o- -o

-X

o

N

IE

I I LD

f\J

X cavi iy 1 cavity 2 o cavi iy 3

Figure 2. Reproducibility ofHe-Ne C^'/^j lasers constructed at

. IMGC. The three servo-systems employed are designated A,

B, and C and the three cavities 1, 2, and 3.

•"^This study deals with the frequency measurements of the hfs com- ponents d, e, f, g of iodine (^^^12) against a fi.xed reference. Only the behavior of the center of gi-avity of these peaks is reported here. The fre- quency, therefore, refers to that of the center of gravity. It is worth mentioning here that the gi'oup of hfs components h, i and ./ behaves similarly to that of d, e, f and g.

69
1.2 Iodine Pressure

Figure 3 is a typical plot of frequency vs. pressure. Our result differs from that of P. Cerez et al. [3]. The

straight line of -9.8 kHz/Pa slope was drawn only

through the experimental points for a cold finger temperature ^ 9 °C. It can be seen that the extrapolation

below this temperature does not fit the data. At low

pressures as the contrast of the peaks becomes poor, the

offsets introduced by the electronics of the servo-system

become important. Nevertheless the behavior shown in Fig. 3 has been obtained consistently.

pressure

Figure 3. Frequency output vs. iodine pressure. Extrapolation

below t(. = 9 ° C seems incorrect for lasers at 0.633 fim.

1.3 Misalignment and Cavity Geometry

The effects of misalignment and cavity geometry are shown in Fig. 4. These curves were obtained by tilting

the mirror on the cell side (Ml). Tilting the other mirror

(M2) did not change the curves significantly. At maximum output power (best alignment), the frequency out- put was not affected by the geometry of the laser cavity,

except for curve d, in which case there was an offset of approximately 100 kHz. Repeating these measurements

with different cavity lengths and laser powers, similar

results were obtained which were independent of the out- put power at best alignment. Shortening the cavity to

N

><

150

b —

iiao -• a i

curve HI M2^ a

a 1000 1000

50. h 4000 1000

c flat 500

d 1000 500 L-340inm 0. 1 -H H 1——1

1 1 h 1 0.0 0.5

W/Wmax

1.0

Figure 4. Effect of misalignment and cavity geometry on fre- quency output. Radiis of curvature of Ml and M2 are in milli- meters. Mirror reflectivity for curve a 99% for both mirrors, and

for a' 99.0% and 99. i% . The effect of reflectivity and hence irra- diance is synall compared to that of cavity configuration.

300 mm using a shorter tube (otherwise similar to the

previous ones), the frequency offset observed with the same mirrors as of curve d (Fig. 4) was 35 kHz. For this cavity length and for mirrors as of curve a and a' output was insensitive to misalignment within ±10 kHz.

Misalignment also had an effect on the slope of the

modulation dependent frequency shift. For a cavity similar to that of curve d (Fig. 4) and at iodine pressure of 17.4 Pa, the slope at Avp.p = 6 MHz was -10.5 kHz/

MHz at best alignment and -4.7 kHz /MHz when the

power was reduced to 35% by tilting one of the mirrors.

1.4 Irradiance

There have been attempts to interpret these results only as an effect of irradiance of the beam traversing the

iodine cell [4]. To this end the frequency shift expected

from irradiance has been recently measured using two

different methods independent of cavity misalignment.

The data shown in Fig. 5 were obtained using a laser cavity similar to that of curve a (Fig. 4), but with mirrors of different reflectivities. The error bars are twice

the standard deviation. Optical surfaces were cleaned

between two measurements using the same mirrors. The

dispersion of the frequency output is well within the

reproducibiUty of He-Ne { I2) lasers.

N

je: < -2

o 99. 0%-gg. 02

99. 0Z-99. 43: - 99. 4Z-99. AX

+- 50 60

irradiance

70

-+ 80 90

[W/mm2]

Figure 5. Frequency output vs. irradimice for a He-Ne f^^^/^j laser at 0.633 (jlwl

In another set of measurements the power was varied

from 44 |jlW to 91 fxW by varying the discharge current from 2.5 mA to 3.5 mA. The variations in the frequency

output were within 1.5 kHz, or 3 x 10"^^v. It can then

be concluded that irradiance had a very small effect on

the frequency output of a He-Ne (^^^12) laser at 0.633 fxm. However the proximity of the neighboring lines causes some spread in frequency through power broadening.

Nevertheless, the modulation-dependent shift in the fre- quency of the center of gravity of a group of adjacent

peaks is not affected by irradiance (see Table 1). That

the pressure-dependent shift is only slightly affected by

irradiance can also be seen from Table 1.

1.5 Temperature of the Cell

Variations in the frequency output of a He-Ne (^^^12) laser with temperature of the iodine cell are shown in Fig. 6. At a pressure of 17.4 Pa and Avp-p = 6 MHz

the frequency shift associated with the cell temperature was found to be approximately 0.16 kHz/K. The effect

70
Table 1. Pressure-dependent and modulation-dependent fre- quency shift at different values of irradiance. The measurements

were taken in conjunction with those of Fig. 5. H

[ W/mm']

Av/P

IkHz/Pal [kHz/MHz]

47 - in p

54 -119

70 -9.8 -10.4

79 — -10.7

87 -10.2 -10.6

2 5 300110 15

' ' 1' —'—h-_,—I—I—1—1—t10 15 20

lodina prassur* [Pa]

Figure 6. Frequency output vs. iodine pressure at different temperatures of the cell. For t^, < 200 °C, data below =

9 °C are not shown.

shown in Fig. 3 disappears at 200 °C and a solid line has

then been drawn for this temperature starting from a pressure of 4.1 Pa. Heating of the cell seemed to improve the line shape as a small decrease in the modulation-dependent frequency

shift, from -9.9 kHz/MHz at 28 °C to -8.4 kHz/MHz at 200 °C at a constant iodine pressure of 17.4 Pa, was

observed.

2. He-Ne (^27|^) Laser at 0.604 ixm

It is known [5, 6] that the 0.612 |xm orange line of HeNe is coincident with iodine transitions which are strong

and easily saturable. Further investigations are in progress at IMGC.

A laser using a commercial plasma tube is under con- struction and with the proper choice of mirrors is ex- pected to give 25 mW output at 0.633 p-m. Using two

mirrors each of 1200 mm radius of curvature and 99.8%

reflectivity, the laser could be operated at 0.604 \x.m [7], with a power output of only 25 (jlW. The effect of an

intracavity cell was the same as that at 0.612 (xm and

the laser had single mode operation over several hundreds of megahertz. Figure 7 shows the emission profile

and the third derivative signal at an iodine pressure of 0.8 Pa and a temperature of 160 °C. The lines seen are

as yet unidentified. A large modulation (nearly 15 mHz

peak-to-peak) had to be applied because of a frequency

jitter of approximately 10 MHz due mainly to the poor

mechanical mount.

Figure 7. Output power and third harmonic signal vs. laser frequency for a He-Ne ('"^'^I-z) laser at 0.60^ \i.m.

3. Conclusions

In spite of the fact that the stability of the He-Ne

C^'^h) lasers at 0.633 |xm approaches 10"^^ v and the

reproducibility obtainable in a single laboratory is better

than ±10 kHz or ±2 x 10"^^ v, the repeatability between

lasers constructed and operated in different laboratories

may fall short of 10"^° v in the absence of an agreement

about the operating conditions and specifications of cavity geometry. Towards using these lasers as practical

secondary frequency standards in the optical region, the

following operating conditions are suggested:

P = (17.4 ± 1) Pa

Avp_p = (6 ± 0.2) MHz

= (200 ± 5) °C

Several cavity configurations give similar performances; however, cavities as of curve d in Fig. 4 should be

discarded. It is probable that for He-Ne (^^'Is) lasers at 0.612 iJim and at 0.604 [xm a similar problem of choosing

the cavity may occur in the future, if the lasers are operated with an intracavity cell.

References

ri] p. Cerez and S. J. Bennet, IEEE Trans. Instrum. Meas.

IM-27, 396 (1978).

71
[2] J. M. Chartier et a/., IEEE Trans. Instrum. Meas. IM-25, 450.

[31 P. Cerez et ai, PTB-Bericht Me-17, Ed. by F. Bayer-Helms

(Physikalisch-Technische Bundesanstalt, Braunschweig, May

1977) p. 71.

[4] F. Bertinetto et al., CPEM Digest, Conference on Precision Electromagnetic Measurements, IEEE Cat. No. 78CH1320-

1 IM, p. 79.

[5] P. Cerez et al., IEEE Trans. Instrum. Meas. IM-29, 352

(1980).

[6] K. Dschao et al., IEEE Trans. Instrum. Meas. IM-29, 354

(1980).

[7] A. D. White and J. D. Rigden, Appl. Phys. Lett. 2, 211

(1963).

72
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Recent Work on 612 nm He-Ne Stabilized Lasers

A. Brillet, P. Cerez, and C. N. Man-Pichot

Laboratoire de I'Horloge Atomique, Equipe de Recherche du CNRS, associee a TUniversite Paris-Sud,

Bat. 221 - Universite Paris-Sud, 91405 - Orsay - France

We report on the metrological properties of 612 nm He-Ne lasers frequency stabiHzed on ^^'^12 saturated absorption lines. We describe the new results of spectroscopic and metrological interest, obtained both with conventional internal cell devices and with the new technique using an external cell inside a Fabry-Perot resonator. A reproducibility of 6 x 10"^^ is obtained with this last technique.

Key words: He-Ne laser; hyperfine predissociation; optical frequency standards; saturated absorption.

1. Introduction

He-Ne lasers stablized by saturated absorption in ^^'^la at 633 nm [1-3] or 612 nm [4] are now widely used in precision interferometry and high resolution spectroscopy at the 10^^*^ - 10"" precision level. We show in this paper how both the high saturation level obtained with

internal cell devices and the very high resolution capability of the external cell technique [5] allowed us to obtain new spectroscopic results on the 612 nm R(47)9-2 iodine

line, which in turn, helped in improving the metrological

properties of these optical frequency standards.

2. Experimental Techniques

2.1 Internal Cell Devices

The cavity of our He-Ne laser is made of three silica rods and contains a commercial He-Ne tube, designed to produce 5 to 7 mW of 633 radiation, a 10 cm long iodine

cell with Brewster windows, and a Littrow prism for wavelength selection. Its free spectral range is about

200 MHz. Because of the low gain at 612 nm, both mirrors are high reflectivity, and the power emitted through

each mirror at low iodine pressure is 300 pW at most,

strongly depending on the cleanliness of the optical surfaces. This corresponds to an intracavity power of nearly

1 W! Even at the lowest iodine pressure available

(0.2 Pa), the laser is single frequency and continuously

tunable over 1-200 MHz, due to a differential saturation

effect [4]. These lasers were used for internal cell saturated absorption studies, with iodine pressures ranging between

2 and 5 Pa. They were also used as sources for the external cell studies, with a much lower iodine pressure (0.1 to

1 Pa). Their emission linewidth, with good passive isola- tion, has now been reduced to 30 kHz (6 x 10"").

2.2 External Cell Technique

2.2.1 Optical Set-up

This new saturated absorption technique was first pro- posed by Cole in 1975 [6], and first demonstrated by C. Pichot-Man et al. [7]. The high sensitivity of this tech- nique is easy to understand with simple arguments. Let

us consider a resonator with an input mirror Mi, an

absorbing cell, and an output mirror M2, having respectively the transmissivities <i, t, and t2 for the field amplitude. If the mirrors are perfect, their reflectivities are

'"i = |1 - and r2 = |1 - i| |^^. At resonance,

the intensity transmission of the cavity is then: T =

(1 - t r^rif

and its variation vdth t is:

dt

(1)

(2) (1 - t rir2)^

For ri = 0.98, = 0.98, we get T = 0.45, and

dT/dt = 40, which represents the gain in sensitivity

compared to the case of a simple saturated absorption ex- periment. Under the same conditions, the energy storage

factor inside the resonator is:

S =

(1 - t rir2r = 22.7 (3)

Thus this technique allows one to perform saturated ab- sorption spectroscopy even vdth very weak laser sources. The numerical values given above correspond approximately to our experimental conditions. Let us note that

Eqs. (1) to (3), which do not include the intensity dependence of t, are valid only for low saturation parameters.

They constitute, however, a valid approximation, since we always tried to avoid power broadening. If the satu- ration effects become important, the sensitivity d T/dt

further increases, up to the limit of bistabihty.

A block diagram of the experimental set-up is shown in Fig. 1. The resonator is made of two invar rods, 50 cm

long. It contains a 20 cm or 35 cm iodine cell, and an

anti-reflection coated lens acting as a telescope. The

beam waist on mirror M2 can be varied between ivo -0.2

and wq = 3 mm, by translating the lens. Feedback to the laser is prevented by a Faraday or an

acousto-optic isolator.

2.2.2 Electronic Set-up

To operate this spectrometer, one wants to keep the resonator frequency locked onto the laser frequency.

This is realized by a conventional servo-loop using fre- quency modulation of the resonator at 20 kHz (kc in the

figure) and synchronous detection. It is important to obtain a very tight lock in order to avoid the excess amplitude noise which would result from a relative frequency

jitter between the laser and the resonator.

73
PZT^ 16.5 kc

Laser/Iodine

Lock

P.S.D.

x3 20kc

Cavity/Laser

Lock

Isolator M, L, u

Figure 1. Block diagram of an external cell system.

The iodine saturated absorption features are detected

by a second frequency modulation, at 5.5 kHz, which is applied simultaneously to the laser and the resonator with the same phase and amplitude so that the signal

detected at 5.5 kHz or its 3''^ harmonic is due only to the

iodine features and not to a residual frequency mismatch

between the laser and the resonator. This signal is syn- chronously detected and recorded, or used as an error

signal to lock the laser frequency.

3. Some Relevant Spectroscopic Properties

of 1271^ at 612nm

3.1 The Saturated Absorption Spectrum

Figure 2 shows the whole spectrum which can be registered with an internal cell device, completed by higher

resolution studies with an external cell. One can easily

assign the 21 hyperfine components of R(47)9-2, the 15

from R(48)15-5, and the 10 highest frequency components

from P(48)ll-3. Furthermore, the spectrum contains over

40 additional lines, shown pointing downwards on Fig. 2. They appear clearly only at high intensity, and we could

assign most of them to "forbidden," = 0 transitions

and to Doppler generated cross-over resonances between

these transitions and allowed ones. Moreover, this as- signment allows one to predict the frequency of some

other cross-over resonances which could not be resolved

because they were too close to strong allowed transitions, like m, q, r (Fig. 3). The relative intensities of the allowed transitions compared to R(47)9-2 are about 0.8 for P(48)ll-3 and 0.07 for R(48)15-5 at room temperature. It is possible to enhance

by a factor 10 the 15-5 line intensity, using a heated cell. At low power, the computed intensity of the saturated

absorption signal is about 10"'* for =0 lines [10] and

10"^ for the cross-overs. With increasing power, however, the allowed lines broaden and their intensity saturates, while the AF = 0 lines keep increasing linearly, so that the intensity ratio is only about 10 in the usual

operating conditions of an internal cell device.

3.2 Linewidths and Hyperfine Predissociation

3.2.1 Instrumental Broadening

In internal cell devices, the problem has been treated

earlier [8] for 633 nm lasers. At 612 nm, the linewidth of

all the iodine components is about 6 MHz (FWHM),

mainly due to power broadening. Changing the pressure

does not affect the linewidth very much, because any de- crease (increase) of the pressure broadening is nearly

compensated by an increase (decrease) of the power

broadening.

With the external cell technique, it is possible to obtain

low saturation parameters. The main source of instrumental broadening is then the pressure broadening which

is about 250 kHz /Pa around 1 Pa. Smaller additional efo

m I k XII

127,

(stu) +

R 47 9-2

+ R 48 15-5

I P48 11-3

t ed

I X o

oLO

14 13 C,5 C,,

g r q p 0 (abc)

q P o o Q a Q a

n m

+. +

I k +, +, (hij) +1 o

o

12 '13 Ci2

11c^C,o 10"9 Cg '8 8 C7 7

c b a

o

o

e.d + 0

(abc)

+.

2:00

+

6 C.

Figure 2. '-7.2 saturated absorption spectrum at 612 nm. The lines shown pointing

downwards are forbidden transitions and Doppler generated level crossings. The Af =

0 lines, and cross-over resonances assignable to R(i7)9-2 are labelled 1 . . . U and Ci

. . . Ci5, respectively.

74
AF=0 1^ Forbidden Lines

Level Crossings ^

Figure 3. RH7)9-2 hyperfine structure. The intersection of the diagonal lines

(which connect some forbidden and allowed resonances) and the frequency axis

give the cross-over positions.

fects come from the laser frequency jitter (30 kHz) and

from the finite size of the laser beam waist (10 kHz-mm).

3.2.2 Radiative Lifetimes and Predissociation The natural linewidth of a given iodine hyperfine transition can be expressed by the formula

1 W = [Fi + Tv + C^JiJ + 1) + Thp], (4)

where Fi is the instrumental broadening, Fv is the radiative Hfetime of the level, Cy J(J + 1) is the gyroscopic

predissociation, and Fhp, the hyperfine predissociation

rate, is given by [9]: rHp(V,J„F) = X \o^(I'F,J) 1

2

2t2 P +

3(1 J)^ + 1 1 J - PJ

(2J - 1)(2J + 3)

(5)

-av Cv I J

where ay and Cy are the predissociation constants. The

projection coefficients a{I,F,J) can be determined by fit- ting the well-known hyperfine hamiltonian with the

measured hyperfine splittings [10]. We specially studied the hyperfine predissociation of the R(47)9-2 line by carefully measuring the linewidth of

all its components, and comparing with Eqs. (4, 5), after

a zero pressure extrapolation. For a pressure of 1 Pa, for instance, the observed linewidths vary between 495 kHz

for the component p and 1070 kHz for s. Choosing ai -

135 10^ S-], Ci = 118 s-\ and Fg + F/ - 2.344 10^ s-\

the rms difference between experimental and calculated

linewidths is only 12 kHz. However, the values of ay and

Cy are probably slightly underestimated because of the

possible presence of a residual background gas in the

iodine cells: since the collision broadening rates are

larger for narrow lines than for wider lines due to velocity changing collisions [11], the zero pressure extrapolations would be biased. We estimate our values of ay and

Cy to be too small by about 10% , leading to a 9 = (150 ±

10) X 10^ s^i and C| = (130 ± 10) 8"^ Our results also confirm the negative sign of ay Cy

.

4. Metrological Results

4.1 Internal Cell Devices

In these devices, the most important criterion for choosing the best frequency reference lines is the ab- sence of neighboring lines or cross-over resonances. In

this respect, only h, 0, and t are suitable in the R(47)9-2

line. The stability of the lasers locked to 0 or t was comparable with the results obtained at 633 nm with heated

cells [12]. Their reproducibility has been checked by comparison with a BIPM laser: both lasers produced the same frequency within 5 kHz (10"^^). The causes which

limit this reproducibility are not understood. They are very difficult to isolate experimentally because all the

parameters are strongly coupled in these compact sys- tems. Anyway, it seems reasonable to use these simple

devices for interferometry at the 10"^" level and to prefer

external cell systems for higher precision measurements.

4.2 External Cell Systems

In this case, most lines are well isolated, because they

are narrow and because the spectrum is simple, the in- tensity of forbidden lines being negligible at low power.

Since the intensity of a saturated absorption feature is inversely proportional to the square of its vddth (see

Fig. 4), the figure of merit,

jS/N)

W

M ' = (6)

is proportional to W"^. It is then strongly recommended

to use the narrowest lines as frequency references: m, n, p, and t are the best choices, j being perturbed by e R(48)15-5.

Using m and n components, we obtained a short term

stability (Allan variance) which varies as 5 x 10

x

t"^, with a flicker floor at 2 x 10"^^. We expect in the next experiments a tenfold improvement since the excess

noise due to the frequency mismatch between the laser and the resonator has been greatly reduced.

The most important step forward is the improvement

of the reproducibility which is 300 Hz (6 x 10"^^) \nth

these systems. The iodine pressure shift is -4 x 10"^- Pa"\ and we can control the iodine pressure with a precision better than 0.05 Pa. A variation by a factor of 2 in

75
Figure 4. First derivative lineshape of R(^7)9-2, h, i, andj components, showing

hyperfine predissociation.

the power from any of the lasers induced no measurable

frequency shift. Geometrical effects due to the mismatch

of the counterpropagating waves [13] are completely

negligible with this technique. The only limitation we

could observe was due to the offsets of the servo-loops,

so we think we might be able to improve this reproducibility which is already the best one ever reported for a

visible optical frequency standard.

5. Conclusions

A detailed spectroscopic study of the ^^^l2 properties at 612 nm allows us to determine the best reference lines to be used in internal and external cell devices, and helps in understanding the limitations of these systems. The new

saturated absorption technique using an external cell in- side a Fabry-Perot resonator proves to be highly effi- cient for high resolution spectroscopy and for use in optical frequency standards.

References

[1] G. R. Hanes and C. E. Dahlstrom, Appl. Phys. Lett. 14, 362

(1969).

[2] A. J. Wallard, J. Phys. E 5, 926 (1972).

[3] A. Brillet, P. Cerez, and H. Clergeot, IEEE J. Quantum

Electron. QE-10, 526 (1974).

[4] P. Cerez and S. J. Bennett, Appl. Opt. 18,1079 (1979).

[5] P. Cerez, A. Brillet, C. N. Pichot-Man, and R. Felder,

IEEE Trans. Instrum. Meas. IM-29, 352 (1980).

[6] J. B. Cole, J. Phys. D 8, 1392 (1975).

[7] C. N. Pichot-Man, P. Cerez, and A. Brillet, in Summaries of Contributions to the 11th Annual EGAS Conference (Laboratoire de Spectroscopie Hertzienne de TENS, Paris,

1979), p. 133.

[8] A. Brillet and P. Cerez, Metrologia 13, 137 (1977).

[9] M. Broyer, J. Vique, and J. C. Lehmann, J. Chem. Phys. 64, 4793 (1976).

[10] J. P. Picque, private communication.

[11] J. L. Le Gouet, J. Phys. B 11, 3001 (1978).

[12] P. Cerez, S. J. Bennet, and C. Audoin, C. R. Acad. Sci. Ser. B 286, 53 (1978).

[13] J. L. Hall and C. J. Borde, Appl. Phys. Lett. 23, 788 (1976).

76
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Iodine and Methane Stabilized He-Ne Lasers as Wavelength Standards

Shen Nai-cheng, Wu Yao-xiang, Sun Yi-min, Li Cheng-yang, and Zhang Xue-bin

National Institute of Metrology, Beijing, P.R.C.

and

Wang Chu

Peking University, Beijing, P.R.C.

The iodine and methane stabilized lasers designed by NIM and Peking University can be used as wavelength standards. We have compared the frequency differences of the lasers between NIM and

BIPM in Paris in April 1980. The relative frequency differences are 2.9 x 10"^^ for iodine and 6.3 x

10~^^ for methane, respectively. When the laser power is given a fixed value, the frequency variation

of the iodine stabilized laser can be very small. The power shift and standard power value are dis- cussed in this paper. Key words: frequency reproducibility; iodine and methane stabilized lasers; power shift; power stand- ard value; wavelength standard.

1. Introduction

Iodine and methane stabilized He-Ne lasers have been

used as wavelength standards since 1973 [1]. In particular, the 633 nm w^avelength of ^^'l2 is very useful and con- venient. International intercomparisons of lasers have

been done many times. We performed the first comparison between NIM and BIPM in April 1980 [2]. The

results of our comparison were satisfactory. Although

the construction of our lasers is different from that of

BIPM, the frequency difference of the iodine stabilized lasers is less than 3 x 10"^^ This result gives strong

proof of the good reproducibility among different 633 nm

iodine stabilized lasers.

2. The Results of the Comparison between

NIM and BIPM in 1980

2.1 The Stabilized He-Ne Laser

Table 1 shows the design characteristics of the ^^^2

stabilized lasers. The measurements were done at BIPM,

April 2-17, 1980. The frequency difference and standard

deviation of the two lasers were each determined from

the small matrix of the measured values from the d to g

components by a least-squares calculation. The mean frequency difference of thirteen measurements was calcu- lated to be

/nim2 - /bipm2 = 13.8 kHz, a = 4.5 kHz

/nimi - /bipm2 = -20.3 kHz, a = 3.8 kHz

In the comparison, all lasers were adjusted to an iodine

pressure of 17.3 Pa (15 °C) and a modulation amplitude of 6 MHz. These are standard operation conditions. The

laser powers are slightly different between NIM and

BIPM. The power regions used in the measurements were

17 |xW < Pnim2 < 27 |jlW,

20 (xW < Pnimi < 28 |jlW,

25 pW < Pbipm2 < 43 pW.

We have also measured the influence of the iodine pressure, modulation amplitude, and the laser power on

the laser frequency. For example, the pressure shift of

the/ component is -9.3 and -9.2 kHz/Pa for NIM2 and

NIMI, respectively, and the modulation shift of the /

component is -7.9 kHz/MHzpp for both lasers. For the

other components, the shifts are slightly different. The

Table 1. Design characteristics of I^-stabilized laser.

Laser tube l2-ceii Reflective mirror Output power

CW301 99.7% 99.5%

BIPM 2 10 cm 40 |jlW 25 cm, 3He-2'^Ne 50 cm oo

made in NIM 99.7% 99.7%

NIM (1,2) 10 cm 25 jiW

11 cm, ^He-TM'Ne 120 cm 120 cm

77
power shifts are -0.05, -0.06, and -0.18 kHz/jjiW for NIM2, NIMl, and BIPM2, respectively.

2.2 The Methane Stabilized He-Ne Laser

The mean frequency difference was calculated to be

/nimchi - /bich46 = 0.56 kHz, a = 1.29 kHz.

The coefficient of frequency variation for NIM-CHl on

the discharge current in the laser tube is -1.39 kHz/mA.

The modulation shift of NIM-CHl is +657 Hz/100 kHzpp.

For the comparison, the modulation amplitude of NIM-CHl was adjusted to 1.2 MHz.

3. The Results of the Comparison Between

NIM2and NIM1 in 1981

The frequency difference between l2-stabilized lasers NIM2 and NIMl was determined to be 34 kHz at BIPM

in 1980. We have measured the difference once again in January 1981. The mean value of thirteen measurements

at various powers was calculated to be

/nIM2 ~ / NIMl 22.9 kHz , CT = 9.8 kHz . The laser tubes of NIM2 and NIMl were replaced with new ones which were filled with the isotope ^*^Ne in April

1981. The other characteristics of the lasers were the same as in 1980. The power of one laser was fixed, but

that of the other was changed from its minimum to its maximum during the measurement, then the power condition of the two lasers was interchanged. When the region

of the power variation was within 9 fxW to 42 ixW, the mean frequency difference varied from 7 kHz to 20 kHz

and the standard deviations were about 6-11 kHz. So the

power shift is too great for good frequency reproducibility. But as the powers of the two lasers were adjusted to about 30-32 |jlW, both the mean frequency difference of the small matrix and the standard deviations were very

small. The value of the difference between NIMl and

NIM2 was only 6 kHz, with a = 2 kHz. This result is better than the intercomparison for NIM at BIPM in 1980. We shall discuss this in the next section.

4. The Selection and Measurement of the

Standard Value of the Laser Power for

Good Reproducibility

In order to get good frequency stability and reproducibility, the absorption peak must have a high contrast and

a narrow width. On the basis of Greenstein's theory [3], when the other operating conditions are fixed, we find the contrast of the absorption signal to be roughly pro- portional to f{X), which is given by:

f{X) = (1 + I/Ia)-'^ - (1 + 27//a)

-1/2 (1) where X = IIIa, I is the laser output power, and Ia is the saturation power of the iodine line. The linewidth

half width at half maximum is well represented by the

relation

7 = 7o(l + I/Ia)''' (2) where 70 and 7 are the linewidth without and with power

broadening, respectively.

If we take I /Ia = 0.6, f{X) = 0.117 = 0.89 /(Z W

(see Fig. 1), the broadening factor (1 -I- I/Ia)^ = 1.26. In this case, the contrast and width of the peak is better

for a stabilized laser. The reason for selecting a standard

value of I /IA instead of I is that the contrast and width

0.132

0.117 0-12 X = l.42,f(X)max = 0-'32

0.6 1.42

Figure 1. f(X) versus X.

depend on I /Ia- At the same time, the power broadening

increases the asymmetry of the line; this is possibly a main source of the power shift. Thus, the power shift depends on I /Ia We have measured 7 and Ia of NIM2 and NIMl with a

third harmonic locking technique. The third harmonic

signal line shape A^ix ) is given [4] by:

Asix) = -16x/m^ + i^/mH)

X {x(3m2 - Ax"^ + 12)(d + 1 + - x"^)^ + (12a;2 - 3m2 - A)[d - (I + m~ - x^)]!^} (3)

where

d = [{\ + - x^f + 4x^],21172 (4) and X and m are the detuning and modulation amplitude

divided by 7. The shape of the third harmonic signal is shown in Fig. 2. We can see from Fig. 2 that the third harmonic signal crosses the axis at three points. Point 0 is the reference

point of stabilization. The points on the two sides can be

also locked, but their frequency values and 7 depend on m. Typical relationships between these quantities are il- lustrated in Fig. 3. We have measured 7 and Ia by using

these relationships and the broadening formula of Eq. (2). The results are that 70 is 2.1 MHz and 1.9 MHz,

IA is 54 |jlW and 50 ixW, and / = (0.6) 7^ is 32 fxW and

30 M-W for NIMl and NIM2, respectively.

X

Figure 2. Profile diagram of As( x ).

78
2 h = /• m

(MHz)

1.0 1.25 1.5 1.75 2.0

Figure 3. Curve showing the dependence of Xp q and y on m.

We have also measured the line width at various pres- sures by this method. The pressure broadening is about

36.5 kHz/Pa [6], so that the total broadening is about

630 kHz at 17.3 Pa, and the natural width is about

1.25 MHz.

The asymmetry of the line was observed in locking

measurements. The amount of asymmetry can be ex- pressed as the difference between the absolute values of Xp and Xq. This asymmetry increased as the power was

gradually increased. This may have been caused by a frequency-dependent gas lens effect [5] in iodine. We

have not found any increase in asymmetry when the

pressure or modulation amplitude was increased. It is possible that the above asymmetry is the main reason for the power shift. Thus we select a small value of / /Ia as the standard value of laser power. Our selected value of

I /IA is 0.6; in the case of our lasers, the power is about

30 to 32 |jlW.

5. Conclusion

The power shift is the main effect limiting the reproducibility of laser frequency. We hope to determine an ap- propriate standard value of the laser power in a manner

similar to that for the pressure and modulation amplitude.

We shall check the method by further experiments [6]. References

[1] CCDM Recommendation Ml (1973); J. Terrien, Metrologia 10, 75 (1974).

[2] N. C. Shen, C. Y. Li, Y. M. Sun, C. Wang, J.-M. Chartier,

and R. Felder, CCDM/82-27 (document submitted to the

1982 meeting of the Consultative Committee for the Defini- tion of the Meter).

[3] H. Greenstein, J. Appl. Phys. 43, 1732 (1972).

[4] R. Lenaour et al., Opt. Quantum Electron. 10, 119 (1978).

[5] G. Kramer et al., Z. Naturforsch. 30a, 1128 (1975).

[6] Shen Nai-cheng, CCDM/82-12.

79

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Spatial Coherence and Optical Wavelength Metrology

p. Boucharelne

Institut National de Metrologie, Conservatoire National des Arts et Metiers, 292 rue Saint Martin F 75141 Paris

cedex 03

We discuss the influence on precise measurements of optical wavelengths of spatial coherence of laser beams, or of classical sources when the "etendue" of the interferometer is reduced by isolating the central fringe at high path differences. The interferometric observation of the phase angle at a

fixed path difference is currently made with an uncertainty less than 10 ~^ of a fringe. With spatially coherent illumination, phase shifts of this order can be given by scratches or dust pai'ticles, or by

mirror aberrations, and cannot be compensated by subtracting two phase angles at two path

differences. If the spatial coherence of a laser beam is destroyed, these errors vanish, but only if the etendue of the interferometer is large enough. At very high path differences allowed by the good

temporal coherence of lasers, compensated field interferometers may be useful to eliminate these

spurious shifts and the uncertainty in diffraction corrections. Key words: diffraction correction: optical wavelength metrology.

1. Introduction

The unit of length is defined in terms of a wavelength.

The definition of the meter will probably be soon

changed, but for a long time, practical measurements of length up to several meters will be best achieved by optical interferometry.

The wavelength is the spatial periodicity of the phase

and of the amplitude of a monochromatic infinite plane wave. If the wave is limited, optical rays are bent by dif- fraction, and a correction has to be made to the apparent

wavelength. The correction can be calculated only in some simple cases, as for instance with a Gaussian beam.

2. Spatial Coherence of Laser Beams

Lasers deliver a beam of high temporal coherence,

which is essential for observing interference fringes over

path differences of several hundred meters. Lasers also have quite perfect spatial coherence which might be a good condition if one wants to have a plane wave and to look at the true wavelength. Nevertheless, optical

wavelength metrology needs interpolation within a fringe

at high path differences. Interpolation within 10"-' of a fringe is often obtained. With a spatially coherent wave,

local phase shifts and errors are produced by any parasitic wave created from the main wave by reflection, dif-. fraction, or diffusion. For example any punctual defect on a mirror or a lens, like a scratch, or a dust particle,

gives rise to a spherical wave which interferes with the

main wave in almost all space (Fig. 1). At the point

where the wave is observed, or where it interferes with

the reference wave of the interferometer, this spurious wave brings a phase shift of the main wave, depending

on the ratio of the amplitudes A„ and A^, of the two

waves, and on the phase difference between them. The

maximum phase error is observed when the phase differ- ence is ± -rT/2 (Fig. 2) and the statistical mean of this er- ror is zero. To be sure that this shift is less than 10"-^ of a

fringe, the energy of the parasitic beam must be less than 10"'' of the energy of the main beam. Such an error

occurs with any spatially coherent wave, plane, spherical, or Gaussian, and to a lesser extent with a partially

coherent wave.

Figure 1. The influence of a local defect of a mirror on a plane

wave. The spherical diffracted wave interferes with the main

wave in all space, giving phase shifts.

II

I

I

I

I

I

Figure 2. Interference of the main beam of amplitude Ag iiitk the parasitic beam A„ gives a phase shift of the main beam. The

statistical mean of this shift is zero, and the maximum value is observed at the points ivhere the waves are in quadrature.

81
3. Compensation of Mirror Shape Defects

If the optical surface of the mirror of an interferometer

is not perfect (plane, spherical, or otherwise) the path

difference between rays reflected by an area dx dy

around the point with coordinates x and y is different

from zV) by an amount hix, y). Aq can be the path differ- ence at the point x = 0, y =0, or the mean path differ- ence over the whole area of the mirror, or any arbitrary

path difference, so that bix, y) is defined with an arbi- trary origin 8 = 0. In interferometric wavelength metrology, wavelengths

and path differences are measured with an uncertainty

much less than the amplitude of hix, y), because a spatially incoherent illumination of the mirror is assumed,

and the intensities of the interference signals of different areas dx dy are added. The phase error resulting from

8 (z, y) in the interference signal is compensated by tak- ing the phase difference between two path differences,

assuming that the energy distribution on the mirrors is kept constant. Thus, fractional phase excesses can be

measured in two-wave interferometry with an uncertainty less than 10"^ of a fringe with mirrors whose shape

is manufactured only within a tenth of a fringe.

If the mirrors of the interferometer are illuminated by

spatially coherent beams, we have to add the amplitudes

of different areas dx dy, and the consequence of the

shape error is no longer the same for a given mirror at two different positions, even if the ampHtude distribution on the mirror remains constant.

4. Compensated Field Interferometers and

Spatially Incoherent Illumination

It is possible to destroy the spatial coherence of a laser beam with one or two moving diffusers, and a sufficiently

long time response of the receiver. If such a diffused

beam is travelling through a classical interferometer

(Michelson or Fabry-Perot), it must be limited at infinity

by the hole isolating the central fringe. The solid angle of

this hole is 277 /p, where p is the interference order. For

high path differences, this hole may give a partial spatial

coherence to the illumination of the mirror area, and the

interferometer is more or less sensitive to the previously

described phase errors. To avoid such spatial coherence, with a diffused laser beam as well as with a spatially incoherent source, it is necessary to work with a large etendue interferometer,

with compensated field, like a spherical Fabry-Perot or a

cat's eye Michelson [1]. The gain in etendue is particularly

high for path differences of 10*' fringes and more, and the

phase errors due to diffraction can become negligible.

However, a fundamental question remains with these

interferometers, in which at least one mirror has to be

changed when the path difference is changed. With in- coherent illumination, the weighted mean of the path

difference is the same for two radiations of different

wavelengths only if the error function weighted by the

incoherent illumination / (a:?, y),

F(8) = f Iix,y)dx dy , •'8=const is an even function of 8, or can be made even by a change

in origin of 8.

5. Conclusion

With stabilized lasers, the main uncertainty in wavelength comparisons will probably lie in the diffrac- tion correction for imperfect mirrors and phase errors due to parasitic waves. The large etendue of field compensated interferometers used with spatially incoherent

illumination reduces the influence of diffraction. The

parasitic waves are also present with incoherent illumination, but the statistical mean of the phase error is zero. When we built our cat's eye interferometer with a compensated field, it was because of the limited luminosity of

the krypton lamp. With lasers this is not a factor, but w^

hope that our apparatus will be useful in studying those

aspects of optical wavelength metrology that were not in sight at the beginning.

Reference

[1] p. Bouchareine, in Atomic Masses and Fundamental Constants 5, Ed. by J. H. Sanders and A. H. Wapstra (Plenum,

New York, 1976), p. 417.

82
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Spectroscopy of Stored Ions

D. J. Wineland

Frequency and Time Standards Group, Time and Frequency Division

National Bureau of Standards, Boulder, CO 80303

The benign environment and long confinement times obtained with ion storage techniques have led to some unique experiments in the area of precision measurements and fundamental constants. This is perhaps epitomized by the single electron g factor measurements at the University of Washington in which a precision of 4 parts in 10" has been attained. Now, use of lasers to cool stored ions has allowed

the experimentalist to approach the goal of unperturbed atomic ions nearly at rest; most recently, spec- troscopy has been performed on single "cold" trapped ions. Stored ion experiments in the area of precision measurements and fundamental constants will be briefly reviewed. These include experiments on

e /e"^ g factors, mass spectroscopy, lifetimes, and atomic spectroscopy. The intent is to emphasize the unique environment provided by ion storage techniques for these measurements. Key words: atomic ion spectroscopy; electron g factor; fundamental constants; ion storage; mass spec- troscopy; precision measui'ements.

1. Introduction

The purpose of this paper is twofold. The first purpose is to outline "trapped ion" experiments of the last decade

which have contributed to the area of precision measure- ments and fundamental constants. The second purpose is to emphasize why the stored ion "environment" is particularly well suited to the area of precision measurements.

The main advantage of the stored ion technique is that

the ideal of an unperturbed species at rest in space is ap- proached to a high degree. Specifically, charged particles such as electrons and atomic ions can be stored for long periods of time (essentially indefinitely) without the

usual perturbations associated with confinement (for ex- ample the perturbations due to collisions with walls or

buffer gasses in a traditional optical pumping experiment). These unique properties were exploited most no- tably by Dehmelt and co-workers [1] before 1970; since

then, that group and others have continued to extend these

techniques with dramatic results. Unfortunately, there is a price to be paid for this property of long storage times with small perturbations—the

number of particles that can be stored is typically small

(slO'' for a "trap" with centimeter dimensions); the

resulting low densities are governed by the competition

between space charge repulsion and the confining elec- tromagnetic forces obtained under normal laboratory conditions. These low numbers, of course, require very sensitive detection techniques and preclude many types of experiments—for example spectroscopic experiments on complex molecular ions where only a small fraction of the

ions is in a given state. Although some effort has been

made to increase the number of ions by space charge neutralization [2, 3] and trap "arrays" [4], these methods in- troduce other experimental complications. In spite of the

low numbers obtained, sensitive techniques have been developed to detect simple species such as electrons and

atomic ions so that single electrons [5, 6] and ions [7, 8] have been observed.

Since the intent is to discuss experiments relating to precision measurements and fundamental constants,

many interesting experiments using stored ion tech- niques are not discussed in this paper. The reader is re- ferred elsewhere to experiments on, for example, photodetachment [9, 10], chemical reactions [11], electron-ion

recombination [12], charge transfer [13, 14], and nonneutral plasma studies [15]. Also, more general reviews

are available [1, 16, 17]. In Sec. 2, methods of trapping are briefly reviewed. In

Sees. 3-6, experiments on electrons/positrons, mass

spectroscopy, atomic spectroscopy, and lifetime measure- ments are described with emphasis on the ion's environment. In the last section, speculations about other fu- ture experiments are made.

2. Ion Trapping Methods

Four types of traps have been useful: the rf (or Paul)

trap, the Penning trap, and electrostatic and magnetostatic traps. These traps and their relevant properties are briefly described here. More detailed descriptions are given elsewhere [1, 16].

2.1 The rf Quadrupole Trap

The ideal rf or Paul [18] trap uses hyperbolic elec- trodes in a vacuum apparatus as shown in Fig. 1. These

electrodes are symmetric about the z axis, so we can

describe the potential in cylindrical coordinates. If an al- ternating voltage of frequency fi is applied between the

endcaps and ring electrode, then the instantaneous potential inside the trap is given by (following the notation

of Ref. [1]):

4)(r,z) = A(r2 - 22^), A - Aq cos f) f. (1) An ion experiences an rf electric field such that its motion (the "micromotion") is 180° out of phase with

respect to the electric force. Because the electric field is inhomogeneous, the force averaged over one period (T =

277/(1) of the micromotion is in a direction of weaker field amplitude (independent of the sign of the chai'ge), i.e.,

83
(a) rf or Penning trap

"end caps'

B = B_ 2 (required for Penning trap)

0(r, z) = A(r2 - 2^)

electrode surfaces generated by t'^U, z) = const.

(b) Orbitron trap

^A(r2- 2z2) + Bln(-^)

Figure 1. Electrode configuration for rf. Penning, and orbi- tron ion traps. All electrode surfaces are figures of revolution about the axial (z) direction. In a practical orbitroyi trap, it is desirable to make the center electrode as thin as possible to reduce ion loss from collisions.

towards the center of the trap. For Ct sufficiently high,

this restoring force gives rise to a pseudopotential

^(r,2) = [{rf- + 4(2)2]

where r and z are the positions of the ion averaged over

T, e is the ion charge, and M is the ion mass. The result- ing "secular" motion is

z - zq cos (coji + ((),) - - , ^ 1, - o- 2V2^o

X — Xq cos ((0,4 + (Pj-) = ^Wj- =

y = yo cos (co,.^ + <^y) In more detail, we have for the z motion:

MCl

20 1 +

CI cos n t cos a),L

where we require fl/ojj » 1. For ease of comparison with the Penning trap, it will be useful to consider the case of a spherical pseudopotential well in the rf trap. This is accomplished by

simultaneously applying a static potential between the

ring and endcaps so that

<^ = iUo + Ao cos nO(r2 - 2z^)

.

If we choose

Ur

^(r, z) =

Ai, then

(2)

In this case:

Z = Zo 1 -I- cos ilt COS co,^

Xq

yo

1 + aCOS fit COS dirt , (3)

where coj — ov —

2eAo

Ma

Approximate experimental values for stored atomic ions are such that ro — 1 cm, Aj is a few hundred volts/cm^,

a/2TT is a few MHz, and n/co^ = 10. If the secular motion is "cold" then we expect the

maximum density of ions to be given when the force from

the space charge potential 4), cancels that due to the

trap. In this case, for the pseudo-potential given by Eq.

(2), the ion cloud is a uniformly charged sphere of charge

density p with internal potential

<\yi = (2/3) Tip (r2

Therefore for a well with 2eAi/Ma^ = 10 V/cm^, the

condition cj), = - ^(r, z ) gives a density n = 3.3 x

10^/cm^. In practice, attainable densities are approximately 1 to 2 orders of magnitude lower due to "rf

heating" —a process which couples kinetic energy from

the micromotion into the secular motion. One way this happens is through collisions with background gas. However, at high vacuum other couphng mechanisms can

occur partly due to the presence of impurity ion s or imperfections in the trap electrodes. This process can

give kinetic energies of ions of as much as electron volts and shortened storage times of less than 1 s. If one uses

a light buffer gas such as He to provide viscous damping

[1] then storage times of many days can be achieved

[19, 20]. However, this increases perturbations due to

collisions; therefore, for experiments on large numbers of ions in an rf trap, there appears to be a practical tradeoff between the temperature of the ions and possible per- turbations due to collisions with a cooling buffer gas. However, this is not a fundamental limitation and one hopes it will be solved in the future. We note that in principle, densities could be increased

by increasing the rf field strengths; in practice, higher

densities (close to 10^/cm^ [21]) have been achieved by de- creasing the electrode dimensions. This, of course, may

not increase the ion number and it also may increase the

rf heating because the field imperfections may be rela- tively larger.

It is interesting to note that the methods proposed for trapping of neutral particles in laser beams [22] are very

analogous to the rf trapping of charged particles. In the

laser beam case, the outer atomic electron experiences

net forces in the (inhomogeneous) laser field. But since

the electron is bound to the nucleus, dispersive effects

84
occur. If the laser is tuned above the frequency of an

electronic transition, similar to an ion in a rf trap, the

atom is forced to regions of low field intensity. If the

laser is tuned below resonance, the atom is forced to re- gions of high field intensity.

2.2 The Penning Trap

The Penning trap [23] uses the same electrode configuration as the rf trap (Fig. 1) but now A in Eq. (1) is a

constant Uo such that the charged species see a static potential well along the z axis. We have =

Uoir^ - 2z^). This causes a repulsive potential in the X -y plane which can be overcome by superimposing a

static magnetic field along z{B = Boz). For a single ion

in the trap, the equations of motion are [1, 24]

Z 4- a)|z - 0, 4e[7o >2 —

r = 1 2 M Miri + 2zi)

'

eBo

where

r — X + iy.

Therefore

z = 2o cos coji and r - r^e'"""' + r,„e~"°"'' where

^+

i(o, _ [(a),/2)2 - co2/2]i'2. |Wm I ~ 2 Some useful expressions are:

o)^ -I- 0)2^/2 = tOcW (oo = o),,, or coc),

tOc -I- (0„, =(»>(.,

a)(.a)^ = a)|/2.

(4)

(5)

(6a)

(6b)

(6c)

Typical operating conditions in a Penning trap are such

that for electrode spacings about 1 cm, Vq is a few volts and Bq is larger than a few tenths of a tesla. The re- quired large magnetic field can either be an advantage or disadvantage, depending on the experiment. It should be

noted that the magnetron motion (the r„,e term) is in an unstable equilibrium in the trap. For example, if collisions with background neutrals occur, the ions will diffuse out of the trap. (When the magnetron velocity is much less than the velocity of the neutrals, r,„ random

walks in the x-y plane with step size ~r,..) This is a potential disadvantage when compared to the rf trap for which all three degrees of freedom {x, y, z) are in stable

equilibrium. In practice however, this is not a limitation

because ions can be stored for days in a room temperature apparatus [25] and electrons for weeks [5] in an 80 K

apparatus. Moreover, the technique of sideband or radiation pressure cooling [6, 21, 25, 26] can reverse this diffu- sion process.

If the voltage applied to the electrodes becomes too

high, then the radial electric field is high enough to over- come the ev X B/c magnetic force and the ions strike the

ring electrode in exponentially increasing orbits (argument of square root in Eq. (5) becomes negative). For

singly ionized atoms, the voltage where this occurs is

given by V, ^ 1200 Bl{r(, + 2zf^)IM where V, is in volts, Bq in tesla, M in u (atomic mass units) and dimensions in cm. This same mechanism limits the densities

achievable in the Penning trap since space charge also gives radial electric fields. If the axial and cyclotron temperatures are low enough, then we can approximate the

equilibrium distribution of the ion cloud [27] by a uniformly charged ellipsoid which has an (internal) potential

of:

(t),(r) = (2/3) TTp (ar^ + hz^)

where a and h are constants. The r motion of an individ- ual ion is now given by Eq. (4) with

V

1 2 _ 4TTepa

2 3M

1/2

(7)

where w, is the axial frequency for a single ion in the

trap. For a spherical distribution, a - b - \; and the

maximum density allowable (argument of square root term kept positive) is given by (using ({), (z ) = -ct)r(2 )):

n -

e <

or < 2.7 X lO^Bi/M

where So is in tesla and M in u. For Bq = 1 T, M =

100 u, n < 2.7 X lOVcm^ which is similar to the rf trap

case. In experiments at the National Bureau of Standards (NBS), densities within a factor of about 3 of the

theoretical maximum have been observed as indicated by

the space charge shifted magnetron rotation frequency.

2.3 Electrostatic Traps

The orbitron trap [28] has the advantage of simplicity

and although lifetimes are observed to be relatively

short, precision lifetime and spectroscopic measurements

have been made by Prior and his colleagues at Berkeley

[29, 30]. The orbitron trap is purely electrostatic and has axial symmetry as shown in Fig. lb. A potential which is useful is given by the general form [31]:

(j) = A{r^ - 22 2) + 51n(r/ri),

because a harmonic well is provided along the z axis. The

equations of motion for the x-y plane are not solvable

analytically, but the motion is basically composed of pre- cessing orbits about the z axis [32]. Storage times in such

a trap should be shorter because a single collision \yth a background neutral is sufficient to cause an ion to collide with the center electrode and be lost. Also the cooling

discussed in Sec. 6 would be more difficult to realize. Nevertheless, with a trap whose electrodes approximate

equipotentials of this function, trapping times of about one second were achieved by Knight [31] at a pressure of

10"'^ Pa ( = 10" * Torr). (The center electrode was a 100

fjLm dia. wire.)

An axial magnetic field superimposed along the z axis

has been investigated theoretically [33]; in this case, storage times should increase dramatically. Ions vdW

eventually be lost by diffusing in towards the center elec- trode; therefore, storage times should be comparable to that of the Penning trap. However, the frequency of the

drift motion about the center of the trap will depend on

the distance from the z axis; for the Penning trap this

drift (magnetron) frequency is independent of radial posi85
tion (neglecting space charge) and may be part of the reason for the slow evolution of the ion cloud under the

influence of the ion-ion coUisions. A possible advantage

over the Penning trap is that experiments may be possible at much lower magnetic fields [33].

2.4 Magnetostatic Traps

Magnetic bottles are of course used extensively for plasma confinement and also in ion spectroscopy. The

disadvantage in terms of spectroscopy appears to be that

rather large magnetic field inhomogeneities are required

for trapping, thus causing inhomogeneities and broadening in paramagnetic ion line shapes. The main application

in terms of precision measurements has been the Michigan eVe~ experiments [34]; these are reviewed by A. Rich at this conference.

3. e /e g-2 Measurements in a Penning

Trap

This experiment has been pursued primarily by the

groups of Dehmeit at the University of Washington, and

Graff at Mainz [35-37]. Only the Washington single elec- tron experiment [6] will be discussed here, since its accu- racy significantly exceeds that of the experiments on

clouds of electrons [35-38]. Experiments on clouds of

electrons are susceptible to rather large and difficult to measure space charge shifts. For example, the fractional

shift of the cyclotron frequency of one electron due to another electron a distance r away (in a direction perpendicular to the magnetic field) is approximately given by

[39]

Acq

coo

1 dr irEr)

where m is the electron mass, Wc is the cyclotron fre- quency, and Ey. is the electric field at one electron from

the other. For a magnetic field of 1 T, and r = 100 ixm,

Aco/ojo = 4.1 X 10"^ which gives a shift of about 4 ppm in g-2.. In 1971, the Michigan group had measured the

electron g - 2 to 3 ppm [40]; this result and the problems

associated with the space charge shifts led the Washington group to develop an experiment based on single

electrons. A complete account of this experiment is given

in Refs. [6, 41]. A simpler account is given in Ref. [42]. Briefly, a single electron can be detected [5, 42, 43] by

monitoring the currents induced in the electrodes by the

axial iz) oscillation at frequency co, . The harmonically

bound electron is equivalent to a series L-C circuit which

shunts the electrodes [5]; this L-C oscillator can be phase

locked to an external oscillator by feeding back on the

trap voltage Vo [6]. It is useful to think of the single

electron bound to the Penning trap as a one electron

pseudo atom whose "nucleus" is the earth (to which the

Penning trap is fixed), thus the name "geonium" [6]. The quantum mechanical solution for the energy levels of

this "atom" are given by:

E/h = mvs + (n + l/2)v; + {k + l/2)v, - (9 + l/2)v,„

where m is the spin quantum number (m = ±1/2) and n,

k, and q are the cyclotron, axial and magnetron quantum

numbers (n, k, q = 0, 1, 2 . . .). (The magnetron energy

is mostly potential energy and is negative.)

In the g-2 and e */e ~ mass ratio experiments it is necessary to detect the occurrence of spin flip and cyclo- tron transitions. This can be accomplished by superimposing a weak magnetic bottle centered on the trap such

that the field is [6]: B = (5o + P(z^ - ^2/2)2 - ^zyy - ^zxx . The spin, cyclotron, and magnetron magnetic moments

interact with this magnetic bottle to shift the axial fre- quency by an amount

bvz = [m + n + 1/2 + (v,„/V(.)g]8

8 — fXfi p/(2Tr^mvJ

For V, = 60 MHz and (3 = 0.012 T/cm^, 5 - 1 Hz, therefore for example, spin flip transitions would show up as 1 Hz changes in or as an accompanying step increase in feedback voltage to the locked axial resonance [6]. For reproducibility, it was desirable to have the elec- tron at the center of the trap. This could be accomplished

by sideband excitation [6, 26, 44, 45] as follows: with the

(damped) axial motion driven by an inhomogeneous rf

field of frequency + v,„ , the electron sees a sideband

at because of the magnetron motion through the rf

field. Therefore the electron absorbs quanta of energy

hiv^ + v,n) and reradiates (by the damping) quanta of energy h . The deficit, h , goes into the magnetron

energy which shrinks the magnetron orbit. In principle,

this technique allows for infinite storage times.

The electron g-2 has now been measured with an ac- curacy of 0.04 ppm [46] and the positron g - 2 to 0.05

ppm [47]. The details of these remarkable experiments

are described in Ref. [6]; rather than give a complete

description of the experiments, only the experimental en- vironment is described here. From Ref. [6], the cyclotron

degree of freedom was radiatively thermalized to 4 K. The axial temperature was somewhat higher. Pressure was estimated to be as low as 10 Pa so that colhsions

with background gas could be neglected. At temperatures near 4 K, axial excursions of 0.1 mm and cyclotron

orbit sizes of —60 nm would be obtained. During cyclo- tron excitation to n — 15, this gave a corresponding relativistic shift of only -6 x 10"-'. Using sideband excitation, r,„ — 14 |xm. From Eq. (6b), there is, of course a large systematic shift (~6 x 10""* in $r - 2) between the

observed and free space cyclotron frequencies. Due to imperfections in the trap, a direct measurement of v,„ (by

magnetron excitation) and the calculated value from Eq.

(6c) can disagree slightly. However, it can be shown that

the error in v,„ is much less [6, 48] than the above differ- ence which was measured to be 10 v,„. Thus the error

in g - 2 was negligible. Errors in the present experiments are estimated from field jitter and drift and dependence of g -2 on appHed spin flip power [47]. Ulti- mately, the experiment will be limited by systematic

shifts associated with the superimposed magnetic bottle. To eliminate this problem, new ways have been proposed

to detect the spin flips [6, 49]. For example, it might be

possible to use a switchable magnetic bottle so that while

the cyclotron and g-2 resonances are driven, the bottle would be off and transitions would be detected by turn- ing the bottle on [50]. Even without a magnetic bottle,

the axial motion is still hnked to the cyclotron degees of

freedom via the relativistic mass shift [6, 51]. This cou- pling is about an order of magnitude smaller than for the

magnetic bottle; the required increase in axial well depth

stabihty might be provided with a reference electron in an adjacent trap whose axial frequency is locked to a particular value [52].

86
4. Mass Spectroscopy

With the long storage times and high vacuums in the

traps, one could expect extremely narrow motional reso- nances. In the Washington electron experiments cyclotron

excitation resonances reproducible to a few parts in 10^ have been observed [53], and in principle the resolution

should only be limited by effects due to inhomogeneous or time varying fields. Using the methods described in Sec.

3, the electron/positron mass ratio has recently been

measured to about 0.1 ppm [63]. This very important ex- periment represents an improvement by more than two

orders of magnitude over previous experiments. In addition, the cyclotron resonances of both atomic ions and

electrons can be measured in the same magnetic field and

in the same apparatus. This rather unique feature has led to experiments designed to make a direct measurement of

the electron/proton mass ratio in a Penning trap.

The three current experiments using this method

[53-55] all basically measure the (shifted) cyclotron fre- quencies of electrons and protons in the same magnetic

field. In the experiment of Gartner and Klempt [54], resonances were detected by electron/proton loss; the

largest uncertainty was due to a space charge shift in the

proton cyclotron resonance. In the Van Dyck and

Schvdnberg experiment, resonances were detected in several (non-destructive) ways [53]; the limit here was

the uncertainty in the relative positions of the electrons

and protons in the trap and also possibly ion number

(space charge) dependent shifts. In the experiment of

Graff, Kalinowsky, and Traut, resonances were detected

by an energy dependent time of flight technique [55]. The

experiment of Van Dyck and Schwinberg has the smallest

uncertainty (0.14 ppm) and substantial improvements can be expected.

With regard to space charge shifts, it is worth noting

that in an ideal trap described by a quadratic electric potential and uniform magnetic field, if only one species is trapped and if the cyclotron motion is excited by a uni- form electric field, then no space charge shifts can occur because only the center of mass mode is excited [26, 56]. Unfortunately, deviations from the ideal trap fields can

occur, impurity ions can be present, and the excitation

usually occurs from non-uniform fields. All of these

things can lead to (partial) excitation of the internal

modes which then results in space charge shifts. These

systematic shifts increase with the mass of the ion and

therefore are much more important for protons and

atomic ions than for electrons. After the initial single electron work, it became ap- parent that it should be possible to observe single ion cy- clotron resonances by observing the induced currents in the ring electrode if it is divided in half [57] or into quadrants [53]; the key advantage here is that space charge

shifts are totally absent. Since the cyclotron frequency

of protons in a field of 4 T is about 60 MHz, electronic

detection should be similar to the detection of the axial resonance of single electrons. Other, related mass spectroscopy experiments should

be possible. First, using techniques as those referred to above, mass ratios of various atomic (molecular) ions

could be measured. It should also be possible to observe

the cyclotron resonance of single atomic ions by laser techniques. For example, if a single cold Be^ ion is local- ized at the center of the trap [8] with a laser beam whose

diameter is approximately 3 |xm, then it should be possible to observe cyclotron excitation which increases the

orbit radius to about 3 jjim by the decrease in fluores- cence scattering. This corresponds to a temperature

(ksT = (1/2) M (a);.)~V;0 of 14 K in a field of 5 T and a rel- ativistic shift of 1.4 x 10 The axial and magnetron

frequencies could be measured in a similar fashion. If the electron cyclotron frequency were measured in the same magnetic field, the accuracy in Mg/Mp from this in- direct experiment would be limited by the accuracy of Ms^/Mp which is about 0.04 ppm [58]. In preliminary experiments at NBS, the axial reso- nance (v^ 200 kHz) of a small cloud (<10) of Mg^ ions

has been measured in this way; however significant

broadening (a few parts in 10^) is observed—presumably

due to impurity ions. For an initial experiment on Be"^, the electron cyclotron frequency can be measured in- directly by measuring the spin flip frequency on the same

ion; since gi (Be"^) may be calculable to as good as a few

parts in 10 [59], the error in M^/Mp could still be lim- ited by the error in Mng/Mp. Ultimately when Mp/Mp is known better by other means, such measurements should

lead to better experimental determinations of gj (Be^)

and MBe/Mp.

We conclude by remarking that the extreme confinement possible with the stored ion techniques considerably relaxes the requirements on magnetic field homogeneity in these experiments.

5. Radiative Lifetime IVIeasurements

Although perhaps not "high precision" in an absolute

sense, the radiative lifetime measurements of Prior and

colleagues at Berkeley should be included in this review

because these measurements, made on simple atomic

ions, can be compared with various theories. Lifetimes of

2^5 0 Li+ (t - 503 |jls) in a Penning trap [60], 2s He+ (t -

1.92 ms) in an orbitron [29], and 2^Si Li+ (t - 58.6 s) in an rf trap [61] have been measured. One should also in- clude the 5'^Dz/2 Ba^ (t — 17.5 s) measurement in an rf trap by Schneider and Werth [62] and the 5^0^/2 Ba^ (t —

47 s) measurement in an rf trap by Plumelle et al. [19]. The key advantage of the traps in these measurements is the very long storage times and the benign environment

which minimizes the effects of other relaxations. For the

very long lifetimes, perhaps no other method is available.

6. Atomic Spectroscopy

As discussed in Sec. 1, the small number of ions in- volved makes it desirable to study simple atomic ions,

but this does not preclude the study of simple molecular

ions [1, 64]. As we will see, the potential accuracy and

precision for atomic spectroscopy using ion storage tech- niques is quite remarkable. This was apparent from the

early experiments on the ^He"^ ground state hyperfine

structure [1, 65] where linewidths of about 10 Hz at 8.7 GHz were obtained. Since that time the 2s ^He"^ hyperfine structure (hfs) has been measured in an orbitron

trap by Prior and Wang [66]. The importance of this measurement is that a comparison between the Is and 2s

hfs suppresses the effects of nuclear structure corrections and allows an accurate check of the QED corrections; such experimental/theoretical comparisons are most meaningful for hydrogenic ions. A significant interest in stored atomic ion spectroscopy

stems from the desire to make a frequency standard.

Stimulated by the He"^ experiments, various proposals

[65, 67-78] have been made for microwave and optical

stored ion frequency standai'ds. For a microwave stand87
ard, considerable attention has been given to the ^^^Hg"^

ion because (1) its ground state hyperfine frequency (—40

GHz) may be the largest of any conveniently usable ion,

(2) its large mass gives a relatively small second order

Doppler shift at a given temperature (~2 x 10 "^'^ at 300

K), and (3) a ^"^^Hg^ lamp source (194 nm) is available for optical pumping. Starting with the work of Major and

Werth [69], groups at Mainz [74], Orsay [76], and at least one commercial company [74] have sought to develop a frequency standard based on ^^^Hg^ ions stored in an rf trap. This work has been developed to a fairly high

level; the group at Orsay [76] has made a working standard whose stability compares favorably with that of a commercial cesium beam frequency standard.

However, the full potential of the stored ion technique

has yet to be realized. Historically, it appears that this has been due to two problems. (1) Because it has been

difficult to cool the ions, second order Doppler or time di- lation shifts can be sizeable (e.g.: —10"^^ in Ref. [74], ~4

X 10~^^ in Ref. [65]). (2) Signal-to-noise ratio has been

poor due to the small number of stored ions. In the past two or three years, both of these problems

have been addressed. In 1978, groups at NBS and

Heidelberg demonstrated [21, 25] that radiation pressure

from lasers [44, 70, 79] could be used to cool ions to temperatures <0.1 K, thereby reducing the second-order

Doppler shift by more than three orders of magnitude

below the room temperature case. As discussed below,

the reduction of kinetic energy is most favorable for very

small numbers of ions (down to one ion), so that there is a trade-off between the maximum number of ions we can

use and the minimum second-order Doppler shift that can be achieved.

With regard to signal-to-noise ratio, in certain opticalpumping, double-resonance experiments, it is possible to scatter many optical photons from each ion for each mi- crowave or optical "clock" photon absorbed [70,

73, 79-81]. This can allow one to make up for losses in detection efficiency due to small solid angle, small quantum efficiency in the photon detector, etc., so that the

transition probability for each ion can be measured with

unity detection efficiency. This means that the signal-tonoise ratio need be limited only by the statistical fluctua- tions in the number of ions that have made the transition

[78, 81]. Therefore the signal-to-noise ratio can be maximized for the given number of ions. The ability to sensi- tively detect the fluorescence from ions is emphasized by

experiments on single ions [7, 8] where it is even possible

to make photographs as shown in Fig. 2. More recently, the narrow linewidths anticipated for the stored ion technique have been observed; a resonance

linewidth of about 0.012 Hz at 292 MHz has been ob- served [82] for the imi,mj) = (-3/2, 1/2)

^

(-1/2, 1/2)

hyperfine transition of -^Mg* at a magnetic field of about

1.24 T where the first derivative of the transition fre- quency with respect to magnetic field is zero. (The Ramsey interference method was implemented by applying

two rf pulses of 1 s duration separated by 41 s.) These narrow linewidths should be preserved with hyperfine

transitions of higher frequency, such as in Hg^, but, of course, more attention must be paid to field homogeneity

and stability. In any case, the above results allow one to contemplate microwave frequency standards with Q >

10^^ and inaccuracies <10^^'' [78]. Of course, in addition

to the interest in frequency standards, very high precision hyperfine and optical spectroscopy on Hg^ [74], Ba^

[20, 82], Mg+ [25, 82, 83], and other alkali-like ions [75] is possible.

Figure 2. Photograph of fluorescence from single Ba* ion in an if trap (T = 10-36 ink; from. Ref. [86]). The schematic di- agram, of the rf trap from same view. Inner diameter of ring is

0. 7 mm..

The possibilities for an optical frequency standard are even more dramatic [70, 71, 77, 78, 84, 85]. For a given in- teraction time, the Q of a transition will scale with the

frequency. Therefore, in principle, an optical frequency

standard would have clear advantages over a microwave

frequency standard. The increased Q would allow one to work with smaller numbers, even down to one ion

[70,77,78], and therefore obtain the "ultimate" atomic

frequency standard. Example optical frequency standard

candidates are the ^Pq <— ^So transition in Tr, Ar or Ga"^

[70], the 3P ^ transition in 5 + [85], the D ^ S

Raman transitions in Ba+ [77], the S ^ D two photon

transition in Hg+ [71, 78], and the ^P3/2 ^Py2 transition

in Pb+ [72]. All have Q > W\ Unfortunately, before the

full potential of an ion-storage optical frequency standard

can be realized, tunable lasers with suitable spectral purity must become available. This problem may be nearing

solution [84]. Also, if an optical frequency standard is to provide time, the phase of the radiation must be measured. This appears to present a much more formidable

problem [84, 85]. Therefore it is expedient to continue to pursue a microwave frequency standard where both of

these problems have been solved. Rather than concentrate on the details of the various

experiments it is useful to examine the environment of the stored ions in order to identify possible systematic

effects. We will be primarily concerned with Stark and

second order Doppler shifts. Usually only second order

Stark shifts will be important; therefore we will be in- terested only in <E^->. As an example, for the strongly

polarizable cesium (neutral) atom, the ground state hyperfine shift [88] is equal to: 8v/v = -2.5 x IQ-^^ (E

in volts/cm); therefore electric fields of volts/cm become

88
important in very high accuracy experiments [90]. Magnetic field shifts due to instabilities and inhomogeneities

are of course important in the Penning trap but can be

made negligible in many experiments. (See for example

Refs. [78,82].) Other effects such as electric quadrupole

frequency shifts could be important in certain experiments [70, 78]. For single ions, laser cooling has already achieved temperatures betw^een 10 mK and 100 mK [7, 8]. Theoretically, when the motional oscillation frequencies 11, (w, and

uir for the rf trap and coz, coc , and w,,, for the Penning

trap) are less than the natural linewidth (Av = y/2'n) of the optical cooling transition, then the limiting "temperature" (governed by photon recoil) in each degree of freedom is given by fc^r- ty/2 [21,25,44,87]. (For a

single ion, the precise minimum temperature depends on

the angle of incidence of the laser beam and on the spatial distribution of recoil photons [87].) For strongly al- lowed transitions as in Ba"^ or Mg^, this limiting temperature is about 1 mK. For more weakly allowed transitions the temperature is correspondingly less but other

limits such as thermodynamic limits can come into play

[44]. When the condition ft,- » -y is fulfilled then the

limiting energy [44, 86] is given by =

{<%{> + 1/2) where <w,> = 577(16flf). Therefore the

limiting kinetic energy is given by Eki — ^ but the

uncertainty in energy (and therefore second order

Doppler effect) is much less [44]. For simplicity we will assume only the case ft, « -y below, however even

better results are potentially obtained for the opposite

condition. For a single ion in an rf trap, when = 0 the

nonthermal micromotion has an average kinetic energy

equal to that of the secular motion [1]; this is also true in the spherical trap. In the Penning trap the kinetic en- ergy in the nonthermal magnetron motion can be much

less than in the cyclotron or axial modes. Therefore, the

minimum second order Doppler shifts are given approximately by:

3

vo Mc

ty/Mc^ rf trap — %y/ Mc"^ Penning trap

For a single ion in an rf trap (assuming the spherical

trap of Eqs. (2) and (3)), <E^> is primarily due to the

oscillating rf fields and is largest for the z motion. A simple calculation gives <E^>^ = M fft.y/e^ for maximum

laser cooling or <E'^>z - 2.M Cl^gT /e^ for a given temperature in the z secular motion. For a single ion in a Penning trap, it is usually possible to make r^, « z

[8, 87], therefore Stark shifts from the static fields are primarily due to the z motion. We find <E'^>^ = M (x)?/{2e^) for maximum laser cooling or <E^>2 = kgTM (x>i/e^ at temperature T. In the Penning trap, a larger effect can be caused by the motional electric field E = V X B/c. We have <E^>m = tyB^/iMc^) (maximum laser cooling) and <E^>m ^ ^kgT B^iMc^). In Table 1 are shown examples of the second order Doppler

shift and <E^> for single ions in rf and Penning traps.

For clouds of identical ions, we first consider the electric fields due to collisions between ions. For the rf trap, we neglect the energy in the micromotion since the ions are driven in phase, therefore ion coUisional effects in the rf and Penning traps are treated the same. <E^>

will of course depend on the cloud density and temperature but some idea of the magnitude can be given by cal- culating the electric field for one ion on another at the

distance of closest approach. Assuming the maximum en- ergy available for closest approach is given by Skg T, we

have E^ = 6.1 X IQ-^ V/cm (y/2'n = 10 MHz and maxTable 1. Fractional second order Doppler shifts, Stark fields, and classical nns axial ampliUides (z„„J for single ions in rf and Peyining traps. When y/2TT is given, we assume maximum theoretical laser cooling (Cl, « y). For both traps we assume M = 100 u. For the rf trap n/2-iT (rf drive frequency) = 1 MHz, Aq =

300 V/cm^. For the Penning trap, ix)J2Ti = 20 kHz, B = IT.T is the temperature

of the secular motion for the rf trap and the temperatu re of the cyclotron and axial motion for the Penning trap. <E^>. is the mean square elecfric field for motioyi alofig the z axis, <E~>m is the mean square "motional" electnc field for the V X B/c force. Note that z„„s .for the Penniyig trap can be reduced at the expense

of increasing <E'^>,.

-y/2TT 10 MHz IkHz

^(K) 300 4 2.4 X 10" 4 2.4 X 10"^

8.3 X 10" 13 1.1 X 10"-14 6.6 X 10'-19 6.6 X 10-23

rf <£'2>^(y2/cm2) 210 2.8 1.7 X 10

-4 1.7 X 10-8

2rms(|J-m) 170 20 0.15 1.5 X 10-3

4.2 X 10" la 5.5 X 10"-15 8.3 X 10"-19 3.3 X 10-^

4.2 X 10

-2 5.7 X 10 3.4 X 10-8 3.4 X 10-12 Pen. (y2/cm2)

4.0 X 10-10 <E^>M 4.99 0.067 4.0 X 10

-6

(y2/cm2)

i z,^Jpm) 1300 145 1.1 0.011

89
imum laser cooling) and E mux = 7.4 V/cm at T = 4 K. Therefore at modest temperatures, ion-ion collision in- duced Stark shifts can be quite small. For clouds of ions, other effects can contribute to Stark

and second order Doppler shifts. We will consider only

theoretical limits and therefore neglect effects such as rf heating which may be the real limitation in a practical

experiment. We will assume that the secular motion in an

rf trap and the axial and cyclotron modes in a Penning

trap have been cooled to negligible values. For both

traps we will assume that it is desirable to maximize the

number of ions A'^. Then in an rf trap we must consider the effects of the

micromotion and corresponding electric fields for ions on

the edge of the cloud. We impose the constraint that the

maximum fractional second order Doppler shift not exceed a certain value (e). Therefore for the rf spherical

well we set ^(r,2) = -({), , and with the condition on e (the maximum second order Doppler effect is due to ions

at 2 - Vi, r =0) we find

A^max = 6.48 X r,Me

when M is in u, and r, is the cloud radius.

For the Penning trap, the maximum second order

Doppler effect is due to the magnetron motion of ions on

the edge of the cloud (r,„ - r,, z = 0). We set <^i{z) -

- (j)r(z ) and for a spherical cloud we find from Eqs.

(5-7),

- 1.96 X

440 M vr

B

'

where B is in tesla, M in u. Negative solutions are unphysical because they correspond to parameters where

the magnetron second order Doppler shift cannot be

made as large as e (for a spherical cloud).

For very small values of e, very low density ion clouds are required which implies very small applied potentials.

From the condition cj); = - "^^t- for the rf trap we have

(for the spherical well):

= 7.2 X 10" V

cm'' and similarly, for the Penning trap we have:

3.6 X 10" V

cm'

We can also calculate the corresponding electric fields. As before, for the rf trap, the maximum fields occur on

ions for z = r, and r - 0 and we have

2MCL^Nr

z (max)

In the Penning trap, the electric fields cancel along the z axis. Along the radial direction,

<^^^>.(nmx) - (2Mc2e/(en))2.

In Table 2 are shown some representative values of max- imum numbers of stored ions, trap potentials, and Stark

shifts for various values of e and r, on clouds of ions. In

certain configurations, second order Doppler and Stark

shifts could still be a problem; however, with small

enough numbers of ions these can be overcome. We note

that in many cases, the electric fields from black-body ra- diation i<E^>bb = (10 V/cm)^) can be much larger than

those due to trapping conditions [78, 89, 90]. Therefore,

operation at reduced environmental temperatures may

ultimately be required.

Table 2. Maximum numbers, pseudopotential well depths for rf traps, applied potentials for Penning traps, and electric fields for "cold" spherical ion clouds in rf and Perming traps. A maximum fractional second order Doppler shift e is assumed. The secular motion for the rf trap and the axial and cyclotron motion for the Penning trap are assumed to be frozen

out (i.e., cooled to negligible values), r^ = ion cloud radius: M = 100 u, Q/Z-n = 1 MHz, B

= IT.

10

-12 10 -12 10 -15 10 -15

rf

Pen.

r , (cm)

Nmax 2eAi

<E^>

5

V

V2

cnT"

V

0.5 3.2 X 10^

0.19

760

4.5 X 10^

1.3

0.14

0.01 6500

470

760

0.5 320

1.9 X 10-4

0.76

1.5x10^

0.043

1.4 X 10-7

0.01 ~6

0.47

0.76

53

1.9

3.5 X 10-4

90
7. Other Possibilities

With the special features of the ion traps in mind, it is worthwhile speculating on possible future experiments.

A very interesting class of experiments has already been

initiated on multiply charged ions. Electron capture

rates of Ne*^ (1 ^ g ^ 10) in an orbitron trap have re- cently been measured [14] and Penning traps are being

developed for high Z ion storage [91]. These experiments

may pave the way for interesting spectroscopy on for ex- ample high Z hydrogenic ions where precise theoretical

calculations are available. Various energy separations be- come higher; for example the -a^u'*^^ ground state hfs transition may be in the visible [92]. In the future, more

exotic species may be available for study such as anti- protons, e^/e" pairs in a Penning trap or e"^/e~, e^/p",

e~/p^ or p'^/p" pairs in an rf trap [92]. The Penning trap may be an ideal place to measure gj

factors in certain atomic ions. First, because very small

samples (<100 ixm) can be used, the effects of inhomogeneous fields are greatly reduced [81]. Second, atomic

gj factors could be measured in terms of the Bohr magnetron by measuring the cyclotron frequency of free elec- trons in the same magnetic field. This technique has al- ready been used to measure the gj factor in [93]. More generally, these features plus the very high resolutions possible [82] should allow precise measurements of

other magnetic-field-dependent structure. For example,

deviations in the Breit-Rabi formula for hfs could be

measured and it may be possible to measure the effects of nuclear diamagnetism [94]. From Tables 1 and 2, it is evident that Stark shifts can be very small on cooled ions; this suggests that the spec- troscopy of Rydberg ions vdth minimal Stark shifts may

be possible. With this in mind, perhaps a measurement of

the Rydberg along the lines of Kleppner's proposal [95]

is possible in an ion. Finally, ion storage techniques might also be used for measurement devices, for example, to measure magnetic

fields [96] or frequencies of lasers [45, 97]. Quite generally, it is now feasible to think of experiments on stored ions where the perturbations are ex- tremely small. Hopefully, this can lead to many new precision experiments other than the ones discussed here.

The preparation of this paper was supported in part by

the Air Force Office of Scientific Research and the Office

of Naval Research. The author thanks W. M. Itano, G. Dunn, J. C. Bergquist, R. E. Drullinger and F. L. Walls

for a critical reading of the manuscript and R. Ray and

B. Barrett for manuscript typing.

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92
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Progress Toward a Stored Ion Frequency Standard at the National Bureau of

Standards*

Wayne M. Itano, D. J. Wineland, J. C. Bergquist, and F. L. Walls

Frequency and Time Standards Group, Time and Frequency Division, National Bureau of Standards, Boulder,

CO 80303

Two fundamental problems with the development of a primary frequency standard based on stored

ions have long been apparent—^the second-order Doppler shift and the low signal-to-noise ratio. Both

problems have been addressed in experiments at the National Bureau of Standards (NBS)—^the first by

the development of the laser cooling technique and the second by the development of laser-opticalpumping techniques with high detection efficiency. Also, a hyperfine transition in ^^Mg'*' has been

observed by rf-optical double resonance w\th a linewidth of 0.0i2 Hz and a Q of 2.4 x 10^^ A possible

microwave frequency and time standard based on a ground-state hyperfine transition in ^''^Hg^ and a

possible optical frequency standard based on the two-photon 5d^ 6s^^D5/2 ^ 5d^° 6s ^Sy2 transition in l99Hg+ or 20iHg+ are described. Key words: atomic frequency standards; atomic hyperfine structure; double resonance; Hg"*"; laser cooling; Mg"*"; optical pumping; stored ions.

1. Introduction

It has long been realized that microwave or optical

transitions of ions stored in electromagnetic traps might

form the basis for new types of frequency standards. The

basic advantage of such devices is that very long interrogation times (up to many seconds) and, therefore, high

transition line Q's can be achieved, while the perturbations which usually accompany the confinement of atoms,

such as collisions with buffer gas molecules or the cell walls, can be substantially avoided. Ground-state hyperfine transitions of ^He^ [1], ^^^g^ [2-4], and ^-"Ba^ [5] ions stored in rf quadrupole traps have been observed

with high resolution by direct or indirect optical pumping

techniques. In fact, Q's greater than 10^'' have been observed on the 40.5 GHz hyperfine transition of '^^Hg^.

The accuracies of the measui'ements were limited largely

by the uncertainty of the second-order Doppler (time di- lation) shift. The signal-to-noise ratios were relatively

low, because the maximum number of ions stored was

about 10^ - 10*' and the transitions were detected with an

efficiency much less than one. Techniques developed at NBS in the last three years address both of these problems. Resonant light pressure has been used to cool stored ions to temperatures below 1 K, thus reducing the

second-order Doppler shift. Laser-optical-pumping tech- niques have been developed which are, in principle, capable of detecting transitions with nearly unit efficiency, so that the signal-to-noise ratio is limited only by the statistical fluctuations in the number of ions that make the

transition.

2. Laser Cooling and Dpuble Resonance of

Mg

Laser cooling of Mg"^ ions stored in a Penning trap has

been investigated in a series of experiments at NBS

[6-10]. Ground-state hyperfine transitions have been

*Supported in part by the Air Force Office of Scientific Research and

the Office of Naval Research.

detected with extremely high resolution by rf-optical

double resonance techniques [10]. Laser cooling of Ba^

ions stored in a quadrupole rf trap has been demonstrated in a related series of experiments at Heidelberg

[11].

2.1 Apparatus

In a Penning trap, ions are confined by a combination

of a uniform magnetic field and a quadrupolar electrostatic potential [12, 13]. The trap used had dimensions

To = 1.64 zo = 0.63 cm in the notation of Ref. [12]. Typical operating parameters were Vo — 7 Y and Bq — IT,

where Vq is the potential across the electrodes and 5o is the magnetic field. The storage time for Mg^ ions in the

ultrahigh vacuum environment (P < 10"' Pa) was about a day.

Narrowband, tunable radiation resonant with the 3s

"Si/2 ^ 3p ^P3/2 transition at 279.6 nm was required for the laser cooling and the double resonance experiments.

This was produced by frequency doubling the output of a single-mode cw Rhodamine 110 dye laser in a 90° phasematched AD*P crystal. Between 5 and 30 |xW of UV

power was generated in a bandwidth of about 1 IVIHz. The dye laser could be long-term stabilized at discrete

frequencies to less than 1 MHz by locking it to saturated

absorption features in I2. Fine tuning could be accomplished by varying the trap magnetic field to Zeeman

shift the Mg+ levels. The UV radiation was focused at the center of the

trap, passing through holes in the ring electrode. Reso- nance fluorescence photons emitted by the ions in the

backward direction were detected by a photomultiplier

tube and counted. The net detection efficiency for a pho- ton emitted from an ion, taking into account the collec- tion solid angle, mirror reflectivity, transmission of the

lenses and filters, and j)hotomultiplier quantum effi- ciency, was about 2 x 10"°.

93
2.2 Laser Cooling

The basic principle of laser cooling of free or harmonically bound atoms [14, 15] can be explained as follows: Consider a monochromatic, directed beam of light whose

frequency is slightly below that of a strong resonance

line of the atoms. Atoms whose velocities are directed to- ward the light source see a frequency Doppler shifted closer to resonance; if their velocities are directed away

from the source, the frequency shift is away from reso- nance. Therefore, the atoms tend to absorb photons when

their velocity is directed toward the source. This slows

them down, since the momentum of the absorbed photons

reduces the atomic momentum. (The photons are re- emitted in random directions.) All velocity components

could be reduced with six laser beams directed along the

±x, ±y, and ±z directions. A single beam suffices to cool

all oscillational modes of an atom bound in a threedimensional harmonic potential, if the modes are nondegenerate and if the beam is not directed along one of the

principal axes [11, 15, 16]. For an ion in a Penning trap, cooling of the axial and

cyclotron modes takes place in the same way as for harmonically bound atoms, if the laser is tuned below reso- nance [16]. Cooling of the magnetron mode, which is an E X B drift of the center of the cyclotron orbit around

the trap axis, can be accomplished by focusing the laser beam so that it is more intense on the side of the trap

axis on which the magnetron motion recedes from the

laser [6-9, 16].

It is easiest to achieve very low temperatures with

small numbers of ions. This is because the radial electric

field due to space charge increases the magnetron velocity. This problem does not exist for a single, isolated ion,, and the lowest temperatures were observed for this case

[9]. A recording of the fluorescence intensity from a small number of ^^Mg"^ ions is shown in Fig. 1. After the

ions were cooled and localized at the trap center, an oven

containing ^^Mg was heated in order to induce the resonant charge exchange reaction (^Mg^ + ^^Mg

^Mg -I- ^^Mg^ ). The resulting ^Mg"^ ions were ejected

from the trap by resonant cyclotron-magnetron rf excitation. We attribute the step decreases in the fluorescence

to individual charge exchange events and the last plateau

above background to the fluorescence from a single ion. The "temperature" of a single ion was determined

from the Doppler width by optical-optical double reso- nance (see Fig. 2). (We define the "temperature" of a

single ion in terms of its average kinetic energy.) When

the light is polarized perpendicular to the magnetic field and is close to resonance vdth the (^Sy2, Mj = -1/2)—

(^Pa/2, Mj = -3/2) Zeeman component, most of the ^Siy2

population is pumped into the Mj = -1/2 sublevel by the

following mechanism [8]: The electric dipole selection

rules allow Mj to change by ±1 in the ^P3/2 transition and by 0 or ±1 in the subsequent ^P3/2 ^Siy2 decay.

Thus, the -1/2 ^ -3/2 and +1/2 +S/2 transitions (we

denote the and M/-values by the first and

second numbers respectively) do not optically pump the

ground state, since the ion must decay back to the same

sublevel. The other allowed transitions, +1/2 -1/2 and

-1/2 —> +1/2, do cause optical pumping, because the ion can decay to either sublevel. These off-resonance transitions are driven in their Lorentzian wings (the Zeeman

splitting is much greater than the Doppler broadening),

at a rate > 1 s"\ which is much faster than any other re- laxation rate between ground-state sublevels. The detuning of the Hght from the -1/2^ +1/2 transition is four

times its detuning from the +1/2—» -1/2 transition.

Hence, the Mj = +1/2 ground-state sublevel is depopulated 16 times faster than the Mj - -1/2 sublevel. In the

steady state, 16/17 — 94% of the population is in the

Mj = -1/2 sublevel. Since the photon scattering rate is proportional to the number of ions in the Mj = -1/2 sublevel, any process that changes this number is detectable

as a change in the fluorescence intensity. This is the

basis of the double resonance technique. To obtain the

data shown in Fig. 2, one laser was tuned slightly below

the -1/2 -3/2 transition, in order to provide cooling

and fluorescence detection, while a low power laser was

swept across the -1/2^ -1/2 transition in order to de- plete the Mj - -1/2 sublevel. The resulting hneshape

reflects both the natural and Doppler broadenings. Also

shown in Fig. 2 are simulated curves for temperatures

T = 0 K and 100 mK. We estimate from this data that

T" = 50 ± 30 mK. Since the light beam was incident at an

angle of 82° with respect to the magnetic field, this is primarily a measurement of the cyclotron-magnetron (x- y) temperature. The axial (z) motion was not cooled as

efficiently as the cyclotron-magnetron motion because of the beam angle. The axial temperature was estimated to be about 600 mK by lowering the well depth and noting

the decrease in the fluorescence due to the decreased

fraction of time the ion spent in the laser beam. According to a theoretical model, it should be possible to obtain

a cyclotron-magnetron temperature of about 1 mK and an

— BACKGROUND

TIME

Figure 1. -'^Mg^ fluoresceyice as a function of time. Each point represents a

10 s integration. The three large steps are due to single charge exchange

events, which remove the '^Mg^ ions, one at a time. The last plateau above

background is due to a single ion. (From Ref. [9].)

94
200

REUTIVE LASER FREQUENCY (MHz)

Figure 2. Optical-optical double resonance of a single ^Mg'^

ion. The fluorescence due to a strong laser tuned to the ('282/2, Mj = -112)^ (~P.3/e, Mj = -312) transition is plotted as a

function of the frequency of a weak laser, which is tuned con- tinuously across the C^Sj/^, Mj = -1/2) fPsi-z, Mj = -112)

transition. Each dot represents a 10 s integration. The dashed

curves are simulated data for ion "temperatures" 0 K and

100 mK. (FromRef. [9].)

axial temperature of about 11 mK for these conditions

[16]. At present, the discrepancy is not understood, but

may be due to the presence of impurity ions in the trap.

2.3 RF-Optical Double Resonance

The microwave Zeeman transition in the ground state

(^81/2, Mj = -1/2-^ +1/2) also can be detected by a de- crease in the fluorescence intensity [8]. This "flop-out"

detection method can be very efficient. In effect, a transition due to a single microwave "photon" interrupts the

stream of scattered optical photons until the ion is pumped back to the Mj = -1/2 sublevel by weak, off- resonance scattering. Since the decrease in the number

of scattered photons can be very large, it is possible to make up for poor light collection and detector quantum

efficiencies, so that the transition can be detected with

nearly unit efficiency. "Flop-in" detection where a large

increase in the number of photons scattered is caused by

absorption of a single microwave photon, is also possible

[17]. Similar detection methods have been proposed

previously [18]. Various ground-state rf and microwave transitions in ^Mg"^ were detected by double resonance. ^^Mg"^ has nuclear spin / = 5/2; in a high magnetic field, the groundstate sublevels can be denoted by (Mi, Mj). If the laser

is tuned to the {^^vi. Mi = -5/2, Mj = -1/2) {^V^,

Ml - -5/2, Mj — -3/2) transition, pumping into the

Mj = -1/2 ground-state manifold takes place in the same way as for ^^Mg"^, which has 7=0. The pumping of nearly all of the population into the Mi = -5/2 sublevels occurs because of hyperfine coupling in the excited state

[7, 8]. The excited-state sublevels which are labeled (Mi

,

Mj - -3/2) all contain small admixtures of states with

lower Ml, except for (M/ = -5/2, Mj = -3/2), which is pure. Hence, if an ion in the (Mi, -1/2) ground-state sublevel is excited to the (Mi, -3/2) excited-state sublevel,

it has a small probability of decaying to a ground-state

sublevel with lower Mi, unless Mi = -5/2. Eventually,

the ions are trapped in the Mi = -5/2 sublevels. After

this pumping takes place, the (-5/2, -1/2)^ (-5/2, + 1/2) electronic spin flip transition and the (-5/2,

-1/2) (-3/2, -1/2) nuclear spin flip can be detected by

a decrease in the fluorescence intensity when resonant rf

is applied. When the laser is tuned to the Mi =

+5/2, Mj = +1/2)^ (2P3^, Ml = +5/2, Mj = +3/2)

transition, most of the population is pumped into the

(+5/2, +1/2) sublevel, and the (+5/2, +1/2)^ (+5/2,

-1/2) and (+5/2, +1/2) (+3/2, +1/2) transitions can be

observed.

The observed resonances were broadened by magnetic

field fluctuations. However, the magnetic field derivative of the (-3/2, +1/2)

^

(-1/2, +1/2) transition fre- quency goes to zero at Bq — 1.2398 T. The transition was

observed near this field with linewidths as narrow as 12 mHz [10]. The following method was used to detect

the transition: The ions were optically pumped into the

(-5/2, -1/2) sublevel. Some population was transferred

to the (-3/2, 1/2) sublevel by saturating the (-5/2,

-1/2)

^

(-5/2, +1/2) and (-5/2, +1/2)

^

(-3/2, +1/2)

transitions with rf. The (-3/2, +1/2) (-1/2, +1/2) transition was driven with the light blocked and the saturat- ing rf off. The laser and saturating rf were turned on

again, and the number of transitions made to the (-1/2, + 1/2) sublevel was indicated by a decrease in the fluores- cence level. The Ramsey interference method [19] was

implemented by pulsing the rf, resulting in an oscillatory

lineshape (see Figs. 3a, 3b). Several transition frequencies were measured in order to determine the hvperfine constant A = -596.254 376(54) MHz and the nuclear-toelectronic g factor ratio gi/gj = 9.299 484(75) x 10^^ [10]. (Estimated standard deviations in parentheses.) The uncertainties of A and gi/gj are dominated by the uncertainties of the field-sensitive transitions, which are due

to fluctuations of about 1 ppm in Bq. Further experimental efforts are directed toward developing a trap with

greatly improved light collection efficiency in order to continue studies of Mg"^ and to initiate similar studies of Be+.

1

i 1 1

0.023 Hz

1

\ 5^/

oV/

\o/ \of

6

291 996

Q/

\i

t 251.899

\/

Hz

kJ \ 1 1 1 1

0

"

1.80 1.85 1.90 L95 2.00 RF Frequency - 291 996 250 (Hz)

Figure 3a. (Mi, Mj) = (-312, +112)^ (-112, +112) ^^Mg^

ground-state hyperfine resonance. Each circle represents the average of four measurements (total fluorescence detection in- tegration time of 8 s). The oscillatoi-y lineshape results from

the use of the Ramsey method. Two coherent rf pulses of duration T = 1.02 s separated by T = 20.72 s were applied. The

solid curve is a theoretical fit. The vertical arroxv marks the central minimum, which corresponds to the resonance fre- quency. The magnetic field Bq was set so that the (—512,

-112) <—> (-512, +112) electronic spin flip resonance was in the range of 36 2Jt8.37i. ± 0.750 MHz, which comsponds to Bn = 1.2398 T.

95
T

291 996 251.899 Hz

1.88 1.89 1.90 1.91 1.92 RF Frequency - 291 996 250 (Hz)

Figure 3b. (-312, +1/2) (-1/2, +1/2) ^^Mg^ kyperfine resonance. The total fluorescence detection integratwn time

ivas 16 s per point and T was 4140 s. The experimental conditions were otherwise the same as for Fig. 3a. The vertical ar- row marks the central minimum. (FromRef. [10].)

3. Proposals for Hg Frequency Standards

Here we outline proposals for a microwave frequency

and time standard based on a hyperfine transition in 2oifjg+ and for an optical frequency standard based on a two-photon transition in ^^^Hg"^ or ^''^Hg"*". These proposals incorporate the laser cooling and laser-opticalpumping techniques demonstrated with Mg^. They use

the Penning trap instead of the rf trap because cooling

(of more than one ion) appears to be easier. Preliminary

experimental work on these devices has begun at NBS.

Further details are published elsewhere [17].

3.1 Microwave Frequency Standard

The basic advantages of Hg"^ for a microwave fre- quency standard are the large hyperfine structure (hence

high Q for a given linewidth) and high mass (hence low

second-order Doppler shift for a given temperature).

Other groups [2-4, 20] have worked with the TMHg^ [F,

Mp) = (1,0) <

> (0,0) transition using a ^°^Hg"^ lamp for

optical pumping and an rf trap near zero magnetic field. For a Penning trap, the best transition appears to be the

2oiHg+ ^jp^ Mf) = (1,1)^ (2,1) transition, which is field-independent to first order at 5o — 0.534 T, with fre- quency —25.9 GHz. If 5o can be controlled to slightly

better than 0.1 ppm over the volume of the ion cloud, the

fractional second-order field shift can be kept below

10"^^. A laser tuned just below a particular hyperfineZeeman component of the 6s 6p ^Fyz 194.2 nm line could be used to cool the ions. All of the ground-state

sublevels would have to be rf-mixed, in order to permit

cyclic interaction with the laser. Essentially all of the

ions could then be pumped into the (F, Mf) = (1,1)

sublevel by leaving it unmixed by the rf. The laser and

mixing rf could then be shut off, and the (1,1)-^ (2,1)

transition driven. The number of ions that made the transition could be determined by turning the laser on, rf-mixing all but the (1,1) sublevel, and counting the

scattered photons. In order to keep the second-order

Doppler shift below 10"^^, the "temperature" must be

kept below 1.45 K. This should be easy for the cyclotron

and axial modes, but the space-charge-induced magnetron velocity is a problem for a large cloud of ions. If the shift is to be kept below 10"^^, and the ions are to be

contained vdthin a 1 cm diameter spherical volume, the

maximum number of ions is about 8.2 x 10^. All other

systematic shifts, such as those due to collisions, the trap

fields, or thermal radiation, appear to be much less than

10"^^. For an interrogation time of 50 s, a Q of 2.6 x 10^" would be obtained using the Ramsey method. With

8.2 X lO'^ ions and a signal-to-noise ratio lim.ited only by

statistical fluctuations in the number of ions that make

the transition, the fractional frequency uncertainty for measurement time, t, is calculated to be o-j,(t) = 2 x

^Qj, ^ ^ jOO s. Actual short-term stability may

be limited by available local oscillators. Various possibilities exist for the required 194.2 nm

source. One is a narrowband, high repetition rate ArF*

excimer laser. Another, which is being pursued at NBS,

is sum-frequency mixing in a KB5 crystal of the output of a single-mode 790 nm cw ring dye laser and the second

harmonic, generated in an ADP crystal, of the output of a 514 nm stabilized, single-mode cw Ar"^ laser. The basic

method has been demonstrated previously with pulsed

lasers [21].

3.2 Optical Frequency Standard

It has previouslv been pointed out that the twophoton-allowed 5d^''^ 6s ^Siy2 5d^ 6s^ ^^5/2 Hg"^ transition, which has a Q of 7.4 x 10^"*, could be used in an

optical frequency standard [22]. The first-order Doppler

effect can be eliminated by driving the transition with

counter-propagating 563.2 nm laser beams. HyperfineZeeman components whose magnetic field derivatives

vanish at particular values of 5o exist in ^^^Hg"^ and

20iHg+. The fractional frequency shifts are much smaller

than for the microwave frequency standard for the same

fractional change in Bq. The second-order Doppler shift can be reduced to the same degree as for the microwave

standard by cooling with a 194.2 nm laser. The two- photon transition can be detected with high efficiency by

a change in the 194.2 nm fluorescence intensity. Taking

full advantage of the high Q transition would require a

laser with a linevddth of less than 1 Hz, which does not

exist at present. However, linewidths — 100 Hz appear

feasible and could be used for initial experiments. The

largest systematic shift appears to be the ac Stark shift due to the 563.2 nm laser beams. If the laser linewidth is less than the natural linewidth, and the transition is near

saturation, this shift is about 2 x 10"^^. Assuming ideal lasers are available, the stability of this standard is cal- culated to be CTj,(t) = 2 X 10-1^ t'^ for 8.2 x 10'' ions and

(TyiT) = 6 X 10"^^ T"^ (t > 1 s) for a single ion. References

[1] H. A. Schuessler, E. N. Fortson, and H. G. Dehmelt, Phys. Rev. 187, 5 (1969).

[2] F. G. Major and G. Werth, Phys. Rev. Lett. 30, 1155 (1973); Appl. Phys. 15, 201 (1978).

[3] M. D. McGuire, R. Petsch, and G. Werth, Phys. Rev. A 17, 1999 (1978).

[4] M. Jardino and M. Desaintfuscien, IEEE Trans. Instrum. Meas. IM-29, 163 (1980); M. Jardino, M. Desaintfuscien, R. Barillet, J. Viennet, P. Petit, and C. Audoin, Appl.

Phys. 24, 107 (1981).

[5] R. Blatt and G. Werth, Z. Phys. 299, 24 (1981); these Proceedings.

[6] D. J. Wineland, R. E. Drullinger, and F. L. Walls, Phys. Rev. Lett. 40, 1639 (1978).

[7] R. E. Drullinger, D. J. Wineland, and J. C. Bergquist, Appl.

Phys. 22, 365 (1980).

[8] D. J. Wineland, J. C. Bergquist, W. M. Itano, and R. E.

Drullinger, Opt. Lett. 5, 245 (1980).

[9] D. J. Wineland and W. M. Itano, Phys. Lett. 82A, 75 (1981).

96
[10] W. M. Itano and D. J. Wineland, Phys. Rev. A, 24, 1364

(1981).

[11] W. Neuhauser, M. Hohenstatt, P. E. Toschek, and H. Dehmelt, Pliys. Rev. Lett. 41, 233 (1978); Appl. Phys. 17, 123 (1978); Pliys. Rev. A 22, 1137 (1980).

[12] H. Delimelt, in Advances in Atomic and Molecular Physics, Ed. by D. R. Bates and I. Estermann (Acadeniic Press, New York, 1967), Vol. 3, p. 53; Vol. 5, p. 109 (1969).

[13] D. J. Wineland, these Proceedings.

[14] T. W. Hansch and A. L. Schawlow, Opt. Commun. 13, 68

(1975).

[15] D. J. Wineland and W. M. Itano, Phys. Rev. A 20, 1521

(1979).

[16] W. M. Itano and D. J. Wineland, Phys. Rev. A 25, 35 (1982).

[17] D. J. Wineland, W. M. Itano, J. C. Bergquist, and F. L. Walls, Proc. 35th Ann. Symp. on Freq. Control (U.S.

Army Electronics Command, Fort Monmouth, NJ, 1981).

[18] H. Dehmelt, Bull. Am. Phys. Soc. 20, 60 (1975).

[19] N. F. Ramsey, Molecular Beams, (Oxford University Press, London, 1956).

[20] M. D. McGuire, Bull. Am. Phys. Soc. 26, 615 (1981).

[21] R. E. Stickel and F. B. Dunning, Appl. Opt. 17, 981 (1978).

[22] P. L. Bender, J. L. Hall, R. H. Garstang, F. M. J. Pichanick, W. W. Smith, R. L. Barger, and J. B. West, Bull. Am. Phys. Soc. 21, 599 (1976).

97

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

High Resolution Microwave Spectroscopy on Trapped Ba Ions*

W. Becker, R. Blatt, and G. Werth

Institut fiir Physik, Universitat Mainz, 6500 Mainz, F.R.G.

We performed an optical double resonance experiment on the ground state of '^''Ba"^ and '''^Ba* ions. About 10'' particles were confined in an rf quadrupole trap for many hours. Hfs state selection by

pulsed laser optical pumping was followed by microwave transitions, which were observed via change

in the ionic fluorescence intensity. Linewidths of the order of the laser repetition frequency (1-20 Hz)

and a complicated line structure were observed in the "field independent" F = 1, m = 0 ^ F = 2, m = 0 transitions. The statistical uncertainty of the line center was below 0.1 Hz. The results for the hyperfine separations, including corrections to zero magnetic and electric field, are

Av(i^'^Ba+) = 8 037 741 667.69 (0.37) Hz, and

Av(i3^Ba+) = 7 183 340 234.35 (0.57) Hz. Key words: frequency standard; hyperfine structure; ion storage.

Microwave resonance spectroscopy on electrodynamically confined ions can achieve very high resolution and

accuracy; the relaxation times between ground state sublevels can be of the order of many seconds and

uncontrolled perturbations can be made very small

(<10"^'*). Thus this method may have applications to frequency and time standards. Best suited for this purpose are heavy ions with large hyperfine separation

to reduce the limitation imposed by the second order

Doppler effect. We performed an optical double resonance experiment

on both stable odd Ba isotopes with masses 135 and 137. Preliminary results on the latter have been published

recently [1]. The ions were created by surface ionization

of a sample of isotope separated Ba on a hot Ft filament

placed close to an endcap of an rf quadrupole ion trap,

whose operation is described in detail in the literature

[2, 3]. At 10 ^' mbar of He background pressure about

2% of the total ions emitted from the filament are trapped and remain inside the trap for many hours. At

typical operating conditions (V^c = 8 V, V = '750 V,

n/2'n- = 500 kHz, = 20 mm) the trap formed a pseudopotential valley of about 30 eV depth. The ions were maintained at a temperature which corresponds to 10% of the potential well depth [4]. Figure 1 shows a block diagram of the experimental setup. Ground state population inversion was achieved by optical pumping

with a linearly polarized pulsed dye laser, tuned to one of

the hyperfine components of the 6S1/2 - 6P1/0 transition

at 493.4 nm (Fig. 2). The laser spectral width (1 GHz)

and the optical Doppler width (3 GHz) were small

compared to the hyperfine splitting (8 and 7.1 GHz,

respectively) for the two isotopes. Taking into account

that the excited P1/2 state partially decays into the metastable 5D3/2 state, whose radiative decay time is as long

as 17.5 s [5], we achieved almost complete population

inversion after 10-15 laser pulses, which were well above

saturation intensity.

To induce the hyperfine transition, microvv^aves were

coupled into the trap by a hairpin antenna, placed

between the ring and one of the endcap electrodes. The

microwaves were created by a temperature stabilized

klystron and phase locked to the 17th harmonic of a quartz oscillator, referenced to a Rb atomic clock.

Figure 1. Setup of the experiment.

I ('3^Ba) = ^2

493,4 nm

6S 1,2

649. 6 nm

Fr 2

8 04 GHz

*Work supported by the "Deutsche Forschungsgemeinschaft. Figure 2. Relevant energy levels of Ba *.

99
Resonance on the possible AF = 1, ^m ^ 0, ±1

transitions was monitored via the change in fluorescence

intensity perpendicular to the primary light either at 493.4 nm or 649.6 nm. In spite of the low quantum

efficiency of our photomultipHer in the red (<1%) we

decided to monitor the D transition to be

independent of any laser stray light. Among the possible

transitions theF=2, m-O^F = l,m-0 line depends only in second order on the magnetic field. At

field strengths of about 100 mG inhomogeneities do not represent a significant contribution to line broadening

effects. The remaining field dependent AF = 1, Am =

0, ±1 transitions were used to determine the magnetic

field strength. The lineshape of the "field independent"

transition is dominated by the pulsed laser excitation

scheme. If T is the time between two consecutive laser

pulses, the well known theory for a two level system

predicts for the transition probability

P(oo) = sin2{r/2((co -000)2 + 72)1/2}

where to is the microwave frequency, ojo the transition

frequency, and y the transition rate, depending on the

microwave power. In our case, however, the diameter of

the ion cloud, which assumes a Gaussian density

distribution [4], is about 10 mm—larger than the laser beam diameter (4 mm). Thus there is a chance for an ion

to escape one or even more laser pulses and extend its free interaction time with the microwave field to multiples of T. Consequently the total transition

probability is given by

P(a))

(to - 0^0)"= + r

X % sinHk r/2((to - coo)^ + 72)1^}.

k =1

The constants become rapidly smaller with increasing

k. Throughout our analysis of the observed transitions we found that n - 5 was sufficient to fit all the data. Normalizing to Sa^ = 1 we found cii = 0.781, 02 =

0.156, as = 0.039, 04 = 0.016, as = 0.008. These values were held constant throughout the whole analysis. In

addition we had to add a small term, b exp{-(to - 100)2/72} to the transition probability to account for spatial variation of the microwave power over the trap volume.

Finally ion loss was assumed to be linear, since the

storage times of many hours far exceeded the averaging

time of a few minutes for one line. With this lineshape

formula we found excellent agreement with all the

observed transitions, within our limits of error. Figure 3 shows an example of a ^^'^Ba'^ line which was taken with

high spectral resolution (0.6 Hz per point) at a laser repetition rate of 16 Hz. The width of the central maximum depends linearly on the laser repetition rate. Even

at repetition rates of 1 Hz we found a FWHM of 0.87 Hz

in the "field independent" transition (Fig. 4) and no

indication of relaxation broadening.

The limitations on accuracy in our experiment were

mainly given by the applied fields and the available

equipment. We varied the magnetic field between 0.2 and

2.5 G. To obtain a decent rate of information we

operated the laser typically at 10-20 Hz, thus limiting

the spectral resolution of the transition. The extrapolation to zero magnetic field according to the Breit-RabiFormula (Fig. 5) shows an uncertainty of 0.23 Hz for

^^''Ba* and 0.51 Hz for ^^Ba"^. Variation of the electric

field changes the potential depth of the trap and accord-

+ 0 +20

i 037 742 033 Hz

+ 40 +60

Frequency (Hz)

Figure 3. F = 1, m = 0-^ F = 2, m = 0 hyperfine transition

in '"Ba^ . Laser repetition rate was 16.6 s'', averaging time

15 s per point, microwave frequency steps 0.6 Hz. The full line represents the theoretical lineshape according to the formula

given, in the text. Statistical uncertainty of the line center (one

standard deviation) is 0.07 Hz.

= 7.183.340.594,908

(0,013) Hz

0 80 160 2 40 3.20

7 183 340 594 Hz Frequency (Hz)

Figure 4. "Field independent" transition in ^^^Ba*. Laser repetition rate 1 s~\ averaging time 30 s per channel, microwave

steps 0.1 Hz.

ingly the ion temperature. This leads to a shift from the

second order Doppler effect combined with a possible

Stark shift. The contribution of the latter however is assumed to be very small. Experimentally we found

8v/v = Di-2.5 ±1.1) X 10 12, if D is the depth of the

trapping potential. Extrapolation to zero electric field results in an error of 0.27 Hz. We operated the trap at He buffer gas densities of about 10"^ mbar in order to trap a sufficient number of ions. Once the ions are trapped, the gas could have been removed. However,

since a possible pressure shift of the hyperfine transition

100
8 037.741 600 Hz

Magnetic field (10'^Tesla)

possible by more careful determination of the electric and

magnetic field dependence of the "field independent"

line. Limitations in line-Q by collisional relaxation may

be estimated from our measurements: Since at 10 mbar we obtained about v/Av = 10^*^, one may easily gain 3 or

4 orders of magnitude by operating at UHV pressures. We did not extend our measurements into that range,

since the stability of our available crystal oscillators was

too poor to obtain the required resolution.

The short term frequency stability (Allan variance

hflf) of this system operated in feedback mode as a fre- quency standard can be estimated. The optimum feed- back signal is obtained if the frequency is chopped

between the opposite points of maximum signal change of

the resonance curve. For this case we have hflf =

iS/4Q)il/SNR) where Q is the quality factor wq/So) of the

line and SNR the signal to noise ratio. Taking Fig. 4 as an example, where we have SNR = 100 at 30 s averaging time, we obtain 8///' = 5 x 10 This is in the same region as the short-term stability of commercially

available Cs standards. Technical improvements on the

signal to noise ratio are easily possible if one considers

the poor solid angle of our optics of 2% and the quantum

efficiency of our photomultiplier at 650 nm of about 1%.

In addition the successful demonstration of optical side- band cooHng [7] indicates that the main source of error, the second order Doppler effect, may be reduced by

several orders of magnitude.

Figure 5. Magnetic field dependence of the F = 1, m = 0 ^ F

= 2, m = 0 transition in ' Ba'^. The magnetic field was deter- mined by several AF = 1, Am = 0, ±1 transitions. Full

line: Breit-Rabi formula fitted to the experimental points.

according to the known data [6] is 8v/v = 6 x 10"^^ and

collisional relaxation rates between the hfs levels obviously are smaller than 1 sec"^ (corresponding to ct < 2 x

10 cm^), we maintained this pressure during our measurements, since it increased the signal to noise ratio by

colHsional deexcitation of the long living metastable Dstate. The final results for both Ba^ isotopes are hsted

in Table 1. Further improvement in accuracy is easily

References

[1] R. Blatt and G. Werth, Z. Phys. A 299, 93 (1981).

[2] H. G. Dehmelt, Adv. At. Mol. Phys. 3, .53 (1976).

[3] J. F. J. Todd, G. Lawson, and R. F. Bonner, in Quadrupole

Mass Spectrometra, Ed. by P. H. Dawson (Elsevier, Amsterdam, 1976).

[4] H. Schaaf, U. Schmeling, and G. Werth, Appl. Phys. 2.5, 249

(1981).

[5] R. Schneider and G. Werth, Z. Phys. A 293, 183 (1979).

[6] H. Ackermann et al, Phys. Lett. 44A, 515 (1973).

[7] W. Neuhauser, M. Hohenstatt, P. Toschek, and H. Dehmelt,

Appl. Phys. 17, 123 (1978).

Table 1. Ground state hfs separation.

Isotope 2A [Hz]

statistical uncert.

of line center

error of magn.

field extrapol. error of pot. depth extrapol.

total error

(Icr)

8 037 741 667 .69 <0.1 0.23 0.27 0.37

i35Ba+ 7 183 340 234 .35 <0.1 0.51 0.25 0.57

101

Precision Measurement and Fmidamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Observation of High Order Side Bands in the Spectrum of Stored He Ions*

H. A. Schuessler and H. S. Lakkaraju

Department of Physics, Texas A&M University, College Station, TX 77843

The magnetic resonance spectrum of the 1 ground state of "^He^ ions was remeasured using a radio-frequency quadrupole ion trap and the ion storage exchange collision technique. In particular the Hne shape of the = ± 1, A?n^ = ±1 hyperfme structure transition was studied. The spectrum was observed to consist not only of the component at the main transition frequency (vq =

8655.662 MHz, Hq = 7.13 G) but also of components at discrete frequencies of up to 400 kHz above and

below the central transition frequency vq. A calculation of the spectrum based on the correlation function formalism reproduced the main features of the experimental spectrum although differences are dis- cernible. The dependence of the side-band spectrum on the energy and energy distribution of stored

ions in both standing and traveling wave fields was also investigated numerically. The present investi- gation has a strong bearing on side-band cooling techniques and on improving the precision of spectroscopic measurements. Key words: Doppler effect of bound radiators; precision spectroscopic measurements; side bands; side- band cooling; stored ions.

1. Introduction

It is well-knovi^n that the absorption and emission of radiation from a moving atom is shifted by the Doppler

effect. In general the change in photon frequency is pro- portional to the velocity component of the radiating atom

along the direction of the photon wave vector. In the special case of a bound atomic system moving periodically

about a guiding center, the Doppler effect generates side

bands in the spectrum of the radiating atom. The first study [1] of these side bands was made for the spectrum

of ions stored in a radio-frequency (rf) quadrupole ion

trap, where an ensemble of ions is bound practically harmonically in three dimensions. It was shown that low

order side bands occur in the hyperfine structure (hfs)

spectrum of the ^He"^ ion at separations determined by

the various characteristic ion oscillation frequencies and

that the spacings are determined by the ion trap operation voltages. In this paper we report on the observation

of high order side bands in the same system and present

a theoretical interpretation valid within the limits of first order perturbation theory.

The occurrence of the side bands in the spectra of harmonically bound systems has significant impact on the

precision measurements of atomic and molecular spectra,

as demonstrated by the side-band cooling experiments of stored ions in the recent past [2], making it possible to improve the precision of ultrahigh resolution spectroscopy experiments by reducing the Doppler effects.

2. Experiments

We have used the ion storage exchange collision tech- nique [3] to observe the A F = ± 1 and A nif = ± 1 hfs

transition of the 1 ground state of '^He"^. The ap- paratus used in our work is illustrated in Fig. 1. ^He"^ ions are produced inside the rf quadrupole ion trap by

electron bombardment of the background helium gas and

*Supported by the National Science Foundation under Grant No. Phy

81-11943 and by the Center for Energy and Mineral Resources at Texas A&M University.

Vaclon pumps

Figure 1. Experirnental ann7igement of the ion trap apparatus

for studying motio7ial side bands in the microwave spectrum of

'^He^ ions.

are subjected to an atomic beam of cesium polarized by

optical pumping. In the ensuing collisions, the ions

quickly assume the polarization of the cesium atoms by

spin exchange and are subsequently slowly lost by charge

exchange. The rate at which the '^He"^ ions are lost is determined by the ion polarization. In this way the spin

dependent charge exchange is used to monitor the

changes in the polarization of the ^He^ ions when magnetic dipole transitions are induced between the hyperfine

structure levels of the ^He^ ground state. Magnetic di- pole transitions are then detected by counting the

number of ions left in the trap after a fixed interaction

103
time with the cesium atomic beam. The ion trap struc- ture is not only used to store the ions but also as a mi- crowave cavity. The microwave field required for the

magnetic resonance is coupled into the trap by a X/2 slot antenna cut into one of the end cap electrodes. The mi- crowave field inside the trap cavity is similar to that of a TE013 mode of an equivalent cylindrical cavity whose ef- fective dimensions are slightly larger than the trap di- mensions. In the weak external magnetic field of 7.13 G

produced by a pair of Helmholtz coils, the transition fre- quency is measured at 8655.662 MHz.

In Fig. 2 a typical experimental spectrum is shown. The

spectrum is a point-by-point plot of the data obtained by

varying the frequency in discrete steps. As is evident

from the data, the spectrum consists of a central peak at the main transition frequency (vq = 8655.662 MHz) and of a series of side peaks spaced at separations of the various

characteristic ion oscillations frequencies. Only the central peak and the components at frequencies higher than

vo are shown in the figure. The additional components in the spectrum are side bands and are the manifestation of the first order Doppler effect in harmonically bound physical systems [4].

OJ <N CJ

\D ID 'D

ID r~- 00

00 00 (0 FREQUENCY (MHZ)

Figure 2. Experimental microwave spectrum of the stored

'^He'^ ions (^F = ±1, Amy = ±1 transition of 1 ^Sy^ ground

state).

3. Theory

In order to quantitatively account for the side-band

spectrum of the oscillating ensemble of ions, a calculation

using the correlation function formalism [5] was made. In

this calculation the power spectrum, or alternately the

transition rate, associated with the frequency wo =

{E m - EQ/h is proportional to the Fourier transform of the correlation function

P (oj, o),,) /__^ G,„^. (t) e d T (1) where E,„ and E^ are the energies of the levels involved

and G„iki'^) is the correlation function defined as the ensemble average of the matrix elements of the perturbation Hamiltonian in the stationary perturbation approximation given by

G,„k (t) = <X,^ (i)3f;tm it')> (2) for the case of no or weak saturation. _ The matrix ele- ments of the perturbation Hamiltonian I'tC involve the mi- crowave field inducing the transition. The microwave

field in turn implicitly is a function of the motion of the

ions. The motion of the ions in a rf quadrupole trap is determined by the operating parameters of the trap and is described by a Mathieu type differential equation in both

the axial and radial directions. The motion of the ions

consists of the large amplitude macromotion at frequencies (x)r and CO; with the small amplitude micromotion at the driving frequency CL superimposed on the macromotion. In the harmonic approximation used in the following, only the dominant mode of oscillation at the macro- motion is considered. The motion of the ions is then

described by

z = ^sin(a)j^ -I- (});) and r = psin(a)r< + c}),.) . (3) The amplitudes ^ and p together with the phase angles 4); and <^r determine the exact trajectory that a particular

ion assumes in the trap. With the motion of the ions in the trap given by Eq. (3) and the microwave field ap- propriate for the TE013 mode of a cylindrical cavity, the

calculation of the intensities of the side bands in the

spectrum of stored ions proceeds in two steps. First an

expression for G„,a(t) is derived and next the Fourier

transform of G,„ki'^) is formed which yields the relative

intensities of the side bands in the spectrum. The power

spectrum thus calculated is obtained as

P(co,a3o) =P(coo)//F i I"TM"/-TM''

III. II =0

p dp dif ip, i)J,f,ik,p/2)Jl{ksO

X [8(a) - (u)o + |(±2TOaj, ±2>;w,) |))

+ 8(co - (0)0+ I + 2m o),. ± 2n ) | ))]

TT ,9 r' "lax c

- max , , ^ + X Jo L p^p^^

k.l =0 " max

X J^Hk 1 p/2)J,K, ik 1 p/2) /|, ^1 (k -iOf (p. O

X [5(0) - (0)0+ |( + (2A: +1)0), ± (2/ +1)0);) |))

+ 8(o) - (o)o+ |(^(2A: +l)o), ± (2/ +l)oj;) | ))] (4) where the constant P (o)o) contains all other constants,

and /(p, ^) is the ion oscillation amplitude distribution

function. In Eq. (4) ki and A: 3 are determined by the

boundary conditions of the standing wave field in the

equivalent cavity with /ci = xoi the first zero of J i(x ), Tc being the radial dimension of the cavity, and k^Zc =

3 tt/2, where Zc is the axial dimension of the equivalent

cavity. Also H[ - 1.73 i/i is the relationship between

the field amplitudes of the radial and axial microwave

fields. The above expression gives the spectrum on the

high frequency side of the central line. The actual spec- trum is symmetric about the central line. The power spectrum given by Eq. (4) not only contains

104
the central line at w = wq but also contributions at the

various multiples of o),- and . For a given side band the

contribution from w,. and w, is such that either both even

or both odd order side bands contribute. There are no

side bands for which an odd harmonic of w,. appears with an even harmonic co, or vice versa. This feature is a consequence of the spatial symmetry of the microwave

field in the trap. Furthermore, it is seen that the pair of integers m and n, with specified values, gives rise to more than one spectral component, with the same intensity. For instance, two side bands are generated from the

combination of the frequencies 2 co,. and 2 o), , one at (wq +

' 2 w,. + 2 o),) and the other at (wq + 2 - 2 to, ). No relaxation process is present in the calculation and

this leads to infinitely sharp lines represented by the

delta functions. However, the observed line shapes have

a finite width. For a comparison between the calculated

spectrum and the observed spectrum, the delta functions are later replaced by Lorentzian functions with an empirically determined value of the linewidth parameter. Fig.

3 shows calculated spectra for a uniform distribution and

a Boltzmann distribution with an equilibrium temperature kT^ = 0.43 eV.

for the other geometries. The calculation of the spectrum

for the traveling wave field was also carried out and the

power spectrum for this case is PtK 0)0) = Pt(coo)//,2j, ^ f ''^'jl"^%dpdif(p, 0

111 =0 " iTM-'<

X [J,f;{kOU^ - (m + wwj)] (5)

where m is an integer, H u is the amplitude and k =

2 -7t/\ the wave number of the traveling microwave field. In order to match the conditions of the experiment it is assumed in this spectrum that the wave travels in the z direction with a plane wavefront. In this case the radial motion does not contribute and the side bands generated

are due to the z motion only. Figure 4 displays the spectra calculated from Eq. (5) for a uniform distribution and

a Boltzmann distribution with an equilibrium temperature of kTg = 0.43 eV. The operating parameters of the

experiment are shown in Table 1, and Table 2 summarizes the side-band intensities computed for a uniform en- ergy distribution and a Boltzmann distribution for dif- ferent equilibrium temperatures.

— 10-

>-

q:<

cr. H

CD<

COz

UJ

10-

10-3

lO-i

10-2

10-3

3 - 3 ^

I 3 CM 3

I 3 3 o

.84x 10

10-5

(a)

(b)

0 100 200

FREQUENCY (kHz)

>-

I r

if) t 10-

tr<

ir5

10" <

lO-i

IQ-

(a)

J I I_LJ I I I I I I

(b)

J I I I 1_LJ I 1 1 I I I 0 100 200

FREQUENCY (kHz)

Figure 3. Side-band spectrum calculated for a pure standmg

wave field in the ion trap cavity arid no relaxatiori processes,

(a) Uniform ion energy distribution, (b) Boltzmann distribution of ion energies for an equivalent temperature kT^ =

O.US eV.

Figure 4. Side-band spectrum calculated for a pure travelirig wave field in the ion trap cavity (propagation along the z direction) and no relaxation processes, (a) Uniform ion energy dis- tribution, (b) Boltzmann distribution of ion energies for an

equivaleyit temperature kTg = O.US eV.

So far the case of a standing microwave field in the ion

trap has been discussed for the particular example of the

TEoi3 mode of interest in the present experiment. Other

field geometries can be handled in a similar way by re- placing the microwave field by the corresponding ones

4. Results

A comparison of Fig. 3 and Fig. 4 clearly indicates that

the spectrum due to a standing wave excitation field is richer in side bands than that due to the traveling wave.

105
Table 1. Operating parameters.

Ti'ap axial dimension ^0 = 2.55 cm

Trap radial dimension = 3.60 cm RF trapping frequency n = 2 -IT X lOS-ad/sec

Axial oscillation frequency to. = 2 TT X 110 X 10^ rad/sec

Radial oscillation frequency w,. = 2 -IT X 57 X 103 rad/sec

RF trapping voltage ^0 = 127 V (Peak)

DC trapping voltage Uo = 0.24 V

Axial well depth eD, = 5.05 eV

Radial well depth eD,. = 2.35 eV

Axial dimension of the equivalent 2, = 2.95 cm

microwave cavity Radial dimension of the equivalent = 4.16 cm

microwave cavity

Microwave power coupled into trap P = 300 mW

Number of trapped ions n = 10^

Table 2. Relative intensities of the sidebands.

Side band: /„,„ Relative Intensity

)n CD,, Uniform Distribution Boltzmann Distribution

}) ^ CO, = lO'' K = 5 X 10^ K = 3 X 10^ K = IK

Standitig Wave Field:

/oo 2.73 X 10-2 1 43 x iQ-i 1.84 x 10"! 2.29 x 10"! 2.70 x IQ-^ III 1-14 X 10-2 g 00 X 10-3 2.42 x 10-^ 1.53 x 10-5 1.75 x lO-^^ 1 20 5.32 x 10-3 X 10-3 ^ x iQ-s 6.70 x 10-*^ 7.74 x 10-"

Iq2 9.79 X 10-3 2.66 x 10-3 9.87 x 10-^ 5.00 x IQ-^ 6.58 x 10-"

Traveling Wave Field in the z Direction: Iq 4.69 x 10-2 1.19 x 10-1 1 31 ^ iq-i ^ 57 x lO-i 1.59 x 10 ^

1 1 2.26 x 10-2 2.14 X 10-2 ^ 35 ^ iq-2 i qq x 10-3 3 70 x 10-6 12 1.73 X 10-2 4.13 X 10-4 X 10-4 5 57 x iQ-e 3.99 x lO^ii /g 1.08 X 10-2 3 37 X 10-4 6.84 x 10-^ 2.23 x IQ-^ 8.32 x IQ-^*^

The origin of tliis is the fact that in the rest frame of the

oscillating ions a traveling wave appears to be frequency

modulated while the standing wave appears to be fre- quency as well as amplitude modulated. A comparison

between the experimental (solid line) spectrum and a cal- culated spectrum (dashed line) is made in Fig. 5. The

dashed curve represents the sum of the calculated spectra, with 95% contribution from the standing wave field and 5% contribution from the traveling wave field, for the case of a uniform ion energy distribution. The calcu- lated spectrum reproduces the observed spectrum rea- sonably well as far as the gross structure is concerned,

although several differences are noticeable. It is evident

that the widths of the experimental lines become larger

for the higher order side bands, and the experimental

spectrum appears to have more structure. This can be

understood by noting that the number of side bands

which are present depends on the spatial symmetry of the exciting radiation. In our calculation it was assumed

that the field geometry is that of a TEoi.s mode, which

yielded a particular combination of either even or odd

harmonics. In view of the off axis coupling of the mi- crowave field into the trap and the low Q of the trap employed, departures from the axial symmetry of the TEom

mode are expected. This departure will result in additional combinations of the harmonics. For example, odd

—• EXPERIMENTAL -— CALCULATED

lOOr ,

50

C

Figure 5. Comparison of experimental (solid curve) and calculated (dashed curve) spectra.

harmonics in the radial direction may weakly couple to the even harmonics in the axial direction and vice versa. In this case the observed spectrum is richer in components. In addition, side bands [6] generated by the cy- clotron motion of the ions about the static magnetic field occur. As a consequence, each side band contains three

106
components, namely, an unshifted one and two components on either side at frequencies of ±00^ , where co^ is the cyclotron motion frequency. In the present experiment the magnetic field dependent side bands have a separation coc =3.6 kHz and are not resolved. Therefore

the magnetic field dependent side bands overlap with the

ion motional side bands and contribute to the width of each line. Assuming that the purely power broadened

part of the line width is 4 kHz and that the height of the

magnetic side bands is about 40% of the central line, a composite line width of about 10 kHz results, in agree- ment with our measurements. Any residual anharmonicity in the trap will also lead to a broadening of the side

bands, particularly of the ones of high order, which are produced when hot ions sample the whole trap.

5. Conclusion

The line shape of transitions on bound radiating sys- tems exhibits Doppler-effect-generated side bands which

were observed in high order in the microwave spectrun^i of stored '^He^ ions. A theoretical interpretation was

developed using the correlation function formalism and is valid in the case of weak and intermediate fields for which the transition of interest is not saturated.

References

[1] H. A. Schuessler, Appl. Phys. Lett. 18, 117 (1971).

[2] D. J. Wineland, R. E. Drullinger, and F. L. Walls, Phys. Rev. Lett. 40, 1639 (1978): W. Newhauser, M. Hohenstatt,

P. Toschek, and H. G. Dehmelt, Phys. Rev. Lett. 41, 2.33 (1978).

[3] H. A. Schuessler, E. N. Fortson, and H. G. Dehmelt, Phys. Rev. 187, 5 (1969).

[4] R. H. Dicke, Phys. Rev. 89, 472 (1952).

[5] C. P. Slichter, Principles of Magnetic Resonance (Harper and

Row, New York, 1963) p. 127; J. H. Van Vleck and D. L. Huber, Rev. Mod. Phys. 49, 939 (1977).

[6] H. A. Schuessler, Bull. Am. Phys. Soc. 18, 532 (1971).

107

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Quantum Limits in the Measurements of e.m. Fields and Frequency

V. B. Braginsky

Department of Physics, Moscow State University, Moscow 117234 U.S.S.R.

The quantum limit on the frequency stability of an electromagnetic (e.m.) autooscillator may be

written in the following form (see Refs. [1, 2] for details):

where Aw^/w^, is the random relative deviation of the frequency during an interval t, % is Planck's constant divided by 2-n-, Y is the Young's modulus of the material from which the e.m. resonator is made,

and V is the volume of the resonator. Equation (1) is valid (i) if the power W of the e.m. autooscillator

is optimal:

where Qg is the quality factor of the e.m. resonator in the autooscillator; (ii) if kT < 'huig ; and (iii) if the thermal relaxation time of the e.m. resonator is less than Qp/wg. A special case of Eqs. (1) and (2) is the well-known Schawlow-Tovraes formula for the quantum limit on frequency stability. Substituting in Eq. (1) F = 4 X 10^^ dyn/cm^ (this is the Young's modulus of sapphire), V = 10 cm^, t = 10 sec, we obtain A(j)g/wg = 1 X 10~^^ This value is five orders smaller than the smallest Aw^/wg currently ob- tained by J. Turneaure [3] at Stanford University.

From a technical point of view one of the most serious obstacles to reaching the level of instability

Aojg/wg — 10 to 10"-' is the factor (l/ojreg )5a)res/5T'—i.e., the dependence of eigen frequency w^es of an e.m. resonator on the temperature. For example a niobium microwave resonator at T = 2 K has

(ly cores ) 3tjjj.es = 4 X lO^^/K due to a strong dependence of the superconductive impedance on temperature [4]; to reach the level Awg/wp = 10"^^ it is necessary to have AT < 1 x 10"^ K. On the

other hand, if it were possible to use only a dielectrical material to make a resonator, then the factor

(l/wres ) 5a)res/5T' would be approximately equal to the thermal expansion coefficient a^. The measure- ments of ay for sapphire [5] have shown that aj- = 5 x 10"^^ at J" = 2 K; in other words this factor is three orders less than for the niobium resonator. It is possible to make a microwave resonator with

high Qg using rings or disks of sapphire exciting modes with complete internal reflection. Simple cal- culations [6] show that the conditions on the shape of the resonator and its impurities are not so severe

as to require great efforts to achieve Qg — 10^. The losses in this dielectric are small (tan 8 < 1.5 x

10~^; see Ref. [4]). Preliminary measurements of Qp in a sapphire disk with a radius 5 cm and thick- ness 2 cm at a frequency 9 x 10^ Hz have shown this approach to be promising: at 7" = 300 K we ob- tained Qe = 2 X 10^, at T = 77 K, = 10^ and at T = 4 K the value > 3.5 x \{f> (see Ref. [7] for

details).

If an experimentalist wants to measure the values of electric field E or magnetic field B in an

e.m. resonator during a short time t, then quantum limitations become important. For example if one makes simultaneous measurements of E or B in the bandwidth A co,, = w,, during t = l/u), then

the smallest error for E and B is

where is the volume of the resonator. The perturbations of E and B due to this measurement are described by the same Eq. (3) (see review articles, Refs. [81 and [9]). In principle it is possible to realize such procedures that beat the hmit given in Eq. (3). These procedures are described in Refs.

[8, 9, 10, 11]. As an example of such measurements (which now are called quantum-non-demolition

measurements) we discuss here a procedure described in detail in Ref. |11|. Suppose that an optical resonator (which has the form of a closed ring of optical fiber) is located in the capacity part of a Klystron-type microwave resonator. Suppose also that the volume of the optical resonator and the volume of the capacitor are approximately equal. If the dielectric constant e = e„ (1 + aE-), then if there are some optical quanta in the optical resonator, one can register a shift of the eigen frequency

of the microwave resonator. If the Q is high enough and the level of losses in the fiber and in the microwave cavity are low enough, it is possible with this method to count optical quanta without

absorption. In this measurement the number and phase uncertainties must satisfy A n A(|) 1/2, but

the value A n may be much less than unity and the phase will be seriously perturbed.

(1)

optimal (2)

References

[1] V. B. Braginsky and S. P. Vyatchanin, Sov. Phys. JETP 74, 828 (1978).

109
[2] V. B. Braginsky, S. P. Vyatchanin, and V. Z. Panov, Sov. Phys. Dokl. 247, 583 (1979).

[3] J. Turneaure, private communication.

[4] V. B. Braginsky and V. Z. Panov, IEEE Trans. Magn. MAG-17, 955 (1981).

[5] V. B. Braginsky, S. I. Vasiliev, and V. I. Panov, JETP Lett. 6, 665 (1980).

[6] V. B. Braginsky and S. P. Vyatchanin, Sov. Phys. Dokl. 252, 584 (1980).

[7] V. B. Braginsky, H. S. Bagdasarov, V. I. Panov, and A. V. Timaschov, JETP Lett. 7, 10 (1981).

[8] C. M. Caves, K. S. Thorne, R. W. P. Drever, V. D. Sandberg, and M. Zimmermann, Rev. Mod.

Phys. 52, 341 (1980).

[9] V. B. Braginsky, Y. L Vorontsov, and K. S. Thorne, Science 209, 547 (1980).

[10] V B. Braginsky and F. Y. Khalili, Sov. Phys. JETP 78, 1712 (1980).

[11] V. B. Braginsky and S. P. Vyatchanin, "On the Quantum-Non-Demolition Measurement of the Energy of Optical Quanta", Preprint No. 9/1981, Physical Department, Moscow State University,

1981.

110
Precision Measuremeyit and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984). SPECTROSCOPY, QUANTUM ELECTRODYNAMICS,

AND ELEMENTARY PARTICLES

Precision Laser Spectroscopy*

T. W. Hansch

Department of Physics, Stanford University, Stanford, CA 94305

Precision laser spectroscopy of atomic hydrogen and deuterium wall be reviewed. The Balmer-a

line has been studied by Doppler-free saturated absorption spectroscopy, polarization spectroscopy,

optical-radiofrequency double quantum spectroscopy, and by laser-quenching of a beam of metastable

atoms. These experiments have led to an eightyfold improvement in the accuracy of the Rydberg constant. Two-photon spectroscopy of the 1S-2S transition has made possible an accurate measurement of the ground state Lamb shift, and further advances in resolution promise new stringent tests of quan- tum electrodynamic theory.

Key words: fundamental constants; high resolution spectroscopy; hydrogen; lasers; precision measurements; quantum electrodynamics.

1. Introduction

Precision laser spectroscopy of simple quantum

mechanical systems can be a powerful tool to determine

better values of fundamental constants and to probe the

limits of quantum electrodynamic theory. At present,

only the hydrogen atom and its isotopes, the heavier

hydrogen-like ions, and certain hydrogen-like exotic

atoms permit sufficiently accurate model calculations for such an approach. Future theoretical advances should

make it possible to add three body systems such as the

neutral helium atom or the hydrogen molecular ion to the

list of promising candidates.

Of these candidates, only hydrogen and deuterium have

so far been studied extensively by precision laser spec- troscopy [1], and we will limit our discussions to the

work on these two simplest of the stable atoms. We will review the experiments reported to date, and we will investigate some of the possible directions for future

progress.

2. Precision Laser Spectroscopy of the

Hydrogen Balmer-a Line and the Rydberg

Constant

The simple and regular Balmer spectrum of atomic hydrogen can be readily observed in the light emitted by a glow discharge. The exploration of this spectrum has

played a crucial role in the development of atomic physics

and quantum mechanics [1]. More than once, seemingly

minute discrepancies between experiment and theory

have led to major revolutions in our understanding of

*Work supported by the National Science Foundation under Grant PHY80-10689 and by the U.S. Office of Naval Re.search, contract ONR

N00014-78-C-0403.

quantum physics. However, no classical spectroscopic ob- servation has ever succeeded in fully resolving the intri- cate fine structure of the Balmer lines. The spectra al- ways remained blurred by Doppler broadening due to the

rapid thermal motion of the light hydrogen atoms.

2.1 Saturation Spectroscopy

Dramatic progress in spectral resolution became possi- ble only with the advent of highly monochromatic tunable

dye lasers together with techniques of Doppler-free laser spectroscopy. The prominent red Balmer line was the

first hydrogen line to be studied by Doppler-free satu- rated absorption spectroscopy [2]. In this technique, two

monochromatic laser beams are sent in opposite directions through the absorbing gas sample. When the laser

is tuned to the center of a Doppler broadened line both

beams can interact with the same atoms, those with zero

axial velocity. The signal is observed as a bleaching of the absorption of the probe beam, caused by the saturat- ing beam.

Even though only a relatively simple pulsed dye laser was available in our initial experiments at Stanford, we

were thrilled by the spectra that could be recorded in a simple Wood type gas discharge. As illustrated in Fig.

1, we were able to resolve single fine structure components of the Balmer-a line, and the n = 2 Lamb shift could be observed directly in the optical spectrum. In

1974, Munir Nayfeh at Stanford [3] completed an absolute wavelength measurement of the strong 2P3 2 - 3D5 2 component of hydrogen and deuterium which yielded an

eightfold improved value of the Rydberg constant, com- pared to the value recommended in the 1973 adjustment

of the fundamental constants [4]. Since then, frequency stabilized cw dye lasers, whose

resolution is not limited by pulse length, have become

111
SPECTRUM OF HYDROGEN

0 500000

I 1 1 I

I I GHz BALMER

SERIES

Ha

DOPPLER

PROFILE

{300°K

SATURATION

SPECTRUM

X 40 000

-| 1-LAMB SHIFT

~l ' 10 GHz

FiGURE 1. Top: Balmer spectrum of atomic hydrogen. Center: Doppler profile of the red Bahner-a line at room te))iperature

and theoretical fine strncture co)npo}ients. Botto)n: Dopplerfree spectrum of Balmer-a, recorded by saturated absorption

spectroscopy with a pulsed dye laser jj, .y|.

readily available. Moreover, new techniques of saturation spectroscopy have been developed, which can reach

shot-noise limited sensitivity despite laser intensity fluc- tuations [5, 6]. In polarization spectroscopy [5], the probe

beam monitors the dicroism and birefringence of the sample induced by a circularly or linearly polarized saturat- ing beam. The technique takes advantage of the fact that

small changes in light polarization can be detected with

higher sensitivity than changes in intensity. Because of

its higher sensitivity, the method permits measurements

at lower atom densities and lower laser intensities, so that pressure broadening, power broadening, and related

problems can be much reduced.

In 1978, J. E. M. Goldsmith et al. at Stanford [7] completed a new measurement of the Rydberg constant,

observing the hydrogen Balmer-a line in a mild heliumhydrogen discharge by polarization spectroscopy with a cw dye laser. As shown in Fig. 2, the resolution of the

weak but narrow 2Sy2 - 2Fy2 fine structure component was at least 5 times better than in the earlier pulsed ex- periments. The absolute wavelength of this component was measured relative to the hyperfine component of the ^2^l2 B - XR(127) 11-5 transition at 632.8 nm, us- ing a near-coincident h line (the i hyperfine component

of the ^^'^12 B - X R(73) 5-5 transition) as an intermediate reference [8]. Systematic line shifts due to the discharge plasma were

studied very carefully in a series of measurements which

yielded several results interesting in their own right

[9, 10]. For instance, anomalous pressure shifts have

been observed which could be explained in terms of colli- sional decoupling of the 3P hyperfine structure [9]. The

final evaluation of all measurements gave a threefold improvement in the accuracy of the Rydberg value. Another twofold improvement could be obtained immediately with a more accurate absolute wavelength measure- ment of the intermediate iodine reference line.

LASER FREQUENCY DETUNING (GHz) V

0 2 4 6 8 10

2S,/2(F-I)-3P,,3 2S,,2(F=0)-3P,,2

0 100 200 300 400

LASER FREQUENCY DETUNING (MHz) 1/

Figure 2. Doppler-free polarization spectrum of the Balmer-a

line, recorded with a cw dye laser in a mild He-H^ discharge

[7].

The two Stanford measurements, as originally published, used values for the iodine reference wavelength

(^°Ne: ^^^l2, i) that differed slightly from the rounded

value of 632.991 399 nm recommended by the Committee

for the Definition of the Meter [11]. The results in Table 1 and Fig. 3 have been adjusted to be consistent

with this rounded value. Specifically, the 1974 Rydberg

value has been adjusted downwards by 0.0024 ppm, and

the 1978 value has been adjusted upwards by

0.0013 ppm. The 1974 result is a composite of two some- what differing Rydberg values for deuterium and hydrogen. The adjusted 1974 deuterium value, Ro, - 109

737.314 8 ± 0.001 0 cm"^ is in excellent agreement with

the 1978 measurement. The adjusted 1974 hydrogen

value, i^oc = 109 737.312 8 + 0.001 0 cm-\ is almost

0.02 ppm smaller. However, the latter includes substantial systematic corrections for the larger 2P3y2 hyperfine

splitting of the light isotope, and the uncertainty of these

corrections in the presence of collisions may have been

underestimated in 1974.

THE RYDBERG CONSTANT

1973 COHEN a TAYLOR

I •—

I

-J I I L _L

.310 .315 .320

(Rco - 109 737 cm-') —

1974 HANSCH ET AL.

1978 GOLDSMITH ET AL.

1979 MORRIS a PETLEY

1981 AMIN ET AL.

_l ! I

I I I I I .325

Figure 3. Recent measurements of the Rydberg constant.

112
Table 1 and Fig. 3 also give the results of an independent measurement of the Rydberg constant, reported in 1980 by B. W. Petley et al. [12]. The Balmer-a line of hydrogen was observed in a Wood type gas discharge by

saturated absorption spectroscopy with a cw dye laser. The result is in good agreement with the earlier Stanford

values.

Table 1. Measurements of the Rydberg Constant

Year Authors

73 COHEN, TAYLOR [6] 109 737.317 70 ± 0.008 30

74 HANSCH

NAYFEH, LEE,

CURRY,

SHAHIN [5]

109 737.314 10 ± 0.001 00

78 GOLDSMITH,

WEBER,

HANSCH [10]

109 737.314 90 ± 0.000 32

79 PETLEY,

MORRIS,

SHAWYER [15]

109 737.315 13 ± 0.000 85

81 AMIN,

CALDWELL,

LICHTEN [16]

109 737.315 21 ± 0.000 11

2.2 Laser Spectroscopy of an Atomic Beam

The accuracy of the best measurements in hydrogen

discharges [7] does not appear to be limited by pressure

shifts or Stark effect in the discharge plasma. Nonetheless, it has long been obvious that a collisionless beam of metastable hydrogen atoms would be a more ideal sample

for precision laser spectroscopy [3]. S. R. Amin et al. of Yale University are the first who

have succeeded with such an atomic beam experiment,

and they report on a new Rydberg measurement else- where in these proceedings [13]. The Balmer-a line of hydrogen and deuterium is observed by exciting the

metastable 2S (F = 1) atoms with cw dye laser beams

which cross the atomic beam at a right angle. IMost of the

excited 3P atoms quickly decay into the IS ground state, and the resulting quenching of the 2S state can be ob- served with a detector for metastable atoms, just as in the

classical Lamb-Retherford experiment [14]. Such linear

atomic beam spectroscopy requires fewer systematic

corrections than nonlinear saturation spectroscopy, and a wavelength measurement by direct comparison with an

iodine stabilized He-Ne laser has yielded a new Rydberg

value accurate to one part in 10'', as shown in Table 1 and

Fig. 3.

2.3 Two-Photon Spectroscopy

Although the accuracy of linear laser spectroscopy of a metastable hydrogen beam appears amenable to further

improvements, the resolution of the narrowest Balmer-a

components will always be limited by the short lifetime of the upper 3P state to no better than 29 MHz.

D. E. Roberts and E. N. Fortson [15] were the first to point out that narrower lines can be obtained if an additional radiofrequency field is applied so that radiofrequency optical double quantum transitions are induced

from the 2Si/2 level to the longer living 3Si/2 and 3Dy^

level. E. W. Weber and J. E. M. Goldsmith [16] have ob- served lines as narrow as 20 MHz by applying this tech- nique to hydrogen atoms in a gas discharge, and they

have been able to measure the small 3P3/2 - 3D3/2 Lamb

shift directly by comparing single- and double-quantum

signals. C. E. Wieman and collaborators [17] have re- cently begun to apply the same technique to a beam of metastable hydrogen atoms, and they expect to reach a

resolution better than 1 MHz, corresponding to the

natural width of the 3S level. The same narrow lines could also be observed by exci- tation with two laser photons of equal frequency. If the

two photons come from opposite directions, first order

Doppler broadening is automatically eliminated without

any need to select slow atoms, because from a moving

atom the two photons have equal but opposite Doppler

shifts, so that their sum-frequency is constant [18]. Doppler-free two-photon spectroscopy of the Balmer-a

transition has so far been stifled by the lack of suitable

highly monochromatic tunable lasers in the near infrared.

But visible dye lasers should make it possible to study

transitions from 2S to high Rydberg levels by this technique.

By comparing the wavelengths of such transitions with

that of the Balmer-a line, one might detect, for instance, some small deviations from Coulomb's law, which may

exist within atomic dimensions, but which may have es- caped detection in the past.

3. Two-Photon Spectroscopy of Hydrogen

1S-2S

There is another, even more intriguing transition in hydrogen which can be studied by Doppler-free two- photon spectroscopy: the transition from the IS ground

state to the metastable 2S state. The 1/7 sec lifetime of the upper level implies an ultimate natural linevddth as narrow as 1 Hz. There is no intermediate near resonant level which

would enhance the two-photon transition rate. However,

even small numbers of excited 2S hydrogen atoms can be

detected with high sensitivity, by monitoring the vacuum

ultraviolet Lyman-a radiation emitted after conversion to the 2P state by collisions or external fields, or by photoionizing the 2S atoms and observing charged particles.

Unfortunately, however, two-photon excitation of 1S-2S requires monochromatic ultraviolet radiation near 243 nm, where there are still no good tunable laser sources available. Intense coherent radiation at this wavelength can be generated by frequency doubling of a pulsed dye laser in a nonlinear optical crystal, and 1S-2S

two-photon spectra have been observed at Stanford with

such sources [19-21]. But the resolution remained limited

by the bandwidth of the pulsed lasers.

3.1 Spectroscopy with Pulsed Lasers

The best 1S-2S spectra so far have been recorded by

C. E. Wieman, [21] who reached a resolution of 120 MHz (FWHM at 243 nm) with the help of a blue

single-mode cw dye laser oscillator with nitrogen-pumped

113
pulsed dye laser amplifier chain and lithium niobate fre- quency doubler. The hydrogen atoms were generated in a Wood type discharge tube and carried by gas flow and

diffusion into the observation chamber, where they were

excited by two counterpropagating beams from the laser system. The emitted vacuum ultraviolet Lyman-a photons were observed through a magnesium fluoride side

window by a solar blind photomultiplier. The spectral

resolution was sufficient to resolve the hyperfine doublets (Fig. 4), although another 100 million-fold improvement should ultimately be possible. But even these crude

spectra permitted a measurement of the 611 GHz H-D

isotope shift to within 6.3 MHz and provided a first qualitative confirmation of the predicted small 11.9 MHz rel- ativistic correction due to nuclear recoil [21].

BOHR DIRAC QED

H 1S-2S D 1S-2S

0.2 0.4 0.6 168.0 DYE LASER FREQUENCY TUNING —

168.2 GHz

Figure 4. Duppler-free two-photon spectrum of the 1S-2S transition in hydrogen and deuterium vnth resolved hyperfine

splittings [21].

In the same series of experiments [19-21], the 1S-2S

energy interval was compared with the w = 2 - 4 interval, by simultaneously observing the Balmer-p Une with

the visible dye laser output. If the simple Bohr theory were correct, the n =1-2 would be exactly four times

the w = 2 - 4 interval, and both transitions would be

observed at exactly the same laser frequency. In reality, this degeneracy is lifted by relativistic and quantum elec- trodynamic corrections, and we expect line splittings and

displacements as illustrated in Fig. 5. By measuring the

separation of the 1S-2S resonance from one of the Bal- mer components, one can determine an experimental

value of the Lamb shift of the IS ground state, which

cannot be measured by radiofrequency techniques,

because there is no IP state which could serve as a

reference.

The predicted 8149.43 ± 0.08 MHz Lamb shift of the

hydrogen IS state has been confirmed within 0.4%, by

comparing the 1S-2S spectrum with a polarization spec- trum of the Balmer-p line, observed with the cw dye

laser output in a Wood type gas discharge [21]. The uncertainty in this experiment was dominated by laser fre- quency shifts due to rapid refractive index changes in the

pulsed dye ampHfiers. Such chirping introduces unknown

phase parameters into the calculation of the line-shape of the two-photon signals. Pressure shifts of the Balmer-(3

spectrum were the next largest source of error. Both problems can be overcome with mere technical

improvements, and the intrinsic narrow natural hnewidths of the observed transitions make it appear likely

that measurements of the IS Lamb shift will eventually

reach a higher accuracy than radiofrequency measure- ments of the 2S Lamb shift [22]. The latter provide one

of the most stringent current tests of quantum electrody-

/S4D,F5,2

2P3/2

IS-2S TWO -PHOTON

EXCITATION WITH

SECOND HARMONIC

FREQUENCY

IS,,,

RESONANCE

(c)

PHOTON 1

1[—IS LAMB SHIFT

X 1/4

. 1 -5 0 5 GHz LASER FREQUENCY DETUNING—^

Figure 5. Top: Simplified diagram of hydrogen energy levels and transitions. The Dirac fine structure and QED correc- tions are shoum on an enlarged scale; hyperfine structure has

been ignored. Bottom: Fine structure spectrum of the Bahner-^

line and relative position of the 1S-2S two-photon resonance,

as recorded -with the second harmoyiic frequency. The dashed

line gives the hypothetical position of the 1S-2S resonance if there were no IS lamb shift.

namics, but they are plagued by a 100 MHz natural

linewidth due to the short lifetime of the 2P state. Persisting discrepancies between experiment and the predictions of different computational approaches [23, 24]

make further accurate Lamb shift measurements highly

desirable.

3.2 Current Efforts Towards Higher Resolution

In order to avoid the limitations of a pulsed laser source, A. I. Ferguson, J. E. M. Goldsmith, B. Couillaud, A. Siegel, J. E. Lawler and other collaborators at Stanford have invested considerable effort into an experiment designed to observe the hydrogen 1S-2S

two-photon transition with low power cw ultraviolet radiation. While known nonlinear optical crystals do not per- mit efficient 90 degrees phase matched second harmonic

generation down to 243 nm, cw ultraviolet radiation at

this wavelength can be produced as the sum frequency of a violet krypton ion laser and a yellow rhodamine 6G dye

laser in a crystal of ammonium dihydrogen phosphate

(ADP), cooled close to liquid nitrogen temperature [25]. With frequency stabilized single frequency lasers of 0.6 and 2.5 watts power, respectively, focused to a waist di- ameter of 100 |xm inside a 5 cm long ADP crystal, about

700 |JL W of tunable cw ultraviolet power have been pro- duced. A frequency locked external passive enhancement

cavity increases this power to several mW at the sample.

Such a power should be sufficient for a resonant signal of several hundred Lyman-a photons per second under the

chosen experimental conditions.

114
Unfortunately, however, the ADP crystal is damaged

within less than a minute under these conditions. We

speculate that the ultraviolet light produces color centers, perhaps associated with heavy ion impurities,

and that these centers then absorb enough of the intense

primary beams to damage the crystal. We are presently

investigating whether mixing of more widely separated

primary wavelengths in ADP near room temperature can

provide a cure for this problem, and we are actively pur- suing alternative approaches. Once the crystal damage

problem is solved, it should be possible to reduce the

bandwidth of such a cw ultraviolet source with the help

of fast servo controls to a few kHz.

However, other causes of line broadening, in particular

pressure broadening, transit broadening, and transverse Doppler broadening have to be overcome before such a resolution can be approached in the 1S-2S two-photon

spectrum [25]. A beam of ground state hydrogen atoms,

cooled close to liquid helium temperature [26], and in- teracting with nearly collinear counterpropagating laser beams would minimize such problems, and it appears

technically quite feasible to observe the 1S-2S two- photon transition with a line width of a few- tens of kHz,

or a resolution approaching one part in 10^^ The line center could then be determined to within 1 part in 10^^ or better, once accurate frequency standards become

available in the visible and ultraviolet region.

Such a precise measurement of the 1S-2S frequency

could, of course, be used to determine a still better value

of the Rydberg constant. However, the current uncertainty of the electron /proton mass ratio [27] (about

0.14 ppm) limits the accuracy of such a Rydberg value to about 1 part in 10^°. Considerable improvements of direct

measurements of the electron/proton mass ratio have

been predicted [27]. Alternatively, a better mass ratio could be determined from a precision measurement of the

611 GHz H-D isotope shift of the 1S-2S frequency. However, uncertainties of the fine structure constant and of the mean square radii of the nuclear charge distributions

would still impose error limits of about "4 parts in 10^^ for the Rydberg.

If the electron /proton mass ratio and the fine structure constant can be measured independently with improved

accuracy, then a precise measurement of the 1S-2S H-D

isotope shift could provide an accurate probe for nuclear

structure and recoil shifts. To determine a precise Rydberg value that is not limited by nuclear structure corrections, one could combine

the 1S-2S measurement with a precise measurement of a two-photon transition from 2S to one of the higher nS

levels. Taking advantage of the fact that the lowest order

nuclear structure corrections scale with the inverse cube

of the principal quantum number n, one can easily construct differences of transition frequencies which are no

longer sensitive to the exact nuclear sizes. Neither a measurement of the 1S-2S frequency nor of the isotope shift by itself can provide a very stringent

test of quantum electrodynamics, because we are free to adjust the values of fundamental constants until the cal- culations agree with the observations. However, if we

form the ratio of the 1S-2S frequency to the frequency of

a different hydrogenic transition, such as a Balmer transition, or a transition to or between Rydberg states, we

arrive at a dimensionless quantity, which, to lowest or- der, no longer depends on the Rydberg constant, and

which can be calculated with very high precision. An ac- curate measurement of such a frequency ratio would per- mit a very interesting test of theory, and if the past is any guide, it may well lead to some surprising discovery.

References

[1] T. W. Hansch, G. W. Series, and A. L. Schawlow, Sci. Am.

2403, 94 (March 1979).

[2] T. W. Hansch, I. S. Shahin, and A. L. Schawlow, Nature

235, 63 (1972).

[3] T. W. Hansch, M. H. Nayfeh, S. A. Lee, S. M. Curry, and I. S. Shahin, Phys. Rev. Lett. 32, 1336 (1974).

[4] E. R. Cohen and B. N. Taylor, J. Phys. Chem. Ref. Data 2, 663 (1973).

[5] C. E. Wieman and T. W. Hansch, Phys. Rev. Lett. .36, 1170

(1976).

[6] R. K. Raj, P. Bloch, J. J. Snyder, G. Camy, and M. Ducloy,

Phys. Rev. Lett. 44, 12-51 (1980).

[7] J. E. M. Goldsmith, E. W. Weber, and T. W. Hansch, Phys. Rev. Lett. 41, 940 (1978).

[8] J. E. M. Goldsmith, E. W. Weber, F. V. Kowalski, and A.

L. Schawlow, Appl. Opt. 18, 1983 (1979).

[9] E. W. Weber and J. E. M. Goldsmith, Phys. Lett. 70A, 95

(1979) . [10] E. W. Weber, Phys. Rev. A 20, 2278 (1979).

[11] Comite Consultatif pour la Definition du Metre, 5*' Session

1973, Bureau International des Poids et Mesures, p. M23.

[12] B. W. Petley, K. Morris, and R. E. Shawyer, J. Phys. B: Atom. Molec. Phys. 13, 3099 (1980).

[13] S. R. Amin, C. D. Caldwell, and W. Lichten, these proceedings; Phys. Rev. Lett. 47, 1234 (1981).

[14] W. E. Lamb and R. C. Retherford, Phys. Rev. Lett. 79, .549 (1950).

[15] D. E. Roberts and E. N. Fortson, Phys. Rev. Lett. 31, 1539

(1973).

[16] E. W. Weber and J. E. M. Goldsmith. Phys. Rev. Lett. 41, 940(1978).

[17] C. E. Wieman, private communication; D. Shiner and C. Wieman, these proceedings.

[18] N. Bloembergen and M. D. Levenson, in High Resolution Laser Specfroscopy (Topics in Applied Physics, Vol. 13), Ed. by K. Shimoda (Springer- Verlag, Berlin", 1976), p. 315.

[19] T. W. Hansch, S. A. Lee, R. Wallenstein, and C. E. Wie- man, Phys. Rev. Lett. 34, 307 (1975).

[20] S. A. Lee, R. Wallenstein, and T. W. Hansch, Phvs. Rev.

Lett. 35, 1262 (1975).

[21] C. E. Wieman and T. W. Hansch, Phvs. Rev. A 22, 192

(1980) . [22] S. R. Lundeen and F. M. Pipkin, Phvs. Rev. Lett. 46, 232

(1981) . [23] G. W. Erickson, Phys. Rev. Lett. 27, 780 (1971).

[24] P. J. Mohr, Phys. Rev. Lett. 34, 1050 (1975).

[25] A. I. Ferguson, J. E. M. Goldsmith, T. W. Hansch, and E. W. Weber, in Laser Spectroscopi/ IV, Ed. by H. Walther and

K. W. Rothe (Springer Series in Optical Sciences, Vol. 21, Springer- Verlag, Berlin, 1979) p. 31. [26] S. B. Crampton, T. J. Greytak, D. Kleppner, W. D. Phillips, D. A. Smith, and A. Weinrib, Phys. Rev. Lett. 42 (1979).

[27] R. S. Van Dyck, Jr. and P. B. Schwinberg, Phys. Rev. Lett.

47, 395 (1981); these proceedings.

115

Precision Measurement and Fundarnental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Atomic Beam, Linear, Single-Photon l\/leasurement of the Rydberg Constant*

S. R. Amin, C. D. Caldwell, and W. LIchten

Department of Physics, Yale University, Box 6666, New Haven, CT 06511

The Rydberg constant has been measured to a standard error of one part in 10^ for the first time

by atomic beam, laser spectroscopy. The results are as follows- R ^ki^Re-'^^Ne: ^^^2, B) = 6.946 263

169 (±1 part in 10^). With the CCDM definition of X(=^He-20Ne: i) rounded exactly to 632

991.399 pm, /Joe = 109 737.315 21 (± 1 part in 10^) cm'^ The results are consistent with recent, less precise measurements. Key words: atomic beam; fundamental constants; hydrogen; laser spectroscopy; Rydberg constant.

1. Introduction

High precision, optical spectroscopy of atomic hydrogen is of considerable current interest [1-10]. It is agreed [1-7] that an atomic beam should be used to eliminate pressure and current corrections of the order of 1 -

2 X 10"^, which occur in gas cell measurements. One at- tempt has been reported [7]. Others are underway at Michigan [8], Bonn, Heidelberg [9], and NBS Washington

[10]; we present a Yale effort in this paper. Efforts to use the two-photon IS <—> 2S transition [11-14] by improving the waveforms [15] by means of cw lasers are current in Stanford [16] and Novosibirsk [17]. Another

proposal is microwave transitions between Rydberg

states of atoms to make a direct frequency determination

of the Rydberg constant [18, 19]. We report an atomic beam, single photon determination of the Rydberg constant, by means of the Balmer-a

line. The four transitions observed are H(2S, F = I) ^

H(3P, J = 1/2, F =1, 0), ^ H(3P, J = 3/2, F = 2, 1); D(2S, F = 3/2) ^ D(3P, / = 1/2, F = 1/2, 3/2), D(3P, J = 3/2, F = 1/2, 3/2, 5/2). (See Fig. 1.)

1.1 Plan of Experiment

The experiment [6] is an optical analogue of the classic Lamb-Retherford paper [20]. A beam of atomic hydrogen

comes from a W oven at 2850 K. A small fraction of the

atoms (~ 10"^) is excited by electron bombardment to the

metastable 2S state and is detected by secondary elec- tron emission. Typical currents are ~ 3 x 10"^'' A, with a

signal-to-noise ratio of 300/1. A chopped, low intensity

(—0.002 W/cm^), tunable, cw dye laser beam crosses the

atomic beam at right angles and quenches the metastable

atoms with Balmer-a (2S - 3P) light. A retroflector eliminates first order Doppler shifts. A lock-in amplifier, re- ferred to the chopper, detects the quenched part of the

atomic beam.

Figure 2 shows the lasers and optics. We use standard

H and D lamps, a Hartridge reversion spectroscope and

quartz etalons (P.P.E.3), 0.1 mm or 1 mm thick, to allow

tuning to within one GHz of the H or D Balmer-a line.

Eu MHz

in -oo h °

L-io H

\

F= I100

-10

50

F= I

F = 0

F=l

NATURAL WIDTH

•^3 Pi/g"* K t ° \f=I/2 -10

50

2 S|/2-*

F = 3/2

F = l/2 . -50

--I00

-50

MHz

F = 0

-150 MHz

*This research was aided by Grants PHY 78-26138 and PHY 80-20465 of the National Science Foundation.

Figure 1. Energy levels of H and D atoyns involved in the present experinwnt. The term separations between fine struc- ture levels and between levels of different principal quantum

numbers are suppressed.

117
TO ATOMIC BEAM

Figure 2. Pian of the lasers and optics of the present experi- ment. CH, chopper; N. D., neutral clensitij filter; B.E beam

expander; M, Mirror; IS, isolator; D.L., dye laser; P.D., photodiode; C.F.E., confocal scanning Fabry-Perot etalon; LH.N.L., iodine stabilized, 'He--'-'Ne laser (primary

wavelength standard; plasma tube Jodon CE-16-1.5 CS);

A.L., argon ion laser; P.l, P:2, pinhohes, 1 mm diameter;

P.P.E., plane parallel Fabry-Perot etalon, 1 and 2 pressure scanned; L, Lens; P.J, pinhole, 2-5 mm diameter; P.It, pinhole, 50 p,m diameter; S.L., atomic wave length standard

lamp; H.R.S., Hartridge reversion spectroscope.

2. Methods

2.1 The Lines: Structure, Width and Shape

Figure 3 shows a dye laser scan of the two fs components of the hydrogen Hne. There are no crossovers or other artifacts characteristic of non-hnear techniques

[2-4, 11-15]. There is no hfs because the excitation proc- ess at ~ 1300 gauss leads to population of only the level F = / + 1/2 of the 2S state [20]; also, the hfs of the 3P

state is unresolvable. (see Fig. 1.) Figure 4 compares the D transitions to the ^P3/2 levels with a theoretical line shape. The extrapolated line width

at zero power is measured to be 42 ± 3 MHz. Early runs

are vdth an unexpanded beam and in a stray field of

3.3 gauss. Later runs, with an expanded laser beam and

magnetic shielding (stray field 0.3 gauss) produce slightly

better line widths, typically power broadened to ~ 50 MHz. The line at zero optical power has a Voight

profile with a natural, Lorentzian width equal to 30 MHz

[21]. The Gaussian component is the root-mean-square sum of laser jitter of 5 MHz, hyperfine and Zeeman

broadening of 13 MHz, and Doppler broadening of 12.5 ±

3/2

Figure 3. Laser scan of the atomic hydrogen line. The plot is from a two pen recorder, with the time axis horizontal. Lower

plot: scanning Fabry-Perot interferometer (free spectral range = 3 GHz). Upper plot: lock-in detector. Since the fre- quency scan is linear with time ( lowerfigure), the upperfigure

is equivalent to a plot of signal vs. frequency. The dye laser in- tensity is reduced by a factor of four to ensure linearity of response with line strength. The theoretical intensity ratio of

211 is in excellent agreonent with experiment.

•50 0 50

RELATIVE FREQUENCY (MHz)

Figure 4. Comparison of theoretical and experimental liyie shapes for D, 2^Sj/2^ S^P^i^. Circles are experimental

points; size of the circle shows error in frequency. The solid

line is a theoretical curve; the only parameter is percentage

quenchiyig.

1 MHz, which arises from spread of the atomic beam

(1.25+ 0.15 mrad). The total Gaussian width of 19 ±

1 MHz, by the tables of Davies and Vaughan [22], gives a

final width of 40 ± 1 MHz, which agrees with the ob- served value of 42 ± 3 MHz [21]. From the measured width at zero power and transition

probability at line center, the line shape (Fig. 4) is cal- culable with no further adjustable parameters. Within

the experimental errors, the theoretical and experimental line shapes agree.

118
Splitting a line to 1% of its width presupposes an

understanding of its shape, especially as to any possible

asymmetry present [20]. We consider our understanding

to be satisfactory.

2.2 Wavelength Measurements

In earlier runs with the etalon (P.P.E.2) spacing at 4.3 cm, we use a simple, manual laser tuning. For later runs, with 12.25 and 31.4 cm spacings, we frequency

modulate the laser with a 15 Hz square wave of peak-topeak shift of 50 MHz, approximately equal to the line width. With the modulating voltage as a reference for lock-in detection, the line shape becomes a derivative

trace. We then hold the laser on the line center by keeping the lock-in output at zero. We compare the wavelength of the laser line in the

etalon (Fig. 2, P.P.E.2) to a ^He-'^Ne laser, stabilized to the i29i2 B 6329 A peak by a third derivative lock [28]. The etalon is pressure scanned, has Cervit spacers, and

has multi-layered dielectric coated mirrors with 99.5%

reflectivity. Minimum finesse is F = 40. The standard is compared once with a portable '^He--°Ne: ^^^lo laser tuned

to the "i" and "g" peaks, set up by Howard Layer of

the U.S. National Bureau of Standards, Washington,

D.C. His wavelength standard has been compared with

standards in other countries [23,24]. The ^"^lo "B" laser

is (5 ± 5) X lO"^*^ parts high in frequency.

Two methods of wavelength measurement are used. By

use of two or more spacings, the method of virtual mirrors [25] eliminates the effects of phase shifts in the mirror coatings. Alternatively, we use the reflection curves

furnished by the vendors to calculate the phase shift [26]. The second method uses only one spacing (31.4 cm).

The integral order numbers of the etalons are made

unambiguous by measurement of the fractional order

with several atomic lines, all obtained in atomic beams:

H 6563, Ne 6533, Ne 6334, Ne 6266 and the standard

6329 A. Wavelengths for the neon lines will be published

elsewhere [27]. Table 1. Correctioyis

Figure 5 shows part of a pressure scan of the etalon. A

typical scan covers 4 orders. Data for the deuterium

3 ^P;i/2 peak is taken at all three etalon spacings. Data for the other three lines is taken at the 4.3 cm and 31.4 cm

spacings. A total of 272 lines is measured.

in

I-

_io

>

E

1 1 1 1 1 30 SECONDS

Figure 5. Pressure scan of Fabry-Perot etalon with planeparallel faces 314 cm apart (FSR = 1^78 MHz). ST, standard

laser line; D^, D2 dye laser lines; each order is doubled by the frequency modulation.

3. Corrections and Results

3.1 Miscellaneous Corrections

To correct for asymmetric excitation of Zeeman levels at 3.3 gauss, we average the strengths and frequencies of

all the Zeeman transition components by use of the

Breit-Rabi equation [28]. We also take into account the

effect of the polarization of the incident light. In

0.3 gauss, where the possibility of Majorana transitions

makes the magnetic sub-level populations unknown, we

use the theoretical line strengths [29] to calculate shifts. Light shifts are negligible at the power levels used in this experiment. We find no observable systematic ef- fects in the recording system and no asymmetry in the

line shapes. Tables 1 and 2 list the results and corrections. Table 3 gives estimated errors.

to (parts in iO^").

Transition

Correction'' H, 3Ply2 H, 3P3/2 D, 3Pi^ D, 3P^

Phase shiff^^ 466 466 463 463

Refractive index^-"^ 34 34 34 34 Non-linear scan 6 9 26 22 Magnetic field'^ -8 -5 -4 3 2S hfs 972 972 299 299 3P hfs 32 -32 10 -10

Optical pumping 0 0 -1 -2

Wavelength standard 5 5 5 5

Diffraction -3 -3 -3 -3

2nd order Doppler 5 5 2 2 Photon recoil -10 -10 -5 -5

Total 999^

1507^^ 941^1446'^

329'^

830^' 311^

805*^

''See Ref. [36]. ''For both sets of data, unless noted.

"^Phase shift data only,

'"'virtual mirrors data only.

119
Table 2. Calculation of the Rydberg Constant R^.

Upper State H, 3P3/2 D, 3Pl;2 D, 3P3^

Wave number''

l/\ - 15 230 cm^i

Phase shift Virtual mirrors

Theory'^

3.256 773

3.256 749

3.257 094

3.365 148

3.365 154 3.365 505

7.401 652

7.401 639

7.401 998

7.510 138

7.510 102 7.510 438

Rydberg constant'' Kx — 109 161 cm '

Phase shift Virtual mirrors

0.315 39

0.315 22

0.315 13 0.315 170.315 21 0.315 11

0.315 55

0.315 27

Combined by atoms: H D

Phase shift Virtual mirrors0.315 26

0.315 19

0.315 38

0.315 19

Grand average:

Phase shift Virtual mirrors Weighted average Rx =

0.315 32 ± .000 09^

0.315 19 ± .000 034

109 737.315 21 ± .000 03 cm"!

^Corrections given in Table 1. ''Based on CCDM \(3He-20Ne: i) = 632 991.399 pm [30].

'^R'YYi = 109 737.3177 cm"V = ^ TH TH '' ^ \.^'^\

Table 3. Emr estimates (parts in 10^^).

Statistical 3 Magnetic field 2 Recording and pressure scanning 6 3P hfs, optical pumping, and

light shifts 5 Wavelength standard 5 Non-uniform mirror coatings 3

Diffraction 1 2nd order Doppler, photon recoil, 2S hfs, index of refraction, and minor effects 0

Root mean square error 10

3.2 Wavelengths

The measured quantities are the four v^^avelength ratios

for the Balmer-a lines: X {"^Ue-^-Ne: ^-H^, B)/\(H or D,

22S1/2 32P3/2 or 32P1/2). These, with the wavelength definitions of the CCDM [30] and the quantum electrodynamical, reduced mass, and nuclear structure corrections

of Erickson [31], give four independent results for the

virtual mirrors and four for the phase shift method. To

average out systematic effects, we take each of the four

results with equal weight. Both sets of data are combined

into a weighted average for the Rydberg constant.

The primary datum of the experiment is the product of the Rydberg constant with a stabilized laser wavelength:

i2xX(^He-22Ne: ''^^2, B) = 6.946 263 169

(± 1 part in 10^)

With the CCDM meter [30], we have

for \(3He-20Ne: ^^'^h, i) = 632 991.399 pm , /Ex = 109 737.315 21 cm"' (± 1 x IQ-S)

(1)

(2)

(3)

To obtain /2x in frequency units, we use the following

chain:

/(He-Ne: CH4) = 88 376 181.616 MHz

(±0.03 X 10-^)

as measured by Knight et at. [32], and

X (He-Ne: CH4)

(4)

X(^He-20Ne: '^^h, k)

5.359 049 260 6

(±0.2 X 10^^) , (5)

as measured by Layer et al. at NBS [33]. This gives

/(^He-^ONe: ^^^2, k) = 473 612 310.7 MHz

(±0.2 X 10"^) (6)

120
which, when combined with the known offset [34]

/(3He-22Ne: B) - /(^He-^ONe: k) = 892.563 MHz , (7) gives

fi^Ue-^Ne: B)nbs = 473 613 203.2 MHz

(±0.2 X 10-9) _ (g)

The chain of Woods et al. [35] gives the frequency of

the (^He-20Ne: d) laser to be 473 612 380.5 MHz

(± 0.6 X 10"^) which, when combined with the offset [34]

109 737.312 .313

I I .314

I .315I

/ (12912, B) -/ (12^12, d) 823.21 MHz, gives

/(3He-22Ne, ^^^2, B)npl = 473 613 203.7 MHz

(±0.6 X 10-9) . (8') A weighted average of the two results is:

/(3He-22Ne, B)av = 473 613 203.4 MHz

(±0.2 X 10-9) _ (3-)

By combining Eqs. (1) and (8"), we have

c i2:c = 3 289 841 951 MHz (±1 x IQ-^) (9) We obtain a different value for /2x if we assume the

tentative definition of the meter [34] in terms of the

velocity of light: c ^ 299 792 458 m/sec; i?. = 109

737.315 37 cm-^ (± 1 x IQ-^).

4. Discussion

The results for the four lines agree very well for the

virtual mirrors; the phase shift calculations agree less

well. The phase shift data carries 11% of the weight of the final result and shifts it upward by two parts in 10^**. Table 4 and Fig. 6 compare our results with other au- thors [2-4, 36].

Table 4. Comparison vrith earlier measurements.

Reference R - 109 737 cm

-1

Taylor et al.^ 0.312 + .005 Hansch et al.^''^ 0.314 .001 Petley et al^ 0.315 13 ± .000 85 Goldsmith et al.^'^ 0.314 90 ± .000 32 Present result 0.315 21 ± .000 11

^Corrected to: (3He-20Ne: ^^^h, i) = 632 991.399 pm

''Ref. [36] 'Ref. [2] ^Ref. [3] '^Ref. [4]

The results of Hansch et al. [2] are in slight disagreement with the present paper and also are internally in- consistent. The results of Goldsmith et al. [4] are in fair agreement; the results of Petley et al. [3] and Taylor

et al. [36] are in excellent agreement with the present

results. The results of the phase shift calculations are en- couraging. Such calculations should be useful for all but

Hdnsch

(P/o/(l974)

Goldsmith eto/ (1978)

Petley efa/ (1980)

This work

X

(3

2<r H K

O virtual mirrors X phase shift calculation

I I I I

.316

I

D 2P3,2-3D5,2

Weighted Mean

H l/2~ 3P|/2

3/2- 3D5/2

1/2" 3P|/2

'l/2- 3D3/2

H

H

HWeighted Mean

} ^ 2S,/2-3P|/;

} H 2S,/2-3P3/

} D 2S,/2-3P|,j

X J

2S|/2 3P3/2

Means

(J) Weighted Mean

tlx 10"^

109 737312.313 .314 .315 .316 R«)(cm"')

Figure 6. Comparison of recent measurements of the Rydberg

constant. Errors for individual yneasurements are statistical. Errors for weighted means include systematic errors. See

Table i. the most precise measurements and help to answer an unsolved problem in metrology; that is, how to use light waves to measure dimensions of objects to very high ac- curacy. Recent attempts to measure objects accurately

have led to inconsistent results [38]. The present phase

shift data deviate by approximately one part in 10^ from

our final value. This corresponds to a phase shift on reflection of 1/6° or an error of 1.5 A (0.00015 ixm) in the

location of the reflecting surface. Further investigation

in this area is desirable [37]. Minor improvements in techniques would improve Rydberg measurements to a precision of 1 - 2 x 10-^°, comparable to that of the most reliable, current definition of the meter by means of optical wavelengths. Further improvements would allow accurate measurement of the

electron-proton mass ratio [6, 8, 39].

We thank Howard Layer for advice on and calibration

of our wavelength standard, and for the loan of an ^-H-? absorption cell; Barry Taylor for helpful conversations:

Munir Nayfeh, Salim Jabr, and Kim Woodle for aid in building the wavelength standard; Zoe Goodman, David

Mariani, and Alain Rostain for aid in the experiment; K. M. Baird, R. L. Barger, J. L. Hall. T. W Hansch, W. R.

C. Rowley, and J. R. Whetstone for helpful discussions;

and W. R. Bennett, Jr. for use of his laboratory.

121
References [21]

[1] For an early review of the subject, see G. W. Series in Precision MeaHurement and Fundamental Constants, Ed. by D. N. [22] Langenberg and B. N. Taylor, Natl. Bur. Stand. (U.S.) Spec.

Pubf. 343 (Aug. 1971), p. 73. [23]

[2] T. W. Hansch, M. H. Nayfeh, S. A. Lee, S. M. Curry, and I. S. Shahin, Phys. Rev. Lett. 32, 1336 (1974); M. H. Nayfeh, [24] Ph.D. Thesis, Stanford University, 1974 (M. L. Report 2318),

pp. 20 and 85. [25]

[3] B. W. Petley, K. Morris, and R. E. Shawyer, J. Phys. B. 13, 3099 (1980).

[4] J. E. M. Goldsmith, E. W. Weber, and T. W. Hansch, Phys. Rev. Lett. 41, 1525 (1978); J. E. M. Goldsmith, E. W. Weber,

F. V. Kowalski, and A. L. Schawlow, Appl. Opt. 18, 1983

(1979). Their definition of the meter differs from Refs. [2, 3] and the recommendations of the CCDM. Our Table 4 and Fig. 6 give values for R,, which are corrected to a single, consistent [26] standard for the meter. After our paper was delivered, Mr. van Dierendonck informed us of his recalculation of the results

of Goldsmith ef ai, based on new measurements listed in Ref.

[34] and in F. Spieweck, IEEE Trans. Instrum. Meas. IM29, 361 (1980). He gives = 109 737.314 88 cm

[5] J. L. Hall, Science 202, 147 (1978).

[6] T. Wik, W. R. Bennett, Jr., and W. Lichten, Phys. Rev.

Lett. 40, 1080 (1978). See Fig. 1 for the basic plan of the [27] present experiment. See also S. R. Amin, C. D. Caldwell, and W. Lichten, Phys. Rev. Lett. 47, 1234 (1981). [28]

[7] J. B. West, R. L. Barger, and T. C. English, IEEE Trans. Quant. Electron QE-11, 18 (1975); R. L. Barger, T. C. English, and J. B. West, in Atomic Masses and Fundamen- [29]

tal Constants 5, Ed. by J. H. Sanders and A. H. Wapstra

(Plenum Press, London, 1976), p. 565.

[8] D. Shiner and C. Wieman, these proceedings. [30]

[9] E. W. Weber and co-workers, University of Heidelberg; B. Burghardt et al., University of Bonn, these proceedings.

[10] K. C. Harvey, these proceedings. [31]

[11] D. E. Roberts and E. N. Fortson, Phys. Rev. Lett. 31, 1539 [32] (1973).

[12] T. W. Hansch, S. A. Lee, R. Wallenstein, and C. Wieman, Phvs. [33] Rev. Lett. 34, 307 (1975).

[13] S. A. Lee, R. Wallenstein, and T. W. Hansch, Phys. Rev. Lett. [34]

35, 1262 (1975).

[14] C. Wieman and T. W. Hansch, Phys. Rev. A 22, 192 (1980). [35] [15] J. N. Eckstein, A. L Ferguson, and T. W. Han.sch. Phys. Rev.

Lett. 40, 847 (1978). [36]

[16] T. W. Hansch, these proceedings. For positronium, see S. Chu

and A. P. Mills, Jr., Phys. Rev. Lett. 48, 1333 (1982). [;^7] [17] V. Chebotayev, private communication. [38]

[18] D. Kleppner, Bull. Am. Phys. Soc. 20, 1458 (1975).

[19] See C. Fabre, P. Goy, and S. Haroche, J. Phys. B 10, L183

(1977). See also Re'f. [10]. [20] W. E. Lamb and R. C. Retherford, Phys. Rev. 79, 549 [39] (1950). See par. 5 for production of a polarized beam of atoms which, at low magnetic fields, becomes a beam in one hf level.

If the non-Lorentzian parts are not Gaussian, we add the variances and set the non-Lorentzian width equal to 2.35 CT, as in the Gaussian distribution.

J. T. Davies and J. M. Vaughan, Astrophys. J. 137, 1302

(1963). W. G. Schweitzer, E. G. Kessler, R. D. Deslattes, H. P. Layer, and J. R. Whetstone, Appl. Opt. 12, 2927 (1973). H. P. Layer, W. R. C. Rowley, and B. R. Marx, Opt. Lett.

6, 188 (1981). H. Buisson and Ch. Fabry, J. Phys. Theor. Appl. 7, 169 (1908); Ch. Fabry and H. Buisson, ibid., p. 417; W. F. Meggers, Bull. Natl. Bur. Stand. (U.S.) 12, 198 (1915-16); K. W. Meissner,

J. Opt. Soc. Am. 31, 405 (1941); D. H. Rank, E. R. Shull, J. M. Bennett, and T. A. Wiggins, il)id. 43, 9.52 (1953); D. H. Rank and H. E. Bennett, ibid. 45, 69 (1955); R. W. Stanley and K. L. Andrew, ibid. 54, 625 (1964); R. L. Barger and J. L. Hall, Appl. Phys. Lett. 22, 196 (1973).

J. M. Bennett, J. Opt. Soc. Am. 54, 612 (1964); C. J. Koester,

J. Res. Natl. Bur. Stand. Sect. A; 64, 191 (1960); D. H. Rank

and H. E. Bennett, op. cit.; P. W. Baumeister and F. A. Jenkins, J. Opt. Soc. Am. 47, 57 (1957); J. Bauer, Ann. Phys.

(Leipzig) 20, 181 (1934); C. F. Bruce and P. E. Ciddor, J. Opt.

Soc. Am. 50, 295 (1960); M. Born and E. Wolf, Principles of

Optics, 2nd ed. (Pergamon Press, New York, 1975), pp. 52-

59.

S. R. Amin, Ph.D. Thesis, Yale University, Dec. 1982; sub- mitted to J. Opt. Soc. Am.

G. Breit and I. I. Rabi, Phys. Rev. 38, 2082 (1931); N. F. Ramsey, Molecular Beams (Oxford University Press, London, 1956), p. 84. E. U. Condon and G. H. Shortley, The Theory of Atomic

Spectra (Cambridge University Press, London, 1935),

p. 243. Comite Consultatif pour la Definition du Metre (CCDM), 5*^ Session, 1973, Bureau International des Poids et Mesures

(BIPM), p. M23.

G. W. Erickson, J. Phys. Chem. Ref. Data 6, 831 (1977). D. J. E. Knight, G. J. Edwards, P. R. Pearce, and N. R. Cross, IEEE Ti-ans. Instrum. Meas. IM-29, 257 (1980). H. P. Layer, R. D. Deslattes, and W. G. Schweitzer, Jr., Appl. Opt. 15, 734 (1976). For a discussion of comparisons of He-Ne lasers, see CCDM,

6^ Session, 1979, BIPM, especially CCDM/78-13.

P. T. Woods, K. C. Shotton, and W. R. C. Rowley, Appl.

Opt. 17, 1048(1978).

B. N. Taylor, W. H. Parker, and D. N. Langenberg, Rev. Mod. Phys. 41, 375 (1969). W. Lichten, to be submitted for publication. E. Englehard, Natl. Bur. Stand. (U.S.) Cir. 581, p. 13 (1957); H. A. Bowman, R. M. Schoonover, and C. L. Carroll, J. Res.

Natl. Bur. Stand. Sect. A: 78, 13(1974); NBS Internal Report

7.5-768. R. S. Van Dyck and P. B. Schwinberg, these proceedings; Phys. Rev. Lett. 47, 395 (1981).

J

122
Precision Measurement and Fundamental Constafits II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Current Work on Two Photon Excitation in a Hydrogen Beam for the

IVIeasurement of the Rydberg Constant and mjmp*

D. Shiner and C. Wieman

Department of Physics, University of Michigan, Ann Arbor, Ml 48109

We have observed the two quantum 28 to 3S transition in a beam of hydrogen in the metastable 2S

state. The transition was excited by 6563 A laser light plus a 315 MHz radio frequency field and is much narrower than the 2S to 2P single photon transition. The transition is detected by observing the

increase in the number of metastable atoms which survive passage through the region containing the two fields. Work is underway to precisely measure the transition energy and the shift between hydro- gen and deuterium. These results will provide more precise values for the Rydberg constant and the

electron to proton mass ratio. Key words: fundamental constants; hydrogen; laser spectroscopy.

1. Introduction

We report on the current status of an experiment to measure the 2S-3S energy separations in atomic hydrogen and deuterium to very high precision. From these

measurements we expect to obtain values for the Rydberg constant and for the electron to proton mass ratio more precise than the current values, and to provide an improved test of the relativistic nuclear recoil correction

to the energies of the hydrogenic states. To measure

this energy separation we excite the 2S 3S two quantum transition in a beam of 2S (metastable) hydrogen us^ ing a dye laser tuned to the 2S 3P transition at 6563 A

and a radiofrequency field of —300 MHz. This two quantum transition requires comparable laser power but has a natural linewidth 30 times narrower than the 2S 3P

one photon transition. Weber and Goldsmith [1] have

detected this two photon transition using polarization

spectroscopy in a Wood' s discharge but the discharge en- vironment resulted in linewidths only slightly less than

the 3P natural wddth. We are presently observing the 2S —> 3S transition and studying the line shape.

2. Background

The theory of double quantum transitions has been dis- cussed by many authors, and Roberts and Fortson [2] consider our particular case of laser plus radiofrequency

exciting the 2S 3S transition in hydrogen. Although

their treatment makes some simplifying assumptions

(primarily low laser intensity) in solving the coupled 2S - 3P - 3S problem, these assumptions, and hence their

results, match our experimental conditions with only

slight changes.

From Eq. 1 (or pictorially in Fig. 2) of their paper one

can see the key features of this resonance. First, when

the rf is tuned to the 3S - 3P difference and the laser fre- quency varied, the transition rate into the 3S state has a sharp resonance which for low rf power has a width limited only by the 1 MHz 3S state natural linewidth. We

have detected this process in our apparatus by observing

*This work supported by National Bureau of Standards Precision Meas- urements Grant #NB79NADA0018.

the Lyman-a photons which are emitted when the 3S

state decays via the 2P state. However, we have found

there is a second way to observe the 2S 3S resonance

which, for technical reasons which will be discussed

below, is far superior. This second approach uses the fact that along with the narrow rise in the 3S transition rate there is a corresponding decrease in the transition rate out of the 2S state. Although this may not be obvious in- tuitively it is quite evident in the results of Roberts and

Fortson. This reduced transition rate means the 2S

atoms will have a greater probability of passing through

the interaction region without undergoing an excitation

and subsequent decay to the ground state. One can

therefore observe the two photon transition by seeing an

increase in the number of metastable atoms detected

downstream of the interaction region. When power

broadening is included the exact line shape depends on

the rf and laser intensities, and the length of the interaction region. To give a specific example however, to quench the beam off resonance to and on resonance to e"^ of its initial value, the powers required broaden the

line to ~2 MHz.

3. Experimental Setup

The present experimental apparatus is shown schematically in Fig. 1. The metastable hydrogen source is similar in design and performance to that used by Lamb [3]. Molecular hydrogen is disassociated in a hot tungsten oven producing a beam of atomic hydrogen. This beam

passes through an electron bombardment region where a small fraction is excited to the 2S state. This provides

~2 X 10^*' H(2S)/sr-sec from a source slit 1 cm high and

0.04 cm wide. When the beam is collimated to give a resid- ual Doppler width in the perpendicular direction of

1 MHz, about 10-* H(2S)/sec remain. We have investi- gated the use of an rf disassociator for the molecular hydrogen, but have found that, at the present stage of the

experiment, the increased Lyman-a background this technique introduces outweighs the advantages of a lower beam.

The H(2S) beam passes through an interaction region

where there is an rf electric field on the order of IV/cm

and a dye laser beam with ~1 mW/cm^ intensity. The rf

field is produced in a parallel plate 50 ft transmission

123
u QniiprF INTERACTION DETECTOR bUUKLb p^giQ^ p^gjQ^

A—^

DYE

LASER

REFERENCE

INTERFEROMETER

Figure 1. Schematic of the apparatus.

In Fig. 3, on an expanded frequency scale, we show a

similar scan of the 2Si/2(i^ =1) ^ 2Pi/2(F =1, 0) component with a 324 MHz rf field present. Although the two

photon transition is clearly narrower than the 3P width it is substantially broader than the 4 MHz residual Doppler

width set by the beam collimation. The additional

broadening is due to the 10+ MHz frequency jitter of the

laser as it is scanned. We have now locked the laser fre- quency to the interferometer and in future work will scan

only the rf frequency to observe the transition. We anti- cipate this will provide significantly narrower lines.

5. Future Work

Our immediate effort is to make a detailed study of the

two photon line shape and the various mechanisms which

cause broadening and shifts. The most difficult shifts to deal with are those which depend on the atomic velocity.

We believe these can be accurately determined however

by appropriate chopping of the laser and atomic beams

and timing the arrivals of the metastable atoms. This can be done quite well since the time response of the electron

gun, the electroptic laser amplitude modulator, and the

detector are all very short compared to the flight time of the atoms, and the time interval between the arrival of individual metastable atoms.

line. The laser beam is ~2 cm in diameter and passes

through the central region of the rf field perpendicular to the atomic beam. The laser can be retroreflected to check for misalignments.

The metastable atoms remaining after the interaction

region go through a narrow collimation slit and are then

detected by quenching them with a dc electric field between two closely spaced Lyman-a counters. These

subtend a solid angle of —rrsr each and see a negligible

amount of background Lyman-a photons. These two fac- tors make this method superior to the direct observation

of the 3S decay. In that method the detected Lyman-a

photons are emitted from a region of appreciable size and

with the additional problem that the requirements on rf

field homogeneity force the detectors to be a substantial

distance from it. This situation results in small detection

solid angle and substantial problems with Lyman-a background suppression.

The Lyman-a photon counters are photoionization detectors of our own design. They have 3 Torr of benzene vapor

contained in a metal tube with a magnesium window on one

side. We measure the current pulses on a wire down the center of the tube to which a bias voltage is applied. The

gain is sufficiently high (~5 x 10'^) that the single photon

pulses can be timed and counted with standard "Nim bin"

electronics. At the present time the detectors have a quantum efficiency of about 15%, which is 30-40% lower than we expect. The difference is probably caused by contamination on the window and in the benzene.

The dye laser is a Spectra-Physics model 380A which is frequency locked to a stable Fabry-Perot interferometer.

This provides a short term linewidth of about 1 MHz.

4. Present Results

In Fig. 2 we show typical spectra obtained when the

laser frequency was not locked to the Fabry-Perot inter- ferometer but was scanned over the 2Si/2 3Py2 single

photon transition when no rf field was present. Note that

the quantity being observed is the number of surviving

2S atoms and thus the resonance is a decrease in the count rate.

2000

4000

6000

- 2S./2(F=0)^3Pi/2(F=l)

1 r 1 1 [ 1 1 1 ^ 100 200 300 400 FREQUENCY (MH?)

Figure 2. Single photon transition spectrum.

500

1000

1500

2000 40 80 FREQUENCY (mi)

120

Figure 3. Two photon 2Sj/., (F = 1) ^ SP^/^ (F = 0,1) 3Si,2 (F = 1) transition.

124
Once these studies are completed we will measure a

series of beat frequencies between two tuneable dye

lasers to compare the 2S 3S energy with I2 lines which

are absolute wavelength references to obtain an improved value of the Rydberg constant. Also we will measure the energy difference between the hydrogen and

deuterium resonances. These can be observed simultaneously in one apparatus. This will provide a greatly improved test of the relativistic nuclear recoil correction to

the hydrogenic energy levels [4], and a measurement of

the electron to proton mass ratio. References

fll E. Weber andJ. Goldsmith, Phys. Kev. Lett. 41, 940 (1978).

12] I). E. Roberts and E. N. Fortson, Phvs. Rev. Lett. 31, l.')89 (197;^). W. Lamb and R. Retherford, Phys. Rev. 79, ,'j49 (19.')0). [4 1 C. Wieman and T. Han.sch, Phys. Rev. A 22, 192 (1980).

125

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Measurement of the 22P3/2 - 22S1/2 Fine-Structure Interval in Atomic

Hydrogen*

K. A. Safinya,** K. K. Chan, S. R. Lundeen, and F. M. Pipkin

Lyman Laboratory, Harvard University, Cambridge, MA 02138

A separated-microwave-oscillatory-field technique has been used in conjunction with a fast atomic- hydrogen beam to measure the 2^P3/2 - 2rSy2 fine-structure interval in atomic hydrogen. The value ob- tained for the 2-P3/2 - 2^Siy2 fine-structure interval is 9911.117(41) MHz. This value is compared with

other measurements of this fine structure interval and the potential precision obtainable in a more definitive measurement is discussed. Key wrords: fast atomic beam; fine structure constant; fine structure of hydrogen; Lamb shift; separated-oscillatory-fields.

At present the accepted value for the fine structure constant is derived indirectly using the equation [1]

-1 a =

^^NBS

M-B

(2e/h)>NBS 7p(l0W)NBS

Here c is the speed of light in vacuum; R ^, the Rydberg

constant for infinite mass; flxBs/i^, the ratio of the National Bureau of Standards (NBS) as-maintained ohm to the absolute or SI (International System) ohm; |xVm-b,

the magnetic xnoment of the proton in units of the Bohr

magneton; (2e//i)xBS) the ratio of twice the elementary

charge to the Planck constant measured in terms of the

NBS as-maintained volt using the ac Josephson effect; and 'yp(low)NBs, the gyromagnetic ratio of the proton

measured in terms of the NBS as-maintained ampere by

the lovi^-field method. The most recent value is [2] a -1 = 137.035963(15) (0.11 ppm).

An alternative value for the fine structure constant of comparable precision can be determined from the meas- urement of g - 2 for the electron with the assumption

that the quantum electrodynamic calculation of gr - 2 is correct. The value obtained in this manner is [2, 3]^ a- 1 = 137.035993(10) (0.073 ppm).

These two values for a disagree by over 2 standard deviations. As an aid for resolving this discrepancy it would

be valuable to have a third equally precise value for the

fine structure constant which was free from theoretical

ambiguities.

A theoretically unambiguous value for the fine-struc- ture constant can be determined from the 2-P3/2 - 2-P1/2

fine-structure interval in atomic hydrogen [1]. This fine- structure interval can be calculated from independent

measurements of the 2-S1/2 - 2-P1/2 (Lamb shift) and the

2^Px,/2 ~ 2^Si/2 fine-structure intervals. At present, the

uncertainty in the P^P3/2 - 2^Pi/2 fine-structure interval is

*Research supported in part by National Science Foundation Grant No.

78-09657. **Present address: SRI International, Palo Alto, CA 95205.

^Editors' note: See also the paper by T. Kinoshita and W. B. Lindquist

in these proceedings.

due primarily to the 4 ppm error in the most precise

measurement of the 2-P3^ - 2-Siy2 interval and the

disagreement between the several measurements. This

limits the precision of the value of the fine-structure con- stant calculated from this interval to 2 ppm [1]. This

paper reports the preliminary results of a new measure- ment of the 2-P3/2 - 2-S1/2 fine-structure interval with the use of a separated-microwave-oscillatory-field technique

[4, 5] which is as precise as previous measurements and

which gives promise of an order-of-magnitude increase in the precision of the determination of this fine-structure

interval. Figure 1(a) shows a schematic drawing of the ap- paratus. Fast hydrogen atoms in the 2-S1.2 metastable

state were formed through charge-exchange collisions in nitrogen gas from a 50- or 100-keV proton beam. The

atoms passed through a state-selecting radio-frequency

field (1100 MHz) set to quench atoms in the 2-S12 (F = 1) state and entered the spectroscopy region. The number of atoms in the 2-S1/2 (F = 0) state traversing the spectroscopy region was observed by quenching the atoms with a radio-frequency field (910 MHz) set at the central fre- quency for the 2-Sir2 (F = 0) ^ 2-Pi^ {F = 1) transition

and observing the number of Lyman-a photons emitted.

Figure 1(b) shows a schematic drawing of the mi- crowave apparatus used in the spectroscopy region to drive the transition. The microwave source was a klys- tron phase locked to the 25th harmonic of a frequency

synthesizer. A traveling-wave-tube amplifier was employed to obtain enough power (700 mW) to drive the

transition. A custom-made magic tee provided two equal

components of the microwave field which were either in phase or 180° out of phase. A high-power three-pole

switch altered the power between the 0°, off, and 180° configurations. The beam passed through 0.634-cmdiameter holes in the narrow side of the spectroscopy

waveguides, which were separated by 2.721(3) cm. Tubes

with a length of 0.953 cm soldered to each side of the

guide acted as waveguides beyond cutoff and eliminated

the leakage of microwave power. Two Schottky diodes in matched crystal holders were used to monitor the mi- crowave power in each waveguide.

Figure 2 shows typical resonance cur\-es for a 49 keV

proton beam. The quenching signal, Q, is the average

percentage change in the surviving number of 2-S12

(F = 0) atoms when the microwave field is turned on

127
(a)

ACCELERATOR CHARGE

EXCHANGE STATE

SELECTION SPECTROSCOPY

DETECTION

POST

H(2S)

(b)

Figure 1. (a) A schematic drawing of the apparatus, (b) A

schematic drawing of the microwave plumbing in the spectros- copy region. A, three-pole switch; B, matched load; C, low-pass

filters; D, precision variable attenuators; E, 50-dB circulators; F, vacimm. windows; G, magic tee; H, interaction waveguides;

I, beayn; J, detector diodes.

T 'I

'

I 'I

' I

'

I

' r

240 -200 m -120 -80 40 0 40

( I'- 10029,0) (MHz)

120 leo 200 240

Figure 2. Typical quenching (Q) and interfereyice (I) signals for a iO-keV beam. None of the corrections for asymmetries in

the line shape have been applied to these data.

with a relative phase of 0° and 180°; the interference signal, /, is the difference in the percentage change in the

surviving number of 2^Si/2 atoms when the microwave

field is turned on with relative phases of 0° and 180°. The

width of the interference signal is less than the 100 MHz

natural linewidth due to the selection of long-lived Pstate atoms by the separated oscillatory fields. In order to cancel errors due to the Doppler shift and

to a possible error in the phase between the two mi- crowave fields at the location of the beam, data were

taken in four configurations for each major run. The

Doppler shift was cancelled by first making measure- ments with the microwaves propagating in one direction

in the spectroscopy region and then reversing the feed so that the microwaves propagated in the opposite direction. The phase error was cancelled by interchanging the

two waveguides and their driving components in the in- teraction region so that the time order in which the beam

passed through the two interaction regions was inverted.

The method of symmetric points was used to determine

the centers of the resonance curves. Table 1 summarizes

the results of the measurements, the average corrections

required to determine the 2^P3/2 ~ 2^Si/2 fine-structure

interval, and the error in the corrections. The data at each point were corrected for the overlap of the residual

neighboring transitions, the variation_of the microwave

power over the line, the incomplete Q subtraction, and

the shift due to the motional electric field arising from

the microwave magnetic field in the waveguide.

Prior to the experiment the two power measuring

diodes and their mounts were calibrated with the use of a 10-dB attenuator and a Hewlett-Packard power meter. During the spectroscopy measurements the average

power in the two diodes was set to be the same in both

the 0° and 180° configurations. The voltages across each

diode in both the 0° and 180° configurations were

recorded for all data points. The sum of the two diode

voltages was monitored with the computer and the data were corrected on line for the small power variation during each data point. Subsequent to taking data the diodes were recalibrated and the reflection coefficients of the

components which could affect the microwave electric

field at the beam were carefully measured. The new

measurements showed that the microwave electric field decreased by 1.9(3)% /(lOO MHz) over the region where

measurements of the resonance line were made. The data were corrected for this variation in the microwave electric field. From computer simulations we estimated that

Table 1. A summary of the measured line centers and the corrections required

to determine the 2-P,i,, - '2- S,/,.) fine-structure interval.

Data Set

Beam energy (keV) 107.5(2.3) 48.7(1.0) 48.7(1.0) Signal size (% peak to peak) 4.5 1.8 1.8 Raw center (MHz) 10028.628(6) 10029.054(14) 10028.966(16) Corrections F = 1 overlap -0.091(5) +0.052(9) +0.052(9)

IQS -0.018(3) -0.005(1) -0.005(1)

V X B -0.098(20) -0.070(14) -0.070(14)

Bloch-Siegert and rf Stark shift -0.025(5) -0.025(5) -0.025(5)

Time dilation + 1.148(24) +0.520(11) +0.520(11)

rf-power slope 0.000(6) 0.000(6) 0.000(6) v(22P3^ (f = 1) - 2%^ {F = 0)) (MHz) 10029.544(33) 10029.526(26) 10029.438(27

Hyperfine structure (MHz) -118.386 -118.386 -118.386

v(2^Fsj2 - ^%/2) (MHz) 9911.158(33) 9911.140(26) 9911.052(27)

128
because of the oscillatory shape of the interference curve

a 1% /(lOO MHz) error in the relative electric field would

shift the center of the interference curve by less than

10 kHz.

Variations in the relative magnitudes of the microw^ave

fields in the two waveguides between the 0° and 180° configurations can result in a spurious interference signal

which persists even when the waveguides are very far apart. This signal, which is termed the incomplete Q subtraction (IQS), was determined by measurements made

with a wide separation for both 50- and 100-keV beams.

They were fitted to straight lines and used to correct the

observed interference signals. The spurious signal had a

slope of -0.22(4)% /(lOO MHz) for the 100-keV data and a

slope of -0.17(5)% /(lOO MHz) for the 50-keV data. The motional electric field due to the magnetic field in the waveguide produces a shift (v x B) which has a weak

dependence on the position of the resonance line. A pro- gram which simulated the resonance was used to calculate

this shift. A simulation program was also usedjto calculate the Bloch-Siegert and rf Stark shifts for the Q and / sig- nals. The second-order Doppler shift was calculated with

use of the result of a separate experiment in which the

first-order Doppler shift was employed to determine the

velocity of the beam [6]. The primary source of the 2.1%

error in the second-order Doppler shift was the error in the measurement of the magnetic field used to set the en- ergy of the accelerator. Since the three independent measurements of the

2^P3/2 - 2-S1/2 fine-structure summarized in Table 1 differ by more than the internally derived errors indicate they

should, we have taken the final statistical error to be the

standard deviation of the mean for the three measurements. The final error was obtained by combining in quadrature the statistical error (0.033 MHz) and the average

systematic error (0.025 MHz). This procedure gave

V(22P3;2 - 22S1/2) = 9911.117(41) MHz.

Figure 3 shows a comparison between this measurement, the results of previous measurements [7-9], and

the theoretical value for this interval. The theoretical

value was calculated using a"^ = 137.035963(15) from

Ref. [2] and Mohr's calculation [10] of the Lamb shift with the radius of the proton [11] taken equal to 0.862(12) X 10-1^ cm. These calculations gave

V(22P3;2 - 22Pi;2) = 10969.044(2) MHz,

v(22Si/2 - 22Pi;2) = 1057.884(13) MHz,

and

V(22P3^ - 22S1/2) = 9911.160(13) MHz.

The use of Erickson's calculation [12] of the Lamb shift would give

V(22P3;2 - 22S1/3) = 9911.114(10) MHz.

Mohr's value has been used in Fig. 3 since it is most

favored by the recent measurements of the Lamb shift

[5]. The new measurement reported here agrees with the

theoretical value for this interval and is clearly competitive with the other measurements.

Table 1 shows that the primary limitation in the meas- urement of the 2^P;v2 - 2^Si/2 fine-structure interval is the systematic error and not the statistical error. Of the

systematic errors the most troublesome is the variation

in the relative phase of the microwave fields in the two

waveguides when the apparatus is taken apart and

reassembled. It is our conclusion from a recently re- ported measurement of the Lamb shift [5] that these sysT I r Theory

99110 1 2 .3 .4 .5 (MHz)

Figure 3. Comparison of the theoretical and measured values

for the 2'^Psi2 ~ 2^S fiyie-structure interval iyi hydrogen. The

-points marked 1, 2, and 3 are the results from. Refs. [7], [8], and [9], respectively. The origin of the theoretical value is described in the text.

tematic errors can be reduced through improvements in the apparatus, the procedure used to take data, and

further theoretical studies of the resonance line shape.

The replacement of the present detector, which consists

of a single solar blind photomultiplier, by the recently

developed large solid angle proportional counter for Lyman-a radiation [13] should increase the detection effi- ciency by a factor of 100 and enable us to work with reso- nance lines with a width of 15 MHz and the same signal to noise ratio as the resonance lines used in the preliminary measurement. This combination of these improvements should conservatively yield an order-of-magnitude

increase in the precision of the determination of the fine- structure interval to the level of 0.4 ppm. This should

provide a value of the fine-structure constant with a precision of 0.2 ppm or, alternatively, with the use of a

determined from other sources, a measurement of the

Lamb shift to 4 ppm.

References

[1] For a general discussion of the determination of the fine structure constant, see B. N. Taylor, W. H. Pai-ker, and

B. N. Langenberg, Rev. Mod. Phys. 41, 375 (1969).

[2] E. R. Williams and P. T. Olsen, Phys. Rev. Lett. 42, 1575

(1979).

[3] R. S. Van Dyck, Jr., P. B. Schwinberg, and H. G. Dehmelt,

Phys. Rev." Lett. 38, 310 (1977); and Bull. Am. Phys. Soc.

24, 758 (1979). , [4] For a report on the use of the separated-oscillatory-field

technique to measure the Lamb shift in H(n = 3), see C. W. Fabjan and F. M. Pipkin, Phys. Rev. A 6, 556 (1972).

[5] For reports on the use of the separated-oscillatory-field tech- nique to measure the Lamb shift in H(n = 2), see S. R. Lundeen and F. M. Pipkin. Phys. Rev. Lett. 34, 1368

(1975); Phys. Rev. Lett. 46, 232 (1981); these proceedings.

[61 K. A. Safinya, Ph.D. thesis. Harvard University. 1979

(unpublished).

[7] B. L. Cosens and T. V. Vorburger, Phys. Rev. A 2. 16

(1970). [i^(2-P3,2 - 2-81,2) = 9911.173(42) MHz.]

129
[81 T W Shyn, T. Rebane, R. T. Robiscoe, and W. L. Williams,

'Phys. Rev. A 3, 116 (1971). [v{2^P3^ - 2%^) =

9911.250(63) MHz.] ^ ^ , [9] S. L. Kaufman, W. E. Lamb, Jr., K. R. Lea, and M.

Leventhal, Phys. Rev. A 4, 2128 (1971). [v(22P3;2 -

2^Sy2) = 9911.377(26) MHz.]

riO] P. J. Mohr, Phys. Rev. Lett. 34, 1050 (1975).

[11] G. G. Simon, F. Borkowski, Ch. Schmitt, and V. W. Walther,

Z. Naturforsh. 35A, 1 (1980).

[12] C. W. Erickson, Phys. Rev. Lett. 27, 780 (1971).

[13] J. J. Bollinger and F. M. Pipkin, Rev. Sci. Instrum. 52, 938

(1981).

130
Precision Measurement and Fufidamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Measurement of the Lamb Shift in Hydrogen, n = 2*

S. R. Lundeen and F. M. Pipkin

Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138

A measurement based on the fast atomic beam separated oscillatory field method of sub-natural Hnewidth spectroscopy gives for the Lamb shift in hydrogen: S {n = 2) = 1057.845(9) MHz. The result

is not in good agreement with theory.

Key words: fast atomic beam; hydrogen fine structure; Lamb shift; quantum electrodynamics;

separated oscillatory fields.

Measurements of the Lamb shift interval (2^Si/2 ~

2^Pi/2) in hydrogen and hydrogenic ions are among the

most sensitive tests of QED at low energies, and in particular of QED's description of the electromagnetic va- cuum state. The Dirac Equation, which describes the in- teraction of nucleus and electron relativistically, but in the absence of vacuum fluctuations, predicts incorrectly

that this interval is zero. More sophisticated theoretical

descriptions of these fundamental bound states reproduce

the measured interval to at least 100 ppm. Improvements

in the measurement precision beyond this level are diffi- cult because the resonance linewidth is normally limited

by the 100 MHz natural width of the 2Pi/2 state which is about 10% of the Lamb shift interval. In the measure- ment described here [1], the method of separated oscilla- tory fields [2] is used to observe resonances that are narrowed considerably below this natural width. The in- creased resolution thus obtained, combined with the re- duced sensitivity to systematic effects and the convenient

availability of systematic checks lead to significant improvement in the experimental precision.

The fast atomic beam apparatus used for this measure- ment is shown schematically in Fig. 1. Hydrogen atoms

in the 2^Si/2 {F = 0) state are prepared by charge ex- change from a fast proton beam, using radiofrequency

hyperfine-state selection techniques. The beam passes

through two separated coherent oscillatory fields and the

surviving number of metastable atoms is monitored by

observing the Lyman-a photons emitted in Stark induced

decay to the ground state. The fractional quenching of the metastable beam by the separated oscillatory fields is measured for relative phases of 0° and 180° between the

two fields. When averaged these two signals give a large

broad signal (Q ) whose width is determined by the natural linewidth and the transit time through the rf

plates. The difference of the two signals (/ = Q(0°) —

Q(180°)) gives the interference signal whose width is determined by the transit time between the two regions.

Measurements are made in zero magnetic field and the resonance observed by changing the frequency of the applied rf field while keeping its amplitude constant. Figure 2 shows a typical line profile for the 2^Si/2

(F = 0) - 2^1/2 (F = 1) transition. For each configuration, line profiles are taken with (a) the direction of the

propagation of the rf field in the interaction region re-

*Research supported in part by National Science Foundation Grant No.

78-^9657

Figure 1. Fast-beam, separated oscillatory field apparatus.

The coaxial magic tee divides the rf power equally betiveen the two separated rf regioyis with relative phase of 0° or 180°.

versed, (b) the time order in which the beam passed

through the interaction region reversed, and both (a) and

(b). The average center for four such lines is free from

shifts due to the Doppler effect and rf phase errors. The

average of the four line scans is found to be very symmetric; the raw data shown no asymmetry larger than

0.1% of the peak to peak interference signal, all of which

is accounted for by systematic effects. Thus the line centers can be determined by the method of symmetric

points. As a check on systematic effects, measurements

are made in eight different experiment configurations

which differ in beam velocity, plate separation, and plate

geometry. Table 1 lists the eight configurations

together with the sizes and widths of the interference

signals.

The raw experimental line centers obtained in each con- figuration are shown in Table 1 with the associated

statistical error (1 standard deviation). These numbers

must be corrected for several small systematic effects,

the total of which is shown in Table 1. The origin of the

corrections, a more detailed tabulation of which is given

in ReL [1], is given below as well as an indication of the

maximum systematic uncertainty (one standard deviation) which each produces in any of the eight configurations of Table 1.

131
Figure 2. Measured interference (I) and average quenching

(Q ) signals for coyifiguration U- The radius of the plotted

points is lOa.

1) Time dilation ± 0.004 MHz): The resonance fre- quency should be measured in the moving atom's

rest frame.

2) Bloch-Siegert and rf Stark shifts ± 0.004

MHz): The shifts produced by the anti-resonant

2-S1/2 - 2''^Pi/2 and non-resonant 2"Si/2 - 2^P;y2

coupling. The net shift depends on plate spacing

through a calculated geometric factor.

3) Plate coupling ± 0.002 MHz): The impedance of the rf plates depends slightly on whether they are driven with 0° or 180° relative phase.

4) F =1 ± 0.001 MHz): There are residual traces of the 22Si/2 iF=l) state.

5) Incomplete Q subtraction ± 0.013 MHz): Systematic errors in the control of relative rf power

between 0° and 180° states may produce a small

spurious signal.

6) Off-axis distortion ± 0.004 MHz): The non- uniform rf-field polarity seen by atoms traveling

through the rf plates off axis may lead to small

distortions.

7) rf-field slope ± 0.003 MHz): Laboratory calibra- tion of the rf electric field strength within the

plates indicates that under experimental conditions

it varies hy AE/E - - 0.08(7)% /lOO MHz.

8) Additional field slope ± 0.010 MHz): The need

for this ad-hoc correction, equivalent to a variation

in rf electric field strength by 0.5(2)% / 100 MHz, is indicated by three independent lines of internal

evidence. This may indicate that an error was made in calibrating the rf fields. On the other

hand, it may indicate that some other systematic

effect is present whose physical origin is unknown

but which, in effect, mimics an error in the rf field calibration. Work is in progress to clarify the ori- gin of this problem, but in the meantime the

proper treatment of the data seems clear. A more

detailed discussion of this correction is given in Ref. [1]. The sensitivity of the line center to individual sys- tematic errors varies widely between the eight configurations. For instance, although correction eight results in a systematic uncertainty of ±0.010 MHz in configuration 1,

it gives an uncertainty of only ±0.002 MHz in configuration 6. Several other potential sources of systematic er- ror have been studied and found to be negligible. They

are: overlapping resonances in high )i states detected through

cascades; stray electric and magnetic fields; and coherent

population of S and P states. The corrected line centers shown in Table 1, constitute

eight independent determinations of the 2-S1/2 (F = 0) -

2^Pi/2 (F = 1) interval, spanning a range of 32 ppm of

the Lamb shift interval. To compute the final result, the

results from 55 keV and 107 keV configurations are first combined in separate weighted averages. We assume

that each systematic error is completely correlated

between different configurations, but that different sys- tematic effects are mutually uncorrected. The results for the two beam energies are in excellent agreement and

when averaged give:

v[22Si/2 (F = 0) - 22Pi,2 (F = 1)] = 909.887(9) MHz

vdth a random error of ±0.004 MHz and a systematic er- ror of ±0.008 MHz. Correcting for the known hyperfine

structure of the 2^Si/2 and 2^Pi/2 states, the Lamb shift interval is found to be

S (H, n = 2) = 909.887(9) + 147.958 "

' = 1057.845(9) MHz.

Table 1. A summary of the configurations in which data were taken. One independent result requires four line profiles.

Config. Beam Energy

(keV)

Plate Spacing

(cm)

Peak to Peak

I Signal (%)

FWHM

(MHz)

Number of Independent

Results

Raw

Center (MHz)

Total Systematic

Correction

(MHz)

Corrected Center (MHz)

Weight

1 55.1(1.1) 2.34 6.08 45 5 909.930(8) -0.030(12) 909.900(15) 0.230 2 55.1(1.1) 3.30* 3.65 41 8 909.887(10) -0.017(9) 909.870(13) 0.230

3 55.1(1.1) 3.30 2.95 38 4 909.893(6) +0.008(6) 909.901(9) 0.344

4 55.1(1.1) 4.30 1.35 32 4 909.847(11) +0.020(8) 909.867(14) 0.142

5 55.1(1.1) 5.08 0.66 29 4 909.844(18) +0.027(14) 909.871(23) 0.054

6 106.8(2.1) 5.08 1.82 35 8 909.818(9) +0.069(7) 909.887(11) 0.590

7 106.8(2.1) 6.62 0.64 28 6 909.807(11) 0.079(9) 909.886(14) 0.365

8 106.8(2.1) 7.60 0.37 25 2 909.788(38) 0.081(14) 909.869(40) 0.045

*For configuration 2, the inner diameter of the cylinder pipe surrounding the rf field region was increased from 3.81 cm to .5.08 cm.

132
The result, shown in Fig. 3, is in good agreement with

previous measurements of Lamb, et al. [3, 4] and Robiscoe, et al. [4, 5] both of which were obtained using

thermal atomic beams and Zeeman tuning for the Lamb

shift resonance, and also with the more recent measure- ment of Newton, Andrews, and Unsworth [6], obtained

with a fast atomic beam in zero magnetic field without

the use of separated oscillatory fields.^ The predictions of QED for the Lamb shift interval have been evaluated

most precisely by Erickson [7] and Mohr [8]. When modified to reflect recent redeterminations of the proton rms

MOHR ERICKSON

.80 .90 J (H.n = 2) - 1057.00 MHz

Figure 3. Comparison of the result of this experiment with pre- vious measurements A (Refs. [3, Jf /), B (Refs. [4, 51), C (Ref.

[6]) and with theoretical calculations of Erickson [7] and Mohr

[8]. The experimental error bars represent ±1 standard de- viation, as do the widths of the vertical bars showing the the- oretical results.

charge radius (r^ = 0.862(12) fm) [9], their results are (in

MHz):

Erickson: S (n = 2) = 1057.930(10), T-E = 0.(J8.5(13) Mohr : Sin = 2) = 1057.884(13), T-E = 0.039(16).

As illustrated in Fig. 3, neither result is in good agree- ment with experiment.^

Based on the results obtained to date, the prospects

for further improvements in this measurement appear

promising. The statistical signal to noise ratio can be improved by a factor 100 by more efficient detection of the

metastable flux. This will allow precise studies of even narrower lines (FWHM < 5 MHz) and better checks of systematic effects. At this time, except for the need to

clarify the origin of correction 8, no fundamental sys- tematic problems are yet evident. It appears reasonable,

therefore, to hope for a measurement with a precision of ±0.001 MHz (1 ppm) using this technique. Full utilization of such a result would require a slight improvement in knowledge of the proton size to 0.4 percent and a considerable improvement in the precision of the QED calculations. References

[11 S. R. Lundeen and F, M. Pipkin, Phys. Rev. Lett. 4fi, 2^2

(1981).

[2] N. F. Ramsey, Molecular Beams (Oxford Univ. Press, London, 1956), p. 124.

[31 S. Triebwasser, E. S. Dayhoff, and W. E. Lamb, Jr., Phys. Rev. 89, 98 (1953).

[41 B. N. Taylor, W. H. Parker, and D. N. Lanjjenberg, Rev. Mod. Phys. 41, 375 (1969).

[51 R. T. Robiscoe and T. W. Shyn, Phys. Rev. Lett. 24, 559

(1970).

[61 G. Newton, D. A. Andrews, and P. J. Uilsworth, Philos. Trans. R. Soc. London, Ser. A:290, 373 (1979).

[71 G. W. Erickson, Phys. Rev. Lett. 27, 780 (1971).

[81 P. .J. Mohr, Phys. Rev. Lett. :34, 10.50 (1975).

[91 G. G. Simon, F. Borkowski, Ch. Schmitt, and V. W. Walther,

Z. Naturforsch. Teil A:35, 1 (1980).

Editors' note: See also the paper by Yu. L. Sokolov in these proceedings.

-Editors' note: Additional nucleai' size corrections proposed by E. Borie

(Phys. Rev. Lett. 47, Xi< (1981)) would reduce the theoretical values of S

by 40 ppm, bringing them into much better agreement with the e.xperi- mental value reported here, but see also the comment by G. P. Lepage.

D. R. Yennie, and G. VV. Erickson, Phys. Rev. Lett. 47". 1()40 (1981)'.

Precision Measurement and Fundamental Constants 11, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Atomic Interferometer Method Measurement of the Lamb Shift in Hydrogen

(n = 2)

Yu. L. Sokolov

Kurchatov Atomic Energy Institute, IVIoscow, U.S.S.R.

The frequency of the (2Si^2. F = 0) - (2Pi^2. F = I) transition in the hydrogen atom has been measured by means of an atomic interferometer. The Lamb shift has been found to be S = 1057.8583 ±

0.0022 MHz, where the uncertainty is the statistical standard deviation of a single observation. Key words: atomic interferometer; atom velocity measurement; hydrogen atom; Lamb shift.

1. Introduction

The measurement of the Lamb shift S in the hydrogen

atom is known to be one of the most important verifica- tions of QED. However, in spite of the fact that more

than a quarter of a century has been devoted to the

measurement of S,^ the increase in accuracy has proved

to be rather modest. The spread in the experimental

values of the Lamb shift measured by current methods is obviously too great to make a definitive comparison with

the theory. As a matter of fact, the theoretical value of S cannot be calculated to an accuracy much better than

0.01 MHz. Thus, if a discrepancy is found to exist

between Sexp and Stheo. it is unclear to wh'at fact it should

be ascribed—most likely to errors in the measurements

and calculations, rather than to a failure in QED.

We performed a measurement of S by means of the

atomic interferometer described in Refs. [2] and [3]. The

results obtained indicate that we may expect a further

increase in accuracy.

2. Experimental Setup

Figure 1 shows a schematic diagram of the "Pamir"

unit which was used for the Lamb shift measurements.

Protons with energies of about 20 keV emerging from the

ion source (1) were passed through the velocity analyser

consisting of a magnet (2) and a 0.02 cm wide slit dia- phragm (3). The magnet dispersion measured in the plane

of this diaphragm is 163 eV/cm; thus the energy spread

of protons passing through the diaphragm does not exceed 3.25 eV, about 1.6 x 10"^ of their energy.

'The classic work by S. Triebwasser, E. S. Dayhoff, and W. E. Lamb

[1] was performed in 1953.

Neutral hydrogen atoms were produced in the charge

exchange chamber (4). The mixed beam was passed

through a weak magnetic field (5), deflecting the proton

component by an angle of about 0.2°; the proton current was measured by the Faraday cup (6). A parallel plate

capacitor (7) was used for quenching 2Siy2 atoms, which was necessary for the background determination. The

atom velocity was measured by means of system (8) which was essentially an ordinary interferometer with a

longitudinal field [2] and a fixed inter-electrode distance

(i.e., with a fixed time-of-flight). The 2Pi^ atoms pro- duced under the action of the nonadiabatic field were

recorded by a fixed detector (8a) and a detector (8b)

which could be moved along the beam trajectory over a range of 1.5 cm. When not operating, the electrodes of system (8) were removed from the beam trajectory.

For monitoring the 2Si/2 atom flux a portion of the

beam was transmitted through the pickup (9) consisting

of a parallel plate capacitor where 2Si/2 atoms were

quenched and a L^ detector. To eliminate the 2Si/2 state component with total angular momentum F - I, the

beam, as usual, was transmitted through rf fields with

frequencies 1147 and 1087 MHz. The intensity of the F =

0 component was monitored by means of a pickup (11)

analogous to pickup (9). The remaining part of the beam

passed through the "double" interferometer (12) with

the detector (13). Pickup (14) had a quenching field and was intended for observing the interference of the 2Siy2 state components formed by the nonadiabatic fields. The

total neutral atom current was measured by the end

detector (15). In the experiments described, the neutral H atom

beam length measured fi'om charge exchange chamber (4) to the inlet slit of the interferometer (12) was 300 cm. Such a length was chosen so that the states with » = 3 -

8a 9 II 14 D D^nii'ii Di_,

ni I \ I lOa lOb 8 8b 13

Figure 1. Diagram of "Pamir" unit (notations in the text).

135

15
7 which are populated rather densely and at the same

time are comparatively long-lived had a sufficient time

for de-excitation since their cascade transitions to the IS

ground state produce the quantum background.

3. Ion Source

The main requirement imposed on the ion source was

to obtain a thin (3^ mm cross section) proton beam

which was only weakly divergent and stable in time.

We developed a source which produced a two- component beam whose total current was 2.5-3.0 mA.

This beam contained a comparatively weak central component with an angular divergence of about 10"^ and a current of 70-100 |jlA. The diverging part of the beam

was stopped by the diaphragm and the "jet" component,

without any focusing, was directed onto the inlet slit (3) of the charge exchange chamber.

Molecular hydrogen was chosen as the gas where pro- ton charge exchange took place. Such a choice may seem

not quite reasonable since the process of proton interaction with H2-molecules can proceed in several channels,

which will result in production of 2Si/2 atoms with dif- ferent velocities. However as the collimation conditions are very stringent, a beam fraction with a divergence not exceeding lO""* came into the interferometer. Such a selection by trajectories ensured a high uniformity in neutral atom velocities.

4. Background Estimation

Let us introduce the following notations:

mi —"natural" detector background;

mz —backgi'ound produced by deexcitation of highlyexcited atoms; m z —background due to atom collisions with the residual

gas; m4 —background due to interaction of the beam atoms

with the edges of the interferometer slits; m5 —background arising because of penetration of quanta to the detector through the outlet interfer- ometer slit. The main contribution to the total background is the

m2 component. As noted above, its intensity may be re- duced if the observation point is put at an appropriate

distance from the charge exchange chamber.

We calculated contributions to the m.2 component from

the (n S - 2P) and (w D - 2P) transitions for 3 ^ n ^

25. It follows from the results obtained that the main

part of the L^, background is due to the (4S - 2P) and

(5S - 2P) transitions with lifetimes of 3.88 x 10"^ sec and 7.76 x 10"'' sec, respectively. The flux of quanta

produced by a beam section with length Ax = 1 cm (in the region 300 cm from the charge exchange chamber)

reaches 5 x 10"^ of the neutral atom flux. The control experiments showed that the measure- ments performed when the quenching field (7) is switched on give the same results with the non-adiabatic field switched either on or off, and, hence, such a field does

not have a noticeable effect on highly excited atoms.

Estimation of the L„ quantum flux arising due to colli- sions of His atoms with the residual gas (which is mainly

hydrogen flowing from the charge exchange chamber)

shows that this background component proves to be

negligibly small (of the order 5 quanta/sec at the gas

pressure ~5 x 10"*^ Torr). Equally as small are the contributions from excited atoms and from 2Si/2 atoms be- cause their flux is about 1.5% of the IS1/2 and the cross section of 2Si/2 atom quenching in collisions is about 100

times smaller than that of the His + H| H2P + R

process. The nii and background components are eliminated

in adjustment of the device.

The total background value was determined in the ex- periment when no fields were on. In this case comparison

of readings of the end meter (15) and detectors (9) and

(13) allowed us to conclude that the background recorded

is proportional to the neutral atom current within the

limits of the attainable accuracy of L^, quantum flux measurements.

In the experiments on atomic state interference the

value

A^i - A^i N9 - N2h was determined, where A'^i and ^^2 are the readings of detector (13) and monitor (11) with quenching field (7) svdtched off and Nih and Nzh are the background values

measured, i.e. the readings of the same detectors with

the quenching field switched on. For the sake of reducing

the measurement time, in some experiments reading of the end meter (15), monitor (11), and detector (13) were

recorded with the quenching field switched off, i.e. the

values A'^i and A'^2 were determined. The corresponding

values Nih and Nzt were found from readings of the end

meter (15) calibrated in advance.

5. Interferometer

The analysis performed by V. P. Yakolev shows that

the Lamb shift may be determined using a two-region in- terferometer consisting of electric fields I and II (see

Fig. 2) changing nonadiabatically on the boundaries [3].

H2S

L-DETECTOR

Figure 2. Diagram of the "dual" atomic interferometer.

When using such a device the interference pattern of the two phase-shifted 2P state components induced by

fields I and II is described by the following formula:

136
F(M) = Acos( M + a)

+ B cos( A/ + p)

X exp (1)

where ?o = v t, v is the atom velocity, t is the lifetime of

the 2P atom; w = 2Trv is the (2Siy2, F = 0) - {2Py2, F =

1) transition frequency; coi = 2-77 vi is the fine struction

splitting frequency; is the change in the distance

between the inlet I and outlet II systems of the interfer- ometer. The A, B amplitudes and phases a, p depend on

the condition in systems I and II and enter as parameters which are not calculated. Equation (1) can be rewritten as:

F'(AO = A' cos(- AO

-I- —B'

cos —^\ \f

A V V^

exp AL

2L

AO

(2)

Here A' is the scale factor which is of no interest; thus in fitting the theoretical curve to the experimental points

the values w/v and B' /A' are determined.

If the field strengths in systems I and II are about 250

V/cm then B' /A' vdll be equal to about 0.1. Thus curve F '(AO vdll represent a superposition of two cosinusoids

modulated by the attenuating exponent; the second

cosinusoid has a frequency a factor of 10 higher and an

amplitude a factor of 100 lower than the first cosinusoid,

and thus produces small, fast oscillations about the

latter. In the experiments the La quantum yield recorded by

detector (13) was measured as a function of the time-offlight T between systems I and II. The strengths of the

nonadiabatic fields Ei and E2 were held constant during

the measurement.

A change in the time-of-flight T was achieved by

changing the distance I between systems I and II. For

this purpose the inlet system I was displaced relative to the fixed system II over a range of 0-2 cm by means of a

precision mechanism. The displacement A/ was read (or

monitored) by a precise scale rigidly connected to system

I. Readings were made by means of a microscope pro- vided with an eyepiece micrometer having a double hair

line. The reading accuracy was 1 x 10"^ cm.

In determining the l2viT)E^,E^ dependence, the fol- lowing procedure was adopted:

1. At given field strengths in systems I and II and a given distance / the number of quanta from the 2P

state decay is determined by detector (13) after atoms

have passed through the interferometer, i.e., the value

h'p{T)E^,E2 =/ (A/) is found.

2. The direction of the field in system II is reversed,

with other conditions remaining constant. The measure- ment gives the value /2P (T)/,. , = /(A /).

3. Difference / - 7 is found.

4. The field E2 is returned to the initial value. Then

the distance I is changed by M and two more measure- ments are made with different directions of the field E2.

5. The procedure is repeated for a chosen series of increments M. If a certain L,, (its value is of no importance) is taken as the initial distance between the

systems, the increments A/ may be both positive and

negative. The dependence F (AO = / (AO - /(A/) ob- tained in this way is described by the theoretical formula

Eq. (1). Prior to the initiation of the measurements, with the

distance I - 0 and the outlet electrode of system II grounded, an appropriate strength of the field E \ in sys- tem I was determined, corresponding to the interference

maximum. Then, with system II on, the field E2 was

similarly chosen. The strengths of both fields were of the

order of 300 V/cm. Under this condition the number of Lc quanta recorded by detector (13) in the interference

maximum for I - 0, was about 5 x 10^ per second. The

measurements were carried out for 0 ^ ? ^ 2 cm at 41

points, i.e., for AZ = 0.05 cm. Determination of the co/v and B' /A' values in Eq. (2) was done by a least squares

fitting procedure.

To check the state of the setup, its resolution relative to the Lamb shift was periodically determined using the

method described in Ref. [2].

6. Measurement of 2S.,/2-Atom Velocity

It follows from Eq. (2) that for determination of the

Lamb frequency S the 2Siy2 atoms velocity must be measured by an independent method. Stabilization and meas- urement of the velocity are the most complicated parts of

this experiment and the source of the main limitations of the method.

The 2Si/2 atoms' velocity can be determined, in principle, by transmitting them through the quenching electric

field and observing the resulting L^ radiation at a small

angle with respect to the beam trajectory. Then the atom

velocity can be found by the Doppler shift of the Lq line. This method was tested on the "Pamir" setup with a one meter spherical diffraction grating having 1200 lines /mm.

However, the results obtained revealed that such a method does not permit one to determine the velocity

with the required accuracy (not better than 10"^), mainly

because of the low aperture ratio of the device. In all the Lamb-shift-experiments the atom velocity was measured by observation of de-excitation of 2Pl2

atoms produced under the action of the non-adiabatic

field. When the detector is moved along the beam trajectory,

the Lo, radiation intensity will vary as / = /o exp(-x /lo), where x is the distance measured from an

arbitrary reference point, lo = vi where v is the atom

velocity, and t is the 2P atom lifetime. Thus, the velocity

(more accurately the value lllo) can be determined by

the slope of the straight line ln(7 ) = const - (.r /lo )

.

The corresponding measurement system must include

two detectors one of which (monitor) is fixed and the

other moves along the beam trajectory.

As has been said, in the experiments on the Lamb shift the determination of the velocity was performed by means of system (8) which has a detector (8b) with a dis- placement mechanism similar to that of the interferometer detector (13). For determination of the atom velocity

the movable system I was also used in combination with

detector (13). In this case the system II electrodes were

grounded.

At first glance it might appear that determination of

137
the value lo using the method under consideration can be

made by measuring the intensity at two points of the

beam trajectory, divided by a distance Xg, which is a typical procedure in the case of determination of an absorption coefficient. However, such a procedure is only valid

when it is known beforehand that the intensity of radiation recorded decreases exponentially with x and that the

experimental parameters do not vary with time. As a consequence the "two point" method proves to be in- valid, in principle, for a measurement of the beam atoms'

velocity since a supposition of the exponential nature of the Izpix ) dependence is accepted a priori and, generally

speaking, may be invalid either because of experimental

defects (e.g., velocity variations) or because of the influ- ence of some unknown factors. Thus the analysis of the

experimental data must include verification of the linearity of the function studied. In addition, if the intensity is recorded only at two

points of the trajectory, an error in the measurement of the distance Xo appears as a systematic error. When, on

the other hand, the measurement is performed at many

points, errors of this type become random, which permits one to make a statistical treatment of the data using the

least squares method.

The following procedure was adopted in accordance

with the considered formulation of the problem: The

radiation intensity was measured at several points of the

beam trajectory, i.e., for a number of discrete values

XiH = 1), 2, 3, . . . . k), n separate measurements being

performed within equal time intervals at each ?-th point.

Then for the set of values = In (/,,,„), (m = 1, 2 . . n) corresponding to the theoretical dependence Y =

a + ^x where Y = In /, a = In /o and p = l/lo, the empirical dependence (regression line) y - a + ^x was

found. Calculation of its parameters allows one to find lo and determine its standard error. Measurements of the 1 2P intensity were made at 26

points for Ax = 0.04 cm at the 1 cm long trajectory section; 20 measurements were performed at each point.

The number of counts at the first point (i = 1) being at a distance of 0.03 cm from the outlet electrode of system

(8) was about 6 x IC* pulses/sec. In the control experiments, in addition to the statistical error A lo , the resolution of the setup (considered as an instrument for atom

velocity measuring) was determined by the method

described in Ref. [2]. In an experiment of this type the

difference in the velocities of His and H2S atoms produced

in the charge exchange chamber was determined. The

measured difference in Ig values corresponds to a differ- ence in energies of about 10 eV, which is indicative, first, of a sufficiently high resolution of the method and,

second, of the circumstance that in the beam one monoenergetic group of 2S1/2 atoms is present.

In the last version of the "Pamir" device the atom velocity is stabilized to a sufficient degree after a 3 hour warm up of the entire system. If no failures occur in the

operation of the apparatus, the velocity stays constant

for about 2.5 hours and then begins to decrease slowly.

The atom velocity changes somewhat after the apparatus

is turned on again.

7. Results

For two years about 200 Lamb shift measurements were carried out. However, it followed from the data ob- tained that only in 34 cases the atom velocity remained

constant within permissible limits when the 1 2viT )e ^, e ^ dependence was determined (the velocity was measured

in the beginning and in the end of the experiment).

Figure 3 shows the histogram for the v values. The

blocks shaded represent the v values obtained in treating

the measurements according to the criterion of selection

by velocity constancy. They form a compact group with a mean value of

V - 909.9003 ± 0.0022 MHz

The error is equal to the standard deviation of a single

observation and represents statistical uncertainty only.

The Lamb shift 1057.8583 ± 0.0022 MHz

corresponds to this v.

909.890 909.900 909.910

i/,MHz

Figure 3. The histogram, of v values.

The value of S given here is somewhat different from

that reported earlier [4]. This is due to the fact that by

the time the present paper had been written, V. P. Yakovlev improved the value of the 2P atom lifetime

(t = 1.596185 X 10-9 gg(, found).

Thus, after an appropriate correction was introduced,

S was found to be in a satisfactory agreement with the

experimental values 1057.862(20) and 1057.845(9) MHz

obtained in Refs. [5] and [6], respectively. However, the

value found differs from the theoretical data available

[7, 8], which was also pointed out in Ref. [6].^ But it seems one need not lend a great importance to this discrepancy since the differences between the theoretical

and experimental S values do not definitely go beyond

the error limits. Therefore one cannot speak with confi- dence about the existence of any difference between S exp and Stheo - It appears that further improvement of the S

measurement and calculation accuracy is needed.

The general impression the author has after many

years of work with the atomic interferometer is that such an increase in the accuracy is quite feasible. The method

must be improved first of all by replacement of the

^-measuring arrangements consisting of scales and microscopes with precision transducers which transform the

linear displacement of movable elements into electrical signals expressed in numerical code.

The author is deeply grateful to V. P. Yakovlev for his constant help and numerous discussions.

References

[1] S. Triebwasser, E. S. Dayhoff, and W. E. Lamb, Jr., Phys. Rev. 89, 98 (1953).

^Editor's note: see also footnote number 2 to the paper by Lundeen and

Pipkin in these proceedings.

138
[2] Yu. L. Sokolov, Zh. Eksp. Teor. Fiz. 63, 461 (1972) [Sov. Phys.

JETP 36, 243 (1973).]

[3] Yu. L. Sokolov, m Atomic Phiifiica (i, Ed. by R. Damburg (Plenum

Press, New York, 1978), p. 207.

[4] Yu. L. Sokolov, Program and Abstracts of this conference, p. 63.

[51 G. Newton, D. A. Andrews, and P.J. Unsworth, Philos. Trans.

R. Soc. London, Ser. A: 290, 373 (1979).

[61 S. R. Lundeen and F. M. Pipkin, Phys. Rev. Lett. 46, 232 (1981); these proceedings.

[7] G. W. Erickson, Phys. Rev. Lett. 27, 780 (1971).

[8] P. J. Mohr, Phys. Rev. Lett. 34. 10.50 (1975).

139

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Measurement of the 4 S^^^ - ^ P-1/2 Lamb Shift in He *

J. J. Bollinger, S. R. Lundeen, and F. M. Pipkin

Lyman Laboratory, Harvard University, Cambridge, MA 02138

The separated oscillatory field technique is being used with a fast 120 keV He" beam and a minia- ture microwave spectroscopy' region to make a zero magnetic field measurement of the n = 4 Lamb

shift interval in He^. The iSyo state population is monitored by observing its decay to the 2P state with a large solid angle VUV photoionization detector. With 10 minutes of counting time, the signal to noise ratio is sufficient to make a 100 ppm measurement of the line center. It is believed that with this signal to noise ratio the systematics can be well enough understood to determine the line center better than previous He"^ Lamb shift measurements and test the Erickson and Mohr calculations of the Lamb

shift. Key words: cascades; He"*" Lamb shift; large solid angle VUV detector; overlapping transitions; separated oscillatory fields.

1. Introduction

Measurements of the Lamb shift of hydrogenic ions

provide sensitive low energy tests of quantum electrodynamics. The sensitivity of these measurements is determined both by the accuracy of the experimental measure- ments and by the accuracy of the theoretical calculations. In hydrogen, a major source of uncertainty is the proton rms charge radius. Because the uncertainty in the a particle rms charge radius [1] is about a factor of five less than the uncertainty in the proton rms charge radius [2] (Ar/r = 0.03% and 1.4% respectively), measurements of the Lamb shift intervals in '^He^ provide, potentially, some of the best tests of quantum electrodynamics in hydrogenic ions. In addition, the absence of hyperfme

structure in *He* provides an experimental simplification over Lamb shift measurements in H. These factors have

motivated us to make a Lamb shift measurement in ^He"^ using the fast beam, separated oscillatory field technique

(SOF) developed in this laboratory [3]. The n = 4 '^Sy^ -

^Pi^ Lamb shift has been chosen as an experimentally

convenient Lamb shift to measure with this technique

and our present 120 keV fast beam.

Figure 1 shows all direct sub 250 ppm Lamb shift measurements in '*He^ together with two recent calculations. Comparisons between measurements of the Lamb

shift intervals n - S1/2 - n ^Pi-? for different n are made by

normalizing all measurements to the n = 2 frequency

scale. The overall agreement between theory and experiment is not good. The precision of these measurements

(approximately 100 ppm) is more than a factor of nine worse than the current best Lamb shift measurements in hydrogen [3, 10]. Significant improvements are needed

in the precision of He^ Lamb shift measurements to reduce the experimental uncertainty to the level of uncertainty in the theoretical calculations and take full ad- vantage of the precise knowledge of the ^He^ nuclear

charge radius.

Although the Mader, Leventhal, and Lamb measure- ment [6] in Fig. 1 is the only sub 250 ppm Lamb shift measurement involving a non-metastable (w ^ 3) "Siy2

*Research supported in part bv National Science Foundation Grant No.

78-09657.

state in He"^, many others have been made much less precisely. A major systematic problem common to most

of the n & 3 experiments has been contributions to the

Lamb shift resonance signal due to overlapping transi- tions in higher n states. For this or any experiment

involving a Lamb shift measurement with n ^ 3 to be

successful, contributions due to overlapping transitions

in higher n states must be identified and taken into account.

2. Description of the Experiment

Figure 2 gives a sketch of the basic experimental lay- out. Excited He^ atoms are created by collision by passM[8] E[9]

S ("^He"") - 14,000 MHz

Figure 1. Previous measurements of the Lamb shifts in ''He'^ [i-7] with tivo calculations by Mohr [8] and Erickson [9]. The

dark circles are n = 2 measurements. The open circle is a?; n = 3 measurement normalized to the n = 2 frequency scale by multiplying by the ratio of Erickson's values for the n = 2 and n = J ^He'^ Lamb shifts.

141
4P LIFETIME

H4S LIFETIME

I 1 1cm

i J L

SOF

SPECTROSCOPY

REGION

EXCITATION

CELL 20 GHz

CAVITY

VACUUM

BAFFLE

4S STATE

LYMAN a DETECTOR

B3-

BEAM

MONITORING

FARADAY

^CUP

// -

II'

Figure 2. Sketch of the experimental arrmigement.

ing a fast (120 keV, 2.4 x 10*^ cm/sec) He^ beam

through a short, 4 cm cell at 10-40 mTorr pressure. Due

to their lifetime, 4^P He^ atoms will decay almost immediately, lasting on the average less than 2 mm down

the beam line. After passing through a vacuum baffle iso- lating the excitation cell chamber from the rest of the vacuum system, the He^ atoms enter several microwave

spectroscopy regions. The second region is the SOF re- gion consisting of identical, 0.7 cm wide, rectangular50ft coaxial transmission lines separated by a small dis- tance (typically 0.00, 0.32 or 0.64 cm). After the micro- wave spectroscopy regions, the 4-Si/2 state population is monitored by observing its decay (121.5 nm) to the 2P

state with a large soHd angle vacuum ultraviolet photoionization detector.

Applying microwave power near the n = 4-Si/2 - ^Pi/2 resonant frequency to the SOF region transfers population from the 4^Si/2 state to the short lived 4"Pi/2 state, resulting in a decrease of the 4-Si/2 state population and

the output of the photodetector. For a fixed phase differ- ence of the two SOF regions the quenching signal of the

4^Si/2 atoms is defined by the percent charge in the out- put of the photodetector when microwave power is applied. The width of the quenching signal is determined by

the natural linewidth and the transit time through one of the rectangular transmission lines. The difference in the

quenching signals when the two SOF regions are driven

in phase and 180° out of phase is the interference signal.

The width of the interference signal is determined by the

transit time between the two SOF regions. In addition to the SOF spectroscopy region described

above, a fixed 20.18 GHz spectroscopy region consisting

of a K-band waveguide cavity is located just before the

SOF region. Applying resonant microwave power to this cavity quenches the 4^Si/2 state via the 4^Si/2 ~ 4:^^3/2 transition (see Fig. 3) and permits the observation of any contributions due to overlapping transitions in higher n states. These contributions are subtracted from the un- quenched Lamb shift signal.

Data are taken in zero magnetic field and as a function

of the frequency of the microwave power applied to the

SOF region across a 300 MHz bandwidth centered on the

1769 MHz Lamb shift transition. In addition, to cancel

first order Doppler shifts and possible phase errors between the two SOF transmission lines, data are taken

with the direction of propagation of the microwave field in the SOF region reversed, and with the time order in which the beam passes through the SOF transmission

lines reversed. The average of these four interchanges

will be free of first order Doppler shifts and phase errors between the two SOF transmission lines. After any

asymmetries in the lineshape are understood and

corrected for, the method of symmetric points is used to calculate the line center. The large solid angle Lyman a (121.5 nm) detector

consists of six NO photoionization chambers with rec- tangular MgF2 windows which totally surround the beam

(see Fig. 4). The detector housing is made out of alumi- num and consists of an inner and outer cylinder, two end

flanges, six perforated chamber separations, and a filling

tube, all aluminum dip brazed together into one piece.

The electrons from the NO photoionization are collected on 0.76 mm diameter probes extending from the rear of each chamber and separated from the chamber housing

by ceramic insulators and a metal guard ring to reduce

leakage currents. The Lyman a detector is operated in the proportional mode with the aluminum housing at negative voltage. The average fractional solid angle subtended by the photodetector over its 3.5 cm length is greater than 0.55. The combined detection efficiency of the 2 mm MgF2 window transmission and the NO photoionization quantum efficiency is estimated to be 35%

.

Consequently the overall detection efficiency for Lyman

a photons is approximately 20% over the 3.5 cm length

of the detector.

3. Discussion

Figure 5 shows some data taken with the plates

separated by 0.32 cm. The contribution to the un- quenched Lamb shift signal due to the overlapping transitions in higher n states is approximately 10% of the

Lamb shift signal height. The dominant overlapping tran142
h/2

P3/2 3,725

11.217

J, 87

; "3.740-

'D3/2

7/2 _1.12

G9/2

•'7/2

FILLING TUBE

22.5 1.5 4.4 14?

"2

^il2 3.64 6

7.304 ^^'2

14.4 .78 2 3 4 6

Figure 3. Fine structure intervals (GHz) and lifetimes (nsec) in the n = i and n = 5 manifolds of ''He'*'. The n = 5 (J =

5/2) - (J = 712) transitions at 1.87 GHz overlap the n = U 1.769 GHz Lamb shift.

MgF2 WINDOW

Figure 4. Isometric ayid cross sectional sketches of the large

solid angle Lyman-a detector.

can be done well enough to permit at least a 50 ppm

measurement of the line center. Over 200 hours of data-taking have been completed.

With the large solid angle detector and short geometry of

this experiment, good signal to noise has been obtained.

With 10 minutes of counting time, the signal to noise ratio is sufficient to make a 100 ppm measurement of the

corrected (with the overlapping transition contributions

subtracted out) line center. With this signal to noise ratio, we beheve the systematics can be well enough understood to determine the line center better than previous

He"^ Lamb shift measurements and experimentally distin- guish the Erickson [9] and Mohr [8] calculations of the

Lamb shift. References

sitions which appear in the Lamb shift resonance signal are the n = 5 (J = 5/2) - (J = 7/2) electric dipole

transitions at 1.87 GHz (see Fig. 3). Subtracting this contribution from the unquenched Lamb shift signal,

shifts the resonance line center by approximately 2 MHz

down in frequency. If a 100 kHz (50 ppm) measurement

is to be made, the subtraction technique to get rid of contributions due to overlapping transitions must work very

well. To test how well this technique is working, data

have been taken at different spacings of the SOF regions

and at different levels of microwave power applied to the

SOF region. Changing either one of these parameters

changes the relative sizes of the signals due to Lamb

shift and overlapping transitions and will therefore help

in determining how well the subtraction technique is working.

In addition to contributions due to overlapping transitions, the calibration of the microwave power applied to the atoms in the SOF region as a function of frequency

will be a major systematic problem. From past work

done in this laboratory, it is believed that this calibration

[1] A. Bertin ef al., Phys. Lett. 55B, 411 (1975); G. Carboni et al., Nuovo Ciniento A 34. 493 (1976).

[21 G. G. Simon, F. Borkowski, Ch. Schmitt, and V. W. Walther,

Z. Naturforsch. Teil A: 35, 1 (1980).

[31 S. R. Lundeen and F. M. Pipkin, Phys. Rev. Lett. 34, 1368

(1975); S. R. Lundeen and F. M. Pipkin, Phys. Rev. Lett. 46, 232 (1981), and these proceedings.

[4] M. A. Narasimham and R. L. Strombotne, Phys. Rev. A 4, 14 (1971) [Sdi = 2) = 14 046.2 ± 1.2 MHzl.

[51 E. Lipworth and R. Novick, Phys. Rev. 108, 1434 (1957) \S{u = 2) = 14 040.2 ± 1.8 MHz].

[61 G. W. F. Drake, S. P. Goldman, and A. Van Wijngaarden. Phvs. Rev. A 20, 1299 (1979) [S{)i = 2) = 14 040.2 ± 2.9 MHzl.

[71 D. L. Mader, M. Leventhal, and W. E. Lamb, Jr., Phvs. Rev. A 3, 1832 (1971) [S{n = 3) = 4183.17 ± 0.54 MHz]."

[81 P. J. Mohr, Phys. Rev. Lett. 34, 1050 (1975), and in Beam-Foil

Spectrvscopi), Ed. by L A. Sellin and D. J. Pegg (Plenum

Press, New York, 1976), p. 89 [S{n = 2) = 14 042.05 ± 0.55 MHzl.

[91 G. W. Erickson, J. Phys. Chem. Ref. Data 6, 831 (1977) lS(n = 2) = 14 044.64 ± 0.54 MHzl.

[101 G. Newton, D. A. Andrews, and P. J. Unsworth. Philos. Trans.

R. Soc. London, Ser. A: 290, 373 (1979).

143
1600 1700 1800 1900

FREQUENCY (MHz)

Figure 5. Typical raw iyiterference (I) signal for a 120 keV beam and a SOF separation of

0.32 cm. The solid dots are taken with no 20 GHz quenching of the iS states. The x's are taken with 20 GHz quenching of the J^S states, and the hollow dots are the differences of the two.

144
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Lamb Shift in the Hydrogenic Ion Cl^^^

E. T. Nelson, O. R. Wood II, C. K. N. Patel, M. Leventhal, and D. E. Murnick

Bell Laboratories, Murray Hill, NJ 07974

and

H. W. Kugel* and Y. Niv**

Rutgers University, Piscataway, NJ 08854

The CI 2 Lamb shift has been measured by resonant laser quenching of a fast {v/c - 0.1) metastable beam. The resu't for the 2Si/2 -2Pi/2 splitting is 31. 19(22)THz in agreement with the calculation of Mohr and with the series expansion in powers of {Za).

Key words: high Z hydrogenic ions; Lamb shift; laser resonance spectroscopy.

1. Introduction

Studies of the h ^ 2 Lamb shift, S, in high Z hydrogenic systems are important tests of the high field limits of QED [1]. In addition, these measurements offer a sensitive test of modern calculational procedures. Since

these calculations are based on expansions in a and Za,

the effects of higher order terms play an increasingly important role in the high Z hydrogenics. By scaling any

potential discrepancy between experiment and theory

with Z, it is in principle possible to identify which

terms may be in error. The best detailed calculations have

been done independently by Erickson [2] and by Mohr [3] and differ by 2% at Z = 17.

The experimental method is a modification of the classic Lamb-Retherford experiment [4], in which an optical

frequency electromagnetic field probes the 2S i/2-2P vz en- ergy level splitting, and has been described in a recent review [5].

2. Experiment

A schematic diagram of the experimental apparatus is shown in Fig. 1. A 190 MeV chlorine beam from the

Brookhaven National Laboratory double tandem van de

Graaff accelerator facility was fully stripped prior to entering a switching magnet to the Lamb shift resonance

.PRECISION LINEAR MOTION MICROMETER

IR POWER

DETECTOR\ NaC2 IR <^

RECTRACTABLE

Fe^S X-RAY CALIBRATION SOURCE

1 MIRROR

Figure 1. Schematic diagram of Cl"''*' experimental syste

*Present address: Princeton Plasma Physics Laboratory; Princeton, NJ.

**Dr. Chaim Weizmann Fellow; Resident Visitor at Bell Laboratories.

145
beam line. A thin adder foil was then used to preferentially populate 2Si,2 CP''* ions by electron pickup. Of or- der 0.1% of the total beam was promoted to this state resulting in an estimated metastable beam current of 400 pA. As the metastable state pickup was probably a surface interaction, and lower states of ionization and

scattered ions were the major source of background radiation, ultrathin carbon foils (0.5 to 5.0 |xg/cm-) were

used. (The x-ray spectrum observed with 300 mm- x

7 mm thick planar intrinsic Ge detectors consisted of two

photon El and Ml radiation from the 2S state, Lyman a

radiation from the 2P levels, and radiation from n =

2 CP''* as well as analogous transitions from other ioniza- tion states formed in the foil and by the scattered beam.)

A 152 |jLm Be window absorbed 85% of the dominant two

photon decay from the CP"* 2Si/2 state. Typical total count rates were 70 kHz per detector.

The CO2 laser consisted of a grating tuned oscillator vdth pulsed electrical excitation and cavity Q-switching

that produced 25 mJ, 150 |xsec pulses (10.6 |xm, P(20)) at a repetition rate of 480 Hz. The oscillator output was

mode-matched into a 13 m folded amplifier with a satu- rated gain of 14. The amplified laser beam had good spatial coherence, and was focused with //3.8 KCl optics to a 150 |jLm spot (FWHM) coincident with the metastable

beam. Frequency tuning was accomplished by the Doppler effect and line tuning of the laser across the 9 ixm

and 10 |xm bands between 28.3 THz and 32.33 THz.

A retractable alignment collimator with position reproducibility of ±25 |jLm was used to precisely define the in- teraction volume and align the system so that both particle and laser beams were focused to the same point in space with a well defined angle, between the x-ray detectors. The collimator system had to be removed during

measurement as any particle interactions caused unac- ceptably high x-ray backgrounds, and even the edges of

the focused laser beam had sufficient power to remove

material from the metal collimator. In separate experiments the particle beam waist was determined to be

0.4 mm and the laser beam waist to be 0.15 mm.

The laser induced fluoi'escence was monitored by

detection of enhanced Lyman a radiation at 2.95 keV

from the interaction volume within a 200 ixsec interval

coincident with the laser pulse. Typical laser induced count rates of 20 Hz were observed at the resonance

peak. The accurate recovery of this small excess count

rate within the 140 kHz background was achieved by us- ing 20 MHz scalars and a 2 MHz clock to subtract the counts accumulated during the time the laser was off (scaled by the laser duty cycle) from the counts accumulated during the time the laser was on. No excess counts were observed when the laser beam was blocked or dis- placed so as to not intersect the CP''' particle beam. To

enhance the signal to noise ratio the laser parameters were varied to maximize an expression proportional to the laser induced count rate divided by the fluctuation in the background counts accumulated during the laser

pulse:

where P is the average laser power, / is the pulse repetition rate, and is the laser pulse duration. The detectors were also collimated to view only the interaction

volume so as to reduce the background rate. The

discrete tuning range of the laser was limited so that

about one half-width of the resonance curve could be

probed at fixed intersection angle and beam energy. An

angle of 130° was chosen for obtaining most data in order

to have highest sensitivity to line shape and center.

3. Line Shape Theory and Curve Fitting

For the case of n = 2, Z = 17, and / = 3/2, one finds

that the hyperfine spHttings involved are of the order of two percent of the natural linewidths of the transition. Since it is known on general grounds that this effect does

not shift the center of gravity of the 2S1/2 -2Pi/2 transition

the hyperfine interaction does not play a significant role

in the Hneshape. In the laboratory the particle beam

makes an angle 9 with respect to the laser beam. Atoms

in the 2Si/2 state are driven to the 2Pi/2 state by the laser and the subsequent Lyman a decay x-ray is detected.

The basic induced transition probability per unit line time |JL is given by the Wigner-Weisskopff theory. In the

rest frame of the atom:

e 1 Om ITi

4 TT where Irl^ is the relevant matrix element squared, S„, is the power density in the moving frame, (i.e., the atom's

rest frame), v,„ is the laser frequency in the moving

frame, F is the linevvidth, and Vq = S. The quantity |jl is a function of 9 because both S,n and

v,n are functions of 9. From the Doppler shift formalism,

vl(1 - v/c cos 9) '

vi - .^/o' • where vl is the laboratory laser frequency. An expression relating the laboratory laser power density Sl with

Sm can be obtained from the Lorentizan transformation

of the field components, - SlJ^H - p cos 9)2,

'

p = v/c . (4) Data is obtained at each laser frequency normalized to integrated particle current and average laser power.

Frequencies are changed in random order approximately

every minute to average out any long term shifts or vari- ations in the system. The signal is proportional to the

product of laser power density, interaction time, and

fraction of particle beam intercepted by the laser. As the

focused laser beam is smaller than the particle beam this product is independent of small variations in laser spot

size. Since data is obtained at fixed angle 9, very far from

saturation of the transition, the signal expression be- comes to a good approximation a simple Lorentizan with

the natural linevddth. The best fit parameters are the

Lamb shift and an amplitude parameter.

A typical set of data resulting from 8 h of measurement, the fitted resonance curve, the final experimental

result for the Lamb shift from all the data, and the calcu- lated values are shown in Fig. 2, About 50 h of data were obtained, most of it at a beam energy of 190 MeV

and 9 = 130°. In order to check for systematic errors some data were obtained at a different energy (150 MeV)

and different interaction angles (8 = 50° and 160°). Also

the foil to interaction volume distance was changed. The

results of these measurements were consistent vdth the

results obtained using the standard parameter values. The final result for the Lamb shift was obtained by

making an overall fit to all the data, with a single center frequency Vq as a free parameter, together vdth free am146
U(THz)

Figure 2. Cl'^'^ excitation spectrum.. The circular point unth

error bar indicates the final experimental value for the Laynb

shift. M and E indicate values calculated by Mohr [3] and

Erickson [2].

quency v^^p, while estimating the error in the mean from

the scatter of the different results around the mean gives

the same 0.7% relative error. A number of identifiable systematic effects have been

considered in detail. These include neglect of hyperfine

structure as noted above along with neglect of effects due to the 2P3/2 level. Also possible errors in ion beam

velocity, intersection angle, and particle beam-laser

beam overlap have been considered. We estimate that

the resultant error from all these effects is less than

0.03 THz. Hence the experimental error is determined

predominantly by statistics.

4. Conclusion

The experimental result for the Lamb shift for the

Z = 17 hydrogenic system Cl^*'* is in agreement with the

calculation of Mohr and with the series expansion in powers of (Za); and is three standard deviations below

the calculation of Erickson. Scaling the discrepancy

recently reported for the Z - I Lamb shift [6] with an

order higher than Z-' is in disagreement with the experimental result and indicates that theoretical corrections of the order of (aZ)'^ must be considered possibly due to proton size, structure, or relativistic recoil effects. References

plitude parameters for each individual resonance curve. This fit yielded a value for the Lamb shift of v^'^^ - 3L 19(22) THz with = 29 for 41 degrees of freedom.

The error stated is the statistical one standard deviation.

Employing a separate fit for each curve alone and

averaging the results thus obtained yields a similar fre-

[1] S. J. Brodsky and P. J. Mohr, in Structure and Collisions of

loyis and Atoms, Ed. by I. A. Sellin (Springer, Berlin,

1978), p. 3. [2] G. W. Erickson, Phys. Rev. Lett. 27, 780 (1971).

[3] P. J. Mohr, Phys. Rev. Lett. 34, 1050 (1975).

[4] W. E. Lamb and R. C. Retherford, Phys. Rev. 79, 549 (1950).

[5] H. W. Kugel and D. E. Murnick, Rep. Prog. Phys. 40, 297

(1977).

[6] S. R. Lundeen and F. M. Pipkin, Phys. Rev. Lett. 46, 232

(1981).

147

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Helium Fine Structure and the Fine Structure Constant

W. Frieze, E. A. l-linds, A. Kponou, and V. W. IHughes

Yale University, New Haven, CT 06520

and

F. M. J. Pichanick

University of Massachusetts, Amherst, MA 01003

A series of precision measurements have been made of the fine structure interval v^^ , in the

state of helium. The results are vqi = 29 616.864(36) MHz (1.2 ppm), vjg = 2 291.196(5) MHz

(2.2 ppm), and vq2 = 31 908.040(20) MHz (0.6 ppm). These results are self-consistent and in good

agreement vdth theory. Experiment and theory can be combined to produce an independent value for the fine structure constant a" ' = 137.036 13(11) (0.8 ppm), in good agreement with the more accurate

value currently accepted a"' = 137.035 963(15) (0.11 ppm).

Key vi^ords: fine structure constant; helium fine structure; optical-microwave spectroscopy; quantum

electrodynamics.

1. Introduction

One of the most important tests of quantum electrodynamics is in the study of simple atomic systems, and in this respect experimental and theoretical work on the 2^P

fine structure of helium has been particularly fruitful. In

particular the helium 2^F fine structure is more favorable

for precise measurement than its counterpart in the 2^P

state of atomic hydrogen. The helium 2^P state has a rel- atively long lifetime, 1.0 x 10"^ s, compared to 1.6 x

10 s for the 2^P state of hydrogen. This gives for the

2^Po - 2^ Pi and 2^ Pi - 2^P2 intervals respectively, a

ratio of fine-structure interval to natural linewidth of 85

and 7 times the ratio for the 2^Pi/2 - 2^Ps/2 interval for hydrogen. The helium fine structure measurements

currently provide an independent measurement of the

fine structure constant with a precision of 0.8 ppm,

where the accuracy is limited by the theory. Although

this is the most accurate value available from atomic

spectroscopy it is about a factor of eight less accurate than values obtained from the ac Josephson Effect [1] and the anomalous gr-factor of the electron [2].

2. Experimental Technique

The energy level diagram of helium is given in Fig. 1 with an inset showing the 2^P fine structure to an ex- panded scale. The 2^P state is inverted, the lowest value

of J lying highest, because the spin-other-orbit interaction makes a larger contribution than does the spin-selforbit. In addition the Lande interval rule is not even

obeyed approximately because the spin-spin interaction is comparable to the spin-orbit.

Early measurements were made using optical spectroscopy [3] with a precision of about 1%. The first precise

experiment (150 ppm) utilizing magnetic dipole transitions between 2^P fine structure levels was performed [4] by exciting the levels by electron bombardment and ob- serving changes in intensity and polarization of the 2^P -

2^8 optical radiation. The precision was improved to about 10 ppm by level-crossing measurements [5], and in

VOLTS

25

24 -

///////////////////////////////A

(Porahelium) (Orthoheliu

J=0

29617 MHz

2291 MHz

electron bombordment

Figure 1. Energy level diagram of helium, with the l^S ground

state suppressed. The inset shows the 2^P fine structure on an

expanded scale.

149
178 cm *28

4

70.4

B Magnet

It23-

5.8

» 60

'! 30

C

.2 0 ®-30

•J _ '-Electron\

o-eO*^ Gun \

f ^Cavity

'- Col I i motor > \ / \-i*o

State Selector^ ^ e

Magnet^

Figure 2. Trajectory diagram illustrating the experimental

technique.

fact the optical level-crossing technique was discovered

in the 2'T state. Level-crossing experiments of higher

precision have been conducted recently [6], but this tech- nique is severely limited by its inability to make meas- urements of fine structure intervals independent of Zee- man parameters.

The method we have employed, the optical-microwave

atomic beam magnetic resonance technique does not have

this limitation. Furthermore the experiments are per- formed on a beam of free atoms, and level shifts due to

collisions are not a problem. The basic experimental set- up is illustrated in Fig. 2. A well-collimated beam of atoms in the metastable 2^S state is obtained by electron

bombardment of the 2^S ground state. Two inhomogeneous deflecting magnets A, B are used, in conjunction with

state-selector stops, to define specific spatial trajectories

for atoms which leave the electron gun in a particular nis sublevel and arrive at the detector in another. Between

the A and B magnets is a region of uniform magnetic

field where the atoms are optically excited to the

state while they also pass through a microwave cavity.

During the process of optical excitation and subsequent

decay back to the 2'^S state a fraction of the atoms will transfer from one m,, sublevel to another, and as illus- trated in Fig. 2 our detector was located to measure the

fraction making the transfer m,, = 0 ^ m.s = -l-l. If during the excitation and decay process a microwave

magnetic-dipole transition is induced between a pair of 2^P (J, mj) sublevels, the fractional 2^S (mj transfer

will be modified. In this way the microwave transition in the 2^P state is rendered observable in the detected 2^S

beam. The fractional microwave signal depends on rela- tive optical excitation and decay matrix elements for the

pair of 2^P {J, mj) sublevels in question. In the fine-structure measurements three transitions were measured involving the 2^P (mj = 0) sublevels for

all possible pairs of / = 0, 1, 2. The raj = 0 sublevels were chosen to minimize the magnetic field dependence

of the transition frequencies. In the presence of a magnetic field sublevels of different / and same ynj are ad- mixed resulting in a slight field-dependence of the

mj = 0 sublevels. This admixture, however, had the vir- tue of enabling us to observe the otherwise forbidden

J = 0 ^ J =2 transition. Thus the J = 0 ^ / =1

and J = 1 —> J =2 intervals were measured as well as

8.64 8.65 8.66 8.67 8.68 8.69 8.70

NMR FREQ. (MHz)

Figure 3. Resonance line shape for a single run on the transi- tion 2^? (J, mj): (0,0) (2,0).

their sum / = 0^/ = 2 to provide a check on the ex- perimental consistency. Other transitions were measured

to obtain independent values of the Zeeman parameters

for the 2^P state. In frequency units the transition natural linewidth was

3 MHz, and optimal signal-to-noise ratio was obtained

when this was power-broadened to about 5 MHz. This

necessitated microwave magnetic fields of between 2 G

and 3 G. The lines were plotted by sweeping the uniform

magnetic field utilizing the slight dependence thereon of the transition frequency. Figure 3 shows a typical reso- nance line for the / = 0 ^ / =2 interval measurement.

The precision of that run was 0.05 MHz or about 1% of the linewidth.

Further details of the experiments can be found in the

literature [7].

3. Results and Discussion

The results for the fine structure intervals v,,, summarized below:

are

vol = 29 616.864

vi2 = 2 291.196

0.036 MHz (1.2 ppm)

0.005 MHz (2.2 ppm)

vol + vi2 = 31 908.060 ± 0.036 MHz (1.1 ppm)

V02 = 31 908.040 ± 0.020 MHz (0.6 ppm)

150
Table 1. Theoretical contributions to the fine structure of t'P helium in MHz.

Second

Interval a^Ry a'^Ry (m/M)a2Ry Order a^Ry ^theory

Vol 29564.587 54.708 -10.707 11.657 -3.331 29616.914 ±0.006 ±0.00044 ±0.042 ±0.0039 ±0.043

(0.21 ppm) (0.015 ppm) (1.42 ppm) (0.13 ppm) (1.44 ppm)

vi2 2317.204 -22.548 1.952 -6.866 1.542 2291.284 ±0.0018 ±0.00088 ±0.081 ±0.0068 ±0.081

(0.76 ppm) (0.39 ppm) (35 ppm) (3.0 ppm) (35 ppm)

It can be seen that the measured vo2 is in excellent

agreement with the sum of the measured voi, vi2, establishing experimental consistency and suggesting that

there were no unaccounted systematic errors. The most

precise theoretical number is for voi, and the most accu- rate experimental comparison for this is the difference

Vol = 29 616.844(21) MHz (0.7 ppm) expt.

Vol = 29 616.914(43) MHz (1.5 ppm) theory.

The theory is summarized in Table 1. The current theory includes quantum electrodynamic terms [8] to order a^Ry as well as corrections to the spin-dependent

operators [9] to the same order. An alternative approach is to use the theory to obtain an independent value for the fine structure constant with

the result

a-^ = 137.036 13(11) (0.8 ppm).

This is to be compared with the more accurate value

currently accepted [1] a-i = 137.035 963(15) (0.11 ppm).

We believe that refinements in the experimental tech- nique could improve the accuracy of the measured interval Vol to 0.2 ppm, making possible a 0.1 ppm determination of the fine structure constant. This would require

the theory to be done to order a'^Ry.

The research was supported in part by the National

Science Foundation under grant No. PHY 78-25655.

References

in E. R. Williams and P. T. Olsen, Phvs. Rev. Lett. 42, 1575

(1979).

|2I R. S. Van Dyck, Jr., R. P. Schwinberji', and H. A. Dehmelt,

Phys. Rev. Lett. 38, 810 (1977).

131 J. P. Barrat, J. Phvs. Radium 20, 541 (1959); 20, 638 (1959);

20, 657 (19.59).

141 I. Wieder and W. E. Lamb, .Jr., Phvs. Rev. 107, 125 (1957).

[51 F. D. Colegrove, P. A. Franken.'R. R. Lewis, and R. M.

Sands, Phys. Rev. Lett. 3, 420 (1959).

[61 C. Lhillier, J. P. Farou.x, and N. Billv, .J. Phvs. (Paris), 37, 835 (1976).

[71 F. M. J. Pichanick, R. D. Swift, C. E. .Johnson, and V. W.

Hughes, Phys. Rev. 169, 55 (19(38); S. A. Lewis, F. M. J. Pichanick, and V. W. Hughes, Phys. Rev. A 2, 86 (1970); A. Kponou, V. W. Hughes, C. E. Johnson, S. A. Lewis, and F. M. J. Pichanick, Phys. Rev. A 24, 264 (1981); W. Frieze, E.

A. Hinds, V. W. Hughes, and F. M. J. Pichanick. Phys. Rev. A 24, 279 (1981).

[81 M. Douglas and N. M. Kroll, Ann. Phvs. (N.Y.) 82, 89 (1974).

[91 M. L. Lewis and P. H. Serafino, Phys. Rev. A 18, 867 (1978).

151

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Preliminary Measurement of the J = 0 to J = 2 Fine Structure Interval

in the 3^P State of Helium*

Mark Feldman^, Tom Breeden, Louis DIMauro, Taiqian Dong, and IHarold Metcalf

Department of Physics, State University of New York at Stony Brook, Stony Brook, NY 11794

We have made time resolved measurements of the level crossing signal from the / = M =2, and

the J = M = 0 sublevels of the 3^1^ state of He near 2277 gauss. A thermal beam of He 2'^S metastables is excited by a pulse of 3889 A dye laser light and a temporal slice of the fluoresence at 90° to the field is recorded by fast electronics as a function of field. This experiment has been carefully

designed to minimize the effects of very many possible systematic errors. We have a S/N of about 15 in each of about 100 data sets (— 1 hour run time). The result has a statistical standard error of 1.4 ppm

and a systematic uncertainty of about 1.4 ppm, resulting in a net uncertainty of 2 ppm. The distribution of the results is approximately a Gaussian of width 14 ppm and the average, corrected for field in- homogeneity, is 9695.013 kHz (NMR, HgO).

Key w^ords: fine structure; helium; level crossing; line narrowing; time resolution.

1. Introduction

Measurements of the fine structure in the 2"^? state of helium can now be compared with theory at the ppm

level [1], and show promise of providing an independent measure of the fine structure constant a. Although a comparably accurate theory for the 3^P fine structure does not presently exist, the possibility of such a calculation makes the measurement of this interval very attractive. Furthermore, line narrowing techniques employing

time delayed spectroscopy [2] offer the possibility of sub-natural width signals that enhance both accuracy and

precision of such measurements.

We have made time resolved measurements of the level crossing signal from the / = 2, M ^ 2 and the ./ = M = 0 sublevels of the 3^P state of He near 2277 gauss

(see Fig. 1). A thermal beam of He 2'^S metastables is excited by a pulse of 3889 A dye laser light and a temporal slice of the fluoresence at 90° to the field is recorded by fast electronics as a function of field. A

frequency-locked NMR system is used to sweep the sta- bilized magnetic field through several linevndths in a

series of 100 steps, repeatable to about 1 ppm. [3] The level crossing field depends on both the zero field splitting and the Zeeman effect. The precise measure- ment reported here can be combined with an accurate

evaluation of the Zeeman effect to determine the fine structure splitting. This can be compared with the relativistic theory of the helium atom, including QED contri- butions, for a test of that theory or as a measure of the

fine structure constant a.

2. Apparatus

A schematic diagram of the apparatus is shown in Fig.

2. The source of metastable triplet He atoms consists of

*Supported by the National Science Foundation

'Present address: Time and Frequency Division, N.B.S., Boulder, CO

80303

0 0.05 OJ 0.15 0.2 0.25 0.3

MAGNETIC FIELD (TESLA)

Figure 1. The Zeeman effect in S^P helium in the field range

of interest. The zero of energy is arbitrary.

153
PHOTOMULTIPLIER/^

MAGNET

POWER

SUPPLY

1 t FREQUENCY

SYNTHESIZER

COUNT AND

DIVIDE BY 30

trigger pick-off

GATED

INTEGRATOR

VETO

first ton

pulses

SIGNAL AVERAGER

>Tnr;Ai u&Mir >- 0PT06ALVANIC GEL L

NITROGEN PUMPED

DYE LASER

PRESSURE TUNED

Figure 2. Partial schematic diagram of apparatus.

two differentially pumped chambers with an exit to a

1.5 m beam line that is maintained at 5 x 10"^ Torr. We use a large jet booster pump on this specially

designed vacuum system enabling a helium flow of several Torr- liters /sec from a microwave discharge

source, and a skimmer aperture that passes only a small

fraction of the total helium flow into an intermediate vac- uum region. A second skimmer allows only the helium

beam to enter into the high-vacuum (— lO"*" Torr) beam

pipe. The resulting high pressure (a few Torr), high flow source produces a thermal beam of more than 10^^ atoms /sec and a density of more than 10^ metastable He

atoms/cm^ in our interaction region 1 m^from the source. This is sufficient for us to detect 3889 A fluorescence of several photons /laser pulse.

Our dye laser has a bandwidth close to the design limit of about 3 GHz, peak output power of about 6 kW, and

is pressure tuned. An airspaced, intracavity etalon en- ables us to resolve the various fine structure transitions,

thus reducing the background (and therefore the noise)

from many possible transitions that contribute to the

fluoresence but not to the signal. Figure 3 shows that

many of these contributions can be reduced or eliminated

by appropriate tuning of the laser. Figure 4 shows the

fluorescence obtained by sweeping the laser through

—30 GHz. The width of the peaks is caused by both the

3 GHz laser width and the 3 GHz Doppler width (45° in- tersection of beams).

Our field measurement and sweep is accomplished by a

series of single steps, each of which is reliable and re- peatable. At the heart of the system (see central part of Fig. 2) is a Hewlett Packard programmable frequency

synthesizer that is stepped by 0.3 kHz by a pulse from a

trigger pick-off photodetector every 30 laser pulses (laser

is free running). An NMR marginal oscillator is locked to the synthesizer with a mixer-amplifier-varactor system

so that it also steps frequency. The NMR signal from the

marginal oscillator is amplified by a lock-in detector

which provides a dispersion-shaped error signal that is fed to the programmable sweeping magnet power supply

so that the magnet is always brought to the center of the resonance. The field is thus locked to the NMR at each of 100 steps corresponding to the 100 channels of the signal averager. Flyback for multiple sweeps is arranged with

appropriate electronics. With this system the accuracy of our individual field measurements is about 1 ppm, and

the overall field calibration has internal precision of about 0.3 ppm.

Fluorescence is detected by a photomultipHer (PMT)

that views about 1 cm of the laser beam path through a telescope of approximately unit magnification. The PMT

0 500 1000 1500 2000 2500 GAUSS

Figure 3. Frequency of allowed optical transitions (AM =

±1) in magnetic fields of interest. The zero of energy is arbi- trary.

154
LASER

OPERATING FREQUENCY

FREQUENCY

Figure 4. Trace offluorescence obtained by sweeping the pres- sure of the laser timing chamber. Compare with Fig. 3.

is well shielded from fringing fields and is located well outside of the magnet. The atomic beam is much larger

than either the PMT field of view or the 5 mm laser beam.

The time-delayed detection of fluorescent light is accomplished by a LeCroy 227 gated integrator that

receives the output of the PMT. The trigger pick-off

pulse that signals the field and frequency stepping is also used to start the delay interval of the integrator (see

Fig. 2). After this delay has elapsed (typically 80-100

nsec) the integration begins for a measurement period

(typically 80-100 nsec). Each of these time intervals is adjustable. The output of the integrator is averaged for 20 laser pulses in one channel of our analyzer, and the

channel is incremented with the field and NMR fre- quency.

The effect of time resolution on the signal shape

deserves further comment. The precision of experiments

on excited states has always been limited by the natural

width of the transition. It is possible to modify the

experiment to reduce the effect of the finite lifetime in limiting the precision of measurements of excited state parameters [2]. The modification is accomplished by arranging the

experimental apparatus to interact with the atoms under

study for two relatively short and well separated intervals. In our level crossing experiments we excite the

species under study with a pulse of light of duration

short compared with the excited state lifetime, observe

the decay fluorescence for a well-defined interval, and

separate these intervals by a selectable time span. For

delay times considerably longer than the lifetime, the

central part of the oscillatory signal [4] is considerably narrower than that from a continuous experiment. What

is more important though, is that because these signals

have high frequency information far into the wings, they

will be much more useful for defining the position of the

center of the level crossing. [2]

3. Data Acquisition and Processing

Each data set consists of the averaged fluorescence

from 8-10 sweeps over 100 channels (corresponding to 100 field points). We veto the first 10 laser pulses of

each step in order to minimize the effect of magnetic field

settHng time. Each sweep step has 30 laser pulses,

resulting in 16,000-20,000 laser pulses averaged during

the 45-60 minute data collection period. We detect 2-4 photons/laser pulse, about 500 in each channel, producing an expected scatter of 4-5% of the total detected fluores- cence. Since the signal is about 40% of the total fluores- cence, the S/N we expect is about 8-10, and the typical

signal shown in Fig. 5 corroborates this estimate.

1-1 3 1 I I I I L 2274 2275 2276 2277 2278 2279 2280 2281

Field ( Gauss

)

Figure 5. Typical signal (unsmoothed) and fitted lineshape. This data set was taken iyi 50 minutes.

Using a standard non-linear least squares routine, we

fit a theoretical lineshape to the data by varying the center frequency and 2-4 other experimental parameters.

From the results of the fit we can determine the NMR

frequency of the position of the central maximum. The

result of averaging over 100 data sets is 9695.013 kHz

(corrected to H2O frequency). The distribution of our

results, shown in Fig. 6, has a standard deviation of about 14 ppm resulting in a statistical standard error of

1.4 pm.

4. Systematic Errors

This experiment has been carefully designed to minimize the effects of possible systematic errors. There is no rf to broaden or shift the signal. It is done in a beam

to eliminate effects of both collisions and stray charged

particles. All the data is taken long after the laser pulse

ends so that there are no optical power effects. The data

averaging system is digital so that the signal-to-noise ratio is not influenced by extraneous signals. The principle

systematic effects under investigation derive from alignment uncertainties and field calibration. In order to test for effects associated with magnetic

field settling between steps, we have swept the field in both directions. In order to test for a variety of effects (alignment, field sensitivity of PMT, etc.), we have re- versed the direction of the magnetic field. These experiments divide our data into four roughly equal populations

providing the following results. Sweep up results aver155
^

PIPKIN a KRAMER

hmthis work

I—I 4 ppm

-3.5-3.0-2.5-2.0 -1.5 40 -0.5 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 DEVIATION FROM AVERAGE

(o- units)

Figure 6. Histogram of results of center frequency and a Gaussian of appropriate area. The results of the measurement reported here, shovm above the histogram, includes both

statistical and systematic uncertainty which combine to 2.0 ppm.

age 5 ± 2 ppm lower than sweep down; field "reversed"

data average 11 ±2 ppm lower than field "normal". We

attribute these differences to shifts associated with time

lags (sweep direction) and misalignment (field direction)

and average over the results. Because these shifts of 2.5 ± 1 ppm and 5.5 ± 1 ppm may not be exactly compensated by averaging, we allow a 1 ppm systematic uncertainty associated with these effects. The magnetic field is measured by a probe located a few cm from the interaction region, and the measured

field value is corrected by the measured difference

between this and a movable probe which is centered in the interaction region before and after each day's run- ning time (5 to 15 data sets). We have found this correction to be constant over a period of a few days, as long as the magnet is left on, but to change by as much as 25 ppm for different magnet cycles. Our uncertainty in measuring this correction on a given day is usually less than 1 ppm, and the errors are expected to be random.

We therefore assign a residual uncertainty of 1 ppm/

VNo of days = 0.4 ppm to this effect. We have mapped the field over considerable distances

a number of times in order to determine the effect of the

field inhomogeneity over the interaction region. This can

cause a systematic shift because the probe does not have

the same shape (spherical) as the region of intersection of

the laser beam, atomic beam, and field of view of the

PMT. Although the inhomogeneity over the interaction

region can be a few ppm, a conservative estimate of the

uncertainty arising from a linear variation of the field is about 0.7 ppm. Any non-Hnear field variation does not

significantly change this estimate.

We have considered the contribution from a number of other sources to possible systematic errors in this meas- urement but have found them to be negligible. These

include laser frequency setting and drift, overlapping

resonances, diamagnetic shielding, time base of syn- thesizer (measured to be 0.3 ppm), and PMT field dependence. The total systematic uncertainty arises from the

field sweep and direction results (1.0 ppm), the probe

displacement correction (0.7 ppm), and the field inhomogeneity (0.7 ppm). The quadrature sum of these contri- butions is 1.4 ppm.

5. Results^

We combine the systematic and statistical uncertainty

(1.4 ppm each) in quadrature to obtain 2.0 ppm for the

'A new possibility of light shifts in quantum beats has been discovered

while this paper was in press. It will be described in the .June 1982, issue of Phys. Rev. A. It may affect these results by as much as -5 ppm.

156
total. Our present result, pending further work, is quoted as the NMR frequency of protons in water at the center of the time-resolved level crossing signal: /

=

9695.013(19) kHz. This is in good agreement with the

work of Kramer and Pipkin [5] that produced

9695.072(33) kHz and the result of Lhuiller et al. [6], 2277.125(15) gauss or 9695.099(64) kHz.

References

[1] W. Frieze et al., these proceedings.

[2] H. Metcalf and W. Phillips, Opt. Lett. 5, 540 (1980); these proceedings.

[3] For more details of all aspects of this experiment, see the

thesis of Mark Feldman, State University of N.Y., 1980.

[4] P. Schenck, R. Hilborn, and H. Metcalf, Phys. Rev. Lett. .31, 189 (1973).

[5] P. Kramer and F. Pipkin, Phys. Rev. A18, 212 (1978).

[6] C. Lhuillier, P. Riviere, and J. Faroux, C.R. Acad. Sci. Ser. B 276, 607 (1976).

157

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Laser Microwave Precision [\/leasurements of 2 S-, and 2 P Term

Splittings in Helium-Like Li

U. Kbtz, J. Kowalski, R. Neumann, 8. Noehte, H. Suhr, K. Winkler, and G. zu Putlitz*

Physikalisches Institut der Universitat Heidelberg, F.R.G.

The hyperfine structure (hfs) splittings of the metastable ls2.s '^Si state of helium-like ^•'^Li"'' have

been measured with combined laser optical pumping and microwave resonance. A low energy Li^ ion beam, optically excited by an intersecting laser beam, passed through a waveguide where radio frequency transitions were induced. The resulting population transfer between the hfs levels of the '^Sj was detected via the change of fluorescence light intensity from a second crossing redon of laser light

and ion beam located behind the waveguide. The magnetic hfs constants A (''Li , 2^Si) and A Chi'^, 2 ^Sj) were extracted from the measurements and compared with theory. The hfs anomaly is in good

agreement with the value obtained from the magnetic hfs constants of the atomic 2 ^Sy2 ground states. In addition rf measurements of the 2 hfs have been performed.

Key words: fine and hyperfine structure; hyperfine structure anomaly; laser-microwave spectroscopy;

singly ionized lithium; two-electron system.

Singly ionized lithium belongs, together with helium,

to the class of simple two-electron systems which may be

regarded as fundamental in the sense that the energy

states of these systems can be calculated with high

precision. The exact knowledge of such energies can

serve either as an experimental test of the underlying

theory or, if this theory is accepted, as a means for the

determination of fundamental constants. The precise

determination of the He fine structure splitting of the

ls2p term [1] has become the source for an exact

value of the fine structure constant a. In this paper

results of radio frequency (rf) measurements of the

hyperfine structure (hfs) splittings of the metastable

Is 2s ^Si term (t = 50 s) and of the short-lived Is 2p ^P2 term (t = 43 ns) of Li^ are reported. The relevant part

of the energy level diagram is shown in Fig. 1.

eV

66.6- 2%,-

64.4

5.390

F

3/2

-2^

-1-

> =5l.85A k

2\ J <

'Li*

7/2

5/2

3/2

1/2

5/2

3/2

1/2

5/2

3/2

1/2

GHz

180 160

li.0 {120

100 + 8060

i.0 20

0

2^Sy^ atom ground state

Figure 1. Energy level scheme of the ls2s ^Sj and ls2p ^P

terms of^'^Li"^, the isotope shift is omitted.

*A]so: Gesellschaft fiir Schwerionenforschung, Darmstadt, Federal

Republic of Germany.

The hfs of the 2 '^Si term of ''Li^ and ^Li+ has been

measured by means of the optical pumping method [2-4]. Li atoms were ionized and excited to the metastable 2 ^Si term by electron impact when leaving the aperture of an

oven, and were focused to a well collimated beam of 200-400 eV (Fig. 2). Selective optical excitation of, e.g., the ^Si, F = 5/2 - ^Pa, F = 5/2 transition with a single mode dye laser leads to a depopulation of the '^Si, F - 5/2 term, which can be monitored by the same laser

in a second interaction region 15 cm downstream in the

ion beam. Between the two laser ion-beam intersection

points the ions pass a waveguide, where rf transitions

^Si, F = 3/2 - 5/2 are induced. This leads to a repopulation of the = 5/2 term, which is observed as an

increase of the fluorescent light in the monitor region.

The linewidth (0.6-1.3 MHz) of the rf signal is dominated

by the interaction time of the ions with the rf field. In order to eliminate the Doppler shift in the rf transition of several MHz the waveguide is terminated to produce a counterpropagating wave. Two signals shifted by

±vo(y/v phase) with respect to the center frequency of the

~ PHOTOMULTIPLIER DETECTION SYSTEM

/ _ WAVEGUIDE WITH MAGNETIC SHIELDING

REE LOCK- IN

Li OVEN

*200V MCA

TO PUMP LOCK-IN

Figure 2. Experimental arrangement of the laser-microwave

spectrometer and ion beam chamber.

159
hfs transition vq are obtained. Both signals are recorded

simultaneously and stored in a multichannel analyzer. A

more detailed description of the experimental set-up is given in Ref. [4]. Figures 3 and 4 show examples of microwave resonance

signals for the 2 ^Si transitions in '^Li^ and ^Li+. The

results of the 2 ^Si measurements are compiled in Table

la and b. From the hfs splittings in ^'•^Li"' the magnetic

hfs interaction constants A ("Li^' and A CU^^ and small

energy depressions SC'Li^' and 8(^Li+' of the F = / hfs sublevels were extracted. Within the experimental errors the resulting hfs anomaly A, defined by the equation A =

i^A/''A) Cgi/^gi) - 1 [5], where ^''gi are the nuclear gSIGNAL

(ARB. UNITS)

^Li* 1s2s H1 F=0— F=1

2999 3000 3001 3002 3003 3004 3005 ^^"^

Figure 3a. Microwave transition F = 0 F = 1 in the 2 ^Sj

state.

6002.3 MHz 6005.0 MHz

Figure 3b. Microwave transition F = 1 ^ F = 2 in the 2 ^Sj

state.

factors, is in good agreement with the value [6, 7] obtained from the magnetic hfs constants of the atomic

2 ^Si^ ground states (Table 2). The equation for the energy splitting between two hfs sublevels F and F ' can be found in Ref. [8], and is written in the following form:

3 m,

g, [(1 + e)

Thereby e is the correction for the second electron [9], and

Yp = (1/4) [F(F + 1) - /(/ + 1) - S{S + 1)].

The equation was used for a preHminary calculation of the A factors which corrects only for the second electron,

the anomalous magnetic moment in lowest order

(~a/2TT), and for the reduced mass. The error for Atheor. in Table 3 is dominated by the uncertainty of e. The

difference from the experimental value is partly due to higher order QED and relativistic corrections. As is well

known from ^He [10] there should exist also a significant

nuclear structure correction.

As can be seen from the experimental splitting values

in Table 1, the ratio of the splittings within one isotope

deviates from the Lande interval rule. For example, in "Li^ this ratio should be 5/3. However hyperfine mixing

of the 2 '^Si term with the 2 ^Sq, with the latter

FWHM=700 kHz

11892.383 MHz

Figure 4a. (a) 2 '^Sj, F = 112 - F = 312 microwave transition

in ^Li^ with fit curves (solid line: "Ramsey-type" function as discussed in [21], dashed line: Lorentzian).

FWHM = 570 kHz

Figure 4b. 2 ^S^, F = 3/2 - F = 5/2 transition in '^Li* with two Doppler shifted signal curves.

160
Table 1. Experimental values of the 2 S; hfs splittings in

^Li'^ (a) and '^Li^ (b). 1s2s 0 F = 3/2

a. F = Q-F = 1 F = l-F = 2 Magnetic hfs constant A (^Li+, 2 %)

F = 1 depression 5(*^Li+, 2 %)

b. F = l/2-F = 3/2 F = 3/2-F = 5/2 Magnetic hfs constant A (^Li+, 2^81)

F = 3/2 depression 8('^Li+, 2 %)

3001.780(50) MHz

6003.600(50) MHz

3001.793(17) MHz

13(37) kHz

11890.018(40) MHz

19817.673(40) MHz

7926.923(14) MHz

366(29) kHz

Table 2. Hyperfine structure anomaly A.

Li+ ion (2 Li atom (2 ^Syz)

6.76(60) X 10-5 6.806(63) X 10-5 [g^ 7]

Table 3. Theoretical estimates of A and 5.

''Li+ ^Li+

Atheor. (in MHz) 3000.81(10) 7925.15(23)

Stheor. (in kHz) 39 511

15339.6 cm

1s2s h,

-1 5/2 -r5A/2

s

I cm

3A/2

_L_

T

3/2

1/2 -L

Figure 5. Diagram of the 2 'S, hyperfine multiplet of 'Li in- cluding the F = J/2 depression (enlarged).

eV

66.6

64.4

F

3/2

2%0,2,1

;\ =5485 A

2^5,

7/2 5/2

3/2

1/2

5/2

3/2

1/2

5/2 3/2

1/2

5.39 -- I'Sn ION ^

0

-L 2^Si/2ATOM

GROUND STATE

represented by the nuclear spin I -3/2, increases the

distance between the 2 ^Si (F = 3/2) and 2^So (F =

3/2) sublevels [11, 12] and causes a depression of the

2 ^Si, F - 3/2 hfs state (Fig. 5). This effect can be calculated. Results based on hfs theory of two-electron sys- tems, given in [13-15] are hsted in Table 3. More refined

calculations using better wavefunctions are in pro- gress [16]. The method of laser-microwave spectroscopy has been

extended to the excited short-hved 2 '^P multiplet of " Li*,

in order to measure the 2 hfs with rf methods. The ex- perimental scheme is the following: The low-velocity Li"^ ion beam passes a microwave cavity. A dye laser beam

crosses the ion beam inside the cavity and excites ions

from a certain 2 ^Si hfs sublevel to one of the 2 '^P hfs

states, e.g., with the transition 2 '^Si, F = 3/2 - 2 ^2,

F = 5/2 (Fig. 6). Microwave transitions (AF = 1) in the

2 ^P9 to one of the neighboring hfs sublevels with F =

3/2 or 7/2 change the decay rate back to 2 '^Si, F = 3/2. The same laser monitors the resulting population change

of the 2 '^Si, F = 3/2 in a second crossing region far outside the cavity. An example of a microwave resonance

signal is shown in Fig. 7. The method can be compared

with the magnetic deflection method used by V. W.

Hughes and co-workers for the very precise 2 ^P fine structure measurement in ^He [1]. In the Li"^ experiment the monitoring laser beam replaces the deflecting

magnetic field. The technique is very successful for ion- beam precision spectrosopy.

Figure 6. Energy diagram of the 2 ^S^ and 2 multiplets of ^Li^ including the optical pumping cycle described in the text.

SIGNAL

(ARB. UN.

FWHM = 8.5 MHz

11750 11770 11790 ]) [MHzI

Figure 7. Signal ciirve of the 2 '^Pg (F = 5/2 - F = 7/2) micro- wave transition in 'Li . For the determination of the larger frequency splittings between different fine structure (fs) levels of the

term a recently tested [17] radio frequency controlled two dye laser spectrometer will be used. It allows one to measure laser difference frequencies continuously up to 18 GHz. Larger energy differences between Li^ ^P fs levels will be measured stepwise by using, for example,

161
I2 lines as references between the corresponding Li"^ transitions to which one laser can be stabilized. These

measurements should be capable of determining the fs with ppm precision.

For the first time experimental data can now be

obtained which allow one to test the very precise hfs [18]

and fine structure [19, 20] theories existing for Li^. References

\l] W. E. Frieze, E. A. Hindis, V. W. Hughes, and F. M. Pichanick, Phvs. Lett. 78A, 322 (19X0).

121 a. D.' Rosner, R. H. Holt, and T. D. Gaily. Phys. Rev. Lett.

35, 785 (1975).

[31 W. Ertmer and B. Hofer, Z. Phys. A 27«, 9 (197(5).

[41 U. Kotz, J. Kowalski, R. Newmann, S. Noehte, H. Suhr, K. Winkler, and G. zu Putlitz, Z. Phys. A 300, 25 (1981).

[51 H. M. Foley, in Afoinic Plit/ftics, Ed. hy B. Bederson, V. W.

Cohen, and F. M. J. Pichanick (Plenum Press, New York,

19(i9) ]). 509.

[(il A. Beckmann, K. D. Boklen, and D. Elke, Z. Phys. 270, 173

(1974) (Earlier 2 -S, _. hfs precision measurements in Li are

cited in Ref. [71. [71 E. Arimondo, M. Inguscio, and P. Violino, Rev. Mod. Phys. 49, 31 (1977).

[8] H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms (Springer, Berlin, 1957).

[9] P. J. Luke, R. E. Meyerott, and W. W. Clendenin, Phys. Rev. 85, 401 (1952).

[10] S. D. Rosner and F. M. Pipkin, Phys. Rev. A 1, 571 (1970).

1^11] I. I. Sobel'man, Introduction to the Theory of Atomic Spectra (Pergamon Press, New York, 1972).

[12] M. M. Sternheim, Phys. Rev. Lett. 15, 545 (1965).

[13] G. Breit and L. A. Wills, Phys. Rev. 44, 470 (1933).

[14] C. Schwartz, Phys. Rev. 97, 380 (1955).

[15] A. Lurio, M. Mandel, and R. Novick, Phys. Rev. 126, 1758

(1962).

[16] R. Herman, J. Kowalski, R. Neumann, and G. zu Putlitz, to be published.

[17] J. Kowalski, F. Mayer, R. Neumann, S. Noehte, R. Schwarzwald, H. Suhr, K. Winkler, and G. zu Putlitz, Abstract Book, p. 1068, European Conference on Atomic

Physics (ECAP), held in Heidelberg, 6-10 April 1981.

[18] N. A. Jette, T. Lee, and T. P. Das, Phys. Rev. A 9, 2337

(1974).

[19] B. Schiff, Y. Accad, and C. L. Pekeris, Phys. Rev. A 1, 1837

(1970) . [20] Y. Accad, C. L. Pekeris, and B. Schiff, Phys. Rev. A 4, 516

(1971) . [21] N. F. Ramsey, Molecular Beams (Oxford University Press, London, 1956).

162
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

An Optically Pumped Metastable Hydrogen Beam

K. C. Harvey

Center for Absolute Physical Quantities, National Bureau of Standards, Washington, DC 20234

A beam source of atomic hydrogen is described which produces metastable atoms in the 2Siy2 state by optical pumping. An effusive beam of atomic hydrogen is generated in the ground state. The atoms

pass in front of a VUV lamp producing Lyman-3 (1026 A) radiation. The atoms are excited to the 3p

level and then cascade to the 2Sy2 state. The metastable atoms are measured by quenching them with an electric field and detecting the emitted Lyman-a radiation. Beams with a flux of 10^ atoms /sec were

obtained. A metastable-beam effective temperature of 100 K was measured. In addition preliminary measurements of the transition from n = 9 to w = 20 in hydrogen using a Woods discharge are presented and directions for improvement are indicated. Key words: atomic beam; hydrogen; metastable states; optical pumping.

The metastable 2Si/2 state of atomic hydrogen has long

been a useful system for measurements in the area of fundamental constants as well as basic physics. With the

need for increasingly precise and accurate measurements, a suitable source of metastable atoms has not al- w^ays been available. For such measurements, the atom

must be isolated from uncontrolled external influences

which means an atomic beam must be used. In such an arrangement the number density is small. In addition the

velocity distribution of the beam should be well characterized so that experimental systematics may be properly

taken into account. An optically pumped hydrogen beam

is a source which offers the possibility of high number

densities and at the same time a well defined, slow velocity distribution.

There are a number of existing and proposed methods

for producing metastable hydrogen beams. Electron bombardment was the original technique [1] and is still fre- quently employed. Typical fluxes are about 10^ atoms/sec

vdth temperatures of about 2000 K, but their usefulness

is limited by the momentum transfer produced in the excitation process by the electrons. This leads to a velocity

distribution which strongly depends on the electron en- ergy and collimation parameters [2]. Charge exchange

between a proton beam and cesium vapor [3] produces

typical fluxes of 10^^ atoms/sec. These beams have, however, a large energy of 500 eV or more and limited state

selectivity since long-lived Rydberg states are simultaneously produced with the 2Siy2 state in the charge ex- change process. Proposals for neutral atomic traps [4], though promising, have not yet been realized. Schemes

employing laser-induced nonlinear processes [5] are now

limited by the low duty cycle of pulsed lasers and the

photoionization which is always present.

The technique of optical pumping offers a way to over- come these limitations. Although the advantages of this approach were early appreciated by Lamb and Retherford [1], its development was inhibited by the absence of an adequate source of radiation [6]. Recently an ap- propriate VUV lamp has been developed [7], In the ar- rangement presented here, a dense, collimated beam of

neutral hydrogen in the ground state is produced and

passes in front of the VUV lamp. The lamp irradiates the

atoms with Lyman-p (1026 A) radiation. The atoms are transferred to the 3P state from which they cascade to

the 2Si/2 state with a branching ratio of about 12% (see

Fig. 1). From the lamp region the metastable atoms

travel to an experimental region or an area where the

metastable flux is measured. Since Lyman-p light may be

readily filtered, it is possible to produce highly state selected beams. The number of metastables produced is limited by the flux of ground state atoms and the flux of Lyman-P photons which suggest that high densities are

possible. In principle, the entire neutral beam can be

converted to metastable atoms.

IONIZATION CONTINUUM

1 6563

61

12

10

15

26

BALMER SERIES

LYMAN SERIES

Figure 1. Energy level diagram of hydrogen.

A schematic of the atomic beam system is shown in Fig. 2. The vacuum chamber is a stainless steel box in

163
Figure 2. Schematic of the atomic beam apparatus showing the hydrogen source, lamp region, and metastable detector.

which various items are mounted on plates supported

kinematically on stainless steel rods. Most of the components in vacuum are gold plated to improve conductivity and reduce charge accumulation. The chamber has

a number of feedthrough ports along its length as well as view ports. All dielectric surfaces on windows are

shielded with metallic mesh to reduce stray fields from

accumulated charges. The apparatus is pumped by a 1000 C/sec diffusion pump which uses a low vapor pres- sure silicone oil to reduce backstreaming. A high conductivity liquid nitrogen trap is located between the

chamber and pumps. The end vacuum of the system is about 10^* Torr. The neutral atomic hydrogen beam is produced by dis- sociation of molecular hydrogen in an rf discharge and

then effusion through a multichannel array. The molecular hydrogen is first purified with a palladium leak. The

hydrogen is introduced into a pyrex bulb 4.0 cm in diameter where it is dissociated by an rf oscillator delivering

25 W at 30 MHz. The degree of dissociation is monitored by observing in the discharge the relative intensity

of the atomic Balmer series and the yellow molecular hydrogen lines. The glass multichannel array is attached to the discharge bulb to coUimate the beam. The array has

capillaries 50 |xm in diameter and 2 mm in length, ar- ranged so that the array has transparency of 50% . The

total area of the array is that of a circle 3 mm in diameter. Supports on the discharge bulb allow the direction of the beam to be adjusted. The hydrogen beam is es- timated to be about 50% dissociated when it enters the vacuum chamber. Using the theory of multichannel gas

collimating arrays [8], the divergence of the beam is found to be 2.4° in the low pressure limit when the pres- sure in the discharge is about 10 |xm. The beam flux is measured using the pumping speed and the vacuum

chamber pressure. A pressure of 10"'' Torr in the

chamber indicates a beam intensity of 3 x 10^^ atoms /sec. The above method for producing a beam was

chosen since it generates an intense beam without dif- ferential pumping and it can be readily modified to pro- duce a beam at cryogenic temperatures [9]. The rf discharge produces ions such as H"^, Ho^ and E.^ as well as neutral H and H2. Since these ions might

quench the metastables in the beam and create fast neutrals by charge exchange, they are swept away by an

electric field produced by the ion plates. This is an electrode structure [10] which creates a uniform field between the electrodes but very weak fields a short dis- tance from them.

The Lyman-(3 lamp has been described elsewhere [7] and only details relevant to the operation of the beam

will be repeated here. This lamp is a relatively high in- tensity source of hydrogen resonance radiation with a narrow spectral width and low self-reversal to better

match the beam's absorption profile. The lamp's window

is a LiF crystal 12 mm in diameter and 1 mm thick

which when cooled will transmit radiation at 1026 A [11]. Electrodes near the window excite an rf discharge in a gas composed of 1% hydrogen and 99% helium. The lamp

is surrounded by a Dewar filled with liquid nitrogen

which cools the window and the gas. With an rf power of 10 W at 130 MHz the lamp^ produces lO^^ - lO^^ photons

sec.~^ steradian"^ at 1026 A. A gold-plated copper box

surrounds the lamp to reduce rf interference and to iso- late the hydrogen beam from the lamp's rf fields. The

VUV radiation passes out of the box through a 2.0 cm by

2.0 cm opening covered with a gold mesh having 40

lines /cm and 82% transparency. The lamp's window is lo- cated 2 cm from the center of the beam and 25.0 cm

down the beam from the multichannel array. At this point, ground state atoms are optically pumped to the

metastable state. The diameter of the atomic beam is about 1.0 cm. In order to reduce scattered Lyman-a

light from the lamp, an aperture is located about 31 cm

from the multichannel array. This is a large gold-plated

copper sheet 3 mm thick with a 1 cm hole for the beam

to pass through.

The flux of metastable atoms is measured by applying an electric field to mix the 2Sy2 and 2Py2 states and then

detecting the emitted Lyman-a radiation when the atom

decays to the ground state. This quenching region is lo- cated approximately 48 cm from the multichannel array

where the atomic beam is 2 cm in diameter. The quenching electrodes [12] are arranged so that a well defined

field occurs in a restricted portion of the atomic beam.

This allows the metastable beam to be unperturbed ex- cept immediately in front of the Lyman-a detector where

the fluorescence may be efficiently detected. The elec- trodes are a pair of bars 2.0 cm long, 0.5 cm wide and

0.15 cm thick. They are located symmetrically about the

beam 2.0 cm apart so that the long axis of the bar is per- pendicular to the beam. Each electrode is mounted on a grounded plane 6.5 cm square with a 0.1 cm insulating

stand-off. The potentials applied to each electrode are equal in magnitude but of opposite polarity. The field at the center of the electrode structure is 0.2 V/cm per volt of applied potential difference. The entire electrode structure is gold plated and highly polished to reduce

stray fields. It is estimated that 95% of the quenching

occurs in 2.0 cm along the beam axis between the

electrodes.

The Lyman-a fluorescence is detected with a benzenefilled, ionization detector. The detector has a cylindrical,

gold-plated, copper body 3.5 cm in diameter and 7.5 cm

long with a 0.05 cm diameter tungsten wire electrode

down its center. On one side is a MgF2 window 4.0 cm by

2.0 cm and 0.2 cm thick and on the other side is a high- vacuum valve which allows the detector to be removed

from the chamber for refilling. The fill gas was 2.0 Torr

benzene, 2.9 Torr argon and 9.8 Torr carbon dioxide.

Under these conditions, the gain was unity to about

100 V and approximately exponential thereafter. The

quantum efficiency of the benzene mixture is 40% and

the transmission of the MgF2 window is 50% for Lymana radiation. The detector is located 3.0 cm from the

164
quench region in a gold-plated copper box with a gold

mesh window. This allows approximately 5% of the

fluorescence to be collected. The final element of the beam line is a VUV photodiode

which facilitates the final beam alignment. It consists of a stainless steel photocathode and a collector electrode

consisting of a stainless steel tube. After all of the ele- ments in the chamber are properly arranged the hydrogen beam can be unambiguously aligned by maximizing

the diode's photoelectric current. The multichannel array

collimates the VUV radiation from the discharge bulb as

well as the neutral atom flux so that one travels the same

path as the other. The flux of metastable 2S1/2 hydrogen was measured

using the arrangement just described. The maximum

metastable flux was obtained with the lamp containing a 1% hydrogen, 99% helium mixture, possibly due to low

self-reversal of the Lyman-p line. For a beam intensity of

1.5 X 10^^ atoms/sec, a quench voltage of ±25 V and a detector bias of 800 V, a metastable signal of 2.0 pA and

a Lyman-a background of 120 pA was measured. When

the beam, lamp and quench field were independently

turned off and on the metastable signal appeared or disappeared as expected. The Lyman-a background comes from the lamp and could be eliminated by improved shielding. Using the previously given parameters

for the transmission of the detector window, the

detector's quantum efficiency, the collection efficiency

for the fluorescence and the measured gain of the detector at 800 V, the current of 2.0 pA corresponds to a metastable flux of 2 x 10^ atoms/sec.

The metastable signal measured as a function of the potential applied to the quenching electrodes is shown in Fig. 3. In order to eliminate the large Lyman-a background from the lamp, synchronous detection was used.

A signal generator supplied a square wave voltage of

25 V and 1 kHz to a wire located between the aperture

CO

^2

CO

I 0 2000 4000

QUENCHING VOLTAGE SQUARED

Figure 3. Quenching curve for the metastable beam. The points

are the measured values and the curve is from a least squares

fit to the Bethe-Laynb theory unth Y = Cifl - exp(-C.2X)]

and Cj = 5.61 and C2 = 0.0019.

and quench electrodes. This caused the metastable flux to be quenched at this frequency while the lamp background was unaffected. The detector current went to a

lock-in amplifier which was referenced by the signal generator. A least-squares fit to the data was made using

the function,

Y = Ci[l - exp(-C2X)] (1) where Y is the detected signal and X is the quench volt- age squared. The choice of this function is motivated by

the Bethe-Lamb theory of the lifetime of the 2S state in external fields [1] [12]! The fit yields Ci = 5.61(14) and

C2 = 0.0019(2). Ci is the asymptotic limit for the satu- ration curve while C-? is directly related to the beam's mean velocity.

An approximate calculation of the number of metastable atoms produced by the lamp may be done by con- sidering the transverse velocity components of the beam.

Since the hydrogen beam is collimated, the overlap of the

Doppler broadened absorption Hne with the emission line of the lamp is reduced. An expression for the fraction of photons from the lamp absorbed by the beam is given

by [13],

An — ikon9\2

(1 + a^)2\l/2 2!(1 + 2a^)^

(2)

where /cq is the atomic beam's Doppler absorption coeffi- cient, C is the absorption depth, and a is the ratio of the

lamp Doppler width to the atomic-beam absorption width

which is reduced by collimation. If the beam source is as- sumed to operate at 300 K and the lamp at 77 K, a = 29,

ko =0.19 cm^^ for a beam flux of 1.5 x 10'^ atoms sec"^ and A a = 6.0 x 10"^. The number of metastables pro- duced is then given by the product of the fractional absorption A„, the number of Lyman-p photons from the

lamp, 10^^ to 10^^ sec"\ and the optical pumping effi- ciency, 0.12. The result is 1.4 x 10^ metastables per

second. This is clearly more than was measured and rea- sons for this will be discussed later. The number of metastable atoms F which decay in the

quenching electrode region can be written as [1] F = Fo[l - exp(-f/VTs)] (3) where Fq is the total metastable flux, [' is the length of the interaction region, v is the atomic velocity and ts is the lifetime of the 2S state induced by the electric field. The Hfetime is given by ts =^ (£'/475)"^<p for weak mixing [15] wliere E is the electric field in volts /cm and tp is the lifetime of the 2P state. The average velocity may be

used for v to a good approximation [14]. The velocity dis- tribution of the metastable atoms produced by optical

pumping can be obtained by first considering rate equations for the IS, 3P and 2S three level system. Assuming

the weak optical pumping limit where most atoms are in the ground state and tiis = -Tri2s one obtains

A )mfB(v)A (4) A32 + A31 V where A and B are the Einstein coefficients, Wi is the

energy density of the lamp photons, r\B the number density of the beam, d the interaction length and /g is the

Maxwell-Boltzman velocity distribution for the beam

[16]. This indicates that the metastable atoms will have a

velocity distribution fzsiv),

/2s (v) = -^vexp(--^)

a'' a"

(5)

where a = \/2kT/m. The mean velocity for the distri- bution /2s is V = VTTa/2. This may be used to relate

165
Eq. (3) with Eq. (1) and the experimentally derived

parameter €2- For C2 = 1.9 x 10"^, a metastable beam

temperature Tg is found to be 100 K. This is much below

the neutral beam temperature of 300 K. Although

Eq. (5) indicates that the optical pumping process leads

to a slower velocity distribution, its magnitude is not great enough to explain the metastable beam temperature. The measured metastable flux at 10^ atoms /sec is con- siderably less than the flux estimated by using Eq. (2). If the assumptions involved in this estimation are valid, a number of effects may be present to reduce the experimental yield, such as quenching due to stray fields, motional electric fields, blackbody radiation, and collisions with the background gas. The stray fields arise from contact potentials and accumulated charges on insulating

films mostly arising from electrons produced by the

photoelectric effect and the VUV radiation from the

lamp. The gold plating reduces the accumulated charge,

and from Eq. (3) it is clear that contact potentials on the

order of 0.1 V are insufficient to explain the discrepancy.

The motional Stark quenching may be readily calculated

using standard techniques [1]. The magnetic field in the

metastable beam was measured to be less than 30 milli Gauss, which leads to ts = 10° sec, too large to explain

the discrepancy. The 2S1/2 state may also be quenched by

blackbody radiation which is present in the vacuum

chamber at T = 300 K. Previous calculations of this ef- fect [17] but ignoring the fine structure and Lamb shift find Ts = 7.0 x lO'* sec, while similar considerations for quenching through P1/2 yield ts = 3.2 x 10° sec and

through P3/2, Ts = 3.3 x 10^ sec, which is again much too

large. Finally, the quenching due to collisions with the

background gas is especially effective at low velocities. Using a cross section of a = 2 x 10"^^ cm^ [18] one finds a loss in beam intensity of only 1%. Thus it seems most

likely that the discrepancy should be explained by an

overestimation of the flux from the lamp or, more likely,

that of the neutral hydrogen beam.

The mean velocity of the metastable beam is anomalously low at T" = 100 K. Quenching due to the motional Stark effect is proportional to the velocity [14]. This would lead to a reduction in the temperature of the

beam. However, as seen above, for the magnetic fields occurring in the vacuum chamber, this is a negligible ef- fect. A more likely explanation involves the spectral

overlap of the lamp at liquid nitrogen temperature with

the absorption profile of the beam. Previous calculations

have assumed that the radiation traveled perpendicular

to the beam. This is not so since the vdndow is only 2 cm

from the beam. Atoms traveling towards the lamp have

their absorption profile shifted up in frequency by the

Doppler effect. However only the slower atoms can ab- sorb the lamp radiation since it is emitted at a lower temperature. Atoms traveling away from the lamp have their

absorption lines Doppler shifted down in frequency, but once again only the low velocity atoms can absorb the

lamp's radiation. Therefore, the low velocity atoms are

preferentially optically pumped and it is not surprising

that the temperature of the beam roughly corresponds to the temperature of the lamp.

The source of metastable atomic hydrogen described

here has a flux which compares favorably with other

1.5M CO, LASER

CO 2 SPECTRUM

ANALYSER

CO 2 WAVEGUIDE

LASER

0 Y

InSb SPECTROMETER

C^P HgCdTe

TO SPECTRUM ANALYSER

DISCHARGE TUBE

CHOPPER

I.L //////

n = 9

7/2

5/2

3/2

1/2

n^20

Figure 5. Schematic of experiment.

U 1__L -150 -100 -50 0 50 V [MHz] —«-

n=2

5/2 3/2

1/2

J

n=9

Figure 4. Energy level diagram for saturated fluorescence in

the Rydberg states of hydrogen. Figure 6. Measured spectra for n = 9 to n = 20 transition.

166
sources which use electron bombardment. Moreover, the

beam spread due to recoil from absorption of the

Lyman-(3 photon is only 0.1 degree and the second-order

Doppler effect is only two parts in 10^^ since a slow beam

with a well defined velocity distribution is used. This

suggests it may find use in precision experiments in optical spectroscopy or level crossing spectroscopy. In addition, there are several avenues for improvement. The

Lyman-a background may be reduced by further shielding. The lamp may be scaled to larger lamps or arrays of lamps vdth higher flux. Laser sources may become avail- able to greatly increase the radiation at 1026 A [19]. Finally, intense cooled sources of atomic hydrogen beams

may be used at liquid nitrogen [9] or liquid helium temperatures [20]. Such improvements could lead to a source which is ideally suited to experiments requiring high

state selectivity and large intensity [21]. A precision measurement which would require the ab- sence of perturbing electric and magnetic fields afforded

by the atomic beam is indicated by Fig. 4. This relies on

a coincidence of the 9 |xm R{20) COo laser line and the

transition from n = 9 to n = 20 in hydrogen. A preliminary measurement using a Woods discharge has been

done. The technique of saturated fluorescence is used

where the transition from w = 9 to = 2 is monitored.

Since the selection rules for dipole-allowed transition

only connect the S, P and D levels of » = 9 to n = 2, the levels observed by the technique are shown in Fig. 4. The experimental arrangement is shown in Fig. 5. A CO2 waveguide laser is used to induce the transitions. It is directed through a hydrogen discharge,

chopped and reflected back on itself. The laser has a power of 500 mW, 1 GHz tunability, and less than

100 kHz passive stability. The n = 9 to n = 2 fluores- cence is filtered by a 1/4-meter spectrometer and

detected by a photomultiplier and lock-in detector. The

frequency of the hydrogen transition is measured by

heterodyning with a 1.5 m positive-column CO2 laser in a HgCdTe diode. The beat signal from the diode is measured with a rf spectrum analyzer. The 1.5 m CO2 laser can be locked with a CO2 cell so that measurement of the

absolute frequencies is possible [22]. The transitions to be expected are shown at the top of Fig. 6 [23]. It is noteworthy that the lifetimes of these states are rela- tively long. The hfetime of 9S is 1.3 x 10*'' sec while

20P is 1.4 X 10"" sec. This gives a transition with a Q

thirty times greater than the Balmer-a line. At the bot- tom of Fig. 6, a trace is shown of the spectrum actually

observed. It is clear that the plasma environment of the

discharge has considerably broadened the lines. The

transition for 9S to 20? which occurs on the right side of the trace is the least broadened. This is possibly due to these levels having the smallest Stark polarizabilities.

Undoubtedly, future measurement which can reduce

external perturbations will yield improved results.

The author would like to thank R. D. Deslattes for valuable support at the initiation of this work, W. L. Williams for numerous suggestions and details of the

quenching of metastable hydrogen, and L. Levy for infor- mation on the ionization detector which he designed and was used in this work.

References

[11 W. E. Lamb, Jr., and R. C. Retherford, Phvs. Rev. 79, .349 (1950).

[2] R. T. Robiscoe and T. W. Shvn. Phv.^. Rev. Lett. 24. .j.'39 (1970).

[31 B. L. Donnally, T. Clapp, W. Sawyer, and M. Schultz, Phys. Rev. Lett. 12, .502 (19(M).

[4] W. H. Wing, Phy.s. Rev. Lett. 45, 631 (19iS0).

[51 H. Zacharias, H. Rottke, ,J. Danon, and K. H. Welge, Opt.

Commun. ;57, 15 (19)^1).

[6] .J. W. Heberle, H. A. Reich, and P. Kusch, Phy.^. Rev. 101, (523

(19.5(5). [71 K. C. Harvey, Appl. Opt. 20, 2883 (1981).

[81 C. B. Lucas, Vacuum 23, -395 (1973).

[9] H. WiLsch, .J. Chem. Phys. .56, 1412 (1972).

[101 S. Czuchlewski, S. R. Rvan, and W. H. Wing, Rev. Sci. Instrum.

47, 1026 (1976).

[Ill W. R. Hunter and S. A. Malo, J. Phys. Chem. Solids 30, 2739

(1969).

[121 R. T. Robiscoe, Phys. Rev. 138, A22 (1965).

[131 M. W. Zemanskv, Phys. Rev. 3«, 219 (1930).

[141 W. E. Lamb, Phys. Rev. 85, 259 (19-52). [151 H. A. Bethe and E. E. Salpeter, Quant iiiii MeclnuiicK ofOiicaiid Ttvo-Elecfroii Atoiiis (Plenum, New York, 1977).

[161 G. Comsa, Vacuum 19 , 277 (19(59). [171 J. W. Farley and W. H. Wing, Phys. Rev. A 23, 2397 (1981).

[181 S. J. Czuchlewski, Ph.D. thesis, Yale University, 1973.

[191 G. Pellegrini, Nucl. Instrum. Meth. 177, 227 (1980).

[201 R. W. Cline, D. A. Smith, T. J. Greytak, and D. Klei)pner,

Phys. Rev. Lett. 45, 2117 (1980).

[211 R. R. Lewis and W. L. Williams, Phvs. Lett. .59B. 70 (1975).

[221 K. M. Eyenson, J. S. Wells, F. R. Petersen, B. L. Danielson, and G. W. Day, Appl. Phvs. Lett. 22, 192 (1973).

[231 G. W. Erickson, -J. Phvs. Chem. Ref. Data 6, 831 (1977).

167

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Precise Determination of the S and P Quantum Defects in Sodium and

Cesium by IVIillimeter and Submillimeter Spectroscopy Between

Rydberg States

P. Goy, J. M. Raimond, G. Vitrant, C. Fabre, S. Haroche, and M. Gross

Laboratoire de Physique de I'Ecole Normale Superieure 24, rue Lhomond, 75231 Paris Cedex 05, France

Well-stabilized millimeter and submillimeter sources in the frequency range 50-500 GHz permit

one to induce narrow transitions between Rydberg states of alkalis with the principal quantum

numbers n in the range 23 =£ h =s 41. The levels are prepared by laser excitation. Detection of the atomic levels is performed through the selective field ionization technique. Precise expei'imental values

for quantum defects and fine structure intervals are reported for the S and P states in sodium and

cesium. The hyperfine structure of Rydberg states n Si 2 and n Pi 2 has been observed for the first time

in cesium. The extension of these e.xperiments with increased accuracy to the spectroscopy of hydrogen

would provide a new way to determine the Rydberg constant in frequency units. Key words: alkali atoms; fine structure, hyperfine structure; millimeter and submillimeter waves; quantum defects; Rydberg constant; Rydberg states.

1. Introduction

Since they have a single valence electron, alkali atoms

are like hydrogen in many ways. The simple energy' spec- trum E — -Ru/n^ in hydrogen becomes E =

—RA/i'n. - e)^ in alkalis, where the energy E is measured

from the ionization limit, n is the principal quantum

number, R is the Rydberg constant and e is the quantum

defect which reflects the perturbing effect of the core on

the valence electron. The quantum defect e depends on

the alkali and on the angular momentum f of the level. For a given species, it decreases with increasing!' values. For a given f, it increases with the atomic number of the

alkali. In first approximation, e is independent of the

principal quantum number n. Only very precise experiments make evident the systematic variation of e with

the binding energy. High resolution experiments per- formed in the optical range can determine the

slight variation of e with n. See for instance Refs. [1, 2] for the case of the quantum defects in cesium, ^f^{n).

In this paper, we present the most precise determination of the S and P quantum defects close to ionization. It

is obtained by measuring the energy difference between

successive excited states. The first state is excited from

the ground state in an atomic beam by laser excitation.

The transition to the second state is induced by a micro- wave source, the frequency of which is carefully measured. Due to the long lifetime of the Rydberg states (about 50 |xs for n around 30), some very narrow

(10 kHz range) resonances can be observed, yielding resonance Q-factors in the 10" range. The best atomic

frequency determination is in the 10~* range and the ulti- mate uncertainty in the determination of the quantum defect can be as low as a few times 10"".

2. Experimental Set-Up

2.1 Rydberg Atoms Excitation and Detection [3] The experiment is performed on an alkali atomic beam

(Na or Cs) propagating in a vacuum chamber with a background pressure of 10^^ - 10 " Torr. From the ground

state (3S or 6S), the atoms are excited to an nS state by

a stepwise process, via the intermediate 'SFm (Na) or

6P3A> (Cs) level, by two pulsed dve lasers. These two dye

lasers (5896 A and -4110 A for Na, 8521 A and -5100^1

for Cs) are themselves excited by the same pulsed pump

laser of the nitrogen or Yag type.

After the pulsed laser excitation, the Rydberg states interact with the cw microwave radiation. The microwave

power necessary to saturate transitions between close levels is very small: 10"^W/cm- or less for two-photon

transitions and 10"^°W/cm^ or less for single photon transitions. The interaction lasts a variable time interval

between 2 and 100 |jls. Then the atoms enter between

the plates of a capacitor. An electric field ramp is applied to the atoms. This ramp ionizes each Rydberg level

at a characteristic field value. Thus each level is ionized

at a specific time. The electrons produced are collected and amplified in an electron multiplier. This well-known

field ionization technique [3] is selective and very sensitive: a single Rydberg atom gives a macroscopic current. The final state population is averaged over a few tens of laser pulses by a PAR 162 boxcar integrator, and

recorded as a function of the microwave frequency.

2.2 The IVIicrowave Set-Up

The microwave set-up is shown in Fig. 1. Among the

microwave sources used were nine carcinotrons

(Thomson-C.S.F. backwave oscillators) working respectively in the frequencv ranges (in GHz): 51-61, 75-77,

77-82, 88-97, 91-100, 107-125, 226-274, 277-314, 417^85.

The frequency F,. of a carcinotron is simply determined

by the voltage appUed to it. In order to count Fc and to

stabilize it, we used as a reference an X-band klystron

(X-13) locked to a quartz oscillator (MOS-5 microwave

system). The frequency of the klystron Fk is directly

measured on a counter, the clock reference of which comes from a long wave receiver for standard frequency

with an accuracy better than 10"^ (Adret 4801 A). A

mixer [4] using a packaged small Schottky diode placed

across a waveguide generates the harmonic of the

klystron (5 « p ^ 38) and mixes it with the carcinotron

169
radiation. The beat note signal at the intermediate fre- quency F if = \Fc - pFic I is amplified in the interval

0-500 MHz, counted and monitored on a spectrum

analyzer (HP 141T, 8552B, 8553B). The beat note is also used to lock the carcinotron. Phase-locking carcinotrons

is easy up to 200 GHz by comparing the phase of the

beat signal with that of the signal given by a synthesizer.

The linew^idth of the free running carcinotron

(—0.5 MHz) is then reduced below the resolution Hmit of the spectrum analyzer (a few Hz). Above 200 GHz, the

phase noise of the multiplied frequency of the klystron

becomes noticeable. Then the frequency-locking arrangement is more convenient. The linewidth of a 460 GHz

carcinotron can thereby be reduced from 2 MHz to

1 kHz.

Above the cut-off of the waveguide the same harmonic

mixer is used for all frequencies. For instance, F ^

80 GHz is measured with an F-band (90-140 GHz)

waveguide. Naturally the conversion loss increases with

increasing harmonic mixing number p, going from 25 dB

with p = 7 atFc =80 GHz, up to 85 dB with p = 38 at Fc — 460 GHz. Nevertheless the signal-to-noise is still reasonable at the upper frequency (^ 20 dB in 100 kHz

bandwidth).

The carcinotron can be very powerful (4 W at 80 GHz). Microwave leaks sometimes make it difficult to desaturate the most sensitive transitions for which

powers in the nW range are sufficient. Instead of the

carcinotron, we have also used the millimeter wave coming out of the harmonic mixer working as a frequency

multiplier (top of Fig. 1). The harmonics F - pF^ of the

klystron frequency Fk are available at the millimeter

waveguide output of the frequency multiplier as soon as

their frequency is larger than the cut-off frequency of the

waveguide. Starting with 100 mW from the klystron near 11.5 GHz, the following optimized microwave

powers have been obtained: 0.1 mW at 80 GHz; 10 |xW

at 115 GHz; 1 jjlW at 230 GHz and so on [4]. Reducing

this microwave power is easily achieved by reducing the

klystron power. Practically all single-photon transitions

vdth F ^ 340 GHz (P 5= 1 nW) and" two-photon transi- tions with F ^ 110 GHz (P ^ 10 m-W) can be obtained

with this very simple arrangement.

For any transition n1 7i 'f' the resonant frequency F

is given by:

stabilized

IMillimeter Wave

Frequency

IMultiplier

Klystron

Carcinotron lotro '7/7/

Power

Supply

Frequency

Stabilizer

Power

Supply

12 GHz

Counter

777?

V

Longwave

Frequency

Standard

Phase

Detector

Figure 1. Microwave arrangement. On the top right is the cen- timeter klystroyi, the frequency of which (Fj^ ) is quartzstabilized. This klystron can be used either of two ways: 1st—

the centimeter poiver is sent down to a harmonic mixer, the carcinotron frequency beats with the p harmonic of Fj^ and a feedback loop is used to stabilize the intermediate fre- quency F if 2ndr--the centimeter microwave is sent up to a fre- quency multiplier, the milliymter frequency F is directly ob- tained at the multiplier output F = pFj^ and the carcinotron is not used.

F/R = [n - e('(n)]"2 - [n' - ef.(n')]-2 (1)

3. Experimental Results

3.1 Two-Photon Doppler Free nS (n -i- 1)S

Transitions in Sodium [5] To cancel the Doppler broadening, it is possible to observe the two-photon absorption of two counter- propagating photons in the standing wave of a microwave

cavity. This cavity is of the Fabry-Perot semi-confocal

type and the atomic beam crosses it in a direction per- pendicular to its optical axis. It is tuned by adjusting the

distance between mirrors while monitoring the transmitted microwave power.

The following n S ^ (n + 1)S transitions have been

observed: n = 32, 33, 34, 36, 39, and 40. With interaction times up to 100 |jls the resonance linewidths are

respectively: ^F, = 36, 37, 28, 25, 11, and 13 kHz. The

narrowest transitions (Fig. 2) are transit-time limited

and the resonance Q is of the order of Q Fjl^Fc -

59 260 221.5/11 ~ 5 x 10^ The uncertainty in frequency

position ±2 kHz includes an "instrumental" uncertainty

of a tenth of a linewidth and a ±1 kHz uncertainty due to Stark shift induced by stray electric fields (a few tens of mV/cm).

Using Eq. (1) and the sodium Rydberg constant

(/^Na = 3.289 763 409(20) lO^^ Hz), we can deduce the

following formula for the S-quantum defect in sodium,

valid for n ^ 17: eI'Mw) = 1.347 969 2(4) + 0.061 37(10) n*-^ (2) where n* — n - 1.348. Any nS n'S two-photon transition in Na can thus be

predicted with good accuracy for n ^ 17. Since the n -

17 level is 6 THz below the ionization limit, a large

number of transitions in the 0-6 THz range could be ab- solutely calibrated, and could serve as frequency markers

in the centimeter, millimeter, submillimeter and far- infrared range.

170
Vc

-30 -20 -10 0 10 20 30 kHz

Figure 2. Two-photon 39S JfOS transition in Na. The two

traces are obtained for a Doppler free tivo-photon transition in sodium with a carcinotron stabilized frequericy around

59 GHz. Increasing the interactioyi time froyn 85 (jls to 100 |xs decreases the resonance linewidth in the same ratio: this suggests that this linewidth is transit-time limited. The arroivs show the unresolved hyperfine stmcture difference of the nSj/2

levels.

3.2 One-Photon nS n'P Transitions in Sodium [6]

Transitions hS n Pj/., and n S —* n Vy., have been ob- served with n = 23, 24, 25, 32, 33, 34, and 36. We have

also observed hS —> in + 1)P transitions with n = 34

and 35. The uncertainty on the frequency measurements,

of the order of 0.5 to 1 MHz, reflects the finite width of resonance (Doppler effect) and residual Stark and Zee- man effects. According to Eq. (2), the S quantum defect is well- known for each transition. The ep quantum defects can

then be obtained. One can deduce the following formula

(which predicts the observed value within ±1 MHz):

eP^^^ = 0.855 424(6) + 0.122 2(2) n* (3) where n* — n - 0.855 4; and = 0.854 608(3) + 0.122 0(2) n* (4) where n* — n - 0.854 6 The observed fine structure separation AF(n) =

^ Pi/2 ~^ P3/2 has an n " '"^ dependence:

AF^Mn) = 5.367(5) W^{n - 0.855) Hz (5) where 0.855 is the averaged Piy2, P3/2 quantum defect.

3.3 Two-Photon nS ^ (n -1- 1)S Transitions in Cs [7]

Due to their higher mass and lower operating temperature of the oven, the thermal velocity of Cs atoms is lower than that of Na. Thus the Doppler effect is less important, although it is not quite negligible (—100 kHz).

The experiments described here are not performed in the

cavity because a large fraction of the slow Cs atoms de- cays during the transit of the 50 mm diameter cavity.

The following n S ^ (n + 1)S transitions have been

observed: n = 27, 32, 33, 34, 35, 36, 37 and 38 with the

following respective linewidths AF = 390, 210, 200, Z,

250, 186, X and 430 kHz respectively {X when unmeasured). The interaction time is in the range 2 to 10 |jls. The frequency determination accuracy is ±100 kHz and

is essentially due to the Doppler effect and Stark effects from stray electric fields. Using Eq. (1) and the value of the Rydberg constant in Cs (i2cs = 3.289 828 299(20) lO^^ Hz), one obtains the

following expression for the S-quantum defects: 4^{n) = 4.049 325(15) + 0.246 2(20) n

-'^ (6) where — n - 4.0493.

3.4 One-Photon nS nP Transitions in Cs [7]

3.4.1 Quantum Defects of the nP Levels

The following transitions nS ^ n Yyo have been ob- served: n = 24, 25, 26, 27, 28, 31, 32 and 38. Also the n S K P3,2 transitions: n = 25, 26, 27, 28, 30, 32, 34, 35

and 38. For all these transitions a large splitting of the resonance signal is evident. It is due to the hyperfine

structure of the n S1/2 initial state (see in Fig. 3 the

23S —> 23P3/2 transition split into two peaks "5" and

"6"). Moreover, one can observe also a second splitting

in the nS n P1/9 transitions due to the hyperfine components of the n Pi/2 levels (see in Fig. 3 the four peaks

"1", "2", "3", and "4"). Taking into account this structure and measuring the frequency at the line center, we obtain from Eqs. (1) and (6) the following expressions: epC^^(n) = 3.591 56(3) + 0.3714(36) 7i' (7) where n* — n — 3.5916; and, = 3.559 06(3) + 0.374(4)11* -2 (8) where n* — n — 3.5591

The observed nS nP transition frequencies agi-ee

within ±3 MHz with the values calculated from Eqs. (1), (6), (7) and (8).

171
T i

1 -//- 1 r

I I I

I I \ I I I \ I i_

426.78 426.80 426.82 455.96 455.98 456.00 Frequency (GHz)

Figure 3. Sirigle photon 23S ^23P transition in Cs. The large

frequency difference (29172 MHz) between the two groups

"l~2-3-i" and "5-6" is due to the fine structure 23Pii2 - 23Pj/2 of the 23P level. Also observed are: the hyperfine structure of

the 23Sii2 level (9.2 MHz) in the separation "5" to "6" and

"1-2" to "3-Jt", and the hyperfine structure of the 23Pii2 level on the splitting (2.2 MHz) "1" to "2" and "3" to The hyperfine structure of 23Pjf2 is unresolved here.

3.4.2 Fine Structure of the nP Levels

Observed on the levels 23, 25, 26, 27, 28, 32 and 34,

the fine structure interval is determined to a good rela- tive accuracy due to its large absolute value: ^F^^n) = 2.138 3(2) WHn - 3.5757)-3 Hz, (9) where 3.5757 is the average Fy^, P3/2 quantum defect.

3.4.3 Hyperfine Structure of the nSi/2 Levels

The large splitting of the resonance traces of the n Si/2 n Pi/2 or n S1/2 n P3/2 transitions gives the fol- lowing experimental values of the hyperfine structure (F = 3 F = 4) of the n Qy> levels with the absolute

accuracy ±0.8 MHz: 9.2, 5.8, 5.7 and 4.7 MHz for n =

23, 25, 26 and 28 respectively (see for instance in Fig. 3 with n = 23 the distance "5" to "6" or "1-2" to "3-4").

3.4.4 Hyperfine Structure of the P Levels

The n P3/2 levels are split in three sublevels giving rise to a pattern more compHcated (unresolved on "5" and

"6" in Fig. 3) than for the n S1/2 «Pi/2 transition. Observing only the double splitting of the n Sy2 n P1/2 transitions (distance "1" to "2" and "3" to "4" in Fig. 3), we obtain the experimental values for the Piy2

hyperfine splitting {F = 3 ^ F =4) with the absolute

uncertainty ±0.2 MHz: 2.23, 1.6, 1.25 and 1.1 for n =

23, 25, 26 and 28, respectively.

4. Conclusion

An improvement by a factor of 10 to 100 of the resolution obtained here is certainly possible. Cooling the

atomic beam apparatus to liquid He temperature should

suppress blackbody effects and cancel stray electric fields

[8]. One can imagine for instance that for the two photon

Doppler free resonance, the natural linewidth ~1 kHz

could be observed. In this way a relative accuracy in the

range lO"^'' on the frequency could be obtained on n S

in + 1)S, nS ^ in + 3)S, nS ^ {n + 5)S . . . transitions. The alkali Rydberg levels would provide a secondary frequency standard for the calibration of elec- tromagnetic radiation from the microwave to the far in- frared domain.

The extension of these experiments to the spectroscopy

of hydrogen Rydberg levels is worth considering [9]. In

this case, the quantum defects vanish and the Rydberg

constant Rfj can be directly measured in frequency units.

Thanks are due to Dr. R. Adde (Electronique Fondamentale, Orsay) and to Dr. A. Clairon (LPTF, Observatoire de Paris) for the lending of equipment. One of the

authors (P. G.) would like to express his gratitude to A. Clairon for many fruitful discussions.

References

[1] C. J. Lorenzen, K. H. Weber, and K. Niemax, Opt. Comm. 33, 271 (1980).

[2] C. J. Lorenzen and K. Niemax, J. Quant. Spectros. Radiat. Transfer 22, 247 (1979).

[3] C. Fabre, S. Haroche, and P. Goy, Phys. Rev. A 18, 229

(1978).

[4] P. Goy, Int. J. Infrared Millimeter Waves 3, 221 (1982).

[51 P. Goy, C. Fabre, M. Gross, and S. Haroche, J. Phys. B 13, L

83 (1980).

[61 C. Fabre, S. Haroche, and P. Goy, Phys. Rev. A 22, 778

(1980).

[7] P. Goy, J. M. Raimond. G. Vitrant, and S. Haroche, Phys. Rev. A 26, 2733 (1982).

[81 F. Witteborn and W. Fairbank, Rev. Sci. Instrum. 48, 1 (1977).

[91 D. Kleppner, Bull. Am. Phys. Soc. 20, 1458 (1976).

172
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

A New Method for Measuring the Fine Structure Constant

Using Stark Spectroscopy

Michael G. Littman

Department of Mechanical and Aerospace Engineering, Princeton University

Princeton, NJ OSMO

and

William D. Phillips

Electrical Measurements and Standards Division, National Bureau of Standards, Washington, DC 20234

An experiment to determine a value for a, the fine structure constant, is proposed. The determination is to be based on a measurement of the Stark effect of hydrogen Rydberg states. Hydrogen atoms

in a uniform field of known strength will be excited to Rydberg levels using intense tunable lasers. The

presence of excited atoms will be detected using the sensitive technique of field ionization. A precise

measurement of the linear energy shift of an extreme Stark state is to be made, and from this measurement, in conjunction with reported values of the Rydberg constant, and 2e/h, the Josephson

frequency-voltage ratio, a value of a will be determined. The estimated accuracy of the determination

is expected to be competitive with or better than the 0.11 ppm accuracy of the current best non-QED

determination of a. ^

Key words: fine structure constant; hydrogen atom; Rydberg states; Stark effect.

1 . Introduction

The need for an improved value for a, now known to an

uncertainty of 0.11 ppm [1], is almost self-evident. The

fine structure constant is the primary expansion parameter used in quantum electrodynamics (QED). An accurate value of a is needed for virtually any comparison of QED

theory with experiment. Currently, there are several ex- periments whose comparison vidth theory is limited by

the uncertainty in a. Of these, one of the most notable is the recent "g-2" determination by researchers at the

University of Washington [2]. Here, the uncertainty in the measurement of the electron anomalous moment is less than the uncertainty imposed by a on the theoretical

prediction. A 2.5 standard deviation discrepancy^ with

QED theory is obtained [1]. The value of a itself is also subject to controversy. While the best QED-independent

value of a is based on measurements of the gyromagnetic

ratio of the proton, ^p, a discrepancy of 6 ppm (12 standard deviations) exists between recent precision measure- ments of "Yp [1,3]. This represents a 3 ppm discrepancy in

a. Such discrepancies point to the need for redundant, in- dependent determinations of a. The motivation for the demonstration of the usefulness

of precision Stark spectroscopy is also clear. In addition

1 Recent calculations by Levine and Roskies and by Kinoshita and Lindquist (elsewhere in these proceedings) have reduced this discrepancy.

2 The field is expressed in units of that field produced by one electron charge at a distance equal to the Bohr radius, and the energy in units of the electrical potential energy of a pair of charges separated by a Bohr

radius.

to the obvious contributions to the area of fundamental

constants, precision Stark spectroscopy can be of considerable importance in the area of precision electrical measurements. Of particular interest are the possible applications to transfer voltage standards and the measure- ment of large potential differences.

2. Theory

In the following discussion we v^dll summarize the

theoretical basis for the measurement and some of the more important experimental considerations. In an electrical field of magnitude F, the non-relativistic energy eigenvalues for hydrogen are given [4] by

the perturbation expansion (atomic units^

)

W + ^ nkF

2«2 2 16

4ir2 n^F (1)

X (17^2 - Sk^ + 9m- + 19)+ .

Here n, m, and k are, respectively, the principal, azimuthal, and electric quantum numbers [m can assume in- tegral values between (» - 1) and -(n -1), and k can assume bi-integral values between (n - | m | - 1) and - in - I m I - 1)]. A map of levels having n = 12-14 and

I m I = 0, 1 according to Eq. (1) is given in Fig. 1. It is clear from this Stark map that the linear effect is dominant

over the range of fields presented.

To derive an expression for a, consider the linear term

ofEq. (1), Wi = SnkF/2. (2)

173
0 S 10 15 20 FIELD [kV/cm]

Figure 1. Hi/drogen Stark fstriictnre according to poiiu-hatioN

theory. The m = 0 levels are iiidicated bij solid li)ies: the \ iii

\ = 1 levels are indicated by dashed li)tes.

Converting to SI units and expressing the energy shift in terms of a frequency, vi, we have:

hvi - ZnkEea o

/ 2 where a,, is the Bohr radius, E is the field in V/m, and e

is the electron charge. Using the relation R^^_ = a/i-nao, we can write

vi = T^nkE i^)aR^-' . (4) 16 h

Although Eq. (4) is expressed in SI units, it is easy to see that the expression is still valid if the electric field, E, is expressed as £'nbs (i-e., is measured in units of

Vnbs /meter where Vnbs is the NBS unit of voltage) and

the Josephson frequency-voltage ratio, 2e /h, is expressed

in units of Hz/Vnbs Since the NBS volt is realized in terms of 2e/h to 0.03 ppm [5] and Ro. is known to 0.001 ppm [6], the determination of a at the 0.1 ppm

level will not be limited by the accuracy of the other con- stants in Eq. (4). It can be shown that higher (A'^*) order terms in the

Stark shift may be written (in SI units) as:

VM ^ C E^i^fa^c^'-'R^'' , (5) h

where C is a numerical constant which depends on the

quantum numbers. Only the speed of light, c, appears as an additional constant. Since c is known as well as the

meter can be realized, the contribution due to higher

order terms in the Stark expansion will not reduce the

accuracy of an a determination as long as they are not

too large. In view of the fact that the proposed precision meas- urement is based upon a perturbation expansion of the

energy, the general convergence properties of the series are of particular importance. Several recent theoretical

studies have considered the asymptotic nature of the per- turbation expansion. Koch, using the tabulation of the

terms of the expansion through 150"^ order by Silver- stone and co-workers [7], observes that if n'^F ap- proaches unity, the lack of convergence or, in some

cases, the convergence to the wrong value, is striking

[8]. He notes that the use of Pade approximants allows

the series to be resummed with improved convergence.

Others have examined the strong field effects on hydrogen Stark states using exact numerical techniques [9]. To

date, there have been few experimental checks of these

calculations; however, the different theoretical ap- proaches appear to be in reasonable agreement.

The lack of convergence of the perturbation expansion

reflects the fact that Stark states are quasi-bound. In any given field there is a finite probability that levels

will ionize spontaneously. In weak fields, levels ionize by

a quantum-mechanical tunneling process, while in strong

fields they ionize classically by rolling "over-thebarrier". In practice, ionization effects are important

only for fields on the order of or larger than the classical threshold field, Fc = (W'^/4) = (l/16t?^) [10]. For fields less than Fc , decay is dominated by radiation with an average decay rate that decreases with increasing n as n [4, 10]. In other words, hydrogen Stark states in low fields are quasi-stable and should be well described

by a perturbation approach. It is primarily for these rea- sons that we restrict our attention to the low field regime.

Due to the precise nature of the proposed measurements, the effects of fine and hyperfine structure, the

Lamb shift and the ac Stark effect cannot be ignored.

Even though we will not discuss these effects here, in the final analysis they will be taken into account. The proposed experiment, which is described in the fol- lowing section, requires the optical excitation of an ex- treme n — 35 Stark state from a low lying n = 2 level. In order to determine the optically accessible states it is necessary to understand fully the symmetry of the states involved. It is well known that the degeneracy in hydrogen is broken by the Lamb shift. In sufficiently weak

fields, that is, fields in which the Stark interaction is small compared with the degeneracy-breaking Lamb shift interaction, the spherical character of a given state is maintained. For the low lying |2 "S1/2 > , the electric in- teraction is important only above 300 V/cm. Below this

field value, the state is nearly spherical. In contrast,

highly excited levels assume a parabolic nature at very

small fields. This situation in hydrogen at low fields in which the lower state is spherical and the upper states are parabolic, is analogous to the situation that exists in

alkali metals in weak or strong fields. In the alkali case

all parabolic Rydberg states of a given m are accessible

from the spherical ground state [11]. We expect, like- wise, that all hydrogen parabolic states, including the ex- treme ones, will be accessible in weak fields. Recent experimental studies of the Stark effect have

dealt primarily with hydrogen-like atoms. Littman and

co-workers have measured the Stark structure of sodium

using tunable lasers and atomic beams [12]. They have

demonstrated the feasibility of selective pulsed laser excitation of each of the Stark states of a given azimuthal

quantum number from a low-lying level. Koch has

measured the position of several Stark states in atomic

hydrogen using quite different methods [13]. In Koch's

experiments a fast excited hydrogen beam is further excited to Rydberg levels using a fixed frequency CO2

laser. The Doppler effect is used for tuning and field ioni- zation is used for detection. Together these experiments

174
demonstrate that indeed one can use conventional tech- niques to excite and detect hydrogenic Stark states.

3. Experiment

The plans for measuring vi and £'nbs are presented

here. These two quantities are to be measured

separately, with the ultimate goal being the determination of the quotient, vi/£'nbs, to 0.10 ppm or better. In the interest of brevity we consider only the more important aspects of the proposed experiment.

A schematic of our experimental configuration is given

in Fig. 2. We require a thermal atomic beam source, a uniform field region, intense pulsed and cw lasers, and a

field-ionization detection region.

ELECTRON

MULTIPLIER

ATOMIC FIELD AND IONIZATION DEFLECTION ION SOURCE INTERACTION REGION REGION DETECTOR ZONE

HYDROGEN LEVELS

243nm

Figure 2. Experimental arrangement. The excitation scheme is depicted in the inset.

The restriction of a maximum field of 40 V/cm has a number of implications. First, the shift between the ex- treme high-w Stark components is limited to 8 cm"^

Since the plan is to determine vi to 0.10 ppm, the shift must be measured to about 20 kHz. This involves splitting the line by 1 part in 10. Second, corrections due to quadratic and higher order perturbation terms can be

made reliably. Here the contribution of the higher order

terms to the total shift is less than a few percent. Finally, this field region is well below the n = 35 ionization

field of 250 V/cm, and thus ionization effects will not be

important.

The difficult aspect of this experiment deals with the

precise measurement of the field strength. To determine

the field we must accurately measure an electrical potential difference and a distance. In addition, we must take

into account a number of effects which would influence

the strength of field that a given atom sees. Effects such

as contact potential, the "patch" effect, fringing field, motional electrical field, and other external stray fields,

such as those produced by photoions and photoelectrons

created during laser excitation, can influence the precision of our determination. Fortunately, we have the

flexibility to work with many different states in many

different fields, so that we have natural internal con- sistency checks on our measurements. If necessary, we

can even use the atoms to map out the field to determine

local inhomogeneities.

The field region will consist of a vacuum-spaced capaci- tor and an associated precision voltage source as shown

in Fig. 3. The field plates will be thin metallic films evaporated onto precision optical flats (X/200). In addition,

three symmetrically-placed, dielectric mirror pads will be

deposited on each of the substrates. These mirror pairs

will constitute high finesse Fabry-Perot interferometers

which, when used in conjunction with a stabilized

The atomic source will provide a ground state density

of 10" atoms /cm; the mean atomic velocity will be about

10^ cm/sec. Excitation will occur in the static field in two

two-photon steps, as shown in the inset in Fig. 2. We

plan to excite Stark states having n — 35 from the

ground state using the perturbed 1 2 ^ S1/2 > level as a resonant intermediate state. Both steps of excitation will take advantage of Doppler-free counter-propagating excitation. The first step, which serves to populate the in- termediate level, will be initiated by a 25 MW/cm^ burst

of radiation at 243 nm (2.5 mJ/10 ns focused to 1 mm^).

For this excitation step we will use a laser system similar to that used by Bjorklund and Freeman in their work

on three photon excitation of atomic hydrogen [14]. The

second step of excitation will be driven by 30 W/cm^ of continuous radiation at 730 nm from a ring dye laser (300 mW focused to 1 mm^ ). The transition linewidth is an important and limiting

factor in the measurement procedure. The minimum

linewidth expected as a result of transit time broadening

in the cw laser will be about 100 kHz. (Because of their

motion, the atoms sample the optical field only for about

1 (jLsec.) Additional linewidth broadening is expected as a

result of the field-induced quenching of the 1 2 ^ Syz >

level. To prevent significant broadening we must restrict the external field to values below 40 V/cm. (At 40 V/cm

less than 1% of the nearby rapidly-decaying |2 ^Fy2>

level is mixed.)

—>l PHOTODIODE

Figure 3. Precision vacuum-spaced capacitor.

175
helium-neon laser, will help determine surface parallelism and relative plate separation. Each of the optical

flats will be attached to a long-range piezoelectricallydriven translator which will be used to effect the change

in plate separation. The field measurement is to be made

as follows: First, the lasers will be tuned to an atomic

Stark resonance. Next, the plate separation will be in- creased carefully by about 1 cm. Finally, the electrical potential difference will be readjusted to bring the sytem

back into resonance. The change in electrical potential di- vided by the change in plate separation is the field strength. We assume, of course, that fringing field ef- fects can be taken into account to this level of precision.

The differential measurement helps to eliminate problems of contact potential and difficulties in the absolute

determination of surface positions. The interferometers

will be used to determine the change in plate separation

to better than 10 A . A precision voltage divider will be

used to determine the change in electrical potential in terms of the NBS voltage standard to better than 4 fiV. Field induced ionization will be used to detect the excited atoms [15]. Since the atoms will drift approximately

5 cm before entering the detection region, we can antici- pate a loss of signal by a factor of 10 due to radiative

decay and blackbody-induced diffusion of excited state population [16]. The remainder of the excited atoms, in- cluding those that have diffused to high-n states, will be

detected with near unity efficiency.

We expect a signal of about 3 counts per laser shot. A

random noise contribution due to several effects including multiplier dark noise of about 0. 1 count per laser shot

is expected. Given that the anticipated signal-to-noise ratio is 30, we should have no trouble in splitting the line to the needed factor of 10. Thus, vi/Enbs should be able

to be determined to the required precision of 0.10 ppm.

References

[1] E. R. Williams and P. T. Olsen, Phys. Rev. Lett. 42, 1575

(1979). (This is the best determination of a in which QED

is not used for interpretation of the experimental results.)

[2] R. S. Van Dyck, Jr., P. B. Schwinberg, and H. G. Dehmelt,

Phys. Rev. Lett. 38, 310 (1977); and these proceedings.

[3] P. Vigoureux and N. Dupuy, Natl. Phys. Lab. Report DES

59 (January 1980).

[4] H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One

arid Two Electron Atoym (Academic Press, New York,

1957).

[5] B. F. Field, T. F. Finnegan, and J. Toots, Metrologia 9, 155

(1973).

[6] J. E. M. Goldsmith, E. W. Weber, and T. W. Hansch, Phys. Rev. Lett. 41, 1525 (1978).

[7] H. J. Silverstone, B. G. Adams, J. Cizek, and P. Otto, Phys. Rev. Lett. 42, 1498 (1979).

[8] P. M. Koch, Phys. Rev. Lett. 41, 99 (1978); H. J. Silverstone and P. M. Koch, J. Phys. B 12, L537 (1979).

[9] R. J. Damburg and V. V. Kolosov, J. Phys. B 9, 3149 (1976);

11, 1921 (1978); 16, 2637 (1979); and references therein.

[10] M. G. Littman, M. M. Kash, and D. Kleppner, Phys. Rev.

Lett. 41, 103 (1978).

[11] M. L. Zimmerman, M. G. Littman, M. M. Kash, and D. Kleppner, Phys. Rev. 20, 2251 (1979).

[12] M. G. Littman, M. L. Zimmerman, T. W. Ducas, R. R. Free- man, and D. Kleppner, Phys. Rev. Lett. 36, 788 (1976).

[13] P. M. Koch, Opt. Commun. 20, 115 (1977).

[14] G. C. Bjorklund, C. P. Ausschnitt, R. R. Freeman, and R. H. Storz, Appl. Phys. Lett. 33, 54 (1978).

[15] T. W. Ducas, M. G. Littman, R. R. Freeman, and D. Kleppner, Phys. Rev. Lett. 35, 366 (1975).

[16] If n is greater than about 10, the apparent decay rate is modified by the presence of room temperature blackbody

radiation, which serves to redistribute the excited population among many nearby levels; T. F. Gallagher and W. E. Cooke, Phys. Rev. Lett. 42, 835 (1979); John W. Farley,

private communication.

176
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Time Resolved Sub-Natural Width Spectroscopy

William D. Phillips

Electrical Measurements and Standards Division, National Bureau of Standards, Washington, DC 20234

and

Harold J. Metcalf

Physics Department, State University of New York at Stony Brook, Stony Brook, NY 11794

A number of techniques exist in both practice and theory for achieving spectral signals narrower

than the width imposed by the natural lifetime. We examine a simple mathematical model for subnatural width spectroscopy, and identify two distinct characteristics of time resolved line narrowing.

We discuss the conditions where line narrowing techniques may be valuable, and comment on some

misconceptions concerning these techniques.

Key words: high resolution spectroscopy; precision measurement; spectral line narrowing; sub-natural

linewidth spectroscopy; time-resolved spectroscopy.

1. Introduction

In 1950 Ramsey [1] introduced the separated oscilla- tory field method for atomic beam magnetic resonance

spectroscopy. This method produces spectral features of width characterized by the total time the atoms spend in the apparatus, rather than the time spent interacting

with the radiation. In 1960 Hughes. [2] suggested that

Ramsey's technique could also be used for short-lived

states to produce spectral features that are narrower

than the natural, decay broadened linewidth. This possibility was particularly attractive because the ultimate

precision of many spectroscopic measurements is limited

by the natural width imposed by the decay of one or both

of the states involved in the transition being studied.

Since 1960, many specific experimental techniques have

been proposed and used to achieve signals narrower than

the natural linewidth [3-16]. An example of particular in- terest in the field of precision measurements and fundamental constants is the determination of the Lamb shift

[5]. Successful applications of sub-natural linewidth spec- troscopy span the electromagnetic spectrum from the

Lamb shift [5] at 10^ Hz to Mossbauer intervals [7] at

10^^ Hz. All of the various techniques achieve line nar- rowing by using time-resolved techniques for preparation

and observation of the system being studied and therefore must be amenable to temporal resolution. These

techniques always involve sacrificing signal amplitude in favor of reduced spectral width. Usually a linear de- crease in width results in an exponential loss of signal. In earlier work [16] we numerically modeled the application of line narrowing techniques to a noisy signal. We

concluded that it was advantageous to trade signal-tonoise ratio (S/N) for increased resolution in the case where the determination of the spectral line center was

limited by the presence of an unknown asymmetry. Errors

due to asymmetries are reduced in proportion to the

amount of narrowing achieved, so that if there is sufficient S/N the accuracy of the line center determination is improved even if the precision is not. (The possibility of just

such a systematic error is a major motivation for the use

of line-narrowing techniques in Lamb shift [5] and level crossing [3] experiments.) If the line is known to be symmetric, or if the exact line shape is known, then line nar- rowing techniques do not improve the center determination. In this paper we use our mathematical model for line narrowing to identify two distinct features of the process

that we have called time-delayed interference (TDI) and

decay envelope masking (DEM). We compare the relative

advantages of these different methods for line center determinations and for interpretation of complex, un- resolved spectra.

2. The Model

Consider a decaying oscillation given by

s{t) socos(ajo^ + ct))e-^'. (1) This might represent, for example, the electric field radi- ated by an atom in free induction decay, or the oscillating

portion of a quantum beat signal observed on a photomultiplier. The spectrum of this signal is obtained by taking

the Fourier transform of s (i):

S(co) = fe -'-'s{t)dt = ^

7 IX 1 (2)

where x = (wq - w)/y and we have ignored the "antiresonant" terms. When cj) = 0 the real part of the spec- trum is an absorptive Lorentzian with full width at half maximum (FWHM) of 2 7 and the imaginary part is a dispersive Lorentzian. The power spectrum, P(co), ob- tained by taking the absolute square of the spectrum

S(a)), is a purely absorptive Lorentzian with FWHM of 2y independent of <j). The signal, the real part of the

177
spectrum, and the power spectrum are shown in Fig. 1(a). Time resolution, and thus line narrowing, is obtained

by applying a "window function" W{t) to the signal

s it ), and then performing the Fourier transform:

S(a)) = f W{t)s{t)dt (3)

Windov/ functions that discriminate against the early

part of the signal in favor of the later part in general will produce line narrowing.

In the following sections we examine the action of two

different window functions: a rectangular window and an

exponentially rising window. We shall consider how the

line narrowing arises in each of these cases, and show

that there are some fundamental, qualitative differences

between the two methods under the same measurement

conditions. The existence of two quahtatively different

types of line narrowing has not been appreciated before.

3. Time Delayed Interference

Consider a rectangular window function W (t ) given by:

0, f < ^o Wit) -

1, t > ^0- Ignoring anti-resonant terms, the spectrum is

(4)

S(a)) = fe~"-''Wit)sit)dt

y ix - 1 (5)

where T = ytg. For (J) = 0, the real part of the spectrum

is given by

Re{S(co)}

S,, - (.rsin.rr - cos-tT) . (6)

This function, shown in Fig. Kb) for T = 4, is a damped

oscillatory function, symmetric about x = 0. The central

W(l) s(l)W(t)

peak of the spectrum is considerably narrower than it would be without time resolution. For T » 1 the

FWHM of the central peak (with the maximum height of the peak measured from zero) is approximately 2/T. The

amplitude of the central peak is reduced by e

"'^ from

that of the unnarrowed spectrum.

Note that the envelope of the spectrum is quite broad even though the central feature of the spectrum given by

Eq. (6) is narrow. This contradicts the notion that the

window function discriminates against that portion of the

signal contributing to the wings of the spectrum in favor

of that contributing to its center. To illustrate this point more clearly, consider the power spectrum for the rec- tangular window:

P(co) = |S((o) |2 = ^

-2T

1 +

(7)

Except for a reduction in amplitude, the power spectrum

is unchanged by this windowing. Furthermore, the shape

is independent of <^ since the process of taking the absolute square destroys phase information. While one often

hears that time resolved line narrowing techniques work

because the detection process looks only at "long-lived"

atoms that exhibit a narrower spectrum, it is clear from

this analysis that late-arriving radiation shows exactly

the same spectral distribution as the entire ensemble.

The narrow features of Eqs. (5) and (6) are not, therefore, due to the selection of late radiation alone, but

rather to interference between two oscillations. The ac- cumulated phase difference between these changes

rapidly for small changes in x when T is large, and ac- counts for the narrow features. The narrowing is lost in the power spectrum because the phase information is lost. In the separated oscillatory fields method, the

relevant phase difference is between the oscillating mo- ment of the atoms under observation and the applied os- cillatory field; in time resolved level crossing [3] the

relevant phase information is preserved by a mixing

between the states involved in the crossing. This description of the narrowing that occurs in such experiments has

sometimes been appreciated but popular and erroneous

explanations of the narrowing have also persisted in the

literature.

Re{S(u))

Frequency - Frequency *

Figure 1. Sigyials and spectra for a decaying oscillation observed through various win- dows. Window functions. Wit); windowed signals, sit)iWit); real part of the Fourier

transform, Re{S (w)}; and power spectrum, P (w) are shovm for no window (a), rectangular

window (b), exponential window (c), and Gaussian window (d). Scale factors indicate the relative expansion of the vertical scale.

178
As a further example to emphasize the importance of phase preservation along with time delay, consider meas- urement of the spectrum of a radiating atom with a slowly swept Fabry-Perot interferometer followed by a detector such as a photomultiplier. By repeated observations of the radiation described by Eq. (1), one would obtain the power spectrum of the emission, a Lorentzian

with FWHM of 2-y. If a shutter, synchronized with the

excitation of the atom, were placed between the atom

and the spectrometer, and only opened after a long delay, the same power spectrum would be recorded, only

vdth smaller amplitude. If, on the other hand, the

shutter were placed between the interferometer and the

detector, a spectrum similar to that of Eq. (6) would be

obtained because light emitted early by the atom is stored in the interferometer, and interferes with light

emitted later. The shutter does not select the light from

atoms which emit late, but rather it selects that time

when the interference between the early and late emissions is observed. Reference [17] gives a more detailed

analysis of a very similar situation involving the

Mossbauer spectrum of gamma rays passed through a

filter and then detected with time resolution.

4. Decay Envelope Masking

We have seen in the previous section that the rec- tangular step window in Eq. (4) has no effect on the

shape of the power spectrum of the signal given by

Eq. (1). This is a special result for that particular window and signal; if the rectangular window function goes

to zero at some time t > to, the power spectrum becomes

broader but the spectrum of Eq. (6) still retains its nar- row features. It is natural to ask whether one can narrow

the power spectrum as well as broaden it. Any window

function that causes the signal to decay more slowly than

e"^* will accomplish this. Consider the exponentially ris- ing function given by

W(t) = (8)

1 ,t^t„-

The signal and spectra resulting from this window function are shown in Fig. 1(c). We choose (|) = 0, ignore

anti-resonant terms, and find that the real part of S (co)

is:

Re{S(co)} = ^e-T smxT _ cosxT - x sinxT

X 1 + x^

(9)

The second term in the brackets is identical to Eq. (6), and comes from t > tg where W{t) = 1. The center of the spectrum is near x = 0, and in this region the first term is dominant when T » 1. Its FWHM is approximately 3.8/r. If we ignore the second term in Eq. (9) and the similar term in the imaginary part, the power

spectrum becomes:

P(a)) = -2T 1 cosxT

(10)

Near the center of this spectrum its FWHM is about

5.6/T for the domain T » 1. For any value of T, Re{S (u))} and P (w) are each narrower than the

corresponding raw spectrum obtained with no window

function. The narrowing of the power spectrum, P (co) in this

case, indicates that an interference effect is not responsible. In terms of the spectrometer analogy of the previous

section, a variable attenuator with a transmission given

by Eq. (8) between the radiating atom and the spectrometer would give a spectrum narrower than obtained

without the attenuator. We might say that the variable attenuator has filtered out light in the wings of the emission spectrum and that this filtering is accomplished by

modifying the decay rate of the radiation as seen by the

spectrometer, 7iot by choosing only the light that arrives

later.

5. Comparison of the Two Types of

Narrowing

It is clear that separated oscillatory fields and time

resolved level crossing rely mainly on time delayed in- terference for the narrowing they achieve. Copley et al. [18] have analysed possibilities that offer an interesting

hybridization of the two methods. If the time resolved

level crossing signal is observed with a rectangular window equivalent to our Wit) given in Eq. (4), a result essentially identical to Eq. (6) is obtained (see Eq. (2) of Ref. [18]). For some applications the extra oscillations in the wings may be undesirable and various "apodizations" have been proposed to remove them by modifying

the window function. Copley et al. have studied ex- ponential and Gaussian windows. Their Gaussian window

has a mixed effect, resulting in both TDI and DEM nar- rowing. Using a Gaussian window proposed by Dodd and

Series [11] which leads to no oscillations, we obtain the

signal and spectra of Fig. 1(d). Since this narrowing is preserved in the power spectrum, we conclude that this type of window function leads to narrowing predominantly by DEM. Choice of a narrower Gaussian leads to oscillations in the spectrum, and the narrowing is then

due more to TDI.

Some recently suggested techniques for line narrowing

propose to populate an exponentially decaying state in an

exponentially decaying manner, either by excitation from

a decaying state [13] or by excitation with a decaying

pulse of radiation [15]. It appears that these methods fall into the DEM category; the predicted line shapes are very similar to those obtained in level crossing with ex- ponential window functions (compare Eq. (13) of Ref.

[18] with Eq. (14) of Ref. [13a]).

Which of the two line narrowing mechanisms produces

better results depends heavily on what the experimenter

wants to learn. For example, to find the center of an iso- lated spectral line TDI is preferable because it suffers

less signal loss for the same degree of narrowing than

DEM. A rectangular window starting at 4 lifetimes gives a TDI spectrum with a central feature about 5 times nar- rower and 50 times smaller than the raw spectrum. To

achieve the same width with an exponential window

would reduce the height of the peak by another factor of

6. But to resolve blended spectral features, DEM might

be better. Figure 2 shows two spectral lines of equal

height and width, separated by half of the FWHM. Vari- ous types of windows with the same characteristic time

(4 lifetimes) were applied in an attempt to resolve the

structure. A Gaussian window chosen according to the

prescription of Dodd and Series [11] does not quite

resolve the structure. The rectangular window resolves

the structure completely, but produces a complicated

pattern that might be difficult to interpret in the absence

of prior knowledge about the structure. The exponential

vdndow partially resolves the structure, with little confusion from the oscillations. The choice of the best narrowing method for a given situation will obviously depend on

a number of factors, including the knowledge of the structure and the available S/N.

179
^^^^"""^'''^

^^^^^^^^^^^^

b / \

c A

d A

Frequency

Figure 2. Calculated spectra of txvo unresolved liyies obtained

with various window fimctions. Dotted curves iyi (a) are the in- dividual lines. The solid curve is the no-window spectrum.

Spectra for Gaussian window (b), rectangular wiyidow (c), and

exponeyitial window (d), are also shown.

6. Summary

We have developed a simple model for time resolved

line narrowing which is closely analogous to techniques

such as time resolved level crossing spectroscopy. We

have shown that within the context of this model two

qualitatively different mechanisms for narrowing exist. We have compared, for some special cases, the relative

utility of the two methods in applications to line center determination and resolution of complex spectra.

References

[1] N. F. Ramsey, Phys. Rev. 78, 695 (1950).

[2] V. W. Hughes, in Quantum. Electronics, Ed. by C. H. Townes (Columbia University Press, New York, 1960), p. 502.

[3] P. Schenck, R. C. Hilborn, and H. J. Metcalf, Phys. Rev.

Lett. 31, 189 (1973).

[4] J. S. Deech, P. Hannaford, and G. W. Series, J. Phys. B 7, 1131 (1974).

[5] S. R. Lundeen and F. M. Pipkin, Phys. Rev. Lett. 34, 1368

(1975); ibid. 46, 232 (1981); and these proceedings.

[6] H. Figger and H. Walther, Z. Phys. 267, 1 (1974).

[7] J. E. Monahan and G. J. Perlow, Phys. Rev. A 20, 1499

(1979), and references therein describe the Mossbauer

work.

[8] T. W. Ducas, M. G. Littman, and M. L. Zimmerman, Phys. Rev. Lett 35, 1752 (1975).

[9] P. B. Ki-amer, Phys. Rev. Lett. 38, 1021 (1977). Here and in Ref. [8] the phase of the transient state is preserved by a

well-defined mixing with another state.

[10] T. Krist, P. Kuske, A. Gaupp, W. Wittman, and H. J. Andra, Phys. Lett. 61A, 94 (1977).

[11] J. N. Dodd and G. W. Series, in Progress in Atomic Spectroscopy, Pai't A, Ed. by W. Hanle and H. Kleinpoppen

(Plenum, New York, 1978), p. 639.

[12] D. E. Casperson et al, Phys. Lett. 59B, 397 (1975).

[13] P. Meystre, M. 0. Scully, and H. Walther, Opt. Commun.

33, 153 (1980); H. W. Lee, P. Meystre, and M. 0. Scully,

Phys. Rev. A 24, 1914 (1981).

[14] A. Denis and J. Desesquelles, Phys. Rev. A 21, 1602 (1980).

[15] P. L. Knight and P. E. Coleman, J. Phys. B 13, 4345 (1980); P. E. Coleman, D. Kagan, and P. L. Knight, Opt. Commun. 36, 127 (1981); P. M. Radmore, P. L. Knight, and D. Kagan, Phys. Lett. 82A, 288 (1981).

[16] H. J. Metcalf and W. D. Phillips, Opt. Lett. 5, 540 (1980).

[17] F. J. Lynch, R. E. Holland, and M. Hamermesh, Phys. Rev.

120, 513 (1960).

[18] G. Copley, B. P. Kibble, and G. W. Series, J. Phys. B 1, 724

(1968).

180
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

The Spectroscopy of Atoms and Molecules in Gases:

Corrections to the Doppler-Recoil Shift*

Mark P. Haugan

Center for Radiophysics and Space Research, Cornell University, Ithaca, NY 14853

and

Frank V. Kowalski

Department of Physics, Colorado School of Mines, Golden, CO 80401

We analyze the excitation by a monochromatic plane electromagnetic wave of a two-level atom or molecule moving through a rarefied gas. This yields a prediction for the Doppler-recoil shift which

differs significantly from the familiar formula derived by studying isolated atoms or molecules in vac- uum. The difference arises from an interplay between the collective and individual particle responses of a gas to an electromagnetic wave and is of fundamental importance for high-resolution laser spectroscopy. We propose an experiment using saturation spectroscopy to observe sodium molecular transi- tions near the atomic D lines that could easily verify our prediction for the first order Doppler shift of spectral features due to atomic or molecular absorption in rarefied gases. Key words: Doppler and recoil effects; gases; high-resolution spectroscopy; macroscopic electrodynamics.

1. Introduction

Consider an atom or molecule w^ith an energy differ- ence Ae between two internal states. The natural fre- quency 0)0 of a transition between these states is given by

Ae = flcoo- However, because of the Doppler and recoil

effects, the observed frequency of the transition depends

on the velocity V of the atom or molecule relative to the

laboratory and on its mass M. The difference between

the observed frequency and the natural frequency of the

transition is the Doppler-recoil shift. Let COL represent the laboratory frequency of an elec- tromagnetic wave that excites a transition in an isolated

atom or molecule. The relativistic Doppler-recoil formula

can then be written [1]

Wo

(1 - yvc^)2\l/2

1 - fc. V/c 1 (1)

where /cl is the wave's propagation direction and M is the mean rest mass of the atom or molecule in its initial and final states. We emphasize that this result is derived

for an atom or molecule in isolation, in vacuum. The initial factor on the right-hand side of Eq. (1) is the Doppler

factor, whereas the second factor represents the recoil

effect. It accounts for the recoil kinetic energy, as measured in the frame in which the atom or molecule is ini- tially at rest, that must be imparted when a photon is absorbed.

An understanding of the Doppler-recoil effect is crucial

for the high-resolution spectroscopy of atoms and

molecules in gases and its many applications. It is generally assumed that Eq. (1) gives the Doppler-recoil shift occurring when an atom or molecule within a rarefied gas

is excited. We show that this assumption is incorrect.

When an atom or molecule moves through a sample gas

that is at rest in the laboratory, the Doppler-recoil shift

is affected by an interesting interplay between the collec- tive response of the gas to an incident electromagnetic wave and the wave's excitation of the atom or molecule.

We find that the Doppler-recoil formula becomes

Wo (1-^2/^2)1/2

1 - n(wL)/fL • V/C

(2)

*Supported in part by the National Science Foundation (PHY80 -07957).

where n (wl) is the refractive index of the gas at the lab- oratory frequency and R is a rather complicated recoil factor given in Eq. (11). This result is fully relativistic.

It differs from Eq. (1) because the interaction between

the excited atom or molecule and the surrounding gas has

been accounted for. In section 3, we describe a proposal

for an experiment on molecular transitions in sodium va- por which could easily detect the differences between the

first-order Doppler predictions of Eqs. (1) and (2). The

predictions can differ by several MHz. Equation (14)

gives a simple approximation to Eq. (2) that is suffi- ciently accurate for most practical purposes. Section 2 of this paper is devoted to an overview of the

derivation of the Doppler-recoil formula, Eq. (2), and to a brief discussion of the physics that underlies it. The

Doppler-recoil shift is understood in terms of the recoil kinetic energy, as measured in the laboratory frame, that

is imparted to an atom or molecule which absorbs a photon. Interactions among the polarizations and polarization currents induced in the atoms and molecules of a gas

by an electromagnetic wave influence the recoil momen- tum and, thus, the recoil kinetic energy that is imparted

to an atom or molecule excited by the wave. Accounting

for the effects of such interactions leads to the Dopplerrecoil formula of Eq. (2). Our calculations treat the exci- tation of a two-level atom or molecule moving through a rarefied gaseous medium. Space limitations require that the details of these calculations and the discussion of a number of important points, including the response of a

181
gaseous medium to the excitation of one of its molecules,

appear elsewhere [2].

2. The Doppler-Recoil Shift for Absorption

by Atoms and Molecules in Gases

2.1 Essential Ideas

A sample gas is at rest in our laboratory. We study the

interaction between a molecule with velocity V and an

electromagnetic wave with laboratory frequency wl prop- agating in the direction. The object is to determine

the frequency for which the wave resonates with a transition of the molecule having natural frequency wq. We

make two assumptions in our derivation of this resonance

condition. Fortunately, in many experimental settings

they are valid assumptions.

First, the gas must act as a continuous medium. The wave we consider should satisfy the macroscopic Maxwell

equations for some index of refraction n (col) when propa- gating through the gas. Though there is absorption in the

gas, the amplitude of the incident wave does not vary ap- preciably over the volume in which a molecule is local- ized. Absorption is, therefore, an inessential complication

of the discussion of interaction between wave and

molecule. We take the index 7i (col) to be real for simplicity. We will suppose, to be definite, that the gas is non- magnetic so that ri^((OL) = e(ioL), the permittivity of the

gas. This is generally the case for real gases. The condition that the gas act as a continuous medium is, given the

density of the gas and the dielectric properties of its molecules, a condition on the extent of the gas sample. If the index n (col) characterizes propagation through an in- finite gas, the same index also characterizes propagation

through a finite sample so long as it is much larger than

the extinction theorem [3] distance, c [col I ^ (^l) "

1 I For the gas we consider in section 3, the extinction theorem distance is roughly 0.05 cm at the frequencies of interest. The second condition which must be satisfied by the

gas is that it be sufficiently rarefied. -Clearly, molecular

collisions must be infrequent enough that collisional broadening of spectral lines does not mask the Dopplerrecoil effect. Significantly, the condition guaranteeing

that this does not occur leads to an essential simplification of the computation of the resonance condition. In

fact, this simplification makes the problem tractable. For collisional broadening to be negligible, the fre- quency of molecular coUisions must be small compared to the molecular transition probability per unit time induced

by the incident electromagnetic wave. When this condition is fulfilled, molecules are excited in a time interval so short that surrounding molecules are unable to ap- proach them during the process. This fact, that for our purposes molecules have a mean separation large compared to their sizes, has two important consequences.

First, in the absence of an electromagnetic wave the

internal structure, e.g., energy levels, of molecules will be the same as if the molecules were isolated. Second,

when an electromagnetic wave is present the molecules

within the gas are influenced only by its macroscopic

electric and magnetic fields. This follows because the microscopic fields within the gas deviate from the

spatially-averaged macroscopic fields only in the vicinity of individual molecules. Since the molecules are well separated, the deviation produced by one molecule is unable to affect any other molecule.

We conclude that analysis of the excitation by an elec- tromagnetic wave of internal transitions in the molecules

of a collisionless gas reduces to the study of isolated

molecules responding to the macroscopic fields of the wave in the surrounding medium. The macroscopic fields mediate all non-collisional effects of surrounding gas on an excited molecule as well as giving the effect of the in- cident wave. The effects of relatively infrequent molecular collisions can be accounted for after the analysis of the Doppler recoil effect for a collisionless gas.

2.2 The Nonrelativistic Doppler-Recoil Shift

In this section we briefly describe our semi-classical

analysis of the excitation by an electromagnetic wave of a two-level molecule moving slowly through a coUisionless

gas. We will describe the analysis for molecules with ar- bitrary velocities in section 2.3. Since we are interested

in the recoil of the excited molecule we include its center-of-mass degree of freedom in the calculation. This

and the use of the incident wave's macroscopic fields are the simple differences between what follows and the fa- miliar analysis of the excitation of an isolated two-level

system.

The molecule we study has a pair of internal states, a and b. Under the conditions described in the preceding

section these states are indistinguishable from those of an isolated, but otherwise identical, molecule. We con- sider a molecule initially in a state A = (a, K), where K

is the molecule's center-of-mass wavevector. The momentum and energy of this molecule are tK = M \ and ^co^ = e„ + 1l^K^/2M , (3) where M is the molecule's mass, V its initial velocity,

and Ca its internal energy in state o. Given a resonant incident wave, the wave-molecule in- teraction Hamiltonian will connect the molecular state A

to a state B = (Jb, K'). Typically, the dipole approximation to the interaction Hamiltonian will suffice,

H; = - P • E(X, t) . (4) The symbol P denotes the molecule's electric dipole

operator. This operator only depends on internal molecular coordinates. The field E(X, t ) is the macroscopic field of the incident classical wave. It depends on the

molecule's center-of-mass coordinate, X. For an incident

monochromatic plane wave, the matrix element connect- ing A and B decomposes into a product of a momentum

5-function with the usual dipole matrix element, P, that connects the internal states a and b. The 8-function

determines the recoil momentum ^(K' - K) that is imparted to the excited molecule. For an incident wave

with macroscopic fields

E(X, t) = xEocosiixii^t - Rl • X) , (5a)

B(X, t) = ?/5ocos((OLi - Rl • X) , (5b)

where Rl = n (col) wl c f, = n{oiiJEo, and the recoil momentum is tki^. The particular choice for the polarization and phase of this wave does not affect our final conclusions.

The state of a molecule under the influence of the wave

of Eq. (5) will be a superposition of states A and B, ^ = Ca\A > + Cb\B > . (6)

The system of equations

Ca = (rP£o/2^)Cge"'''"S4-'^L" ^ (7^) Cb = dPEomCAe"''"'-^"''^", (7b)

182
where wg^^ = wg - coa, with initial conditions C^CO) = 1 and Cs(0) = 0, determines the state of the two-level

molecule we are studying. The condition for resonance is

The differences between Eqs. (7) and the two-state equations usually encountered are that mba includes the

change of center-of-mass kinetic energy that accompanies

the internal excitation of the molecule and that the recoil momentum imparted to the excited molecule is fiki^ —

n (a)L)a)L c /c l, rather than ul c "^^l- The former means

that the resonance condition, wl - ^ba, accounts for the

nonrelativistic Doppler-recoil shift. TThe latter explains

the difference between the shifts for molecules excited

within a gas and for molecules excited in isolation. Referring to Eq. (3), we write the resonance condition as

f?a)L = ieh - Ca) + [1iHK + kO'' - ti-K^V2M , (8) with K = tr^MY, where V is the initial velocity of the

excited molecule. The quantity tr^iej, - Qa) is simply the

natural frequency of the transition. To first order in Vc

and tixio /2Mc'^, which is as far as we can trust our nonrelativistic analysis, we find

in agreement with the general result of Eq. (2) to the

order which we are working.

To conclude this section, it is worth remarking that the

force responsible for imparting recoil momentum to an

excited molecule is easy to understand. It arises from the

interaction between the fields of the incident wave and

the polarization and polarization current they induce in the molecule. Clearly, the semi-classical force on the

molecule is

Fit ) = <^ I (P • V)E(X, O + c -ip X B(X, O I ^ >, (10) where ^ is the solution, Eq. (6), to the two-state Eqs.

(7). Since the fields in this expression are the macroscopic fields of the incident wave within a gaseous

medium, it is not surprising that the recoil momentum

imparted to the excited molecule should differ from that

imparted to a molecule excited in vacuum. Using the

Rabi [4] exact solution to the two-state equations for the

case of a resonant incident wave, it is not difficult to verify that the impulse of the force given by Eq. (10) during

the time interval in which the molecule flops from state A to B, the recoil momentum imparted to the excited

molecule, is n({jOL)^WL c "^/c l- This dependence of an excited molecule's recoil momentum on the properties of surrounding gas is the central reason for the differences

between the Doppler-recoil formulae in Eqs. (1) and (2).

2.3 The Relativistic Doppler-Recoil Shift

We have argued that the analysis of molecular excitation by an electromagnetic wave propagating through a

colHsionless gaseous medium reduces to the analysis of an isolated molecule interacting with the macroscopic

electric and magnetic fields of the wave. Since the physics of an isolated molecule interacting with electromagnetic fields is Lorentz invariant, the simplest way to analyze the excitation of a molecule that moves rapidly

through the gas is to study the excitation process in the

frame in which the molecule is initially at rest. In this frame, the analysis proceeds just as it would in the

preceding section for a molecule initially at rest in the

laboratory. The main difference is that the molecule in- teracts with electric and magnetic fields that are ob- tained by transforming the laboratory macroscopic fields

.

of the incident wave to the molecular rest frame. The

only other change is to employ relativistically correct re- lations between the velocity of the molecule and its momentum and kinetic energy in place of Eq. (3). In this short paper we can only quote the results of

this relativistic analysis. The Doppler-recoil formula of Eq. (2) is -

(1 - V^/c^)^

(1 - nMkL V/c)

1 +

(1 - A

2A

(1 - tiwo/2Mc^)

where

(1 - -kojo/2Mc'^)

(1 +two/2Mc'^)

1 +

(1 -tm/2Mc-)

with

CO = -yaJLWi, (1 - h(col) 3 k l)

(11)

(12)

(13a)

k = n(a)L)7WLC (1 - p

•

+ 2 [d - n-i(coL)Jp /cl

V2

1 - 7l~HioOh^' (13b)

the frequency and the magnitude of the wavevector of the incident wave as measured in the frame in which the

molecule is initially at rest. Of course, p = Vc and 7 =

(1 - P').

3. An Experimental Proposal to Verify the

Doppler-Recoil Formula for Absorption by

Atoms and Molecules in Gases

It is an interesting theoretical exercise to obtain the

fully relativistic Doppler-recoil shift for molecular excita- tion in a gas. However, only the first few terms of the

expansion of equation (11) in the Doppler parameter Vc~^

and the recoil parameter t(jjo/2Mc~ will be of practical in- terest. Including only the first and second order Doppler

shifts and the first order recoil effect, the Doppler-recoil

formula of Eq. (11) reduces to

(OL = cood + nii^Okt V/c + n-Mikh V/c)^ - y2/2c2 - 77,2(a)L)^COo/2Mc2) . (14) We can imagine using this equation to predict the resonant laboratory frequency for a transition of given

natural frequency wq in a molecule of mass M moving

with velocity V, or we can imagine using it to reduce

resonant frequencies observed in an experiment to a natural frequency for the transition being studied. Our

point is that for either of these purposes the use of the

usual Doppler-recoil relation rather than Eq. (14) when

considering the excitation of molecules within a gas

under reasonable experimental conditions can lead to sig- nificant error. In fact, certain experiments are particularly sensitive to differences between the Doppler-recoil

formulae. For example, consider a measurement of the

separation between spectral lines corresponding to two

transitions with nearly equal natural frequencies vo and

vq. We will discuss frequencies rather than angular fre- quencies in the remainder of this section. Observe the

transitions of molecules in a velocity slice of the thermal

183
distribution having "velocity" v - V. Equation

(14) predicts the laboratory frequencies of the observed

line centers,

VL = Vo{l + W(vl)VC"^}

v'h = vo{l + n{v'i)vc~'^} . (15)

We included only the first order Doppler effect because

this is sufficient to illustrate the following general point:

The observed separation between the lines, vl - vl, depends on the difference in the refractive index between

the frequencies vl and vL as well as on the velocity v. Application of the usual Doppler-recoil formula rather

than Eq. (14) in cases where dispersion makes this index

difference significant can produce substantial errors. In our proposed realization of the preceding experiment, to be described below, fractional errors in the line separation of up to 10"^, corresponding to an error of several

MHz in frequency, can occur. To verify at least the first-order Doppler shift prediction contained in Eq. (14), we propose an experiment on sodium molecular transitions which have a common

ground state and which straddle the sodium D lines. The

molecules are pi'esent in a sample of sodium vapor whose

index of refraction is provided primarily by the D lines. Dispersion of the vapor medium will be important.

Using saturation spectroscopy on the molecular three

level system (see reference [5]) a velocity slice of the

thermal velocity distribution can be defined with the

1^2 transition. Light of frequency v£^^' propagates in the /c'l direction through the sodium vapor saturating the

1-^2 molecular transition. The class velocity v = /c l

• V

of the selected molecules is

where v^^^' is the natural frequency of the 1^2 transition. We will consider v to be of the order of the root mean square thermal velocity of molecules in the sodium

vapor when estimating experimental "effects. The reso- nance condition for excitation by a counterpropagating

beam, /cl direction, of the 1-^3 transition in molecules of the chosen velocity slice is vP = v,ji3'[l - w(vp) w-M . (17)

Inserting the expression for v from Eq. (16), we can obtain the relation among observables

a relation between the Doppler shifts of the two molecular lines for the chosen velocity slice. The experiment we

propose would verify that this relation holds. For sodium vapor at roughly 700 K and one Torr mono- mer vapor pressure, and for sodium molecular transitions, ground (Z^S/) to excited (A^S,^) states, (3, 13) -»

(14, 26) and (3, 13) ^ (14, 25), {v ',/')-» {v ", J "), the

deviation from equal Doppler shifts for the two lines predicted by Eq. (18) is two MHz. This can be doubled to

four MHz by replacing observations of the y = 0 slice transition frequencies, V(5^^' and v^^'^\ by observations of

the transition frequencies for the velocity slice opposite

the one selected above in the thermal profile.

The proposed experiment would require two dye

lasers, one to saturate the 1—>2 transition (the 1-^2

laser); the other to probe the 1^3 transition (the 1^3

laser). A thermally stabiHzed cavity would be used as a frequency reference. By scanning the 1^2 laser and

simultaneously recording the cavity transmission fringes one could determine where the zero velocity 1-^2 reso- nance occurs relative to the frequency markers of the

cavity. This laser can then be locked to the 1-^2 transi- tion while the 1^3 laser can be scanned to determine

the zero velocity 1^3 transition frequency relative to the frequency markers. Having thus measured V(j^^' and

the 1^2 laser would be locked to a cavity reso- nance about one GHz away from the zero- velocity 1^2

transition frequency. This laser would then be saturating

a slice of the thermal distribution with velocity v comparable to the root mean square molecular velocity. The

1^3 laser could then be tuned to determine the 1^3

resonance frequency relative to the frequency markers.

Finally, having measured vP-' and v\}'^\ Eq. (18) could be

put to the test since n(v£® ) and r\{v\}^'' ) can be either

measured or calculated with sufficient accuracy.

4. Concluding Remarks

The Doppler-recoil shift is fundamental to our understanding of the interaction between atoms and molecules

in gases and electromagnetic radiation. Our central result

is a demonstration that interactions between gas particles as they respond to an incident wave can affect the

Doppler-recoil shift. Equation (11), approximated in Eq.

(14), accounts for such interactions in a rarefied gas. Our

conclusions emerge from a detailed analysis of the excitation of a two-level molecule within a collisionless gas by a plane monochromatic wave. To test our results, we propose an experiment on

molecular transitions in sodium vapor that could detect

the differences between the first-order Doppler shift predictions of Eq. (14) and the usual Doppler formula,

valid for the excitation of isolated molecules. These

differences can be made small by working with suffi- ciently rarefied gas. However, experiments are often

performed under conditions where they cannot be

neglected. Finally, the experiment we propose provides an example of the practical consequences of the Dopplerrecoil formula of Eq. (14) in the important case where

dispersion of a gas is significant.

References

[1] See for example J. L. Hall, C. J. Borde, and K. Uehara,

Phys. Rev. Lett. 37, 1339 (1976).

[2] M. P. Haugan and F. V. Kowalski, to be published.

[3] J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley,

New York, 1975) and refei-ences therein.

[4] I. I. Rabi, Phys. Rev. 51, 652 (1937).

[5] M. E. Kaminsky, R. T. Hawkins, F. V. Kowalski, and A. L. Schawlow, Phys. Rev. Lett. .36, 671 (1976).

184
Precision Measurement and Fimdamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

The Implications of QED Theory for the Fundamental Constants*

G. P. Lepage and D. R. Yennie

Newman Laboratory of Nuclear Studies, Cornell University, Ithaca, NY 14853

We review the current theoretical status of high precision measurements in quantum electro- dynamics (QED). Theoretical predictions in QED fall into one of two general categories: perturbative,

as for the magnetic moments of electrons and muons; and non-perturbative, as for the properties of atoms. We contrast these predictions and explore their implications for the fundamental constants of nature. Key words: electron anomaly; hyperfine splitting; Lamb shift; muon anomaly; muonium; positronium;

quantum electrodynamics.

1. Introduction

Quantum electrodynamics (QED) is the theory of the

interaction of charged leptons with the quantized elec- tromagnetic field. Given its tremendous success in predicting a wide range of experimental facts, from very

refined details in atomic spectra to leptonic interactions

in the multi-GeV range, it must represent in some measure the true nature of these interactions. It seems inconceivable that any theory which is conceptually less so- phisticated could produce the same results. On the other

hand, QED is almost certainly a part of a larger, deeper

theory involving the weak, as well as the strong, interactions. Many aspects of the electroweak theory seem al- ready well understood. It is even possible that the lep- tons (electron, muon, . . . ) are not elementary particles,

but rather composites of still more fundamental entities. As yet there is no evidence for such structure, dowTi to distances as small as 10"^^ cm (i.e., up to energies of many GeV). In any case, QED should probably be re- garded as a low energy appro.ximation to some new fun- damental theory—i.e., as a phenomenology. These ideas

and other speculations are elaborated in Stanley

Brodsky's contribution to this conference.

If it is "merely" a phenomenology, how can we ac- count for QED's remarkable quantitative success as a theory of electromagnetic interactions? The answer is in- timately related to one of the theory's most characteristic features, its renormalizability. Because it is renormalizable, all short distance, high energy effects in QED

can be absorbed into a finite number of measurable masses and charges. Thus the high energy structure of the ultimate theory is irrelevant to the analysis of low

energy phenomena except insofar as it determines the masses and charges of the particles involved. The renormalization program is also of tremendous practical importance, since the high energy behavior of pure QED leads

to divergent integrals in perturbation theory. Wliether

or not this reflects the incompleteness of QED, or is rather just an artifact of perturbation theory, all divergent integrals are made convergent "simply" by ex- pressing observable quantities in terms of physical

*Supported in part by the National Science Foundation.

masses and charges. The results of this intricate renor- malization procedure have been outstandingly successful. Here we will discuss the high precision, low energy

features of QED—i.e., those quantities which have implications for, or require knowledge of, the fundamental

constants, and particularly a, , and c. Rather than at- tempt to cover everything, w^e will concentrate on a small

number of topics, in the hope of giving an accurate impression of the present status of theoretical work in this

field. The most comprehensive review of QED in the past

decade is the excellent paper by Lautrup et al. [1]. References to earlier work may be found there. The subject naturally breaks down into quite different areas, each requiring its own special types of expertise:

(a) Perturbative. To compute magnetic moments or scattering amplitudes for leptons, it is necessary to evaluate a series of Feynman graphs. Each graph contri- butes to a definite order in a, and the final results are given as a (truncated) power series in a. A thorough

mastery of the renormalization program, with its intri- cate subtractions of nested and overlapping divergences,

is necessary.

(b) Non-perturbative. In bound state calculations, a

definite graph does not correspond to a definite order in a. Often infinite sets of graphs are needed to evaluate a particular order. The method for organizing such a calculation is far from unique, and contributions can be readily

shifted betw-een the structure of the wave function and

that of the interaction kernels. In brief, the formulation

of the problem is more elusive than in perturbative anal- yses, while the actual computational work is in many

ways substantially simpler.

These areas involve complementary aspects of QED,

and both are of considerable importance in furthering our understanding of quantum field theories, renormalization

theory, relativistic bound state theories, and so on.

2. Lepton Anomalies

The 5f-factors of the electron and muon are both very

close to 2, the value predicted by the Dirac theory. To

the present time, it appears that the deviations from this value (i.e., the anomaHes) are explained by QED, to- gether with very small corrections due to strong interac185
tions. Measurements and calculations of {g-2)l2 have been

prominent in the history of QED for almost 35 years. Progress in the two has been well matched. Advances in one have stimulated advances in the other, from

Schwinger's famous calculation of the first order term,

a/2-n-, until the present stage where theory is being

tested at the {al-nY level (i.e., four virtual photons).

These calculations provide the most stringent test of the renormalization program not only for QED, but for quantum field theories in general. The nature of this analysis is illustrated in Fig. 1. Circles are drawn around

pieces of the graph which have divergent integrals. The

renormalization program is a precisely defined procedure

for removing these divergences. In practice, its realiza- tion is highly non-trivial, especially when numerical in- tegration is necessary. Lengthy integrands must be subtracted in intricate ways to eliminate all combinations of

Figure 1. An example of a)i eighth order vertex coutribiitio)i

.

Each dashed circle encloses a divergent graph or subgraph. Note the nesting atid overlapping of these divergences. If the figure is confusing, it conveys the correct inipressio)i.'

nested and overlapping divergences. Another complication is that the separate graphs have spurious infrared

behavior (i.e., sensitivity to very small virtual photon

momenta), which cancels in the results when all diagrams

are combined. The complexity of these subtractions, the

dimensionality of the integrations, and the number of di- agrams all increase with the order in (a/fr). The (a/Ti)^

term (7 diagrams) has been known analytically for about

25 years. The (a/ir)^ term has been reasonably well known for about 10 years, and work continues to improve

its accuracy [2]. Fifty-one of the 72 diagrams have been

calculated analytically; the remainder have been

evaluated by a combination of analytic and numerical

methods.

The theory of the electron's anomaly is summarized in Table 1, and compared with the current measurement of Dehmelt and his collaborators [3]. To give some indica- tion of the recent progress in this area, we quote both

the old (pre- 1980) and the current estimates for the coefficients of (a/ir)-^ and (a/ir)^. The measurements and their

implications are further discussed by A. Rich in his con- tribution to this conference.

Four or five years ago, the order (aZ-n-)-^ calculations

seemed to mark the limits to which theorists and computers could push such an analysis. However, the experimental accuracy was clearly destined to improve substantially. Anticipating these developments, Kinoshita

studied the possibility of computing to order (a/ir)'^. He

decided (he says to his regret) that it was feasible and

embarked upon the project with Lindquist [4]. The scope

of their calculation is enormous, even heroic. By clever

techniques, they group several diagrams into single in- tegrals, thereby reducing the 891 diagrams into "only"

about 100 integrals. A typical integrand has 5000 to 15000 terms and can involve as many as 10 integration

variables. Their results, presented here for the first time, must be regarded as very preliminary. More details can be found in Kinoshita's contribution to this confer- ence.

Table 1. Present .'itatns of the electron anomaly. Arrows indicate changes

a)inoH)iced at this conference (see text).

a/^P = 1 159 652 200(40) x IQ-^^

a^QED = i - 0.328 479 0(-)2

+ (1.183 5(61) 1.176 5(13)) (—f

+ {? ^ - 2.3 (3.6)) i—f

TT

= 1 159 652 566 x 10-^2

^1 159 652 411 X 10-12

5a, (a) = 127 x lO'^^ ; Sa, (theory)^ = 76 x lO^^^ _> i06 x lO'^^

ttg (hadronic + muonic) = 4 x IQ-^^

a/^'P - a/*^ = (-12.6(5.3) -> -7.3 (5.9)) (—)4

^ The error in the older theory makes no allowance for the (a /tt)^ term.

186
Table 2. Present status of the )Nito>i cnioniali/. The experimental

result is from Ref. [6]; the origin of the theoretical results may be found in Calmet et al. [oj.

= 1 165 923(12) X 10-

QED = 1 ^ + 0.765 782 + 24.45(6) + 135(64) Z Tt IT ^ ^

= 1 165 852.2(2.0) x 10""

had 66.7(9.4) X 10-9

a

weak = 2 X 10-9

th _= 1 165 921(10) X 10^

A higher order correction is only useful once lower

order coefficients have been computed analytically or at least to a high degree of accuracy. Our theoretical esti- mate of the (a/-n-)'^ coefficient has been substantially improved by Levine and Roskies, using a combination of analytic and numerical techniques. See their contribution for more details. Current results for the muon's anomaly are given in Table 2. Much of the analysis is identical for electrons

and muons. However, the effects of electrons on the muon anomaly, via vacuum polarization, and vice versa

are quite asymmetric. An electron, being much less massive, cannot readily create a virtual muon-antimuon pair,

and hence muoiis have little effect on a,,. Muons, on the

other hand, can easily create virtual electron-positron

pairs, and indeed in second and higher order graphs con- taining electron loops dominate. For this same reason, strong interactions are much more important in a than

in a^. Their origin is indicated schematically in Fig. 2; clearly their presence is demonstrated by the e.xcellent agreement between theory and experiment. They are presently the most uncertain part of the theoretical

analysis. A careful recent review of the theoretical contributions to a is given by Calmet et al. [5].

3. Bound States in QED

As mentioned in the Introduction, the technical problems encountered in bound state calculations are far dif- ferent in chai'acter from those of the anomalies. To

proceed, one must find a convenient, solvable approximation to the bound state problem, and then a systematic

perturbation theory for identifying and computing corrections. The way this is to be done is certainly not unique

for the relativistic two-body problem, and the optimum

procedure is far from obvious. The calculations themselves are fairly laborious and full of spurious difficulties. One often finds that contributions from different perturbations cancel completely in a remarkable, but not necessarily obvious, way. An approximation made in one place

to simplify the calculation of leading terms can result in a disaster at higher levels of approximation. To make the

calculations relatively compact, and hence more likely to be error-free, requires considerable trial and error. Clearly, further elaboration of these methods is inappropriate here. Instead, we will try to illustrate the

status of the work by examining some of the more important bound state quantities.

3.1 Hydrogen Fine Structure and the Lamb Shift

Figure 3 shows the n = 2 fine structure of hydrogen.

All three separations have been measured to varying de- grees of accuracy. The fine structure i\E has the least theoretical uncertainty; it is given by (in frequency

units) ^

16 [1+1 (Za)-'] meMr

2 me me

nip rriR

-3

-I- 2a.

rrie Mr

- —(Za)- [ln(Z a)-- + hp^]

IT

0850681

Figure 2. The main hadronic contribution to the )nuon ano)naly. The shaded blob represents the hadronic vacuum polarization; it can be determined experimentally from the process

e" + e~ ^ hadrons.

= 10969. 044(3)MHz ,

'Although we are most interested in hydrogen (Z = 1). we frequently

give the formulas for general Z. This distinguishes contributions related

to binding effects {-* factors of Za) from those related to other virtual photons factors of a).

187
where me/mn = (1 + nie/mp) and, according to an esti- mate by Erickson [7], bfs = 11/24 ± 3/2. The uncertainties in od and in bfg each contribute about 2 kHz to the

uncertainty in AE. In searching for theoretical corrections previously overlooked, we found the only one of possible significance is of relative order (Z a)hne/mp.

This should be calculated, if only to demonstrate that it is smaller than the previously mentioned uncertainties.

Table 3. Comparison of experimental and

theoretical values of the Lamb shift.

MHz

Direct measurements [8] Andrews and Newton (A-N)

Lundeen and Pipkin (L-P) Sokolov (S)

1057.862(20)

1057.845(9) 1057.8583(22)

From large interval measurements [9] Kaufman, Lamb, Lea, and Levinthal (K-L-L-L)

Shyn, Williams, Robiscoe,

and Rebane (S-W-R-R)

Vorburger and Cosens (V-C)

Safmya, Chan, Lundeen,

and ripKin (o-O-L-r)

1057.664(30)

1057.794(60)

1057.874(40)

100(.yii((41)

Theory

Erickson (E) [10] Mohr (M) [11]

1057.918(20)

1057.872(20)

0850681

Figure 3. The n = 2 fine structure of hydrogen

(hyperfine structure is not shown).

Figure 4 summarizes the more recent theoretical and

experimental results for the Lamb shift (§ in Fig. 3). Numerical results are given in Table 3. Two theoretical

values are presented since the Erickson-Mohr discrepancy

(see below) of 46 ± 10 kHz has not yet been resolved. The

overall theoretical uncertainty is estimated to be roughly

20 kHz, based on order-of-magnitude estimates of the various, as yet uncalculated, terms to be discussed below.

I •—I (A-N) "1 ^ (L-P) ) J

• (S) J

• ^(K-L-L-L)

(S-W-R-R)' • ' I (V-C) I • '

I (S-C-L-P) I • <J

The values of the Lamb shift determined from measure- ments of the large separation {^E - §) are far less ac- curate than the recent, more direct determinations, but

it should be possible to reduce the errors to a similar level (Safinya et al. [9]).

Id) (e) (f)

(E) (M)

1057

_L .650 .700 750 .800 AE(2S|

.850 .900 .950

Theory

(g)

Fkuire 4. Recent deiermi)iatii)yts of the Lamb shift. Direct ex- perimental determi)iatious are indicated^ by § and determi- nations from the large interval by AE - §. Theoretical values are also shown. References and )runierical values are give)/ in Table 3.

Figure 5. Examples of QED contributions to the Lamb shift, as described in the text. Photon lines terminating in x )-epre- sent individual Coulomb interactions. Doubled lines represent

the complete Coulotnb propagator or wave function.

188
Various QED contributions to the Lamb shift are illus- trated in Fig. 5. These are illustrative rather than

inclusive. The double lines in Fig. 5(a) represent all numbers of Coulomb interactions—i.e., the figures

represent an infinite number of Feynman diagrams, each

of which can contribute to a given order in a. In Fig.

5(b), additional Coulomb interactions in the closed elec- tron loop are unimportant for hydrogen [12], but the

graph yields various powers of Z a because of the properties of the wave function. The result for diagrams (a) and

(b) may be expressed as (E: Erickson [10]; M: Mohr [11]):

where L - a^R -^x/^Tx = 135.643 89(5) MHz. All terms

are known exactly except G {Z a). In contrast to previous

reviews, we advocate including the reduced mass factor

in the order ttZq term. This reduces that contribution by

11 kHz, but this reduction may be offset by uncalculated

recoil corrections.

The difference between the Erickson and Mohr calcula- tions of G{Z a.) is illustrated by Fig. 6. Erickson uses an

expansion of the Lamb shift operator in which he ignores

certain small terms and calculates the important

integrals exactly, as functions of Z a. His error estimate

MHz

S'l^ = Z^L [ln(Za)~2 - 2.207909]

nie 1050.560

+ 2.2962 irZa 7.129

+ {Zaf ln-(Z a)-2 + 3.9184 \n(Z a)"^

+ GiZ a)] -0.374(4)^

-0.420(9)'^

1057.315(4)^'

1057.269(9)^^

is a guess of the size of the omitted terms. Mohr does a numerical integration of an exact expression derived

from a partial wave expansion of the same operator. His

integrals converge well for large Z but not for Z small. Consequently he must extrapolate to obtain G (Z a) for Z = \. Mohr's extrapolation seems inconsistent with

0 10 20 30 40 50

2 0850661

Figure 6. Illustratioyi of the Erickson-Molir discrcpatui/. Val- ues ofGst:(Za) are plotted I'S. Z: GsffZa) is the part ofG(Za)

associated with Fig. 5(a). Erickson evaluates the most important terms for Z ~ 1, while Mohr extrapolates from larger Z

to Z = 1. For agreement , the extrapolation iroiild hare .s-o)/'c- whaf unexpected behavior for Z s lo (indicated

Erickson's result for Z = 1, and his prediction of the

small Z behavior. This conflict may ultimately be

resolved in any one of several ways. Perhaps the terms

neglected by Erickson add up to - 6. Possibly G (Z a) has a more complex structure for 1 « Z « 10. Clearly more work is needed on this important problem. [We know of such work in progress by J. Sapirstein. He expands the

Lamb shift operator, dropping operators of higher order

than a(Z a)''. The remaining expressions are evaluated

numerically using a closed form expression for the

Coulomb propagator. His results are awaited with great

interest.]-.

Figures 5(c) and (d) show sample terms of relative

order a. All such terms have been calculated analytically, and they yield a Lamb shift contribution of

§'2) = 0.3215—L = 0.101 MHz .

Other QED corrections remain to be worked out. Coulomb corrections to diagrams (c) and (d), such as (e), and the photon scattering diagram (f) should be of order

a(Za)L ~ 0.01 MHz .

Finally, higher order vertices and vacuum polarization

Sapirstein' s calculation for the IS state in hydrogen has been complet- ed [Phys. Rev. Lett. 47, 1723 (1981)]; within errors, his result agi-ees with Mohr' s.

189
diagrams, as in Fig. 5g-i, might contribute in order

2L ~ 0.001 MHz .

Thus far we have described contributions for which the

electron is assumed to move in an external Coulombic potential. Proton motion has been included only by intro- ducing reduced mass factors {m(.hnii)~'^ , coming mainly

from the radial scale of the wave function, which depends

on w/f. It is important also to consider more dynamical

recoil corrections. The most important of these arise

when retardation in the photon propagator is accounted

for. Examples of kernels contributing to recoil corrections are given in Fig. 7. In Fig. 7(a), the virtual photon spans any number of Coulomb interactions. The physics here is related to the

leading order Lamb shift in a simple way. For very soft photons {k < Zam), the photon interacts primarily

vdth a particle's convection current, which is proportional to ep/nie for an electron and {-Ze) {-p Imp) for the nucleus. Thus for this range of photon momenta, the

total current is enhanced by a factor (1 + Zme/in,j) rela- tive to leading order. This accounts for the relatively

large coefficient in this correction, which is known

analytically:

^'3) = 4.860Z—L = 0.359 MHz

(the exact calculation reveals a coefficient which is smaller than suggested by the naive argument). The

state also has a small shift which contributes to the Lamb

shift:

§(4) 24 - -0.002 MHz.

Some two-loop recoil corrections are suggested by Fig.

7(b). While these might possibly compensate the change

in the iraZL term discussed above, they should be computed. Their order of magnitude is

ttolZ' L ~ 0.002 MHz .

Additional recoil corrections might come from diagrams

like Fig. 7(c). These are probably somewhat smaller, be- ing of order

n{ —)Z~L ~ 0.001 MHz , n vip

but again they should be worked out. Finally, there are corrections associated with the ha- dronic structure of the nucleon. The finite size correction

illustrated in Fig. 7(d) is

§(5) = ^Z^a^R^cnie^ <R'^>p = 0.145(4) MHz ,

(a) lb)

(d) (e) (f)

Figure 7. Examples of various recoil and nucleon structure cojitributions to the Lamb shift, as described in the text. In (a), the blob represents any number of Coulomb interactions. In (d) and (e), the blob represents the static charge distribution of the proton, while in (f) and (g) it represents non-static features of the proton's .structure.

190
using the most recent value of <R'^>p as deduced from

e-p scattering i<R'^>^ = 0.862 fm). One should view

the quoted uncertainty in this number with some caution,

given the significant change in <R'">p from the previously accepted value (which gave = 0.127 MHz). Radiative corrections to the finite size correction [Fig. 7(e)] are suppressed by an additional factor of order a(Z a),

making them negligible. Proton polarizability corrections, as in Fig. 7(f) and (g), should be completely unimportant for the Lamb shift because the two diagrams cancel for loop momenta much larger than m^. As noted

below, these last diagrams are important for the hydrogen hyperfine structure. In summary then, several theoretical points must be

cleared up before a meaningful comparison with experiment is possible at present levels of accuracy. Most important is to resolve the discrepancy between Erickson

and Mohr. After that, several calculable QED contributions and recoil corrections should be worked out. At

that point, the Lamb shift becomes a combined test of our knowledge of the proton radius and of fundamental

QED.

3.2 Hyperfine Structure in IHydrogen, Muonium, and

Positronium

Table 4. Status of niuotniiDi h/s.

Exp^

Av = 4 463 302.88(16) (0.036 ppm) kHz

= 3.183 345 3(10) (0.31ppm)

Theory'^

ET _ 16 9 p ^

= 4 459 034.5(1.8) (0.4 ppm)

Av = EfT X corrections = 4 463 304.4 (1.8, -.5) kHz

^Presented at this conference by V. W. Hughes.

'^The first error in A v arises from the uncertainty in a and

jjL^: the second is a guess of the uncertainty due to missing

theoretical contributions.

The hyperfine separation in hydrogen is one of the

most accurately determined quantities in physics [13]: Ave^p = 1 420 405.751 766 7(9) kHz

Regrettably, theory is limited by nuclear structure and

by radiative corrections to much less precision [14]. Avt^ = 1 420 406(4) kHz

While uncertainties due to QED radiative corrections will be considerably reduced by current work on muonium,

there is little hope that nucleon structure effects can be

computed beyond 1 ppm without some radical improvement in our understanding of strong interactions.

Here we illustrate the theory more fully for muonium,

a fjL^e" bound state. Theory and experiment are summarized in Table 4. The theoretical terms known before this conference, and their contributions to Av are given by {Z = 1):

The first line is the splitting due to the interaction of two magnetic moments, without refinement. The error is due to uncertainty in measured values of iXp^/fi^, or equivalently in m ^. The second hne incorporates a relativistic correction due to Breit and the leading radiative

corrections excluding those included in Cg, a p.. The third

contribution is again a radiative correction. The coeffi- cient 18(5) was estimated by Brodsky and Erickson [15], and is analogous to GssiZ a.) in the Lamb shift—i.e., the

term disputed by Erickson and Mohr. Experience with

the Lamb shift suggests that the error estimate for this coefficient is reasonable. However, it is most important

that this uncertainty be substantially reduced. The

fourth term is the leading recoil correction. It is illus- trated by Figs. 8(a) and (b). An indication of the complications involved in such calculations is that diagi'am (b) also contains lower order contributions which must be re- moved to avoid double counting. Once this is done, the

^v = Ef\1 + a,

+ + a(Za-)(ln2 - |)

+ — (Za)2[ - 4 ln-(Za)~- - 1.067 ln(Z a)-^

- 3

+ 18(5)]

Z a m/e ln(m^/me )

„ niR In a

^

- 2

We + m

Zor m,

In

nip

4 464 205.4(1.8)

- 72.9

- 31.5(2.8)

- 801.2

+ 11.2

- 6.6

4 463 304.4(1.8,2.8)

191
e- ft* e- ft* e- ft*

(b) (c) (d)

e- fi' (e) ( f

)

(g)

e- fj.* (h)

0850661

Figure 8. Examples of various co)itribiitio)is to the muouimn

hyperfiiie strKcfure, as described in the test. The same contri- butions occur in positroniuni . with p* replaced hi/ e*

.

Similar corrections, but on the muon line, have not yet

been evaluated. To some extent these are incorporated in the value of which appears as an overall factor mEp,

but there appear to be additional contributions as well. There are terms, associated with Figs. 8(c) and (d) and

the like, in which the In a"^ appearing above is replaced

by In mjm^ or by a constant. It has been shown that

the \x\{yyijm,) terms add to zero [19]. The additive con- stant is now being evaluated [20] and will soon be known.

It could easily contribute at the level of a few kHz.

To summarize the situation for muonium: many of the

remaining recoil terms will soon be computed. The uncertainty in the relative order a(Z a)^ radiative corrections

is of a different nature, but it is certainly feasible to reduce it considerably. Hopefully that challenge will be

met in the not too distant future.

All terms contributing to the muonium hyperfine splitting also appear in the positronium spHtting, but with m^^ replaced by m,.. However, positronium is more difficult to analyze to the same relative accuracy because recoil effects are much more important (i.e., terms with high

powers of m,./)nr. cannot be ignored), and because annihilation diagrams add many new contributions. Current

theory and experiment are compared in Table 5. All

Table 5. Statiis of positronium HFS.

typical loop momenta range between and rn^, and

consequently the "one-loop" correction is proportional to the square of the wave function at the origin. The fifth term is relatively recent [16] and comes from two-loop di- agrams such as those in Figs. 8(c) and (d). Again lower

order terms, including those from Figs. 8(a) and (b), must be removed, but here the resulting loop integrations range from - am^ to ~ ni,.. The last term is known

as a "radiative-recoil" correction [17]. It results from ra- diative corrections to the kernels in Figs. 8(a) and (b), such as those illustrated in Figs. 8(e) to (h). In fact only

the vacuum polarization diagrams [Figs. 8(e) and (f)] contribute to \n~{mjm,.). Since the momenta in the

recoil loop [Figs. 8(a) and (b)] tend to be large relative to

ill,,, the photon propagator containing the vacuum polarization develops the extra ln{)ii^/iii,.). The vacuumpolarization contribution has now been computed more

completely with the result [18]

EA^fZ^i -21n^i^ - I TT mu

- f - ^] m.e 3 nie 9 3

The new (underlined) terms yield a contribution of -2.4

kHz [2]. Further radiative-recoil corrections, coming from kernels as in Figs. 8(g) and (h), have also now been

evaluated [18]. They give'^

= (2.3 + ? X 0.12) kHz .

^he vacuum polarization contribution has been verified numerically by

Lepage. He will also evaluate the electron radiative corrections numerically, including the non-logarithmic term.

Experiment

Av = 203 387.0 (1.6) MHz [21]

203 384.9 (1.2) MHz [22]

Theory

+ r^a-ln a-' + Ka^ - K' —(In a-)-

= 203 399.1 + 10.8K - 2. IX' MHz

terms of relative order a and a-lna"' have now been

computed, and some progress has been made on the

order a- corrections (F^ig. 9). It is urgent that all terms

of relative order a- be computed for a proper comparison

between theory and experiment. As indicated in the

table, terms of order — In-a- are also likely and these

too should be evaluated.

4. Summary

Our summary can be very brief. While it is expected

that at some level—very high energies or high orders of perturbation theory—QED will be revealed to be incomplete, clear-cut evidence for this has not yet emerged.

Further theoretical work is needed in most areas in order

to do justice to the analysis of the fine results from many

beautiful experiments. In the various cases, we have in- dicated which work is in progress and where further

work is feasible and necessary. We hope theorists will ac- cept the challenge to match or exceed the quality of the

results from the experimentalists.

192
0850681 Figure 9. Recent and fiitin-e ivork on jxisitrdiiin in hiipcffine structure, (a-c): the logarithmic terms which have been ob- tained in the past few years [ 16, 2Jl. (d) and (e): Contributions

proportional to a^R,x, worked out in Ref. [2Jfl and Ref. [J5J,

respectively, (f): To be evaluated in the near future [ 18/; also

see footnote 2. (g) and (h): The blobs represent an arbitrary

number of Coulomb e.ix-hanges.

References

[1] B. E. Lautrup, A. Petermann, and E. de Rafael, Phys. Rep.

3C, 193 (1972).

[2] M. J. Levine and R. Roskies, Phys. Rev. D 14, 2191 (1976), and references therein.

[3] R. S. Van Dyck, Jr., P. B. Schwinberg, and H. G. Dehmelt,

Phys. Rev. Lett. 38, 310 (1977).

[4] T. Kinoshita and W. B. Lindquist, Phys. Rev. Lett. 47, 1.573 (1981); and Cornell preprints CLNS-424 and 426 (1979); and 508, 509, and 510 (1981).

[51 J. Calmet, S. Narison, M. Perrottet, and E. de Rafael, Rev. Mod. Phys. 49, 21 (1977).

[6] J. Bailey cf nl., Phys. Lett. 68B, 191 (1977).

[71 G. W. Erickson, J. Phys. Chem. Ref. Data 6, 831 (1977).

[81 D. A. Andrews and G. Newton, Phys. Rev. Lett. 7, 1259 (197(5); S. R. Lundeen and F. M. Pipkin, Phys. Rev. Lett. 46, 232

(1981); Yu. L. Sokolov, these proceeding's.

[91 S. L. Kaufman, W. E. Lamb, Jr., K. R. Lea, and M. Leventhal,

Phys. Rev. Lett. 22, 507 (19(i9); T. W. Shvn, W. L. Williams,

R. T. Robiscoe, and T. Rebane, Phys. Rev. Lett. 22, 1273

(19(;9); T. V. Vorburger and B. L. Cosens, Phys. Rev. Lett.

23, 1273 (19(i9); K. A. Safinya, K. K. Chan, S. R. Lundeen, and F. M. Pipkin, these jM-oceedings.

[101 G. W. Erickson, Phys. Rev. Lett. 27, 780 (li)71). [Ill P. J. Mohr, Phvs. Rev. Lett. 27, 10.50 (1975).

[121 E. H. Wichmann and N. M. KroU, Phys. Rev. 101, 843 (195(;). [131 H. Hellwig cf at., IEEE Trans. Instrum. Meas. IM-19, 200

(1970); L. Essen cf at.. Nature (London) 229, 110 (1971).

[141 S. D. Drell and J. D. Sulhvan, Phys. Rev. 154, 1477 (19(i7) and

references therein.

[151 S. J. Brodsky and G. W. Erickson, Phvs. Rev. 148, 2(5 (19(i(i). [161 G. P. Lepage, Phys. Rev. A Iti, X()3 (i977); G. T. Bodwin and

D. R. Yennie, Phys. Rep. 43, 267 (1978).

[171 W. E. Caswell and G. P. Lepage, Phys. Rev. Lett. 41, 1092

(1978).

[181 E. Terray and I). R. Yennie, in ])reparatii)ii. [191 G. T. Bodwin, D. R. Yennie, and M. Gregorio, Phys. Rev. Lett.

41, 1088 (1978).

[201 G. T. Bodwin, I). R. Yennie, and M. Gregorio, in preparation.

[211 A. P. Mills, Jr., and G. H. Bearman, Phys. Rev. Lett. .34. 246

(1975).

[221 P. 0. Egan, V. W. Hughes, and M. H. Yani, Phys. Rev. A 15, 251 (1979).

[231 W. E. Caswell and G. P. Leijage, Phvs. Rev. A 20, 36 (1979).

[241 Vu. K. Cung, A. Devoto, T. Fulton, and W. VV. Repko. Phys. Rev. A 19, 1886 (1979).

[251 Vu. K. Cung, A. Devoto, T. Fulton, and W. W. Repko. Phys.

Lett. 68B, 474 (1977).

193

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Uncertainties in QED Fine Structure Calculations

Glen W. Erickson

Physics Department, University of California, Davis, CA 95616

The most general calculation of precise one-electron atomic energy levels includes uncertainties which ai-e important in the determination of fundamental constants and in the comparison of QED

theory and precision measurements. Besides the usual Lamb shift splitting S between nSy2 and nPy2

states, and the fine structure interval AE between nP^ and nPi/2 states, the large interval between

71 and h Py2 states has been directly measured to high precision, but the uncertainty in its calcu- lated value is often mistakenly taken to be the combination of uncertainties in S and AE. This vdll be

clarified, and the sources of these uncertainties in the low-Z calculations will be discussed. Comparisons will be made with independent calculations at higher Z and with experimental values. The overall picture is one of general agreement, but with small unresolved discrepancies. It will be seen that a ten- tative resolution may be obtained by discarding one of the calculations, the most recent proton and al- pha particle size measurements, and 7 of the 27 most precise measurements of energy level difference. Key words: atomic structure; electron structure; energy level shifts; fine structure; Lamb shift; proton

size; quantum electrodynamics. .

1. Introduction

Although the most precise calculations of quantum

electrodynamical (QED) atomic energy level shifts have

not changed since their results were published [1, 2], the

interpretation has varied, partly due to the use of dif- ferent nuclear sizes [3-6], and partly due to increasing

precision of Lamb shift (n S1/2 - n Fy?) and large interval

inFs/2 ~ nS>y2) measurements [7-33]. The situation has

mostly been considered as unresolved, especially since

the results of the measurements have changed from

agreement with one calculation to agreement with the

other and to disagreement with both calculations. I would like to point out here that a tentative resolution

exists, although it is not especially satisfying since it implies that there are errors, omissions and /or underestimates of uncertainties in one of the calculations

(mine), in the two most recently published values of the

sizes of the proton and alpha particle [4, 6], and in some

of the most precise measurements of energy level differ- ences [8, 15, 18, 19, 21, 27, 29]. The other uncertainty dis- cussions here are not new, simply repeating selected

parts of the comprehensive uncertainty discussion vdth

my 1977 calculation of energy levels of one-electron

atoms [1].

2. Interpretations of Discrepancies

The interpretations of comparisons between calcula- tions and precision measurements of Lamb shifts and

large intervals are highly dependent on the relative uncertainties involved. Because the most precise measure- ments in hydrogen [7-9] now have uncertainties of the same order of magnitude as those of the calculations,

±0.01 MHz = ±10 kHz, we must consider the different sources of those uncertainties before we decide if the

measurements are testing QED theory, are testing the

correctness of a calculational method, are testing higher

order QED effects, are verifying relativistic quantum

mechanics, are measuring the fine-structure constant, or

are measuring nuclear size. Besides finding that all of

these aspects are involved, I should point out that, after discrepancies (between different calculations, between

different experiments, and between theory and experiment) are resolved, a precise agreement between theory

and experiment allows useful Hmits to be placed on alter- native theories.

Before getting into details, let me point out the differ- ence between two apparently contradictory, but not necessarily inconsistent, interpretations of "discrepant"

values: (1) they can indicate that both results support the

underlying correctness of each other, and (2) they can in- dicate that at least one of the results is wrong. The

second interpretation is the most obvious one, but

depends on whether the "discrepancy" is large or small

compared to the combined uncertainty. The first in- terpretation is valid if the "discrepancy" is small compared to the amount of "correctness" involved. Both

interpretations can be valid simultaneously.

For example, my calculation [1] of a Lamb shift remainder coefficient G as -17.1 ± 0.6 and Peter Mohr's

calculation [2] of the same remainder as -23.4 ± 1.2 differ by 6.3 ± 1.3. The first interpretation is that we, by

using extremely different calculational methods to obtain

essentially the same large negative value of a difficult remainder, have verified each others methods to an accuracy, ±6.3/2, which is an order of magnitude smaller

than the remainder itself. The second interpretation notes that we differ by over four times our estimated uncertainty, so that we have either made an error or have

under-estimated the uncertainty of our methods. Both

interpretations are valid. The first allows our results to be used to test QED (or alternative theories) to the level of our 6.3 difference (46 kHz). The second points out the

need to look for errors and /or to decide which result is more likely to be correct if it is to be used to the level of the estimated uncertainty (9 kHz).

The same set of interpretations can be made of the partially discrepant precision measurements of the n = 2 Lamb shift in hydrogen [7-11], which agree at the level of 20 kHz and disagree at the level of 10 kHz. The agree195
ment is sufficient to help decide which calculated value is preferred. The disagreement is sufficient to point out the

possibilities of an error in a measurement [8] or of an

inappropriately small uncertainty [7]. Depending on our purposes, we can treat slightly

discrepant values in two distinctly different ways. The

simplest is to use a weighted average to avoid missing

the correct value by too much. Another way is to tentatively select one of the values as correct (and the other

as incorrect) and then to test our hypothesis by using the

selected value (to its claimed accuracy) and seeing if our

selection will minimize discrepancies in related results. For example, instead of using a proton size obtained by

adjusting scattering data normalization [4] in order to bring discrepant measurements into agreement at 7), -

0.862 ± 0.014 fm, we will consider the possibility that

only the earlier measurements [3], = 0.80 ± 0.02 fm

or 0.805 ± 0.011 fm, are correct. This turns out to yield

much better Lamb shift agreement than is possible with

the larger proton size.

3. Comparison ot Uncertainties

Let us now proceed to compare the sizes of the uncertainties in the calculations of the shift of an individual

energy level. The largest contribution, and usually the

largest uncertainty, is from the electron structure or self-energy contribution [1,2]

^ {Lln(Z a) - + — J 6,0 Strn'* 24 + L„ + -Itt^^ + H{Za)} (1) 8 2/ + 1 whose higher order part has an expansion

H(Zoi) = C-^ZoL + (Za)-[C62ln-(Za)2 + C6iln(Za)-2 + G{Za)] (2)

with leading coefficients Cj, C62. and Cei, that have been

calculated exactly. The remainder G {Z a) has not been

calculated exactly or completely for arbitrary states, but

the contributions of the dominant terms have been carefully accounted for by Erickson [1], who estimated the

net size of the omitted terms for an uncertainty 8G (Z a). Using a different approach, Mohr [2] has numerically

evaluated the complete contribution for individual states and individual large values of Z, and has subtracted the

known leading terms to obtain numerical values of GiZ a). Careful straightforward extrapolation to small Z

yielded Lamb shifts for comparison with experiment and

yielded values of G (Z a) for comparison with Erickson'

s

calculations. These remainder calculations agree quite

well (within small estimated uncertainties) for non-S

states and, as noted before, agree fairly well (within a

fraction of the remainder G) but differ by many times the

estimated uncertainties for S states. For the 2Si/2 - 2Pi/2

Lamb shift in hydrogen {Su), the difference is 46 ± 9 kHz and scales roughly like Z'^/n^ for other S states. The other QED uncertainties for Sh are estimated to be ±6 kHz for uncalculated higher order parts of vacuum

polarization, magnetic moment, and fourth order electron structure contributions, and ±3 kHz for uncalculated

higher order parts of nuclear mass /recoil contributions.

These uncertainties are smaller than the electron struc- ture discrepancy in hydrogen and, because of their Z or m /M scaling, they are neghgibly smaller than the elec- tron structure uncertainty for other atoms.

The new aspect of the greatly improved precision of the Lamb shift measurements is that their uncertainties are now comparable to or smaller than even the ±6 kHz

uncertainty due to proton size uncertainty ±0.02 fm in the nuclear size contribution.

<r^> 2(Za)^mc^

{h lmc)~

(3)

This means that nuclear size can be determined by Lamb

shift measurements as well as, or even better than, by

electron scattering. The primary limitation is from the

uncei-tainties in the QED calculations. The other uncertainties (from Bethe logs, fine- structure constant, nuclear size effect in QED calculations, etc., as in Ref. [1]) are less than ±1 kHz in Sh and

may be ignored for our purposes here. Even for the fine- structure interval AE" or the large interval A£' - S, a

0.11 ppm a uncertainty causes only a 0.22 ppm uncertainty or ±2 kHz.

4. Hydrogen Comparison and a Tentative

Resolution

To illustrate a number of points and to start to make

sense out of the many inconsistent comparisons of theory

and experiment, let us start by considering the Hydrogen

Lamb shift Sh, where the measurements [7-11] lie below

most calculated values. Since this favors smaller calcu- lated S state shifts, it suggests that we consider using

the smaller value of electron structure contribution [2] and the smaller proton size [3]. This turns out to give a reasonably consistent set of comparisons. The calculated

Sh = 1057.867 MHz will be in acceptable agreement with

the weighted average of all the measurements (1057.858

MHz) and with the new measurement with the smallest

claimed uncertainty [7], 1057.8583 ± 0.0022 MHz. The

discrepancies are the fewest possible: The electron struc- ture contribution would be outside my estimated uncertainty [1], the proton size would be outside the Mainz

uncertainty [4], and Sh would exceed the recently improved Harvard measurement [8], 1057.845 ± 0.009 MHz,

by over twice the uncertainty of the latter. If we consider the measurement of the large interval

2P3/2 - 2Si/2, there were three discordant values [13-15]

and there is now a new value [12] somewhat below but in agi'eement with the lowest of the three. If we again use a calculated value based on the smaller proton size and the

smaller upward S-state shift, the result depends slightly on the choice of fine structure constant used, but in any

event lies close to the lowest of the three older values

[13] and thus not far above the newest value [12], not far below the middle older value [14], and only in serious

disagreement with the highest [15] of the older values. In summary, the four measurements by themselves are in- consistent and inconclusive, but our hypothesis requires no more discarding of data than does the data itself, and

requires discarding of the same measurement with which

all of the other three disagree. We will find similar results in the higher Z measurements.

Let me interrupt the evaluation of my hypothesis to briefly discuss the ^•^/o - S1/2 - ^.E - S "large inter- val", which is often treated as the difference between

the fine structure interval (A^") and the Lamb shift (S). However, the large interval calculation does not involve

the Pi/2 level or its uncertainty (which is somewhat larger

than for the P3/2 level). Therefore, there are many disadvantages in trying to use a large interval measurement to obtain a value of either £^E or S. Because of the size of

196
the large interval, its uncertainty due to the fine struc- ture constant (a) uncertainty is larger than for the Lamb

shift. Saying this another way, a large interval measure- ment could be used to determine the fine structure constant a (if the nuclear size and S state QED shift uncertainties were small). On the other hand, if the fine structure constant is accurately known, then the large in- terval has as much to say as does the Lamb shift S about

nuclear size and S state QED shifts. A Fine Structure interval (AE) measurement [16] tells us mostly about the fine structure constant a since the

QED uncertainties are quite small (±2 kHz or 0.2 ppm).

To continue the discussion of which electron structure

calculation is less discrepant with precise measurements,

let us consider deuterium measurements. The two precise

Lamb shift measurements [17, 18] are discordant, as are

the two large interval measurements [19, 20] and do not seem to prefer one calculated value more than the other.

Either way, the smaller of each of the measurements

[18, 19] are discordant.

These comparisons are probably most easily seen in the

form given in Table 1, which considers the difference

between the remainders G (Z a) as calculated by Mohr [2] and Erickson [1]. The differences between measurements

and Erickson' s calculated values are divided by the

difference between Mohr's and Erickson' s calculations,

4a(Za)W^ ^ ,^ ^ _

ZH—fAG X 7.22 kHz. (4) n

In that way, a value of 0 would indicate perfect agree- ment with Erickson' s calculation and a value of +1 shows

agreement with Mohr's smaller electron structure. Erickson' s calculations use the smaller proton and alpha

Table 1. Comparisons of Measuremmts and Calculations

of Erickson [1] a nd Mohr [2]

z

Meas. -

Calc.(M)

Calc.(^)

(Meas. Unc.)

-Calc.(£^)

Quantity Kei.

1 -6.3 1.16 ± 0.05 [7]

1 45 + 0 20 Su [81 1.08 ± 0.44 Sh [9] 0.24 ± 2.19 Sh [10] 1.15 ± 1.38 Sh [11]

-0.26 ± 0.90 {AE - S)h [12] 0.97 ± 0.92 (A^; - S)h [13]

( \ TT — C "1 (l^Ci — o )h [14J 5.46 ± 0.57 (AE - S)h [15]

-0.21 ± 1.41 Sd [17] 6.07 ± 1.41 [18]

-3.40 ± 1.23 {AE - S)d [19] 0.91 ± 2.07 iAE - S)d [20]

2 -5.9 -0.58 ± 0.44 [21] 1.61 ± 0.66 [22] 1.61 ± 1.06 [23]

1.51 ± 0.67 H=3S^^, [24] 0.79 ± 0.59

He

[25] 1.61 ± 2.33 He

[26]

3 -5.64 -0.11 ± 0.71 [27] -0.95 ± 2.36 [28]

8 -4.43 -2.48 ± 0.89 [29] 0.23 ± 1.31 5^7+ [30]

9 -4.15 1.26 ± 2.20 [31]

17 -3.20 1.28 ± 0.39 5^116+ [32]

18 -3.09 1.33 ± 0.78 5^.17 + [33]

197
particle sizes [3,5], so a value of +1 indicates perfect

agreement with the hypothesis being tested. The w = 2 ^He"^ Lamb shift measurements [21-23] are

discrepant. My hypothesis of the small electron structure

contribution [2] (and an older smaller nuclear size [5]) yields a Lamb shift, 14041.9 MHz, which indicates the

discordance of only one of the three measurements [21]

and of a new nuclear size calculation with a rather small

uncertainty [6]. The opposite pair of hypotheses would

favor the other Lamb shift measurement, but requires

the discordance of both of the others. The n > 2 ^He"^ measurements [24-26] of the Lamb

shift and of the large interval definitely favor my hypothesis, although the other electron structure results only lie one or two standard deviations from the measurements.

For the Z > 2 measurements [27-33], some measure- ments prefer the larger electron structure, but there seems to be better agreement with the smaller electron structure [2]. Only two measurements [27, 29] are dis- cordant with that hypothesis, and one of them [29] is discordant with either electron structure calculation. In summary, only seven of the measurements are even

partially discordant with the hypothesis that the smaller

electron structure calculation [2] is correct, none of these

measurements are corroborated by independent measure- ments of the same quantity, and three of them

[18, 19, 29] even disagree with the other electron struc- ture calculation [1].

5. Conclusions

If we accept the resolution given by the tested hypothesis, we can state that: QED seems to be verified to an accuracy of 13 kHz in hydrogen, but the calculation [2] has only been independently checked [1] to an accuracy of 46 kHz. It would be desirable to find errors in the

discrepant calculation and measurements, but there can be serious danger of "intellectual phase-locking", such

as my checking of omitted terms in my electron structure

calculations [1] only until I find enough to add up to -46

kHz. I will avoid this, of course, and trust that experimenters will change their published values only if complete re-analysis demands a correction.

Even if the discrepancies were to be resolved (by verified corrections or by other independent checks to high

precision) it may be seen that the next level of "ulti- mate" verification would be of the order of the 6 kHz

due to the many other uncalculated higher order QED

terms. This would prevent proton size determination

better than ±0.02 fm from Lamb shift measurements, but

could allow size determination for other nuclei. References^

[1] G. W. Erickson, J. Phys. Chem. Ref. Data 6, 831 (1977).

[2] P. J. Mohr, in Beam-Foil Spectroscopy, Ed. by I. A. Sellin and D. J. Pegg (Plenum Press, New York, 1976), p. 89; and Phys. Rev. Lett. 34, 1050 (1975).

[3] = 0.80 ± 0.02 fm: D. Frerejacque, D. Benaksas, and D.

Drickey, Phys. Rev. 141, 1308 (1966):

r = 0.805 ± 0.011 fm: L. N. Hand, D. G. Miller, and R. Wilson, Rev. Mod., Phys. .35, 335

(1963):

'Sapirstein, Phys. Rev. Lett. 47, 172;^ (1981), finds a value ofG for the IS state in good agreement with that obtained by Mohr.

-Includes all measurements accurate to 0.1 MHz y Z'' (2hi)''.

Vp = 0.81 ± 0.04 fm: J. J. Murphy, II, Y. M. Shin, and D. M. Skopik, Phys. Rev. C 9, 2125

(1974).

[4] Tp = 0.87 ± 0.02 fm: F. Borkowski, G. G. Simon, W. H.

Walther, and R. D. Wendling, Z. Physik A 275 29 (1975)' Tp = 0.862 ± 0.012 fm: G. G. Simon, Ch. Schmitt, F. Borkowski, and V. W. Walther, Nucl. Phys. A 333, 381 (1980).

[5] r„ = 1.63 ± 0.04 fm: U. Erich, H. Frank, D. Haas, and H. Prange, Z. Phys. 209, 208 (1968).

[6] r„ = 1.674 ± 0.012: I Sick, J. S. McCarthy, and R. R. Whitney, Phys. Lett. 64B, 33 (1976).

[7] Sh = 1057.8583 ± 0.0022 MHz: Yu. L. Sokolov, these

proceedings.

[8] Sh = 1057.845 ± 0.009 MHz: S. R. Lundeen and F. M. Pipkin, these proceedings and

Phys. Rev. Lett. 46, 232

(1981).

[9] Sh = 1057.862 ± 0.020 MHz: D. A. Andrews and G. Newton, Phys. Rev. Lett. 37, 1254

(1976); G. Newton, D. A. Andrews, and P. J. Unsworth,

Philos. Ti'ans. R. Soc. London

290, 373 (1979).

[10] Sh = 1057.90 ± 0.10 MHz: R. T.'Robiscoe and T. W. Shyn,

Phys. Rev. Lett. 24, 559 (1970).

[11] Sh = 1057.859 ± 0.063 MHz: average calculated in Ref.

[34] from transitions measured by S. Triebwasser, E.

S. Dayhoff, and W. E. Lamb,

Jr., Phys. Rev. 89, 98 (1953).

[12] (AE - S)h = 9911.117 ± 0.041 MHz: K. A. Safinya, K. K. Chan, S. R. Lundeen, and F. M. Pipkin, these proceedings and Phys. Rev.

Lett. 45, 1934

(1980).

[13] (AS - S)h = 9911.173 ± 0.042 MHz: B. L. Cosens and

T. V. Vorburger,

Phys. Rev. A 2, 16

(1970).

[14] (A S)h = 9911.250 ± 0.063 MHz: T. W. Shyn, T. Rebane, R. T. Robiscoe, and W.

L. Williams, Phys. Rev. A3, 116(1971).

[15] (AE - S)h = 9911.377 ± 0.026 MHz: S. L. Kaufman, W.

E. Lamb, Jr., K. R. Lea, and M. Leventhal, Phys. Rev. A 4, 2128 (1971).

[16] (A£')h = 10969.127 ± 0.095 MHz: J. C. Baird, J. Brandenberger, K-I. Gondaira,

and H. Metcalf, Phys. Rev. A 5, 564 (1972); see

also Ref. [34]. [17] Sd = 1059.282 ± 0.064 MHz: B. L. Cosens, Phys. Rev. 173, 49 (1968); see also Ref. [34]. [18] Sd = 1058.996 ± 0.064 MHz: S. Triebwasser, E. S. Dayhoff, and W. E. Lamb, Jr., Phys. Rev. 89, 98 (1953); see also Ref [34]. [19] (AE - S)d = 9912.607 ± 0.056 MHz: E. S. Dayhoff, S. Triebwasser, and W.

E. Lamb, Jr., Phys. Rev. 89, 106 (1953); see also Ref [34]. [20] (AE - S)d = 9912.803 ± 0.094 MHz: Ibid.

[21] S + = 14046.2 ± 1.2 MHz: M. A. Narasimham and R. L. Strombotne, Phys. Rev. A 4, 14 (1971).

198
[22] S^^,

[23] S^^,

[24] w = 3 S

= 14040.2 ± 1.8 IVlHz: E. Lipworth and R. Novick,

Phys. Rev. 108, 1434 (1957); see also Ref. [34]. = 14040.2 ± 2.9 MHz: G. W. F. Drake, S. P. Gold- man, and A. Van Wijngaarden,

Phys. Rev. A 20, 1299 (1979). = 4183.17 ± 0.54 MHz: D. L. Mader, M. LevHe

[25] n = 3

[26] w = 4

[27] S^^,,

[28] 5^.++

[29] S^7+

{AE

S)

S)

He+

He +

= 47844.05 = 20180.6 ± 0.8

enthal, and W. E. Lamb, Jr., Phys. Rev. A3, 1832 (1971). 0.48 MHz: Ibid. MHz: A. Eibofner,

Z. Phys. A 277, 225 (1976). = 62765 ± 21 MHz: M. Leventhal, Phys. Rev. A 11, 427 (1975). = 62790 ± 70 MHz: D. Dietrich, P. Lebow, R. deZafra, and H. Metcalf, Bull. Am.

Phys. Soc. 21, 625 (1976). = 2215.6 ± 7.5 GHz: G. P. Lawrence, C. Y. Fan, and

S. Bashkin, Phys. Rev. Lett. 28,

1612 (1972); ibid. 29, 320 (1972).

[30] S^7+ = 2202.7 ± 11.0 GHz: M. Leventhal, D. E. Murnick,

" and H. W. Kugel, Phys. Rev.

[31] (AE - S)p84

Lett. 28, 1609 (1972). 68853.9 ± 35 GHz: H. W. Kugel, M.

Leventhal, D. E. Murnick, C. K. N. Patel, and 0. R. Wood, 11, Phys. Rev. Lett. .35, 647 (1975).

[32] 5^,16+ = 31.19 ± 0.22 THz: E. T. Nelson, 0. R. Wood, H,

C. K. N. Patel, M. Leventhal,

D. E. Murnick, H. W. Kugel,

and Y. Niv, these proceedings. See also Phys. Rev. Lett. 48, 398 (1982).

[33] S 17+ = 38.0 ± 0.6 THz: H. Gould and R. Marrus, Phys.

^' Rev. Lett. 41, 1457 (1978).

[34] B. N. Taylor, W. H. Parker, and D. N. Langenberg, Rev. Mod. Phys. 41, 375 (1969).

199

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Sixth Order Contributions to g-2 of the Electron

M. J. Levlne

Physics Department, Carnegie l\/lellon University, Pittsburgh, PA 15213

and

R. Z. Roskies

Physics Department, University of Pittsburgh, Pittsburgh, PA 15260

The contributions of 10 graphs to the anomalous magnetic moment of the electron in sixth order

are presented to much greater accuracy than in previous evaluations. These results lead to a revised value for the entire sixth order contribution. When this is combined with the preliminary eighth order

results, theory and experiment are again in reasonably good agreement.

Key w^ords: fine-structure constant; g-factor; magnetic moment; quantum electrodynamics.

Of the 72 Feynman graphs contributing to the g factor

of the electron in sixth order, 46 have been calculated

analytically [1]. Five more have been reduced to one- dimensional integrals [2] and are known to eight decimal

places. Another six (the light-by-light graphs) have been

evaluated [3] to better than three decimal places. We

report here a careful evaluation of 10 of the remaining

graphs.^

The graphs are shovv^n in Fig. 1 and the results in Table 1. We comment below^ on the significance of the

fact that our value for the total is smaller than the best

previous estimate [5]. Details of the calculation will be

published elsewhere. It suffices here to explain that the

contribution of each graph is transformed analytically to

a three-dimensional integral, as in Ref. [2]. Because

further analytic work seemed intractible, we resorted to numerical integration at this stage.

If we write

2 3 a + ae a + ag a

TT TT TT

+ weak interaction effects

7 9 17 19 27

Figure 1. The Feynman graphs for which new values are re- ported in this paper. Graph numbers follow Ref. [ij.

Combining our results with the previously quoted

results, and a numerical evaluation [6] of the five remaining graphs we findae = 1.1765 ± 0.0013 , where the error is essentially all due to the five graphs

of Ref. [6]. If we combine this result with Kinoshita's

estimate reported elsewhere at this conference,

ag - -0.8 ± 2.5 , and use the value [7] = 137.035963 ± 0.000015 , + strong interaction effects ^^(j

+ effects due to jjl's, t's

we know that

aa = 1/2

197 ^ tt2 Wl2 TT-ln2 3^3)

2 4 = -0.328479

'The astute reader will notice that Fig. 1 has only five graphs. In the counting scheme used in the text, a graph is distinguished from its mirror

image.

9-2

(1 159 652 454 ± 127 ± 17 ±73) x 10"

theory where the errors come from the errors in a, ar, and Og,

respectively. (Weak and strong interaction effects, and

the contribution of |x's and t's amount to less than 5 x

10-12.)

The best experimental value for (g -1)1 2 is [8]

(1 159 652 200 ± 40) x 10-12

exp Thus theory and experiment are in moderate agi'eement. Earlier discrepancies between theory and experiment have been reduced in two ways. First, our new

value for the 10 graphs is smaller (by slightly more than

201
Table 1. Contributions of the graph and its mirror image to ag, in the Feynman gauge. Infrared divergences proportional to log \ or

log^ X. have been omitted.

Graph No. 7 9 17 19 27 Total

Our New Values -2.670546 0.617727 0.607660 -0.334698 1.861992 0.082065 ±0.000030 ±0.000121 ±0.000240. ±0.000071 ±0.000240 ±0.000362

Previous Evaluations

Levine and Wright [4] -2.664 0.625 0.613 -0.330 1.854 0.0970 ±0.020 ±0.006 ±0.013 ±0.013 ±0.013 ±0.0230

Cvitanovic and -2.6707 0.6189 0.6097 -0.3182 1.8572 0.0893 Kinoshita [5] ±0.0019 ±0.0064 ±0.0034 ±0.0072 ±0.0086 ±0.0060

one standard deviation) than the most accurate previous

estimate. Secondly, Kinoshita' s estimate for a 8 is negative. Both these effects operate in the same direction and

reduce the disagreement betw^een theory and experiment.

Another w^ay of reporting the results is to assume the

validity of QED, and to estimate a from the experimental

value of g -2 and the theoretical calculations. The result

isa-i - 137.035993 ± 0.000005 ± 0.000002 ± 0.000009 , M^here the errors come from experiment, ae, and ag, respectively. This estimate for a"^ agrees reasonably

with that used above.

It is clear that the most pressing theoretical need is a better evaluation of ag. If its error can be reduced significantly, and if better estimates for a can be obtained, as has been suggested in many vi^ays at this conference,

then we vidll be able to test QED to much greater precision. Conversely, if the experimental precision in the determination of Qe increases as has also been suggested at

this conference, a better evaluation of ag can provide us

vdth an order of magnitude improvement in the present

determination of a, assuming the validity of QED.

References

[1] M. J. Levine and R. Roskies, Phys. Rev. Lett. 30, 772 (1973); Phys. Rev. D 9, 421 (1974); ibid. 14, 2191 (1976); M. J. Levine, R. Roskies, and R. C. Perisho, ibid. 13, 997 (1976); R. Barbieri, M. Caffo, and E. Remiddi, Phys. Lett. 57B,

460 (1975); J. A. Mignaco and E. Remiddi, Nuovo Cimento

A 60, 519 (1969); R. Barbieri and E. Remiddi, Phys. Lett. 49B, 468 (1971).

[2] M. J. Levine, E. Remiddi, and R. Roskies, Phys. Rev. D 20, 2068 (1979).

[3] T. Engelman and M. J. Levine, unpublished.

[4] M. J. Levine and J. Wright, Phys. Rev. D 8, 3171 (1973).

[5] P. Cvitanovic and T. Kinoshita, Phys. Rev. D 10, 40007

(1974).

[6] T. Kinoshita and W. B. Lindquist, Cornell preprint CLNS-374

(1977), unpublished.

[7] E. R. Williams and P. T. Olsen, Phys. Rev. Lett. 42, 1575

(1979).

[8] R. S. Van Dyck, P. B. Schwinberg, and H. G. Dehmelt, Bull. Am. Phys. Soc. 24, 758 (1979).

202
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Calculation of the Eighth Order Anomalous Magnetic Moment of the Electron*

T. Kinoshita

CERN, 1211 Geneva 23, Switzerland**

and

Newman Laboratory of Nuclear Studies, Cornell University, Ithaca, NY 14853

and

W. B. Lindquist

Newman Laboratory of Nuclear Studies, Cornell University, Ithaca, NY 14853

We present a very preliminary result of our calculation of the eighth order QED contribution to the anomalous magnetic moment of the electron. Altogether 891 Feynman diagrams contribute to this term. By a method developed earlier, we have compressed them into about 100 integrals, which are evaluated using adaptive Monte Carlo integration routines. Our result is -0.8 (2.5)(a/-n-)^. Combining

this with the results of lower orders and using a"^ = 137.035 963(15) we find

a^'' = 1 159 652 460 (148) x lO'i^. The difference between experiment and theory is now -251(154) x 10 Key words: anomalous magnetic moment; electron; fine-structure constant; quantum electrodynamics.

1. Introduction

The electron belongs to a special class of elementary

particles whose interaction is predominantly electromagnetic. Since electromagnetic processes are the easiest to measure and can be done so with the greatest of accuracy, it has provided the main testing ground of quantum

electrodynamics (QED). In particular, the magnetic mo- ment anomaly of the electron has been studied both

experimentally and theoretically over 30 years with

ever-increasing precision. A very substantial progress

has been made in the last few years by Dehmelt and his coworkers [1] whose published 1977 value for the magnetic moment anomaly ttg = (^Tg - 2)/2, a/^ = 1 159 652 410 (200) x lO'^^, (1) represents a factor of improvement of 20 over the best

previous measurement. By 1979 [2] the error went dovra

by a further factor of 5:

tte'^ = 1 159 652 200 (40) x lO'^^. (2) In perturbation theory of QED, higher order radiative

corrections to ag can be written as a power series in o/tt:

tte = Ci(a/-rr) + C 2(0/17)2 + C 3(0/77)^ + C4(a/TT)4 + . . . (3)

Thus far the first three coefficients have been calculated

[3];

*Supported in part by the U.S. National Science Foundation.

**1 April-31 July, 1981.

Ci = 0.5,

C2 = -0.328 478 966 ... , C3 = 1.183 5 (61). (4)

If one uses the 1979 value of the fine structure con- stant a [4] a-i = 137.035 963 (15), (5)

the QED prediction, Eq. (4), gives the value

1 159 652 566 x lO-^l (6) To this one must add contributions from other sources. They include the contributions of the muon loop, the t meson loop, and the hadronic effect, which are all small:

ae (muon) = 2.8 x lO'^^

tte (t meson) = 0.01 x 10"^^

Ue (hadron) = 1.6 (2) x lO'^^. (7) The effect of the weak interaction (according to the stan- dard Weinberg-Salam model) is also very small [5]:

ae (weak) - 0.05 x lO'^^. (8) Collecting Eqs. (6), (7), and (8) we obtain the best

theoretical prediction available thus far:

ai'^ = 1 159 652 570 x lO-^^. (9) Comparing this vdth Eq. (2) we see that theory and ex- periment differ by

Aa, = a'/i" -a'J' = -370 x 10" 12 (lO)

203
which is nearly 10 times as large as the experimental er- ror quoted in Eq. (2). In order to decide whether this discrepancy is signifi- cant or not, one must of course examine the errors in the

theoretical value, Eq. (9). The uncertainty in the meas- urement of a quoted in Eq. (5) contributes an error of 127 X 10-12 Eq -phe error due to that of C 3 in Eq.

(4) amounts to 77 x 10^^^ which results almost totally

from the 15 integrals (out of 72) that have been evaluated

only numerically. An additional six integrals (from di- agrams containing photon-photon scattering subdiagrams) have recently had their errors reduced by

several orders of magnitude by a combined analytical-numerical integration technique [6]. This method, currently

being applied to the 15 integrals mentioned above, is ex- pected to reduce the error of C3 to an insignificant level

[7]. The errors in Eqs. (7) and (8) are completely negligi- ble at this stage. Actually, in view of the fact that

(a/TT)'* - 29 X 10-12 (11)

is almost as large as the experimental error in Eq. (2), the most serious theoretical uncertainty arises from the

absence of knowledge about C4 which makes comparison

of theory and experiment rather tentative.

On the other hand, if we assume that the difference

in Eq. (10) is genuine and due entirely to the C4

term, we obtain the "prediction" that

C4 - -13(6), (12) a surprisingly large value. Of course it is entirely possi- ble that the discrepancy, Eq. (10), is mainly of experimental origin and fades away as measurements improve.

Whether this turns out to be the case or not, however,

there is no question that we must know the magnitude

and sign of C4 for a meaningful comparison of the theory

with the present and forthcoming experiments.

Group IV. Vertex diagrams containing a photonphoton scattering subdiagram vdth further

radiative corrections of various kinds. This

group consists of 144 diagrams. The

number of integrals is 13. Group V. Vertex diagrams containing no vacuum polarization loop. This group is comprised of 518 diagrams and is represented by 47 in- tegrals.

All integrands have been generated by an algebraic

program SCHOONSCHIP. A typical integrand is a ra- tional function consisting of up to 15,000 terms, each

term being a product of up to 8 or 9 factors. The integration, over a hypercube of up to 10 dimensions, has been

carried out by adaptive Monte Carlo subroutines

RIWIAD and VEGAS.

The evaluation of the integrals of the first three groups was completed more than 2 years ago. The results are

[9, 10]

Ci = 0.076 6 (6), Ci' = -0.523 8 (10), C{" = 1.419 (16). (13)

The remaining groups are substantially more difficult to evaluate and the results have only recently become avail- able. In view of difficulties encountered in carrying out some of the numerical integrations, the error (90 percent

confidence limit estimated by the integration routines) is

still very substantial and should not be fully trusted. Anyway, our very, very tentative results are [11]:

Cf = -0.78 (48), Cr = -1.0 (2.4). (14)

2. Calculation of C.

4

It is for this reason that we decided to calculate the

quantity C4. In the absence of any clever method which

enables us to estimate it quickly and reliably, we had no

choice but to calculate by brute force the values of all 891

Feynman diagrams that contribute to C 4. However, a

substantial simplification has been achieved by a method

developed earlier [8] which enables us to combine several

integrals into one. In this manner we were able to reduce

the number of integrals to be evaluated to slightly over

100.The 891 Feynman diagrams fall naturally into the fol- lovdng five groups, each of which consists of one or more

gauge invariant sets: Group I. Second-order vertex diagrams containing vacuum polarization loops of second, fourth

and sixth orders. This group consists of 25

diagrams. They are represented by 10

integrals.

Group II. Fourth-order vertex diagrams containing vacuum polarization loops of second and

fourth orders. This group contains 54 di- agrams and is represented by 8 integrals.

Group III. Sixth-order vertex diagrams containing a vacuum polarization loop of second order.

There are 150 diagrams in this group. The

number of independent integrals is 8.

Combining Eqs. (13) and (14) we obtain

Ci = -0.8 (2.5). (15)

The central value may still fluctuate considerably. The

main significance of this result is that we now have finite bounds for C4, although they may be rather soft. Anyway, it appears that the "prediction," Eq. (12), is not borne out by our calculations. From Eqs. (4), (5), (7), (8), (15) and Ref. [7] we obtain

, th _= 1 159 652 460 (148) x 10" (16)

This is consistent with the measurement, Eq. (2), as well

as the new measurement of for the positron [12]. Possible causes of the remaining discrepancy are underestimate of errors in the measurement of and /or the

measurement of a in Eq. (5) as well as theoretical errors. The latter includes not only purely computational errors

in C3 and C4 but also physical assumptions we have implicitly made on the nature of the weak interaction and

the (lack of) internal structure of the electron. An important by-product of our calculation is that it enables us to determine a in a way least dependent on

theoretical ambiguities. We give below the value of a

determined from the measurement, Eq. (2), and one quoted in Ref. [12] and the new calculations of Ref. [7] and this report. For comparison we also list a determined from the muonium hfs [13] and the ac Josephson

effect [4]:

204
a-^ae) = 137.035 993 (10), a-i (muonium hfs) - 137.035 989 (47), a-i (ac Josephson) = 137.035 963 (15). (17)

The error in a"^ (muonium hfs) is mostly theoretical and

requires a considerable improvement of theory to take

advantage of the very accurate measurements. Such

work is in progress at present [14]. Since it is a pure

QED process, comparison of a"^(ae) and (muonium

hfs) is useful for checking the internal consistency of

QED. On the other hand, a"^ (ac Josephson) is based on

a purely solid-state physics phenomenon and is presumably not susceptible to higher order QED corrections

[15].

3. Conclusion

As is seen from Eq. (17) the agreement of a determined by various means is satisfactory at present. But,

irrespective of whether further work establishes the

disagreement of different as or not, it is about time to re-examine the basis of some of the underlying theories

very closely. For instance, is the ac Josephson effect really capable of determining 2e/h to an accuracy better

than 10"*? What is the upper limit of the theoretical er- ror and how does one estimate it? Although the new and exciting way of determining a

discovered by von Klitzing et al. [16] is not yet comparable in accuracy to that of the ac Josephson effect, a

similar question about its theoretical foundation must be

asked in order to make it a viable method for the high

precision determination of a. Finally, I call attention to the very far-fetched, but not completely crazy, possibility that the discrepancy of

aitte) and other a's arises from the electron not being

elementary but having in fact some internal structure which, for instance, may be described as a bound state of

subquarks. The present agreement (or disagreement) of

theory and experiment of ttg suggests that the mass of

such subquarks must be at least of the order of 10^ pro- ton masses [17]. Should this turn out to be the case (which I must say is very unlikely), low energy particle

physics and even soHd-state physics have roles to play in determining the laws of Nature in the domain of ultra- high energy physics.

We should like to thank Brookhaven National Laboratory, the National Laboratory for High Energy Physics,

Japan, and CERN for their generous support of our work.

References

[1] R. S. Van Dyck, Jr., P. B. SchvWnberg, and H. G. Dehmelt,

Phys. Rev. Lett. 38, 310 (1977).

[2] R. S. Van Dyck, Jr., P. B. Schwinberg, and H. G. Dehmelt,

Bull. Am. Phys. Soc. 24, 758 (1979).

[3] C3 is calculated from the results given by M. J. Levine and

R. Roskies, Phys. Rev. D 14, 2191 (1976); T. Kinoshita and W. B. Lindquist, Cornell Preprint CLNS-374 (1977), unpublished, and papers quoted in Refs. [6] and [8]. [4] E. R. Williams and P. T. Olsen, Phys. Rev. Lett. 42, 1575

(1979).

[5] Note that the Weinberg-Salam model has not yet been tested beyond the tree graph level. For discussions of other possibilities, see T. Kinoshita, in New Frontiers in High Energy Physics—Studies in the Natural Sciences (Orbis

Scientiae), A. Perlmutter and L. F. Scott, Eds. (Plenum

Press, New York, 1978), p. 127.

[6] T. Engelmann and M. J. Levine, private communication.

[7] M. J. Levine and R. Roskies, these proceedings. They have

succeeded in reducing the error of 10 of 15 integrals by

several orders of magnitude. They report the new value of C 3 to be 1.1765 (13), where the error is predominantly due

to the 5 remaining integrals.

[8] P. Cvitanovic and T. Kinoshita, Phys. Rev. D 10, 4007

(1974).

[9] T. Kinoshita and W. B. Lindquist, Cornell Preprints CLNS424 and 426 (1979), unpublished.

[10] T. Kinoshita and W. B. Lindquist, Cornell Preprint CLNS508 (1981), unpublished.

[11] T. Kinoshita and W. B. Lindquist, Cornell Preprints CLNS509 and 510 (1981), unpublished.

[12] P. B. Schwinberg, R. S. Van Dyck, Jr., and H. G. Dehmelt,

these proceedings have succeeded in measuring a^. for a

positron. Their result is a.(e+) = 1.159 652 222 (50) x

10-12.

[13] D. E. Casperson et al., Phys. Rev. Lett. 38, 956, and 1504

(1977); P. 0. Egan et al., 8th ICOHEPANS, Vancouver, Canada,

1979, abstract IBIO; V. W. Hughes, these proceedings;

W. E. Caswell and G. P. Lepage, Phys. Rev. A 18, 810

(1977) ; W. E. Caswell and G. P. Lepage, Phys. Rev. Lett. 41, 1092

(1978); G. T. Bodwin and D. R. Yennie, Phys. Rep. 43, 267 (1978); G. T. Bodwin, D. R. Yennie, and M. A. Gregorio, Phys. Rev. Lett. 41, 1088 (1978).

[14] D. R. Yennie and G. P. Lepage, these proceedings.

[15] D. N. Langenberg and J. R. Schrieffer, Phys. Rev. B 3, 1776

(1971); J. B. Hartle, D. J. Scalapino, and R. L. Sugar,

Phys. Rev. B 3, 1778 (1971).

[16] K. v. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett.

45, 494 (1980).

[17] S. Brodsky and S. Drell, Phys. Rev. D 22, 2236 (1980).

! 205

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Experimental Determinations of the Anomalous

Magnetic Moments of the Free Leptons*

Ralph Conti, David Newman, Arthur Rich, and Eric Sweetman

Physics Department, The University of IVIichigan, Ann Arbor, Ml 48109

The ten year period since the First International Conference on Precision Measurements and Fundamental Constants has seen an improvement of approximately one hundred fold in the experimental

determination of the (e", e+ ) and (|jl", ijl''" ) anomalous magnetic moments! Similar progress has been

made in theoretical QED g-2 calculations as well as in the experimental determination of a. The value of the electron gi-factor anomaly (a (e" )) is now knovra to 34 parts per billion (ppb) from measurements on electrons of typical energy 10~'^ eV and may be compared with theory to 146 ppb.

Theory and experiment are in agreement. The value of a (e^ ) has been compared to a (e" ) to 10'' ppb in an experiment done at 0.6 GeV and recent unpublished work on millielectron volt positrons has allowed

a direct determination of a (e"*" ) to better than 100 ppb. All results show that a (e ) = a (e" ) thus con- firming CPT invariance. The values of a (|jl" ) and a (fx* ) are each knovra separately to 10^ ppb

(a (|x~ ) = a (ix"*" ) to this accuracy) and may be compared with the theoretical value (QED plus hadronic

contributions) which is known to about 8 x 10^ ppb. Again theory and experiment are in agreement. In

addition to providing the most direct and precise tests of QED, the g-2 experiments can also be used to

test special relativity, set limits on possible fermion substructure and on electric dipole moments, and

test other aspects of current theory.

The substance of this article will be a review of the experiments mentioned above, a discussion of new work now in progress, and an attempt to predict future possibilities in measuring lepton anomalous moments. The non-QED tests mentioned will also be discussed. Key words: anomalous magnetic moment of leptons; lepton gr-factors; quantum electrodynamics.

1. Introduction

Measurements of the anomalous magnetic moments of leptons have advanced remarkably in the ten years since

PMFC-I. The uncertainty in the electron anomaly has

been reduced by a factor of 180, that of the muon by a

factor of 40, and that of the positron by a factor of 10'*. These measurements test quantum electrodynamics

(QED), are sensitive to the effect of hadronic couplings,

and allovi^ limits to be placed on speculative interactions

involving axion couplings or possible lepton substructure.

Measurements of the lepton anomalies at differing velocities allows precise limits to be placed on the validity of

special relativity and in addition, the muon experiments

provide tests of special relativity involving the tvidn paradox and a direct test of time dilation and set new

limits on the electric dipole moment (EDM) of the muon. Many of these developments (through 1978) have been revievi^ed in the comprehensive article by Field, Picasso

and Combley [1], v^^hile a detailed treatment of earlier

developments (through 1972) may be found in Ref. [2]. The vitality of research on the lepton magnetic moments

is demonstrated by the fact that much of the work to be

discussed in this article is sufficiently new so that it would not have been included in the 1978 Field et al. article. Finally we note that space limitations have re- stricted us to a review which is more a listing and re- ferencing of new developments and their potentialities

than an in-depth discussion of the experiments.

The g^-factor of a particle is defined through the rela- tion of its magnetic moment jx to its spin S, i.e.,

*The g-2 research at Michigan is supported by a grant from the Atomic,

Molecular and Plasma Physics Program of the NSF.

= R.2 moc

S . (1)

Deviations from the Dirac prediction of gr =2, due to radiative corrections and other effects, are called the

'

' anomaly' '

a ^ig -2)/2 . (2) Since measurements of a are always made on leptons

moving nearly perpendicular to a magnetic field, we now

briefly discuss the well-known spin and orbital motion for a particle moving perpendicular to a uniform magnetic

field. In the simple case, the relativistic energy levels for a single lepton are given by:

Win, m) = moc2[(l + {\ + 2n + 2m )x)^ + max] (3) where n is the orbital quantum number (0, 1,2,.. .), m is the spin quantum number (m = ± 1/2), vq - /2'jTmoC

and X = hvQ/moC^ « 1. The level diagram is shown in Fig. 1. The cyclotron and spin precession frequencies vc and vg are determined by the energy difference between successive n and m quantum levels: hvc - W in, m.) - W in - 1, m) (4) hvs ^ Win, 1/2) = Win, -1/2) . (5) For X « 1 Eqs. (4) and (5) yield:

vc = vo/y = fvo + (l + a

Here 7 is the usual relativistic factor 7 = (1 which for x « 1 may be written as [1 -i- (2?; 1)X

(6)

(7)

-1/2 112' Thus the quantum mechanical values for vc and vs are the

207
m = + 1/2 m = - 1/2

Figure 1. Rabi-Layidau levels of an electron in a uniform magnetic field.

same as the classical values (independent of n). The term

(1 - y)vc is the Thomas precession. Both vc and vs are dependent on the velocity of the particle. However, the

difference frequency vq = vs - vc, is independent of 7: VD = avo . (8) This feature allows a precise measurement of the anomaly in experiments where uncertainty in the particle en- ergy would otherwise limit the accuracy. Classically, the

difference frequency is the frequency at which the spin of the lepton precesses relative to its velocity in a magnetic

field. It corresponds quantum mechanically to the energy

difference between quantum levels with a change of both

m and n, or spin-cyclotron transitions: hvD = Win, -1/2) - Win - 1, +1/2) (9)

All of the precision measurements of lepton magnetic moments consist of direct measurements of vq and indirect

evaluations of vq- The value of vq is -obtained either from

Eq. (6) and a measurement of vc^ or from a proton nuclear magnetic resonance measurement as described in sections 2.2 and 3.1.

2. The Electron and Positron g-2

Experiments

2.1 The University of Washington Resonance

Experiment

The most accurate measurement of the electron anomaly a(e) to date is that done by the group at the University of Washington. Using ~10"'^ eV electrons, they

obtain vo by driving, spin-cyclotron transitions using

externally applied rf electric fields. The cyclotron fre- quency is then determined by measuring resonant ab- sorption for Am = 0, Atc = -I-1 transitions at small

values of n. Historically, a resonance technique employed

at Washington was the first method (1958) to directly

yield a result for the anomaly of a free particle.

The model of a uniform magnetic field with no electric

field is not practical for a precision measurement since

the electron must be confined to a small region for an ex- tended period to obtain narrow linewidths. Confinement

is achieved in a Penning trap (Fig. 2) bv an electric quadrupole potential V = iVo/b^) ir^ - 2z^) superposed on a uniform magnetic field Bqz, where, in one version of the

experiment whose parameters we refer to in this section,

Vo =9.1 volts, b - 0.47 cm, and Bq = 18.3 kG. In addition to its cyclotron motion, the electron now executes

axial oscillations along the direction of B at frequency

V, = il/Ti)iy/Voe/mob^) - 59 MHz (10)

and exhibits magnetron motion (a drift of the cyclotron

orbit center about the center of cylindrical symmetry) at frequency

^ 2v|

2

1 - 34 kHz . (11)

Superconducting coil.

51 GHz

cyclotron

drive (2/c')

Axial resonance

signal output

i—'''no't >

"

Magnetic bottle

(in ring i

9.2 V DC =p

(Standard T

cell s

TAxIoI drive input "i ( ^^z - ^s - '

Figure 2. Schematic of the University of Washington ap- paratus used to measure the electron anomaly. This apparatus

allows the measureynent of the cyclotron frequency and the spin-cyclogron beat (g-2) frequency vd = vg — vq on a single electron stored in a Penning trap at ~ 1^ K.

The axial oscillation is driven by an rf signal applied to one of the end caps of the Penning trap, and the motion

of the trapped electrons is observed by detecting the current induced in the other end cap. The two signals are compared in a phase sensitive detector and the difference or error signal is integrated, lowpass filtered, and fed back to Vd as a dc correction voltage, so that the axial

oscillation frequency is locked to the drive frequency.

This correction voltage is proportional to any shift 5 in the natural axial frequency. The amplitude of the

detected signal is used to indicate the number of elec- trons in the trap. The g-2 measurements were made with

only a single electron in the trap at a time, thus eliminating space charge electric fields. In a perfect Penning trap the quadrupole electric field shifts the frequency vc by an amount equal to the magnetron frequency Vm, i.e.,

v'c = vc - Vm- (12) In the non-relativistic limit, vs is unshifted by the electric field, yielding

vb Vs - Vc = Vd + Vm (13) as shown in Fig. 3. A useful expression relating Vm to the

208
Bn = 18.3kG

,_i

hi/j,' = 11(1/3 -y^+yj

n = 0

60MHz vn- 60MHz v„=^35kHz

Figure 3. Energy levels of an electron in a Penning trap.

more easily measured quantities and vc may be obtained from Eqs. (11) and (12) as 2v^v'c = vf . (14)

The shift in [Eq. (13)] is typically several hundred

ppm, four orders of magnitude larger than the final quoted error, and thus requires accurate measurement.

The cyclotron and difference frequencies are measured

by means of their coupling to the axial motion, a coupling

provided by the introduction of a small magnetic inhomogeneity of the form

B, — B() + B'

r

2

(15)

with Bz ~ (120-300) G/cm^. A nickel wire wound around

the central electrode produces this field configuration.

The inhomogeneity causes a shift in the axial frequency

which is dependent on the spin, cyclotron, and magnetron

quantum numbers (m, n, and q respectively):

hvz = [m + n + 1/2 + (vm/vc)<7]8 , (16)

with 8 = [LB^ / 2T^m — 1 Hz. The cyclotron transition (An = 1, Am = 0) is suffi- ciently strong so that changes in n, and thus in 8 , are

easily observed due to random thermal excitation even at the 4.2 K ambient temperature of this experiment. Similarly, the cyclotron frequency is measured by observing even larger changes in n and therefore shifts in when

an external signal at vc is appHed (Fig. 2). Spin-cyclotron transitions {An = -1, Am = -l-l) are

driven by applying an rf voltage to one end cap at fre- quency Vrf. The electron moves axially at the driving fre- quency Prf while moving in cyclotron orbits at vc, so that

it sees the radial magnetic field component associated

with the axial inhomogeneity oscillating at the frequency

y = vq ± Vrf. Since vg = vc + vq, the spin flip transition

is driven only when = vq. The resulting change in m(Am = -1-1) does not produce an immediate change in

8vj because of the simultaneous An = -1 cyclotron

transition. The spin flip is observable, however, as a

shift in the baseline of the cyclotron thermal noise output, since from Eq. (16), 8v^ increases by 8 during a spin

flip. To achieve narrow linewidths, the axial motion is driven alternately at Vrf = vp during the "transition"

phase and then at during the "detection" phase.

The measured frequencies vb, v'c and [from which

can be calculated using Eq. (14)] are then related to the

anomaly by

a(e) - — = (17)

Shifts in a{e) are caused by the non-uniform magnetic

field, relativistic effects, misalignment of the electric and

magnetic field axes, etc. These and other systematic af- fects have been considered by the Washington group but

details have not yet been published. Their final result ob- tained for data runs at 18.6, 32.0, and 51.1 kG is [3]: a(e-) = 0.001 159 652 200(40) . The precision quoted is noted to be due primarily to sys- tematic error and not linewidth as was the case in the

previous (200 ppb) result. This resonance technique can be applied equally well to a positron, whose magnetic moment is expected to be exactly equal to that of the electron [g (e^ ) = g ie' )] if CPT is conserved. The principal new difficulty is the in- jection of positrons from an energetic radioactive source

into a shallow electrostatic well. The Washington group

has overcome this problem by the use of a separate trap

in which the positrons are collected and cooled. Positrons

are later transferred to the more precisely controlled

measurement trap. The result a(e+) = 0.001 159 652 222(50)

has been obtained [4]. Since a(e"^) agrees with a(e")

vdthin the quoted error, the CPT theorem has been verified at the level of 0.05 ppb.

Improvements in the resonance technique, including

the possible elimination of the magnetic bottle, are ex- pected to yield even higher accuracy for both the elec- tron and positron results in the future. Discussion of the new positron experiment and of the uniform field tech- nique appear as contributed papers in these proceedings.

2.2 The Michigan Precession Experiments

The free precession technique is an alternative method

for measuring the anomaly and was in fact the first method to yield the magnetic moment of a free particle

(1953). In this technique, polarized electrons or positrons are trapped in a magnetic or electric well in a magnetic

field (Fig. 4) and, to a first approximation, the spin

precesses relative to the velocity at vd = vg - vc = a vq, as discussed previously. The spin direction is observed

after a known trapping time in order to determine v^. The frequency vq is then determined by measuring the

proton NMR frequency in water vp for the magnetic field in which the particles are trapped. This frequency has

been compared to the electron spin-flip frequency in a hydrogen maser by Phillips et al. [5] who found:

vo = vi>/(1.520993134(21) x 10"^) . A detailed discussion of the precession experiments, in- cluding the various corrections to the formula vd = a vq may be found in Refs. [1] and [2].

209
Polarized Source

Figure 4. Schematic of the proposed University of Michigan

precession experimeyit to measure the electron anomaly. Polarized electrons produced by photoemission are accelerated along a helical path into the central field of a superconducting

solenoid. An electrostatic pulse traps the electrons in a shallow magnetic well. After a predetermined time, the electrons are

ejected and the variation of the longitudinal polarization with trapping time is measured with a Wien filterlMott detector combination.

A variation of this precession technique has been pro- posed by the Michigan group [6]. Electrons with polarization close to 100% are photoemitted from a cesium vapor

(Fano effect) or from a magnetized nickel crystal and will be injected at 120 keV into an extremely shallow (20 to 200 ppb) magnetic trap in the 20 kG magnetic field of a superconducting solenoid (Fig. 4). After precessing for 10

to 20 msec (10'' cycles of vq), the electrons are ejected

and spiral out of the field. The trajectories of these particles have been computed using numerical integration.

We note that the results are in agreement to within 1%

with those obtained analytically on the assumption that

the orbital magnetic moment ixqrb ~ Pi^/B (Pi is the

momentum perpendicular to the magnetic field direction)

is an adiabatic invariant. These studies show that when

the particles reach the position of the Wien filter, their

momentum direction is within 6 degrees of the axis of the

solenoid.

Since, if g = 2, the projection of spin onto momentum

remains constant in a magnetic field, the longitudinal polarization of the electrons at the Wien filter is modulated

at vp. This modulation is independent of the position of

the particle in its cyclotron orbit. The effect of the anomaly {g i= 2) is to cause a calculated 10% depolarization of the ejected beam as it spirals out of the magnetic field. The electrons then pass through the Wien filter which ro- tates the longitudinal component of spin by 90°, and their

polarization (now modulated in the transverse direction)

is measured by Mott scattering.

It is planned to trap approximately 100 electrons per

cycle. With this number of electrons in the trap, vd is shifted by at most 3 ppb due to the space-charge electric

fields. With these parameters and a repetition rate of

30 Hz, a statistical uncertainty of 10 ppb in vd may be ob- tained in only 30 minutes. The overall accuracy of the ex- periment will be limited to about 20 ppb primarily by

systematic effects. The major anticipated systematic uncertainties are due to the current uncertainty in vp/vo

(11 ppb), corrections for the magnetic well (10 ppb), and

radial electric fields in the trapping region (5 ppb).

Major progress toward the realization of this experiment has already taken place. A ten kilogauss 60 cm

diameter by 170 cm long superconducting solenoid,

corrected to sixth order, has been constructed and used

to contain positrons of 650 keV energy in a 60 ppm magnetic mirror trap for times up to 1.2 milliseconds. Using

this solenoid, numerous tests relevant to the proposed

electron experiment were completed. We conclude from

these tests that the system necessary for a 20 ppb measurement of a(e~) can be constructed. In addition, an at- tempt was made to measure a(e^) using the previously

described [6] combination of the resonance and precession techniques. No resonance was seen but the statistics were quite marginal and the time for a thorough search

for the resonance, possibly with a new and stronger

source, has not been taken because of emphasis on the

electron work.

The importance of performing an electron precession

experiment to this accuracy discussed above lies both in

its value as a systematic check on the resonance experiment described previously and because, as a consequence

of the relativistic electron velocity in this experiment, it constitutes a novel test of special relativity (Section 4.2). In addition it would provide the most accurate direct

limit on a possible electron electric dipole moment (3 x

10"^^ e-cm). This is not as precise as limits obtained from

measurements on neutral atoms (10"^ e-cm) but has the

virtue of being independent of the theoretical connection

between an atomic EDM and the corresponding electron

EDM.

2.3 The Novosibirsk Electron-Positron Comparison

Experiment

As stated previously, the CPT theorem has been verified at the 5 x 10"'^ level by separate measurements of the magnetic moments made on 10 "'^ eV electrons and positrons using the resonance technique. A much less precise but still quite interesting experiment has been per- formed by Serednyakov et at. [7], who compared but did not directly measure a(e^) and a(e") for 625 MeV particles in the VEPP-2M storage ring at Novosibirsk. In the

storage ring, the electron and positron beams became polarized parallel to the ring axis, with a time constant of

order one hour, due to the emission of spin-flip synchrotron radiation. The beams are then depolarized by an oscillating longitudinal magnetic field in one section of the

ring (Fig. 5). The appUed frequency v,. is chosen so that

the relative phase between the particle spins and the

depolarizing field remains unchanged after each orbit. A

frequency which satisfies this condition is v,. =vs - 2vc. The change in polarization is then detected as a change in the spin-dependent (e"^e"^) and (e"e") scattering which

occurs within the beam bunches. Both the electrons and

the positrons were seen to depolarize at the same applied

frequency, from which the result is derived:

a(e"^ ) - g (e ) a(e)

1.0 X 10 -5 (95% confidence)

0 Q DEP

Figure 5. The apparatus at Novosibirsk used for the simul- taneous measuremeyit of e , e' depolarization in the VEPP2M storage nng. B: bending magnets; Q: quadrupole lenses; DEP: depolarizer; Cj ~ C^: scintillation counters in It-fold coincidence.

210
2.4 The SLAC High Energy Experiment

A measurement of the electron anomaly at SLAC is of particular interest due to its high energy, 12 GeV (-y -

2.5 X 10^), and its resulting sensitivity as a test of special relativity [8]. This measurement is a by-product of the determination of the polarization of the SLAC longitudinally polarized electron beam [9]. As a polarized electron beam is deflected through a

24.5° bend (9c) in the SLAC switchyard magnetic field,

its precession is observed. The spin rotates by an angle

Od = 0, "y 0c (recall vd = a 7 vc) relative to the momentum,

in this magnet. The longitudinal polarization of the beam

after deflection is a sinusoidal function of the beam

energy E = -inioc^, the magnitude of the initial polarization Po, and the angle between the initial beam momentum and polarization 4)o , i.e., PiE) ^ Pocos(eD + cj)o) - PoCOsi-nE/Eo + c{)o) . (18)

Here the parameter E 0, defined as TTmoc^/a0c, is about

3.2 GeV. Figure 6 shows the longitudinal polarization

measured by Moller scattering at values of 7 between

1.27 X 10^ and 3.81 x 10\ These data points (three ex- trema and one zero-cross measurement) were fit to the

above formula with £'0 (i-e., ct) and Pq as free parameters

and with ^0 assumed to be zero. The result a (e" ) =

0.001 162 2(200), derived from the best fit value of £"0, is in good agreement with the more precise low energy

measurements. A discussion of the g-2 experiments and

this experiment in particular as a test of special relativity may be found in Section 4.2.

/7E/E 0 Figure 6. The results of the SLAC high-energy electron g-2 measurement. The longitudinal component P of the electron beam -polarization is plotted as a function of -aE/E,,, the angle

through which the spin precesses relative to the momeyitmn

during the 2^.5° magnetic bend in the experimental area. The

curve shown is the best fit to the data with a(e) and Po as free parameters.

3. The CERN Muon g-2 Experiment

3.1 The g-2 Determination

The CERN group has performed a series of muon g-2

measurements resulting most recently in the separate

determinations of a {[l^ ) and a {\x~ ) to 10 ppm. This work

tests the theoretical QED prediction of the muon anomaly including an estimate of the eighth-order contribution, checks hadronic contributions calculated to be

67 ppm, sets new limits on the muon EDM, and tests special relativity with regard to Thomas precession and the

twin paradox. In the CERN experiment, pions are

injected into the 7-m storage ring (Fig. 7) with a momen- tum slightly greater than the 3.098 GeV/c design momen- tum of the ring. About 10% of the pions decay before

exiting the storage volume. Muons produced in the top

1.5% of their allowed momentum range are then trapped.

Titanium Electrodes / Figure 7. Plan view of the second CERN m uon storage ring

with cross-sections of the magnets and focusing electrodes. (Dimensions m 7nm.)

They are longitudinally polarized, due to the parity non- conservation in the pion decay, and at the forward angles, where efficient muon trapping occurs, a polarization

of better than 90% is obtained. The muon spins precess around the magnetic field direction until the muons decay

in the ring. The resulting electrons are detected in energy-sensitive shower counters and an energy cut

selects those electrons of higher energy emitted in the

forward direction vdth respect to the muon momentum.

Since the electrons are preferentially emitted along the muon spin direction, the counting rate of high energy

electrons will be modulated at the g-2 frequency as the muon spins precess with respect to their momentum at

this frequency.

High precision was obtained, in part, by the use of a uniform magnetic field in conjunction with an electric

field for vertical focusing. The electric field constitutes an electric quadrupole trap similar to that described for the resonance experiments. As in the electron experiment, the electric field E introduces a shift in cod (the an- gular velocity of the spin vector relative to the velocity)

given by

0)'d = tOD +

moc f - 1 O X E) (19)

where p = v/c (vd = |a>D/2-rT |). The CERN group eliminates the effect of E on cdd by setting the energj^ to 3.098 GeV so that 7 = (1 + 1/a )^ = 29.3, and the electric field dependent term in Eq. (19) vanishes.

The decay data in Fig. 8 are fitted for vd and compared

to the NMR frequency vp for the magnetic field of the

storage ring. After averaging over muon orbit distribu211
10'

sec a.

CM

rOd-

16-59/xsec 59-102 -

102-146 ;

events 10

146-189 189-232 232-275 ; "o

'.„,'•"•'..,"• 275-318 - 318-362

Numbi 10' -

"', ' ''''''1 ' ll'l,'

"

I'll l„

' "ll„l""'"'l„,l ''l^ 362-405 - 405-448

10^ 1

"" lit' 'lllll' ' ' lp^ll

1 1 1

/''ll

' /ill, ' ll'

lllll ,l/

1

1

' I'lr ''i ''ii'' 'ii|''

l|

1 "1 'I l|l

II 'l| III' ll'l

1 1 r 'l

448-491 ; 491-534

10 20 30 40 50 60

Time m microseconds

Figure 8. Decay electron tirne distribution from the CERN

muon g-2 experiment. The distribution contains 14 x lO^

events.

tions and correcting for NMR probe shielding, calibration, and other effects, the result obtained is R = VD(fJL)/vi. = 3.707213(27) x 10" (20)

The ratio of the magnetic moment of the muon to that of the proton (\) was found by averaging the results of Crowe et al. [10] and Casperson et al. [11] for the ratio of the muon Larmor frequency to the proton Larmor fre- quency (or muon moment to proton moment):

X = VL(fJL)/vi> = 3.1833417(39), (21) so that using the relations vl(|jl) = (1 + a (|jl))vo(|j^') and

vd(|Jl) = avoi[x,) we find a(|x) = R/(\ - R). The final result for and data together yields [12]

aiy.) = 0.001 165 924(8) (22)

and for the individual particles

a((x+) = 0.001 165 912(11)

a(|jL-) = 0.001 165 938(12) .

These values are all in agreement with the theoretical

prediction of

a(p,-theory) = 0.001 165 920(10) . The agreement between a(|jL^) and a(|x ) constitutes a

verification of CPT at the 10"^ level.

3.2 Limits on the IVIuon Electric Dipole Moment

The muon g-2 experiments also provide the best limits to date on the electric dipole moment (EDM) of the muon. (See Ref. [1] for a complete discussion of lepton

EDM's and the theories in which they arise.)

A non-zero muon EDM would cause the muon spin to precess about both the electric and magnetic field direc- tions with a resultant motion determined by the E and B

field strengths and by the electric and magnetic moments. One result of this motion would be that the muon

spin would oscillate above and below the plane of the

storage ring with a frequency of vq and 90° out of phase

with the g-2 motion, thus causing the emission of decay

electrons or positrons in a direction along B to be modulated at vd- The data from scintillation counters placed

above and below the median plane were analyzed for this

effect, from which the limit [1] for the EDM (combining

Ijl"^ and data)

-(3.7 was derived.

A comparable limit of

3.4) X 10-19 e-cm

D, 8 X 10-19 g.p^ (95^^ confidence level) can be independently obtained by ascribing any disagreement between the predicted and observed vd to the additional precession of the muon EDM caused by the electric

field, i.e..

VD = Vd 1 +

J4a 2

vdth defined by the relation

/m- e1i 2 2m,, i D,

4. Theoretical Implications of the g-2

Experiments

4.1 QED and Other Contributions to the IVIuon and

Electron Anomalies

The anomalous moment of the electron or muon is predicted as a power series in a, the fine structure constant, plus contributions from other known or speculative

effects:

a = XAnia/i;)" + H (hadron loops)

-I- W (weak interaction effects) -i- X (axions) (23)

where the An represent only QED contributions including muon and tau loops. The coefficients in Eq. (23) are given

in Table 1 with a = e^/tic = [137.035963(15)]-^ [13] used

to evaluate a (theoretical).

Table 1. Theoretic contributions to the electron and muon

g-factor anomalies.

Term^ Electrons Muons References

0.5 0.5 Schwinger [30] A2 -0.328478444 +0.7658582(7)'' Kinoshita [14] Calmet [24] Az 1.1765(13) 24.45(5) Levine [18] Calmet [24] A4 -2.3(3.6) 128(70) Kinoshita [19] Calmet [24] H 1.6(2) X 10-12 66.7(9.4) X 10-9 Calmet [24] W 0.05 X 10-12 2.1(2) X 10-9 Kinoshita [14] Calmet [24] X 0.1 X 10-12 3.7 X 10-9 Leveille [20] Kane [21] 0.001 159 652 416(165) 0.001 165 920(10) Kinoshita [18] Field [1]

^See text for an explanation of symbols used and for discussion. ''Error due to uncertainty in t mass.

212
In calculating the electron anomaly, the uncertainty comes primarily from the 109 ppb uncertainty in the

measurement of a. The non-QED-dependent determination of a is limited by the uncertainty in 7p, the

gyromagnetic ratio of protons in water [13]. Experiments currently under way are expected to improve this measurement by an order of magnitude in the next few

years [15]. The value of a can also be determined from

measurements of the hyperfme structure of muonium

(QED-dependent) which are currently in progress [16]. A

new method for obtaining a non-QED-dependent value

for a from the quantized Hall resistance of a two-dimensional electron gas in a silicon metal-oxide-semiconductor

has recently been suggested [17] and may be capable of producing precise results in the coming decade. A new

result for A 3(e), reported at this conference by Levine

and Roskies [18] A 3(e) = 1.1765(13), reduces the previ- ous uncertainty from this source, 66 ppb, to only 15 ppb.

Evaluation of A 4(e), a monumental task involving a calculation of 891 Feynman diagrams, is in progress and is ex- pected on completion to reduce the uncertainty from this term in a (e) to the order of 1 to 10 ppb. A preliminary

result [19] reported in these proceedings is A 4(e) =

-2.3(3.6). Thus, with these advances in both the QED

calculations and reduction of the error in the value of a,

it is expected that the total uncertainty in the predicted

value of a (e) will be reduced below the level of (a /tt)^ —

30 X 10~^^ in the near future. Contributions in Eq. (23) to a (e) from muon and tau

loops are known accurately. The contributions from the

standard gauge model of the weak interaction (0.05 ppb)

and for hadronic effects (1.6 ppb) are both below the

resolution of electron g-2 experiments for the foreseeable

future. One can also consider the effect on a(e) or a (|jl) of more speculative interactions. For example, the effect of

a light scalar or pseudoscalar neutral particle (axion) on

the anomaly depends on the mass of the axion m.^ and on

the coupling assumed. One standard model (Leveille [20],

Kane [21]) predicts in the limit 0 that a (e) will be

shifted by 0.1 x lO'^^ and a(fjL) by 3.7 x 10"^

The prediction of the muon anomaly a((x) includes siz- able contributions from the A4(|x) term [22] (3.7 ±

2.1 ppm) as well as from hadronic effects [23, 24] (66.7 ±

9.4 ppm) and weak interaction effects [24] (2.1 ±

0.2 ppm). Thus a comparison of the most precise experimental value of a (|j.), derived from the weighted average

of the measurements for and allows a check of the

hadronic contribution to a lepton magnetic moment for the first time. The results are in good agreement both

with QED and with CPT as summarized below:

ajji, theory) - a(|jL, experimental) ^ |._g _^ ai[x, theory)

——a{u,^) — a(ix") ^ , 00 ^ , = (-22 ± 14) ppm . ^

[a(jx+) + a(|jL-)J

If a next generation of muon experiments is undertaken

with error in a (|x) at the 1 ppm level it should be sensitive to the A 4 and weak interaction terms.

If leptons are not point-like, their substructure should

affect a (e) and a (|x) at some level. Using dimensional ar- guments, Brodsky and Drell [25] have shown that one

could expect contributions to a(l) linear in mi/m* (8a(0 ~ mi/m*), where m* is the mass of the

constituent and I refers to e or |x. Present g-2. measure- ments rule out electron and muon constituents with mass

less than -lO^GeV/c^ and ~2xlO^GeV/c2 respectively, corresponding to an effective size no greater than

R ~ 10"^" cm for either particle. This limit on lepton size

is four orders of magnitude smaller than the best current

limits obtained from electron-positron collisions at a 32 GeV center of mass energy. Other models which give

poorer limits on R are discussed in Ref. [25].

4.2 Tests of Special Relativity

The data obtained in the muon g-2 experiments may

also be used to determine the lifetime t of muons at the

accurately known energy (7 = 29.327) with which the muons circulate in the storage ring. This is analogous to a test of the famous "twin paradox" and is a direct test of time dilation. In agreement with special relativity, the

CERN group finds [26]:

'^'^ ~

= (-9 ± 4) X 10-^ . To where tq is obtained from recent measurements of the

rest lifetime accurate to 40 ppm [27]. More recently, new limits of very high precision have

been placed on relativity violations by lepton g-2 meas- urements [8, 28, 29]. As shovra in Eqs. (6) and (7), the

spin and cyclotron frequencies depend on y. As discussed

in Ref. [28], if the y from Thomas precession appearing

in vs is not identical with the 7 arising from electron

dynamics (referred to as 7) in vc, then the difference fre- quency vd would no longer be independent of energy: — = a + (1 - 7/7) • (24)

Comparison of vd/vq measurements at two different velocities then constitutes a test of special relativity which

is independent of QED and which may be parameterized

by the quantity A: A = |[vd/vo]i - [vd/vo]2 I = Ih/yh - [y/y]i\- (25)

The results of the most recent Michigan measurement of the electron anomaly at p = 0.57, when compared with

the low energy (p — 10~^) Washington measurement,

yields A = (5.3 ± 3.5 ) x lO'l The current CERN muon

g-2 measurements at 7 = 29 can be similarly compared to the 1968 CERN result at 7 = 12 to yield A = (236 ±

310) X 10^^. Finally, the agreement between the SLAC

g-2 measurement with 12 GeV electrons and low-energy

results constitutes an ultra-relativistic (7 = 2.5 x 10 ) verification of relativity with A = (2600 ± 20,000) x 10"^

An alternative approach to the problem of characterizing the relativity breakdown is presented in Ref. [29]. In both Refs. [8] and [29] the relativity violation is assumed

to take a specific form, in which the parameter (1 - 7/7)

or its equivalent is expressed as a power series in (7 - 1). The coefficient of the second term in this series

then constitutes the figure of merit for the various ex- periments. Limits approaching 2 x 10"* were obtained by

Michigan and CERN, and 8 x lO"^" by SLAC for the

coefficient in Ref. [8] while limits 1/a times larger are ob- tained using the model of Ref. [29]. The major conclusion

to be drawn from the above work is that the g-2 experiments constitute a novel and sensitive test of special

relativity.

5. Conclusions

The theoretical prediction of the electron and muon

magnetic moments and their experimental determinations

continue to constitute the most direct and precise tests of

213
quantum electrodynamics. Indeed, considering only the

electron gr-factor, the comparison of theory and experiment constitutes the most precise test of any extant

physical theory. Although no new muon experiments are currently underway, the electron g-2 research is very active. We may conclude from a reasonable projection of

what has been presented at this conference that the next

several years could see well established comparisons of

theory and experiment for the electron p-factor anomaly

at the 10 ppb level. A direct electron-positron g-2 comparison by the resonance technique might even exceed

this accuracy. Such results allow the study of a wide

range of physical phenomena including theories with exotic interactions, relativity and symmetry tests, as well

as providing a more stringent test of QED.

References

[1] J. H. Field, E. Picasso, and F. Combley, Sov. Phys. Usp.

22(4) 199 (1979).

[2] A. Rich andJ. C. Wesley, Rev. Mod. Phys. 44, 250 (1972).

[3] R. S. Van Dyck, Jr., P. B. Schwinberg, and H. G. Dehmelt,

Bull. Am. Phys. Soc. 24, 758 (1979).

[4] P. B. Schwinberg, R. S. Van Dyck, Jr., and H. G. Dehmelt,

Bull. Am. Phys. Soc. 26, 597 (1981); and these proceedings.

[5] W. D. Phillips, W. E. Cooke, and W. Kleppner, Phys. Rev.

Lett. 35, 1619 (1975).

[6] D. Newman, E. Sweetman, R. Conti, and A. Rich, in Atomic Masses and Fundamental Constants 6, Ed. by J. A.

Nolen, Jr., and W. Benenson (Plenum Press, New York,

1980), p. 183.

[7] S. I. Serednyakov, V. A. Sidorov, A. N. Skrinsky, G. M.

Tumaikin, and lu. M. Shatunov, Phys. Lett 66B, 102

(1977).

[8] P. S. Cooper, M. J. Alguard, R. D. Ehrlich, V. W. Hughes,

K. Kobayakawa, J. S. Ladish, M. S. Lubell, N. Sasao, K.

P. Schuler, P. A. Souder, D. H. Coward, R. H. Miller, C. Y. Prescott, D. J. Sherder, C. K. Sinclair, G. Baum, W.

Raith, and K. Kondo, Phys. Rev. Lett. 42, 1386 (1979).

[9] P. S. Cooper, M. J. Alguard, R. D. Ehrlich, V. W. Hughes,

H. Kobayakawa, J. S. Ladish, M. S. Lubell, N. Sasao, K.

P. Schuler, P. A. Souder, G. Baum, W. Raith, K. Kondo,

D. H. Coward, R. H. Miller, C. Y. Prescott, D. J. Sherden, and C. K. Sinclair, Phys. Rev. Lett. 34, 1589 (1975).

[10] K. M. Crowe, J. F. Hague. J. E. Rothberg, A. Schenck, D. L. Williams, R. W. Williams, and K. K. Young, Phys. Rev. D

5, 2145 (1972).

[11] D. E. Casperson, T. W. Crane, A. B. Denison, P. 0. Egan, V. W. Hughes, F. G. Mariam, H. Orth, H. W. Reist, P. A. Souder, R. D. Stambaugh, P. A. Thompson, and G. zu Puthtz, Phys. Rev. Lett. 38, 956 (1977).

[12] J. Bailey, K. Borer, F. Combley, H. Drumm, C. Eck, F. J. M.

Farley, J. H. Field, W. Flegel, P. M. Hattersley, F. Krienen,

F. Lange, G. Lebee, E. McMillan, G. Petrucci, E. Picasso, 0. Runolfsson, W. von Ruden, R. W. Williams, and S. Wojcicki, Nucl. Phys. B 150, 1 (1979).

[13] E. R. Williams and P. T. Olsen, Phys. Rev. Lett. 42, 1575 (1979).

[14] T. Kinoshita, paper presented at "Fundamental Interactions' Colloquium in honor of Antoine Visconti, Centre de Physique

Theorique, CNRS Marseille-Luminy (1979).

[15] E. R. Williams, private communication (1980).

[16] V. W. Hughes, private communication (1980).

[17] K. V. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980).

[18] M. Levine and R. Roskies, these proceedings.

[19] T. Kinoshita and W. B. Lindquist, these proceedings.

[20] J. P. Leveille, Nucl. Phys. B 137, 63 (1978).

'

[21] G. Kane, private communication (1980).

[22] J. Calmet, S. Narison, M. Perrottet, and E. de Rafael, Rev. Mod. Phys. 49, 21 (1977).

[23] V. Barger, W. F. Long, and M. G. Olsson, Phys. Lett. 60B, 89

(1975).

[24] J. Calmet, S. Narison, M. Perrottet, and E. de Rafael, Phys.

Lett. 61B, 283 (1976).

[25] S. J. Brodsky and S. D. Drell, Phys. Rev. D 22, 2236 (1980).

[26] J. Bailey, K. Borer, F. Combley, H. Drumm, F. J. M. Farley,

J. H. Field, W. Flegel, P. M. Hattersley, F. Krienen, F. Lange, E. Picasso, and W. von Ruden, Nature 268, 301 (1977).

[27] M. P. Balandin, W. M. Crebenyuk, V. G. Zinov, A. D. Konin, and A. N. Ponomarev, Sov. Phys. JETP 40, 811 (1974).

[28] D. Newman, G. W. Ford, A. Rich, and E. Sweetman, Phys. Rev. Lett. 40, 1355 (1978).

[29] F. Combley, F. J. M. Farley, J. H. Field, and E. Picasso, Phys. Rev. Lett. 42, 1383 (1979).

[30] J. S. Schwinger, Phys. Rev. 73, 416 (1948); and ibid. 75, 898

(1949).

214
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Preliminary Comparison of the Positron and Electron Spin Anomalies*

p. B. Schwlnberg, R. 8. Van Dyck, Jr., and H. G. Dehmelt

Department of Physics, University of Washington, Seattle, WA 98195

A new double Penning trap structure has been built using various techniques developed in our geonium experiment. Primary positrons are captured in a storage trap, centered, and then transferred

into a well-compensated experiment trap where preliminary measurements have yielded the single positron gi-factor anomaly a (e ) = (1159652222 ± 50) x 10~^^. This value was obtained from four runs at a

field of 50.8 kG with the a (e^) values extrapolated to zero spin flip power using the power dependence

observed in the electron geonium experiment. The uncertainty is based on the resonance linewidths and an estimate of the remaining systematic errors. When compared to the electron spin anomaly, we

obtain a positron/electron g'-factor ratio of 1 (22 ± 64) x 10~^ . Key words: compensated Penning trap; fundamental constants; matter-antimatter comparison;

positron-geonium spin anomaly; positron trapping; precision measurements; single isolated positron;

synchronous resonant detection.

1. Introduction

Recent success measuring the electron spin anomaly in a well-compensated Penning trap [1] points out the obvi- ous potential of that device for measuring the positron

anomaly to the same precision. Previously, the positron

had been measured by Rich and Gilleland [2] to an accu- racy of 1000 ppra and indirectly by Serednyakov et al. [3] to 10 ppm. In the first experiment, the anomaly is measured in a magnetic mirror machine by directly observing

the difference between the precessing polarization of the

spin of the positron and its cyclotron rotation and then

calibrating the magnetic field using an NMR probe. However, the crucial weakness of that experiment was the

need for an intense polarized positron source and an effi- cient positron polarimeter which did not exist at that

time [4]. The other previous experimental result was ob- tained using intersecting electron /positron beams in a storage ring which allowed the anomaly frequency of each charge to be directly measured. When combined

with the better known electron anomaly [1], the ratio

aie*)/a(e~) yields the unknown positron anomaly.

In comparison, using well-compensated Penning traps, our resonance techniques can attain accuracies of a few

parts in 10^ for the single positron anomaly, with the

magnetic field calibrated directly in situ with the

positron's cyclotron frequency. In addition, the ratio of the positron and electron gr-factors should be measurable

to even greater accuracy since small systematic errors should be identical to first order. Thus, a very sensitive

test of matter /antimatter symmetry should be possible.

Any violation of that symmetry wiW be a violation of CPT

invariance [5].

2. Experimental Loading Technique

The principal challenge in this experiment was to devise a means of continuously loading positrons into a Penning trap. After careful and tedious construction of this Penning trap, a static scheme that was previously pro- posed by us [6] was found to be adequate to trap many

positrons each hour. This successful scheme was reported

upon previously [7] and is summarized here. A cross section of the Penning trap used to catch the

positrons is shown in Fig. 1. The principal difference

between this positron trap and the compensated Penning

trap used in the electron magnetic moment anomaly ex- periment [1] is the addition of a sealed sodium-22 posi- tron source and a field emission point (electron source)

mounted off-axis in an endcap as shown. During the posi- tron loading process, the positron source is negatively

biased to approximately 300 V. This source bias produces

the potential shown in Fig. 2 along the load-line, which is the line parallel to the z-axis that extends from the posi- tron source into the trap through the center of the hole

in the endcap.

Positron

Figure 1. Positron storage trap. The major changes in the basic compensated Penning trap design are an off-axis sealed sodium-22 positron source, an off-axis field emission electron source (for diagnostic studies) and a dnft channel i?i the end- cap opposite the source for trayisfen-ing positroiis iiito a very

well compensated experiment Penning trap. *This research is supported by the National Science Foundation.

215
Positrons, emitted by the source with energies of ap- proximately 100 keV (most of which is in the cyclotron

motion), travel along the load-line into the trap guided

by the axial magnetic field. Since the maximum potential

along the load-line in the vicinity of the entry hole is less than the far endcap potential (zero), some of the injected

positrons will turn around before hitting the opposite

endcap. Due to the magnetron motion in the trap, the

positrons are moved off the load-line as they travel

through the trap. Hence they see a slightly larger potential (~1 mV) when they return to the entry hole, as illus- trated by the broken line in Fig. 2 (greatly exaggerated).

If the initial axial energy of the injected positron is low

enough, the positron will be trapped for one magnetron

period. During this time, enough axial energy can be ex- tracted by a damping circuit (connected to an endcap)

tuned to the relativistically shifted axial frequency v, , to permanently trap and subsequently thermalize the

positron's 2-axis motion. The positron's cyclotron motion

is thermalized in a few seconds or less by synchrotron ra- diation. Finally, a motional sideband cooling technique

[1] is used to radially center the positrons by applying an

rf drive at + to an asymmetrically placed probe

(also called the SBE probe, shown in Fig. 3), where is the magnetron frequency.

Clearly, the loading technique relies on increasing the

time available to damp the axial motion by temporarily

trapping the positrons via the magnetron rotation. Since

the positrons are trapped for at least one magnetron

period, the axial damping is improved by a factor of vJvm

(5=2000) over on-axis injection. As a result, the trapping

rate obtained was -23 positrons /hr with a 0.5 mCi

source, a 5 V deep trap, and a 51 kG magnetic field. Since the positron loading rate is very sensitive to colli- sions with the background gas during the energy damping phase of the trapping process, the apparatus is usually immersed in liquid helium to obtain pressures less than 10"" Torr within the trapping volume.

STORAGE TRAP

- 0

o

>

c

<uo

o.

<uc

I o

o

o

-4

-8

endcap

surface

z = 0

endcap

surface

Figure 2. Potential almig the load-line. The source is biased at —300 V relative to the endcap in order to produce this potential variation along the line parallel to the z-axis that passes through the hole in the source endcap. The broken line greatly

exaggerates the slightly higher potential (~1 toVJ seen by a

positron on the first return to the entry hole due to the magnetron motion.

SOURCE

EXPERIMENT TRAP

RING

FEP

SIGNAL

ENDCAP

Figure 3. Schematic of double trap configuration. The side- band excitation (SBE) probes are used in the radial centering process for off-axis loaded positrons and position stabilizing in

the experiment trap. Signal endcaps are tuned to the axial fre- quency via an external inductor in order to observe the axial motion driven by an rf signal applied to the opposite endcap.

The dc potential, Vq, is 8.3 V and 10.3 V respectively for the storage and experiment traps.

3. Experiment Trap and the Transfer

Process

A second trap has been carefully designed and compensated to make precision measurements possible. This

trap is similar in detail to the loading trap (but without a positron source and with field emission point now located

on-axis). It is virtually identical to past precision compensated Penning traps [1]. The double trap combination,

shown only schematically in Fig. 3, was constructed be- cause it was initially thought (and later demonstrated)

that the holes required for positron trapping and center- ing would make it nearly impossible to compensate the

storage trap enough to allow precision measurements to be made. By using this second trap, it then became

necessary to transfer the positrons from storage into the

second trap through a channel drilled between the two

adjacent endcap electrodes of the double trap combination. The transfer is accomplished by pulsing the two adja- cent endcaps to the approximately common ring potential

for a few microseconds. Pulses much longer than 10 |jls tend to be less efficient since radial drifting can occur during the passage between traps (for instance via colli- sions with background gas). An accumulated drift more

than 0.025 cm will cause the positron to hit the drive

endcap of the experiment trap (see Fig. 3) since the hole

in that electrode vdll be the smallest one encountered.

Although the transfer is 25 to 50 percent efficient now,

the efficiency of early attempts was greatly reduced by

misalignment of the electric and magnetic axes in addition to higher than expected background pressure in the

drift region between traps. The alignment of electric and

magnetic axes is extremely critical since as little as one degree of misalignment can totally prevent any positrons

from successfully transferring into the experiment trap.

Once the positrons have been transferred, they can be

detected by using a large off-resonance axial drive, similar to that which is normally used when electrons are loaded into the trap, one at a time. This drive has been

precalibrated in order to determine the number of posi- trons transferred, though not immediately after transfer

since the positrons do not enter exactly along the axis of the experiment trap. If time is taken to apply sideband

cooling at Vj -I- Vm on the SBE probe, then, the calibra216
tion becomes reliable. After determining the number

trapped, the excess beyond one are systematically

ejected using intense rf pulses at + again applied

to the SBE probe. Figure 4 shows an example of this process for a case where four positrons were transferred

into the trap and three are ejected. The rf amplitude in the pulse is carefully adjusted such that at least ten con- secutive pulses are required in order to eject one posi- tron from the cloud (i.e., less than 10% chance of driving out one per pulse).

1 1 1 1

ambient =4.2 °K r = l.7sec

* drive

1 1 1 1

time (minutes)

Figure 4. Positron ejection record. An intense rf pulse at +

is applied to the SBE probe and the continuously monitored

driven axial signal registers the loss of single positrons after numerous (> 10) consecutive pulses.

4. Measurements

The signal shown in Fig. 4 is obtained using a preamplifier attached to one endcap and tuned to the axial fre- quency at Vj ~ 60 MHz in order to detect the image

currents due to the positron's driven motion. Once a sin- gle positron is isolated, the much reduced drive signal,

applied to the endcap opposite the signal endcap, is swept in frequency in order to obtain the axial resonance shown in Fig. 5. The compensation of this trap was quite

effective with the result that resonances as narrow as

4 Hz can easily be resolved (resolution approaches

0.01 ppm).

Not shown ,in Fig. 3 are four symmetrically placed

nickel screws, located in the central plane of the ring

electrode within the experiment trap. This magnetic

material will produce a very weak magnetic bottle which

effectively couples the magnetic moment of the charge to the axial resonance in the form of a 1.3 Hz shift per unit

change in the magnetic quantum level via the axial Stern-Gerlach effect [1]. By locking this axial frequency

to a frequency synthesizer, the changes in the magnetic

state are reflected in the lock-loop correction voltage,

making it possible to measure both the cyclotron fre- quency, Vc, and the spin anomaly frequency, Va. Figure 6 shows a typical positron cyclotron resonance

using this coupling scheme. As typical of past electron

FREQUENCY

Figure 5. Driven axial resonance of a single positron. The

phase of the driven oscillator is carefully adjusted to yield this dispersion-type driven resonance as a function of frequency.

The half-line-width (~2 Hz) represents a resolution that exceeds 0.03 ppm.

i/c'(Z^^g=0) = 142,234,736.2 (1.5) kHz

microwave frequency '

Figure 6. Positron cyclotron resonance. The resonance is ob- served in the correctioyi signals of the axial lock loop. The high

frequency tail has an expoyiential decay linewidth of 2k kHz

(see dotted curve) which is significantly larger than expected.

However, the nearly vertical low frequency edge (z = 0) is resolvable to ^1.5 kHz and is independent of lineividth.

traps with magnetic bottles [1], the resonance shows a clear low frequency edge, corresponding to Znns = 0, and

a high frequency exponential tail, indicative of the

Boltzmann distribution of axial states. The dotted curve

represents an exponential decay with a 24 kHz linewidth,

which is four times greater than the narrowest electron

cyclotron resonance obtained in previous traps [1,8].

Nevertheless, the edge is resolvable to approximately

0.01 ppm and is independent of the resonance linewidth. In order to flip the spin, the magnetic bottle is used

again, but now to create a precessing magnetic field in the radial plane at the spin frequency

Vs = Vc + Va .

This process has been described previously [1] and

involves applying the anomaly drive to the endcaps yieldI

217
ing a weakly driven axial motion at Vg. From the

positron's frame of reference rotating at the cyclotron

frequency v'c, the axial modulation will yield sidebands

with Vg as the desirable upper sideband. Figure 7 shows

the corresponding anomaly resonance taken by alternat- ing the detection and excitation in order to yield the best

resolution. The solid line represents an exponential decay

with a 12.5 Hz linewidth. Again, the edge is nearly

resolvable to 0.01 ppm.

I ;—'

1 1 1

1 (Zrms=0) = 164,956, 715.0(2.0) Hz

l^l2.5Hz-l

T/,f -164,956, 700 Hz

Figure 7. Positron anomaly resonance. By alternating detec- tion and excitation, the number of spin flips (out of a fixed

number of attempts) is plotted versus frequency. The solid curve represents an exponential decay linewidth of 12.5 Hz. However, the low frequency edge (z = 0) is resolvable to ±2.0 Hz, which corresponds essentially to the same resolution found for the cyclotron resonance.

5. Results

By measuring corresponding edge frequencies, the g- factor and its anomaly can be determined from the fol- lowing relation:

a{e-) = gie-)/2 - 1 = [v; - v!/2v'Mv'c + v|/2v^] .

This equation has been investigated and found to be

highly accurate as long as v'^, Vc, and are the actual ob- served frequencies of motion in the Penning trap [9]. At

present, four separate runs have been made using a sin- gle positron transferred into the highly compensated

Penning trap. Time-studies of the cyclotron resonances

indicate an effective magnetic field jitter on the order of 0.05 ppm. This may be due to collisions with some resid- ual background gas which causes the radial position of

the particle to vary in time within the magnetic bottle

field. Future work will investigate and seal any possible

leaks found in the trap-tube envelope.

At present, only one systematic effect in the anomaly

(for both electrons and positrons) has been observed: the

anomaly decreases with increasing appHed anomaly

drive. This may be due either to a change in the magnetron radius (change of 2-3 times the minimum radius

observed) or to an uncorrected positive shift in the axial frequency associated with the added rf trapping potential

produced by the strong anomaly drive. The effect was

typically -0.03 ppm for the largest anomaly drive used in the older electron geonium experiments. Thus, from the

four initial positron runs, a preliminary positron spin

anomaly.

a(e+) = 1159652222( ±50) X 10-12

,

is obtained from a weighted least-squares extrapolation

using the systematic power dependence observed for the

electron. Field jitter is incorporated in the individual

anomaly errors, used to produce the weights in the

least-squares adjustment. However, the error (0.043

ppm) represents primarily the uncertainty associated

with the power extrapolation for the positron and only

secondarily the error associated with the least-squares

adjustment.

The result for the magnetic moment spin anomaly of a

single electron [10] was reported earlier to be

a(e-) = 1159652200(40) x lO-^^ where the uncertainty was due to both a time varying

magnetic field and the systematic dependence of aie~) on

the amount of spin power applied. These two results can be compared to theory [11]

ae(theor) = 1159652569(150) x lO-^^ where the uncertainty is almost entirely due to the uncertainty in the fine structure constant [12]. Finally,

combining the results for the measured electron and posi- tron anomalies we obtain a preliminary matter/

antimatter comparison of

gf(e+)/p(e-) = 1 + (22 ± 64) x lO'^^ . The resolution of this g^-factor comparison will significantly improve when measurements of both positron and

electron spin anomalies in the same apparatus are completed, without the pressure effects smearing out the

magnetic field edge.

We wish to thank Don Russell for his excellent machine

work on the traps and glassblower Bob Morley for his su- perb work.

References

[1] R. S. Van Dyck, Jr., P. B. Schwinberg, and H. G. Dehmelt,

in New Frontiers in High Energy Physics—Studies in the Natural Sciences (Orbis Scientiae), A. Perlmutter and L. F. Scott, Eds. (Plenum Press, New York, 1981), p. 337.

[2] J. R. Gilleland and A. Rich, Phys. Rev. A 5, 38 (1972).

[3] S. I. Serednyakov, V. A. Sidorov, A. N. Skrinsky, G. M.

Tumaikin, and Ju. M. Shatunov, Phys. Lett 66B, 102

(1977).

[4] A positron polarimeter with three orders of magnitude more

efficiency is reported by G. Gerber, D. Newman, A. Rich,

and E. Sweetman, in Phys. Rev. D 15, 1189 (1977).

[5] J. H. Field, E. Picasso, and F. Combley, Sov. Phys. Usp.

22 199 (1979).

[6] H. G. Dehmelt, P. B. Schwinberg, and R. S. Van Dyck, Jr.,

Int. J. Mass Spectrom. Ion Phys. 26, 107 (1978).

[7] P. B. Schwinberg, Ph.D. Thesis, University of Washington

(1979); also P. B. Schwinberg, R. S. Van Dyck, Jr., and H.

G. Dehmelt, Phys. Lett. 81A, 119 (1981).

[8] R. S. Van Dyck, Jr., P. B. Schwinberg, and S. H. Bailey, in Atomic Masses and Fundamental Constants 6, J. A.

Nolen, Jr., and W. Benenson, Eds. (Plenum Press, New

York, 1980), p. 173.

[9] G. Gabrielse and R. S. Van Dyck, Jr., Bull. Am. Phys. Soc.

26, 598 (1981); also Lowell Brown, private communication.

[10] R. S. Van Dyck, Jr., P. B. Schwinberg, and H. G. Dehmelt,

Bull. Am. Phys. Soc. 24, 758 (1979).

[11] Based on the best theoretical value of a^ to order a^: ae(theor)= ay2iT - 0.328478445(a/iT)2 + 1.1835(61)(a/Tr)3

(T. Kinoshita, 1979, private communication).

[12] E. R. WilHams and P. T. Oisen, Phys. Rev. Lett. 42, 1575

(1979) [a-' = 137.035963(15)1.

218
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Geonium Without a Magnetic Bottle—A New Generation*

Gerald Gabrielse** and Hans Dehmelt

University of Wasliington, Department of Physics, FIVI-IS, Seattle, WA 98195

Work is now underway to improve the accuracy of the electron magnetic moment measurements by

10 to 100; that is, to an accuracy of 1 part per trillion. At this level of accuracy, the magnetic bottle so

crucial to previous measurements must be removed. In fact, intrinsic bottles due to the paramagnetism

of the trap electrodes must be carefully canceled. A promising replacement for the magnetic bottle is the relativistic mass increase which acts like a magnetic bottle 20 times smaller than the one previously

used but with no distortion of the magnetic field. The relativisitic mass increase has been observed for 0.5 eV electrons. We have also trapped electrons in a new style, compensated Penning trap (described

here) which is simpler to construct, promises to be more reliable when cycled to liquid helium temperatures, and can be easily disassembled. Key words: compensated Penning trap; geonium; intrinsic magnetic bottle; magnetic bottle; Penning

trap; relativistic mass increase.

1. Introduction

The advent of single particle spectroscopy at the

University of Washington began a string of very accurate measurements of properties of elementary particles. The

magnetic moment anomaly of the electron has been measured to an accuracy of 34 parts per billion [1]. Elsewhere

in these proceedings are reports of a comparison of the

electron and positron magnetic moment anomaly to an

accuracy of 64 parts per billion [2], and a measurement

of the ratio of the proton to electron mass to 136 parts

per billion vdth further increases in accuracy expected

soon [3]. We judge there is a reasonable prospect of improving

the measurements of electron and positron magnetic moments by a factor of 10 to 100. This paper is a progress

report on the efforts in the past year and one half to realize this improvement. As in nuclear magnetic resonance

spectroscopy (where resolutions of 10"^^ have been

achieved), we expect that geonium spectroscopy vnW

have similar wide application if we can approach the

NMR resolution. There is also considerable theoretical

interest in the magnetic moments of the electron and positron. Calculations of the (a/7r)^ corrections according to QED are presently underway (see these proceedings). If an independent, more accurate value for the fine struc- ture constant also becomes available (some proposals are contained in these proceedings), then our measurement

will provide a test of the QED calculations. Conversely,

if one chooses to believe the calculation and the appropri- ateness of QED, then our measurement vnW provide the

fine structure constant. To our knowledge, no other techniques can get within orders of magnitude of the accuracies we believe are realizable. The new feature of the next generation geonium ex- periments is that we hope to rely on the relativistic mass

shift rather than using the magnetic bottle which was

crucial to previous experiments. The magnetic bottle is undesirable at this level of accuracy because it makes the

magnetic field inhomogeneous.

*Supported by the National Science Foundation **Chaim Weizmann Postdoctoral Fellow

We divide our program into three phases: the first is completed, the second is mostly completed, and the third

is about to begin. First, we have demonstrated that we

can observe the relativistic mass increase of an electron

of energy 0.5 eV [4]. Second, we have constructed a simpler version of the compensated Penning trap which is demountable and promises to be more reliable as well. We have trapped electrons in it (24 hours before leaving

for this conference) but much testing remains to be done

including a demonstration that intrinsic magnetic bottles have been removed [5]. Third, we shall measure the

electron's magnetic moment anomaly.

2. Observation of the Increase in the

Relativistic Mass, 7/no, of a 0.5 eV Electron

(7 = 1.000 001)

A slightly modified, compensated Penning trap made of

OFHC copper was assembled using electron tube con- struction techniques which have been discussed else- where [6]. For our purposes we need not review the

monoelectron oscillator formed except to observe that

the electron's harmonic oscillation along the magnetic

field lines has a frequency v, (=60 MHz) which is in- versely proportional to the square root of the electron mass. The observed cyclotron oscillation around the magnetic field lines has frequency v\ (=21 GHz) which is in- versely proportional to the electron mass. The energy in the cyclotron motion is increased by

sweeping a microwave drive through the cyclotron fre- quency. Both and v'c thus decrease because the

electron's mass, ymg, (where mo is the rest mass) in- creases. The harmonic and cyclotron oscillation frequencies decrease by 60 Hz and 42 kHz, respectively, for a

1 eV excitation.

A typical measurement is summarized in Fig. 1. We

monitored changes in the harmonic oscillation frequency

(the ordinates of Fig. 1). The sensitivity was calibrated as in Fig. 1(a) by changing by 50 Hz the frequency of the synthesizer to which the electron's harmonic oscilla- tion frequency was locked. This was done repeatedly to get an indication of uncertainty.

219
50Hz

<

(a)

Calibration

500 sec

a>

30Hztime

(b)

Observed

Relativistic Shift in 1/^ sweep direction

lOOkHz-H helectron —

driven out

of trap

cyclotron drive 21.4GHz)

Figure 1. (a) Calibration of sensitivity to changes in v^; (b) ob- servation of relativistic mass increase of 0.5 eV electron.

Sensitivity was never better than 20 Hz out of 60 MHz

with 400 second signal averaging. This sensitivity was

rather poor compared to the 1 Hz resolution reported for earlier geonium measurements with a one second time

constant [7]. However, we were awaiting delivery of a superconducting NMR magnet with a liquid helium

Dewar in its bore and thus were forced to use an old Varian electromagnet and immerse the Penning trap in liquid nitrogen rather than liquid helium. The new ap- paratus (discussed in the next section) will doubtlessly

have much better sensitivity.

A typical mass shift measurement is plotted in Fig. Kb). The ordinate is identical to Fig. 1(a). The

cyclotron drive was swept down through resonance (right to left). Like the anharmonic oscillator, the cyclotron fre- quency is pulled along as the excitation proceeds. The

strength of the microwave drive is very critical. A

slightly stronger drive knocks the electron out of the

trap, presumably via collisions. A further complication accounts for the restricted fre- quency range and the abrupt loss of the electron at the

left of Fig. 1(b). The cyclotron motion of an electron in a Penning trap is actually superimposed on a slow drift orbit (magnetron motion) of frequency v„, = 90 kHz for our

trap. As has been observed elsewhere [7], the electron can also absorb energy at v'c + v,„ or - v,„ , the former

shrinking the magnetron radius, the latter increasing it. We thus sweep down in frequency. When we hit the v'c -

Vm sideband our strong drive pushes the electron out of the trap radially.

2.1 New Demountable Penning Trap

We have successfully trapped electrons in a new style,

compensated Penning trap which differs substantially

from earlier traps. Previous Penning traps [6, 7] were

constructed like glass vacuum tubes. A change in the

trap, however slight, required that the whole trap be

disassembled, reassembled on a new pin base, sealed by

a glass blower and baked to 400 °C. When cycled

between room temperature and liquid helium temperature, the glass-to-metal seals proved to be unreliable vacuum seals. An internal ion pump was thus added to partially compensate. Finally, each trap consisted of five separately machined metal electrodes separated by four

tubular glass spacers each of which was precision ground.

Much precision mill work was required so that construction and assembly times were measured in man-years.

We felt that the rigors of a parts per billion measure- ment would be greatly facilitated by a Penning trap

which was simpler, more rugged, and easily accessible

for modification. The glass vacuum tube technology was

accordingly abandoned in favor of a tubular copper en- velope, sealed to copper flanges with demountable indium

seals. We dispensed with the ion pump entirely since it seems clear that cryopumping alone will easily keep the vacuum inside the trap to better than 10"^^ Torr [8]. Porous glass cryopump cells were mounted on the upper

copper flange to greatly increase the cryopumping sur- face area. The whole system was baked briefly to 100 °C

to drive off water vapor from the electron surfaces and

then a copper pump-out tube was pinched off at a vacuum

of 5 X 10"^ Torr. The trap itself is mounted on a copper pin base flange as shown in Fig. 2. Alignment is provided by a ring of tungsten pins brazed to the copper base. These pins align

1.0 0.5 0.5 1.0 in

3 cm

Figure 2. Demountable Penning trap; % = 30° and 0 = 70° su- perimposed on Penning trap outline to indicate placement of magnetic compensation tnaterial.

220
the trap elements by way of thin machinable glass ceramic

(MACOR) rings on the outside of the endcaps and by

springs. The two endcaps and the ring are made of

molybdenum in this first version. These electrodes are simpler to construct than earlier traps in that no precision

mill work is needed. The four ground glass spacers and the

two metal compensation rings of previous traps have all been replaced by two spacers constructed from MACOR.

Anharmonicity compensation is provided by a "thickfilm" of silver metalized to the MACOR. Each thick film compensation ring is split to make a 1 turn coil which will be used to drive spin flip transitions (at = 180 MHz).

Holes through the spacers admit microwave drives and

small probes are used to apply radio-frequency driving

signals for cooling. MACOR can be machined using con- ventional metal working techniques and the thick film

compensation electrodes can be added in several hours.

Replacing six parts by two makes more precise alignment

possible and also simplifies the machining.

Two demountable ports, also vacuum sealed by indium

0-rings, are provided for admitting microwaves into the

trap. In this first version microwaves will be generated

by multiplying X-band up to 168 GHz in a multiplication

diode inserted from outside the trap into a glass tube

sealed to a copper stem. The electrical feed-throughs

brazed into the copper base are commercially available. They are made from Constantan which is copper brazed

to ceramic. The copper base has a hole on the trap axis

which will allow inserting a ^He NMR probe wdthin 1.5 inches of the trap center. This probe will be used if necessary to stabilize the 60 kG magnetic field.

3. Bottleless Geonium

The small nickel ring, deliberately introduced into pre- vious Penning traps, must be removed to improve our

resolution of the anomaly beyond 10"^. In addition, however, magnetic field inhomogeneities introduced by the

paramagnetism of the trap electrodes must be removed

as well [5]. A ring about the z axis with magnetism M

and small cross sectional area A produces a magnetic

field near the origin at (r, 6) and (p, z)

cc B = XAkr\Pk(cosQ)z - (k + ly' P^ ^,icos BTp]

0 where Pkm is an associated Legendre polynomial. In cgs

units

Ak = 2 7tMA(/c + l)ik + 2)pr-''~^Pi,+2(cosQ)

,

where now the coordinates locate the ring. A symmetrical Penning trap like this one, with po = V^^o thus has

a magnetic bottle with A2 = 30 M. For a molybdenum

trap at 4.2 K and 60 kG this suggests an intrinsic bottle about 20% of that previously introduced deliberately.

The next trap will no doubt be made of copper as a

result (i.e., ten times less susceptibility). In the meantime, compensation rings made of iron will be placed

between 0 = 31° and 0 = 70° where P4 is negative to severely reduce the size of the intrinsic bottle. Figure 2

indicates these angles superimposed on the trap surfaces. Final cancellation will be achieved using the second order

shims of the superconducting magnet.

4. g - 2 in Cooled Relativistic Geonium

We have considered many schemes for measuring g - 2

in bottleless geonium. As might be' expected for an at- tempted measurement accurate to better than a part per

billion, none of these promises to be easy. In the next

few months the new trap will be used to evaluate the

feasibility of these schemes.

The most promising approaches make use of the relativistic perturbation to the cyclotron levels [9, 10], as diagrammed in Fig. 3. The quantum numbers n and m

pertain to cyclotron and spin levels. At 4.2 K the elec- tron spends roughly 90% of its time in n = 0 and 10% in n = 1. The g - 2 transition, between in = 1, m =

- 1/2) and in = 0, m = 1/2), will be driven via single- turn bucking coils made from the compensation elec- trodes. We hope to determine if m = - 1/2 or m = 1/2 and thus whether or not the spin has flipped by ascer- taining whether the first cyclotron absorption occurs at

(1 - R)vc or (1 - 3R)v^. As a first attempt the cyclo- tron drive will be swept towards lower frequencies,

starting between (1 - i?)v,. and (1 - SR)v,.. This start- ing point must of course be determined experimentally.

The successful observation of the relativistic mass in- crease (Section 1) suggests that there will be no major

problem in driving the electron's cyclotron energy high

enough to observe it via the relativistic shift of the harmonic frequency, once the classical region of n = 10 or higher has been reached. Obtaining substantial excitation

probability to n = 10 promises to be more difficult since

the 0. 1 sec lifetime will make it difficult initially to maintain both the excitation and the frequency resolution.

Experiments to establish a feasible detection scheme begin as soon as this conference ends.

m = -l/2 m = + l/2

(l-3R)i/,. (C^

:i-3R)z/^

E=W-moC^

aH(q/2)-l

^ Rshz/(,/2moC'^

E/h 1

n = 0

n = 0

1/2 mahi/

Figure 3. Lowest cyclotron and spin energy levels for geonium.

The new style trap was machined by Ralph Jochim.

Technical assistance was provided by Kim Weimer.

References

[1] R. S. Van Dyck, Jr., P. B. Schwinberg, and H. G. Dehmelt,

Bull. Am. Phys. Soc. 24, 758 (1979).

[2] P. B. Schwinberg, R. S. Van Dyck, .Jr., and H. G. Dehmelt,

these proceedings.

[3] R. S. Van Dyck, Jr. and P. B. Schwinberg, these proceedings.

[4] G. Gabrielse'and H. G. Dehmelt, Bull. Am. Phvs. Soc. 25, 1149

(1980) . [5] G. Gabrielse and H. G. Dehmelt, Bull. Am. Phvs. Soc. 26, 598

(1981) . [6] R. S. Van Dyck, Jr., D. Wineland, P. Ekstrom, and H. G. Dehmelt, Appl. Phys. Lett. 28, 446.

[7] R. S. Van Dyck, Jr., P. B. Schwinberg, and H. C. Dehmelt, in New Frontiers in High Energi/ Physics—Studies in the Natural Sciences (Orbis Scientiae), Ed. by A. Perlmutter and L. F. Scott (Plenum Press, 1978), p. 159.

[8] W. Thompson and S. Hanrahan, J. Vac. Sci. Technol. 14, 643

(1977).

[9] H. Dehmelt, in Atomic Phijsics 7, Ed. by D. Kleppner and F. M. Pipkin (Plenum Press, N.Y., 1981), p. 337.

[10] H. Dehmelt and G. Gabrielse, Bull. Am. Phys. Soc. 26, 797

(1981).

221

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

The Magnetic Moment of Positive Muons

in Units of the Proton Magnetic Moment*

E. Klempt, R. Schuize, and H. Wolf

Institut fCir Physik der Universitat Mainz, F.R.G.

and

M. Camani, F. N. Gygax, W. Ruegg, A. Schencl<, and H. Schilling

Laboratorium fur Hochenergiephysik der ETH, Zurich, c/o SIN, 5432 Villigen, Switzerland

The magnetic moment of positive muons in units of the proton magnetic moment was determined.

A value of ^^/y^ = 3.1833441(17) was found. Key words: diamagnetic shielding; muon magnetic moment; muon chemistry.

1. Introduction

We report on a measurement of the ratio of muon to proton magnetic moments, p.^/|x,j. A preliminary analysis

of part of our data has been published in a previous letter

[1], and a full account of the work has been submitted for publication [2]. The value of M-tiV;) was derived from the

relation shown in Table 1.

Table 1. Basic relation from wh ich

/Lt^/fXp is determined.

1 + a„

1 + (T„ + CT£G

muon spin precession frequency

proton spin precession frequency

diamagnetic shielding of muons

diamagnetic shielding of protons

background correction

The muon spin rotation frequency was measured in a magnetic field of 0.747 T at the SIN superconducting muon channel using a stroboscopic technique. Muons

from the beam with polarization perpendicular to the

magnetic field were stopped in a liquid Br2 target. Their

chemical state depends on the presence or absence of a

small H2O contamination in the Br^ target: with no H2O

present muons form (ix'^e") Br molecules, otherwise a chemical reaction chain places the muons into MuHO

molecules [1, 2]. The two diamagnetic shieldings were

determined from NMR measurements on HBr and H2O

molecules dissolved in liquid Br2, and from quantum

*Supported by the Bundesministerium fiir Forschung und Technologie,

FRG, and by the Schweizerisches Institut fiir Nuklearforschung, SIN.

chemical calculations of the muon-proton isotope shifts

[3]. The spin precession frequency of protons in H2O in the same magnetic field was determined by conventional

methods [4], the proton diamagnetic shielding in H2O is known [5, 6]. Figure 1 sketches the principle of the stroboscopic

method. It is based on the microstructure of the SIN muon beam. Due to the rf acceleration of the primary

proton beam, the muon beam arrives in sequential beam

bursts with a separation in time corresponding to the ac- celeration frequency Cl = 2-n x 50.633 MHz. A magnetic

field of 0.747 T leads to a muon spin rotation frequency

veto counters

master oscillator i

i i i i i of proton accellerator -jj-' ' ' '

ten stroboscopic gates

Figure 1. Schematic representation of the stroboscopic method.

223
of o)^ = 2 n. At this field the polarization of muons from

different beam bursts add up coherently. The polarization is small when muon spin precession frequency and

beam burst repetition frequency are mismatched, i.e., when |a)|j^ - 2fi I ^ The polarization can be ob- served by gating—at the beam burst repetition

frequency—the telescope for detection and identification

of decay positrons. The gate corresponds to a fixed fre- quency "flash light" which detects the polarization in a

rotating coordinate frame. The magnetic field has to be

scanned in order to find a stationary polarization. Ten

contiguous and nonoverlapping stroboscopic gates were

set up covering one complete cycle of the rf period.

A line shape theory was developed which calculates the

fractional number of positron counts, Nj/No, falling into a stroboscopic gate of length kj as a function of the magnetic field. For a periodic muon beam intensity and elec- tronic gates of the same frequency, A'', is given by

ZTiKj 1 -I- .rwith X — {M)„ - 2 as arbitrary phase, and A^ff

and effective asymmetry of the muon decay. The finite width of the muon beam bursts results in an effective

asymmetry of 0.16 compared to an asymmetry of 0.24

found by conventional muon spin resonance techniques at the same beam line.

2. Apparatus and Methodology

The experimental setup basically consists of a target

positioned at the center of an iron magnet, a collimator,

and scintillation counters. The diameter of the pole

pieces is 30 cm with a separation distance of 8 cm. The

target is surrounded by a box of plastic scintillators which serves to identify positrons from muons stopped in the target and to veto muons and unwanted positrons.

The count rate of accepted positrons was 27000 /s for a primary proton beam of 40 |jlA and a muon beam intensity of 2 X 10^ |JL^ through the collimator.

The magnetic field was stabilized by a conventional

NMR system. Maps of the magnetic field over the target

volume of 14 cm'^ were taken regularly by a second NMR

system. Figure 2 shows a histogram of eight different

field maps measured in the course of six days and

corrected for drifts of the magnetic field at the target center (typically 5 x 10"*^). The rms deviation of magnetic field values from the mean is 0.4 ppm. Different

hypothetical |jl ^ stop distributions lead to the same mean

value of the magnetic field with an uncertainty of 0.08 ppm. At the end points of the magnetic field scan

the homogeneity was typically 0.6 ppm. The homogeneity was achieved by a 10 current shim system, the

stability of the field map by thermostating the magnet

cooling water and the ambient air. The NMR measuring

probe was calibrated by comparison with a series of

unshielded calibration probes mounted inside an open

pair of Helmholtz coils modulating the external field. The

final corrections to the NMR frequencies are Hsted in Table 2. Figure 3 shows the stroboscopic signal from 2 out of 10 stroboscopic gates and for a data taking period of 2 days.

The data of all 10 gates were fit simultaneously by use of Eq. (1), but some small modifications for systematic ef- fects were incorporated into the line shape theory:

(i) From incoming muons an electronic veto is derived. Positrons emitted within some 10 ns

Table 2. Corrections to the NMR frequency

of the field measuring probe.

Diamagnetic shielding, cth^o -H 25. 64(3) ppm

Bulk susceptibility - 1.51(4) Dom

Paramagnetic ions + 0.35(5) ppm

Calibration + 0.80(12) ppm

Field map 0.00(8) ppm

Total correction, ct^juj +25.28(16) ppm

B = 0.0 ppm

MAGNETIC FIELD

Figure 2. Histogram of magnetic field values obtained from 8

field maps taken in the course of 6 days.

after a stop are therefore not accepted. In

10 ns the muon polarization vector has precessed

by an angle which depends on the magnetic field. Hence the phase \\>j is different for different values

of the magnetic field. This effect was implemented

into the line shape theory. Test analysis routines

proved that the effect is controlled and had no ef- fect on (o^/o)/,. No free parameter was finally

needed.

(ii) As muons arrive in bursts, there is a periodic

chance that otherwise accepted positrons were

vetoed. This effect was allowed for in the line shape theory and is described by five complex

Fourier coefficients.

(iii) Positrons from muon decay have—due to the magnetic field—curved tracks. This leads to a magnetic field dependent low energy cut in the energy

spectrum of accepted positrons. Thus the effective

asymmetry Aeff is field dependent. This effect was shown to be negligibly small.

(iv) The curvature of positron tracks leads to a magnetic field dependent solid angle for positron

detection. Such a field dependence of v|>/ was al- ready introduced to describe the effect of prompt

vetoes after jx^ stops. The track curvature yields a much smaller effect and was therefore neglected.

(v) Phase jitter of beam bursts and stroboscopic gates

224
850

8.25

iSs.oo

LU

7.50

GATE1 GATE 2 6 00

5.75

5.50 ^

LO

LU

5 25 ^

LU

5.00

31 7 318 319 MHz 31.7 31 8 31.9 MHz

MAGNETIC FIELD (PROTON NMR FREQUENCY

Figure 3. Stroboscopic signal from. 2 out of 10 gates. The dot radii correspond to the statistical error. The data were collected in 2 days.

did not lead to systematic distortions of the line but to an increased variance of the number of posi- tron counts in a stroboscopic gate. Therefore the

errors from individual runs were scaled by

Vx^/A^F, i.e., typically by 1.3.

(vi) The ratio of accepted positrons for target-full tar- get-empty was 14. The target empty rate was

dominantly attributed to muon stops in the quartz

glass ampoule containing the liquid bromine, in the

positron counters, or in the wrappings or surfaces

of the scintillation counters. In target empty runs no stroboscopic background signal was firmly esta- blished. Therefore a background correction of

CfiG = (-0.20 ± 0.25) ppm was deduced from the

largest stroboscopic background signal compatible

with the background data. For contiguous nonoverlapping gates, the 10 gate

lengths Xj and 10 phases 4/, are linked by 10 constraints.

Therefore a total of 22 parameters, w^/o)^, Ae/f, 9 gate

lengths Xj, 1 phase, and 5 complex Fourier coefficients were needed to describe 289 data points {No is fixed).

From the fit, uncorrected values for co^/wp were derived:

V^p = 3.1833212 (20) ((x+e-)Br in Brz, = 3.1833341 (19) (fjL+e')HO in Brg, (2)

(i)

3.1833519 (66) (jji+e -)H0 in H2O.

The latter result was obtained by stopping muons in H2O.

It is only used for a comparison of chemical shifts, not for the final value of |JL^/|Xp. Note that the error due to fluc- tuations of the magnetic field at the target center is con- tained in the above errors.

3. Muon Chemistry

The major advantages of using bromine as a target

substance are the following:

(ii)

(iv)

(V)

fast chemical reactions preserve the full muon polarization; there is one unique chemical compound in which muons finally reside;

(iii) a small water contamination allows the final chemical compound to change;

the compounds allow a measurement of the chemical shift on analogous compounds containing protons; the isotopic shift of the diamagnetic shielding in these compounds can be derived from quantum

chemical calculations; and

(vi) finally, liquid bromine has a high stopping power

and is easy to handle at room temperature.

The discussion of the chemical reactions which un- dergo when they are stopped in liquid bromine and of their diamagnetic shielding is summarized in Fig. 4. Positive muons stopped in liquid bromine neutralize to muonium which reacts with bromine molecules by forming (|jL'^e")Br molecules, (|jL^e") + Byo ~* (|i.^e") Br -l- Br, with an average time of 5 x 10"^^ s. This time is very

short compared to the muonium hyperfine structure period of 2 x 10"^° s. The polarization of the muon is therefore completely conserved. A possible H2O contamination in a pure Br2 target can be removed by adding

P2O5 and by bubbling gaseous HBr through the liquid

Br2. The time constant for the reaction (|jl^^") -I- HBr ^

{[j.^e')li + Br is too long to lead to any sizable fraction

of (|JL ^ e

~ )H molecules.

One bromine target was contaminated with H2O with a concentration of 0.1 mol/1. There the reaction chain

((JL+ e -) Br + H2O -> (|JL^ e ") UzO^ + Br",

(^JL+ e -) H2O+ + H2O ^ (ix+ e -) HO + H3O-.

places the muon into a water molecule within one [is after thermalization.

225
Muons stopping in liquid bromine neutralize to muonium (Mu) atoms which

react with bromine molecules

. . \ ^ • •

Mu • Br — MuBr . Br

Lrquid bromine is contaminated by H3O and HBr

,

. I

, [H,0] > [HBr] = 0

«r . < - # . -q; MuBr » HjO — Br- (MuHjO)*

p. transfer

(MuHjO'.HjO —. MuHO. (HjO)*

p transfer

endproduct of chemical reactions (MuHO)Br,

0 = [H,0) < [HBr]

•'

' < ^ • • HBr » HjO Br- - IH^O)*

p transfer

. - cf' ^ . »-c( MuBr (HjOr 7^ HBr (MuH^O)*

p-^^ excfiange

endproduct of chemical reactions : (MuBr)Br-,

The diamagnetic shielding of muons stopping in liquid bromine has been

derived from

the diamagnetic shielding of protons in H^O at 25''C :

Gh,o = 25 64 (3) ppm

the difference of diamagnetic shieldings of protons in H:.0 and in MuHO and (vluBr dissolved in liquid bromine as obtaimed from high resolution NtviR measurements

°'^^20'Br2

" ° n,o = 3 03(5) ppm ^wbd^^^ - Oh^o = 7 18 (5) ppm

the isotope effect of the diamagnetic shielding which wos colculated by quontum chemical colculotions

(5(MuHO)Br,-°IH2O)Br=-0 28(12)ppm CiMuBDerj " °(HBr)Br/-°^S*^2) Ppm

SUM C(^,^HO)Br, = 28, 39 ( 13) ppm G(MuBr)B,,= 3204(12) ppm

Figure 4. The chemical reactions which jjl* undergo when they are stopped in liquid bromine, and their diamagnetic shielding.

The diamagnetic shielding of muons in (|jL^e~)HO molecules in liquid Br2 has been derived from the shieldings

of HBr and H2O molecules in liquid Br2. The diamagnetic

shieldings of free HBr and H2O molecules are known; in liquid Br2 they are subject to liquid association shifts of

-1.92 ppm and -1.18 ppm, respectively. These were

measured in a conventional high resolution NMR spec- trometer. Liquid association shifts and the difference of the diamagnetic shielding when protons are replaced by muons in a chemical compound were derived from quantum chemical calculations [3, 4]. These isotope shifts of

the diamagnetic shieldings—including the hquid association shifts—were found to be (-0.78 ± 0.12) ppm and

(-0.28 ± 0.12) ppm. The calculation predicted liquid as- sociation shifts of HBr and H2O in Br2 of - 1.93 ppm and - 1.03 ppm, respectively, in striking agreement with the

experimental values, and thus supporting the reliability

of the calculation. The resulting diamagnetic shielding

corrections for in (jx^e )Br and (|x"'e")HO molecules

and their contributions are listed in Table 3. An estimate

given in Ref. [7] is assumed for the isotopic shift of the

diamagnetic shielding of stopped in hquid H2O and

forming ((x^e~)HO molecules.

4. Results

When the corrections listed in Table 1 are apphed to the values of i^Jwp found in the fit the following results are obtained:

fjL^/fjLp = 3.1833434 (20) MuBr in Br2,

ix/|ji^ = 3.1833447 (19) MuHO in Br2,

|x^/|jLp = 3.1833473 (92) MuHO in H2O.

(3)

The three results are in excellent agreement after the

corrections for the diamagnetic shieldings are applied.

This supports the assumption that the chemical reactions

derived from the chemistry of hydrogen atoms also take

place in the case of muonium chemistry. The first two

results are combined, and the systematic errors due to the calibration of the NMR frequency, due to the un- known muon stop distribution, and due to the stroboscopic background are added quadratically. Thus the final result obtained is:

jjL/My = 3.1833441(17) (0.5 ppm).

This value is consistent with previous determinations

[7, 8] of fx/iJLp and vdth the result of a recent high precision experiment on the muonium hyperfine structure in a strong magnetic field [9].

Table 3. Isotopic changes of the chemical shift and total diamagnetic shielding corrections for jj.^ in a ('p.'^e )Br or (\i^e )H0 molecule in liquid Bro (spherical probes), absolute and relative to protons in water.

molecule

-1- environment

Sf^isotope - CTp, - CTp [ppm] Shielding correction in the liquids

Free molecule Molecule in liquid CT^ [ppm]

(MuBr)Br2 -0.70 ± 15% -0.78 ± 15% -H6.40 ± 0.12 4-32.04 ± 0.12

(MuH0)Br2

+ 15%

-0.46 -110%

+ 15%

-0.19 -110%

= -0.28 ± 0.12

-^2.75 ± 0.13 -t-28.39 ± 0.13

(MuH0)h20 - 2.0 ± 2.0^ -2.0 ± 2.0^ -2.0 ±2.0=> -H23.64 ± 2.00^

^Values as assumed in Ref. [7] (only for comparison).

226
References

[1] M. Camani, F. N. Gygax, E. Klempt, W. Rliegg, A. Schenck, H. Schilling, R. Schulze, and H. Wolf, Phys. Lett. 77B, 326

(1978).

[2] M. Camani, F. N. Gygax, E. Klempt, W. Rliegg, A. Schenck,

H. Schilling, R. Schulze, and H. Wolf, Phys. Rev. D 25, 652

(1982).

[3] M. Castro and J. Keller, "The Isotope Chemical Shift of ^+ in HBr and H2O," Report (UNAM, Facultad de Quimica, Mexico-City, 1980), and to be published.

[4] K. Borer, Nucl. Instrum. Methods 143, 203 (1977).

[5] W. D. Phillips, W. E. Cook, and D. Kleppner, Metrologia 13, 179 (1977).

[6] J. C. Hindman, J. Chem. Phys. 44, 4582 (1966).

[7] K. M. Crowe, J. F. Hague, J. E. Rothberg, A. Schenck, D.

L. Williams, R. W. Williams, and K. K. Young, Phys. Rev. D 5, 2145 (1972).

[8] D. E. Casperson, T. W. Crane, A. B. Denison, P. 0. Egan, V. W. Hughes, F. G. Mariam, H. Orth, H. W. Rei.st, P. A. Souder, R. D. Stambaugh, P. A. Thompson, and G. zu Putlitz, Phys. Rev. Lett. 38, 956 (1977).

[9] V. W. Hughes, these proceedings.

227

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Measured g^-Factor Ratio of ^He^(1 ^S^^g) ^He(2^S^)*

H. G. Robinson**

Duke University, Durham, NC 27706

and

C. E. Johnson

North Carolina State University, Raleigh, NC 27607

The grj -factor of the ground state hydrogen-like helium ion C'He^) relative to that of the 2^Si state of helium (^He* ) has been measured using a ^^Rb optical pumping technique. The current experimental

result is srj('*He+. 1 '^^yiVgji^Yie, 2 ^Sj) = 1 - 29.95(10) x 10"^. Using other previously determined

experimental ratios, we find gj(^Y{e'^)/g = 1 - 70.87(10) x lO"*', in excellent agreement vdth

theory. The linewidth budget for the "^He^ Zeeman resonance shows the conspicuous absence of broadening due to charge exchange between the ion and ground state He even though the exchange

rate is >10''^/s. Key words: gyromagnetic ratio; helium; ion; metastable state; precision measurement.

1. Introduction

Precision measurements of the magnetic moment ratios of the simple systems, hydrogen, the free electron spin,

and metastable ^He*(2^Si) have made possible exacting

tests of the theory of the Zeeman effect for such species.

Mass-dependent terms of the theory [1] for hydrogen-like

systems have been checked to within an accuracy of 3 parts in 10^^ through a measurement [2] of the ratio of

5^7 -factors of hydrogen and deuterium. Mass-independent

terms were criticized to 1.3 parts in 10^ by a measure- ment [3] of the ratio of gf-factors of hydrogen and the free electron spin. The theory [4] for the two-electron sys- tems, ^He*, has been compared with an experiment [5] to a precision of 5 parts in 10^. In this paper, we discuss

results of a measurement [6] of the ratio of g'-factors of ^He*, and "^He*. This permits a first precision determination of the gr-factor of a simple atomic ion. It provides

the potential for examination of the Z-dependence of the

theory.

2. Experimental Technique

The experiments were performed in a 100 G, shielded,

electrically shimmed, actively stabilized solenoid which

has a long-term field drift permitting measurements to ~1 part in 10^ and an unaveraged field homogeneity

better than 1 part in 10^ over a typical cell volume of 50 cm^. Initial investigations toward achieving high resolution capabilities for the He* Zeeman resonance used

direct optical pumping of He (1.08 ixm radiation).

Although line widths of -700 Hz (FWHM) were obtained,

a major improvement in resolution was achieved by using

^^Rb optical pumping with a weak, pulsed, electrodeless

rf discharge. A typical pulse width was 50 jxs wdth a re- petitive rate of 240 Hz. See Fig. 1 for a schematic di-

*Work supported by the National Science Foundation.

**1980-81 Visiting Fellow, Joint Institute for Laboratory Astrophysics,

National Bureau of Standards and University of Colorado, Boulder, Colorado 80309.

agram of the experimental apparatus. Both electron

spin-exchange and spin-dependent destructive collisions between Rb and He* contribute to the polarizing/sensing

of the He* resonance. Minimum linewidth for this system was 210 Hz in a cell filled with 14 Torr He.

MAGNETIC FIELD

OUARTERWAVE PULSED DISCHARGE PLATE

Figure 1. Experimental apparatus. A typical "'Rb optical

pumping system is used together urith a pulsed rf discharge to produce the '*He* and '^He" species. Zeeman resonances in- duced by the rf coils are detected as changes in the transmitted

light intensity.

Two effects conspire to degrade resolution when ob- serving the '^He'*^ Zeeman resonance: a lower cell pres- sure and a higher discharge intensity are required to pro- duce and monitor the He^ ion. The upper limit to usable

cell pressure is determined by the three-body collision process

He+ + 2He Heg" + He

.

At a pressure P = 2 Torr, this reaction contributes

—85 Hz to the width of the He"^ Zeeman resonance. However, of even more critical importance is the effect this reaction has on the density of He*. Calculation shows the

equilibrium density ratio [He'^]/[He2*] is proportional to with a ratio of 1:1 occurring at the surprisingly low

229
pressure of about 2.8 Torr. At 6.0 Torr, this ratio is re- duced to 1:10 showing how rapidly signal is lost with in- creasing pressure. Cells with pressures from 1-2 Torr

have been used for the work reported in this paper. The observed He"^ Zeeman resonance line width of 800 Hz is accounted for as follows: —450 Hz, spinexchange with electrons; 148 Hz, diffusion; 85 Hz, threebody collisions with He; 80 Hz, pulsed-discharge lifetime

width; remainder due to field inhomogeneity, spinexchange with Rb, and rf saturation. The known charge

exchange cross section [7] for collisions between He"^ and

He ground state atoms permits calculation of the charge

exchange rate F ~ 4 x 10^ /s. Since the major causes of observed linewidth have been accounted for without in- clusion of charge exchange, we conclude that such colli- sions contribute to the linewidth at most a small fraction

of the collision rate. This is attributed to the absence of a

nuclear moment for the ^He^ species.^

For the same reason, one finds the ^He'' Zeeman reso- nance linewidth (400 Hz in a 2 Torr cell) is accounted for without inclusion of a component due to the known metastability exchange cross section [8]. The He* exchange

rate with ground state He is calculated to be —10^ /s. The

coherence of the rf transitions for the He* and He*

species in the presence of such exchange collision rates is impressive!

Figure 2 shows the resolved resonances of the free electron spin, ^He*, and ^He"^ in a 100 G magnetic field (frequency —280 MHz). In order to display the broad

electron resonance, the rf level used saturated all reso- nances. Under conditions which optimize the electron

spin resonance, linewidths of —320 Hz have been obtained for the electron: in Fig. 2, the electron spin reso- nance linewidth is —5 kHz.

FREQUENCY

Figure 2. Resolved Zeeman resonances for the ^He'^, ^He*, and

free electron spin (e~) species. These resonances appear in the vicinity of 280 MHz in the 100 G applied magnetic field. A sat- urating rf level was used to enhance the broad e" resonance. Separations in ppm are shown.

^In Ref. [7] the Zeeman resonance of "^He"*" was observed with typical linewidths of 800 kHz at pressures of ~0. 1 Torr.

In Fig. 3 a higher resolution scan shows the '*He'^ and

''He* resonances, again in a 100 G magnetic field. Data

acquired for determining the ratio of (/-values for these

species were taken by sampling fifteen frequencies

spread over four linewidths for each resonance. These

thirty frequencies are interleaved so that each resonance

is interrogated alternately. Several full cycles are used

for signal averaging. A nonlinear least-squares routine

fitting to a Lorentzian line shape with baseline and asymmetry yields the line-shape parameters. The ratio of gj

factors is obtained directly from the frequency ratio of the fitted resonance centers.

He (2 )

He + {|2S|/2)

•**•

22 26 30 34 38

FREQUENCY (kHz)-280200kHz

Figure 3. High resolution scan of^He'^ and ''He* Zeeman reso- nances in a 100 G magnetic field. The respective linewidths (FWHM) are 0.8 and 0.k kHz and their separation is 8.37 kHz.

3. Results

Two systematic effects have been identified which tend

to limit the precision attainable in the measurement of

g'j(He"^)/g'j(He*). The first is a shift in the He"^ Zeeman

resonance frequency as a function of the polarization and

intensity of the Rb pumping light. The resonance fre- quency for a" polarization differs from that for a"^ by

—1 X 10"'. However, the average of the data for the two

polarizations as a function of light intensity has zero

slope. Thus such averaging appears to account correctly

for this systematic effect. The shift is interpreted as due

to spin-exchange. Its small size is partially attributed to the low polarization of all the species in the pulsed

discharge afterglow.

The second systematic effect is associated with a small

asymmetry of the He* resonance. This has been a characteristic common to all cells used for our He* work, even

those with He pressure ^ 5 Torr where a He'*' signal was

not observed. Since the He* resonance linewidth is a measure of the discharge intensity, an extrapolation of the

values of 5fj(He'^)/^7(He*) versus He* Hnewidth is used to suppress this effect. However, in the low pressure cells required to observe the He* resonance, the asymmetry

appears to be smaller. This is consistent with the low polarization in such cells and a theoretical explanation [5] of the asymmetry as due to spin-exchange coupling between

the nearly degenerate electron-spin resonance and the

230
He* resonance. No other resonances in the vicinity of He*

have been found.

The current experimental result is gjC^Ee^ l^SyzygjC^He, 2%) - 1 - 29.95(10) x 10"^

The error quoted is non-statistical and is estimated to cover possible remaining systematic effects. By using the previously determined ratios gj(He*)/

QjiRh) [5] and gj{'Rh)/g {e~) [3], the following can be obtained:

gjme\ l^Sy2)/gie-) - 1 - 70.87(10) x IQ-e.

We stress that this is a purely experimental result found

without the use of any theoretical ratios. The theoretical

value [1] for this ratio is

gj{Re+)/g{e-) = 1 - (Za)2/3 + (a/47T)(Za)2 + (m/2M){Zaf + . . . = 1 - (71.002 - 0.124 - 0.015 + . . . ) X 10"*' = 1 - 70.863 X 10 ^

Thus, the experiment confirms the Breit term (Za)^/3 to vdthin 1.4 parts in 10^. We look forward to improvements in the apparatus in- cluding an increased magnetic field and laser pumping

which should permit a precision of at least a few parts in

10^. This will permit a determination of the radiative

correction term (a /4it)(Z a)^ to about 10%

.

References

[1] H. Grotch and R. A. Hegstrom, Phys. Rev. A 4, 59 (1971).

[2] F. G. Walther, W. D. Phillips, and D. Kleppner, Phys. Rev.

Lett. 28, 1159 (1972).

[3] J. S. Tiedeman and H. G. Robinson, Phys. Rev. Lett. .39, 602

(1977).

[4] P. G. Barkley and R. A. Hegstrom, Phys. Rev. A 14, 1574

(1976).

[5] G. M. Keiser, H. G. Robinson, and C. E. Johnson, Phys. Rev. A 16, 822 (1977).

[6] C. E. Johnson and H. G. Robinson, Phys. Rev. Lett. 45, 250

(1980).

[7] M. Pinard and M. Leduc, J. Phys. (Paris) 35, 741 (1974).

[8] J. Dupont-Roc, M. Leduc, and F. Lalbe, Phys. Rev. Lett. 27, 467 (1971).

231

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Determination of the Neutron Magnetic IVIoment

G. L. Greene* and N. F. Ramsey

Harvard University, Cambridge, MA 02138

W. Mampe

institut Laue-Langevin, 38042 Grenoble, France

J. M. Pendlebury and K. Smith

University of Sussex, Falmer, Brighton, BNI 9 QH, United Kingdom

W. B. Dress and P. D. Miller

Oak Ridge National Laboratory, Oak Ridge, TN 37838

Paul Perrin

Centre d' Etudes Nucleaires, 38042 Grenoble, France

The neutron magnetic moment has been measured with an improvement of a factor of 100 over the previous best measurement. Using a magnetic resonance spectrometer of the separated oscillatory field type capable of determining a resonance signal both for neutrons, and for protons in flowing H2O, we

find (ViXp = 0.68497935(17) (0.25 ppm). The neutron magnetic moment can also be expressed without

loss of accuracy in a variety of other units. Key words: flowing water NMR; magnetic resonance; neutron magnetic moment; particle properties.

1. Introduction

The realization that the neutron, a neutral particle,

possesses a non-zero magnetic moment has been of considerable importance in the development of nuclear and

particle physics. Even before any explicit measurement

of |jLn had been made, there was a strong indication that

fXn ^ 0 from comparisons of the proton and deuteron

magnetic moments p-p and ix^. If the neutron and proton

combine in a pure ^Si state to form the deuteron, one

vi^ould expect p.^ = |jLn -I- |i,p. Early measurements of |Xp

and |Xd gave, according to this relation, |jLn ~ -1.8 p,N where p-N is the nuclear magneton.

According to Dirac theory, which assumes structureless, spin 1/2 particles, p-n = 0 and p-p = p-N. The anomolous moments are defined as the differences between the

actual moments and the Dirac moments. The first serious

attempt to explain the anomalous moments of the nucleons was made by Frohch, Heitler and Kemmer [1] in 1938. Their "meson exchange theory" predicted equal

magnitudes and opposite signs for the anomolous proton

and neutron moments [2]. The actual magnitudes of the

moments are much more difficult to obtain using this theory. Though subsequently refined by many others [3] the initial conclusions of Frolich, Heitler, and Kemmer

adequately summarized the state of the theoretical un- derstanding of the neutron magnetic moment until the

introduction of the quark model.

The first explicit measurement of the neutron magnetic

moment was made by Alverez and Bloch [4] in 1940. Us-

*Current address: Gibbs Laboratory, Physics Department, Yale

University, New Haven, CT 06520

ing Rabi's magnetic resonance technique vdth a neutron

beam obtained by deuteron bombardment of Be, they ob- tained a value of | Pn | = 1-93 ± 0.02 px. In general, no

information about the sign of a magnetic moment is ob- tainable from a resonance experiment utilizing a purely

oscillating magnetic field. (In 1949 Rogers and Staub [5] determined the sign of p^ using a resonance technique

that employed a rotating field. As had been thought, the

sign was negative.)

The discovery of Kellog, Rabi, Ramsey, and Zacharias

[6] that the deuteron has an electric quadropole moment

implied that the ground state of the deuteron could not,

in fact, be accurately described by a pure '^Si state but

must have some admixture of D state. Thus the additivity of neutron and proton moments in the deuteron

could not be exact. It was therefore of great interest to determine p.n more accurately, so that it could be com- pared with the well known values of p,p and p-a- In 1947 Arnold and Roberts [7], using a neutron beam

from a reactor, determined p,n in a resonance experiment

with a single oscillatory field. They were able to obtain a more accurate value of |p-n| = 1.913(12)p.N by using

the then novel technique of NMR as a means of field cali- bration. About the same time. Block, Nicodemus, and

Staub [8] also obtained a more accurate value for p-p. They reported their result as the ratio of neutron to pro- ton moment |p-n/M-pl = 0.685001(30).

With these more accurate values for p^, it was possible

to determine the ^Di admixture in the deuteron gi'ound

state by two independent means; through the deuteron

electric quadropole moment, or by comparison between

p,n, p-p and p-d- Gratifyingly, these independent values were consistent.

233
With the introduction of the quark model an appealing

explanation for the value of the ratio of the neutron mo- ment to the proton moment emerged. Sakita [9] and

Beg, Lee, and Pais [10] pointed out that this ratio should

be M-n/jJ^p = -2/3 [11] if the internal symmetry SU(3) is broken only by electromagnetism. In this model, the ratios

of all the baryon moments are uniquely determined

and easily calculated. (Their absolute magnitudes are dependent on the quark magnetic moments. Determining

these would require knowledge of the quark masses.) The

agreement between theory and experiment for |JLn/|Xp is considerably better than for other baryon pairs and is striking for such a simple theory. The approximately 3%

discrepancy between theory and experiment should not be viewed as a significant disagreement. SU(3) (S* SU(2)

(i.e., 3 similar quarks with spin 1/2) is known to be an in- complete description for strongly interacting particles.

At the initiation of the work reported here, the uncertainty associated Vvdth theory far exceeded the experimental error of the best measurement of ix^ [12]. As a

result, theoretical considerations did not provide the

dominant motivation for the current work. Rather, the

possibility of a substantial improvement in our knowledge of a fundamental particle property as well as the opportunity to demonstrate an elegant, new experimental technique prompted our effort. The most accurate determination of ix^, prior to the

work reported here, was that of Corngold, Cohen, and

Ramsey [12]. This experiment involved a thermal neutron beam and benefitted from the use of the separated

oscillatory field method. A result of lix^/M-pl =

0.685039(17) was obtained. The major limitation to the

accuracy of this measurement arose from the method

used to calibrate the magnetic field. An NMR probe of

sm.all dimensions was employed to determine the field at a number of discrete locations. This field map was then

used to give the appropriate field average by a pointwise

integration. The discrete nature of this stepwise integration led to the dominant error in the experimental result. The current work benefitted from several advances in experimental technique. Access to the high flux reactor

of the Institut Laue-Langevin at Grenoble, France, al- lowed the use of an intense "cold" neutron beam. Such

"cold" beams are characterized by low beam velocities. This led to an improvement of a factor of ~5 over the line width obtained by Cohen, Corngold, and Ramsey [12]. An

additional advantage to the use of "cold" neutrons lies in the abihty to employ neutron guides to reduce beam

divergence and therefore increase the flux. Such guides

also allowed a novel field calibration technique which pro- vided the major improvement in experimental sensitivity.

Following a suggestion of Purcell [13] made some time

ago, the field was monitored by obtaining a separated os- cillatory field resonance signal with water flowing

through the neutron guide. In this fashion, the field average taken by the protons in the water is the same as that taken by the neutrons. Since all the relevant di- amagnetic corrections for protons in water are known to high accuracy, a direct measurement of the ratio of the

neutron magnetic moment to the proton magnetic mo- ment was obtained. From the result, the magnetic mo- ment of the neutron can be expressed in a variety of other units.

2. Apparatus

The aim of the experimental technique was to measure

the Larmor precession frequency for neutrons and pro- tons (in water) in precisely the same volume as nearly

simultaneously as possible. With the use of a neutron

guide (a glass tube of circular cross-section) it was a straightforward matter to confine neutrons and water in the same volume. One simply let the water flow through

the neutron guide. This procedure, however, precluded

the possibility of determining the resonance frequency

for neutrons and protons with exact simultaneity. To in- sure that no spurious effects resulted from drifts during

the time required to make a change-over from neutron

beam to flowing water, a second, parallel tube was in- stalled in the spectrometer. Water was continuously sent through this tube to provide a field monitor. By alternat- ing neutrons and protons in the principal tube while pro- tons flowed continuously in the monitor tube, errors due

to field drifts could be detected and reduced to an insig- nificant level. This technique has been discussed in detail elsewhere [14]. The cold source at the high flux reactor of the Institut

Laue-Langevin in Grenoble provided the source of neutrons used in this experiment. The flux through the ap- paratus was typically 1.5 x 10^ neutrons /s. the neutrons were polarized by glancing reflection from a magnetized

mirror. A similar mirror was used to analyze the neutron

polarization at the exit of the spectrometer. The neutrons were detected by a ^Li loaded glass scintillator cou- pled to a photo-multiplier.

The protons in the flowing water were polarized by pas- sage through a chamber placed in a high (~2 kG) pre- polarizing magnet. As the protons spent more than one

longitudinal relaxation time in this field, their polarization

could be approximated by a simple Boltzmann distribution.

While the degree of polarization so obtained was very

small, the enormous flux (by molecular beam standards)

led to an easily detected signal. The proton polarization at exit was detected in a second high field region (~4 kG)

with slightly modified commercial NMR equipment.

The spectrometer was based on a high stability, high

homogeneity, large volume electromagnet. The nominal

field in the magnet was 18 G. The separated oscillatory

field coils (each 3 cm long) were mounted in the magnet

with a separation of 61 cm. Surrounding each of the

separated coils were independently controllable trim coils to adjust the fields at the coils. The magnet and spectrometer were mounted in a two

layer cylindrical molypermalloy magnetic shield to reduce

the effects of time dependent external fields. The entire

assembly was mounted on a large rotating platform. This

allowed the orientation of the spectrometer with respect

to beam velocity to be reversed.

3. Results r

Figures 1 and 2 show typical neutron and proton reso- nance lineshapes obtained in this experiment. The dra- matic difference in line widths (—120 Hz vs —0.8 Hz) for neutrons and protons is due to the great difference in particle velocities for the two cases. The neutron velocity was —150 m/s while the proton velocity was —1 m/s. The pres- ence of many more side lobes in the proton resonance is in- dicative of a narrower relative proton velocity distribution.

The actual quantity measured in this experiment was

the ratio between the Larmor precession frequencies for neutrons in vacuum and for protons in a cylindrical sample of water at 22 °C. To arrive at this quantity, corrections for several shifts and distortions had to be made.

The largest systematic effect resulted from the BlochSiegert effect [15]. This effect may be viewed as arising

234
from the counter-rotating component present in an oscil- lating field. It is a curious feature of the separated oscil- latory field technique that the shift resulting from this effect is not strictly linear in oscillating field power [16]. However, this non-linearity is well understood and can be simply accounted for. Figure 3 illustrates the extrapolation technique which was used to account for the

Bloch-Seigert shift. The data shown in Fig. 3 were taken in the two possible machine orientations. These correspond to a reversal

of the spectrometer with respect to the beam velocity.

The shift seen between the two orientations is presumably due to geometrical imperfections in coil construction. It should be noted that the shifts seen in Fig. 3

arise entirely from the distortion of the neutron reso- nance. The much lower velocity of the protons, and the

consequent reduction in oscillating power, implies negligible shifts in the proton resonance frequency.

150r

Table 1 summarizes the significant shifts and errors in the measurement of the ratio of neutron moment to pro- ton moment in a cylindrical sample of H2O at 22 °C, o>n/a>p (cyl, H2O, 22°). The errors quoted (1 a) in Table 1 are in- dependent and are added in quadrature to yield the error

in Wn/ojp (cyl, H2O, 22°). Our experimental result is

co„/wp (cyl, H2O, 22°) = 0.68499588(16) (0.24 ppm) (1)

11

130

§ 110

90

70

c

D

>xo

o

Ott- resonance counting rate —

O

c

in

i_0)N

a

c

o

9

8

7

6

5

4

CL in

Full polarisation

,Av = 0 8 Hz

\ h i\ / . \ A N

tit/

+ + V

1

+ +

I ^1 No polarisation-

+

J L _L

51.5 52.0 52.5

Frequency (kHz)

53.0 53.5

Figure 1. Typical neutron resonance.

U UQ 50 52 54 56 58 60

v-76400 (Hz)

Figure 2. Typical proton resonance.

X Orientation 1 Orientation 2

° Mean between 1 ,

2

Linear least square best fit

I I I 0 0.1 0.2 0.3 O.A 0.5 0.6 0.7 0.8 0.9 1.0

Figure 3. Plot of neutron to proton resonance frequencies. The quantity

K.(-nI /Alg) (I/Io) is an appropriately scaled oscillating power which yields a

linear fit. (See references in Ref. [16] for a full description). The two onentations correspond to reversals of the machine axis with respect to the beamvelocity.

235
Table 1. Significant shifts and errors in Wn/wp (cyl, H2O, 22°). Statistical error is included in the error assigned to the Bloch-Siegert effect and coil phase errors.

Pmnortinnal X ^\J^J\J^ kj^kjiic^l shift oiii^v FVoportiona,! error

Bloch-Siegert effect +4.94 X 10-6

13 X 10-8

Coil phase errors -1.6 X lO"*' Field inhomogeneity <3 X 10"'^ < 4 X 10 °

Effects of finite -10-^ -10 X 10-8

velocity of distribution External diamagnetic -4.8 X 10"'^ <3 X 10-8

effects of water

The value of the magnetic moment of the neutron can be expressed in terms of the Bohr magnetron (xb with lit- tle loss of accuracy through the relation

cop(cyl,H20,22°)

cop(cyl,H20,22°) gjpCsph, H2O, 22°)

(Op (sph, H2O, 22°) cop (sph, H2O, 35°)

(2)

where the notation "sph" refers to a spherical sample,

and |jLp is the effective proton moment in a spherical sample of H2O at 35 °C as measured by PhilHps, Cook, and

Kleppner [17]. The first ratio in the right hand side of Eq. 2 is our ex- perimental result. The second ratio is given by 1 -

2 ttk/3 = 1 1.505(2) x IQ-*^ from the data summarized

by Pople et al. [18] where k is the volumetric susceptibility of water. The third ratio is taken to be 1 -I- 1.4(1) x

10 ~^ from the results of Hindman as interpreted in a foot- note in Philhps et al. [17]. The final ratio of |jl'p/|jlb is given

by Phillips et al. [17]. From Eq. 2 we conclude. = -1.04187564(26) x 10-^ (0.25 ppm) . (3)

The slight increase in experimental error arises from the

temperature correction in Eq. 2. Using the results of Winkler, Kleppner, Myint, and

Walther [19], we can express (x,, in terms of the free elec- tron moment (Xe and free proton moment |Xp by

|Xe = 1.04066884(26) x 10^^ (0.25 ppm) (4)

= -0.68497935(17) (0.25 ppm).

Our result can also be expressed in terms of the nuclear

magneton In this case, however, the acccuracy is shghtly degraded by the uncertainty in the electron to proton mass ratio. Using the value of m/M obtained by

combining results from Phillips et al. [17] and Cohen and

Taylor [18] we have

M-N = -1.91304184(88) (0.45 ppm) (5)

Note added in proof: The recently reported value for the proton to electron mass ratio of Van Dyck and

Schvdnberg [these proceedings and Phys. Rev. Lett. 47,

395 (1981)] of M/m = 1836.15300(25) can be combined

vdth the results reported here to give the value (Xn/}JLN —

-1.91304308(54) (0.28 ppm).

We wish to express our thanks to the director and staff of the Institut Laue Langevin for their assistance and

hospitality during the course of this experiment. This

work was supported in part by the Department of Energy and the National Science Foundation (U.S.A.) and

Le Department de Recherche Fundamental du Commissariat a I'Energie Atomique (France).

References

[1] H. Frolich, W. Heitler, and N. Kemmer, Proc. R. Soc. London, Ser. A: 166, 154 (1938).

[2] E. Fermi, in his book Nuclear Physics (University of Chi- cago Press, Chicago 1951) gives a simple explanation of the meson exchange model, though he evidently did not think highly of the theory, calling it a "naive little fantasy"

(p. 13).

[3] See G. L. Greene et al., Phys. Rev. D 20, 2139 (1979) for a more complete bibliography.

[4] L. Alverez and F. Bloch, Phys. Rev. 57, 111 (1940).

[5] E. H. Roger and H. R. Staub, Phys. Rev. 76, 980 (1949).

[6] J. M. B. Kellog, I. I. Rabi, N. F. Ramsey, and J. R. Zacharias, Phys. Rev. 57, 677 (1949).

[7] W. Arnold and A. Roberts, Phys. Rev. 71, 878 (1947).

[8] F. Bloch, D. B. Nicodemus, and H. R. Staub, Phys. Rev. 74, 1025 (1949).

[9] B. Sakita, Phys. Rev. Lett. 13, 643 (1964).

[101 M. A. Beg, B. W. Lee, and A. Pais, Phys. Rev. Lett. 13, 514

(1964).

[11] See also R. D. Young, Am. J. Phys. 41, 472 (1973) for a simple vector addition argument which gives the -2/3 result. This argument was first suggested by Beg et al. [121 V. W. Cohen, N. R. Corngold, and 'N. F. Ramsey, Phys. Rev. 104, 283 (1956).

[13] E. M. Purcell, private communication (1952).

[14] G. L. Greene, Ph.D. Thesis, Harvard University (1977), available as Institut Laue Langevin Technical Report

77GR2065; G. L. Greene et al., Phys. Rev. D 20, 2139

(1979).

[15] F. Bloch and A. Siegert, Phys. Rev. 57, 522 (1940).

[16] J. H. Shirley, J. Appl. Phys. 4, 783 (1963); R. F. Code and N. F. Ramsey, Phys. Rev. A 14, 1945 (1971); and G. L. Greene, Phys. Rev. A 18, 1057 (1978).

[17] W. Phillips, W. E. Cook, and D. Kleppner, Phvs. Rev. Lett.

35, 1619 (1975).

[18] J. A. Pople, W. G. Schneider, and H. J. Bernstein, High

Resolution Nuclear Magnetic Resonance (McGraw Hill, N.Y., 1959).

[19] P. F. Winkler, K. Kleppner, T. Mvint, and F. G. Walther,

Phys. Rev. A 5, 83 (1973).

[20] E. R. Cohen and B. N. Taylor, J. Phys. Chem. Ref. Data 2, 663 (1973).

236
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Precision Exotic Atom Spectroscopy

Vernon W. Hughes

Gibbs Laboratory, Physics Department, Yale University, New Haven, CT 06520

Precision measurements by microwave or laser spectroscopy techniques of the energy levels of exotic atoms containing particles such as positrons, muons, or pions not present in ordinary atoms allow the determination of the fundamental constants associated with these so-called exotic but basic particles. Moreover, the simplicity of some of these atoms, such as positronium (e^e") and muonium (ix^e"), which consist only of structureless leptons, allows precise tests of quantum electrodynamics which is basic to our understanding of many of the fundamental constants, especially the fine-structure constant, a. This paper reviews recent work and work in progi'ess on positronium, muonium, and simple

muonic atoms relevant to the precision measurement-fundamental constants field. Key words: muonic atoms; muonium; positronium; quantum electrodynamics.

1 . Introduction

In the context of this "Second International Confer- ence on Precision Measurement and Fundamental Constants" the topic of precision exotic atom spectroscopy

can offer several contributions.

Firstly, since by definition an exotic atom contains particles such as positrons, muons, or pions not present in or- dinary atoms, which consist of electrons and nuclei, study

of exotic atoms will clearly provide the opportunity to determine the fundamental constants associated with

these so-called exotic but basic particles. Hence, for example, the masses of the negative muon and the negative pion

and the magnetic moments of the positive and negative muons are determined from exotic atom spectroscopy.

Secondly, the simplicity of some of the exotic atoms,

such as positronium (e'^e") and muonium (|jL"^e~) which

consist only of structureless leptons, provides an ideal opportunity for precise tests of the fundamental theory

of quantum electrodynamics which is basic to our understanding of the fundamental constants. Muonic atoms can

in general be considered as hydrogenic in the sense that

to a good approximation the atom can be treated as a sin- gle [x" bound to the nucleus because the outer electrons

are relatively unimportant. Thus they provide a simple

system for studying the basic interactions. An outstanding example is the truly hydrogenic atom djL^^He)"^

where a critical test of electron vacuum polarization has

been made. Relatively low Z muonic atoms such as \x~ P

and Mg have also been used to study electron vacuum

polarization and to establish limits to exotic muonnucleus interactions. Since the muon in a muonic atom

has higher momenta than the electron in an electronic

atom, tests of quantum electrodynamics can be extended

to significantly higher momenta or correspondingly

shorter distances with muonic atoms.

Finally, some of the experimental techniques used to study exotic atoms are quite different from those used for ordinary atoms. These include accelerators to produce the

exotic particles, precision measurements of higher energy

x-rays or 7-rays, and particle detectors. These different

techniques broaden the field of precision measurement.

Since this conference is focused on the fundamental

atomic constants and quantum electrodynamics, I will

discuss principally the atoms most relevant to these topics which are positronium, muonium, and simple muonic

atoms. However, brief mention of hadronic atoms and some other matters will be made.

2. Positronium

Since its discovery by Deutsch [1] in 1951, positronium

has served as an important testing ground for elementary

particle physics and fundamental interactions. These

tests include precise spectroscopic measurements of fine structure intervals, measurements of annihilation rates, and studies of the symmetries associated with the

electron-positron system. In this paper we treat only the

precise energy level measurements; a recent review arti- cle by Rich [2] covers thoroughly the topics of annihilation rates and symmetries.

The energy levels of the n — 1 and n - 2 states of positronium, including its fine structure intervals^ are in- dicated in Fig. 1. Theoretical values to order a-R^z are given by [3]:

E («,L,S,J)

+

11

^LSJ 2L + 1 R:

t(L,S=0,7) - 0 .

^{L.S=1,J)

1. 1-8l +

6 2(2L +1)

3L +4

(L + 1)(2L + 3)

1 L (L +1)

3L -1

L(2L - 1)

if/ = L + 1

if/ = L

if/ = L - 1

(1)

where a is the fine structure constant; /2x is the Rydberg

constant for the electron; and L, S, and / are the quan237
turn numbers for orbital, spin, and total angular momentum, respectively.

Gross

St ru cture

POSITRONIUM

F i ne

Structure

'I r< L

9GHz

n = 2

2 I,

2430 A

(5.1 eV )

I

I

I X

T

I

I

I n = I

t 203GHz

Figure 1. Energy levels of the n = 1 and n = 2 states of positronium.

For the n = 1 state the virtual radiative contributions

to the hfs interval Av between T^Si orthpositronium and

l^So parapositronium have been calculated [4] through

relative order a^lna^', as indicated in Eq. (2):

j? a

[6 tr

+ ^a^\na-^ + Kcc^ - K'—ana^f\ . (2)

Some of the Feynman diagrams contributing to the terms

vdth coefficients K and K' have been evaluated, but complete calculations of these terms remain to be done. Using values of the atomic constants given in Table 1, we

obtain:

Av = (203 339.1 + 10.8 K - 2.1 K') MHz. (3) For the n - 2 state virtual radiative contributions to the fine structure intervals of relative order a, or equivalently to the energy levels of order a^i2xC, have

been evaluated [5]. Hence for the only fine structure in- terval that has been measured, the theoretical value is given by:

^(23Si) - E(2^F2) = -^a^cR^a + 3.766 a)

4o0 = 8625.14 MHz . (4)

The measurement of the hfs interval A v in the ground

state involves the Zeeman effect and utilizes the Zeeman

transition indicated in Fig. 2 [6-10]. The energy of the

M = ±1 states is independent of magnetic field H be- cause of the equality of the magnitudes of the magnetic

moments of e" and e"^ as required by CPT invariance.

The energies of the M = 0 triplet and M = 0 singlet states have approximately a quadratic dependence on H,

and the frequency of the indicated transition between

M = 0 and M - ±1 triplet states is given by

foi - ^[l + x^)^ - 1] , (5)

Table 1. Relevant atomic constants.

= 1.097 373 152 1 (11) x 10^ cm"! (0.001 ppm)^

c = 2.997 924 580 (12) x lOio cm s'^ (0.004 ppm)'' = 137.035 963 (15) (0. 11 ppm)*^

m Jm _

e e = 1 ± 1.3 X 10 ' ° m Jm _ = 206.768 259 (62) (0.3 ppm)^ = 1.521 032 209 (16) x 10"^ (0.01 ppm)^ ^^^^

= 3. 183 345 47 (95) (0.30 ppm)^

Oe = 1 159 652 200 (40) x IQ-'^ g = 1 165 924 (8.5) x lO'^ h

^S. R. Amin, C. D. Caldwell, and W. Lichten, Phys. Rev. Lett. 47 1234 (1981). ^E. R. Cohen and B. N. Taylor, J. Phys. Chem. Ref. Data 2, 663

(1973).

"^E. R. Williams and P. T. Olsen, Phys. Rev. Lett. 42, 1575 (1979). ^Ref. [16].

•^V. W. Hughes and T. Kinoshita, in Muon Physics, Vol. I., Ed. by

V. W. Hughes and C. S. Wu (Academic Press, New York, 1977), p.ll.

Average value from foUovidng two references:

(i) E. Klempt, R. Schulze, H. Wolf, M. Camani, F. N. Gygax, W.

Riiegg, A. Schenck, and H. Schilhng, Phys. Rev. D 25, 652 (1982).

(ii) Ref. [22]. SH. Dehmelt, in Atomic Physics 7, Ed. by D. Kleppner and F. M.

Pipkin (Plenum Press, New York, 1981), p. 337; also P. B. Schwin- berg et al., these proceedings.

^'J. Bailey, K. Borer, F. Combley, H. Drumm, C. Eck, F. J. M.

Farley, J. H. Field, W. Flegel, P. M. Hattersley, F. Krienen, F. Lange, G. Lebee, E. McMillan, G. Petrucci, E. Picasso, 0. Runolfsson, W. von Ruden, R. W. Williams, and S. Wojcicki, Nucl. Phys. B

150, 1 (1979).

6.0

Figure 2. Zeeman energy levels of positronium in its ground

state. where x = 2\i.%g'H/(h^v), in which, ^j^' is the g'-value of

the electron in positronium. The annihilation of the ground

'So state results in the emission of two 511 keV photons

and occurs at the rate Xp = 0.80 x 10 10 whereas the

annihilation of the ground ^Si state results in the emission

238
of three photons whose energies sum to 2?wc" and occurs

at the rate Xo = 0.72 x 10'^ s"'. In a magnetic field the

M = 0 triplet state has an admixture of A/ = 0 singlet state and hence can decay either by two photon or three

photon emission, and the annihilation rate is Xio — X„ +

(x-/4)A.p in which the two photon part becomes important even at small x because Xp » \„. Analogously, in a magnetic field the Af = 0 singlet state has an admixture of M

= 0 triplet state but the three photon annihilation mode

is generally unimportant. The principle of the resonance experiment rests on the unequal population of the M = 0

triplet state and the M = ±1 triplet states in a magnetic

field and on the increase in two photon annihilation which

results from induced transitions from the M = ± 1 to the

M = 0 triplet states. The experimental arrangement of the present Yale ex- periment [8, 10] is shown in Fig. 3. A ^Na positron source

of about 15 mCi is placed inside a microwave cavity resonant at 2323 MHz and filled with N2 gas to a pressure

between 0.25 and 3 atm. Eight Nal(Tl) detectors count in coincidence 0.5 MeV annihilation 7 rays emitted at 180°. The magnetic field of about 8 kG is varied across the reso- nance line as indicated in Fig. 4. Analysis of the resonance

line has been based on the Hamiltonian for positronium in a magnetic field Hq + Hisirnxit in which Hq\s the static magnetic field and H 1 sinw^ is the time dependent mi- crowave magnetic field of amplitude Hi and angular fre- quency 00. The annihilation interaction has been treated by

introducing decaying amplitudes for the states. Figure 5

displays the principal pubhshed values of Av since the

original microwave resonance experiment of Deutsch and

Brown, and Table 2 includes the two most recent measured values together with the theoretical value. Agreement between theory and experiment is satisfactory

vdthin the present accuracy of the theory. Clearly values

of the coefficients K and K ' are needed to allow a more

sensitive comparison of theory and experiment.

Recently Rich [11] has pointed out that the neglect of

off-diagonal matrix elements of the annihilation interaction may introduce significant error in the determination

of Av.

Further improvement in the experimental accuracy of determining Av should be possible by straightforward

technical approaches together with improved statistics. Such an experiment is in the stage of final data analysis

at Yale and a reduction of the error to about 3 ppm is anticipated.

Although the ratio of line center frequency to line width is the same for the Zeeman transition which has

been studied as for a direct Av transition at zero magnetic field, it would nonetheless be valuable to measure

the direct A v transition. The A v frequency of 203 GHz is inconveniently high, but adequate microwave power

sources of the carcinotron type are now available. The use of line narrowing techniques may offer a hope

of an improved measurement, but as yet no practical ex- periment has been proposed.

The beautiful measurement of a fine structure interval

in the n = 2 state of positronium by the Brandeis group

[12-14] was made possible by their discovery that slow

positrons (kinetic energy — 100 eV) striking certain solid surfaces in vacuum have an appreciable probability of re- bounding as positronium in the 2S state in vacuum. Fig- ures 6 and 7 show the Brandeis apparatus and relevant

decay channels of the n - \ and n = 2 states. Positrons

of 30 eV are incident on a Cu target, and produce Ps in the 2S state. An electric dipole transition is induced by a microwave field from the 2^Si to the 2^P2 state, which

SCALE IN INCHES Figure 3. Sdiematic diagram of the experimental apparatus.

X

0.210 0.212 0.214 0.216 0.218 0.2 20

I

1

'

1

'

1 r—, 1

1

•

1

680

(coun 660

IGNAL 640

CO 620

_l , I

I 1 . 1 . 1—

7.6 7.7 7.8 7.9 8.0 H(kG)

Figure 4. Typical data for a resoTiance pass fit to the theoreti- cal lineshape. foi - 2323.36i MHz, 0.31 atm N2.

1952 I • i

1 200ppm MIT

1957 I • 1

I 200ppm COLUMBIA

1970 ^j • 1 58ppm YALE

1972 I 1—^ 24ppm YALE

1975 8ppm 8RAN0EIS

1977 1-*-}h 20ppm YALE

1977 w I 6ppm YALE

I 1 1 . 1 . 1 . 1 I I I I 203.28 203.30 203.32 203.34 203.36 203.38 203.40 203.42 GHz

Figure 5. History of measurements of ^.v of positroniim.

then emits a -Lyman a photon of 2430 A in the spontane- ous transition to the PSi state, followed by the three 7

annihilation of this state. The signal shown in Fig. 7 is the increase in the Lyman a photon-annihilation 7 de- layed coincidence counts due to the microwave field. The

experimental error in measuring the line center is largely

statistical, but the Zeeman effect and motional Stark ef- fect also contribute significantly to the uncertainty in determining the 2^Si to 2^P2 interval. Table 2 gives the

experimental and theoretical values of the 2^Si to 2^P2 in- terval. Agreement is satisfactory and constitutes a test

239
Table 2. Experimental results for positronium

fine structure intervals and comparison ivith theory.

= 1 state; Av (l^^Sj - I^Sq)

Experimental value 203 384.9 (1.2) MHz (6 ppm)^

203 387.0 (1.6) MHz (8 ppm)''

Theoretical value 203 400.3 MHz'^

Avth - Avexp = 14.6(1.0) MHz

n = 2 state; v(2^Si - 2^P2)

Experimental value

(8 628.4 ± 2.8) MHz^

Theoretical value 8 625.14 MHz^

vth - vexp = -3.3(2.8) MHz

^Ref. [9]. ''A. P. Mills and G. H. Bearman, Phys. Rev. 34, 246 (1975).

'^P. Lepage and D. Yennie, these proceedings.

''Ref. [13].

<"Ref. [5]. Note Added in Proof (see Ref. [11]): Corrections to the = 1 state Av

values due to off-diagonal matrix elements of the annihilation interaction have been estimated by Mills. His weighted mean value of the two

corrected measurements is Av = 203.388 5 (10) GHz (5 ppm).

LlNES*-"iJL__

TO II

MICROWAVE ^

SYSTEM

Figure 6. Positron target chamber and microwave cavity. G, 95% -transmission tungsten grid; T, copper target; M, aluminized Suprasil quartz mirror; W, Suprasil quartz window; K,

CsTe photocathode; P, support posts; A^, input antenna; Ao, output antenna; Nal(Tl), annihilation detector coupled to an RCA 8575 phototube. Another Nal(Tl) detector (not shoum) is placed above the chamber directly over the detector shown.

MICROWAVE FREQUENCY (GHi) —

Figure 7. The observed Lyman-a signal S (open circles) and

logarithmic first-difference signal S' (solid circles) as a function of microwave frequency. The inset is the schematic term diagram for the n = 2 and n = 1 Ps states indicating the relevant transitions and the lifetimes j for each level.

of the oL^Ry, term to about 1%. An active program con- tinues at Brandeis to improve the accuracy in the meas- urement of the fine structure of Ps in the n = 2 state. Recently in several laboratories there has been an in- terest in the observation of the 1S-2S two photon transi- tion in Ps. The theoretical value of the energy interval is

[3, 15]

p(23Si - r^Si) = V21

= -cR, 3 p i^cR,

719

768 + Oia^RJ , (6)

in v^^hich the Rydberg constant for positronium, Rp ^

R^ini +/im + + m _)], w^here 2'n^'me*/h^c is the usual

e e e Rydberg constant vdth m = m^_. The equahty of the masses of electron and positron is expected on the basis

of CPT invariance and has been established experimentally to 0.1 ppm [16]. The term 0 ia'^Rp) is the lowest or- der radiative contribution and its principal part is the

Lamb shift in the IS state. A complete evaluation of this term has not yet been done.

The experiment would be similar in principle to the observation of the 1S-2S two photon transition in H by

Doppler-free laser spectroscopy [17, 18]. A principal

problem is to obtain enough Ps atoms during the short

laser pulse. The experimental scheme being considered at Yale is indicated in Fig. 8. Detection would be achieved

through observation of the e^ resulting from photonionization of Ps(2S) in the laser beam.

Note Added in Proof: The l^Si - 2'^Si transition in pos- itronium has recently been observed using two-photon

Doppler free excitation. The observed frequency agrees

with theory to 0.5 GHz [S. Chu and A. P. Mills, Jr., Phys. Rev. Lett. 48, 1333 (1982)].

3. Muonium

Muonium is the hydrogen-like atom consisting of a positive muon and an electron. From the viewpoint of elementary particle physics and fundamental atomic

physics, the motivations for studying muonium are to determine the properties of the muon and to measure the 1 muon-electron interaction in this bound atomic state. Indeed muonium provides an ideal simple system for

| testing modern muon electrodynamics and for searching j

240
TRANSPORT

TUBE

BUNCHING REGION

72" ^

BEAM LINE (NOT TO SCALE)

PION-MUON DECAY

+ + . IT — yli. + V.

IT -o—

(out of popar

I DETECTION APPARATUS

CHAWNELTRON

TARGET CHAMBER

E FIELD PLATE

LASER P« BEAM I L PRODUCTION TARGET

E X 8 VELOCITY SELECTOR

Figure 8. Schematic diagram of the proposed Yale experiment

to measure the IS 23 2-photon transition in Ps by laser spectroscopy.

for effects of weak, strong, or unknown interactions on

the electron-muon bound state. Since its discovery in 1960 [19], made possible by the observation of parity

nonconservation in the tt ^ |x e decay chain (Fig. 9), precision measurements have been made of the hyperfme

structure interval Av in the ground n = 1 state and of the muon magnetic moment [20]. These studies of |JL"^e"

have all confirmed that the muon behaves as a heavy

electron [21]. Muonium is an ideal system to study the electromagnetic interaction of two different leptons and hence to

test muon electrodynamics, which includes the viewpoint

that the muon and electron are elementary, structureless

particles; and also to determine the electromagnetic

properties of the muon, including its spin, magnetic moment, and mass. In addition, a precise value of the fine structure constant a is determined from the hyperfme

structure interval of muonium. Also certain aspects of the weak interactions of the muon and electron can be

tested with muonium. A precise study of a basic system

like muonium, for which the measured quantities can be

calculated accurately, provides criteria for testing new

speculative theories.

Important advances have recently been made in both

our experimental [22] and theoretical [4] knowledge of the hyperfine structure interval A v in the ground n = 1 state of muonium. Closely related improvement has been

achieved in the precision of measurement of the magnetic

moment of the positive muon. The present theoretical value" for Av is given in Table 3. Values of the required atomic constants are given in Table 1. The value for \i.^J\i^ is the weighted

average of the value from the LAMPF experiment [22] and from the ,xSR measurement in liquid Br2 at SIN [23]. There are two principal contributions to the estimated

error in Avth- One is the inaccuracy in the calculation of the radiative correction term 63 and uncalculated

radiative-recoil terms, which contribute an error of about

0.7 ppm. The second is the experimental uncertainty in

|jL^/|Xp of about 0.3 ppm.

The classic muonium microwave magnetic resonance ex- periment, which is possible because of parity nonconservation in the it ^ |x ^ e decay chain (Fig. 9), has recently

e +

I-+(^«+c )<i I +-T cos 0

e"^ " s

TT |j > e decay chain.

Table 3. Theoretical value for muonium Av

and comparison with experiment.

Avth=

16— a- cR x(M-p./M-i ) [1 + im^yn^)]-

1 + — + a e

-I- ej -H €2 + 63 - 5^

de = (ffe - 2)/2: ei = a^(ln2 - 5/2)

^2

3-77 Ina Inct - ln4 281 480

; eg = ^ (18.4 ± 5)

5,, 3a mR m

8 , = m —- + Mr

21na + 81n2 - 3^lo a

2 21n2 m^ 31 , ~

TT I2

28 + Jl! - 19

9 3

where yyi^

A^th

Av,exp

= (4 463 303.7 ± 3.4) kHz (0.77 ppm) = (4 463 302.88 ± 0.16) kHz (0.036 ppm)

Avgxp - Avth = (-0.8 ± 3.4) kHz

been redone at the Los Alamos Meson Physics Facility

(LAMPF) using a much more intense source of stopped muons obtained from a so-called "surface" muon beam

[24]. The energy- level diagi-am of the gi-ound state of muonium in a static magnetic field is shown in Fig. 10. The

general technique was the same as in an earlier experiII 241
X

I 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16

H ( kG)

Figure 10. Energy level diagram for muonium in its l^Sj/^

ground state in a magnetic field.

ment [25]. The two Zeeman transitions (Mj, M = (1/2, 1/2) ^(1/2, -1/2) and (-1/2, -1/2)

^

(-1/2, +1/2) were

observed alternately at a strong static magnetic field of 13.6 kG. The very intense surface muon beam had a mean momentum of = 28MeV/c with a 10% momentum

spread and a polarization of close to 100% , and provided

stopping rates of up to 10^ s~^ average in a 0.5 atm Kr

gas target 20 cm in length for a primary proton beam

current of 300 )xA. The experimental arrangement is shown in Fig. 11. The thin scintillation counter SI monitors the incoming muon beam and the counters S2, S3

detect the decay electrons. A central element in this ex- periment is the high precision solenoid electromagnet

which provides a magnetic field over the region of the mi- crowave cavity which is homogeneous to several ppm and

is stable to about 1 ppm. A typical resonance curve is shown in Fig. 12. It was obtained in a period of about 2 h

by varying the magnetic field under computer control. For

each field point the microwave frequency is changed from

the vi2 to the V34 frequency and also the microwave power

is turned on and off with a modulation period of about

0.16 s. The fitted theoretical line shapes are shown. The

result of our data analysis for the hfs interval A v and for the ratio fx^/|jLp of muon to proton magnetic moments is

Av = 4 463 302.88 ± 0.16 kHz (0.036 ppm)

(x^/lJLp - 3.183 346 1 ± 0.000 001 1 (0.36 ppm) , (7)

in which one standard deviation total errors are given.

Figures 13 and 14 give the histories of the principal

measurements of Av and of |i./(Xp, and indicate that the

published values are in reasonable agreement within

their quoted errors. The agreement of the theoretical and experimental

values of A V is well within the error of the theoretical

value. This agreement provides one of the most sensitive

tests of modern muon electrodynamics.

Active theoretical work is now in progress on Av to evaluate further radiative and recoil contributions to the

level of a few parts in 10^. On the experimental side, the next important advance will probably involve the use of line-narrowing techniques in the measurement of strong

field Zeeman transitions. At LAMPF this will require a pulsed muon beam with a time structure of the order of the muon lifetime of 2 |xs, which could be obtained either

by pulsing the primary proton beam, by pulsing the muon

beam in the muon channel, or by using the proton storage

ring now under construction at LAMPF [26].

Scote in cm

Figure 11. Experimental setup. Si is a 0.2It-iyi. thick plastic

scintillator. It was used for optimizing Kr pressure and beam

degrader (not shown) thickness so that the muon beayn stops in

the middle of the cavity. Si—beayn moyiitor, S2, S3—decay positroyi sigyiature.

(0)

/ /t"

-(b)

_ 1/2 01m Kr ^-f\

Pi2 = I5W /I

P34 = 20W /

"'^

70.7 89,5 108.3 r27.i 145.9 114.7

(H -I3500)(G)

Figure 12. Typical resonayice lines and their fits (solid lines). Each pair of resonance lines was obtained in less than 2 h of data accumulation. Data were taken alternately on opposite

sides of each curve to minimize signal asymmetries due to possible time dependent power drifts. Two further researches on muonium that would be of great interest [20] are, firstly, the measurement of the

fine structure, Lamb shift, and hyperfine structure inter- vals in the n - 2 excited state; and secondly, a direct

search for the conversion of muonium to antimuonium.

The energy level diagram for the w = 1 and n = 2 states of muonium is shown in Fig. 15. It would be particularly valuable to make a precise measurement of the

Lamb shift, or 2^Si/2 to 2^Piy2 energy difference.

Although the Lamb shift in H is known experimentally to high precision [27], its theoretical interpretation is lim- ited by uncertainties associated with proton structure.

242
EXPT

I

I

SCALE

IGHZ

1961 1962 1964

1966 1969

1969

1970

1971

1971 1973

1975

1977

1979

1982

0.IMH2

^ STATIC H STRONG HYALE-NEVIS

10 kHz STRONG H

WEAK H WEAK H SOF WEAK H

YALE-NEVIS

CHICAGO

"OLD MUONIUM" YALE-LAMPF SOF

0.1 kHz STRONG H YALE-HEIDELBERG-LAMPF

4 463 303 ^"theory = 4 463 304(4) Figure 13. History of measurements of Av of muonium.

RECENT MEASURED VALUES OF ^t^Z/ip

CHICAGO- 1970 (/i* e-) (4.8 ppm ) —BERKELEY- 1972 (^SR.HjO)

t (2.6 ppm)

Muonium is an ideal system in which to test the quantum

electrodynamic theory of the Lamb shift since both the muon and the electron are believed to be structureless

particles, unlike the proton which has a structure associ- ated with the strong interactions.

A microwave spectroscopy experiment to measure the

Lamb shift in the n - 2 state of muonium can be done in principle by detecting 2P-1S Lyman-a photons or a change in decay positron angular distribution associated

with a change in muon spin direction resulting from an

induced transition. Because of the large cross section for quenching of the 2S state of hydrogen (and muonium) in an atomic collision, such an experiment would seem to re- quire that |x'^e~(2S) be in vacuum. Recently it has been

found that muonium in its ground state can be formed by

passing |x"^ with energies of about 1 MeV through a thin

foil into vacuum [28]. The experimental arrangement is shown in Fig. 16. The spectrum observed in the Nal

detector is shown in Fig. 17 where the high energy portion indicates that muonium has stopped in the dov^mstream target. Formation of the 2S state should also occur in the foil, and an experiment is being undertaken

at LAMPF to observe the 2Siy2 ^^1/2 transition. A search for the direct transition of muonium to an- timuonium might also be undertaken with muonium in vacuum.

YALE-HEIOELBERG-1977 (/i*e-)

( 1.4 ppm)

SIN - 1978 (^SR. Brj ) (0.9ppm)

SIN - 1981 (^SR. ) (0.53ppm)

YALE- HEIDELBERG -1982(M*e"'

(0.36 ppm)

I I I L J I 0 2 4 6 8 10 12 (/i^//j.p » I0''-3I83340)

Figure 14. History of |x^/|Xp values.

FOIL TARGET SWEEPING MAGNET BEAM STOP WINDOW

DETECTOR Figure 16. Schematic diagram of the experimental apparatus;

Si, S2, and A are plastic scintillation counters. The beam

passes through a 125-\i.m Mylar window at the end of the (x channel, and. a 50-\xm Ti window entering the vacuum

chamber, and is collimated to 7.5 cm diameter by polyethylene

collimators.

10.900 MHz

lO.I eV 13 1221 A

I Figure 15. Energy level diagram for muonium in its n = 1 and n = 2 states, including fine structure. Lamb shift, and hfs terms.

160 -

!^ 120 -

Figure 17. Measured Nal spectrum for 25-\im Be foil target. Solid circles are with the apparatus evacuated; open circles are with 6 Torr helium. For each case 2.8 h of data are shoiim. The

dashed curve is a fit to the data as described in the text.

243
Some years ago the muonic helium atom C^Hejji'e") was

discovered [29] from the observation of its characteristic,

muonium-like, Larmor precession frequency. The Bohr

picture of this amusing atom is indicated in Fig. 18. In a

first approximation it can be considered to consist of an

inner core or pseudo-nucleus consisting of ''Heix" and of an outer electron. The pseudo-nucleus has a unit positive

charge and a magnetic moment equal to that of |jl . Hence the muonic helium atom is analogous to muonium

and would be expected to have a hyperfine structure in- terval similar in magnitude to that of fx^e" but inverted

because of the negative magnetic moment of

MUONIC HELIUM ATOM

PSEUDONUCLEUS. ""He**^" [CHARGE +6: MAGNETIC MOMENT /i.^-]

Figure 18. Schematic diagram of the muonic helium atom. The

orbits are not to scale. The radius of the "pseudonucleus" is 1.3 y- 10~^' cm, while the electroyi orbit radius is r^g/,,. = 5.3 x 10~^ cm.

) 1 2 3

X 4 '5 6 7 8 9

1 1 1 1 1 1

' ( Mj, Mfi )

( 1/2. -1/2)

1 'Ml/2, 1/2)

.(F.Mp)

^410)

"(l.-l)"^;^;^

^-1/2.1/2)

4

1 1 (-1/2. -1/2) 5 10 H (kG)

15

The hyperfine structure transition of "^Heix^e" has re- cently been observed at SIN in a microwave magnetic

resonance experiment at zero static magnetic field [30]. The resonance signal is relatively weak because of the

low residual polarization of in '*He|ji~e". A strong magnetic field measurement of greater precision has also been done [31] to determine both Av and |jl _/|Xp. Fig- ure 19 shows the Breit-Rabi energy level diagram and

Fig. 20 shows a resonance curve for the strong field transition. Table 4 gives the experimental values for Av. The theoretical value of Av for ^He|jL~e~ can be v^rritten [32] as:

grel +grad Av Avfil + S""^' + 8'"^« + , (8) in which A v^ is the leading Fermi term, and S""^',

5'"^"^ and

8'"'^'^ refer to relativistic, radiative, and recoil contribu49.150 49.200 49.250 PROTON NMR FREQUENCY (MHz)

~I

\—I—

I—I—

r

Figure 20. Typical resonance curves for the v^^ transition in

'*He\i.~e " obtained with the fonvard telescope at (a) 15 atm and

(b) 5 atm. The data for these curves were obtained in (a) 2i h

and (b) 100 h. For each curve obtained ivith the forward tele- scope there is a corresponding) curve for the backward telescope.

Table 4. Comparison of theoretical and experimental

values for the hfs interval in ^He\x~e~.

Figure 19. Breit-Rabi energy level diagram of''He\i. e .

Method Av

Variation calculation^ 4465.0(0.3) MHz

Perturbation theory'' 4462.6(3.0) MHz

Born-Oppenheimer theory;

global operator technique'^ 4460 MHz

Strong field experiment^ 4465.004(29) MHz (6.5 ppm)

Weak field experiment*^ 4464.95(6) MHz (13 ppm)

^Ref. [32]; ''Ref. [33]; '^Ref. [35]; ''Ref. [31]; ^Ref. [30].

244
tions. Evaluation of Av^^ with high accuracy is rather

difficult because the most important aspect of the

wavefunction involves the correlation of the muon and

electron whereas the binding energy of the atom, the

quantity to which a variational calculation applies, is determined principally by the muon part of the wavefunction. Theoretical values including the correction terms in Eq. 8 [32-35] are given in Table 4. The theoretical values

are in satisfactory agreement with the experimental

value within the relatively large theoretical errors. Improvement in the accuracy of the value of A Vf is particularly needed.

4. Muonic Atoms

Precision measurements in muonic atoms (ordinary nucleus plus plus electrons) have been of value for a sensitive test of electron vacuum polarization in QED and

for a test of the existence of exotic interactions. It should

be quickly remarked that no significant deviations from

QED or exotic interactions have been observed [20]. The simplest muonic atom in which a significant precision measurement has been made is (fx"%e)^ in which

fine structure intervals in the n — 2 state have been

observed by laser spectroscopy. Figure 21 shows the en- ergy levels of the n = 1 and n = 2 states [20]. Note particularly that the 2^Siy2 level lies below the 2^Pi/2 level and that the energy interval 2^'Py2 to 2^Siy2 is much larger

than the fine structure interval 22P3/2 to 2^Fi/2, due to the

large negative contribution of electron vacuum polarization to the energy of the 2^Si/2 state.

Figure 21. Scheme of the lowest energy levels of the muonic

heliuyn ion

The experimental arrangement [36] is shovra in Fig. 22. A fraction of the muons (—5%) stopped in He

gas form (|jL"*He)^ in the 2S state. A laser pulse at the

transition wavelength from the 2^Si/2 to the 22P3^ state of about 8100 A is then applied, and the transition is detected by observation of the 8.3 keV 2P-1S x-ray. Figure 23 shows an observed resonance line. Comparison of the experimental and theoretical [37] values is shown in Table 5. It is seen that the contribution of the nuclear

size of '*He is large compared to the experimental accu- racy of 0.02%. The nuclear size is known from elastic e~ - ''He scattering and the uncertainty in this contribution to the interval is about 0.3%. The dominant contri- bution to the interval is electron vacuum polarization,

Figure 22. Experimental setup for (^He[y.~)^ fine structure measureynent. M, CH^ moderators; 1,3,^,5, plastic scintillators; 2, anticoincidence Cerenkov counter; T, Invar steel tank;

V, useful volume for stopping muons; Aj-Ag, Nal(Tl) counters; LG, light guide used to inject the infrared radiation into the

target; TC, optical telescope; PC, optical fiber supplying the

eyiergy-monitoring signal.

"I I I r I I I o 8105 8110 8115 8120 8125 8130 8135 X (A)

Figure 23. The 2Sj/2 - 2P^/2 resonance signal. Each dafitm

point represents the number of events, yiormalized to the same number of stopped muons, per stepping motor position SaMP;

the scale in wavelength (shown below) is fixed by the calibration procedure (\j and are two liyies of an Ar lamp.) The

full-line curve drami on the data is the result of a best-fit analysis (x^ = 11) with a Lorentzian line (a^ssiiming for the 2P

decay rate T the theoretical value T = 8 A) plus a constant background.

245
Table 5. Muonic helium fine structure: theory and comparison with exn

2^P - ^Sii2 transition energy'^ {[xT^He)

eriynent: contributions'^ to the

Fine

structure Electron vacuum

polarization Vertexcorrections

Recoil Nuclear size'^ Nuclear

polarization Total

(theory) Total(expt.)

0(a) 0{a^,...)

145.7 1666.1 12.0 -10.7 -0.2 -288.9 ± 4.1 -105.5 r2 + 1.403.1 ± 0.6 1527.1 ± 4.2 1527.5 ± 0.3

^Units in meV and fm. "r^ = mean square radius of ^He = 2.802 fm^ *^e energy interval 22Pjy2 - 2^8;^^ has also been measured, and theory and experiment are in good agreement [36], and agreement of theory and experiment to the accuracy

of about 0.3% with which the ''He size contribution is known confirms the radiative QED contribution. Alternatively, assuming that QED is correct, the experiment

determines the most accurate value of the root mean

square radius of '^He. In order to avoid the ''He nuclear size effect which is associated with the S state, an experiment is being un- dertaken at the AGS at the Brookhaven National Laboratory to observe the 3D to 3P transition [38]. Similar measurements of muonic hydrogen (|x"p) fine structure by laser spectroscopy would be useful to test electron vacuum polarization and also to study proton

structure. Indeed, in principle a direct observation of the

hfs transition in the IS state is possible by observation of the accompanying change in the muon polarization. Fig- ure 24 indicates the energy level scheme for the n - 1 and n - 2 states of fji^p. For a fine structure measurement, formation of the 2S metastable state at low gas

pressure is probably necessary because |jl p (2S) is readily quenched in atomic collisions. Figure 25 shows

observations of Lyman lines from |jl"p obtained in H2 gas

at pressures as low as 1 Torr [39]. These observations were achieved by using the low momentum from de- cay of it" of about 40 MeV/c. However, attempts to ob- serve |x"p (2S) directly by collision-induced 2P ^ IS 7

rays have not yet been successful [40, 41].

F = 0

Figure 24. Energy levels of n = 1 and n = 2 states of muonic

hydrogen (|x"p ). Hyperfim structure interval Av in 1 Si/g state

is ~U4 X 10^^ Hz.

(ai dN

'^E, ISO Totr

(CI

ISO To.r

DOO Jl

400 3289evenls '

?oo

1 1 1 1

100- J

/ J ^^vS

Lr 1 ii

200 113 4 dN (b) 600 Torr 200 _

0 dN

1 ? (d)

-200

600 Tofr

l259evenrs

-100 ill -lOO

? .

1 2 J *

1, l>-sec)

eNERGV (Kev)

Figure 25. X-ray energy distribution for jjl stopping in H2 at low pressures.

In recent years precise measurements of spectra of medium Z muonic atoms have been made with a crystal

diffraction spectrometer at SIN [42], where large [x" stopping rates are achieved with a superconducting muon

channel. Figure 26 shows the crystal diffraction spec- trometer. Figure 27 displays some observed 3D 2P

lines for ^^Mgjx". Figure 28 gives the fractional difference

between standard QED theory and experiment. The good

agreement can be interpreted as a measurement of elec- tron vacuum polarization to about 0.3%. Alternatively,

the result places a limit on non-QED muon-nucleus in- teractions; in particular, within the standard electroweak

theory in which the Higgs boson mediates an interaction

between muon and nucleus it places a lower limit of about 8 MeV to the mass of the Higgs boson.

246
counts /250 s«

100

Figure 26. Curved-crystal spectrometer for muonic X-rays at SIN. general vieiv. 1. Muon target (X-ray source). 2. Cooled

copper socket supporting the target holder. 3. Iron-loaded and normal concrete for shielding. J^. Iron shieldiyig. 5. Bending

blocks with (four) crystals. 6. Crystal turning table. 7. Threaded screw, part of the crystal-rotation system. 8. Tube collimator. 9. Phoswich detector. In hadronic atoms the effects of the strong interaction

between the hadron (tt", K", p, S~, etc.) and the nucleus

are important and have not been calculated from fundamental strong interaction theory. However, studies of the spectra of these atoms have yielded values for the masses and magnetic moments of these hadrons [43-46].

5. Conclusion

Exotic atoms consisting of pure leptons such as e'^e" and ix'^e" are ideal systems 'for studying QED and for measuring the properties of the leptons as well as for looking for exotic effects, because conventional theory in- volves no ambiguities. High precision measurements of energy intervals can be made by the methods of mi- crowave and laser spectroscopy, limited ultimately by

the lifetimes of the particles and atomic states. For

exotic atoms involving hadrons the theoretical interpretation must consider hadron structure, just as for electronic atoms. The field of exotic atom spectroscopy

is relatively new and can be expected to develop further

and to continue to contribute to our knowledge of fundamental interactions and fundamental constants. References

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the measured count rates. The circles are the values of the fit function corrected for the fluctuations in the muon stoppingrate.

200

50

0

-50

-100

(ppm) A published

1003)

Mg Si P

^li'^li '^li'^li ^/I'^/i ^k'^ ^I'^h

Figure 28. Relative difference between experimental and

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values are grouped according to the six transitions measured

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247
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F. G. Mariam, P. A. Souder, H. Orth, J. Vetter, and G. zu

Putlitz, Phys. Rev. Lett. 48, 1168 (1982).

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248
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Fundamental Tests and Measures of the Structure of

Matter at Short Distances*

Stanley J. Brodsky

Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94305

Recent progress in gauge field theories has led to a new perspective on the structure of matter and

basic interactions at short distances. It is clear that at very high energies, quantum electrodynamics,

together with the weak and strong interactions, are part of a unified theory with new fundamental constants, new symmetries, and new conservation laws. A nontechnical introduction to these topics is given, with emphasis on fundamental tests and measurements. Key words: quantum chromodynamics; quantum electrodynamics; strong interactions; weak interactions.

1. Introduction

In the past few years, there has been extraordinary

progress in the understanding of the structure and

interactions of matter at short distances. The most important theoretical progress has been in the area of non- Abelian gauge theories, vi^hich are now leading toward a

unified description of the weak, strong, and electro- dynamic interactions. It is now evident that quantum

electrodynamics is just one manifestation of a larger unified theory. In this review, I will emphasize the areas where fundamental tests and precision measurements are

crucial to the development of basic theory. Many previously believed conservation laws, such as baryon and lep- ton number conservation, are now open to question or in fact are predicted to be violated. High precision tests, in- cluding searches for very rare processes, are thus essential in order to definitively test the theories.

2. Quantum Electrodynamics

One of the most fundamental questions in physics is whether we have actually identified the fundamental

constituents of matter [1]. In quantum electrodynamics,

the leptons e, |x, and t are elementary point-like carriers

of the electromagnetic current, each with a Dirac coupling to the electromagnetic field. High energy (V^ = E t.m. ^ 38 GeV) measurements of the reactions e*e~ ^

e^e~, |jL*|x~, t'^t", and yy at the PETRA storage ring

have placed severe limits on any deviation from the

predicted Dirac structure or any internal lepton structure. For example, measurements of electron-positron

annihilation into muon pairs by the Mark-J collaboration

[2] lead to lower limits (95 percent confidence level)

Surprisingly, the strongest limits on possible internal

lepton structure in some models come from the precise

measurements of the gyromagnetic ratios of the electron

and muon—measurements at the limit of zero momentum

transfer. The most precise published value is [3]

1 g = 1.001 159 652 200 (40)

The QED prediction has now been computed through

order (a/Tr) [ninth order in perturbation theory!] by Kinoshita and Lindquist [4]. The result is —

1.001 159 652 504 (182). The (a/TT)^ calculations require

the evaluation of 891 Feynman diagrams. The uncertainty reflects the limit of error on 10-dimensional nu- merical integrations, as well as the uncertainties in the

determination of a. Since there is no a priori reason why

a spin 1/2 system must have ^ ~ 2 in general (vdtness

the nucleons), it is extraordinary that QED correctly

predicts g ^to 10 significant figures!

Let us now consider the possibility that the electron is composite with an intrinsic radius R [1, 5]. The natural

size of the magnetic moment of a charged extended sys- tem is |JL ~ ei2, which would imply a contribution to the

gyromagnetic ratio Agfg ~ m^R (barring cancellations).

Alternatively, one can compute the magnetic moment of any system from the general relationship between the

anomalous moment a = (l/2)(fif - 2) of a system and its excitation spectrum (the Drell-Hearn-Gerasimov sum

rule). For a spin 1/2 system one has

2TT-a''s,, s

(2)

A- > 160 GeV, A+ > 120 GeV (1)

for modifications MQ- -» 1/(Q2 ± \iq- - A|) of the photon propagator or the electron or muon vertex. Alternatively, this result demonstrates that the electron and muon are effectively point-like down to distances R ~

< 2 X lO-i'^ cm.

*Work supported by the Department of Energy under contract number

DE-AC03-76SF00515.

where Aa = ct^ - is the difference between the spin

parallel and spin antiparallel total photoabsorption cross

sections. Barring special cancellations, Aa~0(oLR-) at energies where the compositeness of the target is manifest. For a composite electron, this again gives the esti- mate Aoe ~- m^R- The agreement between theory and

experiment lAa^l < 10"^ then implies that any internal

size scale of the electron is limited to exceedingly small

distances R < 10"-° cm, R-^ > 10^ GeV. (The limits for the muon are comparable.)

249
It is, however, possible to construct specific models for composite leptons that give a smaller correction to the

magnetic moment than the general estimate given above.

For example, in a model in which the constituent fer- mions have mass nip much less than the intrinsic momentum scale R ~^ of the system, one obtains the quadratic

relationship Aa — ntemfR^. In fact, one can evidently

conceive of the electron as a tightly bound composite sys- tem (radius <10"^^ cm) of permanently confined but light mass (rrif < 100 MeV) fermion constituents, without

violating any high energy or low energy constraint. The

dynamics of such models are, however, far from clear. A

light mass fermion constituent also seems to be required

to understand the small mass of the electron. It is also interesting to note that a very complex theory can ap- pear to be simple and renormalizable at low momentum

scales « R even though the particles are composite at short distances.

3. Weak Interactions and Quantum

Electrodynamics

Before the advent of the Glashow, Weinberg, Salam

SU(2) X U(l) theory [7] of the weak and electrodynamic

interactions, there was no satisfactory way of computing

the weak interaction corrections to QED predictions.

Previous models, besides violating unitarity at high energies, gave logarithmically divergent corrections to the

lepton magnetic moments and even quadratically divergent contributions to neutrino charge radii. It is, however, now clear that QED and the weak interactions are

unified as part of a more general, completely calculable,

renormalizable theory. In particular, the weak interaction contributions to the muon moment in the SU(2) x

U(l) theory, as shown in Fig. 1, are readily calculable

[9]:

compared with the present experimental uncertainty of

-11 X lO-l

Figure l. Weak interaction contributions to the muon magnetic nwment.

Let us briefly review the main features of the GWS

"standard model" [7]: In its initial stage, the theory begins with the assumption of an exact internal symmetry

SU(2) analogous to isospin, with doublets of massless

(negative helicity or "left-handed") leptons and quarks,

and triplets of massless vector bosons; e.g.:

u

d Jl' W

w-

(3)

->-

Figure 2. SU(2) couplings of leptons and quarks.

Dirac coupling of QED and preserve the SU(2) rotational

invariance:

(4)

where Wi" - W,^i and the T, are the set of 2 x 2

i =1 traceless matrices. In fact, the entire theory, including

the W self couplings, is invariant under rotations » e

' '^'•^

'v|/, where A(x ) is an arbitrary 2x2 matrix function

of space and time. This (" non-Abelian" ) local gauge in- variance is a generalization of the (Abelian) local phase

or U(l) gauge invariance of QED, and insures the renormalizability of the theory. One can also define an additional conserved "charge" Y {= -1/2 for leptons, 1/6 for quarks), which reflects the fact that the lepton and quark

currents are separately conserved. The coupling of the

weak hypercharge current g^iY v|;7*^il/ to an additional zero mass vector boson fi^, then has an exact v|;^ e''^'^'^v|; lo- cal U(l) gauge symmetry.

The theory discussed thus far has little resemblance to the observed weak interactions. However, if one intro- duces an extra doublet of interacting scalar bosons c|) =

with non-zero expectation value in the ground

state, then the exact SU(2) gauge invariance of the

theory will be "broken," while still retaining the renormalizability of the theory in the ultraviolet domain. What

emerges at low energies is (1) a massive charged vector boson W", (2) a massive neutral vector boson Zl =

sinewfip. - cos%v/^t, and (3) a massless neutral vector boson Ap, = sin0vy5„ -H cos0wW^ coupled to the electro- dynamic current. One can then identify the electric charge e = gzsinQw and the Fermi constant Gf/V2~-

eVSM^sin^Qw, where the weak mixing angle is given by

tan0w = 9\/92- The weak interactions at low momentum

transfer « then have the form:

eff jfZ

with

and

= S [^^(1 + ^5)7^' - 29 sin2 0w]i^

(5)

(6)

(7)

The interactions of this theory (see Fig. 2) generalize the

The form and relative normalization of the charged and

neutral weak interactions predicted by Eq. (7) have now

250
been checked in many neutrino and weak/electromagnetic

interference experiments —in many cases to better than

1 percent precision. One of the most precise experiments

is the SLAC-Yale [10] measurement of parity violation in deep inelastic polarized electron e ~p scattering. The in- terference between the electromagnetic and weak neutral currents (see Fig. 3) leads to an asymmetry (ct/, -

o'ff VCctl + (jr) in the e~p e~X cross section and thus to a determination of sin%w A combined analysis of the

various neutral current experiments gives (see Fig. 4)

[11]:

sin^Ovr = 0.229 ± 0.009( ±0.005) (8)

Figure 3. Interfering weak and electromagnetic contributions

to deep inelastic lepton-proton scattering.

and

^ ^ \, ^exp^ ^ Q_gg2 ± 0.017(±0.011) (9)

[ ^ J predicted

where the error in parentheses indicates errors due to theoretical uncertainties (radiative corrections, etc.). Thus far, there is no discrepancy vdth the predictions

of the SU(2) X U(l) model, although other models (such

as those that are parity symmetric at large momentum

transfer) are not ruled out. The most critical test of this model will be the experimental confirmation of the W*

and Z " vector bosons at the predicted mass

Mw = 82.0 ± 2.4 GeV

Mz = 93.0 ± 2.0 GeV . (10)

T 1.04 -

1.00 -

K

0.96 -

0.92 I ^ ^ ^ ^ \ \ \ \ \ 1 0.16 0.18 0.20 0.22 0.24 0.26

Figure 4. Values of sin Q^f and k from a phenomenological

analysis of netitral current data. The limit curve corresponds

to a confidence level of 68%. From Ref. [11].

The vddth of the Z'' is particularly interesting since it signals the number of neutrinos. Confirmation of SU(2) x

U(l) model wiW also require the identification of the

Higgs scalars, the origin of symmetry-breaking in the

theory. It is, however, possible that these particles are themselves composites [12] rather than new additional

elementary-field degrees of freedom.

The fermions in the higher generations enter the

SU(2) X U(l) multiplets in parallel to Eq. (2); i.e.:

e

" L

J L T~

*. = fei. [^ir. ill- <")

However, as first noted by Cabibbo, the quarks that ap- pear in the weak interaction theory are not necessarily

the mass eigenstates of the full theory, including strong

interactions. In the case of three generations, the mass

eigenstates

are related by three rotation angles and one CP violating

phase cj) to the weak interaction eigenstates. The angle of the mixing between the first two generation quarks is the Cabibbo angle, which has been determined to considerable accuracy: 6c ~ 13.17° ± 0.64° from analyses of strangeness-changing weak interactions.

The fact that the mixing of three generations leads in a natural way to CP and T violation is a very interesting

result [13]. By parameterizing present data for CP

violating effects in terms of the phase angle <{), one can

predict a non-zero value for the neutron electric dipole

moment:

{D/e)n - 10-3^ cm

compared vdth the present limit ~ 10"^^ cm. If the e, |x, and T neutrinos are massive, then one expects a similar

mixing pattern for the lepton sector and a non-zero value

for the electron electric dipole moment [9].

251
4. The Strong Interactions [14]

The successful application of local gauge theories to weak and electrodynamic interactions has led to an even more fundamental advance in the case of the strong

interactions —the development of quantum chromodynamics. In QCD the fundamental degrees of freedom of hadrons and their interactions are the quanta of quark

and gluon fields that obey an exact internal SU(3) "color

symmetry." Each quark "flavor" q - u, d, s, c, b ...

is represented as a color triplet

8 interacting with an octet of gluon fields = ^ Xq,

a =1 where the \a are the set of 3 x 3 traceless matrices. The

interactions

together with the gluon self couplings (see Fig. 5) preserve an exact local gauge symmetry for arbitrary ro- tations ili A(J-

)

\\i in the SU(3) color space. The gluons

are massless and theory is renormalizable. The lowest

energy states are the color-singlet baryons Ig/;^^^^ > and mesons {l/\/3~)\q{{qR + qsqB + QyQy>- In addition,

bound states of gluons \gg > and \ggg >, as well as \qqg>,

states are predicted.

in QED at zero momentum transfer, this choice is in a sense arbitrary since the one-photon exchange interactions can be computed at any Q by including the vacuum

polarization insertions to all orders:

a(Q2) =

a(Qo')

1 + aiQ§)

(13)

For Q2 » m/, tt(Q-) ~ -(1/3tt) log Q^/m,^ so the effec- tive coupling a(Q^) increases at large Q^. In the case of QCD, the vacuum polarization diagrams involving gluon

self-coupling (see Fig. 6) actually reverse the sign of

7t(Q2):

Figure 6. Vacuum-polarization contributions to quark-quark

scattering.

33

2 nf logQ^ (14)

Figure 5. Quark-gluon couplings in SU(3) color.

It now seems possible that quantum chromodynamics is the the theory of the strong interactions in the same

sense that quantum electrodynamics accounts for elec- tromagnetic interactions. It is well known that the general structure of QCD meshes remarkably well with the

facts of the hadronic world, especially quark-based spec- troscopy, current algebra, the approximate point-like structure of large momentum transfer lepton-hadron

reactions, and the logarithmic violation of scale- invariance in deep-inelastic reactions. The theory is particularly successful in predicting the features of electron-positron annihilation into hadrons: the magnitude and scaling of the total cross section, the production

of hadronic jets with a pattern conforming to elementary

quark and gluon processes, and heavy quark phenomena.

The empirical results are consistent with the basic postulates of QCD, that the charge and weak currents within

hadrons are carried by the quarks and that the strength

of the quark-gluon couplings becomes weak at short dis- tances (asymptotic freedom).

Although it is simplest to define the coupling constant

as long as the number of quark flavors (n/) is less than

17. Thus, for very large Q^, the strong interaction cou- pling constant ag = gi/4:T: is given by (Q^ » A^)

asiQ-) =

4-77

11 - T^f log 5

(15)

The fact that the coupling constant decreases at large

(see Fig. 7) implies that we can use perturbation theory

for calculating hadronic processes at short distance. Cona,(Q2)

Figure 7. Variation of the strong and electromagnetic coupling

strengths unth momentum transfer.

252
versely, the increase of the effective coupling at large

distances is consistent with the expectation that the

theory strongly confines particles of non-zero color. The fundamental parameters of QCD are the coupling

constant as and the quark masses. The actual measure- ment of these quantities is complicated by the fact that

only bound states of quarks and gluons are accessible,

and there can be large and uncertain corrections from

nonperturbative, binding, and other higher-order effects. In addition, the specification of as(Q ) depends on the

momentum scale chosen to express the leading order

results and the choice of normalization scheme.

At this time, the most precise determination of the

QCD coupling constant is given by an analysis of the de- cay of the upsilon, Y, the lowest energy bound state of b and b

- quarks with J^^ = l"" at My = 9.46 ± 0.01 GeV.

The hadronic width of the upsilon can be computed in lowest order perturbation theory from the 3 gluon decay

amplitude (see Fig. 8) in analogy to orthopositronium de- cay into three photons. The QCD prediction, including

T —>

T

Figure 8. Leading contributions to the hadronic and leptonic

decays of the upsilon.

first order radiative corrections as recently computed by

Lepage and Mackenzie [15] is

3g lte(0)l

160

1 + c 3s

with

(16)

C

The lopton decay rate is

- 3.8 ±_ 0.5

1

2 1677 3 16 as_ 3 TT a?)

The measured ratio for Fy had /fy ^ ^ +e - then implies

otsiQ^ = M^) = 0.14 ± 0.01, where the error indicates

la accuracy. The determination [2] of as from e'^e~ qqg jets also gives values in the range as(Q^ ~

lOOOGeV^)- 0.15 to 0.23, although there are large

corrections from hadronization and higher order effects.

It is also possible, in principle, to determine a., from exclusive processes [16] such as ratios of meson form fac- tors: F^{Q^)/F^{Q^) and from radiative corrections to deep-inelastic lepton-hadron scattering.

Considering that new phenomena still continue to be

discovered in QED and atomic physics, it is likely that we have only touched the surface of much more complex

QCD phenomena. Thus far, there have been many semiquantitative tests of the theory, including the verifica- tion of scale-invariant quark-quark interactions from the

behavior of meson and baryon form factors at large

momentum transfer. There is, however, only a rough

qualitative understanding of the basic properties of ha- drons, such as their masses, magnetic moments, and

charge radii. Detailed answers to such questions require an understanding of the hadronic wavefunctions as well

as their multiparticle Fock state structure (see Ref.'[5]).

Perhaps the most dramatic confirmation of QCD would be

the observation of gluonium \gg> and \ggg> states. It should also be emphasized that the reported observation

of fractional charge systems by Fairbank et al. [17] reo- pens the fundamental question of whether quarks are

really confined in QCD—i.e., whether QCD is an exact

local gauge theory or is spontaneously broken so that the

gluons have a finite mass and the confinement potential

has finite range. Alternatively, these results could signal

the existence of color singlet fractional charged particles, or bound states of zero charge color anti-triplets Sc with

quarks, or even fractionally charged subconstituents of the quarks themselves.

5. Unified Theories of the Strong, Weak,

and Electromagnetic Interactions [1, 7]

Thus far, our discussion has led to no real understanding of the quark <—> lepton parallelism, especially why

the proton and electron charges are equal and opposite to within parts in 10-". This could be understood if (a) quarks and leptons have common subconstituents

and/or (b) quarks and leptons are in the same representation of a fundamental symmetry gi-oup. The latter possibility is the central motivation for many grand unified

theories, as exemplified by the SU(5) gauge theory model

of Georgi and Glashow [18]. In this model, the' 15 fer- mions of the first generation (see Fig. 9) are identified

members of the 5 and 10 representation of SU(5); e.g.:

% - (18)

and »|jio is an antisymmetric 5x5 matrix with entries for

Ci^'^ , {u, d)L and (u)i. The gauge bosons correspond to the

set of (24) 5x5 traceless matrices

A =

gluons

XR. Y. B w-

(19)

The X and Y vector bosons have color and fractional

253
SU(2)

Doublets

SU(2)

Si nglets

(?)J(:);(:):(:);

I

I Y.Y R .R B.B

Bd , UrCIr UrCIr R^R UrQ- ^R^R

SU(3)

Singlets

SU(3)

Triplets

2nd Generation: e — u—^c, d—*-s

3rd Generation: e —^r, u —^t, d—

Figure 9. The 15 fimdamental fermions of the first generation.

The subscript L and R indicates particle helicity.

SU(5) is an exact gauge symmetry. At lower energies

this symmetry is broken (by a Higgs scalar vacuum ex- pectation value), leaving symmetries corresponding to the subgroups SU(3)c x SU(2) x U(l). Finally, at much

lower energies, > Mt^, Mz - 10^ GeV^, the theory is again broken (as in Sec. 3) and the exact gauge symmetries that remain are the SU(3)c (massless gluon octet) and U(l) symmetry of QED (massless photons).

From analysis of observed mass scales and the fact that

the U(l) coupUng constant is increasing and the SU(2)

and SU(3)c coupling constants are logarithmically de- creasing, one can estimate that the grand unified scale (where the coupling constants coincide ai — a2= a.^ -

as) is ~ 10^^ GeV (see Fig. 11). (This also gives an esti- mate for Mx, My.) In fact, aGUT(Q^ ~ 10^° GeV^) - 1/42 now becomes the fundamental coupling constant, not a —

1/137.

charge and couple ofjquarks to leptons! Since charge is a generator of SU(5), i|>Q4; = tr Q = 0 and the sum of the

charges of the constituents in any representation must be zero—i.e., SQ,i + = 0—this naturally explains the

equality of lepton and baryon charges.

The basic scenario of SU(5) is then as follows (see

Fig. 10). At momentum transfers much larger than

Mx and My, all particles can be treated as massless and

SU (5)

10<5 GeV

SU(3) 0 SU(2) ® u(n

10 ^ GeV

SU(3) ® U(1)

Mx, My

QCD QED

a.

Figure 10. The approach to equality of the SU(3), SU(2), and

U(l) gauge couplings in the SU(5) grand unified theory.

GUT

I 10^ I0'5

Q (GeV)

Figure 11. Symmetry breaking pattern in SU(5).

The SU(5) model leads to two critical predictions:

(1) The weak mixing angle is fixed to be sin^Ovi/ = 3/8 at the grand unified scale. One can, however, use an extrapolation determined by the renormalization group (over

13 decades!) to compute sinH^ = 0.209 ± 0.006 at Q" -

Mi|. This is not far from the experimental value [10, 11] sin^ew' = 0.229 ± 0.009 ( ± 0.005).

(2) Because of the X and Y couplings, baryon number is not conserved, although baryon number minus lepton

number is still a conserved quantity in SU(5). The proton

is thus unstable and will decay into channels such as p

e^Ti'* (see Fig. 12). The predicted proton decay rate is proportional to the grand unified scale to the fourth

power and thus has a large uncertainty: Tp ~ 10^^-^

years. The present experimental limit is Tp > 10^° years;

in fact, four large-scale experiments now under construction vdll have a sensitivity to Xp ~ 10^^ years, so a decisive test of SU(5) appears possible. This is especially true since the most recent evaluation [15] of the QCD

scale constant Ame ~ 100 MeV in Eq. (15) leads to the

lowering of the grand unified scale mass and hence lower

predicted values for Tp

.

In addition to SU(5), other grand unified theories have

been proposed. All such models have profound implications for the evolution of the early universe [19], which

254
Figure 12. Origin of baryon decay in SU(5).

Figure 13. Diagram leading to double ^-decay in nuclei.

would be expected to trace through the various symmetry phases, starting with a hot soup of massless

quarks, gluons, W, X, etc. A possible connection of the

baryon excess - in the universe to CP violation is also possible [20, 7, 11]. The grand unified models also evidently predict large rates for the production of free magnetic monopoles in the early universe. In many grand unified theories, there is no natural or compelling reason why neutrinos should be massless, and

the determination of the neutrino masses is now a topic

of intense experimental activity [21]. In analogy to the

quark case, the mass eigenstates v,„ may be linear combinations of Vg , v^, and Vj. This means that the neutrino

produced at time ^ = 0 in tt"^ ^ ix^v^f' decay will at t > 0

be a linear combination of v^, (chirality nonconservation), and (neutrino "oscillations").

If a neutrino has mass, it also follows that it can have

an anomalous magnetic moment [9]; diagrams such as

Fig. 15 give contributions of order ~ m^nifGF- One

can thus imagine inducing a neutrino helicity flip using

external magnetic fields. If a neutrino is self-conjugate

C \v> = ±\v> and there are charged right-handed

currents so that W ~ e "v^ and W ~ ^ e 'vn are possible, then double (3-decay processes nn pp e~e~ in nuclei are possible (see Fig. 14). In some models n <

> n

oscillations are also predicted.

Figure 14. Weak interaction contribution to the magnetic mo- ment of a massive neutrino.

6. Conclusions

The concept of a grand unified theory has led to an ex- traordinary new perspective on the structure of matter

at short distances. It is clear that, at very high energies, QED, together with the weak and strong interactions, are part of a unified theory with new fundamental

constants, new symmetries, and new conservation laws. The prediction that baryon and lepton numbers are no

longer individually conserved should lead us to question

all of presently accepted conservation laws and symmetries. This new gauge field theory perspective also makes it evident that the numerical value of parameters

such as ttQED at zero momentum transfer is not significant

in itself, considering the dependence of the coupling on

resolution scale Q'^ and normalization scheme. Questions

concerning the possible time variation of the fundamental

constants such as aqED are now seen to be connected with

the evolution from the early universe.

Although gauge field theories and grand unified

theories are leading to answers to many fundamental

questions, one must acknowledge that countless others

remain, such as the origin of a vast ratio of mass scales: Mx/Mw ~ 10^^, the origin of the fermion masses (e.g.,

why is Mr, < M„ ?), the possible compositeness of leptons

and quarks, and the ultimate role of gravity (Mpianck

10^^ GeV). It is clear that high precision experiments,

searches for rare new processes, as well as detailed high

energy measurements, will be crucial for continuing pro- gress in these areas.

I wish to thank my colleagues at SLAC particularly

Boris Kayser, for helpful discussions.

References

[1] For examples of fundamental theories with composite leptons

and quarks, see J. G. Pati and A. Salam, Phys. Rev. D 10, 275 (1974); H. Harari, Phys. Lett. 86B, 83 (1979); and I. Bars, Yale preprint, YTP-81-19 (1981).

[2] D. P. Barber et al., Phys Rev. Lett. 43 1915 (1979). For a review of the PETRA results, see G. Wolf, DESY 80/13

(1980).

[3] R. S. Van Dyck, Jr., P. B. Schwinberg, and H. G. Dehmelt,

Phys. Rev. Lett. 38, 310 (1977). For a review of anomalous

moment measurements and theory, see J. H. Field, E. Pi- casso, and F. Combley, Sov. Phys. Usp. 22, 198 (1979).

255
[4] T. Kinoshita and W. B. Lindquist, these proceedings. For

recent progress in the analytic evaluation of the anomalous

moment through order (o/tt)^, see M. J. Levine and R. Z. Roskies, these proceedings. A review of other precision measurements in QED is given by G. P. Lepage and D. R. Yennie, these proceedings.

[5] The analysis discussed here follows S. J. Brodsky and S. D.

Drell, Phys. Rev. D 22, 2236 (1980). See also G. L. Shaw,

D. Silverman, and R. Slansky, Phys. Lett. 94B, 57 (1980).

[6] S. Dimopoulos, S. Raby, and L. Susskind, Nucl. Phys. B

173, 208 (1980). L Bars, Ref. [1]. [7] For reviews and references to electroweak theory, see P. Langacker, SLAC-PUB-2544 (1980); L.-F. Li, NSF-ITP81-25 (1981); and the proceedings of the Cornell Z° Theory

Workshop, CLNS-81-485 (1981).

[8] See, e.g., S. J. Brodsky and J. D. Sullivan, Phys. Rev. 1-56, 1644 (1967), and references therein.

[9] B. W. Lee and R. Strock, Phys. Rev. D 17, 2410 (1978) and

references therein.

[10] C. Y. Prescott et al., Phys. Lett. 84B, 524 (1979).

[11] J. Ellis, CERN TH.2942 (1980), and references therein.

[12] L. Susskind, Phys. Rev. D 20, 2619 (1979).

[13] M. Kobayashi and M. Maskawa, Prog. Theor. Phys. 49, 652

(1973).

[14] For recent reviews of QCD see, A. J. Buras, Rev. Mod.

Phys. 52, 1 (1980), A. H. Mueller, CU-TP-192 (1981); E. Reya, DESY 79/88 (1979), S. J. Brodsky and G. P. Lepage, SLAC-PUB-2447 (1979).

[15] G. P. Lepage and P. Mackenzie, Cornell preprint CLNS/81-

498 (1981).

[16] S. J. Brodsky and G. P. Lepage, Phys. Rev. D 22, 2157

(1980) . [17] C. S. Larue, J. D. Phillips, and W. M. Fairbank, Phys. Rev. Lett. 46, 967 (1981).

[18] H. Georgi and S. L. Glashow, Phys. Rev. D 6, 429 (1972).

[19] See, e.g., A. Guth and E. J. Weinberg, Phys. Rev. D 23, 876

(1981) . [20] A. D. Sakharov, JETP Lett. 5, 27 (1967).

[21] For reviews, see S. M. Bilensky and B. Pontecorvo, Phys.

Rep. 41, 1 (1978); and B. Kayser, SLAC-PUB-2685 (1981), and references therein.

256
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Experimental Limit for the Charge of the Free Neutron

R. Gahler,*+ J. Kalus,* and W. Mampe^

The neutron charge has been measured to be q^^ = l^e I (~l-5 ± 2.2) x 10"20 {q ^ = electron charge) at a confidence level of 90%. This value„ brings down the known limit by two orders of magnitude. In the experiment slow neutrons of 20 A wavelength passed a strong electric field of 10 m

length. The deflection of the neutron beam was measured with respect to reversal of the field. For an

increase in sensitivity the beam was focused by a neutron lens to a sharp image in the detector plane.

Over a long run time' the deflection of the neutron beam due to the electric field was less than 2 x 10 ^

ixm. The result on the neutron charge is in agreement with the commonly accepted neutrality of the

neutron. Key words: neutron charge; neutron optics; particle properties.

1. Introduction

1.1 Background

Within the framework of modern unified theories even

a tiny electric charge of the neutron q „ would lead to important consequences: assuming charge conservation as being valid, a finite neutron charge implies that a transition neutron-antineutron, as predicted by some theories

[1] is not allowed; the electric charge has to change its sign in such a transition. Furthermore, if it were found

that all baryons have charges slightly displaced from

their usually accepted values by a common amount e, then the conservation of baryons would follow from the

conservation of electric charge rather than being an in- dependent principle [2]. In this case the decay of the proton, predicted by several unified theories, is forbidden

because the leptons into which it could decay carry a dif- ferent charge.

With the present measurement, a new experimental

limit for the charge of the electron-neutrino qj. can be set by regarding the p-decay of the neutron: n ^ p -I- e + V

.

The neutrality of the proton-electron system, i.e., the

neutrality of the hydrogen atom has been measured to

(0.9 ± 2.7) X 10^-1 q, [3]. {q^ = charge of the elec- tron.) Combined with the new value for the neutron

charge, one gets:

q^ = qeilA ± 1.4) x 10-^° (68% confidence).

For these considerations charge conservation has been

assumed to be valid. Whereas newer theories assume the symmetry of the

existing electric charges —this implies qn — 0 —there were some theories in the past predicting a finite neutron

charge. The observed expansion of the universe would

follow from a neutron charge of about 2 x 10"^^ q^ [4] and

the earth's magnetic field was explained by a neutron

charge of about 2 x 10"^^ electron charges [5]. These

theories can be refuted by our measurement.

1.2 Earlier Measurements

Most experimental limits on the neutron charge were

derived in an indirect way by testing the charge neutrality of atoms or macroscopic bodies. For a review of these

experiments see Dylla and King [3]. We have only to add more recent measurements, which dealt vdth the search

for fractional charges in matter [6-9]. The sensitivity of

all these measurements is in the region of 10"-^ to 10""^ electron charges for the sum of^e + q^ + qn- Direct measurements on the charge of the free neutron

are all based on the deflection of a neutron beam in a strong homogeneous electric field. In the latest and most

precise experiment of this type Shull et al. [10] got a

limit on the neutron charge of qn - qei-1.9 ± 3.7) x

10"^^. In that experiment not the lateral deflection of the

neutron beam was sensed, but its angular deviation, us- ing perfect Si-crystals in Bragg-condition. The authors

also review all earlier charge measurements on the free neutron. It should be noted that the sensitivity of all direct measurements including ours is lower than that of indirect methods. But as mentioned by Shull et al. [10]

"the possibility of a free charge being slightly different

in magnitude, at this small level of charge difference

when particles are amalgamated into an atom, does

exist."

2. Measuring Principle

In the reported experiment a beam of slow neutrons

(velocity v - 200 m/sec, wavelength k = 20 A) enters through a narrow slit S3 into a strong transverse electric

field of about 10 m length (see Fig. 1). A neutron charge

qn would lead to a deflection of the beam correlated to the direction of the periodically reversed electric field. The deflection y is given by:

q^Eaiy = —— (see section 3) (1) 4Qo

with £"0 the electric field applied over the length ( and

Qo the kinetic energy of the neutron. After passage

through the electric field the neutrons are counted

behind slit S4, which is positioned in the slope of the neutron beam. A change in count-rate for opposite field directions would indicate a beam deflection and hence a charge of the neutron. The sensitivity ' for measuring a

*Universitat Bayreuth, 858 Bayreuth, Germany

"^Institut Laue-Langevin, 38042 Grenoble, France

257
5 m 5 m

Sa\

\

S2

V

S3 PRISM

S4 _ LENS .

I E-FIELD E-FIELD I

COUNTER

SLIT WIDTH [^m] 150 100 40 30

SLIT HEIGHT [cm] 20 20 20 30 40

1^ MONOCHROMATOR — NEUTRON OPTICS -\

Figure 1. Top view of the instrument (not to scale).

deflection depends on the steepness of the slope and on

the intensity. In order to obtain a narrow beam with high

intensity, we studied different focusing systems and

found a lens to be the most convenient device [11]. Similar to a lens in light optics, this neutron lens relies on the

difference of the indices of refraction in the lens-material

and in air. The severely chromatic lens (focal length /

~

1/X.^) asks for a fairly monochromatic beam. This is obtained by a prism-monochromator in front of slit S.j. The

neutron lens is positioned 5 m downstream from the en- trance slit S3 which is 40 |xm wide. The lens images the

slit S3 into a plane where sHt S4 is positioned. A detailed

description of the optical system is given in a previous

paper [11].

3. Sensitivity for the Charge Measurement

3.1 Deflection of a Charge in the Electric Field

At first it vnW be shown that for our optical system the

beam deflection is only half that without imaging by the

lens. This result can be obtained by a wave optical calculation and even more simply from geometrical optics: A

neutron starting parallel to the optical axis follows a par- abola in the £'-field leading to a deflection y /4 at the position of the lens, y is the total deflection in the case vdthout the lens. Due to the focusing properties of the

lens, the flight path is bent back parallel to the optical

axis. The fi'-field downstream from the lens bends the

flight path to an identical parabola as before, leading to an additional deflection t//4. The loss of a factor of 2 could be compensated by increasing the image-distance

relative to the object-distance. But in this case the re- quired precision for the lens—already a delicate

problem—becomes even more severe. In our case, the electric field £"0 is not appHed over the

full length Co between S3 and S4—the necessary control of

the optical components implied the need to divide the

field into two halves, one between S3 and the lens, the

other between the lens and S4. If the total length of the

electric field is £1, it can easily be shovra that the deflec- tion y becomes:

This is valid if the two electric fields are arranged symmetrically with respect to the optical elements. With the

given parameters of the machine the deflection y for a hypothetical charge of q'n =1 x 10^^** becomes

?/ = 6.5 X 10^9 ^

(£0 = 11.9 X 10^^, t'o - 10 m, fi = 9 m,

Qo = 2 X 10"^ eV (\ = 20 A))

3.2 Choice of the Instrumental Parameters

The parameters which determine the sensitivity of the

apparatus for the charge measurement are the count rate N at slit S4, the width of the image Wq, the wavelength X (energy Qo), the lengths t'o and fx, and the electric field £0. The actual values of these parameters will be dis- cussed now.

3.2.1 Count Rate N and Width of the Image

The statistical error (j^ of the difference in count rate

A'^i - N2ioY the two different directions of E 0, is given

by:

for A'^i X N2 ^ N and neghgible background.

The measurable shift Uy of the image is given by:

dN/dy

where dN is the change in count rate per shift dy;

dy dy

vdth df/dy being the steepness of the slope of the image

at the position of slit S4. It follows that

(Ty = -77- . (6) dy

In first approximation the image has Gaussian shape. At

the steepest point of the slope 1/-^ is proportional to Wq,

the width of the image.

(3)

(4)

(5)

VN'

(7)

In order to minimize a„, a small image Wq with a high

count rate A'^ is desired. The count rate A'^ increases with

the width of the wavelength band and with the beam

divergence. However, imperfections and aberrations of the focusing lens limit the achievable a,^. A good

compromise between high count rate and small image

width Wo is obtained using the following argument: the

258
parallelism of the four axes of the lens system (see

Ref. [11]) is about 10 |xm. At the position of the image

this leads to a broadening of about 20 |xm for each point

of the object, the slit S3. It follows that a slit width

smaller than about 40 fxm is not useful for S3. Further

broadening of the image arises from the spherical and

chromatic aberration. A broadening of each point to about 30 |jLm was accepted for each of the two aberrations. Concerning the spherical aberration, this allows a beam width of about 1.5 mm in the plane of the lens. For

the chromatic aberration we obtain an acceptable

wavelength band of AX = ±0.3 A at a mean wavelength

of 20 A. In reality a wider wavelength band of A \ =

±0.5 A can be accepted because of the following reason: The monochromator in front of slit S3 (see Fig. 1) delivers a neutron-spectrum which is asymmetric over

the beam cross-section, because the deflection angle of the prism is proportional to Therefore, by underfocusing the shorter and overfocusing the longer wavelengths, one slope of the image is broadened, the other becomes

steeper (see Fig. 2). Only the steeper slope was taken for the charge measurement.

The steeper slope corresponds to a width of the image

of about 80 jjim. The width of slit S4 was taken as 40 ixm

for optimal sensitivity. The count rate behind S4 is about

50 n/sec.

o

I I I I I 0 50 100 150 200 240

S4 Position lnw]

Figure 2. Image of slit in the plane of slit S^. The steeper

slope was taken for the charge measurement.

3.2.2 Mean Wavelength \

The sensitivity of the apparatus to detect a neutron

charge gn is independent of the choice of the mean

wavelength Ao, as long as cold neutrons are used. This is due to the chromatism of the neutron lens and to the

reactor spectrum.

The reactor spectrum d ^/d X varies as X"^ for cold neutrons. Integrated over d\—^the wavelength band ac- cepted by the lens—we get <i> ~ d \/ko- The focal length /

of the lens is proportional to 1/Xo^. It follows df/f ~

d X/Xo. Thus, for constant focal length / and constant ac- ceptable width df resulting from chromatic aberration,

the value dX/Xo is constant. Therefore, the usable intensity becomes proportional to l/X"*. The deflection of a possible charge 9n in the £'-field is proportional to X^ and we get with Eq. (7) for a,^, the precision for the meas- urement of a neutron charge:

Since, as shown above, the count rate is proportional to

X""*, (jq^ becomes independent of Xo. The spectrum of the

guide H18 is peaked at 20 A with a width of about 10 A. Shorter wavelengths are cut off because of the curvature

of the guide. X = 20 A as mean wavelength was taken,

because the high intensity simplifies the alignment of the

optical components. The mean wavelength and the

wavelength spread of ±5 A selected by the monochromator was checked using the critical angle of total reflection

from a Ni-mirror. During the charge measurement the mean wavelength was controlled by the width of the image. This control was possible because of the severe chromatism of the lens.

3.2.3 Lengths 1'^, The length Co of the apparatus was limited by the avail- able space inside the reactor hall of the ILL. The best

suited area was that at the neutron guide HIS, where a free length of 13 m was available.

3.2.4 Electric Field The total length of the electric field was 9 m. It reached from the entrance slit S3 to slit S4 in front of the

detector and was separated by the lens into two sections

of equal length. The beam height of the divergent neu- tron beam increased from 0.2 m at slit S3 to about 0.38 m

at slit S4. This leads to rather large dimensions of the

electrodes: The first one has an area of 0.32 x 4.5 m, the

second one 0.4 x 4.5 m. For reasons of high electric field and for mechanical stability each electrode was a stainless steel tube with rectangular cross section. The sur- faces which produced the electric field were milled. The

waviness of each surface was about 0.1 mm, this being

significantly lower than the 2.5 mm gap between the

electrodes. The mean gap between the electrodes which

determines the mean electric field £'0 was measured with

a capacity meter. One electrode of each pair was con- nected to ground and the other one to a HV-supply with

a maximum output of 30 kV at 2 mA. Each pair of elec- trodes was mounted inside a vacuum tube of 0.8 m diameter and 4.6 m length. A vacuum of better than lO"'* Torr was maintained in each tube by a turbomolecular pump.

The tubes were supported independently from the optical

bench and thus mechanically decoupled from the neutron

optics.

The charge measurement was performed with an electric field of ±5.9 kV/mm. The field was limited to this value by breakdowns due to imperfect finishing and

cleanliness of the electrode surfaces. Attempts at more

efficient cleaning by glow discharge had to be abandoned

because of damage to the insulators. Teflon had been

chosen as the insulator material because of its shock ab- sorbing qualities in the case of high voltage breakdowns.

4. Measurements and Results

4.1 Performance of the Measurement

At the beginning of each cycle the high voltage was

raised in about 1 sec to its nominal value; after a waiting

time of 2 sec the measurement, lasting 10 sec was

259
started. The count rate was recorded on magnetic tape

and the high voltage was switched off. Then a measure- ment with reversed electric field was started. About 1000

to 3000 such cycles were combined in one run. At the end

of such a run we regularly checked the steepness of the

slope of the image in order to be sure that the sensitivity

had not changed. The BFs-detector was connected to two scalers, one for each polarity of the high voltage.

Two monitor counters were used, one for monitoring the

incoming flux in front of slit Si and one near slit S4 in order to detect changes in background. In the case of a high-voltage-break-down (about 1 in 5 minutes) the meas- urement was stopped and the actual count rate was

deleted.

4.2 Data Analysis

Each of the 54 runs was analyzed separately. We ob- served a change of the count rate of less than ±10%

within one run, corresponding to a drift of the image of

less than ±5 |xm. This drift is probably due to thermal

effects, though the reactor hall is temperature stabilized

(within ±1 °C). This drift was small enough to take the

slope dN/dy as constant. The difference in count rate for the two field directions AA'^ was compared to the

squareroot of the count rate VW. These values for all runs are shown in Fig. 3, calculated in terms of the neutron charge. A x^-test was made for AZ, the difference in count rate for every measurement cycle, i.e., each pair of count rates vdth opposite field (AZ - Z + - Z _), assuming A Z = 0 as expectation value. For each run the value

of f(}r) = \%C - V2A - 1 [12] (A = number of cycles

per run) was calculated. Considering only statistical er- rors (and no systematic shift of the image), the distribution of the values / (x^) should be a Gaussian centered at zero with standard deviation 1. The agreement with

these expectation values was good; we got a mean x^ per run of X = 1-02. This indicates that systematic errors are small compared to the statistical errors. A second x^-test, performed with the results from all runs (Fig. 3), dehvers a x^ of 1.2. Consequently the error bars from all runs were increased (not shown in Fig. 3) in order to obtain a value of 1.0 for x"- The errors from the determination of the mean

wavelength \, the length 1', and the electric field E can be neglected. We got a final result:

9n = |ge|(-l-5 ± 1.4) X 10-20 (gg% confidence)

Qn = \qe I (-1-5 ± 2.2) X 10-20 (90% confidence).

Run Number

Figure 3. Electric charge of the neutron for each run.

4.3 Test Measurements

To test whether the apparatus produces a difference in count rate which is not due to the interaction of a hypothetical charge with the electric field, we performed

two other types of measurements:

(1) Seven runs were taken with high electric field £"0 for one direction but zero for the other. This test was

done for the following reason: The electrostatic attraction

of the electrodes changes the gap between them (the

change was less than 20 ixm). Neutrons striking the elec- trodes are reflected and can contribute to the count rate behind slit S4. If the E'-field would not be exactlv the same for both directions { \E + \ - | < 10-^£'+),

the reflected intensities might be different for both direc- tions of E. This could produce a misleading shift of the

image. The measurement with zero field in one direction

increases this possible effect by about 3 orders of magnitude. For these test measurements, we got, calculated in terms of the neutron charge:

q„ = \q, I (0.1 ± 4.6) x 10-2°. ', . ^

Therefore, this effect can be neglected for the charge

measurement. The results with zero field in one direction were used together with the other charge measurements

to obtain the final value.

(2) Eight long runs were performed without any electric field. The measured deflection of the image was

y = (-0.8 ± 1.3) X 10-2 1^^^

assuming negligible systematic errors. This result indi- cates that there is no significant beam deflection simulated by possible errors in the counting system.

The deflection from the measurements with, electric

field was:

y = (9.8 ± 8.9) X IQ-^ |xm

including systematic errors.

5. Systematic Errors Due to the Electric

Field

Besides a possible charge ^p, the magnetic moment of the neutron and its inner strcture can give rise to a deflection of the neutron in an electric field. It is not possi- ble to separate these forces experimentally from the

force acting on a neutron charge, because most of them

are coupled to the direction of E. In an extended paper

[13], it was shown that the deflection caused by a neutron

charge of q^, = 1.10 2'' x —the sensitivity of the re- ported measurement—is large compared to all other deflections. The most convenient way is the wave-optical

treatment where the mean potential energy along two

symmetric paths is calculated. It was shown [13] that

AQ, the mean energy difference between two symmetric

paths, is much smaller for all competing effects than that

resulting from a charge q„ = 1.10 x q^. The energy Q

of a neutron in an electromagnetic field is given by [14]:

+ Ci V • E + ^C2E" + p

• E

= I + II + III + IV + V + VI + VII

260
I: charge in a potential U

II: magnetic moment fx in a magnetic field H; jin =

1.21 X 10-^2 v-s-m;

III: "E X v"-term [15]; v = velocity of the neutron

IV: Foldy-term [16] V: intrinsic Foldy term [17]: C 1 = J'r^p(r)dr VI: Polarizability of the neutron;

VII: Electric dipole moment p [18]. Terms II-IV are caused by the magnetic moment jx^. For

an unpolarized neutron beam the terms II and III cannot

give rise to a shift of the image but only to some

broadening. Though the neutron beam was unpolarized it vi^as shown [13] that even for a completely polarized beam

the deflection is negligible. Interactions V-VII are due

to a possible intrinsic structure of the neutron and give

rise to negligible deflections.

We should like to express our sincere gratitude to Prof. E. Liischer for the kind hospitality we received in his institute, to the workshop of the Fachbereich Physik

of the Technische Universitat Miinchen for most of the

mechanical construction, to the directors and the staff of the ILL for efficient help and valuable discussions, and

to Dr. A. Zeilinger for a critical reading of the

manuscript.

This work was supported by the Bundesministerium

fiir Forschung und Technologie.

References

[1] S. L. Glashow, in "Proceedings of Neutrino 79," Ed. by A. Haatuft and C. Jarlskog (Bergen, Norway, 18-22 June 1979), Vol. I, p. 518.

[2] G. Feinberg and M. Goldhaber, Proc. Natl. Acad. Sci. U.S.A.

45, 1301 (1959).

[.31 H. F. Dylla and J. G. King, Phys. Rev. A 7, 1224 (1973).

[4] L. G. Chambers, Nature 191, 1082 (1961) and references therein.

[51 V. A. Bailey, J. and Proc. Roy. Soc. (N.S. Wales) 94, 77 (1960) and references therein.

[6] E. D. Garris and K. 0. H. Zuck, Nucl. Instrum. Methods 177, 467 (1974).

[7] G. Gallinaro, M. Marinelli, and G. Morpurgo, Phys. Rev. Lett.

38, 1255 (1977).

[8] G. S. La Rue, W. M. Fairbank, and A. F. Hebard, Phys. Rev.

Lett. .38, 1011 (1977).

[91 G. S. La Rue, W. M. Fairbank, and J. D. Philipps, Phys. Rev.

Lett. 42, 142 (1979).

[101 C. G. Shull, K. W. Billmann, and F. A. Wedgwood, Phys. Rev.

153, 1415 (1967).

[11] R. Gahler, J. Kalus, and W. Mampe, J. Phys. E 13, 546 (1980).

[121 R- A. Fisher, Statistical Methods for Research Workers (Oliver and Boyd, Edinburgh, 1958).

[131 R. Gahler, .J. Kalus, and W. Mampe, Phys. Rev. D 25, 2887

(1982).

[141 H. Rauch, in Neutron Interferometri/, Ed. by U. Bonse and H. Rauch (Oxford Univ. Press, O.xford, 1979)', p. 161.

[151 J. Schwinger, Phys. Rev. 69, 681 (1948).

[161 L. L. Foldy, Rev. Mod. Phys. .30, 471 (1958).

[17] G. Dattoli, G. Matone, and D. Prosperi, Istituto Naz. Fisica Nucleare, Frascati, Report LNF-77/9(R) (1977).

[18] W. B. Dress, P. D. Miller, J. M. Pendlebury, P. Perrin, and N.

F. Ramsey, Phys. Rev. D 15, 9 (1977).

261

Precision Measurement and Fundamental Constants 11, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984). GAS CONSTANT, X-RAY INTERFEROMETRY,

NUCLIDIC MASSES, OTHER CONSTANTS, AND

UNCERTAINTY ASSIGNMENT

Methods for the Determination of the Gas Constant

A. R. Colclough

National Physical Laboratory, Teddington, Middlesex TW11 OLW, U.K.

Following a brief discussion of the role of the gas constant in physics, a review is made of the possible methods for its experimental determination and of values obtained in the past by direct measurements or by inference from other work. Four practicable methods by which the gas constant might be accurately determined are considered. These are (a) the conventional method of the limiting density of a gas based on the virial equation of

state, (b) the method of sound velocity measurement in a gas depending upon the normal law for the speed of sound in a compressible fluid together with the virial equation, (c) determination via the Boltzmann constant and Avogadro's constant by the measurement of the electrical noise in a resistance invoking Nyquist's formula, and (d) determination via the Stefan-Boltzmann and other constants by the measurement of the power radiated by a blackbody cavity relying on the Stefan-Boltzmann law. The

four methods have very different sources of systematic error. Most of the historical determinations

leading to the currently recommended value of the gas constant have been carried out by method (a). Some more recent determinations by method (b) can now be shown to be in agreement with them if the sound speed measurements are correctly interpreted and processed. An assessment of the sources of uncertainty characteristic of methods (a) and (b) is attempted and an indication given of how they

might be reduced in the future. These uncertainties may be compared with those to be expected from

the relatively little tried methods (c) and (d). The analysis of the four methods suggests that there is scope for a useful reduction in the uncertainty of the gas and the other thermal constants. Key words: Boltzmann constant; fundamental constant; gas constant; limiting density; Stefan- Boltzmann constant.

1 . Introduction

A physical constant is usually called "fundamental" if

it represents the value of some property associated w^ith a theoretically unanalyzable or irreducible entity or process (e.g., a fundamental particle or the propagation of

light). Nontrivial dimensionless ratios of such constants,

being pure numbers and so quite independent of any

metrological conventions as to the size of physical units, are agreed to be even more fundamental [1]. The gas constant, R, falls into neither of these categories and when

it is referred to as a fundamental constant it can only be

a courtesy title, recalling a time when thermal processes

had not been logically reduced to mechanical ones. It is now usually known simply as the "universal" gas con- stant in view of its role in the equations of state of vir- tually all substances. Nevertheless it remains a key

constant in physics because, for sound metrological as

well as historical or cultural reasons, we choose to retain

temperature as a physical quantity s?(i generis ignoring

the possibility of its reduction to mechanical concepts and

assigning it a base unit in the International System of Units. Thus it must appear in all phenomenological equations where mechanical quantities are present together

with thermal ones in order to maintain dimensional homogeneity. Similarly, its microscopic counterpart,

Boltzmann' s constant k = R /N^^ where A'^a is Avogadro'

s

constant, must often appear in the equations of statistical mechanics in an analogous role, for example to convert a numerical measure of molecular disorder into the thermal

quantity, entropy.

Uncertainties in R and k are usually given as 31 and 32

parts per million (ppm) respectively (one standard deviation based on the 1973 adjustment of Cohen and Taylor

[2]). That of Avogadro's constant is only 5 ppm so that,

combining uncertainties in quadrature, a determination

of R from a measurement of A; will have the same uncertainty to within approximately one percent as the measurement, and vice versa. A value for the ideal gas molar volume, Vq, defined

under the conventional reference conditions of temperature and pressure will of course yield information on R of exactly the same fractional uncertainty and vice versa

since they are related in Boyle's law, Po^o = RTo, by

defined quantities. These are the temperature Tq =

0°C = 273.15 K and the pressure Pq = 101325 Pa. (But

note that Vq must be expressed in terms of length units

of the same kind employed in the unit of pressure. Prior

to the recent decisions of the CGPM (General Conference on Weights and Measures) to redefine the liter as equal

to the dm^, it differed from the dm^ by about 28 ppm.)

R and k can in principle be determined also from a value for the Stefan-Boltzmann constant ct through the

relation

a - Z-n^kVilSh^c^) , (1) where h and c are Planck's constant and the velocity of

light, respectively [3]. Again, h and c are relatively accurately known compared to the accuracy with which a may

be measured (see Section 5). Thus R and A- could be

determined to one quarter of the fractional uncertainty

of a a measurement. But in the past the accuracy with

which <j has been measured has been no better than

about 500 ppm [4] so that it has been the practice to use

the best value of k to calculate a value for ct.

263
R and Vq, k, and a form a small group of interrelated

physical constants which, together with the gravitational constant G, contribute little to knowledge of the values of the fundamental constants as a whole because of the diffi- culties of measuring them accurately. Cohen and Taylor

[2] group them under the unprestigious heading of "Other

Less Precise Quantities." In order for them to contribute

usefully to our knowledge of the values of the other con- stants [e.g., through Eq. (1)], their uncertainties would

have to be reduced to about 1 ppm or less and this seems

unlikely in the immediate future. What then is the use of reducing the uncertainties of R, Vq, and A: to 10 ppm and ct to 40 ppm? This would have no bearing on any great question of physical theory or on practical energy calculations

involving, say, efficient fuel use in industrial plant or en- ergy losses from domestic dwellings.

The real need for new measurements of this order of accuracy arguably arises entirely from their role in primary thermometry. Often a primary thermodynamic

thermometer has to be designed to operate in some re- gion of temperature remote from the triple point of water

where thermodynamic temperature is defined to fix the

magnitude of the kelvin {T = 273.16 K). This may have

the consequence that the primary thermometer may not

function well or even at all near 273. 16 K so that it can- not be calibrated at the fundamental fixed point. A good

example is afforded by acoustic thermometry in the

liquid helium range where thermodynamic temperature is determined from measurements of sound velocity c in helium gas. One has for an ideal gas

c{Tf = yRT/M (2) where y is the ratio of the principal specific heats Cp/C

„

and M is the molar mass of the gas. In this case experimental accuracy would be reduced at low temperatures if the instrument employed to measure c had to be designed

to function at 273.16 K as well to yield T =

[c (D/c (273.16 K)]- X 273.16 K. But with an accurate knowledge of R this is not necessary and Eq. (2) can be

applied directly. Thus a knowledge of the gas constant is equivalent to a water triple point calibration. Similar

arguments can be applied in principle to other ther- mometric techniques, although sometimes, as in gas ther- mometry, it is useful to calibrate at the water triple

point for other reasons. For instance they establish the

need for accurate values of ct and k. Generally ct will be

applicable to primary radiometric thermometry employing Stefan's law or Planck's law while k will apply to microscopic methods such as nuclear orientation thermometry or Johnson noise thermometry.

In what follows four practicable methods by which the

gas constant might be accurately determined are considered. These are (a) the conventional method of the

limiting density of a gas based on the virial equation of

state; (b) the method of sound velocity measurement in a gas depending upon the normal law for the speed of sound in a compressible fluid together with the virial equation; (c) determination via the Boltzmann constant and Avogadro's constant by the measurement of the

electrical noise in a resistance invoking Nyquist's for- mula; and (d) determination via the Stefan-Boltzmann

and other constants by the measurement of the power ra- diated by a blackbody cavity relying on the StefanBoltzmann law. The four methods have very different sources of systematic error. Most of the historical determinations leading to the currently recommended value of the gas constant, R = 8.31441 J K^^ • mor' [2], have

been carried out by method (a). Some m.ore recent determinations by method (b) can now be shown to be in

agreement with them if the sound speed measurements

are correctly interpreted and processed. An assessment

of the sources of uncertainty characteristic of methods (a) and (b) is attempted and an indication given of how they

might be reduced in the future. These uncertainties are then compared with those to be expected from the rela- tively little tried methods (c) and (d). The comparison in- dicates that there is scope for a useful reduction in the

uncertainty of the gas constant.

2. Determination of R by the IVIethod of

Limiting Density

The conventional method for determining R and Vq is the misleadingly named method of limiting density. A

quantity

Lip) = {Po/p)p{p), T ^ To (3)

is measured at successively lower pressures p where density p (p ) is determined by weighing the masses m(p) of a constant volume V of gas at temperature Tq. Extrapolating the measured values to zero pressure yields

L(0) = MPo/iRTo) = M/Vo (4) for T - To, where M is the molar mass of the gas. From

this R and Vo are easily determined.

In the past the directly measured quantities p, V, and ni have been determined respectively by mercury

barometry, the weighing of a glass flask when empty and

filled with water, and weighing the flask evacuated and

filled with the experimental gas. Most such determinations have used oxygen as the experimental gas since until 1961 this was assigned a molar mass of 32 exactly on the

chemical atomic mass scale. Occasionally, the extrapolation of L (p ) to zero pressure was carried out with the aid

of a value for the gradient of the L (p ) isotherm taken

from the results of other workers. Usually the approximate range of measurements was from 0.25 to 1.0 atmosphere. Unfortunately, error analysis for the important

historical determinations of R (shown in Table 1) is usuTable 1. Some historical determinations of the gas constant by the method of limiting density."

Source Gas (dni"^/mol) Rq

(J/K-mol)

Baxter and

Starkweather, 1924

[5], 1926 [6]

02 22.4133 * ±0.0002

8.31421 ±0.00007

Baxter and

Starkweather, 1926

[7]

Ns 22.4126 * ±0.0006

8.31395 ±0.00026

Moles et al., 1934-1937 [8] O2 22.4135 * ±0.0002

8.31429 ±0.00007

Batuecas, Casado, and Malde,

1939-1949 [9, 10, 11]

O2 22.4142 * ±0.0009

8.31455 ±0.00033

Pereira, 1978 [12] N2 22.4132 ±0.0007

8.31417 ±0.00026

^Values for Vq in this Table marked * have been obtained from the last recalculation of Batuecas [11, 13] which incorporated a number of corrections for systematic effects not made by the authors and converted to standard gravity and to the physical scale of relative molecular masses. We have'converted liters to dm3. All quoted uncertainties are purely statistical ones arising from random scatter in a set of measurements. They

represent one standard error.

264
ally rudimentary or lacking altogether. Corrections for several systematic effects are made and it is subsequently

assumed that there are no other significant systematic er- rors and that no significant residual error remains arising

from the uncertainty in the corrections. The error finally

quoted is usually determined solely from the random error

in a set of repeated determinations.

In order to obtain some conception of the likely magnitude of the error in a conventional determination,

consider the expression for L{p) in terms of measured

quantities:

Lip) _ Pq m (p ) _ Pj Wf+g Wf

p V

(5)

vi'here Wf is the mass of the evacuated flask, Wf+g. the mass of the flask filled with gas, Wf+w the mass of the

flask filled wdth water, and fKv the density taken for the

water. The approximate fractional uncertainty in L{p)

will be given by

dL ip )/L ip ) - (rfWf+g ± dWfVWg ± (dWf^w ± dWf)/W,,

±dp/p ± dp,yp,v ± dT^To (6) where "±" here signifies linear or quadratic error combination according to one's preferred school of thought

on the question and where Wg and are the masses of gas and water respectively. In what follows, uncertainties

which are arguably systematic or "Type B" (including

values like p^v taken from other work) will be combined

linearly with the root-sum-square values following in brackets. A plausibly estimated detailed error budget is given in Table 2. It applies to a good contemporary

determination rather than to the historical ones. The

notes following the table give the details of the assumptions made for each particular- source of uncertainty.

Perhaps the most widely felt concern about this method, arising to some extent from the experiences of Guildner et al. [22] with gas thermometry, is that sys- tematic effects related to gas sorption have been in- correctly assessed in the past. Consider the following

sequence of operations and processes which occur in the

determination of m (p ) through measurements of Wf and

Wf.g:

(a) Evacuate flask and seal.

(b) Some gas (helium) diffuses into the flask from the

surrounding atmosphere.

(c) Gas (probably H2O, N2, CO, O2) desorbs from inner

surface of the flask.

(d) The flask is weighed yielding a value for Wf.

(e) The flask is now charged with O2 at Tq and p is measured. Here three problems may arise:

(el) Some diffused or desorbed matter is retained in the

flask leading to an error, dMi, in the value assumed

for M ip is taken to be correct).

(e2) Some diffused or desorbed matter passes from the

flask into the gas handling and barometer systems.

This means that Wf as determined in (c) is no longer

the appropriate value to subtract from Wf+g to obtain m ip ). We attribute the error somewhat arbi- trarily to Wf inserting a corresponding value dWn

into the error budget.

(e3) Some of the O2 introduced into the flask will adsorb

onto the wall where it will increase the observed

value of Wf+g without contributing to the value of p. Again somewhat arbitrarily, this effect is expressed

as an error in Wf rather than in p or Wf+g.

(f) The flask is sealed and weighed to determine Wf+g.

A study of the literature of gas-glass sorption reveals

that sorption effects are very variable, depending on many generally ill-defined factors such as details of glass

composition, gas or vacuum exposure times, gas species,

and past chemical treatment. Nevertheless, some plausi- ble guesses as to the nature of sorption mechanisms can be made and it can be shown reliably that the effect of helium diffusion will be negligible [23, 24]. Almost cer- tainly the sorption of water will be far more important at ambient temperatures than that of any other species in- cluding oxygen, especially where flasks have been used

repeatedly for water and gas weighings without being

subjected to baking at high temperature (see e.g., Ref.

[25]). The latter step would almost certainly affect their

volumes at the ppm level of accuracy and would presumably be unthinkable practice between an initial volume

determination and a subsequent gas weighing. If a frac- tion /' of the desorbed gas is lost from the flask in proc- ess (e2), then the resulting error in the gas constant due

to mechanism (el) and (e2) will be

dR/R = n2 lfi2M2 - MO + Mj - MzVu^Mi , (7) where ^2 and tii are the number of moles of desorbed

and experimental gas respectively having corresponding

molar masses Mi and M2 (^2 « ^i). For M2 = 18 (H2O)

and Ml = 32 (O2) it can be seen that this error is only

weakly dependent on / so that a rough error of

n2/(2Hi) ppm in R may be calculated for / = 0.5. Elsey

[26] quotes figures for the outgassing rate of pyrex from

2.1 X 10"^ Pa-m-s"^ (for glass kept for 1 month in air prior to a ten hour high vacuum exposure) to 9.8 x

10"^ Pa-m-s~^ (for new glass following one hour's exposure to vacuum). If these rates were sustained for 24 hrs

in a 1 L pyrex bulb, errors in the gas constant would

result of 4 and 180 ppm respectively. The former figure

shows that this source of error can probably be controlled with adequate pumping at ambient temperature

and the latter shows that it is vital that it is. Unlike water sorption, oxygen sorption effects have re- ceived considerable attention over the years [27, 28, 29],

especially from Moles and collaborators at Madrid. Such

work has led to corrections being made, occasionally ret- rospectively, based on Freundlich adsorption isotherms.

In such work it is essential that adsorption measure- ments are quoted for the correct type of glass and for the

treatment it would receive in use, especially with regard

to the cycle of evacuation and filling [27, 29]. Fortunately, adsorption equilibrium is rapid, being completed in general in less than one hour provided that

problems due to tap gi'ease or surface bloom are absent.

Hartley et al. [29] quote a figure of 4.1 x 10"^ cm^/cm^ of O2 adsorbed on soda glass at 1 atmosphere and 15 °C,

representing an error of about 20 ppm in a density determination made with a 1 L flask. With a silica flask, on

the other hand, their figures suggest an error of only

0.1 ppm. It can be seen, therefore, that the effect of mechanism (e3), like that of mechanisms (el) and (e2), needs careful control, but can be reduced to manageable

proportions. Arguably the best practice to adopt would

be to make repeated determinations of R with a high

temperature bakeout preceding each gas filling and

weighing, and with the volumetric determination being

made last. In Table 2 it is assumed that this is done and

that sorption errors are limited to 5 ppm. It is also of in- terest to note that Hickmott [30] has observed by the use

of a mass spectrometer values of n oM2 for mercury equal

to those for water in an unbaked pyrex system subject to pumping for two weeks at ambient temperatures. The

significance of such an observation for the historical determinations of i? can only be guessed.

265
Table 2. Plausible en-or summary for a moderyi determination of R

by the method of limiting density.'^

Measured quantity x, with its uncertainties dx and

corrections, if made, with their uncertainties

Corresponding

error in R

(ppm)

p—mercury barometry

dpi. Hg temp, meast. (20 ± O.OrC) & corrn. to 0°C(3.6 X 10-3 p ± 2 X IQ-^p)

cip 2- uncerLainiy in g — lu m/s ) dp3:Hg purity

dp 4: resolution dp 5: capillary effect; dpg: scale; dp 7: resolution dp/p

2

1i2

2

0

7(4)

W[—weigh evacuated flask (about 200 g)

dV^fi! sorption (inc. dM i —see main text) dWf2: air buoyancy of flask (see main text) dW[^: pressure distortion of flask (see main text) dW weight calibration, dW{^: weight buoyancy

Wf+g—weigh gas-filled flask (about 200 g)

fiXA/ n . ' Gnvt'xt inn ^iti/^liirtoci in fi\Aj ^ ^ U. YV f-j-o-1* oUI pLlUll Vill^-iUUtrLl 111 (X r

dW{+^: air buoyancy of flask dWf+g3: pressure distortion of flask dWf+g4: weight calibration; dW[+„^: weight buoyancy

(dWf+g ±dWf)/Wg, Wg = Ig at 500 mmHg

5

122

122

15(7)

Wf-i-\^, —weigh water-filled flask rf^f+wi- sorption & buoyancy as in W( and W^+g above

(see main text) dWf^^'. flask load distortion (see main text) (dWf^^'±dW{)/W^., W^. - 103

g

0

11(1)

Pw—density of water: d Pw/Pw 3(3)

To—filling temperature: dT/TQ = ±0.0005 °C/273 °C 2(2)

dL (p )/L (p ) 28(8)

Extrapolation to zero pressure: dL (0 )/L (0 ), 2 ppm 30(9) M—molar mass {dM ^ included in dWfj)

dRIR 30(9)

^Generally it has been assumed that the work is carried out with the cooperation of a national standards laboratory, but not by such a laboratory. Thus

accuracies are those obtainable from fine instruments calibrated against a pri- mary standard within a year of use, say, rather than from the primary stand- ards themselves. The figures estimated for the uncertainties do not necessarily relate to any particular past work, though occasionally such work has been used to provide a plausible value. {dp 1). Assuming a water-jacketed barometer with temperature control and

determined to ±0.01 °C using mercury-in-glass thermometers. The corrections

to 0 °C (or to any other temperature for which the density of Hg is available) can be made to approximately 2 ppm [14, 15, 16]. (dp 2). Uncertainty in the local value of g wiW depend on where the measure- ment is made. At national standards laboratories g is often determined to parts in 10^. In Table 2 it is assumed that g is known to 1 ppm through a comparison made with an accurately determined value of g using a transfer standard. 266
(dps)- Cook [14, 15] has determined the density of seven pure Hg samples

from different sources. Mean results of two methods agreed to 0.45 ppm and

individual values differed by up to 1.7 ppm. We take ±2 ppm for density of any pure Hg likely to be used in the future. (dp 4). It is assumed that the barometer can be read to about 0.2 Pa employing optical methods or that this precision is achieved by taking the mean of many readings.

(dp 5). It is assumed that large bore barometers are used making the capillary effect negligible. A 4 cm bore will have a capillary error of about 0.013 Pa

or 0.2 ppm of pressure at 500 mmHg (68 kPa) [17, 18]. (dpg). Scale calibration, alignment, and expansion errors can be made negligible vdth care. The same applies where interferometric methods are used. (dWf4, dWf5, dWf+g4, and dW[+g^). It is assumed that weighings can be

effected by substitution to 0.003 mg representing an uncertainty of 2 ppm on

one liter of at STP.

(Pw). The density of water is known to about 3 ppm at 4 °C [19, 20, 21]. The

effect of dissolved air and of the dependence of density on pressure and temperature are less important.

(Extrapolation). This wall obviously depend on the range and quality of

data. It is assumed here that extrapolation introduces errors of 20% of the rms deviation of the isotherm fit. (M). This will depend on the gas chosen. It is no longer possible to use a gas with an exact defined molar mass. But Ne may be used with a measured iso- topic composition enabling M to be defined with an uncertainty of less than

0.5 ppm. Some similar problems to those outlined above will arise in the determination of V by measurements of Wf+w

and W{. But here all errors due to sorption are diminished according to Eq. (6) by a factor W^^ — 10'^ compared to W — 1 in the case of m {p ) and so are

negligible compared to dWfi- ll may be of importance, however, to consider how the

weight of the water affects the volume of the flask on fill- ing. If the flask is filled and sealed while suspended in air

it will suffer forces tending to make it pear-shaped so changing its volume. If it is filled under water, resting on

a flat surface or in a ring before sealing different changes

wiW occur. Because the volume thus determined is not

that applicable to the gas weighing we insert an error

dW{+vf2 in the error budget. Crude calculations indicate

that the effect might amount to several ppm in unfavorable circumstances so that it calls for an error assessment.

Unfortunately, reports of the historical measurements fail to do this. Nor do they usually give adequate details of the geometry, wall thickness and material of the flasks employed, or of the method of filling. In order to make a rough practical assessment of this error, a glass sphere of nominal volume 1 L with wall thickness approximately

1.5 mm was furnished with a long capillary tube and filled vdth colored water. A decrease of 1 ppm in volume was

calculated to cause the water to rise in the capillary by ap- proximately 1 cm. Different methods of supporting the

sphere readily produced variation in its volume of 1 ppm.

Changes in flask volume would also be expected due to variations in the pressure of the experimental gas. Ac- companying this would be a change in the buoyancy in air during weighing (this change is not to be confused with

that arising from variations in air density between

weighings—see below). The change in V from zero pres- sure to one atmosphere might typically be 100 ppm and

could in principle be corrected for from a knowledge of the elasticity of the flask material and its geometry as- suming that it retained its shape at all pressures. But in practice variations in wall thickness and details such as taps and support rings make this difficult. Some historical determinations (e.g., Ref. [28]) allow for this effect using a semiempirical formula due to Moles and Miravalles [31]. But the latters' work indicates that it is only

accurate to about 7% , equivalent, very roughly, to an uncertainty of 7 ppm in R. The obvious way to control this

effect is to increase the wall thickness of the flask. But

this increases its weight and a point comes where deli- cate balances can no longer be employed to measure to 0.01 mg or less. Generally flasks weighing less than 200 g

are used for this reason, implying for a volume of 1 L

with a 50 g glass tap a wall thickness no greater than

about 1.5 mm. In a modern determination this effect would presumably be investigated experimentally (e.g.,

by subjecting a water-filled flask furnished vdth a capillary as described above to external pressure changes).

Any error from this source might be expressed as a weighing error dWf^ in Wf. We assume it can be controlled to 2 ppm.

Even with a perfectly rigid flask changes in buoyancy

due to variations in air density from one weighing to another could have a serious effect. Usually a dummy

counterpoise flask is employed to control this source of

error, but even so, the error needs careful treatment. We suppose that the counterpoise has its mass accurately

matched by the aid of mercury filling and that the external volume difference of the two flasks has been determined. It is difficult to determine changes in air density

to better than about 100 ppm of initial density by pressure, temperature, and humidity measurements. This

uncertainty in the air density, which arises mainly from

uncertainties in air composition, will give rise to a comparable uncertainty in Wg. Thus in order to achieve control of this effect to 1 ppm, it needs to be established

that the external flask volumes differ by no more than

about 1% . We assume this is done, and that buoyancy er- rors in R due to all weighings amount only to 2 ppm

(1 ppm each for Wf and Wf+g).

The final uncertainties arising from other aspects of weighing, pressure, and temperature measurement are more familiar and will not be discussed. Their effect is shovra in Table 2 with its notes.

It is of considerable interest to note that an important

projected refinement of the conventional method of limit- ing density is described in these proceedings by Guildner

and Reilly. By using a piston displacement method to determine the volume of the experimental gas, they hope

to avoid the key problems of sorption. The displaced gas

is condensed into a small weighing vessel at 4.2 K so that problems of buoyancy are likely to be less troublesome

too. Their preliminary assessment of errors suggests a fi267
nal uncertainty in a determination of R might be

achieved of only 3 ppm (99% confidence limit with all uncertainties combined in quadrature). This compares well

with a plausible uncertainty of 30(9) ppm by the conventional method based on our rough calculations.

3. Determination of R by Sound Velocity

l\/leasurement in Gases

In this method the velocity of sound c (jo ) at a temperature close to 273.16 K is measured at increasingly lower

pressures and an isotherm of cHp) vs. p is plotted to enable the ideal gas value cHp = 0) to be obtained by

extrapolation to zero pressure. Then Eq. (2) is used to obtain the gas constant. (The velocity of sound in a real gas may be obtained from a virial expansion of pressure

terms whose coefficients are related to those at the virial equation of state [32, 33].)

The uncertainties in a value of R determined in this way are principally those arising from the sound velocity

measurements. Since temperature and pressure have to be accurately defined it is not possible to employ timeof-flight pulse methods which require long paths. Problems can also arise in identifying the edge of the pulse

since the sharper this is, the worse will be the problems

of dispersion. Continuous wave measurements are not used either due to problems of path accommodation and

the elimination of reflections. Thus interferometric

methods have to be employed. This presents the experimentalist with four rather fundamental choices. Shall he

opt for (a) variable path /fixed frequency or variable

frequency /fixed path interferometry; (b) audio or ultra- sonic interferometry; (c) one or two transducer interfer- ometry; and (d) cyHndrical, rectangular, spherical (or

other) resonator geometry? In what follows errors will be analyzed for a conventional interferometric determination of R much as they were for the limiting density

method in Section 2. Some recent unconventional acoustic

approaches will then be mentioned.

In the variable-path, fixed-frequency interferometer a

linear cavity dimension is changed, e.g., by withdrawing

an acoustic reflector, to bring the cavity from one order

of resonance to another at a constant frequency. Thus

one only needs to measure changes in length to obtain

wavelength /2 and hence sound velocity. This is easier

than accurately measuring absolute length as one has to

in a fixed-path cavity though such variable-path cavities,

having moving parts, are more complicated. More importantly, there are two further advantages of variable-path

interferometry of an acoustic nature. Firstly, all the

acoustic parameters which characterize the cavity (sound

velocity with its small boundary layer changes, absorption, reflection losses, reflection phase changes, and

hence effective cavity length, and the real and imaginary

components of transducer impedance) are frequency

dependent and so change from one order of resonance to the next with the variable frequency method. Indeed, to second order, they change over a single resonance. With

the variable path cavity they remain constant at all orders of resonance since it is operated at constant fre- quency. This greatly simplifies the analysis of the acoustic resonances. The fact that with the variable-path

instrument, absorption and reflection losses are constant

confers a second very important advantage on the

method. It means that they can be separated experimentally without recourse to theory by measuring the losses

at two or more orders of resonance. Then one can check

that loss mechanisms,, especially those arising from the

acoustic boundary layer, are as predicted by theory. This

provides an invaluable check on the operation of one's in- strument and, if the instrument appears not to function

normally, aids in diagnosis. Absorption coefficients can

also be used directly to correct the measured sound velocity for the effect of the acoustic boundary layer (see

below). Finally, if it is decided to operate at ultrasonic

frequencies, commitment to a fixed-frequency variablepath instrument allows one to employ a quartz crystal

transducer operated at resonance where its intrinsic impedance is effectively zero. Thus a measurement of its impedance under load is effectively a direct measurement

of the impedance of the loading resonant cavity which

greatly simplifies data processing.

Whether it is best to use audiofrequency or ultrasonic

interferometry is a controversial matter. In discussing

the closely related problems of acoustic thermometry,

the author has argued strongly for acoustic frequencies

[34]. He is still of this view, if less strongly, but the rea- sons have changed somewhat over the years. There used

to be two main reasons why ultrasonic interferometry was thought to be problematic.

Firstly, there was no practical test one could make to ensure that only the plane wave (oo-order) mode propagated in the cavity. In fact it seemed quite probable that one would be measuring not the required plane wave

phase velocity, but a group velocity corresponding to a group of unresolved higher modes. (Calculations of the

likely magnitude of the resulting error in velocity and

temperature measurement are given in Tables 4 and 5 of Ref. [34], but are based only on guesses about transducer

behavior. ) It was at one time assumed that the presence

of higher modes might be detected from changes in the

separation of the resonances with order or from measure- ments of acoustic absorption coefficients. This need not be

the case, as detailed analysis shows. All that can be done

is to assume that the transducer executes "plausible" vi- brations and to calculate the resulting error. Then an un- certainty could be quoted covering this error several

times. However, the error can be made quite small (about

200 ppm perhaps) if very high frequencies are employed so that the various phase velocities of the higher modes are

close to each other. But then acoustic absorption, proportional to the square of the frequency, may increase rapidly

and bring about a loss of sensitivity.

The second reason why ultrasonic interferometry was

thought to be unreliable was that no theory existed giving the tolerances to which the cavity end faces needed

to be in alignment. At high frequencies with very short

ultrasonic wavelengths it is possible in principle for reso- nance widths to approach optical dimensions given ideal geometry. But if the cavity has geometrical errors of

this order, could one be sure that one understood its behavior as is required of a primary instrument? This

question was later treated by Colclough [35] who pro- vided a realizable criterion for adequate cavity alignment

in the ultrasonic interferometer and showed that optical

cavity alignment is generally satisfactory.

However, other problems have become more important

for ultrasonic interferometry since then. It has long been

known that sound propagates more slowly in a tube or resonator than in the unbounded fluid due to viscous

and conductive mechanisms in the acoustic boundary

layer. Similarly, absorption and reflection losses are

increased. These effects are smaller in the ultrasonic in- strument than in the audio interferometer and have generally been assumed to behave as predicted theoretically

so that the necessary corrections can be made for them.

But no absolutely clear explanation has been found for Plumb' s discovery [36] that the boundary layer effect can

268
disappear at low pressures. This can easily be checked by

measuring the acoustic absorption (bearing out the above

point on the importance of measuring one's losses) and

raises the question as to how one should correct for boundary layer effect on the sound velocity. Again, a

clear answer is needed for a primary measurement. Note

too, that it was originally thought that the small boundary layer effect at ultrasonic frequencies constituted an

important advantage of the method. But an ultrasonic

boundary layer effect which is not corrected for or which

is wrongly corrected for in its absence can introduce a

velocity error even larger than that which remains in an

audio measurement after making an approximate correction for the greater audio boundary layer effect. However, it may be the case that in practice the extra work required to control the larger boundary layer effect

at low frequencies, while unproblematic in principle, imposes a considerable burden on the low frequency method

which results in less data and so poorer statistical control. And certainly the availability of quartz crystal

transducers at ultrasonic frequencies is something that

any audiofrequency worker will envy. To summarize: ul- trasonic thermometry is experimentally more convenient

and productive, but can raise serious problems of in- terpretation while the opposite is the case with audiofrequency thermometry. Thus which is the best to pursue is a difficult matter of judgment.

Finally, there is another serious systematic error which the author has encountered in his audiofrequency

work which may occur in ultrasonic interferometry as

well. This is due to the nonlinearity of the transducer

used to excite the interferometer. This error has not been appreciated until recently [32, 37, 38] although it has been common knowledge for very many years that

transducers, when driven sufficiently hard, are nonlinear. Now that its existence is known, it need not pose

a serious threat either to ultrasonic or audiofrequency

methods, but care is necessary to ensure that it is properly controlled. With regard to the choice as to whether to employ one

or two transducers, most workers in acoustic thermometry use one transducer both to excite the acoustic reso- nances and to monitor them. A few use one to excite the

cavity and another to monitor it. The former is structurally simpler but the latter has the advantage that data

processing may be facilitated since a pressure sensitive

(as opposed to a velocity sensitive) transducer at the

opposite end of the cavity to the exciting transducer pro- duces symmetrical resonance peaks like the quartz crystal single transducer. (Single transducer interferometry was assumed in the preceding discussion.)

Concerning the question of geometry, any form of reso- nator can be employed in principle where one of its linear

dimensions can be related to the acoustic wavelength.

Needless to say the number of possible resonators where

no change of this dimension is called for, i.e., variablefrequency resonators, is larger than the number of vari- able path resonators. For example, a spherical resonator

(a Helmholtz resonator) such as that developed by Moldover and Mehl [39] might be employed—this has the

advantage that there are no transmission boundary layer

losses, but only reflection losses. (Such resonators are

discussed in these proceedings by Holdover and Mehl.)

However, if one wishes to operate with a fixed frequency

and a variable path, then one needs a geometry with a

linear axis of symmetry. Resonators both of rectangular

or circular cross section have been widely employed.

Manufacture of the cylindrical cavity is, of course,

simpler. But in both cases great care needs to be taken

to ensure excellent and stable end face alignment at ul- trasonic frequencies and, at all frequencies, that the wall roughness is small compared to the expected acoustic

boundary layer thickness. In the audiofrequency device it has been assumed throughout that the frequency is held

below the first cut-off frequency of the cavity so that

only the plane wave propagates. This requires that

operating wavelengths are greater than 3.42 cavity radii for the cylindrical interferometer.

Table 3 shows a list of relevant systematic effects with some explanatory notes and references. Not all of the ef- fects are problematic in that their influence can be made

negligible. (These are marked with an asterisk.) But all need to be considered in a primary measurement. The

errors are calculated for a typical audiofrequency and a

typical ultrasonic interferometer.

The analysis presupposed in Table 3 is unlikely to be

applicable in any simple way to the free-propagation ul- trasonic Michelson interferometer under development by

Sauder for the measurement of R nor to the spherical

interferometer of Holdover et al. Transmission boundary

layer errors clearly cannot exist in the absence of guiding

boundaries so that one of the potentially most serious er- rors of conventional cavity interferometry is avoided ab

initio. The interested reader is referred to these workers' contributions to these proceedings for further details. From Table 3 it can be seen that the listed effects can

introduce uncertainties into a measured value of R of ap- proximately 35 ppm (16 ppm root-sum-square). Using a conventional audiofrequency instrument, Colclough,

Quinn, and Chandler [32] determined a value of R for which they quoted a systematic root-sum-square uncertainty of 18 ppm, very close to the estimate of Table 3 and a random uncertainty also of 18 ppm (three standard

errors).

Some confusion has arisen in recent years concerning

the compatability of values of R determined acoustically

and by the method of limiting density. This was due

mainly to the systematic effect of transducer nonlinearity

in the work of Quinn et al. [33] which caused isotherm

curvature at low pressures. Misinterpretation of this cur- vature led to an excessively high value of R [32, 37]. At

approximately the same time an ultrasonic determination

of R by Gammon [44] yielded a value rather closer to the

accepted value derived from limiting density measurements. However, when corrected for the effect of the

acoustic boundary layer. Gammon's result appeared more

compatible with that of Quinn et al. Subsequently the

latters' value was corrected for nonlinearity and was

shown, afterall, to be compatible with the accepted value

[32]. Then an indirect audiofrequency determination by

Colclough [37] by comparing the results of acoustic and

gas thermometry in the liquid helium range (4.2-20 K)

confirmed this. The status of Gammon's work now

remains uncertain while two audiofrequency measure- ments agree with each other and the accepted value of R.

These results are shown in Fig. 1.

4. Determination oi R = A/aIc by Electrical

Noise Measurement

Although at least one institute is considering the meas- urement of k through the Nyquist relation, no such

determination has yet been coippleted. This is largely be- cause the development of electronic components, most

notably FETs, has only recently reached a point where

there is a possibility of competing with conventional

methods. Not surprisingly noise thermometry has re269
Table 3. Plausible error summary for a determination of R

by the sound velocity method.

Source of error, size of correction if made

Resulting error in R

(ppm) Note

Audiofrequency Ultrasonic

Higher modes propagate,

dR/R = (X33/6A'ou)2 0 200 1

Acoustic boundary layer, dc/c = agi/k — 3000 ppm (audio) 4 200 2,3

2nd order BL correction to Kirchhoff-Helmholtz 1st order BL correction, 30 ppm (audio)

0 0* 2,4

Molecular slip at wall, 2 ppm

(audio) 0 0* 2,5

Temperature discontinuity at

wall, 4 ppm (audio) 0 0* 2,6,

Finite thermal conductivity

of wall, 1 ppm (audio) 0 0 2,6, 7 Wall smoothness * 0 0 8 Wall elasticity, 3 ppm (audio) 0 0 6,7 Wall adsorption * 0 0 9 Finite sound amplitudes * 0 0 10 Non-uniform cavity bore * 0 0 11 End face misalignment * 0 0 12 2nd order correction for using * 1st order theory = (a/A; )^ 0 0 13

Blatt effect 0 0 14 Cavity length measurement

(variable path assumed)

2 2 15

Resonance processing (geometrical processing of resonance

circles or peaks assumed)

10 10

Frequency measurement 0 0 Pressure measurement (for isothei-m plotting) 0 0

Nonlinearity of transducer * 10 10 16,2 Realization of Tq

±0.0005 °C/273 °C

2 2

Isotherm extrapolation 2 2 17,2 Molecular weight, 5 ppm 5 5 18 TOTAL 35(16)

*Signifies that the error is negligible or eliminable.

1. Error calculated using Colclough [34], paragraph (iii), p. 96. It represents only a plausible guess. Z33 = 14.54, b = cavity radius, /cqq =

2-iT/(wavelength of 00 mode). An error of about 200 ppm might be expected

with Ne gas in a 3 cm diameter cavity at a frequency of 3 MHz.

2. The correction can be made from absorption measurements or from

values of gas transport properties [33]. agL = absorption coefficient due to boundary layer, k = 2-it/( wavelength).

3. See e.g., Ref. [33], p. 353, Eq. (6). It has been assumed that agL is measured to 0.05 percent at audiofrequencies and used to correct c. 4. See reference to Shields, Lee, and Wiley [40] in Ref. [33], p. 375.

5. See Ref. [40], and p. 402 of Ref. [33].

6. See Henry [41] and Weston [42].

7. This error extrapolates to zero with pressure and so does not affect isotherm measurements.

8. Protrusions and interstices on walls must be negligible compared to the boundary layer thickness; see p. 376, Ref. [33].

270
9. See Ref. [33], p. 354.

10. This is easily checked by measuring velocities at various amplitudes.

11. Second order effects due to changes in the boundary layer effect could in principle arise.

12. See Colclough [35], especially Eq. (46). 13. See Appendix B of Ref. [33]. But see also Colclough, Quinn, and

Chandler [32], p. 128. One needs to think clearly whether to apply this correction.

14. See Colclough [37], p. 353. Also Quinn, Colclough, and Chandler [33],

p. 410. Blatt [43] has argued that as the pressure falls along an acoustic isotherm, the ternary molecular collision frequency will approach the sound frequency so invalidating the normal (zero frequency) virial expansion for C-. This point is discussed in the above references.

15. This assumes interferometric measurement with an unstabilized He- Ne laser.

16. 150 ppm corrected to within 20 ppm is typical of the author's experi- ence [32, 37], but with foresight the error might be kept to 10 ppm.

17. This will obviously depend on the range and quality of data. It is assumed here that extrapolation introduces errors of 20% of the rms deviation of the isotherm fit. 18. This value is based on Ar as used in Ref. [33].

I Gammon, 1976. ultrasonic (uncorrected for bouridary layer effect) Ouin

Chandlei

(n<

, Colclough,

1976, acoustic * revised) --t 1

Colclough, 1979,

of acoustic and gas thermometr

I Gammon, 1977, ul (corrected for bounda

ugh, Quinn, Chandler. 1979, acoj^ic

Pereira Cordldo, 1979, lir

I 1

Figure l. Relationship between recent values of the gas constant determined acoustically and the accepted value. Error

bars represent only the random scatter in a set of repeated measurements (1 standard error).

ceived a higher priority than the determination of k and

the recent literature of the former type of measurement

provides a rich source of information by which to make a prima facie judgment of the potential of the latter. Only

thermometry employing ambient-temperature discrete

component electronics (as opposed to microwave or SQUID electronics [45]) need be of concern in this context. In noise thermometry use is made of the classical Nyquist formula

<vi> = AkTrB - <V^>B/f^^ |A(ia)) \ '^df (8)

to determine thermodynamic temperature T in terms of measured values of some instrument output voltage V. This is related to the mean square source noise voltage

<vi> in bandwidth B = f2 - f \hy the transfer function A (jiiy) of the instrument. To determine k, the resist- ance r will be maintained at or close to the triple point

of water (=273.16 K) to yield a value for k from

<V'^(273.16 K)>. Thermometry measurements for constant r can often be referred simply to some reference

temperature, e.g., To, {T/T(, = <VHT)>/<VHTo)>) and

there is no need to determine k just as in gas thermometry there is no need to determine R. Neither is there any

need to determine A ij co) except insofar as it is neces- sary to define any changes in bandwidth. But this is not

the case with a determination of k where an absolute

measurement of <ys"> is required and hence of A( /a)). This demanding additional measurement (which

corresponds to the additional gas weighing in the method

of Hmiting density) is the main reason why determining

k, superficially with the same apparatus, is harder than

determining T. There are four methods of noise thermometry of potential relevance to a /c-determination. These are (i) the

"direct" method; (ii) the correlation method; (iii) the

Garrison-Lawson method [46]; and (iv) the BorkowskiBlalock method [47]. However, method (iii) need not be considered as a relevant option for determining k. Here T is determined

by adjusting the reference resistance tq to achieve the

operating condition where riTj — roTo so that test and

reference measurements present the same source noise to the measuring system. This has the advantage that amplifier noise can be eliminated from the measured noise

levels rather easily. However, it seems unlikely that any- one would choose a method for determining A: which re- quired measurement of a temperature Ti to 1-10 ppm

accuracy (say) as well as a realization of To- Similarly,

method (iv) will not be considered. Here r is eliminated

from the Nyquist relation by determining the noise

power from the product of the measured current noise

and the measured emf noise in the resistance. This would seem to exchange a relatively easy measurement for a

difficult one and so appear to be a bad strategy for determining k. We therefore consider only (i) the direct

method and (ii) the correlation method.

4.1 The Direct Method

Here the noise emf's of two nominally equal noise

resistances Vi and ro at Ti and Tq, respectively, are com- pared by switching an amplifier between them, squaring

or rectifying the outputs, and time averaging them to yield <VHTo) and <VHT)>.

In this method it is necessary to measure or other\\'ise

determine the contribution VetoV arising from amplifier

noise, noise in the electrical leads to the resistors, and

possibly switch noise. Similarly it is essential to ensure

that gain and bandwidth are equal for both ri and ?'o though this is not unduly problematic since nominally

ri = ro. Neither is the stability of the amphfier,

squarer, and averaging circuit since sufficiently rapid

271
switching between ri and removes drift errors in a comparison. Measurement of ri and ro is in principal simple to accuracies approaching 1 ppm, but where two-lead

systems are employed, perhaps at high temperatures

with ensuing problems of insulation leakage, it may be- come problematic.

With this method of noise thermometry it is not possi- ble in practice to calculate sufficiently accurately from

the measured parameters of an amplifying device such as a FET. The obvious method to measure amplifier /lead

noise (usually expressed as an equivalent input series

resistance y\ at Tq, perhaps combined with a noise current source in parallel with the input) is by shortcircuiting the input. But this usually causes the gain of the amplifier to change drastically and so yields an unsuitable value for correcting measured source noise. More promising is the method of injecting a known white

noise current from a high impedance source through Vq at To which yields information enabling y\ to be calculated. Alternatively noise from two or more resistances at To

can be compared if a reliable model is available of the

amplifier performance as a function of input impedance.

This last approach was adopted by Crovini and Actis [48]

when employing the direct method between 630 and

962 °C and affords the best available historical indication

of what might be expected in a A:-determination of this kind.

These workers analyzed their JFET amplifier in terms

of an equivalent circuit of a sufficiently general kind in which the intrinsic amplifier noise is represented at the

input by a series noise resistor generating noise emf

<e/> = 4kTo7\B, and a parallel resistor rb generating a

partially correlated noise current = AkT^Bh^.

The point of employing an equivalent circuit with two

sources is to make the parameters of the equivalent cir- cuit, ra and rb, independent of the thermometric source

resistance 'y\ which is not possible with a single equivalent amplifier noise resistance model. The noise figure for the amplifier may be written as F = 1 + rjr, + r^/rb + 2Re{^) irjr^)^

where Re("y) is the real part of the complex correlation

coefficient between e„ and i„ . Correspondingly, if is small compared to other parallel input impedances the

measured output will be

<y2> ~ 4kToBr, + <e,2> + <i,f>ri + 2Reiy) ie,fi,f)"^r„ (9)

which is a quadratic in Vg. Using Van der Ziel's [49] JFET expressions 7\ ^ 2/3 g,„ and rb = 2kT/eIg, where

g,„ is the transconductance, e the electronic charge, and

I g the dc gate current, Crovini and Actis calculated that

the term in ri would be negligible compared to <e,f> for their amplifier. The same was true for the last term on

the right hand side since the correlation coefficient y will be entirely imaginary. Thus only <e,f> remains as an error. These conclusions were checked by measuring

<V^> for different values of at Tq relative to a fixed

value of rg at To- Such a <VHr)> isotherm shows firstly

the reliability of the assumption that <V^> ~

AkTQBr\ + <e}> and secondly gives some indication of the statistical accuracy which is achievable in a k- determination by this method. Crovini and Actis found

that in a seven point resistance isotherm (rg = 300 -

1000 ohms approximately) an rms deviation of about

100 ppm from a linear fit of <Y'^> vs. was obtained.

No significant statistical improvement could be obtained

with a quadratic fit and it was not possible to observe

consistent quadratic coefficients in successive trials. Thus it could be concluded that the linear approximation

for <y^> is adequate to vdthin the resolution of the

measurements. The statistical accuracy of the line fitted to the points will be approximately 100 ppm/n^'- where n

is the number of degrees of freedom, four in this case. Thus noise measurements of an attainable statistical ac- curacy of approximately 50 ppm would be suggested by

this work. This is comparable with the performance of the limiting density and acoustic methods. Doubtless in an experiment specially designed to measure k, this per- formance might be improved. The fractional statistical accuracy theoretically attainable in a noise measurement

of this kind on a single resistance is given by \i{f 2 - /

1)]"^'^ for an integration time t. Thus a 10 ppm statisti- cal accuracy should readily be achieved in about

28 hours.

While the work of Crovini and Actis indicates that a

statistical resolution comparable with that of the more

conventional methods should be possible, there remains

the problem of the absolute determination of A(ja)).

This might well involve the evaluation of an attenuation

vs. frequency curve to an accuracy sufficient to evaluate

the area beneath the curve to < 10 ppm. The measure- ment of frequency contributes no appreciable error and

the measurement of the curve-height at a given fre- quency should be feasible to an accuracy of about 1 ppm.

Thus the crucial A ( j co) measurement should not present

a source of error significantly larger than that expected

statistically. It is important, however, that the instrumentation employed should remain stable at the 1 ppm

level of accuracy throughout the whole integration

period. Again, this is attainable with careful temperature

control both of gain and bandwidth controlling components and also of those components in the first ampHfication stage determining amplifer noise.

It seems likely, therefore, that a determination of k by

the direct noise method could yield a more accurate

result than hitherto obtained by conventional means.

4.2 The Correlation Method

In this method the source noise voltage is fed via two

separate pairs of leads to two identical amplifiers whose

output signals are multiplied together, squared, and

time-averaged [50]. Since the respective noise voltages in the amplifier input leads and first stages will be completely uncorrelated, they will not contribute to the output. The respective source resistance noise voltages, on

the other hand, will be perfectly correlated and so will in principle yield <V"> without error. The transfer functions A lij w) and A zij w) of each channel have to be

measured and remain constant as with method (i). Storm [51] has undertaken a preliminary error analysis

of this method prior to carrying out a determination of A: and Klein, Klempt, and Storm [52] have described in detail correlation noise thermometry measurements in the

liquid helium range. The latter work produced an iso- therm analogous to that of Crovini and Actis where a noise temperature of about 2.1 K was plotted as a function of resistance in the range 2.5-15 kft. It was found

that a statistical error of 0.25 mK was obtainable

representing approximately 100 ppm resolution on a

noise signal which was many times smaller than that of Crovini and Actis. The reason why Klein et al. plotted a

noise isotherm was not, as with Crovini and Actis, to determine the amplifier noise which arose mainly from

the FET channel noise. It was rather to control other

noise sources which are much smaller, but nevertheless

272
significant at such low temperatures and which the correlation method does not eliminate. These can arise from

current noise in the separate source resistance lead pairs

which, because the amplifier input impedances, though

large, are finite, generate currents in the source resistor. The resulting voltages are presented symmetrically to both amplifiers and so are not eliminated. These sources

of noise were shown to be negligible by Crovini and Actis

for their instrument operating at much higher temperatures. We may assume, too, that they would cease to be

significant for a correlation measurement at 273.16 K. Were it possible to achieve the resolution of Klein et al. at this temperature a statistical uncertainty of about

1 ppm would be implied.

Storm's analysis assumes a measuring time of 5 x lO^s (about 8 weeks) with a 20 kHz bandwidth implying a

3 ppm statistical noise measurement accuracy compatible

with the figure suggested by the practical performance

achieved by Klein et al. To make use of such a resolution

it is necessary that amplifiers, attenuators, and other

electronic units can be calibrated and proved to hold

their calibration at a comparable level of accuracy for

this length of time. Taking these problems into account and combining uncertainties in quadrature, Storm's

analysis suggests an attainable uncertainty in k of only

8 ppm.

5. Determination of k and Hence R by the

Radiometric IVIethod

The history of direct radiometric determinations of the

Stefan-Boltzmann constant suggests that it is the least accurate way of arriving at a value for R. The most re- cent and best completed determination is due to Blevin

and Brown [53] who reported the value a = (5.6644 ±

0.0075) X 10-^W-m^2.K;-4 implying values for k and R

some 230 ppm below those derived from limiting density

or acoustic measurements. The corresponding uncertainty would be about 110 ppm (one standard error). This

is considerably better than previous radiometric determinations, details of which can be obtained from the discussion of Blevin and Brown, but disappointing compared to other methods. However, interpreted differently, the ex- periment yielded useful thermometric information on the

thermodynamic temperature of the gold point by using

the values of the Stefan-Boltzmann constant implied by

the accepted values of A: and R. (See the introductory re- marks of Section 1 where a knowledge of a was argued to be equivalent to a water triple point calibration.

)

These workers employed a near-black cavity source at the gold point because its high radiance at that temperature enabled more accurate intensity measurements to be

made for a given radiometer sensitivity and rendered the

parasitic effect of stray radiation less troublesome.

Moreover at these temperatures most of the energy is concentrated in the near infrared where highly absorbing

surfaces can be prepared more easily than for the far in- frared. The radiant power arriving at the radiometer was

limited by apertures to define its solid angle and measured by the substitution method in which an equivalent

amount of electrical power is substituted for the radiant

power. Measurement of the latter then yields a value for the former. The more important sources of uncertainty were those in the thermodynamic temperature of the

gold point (equivalent to 0.06% in ct), in the realization of the gold point by calibrated thermocouple (0.06% and

0.03% in a due respectively to calibration and thermal

contact uncertainties), in losses from diffraction (0.06%

in a) and in the lateral non-uniformity of the receiver

(0.05% in a). These and other lesser uncertainties were

combined in quadrature.

In a new primary radiometric measurement, Quinn and

Martin working at the NPL are hoping to avoid many of the larger sources of uncertainty incurred in the above

work and so to produce a measurement of a which is competitive with other methods. They give a full description

of their work elsewhere in these proceedings so that only

the main points need be mentioned here. Firstly their ra- diant source is held at the triple point of water so avoiding uncertainties arising from primary thermometry,

thermocouple calibrations, and thermal contact at very

high temperatures. This naturally gives rise to problems

of distinguishing between radiation entering the radiometer from the source blackbody and from other components at ambient temperatures. The method also makes stringent demands on the sensitivity of the ra- diometer since the radiance, varying as T"*, is much

lower at 273.16 K than at the gold point (1337 K). The

problem of stray radiation is met by providing the ra- diometer mouth with a thermally dead hemisphere except

for the source aperture. This is achieved by using a sur- rounding black radiation trap held at liquid helium temperature. Sensitivity is provided by operating the ra- diometer at about 1 K above the temperature of a bath of superfluid helium at about 2 K to which it is connected

by a thermal resistance. The temperature difference across this thermal resistance can be measured to a few

microkelvins using germanium resistance thermometers.

This is facilitated by the extreme temperature stability of the superfluid helium bath attributable to the

anomalously high thermal diffusivity in the superfluid

state. As with Blevin and Brown's instrument, radiant

flux is measured by the substitution method, with the in- cidental advantage in this case that superconducting con- nections can be made to the radiometer's electrical heater, thus minimizing errors due to dissipation in the current leads. Quinn and Martin hope by this means to measure the

Stefan-Boltzmann constant to about 100 ppm equivalent

to about 25 ppm in R and k. If this is achieved, their

measurements will be comparable in accuracy to those

obtained by the conventional method of limiting density.

Another radiometric determination of the StefanBoltzmann constant is being made by Ono adopting what

is essentially the method of Blevin and Brown. He in- tends to avoid their larger thermometric errors by

operating with a source at the 693 K zinc point whose exact thermodynamic temperature has recently been determined by Guildner et al. [22] by means of highly

developed gas thermometry. Ono expects to achieve an accuracy comparable to that of Quinn and Martin. Again,

interested readers are referred to Ono's contribution to these proceedings for details.

6. Current Best Values and Possibilities for

the Future

The currently accepted "best" value for R,

8.31441 J-K~^-mol"\ is that recalculated by Batuecas

[2, 11, 13] from three sets of results (see Table 1). In this recalculation the results obtained by Baxter et al. and

Moles et al. were processed in two ways to yield four

values of R from two determinations. The first method of processing simply applied a linear fit to the L {p ) meas- urements to obtain L (0) by extrapolation. The second

used only Lip) values taken near Pq and corrected to zero pressure by employing" a value for the second virial coefficient of oxygen critically evaluated by Cragoe [54]

273
Table 4. Recalmlation of the gas constant."-

Source Method

Value

(J-K-i • mol-i)

Uncertainty

(ppm)

Baxter et al. [5, 6] LD, O2 8.31463 * 27 Moles et al. [8] LD, O2 8.31465 * 24 Batuecas et al. [9, 10, 11] LD, O2 8.31518 * 66

Pereira [22] LD, N2 8.31417 75 Colclough et al. [32] Acoustic 8.31448 25 Colclough [37] Acoustic

(indirect) 8.31447 81

^In the case of entries marked * an estimate was made for systematic uncertainty

in eacii measured value of L(p) from the considerations of Section 2. This was

then combined in quadrature with the random uncertainty (3 std. deviations) and

used with the extrapolation uncertainty to calculate a weighting factor for the value of L(0) derived from each value of L(p). A weighted mean of these L(0) values then provided a value for R from each of the three determinations which

arguably reflected the variation of accuracy with pressure in the original L(p) measurements. No statistical errors were available for the work of Moles et al. and weights had to be obtained by interpolation between those estimated for Baxter et al. The uncertainties quoted above represent three std. deviations in R

as calculated from the weighted L(0) means. There is no significant difference between the weighted and unweighted means of the above six values of R, which

have respective standard deviations of 9 ppm and 40 ppm. from the literature of relatively high pressure PVT measurements. This value was expressed by Batuecas as Ao = (9.51 ± 0.9) X 10"* in a form which gives the frac- tional correction for nonideality at Pq- It can be seen

that its uncertainty is equivalent to about 90 ppm in. R.

With these four values he combined his own to provide a mean recommended value. The work by Baxter and

Starkweather [7] employing nitrogen was rejected. This

best value of R is open to the criticisms that it over- weights the two earlier measurements compared to Batuecas' s, that it underweights L{p) values taken

below atmospheric pressure, and that better values for Ao are now available than that derived by Cragoe. We

have recalculated a new "best" value using all the L {p ) values of the three sets of results as re-evaluated by Batuecas and correcting each to zero pressure using the

value Ao^ = (9.89 ± 0.13) x lO"* derived from Weber's

critical evaluation of the PVT literature [55]. Weber's

work also shows that the effect of the third virial coeffi- cient is equivalent to a change of only about 1 ppm in R.

To these three values we have added the new limiting

density value of Pereira obtained with nitrogen at p = P 0 and using an external value of A 0 derived from high

pressure measurements. Two acoustic values due to Colclough et al. and Colclough [32, 37] have been taken into account as well. In all cases an attempt has been made to estimate systematic and random errors including those

due to extrapolation. Where past work does not enable

this to be done easily, informed guesses have been made

on the basis of the considerations of Section 2. The calcu- ation is summarized in Table 4 and the final answers are R = (8314.60 d Vo = (22.41433

k = (1.380665 a = (5.67020 :

0.22) J-K~i-kmol-i

t 0.00059) dm^-mol"'

t 0.00037) X 10-23 j.K-i

0.00060) X 10"^ W-m-2-K-4

In every case three "standard error" uncertainties

have been quoted reflecting both random and systematic

errors. It is of interest to note that a value of R some

32 ppm higher than Batuecas' s best value would be obtained by applying our method of estimation to his three

chosen determinations. It seems that personal judgment

in such matters has a much larger effect on best values

than the systematic effects discussed in Section 2. Our

best value o( R is 23 ppm higher than Batuecas' s. In Sections 2 to 5 plausible error analyses have been

given for the four practicable methods of determining the

gas constant. The method of limiting density which pro- duces the currently accepted value was shown to be subject to systematic effects, notably due to sorption, which were not fully assessed in the reports of the historical determinations. Nor can they easily be assessed in retrospect. However, it has been shown that they could well

be as small as 5 ppm or less and the agreement between

values obtained from acoustic and limiting density meas- urements suggests that the accepted value is free from

systematic errors greater than about 30 ppm. It is possi- ble that this may be confirmed in the near future by ra- diometric measurements.

Appreciable reduction of the uncertainty in R will only come from considerably refined limiting density, acoustic, or noise measurements. Of particular interest in this connection are the projected limiting density measure- ments of Guildner et al. at the NBS, and the projected

noise measurements of Storm at the Universitaet

Muenster. Both parties anticipate root-sum-square uncertainties below 10 ppm. In the former case an advance uncertainty of 3 ppm has been estimated at the 99% confi- dence limit. This would be approximately equivalent to the 1 ppm standard deviation which was spoken of as the

uncertainty limit at which these constants could begin to influence the values of the other fundamental constants through Eq. (1). However, before this can happen it v/ill be necessary for a determination of the Stefan-Boltzmann

constant to be made to about 15 ppm as well. (The

current uncertainty in Planck's constant is 5.4 ppm and

that in the velocity of light is 0.004 ppm.) As will have

become apparent from Section 5, this is at the moment

beyond the aspirations of even the most ambitious exper- imentahst. Thus it seems that the promotion of the ther- mal constants from the category of "other less precise

quantities" must be deferred for the present.

274
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R. Duncan and M. Weston-Smith (Pergamon Press, Oxford, 1977), p. 38.

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U, Ed. by J. H. Sanders and A. H. Wapstra (Plenum

Press, New York, 1972), p. 534.

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Chemical Constants, 14th edition (Longman, London,

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Stand. Sect. A: 80, 703 (1976).

[23] G. F. Weston, Vacuum 25, 469 (1975).

[24] R. J. Elsey, Vacuum 25, 299 (1975).

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Hall, London, (1964).

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[28] E. Moles, T. Toral, and A. Escribano, Trans. Faraday Soc.

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[33] T. J. Quinn, A. R. Colclough and T. R. D. Chandler, Philos. Trans. R. Soc. London, Ser. A: 283, 367 (1976).

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[36] H. H. Plumb, private communication.

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(1979).

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(1979).

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275

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

An Ultrasonic Determination of the Gas Constant

William C. Sauder

Department of Physics, VMI, Lexington, VA 24450

Progress on a gas constant determination by means of ultrasonic interferometry is reported. The

acoustic interferometer is a two arm instrument designed to operate in the range 0.1-1.0 MHz. An ap- proximately half-scale model has been constructed in order to establish the data taking algorithm as well as to investigate aberrations. Electrostatic transducers have been designed for the experiment

that vdll allow characterization of the acoustic field, a necessary step in extracting wave length meas- urements from the fringe data. Key words: acoustic interferometry; gas constant; ultrasonic transducers.

1. Introduction

Prior to the middle of the last decade, the value of the

gas constant had been established by applying data from

limiting density measurements on oxygen to the ideal gas

law^. Then, in 1974, Quinn, Chandler, and Colclough [1] published initial results from their acoustical determination of the constant. In this case the velocity of sound in argon was measured, at pressures betv^^een 0.03 and 1.3 MPa using a classical standing wave acoustic interferometer operated at low frequency (5.6 kHz).

In this paper we report progress on an acoustic determination of R to be carried out vdth a new form of inter- ferometer operating in the ultrasonic domain. We wdll discuss the advantages and disadvantages of the method

and describe the still-evolving design of the interferometer. Estimates of the expected performance will be set

forth.

2. The Method

When the propagation of sound through an ideal gas is an exactly adiabatic process, the gas constant can be re- lated to the sound velocity by the well-known relation

MV

RA TA

R

yT

where v is the sound velocity, M and T are the molecular

weight and temperature of the gas, respectively, and y is the ratio of specific heats. We plan to measure v in a variety of rare gases using an acoustic instrument that rather resembles the Froome

microwave version of the Mach-Zehnder interferometer

[2]. A schematic dravdng of the device is shown in Fig. 1. Two transducers, mounted back-to-back on a moving carriage, are excited simultaneously by a sinusiodal voltage

at ultrasonic frequencies. Signals from stationary microphone transducers at either end of the instrument are combined electronically to form a fringe signal that

varies sinusoidally with carriage position, having a

periodicity of approximately X./2 (where \ is the

wavelength of sound in the gas). A plot of the position of

the carriage at fringe peaks vs. the fringe number, when

corrected appropriately for Fresnel diffraction effects and variations of temperature, will be a straight line having a slope of X/2. We vidll operate at ultrasonic frequencies in the range

0.1-1.0 MHz. Thus X vdll be sufficiently short in compariTB RB

mm

Figure 1. Schematic drawing of the acoustic interferometer. TA, TB: sending transducers; RA, RB: receiving transducers; C: movable carriage.

son vdth dimensions of the instrument to ensure free space propagation of the acoustic waves. This will obvi- ate the necessity of making significant boundary layer

corrections to the measured velocities as would be neces- sary in low frequency interferometers characterized by a large wavelength to dimension ratio [3]. On the other

hand, we vdll not operate at high enough frequencies to require the large dispersion corrections that would be

necessary if the wavelength were comparable vidth the mean free path of the gas molecules [4]. In order for us to extract the gas constant from velocity data as implied in Eq. (1), it will be necessary to make runs at several pressures and then extrapolate to zero pressure. The sound velocity will vary vdth pressure

because of non-ideal behavior of the gas sample. In addition, dispersion effects resulting from viscosity and heat

conduction are pressure dependent [5]. We vdll look for other systematic effects by varying

experimental parameters, e.g., by using a variety of

sample gases (including isotopically enriched gases), and

by operating at several different frequencies in order to bring out the frequency dependence of the near field dif- fraction effects and of the virial coefficients that characterize non-ideality.

3. Progress of the Experiment

In order to study the characteristic behavior of the

double arm acoustic interferometer, we have constructed an approximately half-scale model. The model interfer277
ometer operates in air rather than being enclosed in a vacuum-tight tank as will be the case for the final instru- ment. Displacement of the carriage is measured by count- ing turns of a lead screw, whereas in the final apparatus,

carriage displacement will be determined by means of

optical interferometry.

We anticipate that this model interferometer will be

capable of wavelength (and therefore velocity) measure- ments with an accuracy of 100 ppm or better. Though we

will use the instrument to make acoustic velocity meas- urements for air in order to confirm our planned pro- cedure in the final experiment, the principal use of the

model will be for studying diffraction effects and possible errors introduced by standing waves in the interferometer arms. We can already discern some aspects of the design of the

final interferometer. The instrument will be contained

within a 1-meter long by 0.5 m diameter tank. The extent

of carriage travel possible will allow scanning of 200 to 4000 fringes during a run, depending upon the frequency

and the sample gas employed. The instrument will be submerged in an ice bath, so that temperatures will be stable

to within a few mK. Carriage travel will be determined to within 0.01 fringe (for 6328 A illumination of the optical in- terferometer). We expect standard deviations of the order

of 1 ppm in the velocity determinations.

Because our apparatus employs free space propagation

of acoustic waves, it is necessary that we establish with

certainty the shape of the acoustical wavefronts. Otherwise, it would not be possible to translate the data on

fringe spacing into a determination of wavelength. As a

result, we are devoting considerable effort to the design

of our transducers and the characterization of their

vibration.

The transducers are of the electrostatic type investi- gated extensively by Kuhl, Schodder, and Schroder [6]. As shown in Fig. 2, the backing plate of our transducer

is a gold coated infra-red diffraction grating having a grating spacing of the order of 0.1 mm. A 7 |xm gold

coated Kapton^ polyamide film is made to adhere to the

ridges of the grating by applying a dc bias voltage

between the evaporated gold coating on the film and that on the grating. A high frequency voltage superimposed

on the bias then produces mass-limited vibration of the

rectangular strips of film between these lines of contact. The transducers are operated near resonance.

In order to ensure that the transducers are operating

properly, we periodically will carry out frequency cali- brations to establish the fact that the frequency response

agrees with the theoretically predicted response. Furth- ermore, by incorporating a transducer into one arm of an

optical Michelson interferometer by employing the trans- ducer face as one of the interferometer mirrors, we will be able to examine the variation of phase of vibration over the face by studying the photoelectrically detected

interferometer output signal as the optical beam is moved over the transducer face. Finally, knowing the vi- bration behavior of the transducer face, we will perform

a computer integration of Kirchhoff s formula [7] in order

to establish the shape of wave fronts in the acoustic wave

field. Because much information in addition to data on car- riage position and the phase of the acoustical interfer- ometer signals must be recorded, the instrument will be

operated under microcomputer control. During a typical run, the computer will record the amplitude from each arm of the interferometer, the acoustic fringe signal, car- riage position, temperature, pressure, and frequency. A

block diagram of a portion of the microcomputer interface

is presented in Fig. 3. At the completion of the run, these data, having been stored on magnetic floppy disk,

will then be transmitted to our computer center via a telephone link and stored on magnetic tape. Subsequently, the data will be smoothed and then compared to the theoretically predicted fringe pattern in order to determine a velocity for the run.

trademark of E. I. du Pont de Nemours & Co., Inc.

Figure 2. Grating backed electrostatic transducer. G: grating; F: polyamide film; C: evaporated gold coating; S: spring (Schnort

washer); I: insulator.

278
T

i

!

PLL PD

L- PRV

PLL

PRV -I

io Microcomputer

Figure 3. Block diagram of a portion of the interface. T: trans- ducer; P: preamplifier; B: band pass amplifier; PLL: phaselocked loop; PD: phase detector; PRV: peak-reading voltmeter; L interface for microcomputer bus. Work on the model interferometer and its computer

control as well as the development of the grating trans- ducers has progressed sufficiently far that we would ex- pect to be taking data within a year or two.

4. Conclusion

Because the gas constant is related to the body of

atomic constants through the Avogadro constant and

Boltzmann's constant, and since the latter is still a

"weak link"^, the only method available for uncovering

systematic errors in gas constant determinations is to measure the constant by several methods. The acoustic

determination is then a check on possible systematic er- rors in the old physical chemistry gas density determinations. The completion of the high frequency experiment

described in this paper should estabhsh the viability of the acoustic methods when compared with the results of Quinn et al. One would hope that then we could have

considerable confidence in future "best values" of the

gas constant.

This research has been and is being carried out with

funding from a Research Corporation Grant and a National Bureau of Standards Precision Measurement

Grant. I wish to acknowledge the help of my students,

Ken Yates and Kim-Ying Chu, who, as undergraduates,

are learning about research in general and about the joys

and frustrations of precision measurements in particular.

References

[1] T. J. Quinn, T. R. D. Chandler, and A. R. Colclough, Nature

250, 218 (1974).

[2] K. D. Froome, Proc. R. Soc. London, Ser. A: 247, 109 (1958).

[3] D. H. Smith and R. G. Harlow, Brit. J. Appl. Phys. 14, 102

(1936).

[4] A. B. Bhatia, Ultrasonic Absorption (Oxford Univ. Press, London, 1967) pp. 59-64.

[5] A. B. Bhatia, ibid., p. 61.

[6] W. Kuhl, G. R. Schodder, and F. K. Schroder, Acustica 4, 519 (1964).

[7] S. A. Schelkunoff, Phys. Rev. 56, 308 (1939).

But see the paper by T. J. Quinn and J. E. Martin in these proceedings.

279

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Spherical Acoustic Resonators: Promising Tools for Thermometry and

Measurement of the Gas Constant

M. R. Moldover and J. B. Mehr

Thermophysics Division, National Bureau of Standards, Washington, DC 20234

We report progress in our study of the feasibility of using spherical acoustic resonators for primary thermometry and for measurement of the gas constant. Prototype resonators of differing sizes and materials have been tested. Under conditions of metrological interest (argon, 273 K, 0.1-0.5 MPa,

13 cm diameter resonator), the radially symmetric resonances have halfvddths, gp, which are on the order of 2 x lO""* of the resonance frequencies fp . Instrumental errors in the measurement of /„ and

Qp are on the order of 10"'^ fp. The ratios of several measured resonance frequencies, extrapolated to zero pressure, are within ±3 x 10"^ of those expected from our theoretical model. For the same reso- nances, the measured values of Qp exceed those calculated using nonacoustic data by less than 5 x 10~^ fp . These numbers suggest that an acoustic measurement of the thermodynamic temperature or the gas constant could be accurate to 10 parts in 10^. Further improvements seem probable.

Key words: acoustic resonator; acoustic thermometry; gas constant; speed of sound; thermodynamic

temperature scale; thermometry.

1. Introduction

Accurate speed of sound measurements in dilute helium

have been used to study the relationship between the

thermodynamic temperature scale and practical temperature scales in the 4-20 K range [1, 2]. Recently, speed of sound measurements in dilute argon have yielded a value

for the gas constant, R, with an "overall uncertainty" of 25 parts in 10^ [3]. In each of these measurements a

cylindrical acoustic interferometer was used. In the

present work, we explore the feasibility of applying a

spherical acoustic resonator to similar problems in metrology. We shall present evidence that the errors in the theoretical description of real spherical resonators

can be less than 5 parts in 10^. This implies than an

acoustic measurement of the thermodynamic temperature

or the gas constant could be accurate to 10 parts in 10^. Further improvement is very likely. We conclude that

spherical resonators offer significant practical and

theoretical advantages in comparison with cylindrical resonators for applying speed of sound measurements to problems in metrology.

Significant practical advantages arise from the exceptionally high Q of the radially symmetric acoustic reso- nances. For these modes, the gas motion is perpendicular

to the sphere's walls; thus it is not subject to viscous

damping at the walls. When our 12.7 cm diameter reso- nator is filled with argon at ' 273 K in the range

0.1-0.5 MPa, the halfwidths, Qp, of the first four radial resonances range from 0.5 x 10"^ to 2.0 x 10"^ of the resonance frequencies, fp. These resonances are a factor

of 10 narrower than the resonances of a cylindrical acoustic interferometer which were used in a recent measure- ment of the gas constant [3]. Thus the perturbations to the resonance frequencies arising from the gas-resonator

wall interactions are an order of magnitude smaller in the

spherical resonator than in the cylindrical resonator. The strong signals associated with the radially symmetric modes permit the use of very small transducers

*Physics Department, University of Delaware, Newark, DE 19711

which perturb the resonator geometry in a minor and

predictable way.

The symmetry of the spherical resonator facilitates the

detailed calculation of many perturbations. In particular one can show that the frequencies of the radial modes are

insensitive to geometrical imperfections to first order in perturbation theory. Thus for practical purposes, one can

construct a spherical resonator with modest tolerances

and measure its volume; a detailed map of geometrical

imperfections is not necessary. It is possible to model

analytically the elastic response of the shell of the spherical resonator to the vibrations of the gas within it. The application of spherical resonators to the measure- ment of the speed of sound in dilute gases has been dis- cussed by Bancroft [4], Keolian et al. [5, 6], Moldover et

al. [7], and by Mehl and Moldover [8], The latter refer- ence contains the most complete discussion of the theory

of such resonators in the present context. Much of the apparatus and techniques used for this work is identical with that described by Mehl and Mold- over [8]. Here, we wiW emphasize the significant improvements in technique and the tests of our acoustic

model that have been made subsequent to that work.

2. Experimental Techniques

Two resonators have been used in this work. The first

is an aluminum sphere (designated A) vdth an inside di- ameter of 12.7 cm and a wall thickness of 1.27 cm. The

second is a brass sphere (designated B) with an inside di- ameter of 6.35 cm and a wall thickness of 1.27 cm. Both

resonators were contained vidthin a single pressure vessel

and thermostat; thus they were studied in identical gas

samples at identical temperatures.

Resonator A is essentially the same one described in Ref. [8]. A variety of minor changes in its construction were made in an effort to improve its performance. (As a measure of resonator performance we customarily use

the comparatively easily measured quantities ^Qp/fp

where the Agp are the measured resonance halfwidths

minus the calculated halfmdths). Polishing the interior of

281
the resonator did not significantly change its behavior.

Neither did changing its support from a massive post to a weak spring. Lapping the two halves of the resonator together significantly improved the resonator's performance. The secondary source-detector coupling mentioned

in Ref. [8] was drastically reduced by altering all three

holes in the resonator. A new source transducer 0.35 cm

in diameter was fabricated. This transducer was fitted in a snug hole through the resonator so that the active surface of the transducer was flush with the inner wall of the resonator. This eliminated the source transducer's

coupling hole. The detector transducer was coupled to the resonator by a hole 0.08 cm in diameter and 0.64 cm

long. (This required embedding the detector in the reso- nator wall.) Thus the resonant frequency of this coupling

hole was pushed above the range of our measurements.

Finally, the hole used to admit gas into the resonator was partially plugged so that it too had a diameter of 0.08 cm and an effective length of 0.64 cm. Resonator B was also made in two halves. It was

drilled with three holes spaced 90° apart. The outer portion of each hole was 0.31 cm in diameter and 0.95 cm

long; the inner portion was 0.041 cm in diameter and

0.32 cm long. Transducers fit tightly in the outer portions of two holes; the third hole was used to fill the resonator with gas. The data reported below have been taken at various

temperatures between 15 and 25 °C. A platinum capsule thermometer indicated that the temperature of a

specific point within the thermostat drifted less than

0.5 mK in one week. The temperature difference across the two resonators was less than 2 mK.

Both the argon and the propane were used as received

from commercial suppliers. (The supplier claimed the

minimum purity of the argon was 99.9999% and that of

the propane was 99.94%.) We found that passing the

argon through a molecular sieve trap cooled to -70 °C

did not alter the resonators' performance.

To measure accurately fp and g^, for each resonance,

the resonator was excited successively at 11 distinct fre- quencies ranging from fp — Qp to fp + Qp in steps of

Qpl^. At each frequency the in-phase voltage U and the

quadrature voltage V at the receiving transducer were

measured with a suitable tracking lock-in amplifier, scanner, and digital voltmeter under computer control. This process was then repeated with the frequency

decreasing from fp + Qp to fp - Qr,. (The practice of tak- ing data by stepping both up and down in frequency compensates for possible drifts in the gas temperature which

are linear in time.) Then a function of the theoretically

predicted form

U + iV AL + B + Cf (1)

is fit to the data using_ a _program described by Mehl [9]. Here the ampHtudes A, B, and C are complex numbers.

The last two terms in Eq. (1) represent contributions to the complex voltage U + iV from the '

' tails' ' of all reso- nances other than the one of interest. Moldover et al. [7] demonstrated that the lpwes_t 30 resonances in argon are

fully resolved; hence \B + C f \ was usually less than a few percent of \U + iV |. When the resonators are filled with propane the values

of Qp encountered can be a factor of 10 smaller than those

encountered with argon under similar conditions. For

example, in propane at 287.7 K and 0.52 MPa the first radial mode of resonator A is at f i - 2568 Hz and has

the halfwidth gi - 0.0526 Hz. Under these conditions we

made an important check of the entire data acquisition

system and the numerical methods used to fit the data. We excited this resonance, turned off the excitation, and

then monitored the detected voltage as the resonator rang down. The value of g determined from the decay

curve differed from that obtained by cw measurements

by only 0.0004 Hz or 1.6 x 10"V3. Acoustic Model for Spherical Resonators

We will now review the acoustic model for obtaining,

the speed of sound, c, from the values of fp and gp which

result from processing the raw data via Eq. (1). Important contributions to this model were made by Greenspan

and will be published by him elsewhere [10]. (See also Ref. [8].) The starting point is the calculation of the

eigenfrequencies of a lossless fluid confined in a rigid

spherical enclosure by perfectly insulating walls. Then

surface perturbation theory is used to calculate the complex corrections to the eigenfrequencies arising from four

distinct sources. Finally, the bulk dissipation contribution to the resonance widths is considered. In the discussion of these phenomena we will provide numerical values

for them which are appropriate for resonator A filled with argon at 273 K. For the radial modes of a spherical resonator, the eigenfrequencies are found from the roots of the equation:

rfjo(2TT/a/c) _ ^

(2) da

0.

(Here a is the radius of the resonator and jo is the zeroth

spherical Bessel function.) The frequencies of the first few solutions to Eq. (2) for resonator A with c =

308 m/s are: p /(Hz)

1 3469

2 5964

3 8418

4 10859

5 13294

6 15726

The first surface perturbation occurs because the reso- nator wall has both a high thermal diffusivity and a high

heat capacity per unit volume compared with those of the

gas. The thermal admittance of the walls leads to the

perturbation

A/. (3)

(Here Dt is the thermal diffusivity of the gas which is equal to the thermal conductivity divided by the constant pressure heat capacity per unit volume, and 7 is the ratio of the constant pressure heat capacity to the constant volume heat capacity.) For resonator A filled with ar- gon, Eq. (3) is approximately

(A/, + igs)/f - 2.1 X 10-n-l + i)/VPT- (4) (Here we choose to measure frequency in units of / 1 and

pressure in units of 0.1 MPa, thus /* = //3469 Hz and

P* - P/0.1 MPa.)

Note that the, perturbations given by Eq. (3) and subsequent equations are to be added to the tabulated

"ideal" resonance frequencies to obtain predictions for the measured frequencies. This sign convention is opposite to that of Ref. [8]. A second perturbation occurs because the spherical

282
shell has a finite compliance. Greenspan [10] assumes

that the shell vibrates in a perfectly elastic breathing

motion in response to the radial vibrations of the gas

within it. He finds for resonator A: A /el ^ 4.7 X 10-6p*(cT /rx / 1 - (//20.2 kHz)2

(Here p* is the density of the gas measured in units of

the density of argon at 273 K and 0. 1 MPa and c

* is the

speed of sound divided by the speed of argon at 273 K

and 0.1 MPa.)

This expression diverges as the frequency approaches

the frequency of the breathing mode of the shell,

20.2 kHz, vi^here the approximations used to derive it are no longer accurate. Perhaps it is the failure of these ap- proximations which leads to comparatively large pres- sure-dependent discrepancies between the present model

and the data for the 7th and higher radial modes of reso- nator A (see below).

The third class of surface perturbations considered is that resulting from the holes drilled through the resonator wall. Such holes produce losses and frequency shifts which are complicated functions of frequency; however,

the perturbations are very small unless the resonant fre- quency of the hole itself (considered as an open organ

pipe) happens to coincide with one of the resonances of the spherical cavity. The result is:

2 ^fh + = —^cot[2 77 /L/C) + (1 + i)aL]. (6)

[(Here L is the effective length of the hole, ro is its radius, and a is the Kirchhoff-Helmholtz attenuation

parameter. For argon we may use the approximation: a = 3.22V/^/(rc).]

As a final application of surface perturbation theory we

consider geometrical imperfections in the shape of the

resonator. It can be shown that the first order correction

to the eigenfrequencies of the radial modes vanishes for

all shape perturbations [11] which do not alter the

resonator's volume. As an example of a typical second

order perturbation we quote Greenspan's result for a spheroidal deformation such that one axis of the spheroid

becomes a(l -I- e) while the other two axes become

a(l -I- e)"^. The fractional frequency shifts are 1.2 x

(/"e)^ for argon in resonator A. A value of e = 2 x lO""*

is attainable by conventional machine shop techniquesthus, this particular perturbation can be less than 10"^

for the first 5 modes.

Sound waves propagating in a gas are subject to at- tenuation throughout the gas arising from the nonvanishing thermal conductivity and bulk viscosity of the gas. This attenuation is analogous to dielectric losses in a microwave cavity. For a monatomic gas the bulk contri- bution to the resonance widths is approximately:

^ = ^rf^T = 3.5 X 10^V*/P*. (7)

The frequency perturbation associated with bulk losses is proportional to ighlff", thus, it is negligible in these

experiments.

4. Tests of the Understanding of Practical

Resonators

Our understanding of resonators as realized in the lab- oratory has been subjected to four very different kinds of

tests. First, we have measured the resonance frequencies

of several radial modes with both argon and propane in

resonators A and B. We found that under a range of conditions, ratios of these frequencies are within ±3 x 10"'' of the ratios predicted by the model outlined above. Of

particular interest for potential metrological applications

is the fact that the agreement between the data and the

model improve as the pressure is reduced from 0.8 MPa

to 0.05 MPa. For a second test of understanding, we

have compared the measured values of the resonance

halfvddths, g^, with values of gp which we have calcu- lated using literature data for such quantities as the

viscosity and thermal conductivity. We find that the

measured values of Qp exceed the calculated values by

less than 5 x 10"^ /p under an important class of conditions in both argon and propane. Again, the agreement

between the model and the data improves as the pressure

is reduced. A third test was designed specifically to ex- amine the accuracy of the boundary perturbation theory

for holes in the resonator's walls. We deliberately drilled an extra hole, 0.16 cm in diameter, through the wall of resonator A. The changes in gp Ifp produced by this comparatively large hole were measured and calculated for 6

radial modes. In each case the difference between the

measurement and the calculation was less than 2 x lO"*". A final test has been provided by the published measure- ments t)f the speed of sound in argon [7] and ethylene [8] under conditions where the speed of sound is known from

independent sources. In these meaurements it is demonstrated that the combined accuracy of the laboratory pro- cedures and the acoustic model in use at that time was

limited only by the accuracy with which the volume of resonator A was known, namely ±0.02%. In the present

work we have improved the laboratory procedures and

the acoustic model; however, we have not made additional dimensional measurements. Thus we will not com- ment further on absolute speed of sound measurements.

Our first test is concerned with frequency ratios rather

than the frequencies themselves because we are looking

for systematic failures of the acoustic model at the level of a few parts per million. We wish to examine the model

in ways which are not overwhelmed by our imperfect

knowledge of the acoustic virials of the test gases and

the present limitations in our ability to control gas purity

and temperature.

In Fig. 1 we show an example of the predictions from

our model for perturbations of the resonance frequencies

produced by the thermal admittance [Eq. (3)] and

mechanical admittance [Eq. (5)] of the walls of resonator A under conditions where the sum of these perturbations

is rather large. The perturbations caused by the holes are too small to be seen clearly on this scale. The raw data are analyzed by adding the predicted per- turbations to the measured frequencies. Then certain imperfections in the model will be revealed by any apparent

variation of the speed of sound with the frequency of

measurement. Such apparent variations for some of the

argon data at various pressures are shown in Fig. 2. The ordinate is measured _ TM calculated

y. y. calculated

(Here, r^^^^'^'^^ = fplft for resonator A and r^^^^^^ =

fplf \ for resonator B. Also, / and / are the frequencies of the first radial mode at the pressure of interest.)

The data for both resonators show that as the pressure is reduced, Ar/r becomes quite small or, equivalently, that

the frequency ratios approach their theoretical values. For each mode A rlr can be plotted as a function of pres- sure and extrapolated to zero pressure. The results of

this extrapolation are shown in Fig. 3. The data in

283
400

Af

f

(Parts

10^)

10

Ar

r

(Parts 0

10^)

Oi I I I 0 10 20

Frequency (kHz)

Figure 1. Perturbations to resonance frequencies of resonator A with argoyi at 0.05 MPa. The lower cur-ve is calculated for

the thermal admittance perturbation [Eq. (3)]; the upper curve

is the sum of [Eq. (3)] and the calculated elastic respoyise of

the shell [Eq. (5)].

-10

1 1 Argon

1

•

- P = 0 limit o

o

Normalization •

- A B

•

+ + •

•

-

• Resonator A

O Resonator B

1 1 i 0 10 20

Frequency (kHz)

Figure 3. Deviations in frequency ratios in argon as a function of frequency, extrapolated to zero pressure. The ratios for each resonator are based on the first radial mode of that

resoyiator.

40

Ar

r

(Parts

10^)

40

Argon

295 K

Normalization A B

I

10 Frequency (kHz) 20

Figure 2. Deviations in frequency ratios as a functioji of fre- quency at various pressures in argon. The ratios for each reso- nator are based on the first radial mode of that resonator

( + symbols). Solid symbols, resonator A: diamonds,

0.05 MPa; circles, 0.2 MPa; squares, 0.8 MPa. Open symbols,

resonator B: circles, 0.1 MPa; squares, 0.8 MPa. As the pres- sure is reduced |Ar/r | decreases, i.e., the measured fre- quency ratios approach their theoretical values.

Fig. 3 indicate the level of internal consistency that one can expect in the zero-pressure speed of sound, Cq, if Cq

is obtained from several different modes of a siyigle

spherical resoyiator. The first 4 modes of resonator A

yield zero pressure values of Ar/r which are all within

±1.9 X 10"'' of their average. The first three modes of resonator B yield zero pressure values of A rlr which are

all within ±3.1 x 10^*^ of their average. We have used the argon data for the first four modes

of resonator A and the first three modes of resonator B

to obtain an average ratio of the effective radii of these

resonators in the limit of zero pressure. The result is:

1/3 Ob

Volume of Resonator A

Volume of Resonator B

= 1.997156; argon, P ^0. (9)

It is very instructive to examine these frequency ratios

with propane in the resonators. At a given pressure and

frequency the thermal admittance perturbation to the resonance frequencies of propane is an order of magnitude smaller than that for argon, and the elastic response

of the resonator wall to propane is half as large as that to argon. In Fig. 4 we examine the residual errors in the

frequency ratios, Ar/r, for propane at 0.1 MPa. Here the

frequency ratios for resonator A have been calculated as above {rTM'^^^'^''^^ = fpLfi^) while those for resonator B

have been calculated with the definition r^TM^^'^''*''^ =

(fplf i) a^lap,. Upon comparing Fig. 4 to Fig. 3 mode

by mode we see the same pattern of residual frequency

dependence in Ar/r for both gases. Of course the modes

occur at different frequencies for the two gases. This

suggests that the errors responsible for the residuals are

related to the resonator geometry rather than instrumental noise or our imperfect knowledge of the properties of the two gases. If this conjecture is correct, improvements are quite feasible. The analysis leading to Fig. 4 made use of the ratio of sphere radii determined from argon data. Thus the fact that the data for resonator B nearly coincide with those

from resonator A demonstrates that this ratio is essentially independent of the gas used (at a low enough pressure). Indeed, we can use the propane data to obtain

another measure of this quantity:

Volume of Resonator A

Volume of Resonator B

1.997155; propane, P = 0.1 MPa (10)

284
10

(Parts

10^

-10

1

Propane

289 K

0.1 MPa

Normalization

• Resonator A

O Resonator B

10 Frequency (kHz) 20

Figure 4. Deviations in frequency ratios as a function of fre- quency in propane at 0.1 MPa. Here the frequency ratios are based on the first mode of resonator A and the ratio of the reso- nator diameters established with argon data (Fig. 3). The fact

that the two sets of data coincide shows that the ratio of effec- tive radii of the resonators is independent of the gas used in

the measurement.

g

(Hz)

r Argon

295 K

0 05 MPa

10 Frequency (kHz)

20

Figure 5. Calculated and measured, resonance halfwidths in argon at 295 K and 0.05 MPa. The lower curve is calculated for the thermal admittance perturbation [Eq. (3)]. The upper

curve is the sum of the loiver curve and the bulk losses [Eq. (7)].

The agreement of Eqs. (9) and (10) must be considered

somewhat fortuitous because the two sets of data were

taken at temperatures 6 °C apart and adjusted using

handbook values for the thermal expansions of the alloys

used to fabricate the resonators. (The possibility of measuring the thermal expansion of the assembled resonators

in situ using microwave resonances has been demonstrated by Holdover et at. [7].) We now review the data for the halfwidths of the reso- nances. Figure 5 displays the predictions for the

halfwidths of the first few radial resonances in argon at 0.05 MPa in resonator A. The dominant contributions to

Qp come from the thermal admittance of the resonator

wall [Eq. (3)] and the bulk losses in the gas [Eq. (7)]. As one might expect, there are mechanisms for energy

loss which we have not modeled; thus the measured

values of Qp exceed those calculated. In Fig. 6 we compare our acoustic model with

representative linewidth data for argon. The ordinate is A„measured _ „ calculated Q = 9p 9p .... f fp

'

It is clear that Ag/f is less than 5 x 10^^ for the first six modes of resonator A when the pressure is less than

0.2 MPa. It is also clear that Ag/f decreases for both

resonators as the pressure is lowered. This is additional

evidence that the present acoustic model is a sufficiently accurate description of the laboratory resonators to warrant serious consideration of their application to ther- mometry and /or a measurement of the gas constant. As the pressure is increased, the coupling between the

radial oscillations of the gas and the vibrations of the resonator shells increases. At frequencies where the resonator shells have mechanical resonances of their own

(e.g., the breathing mode in resonator A at 20.2 kHz)

this coupling will be especially strong. We conjecture the

increase in Agr// with pressure and frequency is, in large

part, related to such coupling. In future work we will discuss this in greater detail. We have extensive data for ^g/f when the resonators

are filled vdth propane. For pressures ranging from 0.1

20

Argon

1

295 K

o

o

•

• 1

o •

1

• 0

0 10 20

Frequency (kHz)

Figure 6. The measured resonance halfwidth minus the calculated resonance halfimdth , divided by the frequency f, for argon

at various pressures. Solid symbols, resonator A: diamonds,

0.05 MPa; circles, 0.2 MPa; squares, 0.8 MPa. Opeyi symbols,

resonator B: circles, 0.1 MPa; squares, 0.8 MPa. As the pres- sure decreases, the agreement between the measured and calculated values of g improves. There are no free parameters in the calculations.

to 0.5 MPa we find that ^.g/f is less than 5 x lO"'' for the first 6 radial modes of resonator A. At higher pres- sures we find an anomalous increase in Ay/f. This

precondensation phenomenon will be discussed in detail

in a future publication [12]. In Fig. 7 we show the increase in the measured reso- nance halfwidths which occured upon drilling a hole

0.16 cm diameter in resonator A. The area of this hole

greatly exceeds that of the other holes in the resonator. This hole is resonant near the 4th radial mode. The ex- periment confirms the perturbation calculation for the

hole. The halfwidths of the modes other than the 4th are changed by less than 2 x 10"^ /p.

285
1 Argon

1

1

50

0.1 MPa

1

Shole

1«1» -

f

(Parts

in 1 - —

10^)

\

a • •

0

-

1 1 10 20

Frequency (kHz)

Figure 7. Calculated and measured increase in Ifp for a

hole 0.16 cm in diameter in resonator A. There are no adjust- able parameters in the calculation.

5. Conclusions and Prospects

We have demonstrated our understanding of spherical

acoustic resonators using two different resonators whose

radii are in the ratio 2:1. The resonators were fabricated

from different .materials (aluminum and brass). Each

resonator was used with two gases (argon and propane)

whose acoustic and adsorption characteristics are quite

different from each other's. The data indicate that our understanding of laboratory resonators is such that one

could measure the speed of sound of a gas like argon in the limit of zero pressure with internal inconsistencies no

greater than ±3 x 10"^. There is an indication that this number is related to the geometry of each resonator; hence it could be reduced should the control of the other

experimental variables warrant it. We briefly consider

the implications of this conclusion for thermometry and

measurement of the gas constant. The thermodynamic temperature, T, is related directly

to a ratio of speeds of sound of a monatomic gas in the

limit of zero pressure and zero frequency, namely: T —

Toic/cof. (Here Tq is 273.16 K and Cq is the speed of sound in the gas at the temperature of the triple point of water.) Thus, for acoustic thermometry the primary

measurements needed are resonance frequencies at Tq

and T as well as the thermal expansion (perhaps via

microwave resonances) of the resonator between To

and T. The error of any acoustic measurement of temperature

must be at least twice the precision of the measurement

of c/cq. In this context it should be noted that errors in the acoustic model related to resonator geometry cancel out of speed of sound ratios to high order if a single mode

or some specific selection of modes is consistently used to determine such ratios. It is quite likely that the acoustic

contributions to the scatter among measurements of c/cq

will be smaller than the scatter in Fig. 3 or Fig. 4. In the course of an acoustic temperature measurement

the average atomic weight of the gas must not change;

however, it need not be precisely known. Auxiliary meas- urements such as the pressure, viscosity, and thermal

conductivity of the gas and the volume of the resonator need only be made with accuracies in the range 0.1-1%.

The spherical resonator's performance will not change

substantially with temperature if the ratio of the mean

free path in the gas to the diameter of the resonator is held constant. Thus results comparable to the ones above can be expected when acoustic thermometry is carried out at constant density rather than at constant

pressure. The mechanical coupling between the gas and

resonator is proportional to pc^; thus, it will become less important as the temperature is reduced. If ^He were

used instead of argon, the resonance widths at ambient

temperature would remain unchanged if the pressure were increased by a factor of 2.7. The transducers and

other materials we have used will perform well at all temperatures below 100 °C. Thus, the spherical resonator

as it now exists is a promising tool for studying the temperature scale in the range 1 K-350 K. An acoustic measurement of the gas constant will re- quire the measurement of the value of the speed of sound

of a monatomic gas at (or near) To in the limit of zero pressure and zero frequency. The relation used is: R =

Co 3M/(5To)- Thus a measurement of R will require a very accurate measurement of the atomic weight of the

gas used as well as the internal volume of the resonator

after it is assembled. We imagine hydrostatic weighing is a feasible method for measuring the resonator's volume;

however, we have not investigated this option yet.

References

[1] H. H. Plumb and G. Cataland, Metrologia 2, 127 (1966); G. Cat- aland and H. H. Plumb, Low' temperature thermometry: in- terim report, Natl. Bur. Stand. (U.S.) Tech. Note 765(1972).

[2] A. R. Colclough, Proc. R. Soc. London Ser. A: 365, 349

(1979); see also references therein.

[3] A. R. Colclough, T. J. Quinn, and T. R. Chandler, Proc. R.

Soc. London Ser. A: 368, 125 (1979).

[4] D. Bancroft, Am. J. Phys. 24, 355 (1956).

[5] R. Keolian, S. Garrett, J. Maynard, and I. Rudnick, J. Acoust. Soc. Am. 64(S1), 561 (1978).

[6] R. Keolian, S. Garrett, J. Maynard, and L Rudnick, Bull. Am. Phys. Soc. 24, 623 (1979).

[7] M. R. Moldover, M. Waxman, and M. Greenspan, High

Temp. High Pressures 11, 75 (1979).

[8] J. B. Mehl and M. R. Moldover, J. Chem. Phys. 74, 4062

(1981).

[9] James B. Mehl, J. Acoust. Soc. Am. 64, 1523 (1978).

[10] M. Greenspan, to be published.

[11] P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw Hill, New York, 1953), pp. 1052-1053.

[12] M. R. Moldover and J. B. Mehl, J. Chem. Phys. 77, 455 (1982).

286
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Proposed Method for the Determination of the Molar Gas Constant, R

Leslie A. Guildner and Martin L. Reilly

Temperature and Pressure Measurements and Standards Division

National Bureau of Standards, Washington, DC 20234

A new, accurate measurement of the molar gas constant, R, is proposed. The volume occupied by 2 g of neon gas at 273.15 K is to be determined at pressures of 135, 81 and 40 kPa. An error analysis

shows that the derived value of R should have a total uncertainty of 2.9 ppm at the 99% confidence

level. Key words: fundamental constants; molar gas constant; precision measurement; R.

1. Introduction

The molar gas constant appears as the proportionality constant in the equation of state for one mole of an ideal gas, pV ^ RT , (1) where p is the pressure, V the volume, and T the thermodynamic temperature. The present value of R is derived from measurements of the limiting density of pure oxygen gas v^^hich v^^ere carried out during the

period from 1924 to 1941. Batuecas [1] has summarized

and revievi^ed the relevant data from which he recom- mended a "most probable value" for y,„, the molar

volume of an ideal gas at standard conditions. Accepting

this value of for their compilation, Cohen and Taylor

[2] then calculated 8.31441 ± 0.00026 J/(mol-K) for (where R = V,nPo/To, Po = 101.325 kPa and -

273.15 K). The stated uncertainty is one standard deviation. This value of R has been officially adopted by CODATA [3]. Through the Boltzmann constant, k, (where k — R/Na

and is the Avogadro constant), R is related to two

other fundamental physical constants. These are the

second radiation constant, C2, (where C2 = hc/k, h is Planck's constant and c is the speed of light in vacuum)

and the Stefan-Boltzmann constant, a, (where a =

217^^^15/1^0^). Of these four interdependent constants, R

can be most accurately measured. As a consequence the

values for k, C2, and a are derived from R and the ap- propriate atomic constants. The uncertainty in R (31

ppm) is presently so much larger than that associated

with h, Na , and c, that it constitutes almost all of the es- timated error of k (32 ppm), C2 (31 ppm), and a (125

ppm).

The uncertainty in R also materially affects the treat- ment of good pVT data (in that V-,,, is significantly uncertain) and it is a major component of the uncertainty in the realization of thermodynamic temperatures by acoustic interferometry, isothermal gas thermometry, and

noise thermometry.

New values for R have been determined from several recent experiments. The various methods used have been

reviewed by Colclough [4]. The published values are: Quinn et al. (1976) [5], 8.31573 ± 0.00017; Gammon

(1976) [6], 8.31479 ± 0.00035; Gammon (1977) [7], 8.31531

± 0.00035; Colclough (1979) [8], 8.31447 ± 0.00022; Colclough et al. (1979) [9], 8.31448 ± 0.00007; and Pereira

Cordido (1973, 1979) [10], 8.31417 ± 0.00027. While these

results generally tend to confirm the accepted value of R,

the important fact is that the reported estimates of uncertainty are still much larger than most other fundamental physical constants. The importance of determining R with improved accu- racy was recognized by the Comite Consultatif de

Thermometrie, which specifically recommended to the

CIPM at its 1976 meeting and again at its 1978 meeting

that "measurements of the gas constant be carried out"

[11, 12]. We propose a new, more accurate measurement of the

molar gas constant by reliable, state-of-the-art tech- niques which are expected to be free of significant sys- tematic uncertainty. The resulting value of R, with its anticipated total uncertainty of 2.9 ppm (at the 99% con- fidence level) would markedly reduce the uncertainty as- sociated with the constants Vm, k, cz, and a, to a level comparable to that of the other fundamental constants.

2. Proposed Method

The proposed method provides very exact determination of pV /uT and the extrapolation to zero pressure to determine the molar gas constant, R, from the equation

of state of neon gas, pV/inT) ^ R + Bp/T +..... (2) The volume, V, corresponding to an essentially constant number of moles, n, at a temperature, T, of 273.15 K, is to be determined for pressures, p, of 135, 81 and 40 kPa.

The volume will be metered by multiple strokes of a pis- ton whose displacement is accurately knovra. Approximately 2 g of gas will be transferred at constant pressure

from a thermostated chamber into a detachable weighing

flask held at liquid helium temperature. The combined

relative accuracy of V and n is the same for each pres- sure and is optimized when the mass of gas used is 2 g. As a consequence, the extrapolated value of R will have

much greater accuracy than is possible with the usual

pVT methods.

A semi-schematic drawing of the proposed apparatus is given in Fig. 1. (The thermostated chamber and the

weighing flask are shown to scale.) The piston (P) is shown at endstroke in the thermostated chamber (C)". The piston has a nominal diameter of 7.62 cm and a

287
Figure 1. Schematic of proposed apparatus for determining the gas constant.

stroke of 12.3 cm. It will be sealed in the chamber by a

series of three PTFE seals (Si to S3) which will also guarantee its exact alignment [13]. The region between

Si and S3 is filled with neon at the manometer pressure. This arrangement reduces the danger of leakage across the main piston-chamber seal (Si) and "preconditions"

the piston in case there is any significant tendency for neon to adsorb on the surface. The motion of the piston

will be controlled by a hydraulic system (H) which has

stops to limit the stroke. The surface of the lower end of the chamber (Gi) will be lapped flat so that the cylindrical "gauge block" (G) can be wrung to it. The annular disk (G2) will be wrung

to the other end of G. An annular quartz disk (Qr) will be

wrung to the lapped bottom surface of the piston. At the

extremes of each stroke, the position of the piston will be

established by the capacitance between guarded elec- trodes on the parallel upper or lower surfaces of Qr and

the grounded surfaces Gi or G2. The chamber will be suspended from the top (T). The

volume between C and T will be enclosed by a seal so that it can be filled with neon at the chamber pressure to avoid any effects from pressure dilation. The top will be

supported from the base (B) by four symmetrically

spaced vertical rods (R). The space between the bottom

of the chamber and the base will be enclosed by a cylinder which will be sealed to the base and has a sliding

seal around the outer circumference of the chamber. This

volume vidll be also filled with neon at the manometer

pressure. The length of this assembly will be about 50 cm, and its diameter will be about 25 cm. It will weigh

about 100 kg. The entire assembly, including the

diaphragm transducer (D) and valves (Vi, V2, and V3)

will be placed in a bath of ice and circulating water. The detachable weighing flask (F) will be nominally 1 cm in diameter, 10 cm high, and 0.04 cm in wall thick- ness. It will have an empty mass of 18 g with an interior

volume such that the 2 g of neon gas will exert a pres- sure of about 12 MPa at room temperature. The valve

(Ve) will be an integral part of the flask with interior

threads and will seal against a gold gasket. It is designed

to be operated at a temperature near 30 K. (Note details

in inset.) In a typical displacement cycle the chamber, with V2

closed, will be filled with purified neon through Vi and

V3 as the piston is slowly withdrawn from the position in- dicated in Fig. 1. After equilibration ar\d adjustment of the pressure to balance that of the manometer (M), the

null of the diaphragm transducer will be observed and V3

closed. Valves Vi, V2 and V3 will be of "constant

volume" design [14, 15]; negligible error will be pro- duced by their action. The neon gas will be transferred to the weighing flask

in two stages. First, V2 will be partially opened and, as the piston is advanced into the chamber, the neon will flow slowly into a "holding volume" bounded by valves

V2, V4, and V5, and V7. This volume, which will be

mainly that of the intermediate thermostatted vessel

(TV), is to be slightly greater than the room temperature

volume of the displaced neon. When the travel of the pis- ton is complete and equilibrium reestablished, the pres- sure in the system will be once again balanced exactly

against the manometer by adjusting the temperature of the gas contained in TV. After V2 is closed all the neon in the "holding volume" can be quantitatively frozen into the weighing flask while the chamber is being refilled for another cycle.

To displace 2 g of gas, the practical limitations of both

diameter and stroke of the piston make it necessary to use 3 cycles for a pressure of 135 kPa. Correspondingly,

6 and 10 cycles will be required for the other pressures. No degradation of accuracy is expected, however, be- cause of the extra cycles. After the requisite number of cycles has been executed, valves V5 and Ve vdll be closed

and the flask removed for weighing.

The peripheral facilities needed for evacuation, which

include ion pumps (IPi and IP2) and an oil diffusion pump

(DP), for residual gas analysis (RGA), and for gas purification (from Vg to Vio) are described elsewhere [16].

3. Measurements

The experimental measurements require great care and

exactness. Sufficient experience exists for each element

of the process so that it can be expected to function successfully.

3.1 Pressure

Pressures vdll be established with the NBS high precision manometer. The pressure arm of the manometer is to be filled with the high purity neon. The principles and

288
many details of its operation have been described earlier

[17]. A main feature of the manometer is that the height

of the mercury column is accurately determined with

gauge blocks. The crowns of the menisci are accurately

located relative to the gauge blocks by reproducing precisely the capacitance between the mercury surfaces and

fixed electrodes which themselves are located accurately

with respect to the gauge blocks. Ten calibrated gauge blocks of nominally equal length

(101.6 mm) vdll be used in various combinations of three

to produce the lowest pressure, 40 kPa. Combinations of

six blocks vdll be used to produce a pressure of 81 kPa

and all ten will be used for a pressure of 135 kPa. One

experimental sequence will consist of three measure- ments at the lowest pressure and one each at the two

higher pressures. Two sequences constitute one

'

' set' '

.

3.2 Volume

The volume will be calculated as the product of the

number of strokes and the calibrated piston volume ad- justed by the product of the appropriate piston areas and

the various endpoint displacements as determined by

capacitance.

3.3 Mass of Neon Gas

All weighings v^dll be made using a double-beam va- cuum microbalance which has a sensitivity of < 1 fjig at a

load of 20 g. Two platinum weights will be used. A 20 g

weight will serve as a tare for all weighings. A 2 g

weight wiW be added when weighing the empty 18 gram

flask. The mass of the neon gas, MNe> will be obtained

through transposition weighings of the flask filled vdth

gas and of the empty flask plus the 2 g weight. Thus we

have

Mf + MNe + Ml = 20 g (3a)

and

Mf + 2 g + M2 = 20 g. (3b)

The quantity Mf is the mass of the weighing flask. The

quantity Mi is the mean of the small weight increments

associated vdth ten weighings of the filled flask; M2 is the mean of the ten corresponding weighings of the

empty flask. Therefore the mass of the neon gas is MNe = 2 g + M2 - Ml. (3c)

3.4 Temperature

The temperature of the neon will be assumed to be the mean temperature of the chamber as measured by four long

stem standard platinum resistance thermometers. The sensor of each thermometer will be at the mid-depth in one of the four thermometer wells (TW of Fig. 1) which will be ar- ranged symmetrically about the body of the thermostated

chamber. They will be calibrated at the triple point of water between each set of experiments.

4. Assessment of Imprecision of

IVIeasurements

Equation 2 will be fitted to the experimental data by

the method of least squares. The value of R is the inter- cept of the Hne extrapolated to zero pressure. The data

at 135 and 81 kPa will first be adjusted to compensate for the contribution of the third virial term. This is not

greater than 1.0 ppm and can be made with a negligible

error. The standard deviation of the extrapolated value

reflects the combined standard deviations resulting from

the imprecisions of the directly measured quantities. In

addition, there are uncertainties resulting from the imprecision of the auxiliary measurements. Any errors pro- duce a constant bias in the results, the limits of which

are estimable in terms of the imprecisions determined by

other experiments. (We denote the relative standard de- viation of a directly measured quantity by Sa and that of an auxiliary quantity by .s'p,.)

4.1 Pressure

The only significant error in the measurement of the

pressure is expected to be that from the manometer it- self. Resolved into its components, the pressure has er- rors as follows: (1) The density of the mercury will be

uncertain both in its value at 20 °C (sb = 0.33 ppm) and

in the measurement of its temperature (sa = 0.2 ppm).

(2) The relative standard deviation of the acceleration

due to gravity as measured at the site of the manometer

is Sb = 0.2 ppm. (3) The uncertainty in the height of the

mercury column depends on the calibration of the gauge

blocks for which sb ^ 0.2 ppm.

4.2 Volume

The volume displacement of the piston when moved

between the two fiducial limits will be determined by

weighing the mercury so displaced. The calibration will be

performed at the ice point under the conditions of the ex- periment. At the same time, the separation of the fiducial limits will be measured in situ by laser interferometry.

The piston diameter will also be very carefully determined at the locations corresponding to the extremes of

its travel in order to account for small differences in the

volume resulting from slight differences from its refer- ence position. We anticipate that these adjustments do

not involve significant error. The significant components of error are: (1) The rela- tive standard deviation associated with the absolute

value, at 20 °C, of the density of displaced mercury, in- cluding the mass measurement, is Sb = 0.4 ppm. (2) The

relative standard deviation in the density of the mercury

as a result of the adjustment for the thermal dilation between 20 and 0 °C is sb = 0.3 ppm.

4.3 Moles of Neon

The number of moles of neon gas will be determined

from the ratio of the mass of gas to its molecular weight.

4.3.1 Mass of Neon Gas

The single largest uncertainty of the experiment arises

from the mass assignment of the 2 g platinum weight. Its value will be derived in the most direct way possible

from the National Prototype Kilograms. The standard de- viation is estimated to be 1.5 |jLg (sb = 0.75 ppm).

The relative standard deviation of the mass of neon

will , be the sum in quadrature of the relative standard de- viation of the mean for each set of ten weighings of the

full and empty flask. For each mean, Sa 0.35 ppm, and

for their difference in quadrature, sa = 0.5 ppm.

4.3.2 Molecular Weight of Neon Gas

Naturally occurring neon has isotopes of mass 20, 21,

and 22. Neon gas with a concentration >99.95% -"^Ne,

289
and a balance of mostly ^Ne is formed by radioactive decay; it is readily available. The value of the atomic mass

of each isotope is accurately enough known [18] that no

significant error results. To specify the molecular weight

of the neon gas used in the experiment within 1 ppm, the

concentrations of the two rarer isotopes (which are present at a combined level of 500 ppm or less) need only

be determined within 2%. The relative standard

deviation associated with the molecular weight is Sb =

0.3 ppm.

4.4 Temperature

It is estimated that the standard deviation of the mean

of the four values of the chamber temperature will be

<0.05 mK, or correspondingly, the relative standard de- viation in the thermodynamic temperature at the ice point will be sa = 0.18 ppm.

A summary of these relative standard deviations is given in the following table:

QUANTITY TYPE A TYPE B

ppm Source ppm Source

Pressure

density of Hg

9h

0.2 Temperature 0.3 0.2

0.2

Original measurement

Original measurement

Length calibration

Volume 0.2 Imprecision 0.4 0.3 Hg displacement calib. Thermal dilation of Hg

Moles of gas 0.5 Mass of neon 0.75

0.3

Calib. of 2 g weight

Determination of MW

Temperature 0.2 Imprecision 0.2 Imprecision

0.37 X 10-12 1.11 X 10-'2

The error in R can be evaluated from the equation for the variance of the intercept when pV /{nT) vs. p is ex- trapolated to zero. This is

si - + p2/ X (p, - (4) 2=1 where p is the mean of the individual pressures, p;, and z

is the number of points. For a "set" of measurements

extrapolated vs. pressures in ratios of 3:3:3:6:10, the es- timated variance of R is = 0.37 x 10 "^^ x 0.398 =

0.15 X 10-12, or for 5 "sets" si = 0.03 x lO'^^. The combined uncertainty is s tot - 1-07 x 10"^, and the total uncertainty at the 99% confidence level for 5 "sets" of measurements is 2.71 times this number or 2.9 ppm.

5. Sources of Systematic Error

Numerous other sources of error have been examined

(e.g., the uncertainty of the thermomolecular pressure,

variations in the acceleration due to gravity, etc.). The

expected errors from these sources will be insignificant

in every case. The largest potential source of systematic error could

arise from sorption of neon gas on the interior of the

thermostated chamber and on the piston. However, in contrast to constant volume gas thermometry, the neon chamber will not be subjected to variations of temperature.

Hence it can be evacuated at a temperature high enough

that, after cooling, no significant amount of desorption will occur. As is our practice in using helium for the ther- mometric fluid of the NBS gas thermometer, we intend to purify neon so well that no significant adsorption of impurity can occur. At levels of a few ppm the sorption characteristics of neon itself are uncertain. Such sorption as may

occur will surely be very limited in extent and of the physical type (hence rapid). The experimental method is designed to evade any significant effect based on the fact that a stably sorbed layer of clean neon in the chamber will cause no error. Thus the guard section, which contains neon gas at the manometer pressure, will precondition the

surface of the piston to the maximum extent possible before it enters the main chamber. With such precautions in design and procedure, it is our opinion that no significant

sorption error will be encountered.

References

[1] T. Batuecas, in Atomic Masses and Fundamental Constants

i, Ed. by J. H. Sanders and A. H. Wapstra (Plenum Publ. Corp., New York, 1972), p. 534.

[2] E. R. Cohen and B. N. Taylor, J. Phys. Chem. Ref. Data 2, 663 (1973).

[3] CODATA Bulletin No. 11 (Dec. 1973).

[4] A. R. Colclough, these proceedings.

[5] T. J. Quinn, A. R. Colclough, and T. R. D. Chandler, Philos. Trans. R. Soc. London, Ser. A: 283, 367 (1976).

[6] B. E. Gammon, J. Chem. Phys. 64, 2556 (1976).

[7] B. E. Gammon, (1977), private communication.

[8] A. R. Colclough, Proc. R. Soc. London, Ser. A: 365, 349

(1979).

[9] A. R. Colclough, T. J. Quinn, and T. R. D. Chandler, Proc. R. Soc. London, Ser. A: 368, 125 (1979).

[10] J. M. Pereira Cordido, cited by A. R. Colclough [4]. [11] Comite International des Poids et Mesures. Comite Consultatif de Thermometrie. IF Session. {Rapports}. Pavillion de Breteuil, Sevres, France, 1976.

[12] Comite International des Poids et Mesures. Comite Consultatif de Thermometrie. 12'" Session. {Rapports}. Pavillion de Breteuil, Sevres, France, 1978.

[13] R. E. Edsinger, L. A. Guildner, and R. L. Anderson, Rev.

Sci. Instrum. 42, 945 (1971).

[14] R. L. Anderson and L. A. Guildner, Rev. Sci. Instrum. 36, 615 (1965).

[15] R. L. Anderson, L. A. Guildner, and R. E. Edsinger, Rev.

Sci. Instrum. 41, 1076 (1970).

[16] L. A. Guildner and R. E. Edsinger, J. Res. Natl. Bur. Stand. (U.S.) Sect. A, 80, 703 (1976).

[17] L. A. Guildner, H. F. Stimson, R. E. Edsinger, and R. L. Anderson, IVIetrologia 6, 1 (1970).

[18] A. H. Wapstra and N. B. Gove, Nucl. Data Tables 9, 265

(1971).

290
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

A Radiometric Determination of the Stefan-Boltzmann Constant*

T. J. Qulnn

Bureau International des Poids et Mesures, F-92310 Sevres, France

and

J. E. Martin

National Physical Laboratory, Teddington, Middlesex TW1 1 OLW, U.K. A new determination of the Stefan-Boltzmann constant is being ma:de using a blackbody source of thermal radiation at 273.16 K and a detector in the form of a heat-flow calorimeter at 2 K. The black- body is made of copper having an internal coating of 3M's C401 black paint. The beam of radiation entering the detector is defined by a pair of beryllium-copper apertures near 4.2 K. The calorimeter

detector is a thin copper shell, also coated on the inside with 3M's black, made in the form of a black- body and attached by means of a poorly conducting stainless-steel heat link to a reservoir of superfluid

helium at about 2 K. Electrical power may be supplied to the calorimeter through a series of heaters wound on the outside and connected to an external source of power by superconducting leads. The temperature of the calorimeter is monitored by a germanium resistance thermometer.

On opening a shutter below the first aperture, blackbody radiation arrives at the calorimeter, raising

its temperature by about 3 K. On closing the shutter, electrical power is applied to the calorimeter to maintain the same temperature. The applied electrical power is thus a measure of the thermal radiative power passing through the apertures. From a knowledge of the geometry of the aperture system and

the thermodynamic temperature of the blackbody, the Stefan-Boltzmann constant may be deduced. Key words: radiometer; Stefan-Boltzmann constant; thermal radiation.

1. Introduction

In this paper we describe briefly the design and operation of a calorimetric radiometer that has been built at NFL for the two-fold purpose of making a direct determination of the Stefan-Boltzmann constant and for the

measurement of thermodynamic temperature in the

range -50 °C to 230 °C. Preliminary results have been

obtained for the Stefan-Boltzmann constant and work is in progress both to confirm these results and to make

measurements of thermodynamic temperatures. Nothing

further wiW be said here about the latter since they are

the subject of another publication [1]. The Stefan-Boltzmann constant, a, appears in the

equation that describes the total radiant power, E(T),

emitted per unit area by a blackbody at a thermodynamic

temperature T: E{T) = (tT\ (1) where, from a summation of Planck's equation over all wavelengths, a is given by:

in which c is the speed of light, h is the Planck constant and k is the Boltzmann constant which is itself given by

the ratio of the gas constant, R, to the Avogadro con- stant Nj^. The accuracy with which a can be established

using Eq. (2) is limited principally by the relative uncertainty, of 25 parts in 10^, in the most recent value of R.

Taking the most recent values for the constants, namely:

c - 299 792 458 m s"^ (conventional value recommended

by the 15"^ CGPM 1975)

h - 6.626 176 x 10^ J s 1 (1973 least-squares

A^A = 6.022 045 x lO^^ mol"i

J adjusted values)

R = 8 314.48 ± 0.63 J K'^ kmol ^ (Colclough, Quinn,

and Chandler 1979 [2]) we find a value for the Stefan-Boltzmann constant of a = (5.670 51 ± 0.001 7) x 10"^ W m'^ K'S (3) where the uncertainties in both R and ct are given at the

level of three standard deviations.

A direct experimental determination of a will be of in- terest, therefore, if a relative uncertainty approaching 3 parts in 10"* can be achieved, since it would then provide

a totally independent value for R vdth an uncertainty of about 70 parts in 10^. This is the aim of the present

work.

Previous experimental measurements of a have, with

the exception of that of Blevin and Brown in 1970 [3], given values that were very much larger than the

theoretical value, sometimes by more than 2% and on

average by about 1.5%. The work of Blevin and Brown

demonstrated that, unless very specific measures were

taken, errors due to diffraction and scattering could

easily amount to a few percent. Although their work pro- vides a landmark in absolute radiometry, the value of a

obtained by Blevin and BrowTi was subject to a number

of uncertainties that were not radiometric in origin. In

particular, their radiating source was at the temperature

of freezing gold Ues = 1064.43 °C), the thermodynamic

*Work carried out at the National Physical Laboratory. temperature of which is uncertain by about 0.2 K, lead291
ing to an uncertainty in ct of 0.06%. Further uncertainties associated with the temperature measurement of the

blackbody at the gold point combined with this to give an

overall uncertainty, related to the source temperature, of about 0.1%. The final value for a obtained by Blevin and

Brown was (5.664 4 ± 0.009) x 10"^ Wm'^ K'^ The uncertainty is equivalent to 0.16% at the level of three

standard deviations, and the value differs from that

given about in Eq. (3) by 0.11% (see Fig. 7). It was in view of the uncertainties in Blevin and

Brown's result, stemming from their use of a high temperature source, that we decided to design our method

about a radiating source at the temperature of the triple

point of water, for which the thermodynamic temperature is, by definition, 273.16 K exactly. Although in this way we eliminated many of the problems related to the

temperature of the source, it was at the expense of generating others. These were related mainly to diffraction,

scattering, and to the realization of sources which remain

black at very long wavelengths. We were further influ- enced in our design by the work of Ginnings and Reilly

[4] at the NBS. They had already demonstrated the

feasibility of making measurements, with a sensitivity of a few parts in 10^, of the ratio of the total energies emitted by blackbodies at the steam and ice points. They had

used a low temperature calorimeter operating near 4 K

as a radiation detector. Ginnings and Reilly were aiming

at thermodynamic temperature measurement and made

no attempt to make absolute measurements which might

have led to a new value for a. Unfortunately, their work

was already well advanced before the work of Blevin and

Brown had highlighted the problems of diffraction and it was subsequently found impracticable to modify their

design.

2. Principles of Operation of Our Method

The equipment which we have used to determine cr is illustrated in outline in Fig. 1. The principle of operation

of the system is as follows: A blackbody cavity held at a temperature close to 273.16 K irradiates an aperture sys- tem at a temperature near 4.2 K that permits a beam of radiation to enter a second blackbody cavity held initially at 2 K. The absorbing blackbody acts as a heat flow

calorimeter and the incident radiation causes a rise in temperature until the radiant energy is balanced by the

heat flow, down a poorly conducting heat link, to a heat

sink maintained at a constant temperature close to 2 K. The temperature rise of the cavity, about 3 K for a radi- ating blackbody temperature of 273 K, is monitored by a germanium resistance thermometer attached to the

calorimeter. A shutter, at a temperature near 4.2 K, is then closed and electrical power supplied to the calorimeter, via superconducting leads, to maintain the calorimeter at the same temperature as before. This electrical power is a precise measure of the radiant power previously present and, together with a knowledge of the

geometry of the aperture system, it allows the StefanBoltzmann constant to be deduced. There are, of course, many conditions which must be satisfied before such a simple situation as that just described can exist. The more important of these we shall outline below.

3. Sources of Uncertainty

3.1 Diffraction and Scattering

From the very beginning we were in no doubt that the

success of the work would depend upon whether or not we could overcome the problems of diffraction and

Top plate supported on

anti-vibration mountings

Germanium resistance thermometer

Calorimeter (2K-10K)

Upper aperture(4 2K|

Pumping port Ic turbomolecular pump

Port for shutter operating mechanism

Radiator heater

Ion gauge

Liquid nitrogen reservoir I77K)

Liquid helium reservoir

(4-2 K)

Stainless steel heat link

Calorimeter heater

Liquid He reservoir (4-2K)

Superfluid helium

reservoir at 2K

Radiation trap 14 2K)

Lower aperture ( 4 2K ) Shutter (4 2K)

Supports linking radiator and shield to 77K reservoir

Platinum resistance thermometer (one of eight on blackbody)

Blackbody radiator (273.16K<r<504K)

Figure 1. Ciit-away drawing of radiation calorimeter.

scattering. To do this we tried to design a practical sys- tem which approached as closely as possible the nearly

ideal system illustrated in Fig. 2(a). In the ideal system we require that the lower aperture, L, which limits the

radiation coming from the radiating blackbody, the 'radi- ator', be illuminated over a solid angle of 2-n steradian

and that the absorbing blackbody, the ' calorimeter' , subtend a solid angle of 2tt steradian at the upper aperture,

U, which Hmits the radiation entering it. If these conditions are satisfied then there will be no error in the

measurement of the radiation from the radiator resulting

from diffraction at the edges of the two limiting aper- tures L and U. Problems of scattering arise in the space between the

two apertures. The requirement here is that all of the ra- diation emitted by L, with the exception of that within

the geometrical beam passing through U, must be ab- sorbed. Since only about 1.6% of the radiation emitted

by L passes directly to U, the design of the radiation

trap (see Fig. 3) between L and U poses no mean problem and is crucial to the success of the experiment.

For purely practical reasons it is not possible to achieve complete hemispherical illumination of L nor complete hemispherical collection of the radiation leaving U. Nevertheless a close approach to these ideal conditions can be achieved by means of highly polished reflect- ing surfaces disposed in the manner shown in Fig. 2(b). The fraction of the radiation entering the calorimeter

which originates, due to diffraction, outside the geometrical beam and comes from the direction of the reflecting

surfaces may be calculated follovdng the method of

292
Radiator T=273K

Apertures' T=4.2 K

Calorimeter T= 2K

Figure 2(a). Ideal geo7netry for a radiometric measurement of

the Stefan-Boltzmann constant. The aperture adjacent to the radiator we call 'L' and that adjacent to the calorimeter, '[/'.

T_273K

T_A 2y^ Black paint

Calorimeter

X=100mm RL=9or5mm Ry = 13or9mm AX = 60jjr

5x=60fim 8y = 100^m

Figure 2(b). The practical realization of the ideal geometry

shoum in Fig. 2(a).

Figure 3. Detailed drawing of the radiation trap and upper

part of the blackbody radiator.

Blevin [5]. Errors due to diffraction will only arise if the

reflecting surfaces are at temperatures different from

that of the blackbody and at the same time are not perfect reflectors. The effects of diffraction are of course wavelength

dependent. In a measurement of a based upon 273 K

blackbody radiation we have to take account of radiation

having wavelengths extending from about 2 (xm to 300

|i.m. Not all of these wavelengths are of equal importance, since 99% of the total energy is in the wavelength

range below 85 iim, and 99.9% is in the range below 180

|xm. A negligible amount is in the range below 2.5 ixm. An effective wavelength, calculated to best represent dif- fraction losses, is about 19 \xm for 273 K radiation. It is clearly desirable that the dimensions of the apertures be

as large as possible compared to these wavelengths. The

overall size of the apparatus is directly related to the size of the apertures and followed from the initial aim of determining cr vdth a relative uncertainty approaching 1 part in 10 . This led us to choose a minimum aperture

diameter of about 1 cm and a separation of the apertures

of 10 cm. The dimensions of all the remaining components

followed.

Of major importance were the optical and vacuum properties of the black paint with which the inside surfaces of the radiator, calorimeter, radiation trap, and other sur- faces were coated. This paint was C-401 Black, made by

the 3M Company, and its optical degassing properties were the object of a subsidiary investigation fully

reported elsewhere [6].

293
3.2 The Radiator

The blackbody cavity, Fig. 4, that forms the radiator is made of copper and is coated on the inside with a layer,

about 100 (xm thick, of C-401 Black. The temperature of the blackbody is measured by eight capsule-type platinum resistance thermometers, specially made for us by

the Tinsley Co. so that they could be bolted directly to the copper. The blackbody is suspended by thin stainless

steel strips from the liquid nitrogen reservoir and controlled at a temperature near 273 K by a single ringheater placed near the points of attachment of the steel

strips. The emissivity of the blackbody was calculated [7] on the basis that the 3M's black behaved as a diffuse

emitter of 273 K radiation having an emittance of 0.94 at angles up to about 70° from the normal. Beyond this angle the point exhibits a decrease in emittance with a con- sequent increase in reflectance which is almost wholly

specular. The retroreflectance appears to be negligible at

all angles of view [8].

3.3 The Calorimeter

The radiation absorber, or calorimeter, Fig. 5, is made

in the form of a blackbody cavity similar, of course, in shape and internal dimensions to the radiator, and con- nected by means of a poorly conducting heat link to a

constant temperature reservoir. The constant temperature reservoir is maintained at a temperature close to 2 K and just below the helium-4 lambda point where the

thermal conductivity of superfluid helium is very high

and good temperature stability can be achieved. The heat

capacity of copper at these temperatures is about 10"^ times its value at room temperature. This enables adequate sensitivity to be obtained with a relatively large mass [—300 g] so that an efficient absorber can be constructed. And although the thermal conductivity of stainless steel near 2 K is about 1% of its room temperature

value, the time constant of the calorimeter/heat link as- sembly is still only 1% of what it would be at room temperature for a calorimeter of this size. The measured

time constant for small temperature changes at these

temperatures is about 4 minutes. The thermal conductivity of copper near 2 K is about the same as it is at room temperature. Thus the thermal diffusivity is very

high and so temperature gradients are insignificant for the power levels that we use. The dimensions of the heat link were chosen so that the

temperature difference between the calorimeter and the

2 K reservoir was 3 K, for the large set of apertures,

when viewing 273 K radiation. Using a germanium resist- ance thermometer there is no difficulty in measuring the

temperature rise of the calorimeter with a sensitivity of

1 part in 10^ of its value. The requirements for temperature stability of the 2 K reservoir are, however, very

stringent because of the large change in thermal conductivity of stainless steel as a function of temperature in this range. If the temperature of the 2 K reservoir is drifting, constancy of temperature difference between

the calorimeter and reservoir does not indicate constancy

of power input to the calorimeter. It is necessary to maintain the temperature of the 2 K reservoir within

about 20 |xK of a nominal value for the equivalence of successive power measurements to be good to better

than 1 part in 10^. Further advantages of operating the calorimeter at very low temperatures are that the radiation exchange

with the surroundings can be made negligible and the

heater leads can be made of superconducting wire so that

12 cm

Figure 4. Schematic of the blackbody radiator at 273.16 K

showing positions of the eight capsule-type platinum resistance thermometers, each one enclosed in a cylindrical multi-layer radiation shield. The marginal rays of the radiation beam

passing to the calorimeter are shoum.

Figure 5. The calorimeter, showing how it is supported and

linked to the 2 K constant-temperature reservoir. The points

marked AtoG are also shoum in Fig. 6.

294
no heat is dissipated in them. Nevertheless, great care must be taken in the way that the leads to the heater and

thermometer are thermally anchored to the calorimeter

for a proper equivalence to exist between heating by ra- diant power and electrical power (Fig. 6). That this was

achieved was confirmed experimentally by supplying the same electrical power to different heater windings placed

at various positions on the calorimeter. The indications of the germanium thermometer were independent of the

point of application of the power. The size and shape of the calorimeter were chosen so that the calculated ab- sorption was greater than 0.999 for the large set of apertures. The departure from blackness of the calorimeter

leads to a correction of some 3 parts in lO"* to the measured value and is one of the largest sources of uncertainty in the final value of ct.

Thermal radiation Black coating

Heater

Superconducting

leads

Germanium

resistance

[3 thernnometer

19 Heater leads

Poorly conducting

heat link (stainless steel)

4 Thermometer leads

////////////////////

2K Reservoir

Figure 6. A heat-flow diagram for the calorimeter showing the

critical thermal anchoring points. The points marked A to G

are also shown in Fig. 5.

those to be obtained using the small set of apertures

which have lands only about half as wide.

The geometrical factor, g, that enters into the calcula- tion of a is, from standard photometric theory, given by:

g = + R§ + Z)^} + {{R^ + R^ + D^f (4) where Ri, Ru, and D are the radii of the lower and

upper apertures and their distance apart, respectively.

3.5 The Electrical Measuring System

The electrical measuring system includes the controllers and power supplies for the 2 K reservoir; the black- body radiator and its controlled shield; the stabilized power supply for the calorimeter heater and its current and potential measuring systems; the resistance measuring systems for the platinum resistance thermometers,

germanium resistance thermometer on the calorimeter,

and various other platinum and germanium thermometers

placed elsewhere for monitoring purposes; together with

all the ancillary electrical systems for pumps, pressure

measurement, automatic hquid helium and nitrogen fill- ing systems, etc. The resistance of the platinum ther- mometers on the blackbody was measured using a Tinsley ac resistance bridge Model 5840 and that of the ger- manium thermometer on the calorimeter by a Cryobridge

of Automatic Systems Laboratories Ltd. The current and

potential of the calorimeter heater were measured by means of a digital voltmeter, S.E. Laboratories Model

SM215, connected in turn across the heater itself and a standard resistance arranged so that about 0.8 V was

displayed by the voltmeter. The mean was always taken

of the heater power measured before and after reversing

the direction of the current through the heater. The

differences obtained were small and constant amounting

to about 5 parts in 10^. Independent measurements were

also made, as a check, using a Guildline Direct Current

Comparator potentiometer Model 9930.

3.6 Residual Gas

3.4 The Apertures and Geometrical Factor

Two sets of apertures are to be used so that errors originating in the aperture system and the blackbodies and

also those related to diffraction and scattering, are likely

to change and thus become apparent. The arrangement

of the apertures is shown in Fig. 2(b). Each aperture is made from dispersion-hardened beryllium copper, from

the same sample as a specimen for which the thermal ex- pansion coefficient was measured between room temperature and 4.2 K [9]. The radiation trap which supports the

apertures was made from OFHC (oxygen free, high conductivity) copper. The mean thermal expansion coeffi- cients between 293 K and 4.2 K were taken as 3070 x 10"*' for beryllium copper [9] and 3257 x 10"'' for OFHC copper

[10]. The diameters and distance apart of the apertures were measured using standard metrological techniques in the NPL length section. The edges of the apertures do not depart from mean circles by as much as 1 ixm and scanning

electron microscope photographs of each aperture allowed

us to confirm the form and sharpness of the edges, and the

widths of the cylindrical lands, AX in Fig. 2(b). The reflectivity of these lands is an important source of uncertainty in these results but will be rather less significant in

For the energy received by the calorimeter to be a proper measure of the thermal radiation emitted by the

blackbody, the energy transfer by gas molecules must be

either negligible or corrected for. This energy transfer is made up of the kinetic energy appropriate to a gas

molecule emitted from a blackbody at 273 K plus the en- ergy of adsorption in the calorimeter at about 5 K. The

energy of adsorption is solely that of physisorption,

which does not exceed 36 kJ/mol, since the temperature

of the calorimeter is much too low to allow subsequent

chemisorption to take place. Elementary considerations

based upon simple kinetic theory lead to the result that

in our case the total energy transported to the calorimeter by residual gas, mainly composed of water vapor, at a pressure in the blackbody of 1.3 x 10"*^ Pa (1 x 10"** Torr), would be 1 part in 10^ of the thermal radiation en- ergy. Since the pressure measured inside the blackbody

by an ion gauge was of the order of 4 x 10" Pa (3 x 10"^ Torr) we considered the energy transport by residual gas

to be negligible.

4. Results

A value of the Stefan-Boltzmann constant is obtained

by measuring the electrical power, P{T), required to balance the radiant power, E'(T), received' by the

295
calorimeter from the blackbody at a thermodynamic temperature, T, very close to the triple point of water. The

equations used to calculate a are as follows:

E'iT) = e(r)a(c)(l - - /)(1 + s)g aT* (5) E'iT) = P{T) - bE' ^ V{c)V{s)/R(s) - hE '

, (6)

where e(r) is the emissivity of the radiator (0.999 92), a (c ) is the absorptivity of the calorimeter (0.999 7),

' c? is the fraction of radiant energy lost due to diffraction

(2.2 X 10"^),

/ is the fraction of radiant energy lost due to absorption

at the lands of the apertures (2.8 x 10"^),

s is the fraction of the energy added due to scattering in the radiation trap and at the apertures (1.3 x lO''^), g is the geometrical factor at 4.2 K (0.042 771 6), V{c) and V(.s) are the measured potentials (about 0.8

volts) across the heater and an external standard

resistor, R (s ) (572 ohms), in series with the heater,

hE ' is the small change in radiant power entering the

calorimeter from the radiation trap resulting from

the cooling of the radiation trap (from about 6.5 K to 5.8 K) on closing the shutter. The temperature T is deduced from the readings of the

six platinum resistance thermometers placed at different

points over the lower part of the blackbody viewed

directly by the calorimeter. The mean effective temperature is a function of the different weights given to the

radiation from different parts of the blackbody by the

vignetting effect of the two apertures. The temperatures

are evaluated on IPTS-68 and it is assumed that the

differences between IPTS-68 and thermodynamic temperatures are less than 1 mK in the range from -1-2 °C to -2 °C. The results of measurements made using the large set

of apertures, /2l = 9 mm and - \Z mm, are shov.'n in Table 1 and Fig. 7. The size and associated uncertainty

of the various correction factors appearing in Eqs. (5) and (6) are listed in Table 2. The value of a resulting from these preliminary meas- urements is a = (5.669 19 ± 0.002 4) x lO'^ Wm"2 K^^ (7)

where the uncertainty is given at the level of three

standard deviations and is equivalent to 4.2 parts in 10^. This differs from the value given above in Eq. (3) by 2.3 parts in 10'*. This is equivalent to a difference in the gas

constant, R, of 58 ± 105 parts in lO''. We consider this result to be satisfactory pending the repeat measure- ments wdth the small aperture set and further study of the principal sources of uncertainty given in Table 2.

Value calculated from fundamental constants

I 9^ 1

Blevin & Brown (1970)

I 9^ 1

Present preliminary result

I 1 (Standard deviation of 16 measured values, equivalent to 1 part in 10^ of (T

. . . I ..,,[, .

. . L

5.675 5 670 5 665 5.660 5.655

<r HO'^Wm'^K")

Figure 7. A comparison of the vresent preliminary result, (5.669 19 ± 0.002 J^) x 10~^ W K~\ with that of Blevin and

Bromi, (5.66i U ± 0.009) 10 ^ W m K ^, and with the value of CT calculated from, atomic constants usin^ the most recent value for R, (5.670 51 ± 0.001 7) x iO"* W m~^ K'^. The

imcertainty bars are drawn at the three standard deviation

level.

Table 1. Results of eighteen measurements of the Stefan-Boltzmann

constant, a, made using the large set of apertures, R^ ~ 9 mm and

Ru —12 mm..

"

a/ lO-^W m-2K-4 a/lO-^W m-2K-4

5.669 09 272.397 4 5.669 19 275.547 5 5.669 18 273.314 3 5.669 09 275.384 5 5.669 17 272.426 7 5.669 18 273.446 9 5.669 28 272.470 5 5.669 29 272.203 3 5.669 20 272.448 6 5.669 14 273.099 3 5.669 29 272.499 6 5.669 24 273.050 3 5.669 17 272.465 0 5.669 25 273.160 7 5.669 29 272.637 2 5.669 14 273.025 8

5.669 16 275.651 6 5.669 07 272.976 0

^The mean value is 5.669 19 x 10"^ W m''^ K""* with a standard deviation, from the eighteen measurements, of 7 x 10"^'^ W m"^ K"*

equivalent to 1.2 x 10"^ of ct. The value of T^g is the mean temperature

of the radiator at the time of each measurement.

296
Table 2. Summary of corrections and associated uncertainties (evaluated at the one standard deviation level).

Parameter Correction to ct, Resulting uncertainty

parts in 10'' in ct, parts in 10^

Emissivity of radiator, 6(r

)

+0.8 0.4^ Absorptivity of calorimeter, a (c ) +3.0 0.7^

Diffraction, d +2.2 0.7 Absorption at aperture lands, / +2.8 0.8 Scattering in radiation trap, s -1.3 0.3 Geometrical factor, g 0.25 Measured electric power, P{T) 0.1 bE' -0.1 0.1 Standard deviation of 18 measured values 0.1

Total correction +7.4 x 10 ""^ ct Combined standard deviation 1.4 x 10""* u

Overall relative uncertainty at the three standard deviation

level = 4.2 x lO^l

Although these two uncertainties are probably correlated, we have not taken

this into account in combining them.

The authors are very pleased to acknowledge the many

important contributions made in the early stages of this work by J. P. Compton. In addition we would like to ac- knowledge the able construction work carried out by E.

Pinn, M. Rogers, and E. Charles of the NPL workshop,

the aperture dimensional measurements made by S. Poole, G. Severn, and P. Kelley of the NPL length section, the measurements of the thermal expansion coeffi- cient of beryllium copper carried out for us by C. A. Swenson of Iowa State University, the reflectivity meas- urements made on a model of the calorimeter by J. Geist

and E. Zalewski of NBS, and finally the many helpful

discussions which have taken place with members of the

NPL temperature section.

References

[1] T. J. Quinn and J. E. Martin, in Temperature: Its Measure- ment and Control in Scierice and Industry, Vol. 5, Ed. by

J. E. Schooley (American Institute of Physics, 1982),

p. 103.

[2] A. R. Colclough, T. J. Quinn, and T. R. D. Chandler, Proc. R. Soc. London, Ser. A: 368, 125 (1979).

[3] W. R. Blevin and W. J. Brown, Metrologia 7, 15 (1971).

[4] D. C. Ginnings and M. L. Reilly, in Temperature: Its Meas- urement and Control in Science and Industry, Vol. 4, Part I, Ed. by H. H. Plumb (Instrument Society of America, 1972), p. 339.

[5] W. R. Blevin, Metrologia 6, 39 (1970).

[6] J. P. Compton, J. E. Martin, and T. J. Quinn, J. Phys. D 7, 2501 (1974).

[7] T. J. Quinn, High Temp. High Pressures 12. 359 (1980).

[8] The authors are indebted to J. Geist and E. Zalewski of NBS

for measurements of the reflectance of a model of the

calorimeter.

[9] The thermal expansion coefficient of beryllium copper was

measured for us by C. A. Swenson of Iowa State University. [10] F. J. Kroeger and C. A. Swenson, J. Appl. Phys. 48, 853-864 (1977).

297

^Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

On the Radiometric IVIeasurement of tlie Stefan-Boltzmann

Constant at NRLIVI

Akira Ono

National Research Laboratory of Metrology, Umezono Nllhari, IbarakI 305, Japan

Progress at NRLM on the radiometric measurement of the Stefan-Boltzmann constant is described. In the method, the total radiant flux emitted from a blackbody source that is spatially defined by two

limiting apertures is measured by an absolute radiometer. A newly developed absolute radiometer is described that has improved uniform responsivity over the surface of the receiver; the variation of responsivity is less than 0.2% over an area 15 mm in diameter. Emissivities of blackbody cavities are

calculated by the Monte Carlo method taking into account partial specular reflection of radiation on the

cavity walls. An approach to a perfect blackbody cavity is also presented. It is discussed how uncertainties of blackbody source temperature and of diffraction losses of radiant flux are to be minimized.

Key words: absolute radiometer; blackbody total radiation; emissivity of cavity; partially specular reflection; radiometric measurement; Stefan-Boltzmann constant.

1. introduction

Since the Stefan-Boltzmann constant was precisely

determined by a radiometric method at the National

Measurement Laboratory (total uncertainty of 1300 ppm,

99% confidence level) [1], attention has been directed to possible improvements of the method at the National

Research Laboratory of Metrology (NRLM). A radiometric measurement of the Stefan-Boltzmann constant in- volves a combination of radiometric and radiation thermometric techniques. The former is based upon the absolute

measurement of radiant flux while the latter provides a blackbody radiation source maintained at a known temperature. Therefore a radiometric measurement of the

Stefan-Boltzmann constant will contribute to our understanding of the quantitative consistency between ra- diometry and thermometry, as well as to a precise determination of the fundamental constant. The principle of the method is based upon an absolute

measurement of total radiance from a blackbody whose

thermodynamic temperature is accurately known. Total

radiant flux that is spatially defined by two limiting apertures is measured by a thermal detector sensitive over a wide spectral range. The radiation detector is called an

absolute radiometer because it senses flux values on the

absolute scale in units of power (watts). Major sources of error in the method result from uncertainties of the quality and temperature of the blackbody source, of the absolute measurement of radiant flux, and from the diffraction losses of radiant flux between the two limiting apertures.

The purpose of this paper is to describe recent progress

at NRLM on the radiometric measurement of the StefanBoltzmann constant. First, an absolute radiometer is described that has improved uniform responsivity over

the surface of the receiver. The total uncertainty of the

radiant flux measurement has been reduced to less than

100 ppm, one standard deviation. Second, the evaluation

of blackbody cavity emissivity is discussed. Since a major

part of blackbody total radiation is in the infrared, specular reflection of radiation on surfaces can be important. A

Monte Carlo method is applied to calculate apparent emissivity of the blackbody cavity taking into account partial

specular reflection on the cavity walls. Finally, discussion

is presented on how uncertainties associated with the

blackbody source temperature and with the diffraction losses of radiant flux are minimized.

2. Experimental Apparatus

The experimental apparatus shown in Fig. 1 consists of three major components: the upper cell composed of the

blackbody source, the lower radiometer section, and the

middle aperture section which also provides for one of two connections to the vacuum system. A water-cooled

shutter located between the blackbody source and the

upper limiting aperture (source aperture) allows black- body radiation to be incident upon the absolute radiometer. Otherwise, when the shutter is closed, the radiometer views the shutter which emits blackbody radiation at room temperature.

The absolute radiometer is a thermopile that provides an electrical power equivalent to the absorbed radiant

power. The hemispherical mirror above the radiometer

serves to attain a more uniform responsivity over the

surface of the receiver, as well as to increase the ap- parent absorptivity of radiation. Since the two limiting

apertures, both of 10 mm diameter, are spaced 400 mm

from each other, this arrangem.ent provides flux values of 500 |jlW, 160 |xW, and 90 (xW for blackbody source temperatures of the aluminum point (660 °C), the zinc point

(420 °C), and the lead point (328 °C), respectively.

Since the entire system is evacuated, thermal insula- tion by radiation shielding permits the blackbody source and the source aperture to be in close proximity

(150 mm). The aperture system (the lower half in Fig. 1) can be separated physically from the vessel of the black- body source (the upper half in the figure). When the

aperture system is removed, the angular distribution of irradiance at the location of the source aperture is measured. These facilitate reducing and evaluating diffraction losses of radiant flux between the limiting apertures.

299
10 cm

Figure 1. Experiymntal apparatus, l—vacuum gauge; 2—

thermometer well; 3—sheath heater; i—iuater cooling; 5—

water-cooled shutter; 6 and 7—radiation shield; 8—copper

tube; 9—graphite cynicible; 10—cavity 30 mm. in diameter and

200 mm deep; 11—vacuum pump; 12—fixed-point material; 13—cavity aperture 20 mm in diameter; U—water-cooled

shield; 15—gate valve; 16—source aperture 10 mm. in diameter; 17— radiation trap; 18— vacuum pump; 19—detector aperture

10 mm in diameter; 20—absolute radioymter; 21—

hemispherical mirror.

3. The Absolute Radiometer

An exploded view of the absolute radiometer is illus- trated in Fig. 2. The design principle assumes that uni- form responsivity can be attained by permitting regular

heat flow through the thermocouple legs but suppressing

heat losses between the receiver and the surroundings [2]. Suppression of heat losses was achieved as follows. Heat losses by air convection and conduction are minimized by operating the radiometer in a vacuum. Heat

loss by thermal radiation from the front surface of the re- ceiver is minimized by placement of the hemispherical

mirror above the receiver. The hemispherical mirror

causes thermal radiation emitted from the front surface

of the receiver to be reflected back onto the receiver it2

3

4

Figure 2. Exploded view of the absolute radiometer. 1—

hemispherical mirror; 2—black coating; 3—heater; U—receiver disk with thermocouples; 5—gold coating.

self. Heat loss by thermal radiation from the rear surface

of the receiver is minimized by coating the surface with a low emissivity material (gold).

The responsivity distribution over the surface of the

receiver has quadratic behavior with a minimum at the center and a maximum at the periphery. The responsivity variations were less than 0.2% over the receiver

surface of 15 mm diameter [3]. The responsivity and time

constant of response of the absolute radiometer are listed

in Table 1 with construction details. It should be pointed

out that a more uniform responsivity can be obtained by

a thicker disk without serious increase of the time conTable 1. Perfontmnce and construction details of the absolute radiometer.

Responsivity 0.17 V/W

Variation of responsivity 0.2%

Time constant of response 14 s Receiver

area 15 mm in diameter

disk aluminum 20 (xm thick black coating 3M velvet spray

heater material gold evaporated film heater resistance 54 fl insulating film Araldite 985E

Thermocouple

material copper vs. constantan number of junctions 24 pairs copper wires 0.05 mm in diam. x 15 mm long constantan wires 0.32 mm in diam. x 15 mm long

total resistance 6.4 a

Hemispherical mirror

radius 60 mm

concave depth 55 mm

aperture 40 mm in diameter

surface gold coating

300
stant [4]. An uncertainty of radiant flux measurement

caused by the responsivity variation over the receiver

surface can be confined to a few parts in 10^. As such,

the total uncertainty of the radiant flux measurement

will depend entirely on the effective absorptivity of the

absolute radiometer.

4. Apparent Emissivity of the Blackbody

Source

Assuming that the emissivity and the absorptivity are

equivalent for an isothermal opaque cavity, a Monte

Carlo method has been appHed to the calculation of the

apparent absorptivity of partially specular cavities [5]. Radiative properties of cavity surfaces are characterized

by the uniform specular-diffuse model in which radiation

reflected on a surface consists of two components

specular and diffuse—and the directional hemispherical

reflectivity has the same value at all points on the sur- faces considered. Figure 3 illustrates a cylindrical cavity

with a conical bottom where the cylindrical surface is dif- fuse and the conical surface is intermediate between dif- fuse and specular. This latter surface can be characterized by the uniform specular-diffuse model. Such modeling may be a good approximation for a real cavity where

the cylindrical surface is spirally grooved or baffled and

the conical surface is in the as-machined condition.

.000

> 0.999

tom

ill 01 0.998 <

Q.0.<

0.997

1

1 1

1

'

1

"oTe

/O.2 / -

/o.o -

1 1,1,1 , 1 , 0.0 0.2 0.4 0.6 0.8 1.0

(D) (^is/^^ CS) FRACTION OF SPECULAR COMPONENT

Figure 4. Normal apparent emissivity of a cavity having a depth-to-diameter ratio of 10, an apex angle of 120°, and a sur- face emissivity of the cylinder of 0.8. It is plotted against the fraction of specular reflection component for different surface

emissivities of the cone. Two arrows, D and S, on the abscissa

indicate totally diffuse and coynpletely specular reflection, respectively.

Figure 3. Schematic of a cylindrical cavity with a conical bot- tom. The cylindrical surface is diffuse and the conical surface

is intermediate between specular and diffuse.

Normal apparent emissivities of cylindrical cavities

with conical bottoms were calculated for various cavity

depth-to-diameter ratios, apex angles of the cone, surface

emissivities, and fractions of specular reflection component. It was shown that the apex angle of the cone was

an important factor in the apparent emissivity behavior

with partially specular reflection. For apex angles near

180°, 90°, and 60°, the apparent emissivities decreased

rapidly as the fraction of specular component increased.

This behavior is due to specular reflection effects on the

conical surface. For apex angles of 70° and 50°, the ap- parent emissivities exhibited minima at intermediate

fractions of specular component. This behavior is due to the combined effects of diffuse and specular reflections on the conical surface. For an apex angle of 120°, the ap- parent emissivity increased monotonically as the fraction

of specular component increased.

Figure 4 shows apparent emissivities of a cavity having

a depth-to-diameter ratio of 10, an apex angle of 120°,

and a surface emissivity of the cylinder of 0.8. The ap- parent emissivity is plotted against the fraction of specular component on the conical surface for various surface

emissivities of the cone. If the surface emissivity of the

cone and the fraction of specular component are es- timated as ei= 0.8 ± 0.05 and rig/ri = 0.5 ± 0.1,

respectively, then the apparent emissivity is evaluated as

e = 0.99975 ± 0.0001. It was shown that the surface

emissivity of the cylinder affected the apparent emissivity much less than the surface emissivity of the cone. An effort to make the conical surface specular can be an approach to a perfect blackbody cavity even if the sur- face emissivity of the cone is not so high. In closing, it should be noted that the effect of temperature distribution within a cavity has also been studied intensively at NRLM. A new concept of effective

temperature of a cavity has been introduced that is con- venient for the analysis of the radiant characteristics of non-isothermal cavities [6, 7, 8].

5. Temperature of the Blackbody Source

An uncertainty of the thermodynamic temperature

scale affects the determination of the Stefan-Boltzmann

constant according tc the fourth-power law. The thermodynamic temperature scale has been realized with high

accuracy at the National Bureau of Standards by gas

thermometry [9]. The uncertainties are less than 10 ppm

from 0 to 450 °C. It has been also reported that the scale

is being extended to higher temperature, including the

aluminum point (660 °C) [10]. Since our interest is in the

aluminum, zinc, and lead fixed points as described later, an uncertainty resulting from the thermodynamic temperature scale vrill be confined within 50 ppm in the

determination of the Stefan-Boltzmann constant. In order to avoid uncertainties resulting from temperature measurement of the blackbody source, use has been

made of blackbody sources at the fixed points on the

International Practical Temperature Scale by packing the

fixed-point materials around the blackbody cavities. A

preliminary experiment using the zinc point was not suc- cessful because it was difficult to have a reliable airtight

301
crucible to prevent zinc vapor from passing out the cruci- ble wall to the vacuum. Our interest is shifting to the

aluminum and the lead points because these materials

have vapor pressures much less than that of zinc at their

freezing temperatures.

6. Uncertainties

Major factors in the uncertainty of the present meas- urement are: quality of the blackbody source, absolute

measurement of the radiant flux, and diffraction losses. Uncertainties of the first two factors can be reduced to

less than 100 ppm for each, one standard deviation, as re- viewed here and analyzed in the earlier measurement [1]. The third factor is still difficult to estimate with confi- dence at this stage because data are lacking on the angular distribution of irradiance at the source aperture. If this factor could be evaluated with an uncertainty of less than 100 ppm, the total uncertainty in the present meas- urement of the Stefan-Boltzmann constant can be confined to 160 ppm, one standard deviation.

The author would like to express his sincere thanks to Dr. M. Morimura and Dr. S. Hattori of the National

Research Laboratory of Metrology for their helpful dis- cussions, and to Professor D. P. DeWitt and Dr. R. E. Taylor of Purdue University for their help in preparing

the paper during the author's sabbatical leave at their

laboratories.

References

[1] W. R. Blevin and W. J. Brown, Metrologia 7, 15 (1971).

[2] A. Ono, Metrologia 15, 127 (1979).

[3] A. Ono, Jpn. J. Appl. Phys. 18, 697 (1979).

[4] A. Ono, ibid. 18, 1995 (1979).

[5] A. Ono, J. Opt. Soc. Am. 70, 547 (1980).

[6] S. Hattori, Trans. Soc. Instrument and Control Engineers

(Japan) 15, 847 (1979), in Japanese.

[7] S. Hattori, ibid. 16, 539 (1980), in Japanese.

[8] S. Hattori and A. Ono, to be published.

[9] T. J. Quinn, L. A. Guildner, and W. Thomas. Consultative Committee on Thermometry, Document Number 76-2,

International Bureau of Weights and Measures, Paris

(1976).

[10] T. J. Quinn, L. A. Guildner, A. Moser, and W. Thomas, ibid. Document Number 80-5, International Bureau of Weights

and Measures, Paris (1980).

302
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Applications of X-Ray Interferometry

Richard D. Deslattes*

Division of Physics, National Science Foundation, Washington, DC 20550

This review begins by summarizing work at the PTB and NBS on optical interferometry of [220] repeat distances in samples of monocrystalline Si. Distribution of such an initial calibration to other samples and other species is briefly mentioned. The main emphasis is on subsequent applications of these crystals toward determination of fundamental constants, especially A''^^ and extension of the congruent electromagnetic scale to 7-rays as has so far been carried out at NBS. In the last mentioned

case, applications emerge which include tests of QED in muonic atoms, determination of mass values

for the pion and the kaon, and tests of relativistic self-consistent field calculations for inner vacancy

states in atoms. Key words: density; fundamental constants; gamma-ray standards; molar mass; x-ray interferometry.

1. Introduction

This revievi^ aims to give an overall perspective on x- ray interferometry vdth emphasis on its role in the determination of fundamental constants and in tests of basic

theory. As is well known, these roles have been exemplified in the determination of the Avogadro constant and in the extension of the congruent electromagnetic scale to the 7-ray region. In each of these applications, x-ray in- terferometry alone is insufficient. Clearly it first needs

to be combined vdth optical interferometry to obtain an

optically based crystal lattice repeat distance. However

this step is also insufficient to be of general interest

since a crystal sample is a quasi-artifact. Other measure- ment technologies need to be added in order to reach

manifestly invariant physical quantities. In the case of the Avogadro constant, these other technologies involve

density determinations as well as absolute isotopic abundance measurements. In the case of 7-rays, angle meas- urements are required having both high precision and

high accuracy. These additional steps needed to reach in- variant results enter those results just as sensitively as the initial interferometric determination itself. For the

present overview, it will be necessary to give attention

to these other steps as well, at some cost to the level of detail with which the x-ray interferometry can be

outlined. Another kind of auxiliary resource, though not needed

in principle, is a matter of considerable convenience. This

, is a crystal-to-crystal comparison method which permits

j transfer of the calibration initially embodied in one single

crystal specimen to other specimens and other species.

Once again the technique is required to be of high quality

since its results directly influence numerical values for the invariant quantities ultimately produced. Such a technique also permits, as will be seen below, comparisons of work carried out in different laboratories at pre- liminary stages where invariant results are not available

in all cases.

1.1 Historical Remarks

Present day work on x-ray interferometry began with

the pioneering efforts of U. Bonse and M. Hart in the

*Pennanent Address: Center for Absolute Physical Quantities, National Bureau of Standards, Washington, DC 20234

mid-sixties [1]. Their work initially explored fixed

geometry arrangements of crystal diffracting elements

which remained part of the original crystalline sample.

Although such devices have had a considerable range of

applications (see below), the main interest for this Conference attaches to those generalizations in which one

crystal element has freedom of motion with respect to the others. This articulation, accompanied by suitable

optical interferometry, permits measurement of the lat- tice repeat distance of the crystal in terms of an optical

standard. Such structures were described at the first of

these Conferences in 1970 and have been pursued with some diligence in several laboratories since that time

[2, 3].

1.2 Applications of Fixed Geometry Interferometry

Aside from the initial demonstration of phase contrast microscopy, fixed geometry interferometers have found

their main utility in the measurement of refractive in- dices. These have, of course, considerable interest near

critical absorption edges where anomalous dispersion ef- fects become large. Continuously wavelength scannable

photons available from synchrotron radiation sources make their combination with x-ray interferometry a very

natural and significant enterprise. Such work has been

reviewed in a number of places especially by Hart [4], by

Materlik [5], and by Bonse et al. [6]. It is also the case

that the developments of fixed geometry x-ray interfer- ometry presaged the corresponding work with deBroglie waves, i.e., neutron interferometry. This work has al- ready been quite fruitful in demonstrating important fun- damental properties of Fermions and their interactions

vdth gravity and non-inertial motion. The overall situa- tion of neutron interferometry for these purposes has

been well summarized [7].

1.3 Systems Providing for Motion and Optical

Readout

As is well kndvra, when the last optical element in a

three-crystal interferometer is moved uniformly with

respect to the other two elements, x-ray intensity

recorded in a detector, see Fig. 1, is modulated with the

spatial periodicity of the lattice in a fashion that is substantially independent of the wavelength and characteristics of the source of x-ray illumination. Evidently optical

303
OPTICAL SYSTEM

X-RAY SYSTEM

Figure 1. Diagram indicating the x-ray and optical components of a lattice spacing determination.

interferometry of a sufficiently refined character carried out simultaneously permits a reckoning of the crystal lat- tice repeat distance in terms of the visible wavelength il- luminating the optical interferometer. If the source of

optical radiation is Vv^ell-connected with fundamental

scientific or metrological standards, then these standards

are effectively transferred to the crystalline lattice. There were a number of problems needing to be solved

before such measurements became practical and even to- day they can hardly be considered routine. Most evi- dently, a provision for positional noise levels and optical

interferometry near the picometer level and the mainte- nance of nanoradian tolerances in angular orientation, to- gether with similar parallelism of motion, represented

somewhat formidable challenges. Two such exercises

have now been completed and vdll be discussed in some

detail below.

1 .4 Objectives and Cautions

Clearly the physical properties (including dimensions of

the lattice) of a particular crystal, even if it were known

to be perfect, are of very limited scientific interest since

these properties are not calculable from first principles

to any appreciable accuracy. Thus such calibrated crystals are interesting only as "stepping-stones" helping to bridge the large gap between relatively coarse visible wavelengths and the hundred- to million-fold smaller

wavelengths in the x-ray and 7-ray region. Of course, if the particular crystal measured is otherwise well characterized or if the initial calibration is transferred to specimens of well characterized material, one has the additional hope of being able to determine the Avogadro

constant. However, even in this case, our experience (see

below) indicates that each sample has its own macroscopic density and its own isotopic abundance distribution so that only the formally invariant ratio of mean

molar mass to density is effectively constant for a group

of specimens coming from different sources. There is also the additional coupling of nuclidic mass to thermal vibration which leads, at least in high order, to a dependence

of lattice parameter on mean molar mass so that, in the

final analysis, the only truly invariant quantity for even a perfect crystal is the Avogadro constant itself. When one enters the real world of crystalline samples

produced by a variety of techniques and perhaps not fully

characterized as regards density and isotopic abundance,

the complications become even greater. Several explorations of groups of materials from different sources have

shown lattice variability greater than 1 ppm from proc- ess to process and smaller than 1 ppm within a given

manufacturing process [8, 9]. Of course, there can be,

and often are, underlying changes in the manufacturing

process which are not revealed by trade name designations. In summary, although the crystals with which we

work are objects of considerable beauty, they should

probably be treated with some of the cautions that were

properly applied to various attempts at realization of the

meter before it was fixed to be embodied in a single

artifact.

2. Optical Interferometry of the 220 Period

in Samples of Si

Measurement of a crystal lattice repeat distance evi- dently requires provision for parallel translation of one element of the x-ray interferometer with respect to the

remainder together with closely coupled optical displacement sensing. The required delicacy of linear motion, the

need for mutual orientation adjustment, and the essential

parallelism of motion, can be obtained in a number of dif- ferent ways. Several of these were described at the first Conference of this series [2] while others have been ini- tiated subsequently [3]. Although each of these efforts includes interesting strategies, limitations of length and

my lack of detailed current awareness suggest that this report should focus on the two efforts that have been car- ried to completion. One of these has recently been re- ported by a group at PTB [10] while the other earlier

work was carried out at NBS [11]. Each result that has

been obtained has an adequate precision to serve as a

useful first step in a subsequent chain to invariant quantities. Since only the NBS work has led to determination

of invariant quantities, statements regarding consistency

of the x-ray /optical interferometry have to await direct

sample-to-sample comparisons [12].

2.1 Translation Stages

Although Hart had earlier devised an elegant and in- genious means for providing parallel translation within a

single block of silicon which included also the interfer- ometer elements [13], both NBS and PTB have used flex- ure structures made of steel as carriages for separated

interferometer elements. Figures 2(a) and 2(b) compare

the structures used in the two measurements. The PTB

structure, Fig. 2(a), is of highly symmetric design and is driven directly (in its final embodiment) by a piezoelectric transducer [10a]. The NBS structure. Fig. 2(b), is of a less symmetric design consisting of a flexure parallelogram internal to which there is a three-stage motion

demagnifier. Input drive to the NBS structure (applied

at point X) comes from a predominantly mechanical

source, but one which includes a piezoelectric vernier

[lid]. In each case sensitive tuning adjustments are pro- vided so as to secure critical alignment. Considerable

efforts were required in each case to provide a vacuum

environment relatively free of mechanical disturbance. It

is characteristic of the peripheral problems associated I) v^dth these efforts that each began with a seismic survey ]

304
measurements of Si 220 lattice periods.

Figure 3. Optical systerns used in the PTB (a) and NBS (b) m£asurenients . The systern shoum in (a) has polarization en- coded two-beam optics while that in (b) is an amplitude

(derivative) read high finesse Fabry-Perot interferometer.

of available space on the respective sites! In addition to well-chosen environments, there were appreciable efforts required in the way of antiseismic supports and acoustic

shielding.

2.2 Optical Interferometry

Although NBS and PTB measurements were each re- ferred back to iodine stabilized lasers, the PTB effort used a Lamb dip stabilized laser periodically calibrated

by beat frequency measurements with respect to a ^^'l2 laser while the NBS work employed a ^^^l2 laser directly.

The calibration chain for the laser is in each case suffi- ciently well known as to introduce negligible uncertainty

in the final results. For future reference, I note here the

wavelengths assigned to the two lasers: X.(PTB) = 632

991.415 pm (9 ppb); \(NBS) = 632 990.079 pm (4 ppb).

The optical interferometric techniques used in the two

measurements were quite different. In its final version

the PTB work has made use of the elegant polarization

encoding scheme devised by Curtis et at. [2b]. This is shown in the diagram in Fig. 3(a). The NBS measure- ment employed, on the other hand, high finesse FabryPerot optics as indicated in Fig. 3(b) [Ua, d]. Among the many advantages of the optical interfer- ometer used in the PTB measurements are a strong set

of invariance properties enjoyed by the optical system of

I Fig. 2(a). As was shown by Curtis et at. [2b], this struc- ture is immune to bodily motions of the two crystals to- gether and sensitive only to the difference in positions.

As demonstrated in the detailed analysis by the PTB

group, the relationship of optical and x-ray interference

trajectories is sufficiently congruent that offset and

cosine type errors introduce negligible corrections into the final result [10a]. On the other hand, one is here

dealing with essentially two-beam fringes; use of polarization encoding enhances the sensitivity to a very great extent but the arrangement probably remains somewhat

less sensitive than a high finesse Fabry-Perot cavity.

Data-taking strategy involved summation of repeated

scans through a region of fixed polarization azimuth

while information from both optical and x-ray detectors was accumulated in a multichannel memory provided by a computer.

The optical plan of the NBS measurement shown in Fig. 3(b) offers certain advantages and certain disadvantages and was used with a somewhat different data

acquisition strategy. First of all, the hemispherical

Fabry-Perot cavity exhibited an effective optical finesse

of approximately 1200. This meant that the full width at half maximum of an optical fringe was rather close in its shape to that of the (two-beam) x-ray fringe. These rela- tively desirable features came, of course, at the expense

of having to mount separate optical elements which, on

the present scale of refinement, exhibited appreciable

drift with respect to the x-ray elements. In the end this

305
did not appear to be a serious limitation since, after suit- able aging, drift became both slow and uniform. Thus,

by averaging the results of scans taken in opposite directions, we obtained data equivalent to what would have

been obtained in the absence of drift. Another difficulty

with the NBS measurement scheme is evident also in Fig. 3(b). Clearly the trajectory along which optical in- terference takes place is not degenerate with the trajectory along which x-ray interference takes place. The

offset, 2, introduces a cosine error to the extent of

2 5 0/L for a traverse of length, L, accompanied by a

directional change of 8 0. It was thus required to determine path curvature with appreciable refinement. In fact

this problem entailed a correction to the measured lattice parameter of 0.27 ± 0.03 ppm [lib]. Finally one notes

that the resonances of the hemispherical structures shown are not separated by X/2 but contain an additional

Fresnel phase shift whereby intervals between optical resonances are displaced by approximately 1.6 ppm

[11a, d]. Data-taking in the NBS work did not benefit from the

presence of a computer. Instead, incidental frequency

modulation of the I2 stabilized laser (as viewed through

the hemispherical resonator) formed an input to a tradi- tional lock-in amplifier system. At each lock point we

determined x-ray intensities using the count-rate-meter technique or its digital equivalent. From these intensities

and an estimate of maximum and minimum interference

signals, a standard cumulative phase versus optical order

number plot was used to determine the results. Although, as described above, the mechanical and optical technologies employed in the NBS and PTB work

were quite different, it would appear that each path was

entirely adequate to obtain results of the stated precision

and accuracy (see below). There may be, however, some

subtleties especially in the optical interferometry that

have escaped the rather diligent efforts expended by

both groups.

2.3 Experimental Results

The output datum from an x-ray /optical interferometry

exercise is an integer (in this case it is always 1648) plus a fractional part representing the size of the lattice period modulo the optical half-wavelength. This numeric

translates, via the optical wavelength, to a (specimen

dependent) lattice parameter. Robust comparison of the

two results is thus not directly possible at this stage.

Histograms of measurements are shown in Figs. 4(a)

and (b). Figure 4(a) gives the data from PTB while

Fig. 4(b) gives the earlier results from NBS. Each of

these must be corrected for the different visible wavelengths and then referred back to a common refer- ence temperature. (Both groups used a = 2.56 x

10"^ K"^ so that the temperature corrections do not intro- duce noticeable divergence.) The results for the two dif- ferent samples are (referred to 22.5 °C):

PTB: d = 192 015.560 ± 0.012 fm

NBS: d = 192 015.902 ± 0.019 fm

The silicon sample used in the PTB work was obtained

from Wacker Chemitronic and had low concentrations of

C and 0. The crystal used in the NBS work was obtained

in the mid 1960's from Dow, is of high geometrical quality, and contains unmeasurably small concentrations of C

and 0. Earlier discussions of these results [10a, b] were

0.1 ppm

.2814 .2816 .2818

FRACTIONAL PART OF RATIO

1 ppm

.2620 .2640 .2660

FRACTIONAL PART OF RATIO

Figure 4. Histograms of x-ray Ioptical interferometry results from PTB (a) and NBS (b). Note that because of different opti- cal sources, the fractional scales are not directly comparable. In addition (see text), samples used had different sources and

may not be directly comparable.

based on the assumption that the two specimens shared a common lattice parameter so that the entire noted differ- ence (1.8 ppm) was due to inconsistency between the two

efforts. Subsequent efforts toward direct comparison [12] suggest that the two specimens differ in lattice parameter by less than 1 ppm. This preliminary work showed

apparent secular change in the PTB sample which

remains to be further investigated.

306
3. Application to Determination of the

Avogadro Constant

Given an (hkt) lattice repeat distance obtained as described above, one can (for a cubic crystal) proceed to

a unit cell volume, ao by means

ao. The mass contained in such a unit cell may be

reckoned in two different ways. For a macroscopic density, p, the unit cell mass is evidently pao. If there are w = 8 atoms in the unit cell and these have a mean molar

mass, A, then there is a microscopic mass amounting to 8A {u ) also associated with the same unit cell. Connecting the macroscopic mass pao with the microscopic one, SA{u), is the numerical constant, A^^. Thus, Na itself results from: A^A =8A/pao^.

In this section attention is directed toward determination of p and of A. These are exercises of more than pass- ing difficulty because interest in a 1 ppm estimation of Na requires a first-principles determination of density

and absolute isotopic abundance measurements of the

highest available quality. Efforts in both of these areas have been carried out at NBS while considerable progress in the case of density is reported from PTB at this Conference. New efforts toward absolute density meas- urements of high accuracy are also reported at this Conference by groups from IMGC (Italy) [13a] and

CSIRO (Australia) [13b].

3.1 Density

There are several methods available for a first principles approach to crystal density determination. The most

straightforward would appear to be to use dimensional

metrology to fix the volume of an otherwise well- characterized Si sample; since mass determinations are

rather easy even well below the 1 ppm level, one has

thereby a quite adequate density estimate. Such an ap- proach was proposed by Curtis et al. [2b] and is also being considered by the PTB group [10b]. Problems of this approach include difficulty in producing and maintaining

regularity in the case of highly valued samples.

Another approach makes use of whatever regular solid object can be obtained conveniently. Such an approach is also being considered by the PTB group where it will in- volve using low expansion ceramic material both in the

form of balls and cubes [10b]. By contrast, NBS made

use of steel balls of high quality which were readily avail- able as items of commerce. Volumes of these objects were determined by optical interferometry [14] after

tests of sphericity showed that diameter measurements

alone were adequate to yield volume estimates [15]. Unfortunately there remains an unexplained systematic

offset of 1.6 ppm between the results obtained after two

different cleaning procedures [16]^

However one obtains a macroscopic artifact density

standard (other than the Si crystal itself), it is still necessary to make density transfer measurements either

directly to well-characterized Si samples or indirectly

through the use of an intermediate collection of Si objects which need not be well-characterized. Such

transfer measurements are most readily effected by hydrostatic weighing with the evident freedom to use immersion media other than water. The course to be fol- lowed at PTB in this regard is not yet clear. In the NBS work, density values inherent in a group of

steel spheres were transferred to a group of four (there were five but . . . ) 0.2 kg Si samples by hydrostatic

weighing [17]. These weighings were carried out in a fluorocarbon liquid having large density (1.8 gm/cm^)

and low surface tension (one-fifth that of water). Subsequently, these four transfer standards were compared

with three well-characterized Si crystals whose individ- ual isotopic abundances and contamination by C and 0

were also determined. The multiple samples, use of three spheres, permutation of groups of crystals (six

groups of two in the sphere comparisons) together v^dth the four secondary standards gave a significant over- determination from which the precision of this step could

be objectively assessed. The resulting values for the

well-characterized crystals exhibited a transfer contribution to the final uncertainty of approximately 0.3 ppm.

The large discrepancy in ball diameters obtained after different cleaning procedures remains unresolved forcing an expansion of the estimated uncertainty from near

0.5 ppm to about 0.95 ppm [16]. A summary of density values obtained for the three

crystals used is shown in Table 1. For reasons already

discussed, their relative values are considerably better

known than the absolute values indicated. It is signifi- cant that the spread in data among these samples is quite

large, ~3 ppm. As will be seen below, there is a corresponding variability of molar mass which correlates

very closely with these density differences.

3.2 Isotopic Abundances—Mean Molar Mass

Evidently, one can obtain the quantity of microscopic mass in a unit cell by a fractional abundance weighted sum of isotopic masses. The NBS approach to such problems is well established but had to be pressed quite hard

to obtain accuracies below 1 ppm. The rather long chain

of steps has been described in detail elsewhere [18]. Mention of a few highlights is, however, in order. A large Si crystal (not otherwise involved in our meas- urement) was tested for uniformity of its isotopic abundance ratios. These ratios were then established on an "absolute" basis by comparison with synthetic mixtures

of separated isotopes. Complications are entailed by the

Table 1. Density values for specimen crystals as observed and corrected for C and 0.

Sample Observed density Corrected density '^7n (g/cm^) (g/cm^) (ppm) (ppm)

Di 2.3289942 2.3289935 0.22 0.97

D2 2.3289931 2.3289924 0.44 1.04

D3 2.3290028 2.3290028 0.95 1.33

307
Table 2. Abuyidance and molar mass values for SRM-990 and

samples.

Isotope 28Si 29Si 30Si

atomic % 92.22933 4.66982 3.10085

nuclidic mass 27.9769286 28.9764969 29.9737722

weight % 91.872710 4.817963 3.309326 A(SRM-990) = 28.0855258 Sample Molar Mass

D, 28.085400 D2 28.085379 D3 28.085507

need to adequately characterize abundances in the

"separated" isotopes and by the very small sample quantities of the less abundant species which were available

at the time of our measurements. The large homogeneous

boule was sliced into disks which are publically available

through the Standard Reference Materials Program at NBS as SRM 990 [19]. This public availability invites

testing of the reference abundances on which our molar mass values are based.

All subsequent measurements of abundances were car- ried out relative to those in SRM 990. These comparisons were made in a 15 cm, 60° magnetic sector mass spec- trometer having a well characterized transmission function. This instrument uses gas phase samples, in the

present case SiF4, prepared by thermal decomposition of

BaSiFfi. Initial samples of both separated isotopes and

samples of crystalline material were prepared as fluosili- cic acid which was converted successively to Cs2 SiFg and

thence to BaSiFe.

Results obtained for both SRM 990 and the three crystals whose densities were measured are shown in Table 2. Contributions to the estimate of ct in the samples came from chemical synthesis (0.3 ppm), analysis of

abundances in the separated isotopes (0.3 ppm), and uncertainty in SRM 990 (0.5 ppm). Quadrature combination

of these components yields a composite a of 0.65 ppm.

Table 2 also shows the weighted sum values for mean

molar mass associated with each sample. It should be

noted that the spread of values is quite appreciable but

see below.

3.3 Evaluation of

The three quantities required for obtaining Na are the

unit cell volume, ao, macroscopic density, p, and molar

mass, A [lie]. For the cell volume, a correction is re- quired to take account of the fact that interferometry was done in vacuum while density was measured at at- mospheric pressure. This correction was -1.02 ppm for the NBS measurements. Each of the crystals has, of course, its own cell dimension; these were, however,

found to be sufficiently close together that the interfer- ometer result could be used for all three samples. Two of the samples had appreciable concentrations of both C and

0. The numerical values were, however, such that the

expansion due to the latter was closely compensated by

the contraction due to the former.

Concordance is not so evident in the case of the density

and molar mass values given above. If it is assumed,

however, that such density variation as is encountered

follows entirely from the molar mass changes then the

quantity A/p should exhibit a smaller range of variability. That such is indeed the case is evident from Table

3. The range of variation is, in fact, sufficiently small that we have used the weighted mean value also indicated in Table 3.

Table 3. A/p values for the samples studied.

Sample A/p a„i (ppm) (Tf (ppm)

Di 12.0590290 0.52 1.19

D2 12.0590257 0.65 1.25

D3 12.0590268 1.05 1.48

weighted mean 12.0590274 0.45 0.94

The result of combining the above mean value for A/p

and the cell volume (obtained in the NBS interferometry)

is, as has been previously reported [lib, e]: Na -

6.0220978 x lO^^ u/kg. If the PTB result (after correction for the 0.5 ppm offset noted above) were used with

the assumption of a common value for A/p, this A^^ value

would increase by 4 ppm. Pending a separate evaluation

of A/p for the PTB crystal, there is no basis for further

discussion.

4. Optically-referenced Gamma-ray and

X-ray Measurements

Another area where crystals calibrated in the

optical /x-ray exercise of Section 2 find application is in extension of the congruent electromagnetic scale to very

short wavelengths [20]. To accomplish this extension,

the calibration embodied in a single artifact crystal needs

to be transferred to other specimens and species. That

accomplished, these daughter crystals are useful for wavelength determinations via Bragg-Laue diffraction

provided sensitive, absolute angle measurement tech- niques are available. Work at NBS has provided for both

of these facilities, i.e., distribution and absolute

goniometry. In these applications, contact with atomic

reference values, e.g., R^, is more relevant than the

metrological reference needed for Na- Fortunately both

concepts are well connected to molecular iodine stabilized

He-Ne lasers. The resulting instrumentation has been applied to a variety of problems of current interest [lid]. Among

these I note that provision of new values for 7-ray refer- ence lines for muonic atoms spectra has led to an improved test of QED. Also reference lines for pionic

spectra have provided greater concordance among pion mass determinations. In the case of normal atom spectra, our work has revealed a suggestive pattern of systematic

discrepancies between relativistic self-consistent field calculations and experiment. Incidental to such studies, we have participated in a new determination of the kaon mass by remeasurement on an absolute scale of the Er K

absorption edge [21].

308
4.1 Crystal to Crystal Transfers

The particular specimen of Si crystal involved in the x-ray /optical interferometry is not of a size and shape

suited to general diffraction applications. Furthermore,

the large scattering power associated with the Ge crystals is of considerable value at "y-ray energies ^ 0.2 MeV. It is evidently possible to make such

transfers of calibration by direct remeasurement of 7- lines (x-ray lines being rather too broad). Unfortunately

gamma-ray sources provide only low counting rates and

have certain preparation and handling constraints. It is needed, therefore to have a procedure which uses the

high flux available from a conventional x-ray source but

which is not limited by the large widths and asymmetries

of x-ray lines. A family of suitable procedures was introduced by M.

Hart and collaborators. We have made most use of a variant of these procedures shown in Fig. 5 [lid, f]. This

[111] " ^ [224]

Figure 5. Diagram of yion-dispersive lattice comparison. Optically calibrated crystals and others needing such a reference are placed alternately in the second crystal position.

geometry has small dispersion (zero in the case of

"equal" crystals) and is thus insensitive to x-ray line- width. The measurement cycle entails placing a cali- brated and an uncalibrated specimen in the second

crystal position. For all "equal" crystals, a small offset may obtain due to curvature or inhomogeneity in the long

first crystal. Such an offset, however, vdll appear in scans with both standard and sample crystals. Thus such

an offset will disappear in the difference of the differ- ences. This last mentioned quantity is proportional to

the difference in lattice parameters suggesting the al- most null character of the comparison.

In the case of dealing with comparisons between dif- ferent crystals, we have made use of approximate degeneracies which can often be found between (different)

reflections in the distinct species. An example, used pre- viously by Baker and Hart [22] which we have also used

in an important way, is the approximate equality

between the (800) spacing of Ge and the (355) spacing of

Si; the difference is about 200 ppm. As long as equality or near equality can be maintained, angular measure- ments need sensitivity, but only limited accuracy as con- trasted vdth what is required for the direct wavelength

measurements described in the next section. Suffice it to say that we have been able to make all needed transfers

to an accuracy not far from 0.1 ppm. Thus this step is not a limitation at the level of accuracy (—1 ppm) with

which we are able to access invariant results.

4.2 Bragg-Laue Spectroscopy of X-rays and 7-rays

With crystals of known spacing direct wavelength

measurements are possible provided sufficiently robust

angle measurement schemes are at hand. Figure 6 indi- cates a scheme, particularly useful toward shorter

SOURCE

DETECTOR

( + )

Figure 6. Schematic of arrangement for transmission double

crystal measurement of short wavelengths. With aid of sensitive angle interferometry calibrated from first principles, crystals ivith known spacings produce secondary y-ray standards

as well as accurate x-ray line positions.

309
wavelengths, wherein in-plane diffraction obeys exactly

the Bragg-Laue equation; finite vertical divergence gives an asymmetric contribution to the transmission function

in the dispersive configuration but this is quite tractable

if kept small. Adequate angular sensitivity and dynamic range is readily available from two-beam polarization encoded angle interferometry. As is described in detail in our re- view [lid], one obtains s fully servoed system with noise

below 0.1 milliarcsec and a range of either ±2.5° in an

early version or ±15° in the latest model. Placing this precise angular scale on an absolute basis is a separate

exercise. We use a sensitive null-pointing autocollimator

together vdth a 72 or 24 sided optical polygon. With the

autocollimator providing a reference direction, the angle

interferometer is used to measure the intervals between

all adjacent face pair normals. Since the sum of these

external interfacial angles is constrained to be 2 it, one

obtains a calibration constant for the interferometer and,

incidentally, a rather high quality polygon calibration. Resulting calibration constants vary slowly with time but

by interpolating between prior and subsequent calibrations, a constant good to about 0.1 ppm is readily

obtained.

Applications of such measurement technology has been

in two main areas. First, as will be described next, there

are a number of low energy 7-ray lines which have

served either singly or in groups as reference values for muonic and pionic x-ray spectra. We have reported new

values for most of these lines. A second area of application has emerged from some selective redetermination of atomic x-ray spectra themselves. Although these no

longer have their traditional role as intermediates in chains to 7-rays, they appear to have regained a certain measure of intrinsic interest as will be seen below in Section 4.4.

4.3 Applications to Muonic and Pionic Spectra

Outer orbit muonic spectra particularly from medium

to high Z atoms furnish a well-known testing place for QED. Shifts, for example due to vacuum polarization, are

quite large so that even higher order terms become important. At one time, there appeared to be a significant

pattern of discrepancies which were troublesome in that

most approaches trying to account for them Mathin the

conventional framework appeared unpromising while cer- tain exotic possibilities appeared implausible. Revision of the reference lines by the NBS 7-ray measurements combined with certain improvements in the muonic data have

effectively removed the discrepancies.

Measurements of the corresponding outer orbit spectra

of pionic atoms particularly toward lower values of Z are important sources of information on the pion mass. At

one time, different routes to this mass using different

reference lines were in some disagreement. When the

various reference values were corrected to the new scale,

such discrepancies are no longer significant. A related

measurement of the mass of K'"* has now also been completed with results in agreement with alternative earlier

procedures but not quite as accurate because of certain

technological difficulties in the critical absorption

measurement.

4.4 Application to Atomic X-ray Spectra

It is not at first glance obvious whether this venerable

subject should retain interest. The earlier stage where

splittings of 2pi;2 and 2ps/2 vacancy states (i.e., the Ln -

Liii interval) were used to reveal both nuclear size effects and Lamb-shifts [23] have now merely historical interest. Also, as noted above, x-ray lines as intermediate steps between optical and 7-ray wavelengths are no

longer needed; this is a fortunate circumstance since the

large widths, asymmetries, and unmodelable lineshape

are severe metrological disabilities. What has happened,

however, is that significant progress on the theoretical

side has occurred making serious comparisons with term

values and widely separated term differences (as con- trasted with spin-orbit splittings) both practical and in- teresting provided data are adequate. This was not the

case over the full range of Z which has led to significant new work which will be mentioned below.

On the theoretical side, progress both in principle and

in terms of calculational rigor has been considerable

especially within the past 5 years. What is involved are rather complete relativistic self-consistent field (SCF)

calculations for the ground states and for atoms with

inner-shell vacancies. Differences between total energies

are then estimates of x-ray term energies. In some cases these terms can be approached directly from the experimental side, especially for free atoms. In an approximation whose limitations are difficult to quantify, x-ray line values, photoelectron thresholds, and work functions can be combined to give an estimate of free atom terms using

data obtained from solid samples. A somewhat more

robust procedure permitting use of solid samples to get

estimates close to free atom properties is to focus on

inner shell emission lines noting that condensed matter

effects tend to perturb two inner holes equally. The loss of information entailed because only term differences ap- pear can be compensated by systematic studies of discrepancy patterns as one vacancy is held fixed while

the other changes character. Similarly, systematic trends

in Z tend to be more believable than isolated points,

especially if a "run" of Z includes some free atom data. New experimental work has appeared from both the

Institute Laue-Langevin (ILL) and NBS. The ILL work

has concentrated on almost degenerate pairwise comparisons between high and mid-Z spectra brought into near

coincidence by virtue of using different orders of diffraction. The NBS approach has aimed at obtaining a fairly

uniform sampling of Z values limited only by the practicality of targets for electron bombardment and a desire

to generate as few data as would suffice to reveal and

confirm interesting trends.

What appears to emerge thus far is an approximately

linear dependence of discrepancies with respect to Z

which persists until fairly high Z values. The same general trend appears in Kai, Ka2, K^i, and /Cp3 vdth less clear but not dissimilar results for L aj and L az. At high

Z there seems a tendency for the K series discrepancies

to become smaller. Also there appear to be cusps in the

pattern near nuclear magic numbers. This investigation

continues vdth further work on both the theoretical and

the experimental sides.

Many collaborators were involved in the NBS work

described above; most appear as co-authors in the cited

references. In preparing this review I have turned to E.

G. Kessler, Jr. , and A. Renins who also carry out the

work currently in progress. The cooperation of the PTB

group, especially Drs. Seyfried, Becker, and Siegert,

in sample exchange and discussions, is gratefully

acknowledged.

310
References

[I] U. Bonse and M. Hart, Appl. Phys. Lett. 7, 99 (1965); Z. Phys. 188, 154 (1965). For a review, see U. Bonse and W.

Graeff, in X-Ray Optics, H.-J. Quessar, Ed. (SpringerVerlag, Heidelberg, 1977), p. 93.

[2] Precision Measurement and Fundamental Constants, D. N. Langenberg and B. N. Taylor, Eds., Natl. Bur. Stand.

(U.S.), Spec. Publ. 343 (Aug. 1971), contains preliminary

reports as follows:

a. R. D. Deslattes, p. 279.

b. I. Curtis, I. Morgan, M. Hart, and A. Milne, p. 285.

c. U. Bonse, E. teKaat, and P. Spieker, p. 291.

[3] Additional efforts are underway at the following institutions (personal names are suggested contacts):

a. Instituto CoUonnetti, Torino, Italy; G. Zosi.

b. National Laboratory of Metrology, Tokyo, Japan;

K. Nakayama.

c. VNIIM, Leningrad, USSR; Yu. V. Tai-beyev.

[4] M. Hart, Nucl. Instrum. Methods 172, 209 (1980).

[5] G. Materlik, in Neutron Interferotmty-y, U. Bonse and H.

Rauch, Eds. (Oxford Univ. Press, 1979), p. 431.

[6] U. Bonse, P. Spieker, J.-T. Hein, and G. Materlik, Nucl. In- strum. Methods 172, 223 (1980).

[7] J.-L. Staudenmann, S. A. Werner, R. Colella, and A. W.

Overhauser, Phys. Rev. A 21, 1419 (1980).

[8] M. Ando, D. Bailey, and M. Hart, Acta Crystallogr. Sect. A:

34, 383 (1978).

[9] J. A. Baker, T. N. Tucker, N. E. Moyer, and R. C. Buschert, J. Appl. Phys. 39, 4365 (1968).

[10] Spacing measurements on a sample of Si at PTB is reported

as follows:

a. These proceedings, contributions of P. Becker and

H. Siegert; and H. Siegert and P. Becker. See also

P. Becker, K. Dorenwent, G. Ebeling, R. Lauer, W.

Lucas, R. Probst, H.-J. Rademacher, G. Reim, P. Seyfried, and H. Siegert, Phys. Rev. Lett. 46, 1540

(1981). K.-J. Hanssen, G. Ade, W. Lucas, H. Siegert, and

P. Becker, PTB-Berichte APL-14 (PhysikalischTechnische Bundesanstalt, Marz, 1981). Overviews of the PTB effort on A^^ are given at this Conference and previously.

b. P. Seyfried, these proceedings.

c. P. Becker, in Neutron Interferometry, U. Bonse and

H. Rauch, Eds. (Oxford University Press, 1979),

p. 416.

[II] Results of progressively more refined lattice spacing meas- urements have appeai-ed in reports on x-ray wavelengths

(a) and on A^4 (b). The result was applied to gamma-ray

reference lines in (c).

a. R. D. Deslattes and A. Henins, Phys. Rev. Lett. 31, 972 (1973).

b. R. D. Deslattes, A. Henins, H. A. Bowman, R. M.

Schoonover, C. L. Carroll, I. L. Barnes, L. A.

Machlan, L. J. Moore, and W. R. Shields, Phys. Rev. Lett. 33, 436 (1974); cf. Phys. Rev. Lett. 36, 898 (1976).

c. E. G. Kessler, Jr., R. D. Deslattes, A. Henins, and

W. C. Sauder, Phys. Rev. Lett. 40, 171 (1978). Reviews of this work have appeared focussing on x-ray and

gamma-ray energies (d), on A^^ (e), and on the overall pro- gram (f). Several earlier reviews were presented (g). d. R. D. Deslattes, E. G. Kessler, W. C. Sanders, and

A. Henins, Ann. Phys. (N.Y.) 129, 378 (1980).

e. R. D. Deslattes, in Ann. Rev. Phys. Chem., Vol. 31, B. S. Rabinovitch, J. M. Schurr, and H. L. Strauss, Eds. (Ann. Rev. Inc., Palo Alto), p. 435.

f. R. Deslattes, in Metrology and Fundamental Constants, Proceedings of (bourse LXVIII, Summer

School of Physics—Enrico Fermi, 1976, A. Ferro

Milone, P. Giacomo, and S. Leschiutta, Eds. (North

Holland, Amsterdam, 1980), p. 38.

g. E. G. Kessler, Jr., R. D. Deslattes, W. C. Sauder,

and A. Henins, in Neutron Capture Gamma Ray

Spectroscopy, R. E. Chrien and W. R. Kane, Eds. (Plenum Press, New York, 1979), p. 427. R. D. Deslattes and E. G. Kessler, Jr., in Atomic Masses and Fundamental Constants—6, J. A. Nolen

and W. Benenson, Eds. (Plenum Press, New York,

1980), p. 203. R. D. Deslattes, Jpn. J. Appl. Phys. Suppl. 17-2, 1 (1978). R. D. Deslattes, in Atomic Masses and Fundamental Constants—5, J. H. Sanders and A. H. Wapstra,

Eds. (Plenum Press, New York, 1976), p. 552.

[12] Such a comparison was attempted by A. Henins at NBS

shortly after this Conference. The PTB sample was kindly furnished by oui' colleagues from that institution.

[13] a. A. Peuto and A. Sacconi, these proceedings.

b. G. A. Bell and J. B. Patterson, these proceedings.

[14] J. B. Saunders, J. Res. Natl. Bur. Stand. Sect. C: 76, 11 (1972).

[15] D. P. Johnson, J. Res. Natl. Bur. Stand. Sect. A: 78, 41

(1974).

[16] For a discussion, see Refs. [lie and f]. [17] H. A. Bowman, R. M. Schoonover, and C. L. Carroll, J. Res. Natl. Bur. Stand. Sect. A: 78, 13 (1974).

[18] I. L. Barnes, L. J. Moore, L. A. Machlan, T. J. Murphv, and

W. R. Shields, J. Res. Natl. Bur. Stand. 79A, 727 (1975).

[19] Standard Reference Materials are available through the Office of Standai'd Reference Materials, Room B311, Building

222, National Bureau of Standards, Washington, D.C.

20234.

[20] E. G. Kessler, Jr., L. Jacobs, W. Schwitz, and R. D.

Deslattes, Nucl. Instrum. Methods 160, 435 (1979). See

also Refs. [11c and g]. [21] G. K. Lum, C. E. Wiegand, E. G. Kessler, Jr., R. D.

Deslattes, L. Jacobs, W. Schwitz, and R. Seki, Phys. Rev. D 23, 2522 (1981).

[22] J. F. C. Baker and M. Hart, Acta Crystallogr. Sect. A: 31, 364 (1975).

[23] R. L. Shacklett, Rev. Mod. Phys. 30, 521 (1958).

311

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Work Related to the Determination of the Avogadro Constant in the PTB

P. Seyfried

Physikalisch-Technische Bundesanstalt, D 3300 Braunschweig, F.R.G.

• The d 220"l^ttice spacing in a highly pure nearly perfect silicon crystal was measured and found to be ^220 = 192 015.560 fm with an uncertainty of ±6 x 10~^d220 vacuum at 22.50 °C. Thus for this crystal the first of the three quantities, (^220. density p, and molar mass M, from which the Avogadro

constant Np^ can be derived, is known wath sufficient accuracy. The rf220'value given is -1.8 x 10"^(Z220 smaller than that reported by Deslattes et al. and used in his determination of Nf^. Carbon and oxygen

impurities in our crystal cannot explain this large difference. A possible change in Np^ of +5.4 x 10"'' of its value should be taken into account. Density standards in the form of cubes made of a ceramic material with trade name Zerodur have

been prepared. For two (1 kg) standards an uncertainty of A p/p = ±7 x 10""^ has been obtained. These

standards will be used for density measurements of silicon samples found to be uniform (Ad/d < 4 x

10"*) by both high resolution double crystal and Moire-topography.

Key words: Avogadro constant; density standards; lattice spacing; silicon.

1. Introduction

Stimulated by proposals of Bonse and Hart [1] a project was started in the PTB for the precision measurement of the d22o-lattice spacing in a silicon crystal. The work

gained remarkable impetus by the publications of the ex- tremely successful measurements of Deslattes and co- workers [2, 3] at the NBS. Recently, in an extended series

of measurements, the lattice spacing ^220 of ^ highly pure

nearly perfect silicon crystal was determined with a rela- tive uncertainty of 6 x lO"* [4]. Thus, the first of the three

quantities necessary to derive the Avogadro constant,

namely ^220' the density p, and the molar mass M of a sili- con monocrystal, is known with sufficient accuracy.

In a second project density standards are being

developed in order to improve the accuracy of density

measurements. On the way to a final uncertainty of less than ±5 x io~'^, a series of experimental devices has been

investigated, and most recently two standards have been

prepared, which will now be used for density measure- ments of silicon samples.

Close cooperation with external laboratories for mass

spectrometry is necessary for the determination of molar

masses.

2. Density Determination

The PTB-approach to density standards differs from

that of Bowman et al. [5] at NBS in that we use cubes of a ceramic material (trade name Zerodur), instead of steel

balls. Zerodur was chosen because it is a dielectric material (rather than a conductor), is insensitive to chemical effects, has a very low expansion coefficient

(a < 0.05 x lO-s K"^), and can be easily machined. Cubes

were preferred instead of balls, as then the highly

developed experimental methods and instruments used

for measuring gauge blocks can be applied. Taking into account the known errors, the final uncertainty is es- timated to be 6 X lO""^ p for a 1 kg (a = 73 mm) standard.

Corrections due to deviations from the ideal shape are

negligible if the following limits are not exceeded: unevenness € < \/10, non-parallism tt < 2 x 0.1", deviations from the right angle p < 10". The missing volume

should not exceed 2 x lO"'' of the volume.

Twelve rectangular blocks of mass M (called quaders)

(a = 24 mm, 6 = 25 mm, c = 33 mm), six sharp cornered

and six with bevels were made and investigated. After

initial weighing, the deviations from ideal shape were in- vestigated and found to be well within the limits given

above (see Fig. 1 and Fig. 2). The distances at the centers of opposite faces a, b, c were determined both in- terferometrically and by means of a displacement comparator with mechanical contacting [6], and surfaces and

edges were inspected by optical microscopy and scanning

electron microscopy in order to estimate the missing

volume. Additional weighing after each step ensured that no mass was lost during the process.

F

20Hn lOKV 00 006 S

Figure 1. Missing volume near a comer of a density standard

estimated to be less than 2 x 10~ mm , AV/V ~ 10~^^; photograph taken from a scanning electron microscope.

313
Figure 2. Interference of equal thickness pattern obtained for

the six faces of quader No. 8, unevenness below k/13.

Results obtained for two sharp cornered devices and

two with bevels are summarized in Table 1. The indices

TO and a refer to interferometrical and mechanical meas- urements respectively, which differ by about 20 nm. The

missing volume amounts only to Ay = 2.5 x lO'V and is not a restricting factor as was assumed at the beginning

of the work. The following uncertainties have to be as- signed to the various quantities:

s (m

)

±2 X IQ-'^TO

s (a^) s (b„,) = s (c„,) = ±15 nm

s(a„) s(ba) = s(Ca) = ±20 nm

s(AV)f ±2.5 X 10"*' V

±2.5 X 10"'^ V

s(V^) ±1.0 X 10-6

±1.3 X 10-6

(p)f = ±4' X 10-6 p s (p)s = ±2 X 10 The indices s and / refer to sharp cornered quaders and

those with bevels, respectively.

As a consequence of these results, two sharp cornered

cubes (a - 73 mm, m = 1 kg) were made and the density was determined vdth a final uncertainty of s(p) = ±7 x

10-7p.

Several samples will be prepared from sufficiently uniform silicon (examined by Lang-topography), with one from material of which the lattice parameter is accu- rately known. Their density will be determined in a collab- oration between BIPM and PTB [7]. Separate investigations have begun to determine the

density from the mass and the geometrical dimensions of a regularly shaped piece of silicon. A high quality

cylinder (400 g) is being prepared and a hexagonal prism

is under consideration. Missing volume and density

Table 1. Results of measurements on four rectangular

blocks of Zerodur; indices m and a refer to interferometrical and mechanical measurements, respectively.

Quantity With

No. 1 BevelNo. 3 Sharp C

No. 8 orneredNo. 10

m[g] 50.220 669 50.221 167 50.222 750 50.222 609

ttyn [mm]

Ca [mm]

byn [mm]

ba [mm]

Cm [mm]

Ca [mm]

24.031 411

24.031 413

24.927 595

33.083 707

33.083 737

24.031 552

24.031 565

9/1 097 (1/19

24.927 651

33.083 709

33.083 726

24.031 484

24.031 509

9/1 Q97 11/1 cA.Vi.i 111 24.927 118

33.083 597

33.083 613

24.031 421

OA Q97 1 QQ Ci^.V^i loo

33.083 560

y„, [mmS]

y Tmrn^l y [mm^]

19 818.666

19 818 658

19 818.662

19 818.793

19 818 821

19 818.807

19 818.250

19 818 284

19 818.267

19 818.195

19 818.195

Ay [mm=^] -0.719 -0.610 -0.005 -0.005

y [mm^] 19 817.943 19 818.197 19 818.262 19 818.190

p [g/cm^] 2.534 101 2.534 094 2.534 167 2.534 167

changes in machined and polished faces are expected to

limit the accuracy.

All of this work is timed to achieve an uncertainty in determining density of less than 5 x lO"'^ by 1983.

3. Lattice Spacing in a Silicon Crystal

The d220-lattice spacing in a highly pure, almost perfect

silicon crystal, grown and purified by multiple float zone melting by Wacker-Chemitronik, has been measured us- ing a scanning x-ray interferometer according to Bonse

and Hart [1], combined with an optical polarization inter- ferometer described by Curtis et at. [8]. The principle of

the device is shown in Fig. 3. During a translation of the

analyzer crystal A the signals shown in Fig. 4 are observed. A displacement s = m. K/2is enclosed by two zero

crossings of the optical signal. In this displacement. A'' full periods and a fraction / of the x-ray signal are ob- served; / results from the phase ^(yn) and <t)(0) of the x- ray signal at the limits of the displacement.

The n-fold (n = N + f) of d corresponds to a displacement S = TO\/2.

mk/2 = nd , d = {n/mr\/2 . Here n/m is the quantity to be measured. A first approximation is obtained from a measurement over X/2(m = 1). If m is increased in small steps N may be predetermined

and the measured fraction / gives an improved value for n/m. It is then sufficient to measure / alone. In successive steps (e.g., TO = 1, 2, 4, 8 . . . 128) the uncertainty is reduced until N may be absolutely predetermined for

TO = 120. Details are given in the contribution of P. Becker, these proceedings; see also Refs. [9, 10].

314
x-ray

interferometer ^

incoming X-roy beom

''/^-plates

'*-/2-plate

optical interferometer

Pocl<els-cell

stonding wove pattern

polarisation separator

beam splitter

to the ptiotodetector

Figure 3. Principle of the scanning x-ray interferometer of the PTB: LLL-x-ray iyiterferoyfieter according to Bonse and Hart

[1] combined with an optical interferometer according to Curtis et al. [8].

mX/2 = nd

here N = 5 , f = 0,2

n= N*f ; f = ^(9(i)-<P(0)) In

Figure 4. Signals obtained in the scanning x-ray interferometer of Fig. 3 (schematically); modulation of the x-ray signal

shovun, average intensity subtracted.

In a series of measurements in vacuum at temperatures between 22.42 °C and 22.50 °C a mean value nJm =

1648.281 626 (reducedjo 22.50 °C) with a standard deviation o- = ±5.8 X 10"^ n/m and a standard deviation of the mean = ±4.3 x lO"^ n/m was obtained. Measuring

results, corrections, and" uncertainties are summarized in Table 2.

Table 2. Summarized measuriyig results, corrections,

and imcertainties of the d^-^o-lattice spacing measurements.

Measuring results Number of measurements Number of x-ray fringes

in half the wavelength

Standard deviation Standard deviation of the mean

Resulting lattice spacing

T = 22.5 °C. vacuum (P « 10"^ bar.

Corrections and uyicertainties Fresnel correction Cosine-error Abbe-error

Temperature

Total Uncertainty of wavelength

Uncertainty of the lattice spacing

170

n/m = 1648.281 626

CT = 0.000 095 = 0.000 007

(^220 = 192 015.560 fm

X = (632 991 415 ± 5) fm

/ci = (-3.8 ± 0.2) X la-^n/m

A; 2 = (-0.1 ± 0.2) X la-^n/m

A;3 = ( 0 ± 3.0) X 10"8 n/m

/C4 = ( 0 ± 5.1) X 10-8

k = (-3.9 ± 5.9) X 10-8 n/m

AX = ± 0.8 X 10-8 ^ Ad = ± 6 X 10-8

315
The total correction amounts to less than the standard

deviation for a single measurement. The largest contribution is given by the correction for the nonplanarity of the

wavefronts of the laser beam (Fresnel phase shift) and is calculated from a beam radius in the waist Wg = (0.521 ±

0.016) mm. Only a very small cosine-correction is neces- sary as the normal to the mirrors, the direction of movement, and the direction of the laser beam are parallel to the normal to the lattice planes within a few seconds of

arc, due to elaborate adjustment procedures. The largest

uncertainty is given by a ±0.02 K uncertainty of the

crystal-temperature, carefully calibrated thermistors being used. Details are given in the contribution of Siegert,

these proceedings.

4. Discussion

The d 220" value reported here gives for the first time

the opportunity to compare two lattice spacings of silicon crystals measured by combined optical and x-ray inter- ferometry. This is done below:

^2^^ = 0.192 016 724 nm ± 6 x lO'^ ^220 -rf^BS ^ -0.192 017 07 nm ±10 X lO'^ ^220 = -0.000 000 346 nm (at 25 °C = -1.8 X 10"*^ ^220 and 1 bar)

The difference exceeds the stated uncertainties by almost

a factor of 20. There should of course be a difference in the lattice spacings of the two crystals. However, a

difference as large as this has only been reported in the

comparison measurements of Ando, Bailey, and Hart [11] for an "exotic" sample of silicon. The lattice spacings of

all other samples of float zone silicon of different suppliers all over the world differed from the very pure

PERFX by no more than -6 x 10'^ d. This difference was

assigned to residual carbon impurities.

PERFX samples are reported to differ by less than

10^^ [12]. A Lopex crystal was compared by Deslattes

[13] to a PERFX crystal. Due to carbon and oxygen impurities of riQ = 3.9 x iQi*' cm"^ and hq = 6 x 10^^ cm~^,

respectively, a change in the lattice spacing Ad/d = 4 x

10"^ was predicted using (Ad/d)c = -6.5 x 10 "^^ n.Q and

(Ad/d)o = +3.5 X 10"^ HQ. The difference was measured

to be (5 ± 2) X 10-8 and differs only by (1 ± 2) x lO"* from the predicted value. From this we conclude that impurities of C and 0 are the main sources of lattice parameter variation. Since our

Waso-silicon is very pure (hq = hq < 1 x lo^^ cm"-^) and

uniform (Ad/d < 4 x io~^, measured using Moiretopography and high resolution double crystal topography [14]), our material is expected to be closer to PERFX than to any other material used in the comparison of Ando, Bailey, and Hart.

If a crystal-to-crystal comparison confirms our assumption, there will be a considerable discrepancy indicating a systematic error in one of the measurements or both. On

the basis of our d22o-lattice spacing, leaving fixed all other data as given by Deslattes et al. [15], the Avogadro

constant would have to be changed by +5.4 x 10 of its value. Clearly, there is an urgent need for additional

work.

I am indebted to Professor U. Bonse, University of Dortmund for scientific stimulation and critical discussions. I also wish to thank Dr. M. Kochsiek and other

PTB colleagues for their fruitful cooperation.

References

[1] U. Bonse and M. Hart, Z. Phys. 188, 154 (1965).

[2] R. D. Deslattes and A. Henins, Phys. Rev. Lett. 31, 972

(1973).

[3] R. D. Deslattes, A. Henins, H. A. Bowman, R. M. Schoonover, C. L. Carroll, I. L. Barnes, L. A. Machlan, L. J. Moore, and W. R. Shields, Phys. Rev. Lett. 33, 463 (1974).

[4] P. Becker, K. Dorenwendt, G. Ebeling, R. Lauer, W. Lucas,

R. Probst, H.-J. Rademacher, G. Reim, P. Seyfried, and

H. Siegert, Phys. Rev. Lett. 46, 1540 (1981).

[5] H. A. Bowman, R. M. Schoonover, and C. L. Carroll, Metrologia 10, 117 (1974).

[6] Chr. Hoffrogged and R. Mann, Messtechnik 81, 1 (1973).

[7] Arrangement of M. Kochsiek (PTB) and G. Girard (BIPM).

[8] I. Curtis, I. Morgan, M. Hart, and A. D. Milne, in Precision

Measurement and Fundamental Constants, Ed. by D. N. Langenberg and B. N. Taylor, Natl. Bur. Stand. (U.S.)

Spec. Publ. 343 (Aug. 1971), p. 285.

[9] H.-J. Rademacher, P. Becker, H. Siegert, W. Lucas, K. Dorenwendt, G. Reim, K.-J. Hanszen, R. Lauer, R.

Probst, and P. Seyfried, PTB-Bericht APh-13 (Physikal- isch-Technische Bundesanstalt, Braunschweig, July 1980).

[10] K.-J. Hanszen, G. Ade, W. Lucas, H. Siegert, and P. Becker,

PTB-Bericht, APh-14 (PhysikaHsch-Technische Bundesanstalt, Braunschweig, March 1981).

[11] M. Ando, D. Bailey, and M. Hart, Acta Crystallogr. Sect. A:

34, 484 (1978).

[12] J. A. Baker, T. N. Tucker, N. E. Moyer, and R. C. Buschert, J. Appl. Phys. 33, 4365 (1968).

[13] R. D. Deslattes, private communication.

[14] We are indebted to Prof. Dr. U. Bonse, University of Dort- mund, for these measurements.

[15] R. D. Deslattes, E. G. Kessler, W. C. Sauder, and A. Henins, Ann. Phys. (N.Y.) 129, 378 (1980).

316
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Absolute Determination of the (220)-Lattice Spacing in Silicon

P. Becker and H. Siegert

Physikalisch-Technische Bundesanstalt, D 3300 Braunschweig, F.R.G.

The (220)-lattice plane spacing in a silicon crystal was measured using a combined x-ray and optical interferometer. The experimental set-up and important characteristic features of the interferometer

crystals and the translation device are described in detail. The results of several individual measure- ments are discussed to explain the evaluation method applied. The results of 170 measurements are

characterized by a standard deviation la = 6 x 10"^. Key words: lattice parameter; silicon; two-beam polarization optical interferometer; two-crystal x-ray

interferometer.

1. Introduction

In order to determine the lattice parameter of a silicon crystal by x-ray interferometry [1], the PhysikalischTechnische Bundesanstalt^ adopted a course relying on

the proposals of Bonse, te Kaat, and Spieker [2]. The ex- perimental set-up differs significantly from that of the

National Bureau of Standards, where Deslattes first suc- ceeded in measuring the (220)-lattice plane spacing of a

silicon crystal w^ith high accuracy, see e.g., Ref. [3]. The

basic requirement for performing these measurements is the moving of one part of a Laue-type x-ray interferometer with respect to the fixed crystal part in a direction

normal to the reflecting planes.

2. The X-ray Interferometer

The two crystals of the x-ray interferometer were cut from a piece of silicon material (WASO), grown and purified by multiple float zone melting, manufactured by

Wacker-Chemitronic. A drawing of the interferometer

used in our experiment has been given previously, see

e.g., Ref. [4]. It has the following special features: The

front faces of the crystals are polished to form optical

mirrors. By this provision, the mirrors used for the optical measurement of the displacement are a part of the

crystal lattice itself. Both interferometer crystals, each

with three steel balls attached to its base, rest on supports without being fixed to them, as shown in Fig. 1. Thus the crystals may be easily exchanged, and a direct

comparison of crystal materials of different origin is facil- itated. Alignment of the lattice planes of the two crystals,

parallel within 1 x 10"*^ rad, is achieved by means of the

adjusting element shown in the figure. Slight rotations

about the p axis (axis of a rotational moire pattern) and

about the i? axis (axis about the Bragg angle alignment)

can be effected by two elastic pivots. Deflections of the

leaf springs are made with the eccentric shown in the fig- ure (rough adjustment), or with piezoelectric drives (fine

adjustment). More details of the interferometer crystals

and their alignment are described in Ref. [5], Sects. A

and D.

'Interlaboratory project of the Physikalisch-Technische Bundesanstalt: X-ray interferometry: H. Siegert, P. Becker: Length measuring technique: K. Dorenwendt; Mechanics: H.-J. Rademacher; Electronics: P. Seyfried, G. Ebeling, W. Lucas.

Figure 1. Springs for adjitstment of the analyzer crystal. A de- flection of the leaf springs by ± 1 mm effects a turn of the crystal by 5 X io~^ rad.

3. The Translation Device

During the measurement, the analyzer crystal A must

be moved in the direction normal to the lattice planes

with a velocity of 10"^ nm/s to 10^ nm/s. Tilting angles of only 10"^ rad could be tolerated. For this purpose, translation stages with guiding errors of less than 10"^ rad are necessary. A translation stage fulfilling this condition

and used in these experiments is shown in Fig. 2. It is in the form of a double parallel spring Ref. [5-E] manufactured from one plate of high-quality steel. Each of the

four springs consists of two bars. One of them is con- nected to the external fixed frame, the other one to the

movable central bar supporting the analyzer crystal A

and the adjusting element. The frame rests on four supports, two of which are fixed to a steel plate of very high

stiffness; the other two are equipped with a slide bearing. By this means, small deformations of the basic steel plate during movements of the central bar are avoided.

The displacement of the central bar is achieved by a piezoelectric drive as indicated in the figure.

Guiding errors of the translation stage are caused by

inexact mounting and inhomogeneities of the material.

They are avoided by proper adjustment of the connecting

rod with respect to the drive lever and finally, by means

of the magnetic adjusting elements shown in Fig. 3. In

317
order to reduce or increase the elastic force of the

mechanical springs during displacement, an additional

magnetic force AF of a permanent magnet acts upon the

springs of the translation device as indicated in the upper

part of the figure. In order to eliminate the undesired

Figure 2. Translation stage and drive. The piezoelectric drive

acts upon a lever, which is directly coupled to the central bar

of the double parallel spring. The whole drive is fastened to the frame.

permanent magnets

Figure 3. Magnetic compensation of guiding errors. To begin

with, the permanent magnet I is adjusted at a distance x to the spring of the translatio7i stage and acts with a magyietic force Fg (see operating point Pi in the lower part of the figure).

During measurements, x and Fg are changed by Ax and AF

as a function of deflection of the central bar.

"static" force Fg of the permanent magnet, two identical

magnetic elements I and II are used. In this way, the

operating point Pi on the characteristic curve of a single

permanent magnet changes over to the point Pi + u on

the curve corresponding to a pair of magnets, as shown

schematically in the lower part of the figure.

4. The Optical Interferometer

The translation of the analyzer crystal is measured by

the method of two-beam interferometry, using an optical

polarization interferometer according to Curtis et al. [6]. The principles of the arrangement have been described

and illustrated in Refs. [4, 5-J, 7, 8]. A Lamb-dip stabil- ized He-Ne laser is used as a light source. During measurements, the frequency of the laser light is repeatedly

compared with that of a ^^^l2 stabilized He-Ne laser. The

waist of the laser beam is about 1 mm in diameter, therefore only a very small correction due to the nonplanarity

of the wavefronts [9] (referred to as "Fresnel phase

shift") is necessary. Optical feedback is avoided by the

polarization optical components of the interferometer it- self and, additionally, by a light trap of an attenuation of more than 30 decibels (for details see Ref. [5-H]).

5. Apparatus

The instrument described so far is arranged on a granite table of about 500 kg in weight. Measures for at- tenuating mechanical vibrations and stabilizing the temperature to better than lO"'' K/h in the vicinity of the in- terferometers have already been discussed in Refs.

[4, 5-B, 5-F]. A schematic diagram of the apparatus is given in Ref. [10]. If the translation stage does not move

linearly, and if the loci, where the translation is measured by optical and x-ray means are offset (see e.g.. Fig. 1 in Ref. [11]), both interferometers will measure

different displacements, namely, Sopt and s x-ray By

measuring the relative phase of the optical and x-ray in- terferometer signals, the point of impact of the laser beam onto the crystal mirrors is optimized until the condition Sopt = s x-ray is fulfilled. Both interferometers and

the translation device are mounted on the steel plate

mentioned and inside a vacuum chamber with an aluminium hood. The temperature in the vicinity of the

silicon crystals is determined by several thermocouples

and NTC resistance thermometers which have been re- peatedly calibrated with reference to a platinum resis- tance thermometer.

6. Measuring Procedure

The principle of the arrangement is given in Seyfried's

contribution to these proceedings and this can be seen in Fig. 1 in Ref. [11]. During the translation of the

analyzer crystal, periodic intensity variations indicate

the lattice planes moving through the stationary x-ray wave pattern, as can be seen in Ref. [12]. The intensity

variations of the laser light which occur simultaneously

indicate the number of half of wavelengths X of the laser radiation contained in the displacement.

A displacement of an integer number m of X./2 corresponds to n x-ray fringes. Thus the lattice plane

spacing d can be calculated according to

318
with

d = {n/m)-^\/2 (1)

(2) Here, A^^^ is the integer part of n, while fm is the resi- dual fraction. By counting each x-ray fringe over pathlengths of X/2, i.e., m = 1, we obtain the integer part of n/m to be 1648 and a first approximation of the fraction

of the ratio n/m.

When the integer number has been fixed, it is

according to Deslattes' measurements—only necessary to measure the fringe fraction /„, between increasing optical intervals. Thus the accuracy of the lattice spacing

measurement increases vidth larger displacements. The

experimental data, obtained from each d-measurement

enables us to determine the number of lattice planes con- tained in the whole translation length with an uncertainty of one thousandth of an x-ray fringe.

The last stage of our step-wise progression is shown in Fig. 4. In the preceding steps, the expected value

7. Results

In Fig. 5 the reduced values An of x-ray fringes are

plotted versus the optical periods m. Five measurements

with forward and backward translations are shown. They

give an illustration of the reproducibility obtained. One

hundred and seventy measurements have been carried out in a vacuum at temperatures between 22.42 °C and

22.50 °C. The maximum deviations of the measured

values A n, the crystal temperature and the individual ratio n/m, referred to a common temperature of 22.50 °C,

have also been marked. In the upper part of the figure, for example the fractions Aw(m) with m = 30, 60, 90,

120, 120, 90, 60, 30, 0 have been measured in comparison

with the zeroth order. Continuing this way, forward and

backward for each run, an almost straight line results. Only in exceptional cases do the confidence limits of

the measured fractions deviate from the plotted straight

lines. From its slope, the ratio n/m is determined according to the equation given at the ordinate. In all 170 measurements, the deviation from each other of both

m= J

-10 ^

min 25

time

Figure 4. Output signals of the combined x-ray and optical interferometer. The output sig- nal of the optical interferometer is zero when the phase differeyice between the two beams is an integral number of half optical wavelengths.

{n/m)e was ascertained to be 1648.282. The output sig- nals of the x-ray interferometer (upper part of the figure) and the optical interferometer near its zero level (lower part) are recorded simultaneously as a function of time. The record represents only the backward part of

/,„ determinations over a total displacement of 120 optical orders. We shall discuss here only the left-hand part

of the dravdng. In the interval /, the fraction / 120 is determined by moving the analyzer crystal very

smoothly, and simultaneously evaluating several x-ray

fringes. After a fast translation of the crystal over a range of about 50,000 x-ray periods (interval II) and

after a brief period of mechanical relaxation (interval

III), the fractions /go; / eo • • etc. are measured in the same way until the zeroth optical order is reached.

In order to determine the ratio n/m, it is advantageous

to calculate the differences ^n/m. between the mean

value {n/m) measured during a single run forward and

backward, and the expected value {n/m)e according to An = [n/m. — {n/m)e]m . (3)

branches—each for forward and backward translations

is smaller than six hundredths of one x-ray fringe, and

the deviations of each measured value An(m) are smaller than four hundredths of one x-ray fringe. The

different mean slopes of all measured curves actually

limit the reproducibility. More details with regard to the

determination of the relative phase of the x-ray and optical signal have been previously reported [4, 10, 13]. A histogram of 170 measurements is shown in Fig. 6. A

mean value of (n/m) = 1648.281 626 with a standard de- viation of a single measurement ct = 5.8 x 10"* and a standard deviation of the mean cr,„ = 4.4 x 10"^ is calcu- lated from the results displayed. The dotted hne

represents a least squares fit of a Gaussian curve, \^^len

the data represented by the histogram are divided into 17 groups according to the observation time, there is no

significant change in the ratio n/m (as can be seen in Fig. 4 in Ref. [11]). The systematic contributions in calculating the lattice spacing d together wdth their uncertainties are given in Siegert's contribution to these proceedings.

319
CD

GOCD

0,02

-0,02

0,02

•0,02

0906.22 1=^

0905.11

f

A7//7?=164a281744 4 4

/=22.441 °C

/7//7? = 1648.281709

/ =22.446 °C,

Afterwards, by varying the locus onto the analyzer

crystal A, where x-ray-interferences occur, the given

value VnJmj was verified in more than 300 measurements.

From these results it can be concluded that the standard

deviation of a single measurement can be reduced by a

factor of 3 or more. Nevertheless, with respect to the new determination of Avogadro's number and to the

results given by Deslattes et al., there is an urgent need

to progress from the crystal under investigation to other

Si-crystal materials and to compare crystal materials

from different sources.

0,02

0,02

0912.9 nlh?= 1648.28161 0

1 f=22.WX

1,02

1,02

0910.^ /7/w= 1648.281567

/=22.42rc

0,02 -0907.5 nim = 1648.281463

J I I L J I L. /=22.439°C,

30 60 90 number m of optical periods -

120

Figure 5. Determination of the ratio nIm. Total displacement

120 optical orders.

1648.28140 .28150 .28160 .28170 .28180 .28190

ratio nIm

Figure 6. Distribution of the ratio n/m of 170 measurements.

The authors wish to thank Professor U. Bonse, University of Dortmund, and Professor K. J. Hanszen for their

fruitful discussions and scientific assistance. The financial support of the Federal Minister of Economics is gratefully acknowledged.

References

[1] U. Bonse and M. Hart, Z. Phys. 188, 154 (1965).

[2] U. Bonse, E. te Kaat, and P. Spieker, in Precision Measurement and Fundamental Constants, Ed. by D. N. Langenberg and

B. N. Taylor, Natl. Bur. Stand. (U.S.), Spec. Publ. 343 (Aug.

1971), p. 291.

[3] R. D. Deslattes et al.. Ann. Phys. (N.Y.) 129, 378 (1980).

[4] P. Becker, in Neutron Interferometry, Ed. by U. Bonse and H. Rauch (O.xford University Press, London, 1979), p. 416.

[5] H.J. Rademachere/a/., PTB-Bericht APh-13, Sects. A through P (Physikalisch-Technische Bundesanstalt, Braunschweig, July

1980).

[6] I. Curtis et al., in Precision Measurement and Fundamental

Con.stants, Ed. by D. N. Langenberg and B. N. Taylor, Natl. Bur. Stand. (U.S.), Spec. Publ. 343 (Aug. 1971), p. 285.

[7] G. Reim, in PTB-Bericht Me-13 (Physikalisch-Technische Bundesanstalt, Braunschweig, 1977), p. 139.

[8] D. Kind, P.-V. Seances Com. Int'l. Poids Mes. Ser. 2, 45, 23

(1977).

[9] H. Kogelnik and T. Li, Appl. Opt. 5, 1550 (1966).

[10] P. Becker, in PTB-Bericht E-18, (Physikalisch-Technische

Bundesanstalt, Braunschweig, Sept. 1981), p. 217.

[11] P. Becker et al., Phys. Rev. Lett. 46, 1540 (1981).

[12] U. Bonse and M. Hart, Z. Phys. 190, 455 (1966).

[13] K.J. Hanszen a/., PTB-Bericht APh-14 (Physikalisch-Tech- nische Bundesanstalt, Braunschweig, March 1981).

320
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Systematic Uncertainties in the Determination of the

Lattice Spacing c/(220) in Silicon

H. Slegert and P. Becker

Physikalisch-Technische Bundesanstalt, D 3300 Braunschweig, F.R.G.

The origin and magnitude of corrections and uncertainties connected with the absolute determination of the d(220) lattice plane spacing in a silicon crystal are discussed. Contributions of crystal

material, temperature, optical wavelength, alignment parameters, and guiding errors are estimated. A

total correction of -3.9 x 10 ~*n/?n of the mean value n/m = (X/2d) is calculated. The total uncertainty, comprising random and systematic uncertainties, amounts to ±6 x 10~*d(220). The uncertainty

of the mean contributes by less than one percent to this value, while the main part results from the uncertainties of the crystal temperature and the Abbe error. Key words: lattice parameter; precision length measurement; silicon.

1 . Introduction

In order to achieve an accuracy level of less than 1 x

10 "^ for the lattice spacing d(220) obtained by the 170

measurements as reported in Ref. [1], a great number of subsidiary investigations were carried out.^ Each partic- ular correction and uncertainty was reduced below the la

error of a single measurement of the ratio k/2d (X. - optical wavelength).

2. Origin of Uncertainty Sources

According to the basic equation

d = {n/my^m, (1) the lattice spacing d of a crystal is compared with a laser wavelength \, known in meters, by simultaneous phase

measurements m 2 it and n 2 it in an optical and in an x- ray interferometer.

Principal uncertainty sources in the measurement of d

arise from a non-perfect crystal structure and from the

uncertainty of the absolute crystal temperature. The

main errors of the phase measurements may result from

additional phase shifts proportional to the translation s of the analyzer crystal. Such phase shifts cannot be

separated from the linear phase variation to be measured. Therefore, they lead to an erroneous value of the

lattice plane spacing.

Due to Fresnel phase shift corrections, the wavelength

within the interferometer requires special attention. In the following, these effects will be discussed on the

basis of the experimental data acquired at the Physikalisch-Technische Bundesanstalt (PTE) at the end of 1980.

4nterlaboratory project of the Physikalisch-Technische Bundesanstalt:

x-ray interferometry: H. Siegert, P. Becker; Length measuring techniques: K. Dorenwendt; Mechanics: H.-J. Rademacher; Electronics: P.

Seyfried, G. Ebeling, W. Lucas.

^Supplier: Wacker Chemitronic GmbH, Berghausen, Germany.

Tradename: WASO (float-zoned) EPD-freep =1600 0 cm, p-type material.

3. Magnitude of the Uncertainty Sources

3.1 Homogeneity of the Lattice Plane Spacing

The degree of homogeneity of the rf(220) lattice plane

spacing wdthin the silicon material- of the x-ray interferometer can be estimated from a moire topogram [2] of the crystal lamellas. (For a schematic of the apparatus,

see Ref. [3], p. 31.) The topogram is shown in Fig. 1. Over the whole crystal lamella, no moire period smaller

than A = 5 mm is detectable. The relative uncertainty

Ad/d of the crystal material, calculated from Ad/rf -

d/A, is therefore less than 4 x 10"^ (d ~ 0.2 nm).

1 A mm 1 ^^^^^^^^^^ 1

Figure 1. Moire topogram of the x-ray interferometer (for details see Ref. [3], p. 31). As can he recognized, the lattice spacing varies by less than i x iO~* in the given material. The

region iLsed in the experiments is marked.

3.2 Temporal Drifts

Temporal drifts between the signals of the optical and

the x-ray interferometer are unavoidable. (For details see Ref. [4], p. 75.) Figure 2 gives an example of the

magnitude of the drifts occurring in a temporal interval

of one day. For comparison, one single measurement is carried out in one hour. Intervals with linear drift, nonlinear drift, and no drift are indicated. But to a first ap- proximation, the temporal drifts can be taken as linear

and can be explained by a small temperatui'e vai-iation of

321
I15r

= 0,05

examples © no drift (D linear dfilt (3) nonlinear drift mean of

13 values

18 h 24

0 mK

I -1

t-2

Z3as

-3

mean of

13 values

, slope 120^iK/h

12 timeh 24

Figure 2. Temporal drifts: (a) phase shifts between the signals of the x-ray and optical interferometers; (b) temperature variations of the analyzer crystal (those of the beam splitter are

similar).

the crystals. The linear part of the drift can be explained

by assuming a difference of only A T"^ - A =25 fxK/h

of a distance A - ZO mm between the measuring points

of the two interferometers.

Great efforts were made to reduce all avoidable temperature gradients near the crystals. As shown by

Fig. 2, the analyzer temperature varied by only 3 mK in 24 h. This was an essential requirement for the execution of the precision measurements of d. The uncertainty

of the absolute temperature of the analyzer crystal actually amounts to ±20 mK, a value which will be reduced

in the future.

3.3 Elastic Forces of the Translation Stage

Additional phase shifts proportional to the translation

S of the analyzer are caused by the forces within the

double parallel spring of the translation stage, see e.g., Ref. [5]. These forces vary proportionally to S. For

measuring such displacements and tilts, we used additional optical means of high sensitivity (0. 1 nm; 5 x 10"^

rad).

A special arrangement of the translation stage and the

magnetic compensation of guiding errors, already given

in Ref. [1], proved to be very effective. Within the limits given above we realized an almost perfect linear

motion of the analyzer crystal.

3.4 Cosine Error of the Translation Measurement

The translation S of the analyzer as measured by the

optical interferometer (Sl) and by the x-ray interferometer (iS^: ) is only of equal size if the directions of the lattice vector h, the translation vector S, the mirror normal n, and

the direction of the laser beam / are collinear. The ratio (Sl/Sx) is proportional (see e.g., Ref. [4], p. 131) to an

expression containing the deviation angles and their

differences in arguments of cosine functions. Because all these deviation angles were reduced to values of less than

10"^ rad (see Fig. 3), the quadratic expansion of the

cosine terms leads to a relative error A.S/S < 0.5 x 10"^

of the translation measurement.

Figure 3. Residual angular deviations of the translation vector S, the mirror normal n, and the laser beam I form the direction of the lattice vector h.

3.5 Abbe Error of the Translation Measurement

A so-called Abbe error [6] of the translation measure- ment is produced by a combination of two effects;

(i) a tilt of the analyzer during motion, caused by an

angular guiding error of the translation stage; and

(ii) an offset between the interferometric base lines of

length measurement.

The product of the tilt angle and the offset results in the error given in the upper part of Fig. 4. Errors of this kind are detected and eliminated by

phase measurements between the interferometer signals,

by varying the offset or tilt angle with respect to both

the axis perpendicular to S, the axis usually called -d, and the p axis. In this way, the offsets and H'' between the interferometric base lines were reduced to H% = (40 ± 20) pirn and H" = (240 ± 20) pirn; and the

guiding errors to the residual values Ad, Ap = 0 ±

5 X 10"^ rad. In the lower part of Fig. 4, the results of a new

method for a quantitative elimination of the Abbe error are given. Both parameters, the offset and the tilt angle, were varied independently. The guiding error was

changed by feeding an additional voltage to the piezoelement used for the fine adjustment of the analyzer

crystal. The voltage was continuously increased from

zero to a few volts in order to generate a tilt proportional

to S. The point of intersection of the lines drawn in Fig. 4 gives the value (n/m) = 1648.281 700 where the

Abbe error vanishes. In applying this method to the complete elimination of the Abbe error with respect to the p axis, a new error

not yet discussed in the literature has been observed.

The additional error occurs when, during translation the

lattice planes of the analyzer are tilted in relation to those of the beam splitter. This may be explained by a change of the intensity distribution within the moire pattern, causing a small phase change of the x-ray signal

322
lattice scale

Si- crystoi

optical scale

before

1648.2822

20

.1 , ^ = Jl

Sl 38 |im

9 rod(221 runs

slope -3.5»10 /'A7//7?;/^im

Sl 38 iim

1648.281700 -

b)

600 -400 -200 urn 0

loterol offset H''-*-

Figure 4. Abbe error of the translation measurement due to an

angular guiding error Ad/S/^ and an offset Hd: (a) sclieyne; (b) influence of different values of and AO/S/, on the measured ratio ( n/m).

and thus simulating an additional translation. The effect would not be expected in the case of a perfect crystal structure and an ideal x-ray detector.

By keeping the guiding error Ap^/Sx close to zero, disturbing phase shifts of the x-ray signal were avoided.

According to Fig. 5, the translation error is found to be

proportional to the product pA A . The unknown coeffi- cient C in the relation

AS /Sx = A{rUm)/in7m) = C Ap^/Sx (2)

has been evaluated by means of Fig. 5. The result is C =

18 km.

This value is of the same order of magnitude as the

minimum curvature radius R of the lattice planes, assuming the moire period A ^ 5 mm and using R = A^/8d. It

is not yet clear if a direct relation exists between C and

R.

The value (n/m) obtained for the vanishing Abbe error

(intersection point of the almost horizontal line with the

vertical axis of Fig. 5) does not differ appreciably from

the corresponding value (n/m) = 0 (point P in Fig. 4). A small cosine error in the data of Fig. 4 and 5 has not yet been eliminated.

_J L_ -25

1648.2820

^\ 18 - = (28 runs)

Sx 38 |im

1,5-

t^o v,^^ oo ^ A A ^

• io

fnjm, '4PA-13«10'Vad,-,n /\

c TO TM '30 runs) \.

Sk JD (im A A D

J 1 u -L I i .1 1 -15 -5 0 5

till angle o —

15 « 10 'rod 25

Figure 5. Ratio (nlm.) as a function of the tilt angle

between the lattice planes of the crystals. The guiding error Ap^/Sx is the parameter. [For the meaning of Sx, see Fig. U(a)].

A\/X = (\/27tWo)2 (3) We used a Lamb-dip stabilized HeNe laser (Spectra Physics Mod. 119) the frequency of which was controlled

from time to time by a ^^^l2-stabilized HeNe-reference

laser [9].

4. Results and Discussion

The numerical results for the corrections and their in- dividual uncertainties are given in Table 1. They are based on the above-mentioned series of 170 measure- ments of the ratio {nlm) which are given in Ref. [1]. Of all the corrections, the wavelength correction,

although small, is predominant. The Abbe error and the

absolute crystal temperature contribute most of all to the

uncertainties.

A total correction of -3.9 x 10"^ for the ratio {nlm) is obtained by adding up the individual corrections arith- metically. The contributions of the uncertainties were

regarded as independent quantities and summed in quadrature. The total uncertainty amounts to -6 x 10~^ on a

relative scale. The resulting value for the lattice spacing d(220) of our

silicon crystal at 22.50 °C in vacuum, given at the bottom

of Table 1, has been confirmed under measuring conditions where Abbe errors were almost ehminated.

The authors wish to thank Professor U. Bonse, University of Dortmund, and Professor K.-J. Hanszen for fruitful discussions and scientific assistance. The financial support of the Federal Minister of Economics is gratefully acknowledged.

3.6 Wavelength Correction

The laser beam has a vacuum wavelength Xq -

(632 991 415 ± 5) fm and a Gaussian beam profile. The

beam radius in the waist is Wo = (0.52 ± 0.02) mm. Due

to the non-planarity of the wavefronts [7], a small

Fresnel phase shift correction is needed [8]. Its magnitude is given by

References

[1] p. Becker and H. Siegert, these proceedings.

[2] U. Bonse and M. Hart, Z. Phys. 190, 455 (1966).

[3] H.-J. Rademacher et al. PTB-Bericht APh-13 (PhysikalischTechnische Bundesanstalt, Braunschweig, Julv 1980).

[4] K.-J. Hanszen et al., PTB-Bericht APh-14" (PhysikalischTechnische Bundesanstalt, Braunschweig, March 1981)

323
[5] Ch. Hoffrogge and H.-J. Rademacher, PTB-Mitteihingen 83, 79 (1973)

[6] H. Franke, Lexikon der Physik, Vol. 1 (Stuttgart, 1969), see "Abbe-sches Prinzip".

[7] H. Kogelnik and T. Li, Appl. Opt. 5, 1550 (1966).

[8] K. Dorenwendt and G. Bonsch, Metrologia 12, 57 (1976).

[9] J. Helmcke, PTB-Bericht Me-13 (Physikalisch-Technische

Bundesanstalt, Braunschweig, January 1977), p. 13.

Table 1. Numerical results of the corrections and uncertainties for the measured ratio (nim). Since the result of each measure- ment at an individual temperature (2242 °C . . . 22.50 °C) is reduced to a reference temperature (22.50 °C), no temperature

correction for the mean is required.

result of 170 measurements

nTffi =1648.281 626 (U5.8-10' 22.5° C; vQC.

values ' rel. correction uncertainty

(n/m) xlO^ xlO^

statistical

0 i 0.4 error of (n7m)

wavelength

correct! on -3.8 i 0.2

cosineerror

-0.1 i 0.2

Abbe0 < 1 3.0 error

temperoture 0 t 5.1

loser frequency 0 i 0.8

total correction

-3,9 i 6.0

of (n/m)

V

lattice plone ^ ^^•^"^/imnic ccn .nni-)\/

spacing ^ 220

" 92 015.560 ^ 0.012) //n

324
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

High Precision Studies of Pionic X Rays: Some Past Results and Future

Prospects*

G. Dugan/ L. Delker/' and C. 8. Wu

Columbia University, New York, NY 10027

and

D. C. Lu

Yale University, New Haven, CT 06520

The development of high intensity pionic x-ray sources has allowed a high precision crystal spec- trometer measurement of certain pionic x-ray energies to be made. Tliese energy measurements have

resolved the relativistic fine structure and have been used to provide an improved determination of the negative pion mass (accuracy 6.4 ppm). Substantial further increases in pionic x-ray source intensity

have been shown to be feasible at LAMPF. These increases will make possible the use of double flat crystals and can lead to advances in the study of electron screening in pionic atoms, further high preci- sion pion mass and pionic atom fine structure studies, and possibly also high precision muonic x-ray en- ergy measurements. Key words: bent crystal spectrometer; exotic atoms; pion mass; pionic x rays; relativistic fine structure.

1. Introduction

High resolution studies of pionic x rays using a bent

crystal spectrometer have in the past been limited by the

lack of an adequately intense source, which is required

because of the relatively low efficiency of crystal diffrac- tion instruments. However, the recent pioneering work

of Marushenko et al. [1] illustrated how such an intense source could be provided by placing the pionic x-ray tar- get directly next to the pion production target. We have

used a pionic x-ray source of this general design at the

Nevis synchrocyclotron in conjunction with a large-aper- ture bent crystal spectrometer of high intrinsic resolution. The high flux of pionic x rays and excellent crystal

resolution were exploited to make high accuracy meas- urements of the energies of the tt-TI (5g 4f) and (5f —

4d) transitions, and the tt-P (4f 3d) transition, relative

to the energy of a nuclear gamma ray transition in ^^Mo. Using the energy of the calibration ^Mo line measured at the NBS [3], the absolute energies of the pionic x rays were established, leading to a determination of the pionic mass with an accuracy of 6.4 ppm [4]. The energy differ- ence between the tt-TI (5g 4f) and (5f 4d) lines,

which the crystal spectrometer was able to completely

resolve, is due to the relativistic fine structure; the

measured energy difference was found to agree within

the error (—2%) with the predictions of the POeinGordon equation [5]. The details of these measurements

are discussed below.

In an experiment under development at the LAMPF

accelerator, we plan to utilize the 500 |jlA proton beam in

*Supported in part by the National Science Foundation and the U.S. Department of Energy.

"'Present add ress: Fermi National Accelerator Laboratory, P.O. Bo.\

500, Batavia, IL 60510. ttPresent address: Riverside Research Institute, 80 West End Avenue, New York, NY 10023.

an arrangement similar to that of the Nevis experiment,

but with substantially increased signal rates. Results ob- tained during 1980, based on the observation of the tt-AI (4f 3d) X ray at 30 keV, indicate that for the observation of a typical x-ray line, counting rates of 1000/min at a resolution of AE /E = 10"^ can be expected with the use of a bent crystal, or 10/min at A E /E = 10"^ with a double crystal set. With this performance, one may ex- pect that significant improvements can be made in the

precision of measurements of pionic x rays. The details of our preliminary results and some of the physics implications of such higher precision measurements are dis- cussed below.

2. Nevis Experiment

2.1 Experimental Setup

The experimental arrangement is shown in Fig. 1. Negative pions were produced by the interaction of the

560 MeV, 2 (xA extracted proton beam in a copper target

of total mass 44 gm/cm^. Some of these pions stopped in the x-ray target, which was placed adjacent to the pion

production target as shown in Fig. 1. The x-ray target was composed of slabs of titanium or phosphorous. The

atoms formed by the pions stopping in the x-ray target were the source of the pionic x rays observed in the

experiment.

These pionic x rays traveled along a 6.4 m long hole in the shielding wall (the "neutral beam"), passed through

a 0.91 m long tapered multislit collimator (which served

as a background suppression device) and were incident on

a quartz crystal deployed in the Cauchois configuration.

The crystal was cut to the (3, 1, 0) orientation and bent

to a radius of 3.51 m. By use of a new bending technique

[2], the intrinsic angular resolution achieved over an open aperture of 9 cm by 9 cm was 6.6 sec of arc (FWHM). The diffracted radiation from the crystal

325
passed through a narrow (170 fxm) slit at the image

plane, and was detected in one of a pair of intrinsic Ge

detectors. The crystal spectrometer lineshape was mapped out by scanning the slit (using a linear motion

stage of ±0.5 |xm accuracy) across the angular range of interest under the control of a computer, which also su- pervised the collection of the x-ray data from the Ge

detectors. The effective energy resolution of the crystal

spectrometer with this slit thickness was 17 eV (FWHM)

at 40 keV. The efficiency of the spectrometer in this ar- rangement was roughly 0.5 x 10"^. The spectrometer was enclosed in a cave, as shovra in Fig. 1, to reduce the

neutron background.

X-RAY TARGET

TO SPECTRO METER

PLUG

COPPER PION PRODUCTION TARGET

X-RAY TARGET

DETAIL OF TARGET

1 DIFFRACTED

'X- RAYS

.FOCAL

CIRCLE

Figure 1. Experhmntal layout of crystal spectrometer, colli- mator, and pion targets. Inset shows detail of pion production

and x-ray targets. (1) x-ray target; (2) tt^ production target;

(3) collimator; (k) bent crystal; (5) turnable o-ystal mount;

(6) thin slit; (7) slit table; (8) solid-state detector (Ge);

(9) detector shield; (10) Dewar; (11) spectrometer frame.

2.2 Data Collection

The experiment measured the difference in diffraction

angle between the pionic x rays and a source of nuclear

gamma rays (from the decay of ^^Mo). This angular differ- ence was converted into an energy difference using a

scale factor determined from measurements of the elec- tronic X rays Sm{Kai) and EuiKa-z). The pionic x-ray en- ergies were then determined from this energy difference

by referring to a measurement of the absolute energy of the ^^Mo nuclear gamma ray using a double flat crystal at the NBS [3]. Consequently, the data collection involved

measurements of electronic x rays and nuclear gamma

rays as well as pionic x rays. In each case, the effective

source geometry was arranged to be very similar to that

of the pionic x rays. The electronic x rays were produced by fluorescing

powder oxide targets; the scale factor was determined

from these hnes (-0.0890 eV/fjim at 40 keV) to an accu- racy far better than required. The nuclear gamma rays were generated in a 1.5 Ci liquid source. Figure 2, top,

shows the diffraction line from the nuclear gamma rays;

since the natural linewidth is much less than 17 eV, this

is essentially the spectrometer instrumental lineshape.

Because of the large spectrometer vertical acceptance,

the line profile shows a tail on the low-wavelength side. The nuclear gamma-ray peak position, which is indicative

of the spectrometer stability, was found to be constant to wdthin ±1.5 ixm. During the collection of the pionic x-ray data, the

lineshape was scanned typically 30 times during a 24-h

run, to average out any x-ray intensity variations due to small shifts in the proton beam position on the production

target. X-ray intensity variations due to proton beam in- tensity fluctuations were eliminated by determining the

data collection time at each slit position by the number of protons on the production target. Finally, to provide additional background reduction, data from the Ge detectors were suppressed except during the beam spill, which was reduced from the usual 5 ms to 20 |jls for this experiment.

900

800

700

600

500

400

300

200

1000 . 1200

I 100 1000

900

800

700

600

500

400

300

200

100

99Mo(y:40.583 keV)

/ 3

O

l7eV^

"7r-P(4f-3d)

jf

3

O

O

J L 7200 7400 7600 7800 8000

SLIT POSITION (^m)

Figure 2. Bent-crystal diffraction lines for the calibration y ray from ^^Mo (upper panel) and the (if ^ 3d) pionic x ray from TT-P (lower panel).

326
2.3 Data Analysis 2.4 Results

For each slit position, a pulse height spectrum from

the Ge detectors was obtained. Each of these spectra was

fitted to a Gaussian lineshape (FWHM of 1 keV at 40 keV) plus a linear background. This background was

subtracted and the area of the Gaussian for each slit posi- tion was extracted; this procedure removes the ambient

background appearing in the Ge detector spectrum. The

extracted areas for each slit position are plotted in Fig. 2, bottom, for a typical tt-P (4f-^ 3d) run, and in Fig. 3, for the ir-Ti (5g 4f) and (5f 4d) transitions.

The remaining background in these figures is due to a

diffracted soft-photon continuum. The signal-to-background ratio for the tt-P (4f 3d) data shown in Fig. 2, before the ambient background subtraction, was 2.2. The

peak signal counting rates per proton charge on target were 12/|jlA • min for the tt-P data and 3/|jlA • min for the

7T-Ti (5g 4f) data.

600

o 100

(J

7400 7800 8200 8600 9000

SLIT POSITION (MICRONS)

9400

Figure 3. Crystal spectrometer spectrum ( with Ge background

subtracted) of fine-structure components of the 5-U transition

in pionic titanium. Solid liyie is best fit to the data, with a ^ of

28.7 for 35 degrees offreedom.

The line center positions for the various data sets were

obtained by approximating the spectrometer instrumental lineshape by a Gaussian with an exponential tail, in- tegrated over the slit width. The lineshape is dominantly

instrumental for the pionic x-ray data as well as the nuclear gamma ray data. Since the pionic x-ray and nuclear

gamma-ray lines are quite close in energy, any effects due to the line shape approximation essentially cancel out when the difference in line center position is computed.

This line center difference is converted to an energy

difference using the scale factor obtained from the elec- tronic x-ray data. The energy differences between the

pionic x-ray and nuclear gamma ray lines must finally be

subjected to an —0.8 eV correction associated with finite vertical aperture effects. The final results for the energy

differences were: E i-n-Fiif ^ M))-E {Mo, y) = 4.84 ± 0.20 eV

^;(iT-Ti(5g 40)-E{Mo, y) = -118.4 ± 0.28 eV

£:(TT-Ti(5f ^ 4d))-£(TT-Ti(5g 4f)) = 87.6 ± 1.8 eV .

The errors include both statistical errors and systematic

errors associated with the vertical aperture correction.

The measured energy differences between the ir-P and

TT-Ti x-ray lines, and the -"'Mo gamma-ray line, were con- verted to absolute energies using the ^'^Mo gamma-ray

energy of 40583.468 ± 0.17 eV as measured at the NBS

[3]. The resulting absolute pionic x-ray energies for the

TT-P (4f 3d) and tt-TI (5g -> 4f) transitions were then

compared with the predictions of the Klein-Gordon equation, including corrections due to vacuum polarization,

strong interaction, and electron screening effects. The

pion mass is determined by requiring agreement between

the measured energies and the theoretical predictions.

Table 1 shows the results of these calculations and the

resulting pion mass determination. The error in the mass

includes both experimental and theoretical uncertainties.

The major source of error in the predictions arises from

uncertainties in the appropriate electron K and L shell populations at the time of the pionic transition, which af- fects the electron screening corrections. Estimates of these populations were established by a cascade calculation, including both radiative and non-radiative refilling

processes to the K, L, and M electronic shells. The uncertainties in the estimates are reflected in the uncertainties in the electron screening corrections shown in Table 1. The pion mass determination shown in Table 1 represents an improvement over the present world aver- age [6]: = 139566.9 ± 1.2 keV

but is in agreement with it. When combined with the re- cent improved measurement [7] of the momentum in pion decay, and with the assumption that = rn^-, it also allows a slight reduction in the upper limit on the muon neutrino mass to be made (m^, < 0.52 MeV at 90%

confidence level).

Table 1. Computation for pionic mass

(all energies in eV imless noted).

Transition Measured

energy

Corrections: vacuum

polarization

a(Z a) a\Z a)

Strong

interaction Electron

screening

Klein-Gordon energy

Pionic mass (keV)

Average pion mass (keV)

tt-P (4/- 40588.31

-99.3

- 0.7

0.1

3d)

0.26

-0.23 ± 0.06

0.98 ± 0.20

TT-Ti (5^ -

40465.07

-82.6

- 0.6

0.2

0.0

>4/)

0.33

3.99 ± 0.20

40489.16 ± 0.33 40386.06 ± 0.38

139567.8 1.1 139567.1

139567.5 ±0.9

1.3

A comparison of the measured energy splitting

between the tt-TI (5f -» 4d) and tt-TI (5g 4f) transitions

with the prediction of the Klein-Gordon equation may

also be made. (See also ref [8].) This splitting is due primarily to the relativistic fine structure term in the

Klein-Gordon equation. There are significant corrections

327
due to vacuum polarization, electron screening, and the

strong interaction; a summary of the relevant numbers

contributing to the transition energy difference is presented in Table 2, together with the measured result. Although the agreement between the calculations and the

experimental measurement does verify the prediction of the Klein-Gordon equation for this spHtting, it does not

distinguish between that equation and the Schroedinger

equation with relativistic corrections. This is because the

prediction of the Klein-Gordon equation for the relativistic fine-structure splitting differs from that which one would calculate from relativistic corrections to the

Schroedinger equation by considerably less than the ex- perimental error.

Table 2. Pionic titanium fine structure splittiyig E(5f-^W - EfSg-^if), in eV.

Klein-Gordon splitting 59.3 Vacuum polarization:

first order 25.6 higher order 0.2 Strong interaction 3.3 ± 1.1 Electron screening -1.1 ± 0.1 Total predicted splitting 87.3 ± 1.1 Measured splitting 87.6 ± 1.8

3. LAMPF (Los Alamos Meson Physics

Facility) Experiment

3.1 Experimental Setup

At the LAMPF accelerator, a beam of 800 MeV protons

with a current of about 0.5 mA offers the opportunity for a several order of magnitude increase in the effective

pionic x-ray source intensity compared to that available

at Nevis. In order to investigate the feasibility of an ex- periment at LAMPF similar to that of our Nevis experiment, we have taken preliminary data in a relatively

crude setup. Although these data do not constitute high

precision energy measurements, they do indicate what

future, more precise experiments can expect as regards

signal-to-noise ratios and signal counting rates, which reflects directly upon what instruments can be used and

the kinds of physics questions which can be addressed.

The preliminary investigations used a general setup

similar to that shown in Fig. 1. The pion production tar- get was a block of Mo, of total mass 20 gm/cm^; the effec- tive pionic x-ray target was an aluminum housing around

the Mo block. The block was viewed at a 135° angle from

the proton beam direction, through a 15 m long hole in the shielding wall. Because of initial uncertainties re- garding the background, a 1.5 m long collimator with a

2.54 cm diameter hole in the center was located in the

shielding wall aperture near the crystal; this severely

limited the crystal viewing aperture. The crystal and Ge

detectors were enclosed in a cave for neutron background

shielding. To map out the pionic x-ray line, a 170 |xm slit was scanned across the crystal image plane at the ap- propriate Bragg angle, as in the Nevis experiment.

3.2 Results

Figure 4 shows the diffraction line obtained with the

setup described above. No ambient background subtraction has been made. The line width of 14 eV is dominantly

instrumental. No precise energy calibration was per- formed. The signal-to-noise ratio of 2 and the peak counting rate of 12 counts/min are quite good for this crude

preliminary setup. Extrapolating from this signal rate to that obtainable in a future bent crystal experiment with

a larger collimator aperture, a thicker pion production

target, and a more favorable x-ray target geometry, one

can realistically expect signal rates of roughly 1000 counts/

min. Because of various Hmitations present in the preliminary setup, we expect the background to scale up by con- siderably less than the signal, providing an extrapolated

signal-to-noise ratio of 5 or better.

600 -

500

400

300

200

100

Al {4f-3d)

Slit width. 170 ;jm

9/10/80

17000 17500 18000 18500

Figure 4. Bent crystal diffraction line for the tv-AI (Uf^ 3d) x-ray transition obtained at LAMPF.

3.3 Future Outlook

The development of sufficiently intense sources of pionic X rays to allow the use of crystal diffraction spec- trometers has now progressed to the point where, with

counting rates as high as mentioned above, even very

weak pionic x-ray lines can be studied with bent crystals.

Additionally, the application to pionic x-ray studies of lower efficiency, higher resolution crystal diffraction in- struments (such as the double flat crystal spectrometer)

328
appears quite feasible. With such intense sources of

pionic X rays to be expected, one may also anticipate that

the muons from the decay of these pions in flight could

also provide a less intense but still useful source of

muonic x rays which could be studied with crystal dif- fraction instruments.

With the resolution available in a double crystal pair, one may contemplate a direct measurement of the pionic x-ray line splitting due to missing K-electrons in pionic

atoms, providing an experimental determination of the

electron screening corrections and allowing the precision

of pion mass measurements to be improved to about

1 ppm. Higher precision studies of the relativistic fine structure splitting can distinguish the Klein-Gordon

equation from the Schroedinger equation with relativistic corrections. High precision absolute energy measurements, together with an accurate knowledge of the pion mass, can be used to place limits on the polarizability of

the pion. Finally, if muonic x rays are observable in sufficient intensity, a high precision test of vacuum polarization effects in strong fields (high-Z muonic atoms) can be

carried out. References

[1] V. I. Marushenko et al, Pisma Zh. Eksp. Teor. Fiz. 2.3, 80

(1976) [JETP Lett. 23, 72 (1976)].

[2] D. C. Lu, to be published.

[3] E. G. Kessler, R. D. Deslattes, and W. Schwitz, private communication (1980).

[4] D. C. Lu et al, Phys. Rev. Lett. 45, 1066 (1980).

[5] L. Delker et al., Phys. Rev. Lett. 42, 89 (1979).

[6] R. L. Kelley et al. (Particle Data Group), Rev. Mod. Phys.

52, 51 (1980).

[7] M. Daum et al., Phys. Rev. D 20, 2692 (1979).

[8] K. C. Wang et al., Phys. Rev. A 22, 1072 (1980).

329

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Isotope Shifts of K X-Rays of Lead

G. L Borchert, O. W. B. Schult, and J. Speth

Institut flir Kernphysik, Kernforschungsanlage Julich, D-5170 Jlilich, West Germany

and

P. G. Hansen

Institute of Physics, University of Aarlius, DK-8000 Aarhus, Denmark

and

B. Jonson and H. Ravn

CERN-ISOLDE, CH-1211 Geneva, Switzerland

and

J. B. McGrory*

Oak Ridge National Laboratory, Oak Ridge, TN 37830, U.S.A.

During the last years progress in nuclear theory has allowed more detailed predictions for ground

state properties of heavier elements especially in the vicinity of doubly magic nuclei. Experimentally

the change of the mean square charge radius can be determined by a high resolution measurement of the K x-ray isotope shifts. Therefore we performed a study of the isotope shifts of all stable lead iso- topes and compared them to very recent microscopic calculations. Key words: high resolution crystal spectrometer; isotope shifts of stable lead isotopes; precision meas- urements of x-ray energies.

1 . General Considerations

The energy of atomic electrons is determined by the

Coulomb interaction with the central charge of the nucleus and with all neighboring electrons. Therefore a change

of the nuclear charge distribution will be reflected in a change of the electron binding energies. As the wave function of the Is electrons has the largest overlap with the

nuclear volume the K x-ray energies show the most direct

influence.

Therefore the shift of the x-ray energy when the neutron number changes, the isotope shift, yields information about the change of the mean square charge radius

8 <r^> of the nucleus [1, 2, 3]. Of course, there are other nuclear and electronic ef- fects that are knovm to shift the energies of K x-rays. Generally they are related to the mechanism used to create the K hole vacancy and to the chemical environ- ment. For photoionization unresolved satellites due to multihole excitation shift the centroid of the K x-ray line

[4, 5]. The electronic structure of an electron capture decaying element can partly survive during the lifetime of the

K hole. Thus the change of the electronic Coulomb in- teraction (screening) causes an x-ray shift [6]. Statistically a similar effect arises from the chemical bond [7].

*Research sponsored by the Division of Basic Energy Sciences, U.S. Department of Energy, under contract W-7405-eng-26 with the Union Carbide Corporation.

The interaction of the nuclear magnetic moment viath the electron spin of the K electrons splits the Is level. For a statistical population, i.e., photoionization, only a broadening of the line vddth results [8], whereas in the case of allowed electron capture decay [9] or conversion

decay [10], a nonstatistical population can occur that

shifts the x-ray energy.

To study the isotope shift which is often smaller than

these effects it is essential to aim for proper experimental conditions.

2. The Isotope Shift

Usually the isotope shift is expressed as a sum of three

contributions:

8^;^' = 8^(Coul)^' + 8£;(Mass)-'^' + 8£;(Nuclpol)^'. (1)

The last term describes the contribution of the nuclear

polarization effects. It has been estimated to be negligibly small. The second term gives the contribution of the normal

and specific mass shift [11]. To a good approximation it can be expressed as

8 E (Mass)^^ ~ - 2/3

^~ , (2) 1836 A

* where £ x is the energy of the x-ray transition in keV. In the case of lead for A - A' = 1 it is only 0.7 meV.

331
The first term describes the effect of the change of the

volume of the nuclear charge distribution. Seltzer [3] found a rapidly converging expansion that connects it to the changes of nuclear moments:

hE'iCoul) = XCn 8<r2"> (3)

n 6<y2»t> = J ^p{r)r^"dv/ f p(r)dv. (4)

It is common to write this expression as

8£;'(Coul) = C\ X TTT ^<r^"> = C[\, (5) n ^1

with the nuclear parameter \. Neglecting the very small

Coulomb shift of the 2p3/2 level, the Kai isotope shift in Pb can be expressed numerically as hEiKai) - 8£;(Mass) = meV

fm2

1880 5 <r2> 1 - 1. 12 X 10-3 + . . . (6)

By means of the simple nuclear model of a homogene

ously charged sphere with constant density the magni

tude of the nuclear moments can be approximated by

2n „ 2n A - A' h<r^"> ~

2n + 3 ^0 A

(7)

with Ro = 1.2A^ fm

As the ratio €,}/€[ is largely independent of the type

of electronic transition i, the same quantity X. can be

determined in both x-ray and optical measurements.

As it is rather difficult to calculate to high precision for

optical transitions [13] the C\ coefficient and the mass

shift, these parameters can be obtained from comparison

with x-ray results [14]. Because of the problems in the analytical understanding of muonic x-ray data their comparison to electronic

x-ray data seems to be too early [15, 16, 17]. The information deduced from elastic electron scatter- ing experiments generally depends on the model used for the analysis [18]. Recently a new method was developed

[19] which yields model-independent results for the nuclear parameter X..

3. The Experiment

The simple estimate Eq. (7) shows that the shift of K

x-ray lines for Pb due to the isotope effect is on the order

of 10"^ of the transition energy or 10"^ of the natural line width. Therefore only a high resolution device like a

crystal spectrometer [20, 21] is suited to detect such

small energy differences. Even with such an instrument

it is very difficult to perform absolute energy measure- ments with an accuracy at a level of about 10"'^ [22]. For the study of the isotope shifts in Pb we used our DuMond-type bent crystal spectrometer. It is described

elsewhere [23] and permits us to measure independently

at essentially the same time (see Fig. 1) for each angular

setting the intensity of two sources placed very close to each other at the focal point. In this way the difference

of the x-ray energies is determined largely undisturbed

by the uncertainty of the angular setting and reproducibility. In measuring the x-ray reflections at positive and

negative Bragg angles an uncertainty due to a possible

relative displacement of the sources in the direction of dispersion can be eliminated. An effective shift because

of the vertical aberation can be corrected by measuring

the two sources at interchanged positions and averaging

the results.

A

source 1

source 2

Figure 1. Schematic view of the source arrangement of the crystal spectrometer. The inactive metal foils, source 1 and

source 2, are mounted on aluminum blocks which can he ad- justed separately in the direction of dispersion. The radioactive Yb source to excite the fluorescence is located behind a uranium plate with a slit to let pass the fluorescence radiation

of the metal foils. Another movable uranium plate with two vnndows acts as a shutter, which allows the radiation of only one source at a time to reach the detector.

As sources we used isotopically enriched metallic foils che composition of which is given in Table 1. The x-rays were induced through photoionization by means of a 50 Ci ^^^Yb source placed very close to the Pb samples.

Their dimensions, 4 x 2 x 0.12 mm^, were chosen such

that after proper adjustment the line width of the reflec- tions was essentially the natural one: F ~ l.SFnat [24]. A

typical pair of reflections is shown in Fig. 2. In this way the influence of all the competing effects mentioned above could be eliminated. The experimental

results indicate that we measured the pure isotope shifts,

the accuracy of which was essentially limited only by

counting statistics. We measured each combination of the lead isotopes

^o^Pb, 206pb^ 207pb and '^^^Fh between 20 and 40 times. We

used four different mathematical approaches to analyze

the data:

1. A fit of the line shape and position.

2. An estimate of the line position from the center of gravity [25].

332
Table 1. The composition of the sources in %

Source

Isotope

208pb 206pb 204pb

208pb 96.6 2.1 1.3 207pb 5.5 92.4 2.2

206p]3 2.9 6.7 90.4 204pb 10.0 6.4 12.7 70.9

Counts X lO'Amin

3-

, 2- 9

9 8

9

s —1 1 //

// '

5 /a

Figure 2. A scan across the second order reflections of the Ka-, transition in the experiment '*Pb - Pb. The contents of counter 1 measuririg the intensity of the ^'^^Ph source is denoted by circles, the contents of counter 2 yneasurinq the in- tensity of the Pb source is denoted by crosses. The relative geometrical displacement of the two sources is about 1 \xm es- timated from a mean distance of the line centers of about iO

milli-arcsec.

3. An estimate of the shift from the displacement of the four corresponding lines.

4. Determination of the shift from the difference of the area of the overlapping lines. As a typical example the fit results from the pair

204p|j - -°'Pb are shown in Fig. 3. The results of the in- dividual methods were found to agree within the uncertainties. As the final result the weighted mean was taken

of the results of the four methods. The uncertainty is es- timated as the average of the individual errors. In this

>

£ 500

300

-100

Scan

Figure 3. The energy shift of the Ka^ transition in the experi- ment ''"''Pb - ^"^Pb. Displayed are the individual x-ray shifts from 23 scans as given by a fit of the line shape and position

mth the model of a Lorentz curve folded with a rectangular

distribution which cort'esponds to the effective geometrical source width. The full circle and the horizontal liyies denote

the weighted average and its uncertainty, respectively.

way for the six possible combinations of the four lead iso- topes the measured isotope shifts were obtained that are given in the third column of Table 2. All quoted uncertainties are standard deviations. This overdetermined

system was submitted to a level fit procedure taking into account the isotopical composition of the individual sources according to Table 1. The resulting best values

for the measured shifts are listed in the fourth column of

Table 2. From this system the Coulomb shifts for pure

isotopes were computed. The results are shown in column

5 of Table 2. In column 6 the results of Ref. [26] are shown for comparison. The resulting nuclear parameter

according to Eq. (6) is listed in column 4 of Table 3. In the third column we give the results of our theoretical

calculation. For comparison in column 5 the x-ray results are shown from Ref. [26] and in column 6 very recent preliminary results from elastic electron scattering ex- periments [27]. The latter were found using the Bessel

Fourier analysis according to Ref. [18] and they depend,

therefore, on the applied model parameters.

4. Discussion

The isotope shifts determined in this experiment as given in Table 2, column 4, show a significant improveTable 2. Energy (Ka,, 1)—Energy (Ka,, 2) in meV

Isotope 1 Isotope 2 Experim. Level Fit Coulomb Shift Ref [26]

204 208 324 (16) 325 (13) 414 (17) 206 208 200 (16) 209 (13) 228 (14) 186 (18) 207 208 153 (20) 138 (12) 150 (14) 136 (25) 204 207 186 (15) 187 (12) 264 (17) 206 207 81 (15) 71 (12) 78 (14) 50 (20) 204 206 135 (50) 116 (15) 186 (20) 200 (38)

333
Table 3. The nuclear parameter \ in fm '

Isotope 1 Isotope 2 \ (theor) \ (x-ray) X (Ref. [26]) X. (electr. sc.)

208 207 0.061 0.080 (7) 0.072 (13) 0.0654 (20)

208 206 0.116 0.121 (7) 0.099 (10) 0.1283 (39) 208 204 0.226 0.220 (9) 0.205 (22) 0.2363 (71)

ment in accuracy as compared to the previous results

[26]. The agreement is rather satisfactory.

When comparing the corresponding values of the nuclear parameters to those of the theoretical calculation, one has to keep in mind that the latter considers only the mean square radius. But it has been estimated that

higher order moments should contribute only 7% so that

the comparison is justified. Nevertheless the agreement

is surprisingly good. As for the elastic electron scatter- ing data one has to keep in mind that they still depend on

the special choice of model parameters.

As a common feature of all the data, the nuclear

parameter for the ^ospjj _ 206p^j p^jj, jg jggg \]^2Ln twice

that of the pair ^ospij _ 207p]^ T'h.i'i can be understood in terms of a state-dependent effective neutron charge [28], as there are contributions from fs^, P3/2, iqa, ^^12, and i 13/2 two neutron hole states to the ground state of '^'^^Yh. They all contribute with an effective neutron charge that

has been estimated to be smaller than that of the 2pi^^

state. For the pair ^o^Pb - ^"^Pb there is very good agree- ment between all experimental results and the theoretical one. References

[1] F. Boehm and P. L. Lee, At. Data Nucl. Data Tables 14, 605

(1974).

[2] 0. I. Sumbaev and A. F. Mezentsev, Sov. Phys. JETP 22, 323 (1966); 0. I. Sumbaev, A. F. Mezentsev, V. I. Marushenko, A. S. Ryl'nikov, and G. A. Ivanov, Sov. J. Nucl. Phys. 9, 529 (1969).

[3] E. Seltzer, Phys. Rev. 188, 1916 (1969).

[4] G. L. Borchert, P. G. Hansen, B. Jonson, I. Lindgren, H. L. Ravn, 0. W. B. Schult, P. Tidemand-Petersson, and the

Isolde Collaboration, Phys. Lett. 65A, 297 (1978).

[5] T. A. Carlson, C. W. Nestor, Jr., T. C. Tucker, and F. B. Malik, Phys. Rev. 169, 27 (1968).

[6] G. L. Borchert, P. G. Hansen, B. Jonson, I. Lindgren, H. L. Ravn, 0. W. B. Schult, P. Tidemand-Petersson, and the

Isolde Collaboration, Phys. Lett. 66A, 374 (1978).

[7] 0. I. Sumbaev, in Modern Physics and Chemistry, Vol. I (Ac- ademic Press, London, 1976), p. 31; 0. I. Sumbaev, Sov. Phys.

Usp. 21, 141 (1978).

[8] A. S. Ryl'nikov, A. I. Egorov, G. A. Ivanov, V. I. Marushenko, A. F. Mezentsev, A. I. Smirnov, 0. I. Sumbaev, and V. V. Fedorov, Sov. Phys. JETP 36, 27 (1973).

[9] G. L. Borchert, P. G. Hansen, B. Jonson, H. L. Ravn, 0. W.

B. Schult, P. Tidemand-Petersson, and the Isolde Colla- boration, Phys. Lett. 63A, 15 (1977).

[10] A. I. Egorov, A. A. Rodionov, A. S. Ryl'nikov, A. E. Sovestnov, 0. I. Sumbaev, and V. A. Shaburov, JETP

Lett. 27, 483 (1978).

[11] R. B. Chessler and F. Boehm, Phys. Rev. 166, 1206 (1968).

[12] S. K. Bhattacherjee, F. Boehm, and P. L. Lee, Phys. Rev.

188, 1919 (1969).

[13] A. Steudel, Z. Phys. 133, 438 (1952).

[14] P. Brix, private communication.

[15] R. J. Powers, P. Barreau, B. Bihoreau, J. Miller, J. Morgenstern, J. Picard, and L. Roussel, Nucl. Phys. A 316, 295

(1979).

[16] G. A. Rinker and J. Speth, Nucl. Phys. A 306, 360 (1978).

[17] Y. Yamazaki, H. D. Wohlfahrt, E. B. Shera, M. V. Hoehn,

and R. M. Steffen, Phys. Rev. Lett. 42, 1470 (1979).

[18] H. Euteneuer, J. Friedrich, and N. Voegler, Nucl. Phys. A

298, 452 (1978).

[19] L. Cardman, P. Leconte, D. G. Ravenhall, and S. E. Willi- amsson, Proc. Int. Conf. Nucl. Phys. with Electromagnetic Interactions, Mainz, 1979.

[20] J. W. M. DuMond, Rev. Sci. Instrum. 18, 626 (1947).

[21] Y. Cauchois, J. Phys. Radium 3, 320 (1932).

[22] G. L. Borchert, W. Scheck, and K. P. Wieder, Z. Naturforsch Teil A: 31, 274 (1965).

[23] G. L. Borchert, P. G. Hansen, B. Jonson, H. L. Ravn, 0. W.

B. Schult, and P. Tidemand-Petersson, Nucl. Instrum. and

Methods 178, 209 (1980).

[24] G. C. Nelson, W. John, and B. G. Saunders, Phys. Rev. 187,

1 (1969); and Phys. Rev. A 2, 542 (1970).

[25] P. G. Hansen, Nucl. Instrum. and Methods 154, 321 (1978).

[26] P. L. Lee and F. Boehm, Phys. Rev. C 8, 819 (1973).

[27] G. Fricke, H. Miska, and D. Rychei, private communication.

[28] J. Speth, L. Zamick, and P. Ring, Nucl. Phys. A 232, 1 (1974).

334
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

The Measurement of Atomic Masses by Mass Spectroscopic Methods and a

Role for Atomic Masses in the Determination of the Fundamental Constants

Walter H. Johnson

University of l\/linnesota, Scliool of Physics and Astronomy, 116 Cliurch St. S.E., Minneapolis, IVIN 55455

A description of atomic mass determinations is given with tiie emphasis on mass spectroscopic

methods. A review of current techniques is made which includes conventional mass spectrometers and

radio frequency instruments. Precision attained in these measurements is discussed together with the

possibilities of improvements in precision. Finally, the use of atomic mass measurements as input data

for the determination of fundamental constants is considered. Key words: atomic masses; fundamental constants; mass spectrometer; rf mass spectrometer.

1. Introduction

One of the fundamental characteristics of an atom is its mass and as such the atomic mass has been a quantity

subject to measurement for some time. I intend to re- view some of the important considerations dealing with

the experimental determination of atomic masses, to dis- cuss examples of current instrumentation for this purpose, to give a selection of recent results, and finally to comment on the possibility of experiments in which precise atomic mass information may contribute to the

determination of fundamental constants.

2. Systematics of Atoms

The various atomic species may be classified by two

parameters. A'' the number of neutrons and Z the number

of protons in the nucleus or the number of electrons sur- rounding the positively charged nucleus of the neutral atom. Stable or near-stable atoms are represented in a

distribution of A'^ and Z values which range from 0 ^ A/' ^

-160 and 1 ^ Z ^ 92. There are about 280 different

stable or near-stable atomic species and perhaps five or

six times as many unstable atoms which last long enough

to be studied with present techniques.

Determination of the atomic masses of the stable

atomic species has been the main emphasis of mass meas- urement experiments. These stable atom results supplemented by decay energy' information from unstable atoms

allows the construction of atomic mass tables which list the atomic mass m {ZN ) for most experimentally accessible combinations of N and Z. These masses are specified

as the masses of neutral atoms in their atomic and nuclear ground state. Using these tables, one can calculate the binding energy of atoms with Eq. (1) BE = [Zm^ + Zmp + A^mn - m{ZN)]c^ (1)

in which me, nip, and are the masses of the electron,

proton, and neutron, respectively. The binding energy of an atom may be divided into two parts, one involving the

binding of the electrons about the nucleus and a much

larger part involving the binding of the protons and neutrons in the nucleus. Because most of the binding energy

results from the nuclear interactions, atomic masses have

been extensively employed for studies of nuclear stability. In these studies, corrections sometimes were made

to remove the effect of electron binding. In light atoms

the corrections were very small, but in the heavy atoms

the electronic binding energy can approach 0.1% of the

total binding energy.

The masses employed in Eq. (1) are absolute masses,

i.e., in kilograms as the equation is written. Experience

has shown, however, that the use of a relative scale of atomic masses results in information of considerably

higher accuracy and utility. From early in the history of the measurement of atomic masses, a relative scale was

employed which sets the mass of a particular atom and

then makes all other measurements relative to that atom. The current standard mass is the mass of the atom ^'C in

its nuclear and atomic ground state which is assumed to be exactly 12 u. The conversion of atomic mass from rela- tive to absolute units is given in Eq. (2).

mkg(ZA^) = 7n^{ZN)/NA = mud. 660 5655 ± 85 x 10 "2" kg) (2) where NA.is Avogadro's number [1]. Another useful conversion is to the electron-volt en- ergy unit. This conversion is given in Eq. (3) in which F

is the Faraday [1]. lu = ^ = 931.5016 ± 26 MeV . (3)

3. Measurement of Atomic Masses

Although this review will concentrate on the measure- ment of atomic masses by mass spectroscopic techniques,

it is worthwhile at the onset to indicate that there is a second general technique for the determination of atomic masses. This technique employs nuclear reactions in which high velocity nuclei are caused to interact with

other nuclei. Measurements of the energies and momenta

of the colliding nuclei before and after the collision can be

employed to determine the mass difference between the

incident and the outgoing particles. Atomic mass differ- ence measurements may therefore be accomplished with

this technique. Discussion of this, however, is not within

the scope of this review.

Mass spectroscopic determinations of the atomic mass

of an atom employs a mechanical process in which a force

is applied to the atom and the resultant change in path of the atom is dependent on its mass value. Because it is convenient to apply these forces by electric and magnetic

fields, the atom must be converted to an ion and one

335
must bear in mind that the mass measured is now an ion mass rather than the neutral atom. An ion of charge q and mass m moving with a velocity

V perpendicular to a magnetic field B will move in a cir- cular path of radius R such that

R = mv

Bq

(3)

If the velocity of the ;on is achieved by an acceleration

through electric potential difference V, then

V m

Substituting this value into Eq. (4) yields

B^ qm

(4)

(5)

This equation is the basic design equation for magnetic mass spectrometers. Another useful relationship is the

cyclotron period Tc, of a ion of mass-to-charge ratio yn/q

in a uniform magnetic field B. r_ 2tt R _ 217 V B

m (6)

This quantity as was the case for R'^ m Eq. (5) is directly proportional to the mass-to-charge ratio of the

ion. Equation (5) is the basis for the design of a typical sector field mass spectrometer shown schematically in Fig. 1. Ions are formed in an ion source by collision with

a fast beam of electrons. The ions are then accelerated

through a potential difference V and into a sector magnetic field. Here ions of differing m Iq will have different

radii; shown here are two different m /q values. The two

groups of ions emerge on two different paths with the

beam of lighter mass ions being the inner path. The di- agram also indicates the focusing ability of a sector field. Divergent ions emerging from the ion source are re- focused by the sector field at the collector slit. By varying 5 or V a series of ion beams are sequentially allowed to have the correct radius to pass through

the collector slit and be measured as a collector current

or counted as individually arriving ions. A mass spectrum

is produced which is schematically shown in Fig. 2. The

location of an ion group in this spectrum is related to its m /q, and the spacing between ion groups is related to the mass difference between the ions. Small fractional mass differences such as m\ or m2 in Fig. 2 and known

as mass doublets have been the basis for mass determinations since the early history of these measurements.

In order to make accurate mass difference measure- ments a number of conditions are required.

(a) The ion beam shapes should be identical.

(b) The resolution of the instrument should be as high

as possible to produce sharply defined location.

(c) The transformation from spacing to mass difference

should be well known. This requires that the dispersion

of the sector field can be determined accurately.

A great deal of effort has gone into developing ion- optical designs of mass spectroscopes to produce high

resolution. The aberration of an optical system is usually

expressed as a power series in the angular divergence of the ion beam emerging from the ion source and the velocity spread |3 such that the velocity of the emerging ion

V = Vq{\ + p) where fo is the mean velocity. Defining y

as the image aberration, one can express y as a power

series in a and 3 as indicated in Eq. (7):

y ^ aJBia+B2^ + Bua.- + Bi20L^ + B22^^ + . . .), (7)

ELECTRONCOLLECTOR

-ANALYZER EXIT

SLIT PLATE

ION COLLECTOR PLATE

Figure 1. A schematic of a simple mass spectrometer.

Figure 2. A mass spectrum showing hydrocarbon ion peaks

and eleyyient ion peaks. Doublets mj and measure the mass

difference betiveen the knomi hydrocarbon ion peak and the unknoivn element peak. The doublet H is a calibrating doublet for which the mass difference is one hydrogen mass.

where a m is the radius of curvature of the ion in the magnetic field. The design goal has been to arrange combinations of electric and magnetic fields in the proper way to cause as many of the B coefficients to be zero as possible.

Causing the ion-optical system to have smaller aberrations not only will produce high resolution but also will cause the ion beam shapes to be less and less dependent

on conditions in the ion source and thus more and more

similar. One of the limits to the accuracy of early mass differ- ence measurements was caused by inaccuracy in knowledge of the dispersion characteristics of the ion- optical system. Mass spectra were detected using photographic plates located in the focal plane of the instrument.

336
Because ion beams of differing m/q had different paths

and focused at different points along the plate, lack of homogeneity of the sector magnetic field and variations of the fringe field perturbed the ability to calculate dispersion accurately. An excellent review of the early history

of atomic mass measurement has been w^ritten by Mattauch [2]. Let us next consider some specific examples of mass

spectroscopic devices. I wish to start with an instrument

design that was developed in the late 1940s by A. 0. C. Nier and E. G. Johnson [3] at the University of Minnesota. The design of this instrument marked the emergence of electrical detection in mass measurement. Previous to that time, photographic plate detection was generally employed and the devices were known as mass

spectrographs. This instrument was the first mass meas- urement instrument that properly could be called a mass

spectrometer. The instrument was designed as a symmetrical 90° electrostatic analyzer followed by a 60° asymmetrical magnetic analyzer. The analyzer radii and magnetic sector asymmetry were chosen to produce a device

in which the Bx, B-i, and Bn coefficients of Eq. (7) were

zero for ions focused at the collector slit. A schematic

diagram of the configuration of fields is shown in Fig. 3. Mass difference measurements in this instrument were

based on an ion optical theorem by W. Bleakney [4] or an

earlier version by W. F. G. Swann [5]. Given that an ion

of mass M pursues a certain trajectory in an arbitrary

configuration of electric and magnetic fields; then if all the electric fields are changed by a factor A, a charged

particle of the same charge having a mass M/A will pursue exactly the same trajectory as did the particle of mass M before the change in fields took place.

The various electric fields in the ion source and the

electric field in the electrostatic analyzer are 'the two

sets of electric fields which will influence the passage of an ion through the instrument. Because of the velocity

focusing property B2 = 0, the changes in the ion source

electric fields are not influential in changing the location

of the ion at the collector slit. Changes in the electrostatic analyzer are, however, crucial. The dispersion re- lationship may be derived from Bleakney' s theorem by

assuming two ions of mass mi and m2 in the same charge

state and associated electrostatic analyzer potentials Vi

and Vim2 - rrii mi

V\ ^ ^2 Am or m

^v

V

(8)

The measurement of a mass difference A m is then measured in terms of an electrostatic potential ratio. The

potential ratio ^.V /V can be determined by means of a

resistance divider shown in Fig. 4. With R R ' , one

can show that

Am

m

AV

V

AR

R

(9) The alternation of the group of ions at the collector slit is accomplished by switching the relay shown in Fig. 4 from

position A to position B. Changes in the ion source

potentials are made to follow approximately the A.V/V of the electrostatic analyzer. Also the ion beam is swept

across the collector slit by a small saw-tooth perturbing

magnetic field. These wave forms are shown in Fig. 5. Timing of ion currents through the collector slit is shown

in the line labeled electrometer. If this signal is viewed

on an oscilloscope which is synchronized to begin its sweep at the beginning of the magnet sweep, first one

and then the other ion current is recorded. Similarity of

orbit is achieved when the two ion beam waveforms are

Figure 3. A schematic of a Nier-Johnson geometry double- focusing mass spectrometer.

•0+

TO i ANALYZER

Figure 4. The circuit employed to geyierate the electrostatic analyzer voltage wave forms.

MAGNET

ELECTROMETER

OSCILLOSCOPE

Figure 5. Wave fonus employed in peak ynatching.

coincident. This is achieved by adjustment of AR. When

the two ion beams shapes are coincident, the required

value of A/2 can be used to calculate A m. This technique,

known as peak matching, was first used by Lincoln Smith

[6] and adapted by Giese and CoUins [7] to the Minnesota

instruments.

Peak matching offers a precise method for locating the

matched condition between the two ion beam waveforms.

Smith [8] has claimed that matching accuracy can be as high as 1/2500 of the ion beam width. Typical photographic plate accuracies were usually no better than 1 /50

of the ion beam width [9]. Thus, a very large increase in the accuracy of mass measurement was achieved.

A number of instruments have been constructed at Minnesota using this design. The largest had a magnetic

radius of 40 cm and an ion path of about 4 meters. Full width at half maximum resolutions as high as 1 in 200,000 were achieved with this instrument. The instrument has

been employed for mass measurements for more than 20

years, and in that time measurements of most stable iso- topes have been made.

In 1959, Hintenberger and Konig [10] published a series of ion-optical designs for mass spectrometers

337
which were fully second order focusing, that is, all sec- ond order B coefficients were zero. A number of instruments have been constructed according to these designs.

The instrument at the University of Manitoba described

by Barber et al. [11] will be used as the example of this generation of instrument. The geometry of this instrument is given in Fig. 6. A 94.65° electrostatic analyzer

with a mean radius of 100 cm is combined with a 90° magnetic analyzer of mean radius 62.74 cm. A non-normal

entry and exit from the magnetic field is a significant

feature of the design which through Z focusing produces

a high transmission. The total path length of the ion

beam in this design is 4.6 meters. With a source slit of width 2.2 |xm, the expected base resolution is 1 part in 200,000.

Figure 6. Configuration of the University of Manitoba double- focusing mass spectrometer.

Figure 7 is a schematic diagram of the instrument

showing the vacuum system, ion source, and collector details. The various defining slits are individually adjustable in location and width. The instrument is mounted

on a rigid table which is in turn supported by pneumatic

mounts to aid in isolating the instrument from external

vibrations. A plasma ion source of the von Ardenne type

is employed. Ions are accelerated by an accelerating potential of 19.4 kV and focused on the source slit with a

pair of electrostatic quadrupole lenses. The electrostatic

potentials for the accelerator, quadrupole lenses and the

electrostatic analyzer deflection plates can be incre- mented according to Bleakney's theorem requirements.

Incrementing is accomplished by a timing signal which

also starts other aspects of the measurements. Ions which pass through a collector slit are detected by an

electron multiplier.

Mass differences are determined from the incremental

electrostatic analyzer voltage needed to produce

similar orbits. The circuit which provides the electrostatic analyzer potentials is shown in Fig. 8. The relay at the output of the AV supply alternately switches from

one position to the other each time a timing signal is re- ceived. In one position the electrostatic analyzer voltage

is V, and in the other V ± A.V, where V is the sum of the

series mercury battery voltages or about 780 V. Both V

and A V may be precisely measured by the potentiometer

shown. The batteries and potentiometer are housed in a temperature controlled box for stabiHty. Ions are swept

across the collector slit by a sawtooth magnetic field pro- vided by a Helmholtz coil located at the point where the

beam emerges from the magnet. Oscilloscope presentation of the beam shape is used for focusing and for direct

view peak matching.

Figure 7. Schematic of the University of Manitoba double- focusing mass spectrotneter. Electrostatic analyzer radius

ttg = 1.00 m; magnetic analyzer radius = 0.627U m; <t>g =

H.65°; <t>^ = 90°.

338
A recent mass spectrum of mercury chloride showing

the performance of this instrument is illustrated in Fig. 9. A full width at half maximum resolution of 1 in 945,000 was achieved for this spectrum. Although this in- strument is only about 50% larger than the Minnesota in- strument, various advances, particularly the full second

order focusing and improved transmission, lead to a working resolution almost 5 times higher than the Minnesota instrument.

I vdll next consider an entirely different mass measuring device which employs differences in transit time

rather than spatial separation to form a mass spectrum.

In order to describe the current instrument it is neces- sary to review some of the history of this form of mass

spectrometer. In the early 1950s, Lincoln Smith at Brookhaven National Laboratory designed and con- structed an instrument which was based on the dependence of the cyclotron frequency on the mass-to-charge

ratio of the ion. Smith named this instrument the mass

synchrometer [2]. A schematic of this instrument is shown in Fig. 10. In this instrument, ions were produced

and accelerated in an ion source vdthin a uniform magnetic field. After a half revolution they passed through a modulator which was driven by a radio-frequency volt- age. The change in velocity of the ion produced by the

modulator and consequent change in radius of the ion

path cause a spacial modulation of the beam across slit 85. Those ions which pass through S5 continue and again

pass through the modulator. If the phase of the rf volt- age is correct when the group of ions again reaches the

modulator, the group wall again be decelerated and vdll continue and pass through slit Se. The frequency condition is that the radiofrequency /rf = nf q where n is the

order integer (in practice) about 100 and /0 is the cyclo- tron frequency. The process then is repeated for a third

orbit in the magnetic field. Ions which pass through slit S7 are collected. Perturbing the rf frequency by a

savrtooth A/ will sweep the ion beam across the collector

slit and produce an ion beam shape similar to more con- ventional mass spectrometers. Using a variation of

Bleakney's theorem, one can measure a mass difference

by measuring the change in /rf needed to bring first one

and then the other member of the mass doublet through

the collector slit. Smith found three passes through the

modulator were necessary to remove "ghost" signals at the collector. An instrument was constructed according

to this design with a mean orbit diameter of 10 in. and a

slit spacing of 2/3 in. This instrument produced a

fullwidth at half-height resolution of up to one part in 32,000 and a mass measurement accuracy of one part in 2 X 10'^ and was competitive or exceeded the accuracy of more conventional instruments of that time. A summary

of results from this instrument was published in 1958

[13]. The success of this instrument led Smith to develop a

better design which was originally described in 1960 [14]. Design improvements included the following changes:

(a) The ion source and collector were moved outside of

the magnetic field.

(b) The ion beam within the magnet was required to have a small axial velocity producing a helical path.

(c) The rf modulation was employed in a different manner

than in the mass synchrometer in order to produce both

better transmission of ions and higher resolution.

The design was refined and construction began at Princeton University. By the spring of 1966 most of the major

parts of the instrument were completed and the

remainder of that decade was occupied in testing and

performing initial measurements [15].

-i

4 X 97.2 VOLT > 1

BATTERIES ^

POTENTIO- METER

NULL I > DETECTOR 200

n

:200fl

AV

SUPPLY ELECTROSTATIC ANALYSER

4 X 97.2 VOLT > 1

BATTERIES

'—

i

Figure 8. Circuit used to generate the electrostatic analyzer

voltage wave form for the University of Manitoba mass

spectrometer.

945,000

J v_

2°^Hg35ci2 202Hg35c,37j.,

Figure 9. Resolution performance of tlie University of Manitoba mass spectrometer.

IAD ION SOURCE

FROM RF

SUPPLY

Figure 10. A schematic diagram of the ^nass synchrometer.

339
A schematic diagram of the magnetic field region of the

instrument is shown in Fig. 11. Smith decided that in- stead of reducing the velocity of the ions on each passage

through the modulator, as in the mass synchrometer,

higher intensity could be achieved by a reduction of the

velocity on the first pass and a compensating increase on

the second pass through the modulator. The upper part

of this figure shows an axial view of the ion beam path

through the magnetic field. Ions first pass through an

internal electrostatic deflector which changes their motion from radial to tangential in the magnet. The ions are also given a small axial velocity so that the subsequent helical path allows the introduction of various

beam defining and focusing devices without any problem.

After a half rotation, the beam passes through the modulator where its velocity and therefore diameter is modulated sinusoidally. After a second half rotation the beam

passes through a phase defining slit and then back

through a modulator another half rotation later. The

change in diameter A D i at the first modulator is now re- moved by modulation AD 2 which is opposite to AZ)i. If this occurs correctly, then the beam moves another half rotation and passes through a collector slit and is deflected out of the magnet by a second internal electrostatic deflector. A side view in the lower part of Fig. 11 indicates how the path in the magnet proceeds. Fig- ure 12 shows the complete instrument. Emerging ions

from an ion source pass into a pair of 90° toroidal electrostatic analyzers which form an image just outside the

magnet. This image is used as the object for the internal

deflector which forms an image on the entrance slit. The

beam then proceeds through the path in the magnet. If the radio frequency is correct, the ion beam passes

through the exit slit. An image of the exit beam is formed by the internal deflector which in turn is the object for another pair of 90° torroidal electrostatic

analyzers. The detector is located at the focus of this analyzer.

Velocity and therefore diameter modulation is shown in Fig. 13. For the ion to pass through the exit slit, the velocity modulation Vi on the first pass through the modulator must be balanced by the velocity modulation V2

caused on the second pass through the modulator. In this diagram one sine wave describes Vi and the other

describes -V-z- Conditions are correct for transmission at times when the two sine waves cross. If, however, the

two sine waves coincide, then all ions will be transmitted. This will occur when /rf - (n + 1/2)/ 0 where /o is the cyclotron frequency. By sweeping /rf, a mass spec- trum is produced as shown in the middle diagram of

Fig. 13. Smith noted that the tails of the ion beam shapes were produced by ions which received large velocity

changes. The tails could be removed by a phase defining sHt that limits the change in velocity to a fraction of the

maximum. This is shown by the band in the top view in- dicated by dotted lines in Fig. 13. The result of this lim- iting is the triangular ion beam shape shown in the bot- tom view.

As with the mass synchrometer, peak matching is employed in this instrument to determine mass differences.

Two radio frequencies are generated, / 1 and f2 ^ fi + A /. The frequencies / 1 and /2 are phase locked and

linearly swept a small amount so as to present a time

varying ion beam shape as the output of the detector.

These frequencies cover the range from 50 MHz to 500 MHz. The frequency increment. A/, is derived from

a frequency synthesizer which can be set to 0.1 Hz. All voltages which produce dc fields in the system are alter- nated according to the Bleakney's theorem requirements.

ENTRANCE S MAGNET POLE

^—^'^=r^^g_ ^GNETIC FIELD ~^ PHASE MODULATOR

^DEFINING 'M_a^'"^-^:— „ <;i IT ^ SLIT '-^ —^^-^i^-^-^ ^"-'^

EX T*SLIT MAGNET POLE

16 R.F. MASS SPECTROMETER

ORBIT IN MAGNETIC FIELD

Figure 11. A scheymitic diagram of the magnetic orbit for the rf mass spectrometer.

ION SOURCE

DETECTOR

Figure 12. A schematic diagram, of the complete ion path in

the rf muss spectrometer.

Typical measurement conditions were:

Accelerating voltage

order

frequency

rf voltage

resolution

15 to 23 kV

w = 800 - 1200

140 to 360 MHz

100 to 150 V

one part in 200,000.

During the latter part of the 1960s, much effort was ex- pended, mostly by Smith alone, to achieve the theoretical

resolution of the instrument. Many of the difficulties he

experienced were caused by electric fields resulting from

charged surfaces, a difficulty which influences conventional mass spectrometers also. These fields were not svdtched as required by Bleakney's theorem and thus

added systematic errors to the measurement. By use of baffles and insulator shielding, these effects were minimized to the point that measurements of narrow doublets

could be made. These were reported in 1971 [16] in the

paper "Measurement of Six Light Masses." During the

spring of 1972 Smith's health began to fail and his last

340
Au or AD

Velocity or Diameter Modulations

It DC Beam Current

Intensity Distribution Wittiout Phase - Defining Slit

\

l\

»

ll

1 ll ^ II

II ll

1 1 IJL . . , , 1 III (n+ ^)fo (n'+^)fQ (n+|)to (n'+|)to

Intensity Distribution With

Phase - Defining Slit

Figure 13. Velocity and diameter modulation in the rf mass

spectrometer.

working days on the instrument were July 17 and 18. At

this time he was repeating some of the light mass measurements, now with an instrument having a working

resolution of one part in 400,000. Smith died in December

1972, leaving an instrument with a great deal of promise

but with no program or no individuals who had recently

worked with it. Concern for the continuity of this project was great

and fmally a new location for the instrument was found

at the Delft University of Technology in the Netherlands

under the sponsorship and direction of A. H. Wapstra, J. B. LePoole, and E. Koets. The instrument was moved

and was put back into operation officially on May 27, 1976. The Delft University of Technology was a logical

choice for the new location because of previous ion optical experience in the design of electron microscopes. In

the re-assembly of the instrument, a number of major improvements have been made:

(a) The instrument was re-aligned and a number of construction errors were eliminated.

(b) A higher-speed modern vacuum system was added.

This improvement produced better vacuums and an oil- free system.

(c) Pump oil contamination of interior surfaces was

eliminated. This was very likely a major cause of Smith's

earlier charging problems.

^(d) The vacuum housing and magnet were made more

rigid to minimize vibration problems.

(e) Many sensing and focusing elements were added to study and adjust the beam for optimum passage through

the system.

(f) A Telecentric Object Plane Selector was designed to study the beam cross-section. Any object plane from 0 to 2000 mm ahead of this device is focused with constant

magnification on a channel plate array and a magnified

image of the beam cross-section may be viewed on a

screen. By inserting this system at various points in the

orbit, detailed study and adjustment of the ion beam

cross-section is possible.

(g) Improvements in the homogeneity of the magnetic

field together vdth improvements in the stability of the

magnet power supply were made.

(h) Higher rf voltages were used.

The reassembly and the various improvements have

been described by Koets in several papers [17, 18]. The

instrument now is functioning in a much more predictable manner. Charging effects appear still to contribute to

instability but at a much lower level. The resolution

achieved is as high as one part in 10'^ which is the highest

currently reported by any method. I understand that a measurement program will begin sometime this year.

4. Recent Results

Because of the use of hydrocarbon molecular ions and

fragments as known masses in doublets, the mass of hydrogen and a few other masses such as ^^0, ^''N, ^^C, ^H,

^^Cl, and ^^Cl are considered as secondary standards and

for this reason have received a great deal of attention.

Precision attained in these measurements is high and

thus they are a good example of what precision can be

achieved with effort. The masses of selected secondary

standard atoms can be determined from the measurement

of an over-determined set of mass doublets. Smith [16] in 1971 measured the six masses ^H, ^H, ^^N, ^^0, ^^Cl, and

^'^Cl in terms of 14 distinct doublets. The adjusted mass

of for example was determined to be

1.007 825 029 ± 5 u . The hydrogen mass can also be determined by direct

comparison to ^^C. Smith [16] employed the doublet

C9H10 - CioHg while Benson [19] used the doublet

CuH22 - C12H10. In addition to the secondary standard measurements, a

large number of doublet measurements have been re- ported in the past few years which extend from light

atoms to the heaviest. Fractional error estimates, a/M,

for these mass measurements approach one part in 10^ and for some recent measurements surpass this figure.

Fractional errors for the doublets themselves, a /AM, for doublets with AM less than about 0.1 u have ranges from

10 to 20 ppm to those that approach the 1 ppm level.

It has been the practice in atomic mass measurements

to quote an error which represents the precision of the

measurement. Measurements of the same doublet are done a number of times and the error quoted is the

standard deviation of the distribution of these measurements. Although estimates of possible systematic errors are not usually included as part of the quoted error, one

can determine that certain forms of systematic error are small by means of several measurement procedures. One

possibility is to measure a wide doublet which yields a known result such as CsHg - C3H7. The AM here should

be one hydrogen mass and thus any proportional error

will be indicated by a departure from the known value. These proportional errors are often caused by surface

charging wdthin the instrument which produce

unaccounted-for electric fields. Another means is to measure a set of masses by a series of doublets which

341
over-determine the results. Consistency can then be

checked by using a test. The most extensive measure of consistency of mass

doublet measurements has been done by Wapstra [20] in the preparation of comprehensive least-squares mass

tables. These tables incorporate mass doublet information as well as nuclear reaction energy data in order to develop a best-value set of atomic masses. The tables,

which have been revised periodically to incorporate new

information, have become the accepted source of atomic mass information. The comparison of the subset of doublet measurements from one laboratory with output values can measure the consistency of the error estimation from

that laboratory. Wapstra finds consistency factors from

2.5 to 5 for various mass spectrometer measurements

[21]. A new feature in atomic mass determinations is the re- cent effort to apply the techniques developed for the

measurement of stable atoms to the measurement of short-lived radioactive atoms. A group at Orsay, France,

originally under the leadership of Rene Bernas and now

Robert Klapisch, have made many mass spectroscopic

measurements of various nuclear properties including the

atomic mass. Atomic mass measurements have recently

been reported for isotopes of francium [22], rubidium and

cesium [23], and sodium [24]. Progress in the development of measurement techniques in this area has been

reported by Barber [25] and by Smith [26].

5. Use of Mass Difference Measurements

for Fundamental Constants

In the past, wavelength-energy and energy-mass rela- tionships have been employed to transfer -y-ray and nuclear reaction information to equivalent mass differences. In this way a large and independent source of mass

difference information could be added to conventional mass spectrometer doublet results. Most nuclear reaction

and -y-ray energy information has been based on relative

measurements, using a particular disintegration energy

as a standard. For charged particle reactions, a variety

of natural alpha-emitters have been employed as standards. Absolute a-energy measurments by Rytz [27] have

provided energy standards with uncertainties in some

cases as small as 7 ppm. In a similar manner, y-ray en- ergy measurements depend on a few ^-emitters as standards. The need for better 7-ray standards was recognized

by the lUPAP Commission on Atomic Masses and Fundamental Constants in 1971 and a subcommittee was ap- pointed to make standards recommendations. The report

of the subcommittee (Helmer et al. [28]) bases 7-ray en- ergy standards for energies up to 6.1 MeV on the

411 keV line in ^^^Au. The uncertainty of the 411 keV line was 2.7 ppm based on the wavelength measurement of

Kessler et al. [29] using a double-flat crystal interferometer. Most of the uncertainty in this energy results from

the wavelength-energy conversion using the value from

Cohen and Taylor [1] and only a small amount, 0.37 ppm,

from the wavelength measurement. This work has been

reviewed by Deslattes and Kessler [29] and they suggest

that this form of measurement may be extended to the

wavelengths of -y-rays with energies of several MeV.

The availability of this type of measurement suggests

that the energy-mass transformation may be reversed

and measurements of an absolute wavelength and the

equivalent mass difference could be used together to

yield a value for a combination of fundamental constants. This possibility has been suggested by Deslattes and

Kessler [30] and by Taylor [31]. Suppose that \ is the

wavelength in meters of a 7-ray which has an associated mass difference Am measured in u. By assuming the

Planck relationship and the Einstein mass-energy rela- tionship, the combination of fundamental constants given

in Eq. (10) is equal to the product of the experimental

measurements:

^=XAm. (10)

c In order to estimate the precision needed for a useful

XAm product, one can determine the uncertainty in the

fundamental constant product from Cohen and Taylor [1]. Using the method outlined by Cohen and Taylor to determine product uncertainties, the uncertainty for this product is 1.7 ppm. Measurements of both Am and X to part

per million levels appear to be certainly possible.

As an example of this application, consider the thermal

neutron capture in ^^C which produces a 7-ray with an energy of about 5 MeV. Measurements of this -y-ray en- ergy together with the thermal neutron capture 7-ray in hydrogen at 2.2 MeV can be employed for the wavelength

part of the measurement. The mass difference needed

can be found using a series of mass doublets investigated

by Halverson and Johnson [32]. The two reactions are given in Eqs. (11) and (12)

12c + n ^ 13(3 + 71 (11)

(12)

These equations can also be written as energy balance

equations if all elements are stated in the same mass- energy unit:

^1 mC^C) + m(n) = m(^^C) + Q

mCU) + m(n) = mi^R) + Q^^ ,

(13)

(14)

in which Q^^ and Qy^ are the total energies available in each reaction expressed in mass units. The masses employed in this equation are atomic masses with no corrections because the parent and daughter atoms in each

reaction have the same electronic binding. The total en- ergy for the reaction can be expressed in two parts, the

7-ray energy, E y, and a recoil energy, E-r, as in Eq. (15): = + . (15)

If one assumes that the recoiling atom is free, the recoil energy is given by ^ (hc/y)^

2mc^

Er — (16)

For the ^^C emission, this recoil energy is about one part

in 5000 of the total energy Qy. If is written in atomic mass units as required by Eq. (13), we have

1 hc/k

2mc^ ^2 (17)

Although the correction for recoil contains fundamental

constants, the correction can be made with nominal

values because the correction is small. The reaction energy, Eqs. (13) and (14), can be con- verted to a mass difference by rearrangement. Assume

that all elements in Eq. (13) and Eq. (14) are given in unified mass units. First subtract Eq. (14) from Eq. (13):

12c - iH = i^C - 2H + Q

or 12C2H - 13C1H - Q ^1

^1

Q

Q ^2

^2

(18)

(19)

342
Equation (19) can be converted to available molecules by

multiplying by six: i^Ce^He - ''Ce'He = 6(Q„ - Qy,) (20)

These two molecules in this mass doublet are both benzene and thus have the same atomic binding energy

which cancels out of the mass difference. This doublet

with C6H12 is part of an over-determined mass triplet

shown in Fig. 14. This triplet has been employed by

Halverson and Johnson [32] to determine a value of the

i^Ce^He - ^^Cs^He doublet.

The doublet difference A m, measured in u, is one part

of the input to determine the fundamental constant product. The other part, Qy^ - Qy^, can be simphfied by

assuming the recoil correction for reaction ? to be (1 +

Ui). Then

Qxj - Qk^ = he

X.l\2 +

\lA.2 .(21)

Substituting this value of Q^^ ~ ^72 ^"^^^ ^^^^ yields

Am ^2

X.lX.2

02^1

X.lX2

hN,

(22)

The left side of Eq. (22) now involves measured quantities and the right side is the desired fundamental constant product.

In principle any A m could be used for this comparison.

There are, however, a number of practical considerations

that make some choices better than others.

a. The neutron capture target should be an abundant

isotope.

b. The mass doublet that results should be made up of molecular ions if possible.

c. The 7-ray energy should be linked to an absolute

wavelength measurement as closely as possible.

d. The 7-ray energy should be large compared with the

^H(n, ifH reaction energy.

Table 1 lists several possible choices of atoms for this

test. Included is the calculated recoil energy correction.

The large ^^N(n, 7)^^N reaction energy results in a low

fraction error for the A m measurement. A doublet could

be constructed similar to that found for the carbon based

measurement using molecules of hydrazine. The large energy, however, would make the determination of X. for the emitted 7-ray more difficult. Another possibility of choice for a reaction would be

the decay of a long Hved metastable state. The experi- ence developed by groups measuring the masses of

radioactive atoms would be useful for this form of measurement. Some possible choices are listed in Table 2. Of

these the ^^Mo isomeric transition looks most favorable.

The doublet has a spacing of one part in 39,000 and is wide enough so that a 1 ppm mass difference measure- ment is conceivable. This form of reaction would provide

a more direct comparison of A m and X but will still have

a small recoil correction. (The recoil correction for the

^^Mo decay is only 86 eV.)

In conclusion, it appears that from both the massdifference side and the wavelength side, precisions can

be achieved at present or in the near future to produce

useful fundamental constants results.

84 83

A A

CfiHi

•«- M

Figure 14. The mass triplet employed to determine the C/Hg -

^^C/Hg mass difference. The resolution shown in the lower

portion is one part in 10^.

Table 1. Possible reactions for fundamental

constants tests.

Recoil Reaction E y in |xu energy p.u

i2C(n,7)i^C 5310.067 ± 23^ (5 ppm) 1.084 5309.978 ± 63^ (13 ppm)

i3C(n,7)l4c 8777.746 ± 26^ (3 ppm) 2.751 i4N(n,7)i5N 11,629.936 ± 28^ (2.5 ppm) 4.507

^See Ref. [33]. ^See Ref. [32].

Table 2. Possible isomeric transitions for fundamental constants tests.

Isotope Half-life Energy

(IVIeV)

89y 16 s 0.91 93^0 7 h 2.43 48 min 0.40 i37Ba 2.6 min 0.66

I wish to thank Robert Barber, University of Manitoba, and Erico Koets, Delft University of Technology,

for their significant assistance in providing illustrations and other information for this talk.

References

[1] E. R. Cohen and B. N. Taylor, J. Phys. Chem. Ref. Data 2, 663 (1973).

[2] J. H. E. ]Mattauch, Adv. Mass Spectrom. 3, 1 (1965).

[3] E. G. Johnson and A. 0. Nier, Phys. Rev. 91, 10 (1953).

[4] W. Bleakney, Am. Phys. Teacher 4, 12 (1936).

[5] W. F. G. Swann, J. Franklin Inst. 212, 439 (1931).

[6] L. G. Smith and C. C. Damm, Phys. Rev. 90, 324 (1953).

[7] C. F. Giese and T. L. Collins, Phys. Rev. 96, 823A (1954).

[8] L. G. Smith, Phys. Rev. C 4, 22 (1971).

343
[9] H. E. Duckworth, Rev. Sci. Instrum. 21, 54 (1950).

[10] a. Hintenberger and L. A. Konig, Adv. Mass Spectrom. 1, 16 (1959).

[11] R. C. Barber, R. L. Bishop, H. E. Duckworth, J. 0. Meredith, F. C. G. Southon, P. van Rookhuyzen, and P. Williams, Rev. Sci. Instrum. 42, 1 (1971).

[12] L. G. Smith and C. C. Damm, Rev. Sci. Instrum. 27, 638

(1956).

[13] L. G. Smith, Phys. Rev. Ill, 1606 (1958).

[14] L. G. Smith, in Proceedings of the Interyiational Conference on Nuclidic Masses, Ed. by H. E. Duckworth (Univ. of Toronto Press 1960), p. 418.

[15] L. G. Smith, in Proceedings of the Third International Conference on Nuclidic Masses, Ed. by R. C. Barber

(Univ. of Manitoba Press, 1967), p. 811.

[16] L. G. Smith, Phys. Rev. C 4, 22 (1971).

[17] E. Koets, in Atomic Masses and Fundamental Constants U, Ed. by J. H. Sanders and A. H. Wapstra (Plenum Press, New York, 1976), p. 164.

[18] E. Koets, J. Ki-amer, J. Nonhebel, and J. B. LePoole, in Atomic Masses and Fundamental Constants 6, Ed. by J. A. Nolen, Jr., and W. Benenson (Plenum Press, New

York, 1980), p. 275.

[19] J. L. Benson and W. H. Johnson, Jr., Phys. Rev. 141, 1112

(1966).

[20] A. H. Wapstra and K. Bos, At. Data Nucl. Data Tables 19, 175 (1977).

[21] A. H. Wapstra and K. Bos, At. Data Nucl. Data Tables 20, 1 (1977).

[22] G. Audi, M. Epherre, C. Thibault, R. Klapisch, G. Huber,

F. Touchard, and H. Wollnik, in Atomic Masses and Fuyidamental Constants 6, Ed. by J. A. Nolen, Jr., and W,

Benenson (Plenum Press, New York, 1980), p. 281.

[23] M. Epherre, G. Audi, C. Thibault, and R. Klapisch, in Atomic Masses and F^mdamental Constants 6, Ed. by J. A. Nolen, Jr., and W. Benenson (Plenum Press, New

York, 1980), p. 299.

[24] C. Thibault, M. Epherre, G. Audi, R. Klapisch, G. Huber,

F. Touchard, D. Guillemaud, and F. Naulin, in Atomic Masses and Fundamental Constants 6, Ed. by J. A.

Nolen, Jr., and W. Benenson (Plenum Press, New York,

1980), p. 291.

[25] R. C. Barber, K. S. Kozier, K. S. Sharma, V. P. Derenchuk,

R. J. Ellis, V. S. Venkatasubramanian, and H. E. Duckworth, in Atomic Masses arid Fundaynental Constants 6, Ed. by J. A. Nolen, Jr., and W. Benenson (Plenum Press, New York, 1980), p. 257.

[26] R. J. Smith, Ph.D. Thesis, Univ. of Minnesota (1977).

[27] B. Grennberg and A. Rytz, Metrologia 7, 65 (1971).

[28] R. G. Helmer, P. H. M. Van Assche, and C. vanderLeun,

At. Data Nucl. Data Tables 24, 39 (1979).

[29] E. G. Kessler, Jr., R. D. Deslattes, A. Henins, and W. C. Sauder, Phys. Rev. Lett. 40, 171 (1978).

[30] R. D. Deslattes and E. G. Kessler, Jr., in Atomic Masses and Fundamental Constants 6, Ed. by J. A. Nolen, Jr., and W. Benenson (Plenum Press, New York, 1980), p. 203.

[31] B. N. Taylor, private communication (1981).

[32] J. E. Halverson and W. H. Johnson, Jr., Phys. Rev. C 17, 1414 (1978); Errata, ibid., 18, 2450 (1978).

[33] L. G. Smith and A. H. Wapstra, Phys. Rev. C 11, 1392

(1975).

344
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Absolute Determination of the Threshold

Energies of ^Li(p,n), ^°B(p,n), and '"*N(p,n)

p. H. Barker, M. J. Lovelock, H. Naylor, R. M. Smythe, and R. E. White

Physics Department, Auckland University, New Zealand

The present status of the Heavy Ion Source System (HISS) for the measurement of nuclear reaction energies is discussed, and some results are presented. In particular, accurate values for the

threshold energies of '^Li(p,n), '*'B(p,n), and ^'*N(p,n) are given.

Key words: heavy ion calibration; nuclear reaction energies; one volt standard; (p,n) thresholds.

1. Introduction

The study of the theoretical implications of the ft values of super-allowed beta decays would be placed on a firmer basis if the decay Q-values (Qp) were known more

precisely. At present, they carry error bars which, on

the average, are about 0.5 keV in several MeV. Our

prime aim therefore has been to develop a system to improve this situation considerably, to show the reliability of the system, and to use it to determine as many of the

Qp as possible. In the present work a Q p is derived from measurement

of the threshold of the associated (p,n) reaction. The pro- ton beam from our folded tandem accelerator, AURA II,

is momentum-analyzed by requiring it to pass through

very narrow slots at the object and image positions of an Enge split-pole spectrograph. After the image slot, the

beam is used to examine the yield of a particular (p,n)

reaction near threshold by applying a variable offset volt- age to the target. The shape of the yield curve is then

analyzed in terms of the familiar (V = Vo)^ energy

dependency for a thick target, to give Vq, the offset volt- age at threshold.

To measure the mean proton energy, a subsidiary surface ionization source is used to produce ions of a heavy

element, usually thallium, which are then accelerated

through a potential difference to pass down the same

flight path through the spectrograph as the proton beam

had done. This potential difference is measured using a

resistive divider of our own construction and a one volt standard. The magnetic rigidity of the thallium ions, and

hence of the proton beam, is then calculated from this measured potential difference, with small corrections due

to contact potential and thermal effects. Thus the kinetic

energy of the protons, E p, is given essentially in terms

of the thallium accelerating potential and the masses of a proton and a thallium ion. The threshold energy, £"0 is then just given by E 0 = Ep - Vq. The system is described in detail in Ref. [1], with minor modifications and test results in Ref. [2].

2. The ^Li(p,n) Threshold at 1.88 MeV

This reaction is of interest because the accepted

threshold energy, Eq - 1880.51 ± 0.08 keV [3], carries a very small error and it is therefore a good test for our

system. In addition, we will try to determine Eq for

^H(^Li, n) to compare and look for possible atomic effects.

The yield curve of neutrons from ^Li + p was examined

six times vdth the spectrograph magnetic field set to make E p about 1880 keV. A fresh '^Li target was evaporated in situ for each run and the yield of neutrons was

determined using a ^Li - ZnS backed glass scintillator and photomultiplier vdth a perspex moderator. The beam

intensity was monitored by observing the yield of the

480 keV prompt gamma ray from '''Li -I- p. The magnetic

rigidity of the proton path through the spectrograph was

calibrated using rubidium ions of approximately 22 keV.

Thallium could not be used because the required ac- celerating voltage is too low to be compared vdth our

1 volt standard.

In Fig. 1 is shown one of the neutron yield curves, the

solid curve being the (V - Vq)^ fit to the data points.

The quoted errors for Vq and Ep are typical for this ex- periment and are one standard deviation, as are all other

errors quoted in this paper.

^Li(p,n) Threshold

60 -

50 -

UO

30 -

20

10

YiGld

Vo = 2.42 ± 0.02 kV

Ep = I877.89±0.07 keV

0 -1 -2 -3 -U Offset Volts (kV)

Figure 1. A yield curve for 'Li (p, n). The line is a theoretical

least sqvares fit of the form Y = a + b(V - Foj""-. The mean value for the six runs is £"0 = 1880.41 ±

0.05 keV with Xv = l-O- A small correction can be applied to take into account the proton energy distribution

345
(triangular with base width 0.4 keV). This brings the fi- nal threshold energy to 1880.42 ± 0.05 keV.

3. The ^°B(p,n) Threshold at 4.88 MeV

Although the decay of ^''C is one of the pure Fermi beta

decays referred to in Section 1, the uncertainty in the ft value arises almost entirely from uncertainty in the

branching ratio of the decay. Hence there is no strong reason to improve the measurement of the threshold en- ergy. However, knowledge of the branching ratio may

improve and, in addition, the two sub-keV measurements

quoted in the literature, Refs. [4] and [5], disagree by more than 3 keV.

For this reaction, the yield of the 717 keV gamma ray

which follows the ^''C decay was used as a measure of the

reaction yield. The targets were of amorphous ^°B, evaporated with an electron gun on to thick tantalum backings. The experimental arrangement was such that the

proton beam, after passing through the spectrograph defining slots, struck the target for 30 s, (Ti^i^C = 19 s), and then was interrupted upstream by a tantalum

shutter. The target was pulled on a magnetically coupled

plunger out of the spectrograph body, and the accumulated activity viewed with a Ge(Li) detector. The beam

intensity was monitored by detecting protons scattered

backwards from the target.

After a sufficient yield had been accumulated, normally

after 10 beam cycles, the potential on the target was al- tered, and a yield curve like the one in Fig. 2 was taken,

from which a value for Vq was extracted. The proton

beam energy was calibrated using Tl ions of approximately 24 keV. The error in £"0 arises almost entirely

from the error in Vq due to the low yield of the reaction.

30

25

20

15

10

5

10 B(p,n)

10

Yield

Vq =3.0±1.2 kV

12

Ep =4880.5

± o.ikeV

O -3 -6 -9 -12 Offset volts (kV)

Figure 2. A yield curve for "'B(p,n). The line is a theoretical

least squares fit of the form Y = a + b(V - Vq)'"^. The mean of five such values was Eq = 4876.4 ±

0.55 keV with Xv = 1-0. The correction for the beam en- ergy shape (triangular with base width 0.8 keV) and for non-uniform proton energy loss in the target is +0.03 ±

0.03 keV, and so the final threshold energy is 4876.4 ±

0.6 keV, which is in good agreement with the value implied by the ^^C(p,t) Q-value of Ref. [4], 4876.4 ±

0.6 keV.

The error in the result of Ref. [5], the first Auckland

(p,n) experiment performed with an earlier, more primitive set-up, we attribute to having used a target which was not freshly prepared, and which had indeed already

been used by the Harwell group for an earlier determination. That target cleanliness and stability are crucial will become apparent in the following section.

4. The ^^H{p,n) Threshold at 6.35 MeV

This experiment is very similar to that of Section 3. The

yield of the 2.3 MeV gamma ray following the decay of ^'OiTy2 = 70 s) was measured. The proton beam energy was cahbrated using Tl ions of approximately 31 keV. The

beam-on /beam-off times were 100 s and 200 s respectively. Attempts were made to detenrdne the ^^N(p,n)

threshold energy using targets of melamine, nylon, tanta- lum nitride and titanium nitride. Of these, only melamine

gave a yield which was sufficient, which did not diminish

during the course of an experiment, and which did not have too large an associated contaminant activity. In Fig. 3 are shown the results for the years 1979 and

1980 for those experiments for which no obvious reason

could be found for their exclusion. They form a statisti- cally inconsistent set. This we attribute to deterioration

of the targets during the course of the runs. It is difficult to set up tests which are sensitive enough to detect these

effects, and the best way seems to be to look for inconsistencies between runs done with the same target, and

also between runs done with different targets.

5-5 keV

5-0

6354-0

3-5

3-01-

ThG^''N(p,n) Threshold Energy

Year

1979 1980 1981

Figure 3. Attempts in Auckland to measure Eofor "''N(p,n) in 1979, 1980, and 1981. The points in Fig. 3 which are indicated by the year

1981 were taken with targets of sputtered tantalum ni- tride on a tantalum substrate. We are confident that

these do not deteriorate during use. Their yield is reproducible over periods of many hours. When judged visibly

they look unaffected by the bombardment, and they give a self-consistent set of values for Eq.

In Fig. 4 is one of the nine ^^N(p,n) yield curves. As

can be seen, the error in £'o comes mainly from Vq, with

a small contribution from Er,. In Table 1 are hsted the in- dividual results which combine to give a mean of Eq -

6353.02 ± 0.08 keV, Xv = 0.8. After correction for the

beam energy distribution, triangular with base width

1.05 keV, and for non-uniform proton energy loss in the

target, a total of +0.02 ± 0.02 keV, the threshold be- comes Eo = 6.353.04 ± 0.08 keV.

This figure does not include any correction for atomic

effects. Reference [13] reconunends a correction of

346
-131 eV. However, consultation with one of the authors

has indicated the figure is uncertain but should in any

case be considerably smaller [14]. We have therefore

chosen to leave it out.

1/f N(p,n) Threshold

30

20

10

Yield

Vo = -2.57± 0.16 kV

Ep = 6350.57

± 0.07 keV

J L ll ± ±

0 -2 -4 -6 -8 Offset VbltslkV)

Figure 4. A yield curve for ">N(p,n). The line is a theoretical

least squares fit of the form F = a + b(V - Vo)-"-. Table 1. The set of nine threshold dete rminations

for ^^N(p,n).

Vq (keV) E 0 (keV)

6347.89(07) -4.88(39) 6352.77(40)

6345.69(07) -7.68(45) 16353.37(46)

6345.03(07) -7.54(40) 6352.57(41)

6345.03(07) -8.08(21) 6353.11(22)

6347.01(07) -5.52(28) 6352.53(29)

6347.54(07) -5.47(24) 6353.01(25)

6349.65(07) -3.42(25) 6353.07(26)

6350.57(07) -2.57(16) 6353.14(17)

6353.74(07) +0.64(17) 6353.10(18)

Mean (with Xv = 0.77) 6353.02(08)

Taking the excitation energy of the 0"^, 7" = 1 state of

to be 2312.81 ± 0.06 keV [6], and the latest value for the (n - iR) mass difference, 782.332 ± 0.017 keV [7], the Qp for the superallowed decay of ^^0 becomes Qr =

1808.31 ± 0.10 keV. This is compared in Table 2 with

other sub-keV measurements.

Table 2. Q^^ values for the "0

superallowed decay.

Qp(keV) Reference

1809.3

1810.2

1810.5 1808.8 1807.9

1810.4

± 0.7 ± 0.5 ± 0.5 ± 0.4 ± 0.8 ± 0.6 1808.31 ± 0.10

8

9

10

11124

P*resent result

References

[1] p. H. Barker, H. Naylor, D. P. Stoker, R. E. White, and W.

B. Wood, in Atomic Masses and Fundamental Constants

6, Ed. by J. A. Nolen, Jr., and W. Benenson (Plenum,

New York, 1980), p. 233.

[2] D. P. Stoker, P. H. Barker, H. Naylor, R. E. White, and W.

B. Wood, Nucl. Instrum. Methods 180, 515 (1981).

[3] A. H. Wapstra and K. Bos, At. Data and Nucl. Data Tables

19, 177 (1977).

[4] P. H. Barker and J. A. Nolen, Tokyo Conference on Nuclear

Physics, (1977).

[5] D. C. Robinson and P. H. Barker, Nucl. Phys. A 225, 109

(1974).

[6] F. Ajzenberg-Selove, Nucl. Phys. A 152, 1 (1970).

[7] R. C. Greenwood and R. E. Chrien, Phys. Rev. C 21, 498

(1980).

[8] J. W. Butler and R. 0. Bondelid, Phys. Rev. 121, 1770

(1961).

[9] R. K. Bardin, C. A. Barnes, W. A. Fowler, and P. A.

Seeger, Phys. Rev. 127, 583 (1962).

[10] M. L. Roush, L. A. West, and J. B. Marion, Nucl. Phys. A

147, 235 (1970).

[11] R. E. White and H. Naylor, Nucl. Phys. A 278, 333 (1977).

[12] H. Vonach, P. Glassel, E. Huenges, P. Maier-Komor, H.

Rosier, H. J. Scheerer, H. Paul, and D. Semrad, Nucl. Phys. A 278, 189 (1977).

[13] J. M. Feagin, E. Merzbacher, and W. J. Thompson, Phys.

Lett. 81B, 107 (1979).

[14] W. J. Thompson, private communication.

347

Precision Measuretnent and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Preliminary Proton/Electron Mass Ratio Using a Precision IVIass-Ratio

Spectrometer*

R. S. Van Dyck, Jr. and P. B. Schwinberg

Department of Physics, University of Washington, Seattle, WA 98195

A new type of compensated Penning trap has its ring electrode split into equal quadrants in order

to synchronously detect ion cyclotron resonances. Using this device, we have observed intense unshifted resonances with very small relative linevndths (< 2 x 10~^), comparable to that which is at- tainable in high resolution NMR studies. Thus, cyclotron frequencies of both protons and electrons have been measured in the same magnetic field (5 T) and the same trapping volume (< 10"^ cm^).

From the ratio of these frequencies, a preliminary value of / nig = 1836.15300(25) is obtained with a

relative uncertainty of 0.14 ppm. This value agrees well wath previous direct measurements within

their experimental uncertainties. Key words: compensated quadrupole trap; fundamental constants; ion cyclotron resonances; ion trapping; mass spectrometer; precision measurements; quadring Penning trap; synchronous resonant

detection.

1. Introduction

The proton /electron mass ratio is a fundamental constant which appears often in specific theories of atomic or elementary particle physics in the form of a reduced mass

effect or an isotope shift. An illustration of the need for precision in this mass ratio can be found in the recent comparison of theory vdth a precision measurement of the Lyman-a resonances of atomic hydrogen and deuterium [1]. Also, a 0.01 ppm determination of mp /m^ will allow an unambiguous consistency check to be made upon

many other precisely measured fundamental constants

[2]. Previously, the most precise value of mp/Wg was

1836.1518(7), indirectly determined from other precision constants [3]. This can be compared to direct measure- ments of 1836.1502(53) by Gartner and Klempt [4] and

1836.1527(11) obtained by Graff et al. [5] using a timeof-flight measurement in an external inhomogeneous

magnetic field in order to detect resonant cyclotron exci- tation of an ion prior to ejection from a Penning trap.

The only theoretical value of the mass ratio is a speculation by Wyler [6] that suggests that rup /rUe - 6-n^ =

1836.11811 which is many standard deviations away from

several other indirectly measured values of mp / mg [7] as

well. The compensated Penning trap [8] has been demonstrated to be a very precise resonance spectrometer in the measurement of the spin anomaly of a single electron

[9]; thus, it should be possible to use this type of trap to

isolate single ions and directly measure their ion cyclo- tron resonances [10, 11]. The simple result of comparing

respective cyclotron frequencies then yields directly the mass ratio (assuming identical charges). However, the

motion of a charged particle in a Penning trap slightly

modifies this simple picture. The charge oscillates axially

in the electric potential well vdth a frequency while

rotating in the radial plane at the observed cyclotron fre- quency v'c . In addition, the crossed electric and magnetic

*Supported in part by NBS Precision Measurements Grant No.

G7-9023, in part by a M. J. Murdock Charitable Trust Grant of the Research Corporation, and in part by NSF Grant No. PHY-80-15328.

fields produce a slow unbounded, but metastable, rota- tion of the center of the cyclotron orbit at the magnetron

frequency v^. From the equations of motion for a single

ion of mass m and charge e isolated within a Penning

trap,

V, = (1/2tt) ieVo/mZoY\

v'c ^ Vc - ,

where

Vf. — eBo /2iT mc , Bo is the applied magnetic field, Vo/2 is the well depth

of the trap, and 2Zo is the characteristic endcap separation (at r = 0). The magnetron frequency, used to correct the observed cyclotron frequency, can be either calcu- lated using and v'c or measured using a motional side- band [12] such as + v,„ or + v,„ . Such cooling or centering resonances for the magnetron motion have

been observed by us and described previously [9]. For a

single electron in a recent trap, the measured and computed values have agreed to vdthin a few ppm. Such

agreement is crucial in order to justify using this type of resonance method.

2. Quadring Trap

The experimental apparatus, shown in Fig. 1, consists

of a trap tube supported at the center of a 30 cm long,

very stable (< 0.002 ppm/hr), and reasonably uniform

( < 3 ppm over a 1-cm sphere) superconducting solenoid.

The "quadring" Penning trap, shovra only schematically

in Fig. 1, is illustrated in detail in Fig. 2. In addition to the two endcap and ring electrodes vdth characteristic di- mensions Zo = 0.112 cm and Ro = 0.159 cm, the trap has

two guard rings which compensate for the truncations of the main surfaces, one probe inserted into each guard for the centering drive, and four equal cuts in the hyperbolic

ring electrode to form the "quadring." This design [11] allows the cyclotron resonance to be excited on two oppo349
site quadrants and detected on the other pair which are

externally tuned to the cyclotron frequency. The two axial frequencies, Vz{e~) and v^ip'*'), are also detected using

a tuned circuit resonant at the appropriate frequency and

attached to one or the other endcap. Also shown in Fig. 1 is the Schottky multiplier diode which generates the

microwave field required to excite the electron cyclotron resonance at v'de'). Finally, the entire apparatus consisting of trap tube, microwave diode and tuned preamplifiers is constructed to operate in a liquid helium en- vironment in order to yield the required signal-to-noise

and high resolution.

Included in each endcap is a small nickel ring (volume — 10"^ cm^ ) and a source of either electrons or hydrogen.

The electron source is a sharpened tungsten needle which

Figure 1. Schematic of experiment. Both endcaps and the quadring have tuned preamplifiers attached to measure either or . rf axial drives are applied to the endcap opposite the detector and microwave excitation for {e ~) radiates from a

Schottky multiplier diode.

Figure 2. Quadri^ig Penning trap. The standard compensated

Penning trap is altered by splitting the main ring electrode

into four equal quadrants in order to synchroyiously detect ion cyclotron resonances directly.

can be biased to field emit a primary electron beam

through the trap and generate within the trapping

volume both secondary electrons and ions from either

background gas or hydrogen liberated from a hydrated ti- tanium filament in the opposite endcap. The nickel rings

produce a very weak magnetic bottle [9] which generates

a change in the uniform magnetic field given by

where ' - B2 ~ 350 G/cm2 . The electrodes themselves may increase this coefficient by 50% due to an intrinsic bottle field formed by the

paramagnetism of the molybdenum used to make the

electrodes [13]. The net effect of this magnetic bottle is to couple the magnetic moment of the charge to the axial resonance in the form of a small frequency shift, 2-3 Hz

per quantum level for electrons (v^ ~ 86 MHz) and

10-20 |jlHz per quantum level for protons (v^ ~ 10 MHz).

Obviously, this bottle is useful only for detecting the

electron's cyclotron resonance whose frequency is far too high ( ~ 140 GHz) to observe directly. However, this same bottle will also broaden [11] the cyclotron reso- nances of both ions and electrons via the thermal axial

noise appearing in the z ^ term:

Av, /vc ^ iB2Zo^/Bo){kT /eVo).

Thus, for a bottle with coefficient B2 - 500 G/cm^, Bo =

50.5 kG, r = 4 K, Vo - 2 y for electrons and 54 V for protons, Avc /v^. is 2 x 10"^ and 8 x 10"^*^ respectively.

3. Measurements

Preliminary work using electrons stored in this quadring tap indicate that the chosen well depth is too shal- low to fully compensate the trapping field and to allow narrow single electron resonances to be observed.

Nevertheless, using approximately 10 electrons, the ap- paratus was able to yield the two cyclotron resonances shown in Fig. 3. These signals are proportional to axial frequency shifts and are produced by locking the axial resonance to a stable drive oscillator. As typical of past

electron traps with magnetic bottles [9, 11], the two

resonances show a clear low frequency edge, corresponding to Z rms = 0, and a high frequency exponential tail, in- dicative of the Boltzmann distribution of axial states. Repeatability of these edges is typically better than 0.1 ppm for low applied microwave power.

Ambient = 4. 2°K

i^c'le") = 141 461 242 kHz Bo = 5.05T

i lUrt. Vo = 2.04V

lA —( 1— O.lppm / \./WL "0 ^'^'^''O"^'

>.C

<u

OJ

27 February 1981 microwave frequency

Figure 3. Electron cyclotron resonances. Two separate traces show the repeatability of the low frequency edge (Z,^ = 0). As

usual, the high frequency tail is due to the thermal axial

states.

350
Small numbers of protons have also been trapped and,

for the first time, the proton cyclotron resonance has

been rf driven and synchronously detected. An example

of the high resolution available with this technique is shown in Fig. 4. Linewidths < 0.2 Hz can be reliably obtained out of an observed cyclotron frequency v'c (p =

76.4 MHz (resolution ~ 0.001 ppm), limited primarily by

observation time. From the tuned circuit damping, a sin- gle proton would have a linewidth of 0.005 Hz, though

axial noise (via the magnetic bottle) will make such nar- row resonances very difficult to observe. This single ion

linewidth allows one to estimate an upper limit of 40 for the number of trapped protons in this example. Axial

proton resonances are also synchronously detected but

with approximately 100 times less resolution. However,

their relative contribution to the accuracy of vdp'^) is only 2%. In the future, a more stable electric potential

well will be used to trap the protons in order to eliminate

the contribution of the axial resonance to the uncertainty

of the proton's cyclotron frequency.

p"") = 76 365 476.9Hz

1 Ambient = 4.2°K

1 - 5.05T

1— O.OIppm

\ Vq = 54,4V

5 March 1981

trap, with little compensation, yielded a number dependence 100 times as great and with opposite sign suggesting that this systematic error can be eliminated by

proper adjustment of the compensation control. A plausible description of this number dependence which ap- pears to be consistent with the above observations is made in appendix A. The intercept of the linear fit in Fig. 5 corresponds to the preliminary proton /electron mass ratio m„/mg =

1836.15300(7). However, an analysis of bottle-related position dependence of the magnetic field, given in Appendix B, suggests that a systematic error as large as 0.1 ppm may exist if the two charge types do not have the same average position in space. Thus, a preliminary mass

ratio of

nip /me = 1836.15300(25)

reflects this uncertainty in magnetic field location. Future tests will specifically search for the magnetic saddle

point [11] and vary B2 in order to eliminate this uncertainty. Also, a deeper potential well will be used to trap

and detect a single electron in order to yield vde') resolution which is typically parts in 10^ for previous electron

traps. This preliminary result agrees with values measured by Graff et al. [5] and Gartner and Klempt [4] within their experimental errors. However, we expect

320X 10'

frequency

000

nunnber of 2000

protons

3000

Figure 4. Proton cyclotron resonance. The drive phase is adjusted to produce this dispersion curve during synchronous

detection. The lineu/idth is limited primarily by observation

time.

4. Results

It is well known that any breakdown in ideal trap symmetry would lead to a cyclotron resonance which depends

on the number of trapped ions [4, 5, 14]. This dependence

does not result from space charge, unless the cloud contains dissimilar ions, because electrostatic interactions

between like-ions in a cloud whose extent is small compared to the wavelength of the exciting rf field do not

shift or broaden the cyclotron or axial resonances [15]. The procedure of applying intense rf drives at the axial

frequency of the unwanted ions should remove them from

the trap since their axial motion is undamped.

Figure 5 shows the mass ratio rrip I me plotted versus number of trapped protons. This dependence follows entirely from the number dependence of the proton cyclo- tron resonance v^p'^) since no number dependence has

been observed for Ve{e~) within the resolution available.

The slope of a least squares fitted line through the data

is -0.22 X 10"^/ ion. An earher version of the quadring

Figure 5. Mass ratio versus proton number. This dependence

is due entirely to the sensitivity of {p "*") to proton number

and appears to depend on the compensation of the trap.

our techniques, with appropriate systematic checks completed, will ultimately produce a mass ratio with a precision that exceeds 0.01 ppm. Other possible improvements

include using double traps [16] and a variable magnetic

bottle [17].

We wish to acknowledge our gratitude to machinists

Don Russell and Ralph Jochim for their excellent

machine work on the traps and to glassblower Bob

Moerly for his superb and patient work.

References

[1] S. A. Lee, R. Wallenstein and T. W. Hansch. Phys. Rev.

Lett. 35, 1262 (1975).

[2] For instance, see the 1973 Least Squai-es Adjustment of the Fundamental Constants, E. Richard Cohen and B. N. Taylor, J. Phys. Chem. Ref. Data 2, 663 (1973).

351
[3] See review article by B. N. Taylor, Metrologia 12, 81 (1976), containing the values of \xJp / ^, = 2.792 77406(103) {leastsquares—adjusted value of quantities measured by B. A.

Mamyi'in, N. N. Aruev, and S. A. Alekseenko, Zh. Eksp.

Teor. Fiz. 63, 3 (1972) [Sov. Phys. JETP 36, 1 (1973)], and

by B. W. Petley and K. Morris, J. Phys. A 7, 167 (1974)}

and ^J^/^lg = 0.001 520 993 139(18) [temperature-corrected

value of quantity measured by W. D. Phillips, W. E. Cooke, and D. Kleppner, Metrologia 13, 179 (1977)].

[4] G. Gartner and E. Klempt, Z. Phys. A 287, 1 (1978).

[5] G. Graff, H. Kalinowsky, and J. Traut, Z. Phys. A 297, 35

(1980) . [6] A. Wyler, C. R. Acad. Sci Ser. A 269, 743 (1969); 271, 186

(1971).

[7] See also D. 0. Fystrom, Phys. Rev. Lett. 25, 1469 (1970); J. L. Luxon and A. Rich, Phys. Rev. Lett. 29, 665 (1972); H.

Gubler, S. Munch, and H. H. Staub, Helv. Phys. Acta 46, 772 (1974).

[8] R. S. Van Dyck, Jr., D. J. Wineland, P. A. Ekstrom, and H.

G. Dehmelt, Appl. Phys. Lett. 28, 446 (1976).

[9] R. S. Van Dyck, Jr., P. B. Schwinberg, and H. G. Dehmelt,

in New Frontiers in High-Energy Physics (Plenum, New

York, 1978); also, H. Dehmelt in Atomic Physics 7, D. Kleppner and F. Pipkin, eds. (Plenum, New York, 1981)

p. 337.

[10] D. Wineland, Bull. Am. Phys. Soc. 18, 1573 (1973).

[11] R. S. Van Dyck, Jr., P. B. Schwinberg, and S. H. Bailey, in Atomic Masses and Fmidamental Constants 6, J. A.

Nolen, Jr. and W. Benenson, eds. (Plenum, N.Y., 1980)

p. 173.

[12] H. G. Dehmelt and F. L. Walls, Phys. Rev. Lett. 21, 127

(1968).

[13] G. Gabrielse and H. Dehmelt, Bull. Am. Phys. Soc. 26, 598

(1981) . [14] S. Liebes, Jr. and P. Franken, Phys. Rev. 116, 633 (1959).

[15] D. Wineland and H. Dehmelt, Int. J. Mass Spectrom. Ion Phys. 16, 338 (1973).

[16] P. B. Schwinberg and R. S. Van Dyck, Jr., Bull. Am. Phys.

Soc. 23, 40 (1978) and 24, 35 (1979).

[17] P. B. Schwinberg, R. S. Van Dyck, Jr., and H. G. Dehmelt,

Bull. Am. Phys. Soc. 24, 1202 (1979); P. B. Schwinberg

and R. S. Van Dyck, Jr., Bull. Am. Phys. Soc, 26, 598

(1981). Appendix A

A plausible explanation for the observed dependence of

the proton cyclotron resonance on number of trapped

protons A'^ can be found by noting the follovdng result from a paper by Liebes and Franken [14], but adapted

specifically to a Penning trap,

1 +

r dr 0 0 _ where ^ is the potential function vi^hich establishes the

electrostatic field and the derivatives are to be evaluated

at the guiding center of the magnetron motion. If fields from space charge are neglected (since they should not

affect the observed cyclotron frequency), the potential

function vdth the dominant anharmonic terms in cylindrical coordinates (r, z) is

(t) = (Vo /4Zo^) (r2 - 22 2) + Ba (8r^ - 2irh^ + 8z^)

where Ba —0.2 V/cm'' in earlier uncompensated Penning

traps [8]. Upon taking the appropriate derivatives, one

finds that

v; = V, - v,„ [1 + (48 BaZo^/Vo) (r2 - 2z\]

where the average of (r^ - 22^)0 over the charge distri- bution will contain the observed number dependence.

This expression can be carried a step further if one as- sumes a uniform charge density of ions that will repel

each other until the nearest neighbor distance is R, at which point radial spreading becomes insignificant or is balanced by the centering drive. In addition, if one as- sumes that no axial stacking of ions occurs (i.e., we can

thus neglect the z^ term), and that the area il/2)N'nR^

can represent the average of ^nir^ )o, then

where 7 = 24 BaZo^R^/Vo. The explicit appearance of

the anharmonic coefficient Ba indicates that a well compensated trap should not depend on A^. Also, since v„i is independent of mass, the relative correction to v'de') ~

140 GHz is insignificant compared to the correction to

v'cip^) ~ 76.4 MHz. Finally, if one used the observed

proton cyclotron number dependence of 0.0093 Hz /ion

and Ba ~ 0.2 V/cm\ one finds that R ~ 0.0035 cm. Of

course, if the trap is 100 times better compensated, then

R is 10 times smaller.Appendix B

Since the nickel rings produce a magnetic field that

depends on position in the trap, we must consider effects which will cause both charge types to see a different

magnetic field. Residual contact potentials on the electrodes, which

are expected to be much less than 100 mV, can be

resolved into constant electric fields in one of the three

independent directions. On the average, such fields will give rise to an average relative magnetic field shift given

by

8v,/v, = {B^Zo^/B,) {^V /2Vof , which for the parameters of our trap is much less than

7 X 10"^ and 1 x 10"^^ respectively for electrons and protons. Implicit in this calculation is the assumption that

the bottle and trap centers coincide. By construction,

this is valid to ± 0.001 cm which reduces the corresponding shifts [11] to much less than 1 x 10"'^ also. The particles' axial and radial motion also causes aver- age field shifts. The axial motion causes an average shift equal to hvc/vc = {B^Zo^/Bo) (kT/eVo)

which for r = 4 K corresponds to 2 x 10"^ and 8 x

10"i*' for electrons and protons respectively. For Vc(e"), the

low frequency edge is independent of this effect. However, the cyclotron frequency of both protons and elec- trons depends on the magnetron radius Rm according to 8ve /Vc = B2Ri/2Bo . For the case of a well centered electron motion in a pre- vious trap, R„i < 0.002 cm corresponds to 8 Vc /Vc < 2 x

10"^. If the centered magnetron radius for both protons

and electrons is the same, then the mass ratio does not depend on R„i. However, in order to verify that nip /mg

does not depend on any of these bottle related effects, mass ratio measurements will be made versus 5 2 by ad- ding an external magnetic ring to the present trap tube.

352
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

A Direct Determination of the Proton-Electron Mass Ratio

G. Graff, H. Kalinowsky, and J. Traut

Institut fur Physik der Johannes Gutenberg-Universitat D-6500 Mainz, West Germany

The cyclotron frequencies of free protons and electrons have been measured in a magnetic field of

5.81 tesla with a superimposed electrostatic quadrupole field. The increase of energy connected with a

transition at the cyclotron frequency is detected by the measurement of the time of flight through an inhomogeneous magnetic field. From the ratio of the measured cyclotron frequencies of both particles, the proton-electron mass ratio is deduced. The result, yn^lriif, = 1836.1527(13), agrees within the limits of error (0.69 ppm) with the value of the indirect determmation.

Key words: fundamental constants; proton-electron mass ratio; Penning trap.

1. Introduction

The determination of the proton-electron mass ratio is possible by both a direct and indirect method. The in- direct method requires two different types of measurements: the determination of the magnetic moment of the

proton in units of the Bohr magneton |Xp/fjLB and the

determination of the magnetic moment of the proton in units of the nuclear moment jJip/fXN. The combination of both results leads to a proton-electron mass ratio

mp/me = 1836.15152 (70) [1]. The direct determination is based on the measurements

of the cyclotron frequencies of proton and electron in the same magnetic field: mp/We - /c(Electron)//c (Proton)

2. Apparatus

2.1 Ion Trap

Protons as well as electrons are trapped alternatively

in a homogeneous magnetic field with a superimposed

electrostatic quadrupole field, the potential of which is V{x, y, z) = U/RoHx^ + y^^ - 2z^).

In this configuration the ions behave within the nonrelativistic limit like a three-dimensional harmonic oscillator with the follovdng frequencies:

/, - {2eU/mRo^)^/2TT

fB = fc/2 + fo

fsB = fc/2 " fo

with fo = (/c^/4 + fz^/2)^ and the cyclotron frequency

fc = eB/ym2Tr, y = {1 - v^/cY^.

Under the conditions chosen in our experiment (B =

5.8 T, Ro = 6 mm, and f/ = 1 V) /, is about 16 MHz

(367 kHz), /g = 163 GHz (89 MHz), and fss = 760 Hz

(760 Hz) for electrons (protons). The trap is made of

OHFL copper. Additional correcting electrodes provide a well-shaped quadrupole field. The central ring is split in the X, 2 plane to allow application of an rf-field perpendicular to the static magnetic field. At one side of the

quadrupole trap, 40 cm away from it but near the magnetic field axis, a heated tungsten wire serves as the

electron source. These primary electrons impinge partly

on the interior trap surface through the entrance hole of the endcap (1 mm diameter) producing secondary elec- trons or protons by secondary emission or residual gas

ionization, respectively. A few of them are trapped. Due

to synchrotron radiation in the strong magnetic field the

electrons loose their initial energy of a few eV and cool down to thermal energies of about 30 meV within a second.

2.2 Drift Tube and Detector System

The trap is installed in a 37 cm long copper drift tube

of 30 asm diameter. The surface of the tube was cleaned

by a glow discharge in an atmosphere of hydrogen. To

obtain a vacuum of 10"^ Torr the complete apparatus was

baked out at a temperature of 600 K for two days. At the

open end of the tube a channel plate detector is mounted

to count alternatively electrons or protons after ejection out of the trap (Fig. 1). The complete apparatus is in- stalled within a electromagnetically shielded enclosure to avoid any disturbance of the measurement by unwanted

electromagnetic fields. The data transfer between the ap- paratus inside the cabin and the data acquisition computer is made by fiber optic transmission lines.

magnetic field [t]

Figure 1. The experimental arrangement for the time offlight measurement.

2.3 Magnetic Field

A superconducting magnet is used in this experiment.

Its maximum field strength is 6.4 tesla. Room temperature access is provided along a horizontal bore with

353
52 mm diameter. To ensure a sufficient homogeneity

over the trap volume a series of coils is mounted on a vacuum enclosure. They allow a systematic correction of

linear field gradients in three directions and of the quadratic term along the axis of symmetry. These coils are run at room temperature. The drift of the magnetic field was less than 0.02 ppm per day when the magnet was

run at 5.28 T and about 1.0 ppm per day when run at 5.18 T (Fig. 2).

time [msj

electron -source

' m iOO 300 400 1500 1700

ring

AAAAr

time [h]

Figure 2. Proton cyclotron frequency shift as a function of time caused by the magnetic field drift (error bars correspond

to FWHM).

3. Method

There are three possibilities to obtain the cyclotron

frequency:

a) A measurement of fs at different trapping potentials U and an extrapolation to zero electrostatic

field strength.

b) A measurement of /b and fsB, the sum of which

amounts to the cyclotron frequency fc . c) A direct induction of a transition at frequency

C/b + /eb), which can be enforced at low electric

field strengths.

We selected the latter method since this transition is most insensitive to a misalignment of the trap, to space

charge effects, and to surface potentials. To detect the

cyclotron frequency the following procedure is used

which was proposed originally by Bloch [2]. After creation of a few electrons the potential is kept constant for one second during which the electrons cool down

(Fig. 3). Then the cyclotron frequency is applied for about 500 ms. Finally the trap is cleared by a linear

sweep superimposed by a sequence of pulses which define

the starting time of the ejected particles. With the same

pulses a series of consecutive gates is generated, which

open 24 fast counters one after the other. By this method

the time of flight spectrum is received. The time of flight

is determined by the initial energy of the particles along

the magnetic field axis (about 10 meV for electrons and some 100 meV for protons) and by their transverse en- ergy incorporated in the cyclotron motion. The particles are accelerated by the inhomogeneous magnetic field pro- portional to their transverse energy. When the cyclotron

frequency fc

frequency f;

Figure 3. Trapping potential.

frequency is induced, the trapped particles gain transverse energy. The cyclotron orbit of the particle

and consequently its magnetic moment increases as well. When these particles are expelled out of the trap they

get an additional acceleration in the inhomogeneous part

of the magnetic field corresponding to a decrease of their

time of flight. Thus the cyclotron frequency can be

detected.

4. The Measurement of the Cyclotron

Frequencies

4.1 The Proton Cyclotron Frequency

The cyclotron frequency of the proton has been determined as outlined in section 3. Choosing an initial energy

of 300 meV the mean time of flight is about 30 |xs. At

resonance it decreases by a few percent depending on the

rf-amplitude (Fig. 4). At each frequency the whole cycle

is repeated 30 times. The resulting time distribution

stored in the 24 counters is then read by a PDF 11/20

computer which calculates the mean time of flight and

the statistical error. To study the lineshape and possible

line shifts the following parameters have been varied: the

rf-amplitude, the trap position relative to the magnetic

field, the magnetic field strength, the trap potential, and

the space charge. In addition, the cyclotron frequency of the molecular ion was measured.

80620610 30 50 70 90 710 30

V [Hz] frequency

Figure 4. Proton time offlight as a function, offrequency.

4.1.1 Electric and Magnetic Field Dependence

To the extent that a pure quadrupole field is realized,

the transition frequency (/g + fss) should not depend on

the electric field strength. Within the statistical error

354
this appeared to be correct. At very low voltages

(100 meV), however, when surface potentials become

dominant, the center of ion motion was evidently shifted

along the 2-axis. This leads to observable shifts of the cy- clotron frequency depending on the position of the trap

and the magnetic field inhomogeneity. As expected the

transition probability appeared to be inversely proportional to the electric field strength. Measurements were

performed at 5.28 T and 5.81 T. The mass ratios m-plm^

resulting from both experiments agree within their statistical errors.

4.1 .2 Space Charge Effects Depending on the charge distribution in the trap the

two frequencies /b and /eb might be influenced dif- ferently. Therefore the cyclotron frequency fc may also depend on the charge distribution. Indeed, the cyclotron

frequency fc is both shifted and broadened proportionally

to the number of stored ions. Since the protons are generated by electron bombardment of the residual gas, other ions of heavier masses such as H20"^ and are

also produced and stored inside the trap. These ions contribute, of course, to the space charge. Therefore the

trap was cleared of all other ions before inducing the pro- ton cyclotron frequency. This was achieved in the follow- ing way: After ion creation the trap voltage is decreased

linearly down to 1 V. Simultaneously, a rf-field is ap- phed, the frequency of which is chosen so that all ions

heavier than the protons experience their resonant fre- quency (Fig. 3). The amplitude of this rf-field is high

enough to eject all unwanted ions. The cyclotron fre- quency was then measured for different numbers of trapped protons. Figure 5 shows linewidth and transition

frequency as a function of the number of stored protons.

cyclotron frequency

v.„ [Hz]

1 = 1 10-^

30 50 H-^- Ions Figure 5. Proton cyclotron frequency as a function of the number of trapped protons (error bars correspond to FWHM).

4.2 The Electron Cyclotron Frequency

Within a second after their generation the trapped

electrons are in a thermal state which shifts the electron

cyclotron frequency by 0.05 ppm due to their relativistic mass increase. By applying a stepwise increasing potential at an electrode located at the end of the drift tube,

the energy distribution of the electrons has been measured. Since the magnetic field strength is low at the position of this electrode this method yields the sum of the

energies in the longitudinal and transverse degrees of

freedom. As a result the width of the energy distribution

appeared to be less than 50 meV. The mean value of the

time of flight through the magnetic field for electrons is 3 |xs. At resonance the time of flight decreases by about

10 percent. An increase in microwave amplitude results

in an energy absorption by the trapped electrons, which

leads to a relativistic mass increase and therefore to a

shift and broadening of the cyclotron frequency (Fig. 6).

3.8

3.6 -

3 4

3 2

time of flight [|is]

4 10-^

1 059 083.4 4.2 5.0 5.8 6.6 v [Hz] frequency

Figure 6. Electron time of flight as a function of frequency

and microwave amplitude.

By means of measurements at different microwave amplitudes, this dependence was determined and extrapolated

to zero energy absorption. Smaller linevddth can be ob- tained by increasing the sensitivity for the detection of energy changes [3]. The sensitivity is presently limited

by surface potentials on the inner side of the drift tube. The transition frequency is influenced by power broadening, relativistic mass increase, and cooling effects due to synchrotron radiation. The resulting line shape could not be explained theoretically. Within the statistical error the line shape appeared symmetrical and was fitted by a Lorentzian function.

Again at very low trapping potentials a shift of the cy- clotron frequency was observed. Space charge effects were negligible.

5. Result and Discussion

The measurement of proton and electron cyclotron fre- juencies was performed alternatively at two different

nagnetic field strengths of 5.28 tesla (run duration

120 h) and 5.81 tesla (run duration 60 h). The drift of the

magnetic field during the measurement was determined

by the variation wdth time of both the proton and the

electron cyclotron frequencies. The values of the magnetic field drift calculated separately from the proton and

the electron cyclotron frequencies agree vdthin their statistical errors. The proton electron mass ratio m^lnie

resulting from both experiments agree within their statistical error (0.38 ppm). Table 1 summarizes all corrections to the cyclotron frequencies and Table 2 the error

contributions. The errors correspond to one-standardTable 1. Corrections for the determination of the cyclotron

frequencies of electron and proton.

fc -electron fc -proton

Space charge effect -0.05 ppm

Trapping voltage extrapolation -0.19 ppm +0.12 ppm

Zero kinetic energy extrapolation -HO. 05 ppm

Zero niicrowave amplitude extrapolation +0.09 ppm

Frequency generators difference +0.14 ppm

355
Table 2. Error contribution to electron and proton cyclotron frequencies and

to proton-electron mass ratio.

fc -electron /e -proton mp/rWe

Statistical errors Magnetic field drift extrapolation

Zero microwave amplitude extrapolation

Frequency generators difference

0.07 ppm

0 14 nnm

0.30 ppm

0.01 ppm

0.02 ppm

0 04 nnm

0.09 ppm

0 18 nnm

0.30 ppm

0.01 ppm

Systematic errors Zero kinetic energy extrapolation

Magnetic field inhomogeneity Zero microwave amplitude extrapolation

0.05 ppm

O.ZO ppm

0.40 ppm

0.20 ppm

0.05 ppm

0.40 ppm

0.40 ppm

0.69 ppm

deviations. Our final result is the mean value of both

measurements:

Wp/me = 1836.1527(13) (0.69 ppm) . The result of this direct determination of the protonelectron mass ratio agrees within the limits of error vdth

the most recent least-squares adjusted value [1], mp/me = 1836.15152(70) (0.38 ppm).

References

[1] E. R. Cohen and B. N. Taylor, J. Phys. Chem. Ref. Data 2, 663 (1973).

[2] F. Bloch, Physica 13, 821 (1953).

[3] R. S. van Dyck, P. B. Schwinberg, and H. G. Dehmelt, Phys. Rev. Lett. 38, 310 (1977).

356
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

A New Determination of the Atomic Weight of Silver

and an Improved Value for the Faraday

L J. Powell, T. J. Murphy, and J. W. Gramlich

Center for Analytical Chemistry, National Measurement Laboratory, National Bureau of Standards,

Washington, DC 20234

The atomic weight of a reference sample of silver has been determined by mass spectrometry uith

an uncertainty of one part in 10^. Accurately knowTi quantities of chemically pure ^"^^Ag and ^'^^Ag were mixed to produce standards of known isotopic composition for calibration of the mass spectrometer. The absolute isotopic ratio of the reference sample of silver is i^'Ag/i^SAg = 1.07638 ± 0.00022

yielding an atomic weight of 107.86815 ± 0.00011 [95 percent confidence level (CL)]. Recalculation of the Fai'aday using this atomic weight and the electrochemical equivalent of silver as determined by

Bower and Davis leads to a value of (96486.18 ± 0.13) Ax^igs'S -mol "-^ (68 percent CL).

Key words: atomic weight; chemical assay; Faraday; isotopic composition; mass spectrometi-y; silver. 1. Introduction

The redetermination of the atomic weight of silver was

undertaken in conjunction with the determination of a more accurate value for the Faraday constant. The Fara- day is directly related to other physical constants including the Avogadro constant, the proton gyromagnetic

ratio, the magnetic moment of the proton in nuclear

magnetrons, and the ratio of the NBS as-maintained ampere to the absolute or SI ampere. Over the years, a problem arose in assigning a recommended value for the

Faraday due to apparent discrepancies between the

Faraday determined from electrochemical experiments

and the Faraday calculated from other fundamental constants. Unfortunately, earlier electrochemical experiments were not sufficiently precise [6.8 ppm, 68 percent

confidence level (CL) estimate, for the best silver determination] to either prove or disprove the existence of

this discrepancy. As a result, the Faraday was excluded

from the 1973 CODATA least-squares adjustment of the

fundamental constants [9]. In 1980, Bower and Davis [1], using the same source of

silver as was analyzed in this work, NBS Standard

Reference Material (SRM) 748, published a new value for the electrochemical equivalent of silver with an uncertainty of 1.28 ppm (68 percent CL). The calculation of the

Faraday using this value and the atomic weight of silver determined in 1962 at NBS [2] indicated that significant

differences between the various Faraday calculations

might indeed exist. In order to assess the significance of these differences, however, it became necessary to significantly reduce the uncertainty in the atomic weight of

silver.

2. Approach

The method used for the determination of atomic

weights at NBS combines the techniques of high precision chemical assay with high precision mass spec- trometry. The mass spectrometers to be used for the

isotopic abundance measurements are calibrated for bias

using synthetic mixes of knowm isotopic composition,

prepared from nearly pure, separated isotopes. Extensive research has demonstrated that this bias is due to non-linearities in the measurement circuit and mass

dependent isotopic fractionation [3]. In the case of isotopic ratios near one (as for silver), this bias reduces to iso- topic fractionation. These measured biases are then used

to calculate the absolute isotopic abundance ratio and, ul- timately, the atomic w^eight of the reference sample. A

block diagi^am characterizing the atomic weight method is shown in Fig. 1. The chemical research requires the development of an

assay procedure for the pure element or reference sample

precise to at least one part in 10"*, which will ultimately

be used to assay the separated isotope solutions. In addition, a procedure must be developed to purify the

separated isotopes to eliminate impurities which could in- terfere with either the chemical or mass spectrometric

procedures. Once these methods have been thoroughly

Mass Spectrometric Atomic Weight Method

Initial Mass

Spectrometry Procedure

Separated Isotope

Purification & Purity Verification

identification and

Control of Error Sources

Calibration Mixes

Precise Mass

Spectrometric Procedure

Comparison of Mixes and SRM

Accurate

Isotope Ratio

Accurate Atomic Weighit

Figure 1. Block diagram of the -mass spectrometric atomic

weight method.

357
tested using the reference sample, the separated isotopes are purified and dissolved. Aliquots of each isotope are then taken to produce calibration mixes which are blended to bracket the natural isotopic abundance ratio of the element. In addition, aliquots of each isotope are taken for assay so that accurate solution concentrations can be ascertained. A knowledge of the solution concentrations combined with aliquoting data allows the actual

ratios of the calibration mixes to be calculated. The mass spectrometric research involves the development of a high precision method for the analysis of the

relative isotopic abundance ratios of the reference sample, followed by a comprehensive study to identify and

control any sources of bias which could affect the final measurements.

After the above tasks have been accomplished, two

complete sets of analyses of the calibration mixes and the

SRM are made by two different operators using two dif- ferent mass spectrometers. The calibration mixes are randomly selected and are analyzed in an alternating pat- tern with the reference sample. A comparison of the

relative isotopic abundance ratios obtained for the cali- bration mixes with the calculated or true value produces

a calibration factor. This factor is applied to the relative

isotopic ratio obtained for the SRM to produce the absolute isotopic abundance ratio. The atomic weight is then

obtained by multiplying the fractional abundance of each

isotope by its nuclidic mass^ and summing the resultant

products.

For the redetermination of the atomic weight of silver, a high-precision chemical assay procedure was developed

so that accurately known quantities of ^'^^Ag and ^°^Ag

separated isotopes could be mixed to produce standards

for calibration of the mass spectrometer. This method was based on the gravimetric determination of the major

portion of the silver (>99.7%) as silver iodide and the

determination of the soluble residual silver by isotope di- lution mass spectrometry. The relative isotopic composition of the calibration mixes and the high purity natural

silver were determined by surface ionization mass spec- trometry using a platinum single-filament silica gel tech- nique. This technique, which involves loading sequential

layers of silica gel, sample, and phosphoric acid onto the

filament surface, has not been traditionally used in high

accuracy atomic weight determinations because the

method of ionization enhancement is little understood.

However, in the case of silver, it proved to be the key to achieving levels of precision not possible with other

modes of ionization. Further details of these procedures

and the associated data are published elsewhere [5].

3. Results

The atomic weight of the reference sample of silver was determined to be (107.86815 ± 0.00011) g-mol"^

(95% CL). This reference sample will be issued as both an isotopic and assay Standard Reference Material and

will be designated as SRM 978a. Its isotopic ratio was

found to be experimentally identical to SRM 978, which was based on the 1962 determination of the atomic

weight of silver [2], the major difference being the uncertainty placed on the measurement.

^Nuclidic masses, as published by Wapstra and Bos [4], are known to an

accuracy of a few parts in 10 .

The redetermination of the atomic weight of silver has

allowed the recalculation of the Faraday, yielding a value

of (96486.18 ± 0.13) ANBs-s-mor^ (68% CL) [6]. Figure

2 shows the value of the Faraday calculated from this work as well as other values of the Faraday. Point D is the present CODATA recommended value of Cohen and

Taylor [9], which is calculated from other physical con- stants via least squares. It may be noted that not only

has the overall uncertainty in the Faraday constant been

reduced through this determination, but the value of the

Faraday is now essentially the same as that calculated

using the proton gyromagnetic ratio measured by Kibble

and Hunt [11]. The ultimate impact, however, is that the

Faraday will no doubt be included in the 1983 least- squares adjustment of the fundamental constants presently being carried out by Cohen and Taylor under

CODATA auspices.

Comparison of Recent Detenninations of the Faraday

A

1

I 1 E

I •

1 F I—

i

H

I -

I

I I I I 1 I I 96483 96484 96485 96486 96487 96488

Figure 2. Comparison of various values of the Faraday constant. A, Craig et al., [7]; B, Mamienko and Taylor [8]; C, Marinenko and Taylor [8]; D, 1973 Recommended [9]; E,

Koch and Diehl [10]; F, Kibble and Hunt [11]; G, Deslattes et

al., [12]; H, Bower and Davis, 1960 Ag. [1]; I, Bower and

Davis, 1980 Ag [this work].

References

[1] V. E. Bower and R. S. Davis, J. Res. Natl. Bur. Stand. 85, 175 (1980).

[2]. W. R. Shields, E. L. Garner, and V. H. Dibeler, J. Res.

Natl. Bur. Stand. Sect. A: 66, 1 (1962).

[3] E. L. Garner, L. A. Machlan, and W. R. Shields, Natl. Bur.

Stand. (U.S.), Spec. Publ. 260-27 (1971).

[4] A. H. Wapstra and K. Bos, At. Data Nucl. Data Tables 19, 177 (1977).

[5] L. J. Powell, T. J. Murphy, and J. W. Gramlich, J. Res.

Natl. Bur. Stand. 87, 9 (1982).

[6] V. E. Bower, R. S. Davis, T. J. Murphy, P. J. Paulsen, J. W. Gramlich, and L. J. Powell, J. Res. Natl. Bur. Stand.

87, 21 (1982).

[7] D. N. Craig, J. I. Hoffman, C. A. Law, and W. J. Hamer, J. Res. Nat. Bur. Stand. Sect. A: 64, 381 (1960).

[8] G. Marinenko and J. K. Taylor, Anal. Chem. 40, 1645 (1968).

[9] E. R. Cohen and B. N. Taylor, J. Phys. Chem. Ref. Data 2, 663 (1973).

[10] W. F. Koch and H. Diehl, Talanta 23, 509 (1976).

[11] B. P. Kibble and G. J. Hunt, Metrologia 15, 5 (1979).

[12] R. D. Deslattes, A. Henins, H. A. Bowman, R. M. Schoonover, C. L. Carroll, I. L. Barnes, L. A. Machlan, L. J. Moore, and W. R. Shields, Phys. Rev. Lett. 33, 463 (1974); and these proceedings.

358
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

High Resolution Magnetic Measurements on Rotating Superconductors to

Determine h/niQ*

B. Cabrera, S. (Benjamin) Fetch, and J. T. Anderson

Physics Department, Stanford University, Stanford, OA 94305

We have begun a new experiment to determine k/m.g (Planck's constant divided by the free elec- tron mass) to an accuracy of several parts per million using rotating superconducting rings. This resolution wiW allow for the first time investigation of recently predicted relativistic corrections at a level of 100 to 200 ppm. Here we present initial experimental measurements of h/jyi^. These include high

precision measurements of the induced magnetic lield from the rotating superconducting ring, showing

an improvement in resolution by two orders of magnitude over previously reported observations, and a

clear demonstration of flux quantization in the 50 mm diameter ring.

Key words: electron mass; flux quantization; fundamental constants; London moment; Planck's constant; superconductivity.

1. Introduction

A precise determination of /i/mg using a rotating

superconducting ring can be based on several of the re- markable macroscopic quantum properties of the super- conducting state. Becker [1] and co-workers first predicted that a resistanceless conductor rotated from

rest w^ould exhibit a magnetic moment proportional to its spin speed. The effect came to be know^n as the London

moment after the phenomenological theory of F. London

and H. London [2] was appHed by F. London [3] to a ro- tating superconductor. London predicted a uniform magnetic field

within the rotating lattice. It was first observed by Hil- debrandt [4] and has been verified several subsequent

times to an accuracy approaching a few percent [5-9].

Flux quantization through a superconducting ring, also predictd by F. London [3] (as h/e instead of the correct

h/2e), was first experimentally verified by B. S. Deaver

and W. M. Fairbank [10] (observing h/2.e) and R. Doll and M. Nabauer [11]. More recently, high precision meas- urements of the flux quantum h /2e have been based on

the ac Josephson effect [12-14]. A resolution better

than 0.1 parts per million (ppm) has been achieved, which

is by far the most accurate determination of any property

of a solid state system. At that time several suggestions were made for measurements of h/yn in rotating super- conducting rings [15]. These are based on balancing the

magnetic flux from an integral number of flux quanta

n{h/2e) against the London moment flux 2mto„S/e,

where S is the area bounded by the ring. Thus, we obtain

h/m =4S((o„ -a)„_i). (2) Several experiments have measured h/m in this way

[16, 17], with the most accurate result reported by Parker

and Simmonds [17] in 1970 at PMFC-I with an accuracy of 400 ppm. It is in agreement with the accepted value for the free electron at rest obtained by other techniques.

*This work was supported by NBS Precision Measurement Grant GB9026, NASA Contract NAS8-32355, and NSF Grant DMR 80-26007.

Technological advances in magnetic shielding [18],

SQUID magnetometry [19], cryogenic rotors [20], and

high precision dimensional metrology [21] during the past

decade, primarily associated with the Relativity Gyroscope Experiment [20], now allow an improvement in this determination to the several ppm level or better. For the

first time this will allow investigation of relativistic corrections in a solid state system. Recent theoretical es- timates by B. Cabrera, H. Gutfreund, and W. A. Little

[22] predict corrections of 100-200 ppm in many superconducting materials. In Section 2 we derive the exact non-relativistic ex- pression given by Eq. (1) and summarize the relativistic results. Section 3 outlines the measurement scheme and

describes the apparatus. Recent laboratory measure- ments including high resolution London moment and flux quantization data are presented in Section 4. We con- clude in Section 5 with a discussion of the prospects for future corrected values below the parts per miUion level.

2. Theory

The non-relativistic basis for the determination of h /m

can be understood in several ways. In Fig. 1 the Sommerfeld quantization rule is applied to the canonical

h/m = 4SAai

Figure 1. The Wilson-Sommerfeld quantization rule applied to Cooper electron pairs in a rotating superconducting ring leads

to a simple derivation of Eq. (2).

359
momentum of a Cooper pair around a superconducting

ring. If the ring is many London penetration depths

thick, a line integral can always be found where the su- percurrent is negligible and thus v can be replaced by

0) X r, the rigid body lattice rotation.

A more formal approach is obtained through the

Ginzburg-Landau supercurrent [23]

J (il»*Vii. - ijiVil;*) - -^i|i*i|jA,

7n

(3)

where the mass and charge e" are equal to 2m and 2e,

respectively, and \\i is proportional to the superconducting order parameter with phase cj), v|i = | ili | e ' *. Transforming Eq. (3) into a rotating coordinate system

using the Galilean relation v' = v - w x r, we obtain

ef! I , 2e2

ni V(t) -

ni 1^ \HA + A^), (4)

where we define an effective vector potential due to the

rotation

A„, - — (1) X r

.

(5)

Integrating around a closed path T contained within the

superconducting ring, which rotates at an angular velocity (o, we obtain

m (6)

where n, = 2 | v}; | ^ and S r is the area bounded by P. As

usual we have required \\) to be single-valued. In Fig. 2 we plot the flux through the path F given by the second

term on the right of Eq. (6) against the angular rotation

o). For CD = 0 Eq. (6) is exactly the fluxoid quantization

equation, and for w = 0 it gives the London moment rela- tion for the magnetic flux versus spin speed.

Figure 2. The magnetic flux versus angular velocity for a ro- tating superconducting ring. The magnetic flux nulls, where an integer number of flux quanta cancels the London moment

flux, are shoum.

If we assume the only source of B to be the super

current j (any constant background field will cancel in Eq. (7)), then there exists an m,, for each number of flux quanta n such that j = 0 and B = 0. These correspond to the (|) = 0 points on each parallel line in Fig. 2. We then

obtain the simple relation

/j/m = 4S Aw, (7) where Aoo = co„ - 00,1-1 is the angular velocity difference

between successive nulls. Thus, by measuring a macroscopic area and a frequency difference, a high precision

determination of h/m is possible.

The theoretical analysis [22] for the relativistic corrections to Eq. (6) can be summarized to order (v/c)^ by re- placing m by an effective mass m ' in Eq. (7) (not to be

confused with the soHd state effective mass), where now m m 1 + <T>

mc mc

(8)

The first correction term is due to the mass-velocity shift and for an isotropic superconductor is given by the ex- pectation value of the electron kinetic energy <T> aver- aged over all Fermi surface states (actually states within

A, the superconducting energy gap, of the Fermi surface). It is a factor of 5 or more larger than the second

correction term, which depends on the inner crystal elec- trostatic potential 4>in. Net mass increases of 100 ppm

are predicted for many superconductors. Detailed solid state band structure calculations using existing tech- niques are needed and are expected to give 1% or better

estimates for the correction terms in Eq. (8).

3. Measurement Scheme and Apparatus

The measurements are made with a cryogenic helium

gas bearing constructed entirely of fused quartz (see

Fig. 3). The rotor is a 25 mm radius sphere truncated

1 mm above its equator. A precision 25 (xm wide by

50 nm (500 A) thick superconducting ring will be deposited onto the equatorial plane using photoresist masking

techniques. Gas bearings in the flat top and spherical

bottom of the housing support the rotor with clearances

of 15 |jLm. Helium spin-up channels are located in the

sides of the housing, and spin speeds of up to 150 Hz in

GAS INLETS MAGNETOMETER

DETECTION LOOP

SUPERCONDUCTING RING ON EQUATOR

HELIUM GAS BEARINGS

-UJ SPIN UP CHANNELS —EXHAUST PORTS

+ CD SPIN UP CHANNELS

SPIN UP CHANNELS

Figure 3a. The allfused-quartz helium gas bearing used for

the hlm^ measurements shown schematically.

360
0

Figure 3b. The all-fused-quartz helium gas bearing used for

the hlrrig measurements photographed just prior to assembly.

both directions will be possible. The rotor spin speed is measured using a fiber optic readout mounted through

the bearing top flat. The intensity of the reflected light

is modulated by a frosted radial line in the rotor top surface as it passes under the fiber bundle.

The magnetic flux nulls are measured using a superconducting pick-up loop placed in a groove cut into the top of the lower housing. This loop is coupled to a SQUID magnetometer and is capable of resolving 0.01 of a flux quantum through the rotating ring in a bandwidth of several

hertz. To obtain the necessary stability in the SQUID detection circuit, an ultra-low magnetic field environment is essential. A technique developed by one of us (Cabrera)

uses expandable, superconducting, cylindrical shields

[18]. In bootstrap fashion newly folded shields are carefully cooled inside previously expanded ones until the

desired field reduction is achieved. Shields with absolute

fields below 10"^^ T are produced routinely. No observable changes occur in the field levels even after several

years of use. The determination of the frequency spacing between

nulls A oj is accomplished by counting the number of magnetic field nulls (integer and fractional parts) between

the top spin speeds in the two opposite rotation directions (see Figs. 4 and 5). The spin speed is slowly varied

in the pattern of Fig. 4 (upper) over several hours as the

ring is continuously modulated through its transition

temperature every 1 to 10 seconds (Fig. 4 (middle)) by

controlling the spin-up gas temperature. Since for our

large diameter ring <1^/2L is smaller than one half kT,

after many heatings and coolings a Gaussian distribution

of quantum states results around the equilibrium value

(Fig. 4 (lower)). The rms is about 3 (jjo- However, each

time the ring becomes superconducting a single well- defined quantum state from the distribution is measured

(Fig. 5). A change in frequency of 1 hertz corresponds to passage through 70 nulls. Thus, for our initially cautious

spin speed maximum of ± 50 hertz, 7000 nulls will be in- cluded. Every null need not be observed; only a sufficient number so as to unambiguously determine the total (about every tenth one).

Our initial data acquisition scheme is shown schematically in Fig. 6 with the timing sequence given in Fig. 7. The bearing spin speed is controlled using two Reedair

T

$ = 0

Figure 4. The spin speed is slowly varied (upper) as the ring

ternperature is continuously modulated through T^ (middle).

The large ring inductance causes a Gaussian distribution of

states about <\) = 0 at any w (lower).

•• V V ^ r

NCREASING U)

n + 2

n + i

n

Figure 5. During the ramping phase the nulls must be unambiguously counted (upper) with sufficient integration times at

^niax ^minfor a 1% resolution between nulls (loiver).

361
valves placed in the room temperature portion of the

spin-up gas lines. These valves are miniature, on-off,

reed valves. The states of the two valves are complementary and are controlled by an oscillator with fixed fre- quency and variable duty cycle. The pulse width can be

set using a 10-turn potentiometer or an external ramp

generator. Since the valve operation is complementary,

the total gas flow is constant. However, the distribution

of gas between the -t-w and —oj spin-up channels varies

linearly with the pulse width, thus providing a straightforward method of spin rate control. The constant gas

flow simplifies temperature stabilization.

PICK-UP SQUID Y X-Y LOOP MAGNETOMETER RECORDER

1

FOTONIC FREQUENCY D/A SENSOR COUNTER CONVERTER

HELIUM GAS

BEARING AND ROTOR

SPIN GAS HEATER PULSE PULSE HEATERS GENERATOR SEQUENCER

SPIN GAS VALVES

VARIABLE PULSE WIDTH GENERATOR

RAMP GENERATOR

Figure 6. Block diagram of data acquisition scheme.

I

HEATER

PULSE

COUNTER

INPUT - GATE

COUNTER

GATE

PLOTTER

GATE

PLOTTER.

PEN 30

TIME (sec)

Figure 7. Data acquisition timing sequence.

Temperature modulation of the bearing is accomplished

using manganin wire heater pellets located in the spin-up

gas lines. Heater power is supplied by the heater pulse

generator, which operates in a short-pulse mode or at a 50% duty cycle. The oscillator in this unit is the clock for the heating cycle (typically 15 seconds), as is shown in Fig. 7. The heater pulse generator provides heater

power pulses of selected amplitude and length (up to one

second) and an adjustable dc bias power level by which

the base temperature can be set to just below the ring

transition temperature. The heater power going to each

spin-up line is apportioned according to the amount of gas flowing in that line. The heater has power ranges of 50 mW and 500 mW.

To determine Aw in Eq. (7), the bearing spin speed and

the magnetic flux in the ring are measured simultaneously (Figs. 5 and 6). The spin speed signal from the

KD 100 Fotonic sensor fiber optic output is read using an HP 53 16A frequency counter. The counter digital output

signal is then converted to an analog voltage and plotted on the X-axis of an x-y recorder. At the same time, the

magnetic flux is measured with a SQUID magnetometer

coupled to a pick-up loop embedded in the bearing housing and is plotted on the y-axis of the recorder.

The data acquisition process is controlled with the

pulse sequencer (Fig. 6). The clock period is given by the

heater pulse generator (Figs. 6 and 7). A short time after the heater cycle begins (typically 2 seconds, allowing the return to thermal equilibrium) both the external counter

input gate and the plotter gate open. The next Fotonic sensor pulse triggers the counter and opens its internal

gate. An integral number of pulses are then timed; the

first pulse that occurs after the nominal internal gate

time has elapsed simultaneously closes the internal and

external counter gates. During the next second the

plotter pen is lowered and raised, simultaneously recording the frequency and magnetometer readings. The complete sequence repeats with the next heater cycle. Wlien

the 50% heater duty cycle is used, two data points are obtained during each heater period, alternately measuring the rotating ring in its normal and superconducting

states, as shown schematically in Fig. 5. The pulse sequencer has also been designed for operation under microcomputer control, and this will soon be

implemented. The clock signal will then be supplied by

the computer to the pulse sequencer. In addition to the x-y plotter, the data will be recorded by the microcomputer using the digital interfaces on the counter and the

magnetometer.

Our frequency resolution of 1% between adjacent nulls together with spin speeds corresponding to 7000 nulls will allow a measurement of A co to an accuracy of 2 ppm. Since

improvements in this accuracy to better than 0. 1 ppm ap- pear relatively straightforward, for the foreseeable future

the dominant error in the experimental determination of h Im through Eq. (7) is the area measurement.

In a collaboration with the National Bureau of Standards in Gaithersburg, MD, we have determined the equatorial plane areas for our two rotors to a Ict accuracy of

2 ppm. This work has been reported in detail elsewhere

[24] and will only be summarized here. The thickness

(50 nm) and width (25 fjim) of the superconducting ring

deposited on the equatorial plane of the rotors are chosen

so that all possible cross-sectional areas bounded within

the superconductor have the same value to a resolution of 2 ppm. Thus, a determination of the ring area to this ac- curacy requires only measurements in the equatorial

plane of the bare rotor. The area of a plane figure cannot be determined by any

number of diametral measurements, since there exist fig- ures of constant width other than a circle, all having the same circumference but varying in area [24]. To avoid

this ambiguity the radius of the figure as a function of angle must be measured with respect to a fixed point. The

area uncertainty in diametral measurements is eliminated

by using the rotor sphericity data to correct one absolute

diameter measurement to the true mean diameter.

A computer-aided Talyrond roundness measuring sys- tem is used to determine both the rotor profile and its least squares reference circle with a precision of better

than 5 nm (50 A). If matched by corresponding accuracy

in the individual diametral measurement, the area could

362
be determined to 1 part in 10'^. However, at present, absolute diameter measurements are typically limited to 1 or 2 ppm, although significant advances seem likely in the near future. The NBS is assisting us in these meas- urements using a horizontal diameter measuring machine

based on the HP Laser Interferometer.

To compensate for the expansion of the quartz rotor from 20 °C to 4 K (about 17 ppm), measurements made

through an additional collaboration with the NBS will be

taken on samples from the original rotor blanks.

4. Measurements of the London Moment,

Flux Quantization, and h/niQ

Our inital cryogenic testing of the complete system as described in Section 3 has been very successful. The

helium gas bearing performed well at STP with nominal

gas flows of 500 ml per minute through the bearings. A

spin gas flow of 2000 ml per minute easily allowed spin

speeds up to 15 hertz, our self-imposed initial maximum.

For these first runs, a 40 jxm diameter niobium wire ring was glued to the top of the hemispherical rotor. Its nominal diameter of 49.49 mm allows absolute area measure- ments to only several parts per thousand, thus limiting our present determination of h /m.e through Eq. (2) to

this level. However, this ring has allowed us to successfully test the measurement scheme and demonstrate

resolutions for Aoo approaching parts per million. Using

the SQUID magnetometer readout system and our superconducting ultra-low magnetic field shield (5 X 10"^^ T), we were able to obtain high resolution data of the magnetic moment induced in a superconductor by rotation

(the London moment, Eq. (1)) and to observe clearly flux quantization through the ring (w = 0 in Eq. (6)). These

data, presented below, were taken with the semiautomatic data acquisition system described in Section 3. Flux quantization through this 50 mm diameter ring is observed most convincingly by biasing the temperature

within several millikelvin of the ring's transition temperature - 9.2 K and observing the thermally induced

transitions between neighboring quantum states. A time

plot is shown in Fig. 8, where the temperature is slowly

varied around T^. with about a one minute period. A one

hertz bandwidth for the SQUID output is used. The

quantized flux steps are shown and labeled on the right

side of the figure, whereas the equivalent response at the

SQUID is labeled on the left side. As far as we are aware, these measurements represent the largest area

for which flux quantization has been demonstrated.

Measurements of the magnetic moment of the niobium

ring versus its spin speed were obtained at a constant temperature of 4.5 K. The heaters were disconnected

from the heater pulse generator, which continued to pro- vide the clock frequency for data acquisition. One data

point was generated every 5 seconds. Figure 9 is a plot

of the magnetic flux through the ring, in units of the flux quantum ^o, versus spin speed. It was taken with the

spin speed stabilized to within several millihertz using

feedback. This feedback scheme was implemented by re- placing the ramp generator signal to the variable pulse

width generator by the sum of the ramp generator signal

and the output of the frequency counter after it had been

converted to analog and inverted for negative feedback

(refer to Fig. 6). Note that the zero intercept of the plot

is not arbitrary but was determined by recording the

magnetometer reading at zero velocity.

200

400

600

Figure 9. Rotationally induced magnetic field (London moment) versus spin speed.

The high resolution data of Fig. 10 shows both the flux quantization and the London moment and is exactly an

experimental plot of Eq. (6) and Fig. 2. These data were

taken in a one hertz bandwidth with the temperature

held 100 millikelvin below T(. for each parallel line. After

each line was recorded, repeated heatings and coolings were performed to obtain successive quantum states to- ward each end of the plot. From the data of Fig. 10

alone. Aw of Eqs. (2) and (7) is determined to several

parts per thousand. This resolution per null demonstrates a statistical error in A oj of a part per million

when 7000 null crossings are counted (±50 hertz). The

quantum state labels such as 208 are not arbitrary but

are referred to the flux zero as defined when the ring is in its normal state. The ability to resolve several millihertz changes in the rotational velocity of a superconductor as seen in a magnetic flux readout (Fig. 10)

represents an improvement by two orders of magnitude

over all previously reported measurements.

3.100 3.150 3.200 TIME (minutes HERTZ

Figure 8. Thermally induced transitions between quantum

states clearly demxmstrating flux quantization.

Figure 10. Null crossings for several adjacent quantum states allowing a determination of Aiiito high precision.

363
Our determination of h/m-e from these data (Fig. 10) is limited to 3 parts per thousand because of the difficulty in absolutely measuring the area bounded by our preliminary wire ring. At this level it is in agreement with the

previous work of Parker and Simmonds and with the ac- cepted value obtained from measurements on phenomena

not involving superconductivity. We are now working on

depositing a precision ring on the rotor equatorial plane

to obtain accuracies at the several parts per million level.

5. Conclusions

In initial cryogenic measurements of the induced magnetic flux from a rotating superconducting ring (London

moment) using a precision helium gas bearing, we have

demonstrated an improvement in resolution by two or- ders of magnitude over previously reported observations.

Magnetic moment signals from angular frequency

changes of 3 millihertz were detected. We have also clearly shown the flux quantization levels for our 50 mm

diameter ring. To our knowledge this represents the

largest area in which quantization has been observed.

The combined measurements determine h lm^ to 3 parts

per thousand. The value is in agreement with the ac- cepted value (known to several tenths of a part per million) and is presently limited by our preliminary wire

ring area uncertainty. Using this apparatus with minor

improvements, we will next make measurements on a

precision niobium ring and expect to reach a resolution of 2 ppm in the determination of A co during the next year. Combining this result with our area determinations

corrected to 4 K, we expect an overall uncertainty of 2 ppm on the right side of Eq. (7). Through comparisons with the presently accepted value

of /i/W e for the free electron at rest, we will for the first time be investigating relativistic corrections to the elec- tron mass within the superconducting lattice. Detailed

solid state calculations based on the relativistic theory of

Cabrera, Gutfreund, and Little are expected to match the

experimental accuracy. Various elemental superconductors will be measured to verify these calculations. After the consistency of the calculations and measure- ments has been established, an independent value ofh/rrie

for the free electron can be obtained. Future improvements in the determination of areas together with more

detailed calculations should provide a sub-ppm value, possibly approaching 0.1 ppm. Then the simple relation

a'^ = (h/Trie)

c would provide an independent value of the fine-structure constant a approaching an accuracy of 0.05 ppm, where

the Rydberg constant and the speed of light c are known to several parts per billion^ Then these measure- ments of h/nig would provide a significant input datum

for the next least-squares adjustment of the fundamental

constants.

at the University of Virginia, and its construction was

due to D. E. Davidson. The authors also wish to thank

the Stanford Relativity Gyroscope group and, in particular, C. W. F. Everitt and W. M. Fairbank, for technical

assistance and much encouragement.

References

[1] R. Becker, F. Sauter, and C. Heller, Z. Phys. 85, 772 (1933).

[2] F. London and H. London, Proc. R. Soc. London, Ser. A : 147, 71 (1935).

[3] F. London, Superfluids, Vol. 1 (John Wiley & Sons, Inc., New York, 1950).

[4] A. F. Hildebrandt, Phys. Rev. Lett. 12, 190 (1964).

[5] A. F. Hildebrandt and M. M. Saffren, in Loiv Temperature Physics, LT9, Ed. by J. G. Daunt et al. (Plenum Press, New York,

1965), p. 459.

[6] C. A. King, J. B. Hendricks, and H. E. Rorschach, ibid., p. 466.

[7] M. Bol and W. M. Fairbank, ibid, p. 471.

[8] N. F. Brickman, Phys. Rev. 184, 460 (1969).

[9] K. Oide and H. Hirakawa, J. Phys. Soc. Jpn. 43, 1087 (1977).

[10] B. S. Deaver and W. M. Fairbank, Phys. Rev. Lett. 7, 43

(1961).

[11] R. Doll and M. Nabauer, Phys. Rev. Lett. 7, 51 (1961).

[12] W. H. Parker, B. N. Taylor, and D. N. Langenberg, Phys. Rev. Lett. 18, 287 (1967).

[13] R. F. Dziuba, B. F. Field, and T. F. Finnegan, IEEE

Trans. Instrum. Meas. IM-23, 264 (1974).

[14] V. Kose, IEEE Trans. Instrum. Meas. IM-25, 483 (1976).

[15] B. N. Taylor, W. H. Parker, and D. N. Langenberg, Rev. Mod. Phys. 41, 375 (1969).

[16] J. E. Zimmerman and J. E. Mercereau, Phys. Rev. Lett. 14, 887 (1965).

[17] W. H. Parker and M. B. Simmonds, \x\ Precision Measurement and Fundamental Constants, Ed. by D. N. Langenberg and

B. N. Taylor, Natl. Bur. Stand. (U.S.), Spec. Publ. 343 (Aug.

1971), p. 243.

[18] B. Cabrera and F. van Kann, Acta Astronaut. 5, 125 (1978).

[19] B. Cabrera and J. T. Anderson, in Future Trends in Superconductive Electronics, Ed. by B. S. Deaver, Jr., et al., AIP Conf. Proc. No. 44 (American Institute of Physics,

New York, 1978), p. 161.

[20] C. W. F. Everitt, in Experimental Gravitation, Ed. by B. Bertotti (Academic Press, New York, 1973), p. 331; J. A.

Lipa, Proceedings of the International School of General

Relativity Effects in Physics and Astro-Physics: Experi- ments and Theory (3rd Course), Document MPI- PAE /Astro 138, Max Planck Institute for Physics and Astrophysics, Munich, November 1977, p. 129.

[21] G. J. Siddall, SME Tech. Pap. Ser. IQ78-167 (1978).

[22] B. Cabrera, H. Gutfreund, and W. A. Little, Phys. Rev. B 25, 6644 (1982).

[23] For example: P. G. deGennes, Superconductivity of Metals and Alloys (W. A. Benjamin, Inc., New York, 1966); also A. L. Fetter and J. D. Walecka, Quantum. Theory of Many-Particle Systems (McGraw-Hill, San Francisco,

1971).

[24] B. Cabrera and G. J. Siddall, Precis. Eng. 3, 125 (1981).

We greatly appreciate the assistance given to us by the

National Bureau of Standards in performing the measure- ments of absolute diameters and for many helpful discussions. At Stanford previous work on the design of the

helium gas bearing was contributed by G. B. Hess, now

Refer elsewhere in these proceedings to the many excellent papers on

these measurements.

364
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

High Precision Measurement of the Electron Compton Wavelength (hlm^)

Using Cryogenic Metrological Techniques

J. C. Gallop, B. W. Petley, and W. J. Radcliffe

Division of Quantum l\/letrology, National Physical Laboratory, Teddington, Middlesex TW11 OLW, England

This paper reports progi'ess on the NFL measurement of the Compton wavelength of the electron, which relies on the unique properties obtainable in cryogenic environments. The method is to trap a magnetic field in a superconducting tube. The experimentally measured quantities are the spin precession frequency of the ^He nuclei in the trapped field, and the cross sectional area of the tube. Novel

cryogenic metrology is used for both of these measurements. Key words: cryogenic metrology; free precession; fundamental constants; SQUIDS; superconducting

cavity.

1. Introduction

This paper reports progress on an experiment to determine the Compton wavelength of the electron using a number of novel cryogenic techniques. It is one of a number of interesting fundamental constant determinations vi^hich rely essentially on the phenomenon of fluxoid

quantization in superconducting rings [1]. These measI urements involve a length determination at liquid helium

I temperatures, and the difficulty of carrying out such a

j measurement v^dth high precision has so far inhibited

! serious attempts in this area. The work reported here at- tempts to use the liquid helium environment as an asset

j to the measuring process, rather than a hindrance. A

' measurement of him,, by a rotating superconductor

I method is being undertaken at Stanford University [2].

j This involves an entirely different dimensional measuring

technique and also a relativistic correction. If both meas- urements of hlm^ can be taken to sub-ppm accuracy the

tvio results should allov^r an interesting comparison of

j their radically different methods.

2. Basis of the Experimental Technique

j If a uniform axial magnetic field is applied to a superj conducting tube of inner cross-sectional area A as it is cooled through its transition temperature and is then re- moved, the magnetic field wall remain trapped in the

tube. The single valued nature of the macroscopic vi^aveI function describing the superconducting electron pairs

requires that the magnetic flux linking the tube must be

quantized in units of <I>o = hl2e. If a sample containing

polarized '^He nuclei (gyromagnetic ratio ^He) is inserted

into the uniform field region at the center of the tube the

free precession frequency cone of the nuclei can be measured [3]. It is related to the magnetic field by the

expression

^He = "YHe -^o . (1) Now the flux Hnking the tube is related to B by the

expression

JB • dA = n$o,

where n is an integer. As a result of the variation of the

circulating current density along the axis of the tube the

section of the tube than would be the case for a uniformly

wound solenoid of the same dimensions. This means that

to a very good approximation the above expression may

be replaced by the much simpler relationship

Bo = n^jA . (2) One of the novel aspects of this work is in the method

used to measure the cross sectional area A of the tube.

The tube is converted, by the addition of niobium endplates, into a high Q superconducting microwave cavity.

A measurement of a number of the resonant frequencies

of the cavity allows both its leading dimensions and cer- tain aspects of its irregularities in shape to be calculated

with high precision. If the deduced cavity radius is r then

A = Trr^ = T^C^ {fhnn} , where g{fimn} is some function of the set of measured

resonant frequencies {fimn} [4]. Combining Eqs. (1) to (3) we have then

7He <I>o = TT COHe(c ^ Q

^ \flmn })/" • (4) The ratio yuehp has been measured rather accurately [5], to about 0.02 ppm, and the left hand side of Eq. (4) may

be written as (7He/7p)7p^o = iyue/ypXi^/ i^B)h/2mg, (5) where [i-p/i^s is the proton magnetic moment expressed

in Bohr magnetons. Thus we have finally

h/m^ = 2 TT CJHe / W (C 2

5- 2 [(^/ (Xfi ) (7He / 7p )] , (6) where the parameters outside the square brackets are to be determined experimentally and those inside are al- ready known with an accuracy much better than 1 ppm.

Thus the complete experiment requires three main goals:

(1) to produce a uniform magnetic field in a superconducting cylinder; (2) to observe nuclear magnetic reso- nance of ^He in this uniform field; and (3) to determine

the cross sectional area of the cyhnder by a microwave

resonance technique. The progress made towards fulfill- ing each of these requirements will be reviewed in the

following three sections.

3

3. Observation of NMR in Liquid He

The reason for choosing ^He as the NMR probe to measure the trapped field is that it is one of a very few

systems with non-zero nuclear magnetic moment which

365
are liquid or gaseous at superconducting temperatures

and thus allows a narrow resonance line to be detected.

The ^He free precession frequency has been measured

with the prototype system with a precision of about

±0.02 Hz. When the full field uniformity has been

achieved this should be improved by at least a factor of

10^, so that it seems reasonable to expect that the fre- quency will be measured to within ±0.0002 Hz,

corresponding to an uncertainty in the magnetic field measurement of dB rfwHe/^He = 5 X 10-12 T

A precision of 0.3 ppm in the frequency measurement

would then require a trapped field of only 2 x 10"'^ T. At

a temperature of 3 K and using a SQUID as a low noise

detector of the free precession signal, it is estimated that

the number of spins required to provide adequate signal

to noise ratio is not more than 10^^, corresponding to a sample volume of less than 1 mm^.

A prototype system has been built which has been used

to detect, for the first time, free precession signals from

the polarized ^He spin system in low fields, using a SQUID. A schematic diagram of the system is shown in Fig. 1. The ^He sample is contained in a pyrex bulb,

around which is wound the primary coil (of high conductivity normal metal). This coil is connected to a multi- turn coil made of superconductor which serves as the

signal coil for an rf SQUID, which has a bandwidth in excess of WHe- The sample is then polarized in a trans- verse magnetic field of around lO'^ T provided by a coil arrangement external to the superconducting tube. When

sufficient polarization of the '^He spins has been attained

this field is removed and the sample is moved axially into

the uniform field region of the tube. At present the NMR

signal has been observed in a magnetic field in the range

of 10"^ - 10"'^ T, trapped in the outer, lead, superconducting tube (see Fig. 1). The decaying free precession

signal detected with a SQUID is shown in Fig. 2. This

corresponds to a beat signal between the SQUID output

signal and a clock frequency of 1.0 kHz. The decay time

of the signal is believed to be set by the homogeneity of

the trapped magnetic field, a Tl time of ~1 s., corresponding to a field uniformity of around 3 x

10"^ T/m. This is compatible with the value of the gra- dient component dB^ Idz, measured with a flux-gate magnetometer which could be moved along the axis of the

tube.

4. Measurement of Tube Diameter by a

Microwave Resonance Technique

The superconducting tube must be circular in cross sec- tion and of uniform diameter, at least over a region near

the center of length comparable with the diameter. This

requirement stems from the need for the trapped field to be uniform enough to allow resolution of the NMR signals

corresponding to integer values and (n -I- l)<I>o of

the quantized flux. The maximum diameter of the tube is set by the criterion that n must be determined for the

minimum value of B^ consistent with an accurate determination of (DHe- That is if the center of the resonance

line can be determined to an accuracy of, say, 3 x 10"^

then n £ 10^ Taking rf5 = 5 x lO^^^ t (section 3) r < {^jT^dBY'^ = 10 mm . Boring and lapping techniques available at NPL have al- lowed a circular hole of 23 mm diameter to be produced

in a solid block of niobium (about 120 mm in length) with

a tolerance of 0.2 |xm over a 50 mm length. It is beSuperconducting

lead shield

Polarising coils

El

Figure 1. A schematic diaugram of the apparatus for detecting

the ^He free precession signal using a SQUID.

[ ^

Figure 2. Free precession signal from ^He sample at i.2 K in a field of 10~^ T, detected with a SQUID.

lieved that at any point on the axis the deviations from

circularity are an order of magnitude less than this. (A

check on the circularity should be available from the

measured frequencies of such microwave mode families as the TEsin , which do not possess complete rotational symmetry about the tube axis [4].) Already the general technique for measuring the mean

366
diameter of a tube at room temperature by this method

has been demonstrated. It has also been possible to show that changes in mean radius of the tube with axial position can be measured. A brass cylindrical cavity was

used, which had a step change of about 30 |xm in radius over its central region [4]. Work is now proceeding on measuring resonant modes of the superconducting nio- bium cavity. About 30 microwave resonances in the range

7 to 20 GHz have been detected with this cavity, in vac- uum, from room temperature down to about 2 K. These

measurements indicate that the shape of the cylinder

does not distort significantly as it is cooled down. The

thermal expansion coefficient has been measured and

compared with other experimental values, giving agree- ment to within about 10%. The highest Q values so far observed are of the order of 2 x 10 , which is quite suffi- cient to allow the resonance line center to be determined

to 1 part in 10^, the required precision for an overall ex- perimental accuracy of 0.3 ppm. In view of the large

number of resonant frequencies existing in the micro- wave frequency range available to us (7-20 GHz) a semiautomatic system is being developed to determine

frequency, linewidth, and amplitude of the resonances as a function of temperature.

5. Production of a Uniform Trapped

Magnetic Field

One of the crucial requirements for the proposed high

precision determination is that it should be possible to trap a uniform magnetic field through the superconducting tube such chat the fluxoid quantization number is constant along the length of the tube. That is to say no

flux lines must leave the tube through the superconducting wall. Preliminary experiments, in which a tube of

lead was cooled through its superconducting transition in the earth's magnetic field by cooling it in a cryostat by

immersion in liquid helium, showed that although rather

uniform magnetic fields can be trapped in this way

{dbo/dz ~ 3 X 10~'^T/m), this uniformity is about two

orders of magnitude worse than would be expected if no

flux lines left the tube through the wall. The experimental arrangement now being used consists of an outer

lead tube, which is closed at both ends with non super- conducting plates, which serves as a vacuum can as well

as a superconducting shield. Inside this is suspended the

precision niobium tube. The support structure is of low

thermal conductivity and the niobium tube is supplied

with a heater so that after the lead tube has become su- perconducting the niobium tube may be heated through

its transition temperature before being allowed to be- come superconducting again in a controlled way via adjustment of the heater current. The uniformity of the

trapped field can be measured (although not its absolute

value) by moving the pick-up coil of a SQUID magnetometer along the axis of the niobium tube. In this way it has been shown that the "heat flushing" technique just

described can improve the trapped field uniformity by at least two orders of magnitude and the effect of single

flux lines leaving the tube through the walls can be

clearly seen. A region of about 20 mm in length has been

produced over which constant flux quantum number has

been achieved, and the axial field gradient dB^ Idz < 6 x

jQ-io

'p/jyi -pjjg ^Q^a,l change in trapped field over this length is less than 2 parts in 10^. It is planned to extend

this region over the whole length of the tube, by improved heat flushing or perhaps by high temperature an- nealing of the precision niobium tube.

6. Summary

A novel method is described here for measuring hlm^

(or equivalently the electron Compton wavelength

hlm^c) based on magnetic flux quantization and dimensional measurement using a high Q superconducting cavity. Although a number of questions remain to be

.answered it is hoped that the method will allow an ulti- mate precision of better than 1 ppm. At this level of precision a value for hlm^ would provide a useful input

datum for the next evaluation of the fundamental constants. When combined with the Rydberg, it provides a value for the fine structure constant with an

accuracy twice that of hhn^. This constant also provides

a direct link between /?/2e, the unit in which the maintained volt is measured, and the unit in which the maintained ampere may be monitored. Such a measurement

is of interest in its own right but perhaps more importantly it allows scope for investigating new techniques in cryogenic metrology. It is believed that the progress

made so far with this work suggests there are a number

of interesting applications of the new metrology involved.

These include a SQUID free-precession magnetometer

[6], a novel form of nuclear gyroscope [7], and the application at room temperature of the microwave dimensional

measurement technique [4]. References

[1] B. N. Taylor, W. H. Parker, and D. N. Langenberg, Rev. Mod. Phys. 41, 375 (1969).

[2] B. Cabrera, S. Benjamin, and J. T. Anderson, these

proceedings.

[3] J. C. Gallop, J. Phys. B 11, L93 (1978).

[4] J. C. Gallop and W. J. Radcliffe, J. Phys. E 14, 461 (1981).

[5] Y. I. Neronov and A. E. Barzakh, Sov. Phys. JETP 48, 769

(1978).

[6] J. C. Gallop and W. J. Radcliffe, Physica B C, 107, 621

(1981).

[7] R. Adams, S. P. Potts, J. C. Gallop, and W. J. Radcliffe in SQUID '80—Superconducting Quantum Interference Devices and their Applications, Ed. by H. D. Hahlbohm and

H. Lubbig (Walter de Gruyter, Berlin, 1980), p. 509.

367

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Preliminary Determination of h/m^

E. Krliger, W. Nistler, and W. Weirauch

Physikalisch-Technische Bundesanstalt, Bundesallee 100, D-3300 Braunschweig, F.R.G.

A precisely measured value of the ratio /m„ (Planck constant divided by the neutron mass) would

be a new input parameter for the least-squares adjustment of the fundamental constants. /i/m„ can be

determined by measuring the wavelength and the velocity of reactor neutrons. It is expected that this measurement can be carried out with a relative uncertainty of about 1 x lO-^, using a method

described in this paper. The method has been successfully tested in a preliminary determination of h/m^ and the results are reported.

Key words: fundamental constants; polarized neutrons; precision measurement.

1 . Introduction

A measurement of the ratio /i/m„ (Planck constant di- vided by the neutron mass) vi^ith a relative uncertainty of

about 1 X 10"^ is being prepared in the PhysikalischTechnische Bundesanstalt. This quantity has not so far been precisely measured.

The last least-squares adjustment of the fundamental

constants was carried out by Cohen and Taylor [1] in 1973. If h/m„ is calculated from the values of the constants recommended by these authors, the relative uncertainty of the result is 1.7 x 10"^. This is about the same as expected for the measurement described here.

Nevertheless, the experiment would be of great importance for altering the status of /i/m„ from that of a deduced quantity to that of an additional input parameter

of the least-squares adjustment. The reliability of the adjustment could thus be clearly improved.

The wavelength X. and the velocity v of a neutron are

related by the de Broglie equation:

h/mn = \v . (1) Hence, h/m-n can be determined by measuring both the

wavelength and the velocity of neutrons. Several methods have been proposed for this experiment, in particular by Stedman [2], Mezei [3] and

Weirauch [3-7]. In all of them, A. is determined by Bragg

reflection of the neutrons in a single crystal, and v is measured by a time-of-flight method. However, various

techniques have been proposed for carrying out the velocity measurement with the desired high accuracy. The method finally proposed by Weirauch [5-7] is briefly described in this paper. Furthermore, a preliminary measurement of h/mn carried out to test the feasibility of the experiment is reported.

2. The Measuring Method

2.1 The Principle of the Measurement

For measuring the velocity of the neutrons, a beam of polarized neutrons is modulated by periodically changing

the direction of the polarization vector. After a path of several meters, the neutrons are back-reflected by a sili- con single-crystal. They thus pass the modulator for a

second time, and the beam is again modulated. The

modulator defines the beginning as well as the end of a

flight path. The flight time follows from the phase differ- ence between the two modulations.

The wavelength of the neutrons is determined by the

Bragg reflection in the silicon crystal. Silicon should be

used because almost perfect crystals can be grown from

this material and its lattice parameter is known wdth the

required small uncertainty [8-11].

2.2 The Beam of Polarized Neutrons

A polarized and monochromatic neutron beam is pro- duced by Bragg reflection in a magnetized crystal of Heusler alloy (see Fig. 1, HI). The polarization of the

reflected neutrons is characterized by the polarization vector p, which is antiparallel to the magnetic induction

Bh in the crystal. In the ideal case of a completely polarized neutron beam, the spins of all the neutrons are

parallel to p. The polarized neutrons move in a magnetic guide field (magnetic induction Bo) which is parallel to Bh. The

guide field is required in order to avoid a depolarization

by magnetic stray fields. The beam of the reflected neutrons and Bq define the y

and z coordinates given in Fig. 1. The neutrons traverse a flipping coil F which reverses

p. It produces a constant magnetic induction Bp. p

precesses in the coil around the resulting magnetic induction Br = Bo + Bp. Bp is chosen so that Br is parallel to the X axis and has just the value which causes a precession by the angle it during the passage of the neutrons. The neutrons then hit a second Heusler crystal H2,

magnetized in the same direction as the first one. The

probability that a neutron is reflected by this crystal

depends on the direction of p. It is zero if p is parallel to Bh and maximal for the opposite direction. As p is now

parallel to Bh, the neutrons traverse the crystal without

reflection. Coil CI, which the neutrons pass next, produces a con- stant magnetic induction Bci parallel to the x axis with

Bci = Bq. The resulting magnetic induction forms an angle of 45° with the x and the z axis, p precesses ai-ound

369
this direction. Just at the moment when it is parallel to the X axis, the neutrons leave the coil. Coils of this type were first used by Mezei [12]. They are called tt/2 coils because they change the direction of p by the angle 11/2. The neutrons traverse the modulating assembly M and

a second Tr/2 coil C2. Both these arrangements should be

disregarded for the moment.

The neutrons are then reflected by the silicon crystal

mentioned above and return to coil CI. Along the entire path, from leaving coil Cl to reaching

it again, p precesses in the guide field. But the distance

between coil Cl and the silicon crystal is such that the

entire precession angle is an integer multiple of 2 it. Hence, p is again parallel to the x axis when the neutrons enter coil Cl for the second time. In coil Cl, p is turned back into the direction of the z

axis. The neutrons are then incident on the Heusler

crystal H2 for a second time. Because p is parallel to Bh, the neutrons are again not reflected. Thus, they can- not reach the detector D.

2.3 Modulating the Neutron Beam

The assembly M used for modulating the neutron beam

consists of an aluminum foil, bent to form a meander. It is therefore referred to as a meander coil in the following.

The meander coil forms a close sequence of single coils,

each with one turn. If an electric current is passed

through them, the magnetic inductions in neighboring

coils are antiparallel. If alternating current is used, there

is a phase difference of tt between the magnetic inductions in neighboring coils. An arrangement of this type,

supplied with direct current, was used by Drabkin et al. [13] for the construction of a velocity filter for polarized neutrons. To modulate the direction of p, the meander coil is supplied with, high-frequency alternating current. The coil field is parallel to the guide field. The thickness s of the

single coils is such that the neutrons pass each of them in half a period of the alternating current. As a conse- quence, the neutrons find the same variation of the magnetic induction in all the single coils.

p precesses in the meander coil due to two parallel

magnetic fields, the guide field and the coil field. The

first part of the precession has already been taken into account. Hence, only the precession caused by the coil field is considered in the following.

The magnetic inductions in the single coils are assumed

to vary sinusoidally with time. As a consequence, the

Larmor frequency of p also varies sinusoidally vdth time,

while the neutrons traverse a single coil. The angle by

which p precesses during the passage depends on the

time at which the neutrons enter the coil. The precession angle is the same in all the single coils. In the whole meander coil it adds up to the angle = <i> cos (2ttvO , (2) ^ = A^nmN/hv (3)

(v frequency of the electric current, t time, magnetic

moment of the neutron, amplitude of the magnetic in- duction in the coil, number of single coils of which the

meander coil consists). In principle, the modulation could be produced by a

single coil. This case has been discussed in a previous pa- per [14] but follovidng from Eq. (3), it cannot be appHed

here: in the proposed experiment 4> > 0.61 tt and v ~

1 MHz is required. Because J§m is limited for technical reasons to a few millitesla, these values can only be

achieved vdth A'' » 1.

2.4 Measuring thie Velocity of the Neutrons

When the neutrons leave the meander coil for the first time, the direction of p is modulated as given by Eq. (2). p is at all times in the xy plane (see Fig. 1). Returning

from the silicon crystal, the neutrons traverse the

meander coil for a second time. The direction of p is again modulated, this time by the angle

^{t + 2Z/W) = <|) cos [2'iTv(^ +21/V)]. (4)

I is the distance between the meander coil and the silicon crystal.

The two modulations add up to the total modulation

370
<I>tot(^, O = <I>(0 + ^it + 21/v) = 24) cos(2 uvl/v) cos[2 irvit + l/v)]. (5)

$tot is a periodic function of both the time and the dis- tance I

.

After the second modulation, the neutrons traverse coil CI in which p is turned from the xy plane into the yz

plane. Otot it, I ) is then the angle between p and the z

axis. The neutrons again hit the Heusler crystal H2. Because the direction of p varies, the probability of reflec- tion varies, too. Hence, the time-dependent neutron

current /(^ O = (/o/2) [1 - cos<I)tot(^, O] (6) reaches the detector, /q is the neutron current if p is antiparallel to the z axis. The mean neutron current 7 is measured in the detector. It is given by

7(1) = (/o/2) {1 - Jo[2<J' cos(2'rrvi/v)]}. (7) Jo is the zero-order Bessel function.

Figure 2 shows as calculated for 4> = 0.61 tt and q> = 1.22 IT. In the experiment, <l> will probably be

somewhere between these values.

Figure 2. Calculated ratio I/Iq as a function of the distance I. Parameter: (a) <f> = O.Sl-a, (b) <i> = 1.22 tt.

The velocity of the neutrons is given by

V ^ 2M V, (8) as follows from Eq. (7). A.I is the difference between

values of / belonging to neighboring primary minima of

I{1)IIq (see Fig. 2). The secondary minima which appear

in curve b should not be used for determining v. In order to determine v with high accuracy, an integer

multiple of_A I must be measured. For this purpose, two

minima of I{1)IIq should be determined, the I values of which differ by several meters. To change I, the meander

coil must be moved, not the silicon crystal; a displacement of the crystal would also change the angle by which

p precesses in the guide field.

2.5 Eliminating Long-Term Instabilities of the

Magnetic Guide Field

In section 2.2, it has been assumed that the polarization vector precesses by an integer multiple of 217 on the

way from coil CI to the silicon crystal and back again. If this condition is not met, the maximum and minimum

values of 7(0// 0 differ in general by less than shown in Fig. 2. The precession angle and thus the guide field

must therefore be constant during the entire measuring

time, which will run into many weeks. This difficulty can be avoided by means of a technique which is similar to the spin-echo technique first used by Mezei [12]. To eliminate the instabilities of the guide field, the additional 7t/2 coil C2 is required. It is adjusted in the same way as coil CI. On the way from coil CI to coil C2, the polarization vector p precesses by the angle

(j)i = 2mT + a (-TT < a ^ tt) (9) due to the guide field, n is an integer, a gives the angle

between p and the x axis. When the neutrons enter coil C2, p has the direction shown in Fig. 3a. In coil C2, p is turned into the yz plane. Its component

Tpy is reversed, a becomes the angle between p and the z axis (see Fig. 3b).

z

Figure 3. Polarization vector p at the surfaces of the tt/2 coil C2, (a, b) before the reflection in the silicon crystal Si, (c, d)

after the reflection.

Coil C2 is positioned at such a distance from the silicon crystal that p precesses by a small odd multiple of tt while the neutrons traverse this distance twice. The pre- cession results in the changing of the sign of a (see

Fig. 3c). While the neutrons traverse coil C2 for the second

time, p is turned back into the xy plane. Py is once more

reversed (see Fig. 3d).

Jn Fig. 3d a has the opposite sign as compared with

Fig. 3a. Hence, <f)i has been replaced by

(})2 = 2mT - a (10)

On the way from coil C2 back to coil CI, p again

precesses by 4>i. Between leaving and entering coil CI, it therefore precesses altogether by

<t> - 4>2 + <t>l - 4 W 17 . (11)

Hence, p has the same direction at the beginning and at the end of the path, independent of a.

371
Bo must be constant only during the time the neutrons

need for twice traversing the distance between coil CI

and the silicon crystal. Long-term stability is needed

only where the tt/2 coils are positioned and in the space

between coil C2 and the silicon crystal. However, p

precesses in these regions solely by small angles. Thus,

the absolute change of the precession angle caused by a

deviation of from the correct value is also small. Because this is the only important quantity, the requirements on the stability of 5 o are rather low.

3. Preliminary IVIeasurement

A preliminary determination of h/nin was carried out

at the reactor of the Physikalisch-Technische Bundesanstalt in Braunschweig. The arrangement was as schematically shown in Fig. 1. The (331) reflection of silicon was used, corresponding

to the wavelength \ =0.25 nm. The meander coil was made of aluminum foil, 60 |xm

thick and 90 mm wide. It consisted of =99 single

coils, each of which was s - 1.065 mm thick. From s and the value of the neutron velocity known beforehand, it followed that the modulation frequency had

to be V = 745 kHz. Measurements showed that the

meander coil actually operated best at this frequency.

The amplitude of the electric current in the meander coil was 40 A. Under these conditions, the modulation amplitude is <i> = 0.69 IT, as determined in preparatory

measurements.

The meander coil could be moved by 13 cm in order to vary I. The displacement was measured by a laser inter- ferometer.

The guide field was produced by Helmholtz coils of rec- tangular cross-section. Its magnetic induction was 5o =

1 mT.

Figure 4 shows the mean neutron current 7 measured

as a function of the displacement L of the meander coil.

Each point was measured for a period of 8 hours. 110

minima have been omitted between the two parts of the

measurement, following from the previously known value

of h/run.

A curve of the theoretically expected shape was ad- justed to the measured values by means of a least- squares method. From the positions of their minima it follows that: h/mn = (3.9519 ± 0.0040) x lO'^m^s-^ (12)

The given uncertainty is the standard deviation. In ac- cordance with the preliminary character of the measurement, no systematic uncertainties have been considered.

The measured value of h Im-n, is in good agreement with

the more accurate one calculated from the recommended

values of h and m„ [1], which is

hlrrin = (3.956034 ± 0.000007) x IQ-^m^s-i. (13)

4. The Final Experiment in Preparation

The preliminary measurement of hlntn confirmed the

feasibility of the experiment. However, to achieve the

desired accuracy, a high-precision apparatus must be

constructed. In the final measurement, I will be varied by about

10 m instead of about 10 cm as in the preliminary one. The other data will probably not differ very much from

those given above.

In order to measure hlrrin with a relative uncertainty

of about 1 X 10"^, the positions of the minima of / must

be determined with an uncertainty of a few percent of

the width T at half-minimum (see Fig. 2). This is possible, but the uncertainty in measuring 7 must be much

smaller than in the preliminary experiment (see Fig. 4). To lower the uncertainty of 7, more neutrons must be

counted. The final measurements will therefore be car- ried out at the high-flux reactor of the Institut LaueLangevin in Grenoble.

7

0.2

0

J \ L J L J \ \ L

1.0 3.0 4.0 120.0 121.0 122.0 mm 123.0

L

^

Figure 4. Mean neutron current I measured as a function of the displacement L of the meander coil.

372
5. Conclusion

A preliminary measurement of /i/m„ has been carried

out. It showed that there is a good chance of determining

this quantity with a relative uncertainty of about 1 x

10"^. However, a high-precision apparatus is required,

and the measurement must be carried out at a high-flux

reactor. Such an experiment is in preparation.

The authors would like to thank M. M. W. Berne and

H. Schneider who helped to construct the apparatus for the preliminary measurement.

References

[1] E. R. Cohen and B. N. Taylor, J. Phys. Chem. Ref. Data 2, 663 (1973).

[2] R. Stedman, J. Sci. Instrum., Ser. 2 (J. Phys. E) 1, 1186

(1968).

[3] F. Mezei and W. Weirauch, research proposal No. 01-003R

(1975), Institut Laue-Langevin, Grenoble.

[4] W. Weirauch, Nucl. Instrum. Methods 131, 111 (1975).

[5] W. Weirauch, in Fundamental Phi/sics with Reactor Neutron h and Neutrinos (Inst. Phys. Conf. Ser. No. 42), Ed. by T. von Egidy (The Institute of Physics, Bristol and London, 1978),

p. 47.

[6] W. Weirauch, E. Kriiger, and W. Nistler, in Neutron Spin Echo

(Lect. Notes Phys. 128), Ed. by F. Mezei (Springer, Berlin,

1980), p. 94.

[7] W. Weirauch, to be published.

[8] R. D. Deslattes and A. Henins, Phys. Rev. Lett. 31, 972

(1973) . [9] R. D. Deslattes, A. Henins, H. A. Bowman, R. M. Schoonover, C. L. Carroll, I. L. Barnes, L. A. Machlan, L. J. Moore, and W. R. Shields, Phys. Rev. Lett. 33, 463

(1974) . [10] R. D. Deslattes, A. Henins, R. M. Schoonover, C. L. Carroll, and H. A. Bowman, Phys. Rev. Lett. 36, 898 (1976).

[11] P. Becker, K. Dorenwendt, G. Ebeling, R. Lauer, W. Lucas,

R. Probst, H.-J. Rademacher, G. Reim, P. Seyfried, and

H. Siegert, Phys. Rev. Lett. 46, 1540 (1981).

[12] F. Mezei, Z. Phys. 255, 146 (1972).

[13] G. M. Drabkin, V. A. Trunow, and V. V. Runow, Zh. Eksp.

Teor. Fiz. 54, 362 (1968) [Sov. Phys. JETP 27, 194 (1968)].

[14] K. Weise, A. Mehl, and W. Weirauch, Nucl. Instrum. Methods 140, 269 (1977).

373

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

One has to remember that some errors are randomfor one person and systematic for another. P. Vigoureux

The Assignment of Uncertainties to the Results of Experimental

Measurements

Jbrg W. Mliller

Bureau International des Poids et Mesures, Pavilion de Breteuil, F-92310 Sevres, France

After a brief review of some of the current ways of indicating in a quantitative manner the

uncertainties which are associated with an experimental measurement, the main shortcomings of some

of the usual practices are indicated. The present situation is unsatisfactory and results in frequent

misinterpretations, rendering, for instance, a critical data evaluation a cumbersome task. The traditional practice of distinguishing between "random" and "systematic" uncertainties is one

of the main causes of confusion in the statement of uncertainties. In fact, this separation is in no way

absolute since it is likely to be changed by a subsequent user. A subdivision, if required, should only

depend on the way the numerical value of the uncertainty was arrived at. As for its effect on the final result, due care is automatically taken by a correct application of the well-known general law of error

propagation, for which the only input data needed are the best available estimates for the variances and covariances of the measured quantities.

BIPM recently organized a meeting of experts, chosen by the national standardizing laboratories, for discussion of the relevant problems. This resulted in the establishment of five recommendations

concerning the statement of uncertainties which are included in an appendix.

Key words: covariances; experimental uncertainties; variances.

1. Introduction

A conference on fundamental constants is clearly not

the place where anybody will expect me to explain at length why the presentation of an experimental result should alv^^ays include information on its supposed precision or accuracy. Indeed, from a scientific point of view a numerical result of a measurement which is reported

without any indication concerning the claimed precision is nearly useless as it does not allow a meaningful comparison either with other determinations of the same quantity or with theory, where this is possible.

There are certainly many examples from daily life for which this strict rule can be relaxed with little or no loss

of information. Thus, if we hear that 26 people have been

killed in an air crash, nobody expects an uncertainty to follow the statement; on the other hand, if we are told that a Roman amphora is 2000 years old we will not take

this number at its face value. However, these are limit- ing cases and they are in no way typical of the usual situ- ation prevailing in physical measurements.

Since precise measurements, at least in a field such as astronomy, have been performed for more than 3500

years, as is evidenced, for instance, by the famous Venus

tablets of Ammizaduga and the very sophisticated Babylonian theories for the apparent movement of the moon

and the planets [1], it is quite astonishing that no explicit

mention was ever made of the limited precision inherent

in the observations. In fact, even the closely related

basic notion of probability is completely absent in the

whole thinking of Antiquity and (less surprisingly) in the

Middle Ages, although the zeal of the populations for all sorts of dice-playing and similar random games is well at- tested. Historians may have a plausible explanation for

this strange situation (perhaps religion), but the fact remains quite remarkable.

Obviously, the situation today is basically different

even if compared to the one prevailing, say, only 50

years ago when a scientist could still know personally

most of the people working in his field and therefore base

his judgment on the reliability of results on the appreciation of the personality of his fellow scientists. I ignore

whether in practice this system always worked perfectly

well, but in any case it could not be applied today when we may have to use data of hundreds of workers about

whom we normally know nothing at all. It therefore has

become necessary to replace the possible prestige of a name associated with the result of a measurement by an

"objective" measure of its value, and this is done by the

assignment of uncertainties. This practice is not necessarily an improvement, but rather an unavoidable

development which we have to accept. The old system,

where applicable, may still give useful hints.

2. Generalities

Unfortunately, the assignment of uncertainties has been

plagued for many decades by a number of problems, some

of which may be difficult to avoid, but to a large extent by others which just result from a lack of agreement on what we wish to describe and the manner in which we do

so. The main trouble is that, although most experimenters maintain the practice of assigning an uncertainty

to their measured values, it is often far from obvious to the reader what this number is exactly supposed to mean. This continues to cause serious headaches for any

compiler or critical evaluator of data who is obliged to put the various indications on a common footing, for in- stance in order to assign appropriate statistical weights

to data which he is comparing among themselves or with some theory. Misinterpretation of published error assignments is frequent (and often unavoidable for lack of in375
formation) and may result in the recommendation of

doubtful "best values."

Let us first try to understand what the main problems

are and where they come from. On the basis of this knowledge, we shall then suggest a possible way to over- come some of the difficulties, at least for the most

current practical cases. It will be appropriate to mention

right at the beginning that in a field such as this upon

which so many words have already been expended, it is hardly possible to say anything which is really new; perhaps this is not even wanted, because there is no lack of a variety of well-intentioned advice. Rather, what we

need is a process for sorting out signal from noise, possibly by using as guidelines such old stand-bys as common

sense, simplicity, coherence and usefulness, while taking

care not to violate basic laws of physics and statistics,

where applicable. It is hoped that by proceeding in such

a manner we can see more clearly the main issues and

then, possibly, indicate some way out of a situation which

seems, at present, inextricable. The literature on error statements is enormous, but usually quite unrewarding.

Some of the more interesting opinions can be found in the

proceedings of the first PMFC conference [2], especially

in the informal discussions from which Vigoureux' quotation is borrowed.

Our task is rendered difficult by a number of cir- cumstances which, one might think, are quite irrelevant

to the real problems as they belong to psychology rather

than to physics, but unfortunately this is not so. First we

have to cope with the fact that everyone considers oneself to be an expert. Secondly, no one likes to change

opinions and habits, especially if these are available in printed form—and they are numerous. Finally, there exist quite a number of recipes issued by groups of experts,

most of which are clearly contradictory to each other and

ill-founded, but sometimes backed by influential national or even international organizations. They have in com- mon that they take a narrow and simplistic point of view,

ignore the similar problems met in other fields, and

arrive at descriptions which allow the evaluation of uncertainties with an amazing precision (at least three sig- nificant figures) even for the poorest data. Their world

is invariably Gaussian and continuous. The fact that an

evaluated uncertainty also has an uncertainty is com- monly overlooked.

In view of this state of affairs, which was no secret to the well informed, it was certainly a bold initiative for Dr. E. Ambler, a member of the International Committee of Weights and Measures, to suggest that the Inter- national Bureau of Weights and Measures (BIPM) should

make a serious attempt to clarify the situation and, if possible and with the help of the national laboratories,

arrive at useful and generally acceptable rules for the

statement of experimental uncertainties. Equally gratifying was the readiness of Dr. P. Giacomo, then director

designate of BIPM, to undertake this mission. It was felt that the first need was to know the position of the national laboratories in this controversial field.. This was

done by setting up a "Questionnaire on uncertainties"

where, after an introduction to the problems, eight questions were asked, each preceded by a comment which

placed the problem in the appropriate context and re- called some of the difficulties involved. The main purpose

of this enquiry was to choose carefully the subjects which

could be usefully discussed at a later stage by an ad-hoc

Working Group. Such a preliminary sorting out was con- sidered to be unavoidable in view of the divergence of prevailing opinions. It was thought advisable to avoid

purely philosophical or mathematical discussions as well

as topics related to terminology, although the border line

is often a vague and subjective one. In February 1978

the questionnaire was sent to 32 national standardizing

laboratories inviting them to submit answers to the questions, and to 5 international organizations for information. By the end of the year BIPM had received

responses from 21 countries, included in which were re- plies from all the large national laboratories. Their

analysis has been the subject of a detailed report [3].

3. A Look at Some of the Main Problems at

Issue

Instead of listing the questions asked and the replies

received, which would be a rather tedious exercise, we

prefer to mention some of the basic items under discussion. In addition, various details in the replies of the lab- oratories, as a result of subsequent internal discussion

and better understanding, no longer correspond to their

present point of view. Hence the document [3], although

still of historical interest, is now superseded by later developments which it has no doubt initiated. The

present situation will be discussed in Section 5. It was clear from the beginning that the traditional

subdivision of uncertainties (or "errors," as they used to be called) into "random" (or statistical) and "systematic" contributions would become one of the most

controversial topics. This grouping is so strongly en- trenched in the thinking of physicists and engineers that

at first most of them may have considered it a waste of time and effort to question its good foundation. And yet a

careful enquiry into the origins of the embarrassing con- fusion invariably pointed to this problem as a likely source of trouble. In the prevailing opinion a systematic uncertainty (or

"error") is characterized by the fact that its influence on

the final result is independent of the number n of meas- urements taken. This is in clear contrast to the random

contributions which diminish with the square root of n. This seems to be such a striking difference that a clear

distinction between these two types of uncertainties ap- pears to be a necessity. On the other hand, this view is somewhat difficult to reconcile with the observation that

there also exist cases where the type of an error ap- parently changes, depending upon the use to which it is put. This is such a disagreeable situation in the eyes of an error-class purist, that the only way to cope with it was to ignore it. However, the fact exists and it is quite

easy to find situations in which changes can take place in either direction. For those who believe in the need to combine uncertainties differently according to their type,

this observation is disastrous. How does this come about

and what can we do? The diagnosis of the trouble is simple: it is due to the fact that the traditional classification of uncertainties depends upon the further use we intend

to make of them, and in general this cannot be known in advance. Any classification, if needed at all, should obviously only depend on the past, not on the future. Although many people may be well disposed to accept

the above arguments, they are not wiUing to change their

habits. After all, the system worked quite well for most

practical cases and how would we know whether or not one has to divide an uncertainty by the square root of n? This is an eminently practical question for which we have

to find a convincing answer. It has been given before [4], but since this is a crucial point it may be justified to re- peat the main arguments, perhaps best by means of a

practical example. We assume the reader to be familiar

with the general rules of error propagation, the elements

of which are summarized in Appendix A

376
Let us consider a series of weighings made with a bal- ance, the numerical results of which are represented by

the n results Xj. There may be some doubt as to the ac- curacy of the calibrated masses involved in these measurements. If such an error should exist, it would clearly

affect all the measurements. Let us assume, for the sake

of argument, that the effect is such that instead of the

measured values Xj one should consider the corrected

quantities

Xi + a (1) where a is a possible common shift (other types of corrections might in fact be more realistic, but this is the

simplest situation). Since we assume that all known

corrections have already been applied, the most probable

value of a is zero, but there remains an uncertainty

So, > 0. The experimental uncertainty of a single measurement

Xj is denoted by Sj. and we assume all measurements xj to be independent of each other. For the "function /" (cf. Appendix A) we simply

choose the mean value, hence

n j w j The quantity looked for is the uncertainty of y. This can be evaluated in various ways which depend on the choice

of the variables.

Let us first put

y = fiixi, X2, . . .

, Xn; a) . (3) Since Xj and a are not correlated, all covariances vanish.

As df i /dXj — \ln and df i/da = 1, we readily obtain from

Eq. (A3) for the variance of y

2 —1

S;, n + (1SJ2 = ls,2 +

n

(4)

Another possible choice for the variable is y = fziyi, y2, , Vn) (5) In this case the quantities needed in Eq. (A3) are the

variances

s/ = war (yj) = s/ + s|

and the covariances

Sjk = coviyj, yi,) . For the latter it can be shown (see for instance Ref. [5]) that, since both E(a) and cov(x, , x^.) vanish, we simply

have

S 2 " a As furthermore dfz/dyj = one finds by means of

Eq. (A3)

s2 = -L

y 2 {n{s^ + s|) + n(n - l)sf

]

= J- 2 n + s,

which is identical with Eq. (4). Nobody will be astonished by this result as it corresponds exactly to what we would have expected.

Yet, it is remarkable for several reasons. First, we note

that the outcome does not depend upon the choice of the

variables and that there was no need to decide whether an

uncertainty is of a certain type. All we have to do is to describe correctly the experimental situation and then to apply the general rules of error propagation. This then

yields automatically the correct answer [Eq. (4)]. Of

course, in interpreting the result one can now say that

behaves as a "random" uncertainty and .s„ as a "systematic" one, but this behavior is a result of the application made. If the experiment had consisted of a compari- son of n balances of different types among themselves, the

contributions s„ would have been treated as random uncertainties. An additional piece of information is contained

in Eq. (4) and should not pass unnoticed, namely that both

the "random" and the "systematic" contributions are added "in quadrature" (and not linearly or otherwise), and

finally also that the "systematic error" appears in a form which corresponds to a standard deviation. While all these things are obvious here and come out naturally,

they were hotly debated items in those approaches that are not explicitly based on Eq. (A3). The above discussion

reveals them as ill-founded. In fact, if variances and co- variances are properly taken into account, there is no need to introduce different types of uncertainties.

4. On the Maximum Limit Concepts

There can be little doubt that there are cases where

the numerical assessment of an uncertainty poses a real problem. This is in particular the case when it cannot be

deduced from repeated measurements, because it is not

possible to perform them. In such a situation of incomplete knowledge—which in fact is not an exceptional, but

rather the normal case—the physicist has to make use of

his experience and imagination. With the help of his knowledge of the theoretical basis and the practical reali- zation of a given experiment he has to examine carefully

all possible causes of error, to eliminate them or to apply

corrections. Usually this can only be done in an approximate way; the possible residues have then to be taken as uncertainties. One always hopes that nothing of real importance was forgotten, but there is obviously no

guarantee. As an alternative (but somewhat jocular) clas- sification it has therefore been suggested to distinguish

only between those uncertainties which have been taken

into account (although perhaps not adequately) and the

other ones which have been overlooked. If this were a

realistic approach, it would be excellent. In practice, unfortunately, the second category can only be grasped at the very end, usually as a result of some disturbing

discrepancy. Yet the idea should be kept in mind as it recalls that from a scientific point of view this is certainly

the most interesting class: it permits one to uncover a previously unsuspected influence. It is to a large extent by the discovery of such errors that experimental physics

advances. In any case, the challenge to resolve inconsistencies is a most important stimulus for new experimental or theoretical work, as has been illustrated with numerous examples by Taylor [6]. See also Ref. [7]. This

also reveals the danger implied by the practice occasionally suggested of just enlarging a stated uncertainty in order to avoid possible difficulties later. But obviously, it is not by concealment that problems are solved. In the general confusion on the assignment of uncertainties it has sometimes been taken for granted that

"systematic errors" automatically belong to the class where an objective numerical value is not available, since

the personal judgment of the experimenter comes in. As

a subjective evaluation was ruled out as unscientific, it was claimed that the best and only thing to do was to as- sign "maximum" uncertainties which were chosen large

enough to guarantee that the "true value" lies practically always within the corresponding limits. This is clearly a recipe which is simple to apply, but its useful- ness is questionable. Since a "maximum limit" becomes

"safer" when enlarged, it is an ill-defined quantity for which, for instance, no useful propagation rules can be

377
indicated. Nevertheless, repeated attempts have been

made to save this concept, in particular by associating it with a specific probability density. By interpreting the

upper and lower limits as bounds between which the

"true" value must be located and by further assuming

that, as a result of our ignorance, any point in this inter- val has the same chance to be the "good one," one

arrives at a constant probability density between the

limits. Since the standard deviation for a rectangular

density of total width 2a is given by a/VsT, it has been

suggested that we consider this as the corresponding uncertainty. Indeed, some have taken this approach seri- ously enough to produce detailed numerical tables for the

convolution of a rectangular density with a Gaussian

function. Unfortunately, there is no sound basis for such

an assumption, which is also known as the postulate of Bayes. On the contrary, thanks to Jaynes [8] some

practical counter examples are now known for a priori

densities corresponding to "complete ignorance," in particular the case of a binomial distribution with the probability p as unknown parameter [9]. The appropriate shape

of the density is so far from a rectangular one that this approach is best abandoned.

5. The Present Situation

All the laboratories which had submitted a reply to the

BIPM questionnaire mentioned in Section 2 were invited

to send a participant to the meeting of a Working Group

which had the task of discussing the most important

problems related to the assignment of experimental uncertainties. Experts from 11 national laboratories at- tended the meeting at Sevres from October 21 to 23, 1980. Dr. P. Giacomo, the Director of BIPM, acted as chairman.

Let us have a quick look at the results of this meeting.

For more details the proceedings [10] should be consulted. The case of random uncertainties that can be

evaluated by means of a series of repeated measurements

raised no serious problem and the Working Group recom- mended that they be expressed in terms of estimated

sample deviations s, , adding the corresponding degrees

of freedom v, . This does not imply any assumption on

their distribution. For measurements which are expected

to be correlated, one should also determine and indicate

the respective covariances, provided that the importance

of the problem justifies the additional work. Little need was felt for the evaluation of confidence intervals. If the

population can be taken as normal, as it is frequently as- sumed to be, Ihe correspondence is well knovm and its inclusion does not give any new information. Such an in- dication may be needed, however, for making a statistical decision or for comparison with legally imposed tolerance

limits. More difficult is the situation where uncertainties cannot be obtained by statistical methods; in this case

their estimation must rely on "other methods" which

inevitably imply some element of subjective appreciation.

However, it should be clearly recognized that this fact is neither new nor has it any connection with the previous

error classification: an uncertainty obtained by "guesstimation" may affect a final result in the same way as was

till now considered to be the effect of a "random" or a "systematic" influence. This depends entirely on the

way the quantity under discussion enters our equations

describing the physical situation. The resulting uncertainty is in any case of a random nature. This has led the Working Group to suggest—for purely

practical reasons and in a provisional way, especially as regards terminology—two groups or classes of uncertainty components, namely

—^group A: those which are evaluated by applying statistical methods, and —^group B: those which are evaluated by other means.

This is neither a very strict nor a very profound classifi- cation, but it tries to come close to reality. Since, as we

shall see in a moment, all contributions will be used from now onwards on an equal footing, a possible misgrouping

is of no consequence. There is therefore little need to worry about further details concerning these groups. In

fact, the difference between the groups is not of a "fundamental," but only of a practical nature and should be

helpful to the experimenter. Whereas for group A the

usual methods of evaluation normally lead in a straightforward way to a sample variance or an estimated standard deviation, this is usually not so with group B. However, this difficulty is not peculiar to the new approach;

it has always existed but it used to be masked by con- cepts such as "maximum Hmits" or the like which then

could not be usefully related to the other components of

class A with a better-defined statistical significance.

It would be quite misleading to think that the new clas- sification is, in essence, just the old one presented in another dialect. Surprisingly enough, it was indeed sug- gested in one of the replies to the questionnaire that

replacing the word "error" by "uncertainty" would

automatically solve all the major problems. We have

somewhat less confidence in the power of words and

think that the first concern should be the concepts, and

that the words attached to them are of minor importance.

The old and the new treatment of uncertainties differ in the basic approach and the transition therefore cannot be

performed by means of a dictionary.

Which meaning do we wish to assign to the uncertainty

components of group B? The answer comes from a look

at the general law of error propagation [Eq. (A3)]. Since

this is the only firm basis available, the choice is imposed. If we want to make use of it, we need for all uncertainties quantities which can be considered as valid

approximations to the respective standard deviations. To

claim that such an estimate is impossible to obtain leads

to a deadlock, for this attitude implies that we deny any

possibility of arriving at a value for the uncertainty of a

final result which includes contributions from both

groups A and B. This is obviously a possible point of

view, but practical experience shows that it results in situations which are most unsatisfactory, for nobody

really knows what to do with more than one uncertainty

statement for a final result. It is then left to an evaluator

to combine the various (and sometimes ill-defined) uncertainty components in some way, often to the irritation of

the experimenter. It is our contention that this important step should be performed by the person who is best

acquainted with the details of the measurements, and

this is clearly the experimenter himself.

It should be clear, therefore, that if we want to give

"useful" measures of uncertainty components, they have

to be in such a form that they comply with the usual for- malism. Hence we have to choose something which can be

considered as the best available approximation to the

corresponding standard deviations. This is the essence of the third draft recommendation (cf. Appendix B). We

realize that the text restricts itself to indicating the goal

to be achieved, vvathout giving further advice on the important question of how this can be done in practical cases. This vdll be clearly an important topic for a later document which must be more concerned vdth the practical implementation of the new approach than vdth its background. As the number of possibilities is quite limited, we can already sketch a few of them. The choice

378
depends on the personal preference of the experimenter

and, in particular, on his previous habits. Thus, some may be courageous enough to guess the required quantity, say Uj, directly; this requires a feeling for what

might correspond to something like a 2/3 or 70% confi- dence interval. Others may be more at ease in trying to estimate a confidence interval of 50% (also known as "probable error"); then Uj may be taken as about 1.5 times this quantity. Finally, for those used to starting

with a "safe" confidence interval, the problem is to decide to which multiple of the corresponding "standard

deviation" they assume this to correspond, and to divide

accordingly.

It may be objected that these correspondences depend

on the form of the assumed probability density of the

underlying population. In principle this is true, of course,

but, for the modest precision aimed at, the assumption is usually not a critical one. In any case, it would clearly

make httle sense to replace our ignorance by a firm

prescription. It seems best to leave this matter to the

personal preference of the experimenters, some of whom

may choose for themselves, as an intermediate personal

guide, a Gaussian or a rectangular probability density,

whereas others will feel able to dispense with such a

construction.

Once the estimates are obtained for both classes, they

can be readily transferred to the forms needed in Eq.

(A3) by multiplying them with the corresponding partial

derivatives. It is the squares of these new quantities

which are, for the sake of simplicity, denoted in the draft recommendations by s/ and m/ for classes A and B,

respectively. To this may be added the expressions for the estimated covariances, where required and available.

We are now in a position to substitute these quantities

into the general formula of error propagation. In the ab- sence of correlations between the various n uncertainties

this simply amounts to forming the sums

n rig

i=i i=i

for obtaining the approximate ' ' variance' ' of the required

function y = f (xi, X2, .

, x„). The square root of this quantity has been called "combined uncertainty." This

provisional terminology may provoke opposition, but the

possible adoption of a more suitable name is left to later

deliberations. However, the Working Group was certainly well advised not to use expressions like "standard

deviation' ' or " variance' ' for quantities involving contri- butions from group B in order to avoid justified objections of mathematical statisticians who are used to attri- buting to these names a well-defined meaning which is no

longer pertinent if, as in our case, subjective elements

have to be included.

An important point concerns the relation between this combined uncertainty and the notion of a confidence in- terval. For the contributions from group A the link is usually established by adopting the assumption (which, even here, is often difficult to justify) that all contributions can be taken as a sample belonging to a normal population, perhiitting us thereby to use the corresponding

Student' s t factor for passing from a sample standard deviation to a confidence interval of a given probability

level. For group B, and therefore for the combined uncertainty which includes such contributions, this is much

more difficult to justify. However, it would be quite

naive to think that this problem has been created by the new approach sketched above. Rather, the difficulty has

always existed, but it used to be concealed by fuzzy concepts, whereas now the problem is clearly brought to

light. If we want to be quite honest, we have to admit

that a definite hnk with a confidence level cannot be es- tablished. Of course, one may assume (or rather hope)

that for the contributions belonging to group B, the rela- tion is approximately the same as the one which is valid for a normal population, but this cannot be assured and is possibly just wishful thinking. Invoking the Central

Limit Theorem is normally of little help and seldom justified, at least for situations with few contributions.

Confronted with this difficulty, the Working Group

found it necessary to adopt a somewhat different point of view. While an evaluation of confidence limits in the usual sense is not feasible, we are still free to choose an arbi- trary multiple of the combined uncertainty for obtaining a wdder range and therefore a higher confidence level. The

only difference from the usual procedure now is that this multiplication factor, called k, has a conventional value, as

for instance k = 2 or 3. There is little doubt that this has

in fact often been done before, and not with more justification than now; the real difference therefore only consists in the explicit confession that this procedure is to some extent an arbitrary one. The quantity thus obtained

has been termed an overall uncertainty, which is again to be taken as a provisional denomination, and recommendation 5 says that whenever its application is felt necessary, the multiplying factor should be clearly stated. The need

to use such multiples (which obviously carry no more in- formation than the combined uncertainty) hardly arises in scientific measurements, but is often imposed in industrial or legal metrology, where such requirements are demanded by people who believe that all random deviations necessarily follow the Gaussian law.

6. Final Remarks

The problems involved in the assignment of uncertainties to experimental results, as we have seen in the

above analysis, are of a type which makes their treat- ment at the same time simple and difficult. Their ambiguous nature, it seems to me, is due to the fact that on the one hand they have an obvious relationship with such

fields as mathematics and statistics where one is used to dealing with theoretically well-defined concepts, but on

the other hand they are also intimately linked to the

practical world of laboratories and workshops vdth all their hazards and compromises which are typical of the

daily hfe of a practitioner. As uncertainties are indicated

or used by virtually everybody and in very different

fields, their statement has to be acceptable in a far larger domain than is usually required for more specialized quantities. In addition, the various users impose a number of constraints on format and content which have,

as far as possible, to be taken into account. It is not

surprising, therefore, that there exists a large variety of proposals for achieving this goal.

Instead of taking the various and often contradictory

requirements as a mere nuisance, one can also try to con- sider them as guides permitting us to narrow dowTi the

number of useful proposals. Bearing in mind that we may

choose only those quantities that are compatible with the

conditions imposed by the general propagation law for uncertainties, the possibilities are so strongly restricted

that there remains practically no place for arbitrariness.

The result—and I hope the majority of you can agree

with it—is the proposal outlined above and described in a more comprehensive form by the five recommendations

set up by the Working Group. There is no doubt that

various details are capable of improvement and may

379
eventually have to be changed (for instance notation),

but taken as a whole the present proposal, which is believed to form a reasonably logical and coherent unity,

offers little or no room for fundamental changes: if it is considered unacceptable, it will have to be dropped completely and the problem must be tackled again from

scratch. I cannot l3ut hope that this will not happen, but

the decision is now with those who will try to apply this simple scheme for their own measurements. To my mind,

the questions related to the assignment of uncertainties

do not provide a useful playground for purists or fundamentalists. This is not the sort of problem which can always be solved by a yes or no answer. Some compromises and arbitrary conventions are difficult to avoid and

should be accepted in the interest of a wider applicability. The new scheme described in the draft recommendations is already applied in several national laboratories

and its practical usefulness is going to be tested during a

trial period. If the results are considered satisfactory,

these procedures may then be recommended for general

use. In this way it is hoped to arrive, in due course, at an acceptable uniform way of assigning uncertainties to experimental measurements.

In the realm of fundamental constants, the main subject of this conference, uncertainties occupy just a very

small corner. The more topical and interesting problems

are on another level and concern the most appropriate

way to actually perform the necessary least-squares

adjustment when it is realized that both the mean value

and the associated uncertainty, as indicated by the

experimenter, may be in error. This important subject

will not be discussed here at all (for recent reviews, see

Refs. [7] and [11]), but it may be useful to bear in mind

that it is essential in such an enterprise to dispose of reli- able and realistic estimates for the uncertainties. In the form outlined above, their treatment merely

requires the use of some elementary and old-fashioned

mathematics, and there is in all probability no need at all for anything much more sophisticated (although proposals to the contrary are not lacking). The rest is essentially common sense. It will be noted that special care has been taken to refrain from adopting any hypothesis

which is not really needed, as this would have rendered

the result less general and therefore narrowed its range

of useful application.

Even if we adopt the optimistic point of view that in the future the reporting of uncertainties will follow

essentially the proposals outlined above, it would no doubt be unrealistic to expect a rapid convergence

towards uniformity. Habits are persistent and usually

lifelong. This, therefore, reminds me of the story about

the philosopher who was celebrating his 90th birthday.

One of his young adherents expressed admiration for the

fact that his opinions, once bitterly attacked, are now

quite generally accepted and he asked the scholar how he

had succeeded in convincing all his enemies. The old man

gently smiled and then replied: "The truth is that

nobody has ever changed his opinion, but it just happens

that I have survived them all." There are moments when I wonder if we are not in a somewhat similar situation. Note added in proof: The final French version of the

Recommendation INC-1 (1980), which corresponds to the

draft version given here as Appendix B, has now .been published in the document mentioned in Ref. [10]. In

addition, the Recommendation 1 (CI- 1981) of the CIPM

which concerns the practical implementation of INC-1

(1980) is reproduced on page 26 of the same pubhcation.

The author is very grateful to several staff members of the BIPM for helpful remarks made on a draft version.

He alone, however, is responsible for possible controversial statements expressed in this review.

7. Appendices

A. A Reminder on Error Propagation

As it still forms the undisputed basis for any serious

discussion on uncertainties, it may be useful to recall briefly some simple facts concerning the well-known

"general law of error propagation," as it is called in the

traditional terminology. This seems also justified by the

conclusion, for which the underlying reasoning is sketched in Section 3, that its careful application essentially solves all of the much-discussed problems in this

field which, to a large extent, have been artificially

created by a simple neglect of some basic elements.

If 1/ is a known function of n variables Xj, thus y =

fix I, X2, . . . , Xn), then small changes Axj affect the

' ' result' ' y approximately by an amount

if the terms of higher order in the Taylor expansion of

the function can be neglected. The square of this quantity is given (again in first approximation) by

n

dXj

nX

JL

dXj

Axj

Ax, X

,k=l

(k #j )

dXic Axk (Al)

It is practical to characterize the "random displacements," or "errors," by their variances and covariances,

which can be achieved by putting (E stands for mathematical expectation)

E{(Axj)2} = af and E{^Xj Ax/c} = Ujit . (A2)

Then Eq. (Al) can be written in the more compact form - X(f'j ^jf + X f)f'k<^jk , (A3)

j j i=k with f j = df /dXj and summations from 1 to n. Equation (A3) is the well-known general formula for "error propagation." If one agrees to identify with

<jf, then Eq. (A3) can be further condensed into

Xf 'i f 'k j.k (A4)

For most practical applications one has to replace the

(unknown) population parameters af and Ujk appearing in Eq. (A3) by their corresponding (unbiased) sample esti- mators. Hence, if we assume that for a quantity Xj we

have

Xjz, . . .

, may use

available the m results of measurements Xji,

, and likewise for Xk, then instead of ct| we

.2. = m - 1 XXn Xi and instead of aj^ we may use m 1 X^ Xji X/cj m Xi X^ki

i where all summations over i go from 1 to m. There exist

380
equivalent forms which at times may be more practical;

they can be found in any textbook on elementary statistics.

It is a fairly popular, but clearly unjustified belief that

in most "practical cases" the various contributions to the uncertainty can be taken as independent and that the

terms involving the covariances may therefore be

dropped. This unfortunate prejudice has led to the rather

confusing error classification which hides the fact that, for example, the so-called "systematic" uncertainties are

in no way different from the "random" ones, except that

they are correlated to the measurements in another way.

This point is also discussed in Section 3. B. Draft Recommendations on the Statement of

Uncertainties

These recommendations are presented here in the form

in which they were elaborated by the Working Group at the October meeting. They will be presented, possiblj^

with minor changes, to the Comite International des

Poids et Mesures. They have also been pubHshed in Ref. [12].

1. The uncertainty in the result of a measurement generally consists of several components which may be

grouped into two categories according to the way in which their numerical value is estimated:

A—those which are evaluated by statistical methods,

B—^those which are evaluated by other means. There is not always a simple correspondence between

the classification into categories A or B and the previously used classification into "random" and "systematic" uncertainties. The term "systematic uncertainty" can be misleading and should be avoided.

Any detailed report of the uncertainty should consist of a complete list of the components, specifying for each the

method used to obtain its numerical value.

2. The components in category A are characterized by the

estimated variances, s,^ (or the estimated "standard deviations" Si), and the number of degrees of freedom, v,. Where appropriate, the estimated covariances should be

given.

3. The components in category B should be characterized

by quantities u^, which may be considered as approximations to the corresponding variances, the existence of which is assumed. The quantities may be treated like variances and the quantities Uj like standard deviations.

Where appropriate, the covariances should be treated in a similar way.

4. The combined uncertainty should be characterized by

the numerical value obtained by applying the usual

method for the combination of variances. The combined

uncertainty and its components should be expressed in the form of "standard deviations."

5. If, for particular applications, it is necessary to multiply the combined uncertainty by a factor to obtain an

overall uncertainty, the multiplying factor used must al- ways be stated.

References

[1] B. L. van der Waerden, Science Awakening II, The Birth of Astronomy (Noordhoff, Leyden, 1974).

[2] Precision Measurement and Fundamental Constants, Ed. by D. N. Langenberg and B. N. Taylor, Natl. Bur. Stand.

(U.S.) Spec. Publ. 343 (Aug., 1971), p. 493.

[3] The BIPM enquiry on error statements, Rapport BIPM-80/3

(1980), 50 p. A French translation is being published in Proces-Verbaux Seances Com. Int. Poids Mes. 48 (69th

session) (1980).

[4] J. W. Miller, Nucl. Instrum. Methods 163, 241 (1979).

[5] Les incertitudes de mesures, in La Physique, vol. 4 of the Encyclopedic Scientifique de I'Univers (Gauthier-Villars,

Paris, 1981), p. 11.

[6] B. N. Taylor, in Precision Measurement and Fundamental

Constants, Ed. by D. N. Langenberg and B. N. Taylor, Natl. Bur. Stand. (U.S.) Spec. Publ. 343 (Aug. 197U, p. 49-5. [7] E. R. Cohen, in Metrology and Fundamental Constants,

Proceedings of the International School of Physics "Enrico

Fermi," Course LXVIII, Ed. by A. Ferro Milone, P. Giacomo, and S. Leschiutta (North-Holland, Amsterdam,

1980), p. 581.

[8] E. T. Jaynes, IEEE Trans. Syst. Sci. Cyb. SSC-4, 227

(1968).

[9] J. W. Muller, Un nouveau regard sur les probabilites a

priori, Rapport BIPM-80/6 (1980), 16 p. [10] Report on the Meeting of the BIPM Working Group on the Statement of Uncertainties. The final French version will be published in Proces-Verbaux Seances Com. Int. Poids Mes. 49 (70th session) (1981).

[11] W. Wdger, PTB-Mitteilungen 89, 401 (1979).

[12] P. Giacomo, Metrologia 17, 69 (1981).

381

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

On the Statement of a Total Confidence Interval Based

on the Concept of Randomization of Systematic Errors:

Large and Small Sample Sizes

W. Wbger

Physikalisch-Technische Bundesanstalt, Bundesallee 100, D-3300 Braunschweig, F.R.G.

Interpreting the systematic error in a measurement series as the realization of a random variable, confidence intervals of the mean for the special cases of large and small sample sizes are constructed by

statistical methods, assuming normal distributions. These confidence intervals combine the effects of "random" and "systematic" errors and are extensions of the commonly used formulae. Key words: combination of random and systematic uncertainties; measurement uncertainty; randomization of systematic error; total confidence interval; uncertainty assignment.

1. Introduction

If a physical quantity Y is to be measured, then in the statement about the true value of this quantity the ef- fects of systematic errors introduced by the measuring

process has to be included. Apart from this problem still under discussion, the situation could get even more complex, when a conclusion is to be drawn on the true value

of a quantity Z, which depends not only on Y, but also on some additional quantities, the determination of which

*are accomplished in separate experiments.

In the following it is assumed that a general relationship between Z, Y and the other quantities exists and

can be linearized by an appropriate expansion, so that we

start from

z ^ y + G{Au ... , Am) . (!)• For y we assume a series of n measurements, the

values drawn from a normal distribution with variance

fff. The quantities Ai, . . .

, may consist of influence

quantities on the measurement of y, fundamental constants and auxiliary quantities. We assume further that we know, from estimation or other considerations, inter- vals within which the possible values of these quantities

are situated. The values are constant, but unknown, for the above mentioned series of measurements with

respect to y. Repeating the measurement series under

the same nominal conditions, the function G(Ai, . . .

, An) may very well take a different constant value, which

again is unknown and unpredictable. This is especially

clear for the influence quantities.

Therefore one is led to the idea to interpret the constant value oi Ajij = 1, . . .

, AO in one special measure- ment series as the outcome of a random variable. Obviously the sample space for A, is different from that for y. One can visualize an experiment to determine the true

value of z in the followdng way: There are N + 1 urns, one urn for the y-va\ue, one urn for the A i values and so

on. The experiment then consists of drawing n values at random out of the urn for the ^/-values and simultaneously one value out of the Ai-urn, one value out of the

A2-urn and so on.

With this information at hand, mathematical statistics give the methods to handle the problem of construction

of a confidence interval for the true value of z. This is shown in the following parts of the article.

2. Simplification of the Problem

To each component Aj{j - I, . . . , N) corresponds an

(estimated) interval

(A,), ^ AJ ^ {AJj.

Linearizing the function G(Ai, ... , N) around Aj =

^{(A,)j + iA,)j) ij =1, ... ,N) and calling:

yc = y + G(Ai, . . . , An)

one gets from Eq. (1):

2/c = 2 + i /, = 2 + / ; (2)

fj takes values in the interval

-Axj ^ fj ^ + Ax/

^Xi = \G'(A) I ((A J, - Aj).

We now assume that not only yc , but also / is normally

distributed, the expectation value being zero and the

variance Ug. This variance is assumed to be known from

the width of the intervals for the A/'s, This means that

(Ts is some function of the Aa'/s. In this article we shall not discuss which function should be chosen but only

quote the proposal of S. Wagner [1]:

The proposal is based on the assumption that all ran- dom variables /, are stochastically independent. (This as- sumption may be weakened.)

If we measure several times yc in an experiment with

fixed but unknown /, then the scatter of these values is described by af, the variance of the normal distribution.

383
If we build yc, the arithmetic mean of 7i measurements,

and repeat the experiment again and again, then the

scatter of the yc values is given by af />?, if for each ex- periment / takes the same fixed value. But this is not

true: / also varies from experiment to experiment and its scatter is described by the variance ai. So the total vari- ance of yc is given by: = (T^/n + ct| . (4) Note that af/H usually is not known, but only the

estimate

1 nin - 1) i =1

T,\2 y) (5)

which of course is independent of/.

From Eq. (2) we know that for each experiment, Ijc is an estimate for |i. + /, where |jl is the true value of z and

/ is fixed but unknown. Repeating the experiment again

and again and averaging the resulting ^/s we will get

the average because the average of all the /'s in the

long run in zero. From this we conclude that Tjc is nor- mally distributed with the parameters [jl and =

u'xin -\- aif. (The rigorous proof will be given elsewhere.)

Therefore {yc — |J-)/o- is normally distributed with mean

zero and variance 1.

3. The Confidence interval for the True

Vaiue |x of z

To draw a conclusion on the true value (x of 2 from the

observed values yc and^s|^, consider the random variable

T =

- 1 {yc - |jl)/oV CTf

- 1) V

(6)

a!

The variable T is Student-distributed with {n - 1) de- grees of freedom. But this knowledge is of little help as long as CTf or the ratio al/af is not known.

(a) If the ratio a^/af is known, then from Eq. (6) for a given confidence level P the confidence limits for |jl are given by:

yc t„{n 1) I + n -IT

1/2

(7)

(b) If n is large, then af/« may very well be approximated by s|. Therefore can be regarded as known.

Thus

yc

with the resulting confidence interval

Vc Remark: kp is defined by

12^

^ dx

Inserting Eq. (2) for a| yields Wagners' proposal [1]. (c) Small sample size n.

In this case s | cannot be assumed to be an acceptable ap

proximation for u'i In. If we formally let ^\ In to be s| ir Eq. (6), then we have the variable

T. =

yc 2 2

which is not Student-distributed.

However, using a procedure proposed by Welch [2], it can be shown that the distribution of T'2 may be approximated by a Student-distribution with an effective number

of degrees of freedom:

Veff = {n + 1) 1 +

s

2 From this we get the confidence interval for a given con- fidence level P:

Vc ± tp (Veff) VS^^r^

4. Conciuding Remarks

The confidence limits given above depend on the as- sumption of normal distribution of the random variables

under consideration. This assumption may not always be

true. But then the construction of confidence intervals is rather tedious, if possible at all. Nevertheless, the ran- domization of the "systematic" error, represented here

by the function G(Ai, ... , A,v), still is a useful concept,

showing how the effect of several "systematic" error sources has to be handled, namely by summation of vari- ances, and this is independent of the form of the distribution functions as long as there exist the second moments

of the distributions.

It should be emphasized that the randomization of "systematic" errors is not just a trick, to solve a well known

problem, but has a reaUstic background. This may be documented, for example, by the use of this concept for years in statistical quality control of measuring instruments. Physicists seem to have overlooked a part of the

available information about their so called "systematic"

errors. Of course, they are (at least nearly) systematic for one single experiment conducted under repeatability conditions, but their statistical nature immediately appears if one takes into account the impossibility to reproduce for example exactly the same conditions imposed on the ex- periment, when repeating it some time later. Accepting this statistical behavior of the "systematic"

error, one is left with the problem how to describe it. At

the moment, for this one has to refer to plausible models

like the assumption of a normal distribution with a vari- ance given, for example, by Eq. (2). But this is by no means obligatory and is still under discussion.

References

[1] S. Wagner, PTB-Mitteilungen 89, 83 (1979).

[2] B. L. Welch, Biometrika 34, 28 (1947).

384
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Measurement Assurance

A. F. Dunn

National Research Council of Canada, Ottawa, Canada K1A 0R6

Measurement assurance involves combining intrinsic uncertainties of instruments and standards

utilized in a measurement procedure with uncertainties associated with actual use of the instruments,

in order to provide a meaningful statement of a total effective uncertainty of the measurement

procedure.

Key words: accuracy; (equivalent) standard deviation; measurement assurance; uncertainty.

1. Introduction

The effective accuracy of an experimental determination is made up of contributions from a number of

sources, such as: Random uncertainties in the primary experiment and

subsidiary experiments;

Systematic uncertainties from instruments used (cali- brated and uncalibrated), reference standards used,

and linearity and sensitivity of detectors; and

Influence quantity effects and measurements made to account for, or eliminate, the influencing effects. Systematic errors due to design of measurements.

It is conventional to assess a random uncertainty of

repeated measurements as the standard deviation of ob- served data and attribute to it a connotation of probability of occurrence. Because of the inherent indeterminacy

of the calculation of a standard deviation from a rela- tively few (<50) occurrences, there is little to choose

between an implication of 99.99% certainty (±3. 9s), 99.73% certainty (±3.0s) or 99% certainty (±2.6s).

As a working basis, a band of ±2s (95% certainty) has

been adopted as a statement of the "accuracy" of a re- peated measurement.

In those cases where only random uncertainties of re- peated measurements are to be combined, it has also been conventional to combine them in a root-sum-square

(RSS) combination of standard deviation multiples which

have similar meaning.

From statistical theory of random variations in a process which is a function of several variables

Z = F{Xi, X2, Xs, ),

the combined variance (or standard deviation) is

2 _

variance

dF

dXa covariance {Xp, Xq). (1)

For independent variables, the covariances (Xp, Xq) are

zero and the conventional root-sum-square combination

remains, as

axi 1 dx2 ~ + i4^)Hs.f +

dX'. (2)

Thus to combine uncertainties it is necessary to find a system of assessing equivalent standard deviations to represent the contributing sources of uncertainty in a measurement.

In the absence of any other specific information one

can usually assess an upper limit to the magnitude of an

estimated uncertainty (from manufacturer's literature,

calibration reports, experience, etc.) but generally nothing can be said about the frequency distribution within

the band of possible values. It seems reasonable to as- sume that any value between the estimated limits is equally probable, leading to a rectangular frequency dis- tribution of width 2a and height l/(2a), from which the

variance is Veq^ = a^/3.

The experimental process of estimating the limits (±a)

of the observed deviations is found to be easier than

other techniques tried and, when combined with an as- sumption of equi-probability of occurrence, leads directly

to an estimated equivalent standard deviation, =

0.58 a. {v is the symbol adopted to indicate a derived

equivalent standard deviation rather than a standard de- viation calculated from repeated measurements.)

It is comforting for the experimentalist to realize that

any departure from a rectangular distribution toward a Gaussian distribution means that the derived equivalent

standard deviation is conservative.

An analytical approach to the problem of the combination or convolution of various types of distributions with

each other and with more typical Gaussian distributions

has been given by C. F. Dietrich [1]. In summary, he

finds that for a combination of different types of uncertainties, using the equivalent standard deviations

described, one can be satisfied that the probability of an

uncertainty greater than ±2s is always less than in the

corresponding Gaussian case, although not much less. The work of Dietrich has been accepted, and the combination of uncertainties in experiments has been based on

this sort of interpretation of uncertainty bands converted

to equivalent standard deviations and combined to give a

total uncertainty of the experimental determination

which is further assumed to have approximately the same

probability connotations. It is also assumed that the ef385
fective accuracy of a measurement is represented by

±2s total with the anticipation that no more than 5% of the

individual measurements will exceed this band.

2. Application

As an example, an assessment of the uncertainties in the Josephson-effect determination of voltage in the National Research Council of Canada is shown in Table 1. In summary, the magnitudes of the estimated uncertainties for each of the five sub-divisions of the experimental procedure are: = 41 X 10"^ [random observed - 20 to 80]

V2 = 50 X 10"^ [cell comparisons (TG-NRG)]

V3 = 26 X 10"^ [cell comparisons (NRG)]

Vi = 29 X 10"^ [cell value maintenance]

t;5 = 96 X lO"^ [voltage divider calibration]

The accuracy of a typical Josephson-effect determina

tion of the emf of the cells in the local transfer group

(TG) is 2stg = 2(vi2 + vi)^ = ±0.20 X 10"^ = ±0.20 ppm

The accuracy of the maintenance of a cell in the national reference group (NRG) at its observed working

temperature is 2snrg = 2(^1^ + vi + vi + vi + vi)^ = ±0.24 ppm

Figure 1 is a plot of the mean value of the national

reference group obtained from Josephson determinations

for a 3-year period. It is clear the group mean is decreasing with time, at the rate of 0.374 ppm/a from the

data presented. The dashed lines indicate the boundaries

of the accuracy of determination of the mean value, i.e. displaced ±0.24 ppm from the solid curve representing

the best fit to the Josephson determinations. Of the

20 Josephson determinations plotted 15 lie within

±0.12 ppm, or 75% are within the band represented by

±snrg, in good agreement with the intended interpretation of the equivalent standard deviations used in assess- ment of the total uncertainty.

References

[1] C. F. Dietrich, Uncertainty, Calibration and Probability

(John Wiley & Sons, New York, 1973), Chapters 3 and 4.

7.2-

5 . 8

X = VALUES DIRECTLY FROM JOSEPHSON DETERMINATIONS

B = VALUES EXTRAPOLATED FROM JOSEPHSON DETERMINATIONS

43400 43600 43800 44000 44200 44400 44600 43300 43500 43700 43900 44 100 44300 44500 TIME cmjd:>

Figure 1. Plot of the mean emf of the Canadian national reference group of standard cells as obtained from Josephson effect determi- nations over a three year period.

386
Table 1. Uncertainties in Josephson-volt determination

Spread of Equivalent

Deviation Divisor Standard

Source of Uncertainty (x 10"^) (1 or V^) Deviation

(1) Junction balanced against cells Tl or T2

(a) Junction balance against 0.25-0 resistor of voltage divider

: stability of 10 GHz frequency observed on counter ±20 VS~ 12

: reference frequency for counter ±0.001 \^ —

: estimation of from recorder trace with reversals of galvanometer and

calibration of recorder deflection sensitivity (A/) 1 /A v) <±40 V§~ 23

: estimated systematic error due to assumed linearity of galvanometer,

:

pre-amplifier, elimination ofand

thermal

recorder emf in

deflections, dc leads fromequivalent

junction

to

at 4 K to resistor at ±515 293 K by reversal of junction bias current, with N-R-N treatment of suc- cessive balances <±30 V3~ 17

: voltage indeterminacy due to instability of junction bias current and » non-vertical slope of current steps, equivalent to ±3 "V^ 2

(b) Standard cell balance against lOO-fl resistor of voltage divider

: estimation of A e from recorder trace with reversals of galvanometer and

calibration of recorder deflection sensitivity (AZ)2/Ae)

: estimated systematic error due to assumed linearity of galvanometer,

pre-amplifier, and recorder deflections, equivalent to

: temperature of TG cell enclosure (±0.2 mK) and calibration of recorder

sensitivity, equivalent (-56 x 10"^/K) to

: estimated systematic error due to assumed linearity of pre-amplifier and

recorder deflections, equivalent to

: estimated systematic error due to assumed temperature coefficient of standard cells (±1 x 10"^/K), equivalent to

<±20 12

± 5 1 5

±15 9

± 5 1 5 <± 1 1 1

(c) Estimated systematic error due to assumed equivalence of voltage ratio and resistance ratio, i.e., the assumption that /joo = ^o.25 ±10 1 10

(d) Voltage ratio divider

: temperature change during measurement (<±10 mK), equivalent (1 x

10"^) to ±10 \^ 6

: calibration of temperature-measuring thermistor (±2 mK), equivalent (1 X 10"*^) to ± 2 V3" 1

: measurement of temperature coefficient of voltage ratio (±1 x 10~^/K)

equivalent (<±10 mK) to ±10 y/s 6

: estimated systematic error due to effect of barometric pressure

(±0.7 kPa = ±5 mm Hg) on resistance, equivalent (±20 x 10"^/kPa) to ±15 1 15

V 1 = equivalent standard deviation (square root of sum of squares of individual components) 41 NOTE: In practice, the series of 6-8 measurements made on each standard cell gives a mean value wath a standard deviation

of the mean ~ ~ ±(20 — 80) x 10 ^ which is in good agreement with the estimate obtained above. For each day's

evaluation, the standard deviation of the mean value (sj) is taken as the estimate of the random uncertainty which takes account of the actual combination of the various component uncertainties applicable to the junction-cell balance applicable to that set of measurements. For general summary reporting, the estimated value of v i is used.

(2) Intercomparison of cells Tl, T2, T3 vdth Nl, N2

: sensitivity of each cell-pair comparison (±0.02 |xV), contribution to mean

of sub-group of five cells

: standardization of comparison potentiometer (±0.01%) equivalent (for emf differences of <±10 |jiV) to

: estimated systematic error due to assumed linearity of potentiometer

(±0.01%), equivalent to

: thermal emf due to selector switching of cells

: temperature of TG cell enclosure (±0.2 mK) and calibration of recorder

sensitivity, equivalent (-56 x 10~^/K) to

: estimated systematic error due to assumed linearity of pre-amplifier and

recorder deflections, equivalent to

387

± 6 (Is) 1 6

<± 1 vr 1

± 1 1 1 ± 2 1

±15 9

± 5 1 5
Table 1. Uncertainties in Josephson-volt determination (cont.)

Spread of Equivalent

Deviation Divisor Standard

Source of Uncertainty (x 10"^) (1 or V^) Deviation

: estimated systematic error due to location of temperature sensor and

controlling sensor in standard cell enclosure—^under stable ambient conditions estimated at ±0.2 mK but under slowly varying conditions estimated at ±0.5 mK, equivalent (-56 x 10 ^/K) to ±30 1 30

estimated systematic error due to assumed temperature coefficient of standard cells (±1 x 10"^/K), equivalent (1 mK) to ± 1 1 1 temperature of NRG cell enclosure (±0.3 mK), equivalent (-56 x

10-*^/K) to ±17 10 estimated systematic errors for NRG, as for TG above —location of sensors ±30 1 30 —temperature coefficient of cells ± 1 1 1 determination of values of Nl, N2 from mean value of sub-group of five

cells ± 6 (Is) 1 6

stability of cells during 5-hour period from before to after junction-cell balance ±30 - 17

: estimated systematic error due to 50 m of cable betvi^een two laboratory rooms, decreased by switching techniques used to eliminate thermal

effects ±10 1 10 V2 = equivalent standard deviation = 50

(3) Intercomparison of cells Nl, N2, N3, N4

sensitivity of each comparison (±0.02 |xV), contribution to mean of sub- group of four cells ± 8 (Is) 1 8

standardization of comparison potentiometer (±0.01%) equivalent (for emf differences of < ± 10 fi-V) to <± 1 1 estimated systematic error due to assumed linearity of potentiometer

(±0.01%), equivalent to ± 1 1 1 thermal emf due to selector switching of cells ± 2 \^ . 1 temperature of NRG cell enclosure (±0.3 mK), equivalent (-56 x

10"VK) to ±17 10 estimated systematic error due to scaling of temperature sensor dial for reducing cell values to a fixed reference temperature of 30.0000 °C ±20 1 20

estimated systematic error due to temperature coefficient of cells used

for reduction to 30.0000 °C (±1 x lO'^/K) for reductions of as much as

10 mK ±10 1 10

vz = equivalent standard deviation = 26

(4) Maintenance of NRG (8 cells)

: sensitivity of each comparison (±0.02 |xV), contribution to mean of group

of eight cells ± 4 (Is) 1 4

: standardization of comparison potentiometer (±0.01%) equivalent (for emf differences of <10 |jlV) to ±2 . 1

: estimated systematic error due to assumed linearity of potentiometer

(±0.01%) equivalent to ±1 1 1

: thermal emf due to selector switching of cells ± 2 V3" 1

: temperature of NRG cell enclosure (±0.3 mK), equivalent (-56 x

10-*'/K)to ±17 y/3 10

: extrapolation of mean value by drift rate determined from previous

Josephson determination ((-0.374 ± 0.04) x 10"^/year), for periods as long as 3 months <±10 (Is) 1 10

: conversions of cell values at reference temperature to working values at observed enclosure temperature (±0.2 mK) equivalent (-56 x 10"^/K)

to ±17 V3~ 10

: estimated systematic error due to scaling of temperature sensor dial for • . converting cell values from reference temperature to observed enclosure temperature ±20 1 20

: estimated systematic error due to temperature coefficient of cells (±1 x

10~^/K) for reductions of as much as 10 mK ±10 1 10

^4 = equivalent standard deviation = 29

388
Table 1. Uncertainties in Josephson-volt determination (cont.)

Spread of Equivalent

Deviation Divisor Standard

Source of Uncertainty (x 10~^) (1 or \^) Deviation

(5) Calibration of 400:1 voltage divider

[the divider is composed of two resistors (100 fl and 0.25 n); resistance ratio is measui'ed by a direct current comparator bridge in terms of a Hamon step-up

transfer standard (100 fi in series, 1 0 in parallel) by a reversed 1:1 ratio meas- urement at 100 n and a 4:1 or 1:4 ratio measurement at 0.25 fi]

(a) 100: 100 ratio measurement

: mean value of R loalRs and Eg IE loo measurements ± 50 Vs" 29

: estimated elimination of offset of comparison bridge ±20 1 20

(b) estimated systematic error in series /parallel ratio of step-up standard ±20 1 20

(c) 1:0.25 and 0.25:1 ratio measurement

: mean value of i2p/0. 25 and 0.25 /i2p measurements ±100 VS" 58

: estimated elimination of offset of comparison bridge ±50 1 50

(d) temperature measm'ement

: temperature measuring sensitivity (±1 mK), equivalent (1 x 10~^/K) to ± 1 1

, : calibration of temperature-measuring thermistor (±2 mK), equivalent

(1 X lO-^'/K) to ±2 1

: measurement of temperature coefficient of resistance ratio (±1 x

10"6/K), equivalent (<±20 mK) to <± 20 Vi~ 12

(e) estim.ated systematic error due to effect of barometric pressure on resis- tance (±2 kPa = ±15 mm Hg), equivalent (±20 x 10"^/kPa) to ±40 1 40

^5 = equivalent standard deviation = 96

389

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

An Extended-Least-Squares Treatment of Discrepant Data*

E. Richard Cohen

Rockwell International Science Center, Thousand Oaks, CA 91360

In the usual least squares analysis, the weighting of the data is proportional to the inverse of the assigned variance. Since this variance is itself the result of measurement, its value is uncertain. The

observed residuals in the least squares adjustment provide a posteriori estimates of the variance. Linear, unbiased, minimum-variance estimators (LUMVE) are derived which provide a means for as- signing weights to the experimental data. Using this estimator, data treatment algorithms can be for- mulated which are a significant improvement on the "traditional" Birge-ratio procedures.

Key words: adjustment of data; discrepant data; least squares; minimum variance estimators.

1. Introduction

In an earlier report [1] we attempted to include the uncertainty of the assigned errors of a set of data by including in the analysis an assumed distribution function for a universe of experimental standard deviations, and con- sidered the assigned weight of an experimental result to be a random variable drawn from such a distribution,

rather than an exact value. We then introduced a max- imum likelihood criterion as the basis for determining

"best" estimates of the variance of the data as well as "best" estimates of the parameters of the system.

The approach in this paper is to return to the basic

concept of least squares as embodied in the GaussMarkov theorem [2, 3] which defines the "best" estimate

of any parameter as the estimate with minimum vari- ance. The difference here is that an estimate will be

sought for the variance of the input data as well as for the values of the output parameters.

2. Formulation of the Problem

As is the usual situation in a least squares analysis of

the data bearing on the values of the fundamental physical constants, it is assumed that the experimental data

consist of sets of quantities {yt, s,^) which represent a numerical result of the i^^ measurement and the best a

priori estimated of its variance. The experimental meas- urement can be represented by the expression

Vi = ViiXa) + e, = y,'^ + %Ai^x^ + ei , (1)

a where i runs from 1 to M and a runs from 1 to A'', with

M > N, so that the system of equations is in fact over- determined. In addition, the rank of A must be equal to N in order that a full solution may be found (if the rank

of A is less than only a projective subset of the can

be determined). In Eq. (1) the parameters yt^ and A,„

are fixed quantities, the set of parameters {xj are the

unknowns whose values we vdsh to determine, and is the true error in the measurement.

The true error is assumed to have a variance but

one knows only an estimate of the variance, s;^, and a

confidence parameter, v, , associated with that estimate.

The basic ansatz of this development is to postulate the

relations'

<€i > - 0 , <€, €y > - (T? bij ,

<e, ej efcef> = afaf 8y 8;,<. + afaf6^^8<,^- (2.1) + (T? (jf 8,f Sj-fc

<s,-2> = (jf , <eiSj^> = 0 ,

1 + (2.2)

In Eq. (2.2) a factor of 2 has been introduced into the

definition of <s;^s/> so that one may identify vdth the

number of independent observations from which the mean and variance are deduced. However, it should be

emphasized that Vj need not be given that strict an in- terpretation; we use it here as a parameter (the "effective" number of independent observations, if one vdshes

to so identify it) defining the variance of our estimation

of the variance, or less precisely as a parameter giving a measure of our confidence in the stated value, s;^, of the

assigned variance of the i^^ measurement. Equation (2.2)

implies that the variance of the assigned variance is ex- pressed as

var (s;^) - — CTi* Vi or var(Si) = af/2vi (2.3)

without specifically relating Vi to the number of samples

drawn from a Gaussian distribution.

*Work supported in part by NSF Grant PHY78-26467.

The third component of Eq. (2. 1) assumes that the distribution law for the errors is approximately Gaussian to the extent that the ratio of the

fourth moment to the square of the second moment is the same as for a Gaussian, (Serf). If the fourth moment is given by (3 + 7j)cT-^ it will be necessary to add to this expression the additional term

yk S<m Sm„ O'n but we shall omit it in this discussion since in general we will not have the information required to evaluate the higher moments of the probability

distribution functions. We note that this is the only point at which the

specific form of a probability distribution, beyond the specification of its mean and variance, is required.

391
We adopt as the estimators of Xa and af the linear

combinations

where

J

Yj — YjiXa) — yj^ + ^ AjaXoi

(3.1)

(3.2)

(4)

The present approach to modifying the weights of

discrepant data is embodied in Eq. (3.2). Other expressions for estimating af could certainly be written. The

form adopted here is justified on the basis that it is a

linear combination of the available estimators; s,^ is an a

priori estimate while the discrepancy between the experimental value and its adjusted value is also a measure of

the accuracy of the measurement. Although we could

v^rrite df = a,s,^ + b,{y, - Y,f , the more general form adopted in Eq. (3.2) is to be pre- ferred (and perhaps almost required) because the adjusted

values Yj are themselves dependent on all of the experimental data as expressed by Eqs. (4) and (3.1). Conversely, therefore, the estimate of the variance of the i"^ measurement should depend on the observed variances of

all of the experimental data, and since these are correlated quantities it is appropriate to include a full coeffi- cient matrix 5 rather than the scaler coefficient 6, . In Eqs. (3.1) and (3.2) the vector a, and the tensors C aj and Bji^ must be determined to satisfy the conditions

that the estimators shall be unbiased, minimal variance

estimators. The requirement that the estimators be un- biased places conditions on these coefficients. Since one must have <Xa> = one is led to the conditions

>

or

jP

X C'ajA/p — Set 3

(5)

(5.1)

Since we also require <df> - (jf, one obtains the condition,

<uf> = aiXjf + ^ Ejry,BihEkn<{ym-ym)iyn-yn)> (6)

where

jkmn

Ejffi ^ -^J P^ pm

P

(6.1;

To simplify this, one first writes

"^(^/m ~ ym)iyn ~~ yn)^ ~ ^ A.,fiy X y X f. + Um^mn (7)

and notes that ^ EjmAmy — 0

by virtue of Eq. (5.1). Equation (6) is then reduced to af(l - a,) = X BikEj^Ek^n^l = X Bk^jk • (8) jkm

The coefficients C ai and B-^ arg to be defined by

minimizing the variances of x^ and d, ; these are

var(se<,) = X Cijtrf (9)

var (df) = 2a,2^ + % B;,Bl,,[<FjFi,FpFq> (10)

with

- <FiF,><FpF,>]

Ej yj Yj ^ E (10.1)

Equations (5.1) and (9) are the same as in the usual

least-squares problem [2] and hence lead to the familiar result

Caj rS^iP^P" ' (11)

where the square symmetric matrix W is the variance

matrix of the parameters {x „}, the inverse of the weight

matrix, W.

W„p = W^ = X A;„A;p/af , (11.1)

X ^.pWp, = . (11.2)

P We now turn to the more difficult algebra, that of defining the coefficients B'ji, so as to minimize the variance

of df. We obtain, directly from Eqs. (10.1) and (2.1), <F,F,> = Tjk - (12.1)

<Fj Etc FpFq> = Tjk Tpq +Tjp Tkq + Tjq Tpk (12.2)

With these expressions and Eq. (8), Eq. (10) becomes

var(d5 = -^(af - % B;,Tj,f

+ 2 X X -^jfc Bpq Tjq Tpk

jli PI (13)

Making use of Eq. (11) allows a simphfication of these

expressions. One introduces Eq. (11) into Eq. (6.1) and

finds

Ejm = ^jm \- X ^iaA„,pWa[3 . (6.1a)

Then it follows,

Tjk = <Jj Gk hjk - X Ajo^Ak^W^^ . (6.2a)

aP We define a new matrix, tj^,

tjk = Tjk/fjj (Jk = ((JkJ<Jj)Ejk = ^jk —X^j-^k^^-^ ' (6.3)

and note that tjk is idempotent, i.e., X I'jm imk ~ tjk • n

Finally, we introduce a redefinition of the coefficient

Bji^ by writing

B]k - isyf/^j^k)?>]u

392
and (3.2) reduces to af - Si^ (1 ~ X + % ^ ^I'i Pjfc ^i^rn , (14) ml ml jk where

(2/m - ?/to)/o-« while (13) becomes

var

2ut

i&f) = ^ [(1 - x^j.t,,)'

^1 jk

jk pq

(13.1)

(15)

The derivative of this expression with respect to Pj^

gives

(1 ~ ^pqtpq)^jk ~ ^ Ppg ^jq ^pk

pq pq The trace of Eq. (15) gives

(1 - ^^^qtpq)'^t,, = V,X^;,t"pq pq pq But

-^t.j ^ M - N =f

where / is the usual "degrees of freedom" of the adjust- ment. Then

a,- = v,/(Vi + /)

pq pq With these expressions Eq. (14) becomes

(17.1)

(17.2)

1

V, +/

and hence^

a2 8

^jk r ojk

(Tj(Tk{Vi + f)

(18)

(19)

In this equation we express the estimate of the vari- ance, df, in terms of the unknown variances af, so that

this as it stands is hardly a solution; Eq. (18) gives us an

estimate of the variances only if we know the variances,

but if these are known estimates are not needed. However, Eq. (18) can be reinterpreted as a consistency condition on the best values of the variances and used as an

interactive algorithm.

3. The Adjustment Algorithm

Before we can go further it is necessary to define more

clearly what is meant by v,; specifically, of the three

quantities af, v, , and var(s,), which two are to be con- sidered as independent estimates? If we consider af and

V, to be independently estimated one should replace af in

-It is clear from Eq. (13.1) that only XB,k can be, or need be, determined; Bjf, can not be uniquely defined since tji, does not have an in- verse. This definition for Bji; is the most convenient one.

Eq. (18) by its estimate and write

VisMvi + f

- x^) .

, 2 a. (18.1)

On the other hand, if we consider af and var(Si) to be in- dependently estimated, one must also replace v, in Eq.

(18) by its estimate, df/2var(Si), and hence

&f = sf + 2(x^ - /)var(s, = sf [1 + (x^ - /)/vJ. (18.2)

Since the estimates of af must of necessity be positive, it follows that using Eq. (18.1) would require < / +

min(vj) and this constraint has the disadvantage that the

overall accuracy of the solution (as measured by x^) is controlled by the uncertainty assigned to the least-well- defined experimental error, independent of the value of the associated measurement and whether it is discrepant

or in good agreement with the remainder of the data. Equation (18.2) also provides a constraint, x^ > /

-

min(Vi), but one that is much more realistic since it states that in the presence of data uncertainties which are not

well defined there is a minimum overall uncertainty that

may be assigned to the adjusted values even if the data

are highly consistent. In more cases than not, the situa- tion is one of treating discrepant data, for which one ex- pects x^ > /.

Adopting Eq. (18.2) as the more appropriate choice, one then must carry out a least squares adjustment using

weights

Wi (20)

s^Hv, + x' -/)

from which one determines the adjusted values 7, and x^

(21) SrHvi + -/)

Equation (21) is an imphcit equation for x^ which may be

solved by iteration. As it stands, however, Eq. (21) does not represent a convergent algorithm. Convergence can be achieved by

using Newton's rule: F(z) ^ z - '^wAz){yi - F,)2 - 2 - ^Wi{z)n^ (22)

with

Wi{z) = v,/SiHvi - f + z) .

In carrying out the differentiation F'iz) one may

neglect the dependence of F, on z since it may be shown

that, by virtue of Eqs. (11) and (5.1), dfi IdWj = 0 . Then, if z^ be an estimate of x^, an improved approximation may be found from

Zn+l ^ Zn - F{Zn)/F'{Zn)

or ^ Wj {Zn )ri {Zn )^ + 2« X ^' ^^^i^i (^n )^M

1 + ^Wi{Zyi)hih-i(Zn)yVi

At convergence, obviously, z = ^w,ri^ = x^.

(23)

4. An Example of the Use of the Algorithm

As an example we shall look at a simple one- dimensional case: the spectroscopic data- bearing on the

value of ol'^ used by Cohen and Taylor [4] in the 1973 ad- justment of the physical constant. The data are given in Table 1.

393
Table 1: Determinations of a. ^

Author (a-i - 137) X 10^ a/ppm cTA/ppm v' V

Baird et al. 3544(54) 3.94 3.4 83 13.9 3.9 Kaufman et al. 3416(20) 1.45 1.3 2 2.7 2.7 Shyn 3508(46) 3.36 2.6 138 6.1 5.1 Cosens 3563(31) 2.26 2.1 4 4.8 4.8 Wesley, Rich 3563(42) 3.06 2.7 46 16.2 9.1 Muonuim hfs 3634(21) 1.53 0.2 27 1.04 1.04 Hhfs 3597(22) 1.60 1 1

In order to define values of v, for these data we at- tempt to separate the total uncertainty into the so-called A and B components. The A-component is that portion of the variance w^hich is based on a statistical analysis of re- peated measurements and hence one to which the concept

of degrees of freedom may be used in order to evaluate

the accuracy with which the variance is determined. The

B-component is then simply the remaining variance. The

A-component is loosely identifiable with the "random"

component and B with the "systematic" component.

More precisely A refers to that component of the vari- ance which is due to

'

' statistical fluctuations only.

'

For the data of Baird et al., Shyn, and Wesley and

Rich the value of va is the degrees of freedom of the observations; for Kaufman et al. and for Cosens the observations were statistically inconsistent and the data for different transitions had statistically discrepant means.

In these cases the degrees of freedom were taken to be

the number of transitions measured. The muonium data

from Yale and Chicago were treated as a single set and

the observations were analyzed for a single value for the

hyperfine transition frequency and the density-shift coefficients of the buffer gases (see Ref. [4]). In this case va

is the degrees of freedom of that statistical analysis. For

hydrogen hfs the observation uncertainties are completely negligible and the total uncertainty listed is that

coming from the theoretical uncertainty in relating Vhfs with a. The values of v, the "effective" degrees of free- dom of the observation, were calculated assuming that

the B component of variance should be assigned 1 degree

of freedom. Then we have

g

V "A

VA 'A- vs = 1 ; (24)

or

(24)

The column labeled v' in Table 1 is calculated from

Eq. (24) using the given values of va. It can properly be

argued that the large values of va do not realistically

represent the precision of the observation and that mere

consistency could be hiding a " systematic" error that was

not considered and hence not represented in the estimation of the B component of the variance. On this basis the

values of va were arbitrarily restricted to the condition VA ^ 10 (25)

and the values of v given in the last column of Table 1 are

calculated with that constraint.

The iteration algorithm of Eq. (23) was applied to the

data of Table 1 with an initial estimate zq = f =6 and

hence w, - from Eq. (22); the iterations to conver- gence are summarized below:

2 X = (a-1 - 137) X 10^

6

7.524 10.018

12.525

13.458

13.523

13.523

66.318 36.000

21.141

15.059

13.614

13.524

13.523

3544 3527

3527

3526

3526 3526

3526

We see that adequate convergence has been achieved in only 4 or 5 iterations and the adjusted value is a"' =

137.03526(21) with = 13.52 (in comparison with =

66.32 based on the initial uncertainties). Quoting an error by "external consistency" (i.e., applying the Birge ratio to this error) yields a'^ - 137.03526(32). This value of

for 6 degrees of freedom has a pro forma probability

Pix^ > 13.52 I 6) = 0.04. The data are shown in Fig. 1 where the triangular error-bars represent the assigned uncertainties of Table 1. The expansion factors, by which the

SC81-13149

0.037

0.036

0.035

0.034 -

0.033 Figure 1. The 1973 data pertaining to the fine-structure constant, a~' (from Ref. It). The data of Kponou, Hughes et al. have been omitted. See Table 1. Using the a priori variances

gives = 137.035i5(3i) (external consistency). Using ad- justed variances (expansion factors indicated) gives a' =

137.03526(32).

394
algorithm modifies the errors, is given below each entry

and the dotted lines indicate the magnitude of the ex- tended errors. In comparison with this result, the fitting

using v' in place of v gives a negligibly different value, a

-1 =_ 137.03526(20) with = 13.53. An adjustment with

the Kaufman et al. result omitted would lead to ct"^ =

137.03583(16), = 6.15 with 5 degrees of freedom.

If one uses Eq. (18.1) as the condition for the estimation of the variances, Eq. (23) is replaced by

in +1

1 + ^ViizJ^/ViSi^

(23.1)

However, application of this algorithm leads to a nonphysical solution that implies negative weights (i.e., negative

values for some of the df ), and hence must be rejected.

5. Efficiency of the Estimators

The efficiency of an unbiased estimator in comparison

vdth other unbiased estimators is measured by the re- ciprocal of its variance; therefore, we want to evaluate

Eq. (13.1) and compare it with the a priori variance

given in Eq. (2.3). Substituting Eqs. (17.1) and (17.2) into Eq. (13.1) gives

var(a,) = [-———

g

V,- (v, + fr

jk Vi + f

] = 2aMv, + / ). (23)

This result is quite consistent with the interpretation

of V, as the number of

'

' effectively independent observations' ' which enter into the estimate of af. The a priori

estimate was attributed to p, effectively independent ob- servations; when the residues yi - Y- are included as part of the information used in obtaining an improved es- timate, the effective number of independent observations

is increased by / (not by M, since the M residues are not independent statistical quantities).

6. Conclusions

We may conclude from Eq. (18) that this procedure will lead to the converged value = f in only two cases: if

V, = 0 for all i (and the algorithm reduces to the Birge

algorithm), or if the a priori estimates s,^, are equal to the converged values. In general, however, this algorithm will yield a converged value of x^ which will be

between /, the degrees of freedom, and xo> the initial value calculated using the a priori estimates sf for df. This will be true whether the initial estimate xo is larger or smaller than /. The algorithm will push x^ towards its expectation value and expand the variances of xo > /> Xo < /• This is what a statistically effective algorithm

would be expected to do. If the converged value of x^ is significantly different from / this information is important and should not be ignored, and the reasons for a statistically improbable result should be explored. It can come from several possible sources: (1) there may be

discrepant data which should be rejected from the

analysis because they are afflicted with unknown sys- tematic errors, (2) the a priori estimate of the variance

of an experimental result is in error and should be modified, (3) the accuracy of the estimate of a priori variance

may be in error and the corresponding value of v, should

be altered.

It is almost impossible to identify an individual meas- urement as being "too good," and it is only when xo is improbably small that one would be justified in decreasing the variance measures or increasing the initial values

of V,; on the other hand large discrepancies are more

easily identified and an item which gives a large normalized residual, \ iy,„ - ¥,„)/&,„ |, should be critically examined. Although its assigned variance can be increased in order to reduce the contribution of this measurement to

X^, it will in general be found that an increase in d^

which is sufficiently large to reduce the normalized resi- due to an acceptable value would reduce the effect of this measurement on the values of the adjusted parameters,

Xa, to such a degree that very little would be lost by

eliminating this measurement completely from the

analysis.

References

[1] E. Richard Cohen, Maximum Likelihood Criterion for Adjusting Weights in a Least-Squares Analysis, Rockwell International Science Center, Report SCTR-78-9 (August 1978).

[2] E. R. Cohen, K. M. Crowe, and J. W. M. DuMond, The Fundamental Constants ofPhysics (Interscience, New York, 1957), pp. 237-240.

[3] E. R. Cohen, Rev. Mod. Phys. 25, 709 (1953); see also A. C. Aitken, Proc. R. Soc. Edinburgh 55, 42 1934-35; and R. L. Plackett, Biometrika 36, 458 (1979).

[4] E. R. Cohen and B. N. Taylor, J. Phys. Chem. Ref. Data 2, 663 (1973) (Table 28.1, p. 697).

395

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984). GRAVITATIONAL ACCELERATION, MASS, AND

ELECTRICAL QUANTITIES

Present Status of the Absolute Measurement of Gravitational Acceleration

A. Sakuma

Bureau International des Folds et Mesures, F-92310 Sevres, France

This paper reviews the recent work on the absolute measurement of gravitational acceleration g, covering the last decade since PMFC-I in 1970. The single principle involved in the precise absolute measurement of g to better than 1 part in 10^ is the observation of free fall in the gravity field: All the

ten or so laboratories presently engaged in this measurement employ only this principle of free fall and

no longer the traditional reversible pendulum. A large number of new absolute gravity stations (about

50 by the end of 1980) have been created in Europe, North America, Asia, and Oceania by

transportable absolute gravity meters of several laboratories. These stations are aimed at improving

the accuracy of the world gravity network, IGSN 1971, and also at monitoring in the future the secular

variation of the network. Following the production of a successful prototype apparatus constructed

jointly by IMGC (Torino) and BIPM, two examples of a modified commercial absolute gi'avity meter

have been manufactured (1981) by Jaeger Industries S.A. (France). One is in operation at the Geographical Survey Institute, Japan, and the other at BIPM. These instruments employ the symmetrical free rise and fall of a cube-corner reflector together with a novel "multi-station"

interferometric measurement procedure. An accuracy of better than 1 part in 10^ is currently obtained. The principles together with an outline of the performance of these new instruments are described. Thus absolute gi'avimetry is now beginning to provide a new means for several metrological and

geophysical studies, accessible to all who are interested in this field. Key words: absolute gravimetry; acceleration due to gravity; earth tide; free fall; IGSN 71; secular

variation.

1. Introduction

As I pointed out in my paper at the PMFC-I in 1970

[1], it was in the late 1960's that absolute gravimetry

(absolute determination of acceleration due to gravity)

became established as an important new means for the

study of the earth's gravity field [2]. This new age of absolute gravimetry was opened by a remarkable improvement in the relative accuracy of this measurement from 1 part in lO*' to 1 part in 10^. This jump in accuracy had

been achieved by several new experiments carried out in

this period [3-5] with the methods of "free fall" using

modern techniques, namely: interferometric length measurements with laser sources, high speed chronometry,

and improved vibration control. Based on these new absolute measurements (ten results, of which eight were

obtained by the first transportable absolute gravimeter

of Hammond and Faller [5]), combined with a large

number of gravity data collected during the previous

twenty years by relative measurements (—1200 values by

pendulums, —24000 values by spring gravimeters), a new

world-vdde gravity network was established in 1971. It is called the International Gravity Standardization Net

1971, "IGSN 71" [6].

. This IGSN 71 has been approved and adopted by the

15th General Assembly of the International Union of Geodesy and Geophysics: lUGG (Aug. 1971, Moscow) as the new international gravity reference, replacing the

Potsdam gravity net. IGSN 71 contains 1854 gravity

stations in 494 towns where the gravity values are determined by a single least-squares adjustment of the absolute and relative data. The accuracy of this gravity net is estimated to be better than ±1 part in 10'^ (±1 jxm-s'^) at each station of the net.

Since the establishment of IGSN 71, the principal task

of absolute gravimetry has been to check and improve

the accuracy of this gravity net and also to monitor in the

future its secular variation. All the ten or so laboratories

that have been engaged in modern absolute gravimetry

employ only the method of free fall which is the single

method which can lead to an accuracy better than 1 part

in 10^ for g. Thus, the traditional pendulum method has

ended its long career of three centuries (since the epoch

of I'Abbe Picard in —1670) with a final best accuracy of

0.3 ppm (3 ^JLm -s^^) [7 gj

The present report is intended to summarize the evolution of absolute gravimetry since 1970, developed in ten

laboratories throughout the world. Brief remarks are

also given on each of the particular devices, and the ad- vantages and disadvantages of the different methods are

discussed. It is very encouraging for absolute gravimetry

to note that the local values of gravity are required, in some laboratories of metrology, with more and more ac- curacy for electrical standards experiments and it is said

[9] that in the future, the stability of the kilogram with

respect to time may be monitored by using the stability

of the Josephson voltage standard and the calculable capacitor with an accurate knowledge of the local value of the gravity field.

2. Transportable Absolute Gravimeters

Absolute gravimetry has been one domain of precise

physical measurements which has been particularly

favored by the recent development of the stabilized laser

light source. The characteristics of such sources, accu- racy of wavelength, high light intensity, and smallness of beam divergence, make them ideal for modern gi-avime397
try by the free fall method which requires an ultimate ac- curacy in the length (distance of fall) measurement in vacuum. This requirement is almost perfectly satisfied by

a laser interferometer system in which the falling object

is used as a reflector of the laser beam. In addition, the

continuous observation of free fall by the laser fringecounting method permits the establishment of a large

number of intermediate "multiple stations," e.g., every

2048 fringes counts, to increase the accuracy of the meas- urements by using a statistical treatment of the large

amount of data obtained. It also has need of a very sharp

timing signal during the free fall which can be satisfied by modern digital electronics.

For reasons such as these all ten laboratories presently

engaged in absolute gravimetry employ only the method

of free fall combined with the Michelson type laser inter- ferometer [Fig. la and b].

MAAATWVl

Figure 1. (a) Michelson interferometer system with symrmtri-' calfree rise and fall method, (b) Same system with simple free

fall method.

2.1 Transportable Apparatus With Free Fall Method

(a) Air Force Geophysics Laboratory (AFGL), Hanscomb AFB, Mass. [10]. This apparatus has been called the second generation of transportable gravimeter

developed from the first generation apparatus invented

by Hammond and Faller and used at a number of international sites during the period 1968-69 [5], and which also greatly contributed to the establishment of the new

world gravity net, IGSN 71. This is also the first ap- paratus which introduced the method of multiple stations

in the free falling trajectory (60 cm.) of a cube-corner

prism in vacuum. A high-resolution time digitizer

(^0.125 ns) combined with a minicomputer measures and

memorizes the consecutive time intervals at a large

number of stations (500 ~ 700) during the free fall and

performs a least-squares average. This averaging by the

multiple-station method has made it possible to determine the vertical gradient of gravity and also to detect a systematic in-phase vibration of a few nanometers amplitude of the apparatus due to the triggering of the falling

object.

An apparatus using the simple free fall method is very

sensitive to residual air resistance. This air resistance can be calculated and can also be measured; it is typically

~1 X 10"^ of g for a pressure of 1 Pa after aim fall. Therefore a low pressure, of order 10"^ Pa, is necessary

for the falling object if the error due to air resistance is to be less than 1 x 10"^ of g. A low pressure of this order

is not so easy to realize, especially in the case of the gra- vimeter. Because the vacuum vessel contains some movable elements, among which is the falling object, the vac- uum may change during the free fall. In order to reduce

this resistance effect, the falling object is dropped with

its container; Fig. 2 shows this in a schematic way. At

the top of the chamber a soft iron block is attracted by

an electromagnet and just before release of the block the

ballast mass drops about 1 cm. Upon release, the springs

pull the inner chamber and the ballast mass together,

resulting in the cube corner assembly being roughly

6 mm above its resting place within the inner chamber.

This configuration has resulted in approximately a 100 to 200 times reduction in the effect of air resistance. Thus

with this "chamber-in-a-chamber technique," a vacuum

of 10"^ Pa will become sufficient. One other advantage of

this method is to have a free-fall object which has no magnetic material on it, while still employing the

mechanically simple technique of releasing an object with an electromagnet.

Inner Chamber

Soft Iron

Spring

Spring

Retroreflector Assembly

Cube Comer

Reflector

Figure 2. Free-falling unit in AFGL second generation ap- paratus (schematic diagram).

398
This apparatus is completely automated and the data

are analyzed and corrected for gravity tide in real time.

The apparatus has a total mass of about 700 kg when

packed for air transport and is contained in ten boxes

which can be handled by one or two people. It has been

used at twelve new sites in the United States in the last few years and a final accuracy of a few parts in 10^ is ex- pected. The details of this apparatus will be presented at

this conference.

(b) JILA/NBS, Boulder, Colo. [11]. This is another

transportable laser-interferometer gravimeter which employs the simple free fall of a cube-corner prism and a fi- nal accuracy of a few parts in 10"^ is aimed at. In order

to reduce the effect of air resistance, this apparatus also uses the chamber-in-a-chamber technique with a servocontrolled follower system. Figure 3 shows schematically

the functioning of this falling system. The chamber of the

falling cube corner is equipped with an optical position

detector. This chamber is attached to a servo amplifiermotor system so that the vertical displacement of the

chamber is controlled by this servo system. A sapphire

sphere fixed on the falling cube corner serves as a colli- mator lens for the optical position detector. To start the

free fall of the cube corner the chamber is displaced rapidly downward about 2 mm. Thus the sapphire sphere

enters the horizontal light beam of the position detector

to activate the servo amplifier-motor system so that the

chamber follows the free fall of the cube corner.

Motor

Sapphire

Sphere

Vacuum

-—Chamber

Co-Accelerating

Chamber

Servo Amplifier

Falling

Corner Cube

Figure 3. Servo-driven "chamber-in-a-chamber" JILA free- falling unit (schematic diagrayn).

The window in the bottom of the chamber is made of a very thin (~5 \x.m) transparent membrane called a pellicle. This membrane ensures that the slight inclination or

rotation of the chamber will only negligibly vary the optical path length during the fall of the object. This falling

system is used at present at a pressure of about 1 Pa and

by means of small holes the inner chamber is also maintained at about the same pressure. Therefore, this chamber-in-a-chamber system must cancel out the effect

of air resistance with a factor of about ten thousand in order to reduce the error of this effect sufficiently to as- sure a final accuracy of 3 parts in 10". Apparently it is not likely to attain such a high degree of cancellation so

it will be necessary to determine experimentally the coefficient of this resistance by changing the pressure or by

comparison with another type of absolute gravimeter to determine the correction to the value of gravity. This

gravimeter is equipped with a long-period suspension

system (7" ^ 40 s) for the mechanical isolation of the

reference cube corner from background disturbances. It

is also completely automated and gravity measurements

can be repeated quickly (150 drops in ten minutes). The

recent results show that an accuracy is now obtained of better than 1 part in 10^ and automatic measurements

continued during several days permitted the form of the

perturbing waves of the gravity due to the earth tides to be clearly recorded. The details of this apparatus will also be presented at this conference.

(c) Institute of Automation and Electrometry (lAE),

Novosibirsk [12, 13]. This transportable gravimeter has

been in operation since the beginning of the 1970' s and

also employs the simple free fall of a cube-corner prism

with a stabilized-laser interferometer. The apparatus has no anti-vibration suspension. A set of measurements is carried out to obtain an arithmetical mean value of about

150 drops of the prism. The measurements are completely automatic and after each set of 150 drops (which

takes about 30 minutes) the alignment and adjustment of the apparatus are checked. This apparatus uses a large

electromagnet to reset the falling object after each drop

so the uncertainty in the measured value of g due to this magnetic effect had to be evaluated in order to attain an

accuracy of about 2 parts in 10^. However, a direct comparison of this gravimeter at the BIPM gravity station

carried out in September 1977 demonstrated that the uncertainty due to the magnetic effect had been over- estimated (it was found to be less than a few parts in

10"). The estimated total uncertainty of this apparatus is ±1.6 X 10"^ in which the three major contributions are: the error of vertical alignment and collimation of the in- terferometer: 1.0 X 10 error of laser wavelength: ±8

X 10""; and error of air resistance correction: ±8 x 10"". This apparatus has been used, since 1976, to determine

the gravity at several international stations in Levodo,

Potsdam, Sevres, and Singapore. The measurements in Sevres were aimed at finding the possible systematic er- rors of this apparatus. After about 4000 drops, made

over a period of one week, the value of gravity by this gravimeter was in good agreement with the BIPM value

(to within 4 parts in 10"). (d) National Institute of Metrology (NIM), Beijing [14]. This transportable gravimeter, constructed in 1974, also employs the simple free fall of a cube-corner prism, to- gether with a Lamb-dip-stabihzed laser interferometer.

In this apparatus the falling object assembly (380 g) contains, at its top, a cylinder of soft iron (—45 g) required

for the electromagnetic release of the object. In order to check the perturbing effect of this magnetic material on

the gravity value, a similar object was made with a fer- romagnetic material for which the magnetic moment is about 36 times stronger than that of the usual object.

Experiments made with this object could not detect any

evidence of systematic error. Therefore the error due to the soft-iron piece is considered to be less than 1 part in 10^ of g. In April 1980 this gravimeter was transported to Sevres to compare with the BIPM apparatus. The results agreed to better than 1 part in 10^ with an uncertainty of

399
±1.6 X 10 ^. Details of this apparatus will be given at

this conference.

(e) Earthquake Research Institute (ERI), Tokyo

[15, 16]. This transportable laser-interferometer gravimeter uses a "cat's eye" as the falUng object instead of a cube corner. The rotation of a cat's eye during free fall causes a significant variation of the optical path in the in- terferometer. So in thib apparatus the tolerance of rotation rate of the falling cat's eye is very small, only

±0.3 mrad-s~^ To realize a non-rotating free fall, a piezo-ceramic base is used to support the cat's eye. At

the moment of the beginning of free fall the cat's eye

stays momentarily fixed in space due to a rapid piezoelectric contraction of the ceramic base, and during this short time interval (—0.25 ms) the ceramic base is rapidly

moved away from the fall path of the cat's eye. This device permitted the realization of a very small angle of ro- tation (10 sec of arc during 30 cm fall). The gravimeter

employs also a multiple-station method, the interference

fringes being sampled at millisecond intervals by means

of a photographic recording. The uncertainty of the

measured value of gravity is presently about ±4 x 10"*

of which the two major components are the microseismic

perturbation: ±2.6 x 10"*; and the uncertainty of the

He-Ne laser wavelength: ±2.0 x 10"*.

2.2 Transportable Apparatus With Symmetrical Free

Rise and Fall

(a) Istituto di Metrologia G. Colonnetti (IMGC), Torino

[17, 18]. The method adopted in this transportable gra- vimeter is the observation, from an inertial reference

point, of the symmetrical free rise and fall motion in a vacuum.

This method [18] is the most promising due to its in- herent high accuracy and its great freedom from sys- tematic errors such as residual air resistance, timing

errors, Doppler effect, and so on [1]. In this sjTnmetrical

method, a cube-corner reflector (mirror) is projected

vertically upwards and crosses two defined horizontal

stations whose separation, H, is known. Two independent time intervals corresponding to the upward and

downward passage across the lower station T\, and the

upper station Th, are measured and the value of gravitv

is obtained from: g = m/{Tr - Tk). This method is particularly advantageous in the case of a transportable

apparatus due to the fact that the effect of the residual

air pressure on the falling object is cancelled out to first order since on the upward portion the air resistance or drag acts in the same direction as gravity while during

the falling part this air resistance force -is in the opposite

direction. Thus the vacuum system of the gravimeter can be made very simple: it consists only of a primary pump,

of small capacity, which facilitates transportation and

saves preparation time for the gi'avity measurements.

This first transportable gravimeter using the symmetrical method was developed in 1974 at the IMGC, Turin

after a long technical collaboration with the BIPM,

Sevres. The two essential parts of this apparatus are: a Michelson interferometer in a pressure-tight case and a vacuum chamber in which a cube-corner mirror, forming

a reflector of the interferometer, is launched vertically

by a catapult (Fig. 4a). The distance H and the time of

flight of the reflector are measured simultaneously by a

bi-directional fringe-counting method (Fig. 4b) using a Lamb-dip-stabilized He-Ne laser, the wavelengtn of which, 633 nm, is periodically calibrated

A specially-designed catapult (see next paragraph)

throws the cube-corner mirror of 70 g about 50 cm in a

— Th ^ — Tl

Figure 4. (a) Optical scheme of the IMGC gravimeter: Ph: photomultiplier; Hg: mercury-pool (for levelling); Spl: beam

splitter, (b) Distance ayid time determinations by laser in- terference fiinges (see text). vacuum while giving it a rotation of less than 0.01 rad

per second and a vertical deviation of the trajectory of about lO"'* rad maximum. Thus the acceleration due to the rotation and the horizontal velocity of the projectile

remains within 5 parts in 10^ of g. Two kinds of anti- vibration devices are employed in this apparatus to at- tenuate the perturbation of gravity due to ground motion

and the mechanical shocks caused by launching the pro-- jectile. One is an inertial suspension of the reference

cube corner by a long period (^20 s) vertical seismometer in a rigid pressure-tight case (Fig. 4a). The other

consists of a set of active compensators of vertical vibrations, namely, three piezo-electric supports on which the

interferometer is installed. By these means the perturbation of gravity is reduced over 20 times and the residual

acceleration on the reference cube corner is normally less than 1 part in 10* of g. Because of the complexity of the

manipulation of the launching device, it was difficult to completely automate this gravimeter. However, one

operator can repeat the launching of the cube corner and

can calculate a gravity value every 3 to 4 minutes, result- ing in about 20 measurements during one hour and. owing to the high repeatability of each result, one set of 20

measurements is quite sufficient to determine a mean

value of gravity accurate to 1 part in 10* after the application of the tidal correction. The coefficient of the resid- ual air resistance was determined experimentally by

changing the pressure in the vacuum chamber in the

range 0.01 Pa to 100 Pa. The global coefficient of air resistance includes two other effects, namely, the effect

| of buoyancy amounting to about -1 x 10"* of g per pascal !( in this case; and the effect of the diminution of the laser wavelength due to the refractive index of the residual

air: +3 x 10"^ /Pa; together these lead to -32 x 10"'^/Pa

in a free rise and fall of 0.5 m. Practically, this global ef400
feet is almost negligible in this apparatus because during

gravity measurements the vacuum is maintained nor- mally at 0.02 Pa. It is worth noting that the global coeffi- cient of air resistance in this symmetrical method is about 3000 times smaller than the coefficient of air resist- ance in the simple free fall method.

This apparatus has been used since 1976 to create a number of new gravity stations of 1 x 10~^ g accuracy, about 30 stations in Europe and about 10 in North America, and some 12 stations are projected in Asia during

1981. Not only have these expeditions contributed to an improvement in accuracy of the IGSN 71 but also the

absolute value of gravity obtained at each station will permit the monitoring and detecting of future secular

variation of gravity by means of the periodic repetition of the measurements.

The first absolute calibration of IGSN 71 was made at the 17 stations of the European calibration line in 1976-78 [18]. The results revealed that the uncertainty

of IGSN 71, estimated to be 1 part in 10^, was in general

much lower (^5 x 10"*) than expected.

(b) Jaeger Aviation Division, Levallois-Perret

(France) /BIPM. Following the successful absolute gra- vimetry results obtained by the first prototype transportable apparatus just described, work has been continued at BIPM aimed at improving the performance of this type of apparatus. At the same time the Aviation Division of Jaeger S.A. (France) showed an interest in developing a commercial version of the improved transportable absolute gravimeter. The first such apparatus was manufactured in 1980-81 by Jaeger based on the studies carried

out by BIPM. Of the two instruments manufactured so

far, one is in operation at the Geographical Survey Insti- tute, Tsukuba (Japan) and the other has recently been

delivered to BIPM. This new absolute gravimeter employs

the symmetrical free rise and fall of a cube-corner mirror

with the multiple-station interferometric measurement

procedure: during the free rise and fall, the position of the

cube corner is observed continuously by means of an

iodine-stabilized laser interferometer using a subnanosecond (^0.1 ns) time digitizer. A large amount of data (a total of about 1300 measured relative positions and

times) thus obtained is recorded and used to calculate the

following values by a microprocessor integral with the

gravimeter: a value of gravity at a well-defined height,

the value of the vertical gradient of gravity throughout

the trajectory, and the proportional factor that links deceleration force (due to residual pressure) to the velocity

of the falling object. Figure 5 shows the general view of

the apparatus: the total mass of all assembled components: —400 kg; the overall dimensions in the operating

state: 1.95 m high, 0.9 x 0.9 m at base level. This trans- portable gravimeter requires one day preparation for as- sembly and one half day for disassembly by two operators,

and about 20 measurements of gravity can be made by a semi-automatic procedure during one hour. Because of the

high accuracy and good repeatability of the measurements

(^1 part in 10*), there is no need to make a large number

of measurements. Normally, at one station a total of about

100 measurements made over one or two days, composed

of five sets of 20 measurements at different phases of the

tide correction, are quite sufficient to give an accuracy

better than 1 part in 10* of g. The most important advantage of the symmetrical free

rise and fall is the possibility of the "self-checking" of the reliability of each measured value of gravity. This

checking can be made by comparing the two different

values of gravity obtained in the same launching; one is the value of g obtained during only the upward motion,

Figure 5. Generalviewofcommercializedgravimeter"Jaeger."

M: Michelson interferometer in a vacuum chamber; C: cata- pult mechanism, and corner-cube reflector iyi a vacuum

cylinder; V: vacuum-pumping system; S: anti-vibration suspension; L: iodine-stabilized laser; B: chronometer, computer, and printer.

the other is the value during only the downward motion.

This difference is influenced by the residual air resist- ance, Doppler effect, and electronic response of the

chronometer, but it can be well determined experimentally and the mean of the two values of gravity must be

equal to the integrated value of g obtained by both up- ward and downward motions. Therefore this self check- • ing permits the determination of the statistical weight of each experiment and if a discrepancy is identified by this checking, the result must be rejected. Because of this self-checking feature, a gravimeter using this symmetrical method can be regarded as a veritable absolute ap- paratus which has essentially no need to be compared

with other types of gravimeter. In spite of these important advantages, the symmetrical method has been util- ized only at a few laboratories: NPL, Teddington [3] and

NML, Sydney [20] for stationary gravimeters of 1 x 10"^ accuracy level. The reasons that the symmetrical method

is less popular than the simple free fall, are: (i) the freefall method is incomparably easier to realize than the

symmetrical method; (ii) even with free fall, a final precision of a few parts in 10^ can be obtained if the apparatus

is compared with or calibrated by other symmetrical ap- paratus; (iii) once such a calibration is done, the free-fall apparatus in its highly-automated version will become a very handy field-usable apparatus (e.g., the JILA/NBS

apparatus) replacing in future the spring-type relative

gravimeters (this is a very important trend in modern

absolute gravimetry); and (iv) in the symmetrical method, if the design of projectile and catapult is not quite

satisfactory, the additional perturbing effects, e.g., mechanical shocks, outgassing, rotation, oscillation of the

projectile, etc., may easily vitiate the inherent advantages of the method.

The catapult used in this commercialized gravimeter

has been described [21] but it seems useful to recapitulate schematically with Fig. 6. The projectile (1) is a cube-corner reflector, 5 cm high with a mass of 70 g, designed so that the optical center of the reflector coin- cides with the center of gravity to within 10 |xm in order

to minimize the effects of any rotation which might oc- cur. The three aluminized mirrors of the corner reflector are fixed on a duralumin support in which six holes of 6 mm diameter are made every 60 degi'ees around its

401
Figure 6. Schematic drawing of the catapult with corner-cube

reflector in flight.

vertical center axis to let the laser beam (2) of the inter- ferometer pass freely. This duralumin support also has a

central hole of 11 mm diameter in the vertical direction

and the top of this hole is slightly tapered. This tapered

top of the reflector rests before launching on a conical

piece (4) containing a shock absorbing device, attached at

its top to the elastic cord (3) of the catapult and at its bottom to a cylindrical releasable piece (6) by a nylon

wire (5). The piece (6) is also attached to a weak elastic cord (7) pulling it downward so that the release piece (6) moves always only in the vertical direction. Before

launching, the reflector (1) sits on an elevator table (8) and is turned around its vertical axis so that the laser beam (2) falls correctly in a specified hole. Then the table

(8) descends slowly and leaves the reflector (1) hanging

on the conical piece (4). An annular piece (9) placed coaxially with the wire (5) can be displaced vertically by ro- tating a shaft with clutch (10) and by a pulley (11). By

these mechanisms the piece (6) guided by the ring (9) is- engaged into a hole of the releasing piece (12) by the

pressure of compressed air. Then the ring (9) rises to be

locked into a fixed stage (13) and the shaft (10) is de- clutched to Hberate the pulley system (11) from mechanical friction. When the piece (6) is released, the reflector

(1) is accelerated upwards by the traction of the elastic cord (3). After an accelerated rise of about 15 cm, the

released piece (6) is stopped by the fixed stage (13). Likewise, the conical piece (4) which has accelerated the

reflector is also stopped so that the reflector (1) begins

its free rise. By the shock given by the released piece (6)

to the fixed stage (13), the locked ring (9) is liberated

and is pulled onto the piece (6) by the downward traction

of the spring (14) which opposes and equilibrates with

the upward traction of the piece (6) due to the elastic cord (3). After free rise, the reflector (1) falls and sits again on

the conical piece (4) so that the elastic cord (3) is stretched. During this braked fall, the two pieces (6) and

(9) follow this downward motion of the reflector by means of the traction of the spring (14). During the de- celerated motion, a friction disc in the pulley (11) absorbs

the energy so that the reflector is stopped smoothly

without bouncing. This catapult throws the reflector while giving it a rotation of less than 0.01 rad-s"^ and

with a random horizontal velocity of 0.2 mm-s"\ result- ing in a random perturbation on the value of gravity of —2 parts in 10^ if these quantities are not measured in each throw. The mass of the moving part of the catapult

is only several grams and this mechanism is used as a

catcher and a soft brake as well as the catapult. There is no other element moving or oscillating during the free

rise and fall of the reflector and the effect of outgassing

by this catapult is negligible. In addition, the catapult is mounted on a vibration absorber, therefore the mechanical perturbation due to the operation of the former is in general much smaller than the level of the ground-motion

noise. The elastic cord (3) generates some electrostatic charge

by its deformation in vacuum. The effect of this on the

reflector is eliminated by a metallic tube (15) placed

around the elastic cord. This electrostatic shield tube (di- ameter 5 mm) maintains the capacitance (—10 pF) con- stant between the reflector and ground during the flight.

Due to this constancy of the capacitance the effect of

electrostatic charge on the reflector in flight, it was con- firmed, was negligible, even when the reflector was

charged artificially up to one hundred volts.

(c) Institut fur Angewandte Geodasie (lAG), Frankfurt

[22]. Here, there is in progress another attempt at con- structing a transportable gravimeter using the symmetrical rise and fall method. The projectile consists of a cube

corner made of quartz while a second cube corner serves

as a mass of a vertical seismometer. The cube corners

are parts of a Michelson interferometer for measuring the

distance. The times are measured by a 2 ns counter and

the interference fringes due to the rise and fall of the

projectile are counted simultaneously. A multiple station

method will also be employed.

2.3 Remarks on the Transportable Apparatus

The above-mentioned eight transportable gravimeters

can be classified into the following three types:

(i) Straightforward free-fall type' (lAE, NIM, ERI).

This type of apparatus has no protection against mechanical vibrations and therefore drop-to-drop scatter is very

important (~5 x 10"^ g). Nevertheless, the mean value

of a large number of drops is accurate up to about one

part in 10^ after application of several corrections due to systematic effects. For this kind of apparatus, automation of the measurements is indispensable to obtaining a

sufficient number of drops.

(ii) Sophisticated free-fall type (AFGL, JILA). Use of the chamber-in-a-chamber technique, anti vibration devices, and the multistation method permit one to obtain an accuracy of few parts in 10^. Once compared with a symmetrical apparatus, this type will be a very useful apparatus for field operation.

402
(iii) Symmetrical free-fall type (IMGC, Jaeger, lAG).

The advantage of this type is well known and its commercialization is very important. Future efforts are to be

made for automation, or miniaturization for ease of transport, or for the improvement of accuracy beyond 10"^.

3. Permanent Stations

In August 1975, at the 16th General Assembly of the

lUGG in Grenoble, the International Association of Geodesy (I AG) adopted the following (partial) resolution [23]

about the future extension of the permanent absolute

gravimetry station net: "The I AG recommends that absolute gravity measurements should be made both in stable regions and in regions where secular or longperiod gravity variations are expected and should be re- peated after sufficiently long intervals. To this end:

(a) About 10 permanent stations (or observatories)

should be established where absolute gravity can be ob- served with microgal accuracy {~-10~^ g) to investigate

possible global variations and correlation in these longterm changes of .9 ..."

.

At that time, one station of this kind already existed,

at BIPM Sevres, having been in operation since 1967,

while at the International Latitude Observatory (ILO),

Mizusawa (Japan) construction of a similar apparatus had

just started in collaboration with BIPM.

Since then, however, no new permanent station with a

stationary absolute gravimeter has been established.

The explanation of this fact is two-fold: first, the construction of such a permanent station takes, in general, a very long time—several years at least, with an important

investment in personnel as well as in finance, and it is also necessary to obtain the technical assistance of ex- perienced laboratories in this field. Second, it is during

this interval that transportable absolute gravimeters

have entered upon the scene (IMGC, AFGL, lAE, NIM)

and demonstrated that a new absolute gravity station of ~1 X 10~^ accuracy can be created very easily within a few days. Thus, all laboratories newly interested in absolute gravimetry have adopted the transportable type to create their absolute stations and the measurement of

gravity, although not in a permanent installation, can be

repeated periodically. In such circumstances, the existing

permanent station at BIPM as well as the future station

of ILO, Mizusawa (presently in preliminary operation)

are regarded as special stations where gravity measure- ments can be made with sufficient accuracy (~10"^ g) for the comparison and calibration of different kinds of trans- portable absolute gravimeters.

3.1 Earth Tide Measurements at an Absolute Station

The gravimetric earth tide is known as the most important (periodical) perturbing effect of the Earth's gravity

field (-0.3 ppm, max. at mid-latitude). In order to improve the accuracy of the tide correction an earth-tiderecording spring gravimeter was constructed in 1973 at BIPM. This gravimeter employs a nulling method (the

spring length is kept always constant by an electrostatic

feedback system) to eliminate the effect of the spring's

hysteresis.

This combination of the tide gravimeter (relative ap- paratus) with the absolute apparatus presents mutual ad- vantages: the absolute apparatus permits one to evaluate

the drift of the relative apparatus and to calibrate it, while the relative apparatus provides continuously for the absolute apparatus the true, observed correction of the earth tide which is needed to determine the mean

value of gravity. The instrumental drift of this tide gra- vimeter was found to be very small (less than 5 parts in 10'" per day) and this permitted ready detection of per- turbations of gravity connected with atmospheric pres- sure changes, the coefficient of which varies according to different meteorological conditions from -4nm-s"^to -2

nm-s^ per millibar (100 Pa). Thus the recording of the earth tide is now becoming an indispensable means

for accurate absolute gravimetry and reciprocally, in the near future, absolute gravimetry of ultimate accuracy

(^10"^) will become the best way to calibrate the amplitude and the phase of the tides without perturbing the

tide apparatus.

A new type of earth-tide gravimeter has been

developed since 1972 [24]. In this apparatus, called a su- perconducting gravimeter, the mechanical spring is re- placed by a magnetic field generated by the persistent current flowing in two superconducting coils. The ex- tremely low drift of this gravimeter is a result of the

inherent stability of a persistent current in a superconductor. The only moving part in the gravimeter is a superconducting hollow lead sphere of 3.6 g which is levi- tated by the magnetic field and the position of which is measured electronically. This gravimeter has been re- cently produced commercially [25] and an extremely low

rate of drift, ±6 parts in 10^ of g per year, is reported. It may not, of course, be prudent to use only this gravimeter for the study of the secular variation of gravity, due

to possible instrumental perturbations, e.g., liquid

helium refilling, but the combination of this superconducting gravimeter with an absolute gravimeter will be- come a powerful new means for the study of secular

changes of gravity.

3.2 Long-Term Stability of Gravity

At the BIPM station, a significant variation of gravity

has been observed: +A parts in 10^ during the last 14 years since 1967. A significant portion of this long-term

variation of gravity was observed during relatively short

periods in 1969-72 [26] but the causes have not been

identified up to the present. One defect at this station is that, at present, there is no means to measure the variation of the underground water table. In order to clarify this kind of variation of gravity, it is projected to create

several satellite stations around the BIPM station by means of the transportable absolute gravimeter. The

first satellite was selected at the Geophysical Institute of BRGM at Orleans, about 130 km south of Sevres, where

the variation of the underground water table has been

monitored by means of a piezometer. The difference in gravity of about 1.05 mm-s"^ between Sevres and

Orleans will be calibrated and its constancy will be monitored by the transportable absolute apparatus. Such

gravity liaison with a satellite station will in the future

supply us with information on the question of whether

the observed variation is a local phenomenon or is occur- ring on a regional (even global) scale.

4. Conclusion

The present world-wide activities in absolute gravimetry have been summarized. Transportable absolute gra- vimeters have demonstrated their capabilities at the

level of 1 part in 10^ oi g, corresponding to a resolution of 3 cm height change, by use of modern technologies: sta- bilized lasers, sub-nanosecond electronics, computer proc- essing of data, and automation of measurements. A

large number of new absolute gravity stations (about 50

up to 1980) has been established with international colla403
boration and has verified the accuracy (^1 x 10"') of IGSN 71. This work, that is, extension of a precision ab- solute world gravity net, will be continued with these

gravimeters more intensively and more efficiently in the

1980' s, covering new areas (e.g., Africa, Antarctica)

where the gravity data are not yet sufficiently abundant.

It is notable that, for the first time, a transportable ab- solute gravimeter has been manufactured commercially

(Jaeger 1980-81) using the best principle, namely, the

symmetrical free rise and fall of an interferometer reflector. The permanent gravity station at BIPM has been

used frequently for comparisons and calibrations of vari- ous types of absolute and relative gravimeters. Because

of its necessity and uniqueness, this permanent station

has been maintained and improved with instrumental

reinforcements: earth-tide recording gravimeter, iodinestabilized laser, transportable absolute gravimeter

(BIPM-Jaeger) for establishment of satellite stations. During the 1980' s the principal subjects of research in absolute gravimetry will be: (i) trials for the ultimate

gravimeter working at the 10"^ level or below, e.g., by

use of a multiple-beam reflection interferometer; (ii) nu- merical assessments of global secular variation of the

gravity field; and (iii) studies of the relation between

gravity variation and crustal movements including vol- canic and earthquake phenomena.

References

[1] A. Sakuma, in Precision Measurement and Fundamental

Constants, Ed. by D. N. Langenberg and B. N. Taylor,

Natl. Bur. Stand. (U.S.), Spec. Publ. 343 (Aug. 1971),

p. 447.

[2] J. J. Levallois, Bull. Geodesique 99, 111 (1971).

[3] A. H. Cook, Philos. Trans. R. Soc. London, Ser. A: 261, 211

(1967).

[4] A. Sakuma, Proces-Verbaux Seances Com. Int. Poids Mes.

37 (58th session), 55 (1968).

[5] J. A. Hammond and J. E. Faller, J. Geophys. Res. 76, 7850

(1971).

[6] C. Morelli et al, Publ. Spec. No. 4, Assoc. Inti. Geod.

(1974).

[7] T. Schiiler et al., Ver'dffentlichungen des Zentralinstituts Physik der Erde, No. 10, Potsdam (1971).

[8] E. Hytbten, Publication No. 75 of the Finnish Geodetic In- stitute (1972).

[9] P. T. Olsen, M. E. Cage, W. D. Phillips, and E. R. Williams,

IEEE Trans. Instrum. Meas. IM-29, 234 (1980); and these

proceedings.

[10] J. A. Hammond, Boll. Geofis. Teor. Appl. 20, 401 (1978); and

these proceedings.

[11] J. E. Faller, R. L. Rinker, and M. A. Zumberge, Boll. Geofis. Teor. Appl. 20, 355 (1978); and these proceedings.

[12] G. P. Arnautov et al., Sov. J. Quantum Electron. 9, 333

(1979); Appl. Opt. 13, 310 (1974).

[13] J. D. Boulanger, Publication No. 89 of the Finnish Geodetic

Institute (1979).

[14] Fang Yong Yan et al., Wu Li (Physics) 8, 1 (1979) [in Chinese].

[15] I. Murata, Bull. Earthquake Res. Inst. Univ. Tokyo 53, 49

(1978) [in Japanese].

[16] H. Hanada, J. Geodetic Soc. Jpn. 24, 191 (1978).

[17] L. Cerutti et al., VDI-Berichte No. 212, 49 (1974).

[18] L. Cannizzo, G. Cerutti, and I. Marson, Nuovo Cimento C 1, 39 (1978).

[19] C. Volet, C. R. Acad. Sci. 224, 1815 (1947).

[20] G. A. Bell, D. L. H. Gibbings, and J. B. Patterson, Metrologia 9, 47 (1973).

[21] A. Sakuma, Publ. Internat. Sympos. on Earth's Gravitational Field and Secular Variation in Position, Sydney,

Australia (1973), p. 674. See also Reel. Trav. Bur. Int. Poids Mes. 4, Sect. 13 (1973-1974).

[22] R. Brein, Some Possibilities to Improve the Accuracy of Absolute Gravity Meters, presented at the 17th Gen. Assembly of the Intl. Union of Geodesy and Geophysics (lUGG),

Canberra, Australia (1979).

[23] Resolution No. 16, Bull. Geodesique 118, 377 (1975).

[24] W. A. Prothero and J. M. Goodkind, J. Geophys. Res. 77, 926 (1972).

[25] The GWR Superconducting Gravimeter, GWR Instruments, San Diego, Calif 92121 (1980).

[26] The International Bureau of Weights and Measures 1875-1975, Ed. by C. H. Page and P. Vigoureux, Natl. Bur. Stand. (U.S.), Spec. Publ. 420 (May 1975), p. 114. (Translation of the BIPM

Centennial Volume Le Bureau International des Poids et Mesures 1875-1975).

404
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

A New, Portable, Absolute Gravimeter

M. A. Zumberge, J. E. Faller,* and R. L. Rinker

Joint Institute for Laboratory Astrophysics, National Bureau of Standards and University of Colorado,

Boulder, CO 80309

We report on the performance of a new and easily portable apparatus for the absolute measure- ment of the acceleration of gravity. Rapid acquistion of data and high accuracy result from the use of a

drag-free dropping chamber that descends with the falling object whose acceleration is measured inter- ferometrically. Preliminary results indicate an absolute accuracy of 6 parts in 10^. Key words: absolute gravity meter; freely falling corner cube; stabilized laser interferometer.

1. Introduction

The absolute determination of g, the acceleration due

to the earth's gravity, has long been a measurement of considerable importance. While g continues to play a sig- nificant role in the determination of certain physical constants and standards, vdth the accuracies obtainable

today, this measurement now has broad applications to geophysics. The gravimeter that has been developed at JILA is based on the free-fall interferometric method

which has undergone many refinements since it was first used in the 1960s [1-4]. A freely falUng mass that contains a corner cube serves as the mirror in one arm of a Michelson-type optical interferometer. The times of oc- currence of selected interference fringes give its position

as a function of time, and from this the acceleration of

gravity, g, can be calculated. Over the surface of the earth the value of g varies by

less than 0.5% from its nominal value of 980 cm/sec . This variation is due primarily to the earth's rotation

which produces a centrifugal term causing g to decrease

toward the equator. This rotation also causes the earth's

shape to deviate from a sphere which in turn produces a

further dependence on latitude. Local topography and

density variations are two other important factors. The vertical gravity gradient near the surface is ap- proximately 3 p,Gal/cm (1 Gal = 1 cm/sec-). Because of

this, cm-sized vertical motions of the earth's crust vdll show up as variations in of a few parts in 10^. Since

such motions often occur very slowly, the long-term stability in any instrument—^if it is to detect them—is of

utmost importance. Relative gravimeters which are sensitive to ixGal level changes in gravity have existed now

for a decade or more; however, they must be operated

continuously at a single site if unambiguous results are to be obtained. Furthermore the sensitivity of relative

measurements is often limited by instrumental drifts that

are not simply linear in time. For these reasons, a portable absolute instrument whose measurements depend on

atomic standards and whose absolute accuracy ap- proaches 3 parts in 10^ would be a very useful tool in the

study of tectonophysics. We have successfully built an in- strument that appears to approach this accuracy while

maintaining the highly portable character necessary for

field applications.

2. Design

Figure 1 is a schematic representation of the system. A

Michelson interferometer is arranged in such a way that one of the corner cubes is allowed to fall freely. An in- terference fringe results from each half wavelength of the laser light traversed in the dropped object's descent.

REFERENCE

MIRROR

ISOLATION SPRING

REFERENCE

CORNER

CUBE

*Staff Member, Quantum Physics Division, National Bureau of Standards.

Figure 1. Scheynatic of absolute gravity apparatus.

Over the 20 cm length of free-fall during which the meas- urement is made, 600 000 fringes are generated. The

zero-crossing time of every twelve-thousandth fringe is carefully measured and then stored in a computer. The

resulting 50 data points are fitted to a quadratic curve which then gives the dropped object's acceleration. This

process, which requires only a few seconds, can be re- peated many hundreds of times in one hour.

The quantity actually measured is the second time

derivative of the optical path difference between two optical elements, on-e of which is in some way attached to the earth's surface while the other (the falling one) is ac- celerated by gravity as well as by the vector sum of all other forces acting on it. The interpretation of this meas405
urement as the absolute value of local gravity requires

that we fully understand the limitations set by all other

forces and processes involved. This entails four fundamental considerations: one, non-gravitational forces on

the dropped object must be eliminated or reduced to an

acceptable level; two, the optical path difference must be

translated into a physical length based on an accurate knowledge of the laser' s wavelength and the index of re- fraction appropriate to the residual gas at the operating

level of vacuum; three, the acceleration of the reference

frame from which the measurement is made must be

close to zero; and four, the electronic system that counts and times the occurrence of interference fringes must do

so without introducing any systematic errors. The design

of the instrument proceeded with these four considerations as the underlying guidelines.

Three recent innovations have contributed to the successful development of this instrument: One, a drag-free dropping chamber has been built to shield the dropped

object from non-gravitational forces; it is described in detail below. Two, a long-period isolation spring developed

at JILA [5] has been used to isolate the interferometer's

reference corner cube from vertical motions in the lab- oratory that would otherwise accelerate the reference

frame with respect to which the g measurement is made.

[Note that with a freely falling corner cube and a vertical

inertially suspended reference corner cube, the inter- ferometer input is differential in that vertical motions of the rest of the instrument cancel out.] Three, a newly

implemented -Zeeman stabilization scheme also devised at JILA [6] has been used to stabilize the He-Ne laser that serves as the light source in the interferometer. Its wavelength is periodically (for example on a monthly

basis) compared with an iodine stabilized laser developed

and built for us by the Length and Mass Measurements

and Standards Division of NBS. This absolute

wavelength standard, taken together with a rubidium

frequency standard (used as the basis of our measure- ment electronics), lets us base our gravity measurement

directly on secondary length and time standards whose

accuracy provides the requisite stability for the study of long-term changes in gravity resulting, for example,

from tectonic deformation of the earth's crust in seismically active regions.

3. The Instrument

Air drag makes it impossible to approach any reasonable accuracy without dropping the corner cube in a vacuum. Some years ago at NBS, Tate [7] developed a technique in which the dropped object fell in a vacuum

chamber that was dropped simultaneously, minimizing

the force due to momentum transfer between the falling

object and residual gas molecules in the chamber. To

avoid the vacuum welding and other materials-related

problems associated with ultrahigh vacuum systems, we

have invoked a similar technique. In our instrument the

dropped object is contained in a servo controlled motordriven elevator which moves inside the main vacuum sys- tem. This elevator effects the release and then tracks the

falling object—vdthout touching it during the measurement. As a result, the object falls with the residual gas

molecules rather than through them, while the differential motion between object and elevator is minimized.

This results in a substantial reduction in air drag and al- lows the g determination to be carried out in a more

moderate vacuum.

Figure 2 is a schematic representation of the drag-free

dropping chamber. The dropped object rests in

VACUUM CHAMBER —

DRAG FREE CHAMBER 1

^ PELLICLE

GUIDE RAILS - DRIVE BELT

~1 rFlGURE 2. Schematic of dropping system.

kinematic mounts on a chamber that can be driven along

vertical guide rails by a thin stainless steel belt con- nected to a dc motor. The position of the dropped object

relative to this drag-free chamber is measured by focus- ing light from an LED through a lens attached to the

dropped object onto a position sensitive photodetector.

The error signal thus derived is used to control the motor

that accelerates the chamber downward, leaving the

dropped object floating inside. Near the bottom of the

drop the chamber is first servoed to gently arrest the dropped object's fall, and then used to return the

dropped object to the top of the track for the next measurement. Figure 3 is a photograph of this system.

In addition to shielding the dropped object from drag

forces the falling chamber also serves to remove other

non-gravitational forces. The chamber provides an electrically conducting shell to completely surround the

dropped object so that external electrostatic fields do not

affect the measurement. Also, the purely mechanical

character of the release removes the necessity for having

any sort of magnetic support or release mechanism that

might subsequently result in an unwanted magnetic

force. Finally, buoyancy effects are removed because a pressure gradient cannot exist in a zero-g environment.

The detection of the interference fringes using a photomultiplier results in a sinusoidal signal whose fre- quency is proportional to the falling object's velocity. A

zero crossing detector and a digital scalar are used to convert this signal into a series of about 50 pulses, each

of which corresponds to the dropped object having fallen 6000 wavelengths (12,000 fringes) or about 0.38 cm. The

times of occurrence of these pulses referred to an arbi- trary but common zero are measured to within 0.2 nsec by commercial electronics and stored in a minicomputer.

A quadratic least-squares fit to these data determines g. Analysis of the residuals indicates that the length meas- urement errors are about 0.001 wavelength. The analysis

406
Figure 3. The dropping system.

currently requires about 4 sec so 150 drops can be made

in 10 minutes.

Figure 4 is a photograph of the apparatus. The dropping mechanism is inside a vacuum chamber which is sup- ported by three folding legs. Beneath this is a base that

supports the long-period isolation spring [5] and contains

the associated optical components that comprise the in- terferometer. The small computer and auxiliary electronics occupy two half racks. One person can disassemble

and load the entire system into a small van in about one

hour. Reassembly takes one to two hours. This ease of

portability should facilitate rapid field measurements.

Figure 4. Complete apparatus.

4. Results

Despite the fact that the instrument is portable, it has

been moved only a few times thus far in its development.

Until now, the primary emphasis has been the detection,

understanding, and elimination of systematic errors.

Most of the data have been taken in the sub-basement of

the laboratory wing of JILA on the University of

Colorado campus in Boulder. Some data sets, however,

have been taken in a building on the outskirts of the National Bureau of Standards grounds in Boulder. This

latter site is significantly quieter in that it has less vibration than the JILA sub-basement, but is less convenient

in terms of modifications. We plan to reoccupy the NBS

site, as well as several others around the Western United

States, in the near future. Figure 5 is a histogram of 150 drops which comprises one 10-minute data set. The standard deviation in such

sets varies from as low as 20 lo-Gal to as high as 70 (xGal

during unusually noisy periods caused by poor weather or increased human activity nearby.

II

II

11

II

II

IIIII

III III

III III

ill

III

III

III III nil inn

mill

mill mm

mmmm

iiiiiii

iiimi mum

iiiiiiii

iiiiim

i'L' i o-j 0 I o.;

Figure 5. Histogram of one run. u = 38 [iGal; cy/VN =

3.1 [i.Gal; start time = .5:15; stop time = 5:25.

Figure 6 is the result of two days of continuous operation at about 70% of the maximum possible data acquisition rate. The tidal effects of the sun and moon can easily

be seen. Subtraction of the theoretical tides results in an

r.m.s. deviation of about 6 fxgal. Meteorological effects,

which can be significant at this level, have not yet been

removed.

5. Errors

The high repeatability of a measurement (i.e., the precision) is unfortunately riot always an indication of the ac- curacy. Opportunities for systematic errors abound in this type of measurement and only by varying certain (at

first glance) inconsequential experimental parameters

can their effects, if any, be determined wath confidence.

Even then it is always possible to overlook some signifi- cant processes. Because of space limitations we cannot

give an in-depth error analysis here, but we wall briefly

407
100

•100 THEORY

DATA

RESIDUALSMAY 3 MAY 4

Figure 6. Earth tides.

discuss sources of error that have been considered to date. The effects of possible magnetic and electrostatic

forces on the dropped object are as yet undetectable. The

addition of magnetic fields larger than those normally en- countered by the dropped object have been found not to

affect the value of g. The result of changing the separation of the dropped object and the co-falling chamber

does not cause a shift in the measured value of g, indicat- ing again that electrostatic or magnetic forces betv/een

them are not significant.

Initially we had hoped to be able to work at a pressure

of 10"^ mm of Hg and were encouraged by measurements

in which we purposely introduced a relative velocity

between the falling chamber and the dropped object. No

significant shift in g resulted from rather large relative

velocities (2-4 mm/sec). However, onset of rapid temperature changes produced a several hundred |xGal transient. Rough calculations of the magnitudes of forces due

to temperature and pressure gradients across the

dropped object as well as the effects of surface and cavity outgassing—all of which would occur during times of departure from thermal equilibrium—suggested that a lower vacuum was not only prudent but necessary. In our

present operating vacuum range of 10"*^ to 10"^ mm of

Hg, the problems introduced by changing temperatures were found to be greatly reduced.

Errors that stem from rotation or translation of the

dropped object [8], the effect of small rotations in optical

elements in the interferometric path, the consequences of

optical misalignment, and any effects of optical feedback on the laser, are all believed to be less than 10"^ of g. Other considerations, which have thus far yielded no in- dication that errors are present, include nonrandom

mechanical vibrations introduced systematically by the

measurement and frequency-dependent time delays in the

electronics.

6. Conclusions

Although several of these points deserve further examination, we feel that most of the sources of systematic er- rors have been recognized and are well in hand. It is, however, always possible that further investigation may

reveal other difficulties, but it seems unlikely that our quoted value will change by more than 10 iJiGal. To date, we have made 400 sets of 150 drops during which the sys- tem appeared to operate correctly (e.g., we were not subjecting it to special test conditions) over a one-month

period, yielding a value (reduced to the floor using a gradient of 2.3 |xGal/cm) of 979.608557 Gal. The standard deviation of the results of these 400 sets is 5.9 \i.Ga\ or about 6 parts in 10^ (see Fig. 7), while the standard deviation of the mean of a single 150 drop set ranges from 3

to 7 ixGal. Recognizing the as yet uncompensated meteorological effects, the results are statistically consistent. We

feel the one sigma uncertainty of 6 |xGal is a conservative

representation of the instrument's capabilities. Further

tests of the instrument's accuracy are planned, including

direct comparison with other absolute instruments.

II III III liill mil mil mil mil mil mil mil mmmm

iiiiiii

iiiiiii

IIIIIII

I

I I M

I

I I I I

I M M

I

I I I I

I

I I I I

I

I I I I M I I I

I

I

i I ;

I '4iJ 30 -JO -lO 0 10 20 30 40

Figure 7. Histogram ofi 19 data sets. Mean = 979.608557 Gal;

(T = 5.9 [i.Gal.

This work was supported in part by the Air Force Geophysics Laboratory and in part by the National Bureau of Standards as part of its research program on improved

precision measurement techniques for application to basic

standards.

References

[1] J. E. Faller, J. Geophys. Res. 70, 4035 (1965).

[2] L. Cannizzo, G. Cerutti, and I. Marson, Nuovo Cimento C IC, 39 (1978).

[3] J. A. Hammond and R. L. Iliff, in Applications of Geodesy to Geodynamics, Proceedings of Ninth Geodesy/Solid Earth

and Ocean Physics (GEOP) Research Conference, Ed. by I.

I. Mueller (1978), p. 245.

[4] A. Sakuma, in Proceedings oftiie International Symposium on

the Earth's Gravitational Field and Secular Variations in Position, Ed. by R. S. Mather and P. V. Angus-Leppan (Dept.

of Geodesy, Univ. of New South Wales, Sydney, 1973), p. 674.

408
[5] R. L. Rinker andJ. E. Faller, these proceedings.

[6] T. Baer, F. V. Kowalski, and J. L. Hall, Appl. Opt. 19, 3173

(1980).

[7] D. R. Tate, J. Res. Natl. Bur. Stand. Sect. C: 72, 1 (1968).

[8] A. Sakuma, in Precision Measurement and Fundamental Constants, Ed. by D. N. Langenberg and B. N. Taylor, Natl. Bur. Stand. (U.S.), Spec. Publ. 343 (Aug. 1971), p. 447.

409

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

"Super Spring"—A Long Period Vibration Isolator

R. L. Rinker and J. E. Faller*

Joint Institute for Laboratory Astrophysics, National Bureau of Standards and University of Colorado,

Boulder, CO 80309

We have devised a new mechanical isolating device which we call a "super spring." The super

spring isolator makes use of the fact that a mass suspended by a long spring is effectively isolated (from vibrations) for all frequencies higher than the system's natural resonance. We have developed a method of electronically terminating a 30 cm-long spring in such a way that the mass suspended from it behaves as if the spring were one kilometer or longer in length. This permits us to provide isolation for frequencies as low as 0.02 Hz. We will discuss the principle, the results of shake-table tests, and the

implications of this technique for measurement science. Key words: inertial reference; long period seismometer; mechanical vibration isolation.

1. Introduction

This preliminary determination I generally make

the night before the deflections and periods are determined, which in Oxford is best done on Sunday

night between midnight and 6 a.m. The daytime, of course, is out of the question, owing to the rattling

traffic on the stones in St. Giles' , about a quarter of a mile away; and all nights except Sunday night the railway people are engaged making up trains and

shunting, which is more continuous and disturbing to the steadiness of the ground than a passing train. Even these come through at intervals on a Sunday

night, and this limits the accuracy with which the

periods can be observed. . . The traffic and the trains are not the only causes

of disturbance. Wind, by pressing upon the building

and neighboring trees, of course shakes the ground; but on Sept. 9-10, a particularly quiet night, I had

to leave, owing to a sudden disturbance producing a pendular motion of 15 divisions, or 150 units, and for some time there was not quiet. As the motion was

clearly produced by a lurch of the whole instrument

and table carrying it, and was gi'eater in amount

than any traffic in the busiest part of the day had

ever produced, and was moreover free from the high

period tremor characteristic of human disturbance, I at once set it down to an earthquake. . . I had hoped to have made a gi'eater number of ex- periments under more widely differing conditions, but the strain which they entail is too severe, for not only have I had to give up holidays for the last three

years, but to leave London on Saturdays and occasionally to sit up all Saturday and Sunday nights at the end of a week's work. The conditions, therefore, are too difficult for such an extended series as I should like to make to be possible, and I must after one or more effort, leave the problem to others who

have leisure, and what is of far greater consequence,

a quiet country place undisturbed by road and rail- way traffic, and who possess the knowledge and

manipulative skill which the experiment requires. . .

Professor C. V. Boys "On the Newtonian Constant

of Gravitation" Philosophical Transactions of the Royal Society of London for the Year MDCCCXCV. Volume 186.

*Staff Member, Quantum Physics Division, National Bureau of Standards.

Isolation of experiments from mechanical vibrations

has been a long-standing problem for experimental physicists. Since the time of C. V. Boys, experiments have be- come increasingly more sophisticated and sensitive and

this has served only to increase the problems resulting

from vibration. And while measurements have increased

in accuracy and complexity, the approaches to vibration

isolation have remained at a rather simple level. Today,

commercial isolation systems are available which provide

isolation from frequencies greater than 2 Hz. However,

for experiments sensitive to frequencies of 1 Hz and

below, these systems are not helpful. In particular,

today's state-of-the-art absolute gravity measurements

require isolation at frequencies much lower than 1 Hz. Further, the low frequency limit of earth-bound gravity wave experiments will be limited by the character and

the quality of the isolation that can be afforded. While at

first glance it might seem that the problem of lowering

the isolation regime ("corner" frequency of an isolator)

is a simple matter of getting weaker springs and larger masses, there are very real practical limitations. A simple calculation shows that the resonant frequency of a mass m which stretches (by a distance x) a spring of spring constant k is equal to the frequency of a simple

pendulum whose length I is equal to the amount of this stretch: mg = kx = kl

1/2 V2

0) = Jc_

[ m ^ (1)

Thus a spring vdth a period of 60 sec would have to be

stretched about 1 km—which rules out its use for normal

laboratory applications. Since the frequency of a massspring system depends on g and the stretch length of the spring, it is clear that in order to achieve lowfrequency isolation one must do something other than

attempt to straightforwardly utilize a simple spring

system.

To overcome these practical limitations, we have

developed what we call a "super spring" —a mass sup- ported by a finite length spring but whose period—and

isolation characteristics—is that of a spring very much

longer. In order to understand how one might get a simple spring system of tractable size to behave as if it

411
were, say 1 km in length, imagine a mass hanging from a spring stretched 1 km. As the mass oscillates up and

down all the coils of the spring oscillate with the same

period. The coils near the bottom would have an amplitude nearly equal to the amplitude of the mass, while the

coils near the top would have an amplitude that is small

compared to the amplitude of the mass. At the half-way

point, the amplitude would be exactly half. If we were to study the motion of a coil 30 cm above the mass as the

rest of the spring was lengthened, we would find that the

longer the spring got, the closer this coil would track the mass. In fact, if the spring were infinite in length, the

coil's motion would track the mass exactly. It is this behavior of spring-mass systems that gave us the clue as

to how to terminate the top portion of a spring at 30 cm

so as to make the supported mass act as if it were

suspended from a spring 1 km in length. If one were to grasp the coil 30 cm above the mass and then somehow

continue to move it up and down in exactly the same way

it was moving with a km of spring above it, one could

then cut off the top km and still have a spring that would

act if it were 1 km long. Since this involves causing a

particular point on the spring to track the bottom, a servo approach can be used to transform an ordinary

length of spring into a super spring.

Figure 1 is a schematic representation of the super

spring. The system consists of a bracket supported by

two springs which supply the dc force to support the

bracket and the attached mass-spring system (or in terms

of the previous discussion they supply the force that the

balance of the 1 km of spring would have supplied). From

this bracket, the main spring and mass are hung. The top

of this spring is made to track the motion of its bottom

utilizing a position detector which measures the length of the main spring with respect to the bracket. Light from a

light emitting diode (LED) is focused by a ball (serving

as a rotation insensitive lens) onto a split photodiode.

The currents from the two halves of the diode are ampliLOUD SPEAKER MAGNET

SPEAKER

VOICE COIL

SUPPORT SPRING

MAIN SPRING

BRACKET

POWER

SUPPLY

LED' TRANSPARENT SPHERE' TEST MASS'

^DIFFERENTIAL PRE-AMP

^SPLIT PHOTOCELL

SUPER SPRING

Figure 1. Schematic diagram of the super spring.

fied, by a current controlled voltage source, and the

resulting voltages are then differenced. This supplies an analogue voltage that is a measure of the position of the mass with respect to the bracket. This analogue voltage

is passed through a servo compensating amplifier which

drives a current through a voice coil centered in a loudspeaker magnet, which results in a force on the

bracket. This forces the bracket (and thus the top of the

main spring) to track the mass. The degree of tracking is simply related to the gain of the servo system. The

higher the gain the better the tracking—i.e., the closer the coils track the motion of the bottom mass—and therefore the longer the system period.

At this point a brief discussion of the isolation characteristics of a simple spring-mass system is in order.

2. Theory

Figure 2 shows two simple spring-mass circuits: (a) is called the absolute damping case, and (b) is the relative

damping case. By writing the equations of motion for these systems and solving for X-ziS VXiiS) (the system

transfer function) we get for (a) ^2(5)

and for (b)

X,{S)

X,(S)

(2) m

Wo" m

m

(3)

where S is the Laplace transform variable (ct -I- j

co), p is the system's damping coefficient, k is the spring constant, m is the mass, and = k/m. _ The isolation characteristic is found by evaluating the magnitude of the transfer function for S = j w. For (a) we get

Z2(iw)

^i(ico)

Wo 0)2)2 +

and for (b)

X2(i0))

Zi(io))

4 _L 0)0 + 4'

(a)o2 0)2)2 + _^ ^2

1/2

1/2

(4)

(5)

x,(t)

x,(t

)

x,(t)

X2( t)

(b)

Figure 2. Spring-mass circuits (a) absolute damping case; (b) relative dampitig case.

412
Since spring-mass systems isolate only for frequencies w

that are greater than wq, the two expressions can be simphfied for w > wq. Examination of these asymptotic

forms reveals a basic difference between the two sys- tems. For (a), the "absolute" damping case, we get

1

OJ » 0)0

for (b), the "relative" damping case, we get

X2{j O)) CO » 0)0 .

(6)

(7)

This difference in the asymptotic behavior leads one to the

conclusion that, for an isolation device, the "absolute"

damping case is preferable. The problem, however, is that

the body of the dash pot has been assumed to be connected

to an inertial reference. This being available, however, as- sumes that we have already solved the problem.

The simple spring-mass circuit that models the super

spring is shown in Fig. 3 where Ki is the spring constant

of the support springs, Pi is the damping coefficient of the support springs, nii is the mass of the bracket, K2 is the spring constant of the main spring, ^2 is the damping

coefficient of the main spring, ma is the mass of the test mass, and fit) is the force supplied by the voice coil and

speaker magnet.

Figure 3. Spring-mass circuit for super spring.

The equations of motion are two coupled second-order

linear differential equations:

mix'i = Kiixz - Xi) + Pi(x3 - Xi) + K2{x2 - Xi) + - X,) + fit), (8) m2X2 = K2ixi - X2) + N^i - X2). (9)

The key to the modification of the system's behavior is contained in the term / it ). The exact form of / it ) is determined by the transfer function of the electronics. For this discussion we will use a simplified form for / it ): fit) ^ GiX2 - Xi) +.-isiX2 - Xi) . (10)

This is the statement that the servo action is proportional plus derivative. The most convenient method for the analysis of the system is to work in the Laplace

transform domain.

Substituting Eq. (10) into Eq. (8) and then taking the

Laplace transform of Eqs. (8) and (9) we get

m,X,iS)S^ = iK, + ^,S)[X;iS) - X,iS)] + [K2 + G + O2 + fo)S] X (Z2(S) - Zi(S)) , (11) m2X2iS)S^ = iK2 + p2S)(Zi(S) - X2iS)) , (12)

where XiiS), X2iS), and ^3(5) are the Laplace

transforms of Xiit), X2it), and x^^it), and S is the complex transform variable (a -I- jw). These two equations can be solved for the transfer function of interest: ^= (13)

XiiS)

(7lS +u)f)(72g +a)|)

where 71 = pi/mi, -72 = 712 = is = fo/^^i,

ojf = Ki/mi, o)| = K2/m2, 0)^2 - K2/mi, o)| = G/mi.

For this particular system we can approximate some of these terms for high servo gain values:

,2 71 « ys

72 « 7s

712 « 7s and Eq. (13) becomes

and

0)1

oj|

wf2

«

«

« 0)|

X2iS) ^

XsiS)

iyiS + cof)(72S + o)|)

S ^ + S Ss + S W + S (7iw| + 72wf) + wiW (14)

If we factor the quartic in the denominator into two quadratics of the form

^2(5) ^ iyiS + o)f)(72g + 0)1) ^3(5) iS^ + AS + B)iS- + CS + D)

(15)

for large values of ojs and ys we get

A - , 5 = 0)1 , C =

-yiO)| -I- 72wf

0)| 0)1 D =

OJl^OjJ

The quantity of interest is of course the isolation characteristic \X2ij M)/X2ij (ii) |. For the frequency range o) «

(larger of oji/-yi, 0)2/72) we get

X2ijo,)

X-iiji^)

Wll"0>f/l0)|

2 0)fO)| 2 O)^

1/2 (16)

By comparison to Eq. (4) we see this is exactly the isola- tion characteristic of a simple spring with resonance o)o =

o)iaj2/o)s and damping constant C. This analysis shows

that one can in fact lower the "corner" frequency of a mechanical system with feedback, without the dimensions

of the apparatus getting out of proportion. There is, however, a difficulty with this technique. Recall that C = ^(7iw| + 72W1) 7S

0)fO)f (17)

Since we started with a high "Q" mechanical system, 71 and 72 are small, and since the servo gain is large the

negative term in the expression for C can dominate.

When this is the case the damping factor for the low fre- quency pole in Eq. (15) [S'^ + CS + (wfo)f/o)|)] is negative. This leads to a transient response that is an

exponentially growing sinusoid with frequency ojo. which

is clearly not acceptable.

The obvious means of damping the long-period motion

is to connect a dash pot from X2 to x-s- In the development of super spring, two types of dash pots were tried. The first was a 4 x 10"^ cm tungsten rod immersed in a light silicone oil. While this damped the system's motion

413
quite well, the creep of the fluid up the rod increased the mass of the test mass to a degree that the system simply

continued to m.ove down until the rod hit the bottom of the oil container. Remember that even at dc the spring

behaves as a very long one (and is therefore very soft): The addition of only a few milligrams of mass to the

500 g mass results in a substantial lowering of the equilibrium position.

To eliminate the creep problem associated with the

fluid, a magnetic dash pot was tried. The test mass was

placed in a magnetic field with the result that as the mass moved, eddy currents were induced giving rise to damping forces. This method, though quite successful for low servo gain settings, suffered as the servo gain was

increased because the interaction of the fringing field of the magnet and the magnetic spring tended to add a negative spring constant, which either canceled or dominated the positive spring constant, making the system

unstable.

The damping method we finally decided on modifies the

low frequency (w < 0.628) response of the servo in such a way as to damp the long-period motion. In order to understand how this method works we will consider the

simple spring-mass system shown in Fig. 4 where K is the spring constant of the spring, p is the damping coeffi- cient of the system, m is the mass, and / (<) is a timedependent force which is derived electronically from the

displacement [xi — X2)-

x,(t)

Figure 4. Simplified spring-mass circuit for the damped super

spring. In the Laplace domain the equation of motion can be

written as mS^X2(S) = K[X,{S) - X2iS)] + |BS[Zi(S) - ZaCS)] + G(S)[Zi(S) - Z2(S)] , (18)

where G{S) is the transfer function of the electronics. Equation (18) can be solved for the transfer function

Z2(S )/Zi(S)

X2iS)

Xi(S)

m m

GiS

S- + -^S + 0)2 + —GiS)

m. m

m

1/2 (19)

From this transfer function we can see that the time

domain behavior of the system depends on the mechanical parameters p and w and also on the transfer function

of the electronics G iS ). If we choose

GiS) = yS , (20)

which is equivalent to saying the force is proportional to the time derivatives of (cci - X2), then Eq. (19) becomes

X2iS)

XiiS)

,2 + i§_±^S

m

52 + Sl±^s + 0)2 m

(21)

This is easily identified as the transfer function of a simple spring with a variable damping coefficient (7 -I- 3). If 7 < 0, the "Q" of the system is increased. If 7 > 0, the

"Q" is decreased. The case 7 > 0 is an e.xample of using

electronic feedback to damp a mechanical system.

While the above analysis demonstrates the concept of electronic damping, there is a practical limitation that is not taken into account in Eq. (20). The limitation comes

in the fact that we are not free to choose G iS ) in a totally arbitrary manner. The constraints on our choice of G iS ) stem from the fact that GiS) is the transfer function of an electronic system, which must be realized us- ing amplifiers having a finite band width. We can at best

achieve a transfer function G iS ) that acts as a differentiator for a restricted frequency range. The optimum

G iS ) that can be realized is the transfer function of a second-order band pass filter GiS) =

S^ + AS + 0)|

(22)

This is simply the transfer function of a perfect differentiator with two poles added. One pole must be added for reasons of op-amp stability, the second pole is added to reduce high frequency noise. Using this GiS) in Eq. (19) we get

X,iS) ^

X2iS)

S^ + AS + o)| As +0)2

m

+ ^S

m

52 + A5 + ^2 m

[52 + AS + a,|] + -^S

I J m

(23)

To understand how the response of this system is changed as we increase the value of 7, we utilized root

locus analysis. Figure 5 is a rough sketch of the root

locus of the transfer function Eq. (23) as a function of 7. As 7 is increased we see that the mechanical poles move

toward the a axis indicating a decrease in the "Q" of

S - PLANE

Figure 5. Sketch of the root locus of the transfer function,

Eq. (23), as a function ofy.

414
this resonance. But as 7 is increased, the electronic poles move toward the jw axis, which indicates an increase in the "Q" of the electronic poleg. If 7 is increased enough

the electronic poles will go into the right half plane, at which point the system becomes unstable. While these

poles are electronic in nature, they will nevertheless

cause the mechanical system to oscillate. The total effec- tive damping one can achieve is determined by the tradeoff between the increase in the "Q" of the electronic

poles and the decrease of the "Q" of the mechanical

poles. In the derivation of Eq. (13) we used for the feedback

force f (t)fit) = G{X2 - Xi) + 7s(i2 - ^1) (24)

This is equivalent to saying the transfer function of the servo is GiS) = (G + ysS) . (25)

This of course cannot be the transfer function of any real system, since all real systems have finite gain and band

width. When we assume a general form for the transfer

function G (S ) of the electronics, then Eq. (13) becomes

X2iS)

(71S + a)f)(72S + 0)1)

iS^ + 71 + ^i)(S^ + 72S + 0)1) + S-— GiS)

(26)

For the servo transfer function used in the super spring,

which takes into account the finite band width of the servo amplifiers and the low frequency differentiator

(used to damp the long period motion), G iS ) has the

form

GiS) = G

0)0 + a2TioO)oS

+ P

S^ + 2-noO)oS + 0)1

2tilo)lS

iS^ + 2tu,o)lS + o)f (27)

Here o)o, tiq are the band width and damping ratio for the

servo amp, a is the gain of the second order bandpass

filter used as the differentiator to damp the servo, and

(x)L, thl and p are the break point, damping ratio, and gain

of the second-order bandpass filter used as the low fre- quency differentiator which damps the long period

motion.

The complete analysis of Eq. (26) with this G iS ) in- volves finding the roots of, and eighth-order polynomial

for, various values of G, a, p, ino, o)o, r\L and o)^,. This

analysis has been done using a digital computer and a FORTRAN program. The results were in no way profound, so they will not be discussed at length. It is suffi- cient to say that the optimum values for ojq, (ol , and p were found to depend on the value of G. This value of course is determined by the desired period of the "long

spring." Since we wanted this to be an easily variable

parameter the electronics was designed so that o)o, o)^, a, and p were also easily varied.

3. Mechanical

There are three basic parts to the mechanical system;

the bracket assembly, the flexures which constrain the

bracket to have only a vertical degree of freedom, and

the main and support springs.

The design requirements on the bracket are that its mass must be kept to a minimum while its stiffness be as high as possible. The low mass requirement results from

the fact that the "gain" in Eq. (23) of the electronic

transfer function GiS) is divided by the mass of the

bracket m. Stiffness minimizes the tendency at high servo gain of exciting the mechanical modes of the

bracket. One can also address this problem by applying

damping material (e.g.. Dux Seal) to lower the "Q" of these modes and thus making them harder to excite. The

bracket (see Fig. 6) consists basically of a long tube

which is centered in the external housing by means of a flexure at each end. These flexures constrain the bracket

to move in only the vertical direction. The housing con- taining the position detection system, which consists of a

light emitting diode and a split photodiode, is attached to the bottom of the tube. The voice coil is attached at the

bottom of this housing. The main spring hangs down

through the center of the tube which results in the test mass being centered in the position detection housing.

The weight of this assembly (tube, housing, voice coil, main spring, and test mass) is carried by the three auxili- ary support springs. The voice coil fits into the gap of the speaker magnet—the latter being attached to the

external housing. The external housing rests on three

leveling screws.

FLEXURE

TUBE

MAIN SPRING

FLEXUREDIODE

HOUSING - TEST MASS

VOICE COIL

MAGNET

a

m

Figure 6. Schematic diagram of the overall super spring

system.

415
The flexures are one of the most critical parts of the

mechanical system. They must be soft to vertical motion,

still to sideways motion, and precise enough so as not to cause the tube to tilt as it moves up and down. Several

flexure designs were tried. Figure 7 shows the design

that ultimately worked the best. They are first machined

from 17.5 x 10"^ cm thick, 10.8 cm diameter disks of beryllium copper, and then heat treated. These flexures

give about 2 arcseconds of tilt for a bracket motion of

0.6 cm.

Figure 7. Photograph of one of the flexures.

Since the '

' long' ' spring acts exactly like a spring 1 km

long, minimizing the temperature coefficient of the

spring material is very important. The effect of a change

in temperature comes in chiefly through the temperature

dependence of Young's modulus rather than through the

thermal expansion of the material. Springs made from

beryllium copper or fused silica have Young's modulus

temperature coefficients on the order of 10 Our

springs are made from Ni-Span-C alloy 902 which is manufactured by Huntington Alloys. Ni-Span-C has the

unique property that cold working drives the thermal

elastic coefficient negative while heat treating drives it positive. With proper treating of springs made from NiSpan-C one can get a very low coefficient. The springs we use have a thermoelastic coefficient of about 6 ppm.

Light from a light emitting diode (LED—Texas Inst. TTES16 series) is passed through a cyHndrical lens. This

results in a rectangular image. This image is focused by

the transparent sphere attached to the test mass (see

Fig. 1) onto a split photodiode (UTD Pinspot 2DJ). As a

result of this geometry the amount of light on the two

halves of the diode changes when the test mass moves up

or down. This changes the amount of current from the

diode halves. These currents, amplified by a current-to- voltage converter, are differenced to provide an analogue

signal proportional to the displacement of the test mass.

This signal, processed by a servo-compensating amplifier,

drives a loudspeaker voice coil which in turn supplies the

needed force to cause it to (nearly) track the motion of the test mass. With an LED current of 200 mA the vertical position sensitivity of this detector is 0.7 V (xm^^

The noise level in a 5.0 kHz bandwidth is 5 mV P-P while

the noise level in a 1 Hz bandwidth is about 0.5 mV P-P.

4. Test Results

Once the construction of super spring was completed we tested its isolation characteristics. To do this, we con- structed a "shake table" on which we could place the

spring. This table could produce an amplitude of 25 x

10"^ cm at frequencies of less than 5 Hz, and about 2 x

10"^ cm at 25 Hz. The table surface, driven by a system

of levers and a speaker magnet-voice coil system, is con- strained so as to tilt less than one arcsecond—the accomplishment of which proved to be a nontrivial task. The

power amplifier used to drive the coil could deliver

±12 A into the coil. An LED photodiode position detector was used to monitor the table motion. The isolation measurements were made using a HP 3582A Spectrum

Analyzer. The noise source from the 3582A was used to drive the table. The output from the table's position

detector and the position of the test mass with respect to the floor ("inertial space") were applied to the two in- puts of the spectrum analyzer which computed the

transfer function. Figure 8 is an example of such a transfer function. This particular one was taken with the

spring period set at 12 s (lower than usual) so that the

1/12 Hz resonance could be seen.

2 3 4 5 FREQUENCY (Hz)

Figure 8. Transfer function obtained during ''shake table" test- ing of super spring.

Further evidence that the spring indeed isolates is seen when the test mass is used to hold the reference

cube corner in a free falling absolute gravimeter (Zumberge et al., in these proceedings). Figure 9 shows two

histograms of 150 g measurements (one set with the

inertial reference and one without) which demonstrate a reduction in scatter by a factor of 20.

416
5. Conclusions

Toward the end of the same article quoted in part at the beginning of this paper, C. V. Boys states: "I would

strongly urge that in such a case, a room more uniform in temperature than the one at Oxford should be employed. . . I do not think any ready-made room is likely to be found available. A disused adit, at a great distance

from existing mining operations, would be perfect." The

use of the super spring concept as we have applied it to isolate in the vertical, or as it could be adapted to isolate in other directions, offers an alternative to Boys' recommendation. Clearly the ideal situation would be to both

isolate and operate at a quiet site. In many cases this simply is not practicable. However, in all cases the principles of the super spring can still be used to effectively

get new labs for old.

I 11

I II I I I I

I II I I II Hill II I III I I II mill iiiiiiiiiiiiii I iiiiiii

400 iOO 200 100 0 100 iOO iOO JC

fjL gals

WITHOUT

I

I

I

I

I

III ill

III

III

III

III

III

III III

nil

nil

nil

Bill mil mil mil mil mil mil mil mil

mill

imiii mini

minii inniui

-I inimimi

|IIII|IIM|IIII|IIII|IIII|IIII|IIII|IIII|

4ee 300 TOO loo a loo zoo 3oo 4oe gals

WITH

Figure 9. Histogram of yneasiirements of the gravitational acceleratioyi g inthout and with a super spring isolated reference comer cube.

417

Precision Measurement and Fundamental Constants 11, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Transportable Gravimeter for the Absolute Determination of Gravity

Guo You-guang, Huang Da-lun, Li De-xi, Zhang Guang-yuan, Gao Jing-long,

Fang Yong-yuan, and Huang Cheng-qing

National Institute of Metrology, Beijing, P.R.C.

At the National Institute of Metrology in Beijing, a transportable gravimeter using the method of

free fall has been constructed. The instrument consists of an optical interferometer illuminated by light

from a stabilized He-Ne laser, in which one of the mirrors, a corner-cube reflector, falls freely. The

time standard is obtained from a highly stabilized rubidium clock. The methods of time and distance measurement are described. The effect of the verticality or collimation is discussed. A positive correction has to be included. Some recent (1979-1980) results are presented. The accuracy achieved with

this apparatus is about two parts in 10^. Key words: acceleration due to gravity; coincident method; transportable gravimeter.

1. Introduction

High precision absolute determinations of gravity contribute to our knowledge about the earth's gravity field and its secular variation. It is of considerable significance

for research on earthquake prediction and is important in geophysics and geodesy [1]. At the end of 1977 we developed a transportable gra- vimeter. It is based on the original system [2] but with

improved electronic, vacuum, and mechanical components. The apparatus consists of a Michelson interfer- ometer illuminated by light from a stabilized He-Ne laser locked to the Lamb-dip. A diagram of the optical system

utilized is illustrated in Fig. 1. The apparatus cycles au- tomatically once started and makes 50 drops per hour.

MAIN

BEAMSPLITTER

REFERENCE

CORNER CUBE

PHOTOMULTIPLIER

FALLING

CORNER CUBE

COLLIMATOR

'-^'^^ LENS

1

T

PINHOLE

^

LASER

MERCURY POOL

Figure 1. Diagram of the optical system.

2. Measurement of Distance and Time

The principle of simple free-fall has been adopted. The

initial velocity and position of a freely falling object are

difficult to determine precisely in this method. In order

to avoid the measurement of initial velocity, two successive times of passage of the object, Ti and T2, must be

measured through three horizontal stations hi, hz, and

/i3, for which the vertical separations Hi and H2. are known. The value of g is obtained as follows [3, 4]:

9

Ti

{H2/T2 - Hi/Ti) (1)

The two time intervals Ti and T2, and the distances Hi

and H2, are precisely measured by a coincidence method,

which avoids the problem of measuring fractional clock

cycles or interferometer fringes. A special electronic sys- tem is designed for this purpose. Figure 2 shows a block

diagram of the system.

RUBIDIUM I ^1 AMPLIFY I

IGGER TIME DELAY QNAL " GENERATOR

PHOTOMULTt -

I J AMPLIFY I PLIER I ~ SHAPING

|

T, COUNTER

T2 COUNTER

Figure 2. Block diagram of the electronic system.

The width of input pulses in our instrument is 4 ns. The selected times for the three horizontal positions hi,

hz, and h^ are about 0.09 s, 0.24 s, and 0.39 s after release of the falling object at the top of the chamber.

The determination of accurate time intervals depends on

coincidences between the pulses of the interference

fringes and the signals of the frequency standard.

Theoretical calculations and computer models have showoi

that the probability of such coincidences is sufficient; our experiments have corroborated these calculations. The measured distance H is directly dependent on the

accuracy of the laser's wavelength. Our laser wavelength

is periodically calibrated in terms of the NIM iodine sta- bilized He-Ne laser. A relative accuracy for the wavelength of 10"^ is thus reached. The time standard is a rubidium frequency standard made in China whose accu- racy is better than 1 x 10"^'^.

3. Effect of Verticality or Collimation

If the laser beam is at an angle, the distance measure- ment vdll be in error. The effect is to lengthen the dis- tance corresponding to one fringe and to make the measured value of g lower than the true value. This can be

shown by the variation in the actual optical path when

419
the corner-cube prism moves. It can be deduced directly

from the following viewpoint. Only the motion of the optical center of the corner-cube in the direction of the

light beam will produce a variation of the optical path

difference. We assume that when an object falls freely in the plane M, which consists of the vertical line and the

main light beam passing the optical center A of the

corner-cube (Fig. 3), there will be a small angle a between the main light beam and the vertical line. The

projection of the trajectory of the optical center of the

falling object on the plane M is the curve AB, and the er- ror between the vertical distance AC (h) travelled during

time T and the distance AB' measured by the interferometer is A h. Thus the corresponding error in g is

^9

1 ^d'h dM

2" dt^

1 .

or

Act 1 2 9 2

(2)

(3) From this we see that the error is independent of the

translation value d, as the translation acceleration is al- ways very small in a free fall experiment.

Figure 3. Diagram used to analyze the effect of a nonvertical

laser beam.

An experiment was carried out to verify the effect of

this error. By making the light beam deviate from the

vertical by the small angle 147 seconds of arc, the differ- ence between the correct value and the experimental

result is found to be

^9 -0.29 ± 0.03 mGal,

which agrees with the value calculated by the above

formula:

-0.25 mGal

Therefore, based on a careful measurement of the angles, we must include a positive correction.

4. The Results

Table 1 presents a compendium of results obtained

with the transportable instrument during the last two

years. Table 2 is a summary of the corrections and estimates

of uncertainty for the various possible systematic effects,

both natural and instrumental.

Figure 4 is a typical histogram of the data obtained at a quiet site. It shows the results of 51 drops taken on 26

October 1979 at Shiang Shan laboratory which is located

in the suburbs of Beijing. There, the microseismic-noise

level is very low. The standard deviation of a set of measurements, which is 18 drops, is usually about

70 ixGal

Figure 5 shows the results of five sets of data taken in the time span 19-20 September 1979 at Shiang Shan

Laboratory. Each point represent 17-26 drops with the

value obtained by averaging. Each data point corres- ponds to a 1/2 hour time span. The results of the five sets of data are compared with the computed earth tide adjusted for a best fit. In April 1980 we determined the gravitational acceleration at Sevres A3, BIPM. The result obtained with this transportable gravimeter was 901 925 914 ± 16 ixGal. At

the same time, the value given by A. Sakuma was 980 925

908 ± 8 (xGal. The difference between the two results is within their assigned measurement uncertainties.

Table 1. Summary of data obtained with this NIM transportable

gravimeter (all values in [xGal)

Number of Standard Net Estimated Final

Site Date data sets error correction error value

NIM-1 June-Sept,

(Beijing) 1979 NIM-1 January

(Beijing) 1980 Sevres A3 April

(Paris) 1980

29 12 -299

5 20 -157

29 10 -360

22 980 125 887 ± 25

14 980 125 830 ± 24

13 980 925 914 ± 16

420
Table 2. Corrections and systematic effects and their estimated

uncertainty (for NIM measurements at the BIPM site Sevres A3) Ag ( /X Gal)

Source Correction

(ixGal) Estimated

uncertainty

(iJiGal)

1. Laser wavelength 0

2. Electronic circuit —22

3. Electrostatic and magnetic 0

4. Air resistance 0

5. Velocity of light +25

6. Gravitational gradient -363

Net correction -360

4

5

8

8

0

3

13

80

40

15

10

40 fj. Gal

20 2 8 14 20 19 Sept. 79 20 Sept. 79

Figure 5. Results obtained at Shiang Shan Laboratory. The

points are measured g values, the curve the computed earth

tide.

References

[1] Fang Chun, Measurement of Gravity and the Geography of

the Earth (in Chinese), 1965.

[2] Fang Yong-yuan et al., Physics (in Chinese) 8, 53 (1979).

[3] A. Sakuma, in Precision Measurement ayid Fundamental

Constants, Ed. by D. N. Langenberg and B. N. Taylor,

Natl. Bur. Stand. (U.S.) Spec. Pub 343 (Aug. 1971), p. 447.

[4] J. A. Hammond and J. E. Faller, ibid., p. 457.

Figure 4. Histograrn of data from 51 drops. The standard devi- ation is 61 ix.Gal.

421

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

New Techniques for Absolute Gravity l\/leasurement

James A. Hammond, Robert L. Iliff, and Roger W. Sands

Air Force Geophysics Laboratory, Terrestrial Sciences Division, IHanscom AFB, MA 01731

In the 11 years since PMFC-1 a number of new techniques have been put into practice in the Air Force Geophysics Laboratory's transportable system for measuring the acceleration of gravity. The im- proved system in use at the present time incorporates an earlier vacuum chamber with some modifications and includes new electronics, data analysis, and optical subsystems. The electronics now produce

time measurement at a large number (500) points during the free fall of the object. These time values

are analyzed with a least-squares fit to a second-order polynomial to obtain the average acceleration. The correction for air resistance is now made by monitoring the pressure and making a correction based on extrapolation from high pressures to the low operating pressures. The use of an iodinestabilized laser as a reference for the length measurement has significantly reduced the uncertainty

due to the wavelength of the Lamb-dip stabilized laser. Key words: absolute gravity; acceleration of gravity; gravity; measurement of gravity; precision measurement.

The Air Force Geophysics Laboratory (AFGL) has long

been involved in the development and support of instruments for the absolute measurement of gravity. Until

1975 we were known as the Air Force Cambridge

Research Laboratories (AFCRL) and as such supported

the development of the first transportable absolute gravity system (see Fig. 1) which contributed greatly to the

establishment of a new international gravity standardization network. This instrument and the results obtained

with it have been discussed in the literature [1^].

Since that time AFCRL supported the development of

a second generation instrument which used a chamberin-a-chamber technique. This system produced some rea- sonably good data [5, 6], but several systematic effects presented problems which encouraged us to backtrack

somewhat. We decided, then, to modify the first generation system to use the same electronics as were

developed for the second generation system. Thus, at the

beginning of 1978 we started assembling the system

which exists at this time. Most of the "new techniques"

which we will discuss in this paper are the result of an

innovation first put into operation with the second generation system. The electronics system developed by

JILA (Joint Institute for Laboratory Astrophysics), with

AFCRL support, performed many time measurements

during the fall of the object, and these time measure- ments were used to compute g. This is in contrast to the

three-position technique used in the first generation

system.

When the decision to use the mechanical parts of the

first generation system was made, we decided to reduce

the height of the chamber by about 0.5 m to simplify the

transportation and operation of the system. The first data obtained with this new system have been described

earlier [7]. There was reasonably good agreement

between the new measurement and previous measure- ments at the AFGL site. The instrument, shown in Fig. 2, was used in 1979 for a series of measurements along a calibration line from

Great Falls, Montana to Ft. Davis, Texas. Data with an

. estimated accuracy of 0.010 mGal (one mGal - 10"^ m/s^)

were obtained at most sites, and the operation time re- quired for seven sites was 28 days. At this time and until early in 1980 our timing electronics made a direct

measurement of the time of occurrence of 350 fringes

during the drop. Then a least squares fit of the time and

distance values to the second order equation describing

the distance as a function of time was made. The result- ing g value was the measurement of the drop. The data were acquired with an on-line computer, and a g value was calculated for each drop. Data were averaged to- gether for up to 24 hours of total operating time at each

site. The appropriate corrections were made to arrive at the final value. This measurement technique enables a unique method

to determine the value of the corrections required be- cause of two phenomena which are important in this kind

of measurement. These two phenomena are the vertical gradient of gravity and the finite value of the velocity of

light.

Both effects produce errors in the measured value

when the calculation assumes uniform acceleration. Since

the least-squares method is an averaging process, the

result is some kind of average value of gravity. A spatial

gradient becomes a time-dependent value of g because

the falling body is moving in space and the least-squares

process will obtain this time-averaged value. If the variation in g is small compared with its magnitude, the effec- tive point at which the acceleration is measured is then

the time-averaged position of the falling body. That point can be determined analytically in a straightforward manner by computing a relatively simple integral.

Alternatively, we have used an essentially numerical

method to determine this effective measuring point. We

calculate a set of synthesized times using the formula:

t = i2X /go)"^ - (5/12)A (2Z/^J^, (1)

where A is the gravity gradient, X is the distance the object has fallen, and go is the value of gravity at the top of the chamber. This expression may be derived by solving

the differential equation describing the motion of a body

423

in a gradient field and expanding the result assuming the

gradient is small compared with g. The result of fitting

these time values is to produce a measured value of Qm

which is different from Qo- That difference can be used

to calculate the effective measuring point :

X,n = {gm - go)'A (2) In our case (0.075 m fall before the 0.575 m measurement

interval begins) the result is an effective measurement

position of about 0.32 m from the top of the chamber.

The effect of the velocity of light being finite is to re- quire corrections to the time values which depend upon

where the falling body is in the chamber. A set of

corrected time values can be calculated and then

analyzed by the least-squares technique and the effect on

the measured g value thereby determined. The result of such an exercise is a correction of 0.024 mGal.

The residuals obtained from our least squares fit would

be expected to show effects which could indicate how

well the system is operating. The residuals should show

the face that the gradient is not included in the curve fit- ting. Thus, in plots of residuals we show a more or less smooth line which results from the fit of the synthetic

data described above with an assumed gradient of 0.3 mGal/m. The actual residuals should more or less track

the smooth curve as shown in Fig. 3. The extent that it

GRAVITY RESIDUALS (ANGSTROMS) CHAMBER SUSPENDED

REAL DATA (150 DROP AVERAGES)

THEORETICAL DATA 3 MGAL/M GRADIENT

50 CM

Figure 3. Residuals in position for least squares fit of real and synthesized data.

30

GRAVITY RESIDUALS (ANGSTROMS) NEW BASE

50 DROP AVERAGES

-20

-10

'

. . lO^M 20 CM 30 CM 40 CM 50 CM

-10

«

-20

-30

Figure 4. Residuals showing systematic vibration of reference reflector.

425
does not indicates vibrations or other effects that are in- fluencing the measurement. The earliest such data

showed a strong vibration (Fig. 4) which was removed by

suspending the dropping chamber above the optics box.

More recently such graphs show vibrations of

60 and 120 Hz, the exact source of which has not been

determined.

Other corrections must be made for the effect of air resistance on the falling body and for the difference

between the assumed (for the purposes of computation)

wavelength of the stabilized laser light used as the

length standard and its actual value. Air resistance

corrections are determined periodically by taking data in the pressure range of about 2 x 10"'' to about 5 x 10"^ Torr and extrapolating the variations to zero pressure. The resulting error at 2 x 10"^ Torr appears to be only a few microgals, and thus the linearity is not checked

closely. We have been periodically checking the

wavelength of our laser against iodine-stabilized lasers

built by the National Bureau of Standards. We have been

continuously using the same Lamb-dip stabilized laser for almost 3 years and we have observed the rather typical

drift of the wavelength of about one part in 10^ per year. A considerable number of absolute measurements of gravity have been made over the last few years and (up

until about a year ago) they have generally corroborated

previous measurements. Recently, some systematic ef- fects have perturbed the results and produced some

rather disappointing comparisons with previous measurements. It is hoped that the sources of these problems can be discovered and that the AFGL system will continue to be significant in establishing and maintaining gravity

standards for precision measurements, geodesy, and geophysics.

References

[1] J. A. Hammond and J. E. Faller, IEEE J. Quant. Electron. QE-3, 597 (1967).

[2] J. A. Hammond, A Laser Interferometer System for the Absolute Determination of the Acceleration of Gravity, JILA

Report No. 103, Joint Institute for Laboratory Astrophysics, Boulder, CO (1970).

[3] J. A. Hammond and J. E. Faller, in Precision Measurement and Fundamental Constants, Ed. by D. N. Langenberg and

B. N. Taylor, Natl. Bur. Stand. (U.S.), Spec. Publ. 343 (Aug.

1971), p. 457.

[4] J. A. Hammond and J. E. Faller, J. Geophys. Res. 76, 7850

(1971).

[5] J. E. Faller and J. A. Hammond, A New Portable Absolute

Gravity Instrument, Bulletin D' Information, International Gravimetric Bureau, No. 35 (November 1974), p. 1-43.

[6] J. A. Hammond, Bollettino Di Geofisica Teorica ed Applicata

Vol. XX (December 1978).

[7] J. A. Hammond and R. L. Iliff, The AFGL Absolute Gravity

System, Proceedings of the 9th GEOP Conference, October

2-5, 1978, Dept. of Geodetic Science Report No. 280, The

Ohio State University, Columbus, Ohio (1979).

426
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

The Mass Unit "Kilogram," Precision IVIeasurement of Mass, Attainable

Uncertainties, and Possibilities of a New Definition

M. Kochsiek

Physikalisch-Technische Bundesanstalt, 3300 Braunschweig, F.R.G.

In this review, the present state of the dissemination of the unit of mass and of the uncertainty at- tainable in mass determination is described. The establishment of a mass scale (hierarchy of mass

standards, multiples and submultiples), requirements for mass standards, important designs of weighing machines, and the predominant influencing parameters such as air density, in mass determination

are discussed. Lately it has become possible to determine air density with smaller uncertainty by way

of calculation and experiment. Possibilities are shown for a new definition on the basis of fundamental

constants. Key words: dissemination of mass unit; influences and uncertainties of mass measurement; mass standards; possibihties of new definition; prototype balance.

1. Introduction

In physics mass determinations are carried out over a

large range following various measuring procedures.

Within the framework of this Conference weighings only

are of interest and these are carried out mainly in the

range from 10"^^ kg to 10^ kg. Weighing means using a measuring device, or balance, to compare an unknown

mass with the base-unit "Kilogram," represented by

mass standards or weights.

In the effort to base the definitions of the base units of the SI system on fundamental constants, that is, on constants occurring in nature, the values of which are

ideally independent of the experimental arrangement,

the unit of mass is an exception. According to available

historical data, to this day masses have been compared

using balances vdth a particular mass defined as the primary standard. The linkage of the mass unit to a fundamental constant has not as yet been achieved with the

required precision. In this sense the history of the unit of mass has at no time been spectacular. Over the last hundred years, however, there has been a notable development towards better precision (stable primary standard,

better balances, improved weighing procedures), so that

the smallest relative uncertainties achievable today for mass comparisons are of the order of 10"^ to 10"^, thus

being worthy of belonging to the best in metrology. This

is all the more astonishing as the physical phenomenon

"mass" is still not yet completely understood. So mass

metrology has developed pragmatically without full in- sight into the theoretical problems associated with mass

[1,2].

2. The Unit of Mass and the Hierarchy of

Mass Standards

The unit of mass is the kilogram, which was defined by

the first General Conference of Weights and Measures in 1889 as the mass of the International Kilogram Prototype, maintained at BIPM (Bureau International des

Folds et Mesures) in Sevres, Paris:

The International Kilogram Prototype is a cylinder

39 mm In height and 39 mm in diameter of a 90% platinum with 10% iridium alloy (Pt-Ir); its density is about

21,500 kg-m-l

This means that the unit of mass can never be

transferred with a precision better than permitted by the mass comparison with the International Prototype at BIPM. The consequence is a hierarchy of mass standards

to guarantee the dissemination of the unit of mass vdth

the highest possible accuracy. Since 1889 more than 60

platinum-irldlum cylinders of the same dimensions and

quality have been distributed as national prototypes.

Fig. 1.

Figure 1. National Kilogram Prototype No. 52 at the PTB,

FRG, kept U7ider two bell-jars. In the background are further primary and reference standards.

427
At the top of the hierarchical ladder for the dissemination of the unit of mass stands the International Kilogram

Prototype at the BIPM, Fig. 2. Using a prototype bal- ance, the national prototypes are compared with primary

standards of the BIPM which are themselves compared

with the International Prototype. Thus the International

Prototype only had to be used for comparisons in the

years 1889, 1939 and 1946 and has, therefore, been nearly

protected against wear and tear and possible damage.

Most national laboratories disseminate the unit with

the aid of primary standards, which are made of stainless

steel (density around 8000 kg-m"^). The comparison of these primary standards with the national prototype is the most significant from the point of view of measuring

technique since for the necessary transition from density

21,500 kg- m"-^ (Pt-Ir) to about 8000 kg-m'^ (Fe), the

present uncertainty of the air buoyancy correction is larger than the uncertainty of the balance and of the

other influencing parameters. The reference standards

of the verification boards are then compared with the primary standards, as well as the reference standards of firms and other institutions which can be assigned to the

best classes of mass standards of International OIMLRecommendation No. 20. The prototypes, and primary and reference mass standards are standards of the highest precision; handling means taking the risk each time of a change in the mass

(wear and tear, contamination) and of possible damage.

Hence the time intervals between comparisons should, on

the one hand, be as long as possible, but nevertheless

short enough to allow timely recognition of mass variations. The right-hand column of Fig. 2 gives some indica- tion of the time interval between two comparison meas- urements as well as the measurement site. Since working

International Kilogram

Prototype

Primary standards

of tt)eBIPt^

ottheBIPM'-^;f^l^,

National Kilogram P roto type e g No 52afFRG

e g 10a at iasti97i. the BIPM """9«'

Primary standards of thiePTB

e.g.Sa at last 1977 thePTB next198i

Reference Reference

standards Reference standards

of the standards of the PTB

verification of firms ore ther

c entres institutes

—<.10aat thePTB

Control standards

45a on site

Working standards

41a on site

standards and verification standards can be replaced more

easily, they should be changed depending upon their fre- quency of use [3]. The next step—starting from a 1-kg standard mass—is to realize submultiples and multiples of the unit in the

form of mass standards in such a way that the mass of any arbitrary body can be determined. For example

150.012 g using standards of 100 g, 50 g, 10 mg, and

2 mg. This is done by realizing the values 1 to 10 in every decade, which necessitates at least four standards

for each decade: For example, the subdivision 1,2,2,5 is commonly used. That is, for the decade from 100 g to

1 kg the standards have nominal values 100 g, 200 g (two

weights), and 500 g. Of these weights in the 100 g to 1 kg

decade, two suitably chosen groups of mass standards of equal nominal sum value are all compared with each

other. In this way the well-known overdetermined sys- tem of linear equations is obtained. Many schemes for the intercomparison of sets of weights are available

[4-7, 44]. Since an optimum scheme (few weights with small uncertainty) depends on the balances available, the subdivision of the mass standards, the correlation between

weighings, and the effort put into weighing, many

schemes have been developed. Therefore an optimum

scheme cannot be given.

Figure 3 shows a simple scheme. The 1-kg standard is known; one 500 g, two 200 g, and two 100 g standards

are unknown. For five unknown standards seven determinations are carried out. According to the scheme on one side we have the 1-kg (positive sign), and on the

other side 500 g -H 2 x 200 g + 100 g (also of sum 1 kg

with negative sign). For the seven series of weighings we

obtain observations x(l) to x(7), from which the unknown

masses may be calculated.

Calibration Scheme N=7 observations

K:5 unknown weights

Decade lOOg - 1kg

Observations M kg) (sOOg 200g 200g lOOg lOOg

x(1) + x(2)

x(3)

X (A)

x(5)

X (6)

X (7)

Decade lOg - lOOg

Observations (l00g)(50g 20g 20g lOg lOg

yd )

Decade Ig-lOg

Observations ("lOg) ( 5g 2g 2g Ig Ig 2(1)

Figure 2. Example from PTB, FRG, of a hierarchy of mass

standards. Figure 3. Simple calibration schema with 5 unknown masses

(K) and 7 weighings (N).

428
In the next decade from 10 g to 100 g the 100 g standard is known and the 50 g, 20 g, and 10 g standards are unknown. In this way every decade, and finally every set

of standards, can be consistently determined.

3. Mass Standards

Mass standards must guarantee long-term stability of

their mass. This can be achieved by the choice of corrosion resistant, non-magnetic material, suitable geometry,

and specified procedures for handling, maintaining, and

cleaning the mass standards [8-10]. Forces due to magnetic fields and electrostatic charging may not act on the

standard. This is achieved for primary standards when

the magnetic permeability is smaller than 1.01, the surface resistance is smaller than 10^^ ohm, and the corrosion resistance is smaller than 10"'^ g-m""-A"^ [11]. The

geometry should be chosen for best handling and cleaning; for example, a cyhnder according to OIML-IR No. 20

[12]. The surface should be polished with roughness value

Ra =0.1 |JLm, according to ISO-R 468. Two examples will show what else one has to consider:

In the BIPM it was found that the mass of national

prototypes increases after cleaning by 2 to 5 ixg per

year [13]. This effect was also observed in standards of other institutes. After special measurements it is possible to take this into account by correction [14]. Extensive investigations showed that sorption layers on the surface should be considered when the environmental conditions change [15-17]. For example, the

change in the water sorption layer of a 1-kg mass

standard for an increase in humidity from 40% to 80%

is about 20 |jLg. Figure 4 shows the specific mass of the

water sorption layer calculated using this value

(0.14 ixg-cm"2 X 134 cm^ = 19 |jLg).

So one sees that it is important to carry out mass comparisons under fixed environmental conditions. According

to all these criteria, besides platinum-iridium, high-alloy

steels are also particularly useful for mass standards.

Composition, density, attained magnetic permeabihty,

and roughness of a stainless steel standard are given as an example in Table 1 [18].

Table 1. Stainless steel for mass

standards (example)."

m

0.8

0.6

OA

Al

Material

/ Brass

/ Pt-lr Steel

1 1 20 40 60 80 °/o 100

1. Chemical composition

v^arDon U.UdU%

ivia^llcalufil (.IVijly x.iiW/o

r llOopilOi Us V-T/ A A01 C/ Sulphur (S) 0 004%

Silicon (Si) 0M0%

Chromium (Cr) 19.900%

Nickel (Ni) 24L^^t 900% U\J\J /C/ Molybdenum (Mo) 2.850%

Copper (Cu) 0.100%

Iron (Fe) 50.600%

2. Density at 20 °C P = 7998 kg-m

3. Roughness (ISO-R 468) /2a < 0.11 |jLm

4. Magnetic Permeability fJL < 1.01

^From Ref. [18].

Figure 4. H20-sorption layer m" of different materials as a

function of relative humidity. Surface: polished, = 0.2 [im;

cleaning was with alcohol in an ultrasonic hath.

4. Prototype Balances

Prototype balances and precision balances vdth resolution (i.e., maximum load divided by scale division) higher

than 10^ are today still chiefly being built as beam bal- ances, where either the knife-edge and flat bearing or a

flexible strip acts as a central component. The quality of

all these balances depends on the construction, setup,

and adjustment accuracy, and in particular on storage

and sensitivity as well as on auxiliary apparatus such as the locking device, the device for placing the masses on

the pans, and the arrangement to compensate for small mass differences. In addition, the experience and skill of the operator is of no little importance. Figure 5 shows a

typical prototype balance of the Rueprecht firm of about

1960, built as an equal-arm balance with three knife- edges and two pans, which is still used in numerous state

institutes. The standard deviation achieved is up to 15 |jLg for a maximum load of 1 kg. Based on this construction, further developments over the last 20 years were the following:

During a weighing series the knife-edge and pan are not separated [19, 20], the oscillation of the beam is fol- lowed photoelectrically [21, 22], the balance can be

loaded with up to four standards for substitution

weighing, and it can be remotely controlled [23] (at- tainable standard deviation as small as 1 \x.g). A new design was developed by the National Bureau of Standards (NBS), namely a single-pan beam balance

with two knife-edges and short, unequal arms: see

Figs. 6 and 7. The knife-edges and pans are not separated, and up to six standards can be compared

with each other in a single series [24, 25] (attainable

standard deviation as small as 1 jig).

Praiseworthy is the further development of commercial

balances, e.g., when the beam of a knife-edge balance

is brought into reference position by magnetic compensating forces [26] (see Fig. 8; attainable standard deviation as small as 10 |jLg).

Currently, as far as I know, there are two new developments in the prototype balance field:

In the National Physical Laboratory (NPL). Teddington, and BIPM, Sevres, a beam balance is being

429
t

i

ill- 1

Figure 7. l-kg NBS prototype balance. 1: mass standard; 2: beam with counterweight 3; I: suspension with paji; 5: roundabout for mass standards. (Source, NBS.)

Figure 5. l-kg Rueprecht prototype balance. 1: mass standard;

2: beam; 3: mechanism for loading additional weights; U: device for changing the standards.

UPPER UNIVERSAL POINT COINCIDES WITH E PS

INTERMfDlfiTE FLAT

OPERflT:'JG axis OF INTERVECIATE KNIFE LOAD KNIFE

CENTER KNIFE

ARM LENGTH

, CTRWT z.^' ARM t~

Figure 6. Exploded sketch of the oscillating system of NBS 2 mi.

Figure 8. Block diagram of a magnetic force compensation bal- ance [26]. A: balance; B: control instrument; C: digital readout unit; 1: weighing systern; 2: magnetic compensation

system; 3: inductive beam, scanning; i: temperature compensation for sensitivity; 5: variable-gain amplifier; 6: range selector; 7: measurement amplifier; 8: analog output signal; 9: digital detection of dialed weights; 10: digital readout; 11: au- tomatic tare; 12: digital output signal.

developed with flexible strips instead of a knife-edge

and flat [27]. Figure 9 shows in an exploded sketch the

beam, 1, the flexible strips, 2, and the suspension, 3, of the pan. At the PTB, Braunschweig, a liquid mass comparator

is under construction, where differences in mass are determined by hydrostatic force compensation

(Fig. 10). The buoyant force on a ring-shaped body, 1, submerged in a liquid, 2, is transmitted via three thin

submerged rods, 3, to the suspension, 4, 5, of the bal- ance which is located underneath the liquid container,

6. Owing to the change in upthrust due to the submersion depth of the rods, the vertical equilibrium depends

on the applied load, 8. Using a weight changing device,

430
Figure 9. Enlarged sketch of the beam and flexible strips of BIPM balance under construction [27]. 1: beam; 2: flexible

strip; 3: pan suspensio7i.

Figure 10. Liquid mass comparator at PTB. 1: ring shaped

floater; 2: liquid; 3: rods; \: upper part of suspension; 5: suspension vnth load pan; 6: liquid container in water bath; 7: laser interferometer; 8: mass standard. -

up to six 1-kg weights can be placed consecutively in suspension, whereby the latter is kept constant by

locking. The vertical difference in upthrust for the

various weights is measured using an optical interfer- ometer, 7, and with the aid of appropriate small mass

standards the determined sensitivity is calculated as a mass difference. At this moment first measurements

show a sensitivity of 10 |jLg.

5. Mass Hierarchical Linkup and

Determination of IVIass

According to the basic axiom of the 3rd General

Conference of Weights and Measures in Paris, 1901, the

weight of a body represents a quantity of the nature of a

force and is the product of the mass of the body and the

local acceleration due to gravity [28]: F = mgioc

This is the fundamental formula underlying the comparison of masses with one another. We regard the masses of two bodies to be equal when they exert the same force due to weight for the same value of the acceleration due

to gravity (i.e., at the same place). Nevertheless, it is clear that we can only compare weight forces in vacuum.

For weighing in air, as is generally the case, the sum of weight forces and buoyant forces are compared with each

other.

To prove the equality of the masses of two bodies, we

utilize the balance. WTien forces act at both ends of a pivoted lever of an equal-arm balance, the lever remains

at rest if the resultant torques sum to zero. The equation

of weighing in air is LhimiQioc - 'V^lpLfi'loc) = LRimzgioc - V-zPrQloc) , where

mi, m2 = mass of bodies 1 and 2; Vi,V2 = volumes of bodies 1 and 2;

Pij P2 = density of bodies 1 and 2 from m/V;

PL = density of air during weighing;

g loc = local acceleration due to gravity; and

Lh, Lr = length of the two effective lever arms.

If Ll - Lr and if gioc is considered to be constant within

the immediate vicinity of the balance, then with p -

m/V, we obtain

1_ Pk

Pi m,2 - m\ •

1 - ^

P2 This formula shows that for a comparison of mass standards the densities of the test object and the standard (if

Pi¥= p2 ) as well as the momentary air density have to be

1 431
taken into account. In addition, there are several other

influencing parameters:

5.1 Influences Due to the Mass Standard

A mass standard which satisfies the conditions already

mentioned, protected from dust under bell-jars, of which

the contamination and sorption layers are known and special forks or tongs are used for handling, has a mass stability of a few micrograms over several years in the case

of a 1-kg standard.

5.2 Influences Due to the Balance

The uncertainty values achievable with prototype bal- ances, mainly a few micrograms, have already been mentioned in Section 4. Table 2 shows the relevant data on five chosen balances, including the achievable

uncertainty.

Three balances suitable for comparing mass standards

have relative standard* deviations of 10"'' to 10~*, but

only a few prototype balances like the NBS-2 reach 10"^. One of the smallest known balances is a balance with

which mass variations up to 20 pg are measured by the

frequency variations of a resonating quartz crystal. A

typical railway scale for special purposes in steel mills has a maximum capacity of 1800 t, which is taken up by

16 load cells.

5.3 Influences Due to the Weighing Process

The precision weighing described here takes place in air and so the air buoyancy has a greater effect the more

the density of measurand and standard differ. For the

comparison of 1-kg standards of platinum-iridium with

those of steel, the air buoyancy correction is around

100,000 |jLg. The density of air can only be determined

with a relative uncertainty of 1 to 2 x 10~\ because the

universal gas constant R (or equivalently, the density of

dry air) and the composition of the air are not known to the required precision. In 1979 at an international meeting on mass determination, agreement was reached over a uniform formula to calculate the density of moist air as well as tables and

auxiliary formulae to be used for the application of this formula [29]. Changing environmental conditions further influence

the balance and mass standards during weighing. Here

one should take note of temperature fluctuations (Ai =

0.02 K); atmospheric pressure variations (Ap = 20 Pa);

changes in the relative humidity (A/i = 1%); the composition and purity of the air; inhomogeneities of the gravitational field; stability of the balance site against

gradients and vibration; effect of forces due to air tur- bulence, electrostatic charging, and magnetic fields between the balance, standard, and surroundings; and

change of the temperature gradient along the balance arm.

Table 2. Data on some balances of high accuracy.

Weighing Maximum Standard

Instrument Load Deviation

Mass Standards

Electronic balance with additional weights 5g 0.2 (xg (Srel = 4 X 10-8)

Prototype balance NBS 2 1 kg 1 v-g (Srel = 1 X 10"^)

Mechanical equal arm balance 5t 100 mg (Srel = 2 x lO"")

Research

Resonating crystalline quartz balance 2 mg 20 pg (Srel = 1 x 10"^)

Industry

Railway truck scale with 12 load cells 1800 t 200 kg (Srel = 1 ^ iQ-^)

432
Here are 2 examples:

If the temperature of one entire lever-arm changes by

10"'^ K compared with the other, then the result is a weighing error of 20 |jLg for a 1-kg weight and a 200 mm-long brass lever-arm.

Let the center of gravity of a 1-kg mass lie 50 mm

higher on one side of the balance than the center of gravity of the standard on the other side. The correction for a 50 mm difference in the distance from the

center of mass of the earth is about 15 |xg.

5.4 Influences Due to the Weighing Procedure

Certain effects such as unequal lever arms or an inhomogeneous gravitational field are eliminated when tare weighing according to Borda or exchange weighing ac- cording to Gauss are carried out. Whereas both weighing

procedures can be applied using the three knife-edge bal- ance such as the Rueprecht (Fig. 5), only tare weighing

is possible with the other balances such as the two knifeedge balance. As an example of the problem of unequal

arms, we note that for an effective armlength of 200 mm,

a maximum load of 1 kg, and a specified standard deviation for the balance of 1 fxg, differences of only 0.2 nm in armlength will affect the result. Under optimum conditions, using the best prototype balance to compare two

1-kg standards of equal density, one can achieve a standard deviation as small as 1 |jLg.

6. Attainable One Standard Deviation

Uncertainties

Taking the quadratic sum of the component uncertainties, namely the uncertainty of the national prototype

compared at BIPM: 8 |xg; the uncertainty of the mass

stability of the standards: 5 |xg; the uncertainty of the air buoyancy correction: 10 |jLg; and of the weighing pro- cedure itself: 2 \xg; we obtain for a 1-kg standard of corrosion-resistant, non-magnetic steel of density

8000 kg-m~^ an empirical one standard deviation of uncertainty of about 15 |jLg (Table 3). Based on this value, a mass determination of a measurand with a density of

8000 kg-m~^ can be performed with the uncertainties

shown in Figure 11.

-O

>-

o

100 1 10 100 1 10 100 1 10 100 1 10 100

Jjg I mg I g I kg I t mass »>

Figure 11. Attainable uncertainties (one standard deviation) for rnass determination.

7. Mass Determination for the Realization

of Other Units or Fundamental Constants

Three examples demonstrate the demands made on mass determinations:

For the determination of the Newtonian gravitational constant according to the setup by de Boer [30], weights of mass 125 g and density 2500 kg-m"^ must be

determined with an uncertainty of less than 10"^ which, even taking into account the air buoyancy cor- rection and the standard deviation of the balance, pre- sents no problems (point 1 in Fig. 11). For the determination of the Avogadro constant the

density of silicon is required and for this the mass

determination should have an uncertainty of less than

1 X

10"''^ [31, 32] (mass of the measurand 1 kg, density

2400 kg-m"^, point 2 in Fig. 11). With a certain amount

of effort this requirement can be satisfied. For the realization of the unit of electric potential us- ing the voltage balance, with a dead load of about 3 kg, a mass determination of 30 g with an uncertainty of

less than 2 x 10"^ is required. Whereas the mass

Table 3. Attainable uncertainty in the naass of a 1-kg mass standard of density 8000 kg-m~'^

Sources of Uncertainty One Standard Deviation Uncertainty

At Present Near Future

(M-g)

National standard 8 2 Instability of mass standards 5 2 Air buoyancy 10 2 Prototype balance, weighing procedure 2 2

Root sum square 14 4

433
determination is possible (point 3 in Fig. 11), the reali- zation of a standard deviation of 2 x 10"^ for the volt- age balance presents considerable difficulties [33].

8. Present Activities in the Area of Mass

Determination

Three International working groups under the leadership of BIPM are currently concerned with the analysis

and reduction of sources of error in mass measurement

arising, for example, from determination of air density,

volume of mass standards, sorption layers, cleaning, and

handling. The first result of the work is a report on the

air buoyancy correction with formulae and tables, etc., to calculate the air density [29]. A reduction of uncertainty

in air buoyancy correction is equivalent to a more precise

determination of the air density. A relative uncertainty

of a few parts in 10^ should be aimed for here [13]. In particular, the reference density of dry air, the ac- tual composition of the air, and the degree of contamination due to suspended particles should be determined more accurately [34]. Recent measurements by Schoon- over et al. [28], Prowse [35], and Kobayashi [36], show

that a fundamental determination of the air density vdth

a relative uncertainty of about 10""^ is possible. The

method used is the measurement of the apparent mass

difference of two samples, each of similar surface and mass but differing in volume. Further work in progress

includes investigation of sorption layers on mass standards and the specification of cleaning procedures and

environmental conditions for the maintenance and comparison of standards.

New developments concerning prototype balances have

already been discussed in Section 4. However, weighing

in vacuum is a different problem. As soon as the question has been clarified for which state the mass of the

prototype has been defined, and further knowledge on

the adsorption and desorption of mass standards and bal- ances have been obtained, weighings in vacuum will recommence. With the activities described here it may be possible over the next several years to achieve a relative uncertainty of less than 4 x 10"^ for relating a 1-kg standard

with density 8000 kg-m"^ to one of density 21,000 kg-m~^;

see Table 3, righthand column.

9. New Definition of the IVIass Unit

"Kilogram"

A favorite daydream of metrology, analogous to the current representation of length, is to break away from

the embodiment of the mass unit by the International

Prototype and to define and realize it using fundamental

constants. The following three possibilities are fre- quently discussed in the literature: The kilogram is defined by a fixed number of atoms or molecules, for instance 2.741 x 10^^ particles of the

orange-red radiation of krypton 86 for the transition

between 2pio and Sds states [37]. However, since the

authors never mention how this can be realized, this and other definitions based on counting particles need

not be discussed at this time.

A second possible definition comes from the Avogadro

constant if known with sufficient accuracy and using

the following relation: The kilogram is

(A^a) kmol-^

12

times the mass of an atom of ^^C [38]. The Avogadro

constant itself is measured with a relative uncertainty

of 1 X 10"" using a silicon single crystal via density,

lattice constant, and amount of substance [39, 40]. Compared with the present system where the relative uncertainty in the kilogram is 2 x 10"^ the Avogadro

constant would have to be determined more precisely

by 2 powers of 10; only then would a new definition be

metrologically justified.

A third possibility is to determine the kilogram by

electrostatic forces using a voltage balance [41]. The

three mechanical base units meter, second, and kilo- gram are related to the unit of potential and to the

electric constant by way of the force due to an electric

field. In this case the kilogram would be metrologically

defined by the electrostatic forces of a voltage balance.

All base units, the kilogram too, could be derived from

quantum transitions. However, this would only be a transposition of the least accurately realizable unit

from the ampere to the kilogram. The relative uncertainty of the realization to date would be about 2 x

10"", which is also 2 powers of 10 worse than the

present arrangement. Another problem is the influence

of the gravitational acceleration (see Fig. 12). These considerations show that a new definition cannot be expected in the near future. Nevertheless, the search

should continue for ways of realizing a new definition us- ing fundamental constants, like the Avogadro constant,

in a consistent set of units and constants (Fig. 13).

Maxwell^s Equation

rS Voltagebalance s = 2-10

Figure 12. New definition of the unit of mass based on a volt- age balance [Jtl].

Consistent set of fundome ntal constants

e, F, R. .rrig, .

Cq or ? he

kg V

Figure 13. New definition of the unit of mass based on a consistent set offundamental constants.

434
References

[1] F. S. Sawelski, Die Masse und die Messung (Verlag Harri

Deutsch, Frankfurt/Main, 1977) [original in Russian].

[2] L. E. Hewlett, Z. Angew. Phys. 15, 463 (1963).

[3] S. German and M. Kochsiek, Bull. OIML No. 68, 18 (1977); No. 69, 7 (1977) [Organisation Internationale de Metrologie

Legale, Paris, France].

[4] D. B. Prowse and A. R. Anderson, Metrologia 10, 123 (1964).

[5] P. E. Pontius, Natl. Bur. Stand. Tech. Note 288 (1966).

[6] M. Grabe, PTB Mitteilungen 87, 223 (1977).

[7] J. M. Cameron, M. C. Croarkin, and R. C. Raybold, Natl. Bur. Stand. Technical Note 952 (1977).

[8] A. Bonhoure, Microtecnic 6, 151 (1952).

[9] H. E. Aimer, Weight cleaning procedures, Natl. Bur. Stand. Report 9683 (Feb. 15, 1968).

[10] G. Girard, BIPM Report RM76/5 (Bureau International des Poids et Mesures, 1975).

[11] M. Kochsiek, Wagen u. Dosieren 9, 4 (1978).

[12] OIML-IR 20: Poids des classes de precision E^, E2, Fj, Fg, Ml des 50 kg a 1 mg (Organisation Internationale de

Metrologie Legale, Paris, France).

[13] Rapport sur la reunion concernant les masses, BIPMCIPM/77-1, p. A3 (Bureau International des Poids et Mesures 1977).

[14] M. Kochsiek, PTB Mitteilungen 89, 421 (1979).

[15] Y. Kobayashi, these proceedings.

[16] M. Kochsiek, PTB Mitteilungen 87, 478 (1977).

[17] T. Yoshimori et al., Trans. Jpn. Inst. Met. 14, 396 (1973).

[18] ASMW, Germ. Dem. Rep., Communication CCM/81-13E

Consultative Committee on Mass (Bureau International des Poids et Mesures, 1981).

[19] F. A. Gould, Proc. R. Soc. London, Ser. B: 62, 817 (1949).

[20] R. Spumy, Communication CCM/81-22, Consultative Committee on Mass (Bureau International des Poids et Mesures, 1981).

[21] W. Lotmar and J. G. Ulrich, Metrologia 12, 159 (1976).

[22] M. Kochsiek et al., Bull, OIML No. 70, 7 (1978) [Organisation Internationale de Metrologie Legale, Paris, France].

[23] Y. Kobayashi and K. Uchikawa, INSYMET (1974), Bratislava.

[24] H. A. Bowman and H. E. Aimer, J. Res. Natl. Bur. Stand.

Sect. C: 67, 227 (1963).

[25] H. E. Aimer, J. Res. Natl. Bur. Stand. Sect. C: 76, 1 (1972).

[26] Mettler Instrument Corporation, specifications for model

H315 spec.

[27] Rapport du directeur sur 1' activity et la gestion du BIPM,

corrections et additions, 30 Sept. 1980, document

CIPM/80-11 (Comite International des Poids et Mesures).

[28] The International Bureau of Weights and Measures 1875-1975, Ed. by C. H. Page and P. Vigoureux, Natl. Bur. Stand. (U.S.), Spec. Publ. 420 (May 1975). [English translation of the BIPM

Centennial Volume Le Bureau. International des Poids et Mesures 1875-1975.]

[29] BIPM Report 79/10 (Bureau International des Poids et Mesures, 1979).

[30] H. de Boer, these proceedings.

[31] R. D. Deslattes et al., Phys. Rev. Lett. 33, 463 (1974); Ref

[40]; and these proceedings.

[32] P. Seyfried, these proceedings.

[33] V. Sienknecht, PTB-Bericht E-12 (Physikalisch-Technische

Bundesanstalt, 979), p. 47.

[34] R. M. Schoonover et al., J. Res. Natl. Bur. Stand. 85, 27

(1980).

[35] D. B. Prowse, these proceedings.

[36] Y. Kobayashi, Communication CCM/81-12B, Consultative Committee on Mass (Bureau International des Poids et Mesures, 1981).

[37] S. K. Mukhftrjee and A. Choudhury, ISI Bull. 24, 288 (1973) [Indian Standards Institution, India].

[38] P. Draht, PTB Mitteilungen 85, 44 (1975).

[39] Dimensions/NBS 58, 219 (1974) [Dimensions is the former Natl. Bur. Stand, monthly information magazine].

[40] R. D. Deslattes, in Metrology and Fundamental Constants,

Proc. of the International Summer School "Enrico Fermi,"

course LXVIII, Ed. by A. Ferro Milone, P. Giacomo, and S. Leschiutta (North Holland, Amsterdam, 1980), p. 38. [41] H. J. Schrader, PTB Mitteilungen 85, 29 (1975).

[42] S. I. Toropin and V. C. Snegov, Izmer. Tekh. 1975, No. 12, 75 (1975). [EngUsh transl.: Meas. Tech. 18, 1847 (1975).]

[43] W. F. Koch et al., J. Res. Natl. Bur. Stand. 83, 407 (1978).

[44] M. Romanowski and G. Mihailo, New developments in the metrology of mass standards, unpublished internal report, National Research Council, Canada.

435

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Measurement of Air Density for High Accuracy Mass Determination

D. B. Prowse

CSIRO Division of Applied Physics, Sydney, Australia 2070

A description is given of the air density balance developed for the fundamental determination of the density of air. The method consists of measuring the apparent mass difference betw^een a sphere

and a ring of similar mass and large known difference in volume. The two-arm balance used has been

modified to give a continuous electrical readout of the air density. The uncertainty (3 standard deviations) in air density given by the method is 3.5 x 10"^ kg • m"^. Preliminary results indicate that the

air density balance agrees with the values obtained by the equations developed to calculate air density

. to 2 parts in 10^. Rapid fluctuations up to 1 in 10^ of the air density are observed in an air-conditioned

laboratory.

Key words: air density; apparent mass difference; continuous electrical readout; sphere and ring; two- arm balance.

1. Introduction

All measurements of mass are based on Pt-Ir prototype

kilograms. For practical use these prototypes need to be

compared with stainless steel kilograms and the special

standardizing balances used in these comparisons usually

have a discrimination of 1-2 fxg. Because the masses

have different volumes (approximately 80 cm^ disparity)

the density of the air must be accurately known to enable

corrections to be made for the difference in air buoyancy

on the masses. Recent work by Jones [1] has shown that current state of the art measurements for pressure, temperature, relative humidity, and composition give an

overall fractional uncertainty in air density of 3 x 10"^. This corresponds to an uncertainty of 29 |xg when comparing kilogram artifacts of Pt-Ir and stainless steel. To be able to transfer the mass unit from the Pt-Ir to the stainless steel artifacts with an uncertainty approaching the discrimination of the balances, work at the National Measurement Laboratory has been aimed at a

direct measurement of the air density vdth an uncertainty approaching 1-2 x 10"^. The method used is the measurement of the apparent mass difference between a glass sphere and ring, each of

similar surface area and true mass but differing in volume by just over 1000 cm^. The sphere is fitted with a

small teflon sealed valve to enable it to be evacuated.

Because the sphere is not permanently sealed it is possible to measure the necessary properties to enable the air density to be determined wdthout any assumptions or further calibration.

2. Volume Measurements

Two spheres were made from 1 liter flasks by removing the neck and adding a small teflon-sealed vacuum

valve. The ring was made from glass tubing of diameter

and thickness chosen so that the surface areas of the ring

and spheres were as similar as possible—within about

10% . No effect of the small difference in surface area has

been observed.

If the arms of the balance used for measuring the air density are exactly equal only the difference in volume,

and mass, between the sphere and ring would need to be

measured. Because the arm lengths are not equal, the in- dividual volumes and masses must be known.

The volumes were calculated from weighings in air and

water. The density of water in equilibrium vrith air is known to at least 2 in 10^ [2]. If the density of air can be

calculated to 2 parts in 10'* using standard equations [1, 3] then, except for weighing the sealed spheres in air, the

largest uncertainty in any individual weighing due to the

air density is 4 parts in 10^. Sufficient mercury was added to the spheres so that

they just sank in water. The spheres were then

evacuated—a necessary procedure because the change in volume on evacuation was found to be 1 part in 10 . In use they are always evacuated to a pressure of less than

1 Pa. The weighings were made with a hydrostatic balance.

The vessel containing distilled water was placed in a water bath which was stirred, but not temperature controlled as the temperature remained sufficiently constant and uniform during each measurement.

By weighing the ring and two spheres in air and water

and against each other, 10 equations in 6 unknowms of the form

Msi - Vsip = mid - a/A)

Mr - Vrp - Msi + Vsip = msd - a/A)

,

were obtained. In matrix notation these may be written

(1)

(2)

-1 1 P —a a -p p

Mr

V,.

mi(l - a/A)

miod - a/A)

(3)

437
where

Msi, - mass of sphere 1 and sphere 2, respectively;

^si> ^s2 = volume of sphere 1 and sphere 2, respectively;

My. - mass of the ring;

V,. = volume of the ring;

A = nominal density of stainless steel masses;

p = density of water at the time of each

weighing;

o- - density of air at the time of each

weighing (calculated from measure- ments of pressure, temperature and

relative humidity); and

TOi TO 10 - stainless steel masses required to bal- ance each weighing.

These 10 equations were solved by the method of least-squares for the 6 unknowns, of which only Vsi, Vgz

and Vr are relevant. From the variance-covariance matrix a standard deviation of 3.5 x 10"^ cm^ (3.5 parts in

10^) was obtained for the volume difference Vs -Vr3. Mass Measurements

By adjusting the nichrome wire suspensions the true mass of the spheres and the ring were made equal to within a few mg. The mass difference between the material in the ring and the spheres was determined ini- tially by weighing them in evacuated aluminium cans. This however gave an uncertainty of 0.5 mg, due mainly

to the size of the cans and the difficulty in manipulating

them on the balance. The ring and the spheres were then

compared in air with the valve on each sphere open.

Static electricity was found to cause discrepancies of up

to 1 mg for both sets of weighings. When this was eliminated the standard deviation of the mass difference

between the spheres and the ring was found to be 10 |jLg.

Within the uncertainties the values were the same for weighings made in air and in vacuum.

4. Air Density Balance

The actual measurement of air density is made on a two-arm hydrostatic balance which has been modified by

the method described by Kibble [4] to give electrical readout. A second order filter was added to smooth out

noise and any fluctuations caused by air turbulence. The

sphere and ring were suspended beneath the balance in an enclosure to protect them from air currents. A special

mechanism enabled them to be lifted clear of the pans

without arresting the balance. This permitted the bal- ance zero to be easily determined so that small drifts due

to temperature, etc., could be eliminated. The beam com- partment of the balance was thermally insulated in an attempt to ehminate temperature gradients, but the bal- ance still has a small zero drift of about 80 ixg/K. The

standard deviation of the balance with the Kibble system

is 5 |xg, which in terms of air density is equivalent to 5 X 10"^ kg m"^. Tests to determine the sensitivity of the balance itself to changes in air density were made by filling the balance

case with argon and then helium. Within the uncertainty

of measurement the balance appears to be unaffected by

changes in the ambient atmosphere.

The coil in the system was adjusted to give the balance

a range of ±20 mg which is adequate to cover the change

of air density under ambient conditions in the laboratory.

However, vdth such a large range, the response of the

system is not quite linear so that as the mass difference

measured by the balance increases, the light spot on the

null detector moves further off center. This can be over- come in two ways: working near the balance null by

adjusting the rider associated with the balance, or preferably incorporating an integrator in the feedback

circuit of the operational amplifier to reduce the feedback

current at null to zero. The first method is being used

while the apparatus is being modified to incorporate the

second method.

If the balance is assumed to have equal arm lengths

then the air density is given by ^

- + W - (g - 6)s - + W/A '

where W is a known stainless steel mass of approximately 1.2 g associated with the sphere; a, 6 are the

reading and zero reading of the balance, respectively;

and s is the sensitivity.

For a balance in which the arm lengths are unequal,

the air density is given by an equation of the form: a =

{M,L 1 - M^La) + (m^L i - tyirL-^ + WL i - (a - & )s iy, + wl/A + W/A)L 1 - (F, + TOfi/A)L 2

'

where m^, are the mass of the left and right

hangers, respectively (these hangers are lifted off with

the sphere and ring when the zero reading is determined); Li, Lz are the left and right arm lengths, respectively, of the balance; and L\ and Li are determined by

weighing two known stainless steel masses of the same mass as the spheres and ring (250 g). Equation (5) in- cludes the term (M^Li - MrLi), however, this can be

transformed so that

MsLi - MrL2 = MAL, - Lg) + (M, - M,)Li. (6) As Lj - L2 is small, Mr needs only to be known to

0.1 g. Equations (5) and (6) assume that the sphere is on

the left hand pan of the balance.

5. Conclusion

The uncertainty in air density (3 standard deviations)

given by this method is 3.5 x 10"^ kg • m"^.

The equations developed by Jones [1] and the BIPM [3] for calculating the air density from measurements of pressure, temperature, relative humidity, and composition agree with each other to 2 parts in 10^. The results from these equations have been compared with the density as given by the air density balance for an assumed

standard composition of the air. In general these agreed

to 2 parts in 10^ with any closer agreement being limited

by the uncertainty in calibration of the barometer. Future work vdll improve the pressure measurements and

allow for variation in the composition of the air. The balance shows that the air density in a normal air- conditioned laboratory fluctuates too fast for the level of accuracy obtained from the balance to be used effec- tively. Rapid fluctuations in air density with periods

ranging from seconds to minutes and of magnitude up to

1 in 10^ have been observed. These fluctuations are mainly due to pressure changes, for example, opening the

door of the room causes a density change of 2.3 x

438
10~^ kg • m ^, and are confirmed by observations with a commercial gas density meter (see Fig. 1). Thus for

Figure 1. Recorder trace of the output of the air density bal- ance (a) and a commercial gas density meter (b).

measurements of mass of the highest precision, either

the balance case or the balance room should be sealed

against rapid pressure changes and the air-conditioning

turned off. One obvious application of the air density balance is to monitor the output of both the standardizing and the air density balance with a small computer so that each turn- ing point of the standardizing balance swing can be

corrected for the value of the air density. Another possi- ble application is to obtain a value for the gas constant.

If the balance case and sphere and ring enclosure are

sealed, then by filling them with pure gas it should be

possible to measure the gas constant R to a few parts

in 10^

I would Hke to thank Mr. G. A. Bell and Dr. N. Bignell

for their contribution to this work.

References

[1] F. E. Jones, J. Res. Natl. Bur. Stand. 83, 419 (1978).

[2] G. A. Bell and J. B. Patterson, these proceedings.

[3] Bureau International des Poids et Mesures, Rapport BIPM

79/10, October, 1979.

[4] B. P. Kibble, Metrologia 11, 1 (1975).

439

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

On a More Precise Correction for Buoyancy and Gas Adsorption in Mass

Measurement

Y. Kobayashi

National Research Laboratory of Metrology 1-4, 1-Chome, Umezono, Sakura-Mura, Nlihari-Gun, Ibaraki 305

Japan

Using Pt-Ir Kilogram Prototypes, mass standards made of other materials, for example stainless

steel, are usually calibrated in the atmosphere. In this case, the precision is limited by various conditions of the measuring environment. In this study, the increased precision of the correction for buoy- ancy and the introduction of a correction for water vapor adsorption made it possible to decrease, by

one order of magnitude, the errors caused by the measuring environment.

Keywords: buoyancy correction; gas adsorption; kilogram prototype; mass standards; precision

weighing.

1 . Introduction

It is essential to be able to compare a Pt-Ir Prototype

with standard weights made of other materials such as

stainless steel, etc. In the International System of Units

(SI), the International Prototype Kilogram defines the

unit of mass, which is disseminated as a practical quantity through National Prototype Kilograms. In this case, comparisons are carried out in the atmosphere in order to prevent any change in mass due to diffusion of the gases

included in the Prototypes and other such effects. Consequently, in order to ensure an exact link of practical mass standards to the SI unit, one must take into ac- count appropriate corrections arising from the difference

in materials, shapes, and surface conditions between the

Prototype and standard weight. Additional corrections

for the difference between the weighing environment at the time the National Prototype was calibrated by the

Bureau International des Poids et Mesures (BIPM) and

the environment at the time of the calibration of the

standard weight in terms of the Prototype must also be

considered.

It is therefore necessary to overcome the problems

which limit calibration precision and, ideally, to eliminate

them. These problems can be classified into two

categories: precise correction for buoyancy effects; and

possible corrections for gas adsorption.

2. Precise Correction for Buoyancy

When weighing is carried out in the atmosphere, the

measured value is an apparent mass difference, which in- cludes an influence due to the conditions of the weighing

environment. Apparent mass must be converted to true mass by means of a correction for air buoyancy. This

correction can be determined on the basis of the pressure, temperature, and humidity of the air in the vicinity of the weighing (e.g., in the balance case). Such a determination carries with it 30 to 50 |jLg of uncertainty in the

buoyancy correction when comparing a Pt-Ir Prototype

and a 1-kg stainless steel weight due both to insufficient precision in characterizing the ambient air, and to the

difference between the composition of the air in the vi- cinity of the weighing and that of

'

' standard air' ' . It is impossible to improve this precision by using the above

conventional method.

To obtain better than 10 jig accuracy for the buoyancy

correction, the National Research Laboratory of Metrology (NRLM) has developed a direct measurement method

for air density and for the buoyancy correction itself.

. 2.1 Direct Measurement of the Air Density

When the true mass difference m = (mA - '"^b) between two sinkers A and B, made of stainless steel,

and the volume difference V = (Va ~ Vb) are knovra,

the density of air, da, is determined by the measurement

of the apparent mass difference M in air between the two

sinkers as follows: da = {m - M )/V (1) Figure 1 shows the forms of the sinkers, made so that

their surface areas are similar and their volume differ- ence large. Their mass difference was measured by

weighing under vacuum with a prepision balance having

knife-edge bearings [1]. The volume difference and the

coefficient of volume expansion were determined by hydrostatic weighing.

CO

N.

76.0-

////>

77.4

Figure 1. Forms of the sinkers A and B (side vieiv). The di- mensions are given in mm and they are cylindrical.

Table 1 shows the measured value of the volume and

the coefficient of expansion of the sinkers, and Table 2 shows their mass difference.

The experiment was performed by a substitution

weighing method using the sinkers. Table 3 shows the

result of a N2 density measurement which was carried out as a test of the measuring apparatus. The measured

value of N2 density was in agreement with accepted

441
Table 1. Measured volumes and expansion coefficients of the sinkers.

Sinker Volume at 20 °C

(cnr*) Expansion

coefficient Relativeuncertainty oi volume

A 343.408 44.7 X 10-6/°C 6 X iQ-^rc

B 126.890 44.9 X IQ-^rC 9 X 10-6/°C

Table 2. Mass difference of sinkers A and B a - {W + Q') - {K' + P ') determined by weighing in vacuum. 0—u — nur > n '\ i r\i\ (W + r ) - \K + Q), (2)

Time under Mass difference vacuum (mg)

(days)

83 266.417

93 266.412

99 266.416

Table 3. Measured values of the density of N2

Run No.

Conditions Density

(kg/m^)

P (mmHg)

t CC)H

(%)

As

measured

Converted

to STP

1 776.53 16.05 54 1.20243 1.25063 2 769.00 16.02 1.19515 1.25061

values and had a precision better than 1 x 10"^. These

results suggested the capability of performing a highly

precise measurement of air density, vdthout relying on measurements of the state of the air. Table 4 shov^^s the

results of the air density measurements. Repeated meas- urements of the same air had a precision of 1 x 10"^.

Table 4. Measured values of the density of air.

Run No.

Conditions Density

(kg/m^)

P (mmHg)

t CO

H

(%)

As

measured

Converted

to STP

1 763.79 16.86 32 1.22114 1.29321 2 764.72 17.34 29 1.22085 1.29328 3 757.29 17.41 55 1.20602 1.29296 4 751.31 17.28 52 1.19743 1.29310

2.2 Direct Correction of the Buoyancy

When the apparent mass difference a and b betv^^een two combinations of the prototype K, stainless steel weight W, and auxiliary pans P and Q are measured

under stable air conditions, a and b are obtained as

follows:

where W, K' , P ' , and Q

' are the apparent mass of each

of the weights and auxiliary pans. If the true mass differ- ence m = {P - Q) and the volume difference V =

(Vp - Fq) between the two auxiliary pans are known,

and the volume difference v = (Vk ^ ^w) is also known,

then the mass of W is determined by

W = K + (a + fe)/2 + [(a - 6)/2 + m]{v/V). (3) According to this method, it is possible to carry out a

correction for buoyancy which is more precise than 1 x

10"^ under the condition that v <V and the surface area

of the auxiliary pans is the same as that of the present

sinkers.

3. Correction for Gas Adsorption

Although the mass variation of mass standards caused

by gas adsorption can not be neglected, no method has

been developed to cope with it. In order to achieve

higher accuracy, it is necessary to develop a method of correction for gas adsorption so as to reduce its influ- ence. It appears that mass variation due to gas adsorption is mainly caused by water vapor. Figure 2 shows

the relation between adsorbed gas and humidity as determined by using Ni foil. These measurements, as shown

in Fig. 2, support the claim that humidity is the only important variable. In this case, adsorbed amount per unit

20 40 60 80

Humidity (%)

Figure 2. Relation between the amount of gas adsorbed and

relative humidity.

U2
area, ^, is described as follows: i^y]H + C , (4) where H is relative humidity in percent and C and I, are

constants. Table 5 shows the result of tests on a Ni sample having a surface area of 200.0 cm^ which was measured between limits of humidity of 20% to 70%

.

Table 5. Coefficients of Eq. U).

Sample t (°C)

r\ (|xg/cm2)

C

(fjLg/cm^) H

i%)

17.5 0.009 -0.03 Ni 20 to 70

27.0 0.010 -0.06

Several measurements were carried out on a Pt-Ir Prototype and a stainless steel weight. They were performed

in vacuum as well as in air of various conditions. The

results are as follows:

(S2-1) - (E59) = 95.464 mg in vacuum

(S2-1) - (E59) = 95.489 mg in air , (5)

where (S2-1) is a stainless steel 1-kg weight and (E59) is a Pt-Ir Prototype. The vacuum measurement was made

after 30 days at a pressure of less than 67 Pa; the values

in air (H = 52%) were corrected for buoyancy by means

of the sinkers described above. The difference between

the result in air and in vacuum is 25 |jLg. It seems that

this discrepancy is due to the influence of adsorption.

The equation obtained for the correction due to gas adsorption, derived from the mean values of several measurements, is as follows: Ac = (O.ODSk - (0.0092// - 0.10) AS (6) where Ac is the correction in micrograms, Sk is the sur- face area of the prototype in square centimeters, H is the

relative humidity in percent, and AS is the surface area

difference between the prototype and the stainless steel weight in square centimeters.

4. Conclusion

The increased precision of the correction for buoyancy

and the introduction of a correction for gas adsorption

have made it possible to decrease by one order of magnitude the error caused by the weighing environment.

References

[1] Y. Kobayshi and K. Uchikawa, Bull. Natl. Res. Lab. Metrol. (Tokyo) 32, 6 (1976).

443

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Density Standards—The Density and Thermal Dilatation of Water

G. A. Bell and J. B. Patterson

CSIRO Division of Applied Physics, Sydney, Australia 2070

A hydrostatic weighing experiment has been done to measure the density of water samples of differing isotopic composition from which values have been derived for the density of SMOW (standard mean ocean water). Measurements are also being made of the dilatation of water in the temperature

range 0 to 40 °C.

Key words: density standards; water; weighing.

1. Introduction

Pure water became the standard of density when the

kilogram was first defined as the mass of one cubic de- cimeter of water at its temperature of maximum density

under a pressure of one atmosphere. However the mass

of the kilogram as fabricated by Lavosier was not equal

to the mass of a cubic decimeter of water and this fact was demonstrated by the very careful work at the BIPM

of Guillaume, Chappuis, and of Mace de Lepinay, Buisson, and Benoit. All of this work is reported in the Tra- vaux et Memoires du BIPM [1]. The dilatation of water was measured in the first decade of this century in the temperature range 0° to 40 °C

by Chappuis of the BIPM [2] and by Thiesen, Scheel, and

Diesellhorst of the PTR Berlin [3]. These two sets of

results differed considerably above 15 °C and have been

the subject of numerous re-workings, the first of which was published in 1937 and the latest in 1975 [4] [5]. Although the experimental work of both Chappuis and

Thiesen was of very high quality, it is of little value to- day as the isotopic composition of the water used is not known and the relationship of the temperature scales used by them vdth the modern IPTS is not known vdth

any certainty. By the end of the I960' s it was clear that

something would have to be done to improve the density

standard and two different approaches have been made.

In one approach, measurements are being made with

the object of defining the density standard by means of a

single crystal of a solid, viz. silicon. Whether single crystals of silicon can be produced sufficiently free from impurities, vacancies and dislocations to serve as a natural

standard is not yet established.

The other approach aims at establishing pure air-free water of known isotopic composition, at its temperature

of maximum density, as the density standard, and one experiment to this end is described here.

2. Principle of the Experiment

The measurement of the density of water involves the

measurement of the mass of water displaced by a body of accurately known volume. In order that the volume of the immersed body be readily determined to a high order

of accuracy from its linear dimensions, a body of exact

geometrical form must be used. Practical considerations

limit the choice to a sphere, a rectangular prism or a

cylinder, and for this experiment a sphere was chosen as

it is much less susceptible to damage.

The sphere, which is hollow and made of a low expansion glass, has been described by Bell and Clarke [6]. The experimental set-up is shown in Fig. 1. The sphere

is suspended in a cage from the pan of a two knife substitution balance and provision is made for lifting the

sphere from the suspended cage.

3. Measurement of the Sphere

Measurements using a "Talyrond" roundness measuring instrument indicated that the departure from sphericity did not exceed 100 nm in any measured diametral

plane.

Using a He-Ne stabilized laser calibrated in terms of the krypton 86 standard as a light source, the diameter

SINGLE PAN BALANCE

PRINCIPLE OF EXPERIMENT

Figure 1. Principle of expenment.

445
of the sphere was measured by means of an interferometer based on the design of Saunders [7]. The sphere was

placed between the parallel plates of a Fabry-Perot in- terferometer and the radial separation of the spherical

surface from each plate was measured using Newton's

rings. The interferometer plates were fixed to two steel plates, the separation of which was fixed by three invar

spacing rods. Parallelism of the plates was achieved by

deforming the ends of the rods with a differential screw mechanism. The sphere and parallel plates were mounted

in a temperature controlled vacuum chamber with provision for making all the necessary adjustments externally.

The sphere could be rotated to any orientation and was

able to be lowered away from the interferometer path to

MEASUREMENT OF SPHERE

Figure 2. Measurement of sphere.

measure the separation of the plates.

Twenty small spots of aluminium were evaporated on

to the surface of the sphere at the extremities of ten di- ameters to enhance the visibility of the Newton rings and

to eliminate unwanted interference in transmitted light.

The interferometer is shown diagramatically in Fig. 2. The correction for phase loss on reflection and for the

thickness of the aluminium film was determined by meas- urement in an interference microscope.

The compressibility of the sphere was determined by

using the differences in diameter values at both atmospheric pressure and under vacuum conditions, at a temperature of 20 °C. In all, three complete sets of measurements of the

sphere diameter were made, between two of which the

interferometer was taken down and reassembled. Measurements were made at a number of temperatures in the

range 6 to 40 °C and the results fitted to an expression

quadratic in temperature. The results are shown graphically in Fig. 3.

4. Measurement of the Mass of the Sphere

The mass of the sphere was measured on a 1 kg capacity balance of which the measurement uncertainty is known to be not greater than 50 |xg. The mass was measured using the Gauss double weighing procedure with

standards knovro to within ±1 part in 10'' or ±5 ixg. The

sphere was cleaned with AR grade tri-chloro ethane and

allowed to stand in air for some hours prior to the weighing. The air buoyancy correction amounts to approximately 230 mg and was determined by measuring the

temperature, pressure, and relative humidity of the air, to calculate its density [8]. The initial value of the mass obtained in 1975 was

329.616 5 g. Subsequent measurements showed a slow in- crease in mass but the value appears to have stabilized at a value of 329.617 80 g at which it has remained constant over the past 3 years.

5. The Balance

6

5

4

3

2

1

228.518 0

9

8

7

6

5

228.517 4

1 in

28.5 18 5 .0,

o

>

n

o MEASURED VALUES

10 20 30 40 50 TEMPERATURE "C VOLUME OF ULE SPHERE

Figure 3. Volume of ULE sphere.

60

The balance used for the hydrostatic weighings was a Mettler Model H51 having a maximum load of 160 grams

and a readability of 10 ixg. The minimum increment in mass attainable with the mechanically loaded weights

as supplied was 100 mg, corresponding to a change in the depth of immersion of the suspension of several

millimeters.

To reduce to a minimum the change in the depth of immersion of the suspension between weighings of the

loaded and unloaded cage the balance was modified to in- corporate an additional two decades (1-10 mg) and (10-

100 mg) of mechanically loaded weights. By suitable

choice of the additional weights the balance was tared to produce the same rest point (within 1 mg) for both conditions of loading.

The balance was calibrated over its full range using mass standards of which the values were known with an

uncertainty of ±1 in 10^ or ±5 fjig. It is estimated that the uncertainty associated with the corrected value of any mass reading does not exceed ±5 in 10^ or ±20 ixg

whichever is the greater.

6. Temperature Measurement

The sample of water being measured was contained in either of two glass vessels, one in which the water was in equilibrium with the ambient air, and the other from

446
which the air could be extracted. The vessel used was

immersed in a 50 liter tank of water, the temperature of which was contolled by an electronic proportional controller with a resistive temperature sensor. The water in the outer tank was mixed with a propeller type mechanical stirrer. The temperature of the water sample is measured with a platinum resistance thermometer and an ac resistance bridge designed by Thompson [9]. The ther- mometer was calibrated in terms of the IPTS and regular

measurements were made of the resistance at the triple

point of water. The temperature of the water sample in the inner vessel remained constant within ±1 mK over a period of several hours corresponding to a change in the

density of water of 2x10"^ kg/m^ at 20 °C and 4 x

10-4 l^g/jjj3 at 40 °C.

7. Hydrostatic Weighing Procedure

The vessel containing the water sample was placed in the large water tank, the temperature of which had been

set at the required value. After a suitable time for stabilization of temperature, the balance readings were made

in the following sequence:

1. Balance scale reading with the cage unloaded, 2 readings Si, 52-

2. Balance scale reading with the sphere loaded on

the cage, 4 readings S3, S4, S5, Sq.

3. Balance scale reading with the cage unloaded, 2 readings S 7, S g. 4. Reading of the water temperature with the IPTS

thermometer.

The weight of the sphere in water (W) is then calcu- lated using the equation

W = (Wi - Wo + m) + (S3 + S4 + S5 + S6)/4 - (Si + S2 + S7 + S8)/4

where

Wi — tare weight on balance for sphere, cage, and

suspension

Wo - tare weight on balance for cage and suspension

m = increment weight required to equalize the bal- ance scale readings for both load conditions.

The next set of readings was taken using the values

S7, Sg as new values for Si, So so that any tendency for zero drift of the balance was compensated. At least nine

sets of readings were obtained at each temperature, during which three sets of readings were made of the temperature, pressure, and humidity of the air.

8. Results

The density pr at a temperature T °C was calculated

using the equations

PT = [M -Wa - (T/D){gw/gs)VVT , where

M is the mass of the sphere and Vt its volume at T °C,

W is the observed weight of the sphere in water, a is the air density and D the density of the weights,

Qs is the gravitational acceleration at the level of the

sphere,

gw is the gravitational acceleration at the balance

weight level,

and

is the measured density of water at temperature

r°c,

p, = 999.974 5 -

pr^ = pHl + oiTiT - Tom + Pr(Po - P)],

where

To is the reference temperature and {T - To) ^

o.orc,

OLT is the thermal dilatation of water at temperature T,

Pt- is the compressibility of water at temperature T, Po is the reference pressure (101325 Pa), and

P is the measured pressure (Pa).

Measurements were carried out on various samples of water at different temperatures, either in equilibrium

with the air, or air free, and of different measured isoto- pic compositions.

After adjusting the absolute values to SMOW (standard mean ocean water) values by using the data of Menache

[10], the results below were obtained.

The temperature at which the density of water is a maximum under a pressure of 101 325 Pa (760 mm Hg) is 3.989 ± 0.004 °C. The density of SMOW at the temperature of maximum

density and a pressure of 101 325 Pa is 999.974 5 ± 0.0010 kg/m^

The equation for the density of SMOW in kg/m^ as a function of temperature at 101 325 Pa is

{t - 3.989)^(^ + 335.1596)

558.7666(i + 72.2553)

It is estimated that in the temperature range 0° to 40 °C the uncertainty 1 1 associated with the value of pt is given by

C t = ±(0.001 + 0.000 03 X \t - 4 I )

.

These results are quoted as provisional results and the

uncertainties quoted for the density of SMOW at its temperature of maximum density and for the value at temperature t make allowance for the possibility of a sys- tematic error in the diameter of the sphere. A new determination of the diameter of the sphere is in progress and

the results will be reviewed when this work is completed.

References

[1] C. E. Guillaume, Bur. Int. des Poids et Mesures Trav. et Mem.

14, 1 (1910).

[2] M. P. Chappuis, Bur. Int. des Poids et Mesures Ti'av. et Mem. 13, Dl (1907).

[3] M. Thiesen, Physik Tech. Reichs, Wiss. Abhandl 3.1; 4.1

(1900).

[4] L. W. Tilton and J. R. Tavlor, J. Res. Natl. Bur. Stand. 18, 205

(1937).

[5] G. S. Kell, J. Chem. Eng. Data 20, 97 (1975).

[6] G. A. Bell and A. L. Clarke, in Atomic Masses and Fundamental Constants 5, Ed. by J. H. Sanders and A. H. Wapstra (Plenum Pub. Corp., New York, 1976), p. 615.

[7] J. B. Saunders. J. Res. Natl. Bur. Stand. Sect. C: 76. 11 (1972).

[8] F. E. Jones, J. Res. Natl. Bur. Stand. 83, 419 (1978).

[9] A. M. Thompson and G. W. Small, Proc. lEE 118, 1662

(1971).

[10] M. Menache, C. Beauverger, and G. Girard, Annales Hydrographiques 6, 41 (1978).

447

Precision Measurement and Fundamental Constants 11, B. N. Taylor and W. D. Phillips, Eds.

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Precision IVIeasurements on Solid Artifacts for a Redetermination of the

Density of Water

A. Peuto, A. Sacconi, R. Panciera, W. Pasin, and M. Rasetti

Istituto di Metrologia G. Colonnetti, Strada delle Cacce,73 -10135 Torino, Italy Mass and volume of four spheres made of low thermal expansion (a — 10"^/°C) glass-ceramic were

determined. Mass was measured against stainless steel standards, with ± 2 x 10"^ relative uncertainty. The volume of two of the spheres was obtained through correlation of measured diameters and

roundness data. Diameters were measured with a two-step interferometric method. Roundness data were taken on nine sections 20° apart. Estimated uncertainty of volume values is ± 1 x 10~^. The

volume of all four spheres, at 20 °C, was also measured by hydrostatic weighing in water with ± 3 x

lO"*^ relative uncertainty. The comparison between volumes determined through both methods yields results in agreement within 1.5 x 10"^. Volume ratios agree within 4 x 10""^. Key words: density of water; density standard; hydrostatic weighing; interferometric measurements;

spherical volume; volume measurements.

1. Introduction

The inadequacy of water as a density reference standard has stimulated research work in order to establish a

solid density standard [1-3]. Characterizing such a standard implies interferometric measurements of its linear dimensions, precise mass determination and hydrostatic weighing. The solid chosen should possess simple geometrical shape, mass dimensional stability, low

coefficient of thermal expansion, and mass and density

close to the reference standards. In order to find artifacts meeting the above, often contrasting, requirements, a compromise must be made. The material

selected was Zerodur, a glass-ceramic with a = 10"^/°C

and inert in water and air. Four spheres of Zerodur were

used (henceforth designated as SP, Si, S2 and S3) of 90 mm dia. and 1 kg mass each.

All data were taken at a temperature within ± 1.5 °C

of 20 °C and no correction was made for thermal expansion of the spheres. The difference between the phase

change on reflection from Zerodur and that from the

etalon plates (quartz) was considered to be negligible.

The lasers used were wavelength calibrated against an

iodine (^^'^12) stabilized He-Ne laser with an uncertainty

better than ± 1 x 10"^. All individual uncertainties, ex- pressed as standard deviations, are combined in quadrature, unless otherwise specified.

2. Volume Determinations through

Dimensional Measurements

For dimensional measurements only SP and S2 were

examined. In order to determine the diameter, inter- ferometric techniques were used. Measurements were

performed in two steps. First, diameters were measured

by interferometric counting and mechanical contact (subsequently referred to as "first step" measurements) giving the exact integral part of the fringe order number. In

the second stage the corresponding fringe fraction was

obtained from single-wavelength interferometry. The initial estimate of the diameter was thus corrected to obtain a value with a smaller uncertainty, the correction

depending on the initial estimate and the measured frac- tional fringe. Repeating the same procedure for different

diameters, a mean diameter can be obtained and the

volume determined. This method suffers from tedious

alignment procedures and stability problems characteristic of a triangular interferometer. To avoid these and

also to have a better knowledge of the shape contribution, a method resorting to roundness measurements was

adopted in this work. Roundness measurements yield

the variations in the diameter rather than its absolute

value. Nevertheless it is sufficient to know a single diameter of the explored profile for obtaining a mean diameter representing that profile. Then from the measured di- ameter and the roundness data for a family of planes

passing through this diameter, the average diameter

representing the sphere can be obtained.

First step measurements were made using a Universal

Measuring Machine (Moore M18) equipped with a laser in- terferometer (HP 5500). A linear variable differential transformer (LVDT) bidirectional transducer (GARY I- DIM, with a resolution of 0.01 \x.m) was used as a null detector. To limit systematic errors arising from the

mechanical contact of the feeler, its constant (spherical

tip dia. and bending of the stylus) was taken into account by measuring, in a symmetrical cycle, the length of a dummy gage (interferometrically calibrated gage block).

The reproducibility in a set of 20 measurements was

represented by a standard deviation s = 0.02 iJim and

the systematic uncertainties (due mainly to: the dummy

gage-block calibration, geometrical and mechanical

corrections, instrumental errors in pressure-temperature-humidity measurements, and air composition) bv an

s =0.03 |j.m". The fringe fractions were measured with an inter- ferometer (Fig. 1) similar to that used by Saunders [4] but the fringes were scanned and detected with a rotat- ing mirror and a PMT [5]. The interference ring diameters were then least-squares fitted to yield the fringe fraction. In the relation D = E - (di + ^2), given thf integral part. Id, of the order number of D, the fringe

fraction, fo, can be determined from the fractional

fringes fs, f\, and f-z only, as fo = fs - ^Cfi + /2),

449
PUT

CZ!-

SF X:633 nm

D-.E-(d,+d^)

Figure 1. Optical layout of the interferometer for fringe fraction measurem,ents. PMT, photomultiplier tube; SM, scanning mirror; BS, beam splitter;

Ml, Mi, mirrors; E, etalon length; D, diameter under examination; L, focusing lens; SF, spatial filter; s, screen; m,, to^. auxiliary mirrors.

where F indicates the fractional part, the knowledge of the integer Ie being unnecessary.

Each series of measurements consisted of eight runs. A

sequential scan in which beams 1 and 2 were interrupted

in the order 2, 1, 1, 2 was adopted. After measuring /i

and /2, the sphere was removed and the same procedure

repeated for the etalon alone. The fs's were then re- duced to the same environmental conditions as for / 1 and

/2 and a single D computed for each series using

the relation D = Ho +/z))Xa/2, \a being the laser wavelength in ambient air. In order to randomize the

systematic errors arising from the optical alignments of

the interferometers and of the sphere, the system was

aligned for each series. The results are summarized in Table 1. Estimated systematic uncertainty was ± 0.01 [xm. Repeatability tests (A^ 20) on fg and on

if 1 + /2) yielded an s of 0.003 and 0.01 fringe fractions,

respectively.

Table 1. Results of the diameter measurements obtained from

the two-step method.

Measured Standard Number

diameter deviation of meas. Sphere D (fJLm) S (fJLm) N

SP 89 703.158 0.030 18 S2 90 389.326 0.026 9

It is worth mentioning that first step measurements

show a systematic difference from these results of about

0.1 (Jim, which exceeds the previous estimated uncertainties. This discrepancy may be due to some unidentified source of error. However the root-sum-square combination of uncertainties might be inadequate for assessing

the reproducibihty of measurements performed by dif- ferent methods.

For roundness measurements the previously mentioned

Universal Machine was used. The sphere was rotated

around the measured diameter in equal steps by means of a specially designed mechanism equipped with an incremental encoder. A set of nine profiles, spaced 20° apart, was investigated: Fig. 2 is a typical profile with a radial magnification of 100 000. Eccentricity terms were calcu- lated and subtracted from the transducer signal using the

limacon approximation as criterion for best fit [6]. Each

profile was tested for eight revolutions and time aver- aged to smooth accidental errors. Time sampling of the

profile was taken over 625 angular points per revolution.

Analysis of the data yielded the mean diameter for each

section. A reproducibility test carried out for eight meas- urements of eight runs each, resulted in a standard deviation of 0.01 (xm for a single measurement of the mean

diameter, the std. dev. for maximum diameter variations

(AZ)) being 0.006 |xm. Systematic uncertainty is due

mainly to the spindle errors which, when evaluated by

the reversal method [7], had an s of 0.014 fxm on AZ)

evaluation.

The average diameter Day, i.e., mean diameter aver- aged over the nine profiles, is given by:

(Dav - i5meas)sP = +0.054 ^JLm

(Z)av - -Dmeas)s2 = "0.184 Jim.

200.

-200 180°

200.

-200 360°

Figure 2. A typical roundness profile. A/i and AD are max- imum radial and diametral variations respectively. Both polar

and cartesian plots are given.

450
From these relations and Table 1, the average diameters

of SP and S2 are 89 703.212 jjtm and 90 389.142 ^JLm, respectively. Random uncertainty of each £)av is given

by s = 0.03 |jLm; the systematic uncertainty is given by

s =0.02 |xm. The corresponding volumes are given in Table 3. It is worth mentioning that the standard deviations of the distributions of the nine profiles giving the mean

values (Z)av ~ -Dmeas) are unequal for SP and S2: sgp =

0.02 |jLm and Ss2 = 0.05 ixm. This indicates different

geometrical quality (i.e., deviation from perfect spherical

shape) of the two spheres and can be compared with the max. diametral variation (0.25 ixm and 0.8 jjim, respectively) measured on single profiles.

As a further check of the method outlined above, a second family of nine profiles around another measured

diameter was investigated for the same two spheres.

This should yield the same average diameter as that

obtained previously. Moreover, from the roundness

measurements the difference between the two measured

diameters (A£>/e) can be obtained and compared with the

difference measured directly (A £>>/). For the latter we

had only the data from first step measurements, as only one diameter of each sphere was measured with the tri- angular interferometer. This however poses no problem

as only the differences are of interest here. This comparison is substantially unaffected by any differences in the

geometrical quality of the spheres. The maximum deviation {^.Djt - ^Dm) of 0.03 |xm obtained in our case indi- cates a good agreement between the roundness and

direct measurements in addition to the validity of the

former.

3. Mass Measurements and Hydrostatic

Weighings

The mass of the sphere SP was 959 g, that of SI, S2,

and S3 being 979 g each. The masses were measured by

comparison against a 1 kg stainless steel standard, cali- brated with an uncertainty of ± 100 [xg. For this comparison a polished 20 g Zerodur support (calibrated

within ± 20 ixg) was used for the spheres. Stainless steel standards were then added to equalize loads. A singlepan damped balance of 1 kg max. capacity was used. Its sensitivity, approximately 10 div/mg, was determined at each weighing. Care was taken so that the temperature

in the balance case did not vary by more than 10 mK during weighings. The standard deviation over one weighing

cycle was typically 50 ixg.

The air buoyancy effect on the spheres (density

2 500 kg • m"^) differs from the effect on the 1 kg standard

by about 300 mg. Air density thus becomes critical; it was

evaluated using the relation given in Ref. [8]. The

uncertainty of the air buoyancy correction is estimated^

to be ± 180 |jLg; the overall uncertainty of the mass meas- urements amounts to 200 |xg.

The volumes and the ratios between volumes of all four

spheres were determined by hydrostatic weighing, taking

water as a reference standard. The balance was identical

to the one used for weighings in air, modified so that the

load could be suspended under the pan. A stainless steel device allowed three spheres to be weighed in the same

water bath. It could rotate and move vertically, letting

the sphere under test rest on a hook-shaped suspension

attached to the balance. The suspension wire (dia. =

To check this estimate of air buoyancy uncertainty, several weighings were performed using different instruments for measuring environmental

variables. The experimental standard deviation was found to be 150 )i.g.

0.25 mm) was made of stainless steel covered with platinum black. The water was de-ionized and bi-distilled in a quartz apparatus and subsequently degassed at low pressure. Tests for conductivity and dissolved oxygen were

carried out before and after each series of weighings. For

volume computation (not for ratio determinations) those

weighings conducted in water of conductivity greater

than 0.3 mS • m'^ were disregarded. The temperature of the bath was measured with two platinum resistance

thermometers and an ac double bridge (Automatic Systems Lab., 7 decades). Bigg's dilatation tables [9] were

adopted for the evaluation of the density of water. Corrections for isotopic composition [10] and dissolved

gases were applied according to the relations of Ref. [11]. Four series of weighings, each involving three spheres, were performed, following different weighing sequences.

Each sphere was usually weighed eight times per series. The standard deviation over one series was typically

150 |jLg (4 X 10"^ on volumes). The masses and volumes

of the four spheres are summarized in Table 2. Typical

reproducibility (A'' = 20) is represented by an s =

180 [xg (5 X 10"'''). The systematic uncertainty of the

volume measurements was estimated to be ±1.3 mm^

(3.3 X 10"^).

Table 2. Results of mass measurements and hydrostatic weighings.

Mass s Volume s Sphere (g) N (fxg) (cm^) N (mm^)

SP 958.567 66 8 100 377.939 22 15 0.16 SI 979.351 39 8 150 386.521 87 10 0.20 S2 979.682 15 7 80 386.675 59 19 0.07 S3 978.886 91 8 140 386.356 51 19 0.18

Volume ratios obtained from weighing sequences Si, Sj, Si in the same water bath do not differ from ratios computed from average volumes by more than 4 x 10"^. The comparison between volumes obtained through di- mensional measurements and hydrostatic weighings is shown in Table 3. The agreement of the results lies well

within the uncertainty of the hydrostatic weighings, the

difference being 1.5 x 10~^. The ratios V'sp{^s2 obtained

by the two methods differ by only 3.5 x 10"", account being taken only of those volume ratios obtained from

direct hydrostatic comparisons, i.e., from comparisons in the same bath.

Table 3. Comparison of volumes obtained from diameter measurements with those determined by hydrostatic iveighings.

Volume from

dimensional meas.

Difference from Vh

Vd - Vh

Vh

Sphere Vd (cm^) (cm^)

SP 377.939 78 + 0.000 56 + 1.5 X 10~^

S2 386.676 18 + 0.000 59 + 1.5 X 10"^

From the mass values (Table 2) and dimensionally

determined volumes (Table 3) the densities of SP and S2

are found to be 2536.301 kg • m'^ and 2533.750 kg • m'^,

respectively. The difference between these values is not surprising since the spheres come from different lots of

451
Zerodur. It should be noted that the aim of this experiment was not to pursue a "natural" standard. In our

case only the objects and not the material were to be

characterized as standards. The uncertainty of the density values is ±3 x 10"-^ kg m"^ (1.1 x 10"*^).

4. Conclusions

The reliability of a roundness-based representation of the volume of the spheres was demonstrated experimentally. The intercomparison between dimensionally and hydrostatically determined volumes was shown to yield results in quite good reciprocal agreement. Since different

measurement techniques were involved in this intercomparison, it can be concluded that (within ± 1 x 10"^ limits)

there exist no significant systematic errors in the adopted

methods. Furthermore, the excellent agreement in volume ratios between the two methods assures the feasibility of a dissemination program based on hydrostatic

weighings of a series of secondary density standards.

References

[1] H. A. Bowman, R. M. Schoonover, and C. L. Carrol, J. Res.

Natl. Bur. Stand. Sect. A 78A, 13 (1974).

[2] G. A. Bell and A. L. Clarke, in Atomic Masses and Fundamental CoJistants 5, Ed. by J. H. Sanders and A. H. Wapstra

(Plenum Press, New York, 1976), p. 615.

[3] K. lizuka et al. in Acta IMEKO 1979, Ed. by Gy. Striker (Aka- demiai Kiado, Budapest, 1980), Vol. 3, p. 827.

[4] J. B. Saunders, J. Res. Natl. Bur. Stand. Sect. C 76C, 11

(1972).

[5] A. Sacconi and S. Fontana, Metrologia 11, 33 (1975).

[6] D. J. Whitehouse, J. Phys. E. 6, 921 (1973).

[7] R. R. Donaldson, CIRP Ann. 21, 125 (1972).

[8] BIPM Report 79/10, Sevres (1979).

[9] P. H. Bigg, Br. J. Appl. Phys. 18, 521 (1967).

[10] G. Guzzi, Centro Comune di Ricerca, Euratom, Ispra, private communication.

[11] M. M.. Menache, M.. C. Beauverger, and M. G. Girard, Ann.

Hydrogr. 6 (3), 37 (1978).

452
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

The Influence of Dissolved Air on the Density of Water

Georges Girard and Marie-Josette Coarasa

Bureau International des Polds et Mesures, F-92310 Sevres, France

A knowledge of the density of water to an accuracy of 1 x 10"^ kg-m~^ (1 part in 10^) has been of metrological importance for a long time. The basic work on the density of water dates from the beginning of the century and at the time included the absolute value itself together with the variations

of density as a function of temperature and of amount of dissolved air. In recent years a number of laboratories, BIPM among them, have once again taken up this work.

At BIPM, studies have been undertaken of the influence of variations in isotopic composition on the

density. A relation has been derived which is valid for all samples of water likely to be encountered in any of the various national metrological laboratories. Recent work at BIPM has been devoted to the

question of the effect of dissolved air on the density. The density of samples having different levels of saturation of dissolved air has been measured at various temperatures between 4 and 22 °C. The levels of saturation of dissolved oxygen were within the range 0.1 to 0.95. Key words: atmospheric dissolved gases; compressibility; density of water; fundamental value; hydrostatic weighing; isotopic composition; thermal expansion.

1. Introduction

Pure water is used for precision measurements of density and volume. The knowledge of its density depends

upon a number of factors, the majority of which are not known with sufficient precision:

(a) the fundamental value: the accepted value,

999.972 kg-m~^, is the density of air-free water at 4 °C

under a pressure of 101 325 Pa, determined between

1895 and 1905, a time when the existence of isotopes was not suspected;

(b) the thermal expansion coefficient: the available data

between 0 °C and 40 °C are not consistent, particularly above 16 °C;

(c) dissolved atmospheric gases: their influence is not

sufficiently well knowTi;

(d) isotopic composition: this factor was not studied until recently;

(e) compressibility: the only factor that is known,

around a pressure of one atmosphere, with sufficient precision.

2. Redetermination of the Density

In 1967, Menache drew attention to the fact that for modern requirements, particularly in physical oceanography, there was a need for a better knowledge of the density of water. At that time it was not known to better

than 0.01 kg-m"^ due to the cumulation of uncertainties

stemming from different parameters.

In 1969, the Joint Panel on Oceanogi'aphic Tables and

Standards (sponsored by the International Association

for the Physical Sciences of the Ocean (lAPSO), the

International Council for the Exploration of the Sea

(ICES), UNESCO, and the Scientific Committee on Oceanic Research (SCOR)), prepared a Recommendation that was adopted by lAPSO. The International Union for Geodesy and Geophysics (lUGG) made a recommendation for a new- study of the density of water, of which the text was passed to the Comite International des Poids et Mesures (CIPM).

In parallel. Commission 1.4 (Commission on Physicochemical Measurements and Standards) of the International Union of Pure and Applied Chemistry (lUPAC)

had been looking at the problem since 1965. In the course of successive meetings, in particular those of 1969

and 1971, this Commission accepted concrete proposals

from J. Terrien, at that time Director of BIPM. In 1973

it prepared a Recommendation on the re-determination of the density of water and the study of the various determining factors, which was sent to a certain number of national standards and oceanographic laboratories. In about 1972 preliminary work began in various laboratories and in 1975 IUCjG approved the Recommendation

of lUPAC.

2.1 Fundamental Value—Thermal Expansion

Coefficient

At present, work on the fundamental value and ther- mal expansion coefficient is in progress in Australia,

Italy, Japan and the Federal Republic of Germany.

Briefly, it consists of a determination of the volume of a body by means of geometrical measurements, weighing

the body in air, and then, by hydrostatic weighing,

determining the density of the sample of water in which

it is immersed. If these hydrostatic measurements are repeated (using the same sample of water) at different

temperatures, the thermal expansion coefficient can be

obtained. This same parameter can also be studied by

using a solid body having a very well knowTi thermal ex- pansion coefficient, and carrjdng out hydrostatic weighings at a number of temperatures in the chosen range.

2.2 Ck)mpressibilfty

For the compressibility of water we already have an

equation that is perfectly satisfactory [1].

2.3 Isotopic Composition

The influence of the isotopic composition on the density

has been studied theoretically by Menache who, in 1967,

453
published a formula relating the density to its content of oxygen-18 and deuterium. A later, more complete equation was pubUshed in 1971 [2]. The isotopic composition of ^^0, ^^0 and D of a sample

is compared with that of a reference water which is taken, by the specialists, as a standard of isotopic composition: Standard Mean Ocean Water, now called ViennaSMOW (V-SMOW). With respect to V-SMOW, the rela- tive departures of the density, 8i8 and 8d, for a given

sample of water are defined by the following expressions:

(sample)

[^^0] (V-SMOW)

(V-SMOW)

8d

(sample)

[D]

[H]

(V-SMOW)

[D]

[H]

103 (1)

10=^ (2) (V-SMOW)

where [^^0], [^^0], [D] and [H] are the concentrations

(amount of substance divided by volume) of the different

nuclides present in the samples.

The effect of the isotopic composition on the density of

natural waters is given by the follovdng formula, valid

between 0 °C and 30 °C:

(Psample " Pv-SMOw) X 10^/(kg-m ^)

0.233 8i8 + 0.016 6 8d ±0.1

where 8i8 and 8d are defined above in Eqs. (1) and (2). This theoretical formula has been confirmed by experimental work at BIPM, the preliminary results of which were presented at the lUGG General Assembly in Grenoble in 1975. The measurements were made by weighing a body, of known thermal expansion coefficient, in the

various samples of water of known isotopic composition, over a range of temperatures and pressures. From these

measurements the buoyancy and hence the density of the

water may be deduced.

The samples of water used in these measurements came from: (1) melted Antarctic ice (one of the lightest of the natural waters); (2) the Mediterranean; (3) BIPM tap water. After double distillation, the difference between

the maximum and minimum densities was found to be

19 X 10-3 kg-m^l

3. Influence of Dissolved Atmospheric

Gases

Marek [3], in 1891, was the first person to attempt to measure the difference in density between a sample of

air-free water and the same water saturated with air at the same temperature. Between 0 and 14 °C he found

that this difference was of the order of 3 x 10"^ kg-m'^

and diminished at higher temperatures. Chappuis [4], in 1910, made similar measurements between 5 and 8 °C

(3)

Figure l. The BIPM hydrostatic balance. C, upper enclosure containing the balance; C, lower enclosure containing the hydrostatic bath V; P, two stage pan; E, scale; L, projector; 0,

control console.

Figure 2. Detail of the hydrostatic bath and the suspension

and measurement mechanism of the cylinder in the water sample. Z, cylinder of Zerodur; E, stirrup; F, stirrup suspension

unre; B, cradle; A, stirrer; T, entry of temperature-controlled

water; R, recipient.

454
and obtained results very close to those of Marek.

The difficulties with Marek' s results are that: first, they refer to air-free and saturated states that were al- most certainly far from being realized; second, they pro- vide only the difference in density between two extreme

conditions; and third, they imply the notion of "dissolved

air" that was not sufficiently well defined.

We have therefore repeated the work of Marek, but

taking all the precautions, and using the equipment

(Figs. 1 and 2), that was developed during our work on

the variation of density with isotopic content. The

method is similar: the variation in the buoyancy is measured for a body totally immersed in water for which the

quantity of dissolved atmospheric gas is varied. These

measurements were repeated at various temperatures.

The quantity of dissolved gas was determined from samples taken from the water just after each hydrostatic

weighing.

3.1 Hydrostatic Measurement of the Density of a Sample of Water

Figure 3 shows the two phases, A and B, of a hydrostatic weighing. In Fig. 3(a), the tare mass t balances

the standard of mass and the ensemble of the stirrup

and suspension wire of mass K. In Fig. 3(b), the equilibrium with the tare mass is re-established by the body of mass mi, replacing the standard of mass m2, the immersed cylinder of Zerodur (mass m, volume V, density

p), now being suspended on the pan of the balance.

The equations at equilibrium are as follows:

t = mad - a/k) + K (4)

t = m - Vp + mid - a/X) + K (5) from which

m - Vp - (m2 - mi)(l - a/X) (6)

Figure 3. Schematic showing the use of the hydrostatic bal- ance. The masses of the auxiliary pans P, and Pj are elim- inated by a second weighing in which masses m., and m, are

associated respectively with Pj and P,.

hence

p = [m - (m2 - m])(l - a/k)W H)

where a is the density of air and X. that of the bodies of mass mi and m2. A complete weighing comprises three

weighings under conditions B and two under conditions A. The temperature of the water, at the level of the

Zerodur cylinder, is measured by means of a platinum

resistance thermometer.

Particular care is taken in the degassing of the water. This is carried out by maintaining the doubly-distilled

water for three hours at a temperature of about 68 °C

under the corresponding partial pressure, about 28 kPa,

so as to eliminate all traces of air. In preparing for the

weighing, with the Zerodur cylinder and all its supports

in place, the pressure in the water bath is reduced to a few hundred pascals so that the air-free water can be

transferred to the water bath without air being added.

This low pressure is subsequently maintained for many

hours in order to remove all traces of air from the walls. The following morning, after having let air into the bath,

the absence of air bubbles is checked, a process made

easy by the transparency of Zerodur. The first weighings

are made as soon as the temperature is stabilized (i.e., two hours later) with only a very small amount of air dis- solved in the water. The measurements are then pursued

for a number of days, the quantity of dissolved air slowly

increasing, either naturally or with external assistance.

3.2 The Measurement of the Quantity of Dissolved

Air

In a study of the solubility of the principal atmospheric

gases in sea water (pure water can be considered simply

as sea water having a zero salinity), Menache et al. [5] show that for the four atmospheric gases, O2 N2, Ar, and

CO2, argon and carbon dioxide have very low solubilities for which the sum does not exceed 4% by volume of the

solubilities of the four together. In a parallel study on the rate of solution of the same

gases, made under conditions similar to those existing in our hydrostatic weighings, Menache et al. [5] show that

between 4 and 22 °C the levels of saturation x (N2) and

X (O2), as a function of time of exposure to air, are practically identical. We can thus write, with a good accuracy: X (N2) - X (O2) (8) from which:

cm ^ S (N2)

cm s(02)'

where for a given sample C (N2) is the content of nitro- gen and s (N2) is the solubility of nitrogen or the content

of nitrogen corresponding to the partial pressure p(N2)

of the gas at the time of the experiment.

It is therefore possible, as in the present case when

the sample is not saturated, to limit the analysis to a chemical titration for oxygen. The level of saturation of oxygen alone may then be calculated, which will represent the level of saturation of the sample for dis- solved atmospheric gases. The chemical titration of oxy- gen is carried out by the Winkler method as improved by

Carpenter [6]. This titration is conveniently made with

the 50 cm^ samples taken at the end of each weighing. It was possible to deduce that when air was let into the

hydrostatic bath, the level of saturation was of the order

of 0.03.

455
4. Results^

From the analysis it was possible to associate with

each hydrostatic weighing, which gave a relative value

for the density of water, a value for the quantity of dis- solved atmospheric gas. Measurements were made at temperatures of 4, 10, 16, and 22 °C. Account was taken

of the variation in the isotopic content of the samples

during the course of the weighings. Figures 4 and 5 give

examples of the result obtained at 4 and 22 °C. One can

Figure 4. Variation, at k °C, of the density of a sample of water as a function of dissolved atmospheric gas.

Figure 5. Variation, at 22 °C, of the density of a sample of water as a function of dissolved atmospheric gas.

Note added in proof: In view of certain defects subsequently found in the platinum resistance thermometer used to measure the temperature of the water in the hydrostatic bath, the results given here at 22 °C should have twdce the uncertainty given in the text.

deduce that the variation of density is proportional to the

quantity of dissolved gas at all temperatures studied.

Figure 6 shows the variation of the difference in density between air-free and fully saturated water as a function of temperature. It is planned to carry out further

measurements at other temperatures. The values of Marek and the mean value of Chappuis are also given in the Figure.

5 +

Figure 6. Variation, as a function of temperature, of the difference in density between air-free and fully saturated

water. • , this work (mean value mth uncertainty shown at the 3 CT level); M, curve given by Marek; C, mean value of Chappuis.

We would like to thank M. Menache for helpful advice.

References

[1] G. S. Kell, J. Chem. Eng. Data 15, 119 (1970).

[2] G. Girard and M. Menache, Metrologia 7, 83 (1971).

[3] W. J. Marek, Ann. Phys. Chem. 44, 171 (1891).

[4] P. Chappuis, Trav. Mem. Bur. Int. Poids Mes., 14, Dl (1910).

[5] M. Menache, G. Beauverger, and G. Girard, Ann. Hydrog. 6, 37 (1978).

[6] J. H. Carpenter, Limnol. Oceanogr. 10, 135 (1965).

456
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

The Helium Melting Curve and the Linkage of Fundamental Constants,

Pressure, Density, and Mass

Craig T. Van Degrift

Temperature and Pressure Measurements and Standards Division, National Bureau of Standards,

Washington, DC 20234

Recent progress made toward the calculation of the ground states of liquid and solid helium from

first principles suggests that a future pressure standard might be based on the calculated phase di- agrams of ^He and ^He. The absolute reference pressures of the zero temperature intercepts and

minima of the melting curves of these systems would be transfered to room temperature gages and pro- vide a new connection between fundamental constants and density or mass. Already, present experi- ments suggest that the helium melting curves can provide pressure reference points which are precise vnthin 1 ppm. A summary of the current state of theoretical work is given followed by a discussion of the practical difficulties in realizing the linkages between the melting curve and room temperature

pressure, mass, and density.

Key words: density; fundamental constants; helium; many-body theory; mass; pressure; quantum fluids; quantum solids.

1. Introduction

The helium melting curve is uniquely well-suited to be

used as a set of pressure reference points. Figure 1 shows the region below 1.5 K of the melting curves of the

two stable isotopes of helium—^He and "^He. Both curves

exhibit minima (B and D) and have non-zero intercepts

(A and C) at zero temperature. For a given isotope, the

region above the curve is solid and the region just below

is liquid. Not shovra in the figure are the superfluidordinary fluid and liquid-vapor curves. There is no ordi- nary triple point, as the liquid-vapor equilibrium curve never intersects the melting curve. The liquid-solid curves are so easily measured by low temperature physicists that the region between A and B on the ^He curve

is now used for thermometry [1]. Pressure transducers,

necessary for these measurements, work particularly well at low temperatures because mechanical creep, ther- mal expansivity, and thermal relaxation times of materials are generally many orders of magnitude smaller than

at room temperature.

In this paper, we discuss the key linkages which must

be mastered to fully utilize the potential standards applications of these melting curves. We first discuss the most

difficult linkage—^the theoretical calculations of these

pressure points from fundamental constants. We then

describe the relatively easy task of transferring the melting curve pressures to a room temperature piston gage.

Finally, we briefly summarize the present limitations in manometry and piston gage technology and hence in the

linkage of pressure to density or mass.

It is perhaps optimistic to expect to achieve 1 ppm ac- curacy in all linkages by the turn of the century; theoretical calculations of helium melting pressures are

currently in error by as much as 7% but would contri- bute to pressure standard work only when performed at the 1-5 ppm level. However, the transfer between the

melting curve and room temperature can be expected at the 1 ppm level with straightforward improvements in present technology. The linkage between pressure and

density of mercury is expected to be done at NBS at the

2 ppm level within a year, while the linkage to mass may

reach 2 ppm within a decade when foreseeable improvements are made in dimensional metrology and the

theoretical modeling of piston gage physics. Thus, rather

substantial improvement in the accuracy of theoretical

calculations is needed to provide absolute helium pres- sure standards.

Even in the absence of an accurate theory, however, the

linkage between the melting curve and room temperature

piston gages can already be useful for transferring and

maintaining the accuracy of pressure measurements.

Although the best current piston gage calibration [2], per- formed at 0. 1 MPa against the density of mercury, has an

inaccuracy of 5 ppm, the best working standards are two

to five times poorer in accuracy. At present, the hehum

melting curve and associated low temperature transducers can provide two reference pressures (points B and C in Fig. 1) with an overall precision of about 1 ppm. These

5

4

(0 , Q. ' s

s

^ 3

(A»

a. 2

1

0

1 1 1 1111/

A

Solid •^^^^

''He Melting Curve ? ® C D

Liquid

1

1 1 1 • 1 1 0 200 400 600 800 1000 1200 1400 Temperature (mK)

Figure 1. The melting curves of "He and ^He below 1.5 K.

Points A and C are the zero temperature intercepts ayid points

B and D are minima.

457
points can thus be used immediately to maintain piston

gage reproducibility at these pressures. Such a calibration

method would help to compensate for the fact that piston

gage precision is better by more than a factor of ten than

its accuracy [3].

2. Linking Fundamental Constants to tlie

Tielium IVIelting Curves

The accurate calculation of the thermodynamic properties of a condensed substance from first principles should no longer be considered impossible. Helium-4, being a noble gas with no nuclear spin, may be the object of the

first such calculation with the zero-temperature intercept

of its melting curve serving as a convenient test point.

The problem naturally separates into two similar manybody problems: (1) the calculation of inter-atomic potentials from the interaction of pairs or small clusters of

alpha particles with each other and their electrons; and

(2) the calculation of the thermodynamic properties of an ensemble of atoms interacting via those interatomic

potentials.

The theoretical calculation at zero temperature Mall give the pressure of point C in Fig. 1 primarily in terms

of the follovdng equivalent combinations of microscopic

constants

r, ahc , /IS P oc Gc — cc he —Y , (1)

where e is the charge on an electron, cq is the permittivity

of free space, ao is the Bohr radius, h is Planck's constant, c is the speed of light, a is the fine structure constant, and is the Rydberg.

The first expression in Eq. (1) may be readily recog- nized as the electrostatic energy per unit volume. The

calculation also requires quantum mechanical corrections

of 20% involving the ratio of zero point kinetic energy to the binding energy and thereby bringing in the electron- to-alpha particle mass ratio [4]. At eleviated temperatures, evaluation of kT will be required with quantum

corrections proportional to the ratio of the thermal

wavelength, /j/V2tt MkT , divided by the cube root of the mean atomic volume, ~ ao, where M is the atomic mass.

In recent years considerable progress has been made

toward the formulation of a practical mathematical algorithm which is capable of accurately calculating thermodynamic properties [5]. The method is called the Green's

function Monte Carlo (GFMC) method. For a given in- teraction potential, it is claimed to have an accuracy

limited only by computational effort. The rapidity of con- vergence of the algorithm depends upon the choice of a

function called the "importance function," which approximates the many-body wave function. An imperfect choice

results in a slower convergence to the same final solution. The determination of inter-atomic potentials is pres- ently the weakest part of the calculation. The heliumhelium potential traditionally used is the Lennard-Jones

(12-6) potential with its adjustable coefficients fitted to the second virial coefficients of helium gas. Aziz et al.

[6] have recently proposed an improved potential less dependent on experimental data. In calculating the sec- ond virial coefficients and thermal conductivity for hehum gas, they achieved agreement nearly within ex- perimental error. Potentials describing the interaction of more than two helium atoms, important in the condensed

phases, are just beginning to receive attention.

Whitlock, Ceperley, Chester and Kalos [7] have used

the GFMC method with a Lennard-Jones potential to calculate the pressure dependence of the energy, density

and velocity of sound at zero temperature for Hquid and

solid ''He. They calculated the melting pressure to be

70% above the experimental value. The subsequent use

of the Aziz potential by Kalos, Lee, Whitlock, and Chester [8] improved the pressure calculation to just 7%

above experiment. The results were very sensitive to the

precise shape of the potential. Addition of an approximate three-body potential (Axilrod-Teller) did not improve the agreement. The determination of the proper

interaction potential ab initio may require the direct application of the GFMC method to an ensemble of alpha

particles and electrons. Nevertheless, one can expect improved ab initio calcu- lations of the melting pressure of ''He. For a given importance function, the algorithm has an accuracy which

increases with the square root of the computational ef- fort. Sheer increase in available computational capacity over the next decade may yield a 100-fold improvement

in accuracy [9]. Monte Carlo calculations are particularly

well-suited for computers with parallel processors. Im- provements in the importance function based on physical

insight would be similarly advantageous. The task of cal- culating correct helium-helium interaction potentials

from alpha particles and electrons and that of calculating

the thermodynamics of condensed helium from hehum

atoms are very similar and, if not done as one large computation, can both benefit from machine and algorithm

improvements.

The problem of calculating the zero temperature inter- cept of the melting curve of^He is made more difficult by

its nuclear spin. Lee, Schmidt, Kalos, and Chester [10] have recently reported on their extension of the earlier

^He work and calculated the correct ground state binding

energy of condensed ^He within 12%

.

Calculations of thermodynamic properties at non-zero temperatures (specifically the minimum of the ^He melting curve) are expected to be relatively straight-forward once the ground state calculation can be performed.

It is clear from the foregoing discussion that a great

deal of progress must be made before theoretical calcula- tions of the helium melting pressures reach a level of ac- curacy to be useful for standards work in pressure. Nevertheless, the path to progress is clear, and there are

useful goals to achieve.

3. Uniting IVIeasurements of tlie Helium

l\/lelting Curve to Room Temperature Piston

Gages

The transfer of the melting curve pressures to room

temperature piston gages is best performed in two steps:

(1) the measurement of the melting curve with an in situ (low temperature) pressure transducer; and (2) the caH- bration of the transducer at a higher temperature (typically 4 K) against the room temperature gage. When using the transducer to measure the melting

curve, there is no pressure head correction because the

transducer is in situ. Furthermore, the cubic structure

of solid helium, the rapid annealing of strains, and the

self-regulating nature of a phase boundary make the

melting curve relatively easy to measure, given the

proper cryogenic apparatus. Impurity effects are unimportant since ^He and ''He may be made isotopically pure

to 1 ppm. Isolation of the melting curve measurement

from the filling line can be obtained by the installation of a low temperature valve, but even without a valve, a

458
solid plug of helium fills the line when measurements are made at temperatures below the melting curve minimum.

Recent measurements by Greywall and Busch [1] exemplify the state-of-the-art in measuring the melting curve

vdth a transducer. They present a relation describing the

temperature dependence between 8 and 330 mK of the

melting pressure of ^He relative to the pressure at point B

of Fig. 1. The residuals of their data have a standard deviation of 30 ppm. These are completely dominated by uncertainty of the temperature scale. Their pressure meas- urements have a resolution of 0.2 ppm and an accuracy of

100 ppm limited only by the accuracy of their room temperature piston gage.

Earlier workers [11, 12] measuring the minimum of the

melting curves were similarly limited by the accuracy of room temperature pressure gages. We are presently [13]

measuring the melting curve minima of ^He and "^He us- ing room temperature gages of better accuracy. Aside

from the inaccuracy of the room temperature gage, the

most serious source of error in transferring pressure ac- curacy between a room temperature piston gage and a low temperature transducer is in the pressure head

correction. In our initial work, we simply used a vertical

tube to transmit pressure between low and room temperatures. The pressure head correction was 680 ±

20 ppm. Fortunately, the fill line may be brought out of

the dewar horizontally and the pressure head errors thereby reduced by a factor of 1000. Another possible er- ror source, the thermomolecular pressure effect,

presents no difficulty. For our present 0.075 cm diam.

capillary, its effect is smaller than 0. 1 ppm for pressures

greater than 240 kPa.

4. Linking the Piston Gage to Density or Mass

Peggs, Elliott, and Lewis [2] have made an evaluation

of the accuracy of a piston gage at 0. 1 MPa. To link pres- sure to density, they used a manometer filled vdth

Cook's mercury [14] and attained an inaccuracy of

4.6 ppm. The primary error sources were in the measure- ment of mercury column height (2.9 ppm) and temperature (3.1 ppm). Guildner and Welch [15] are presently

performing a similar experiment with the goal of making

the dominant error that of Cook' s absolute density determination (1 ppm). Peggs et al. also measured the dimensions of their pistons and cylinders to link pressure to mass with an error of 15 ppm.

The transfer of these calibrations from 0. 1 MPa to the

pressures of the helium melting curves produces little loss of accuracy. Uncertainties in the pressure dependence of the geometry of the gage become greater than

0.1 ppm only at pressures above 10 MPa [16]. Although piston gage technology [17] has a long history

and in recent decades has advanced by less than a factor

of two in accuracy per decade, an order of magnitude improvement is possible for low pressure gages during the next decade. When used to link pressure to mass,

present piston gages are limited by uncertainties in the

piston and cylinder areas and by an imperfect understanding of the detailed kinetics of fluid flow in the gap

between the piston and cyHnder [2, 16, 18, 19]. Foreseeable advances in measurement technology could allow the

uncertainty of the piston and cylinder areas to be pushed

below 2 ppm in the next decade [20]. More elaborate

modeling of the fluid dynamics in the gap, tested against

fall-rate meaurement and varied piston and cylinder con- figurations will be required to properly assign a value to the effective piston area for imperfect, but well-measured geometries. Ultimately, less tractable effects such

as those associated with surface roughness and adsorbates may inhibit progress beyond the 2 ppm level. The availability of a stable low-temperature pressure

transducer to be used in conjunction with the helium melting curve to provide pressure fixed points can help to reduce the uncertainty of piston gage calibrations in the

standard laboratory. Furthermore, such reference standards can assist the pressure metrologist in understanding

the limitations in precision of pressure transducers. Finally, there is hope that, one day, hehum melting curve

pressure data derived from first principles calculations can compare in accuracy with those derived from artifact standards.

The author is grateful for numerous discussions with

interested persons within NBS concerned with pressure

and fundamental constants and with theorists working on

the helium many-body problem. Particularly valuable was

the advice of L. A. Guildner which resulted in major re- visions. Also greatly appreciated were conversations

with V. E. Bean, R. J. Hocken, J. C. Houck, M. R. Moldover, R. D. Mountain, W. D. Phillips, J. F. Schooley, and B. E. Welch of NBS and theorists C. E. Campbell, D. M. Ceperley, M. H. Kalos, and R. A. Guyer.

References

[1] D. S. Greywall and P.A. Busch, J. Low Temp. Phys. 46, 451

(1982).

[2] G. N. Peggs, K. W. T. Elliott, and S. Lewis, Metrologia L5, 77

(1979).

[3] John E. Daborn, Meas. and Control 10, 377 (1977).

[4] D. M. Ceperley, private communication.

[5] For a review of Monte Carlo methods as applied to the many-body problem see M. H. Kalos, Nucl. Phys. A 328, 153 (1979).

[6] R. A. Aziz, v. P. S. Nain, J. S. Carley, W. L. Taylor, and

G. T. McConville, J. Chem. Phys. 70, 4330 (1979).

[7] P. A. Whitlock, D. M. Ceperley, G. V. Chester, and M. H.

Kalos, Phys. Rev. B 19, 5598 (1979).

[8] M. H. Kalos, M. A. Lee, P. A. Whitlock, and G. V. Chester,

Phys. Rev. B 24, 115 (1981).

[9] M. Kalos, private communication.

[10] M. A. Lee, K. E. Schmidt, M. H. Kalos, and G. V. Chester,

Phys. Rev. Lett. 46, 728 (1981).

[11] A comprehensive review of the melting curve of ^He was made by W. P. Halperin, F. B. Rassmussen, C. N. Archie,

and R. C. Richardson, J. Low Temp. Phys. 31, 617 (1978).

[12] E. R. Grilly, J. Low Temp. Phys. 4, 615 (1971).

[13] C. T. Van Degrift and W. J. Bowers, Jr., to be published.

[14] A. H. Cook, Philos. Trans. R. Soc, London, Ser. A: 254, 125 . (1961).

[15] L. A. Guildner and B. E. Welch, private communication.

[16] R. S. Dadson, R. G. P. Greig, and Angela Horner, Metrologia

1, 55 (1965).

[17] P. L. M. Heydemann and B. E. Welch, in Experi)yienial Ther- modynamics, Vol II: Experimental Thermodynamics ofNonreacting Fluids, Ed. by B. LeNeindre and B. Vodar

(Butterworths, London, 1975), p. 147.

[18] A. H. Bass, J. Phys. E. 11, 682 (1978).

[19] C. M. Sutton, Report No. 562, Phys. and Eng. Lab, New

Zealand (1977).

[20] R. J. Hocken, private communication. These improvements will come from advances in diamond turning machine technology such as described by R. R. Donaldson, Lawrence Livermore

Lab. Report # UCRL-528-12, Sept. 1979.

459

ij Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Realization of the Electrical SI Units

B. P. Kibble

Division of Electrical Science, National Physical Laboratory, Teddington, England

The ampere is unique in the set of SI units in that the accuracy with which it can be reaUzed is

still insufficient for accurate measurements of the fundamental constants. Possible ways of improving

this situation are described. In contrast, the ohm is in good shape, and a new cryogenic phenomenon

concerning a quantized Hall effect in a MOSFET structure enables a more accurate non-QED measure- ment of the fine structure constant to be made in terms of it. A change of base units to eliminate the

artifact-based definition of the kilogram is a more speculative possibility.

Key words: ampere; fine structure constant; kilogram; moving coil experiment; ohm; quantized Hall

effect.

1. Introduction

Although the title might imply that the realization of

all electrical units is to be discussed, in fact we need only

be concerned with the ampere and the ohm, with its associated impedance units, the farad and the henry. The

magnetic units, the tesla and ampere /meter are dealt with indirectly in the paper by Williams et at. [1] because

magnetic flux densities are measureable with all the precision desired by the usual NMR techniques once the

gyromagnetic ratio of the proton is known in SI units. The volt and watt are directly derivable from the ampere

and ohm, and the SI base units of length and time lead to the coulomb and units of electric field strength.

2. The Ampere

At the time of the last conference ten years ago the

ampere was thought to be realized by various current

balances in different laboratories with sufficient accuracy

for measuring the closely related fundamental constants such as the Faraday F, the gyromagnetic ratio of the

proton y'n and Avogadro's number Np^. Later more accurate work on the measured values of these constants re- vealed discrepancies in relationships between themselves

and other constants which were greater than the uncertainty with which the ampere was thought to have been

realized [2]. A further current balance measurement at the National Physical Laboratory, England (NPL) [3] and

refinements and additions to the constants measurements

have now turned this healthy situation on its head; the

realization of the ampere is now so uncertain that instead

of serving the needs of measurement of the fundamental

constants, measurements of F [4, 5], y'p [1, 3] and A^a

[6-10] are being used to realize the ampere, with a con- sequent loss of physical information.

It has become usual [2] to express the results of ampere realizations as a ratio, K, of a maintained unit of current to the SI unit. The present situation is summarised in Fig. 1. In ascending order in the diagram are

plotted the results of:

1) The latest NPL current balance realization [8]. 2) A mean of previous current balance determinations

[2].3) The result from an attracted disc electrometer

[11, 12].

:::inniiiiiiin:::® ---nmMn---m

I I I I ' : : ®

;

I I I I nil III II 1 1 I I I : : : ®

iiinnmin:::©

I I I I I I I I I II I I —

I

' ';@ "" I

I I I I 1 1 1 iiiiiiiiiiiiiiiiii I I I —I ®

! ® I I I I I I I I I I

-10 -8-6-4-2 2 4 6 8 10

Mointoined omperf K.I* X 10"' IJosephsofi vol 1 / 51 OHH I

Figure 1. The approximate present state ofthe direct and indirect determinations of the ampere (mth two-sigma uncertainties).

4) The geometric mean of the NPL strong-field 7p and

National Bureau of Standards, USA (NBS) weak-field

measurements. This combination is equivalent to the use

of a current balance, and also yields -yp in SI units [2]. 5) The same procedure, applied to the results from the

National Institute of Metrology, Beijing, China [13]. 6) A realization based on a relationship amongst the constants where the least accurately known member is F

[4, 5]. 7) Similarly for A'' a, based on the result of an NBS

measurement [6, 7]. 8) Partly based on results towards a Na measurement

obtained at the Physikalisch-Technische Bundesanstalt,

W. Germany (PTB) [8-10].

The customary error bars, which have the disadvantage

of visually suggesting a limit of error, have been replaced

by shading whose density at a given abscissa represents

the probability, according to the realization in question, of

the corresponding value of K being correct. The shading is terminated at points two standard deviations away from

the mean, as hindsight applied to the results of previous

precision measurements suggests that a subsequent much

more accurate result lies within two, rather than one, standard deviations of previous measurements. Even

with this generous latitude however it is clear in the

present instance that there are inconsistencies in that no

single vertical line can be drawn which intersects all results, even between the ± 2 a limits. More seriously, the

461
true result might lie anywhere within a range of more

than five, or more than twenty, parts in a million depending on whether or not one discounts the "direct" current

balance determinations. The use of a current balance does

not seem favorable for improved accuracy. Electrometers

[11, 12, 14] of various kinds are a possibility, but the pace

of ever-increasing accuracy of measurement of the con- stants associated with the ampere means that the target is receding.

A conventional current balance is prevented from

achieving part per million accuracy by the smallness of the force generated between the fixed and suspended

coils. Moreover, disturbing random convective forces are generated by the considerable heat dissipated in the coils in order to generate even this small force, so there is a several parts per million random component in the

results which makes it difficult to seek systematic errors

at or below this, level. Furthermore it is an exceedingly

difficult and tedious instrument to use, particularly with

regard to the many hundreds of sub micrometer mechanical measurements which must be made to establish the

relative positions of the conductors between which the

force \is generated and even then assumptions must be

made about how these measurements relate to the actual mean current paths.

A possible alternative approach [15] which has been

much discussed during the past five years and which is being implemented in at least two laboratories (NPL and

NBS) [16, 17] uses a coil suspended from a balance in a non-uniform magnetic field. The force F newtons (equal

to an opposing mass times the gravitational acceleration) can be measured when a current / flows in the coil. If, in a separate measurement, the coil is moved so that it has

velocity v meters per second at the instant it passes

through the position it occupied for the weighing operation an emf V is then generated. The equality of mechanical and electrical watts implied in the ampere definition

and electromagnetic theory ensures that the relation

VI = Fv

holds exactly. Because of the careful SI ohm determinations carried out by several laboratories possessing cal- culable capacitors [18-21], resistances R whose values

are accurately known in SI ohms are available, so V can be equated to the potential drop produced by / in and

hence

/ = VWvTRT si amperes

By shaping the coil and magnetic field distribution ap- propriately the force and emf can be made substantially constant over a moderate distance of uniform travel of the coil, and the problem of measuring an instantaneous

velocity is replaced by the easier requirement of obtaining an average of the force and emf. The photograph

(Fig. 2) shows the NPL version of this idea. It is based

on a very large permanent magnet which provides a field of almost 0.7 T in a 58 mm gap between 300 mm square

pole faces. In this gap an astatic coil (wound as a figure

of eight) of 3362 turns having a mean distance of 240 mm

between its vertical sides, is suspended from one end of a long balance beam. The total travel of the coil, obtained

by allowing the beam to rotate, is ± 25 mm centered on

the beam's horizontal position. Voltage measurements

will be made during the traversal of the central ± 20 mm

with the velocity servo controlled to better than 10 ppm.

Force measurements will be obtained by weighing at several of the positions the coil passes through, by

elevating the whole platform on which the balance case

rests. A current of 10"^ A, dissipating a power of 10~^ W

Figure 2. The prototype NPL moving coil absolute ampere

apparatus.

in the coil will generate just over 1 kg weight of force change when its direction is reversed and a velocity of about 2 mm per second will produce 1 V of emf. Figure 2 gives an impression of the whole apparatus before the

thermostatted cabinet which will enclose it is erected.

The moving and weighing processes will be remotely

operated to avoid the thermal effects of a human operator,

and will be largely under automatic computer control, so that the tedious repetition of obtaining results in the vital search for systematic errors will be bearable. We expect

to obtain results with this apparatus in 1982. The final NBS version of this idea [16, 17] is based on the coil geometry used in Campbell mutual inductors and the Ayrton-Jones current balance, where the uniformity of dMIdz

of a coil in a region of the midplane of symmetry of two op- posed cylindrical coils is exploited to give the desirable

uniform generation of emf with rate of displacement.

At NBS, preliminary trials and measurements have

also been carried out using the geometry of the Pellat electrodynamometer where the torque and induced emf

on rotation of a cylindrical coil pivoted about an axis in the plane of symmetry in a uniform magnetic field are measured in turn [16]. The torque was determined from

the product of an opposing mass, the gravitational

acceleration and the length of the balance arm. The rotational rate was found from the linear velocity of an inter- ferometer mirror at the end of the same arm so that the arm length was eliminated when the results of the two

measurements were combined.

In yet another approach to what is essentially the same

fundamental idea, K. Hara et at. [22] are pursuing a cryogenic method first suggested by Sullivan and Frederick [23] of equating the change of electromagnetic energy

of a superconducting body levitated in the magnetic field of a current carrying coil to the change in gravitational

462
potential energy when the body is moved. The situation

could be thought of as the inverse of a moving coil meas- urement as the coil is now fixed and the field threading it is modified by the movement of the levitated body. The

use of cryogenic equipment makes it convenient and at- tractive to measure the currents directly in terms of an

SI resistance and a Josephson effect voltage to provide a determination of the Josephson effect constant in SI

units but the method suffers from difficulties with vibration, in common with room temperature moving coil measurements, and in addition would have the problem

of ensuring that the levitated cryogenic mass was

correctly related to mass standards at room temperature.

3. The Ohm

The farad having been realized in SI units by finding

the SI value, C, of a fixed capacitor from a calculable

capacitor, the ohm can in turn be set up by using

frequency-dependent ac bridges which realize the rela- tionships R — l/a)C or L = 1/(0" C to find the SI resis- tance R in ohms of a resistor at an angular frequency w

or the SI inductance L in henries of an inductor

[13, 18-21]. Such realizations are, at any rate by comparison with the ampere, in good shape, wdth at least five different laboratories' determinations agreeing to vdthin

0.2 parts in a million for the ohm.

As the number of laboratories in various countries pos- sessing Thompson-Lampard capacitors has increased

greatly of late, we may shortly expect yet more results

at about this level of accuracy. At present, it would ap- pear to be very difficult to improve the accuracy much

beyond a few parts in 10®, but the discovery of the quantized Hall effect in two-dimensional electron flow in MOSFET's cooled to liquid helium temperatures [24]

does make it worthwhile to pursue greater accuracy. Figure 3 shows the device schematically; for a suitable

combination of strong magnetic field B and gate voltage

Vq the potential drop V dovra the device is vanishingly

small. Under these conditions the Hall resistance i^n

which is the ratio of the transverse Hall voltage Vh to the current / through the device, appears to be very

closely equal to e'^/nh = 1/n x 25813 ohms, where n is an

integer. The full potential of this discovery is not yet

certain, but the maintained standard of resistance generated by this effect could be useful at any one of three

possible levels.

1) Even if Ry{ depends slightly on the type of device

and its environment, it would still be very useful for

Figure 3. A schematic diagram of a MOSFET device for exhibiting quantized Hall resistance effects.

maintaining a resistance unit locally by specifying the

conditions and type of device.

2) If Ru is device, and within Hmits, environment in- dependent, it could maintain a unit universally so that

resistors need no longer be continually exchanged

between laboratories to compare their results.

3) If Rh, without significant correction (or the corrections are calculable and subject to experimental investi- gation) is exactly an integral fraction of eVh, it provides

a potentially very accurate determination of the value of the fine structure constant. There is therefore an urgent need to make theoretical

and experimental investigations to find out at which level the device is useful, and the contributions relating to this effect are especially welcome [24-31].

4. Possible Developments in the SI Units

Examination of history shows that the SI units and

their predecessors are not a fossilized entity; the defini- tions of the base units have been changed several times

as part of a natural evolution towards higher accuracy. Naturally, care was taken that, at each redefinition, continuity with the old units was ensured to within their

reproducibility. One can also, more radically, contemplate a different choice of base units, and it is interesting

to see whether in any sense one could make an improvement, bearing in mind that the sole purpose of a unit is that the realization of its definition should be more reproducible, both from time to time and place to place, than

any measurement we might seek to make in terms of the

unit. We have seen that the definition of the ampere at present fails this test by two or more factors of ten, and moreover although the definition quite properly specifies

the realization of the unit without regard to the correct- ness of any particular physical theory (in this case elec- tromagnetic theory), unfortunately, to make any sort of

practical apparatus that theory has to be used.

The kilogram is also unsatisfactory in a different sense,

as there has been a rapid evolution in recent years to- wards units based on universal physical properties which

can be shown to high accuracy to be free from secular

change, and are freely available to any laboratory that cares to realize their definition. If the quantized Hall ef- fect and the moving coil experiment fulfil all their early

promise they could be key links in a system of units

whose basis is completely free of material standards. A

possible basis would be to define four units somewhat as

follows: Whizz. 1/299792458 of the velocity of plane electro- magnetic radiation in free space. Accuracy of realization

(AR), 10"^ ppm.

Second. As at present (and implying the hertz). AR

10"^ ppm.

Volt. That potential step which would correspond to radiation of 4.83594 x 10^"^ Hz in a Josephson junction.

AR 10-2 ppm.

Ampere. The current flowing through a quantized Hall

effect device which would give rise to a Hall potential of 6453.2 V. From these, the kilogram would be a derived unit, with

a definition somewhat as follows:

Kilogram. That mass which, when moving with a velocity of one whizz, possesses mechanical energy equal to an

electrical energy of one half volt-ampere-second.

Although this scheme has the merit of providing

universal units, it will only become attractive if it can be

463
shown to yield units at least as reproducible as the

present ones. From an electrical point of view there is no problem because, as we have seen, the reproducibility of the ampere would certainly be improved. No fundamental

constants such as the permeability of free space, need

have their values defined, which is arguably as it should

be, and some apparatus such as the moving coil ap- paratus would realize the kilogram. To approach the

present reproducibility of the kilogram would require uncertainties of a few parts in 10^. It may be pointed out

that there are some problems at this uncertainty level connected with the present kilogram. Presumably the

definition implies that only the platinum-iridium material

is to be included in the mass, in which event the prototype kilogram should be compared with others with all the surface films removed and kept absent, in vacuo. Ordinary cleaning, as at present practiced gives rise to ambiguities of a few parts in 10*. Clearly we do not know now whether the target of a few in 10* is attainable, but if past progress is a reliable guide to the future, the next ten years should give us a very good idea. References

[1] E. R. Williams, P. T. Olsen. and W. D. Phillips, these

proceedings.

[2] B. N. Taylor, Metrologia 12, 81 (1976).

[3] P. Vigoureu.x and N. Dupuy, Nat. Phys. Lab. (NPL) Rep.

DES 59, (1980).

[4] V. E. Bower and R. S. Davis, J. Res. Natl. Bur. Stand. 85, 175

(1980).

[5] L. J. Powell, T. S. Murphy, and .J. W. Gramlich. these pro- ceedings.

[6] R. D. Deslattes, Annu. Rev. Phys. Chem. 31, 435 (1980).

[7] R. D. Deslattes, these proceedings.

[8] H. Siegert and P. Becker, these proceedings.

[9] P. Becker and H. Sieger, these proceedings.

[10] P. Seyfried, these proceedings.

[11] N. Elnekave et al. Rapports Laboratoire Central des Industries Electrique (France), No. 749-677-629-583 (c. 1979).

[12] N. Elnekave and A. Fau, these proceedings.

1^13] Wang Zhu-Xi, these proceedings.

[14] G. J. Sloggett, W. K. Clothier, D. J. Benjamin, M. F. Currey, and H. Bairnsfather, these proceedings.

[15] B. P. Kibble and I. A. Robinson, NPL Report DES 40, (1977).

[16] P. T. Olsen, W. D. Phillips, and E. R. Williams, these

proceedings.

[17] P. T. Olsen, W. D. Phillips, and E. R. Williams, J. Res.

Natl. Bur. Stand. 85, 257 (1980).

[18] R. D. Cutkosky, IEEE Trans. Instrum. Meas. IM23, 305

(1974).

[19] A. M. Thompson, Metrologia 4, 1 (1968).

[20] B. P, Kibble et al. (unpublished), (1979).

[21] ETL News, No. 328 (1977) (in .Japanese).

[22] K. Hara, F. Shista, and T. Kubota, these proceedings.

[23] D. B. Sullivan and N. V. Frederick, IEEE Trans. Magn.

MAG-13, 396 (1977).

[24] K. V. Klitzing, these proceedings.

[25] E. Braun, P. Guttmann, G. Hein, P. Warnecke, and S. Xue,

these proceedings.

[26] C. Yamanouchi, K. Inagaki, J. Moriyama, S. Baba, S. Kawaji, T. Endo, M. Koyanagi, K. Murakami, T. Igarishi,

and A. Nakamura, these proceedings.

[27] M. E. Cage, R. F. Dzuiba, B. F. Field, C. F. Levine, and R.

J. Wagner, these proceedings.

[28] A. Hartland, these proceedings.

[29] A. C. Gossard and D. C. Tsui, these proceedings.

[30] L. Bliek and G. Hein, these proceedings,

[31] S. M. Girvin and R. W. Rendell, these proceedings. -

464

I
\

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

An Absolute Determination of the Volt at LCIE

N. EInekave and A. Fau

Laboratoire Primaire Electricite-Magnetisme, Laboratoire Central des Industries Electriques,

92260 Fontenay-aux-Roses, France

The absolute determination of the volt using Kelvin's electrometer which was carried out in 1978 is now repeated with an improved volt balance. The new instrument includes a larger active electrode and a reduced gap between it and the guard ring. Second order corrections due to gap effects have

been calculated by finite element methods and checked through rheographic mapping. It is estimated

that random errors associated with the use of the new instrument will amount to less than 4 ppm. Key words: absolute volt; electrical standards; International System of Units; Kelvin electrometer;

precision measurements.

1 . Introduction

Determination of the SI volt is one of the main problems of present electrical metrology. We have chosen to make such a determination with a Kelvin-type plane elec- trometer, thus relating the value of emf standards (such

as standard Weston cells) to the SI units: meter, kilo- gram, and second (Fig. 1). From such a determination we

can deduce the fundamental constant ratio e /h (e the

charge of the electron, h Planck's constant) related to emf and frequency by the Josephson effect. The absolute value (f/a) of the applied difference of potential to the electrometer is evaluated in terms of four

main parameters which are: direct capacitance C; dis- tance of electrodes; mass in that equilibrates the electrical attraction between the plates; and the gravitational

acceleration g: mg -^U'-C/d

A slight correction should be made to the calculated

force, to take account of the airgap effect (see proof

below) [1]. An electrometer built in 1975-1978 was used for a first determination of the volt, yielding its absolute value wdth an estimated uncertainty of 9 ppm. The force on the elec- trometer was 0.035 N. A new electrometer with enlarged

electrodes provides now a higher electrostatic force (approximately 0.05 N) with the same applied voltage

(8.4 kV) and electrode distance (1 cm).

2. The Absolute Electrometer

The electrometer (Fig. 2) is of the balance-type, built according to Kelvin's guarded electrode design [2]. It is

frequency

absolute

I capacitance

I (Lampard

Thompson

)

Josephson

effect

absolute

I electrometer

national

^ c

(Ok

to the balance

®

(D A

7m

2e

h

divider divider a®

®

Figure 1. Absolute determination of the volt and 2e /h and the

role of the SI units.

Figure 2. Schematic of the LCIE Kelvin electrometer: (1) x-y

table; (2) base plane; (3) glass slab; (i) fixed electrode; (5) sili- ca spacers; (6) guard ring; (7) movable electrode; (8) micrometer screw; (9) corindon planes; (10) gimbal suspension;

(11) screening; (12) high voltage terminal; (13) additional ring.

I 465
in fact a 3-terminal plane condenser whose guarded elec- trode is suspended from the balance. Electrodes are

thick, gold-chromium plated silica discs (total metal

thickness 0.4 |xm). Flatness is within X/20 for the movable and high voltage electrodes and within \/10 for the

guard ring (\ = 0.6 \j.m). The movable electrode diameter is 14 cm and the airgap is reduced to 0. 16 mm on the new instrument. The edges of the airgap are rounded

with a radius of 0.05 mm and 0.09 mm on the movable

and guard electrodes, respectively.

The distance between the electrodes is defined through

four silica spacers, with an approximate height of 10 mm

(working voltage 8.4 kV) or 12 mm (working voltage

10 kV). The movable electrode suspended from the bal- ance beam is maintained coplanar with the guard through

the use of three stops.

2.1 Adjustment

Parallelism between the high voltage and guard elec- trodes is achieved by the equal height of the spacers. A

mechanical system can then act on the stops to provide

the necessary displacements of the movable electrode in order to bring coplanarity of the latter with the guard.

The first adjustment is to make the guard electrode

horizontal. This is done with reference to an already horizontal plane by auto collimation. A mirror fixed above

the guard disc will henceforth be used in conjunction

with the auto collimator as a horizontal reference for the

electrometer structure. Coplanarity of the guard and movable electrodes is checked by means of a Michelson interferometer (Fig. 3). A first test with red light (He-Ne laser) is made for parallelism. A second test with white light (the unique

black fringe on the movable electrode should be in Hne

with the black fringe on the guard electrode) is then

made to prove exact coplanarity of the surfaces (Fig. 4).

GENERAL CONFIGURATION

A' A

PARALLEL ELECTRODES

'///////A \yyyyyyyy/^y'A \yy//y//y/yy.

A'

COPLANAR ELECTRODES

A and A': BLACK FRINGES WHEN LIGHTING THE

INTERFEROMETER WITH WHITE SOURCE

Figure 4. Fringe "pattern obtained from the Michelson interfer- ometer of Fig. 3.

Figure 3. Michelson interferometer arrangement used to achieve coplanarity of the guard and movable electrodes.

3. Voltage Measurement With Electrometer

Equilibrating the balance with no voltage applied is the

first step in the measurement. Applying the voltage

(8.4 kV or 10 kV) to the electrometer and withdrawing the

calculated weight on the movable electrode side of the bal- ance would then restore equilibrium. It is known, however, that no stable equilibrium can thus be obtained with

a quasi-linear restoring force. Stops are therefore pro- vided which limit the motion of the movable electrode in the downward direction. Decreasing the voltage slightly results in an upward motion of the movable electrode. Electrometer balance is effected with a linear controlling force, through the action of a current carrying coil system (fixed) on a magnet on one of the balance pans. The whole balance system is within a nitrogen filled enclosure at a pressure slightly above atmospheric.

Measurements of capacitance and force are thus made in identical conditions. Capacitance measurements are made

with an inductive-ratio transformer bridge at a frequency

of 795 Hz. Standards used are calibrated with respect to a 5-cylinder, Lampard-Thompson type, cross capacitor.

4. dc Voltage Divider

The divider is made up, on its high voltage side, of one hundred 100 kfl resistors in series, and on its low voltage

side of a single 1000 fl resistor. The parallel setting of the 100 high-voltage resistors results in a 1000 fi resistor

which is compared to the low voltage side resistor by a substitution method.

466
5. Results of the 1978 Determination

An absolute determination of the volt was performed in August 1978 at the LCIE yielding for the as-maintained

French unit of potential Vf

1 Vf = (1 - 4.5 X 10-'^) V , v/here V stands for the SI unit. The French volt being

defined in terms of the Josephson effect with an adopted

value of 2e/h equal to 483 594.64 GHz/Vf, the corrected

SI value for the combined constant is 2elh = 483 596.82 GHz/V.

The total uncertainty was estimated not to exceed

9 ppm.

6. Expected Accuracy of the New

Electrometer

The following are the estimated uncertainties related

to the independent evaluations which are included in the

absolute measurement:

Electrode separation

Capacitance

Gap correction

Mass

2.5 X 10-*5

1 X lO"*' 0.5 X 10"*^ 3 X 10"^ Acceleration (gravitational) 0.2 x 10 ^ The total uncertainty with the new instrument is thus ex- pected to be smaller than 4 ppm.

7. Appendix

7.1 Gap Correction for Axially Symmetrical

Electrometer

Notation: R: inner radius of guard ring

r: outer radius of movable electrode

d\ distance between fixed and movable electrodes

e: permittivity of gaseous medium

C: measured direct capacitance between fixed and

movable electrodes

(j{x ): charge density on fixed electrode as a function of radius x The airgap is limited by two coaxial cylindrical surfaces

(radii R, r). Slightly rounded edges join these cylindrical

surfaces with the plane surfaces of the movable and

guard electrodes (Fig. 5). A so-called "neutral" tube of force separates those

lines of force which end on the movable electrode from

those ending on the guard electrode. In between the

cylindrical surfaces, this tube of force tends to be

cylindrical vdth a radius equal to V/2r . The intersection

of the neutral tube and the fixed electrode plane is a cir- cle with radius R q. Charge density a(x), total charge Q, capacitance C,

and potential difference U are related by

Q = CU = 27: f

u{x)xdx = -nRi a . (1) An average charge density a is thus defined on the area

IT 0^ of the circle of radius R o- The electrostatic force between the fixed and movable

electrode is thus

F = — f V(x)xdx

: M.E

::U=0

\/Rr

o-(x) = e|?(x)|

E(x)

G.R : U=0:

'neutral" tube

CCmin)

F.E(H.V): U = Un

Ro

M.E Movable Electrode G.R Guard Ring

F.E (H.V) Fixed Electrode

U Potential T(x) Electric field

Figure 5. Schematic of the electrometer shotving the quantities

which enter the calculation of the gap correction.

Let us define a function hix ) such that

(t{x) = a[l -I- hix)]

The above relation implies

h(x) xdx = 0 . Jo

It is then easily shown that

2e

2 r 0 Rq •'o

(3)

(4)

(5)

Let

Then

Let

Cr

U^C c

2d Co

^ + ^^'(x)xdx

2Ri

r 0 J b^x ) xdx ,

_C_

Co

The force F can then be written

F = a + m - y)

U^C

2d

(6)

(7)

(8)

(9)

(2)

The radius R o and the corrections p and 7 can be calcu- lated in case of cylindrical gaps vdth sharp edges [3, 4]. For rounded edges a 2-dimensional rheographic plot, or a 3-dimensional finite element calculation that takes into account the cylindrical symmetry of the electrometer,

have been developed.

467
References

[1] C. Snow, Bur. Stand. J. Res. (U.S.) 1, 513 (1928).

[2] W. Thomson (Lord Kelvin), Reprint of Papers on Electrostatics and Magnetism, 2nd edition (Macmillan & Co., London,

1884), Sect. 360.

[3] C. Moon and C. M. Sparks, J. Res. Natl. Bur. Stand. 41, 497

(1948).

[4] H. Abraham and J. Lemoine, J. Phys. Theor. Appl. 4, 463

(1895).

468
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

The CSIRO Absolute Volt Project

G. J. Sloggett, W. K. Clothier, D. J. Benjamin, M. F. Currey, and H. Bairnsfather

CSIRO Division of Applied Physics, Sydney, Australia, 2070

A liquid electrometer of unique design has been constructed to provide a precise absolute voltage

standard. In this progress report the instrument is described and critical aspects of its performance

are assessed. Known sources of uncertainty are consistent with a volt determination at or below the

1 ppm level. The principal areas of remaining work are discussed. Key words: absolute voltage standards; interferometry; liquid electrometer; mercury; precision electrical measurements; SI units.

1. Introduction

The volt may be maintained with a precision of about a

part in 10'^ by means of the ac Josephson effect [1], but

neither the volt nor the ampere is known in absolute measure to better than a few parts in 10^. Taylor [2] showed in 1976 that the best available values for the absolute ampere differed by 4 to 10 in 10^ from three in- direct values calculated from fundamental constants, and

subsequent ampere [3, 4] and volt [5] determinations

have not resolved these disagreements. In 1979 the 16th

Conference Generale des Poids et Mesures recommended

"the continuation and intensifying of research on both

the direct realization of the electrical units and their in- direct realization through the determination of physical constants" [6]. In the CSIRO Division of Applied Physics a direct

determination of the dc volt is being undertaken using a

liquid electrometer [7]. The method entails elevation of a body of conducting liquid by electrostatic attraction towards an electrode at a high potential V a distance d

above it (Fig. 1). The elevation h near the center of the

raised plateau is determined solely by the balance of

electrostatic and gravitational forces, and in principle the

applied voltage V is calculable given a knowledge of h, d,

and certain other physical properties of the system. Surface effects are best eliminated by using at least two

voltages Vi and V2 with corresponding spacings di and

d2 chosen so that h remains nearly constant. One obtains

^2/12"^ - Vih^-"^ = (2pg/eo e.)^ {d. - rfi) (1) where

hi ~ h2 ~ h (2) and p is the density of the liquid, g the acceleration due

to gravity, the permittivity of free space and the

relative permittivity of the medium. If these constants and the ratio V2/V1 are known, the absolute determination of voltage reduces to the measurement of a few

linear dimensions.

Figure l. General schematic of the liquid electrometer and its optical measuring system.

469
2. Electrometer

The chosen liquid is mercury, whose chief advantages

are a high reflectance and a density known to about 1 in

10^, contributing an uncertainty of 0.5 ppm to a volt determination. If the overall uncertainty of the experiment warrants it, a more accurate density determination

will be made, most likely by comparison with a sample

held in this laboratory whose density was determined by

Cook with an uncertainty of 3 or 4 in 10^ [8]. Of the

remaining constants in Eq. (1), g has been determined at the site of the experiment to 1 in 10* by gravimeter

transfer from the site of an absolute g determination in

this laboratory [9]. Cq is known to 1 in 10*, and should

be within 10"* of unity if the experiment is performed, as planned, in a vacuum of order 0.1 Pa. The linear dimensions required for Eq. 1 are obtained

from three interferometric spacing measurements. As

shown in Fig. 1, the mercury is contained in three inter- connected pools, above which is located an optical flat vidth three sections of thin chromium film on its lower

surface. With the measurement voltage V applied to the

central section of film and the outer sections grounded,

the mercury level in the central pool is elevated relative

to that in the outer pools. The three sections of film are

optically semi-transparent and of such a thickness that

reflection fringes of good contrast are obtained in con- junction with a mercury surface. Spacings Ca, d and ej are measured at the three equispaced pool centers, and a raw value for the elevation h is calculated using

h - (1/2) (e„ + eb) - d. (3) This value must be corrected for aplanarity of the three

sections of film and for the gravitational curvature of the

interconnected liquid surfaces. The necessary correction

is determined from spacing measurements made vdth

zero applied electric field. The approximate value of h is 0.662 mm and, for rea- sons related to the interferometry, it is convenient to make spacing d an integral multiple of h:

d ~ nh, (4) where, for stability, n must exceed 2 [7]. A range of

spacings d corresponding to values n from 3 to 7 is avail- able by vertical movement of the optical flat. In all, ten

distinct measurements may be made with the electrometer (five spacings for each of two voltage polarities),

considerably in excess of the minimum number (two) re- quired for the elimination of surface effects. This additional data should be useful in the reduction of measure- ment uncertainty.

A general view of the electrometer is shown in Fig. 2. The three 75 mm diameter pools are located in the upper

surface of a stainless steel block, and may be interconnected and /or connected to a reservoir within the block

by a system of pneumatically-operated valves. The rate

of filling or draining of the pools is controlled by the

pressure of N2 gas admitted to the reservoir.

The pool diameters are sufficient that capillary depression effects are negligible. For a sessile drop of diameter

75.6 mm (a conservative model for mercury in a

vertical-walled pool of diameter 75 mm) the capillary

depression, calculated from Blaisdell's expressions [10],

is 0.05 nm at the pool center and 0.24 nm at a radius of

5.7 mm. These effects are further reduced by means of meniscus rings fitted to the pool rims. These have a tri- angular cross-section with an internal angle (32° to the

horizontal) which is within the range of contact angles

between mercury and stainless steel [11], so that the surFlGURE 2. Absolute liquid electrometer.

470
faces can, in principle, be adjusted to be entirely free of meniscus curvature. In the case of the center pool, however, some curvature is to be expected at the periphery

due to the weakening of the electric field, and there may

also be a modification (most likely a reduction [12]) of the

surface tension due to surface charges.

The optical flat is supported in a carrier which may be

moved vertically by means of a precision crankshaft.

Parallelism between the optical flat and the mercury sur- faces is maintained to within about 10"^ rad over the

range of vertical movement by accurate matching of the

crankshaft eccentrics, by differential adjustment of spring counter-balancing systems at either end of the

carrier, and by the geometry of four spring strips con- necting the flat carrier to pedestals fixed to the baseplate

of the instrument. The chief components of this system

are made of Invar. The instrument is housed in a vacuum chamber which

may be tilted hydraulically as a means of fine adjustment

of the parallelism of the optical flat and the mercury surfaces, and stands on an antivibration table (see Fig. 3). The table has three fine vertical legs each of which

carries a compressive load almost sufficient to cause

buckling. This system, properly damped, attenuates hori- zontal components of vibration above 0.5 Hz. Lead

weights in elevated trays raise the center of mass of the

table to the plane of the pool surfaces, so that any rotational vibration of the assembly does not cause substantial wave motion if, as mentioned, the mercury surfaces are horizontal at their edges. The use of separate pools

for the three spacing measurements, each having local minima of wave motion near its center, and each fitted with a perforated damping plate a few mm below its surface, gives further attenuation of vibration.

Figure 4 shows typical noise and drift in the interfer- ometric spacing measurements, obtained with the three

pools interconnected. The drift is principally due to the

evaporative loss of mercury from the pools, and the noise

to vibration on the pool surfaces. The noise level for spacing d (central pool) is satisfactory but spacings Ca and Cb exhibit a high level of antiphase noise, apparently

due to the vibrational interchange of mercury between

the outer pools. A fourth trace in Fig. 4 shows that the

noise in the mean of Ca and , which is the quantity re- quired for the determination of h via Eq. (3), is similar to that for d. Preliminary results obtained under zerofield conditions indicate that h may be measured with

a repeatability of about 0.3 nm, or 0.5 ppm. This type of random uncertainty can be reduced by repeated

measurements.

3. Interferometry

Under the condition of Eq. (4) all three spacings take

values which are close to integral multiples of h. Figure 1 shows one of the three identical interferometry systems.

A spacing is measured in two steps. First, the unknown

spacing is compared by optical multiplication with the

spacing (nominally h) of a Fabry-Perot reference inter- ferometer and, second, the Fabry-Perot interferometer is calibrated against a stabilised He-Ne laser. The FabryPerot interferometers, which are of critical importance in these procedures, have been described in detail else- where [13]. Their spacings are electromagnetically ad- justable over a range of ±0.05%, the amount of the

adjustment being precisely calculable from the adjusting

current. In the optical multiplication step a white light source is used and the Fabry-Perot spacing adjusted to set on a superposition fringe of order n (for spacing d) or n + 1 (spacings Ca and e^). To calibrate the interferometer a retroreflector is substituted for the electrometer

and the interferometer spacing is adjusted to set on the nearest monochromatic fringe (of order 2093). If the ad- justments in the calibration and optical multiplication

steps are fc and /„, fringes respectively, the electrometer spacing d (for example) is given by

d =

n \

(2093 - /e + /„.), (5)

Figure 3. General view of apparatus, shomng (from top) the

optical bench, the vacuum chamber (front cover removed) containing the electrometer, and the antivibration table.

where k is the laser wavelength (633 nm).

The chief advantage of this method over one in which

the electrometer spacings are measured directly in monochromatic light is that the precision with which the

spacings need be adjusted is greatly relaxed. The present

requirement of a few parts in 10^ is readily achieved with

the aid of capacitance probes mounted in the optical flat carrier (see Fig. 2). The Fabry-Perot interferometers

also provide a convenient means of scanning, which is essential for fringe setting to an accuracy of 10"^ fringe.

Some of the systematic effects limiting the interferometric precision have been discussed elsewhere [13] and the

results of recent work, on the errors due to superposition

fringe asymmetry, will be published shortly. Total uncertainty in the interferometric measurement of any spacing

should not excede 10""^ fringe (0.3 nm). or about 5 in 10' of h.

471
Figure 4. Drift and noise in electrometer spacings. The multi-pen recorder displaces the traces by one small time-scale divi- sion (10 s), in the order indicated by the labelling/.

The components of the optical system are mounted on a

triple optical bench supported on a bridge over the vac- uum chamber, as show^n in Fig. 3.

4. Electrical

For constant h the voltage V must increase in proportion to d, and, for the same value of n as in Eq. (4), should satisfy

V = nVo (6) for some constant voltage Vq. Vg is uniquely determined

by h and has the value 2953 V, which is 290 times the

voltage of ten standard cells in series. A multiple-output

poM^er supply using standard cells as a voltage reference

has been constructed along the lines indicated in Fig. 5. The outputs of 14 transformer-rectifier modules, each

producing a voltage Vo/2, are connected in series and

used to supply current to a 14-section resistive divider.

The electrometer is connected to the (2ri)th divider tap

(at voltage nVn) via a potentiometer control which en- ables the electrometer voltage to be adjusted smoothly,

the selected tap being shorted to the corresponding

power supply output to ensure a low output impedance.

The first section of the divider has an additional tap at 10.18 V, and the amplified difference between this volt- age and that of the reference cells is used to regulate the

ac voltage supplied to the transformer-rectifier modules.

With a loop gain of 10^ the power supply is stable to within a few parts in 10^ over several hours, determined

essentially by the stabihty of the cell e.m.f.s and the di- vider ratios. A small known balance voltage may be in- serted in series with the cells for fine adjustment of the

output voltage.

The standard cells, in a portable temperature-controlled enclosure, will be used to transfer the absolute

volt to the CSIRO as-maintained voltage standard [1]. The ratio of the (2ri)th divider output to the voltage at the 10.18 V tap, nominally 290w, is determined using two

bridges which are housed with the divider in a temperature-controlled airbath. A series-parallel bridge uses two

sets of 12 resistors to establish the ratio (approximately

145 or 12^ + 1) of the first section to its 10.18 V tap. A

movable equal-arm bridge is then used to determine the

ratios between adjacent divider sections, and from the 13 balances so obtained cumulative ratios are calculated. Study of the performance of the two bridges indicates

that the overall divider calibrations should be accurate to within 1 or 2 in 10^.

5. Discussion

With minor exceptions construction of the apparatus is complete and performance testing is well advanced. We

are currently investigating certain systematic effects in the operation of the electrometer.

The electrometer responds as a manometer to pressure

differences between the inner and outer pools, and differ- ences as small as 10"'* times the saturation vapour pres- sure of mercury (0.16 Pa at 20 °C) may be significant in the determination of h. Preliminary measurements sug- gest that such a pressure difference exists and, as might

be expected, has a magnitude dependent on the spacing d

and the pumping history of the vacuum system. A deci- sion on whether to attempt to eliminate this effect or merely to correct for it has been deferred until more reli- able data on its magnitude and repeatability are ob- tained.

472
ELECTROMETER

VOLTAGE

CONTROL

66kHz

OSCILLATOR

POWER

AMPLIFIER

ANALOGUE

MULTIPLIER

TRANSFORMER - RECTIFIER MODULES

(14)

RESISTIVE

DIVIDER SELECTOR

10.18 V

VARIABLE

GAIN

AMPLIFIER

DETECTOR

AMPLIFIER

STANDARD

CELLS

(10)

BALANCE

VOLTAGE

SUPPLY

+ ImV

Figure 5. Schematic of power supply.

The rate of evaporation of a free mercury surface into vacuum is sufficiently high as to make accurate interfer- ometry impossible, and causes unacceptable cooling of the liquid. The rate is reduced by using a low pumping

speed and by forming oxide films on the pool surfaces,

under which conditions the rate of surface movement is of the order shovm in Fig. 4. These measures also tend to reduce the vapor pressure gradients across the pools. A

possibly significant systematic error arises from the oxide films. If the films on the outer pools differ in thick- ness from that on the central pool, as could be caused by

stretching or compression during adjustment of the mercury levels, the resultant differences in surface effects may be significant in the determination of h. This question will be investigated experimentally.

Another small correction to h is yet to be determined.

This arises from the absence, in the zero-field measurements, of the deflection of the optical flat normally

caused by the electric field. The thermal stability of the pool block is of great importance and remains to be fully evaluated. At 20 °C

mercury has a temperature coefficient of expansion of

1.81 X 10~''K"^ and its vapour pressure varies by

0.014 Pa K"\ hence the pool temperatures must be equal

to within a few mK. Temperature uniformity is achieved

by completely excluding significant heat sources from the vacuum chamber and, if necessary, by surrounding the

chamber and antivibration table with an insulating enclosure. Thermal gradients in the apparatus will be monitored by thermocouples and the mercury temperature

measured using two platinum resistance thermometers

installed in holes in the pool block.

Many members of this Division have contributed significantly to this project. We particularly wish to thank D.

Bicknell, F. J. Bramall, M. Jenkins, and H. W. King for their participation in the design or manufacture of the

apparatus.

References

[1] I. K. Harvey, J. C. Macfarlane, and R. B. Frenkel, Metrologia 8, 114 (1972).

[2] B. N. Taylor, Metrologia 12, 81 (1976).

[31 W. Chiao, R. Liu, and P. Shen, IEEE Trans. IM-29, 238

(1980).

[4] P. Vigoureux and N. Dupuy, in CPEM Digest 1980

(abstracts of the Conference on Precision Electromagnetic

Measurements, Braunschweig, 1980, IEEE cat. no. 80CH1497-7 IM), p. 47.

[5] A Fau, Bull. Bur. Nat. Metrologie (Paris) 10 (36), 9 (April

1979); N. Elnekave and A. Fau, these proceedings.

[6] P. Giacomo, Metrologia 16, 55 (1979).

[7] W. K. Clothier, Metrologia 1, 181 (1965).

[8] A. H. Cook, Philos. Trans. R. Soc. London, Ser. A: 254, 125

(1961).

[9] Yu. D. Boulanger, private communication.

[10] B. E. Blaisdell, J. Math. Phys. (Cambridge, Mass.) 19. 186

(1940).

[11] D. J. Wright, Proc. Phys. Soc. London, Sect. B 68. 297

(1955).

[12] N. K. Adam, Tfw Physics and Chemistry of Surfaces (Oxford University Press, London, 1941), p. 336.

[13] W. K. Clothier, G. J. Sloggett, and H. Bairnsfather. Opt.

Eng. 19, 834 (1980).

473

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Status of the Measurement of the NBS Ampere in SI Units

P. T. Olsen, W. D. Phillips, and E. R. Williams

Electrical Measurements and Standards Division, National Bureau of Standards, Washington, DC 20234

We report on the status of a preliminary measurement of the NBS as-maintained ampere in terms

of the SI, or absolute, ampere by a method which avoids the usual determination of dimensions of current-carrying coils. As a result, a major source of uncertainty is removed from the experiment. The

preliminary work reported here has a statistical uncertainty on the order of 1 ppm. Systematic errors have not been fully evaluated, but we believe them to be on the order of 30 ppm or less. Key words: absolute ampere; current balance; fundamental constants; NBS ampere; Pellat balance; SI ampere.

1. Introduction

Electric current is routinely measured in the laboratory in terms of the voltage it produces across a standard

resistor. The United States practical or NBS ampere is that current which vi^ould cause a voltage drop of one NBS volt across one NBS ohm. By contrast, the SI ampere is defined in terms of forces betv\^een current car- rying conductors. Since electrical experiments are invariably performed using the practical electrical units, the

ratio, Ka., of the practical ampere to the SI ampere is an

important, albeit artificial, fundamental constant [1]. Values of Ka obtained indirectly from measurements of

other fundamental constants are in significant disagreement vdth each other and with direct measurements us- ing current balances [2, 3]. A major improvement in the

accuracy of a direct determination of Ka would therefore

shed Hght on the possibility of errors in the present

determinations of such constants as the gyromagnetic

ratio of the proton, the Avogadro constant, and the Fara- day constant, all of which can be used for indirect determinations of KaThe various direct and indirect values of Ka differ by

about 10 ppm and the 1973 recommended value of Ka

may be in error by that much [2]. The measurement re- ported here has a precision of about one ppm and is itself only the first step in a program which may ultimately

yield Ka to an accuracy of 0.1 ppm. This effort, then, is an attempt at improving the present knowledge of a by

as much as two orders of magnitude.

2. Theory

In all previous direct determinations of Ka (absolute

ampere experiments^), the largest contribution to the uncertainty was from the measurement of the geometrical

dimensions of some physical object, most commonly the

diameter and pitch of a coil. A new approach [4] avoids

such measurements and may be envisioned as a comparison of electrical work measured in NBS electrical units to mechanical work measured in absolute or SI units. In practice this comparison is realized by a two part

experiment.

Part one: A coil in motion in a magnetic field generates

a voltage across its terminals resulting from the time

summary of recent absolute ampere experiments is given in Ref. [1].

rate of change of magnetic flux in the coil (d^/dt). The

coil motion is kinematically constrained to follow a fixed

path. A laser interferometer measures its z displacement. An active servo system forces d^/dt to be nearly constant from time to t-z enabling the magnitude of the

voltage to be measured accurately in terms of the NBS

electrical standards:

^^bsKaKq ^ d<^/dt (1) where KqIs the ratio of the NBS ohm to the SI ohm.

Part two: If at a given position z the coil carries a

current, expressed in NBS units as /nbs(2). it will ex- perience a force in the z direction which can be measured

in terms of a known mass m, and the local gravitational

acceleration g. The relevant expression is: F ^ mg ^ KAlNBsid<^/dz) . (2)

If the motion of the coil system is such that time 1 1 oc- curs at position Zi and t2 occurs at position Z2, then from

Eqs. (1) and (2)

A^I) - KAKal'^%NBs(t)dt

mg

K, X

1 /nbs(z)

dz (3)

or

K.

mg

'2 1 /nbs(2 ) dz

Ka I ^(^NBsCOdi

1/2

(4)

3. Experiment

3.1 Apparatus

The apparatus used is the Pellat-type "electrodynamometer" or current balance showoi in Fig. 1. A

central coil (B) with its axis vertical is mounted on a bal- ance beam. The coil and balance are placed in a uniform

horizontal magnetic field (in this case, the magnetic field of a long solenoid). The magnetic field direction (H), the

axis of the coil, and the edge of the balance knife (C) are

475
Figure 1. Schematic representatmi of Pellat-type ampere bal- ance. A: Torquing coils; B: Main coil; C: Central kyiife edge; D: Corner reflector (moving arm of interferometer); E: stand- ard mass; H: Applied stationary magnetic field (supplied by

surrounding solenoid)

.

approximately mutually orthogonal. When free to swing,

the balance rotates in a vertical plane as does the axis of the central coil, so that the flux in the coil varies as the

sine of the angular displacement from the vertical. A

current in the coil produces a torque on the balance

which is proportional to /(j x H) where j is the coil axis vector and H is the applied magnetic field vector. The

torque produced is then countered by an appropriate mass E placed on the scale pan of the balance. The

torque due to this mass m is proportional to m(l x g)

where 1 is the balance arm vector connecting the central knife to the scale pan knife and g is the acceleration of gravity. Thus, if the central coil and the solenoidal field are properly aligned and the solenoid field is uniform, the mass needed to balance the electromagnetic torque will be independent of the balance angle.

A corner reflector attached to the scale pan assembly

is part of a laser interferometer system that is used to measure the vertical displacement z of the scale pan as the balance rotates. The interferometer axis is adjusted

to within 10 ^"^ rad of vertical by autocollimation from a mercury pool.

3.2 Evaluating the Voltage Integral

A block diagram of the system for the voltage measure- ment is given in Fig. 2. When the balance is in motion,

the voltage generated in the central rotating coil is compared to a 10 mV reference voltage. The voltage differ- ence, or error signal V^, is amplified by a null detector

and a current proportional to Ve is applied to three other

A/D

Figure 2. Block diagram for voltage integral determination. T: Torquing coils; M: standard mass: ROT: Main rotatable coil; N: Null detectors; A: servo amplifier.

coils attached to the balance beam.^ These three torquing

coils apply a torque to the balance which changes the ro- tational velocity, completing the servo loop so that the

generated voltage is nearly equal to the reference volt- age. The torquing coils [two of which are seen as A in Fig. 1, the third being inside B] are arranged so that currents in them induce no flux into coil B. The reference voltage is generated from the voltage

drop across a 10 H resistor (Vio). A constant current supply delivers a current which passes through a 1 kft resistor and the 10 fi resistor. The voltage drop across the 1 kfi resistor, Fik, is compared, using a null detector,

to the voltage of a calibrated standard cell, Fstd. and the current is adjusted to null (Kstd ~ ^ik)- By knowing the

ratio of the two resistors and measuring the error volt- age Ve, the voltage generated in the rotatable coil can be

determined in NBS units. When the scale pan gets near

the end of its travel, an optical detector switch reverses

the direction of the current through the 1 kO - 10 0

resistor chain thus reversing the polarity of the reference

voltage. The servo then reverses the rotation direction of the balance, and data is recorded for the opposite sense

of rotation. The total distance between turn-around

points is about 2 cm. The laser interferometer, mixer, and fringe converter system produces a pulse for each X/4 (0.15 |xm) of vertical travel of the scale pan. The pulse appears on one of two channels depending on whether the pan is moving up

or down. An up-down counter is used to determine the

relative position of the pan. A divide-by-32 counter pro- vides a trigger once every 32 pulses which initiates the

reading of a 1 MHz-clock-driven scaler and a reading of the error voltage by the analog-to-digital converter. The

computer, interfaced to the measurement instrumentation involved, typically stores over 100 averages of such

readings. Each average value of Vf.{i), z{i), and tii) is the result of 50 individual measurements. One voltage in- tegral determination consists of 10-12 consecutive re- petitions of such measurements, with alternating sense of rotation, with the region near the turn-around points

(where the balance rotation changes direction) discarded.

At a later time the voltage integral is computed numerically from the stored averages V^ii ) and t{i).

3.3 Evaluating the Force Integral

The torque produced by -3.3 mA in the rotatable coil will support a mass of ~5 g on the scale pan. Thus, when

the current is in such a direction as to produce a downward force on the scale pan with the corner reflector on

it, the opposite scale pan must have a 5 g counter mass

on it for equilibrium. When the current in the rotatable

coil is reversed, a calibrated mass of —10 g is placed on

the pan with the corner reflector. The weight of the 10 g mass gives us the force nig in Eqs. (3, 4).'* It remains to determine I sssi^ ), the change of current in the rotatable

central coil which balances the change of a 10 g mass on

the scale pan. As shown in Fig. 3 the computer reads from, the

up-down counter the position of the corner reflector attached to the scale pan. This is compared to a preset

position, z,, and the computer generates, via the digital to analog converter, an analog correction signal which is

-For stable servo operation, currents proportional to the time integral and derivative of are also applied. ^The acceleration of gravity is measured by a portable absolute gravimeter. See the paper by J. A. Hammond et al. in these proceedings.

476
-(n) 'liool

FRINGE CONV

Figure 3. Block diagram for force determination.

used to drive current through the rotatable coil, completing a servo loop which holds the scale pan position constant to about X/4. The current in the coil is measured by

comparing the voltage drop it produces in a 300 fl resis- tor with a standard cell voltage Fstd- The difference volt- age is read by the calibrated null detector (N) and

recorded by the computer. The 10 g mass is removed, the

current reverses to maintain the scale pan at the

preselected position 2,, and the current is measured

again. Ten to eleven current reversals are accomplished

in 30 minutes and constitute one "force measurement."

In principle one should determine the balancing current

I \Bs(2 ) for a number of positions so as to calculate the

integral of Eq. (4). We have found it more convenient,

however, to infer the z dependence of the current from

the z dependence of the voltage integi'al, evaluated over segments shorter than the full path length z-z - Zi. Thus,

at the present time, we directly determine /nbs(2 ) at one

point only.

4. Analysis, Results, and Discussion

The voltage integral measurements for both senses of rotation and the force-current measurements for mass on and off are combined so as to cancel the effects of linear

drifts (for example, drifts in magnetic field or rotating

coil dimensions), thermal emf s, and null detector zero

offsets. This procedure yields a set of alternating voltage

integral and force-current determinations taken over a period of several hours on any one day. These are further

combined to eliminate the effects of drift and used to cal- culate Ka according to Eq. (4). Figure 4 shows 16 such determinations of K^, plotted

with respect to their unweighted mean, taken on four dif- ferent days. (We specifically do not quote a value for Ka

because of the possibility of a large systematic error as discussed below.) Each point represents about an hour

of running time. The individual uncertainties are based

on the standard deviation of the measurements (after

correction for drift and offsets) from which each value of Ka was calculated. The dotted line shows the one standard deviation limits for the entire set—less than 2 ppm,

and consistent with the individual uncertainties. These

results indicate that a statistical uncertainty better than

1 ppm is easily achievable in a reasonable period of tim.e. In addition, some of this scatter is probably due to instability in the current sources and ambient temperature,

and to poorly damped swinging of the balance pans, problems which can easily be corrected in future work.

Figure 4. Individual values of K'X = - K^^"'. Solid error bars indicate standard deviations of individual points; dotted

lines indicate the standard deviation of the entire set. Points

are plotted in chronological order. Systematic uncertainties due to calibration of the mass,

resistors, standard cell, laser wavelength, time standard,

as well as uncertainties in the determination of the ac- celeration of gravity and the buoyancy correction are all smaller than the statistical scatter and can all be made

smaller than 0.1 ppm. The largest source of systematic

uncertainty in the present work is in the correction for the bending of the balance beam under the load from the

10 g mass. This effect has been considered in previous

ampere determinations [5]. The bending results in an

error because the gravitational force of the 10 g mass is applied at the balance pan, while the magnetic force is applied near the center of the balance. The distortion of the balance is different in the two cases, so that a weighing error results. We have used data from earlier meas- urements [5, 6] on the balance distortion to estimate this. The correction was on the order of 20 ppm, but it is diffi- cult to assess an uncertainty since the mass and mass dis- tribution of the present balance are quite different from

those existing when the original distortion measurements were made. The corrected value for Ka is within about

30 ppm of any of the previous high precision direct and

indirect measurements as summarized in Ref. [2]. Future

measurements of the ampere by this method will be

designed so that a direct substitution of weight for elec- tromagnetic force is carried out, thus eliminating the dis- tortion error described above.

5. Conclusions

We have obtained preliminary results from a new, dynamic method for realizing the ampere. The statistical scatter of a single determination requiring about one hour of running time is less than 2 ppm. The statistics indicate that a realization of the ampere with sub-ppm

accuracy is achievable. We are now designing [7] and

constructing [8] an ampere balance using superconducting coils to generate the magnetic fields. The geometry

will be such that the major part of the balance distortion

problem will be eliminated. Forces and voltages will be

larger by a factor of 100, so that relative statistical scatter due to measurements of these quantities, based on the experience with the present balance, should be at the parts in 10^ level. We believe that this superconducting version of the experiment will lead to a final uncertainty in Ka on the order of 0.1 ppm.

477
References

[1] B. N. Taylor, W. H. Parker, and D. N. Langenberg, Rev. Mod. Phys. 41, 375 (1969); and E. R. Cohen and B. N. Taylor, J. Phys. Chem. Ref. Data 2, 663 (1973).

[2] B. N. Taylor, Metrologia 12, 81 (1976).

[3] B. P. Kibble, these proceedings.

[4] B. P. Kibble and I. Robinson, NPL Report DES No. 40 (National Physical Laboratory, 1977).

[5] R. L. Driscoll, J. Res. Natl. Bur. Stand. 60, 287 (1958).

[6] R. L. Driscoll and P. T. Olsen, unpublished (1968).

[7] P. T. Olsen, M. E. Cage, W. D. Phillips, and E. R. Williams,

IEEE Trans. Instrum. Meas. IM-29, 234 (1980).

[8] W. Y. Chen, J. R. Purcell, P. T. Olsen, W. D. Phillips, and

E. R. Williams, presented at the 1981 CEC/ICMC Cryogenic Engineering Conference, August 10-14, 1981, San

Diego, California.

478
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

A Feasibility Study of an Absolute Determination of the Magnetic Flux

Quantum*

Ko Hara, Fuyuhiko Shiota, and Teruhiro Kubota

Faculty of Engineering, University of Tokyo, Bunkyoku, Tokyo, Japan

The principle and an experimental study of a new method to determine the magnetic flux quantum

(bo is described. Magnetic energy proportional to <i)(f is substituted for and measured in terms of gravitational potential energy. A superconducting magnetic levitation system consisting of a persistent current coil and a superconducting floating body is employed for this energy substitution. Key words: absolute detei'mination; cryoelectronic measurement; magnetic flux quantum.

1. Introduction

The magnetic flux quantum c})o is essential in defining

the unit of voltage in terms of the Josephson effect, yet to date its value is only known to an accuracy of a few ppm.

This situation is one of the major difficulties in the adjustment of the fundamental physical constants [1]. An entirely new method to determine (±)o, hopefully with

—0.1 ppm accuracy, is proposed and the authors now have

under way an experiment to investigate its feasibility.

A superconducting magnetic levitation system consisting of a persistent current mode coil L and a superconducting floating body M is an energy conserving system.

Such a system with a current source z's and a superconducting switch Sw is shown in Fig. 1. The conservation

relation is E (z) ^ EJ^z) + mgz + mz

constant

.

(1)

Here, z is the position of the center of mass of M along

the direction of gravity, E miz ) is the magnetic energy of the system and depends on z, m. is the mass of M, and g

is the acceleration of gravity.

Figure 1. Schematic diagram of flux-up and superconducting

magnetic levitation in a cofistant magnetic flux mode.

*Work supported by a Grant in Aid for Superconducting Quantum Electronics from the Ministry of Education.

When Sw is opened during the time interval 0 to t with

the initial flux zero, the magnetic flux d) linking the sys- tem is <^^j\jdt, (2) where Vj is the voltage across the coil terminals measured in terms of the Josephson voltage reference. ^ is conserved after Sw is closed. Equation (2) describes the

"charge up" of the system or what may be called the

"flux-up" process. The current i through the coil is measured by a SQUID ammeter throughout this flux-up

period and also in the persistent current state. The equilibrium condition of M at 4) is

d E,„iZe)

dZf. 2 dz.

i (Z e)

2 d

dZf. L(2e)

-mg (3)

Here, z^ is the equilibrium z value of M and Eq. (3) can be understood as a functional relation between 4) and z^. Thus, this magnetic levitation system in equilibrium (i =

0) can be used to substitute magnetic energy for gravitational energy.

2. Measurements with Josephson

Electronics

Josephson electronics are used to make two important

measurements:

(1) I'j is measured in terms of a Josephson voltage

reference and is expressed as

Vj = /ct)o, (4) where / is obtained from the ratio (measured uj /(defined

(j)o value). Thus, its uncertainty merely depends on that

of the Josephson voltage standard, i.e., <0.1 ppm.

(2) The current i is measured by a SQUID ammeter

and is expressed as

i = kn 4)0 . (5) Here, it is assumed that the SQUID is calibrated by a Josephson voltage reference and a resistor calibrated absolutely. The integer n is the counted order of the

SQUID and k is the current to flux conversion factor of the ammeter.

479
Combining Eqs. (2) and (4), cj) fluxed-up over a time in- terval T is . T T

<i> = Vj{t)dt =4)/^ fit)dt ^ N<^o, (6) Where A'^ is tlie number of flux quanta linking the sys- tem. The self inductance of the system is thus simply

L — <^/i = NIkn and the magnetic energy is £'m =

<^il2 = knN<i)i/2.

3. Methods of <\>q Determination

With the above preparation, the following three

methods to determine (t)o are at least possible.

(1) Oscillation method: When the body M is in an oscil- latory motion with upper and lower stationary points 22 and Zi, respectively, where z = 0, we have

Nk

[n{zi) - n{z2)]

mg{z2 - zi)

,

and hence

2mg{z2 - Zi)

Nk[n{zi) - n{z2)]

(7)

The underlying principle of this method is easy to understand but it suffers from damping due to the viscosity of the ambient gas and also to eddy current losses in normal

metals close to the system.

(2) Energy injection method: Prior to the experiment,

the system is assumed to be in an equilibrium state characterized by (t)i = N, cj),, and /; = A' c^,, (^i). At t =

fj, Sw is opened and the system is fluxed-up until t = tf and the system reaches the state <i){ = Nf (bo and if =

k (t)(i n (Zf). During the flux-up period, Vj{t) = 4)0/(0 and

i it) = kn (t) (to are measured simultaneously and the en- ergy injected into the system is E,^j = Iri) it)i it)dt. The total energy difference between the initial state (energy Ei) and the final state (energy E t) is equal to E m), provided that the motion of M is so slow that the kinetic

energy contribution during this period is negligible. Thus

^0 is determined by

<txT k C/f{i )n(t)dt + [Nmizi) - NftHztm ^ 1

(8)

This method is thus exact only at the limit of slow flux- up, but is more convenient at the present preliminary

stage of the experiment.

(3) Equilibrium trajectory method: Integrating Eq. (3) over Zg^to z g.^, we find after some manipulation

I'' i\ z ,

" (z, n Niz,

(9)

This method requires knowledge of 6(2e) to sufficient accuracy that the accuracy of the numerical integration

in Eq. (9) is adequate. It is, however, not bothered by

dissipation.

One thing must be added about the uniqueness of

<^{Ze). M has six degrees of freedom of motion, among

which rotation about the vertical axis has no stable

equilibrium when the system is axially symmetric. When

it is slightly anti-symmetric, there may be multiple

equilibrium positions separated by low barriers. In this case, equilibrium trajectories of M can interchange due to a slight shock and the (t) {z e) relation shows multiple

values.

4. Tlie Coil and Floating Body

The coil and the body are illustrated in Fig. 2 with

their dimensions in mm. The conic body is made of Al

coated with Pb. Inside the cone, a solid corner cube re- flector is mounted for interferometry. The mass of M is 25.24 g.

Figure 2. Coil and floating body.

The coil contains 720 turns in 18 layers of 0.25 mm di- ameter Nb wire on a phenol resin form, the bottom end

of which is closed to prevent liquid He from getting in- side it. The self inductance of the coil without M is 48.6 mH.

The fluctuating current through the coil in the persistent current mode without M is a measure of the

ambient magnetic field fluctuations. This was measured

using the SQUID and is shown in Fig. 3. The fluctuations are on the order of 10 ppm of the working current which

is about 1 A.

2 min.

> time

Figure 3. Current fluctuation due to ambient magnetic field fluctuation.

480
With M floating and a persistent current of about 1 A,

the fluctuation current was measured and is shown in Fig. 4. It is about 100 ppm and corresponds to a vibration of M due to external vibrations on the order of

1 |jLm. This test is a good means to examine the Meissner

state of both the coil and the material of which the levi- tated body is composed.

-> 2 min.

e

-> t i m e

Figure 4. Current fluctuation due to ambient meclmyiical

vibration.

5. Experiments

(1) Experimental set up, Figure 5 shows the experimental arrangement. The voltage I'j across the terminals

and the current i through the coil were measured by a microvoltmeter (2% accuracy) and a SQUID ammeter,

respectively. The coil was operated both in the flux-up

mode and the persistent current mode by switching the

thermal superconducting switch Th Sw which was

shunted by a resistor R of 0.97 mfl and which had a nor- mal resistance of 30 m Q. Accordingly, the flux-up time

constant was about 30 s. The amount of flux was obtained

by gi'aphically integi'ating the recorded I'j. The SQUID

ammeter was operated in digital and analog modes simultaneously and its sensitivity was 0.916 mA/count which

could be interpolated to 10"^ counts. The laser inter- ferometer was operated in both digital and analog modes,

and its sensitivity was 2.56 fjim/count in digital and

0.16 |jLm in analog.

^ Laser interferometer

crp

Figure 5. Schematic diagram of experimental set-up.

(2) Uniqueness of the equihbrium floating state. The

equilibrium floating state can be specified by a set of values of <j), z g, and i g. Figure 6 shows the uniqueness of the floating equilibrium for one down and up round trip.

1.0

0.9

0.8

0.7

X I DOWN GOING

cp J

^ j UP GOING

t 1 TURNING

J I I L

10

20 30 [mWb]

Figure 6. Eqidlibrium floating state dependence on d).

In addition, the uniqueness of the state was examined in- crementally with M —20 )jlA and A2 ~ 2 iJim. In this examination, the system was operated in a constant current mode and a small ac current was superposed. The

resultant oscillation of M and the accompanying oscilla- tion in i were observed to be uniquely correlated to the

limit of the ambient fluctuations.

(3) Energy balance in energy injection method. The

stored magnetic and gravitational energy during flux-up are denoted by Afi'm and ^.Eg, respectively. The ratio (AE'm + A£'g)/A£'inj for the successive equilibrium

points was computed for the data in Fig. 6 and was found

to be 93.5, 93.7, 94.4, 97.9, and 96.4%, respectively. In

this test, it was estimated that the kinetic energj' loss was negligible.

To confirm the overall conbo was computed by the equilibusing the data of Fig. 6. The

result, cj)o = (2.07 ± 0.04) x 10"^^ Wb, is in good agi'eement with the presently recommended value.

(4) Estimation of cbo. sistency of the method, c rium trajectory method

6. Discussion

According to the above, the authors beheve the new

method to be a hopeful way for measuring 60 to high ac- curacy. The main advantages of the method are:

(1) There is no need to machine the floating system

precisely nor to measure its dimensions precisely. Only

its stability during the experiment is required. In this sense, the system does need a unique floating state characterized by a set of values 6, z^. and i^. Accordingly, the system may be flexible as long as the unique- ness of these parameters is satisfied. This feature comes

from utilizing the unique properties of Josephson elec- tronics in contrast to Sullivan" s proposal [2] in which precise measurement of the self inductance of the floating

system is necessary.

481
(2) In this floating system, there is no source of static friction and so it is ideal for substituting magnetic energy

with gravitational energy.

(3) In analogy with an ampere balance, the method can be viewed as a weber balance but with a mass far greater

than that used in the ampere balance. This is a positive

feature which should overcome the difficulty pointed out by Petley [3], if it exists. There are many problems to be solved to reach the

goal of an accuracy of a few tenths ppm;

(1) The most important is to find a superconducting

material of ideal high H^^.

(2) A means should be provided to protect the apparatus

from ambient magnetic and mechanical fluctuations.

(3) The reference corner cube reflector should be at- tached to the coil form. At present, it is attached to the

top of the cryostat.

(4) The electronic measuring system should be improved. It is especially important to introduce a Joseph- son voltage reference and a resistance standard cali- brated against a cross-capacitor.

Finally, the attainable error limit vdll be discussed

briefly. Among the relevant quantities, the limiting fac- tor is the resistance measurement which is restricted to

0.1 ppm. In principle, it is possible to exceed this limit for the other quantities. However, measuring the mass of M may be a great source of trouble because M has a large

surface area which makes it a rather poor mass standard.

References

[1] E. R. Cohen and B. N. Taylor, J. Phys. Chem. Ref. Data 2, 663 (1973).

[2] D. B. Sullivan and N. V. Frederick, IEEE Trans. Magn.

MAG-13, 396 (1979).

[3] B. W. Petley, NPL Report Qu52 (National Physical Laboratory, March 1979).

482
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

The Work Done at the Mendeleyev Research Institute

of Metrology (VNIIM) to Improve the Values of the

Fundamental Constants

Yu. V. Tarbeyev

Ail-Union Researchi Institute of Metrology (VNIIM), Moskovsky prospect 19, 198005, Leningrad, U.S.S.R.

The work on fundamental constants carried out at VNIIM is a part of the overall effort in metrol- ogy aimed at improving measurement standards and systems. The results of work on the Avogadro

constant, the gyromagnetic ratio of the proton, the Rydberg constant, and in associated research fields are reported. Improvements in the tec4iniques for adjusting the values of fundamental constants as well

as nuclear spectroscopy reference lines are discussed. Key words: Avogadro constant; discrepant data; fundamental constants; gyromagnetic ratio of the proton; least-squares adjustment of the constants; Rydberg constant.

I 1 . Introduction

The transfer from a system of standards based on the

i SI units to a system of standards based on the fundamental physical constants, i.e., to natural standards, should

j be considered as a major problem in the progress of

metrology. Among the many tasks essential to the solu- tion of the above problem, the decisive one is the task

that involves the determination and improvement of the

values of the fundamental constants. On the one hand the

accuracy of these constants characterizes the depth of our understanding of physical phenomena and the laws

governing their interrelation; and on the other hand, the

development of new standards based on fundamental constants is the best way to carry out a qualitatively new

realization of the units for physical quantities.

Advances in modern physics have already made it possible to develop standards for the second and the meter

based on the method of frequency counting. Thus, a new

standard for the meter based on a He-Ne laser stabilized using saturated absorption in ^^l2 was developed and put

into operation at VNIIM in 1980. Additionally, the high

accuracy of the experiments for determining the velocity

of light in vacuum has enabled the metrology community

to raise the question of how best to use this accurately

known constant and of fixing its value by international

agreement. This could eventually permit the adoption of new and more accurate definitions of the meter and the

second based on the wavelength and the period of the same radiation. At present the problem of developing a "natural" standard for the kilogram is on the agenda.

Indeed, in recent years such a possibility has opened up

as a result of precision experiments on the determination

of the Avogadro constant.

2. Improvement in the System of Standards

for Electrical Quantities

Primary emphasis is being given at VNIIM to the improvement of the system of standards for electrical units

[1] which are essential not only for the metrological as- surance of electrical instruments, but also for refining

the methods and measuring instruments used for a

number of non-electrical quantities by converting them

into electrical ones. The precision of the absolute realizations of the major

electrical units depends to a great extent on the accuracy

of the determination of absolute values of the fundamental physical constants. For example, the accuracy of the

USSR national primary standard for the volt based on

the Josephson effect which has been newly developed at VNIIM depends mainly on the accuracy of the flux quantum h/2e as determined at VNIIM in absolute (SI) units on the basis of its physical connection with other fundamental constants of a "mechanical" nature. Values of

these constants are determined in terms of the basic SI

units, with no electrical units used. The relevant equation is:

h _ c 2e SI 4 me a Mp

wip i2x A^A

where c is the velocity of light in vacuum; mjm-p is the

electron-to-proton mass ratio, equal to (|Xp/|jlb)/(|Xp/m.n) (|J-p is the proton magnetic moment, |Xb is the Bohr magneton,

and |Xn is the nuclear magneton); Mp is the atomic mass

of the proton; a is the fine structure constant; is the

Avogadro constant; and R is the Rydberg constant. The

accuracy of hJ2e depends on that of the above constants with the value of the magnetic constant ixq =

4-n- X 10"'^H/m following from the definition of the base

SI unit for current, the ampere.

In recent years theoretical and experimental studies

have been carried out at VNIIM for improving the above

fundamental constants, with the result that the magnetic

flux quantum h/2e = 2.0678410(10) x lO'^^ Wb was ob- tained for the realization of the absolute volt using Eq.

(1), the relative standard deviation in this case being

5.6 X 10-\ The new value of h/2e defined at VNIIM

differs from the previous one adopted by BIPM on the

basis of the adjusted (CODATA 1973) value of h/2e =

2.0678506 x 10"^^ Wb with a greater uncertainty (standard deviation), namely 2.6 x 10"''. The differences in the values of h/2e adopted in the dif- ferent countries which use the Josephson effect to realize their units of voltage is reflected in the differences in the

483
various national units of emf and voltage. A comparison

of the primary standards maintained at the NBS and

VNIIM (carried out at VNIIM in 1975) showed the mean

difference to be 4.25 \xV. The result agreed with the

difference in the national units of the USSR and USA ob- tained from the 1973 international comparison at BIPM

(Vnbs - VvNiiM ^ 4.22 (xV). The latest comparisons

between the VNIIM standard cells and the NBS standards (carried out at NBS in 1976) gave a greater differ- ence, i.e., 9 fxV. Such a large discrepancy in the 1976

results was probably due to unfavorable conditions of transport of the VNIIM cells to the USA. The compari- son of the VNIIM standard cells carried out at BIPM in 1980 gave a much better result: the mean difference in the emf values obtained at VNIIM and BIPM was

2.15 fxV. It is obvious that further theoretical and experimental studies for the improvement and adjustment of the major fundamental physical constants should be con- ducted to provide international uniformity for the electrical and magnetic units. The importance of this work,

which is to be carried out on a world-wide scale, is con- firmed by the recommendations (El 1976 and El 1978) of

the BIPM Consultative Committee on Electricity put for- ward by the Soviet delegation. The work carried out in this field at VNIIM is briefly outlined below.

3. Work on the Improvement of the Values

of the Fundamental Constants

In order to improve the system of USSR national

standards, work on the redetermination of a number of fundamental constants is currently underway at VNIIM.

3.1 The Proton Gyromagnetic Ratio, y'p

In 1980 a cycle of measurements for a new determination of y'p using the low magnetic field method was completed. The new data were obtained as a result of the

work to determine y'p conducted from 1958 to 1968 [2, 3]. The proton gyromagnetic ratio y'p is determined from

the ratio of the proton free precession frequency Wp for a

spherical sample of distilled water placed in the magnetic

field of a standard current carrying coil, to the magnetic

induction of this field. The magnetic induction is found

using the value of the coil constant K as determined by

measuring the diameters of and spacings between the coil windings as well as from the strength of the current /

flowing through the coil. The current / is measured by

balancing the emf of a standard cell E against the voltage

drop across a standard electrical resistor R connected in series with the windings of the standard coil. Thus, the

value of 7p is found from the expression

~KE

For the redetermination of 7p over the period 1977 to 1980, a new apparatus was developed. It is comprised of a new type of standard magnetic induction coil which per- mits the magnetic induction to be increased by a factor of 5 to 10, the volume of the region with a magnetic field of the required uniformity by a factor of 50, and the effect of the turns diameter measurement uncertainty to be more than halved, in comparison with the Helmholtz coils used before. New equipment was also developed for the contact measurement of the turn-to-turn spacing of the standard

coil, for stabilizing and measuring the strength of the current / for the nuclear precession frequency measurements, and for compensating the earth's magnetic field.

All this made it possible to reduce the corresponding

components of the uncertainty in the determination of 7^ by a factor of 3 to 5 in comparison with the data obtained

in the previous VNIIM 7p experiment.

The new standard magnetic induction coil [4] which

comprises a four-section solenoid on a 300 mm diameter,

500 mm long cylindrical quartz form sets up a 5.7 x

lO"'' T magnetic field with a current through the coil of

1 A. The experimentally determined temperature coeffi- cient of the constant K is 0.56 x 10^^//°C and the nonuniformity of the magnetic induction within the working

volume (a 55 mm diameter sphere) is no more than 1 x

10"^. The coil turn-to-turn spacing was measured by a

laser length measuring device for each pair of turns symmetrical with respect to the coil center along eight generating lines. The systematic error bound is no more than

0.2L fxm, where L is the measured distance in meters. The measurements were computer-controlled with 100

observations made at each point. The systematic error in the turn diameter measurements which were performed

over each section for each turn is no more than 0.8 \xm.

The wire diameter measurement error is no more than

0.1 \xm. Temperature corrections to the standard coil constant K were determined from resistance measure- ments of auxiliary windings wound on the quartz form.

The nuclear magnetic resonance frequency was measured by the free nuclear precession method using

separated receiving and polarizing coils. The spherical

sample of distilled water was 40 mm in diameter. The

time required for a single measurement was 2 s and the

relative standard deviation of the observations was no more than 2 x 10"^. The effect of the magnetic properties

of the nuclear precession converter components, of the

attenuation of the nuclear precession signal amplitude,

and of the screening effect of the windings and forms on

the accuracy of the precession frequency measurements, was determined experimentally. The results of the

analysis of the above error components in the determination of 7p and other components estimated to be 1 x 10"'^ or larger are given in Table 1. The proton gyromagnetic

ratio obtained in terms of the USSR national volt and

ohm units is found to be

7'p = 267512.57 x 10'^ s TyiiuM

The total error bound 0 of the proton gyromagnetic ratio determined in terms of the USSR national units with a confidence level of P = 0.99 is no more than

0 = 1.4 (S 02)^ = 1.5 X 10"^

Estimated in terms of a standard deviation (P = 0.67),

the same error is 0.6 x 10~^.

3.2 The Avogadro Constant, A/^

One of the constants governing the accuracy of hl2.e as obtained from Eq. (1) is the Avogadro constant. The lat- est advances in the measurement of A^a- which made it possible to reduce the measuring error by a factor of over 10, hold much promise for the further refinement of

its value. Impressive progress in atomic mass measuring

techniques and a new approach to the solution of the density problem reduced the uncertainty of these measure- ments to the level of ~5 x 10"''. These achievements as well as the development of an x-ray-optical interferometer, which permitted the monocrystal lattice constant to be measured with an accuracy of better than 1.5 x 10"^,

provided the basis for the precision measurement of Np^. Using this procedure, the Avogadro constant is deter- mined from

484
Table 1. Error bounds for the proton gyromagnetic ratio (P = 0.99).

Relative uncertainty

Contributing factors (X 10-'^)

1. Standard coil turn-to-turn spacing measurement 3

2. Standard coil turn diameter measurement 6

3. Standard coil wire diameter measurement 1 4. Uncertainty in the shape of the coil winding 3

5. Correction for the current distribution over the wire cross-section 1

6. Correction for positioning the connecting leads and return le^d 2

7. Magnetization of standard coil form 2

8. Standard coil temperature measurement 2

9. Measurement of precession decaying signal frequency 2

10. Correction for screening effect of windings 2

11. Calibration of standard cells 2

12. Instability of standard cells 2

13. Standard cell temperature measurement 2

14. Spurious emf s 1 15. Standard resistance calibration 3

16. Standard resistance calibration measurement 2

17. Correction for load dependence of standard

resistance 2

18. Correction for the influence of the standard coil on the earth' s magnetic field compensation system 1 19. Undercompensation of the earth' s magnetic field 1

where M is the atomic mass of the substance (sihcon), / is the number of atoms per unit cell, p is its density, do is the crystal lattice constant, and $ is the unit cell geometric factor which is unity for cubic crystals such as

silicon. Higher accuracy in determining the atomic mass and

density can be achieved with monocrystal silicon enriched

by ^Si up to 99.99 percent and grown by a special process that produces silicon free from defects, dislocations,

and impurities. The development of new density standards produced from monocrystal type substances featur- ing a stable structure is the basis for higher accuracy

measurements of p. Solving these problems as well as obtaining a higher

accuracy in the optical and x-ray measurements will lead

to a decrease in the (relative) uncertainty of the Avogadro constant to about 1 x 10"'. A still higher accuracy in the determination of can be obtained from the accu- mulation of data gained in equally precise experiments

carried out in different laboratories and by different

methods. This will make it possible in the final analysis

to reduce the uncertainty in Na by approximately one more order of magnitude. To solve the problems specified

above, work is being carried out at VNIIM and a number

of other institutions on the development of monoisotopemonocrystalline silicon, x-ray optical interferometers,

and density standards based on samples perfect in shape

(sphere, cube) and made from quartz or glass. It is antic- ipated that the use of monocrystaUine structures and an

x-ray optical interferometer will make it possible to eventually transfer to a natural standard for the unit of mass.

3.3 The Rydberg Constant,

Though further refinement of the Rydberg constant is not immediately required in order to use Eq. (1), it is of great importance for improving our knowledge of the

whole set of fundamental physical constants. The Rydberg constant, R^, describes the binding energy between

an electron and the atomic nucleus and as such is a major

fundamental constant. A promising method of precisely

measuring i2oc is the two-photon absorption technique

used with the IS —> 2S transition in hydrogen [5] in a standing wave field. The elimination of first order

Doppler line broadening enables one to obtain narrow

resonances whose relative width is ~ 10"^^ This opens up

the way for a measurement accurate to < 10"^'' (the un- certainty of the present value of /? x is 1 x 10"^). In order to make such an experiment possible, a number of technical problems must be solved: (1) development of a 243 nm ultraviolet continuous-wave

tunable laser featuring a single frequency and fair radiation power; (2) development of a system for frequency

measurement over the optical range corresponding to the

transition selected; (3) development of a hydrogen atomic

beam source featuring a high intensity in the region of interaction with the radiation, a greater extent of dissociation, and as low noise as possible; and (4) development

of a system for recording metastable hydrogen atoms in the 2^Siy2 state which is essential to obtain a maximum

signal-to-noise ratio. The above method for measuring R^ds being employed

at VNIIM and the Institute of Thermal Physics (Siberian

Branch of the Academy of Sciences of the USSR). A

thermal-type hydrogen beam source whose intensity is ~ 10^^ atoms /(cm^-s) at 10 to 15 cm and a degree of disso- ciation of about 95 percent at some 2700 K, has been

developed at VNIIM. Such parameters for the source make it 'possible to reduce contributions to the two- photon resonance line width due to the second-order

Doppler effect and line broadening due to the time of interaction between the beam atoms and the laser radiation. This enables us to obtain the resulting relative reso- nance hnewidth of about 8 x 10"^". A further increase in the atomic beam density over the interaction region us- ing multi-channel shapers (capillary arrays) and through

a decrease in the effective temperature of the beam will

485
make it possible to obtain a two-photon absorption

linewidth of no more than 3 x 10"^^ This will open up

the way for the determination of the Rydberg constant with an accuracy of the same order.

3.4 Magnetic Flux Quantum, h/2e

To improve the value of the magnetic flux quantum

h/2e, a levitated superconducting mass method which in- volves a comparison between mechanical and electrical energy is being experimentally studied [6, 7]. An elec- tromechanical system under development at VNIIM is an attempt to develop this method for measuring h/2e in absolute units to an accuracy of about 10"^. The levitated body, a thin-walled niobium cone [(1) of Fig. 1)] with a vertex angle of 15°, is mounted on the

thin-walled niobium tube guide (2). The cone can move

together with its guide in the direction preset by the axes of the two superconducting magnetic bearings [(3)

and (4)]. The bearings comprise superconducting coils wound on niobium split forms. The bearing axes are vertically oriented and the 300 mm spacing between the

bearings ensures that the vertical deflection of the guide

is no more than one minute of arc. The upper bearing and

part of the guide are furnished with special slots designed to prevent the body from rotating around the

vertical axis. The windings of the support coil (5) located

inside the cone are composed of Bitter-type disks. The

disks, made of 50 |xm thick niobium foil, are insulated

from each other with mica spacers. The coil outside the

conical surface has the same cone angle as the levitated

body. Located inside the upper part of the guide tube is

\

Figure 1. Schematic diagram of the levitated superconducting

body portion of the apparatus for the new VNIIM experiment

to determine the flux quantum h/2e in SI units. 1: levitated body of thin-walled niobium; 2: thin-walled niobium guiding

tube; 3 and k'- superconducting )nagnetic bearings; 5: support

coil uAnding.

a corner-cube reflector of a laser interferometer, and its lower part accommodates an electrodynamic device made

up of a fixed feedback coil inserted in the gap of a per- manent magnet. This apparatus enables the experiment

to be carried out by the method put forward by B. P. Kibble [8]. The electromechanical system is placed in a niobium

superconducting shield located in an evacuated chamber

inside a helium cryostat which is installed on a shockmounted platform. The levitated mass is about 1 kilo- gram and the support surface area is —180 cm^. Current

from a SQUID-regulator and the feedback current from

the laser interferometer electronic circuit are fed to the

support coil, which makes it possible to fix the levitated mass at distances from the support coil that are an in- tegral number of interferometer fringe spacings.

A standard resistance is switched into the support coil circuit and the voltage across its terminals is measured

in terms of hfl2e using a Josephson voltage standard.

With the levitated mass moving, the emf across the sup- port coil is measured in the same fashion. The principal

uncertainty components have been discussed in Refs. [7] and [8].

3.5 An Improved Method for Measuring the

Combination of Fundamental Constants TpCm-'p/m-n)

Using a Static Mass Spectrometer

The procedure under development at VNIIM is based

on the mass spectrometric method for measuring the

above constants worked out at the loffe PhysicoTechnical Institute (PTI) [9]. Unhke the PTI method,

where a calibrated increment of energy was imparted to ions and the corresponding change in the diameter of their orbit was measured, the VNIIM procedure involves

the measurement of the total energy of the ions and the

total diameter of their orbit. In the method,

measurement of these basic parameters is based on full deceleration of ions in a gap with a calibrated potential

difference taking into account the distribution of ions over the width of the analyser slits. Calculations show

that the method under development provides a more

accurate measurement of the above parameters in comparison with the PTI method [9], "and will allow

measuring the set of fundamental constants, ^pdJi^M-N) to better than 0.5 x 10"^. Since to date the proton magnetic

moment in nuclear magnetons, \i^y\y.A, has been measured

to 0.43 X lO"'', the method under development will make

it possible to obtain a new value of 7p accurate to 0.66 x

10~^. The units for ^J, obtained in this method correspond

to those of "Yp as obtained in the low magnetic field method while phsyically the experiment is carried out in a strong field. This distinguishing feature of the mass- spectrometer method should help to eliminate some of the discrepancies concerning the proton gyromagnetic

ratio data [10] when adequate accuracy is attained.

4. Refinement of the Data Handling

Procedure for the Adjustment of the

Fundamental Constants

The growing importance of the fundamental constants

for many problems in science and technology, especially

for metrology, places more stringent requirements on the

reliability of a self-consistent set of their values. Considerable attention has always been given to this particular

problem and it was extensively discussed at the First

International Conference on Precision Measurement and

486
Fundamental Constants [11]. It is obvious that the relia- bility of the adjusted values of the fundamental constants

is determined not only by that of the results obtained in the different experiments which provide the basis for the

adjustment, but also by the procedure used for handling

the data that culminates in a self-consistent set of values

for these constants. A distinguishing feature of the data

handling procedure employed for the fundamental physical constants is the necessity to solve the problem of

their inconsistency. Investigations [12] show that the reliability of a final set of adjusted values for the fundamental constants can be increased appreciably by the

choice of an optimum way for solving this problem.

Analytical studies carried out at VNIIM of the generalization of the methods for eliminating inconsistencies

among the data on the fundamental constants (methods

which have been improved over the years by R. T. Birge

[13], and E. R. Cohen and J. W. M. DuMond [14], B. N.

Taylor, W. H. Parker, and D. N. Langenberg [15], and

E. R. Cohen and B. N. Taylor [16]), have made it possi- ble to draw a number of conclusions.

There are several methods at present for eliminating

inconsistent data, among which two main approaches may

be distinguished. One method consists of increasing the

variances for the whole set of data by the same general

scale factor equal to the Birge ratio [13]: R 2 (2) where is the calculated value of and F is the

number of degrees of freedom for the set of data. The

use of the Birge ratio impHes that the data variances' are assumed to be known with equal accuracy (i.e., all are

equally reliable), and that the general scale factor is equally well known.

The other method consists in the analysis of variance

[14] which leads to the choice of a particular sub-group of consistent data out of the entire set of input data. In the

1973 adjustment a combination of these methods was

used, that is, part of the data was excluded from the treatment and for individual data groups the uncertainties were expanded by the proper Birge ratio. Introduction of a set of expansion factors Rf = a,' ^ / af for the a

priori uncertainties af is common to the above methods.

These expansion factors are introduced so as to meet exactly or approximately the condition

(3) for the new uncertainties ^. Since there is no objective way of selecting specific

values for the factors Ri^ (at that stage of the adjustment

when the data have already been subjected to preliminary analysis and are ready for statistical treatment), all values should be considered as more or less equally probable. Thus the main idea underlying the improvement in the adjustment procedure consists in the randomization

of a set of possible values for the factors Ri^ or

equivalently, the a posteriori variances ctJ^. If no rigid assumptions are made as to the statistical mechanism governing the measured values x? and uf, the

method for their generalization becomes evident when

their common distinctive features are considered. This

method can be expressed by Eq. (3) and by

MiRh = max,

where ^{R^^) is the probability density function of dif- ferent values toY.Rx^. Choosing $ in the form of different

distributions, one can easily derive both the usual adjustment methods and their generalizations. For example,

the method characterized by Eq. (2) corresponds to the

uniform distribution; the analysis of variance method to the delta function shaped distribution; and so on. The

algorithm

S(l 2 \2 Vaf) mm (4)

corresponds to the assumption (most natural from the

physical point of view) that within the body of data under

treatment, lower probabilities correspond to greater rel- ative errors in the a priori variances [17], with the nor- mal distribution adopted for <I>. The results obtained using this algorithm [i.e., Eqs. (3) and (4)] and some modified versions have been studied at VNIIM and NBS within the framework of their joint

cooperative program in metrology. It was shown that the

application of this algorithm to the input data of the

three recent adjustments of the constants (1963, 1969,

and 1973) changes the adjusted values by more than the

standard deviations specified for them, with the changed

values in good agreement wdth the improved adjusted

values found by taking into account the most recent ex- perimental results. More specifically, it was found that

the relative deviations of the adjusted values calculated

using Eqs. (3) and (4) from the improved (i.e., most re- cent) values of the constants (also calculated from these

equations) are considerably less and have a smaller

spread than those obtained by the analysis of variance

(Fig. 2). Thus, the results of the studies carried out at VNIIM show that the suggested algorithm, which generalizes the usual methods for adjusting the constants,

substantially increases the reliability of the final set of the adjusted values for the fundamental physical constants.

1963 1969 1973

Figure 2. Average relative deviations t - Mmpr I ^ad] adjusted values of the constants. Here Aadj

is the adjusted value of the constant; dadj is its one standard

deviation uncertainty resulting from the adjustment; and

Aiynpr is the current improved adjusted value resulting from

the application of Eqs. (3) and (h) (i.e., tJie VNIIM algorithm)

to the best data currently available_ on the fundamental constants. The curve marked I is for Aq^^j taken to be the results of the 1963, 1969, and 1973 adjustments, respectively

[U, 15, 16]. The curve marked II is for A^dj taken to be the results of the application of the VNIIM algorithm to all of the data available at the time of the 1963, 1969, and 1973 adjustments, respectively. The numbers in parentheses give the spread in t.

487
5. Nuclear Spectroscopic Reference Lines

The last five years have been marked by a substantially increased precision in the determination of energies

and relative line intensities of nuclear electromagnetic

radiation. Therefore, the problem of choosing the energy

reference lines (-y-ray standards) needed to make the rel- ative measurement process convenient and sufficiently

precise has become particularly pressing. However, too great a number of reference lines could make numerous

intercomparisons necessary, which would lead to difficul- ties in evaluating the situation at any instant of time. An

obvious way out is to establish a system of reference

lines as first suggested by Associate Member of the

Academy of Sciences of the USSR, B. S. Dzhelepov in his book Methods for Developmeyit of Complex Decay

Schemes (1974). In such a system, a particular reference

line is taken as the main line and other reference lines are compared primarily to the main one. A classification of energy reference lines developed at VNIIM is based on the principle that the main metrological line (presently ^^Kr, X = 6058 A ) should be the first order reference line. Draft rules for introducing the

second, third, and fourth order reference lines for nuclear spectroscopy have also been proposed. Simultane- ous analysis of all of the requirements which a reference

line of the second order should satisfy, has led to the conclusion that the best choice is the 411.8 keV, ^^^Hg 7-ray

line though there are a number of ^-ray transitions of

^^^Yb, ^^^Ir, etc., which are now measured with nearly

the same accuracy. However, reasons why the x-ray line WiiCal should be temporarily regarded as the reference

line of the second order have been discussed.

Reference lines of the third order should be determined by direct comparison with those of the first and

second order. They should be compared at the current

level of precision by methods based on well known physical laws. Concurrently they can be introduced using the

two reference lines of the third order already adopted

and Ritz's rule: Ei+E2 = Ez- Based on these rules,

148 transitions were classified as reference lines of the

third order in 1980. Rules for introducing reference lines of the fourth

order have made it possible to introduce 87 reference

lines, and this number is growing rapidly. The main body

of information is provided by Ge(Li) spectrometers used

for measuring small energy differences.

Another area of interest is reference ratios of intensities. In nuclear spectroscopy there are methods that

allow the spectral line intensity ratios to be measured

with an accuracy of 2-3 percent. However, there is an in- direct method which permits these ratios in some cases

to be determined to 2 x lO"-" percent. At VNIIM 46

reference ratios have been calculated to better than one

percent [18]. The experience gained at VNIIM shows that the scope

of work carried out on the fundamental physical constants makes it necessary to pose the question of coordinating these studies within the framework of the various

international metrological organizations. First of all, we

would like to suggest for discussion a system of energy

reference lines (7-ray standards) developed in the USSR

with a view to working out an international system of 7- ray standards.

In conclusion, the author would hke to express his gratitude to the following researchers at VNIIM who pre- pared material for this review: K. A. Krasnov, A. B. Mamyrin, T. B. Rozhdestvenskaya, N. V. Studentsov, and

S. A. Shestopalova.

References

[1] Yu. V. Tarbeyev, T. B. Rozhdestvenskaya, I. N. Krotkov,

and 0. P. Galakhova, in Measuretnent for Progress in Science and Technologii, ACTA IMEKO 1979, Vol. S, Ed. by Gy. Striker, J. Solt, and T. Kemeny (North Holland,

Amsterdam, 1980), p. 795.

[2] B. M. Yanovskii and N. V. Studentsov, Izmeritel. Tekhn. 6, 28 (1962). [English transL: Meas. Tech. 1962, 482 (1962).]

[3] N. V. Studentsov, T. N. Malyarevskaya, and V. Ya. Shifrin, Izmeritel. Tekhn. 11, 29 (1968). [English transl.: Meas.

Tech. 1968, 1483 (1968).]

[4] N. V. Studentsov and V. N. Khorev, Abstracts of Reports

for the All Union Scientific and Technical Session on Prob- lems of Magnetic Measurements and Magnetic Measuring

Instruments, VNIIM, Leningi-ad, 1972 [in Russian].

[5] L. S. Vasilenko, V. P. Chebotayev, and A. V Shishayev,

Pis' ma Zh. Eksp. Teor. Fiz.' 12, 161 (1970). [English

transl.: JETP Lett. 12, 113 (1970).]

[6] D. B. Sullivan and N. V. Frederick, IEEE Ti-ans. Magn.

MAG-13, 396 (1977).

[7] E. T. Frantzouz, Metrologiya (USSR) No. 1, 43 (1979) [in Russian].

[8] B. P. Kibble and I. Robinson, NPL Report DES 40, (National Physical Laboratory, Sept., 1977).

[9] B. A. Mamyrin, N. N. Aruyev, and S. A. Alekseenko, Zh. Eskp. Teor. Fiz. 63, 3 (1972). [English transl.: Sov. Phys.

JETP 36, 1 (1973).]

[10] B. A. Mamyrin, Metrologiya (USSR), No. 4 (1981) [in Russian]. Ill] Precision Measurement and Fundamental Constants, Ed. by D N. Langenberg and B. N. Tavlor, Natl. Bur. Stand.

(U.S.), Spec. Publ. ;343 (Aug. 1971), pp. 49:3-525.

[12] V. S. Touninsky, Metrologiya (USSR), No. 1, 13 (1978) [in Russian].

[13] R. T. Birge, Phys. Rev. 40, 207 (1932).

[14] E. R. Cohen and J. W. M. DuMond, Rev. Mod. Phys. 37, 537

(1965).

[15] B. N. Taylor, W. H. Parker, and D. N. Langenberg, Rev. Mod. Phys. 41, 375 (1969).

[16] E. R. Cohen and B. N. Taylor, J. Phys. Chem. Ref. Data 2, 663 (1973).

[17] V. S. Touninsky and S. V. Kholin, Metrologiya No. 8, 3

(1975) [in Russian].

[18] B. S. Dzhelepov and S. A. Shestopalova, Nuclear

Spectroscopic-Ray Standards (Atomizdat, Moscow, 1980)

[in Russian].

488
Precision Measiirement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Realization of a Josephson Potentiometer

M. Koyanagi, T. Endo, and A. Nakamura

Electrotechnical Laboratory, 1-1-4 Umezono, Sakura-Mura, Niihari-Gun, Ibaraki, 305 Japan

Preliminary experiments have been done on a new Josephson potentiometer. The emf of a standard

cell is determined with an uncertainty at the 10"* level by using a Josephson voltage of 100 mV which

is generated by multiple Josephson junctions.

Key words: ac Josephson effect; Josephson junction array; Josephson voltage standard; microwave strip

line; tunnel junction.

1. Introduction

The constant voltage produced by the ac Josephson ef- fect has two remarkable features: (1) universality as a reference voltage; and (2) its variability. Therefore, it is possible to make a high precision potentiometer using the

ac Josephson effect. In fact, the Josephson voltage standard systems used at present in several countries are based on these characteristics [1]. However, as the

Josephson voltage used in these systems is in the range

from 1 to 10 mV, a voltage divider which has a fixed di- vider ratio is necessary for obtaining the standard volt- age of about 1 V. If we can generate a Josephson voltage

as high as 1 V, we will be able to make a high precision

potentiometer vdthout depending on the divider which

may be the main origin of the uncertainty for such a high

precision potentiometer. Such a potentiometer will also serve as a Josephson voltage standard system with

higher precision than those standard systems used at present because its uncertainty will simply be attribut- able to the Josephson voltage and the voltage sensitivity

of the null detector.

A few attempts have been reported on the generation

of Josephson voltages where the aim was 100 mV or higher [2 to 5]. However, none of them were completed

as a system. In the present paper we report preliminary

experiments on determining the emf of a standard cell by

using a Josephson voltage of 100 mV. In the experiments

the divider ratio is calibrated automatically in the determination of the standard cell voltage.

2. Experiment

The present Josephson potentiometer consists of the

Josephson voltage generator of 100 mV, a voltage divider

which is composed of ten resistors of 45 fi, a constant

current supply, and two photo-cell type galvanometers as

null detectors. In order to understand the principle of the potentiometer, let us explain the method for determining the emf of a standard cell (see Fig. 1). At first, using galvanometer 1, the current from the constant

current supply is adjusted so that the voltage across the

divider becomes equal to the unknown voltage such as the emf of a standard cell of nearly 1 V. The standard

cell voltage is simply given by the sum of the voltages

produced in each resistor of the divider. The voltage

across each resistor (about 100 mV) can be determined in terms of the Josephson voltage by adjusting the micro- wave frequency. The uncertainty of the present potentiometer would be attributed to the Jogephson voltage of 100 mV, the voltage sensitivity of the galvanometers,

and the stability of the voltage across the divider (dependent upon the stability of the constant current supply) in the time interval necessary for determining the voltage. In this system, the divider ratio is given automatically in the course of the voltage measurements as long as the current through the divider is kept constant. An un- known voltage lower than 1 V is determined by using a smaller number of resistors with the same procedure as mentioned above.

Figure 1. Schematic diagram of the Josephson potentiometer for determining the emf of a standard cell. The principal part of the Josephson voltage generator

is an array of twenty Josephson junctions connected in series for generating 100 mV. Microwaves are supplied

to the twenty junctions simultaneously from a single

microwave generator, although we need twenty dc current sources for the independent biasing of each junction. The biasing system contains a monitoring system

which has an oscilloscope, an ac current supply, and a

digital voltmeter. The step structure around a dc bias

point of any one of the twenty junctions is monitored on

the oscilloscope, and the total dc voltage is read by the

digital voltmeter. The monitoring system is disconnected

when the comparison is made between the Josephson

voltage and the voltage across a resistor. The junctions of Pb(In, Au)-o.xide-Pb(Au) are fabri- cated on a silicon wafer by the procedures of photolithography, vacuum evaporation, rf-sputter cleaning, and

rf-oxidation. The typical size of a junction is 0.9 x

0.05 mm", where the 0.9 mm length corresponds approximately to half of the wave length at 10 GHz.

489
Microwaves of 9.3 GHz are fed through a 50 ohm strip

line made of evaporated superconducting film, and a copper plate attached to the waveguide wall as shown in Fig. 2. The array of twenty Josephson tunnel junctions is incorporated into the superconducting film. One of two

ends of the strip line is connected to an antenna inserted

into the X-band waveguide. Another end is terminated to ground through a resistor of 50 ft so that the reflected

Josephson

junction Silicon wafer

Microwave

Shielding case

Figure 2. Sample mount. microwave power from the end of the strip line becomes

small and all junctions are irradiated with nearly equal

microwave power without a dependence on junction position. The X-band strip line filter is inserted between the

junction and the terminal for the dc lead as shown in Fig. 3. With this filter, mutual interactions are considerably reduced between the strip line for the microwaves

and the leads for the dc supply.

For the potentiometer experiment, one needs large

widths of the constant voltage steps at reasonably high

voltages. In Fig. 4, the width and the voltage of the step

used are shown for each junction at various positions

along the strip line when the incident power to the cryostat is 210 mW and the total dc voltage is 101.9 mV. The

closed circles and solid bars show the bias voltages and

the current widths. Two junctions. No. 17 and 18, are missing because they showed poor coupling to the

microwaves.

In spite of our intent of irradiating all the junctions

with uniform microwave power, the microwave standing wave in the strip line prevents such uniform irradiation. This may be due to the existence of constrictions in the

superconducting strip line near the Josephson junctions.

The divider, which consists of ten resistors with values

of resistance within 14 ppm, is placed in an oil bath which

is temperature regulated within 0.005 °C at 23 °C. The

temperature coefficient of the resistors is 0.6 ppm/ °C. The self heating effect of the resistors is negligible. The

current supply to the divider is made by a method similar

to that of Finnegan and Denenstein [6] and has a drift of 2 to 3 ppm/hr.

The working standard cell used is contained in a temperature regulated air bath which has temperature

stability of 0.001 °C at 30 °C as long as the room temperature is kept at (23 ± 0.5)°C. The air bath is placed in another shielded room. The comparison between the volt- age across the divider and the voltage of the standard

cell is performed through a cable of about 15 m. Before

and after the experiments, the standard cell is calibrated

against three other standard cells in the same air bath.

Figure 3. The Josephson device pattern for 100 mV generation.

(A) Photograph of the pattern on the wafer; (B) microscope

photograph of the Josephson junctions.

10

mV

0;o• o

>n

T T

junction number 5

\ 3^

2

bias "

point

20

P ; 210 mW

f : 9-293 GHz Vtot^j : 101-9 mV

I L_ 0 10 20mm

position of the junction

Figure 4. The bias voltages and the current widths of the constant voltage steps used for voltage generation. The closed

circle and solid bar show bias voltage and current width,

respectively.

The microwave frequency is read on the frequency

counter along with the rubidium frequency standard. The

drift of the present rubidium frequency standard is below

10~iVmonth.

490
For determining the voltage across each resistor of the

divider, four comparisons are usually made wdth the current direction through the divider and the junction

' H f- in order to compensate for thermal emf's and

the drift of the observed voltage difference. Therefore, one run of the experiment consists of 40 comparisons bei cause we have to determine the voltages across ten resisI tors for the measurement of the standard cell voltage.

As the present constant current supply to the divider

is not stable enough to determine the emf of the standard

cell vdth an uncertainty at the 10"^ level, the preliminary

experiment is performed assuming that the voltage of the

standard cell is constant through the experimental time

interval which is about ten hours, and the constant

current supply is adjusted in each comparison so that the

' voltage across the divider is balanced to the working

! standard cell.

I 3. Results

Sources of uncertainty in the determination of the emf

of the standard cell are given in Table 1. Random uncertainty for determining the voltage of one resistor in the

divider is estimated from the scatter for seven runs. The

random uncertainty is about 4.5 x 10^^. This uncertainty

reduces to 1.5 x 10"* for the determination of the emf of the standard cell because ten equivalent measurements

are contained in that determination. However, the con-

' stant voltage steps of some junctions have small but

non-zero resistance on the steps which may be due to

^ external noise. By taking into account the uncertainty

due to this non-zero resistance of the constant voltage

steps, the uncertainty of the present experiment would

be at the 10"* level, though each junction is biased near

1 the center of the step.

Table 1. Sources of uncertainty in the determiyiation of

the emf of a standard cell.

Relative uncertainties Sources (one standard deviation)

(1) Standard cell comparison

Temperature fluctuation of voltage 0.3 X 10"*

divider Leakage resistance o.i Resolution of standard cell 0.5 balance

Calibration signal of standard 0.5

cell balance

(2) Josephson junction comparison

Resolution of frequency counter 0.01 Resolution of junction balance 0.3 Calibration signal of junction ^ 0.3 balance

(3) Drift of the junction bias current 3.0 combined with inclination of the step

(4) Random uncertainty for determining 1.5 the voltage of one resistor in the

divider

Total (RSS) 3.5 X 10"*

In Table 2 we give values of the average emf of three

standard cells in the same air bath mentioned above,

which were determined by the present preliminary

experiments.

Table 2. The emf of a standard cell determined in tfie present experiment.^

Date

(1981) (V)

April 30 1.01816963 May 2 1.01816965 May 7 1.01816962 May 14 1.01816955

^One standard deviation is O.OSS ppm.

In the course of determining the emf of the standard

cell, the resistance ratios {R\/Ri, i = 2 to 10) are measured automatically. Comparisons between resistance ratios determined by the present Josephson potentiometer

[{R[ - R\)/R\]i,v, and by the current comparator type

potentiometer [{Ri - R\)/Ri]c.v. are given in Table 3. They agree vdthin the uncertainty of 0. 1 ppm.

Table 3. Comparisons of the resistance

ratio between two measurements, one by the Josephson potentiometer and one by a current comparator type potentiometer.^

Number of [(i? ; - i)/i2 Jj p [{R , - R {}/R {\q y>. resistor (ppm) (ppm)

12 5.56 5.54 3 11.85 11.87 4 7.71 7.79 5 5.76 5.71 6 -1.92 -1.84

7 3.71 3.67 8 -0.99 -0.99

9 2.54 2.48 10 3.34 3.32

^Standard deviations are below 1 x 10 ^ for both measurements.

4. Conclusion

As a demonstration of the Josephson potentiometer,

the emf's of standard cells were determined with an uncertainty at the 10"* level by using a Josephson voltage

of 100 mV. Further improvement of the uncertainty will be possible by using a stable constant current supply

with an uncertainty at the 10"^ level instead of the

present current supply used, and by removing the external noise which may be the cause of the non-zero resist- ance of the constant-voltage steps.

The 'authors would like to express their thanks to Toshiaki Sakuraba for measuring the temperature of the

air bath.

491
References

[1] B. W. Petley, in Atomic Masses and Fundamental Constants

5, Ed. by J. H. Sanders and A. H. Wapstra (Plenum Press, New York, 1976), p. 450.

[2] M. Koyanagi, T. Endo, and A. Nakamura, in Future Trends

in Superconductive Electronics, Ed. by B. S. Deaver, Jr., C. M. Falco, J. H. Harris, and S. A. Wolf, AIP Conference

Proceedings No. 44 (American Institute of Physics, New

York, 1978), p. 187.

[3] R. Joly, E. C. Hannah, J. E. Opfer, and L. S. Cutler, IEEE

J. Solid-state Circuits SC-14, 685 (1979).

[4] R. L. Kautz, Appl. Phys. Lett. 36, 2289 (1980).

[5] R. L. Kautz and G. Costabile, IEEE Ti-ans. Magn. MAG-17,

780 (1981).

[6] T. F. Finnegan and A. Denenstein, Rev. Sci. Instrum. 44, 944

(1973).

492
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

A Transportable Josephson Voltage Standard

K. Lahdenpera, H. Seppa, and P. Wallin

Helsinki University of Teclinology and Technical Research Centre of Finland, Otakaari 5, 02150 Espoo 15,

Finland

A transportable cryogenic Josephson effect emf standard has been developed in which all precise measurements are carried out in a helium bath (4.2 K). This standard uses a superconducting current comparator and 0.1 Hz measurement current for the calibration of a resistive potential divider. The

cryogenic emf standard has a measurement uncertainty of 2 x 10^ (one standard deviation).

Key words: ac Josephson effect; Josephson voltage standard; superconductivity.

1. Introduction

Standards of emf are nowadays maintained in a number

of laboratories using the ac Josephson effect, which relates potential to frequency by the equation

(1)

in which h is Planck's constant and e the electron charge

[e.g., 1, 2]. The development of Josephson voltage standards has lately split into two branches. On the one hand

the goal is better repeatability, on the other hand reliability and easy use. The former goal leads often to complex solutions, and difficulties with reliability in operation, while to reach the latter requires compromises with

regard to accuracy. The components of Josephson voltage standards

developed in various countries have been described in several publications. We describe here the main features

and novel parts of our instrument. The aim of this work

was to develop a transportable, small sized Josephson

voltage standard.

2. Apparatus

A cryogenic Josephson effect emf standard has been

developed in which precise measurements are carried out

in a helium bath (4.2 K). The cryogenic system includes

the Josephson junction (a Nb-Nb point contact or a Nboxide-Pb junction), a cryogenic voltage divider which is calibrated during measurements using a cryogenic

current comparator, a SQUID as a null detector, and a superconducting switch. The diameter of the cryogenic

unit is 48 mm.

The principle of the instrument is shown in Fig. 1 and

a photograph in Fig. 2. The photograph does not show

the microwave frequency source for the Josephson junction. In the circuit of Fig. 1 a standard cell is calibrated

using a Josephson junction. The comparison is made with

a potentiometer (resistor.'; 200 Cl and 0.4 Q). The null detector of the potentiometer is a SQUID. The measure- ment of voltage is made with the switch S in position 2, and calibration of the resistors in the potentiometer with

the switch in position 1. The signal obtained from the

SQUID controls the current source that balances the po- tentiometer. The galvanometer used in the measurement

of the emf of the standard cell is balanced by adjusting

the 10 GHz microwave frequency.

Calibration

circuits

for cryogenic

potentiometer

I Squid

electronics

Squid comporotor

i^2Kj

Figure 1. Schematic diagram of the transportable cryogenic

Josephson voltage standard.

Figure 2. Photograph of the voltage standard. New methods of coupling microwave radiation to Josephson junctions have been developed. The junctions are shown in Fig. 3 and the microwave frequency source

in Fig. 5. In the case of a thin film tunnel junction the

microstripline coupling technique was used and a mismatch of 11 dB was obtained between the signal source and the junction at x-band. The coupling was

493
made using a linear taper. Because of the high impedance

difference between the signal source (50 fi) and the

resonant type tunnel junction (0.1 fl), the linear taper was made using both a coplanar stripline and an ordinary

stripline. In the case of a point contact the mismatch was

6 dB. The point contact was set on the end of the coaxial

line and a 0.57 \ transformer was used to match the

50 fl-line to the junction [impedance (0.1 + 2j)Q.]. Voltage steps at the level of 2 mV were used and current

steps were about 50 to 250 |xA (point contact) and about

20 |jlA (tunnel junction). The SQUID uses a thin film Nb-Pb tunnel junction, of which the resolution is about 6 X 10-5(I)oVHz.

A voltage divider resistance ratio of 1:500 was used. A

block diagram of the resistance ratio measurement sys- tem is illustrated in Fig. 4. The resistance bridge is based on a cryogenic current comparator [3]. A low fre- quency alternating current was used to avoid the disturbing influence of 1/f-noise from the SQUID. The form of the ac-current can be seen in Fig. 4. There is a transient

in the output of the SQUID (see Fig. 4) because of the

large inductances of the cryogenic current comparator.

The effect of this transient is avoided by using a sample

and hold circuit. The sample and hold circuit is in the

"hold" mode and both the switches in the phase sensi- tive detector are open when the measurement current is

Point contact

Cooxial line

50pm

in <

-0.57 \

z= son in

R = o.in

x = 2n

Mismotch opprox. 6clB Current steps ot the Level

of 2 mV were 50- 250 p A

Figure 3(a). Schematic of scheyne used to couple microwave ra- diation to point contact junctio7is.

Nb-Pb -tunnel junction

Taper Josephsonjunction

CopLanor

stripline Stripline

50n Linear

taper -o.in

Mismatch approx. 11 dB Current steps at the

level of 2 mV

were approx. 20 pA

Figure 3(b). Schematic of scheme used to couple microivave ra- diation to thin fibn tunnel junctions.

I CRYOGENIC

t-CURRENT 4.2 Kj COMPARATOR

Figure 4(a). Calibration circuit for the cryogenic voltage di- vider.

494
MEASUREMENT

INTERVAL

SWICH 1 ON

SWICH 2 ON'

Figure 4(b). Waveforms of sigymh.

Frequency

standard \ Resonont Phose

cavity lock

Sunn 10 GHz

oscillotor

I

T

1

T

Figure 5. Microwave frequency source. The signals from the Gunn oscillator and from the stable tunable resonant cavity are fed to the harynonic mixer, The intermediate frequency is phase-locked to the frequeyicy standard. The frequency is monitored using a microwave counter.

3. Measuring Results

The uncertainty of the measurements made using this voltage standard is about 2 x io~^ (standard deviation),

due to the uncertainty concerning the extent of voltage

divider self-heating during calibration. A comparison between the Josephson standard and a standard cell group calibrated at the BIPM has been car- ried out. The accuracy of the comparison was limited by

the instability of the standard cells. The measured differ- ence was 2 X 10"'^. The Josephson standard was later compared to the

PTB (Federal Republic of Germany) voltage standard at Braunschweig. The mean values of the results obtained

using the two standards differed by about 1 x lo^^ and

the standard deviation was 5 x lO"*^. The comparison was made through an intermediate working group of standard

cells. Their instabihty limited the measurement accuracy. The systematic uncertainty due to this limitation and

due to an unreliable contact was estimated to be 1 x 10"''. The measurement results are seen in Fig. 6.

U(PTB)-U(VTT)

0.20

9 September

Figure 6. A comparison between U(PTB) and U(VTT) in September 1980.

4. Reliability

The Josephson thin film tunnel junction and the SQUID

are very reliable. During several years of use no service

has been needed. The Josephson point-contact has proved

to be the most unreHable component in the instrument.

It has to be tuned at intervals of about six months.

Between the tunings the junction is reliable and requires no adjustments.

References

[1] B. F. Field, T. F. Finnegan, and J. Toots, Metrologia 9. 155

(1973) . [2] I. K. Harvey, Metrologia 12, 47 (1976).

[3] D. B. Sullivan and R. F. Dziuba, Rev. Sci. Instrum. 45, 517

(1974) . [4] H. Seppa, CPEM Digest 1980, IEEE Catalog No. OCH 1497-7 IM. p. 172 (1980).

changing. Balance of the bridge is achieved by variation

of the resistor R p. The resistance ratio can be calculated

by determining the values of R p and R x- More details can be found in Ref. [4]. The voltage divider was calibrated using different currents and the result of the ratio determination was

obtained by extrapolating the result to zero current, thus

minimizing the effect of self-heating. This measurement

is laborious because of the requirement for many measurements. However, this part of the measurement is quite easy to automate.

495

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

The Proton Gyromagnetic Ratio in H2O—A Problem in Dimensional

Metrology*

E. R. Williams, P. T. Olsen, and W. D. Phillips

Electrical Measurements and Standards Division, National Bureau of Standards, Washington, DC 20234

The last 10 years of work on the proton gyromagnetic ratio in H2O, y'p , is reviewed. Results from

both the low and high field methods are summarized along with short descriptions of several experi- ments. However, the emphasis is on the results from the low field method and the discrepancies among

them. The values of the fine structure constant, a, obtained from the low field experiments are compared with several values of a obtained from various atomic physics experiments which to varying de- grees require quantum electrodynamic theory (QED) for their interpretation. There is good agreement

between the values of a obtained from two 7p determinations and these QED experiments as well as with the values of a obtained from quantized Hall resistance measurements. Two other y'p determinations disagree drastically: one differs by (5 ± 0.6) ppm and the other by (7 ± 0.5) ppm. We have no ex- planation for a disagreement this large. •Key words: fine structure constant; fundamental constants: proton gyromagnetic ratio; QED tests.

1. Introduction

The past decade has been very eventful for proton

gyromagnetic ratio experiments. On the one hand, the

best value of 7p(low) has improved by a full order of magnitude with a corresponding improvement in the

resulting value of the fine-structure constant, a. This

value for a is consistent with that obtained from the

anomalous electron moment using quantum electro- dynamic (QED) theory. On the other hand, the value of a

obtained from t^NO other y'p{\ovi) experiments completely

disagrees with the QED-consistent value. The high field measurements were not very consistent at the beginning

of the decade, but significant improvements have been

made in both the quoted accuracy and the consistency

among different laboratories.

When we first started this review paper, we felt that

much of the paper would be spent describing the successes

(i.e., the tests of physics) which have come about over the

past decade, in part due to the increased accuracy of measurements of 7^. Indeed, we will show data which are

in excellent agreement and represent some of the most

precise tests of QED, of the ac Josephson effect, and of a new effect, the quantized Hall resistance. But it has be- come clear that there also exists a large discrepancy

among various low field 7], determinations.

Because there exist these large discrepancies among

recent values and because the low field value has a dramatic impact on the value of the fine-structure constant, the low field 7^ experiment requires the most dis- cussion. There are four recent values of 7p(low); the

spread in values is over 8 ppm, a discrepancy of over 10 standard deviations (see Table 1). At the same time, the

value of 7p(low) claiming the highest accuracy is in re- markable agreement with QED predictions and with the

latest quantized Hall resistance results.

Table 1. Low field y'p measurements since 1970.

Authors (year), and

Laboratory

7; (low) - 26751.00 X 10"" s'^ TbY- 76 Uncertainty'^

(ppm)

Olsen, Driscoll (72>NBS'

Olsen, Williams (75) NBS''

Vigoureux, Dupuy (78) NPL*^

Williams, Olsen (79) NBS''

Chia, Shen (80) NIM^

Forkeit, Schlesok (80) ASMW''

Tarbeyev (81) VNIIM'

0.384(54)

0.354(11)

0.178(13) 0.3625(57)

0.391(21)

0.55(13) 0.228(16)

2.0

0.4

0.5

0.2

0.8

5.0

0.6

^All values have been expressed in terms of the BIPM 1976 unit of current. '^One-standard deviation (68% confidence level) estimates. •^Dimensions of solenoid measured mechanically; see Ref. 1. ''This value is by the same technique as in footnote f (i.e., not an independent value). Ref. [2]. eRefs. [3, 4]. lief. [5]. SRefs. [6,7]. ^Refs. [8, 9]. 'Ref. [10].

*Supported in part by the Department of Energy

497
The proton gyromagnetic ratio in H2O, y'p, is defined to be the proton precession frequency Wp of a spherical sample of H2O at 25 °C when placed in a magnetic field B, di- vided by the magnitude of the field if the sample were

not present:

y'p = <^'p/B . (1) In both the low and high field methods, Wp is measured

by conventional NMR techniques. In the low field method, B is calculated from the measured dimensions of a current carrying precision solenoid or a precision

Helmholtz coil: B - \s.JlabG, where ijlq is the permeability of free space, /lab is the current measured in some

laboratory unit, and G is the coil geometry factor which

has units of inverse length. Equation (1) becomes

7p(l0W)LAB = Wp/(|Xo/labG) = Ka^'p , (2) where K^, = Alab/A is the ratio of the laboratory unit of current to the SI unit of current. As we shall see, the

value of y'p (low) in laboratory units is particularly important to the fundamental constants field because /lab can be realized in terms of a voltage using the Josephson ef- fect and a resistance which can be realized in SI units us- ing the calculable capacitor. The measurement of y'p in SI

units is limited to the precision with which Kp, is presently known. In the high field method of determining

7p, one calculates B from the force equation d F = I {d I X B) where dF is the elemental force experienced by a current element I dl in the presence of a magnetic induction B. In a uniform field, with a wire of length L per- pendicular to the field, one can measure the force, F,

perpendicular to both and thus

7;(high)LAB = co; /lab L/F = y'p/KA . (3) The distinguishing feature between the high and low field methods is not the field that is used, but that the two

quantities are measured in different units. For example,

to compare a high and low field measurement one must

introduce another experimental quantity which involves a mass measurement (i.e., an absolute ampere, a Faraday,

or an Avogadro's constant experiment). Alternatively,

Eqs. (2) and (3) can be combined to obtain or y'p in SI

units from the ratio or product of 7p(low) and 7p(high):

-11/2 7p y'o ( high)LAB y'p ( 1ow)lab , (4)

Ka = [7p(low)LAB/7p(high)LAB)] 1/2 (5)

2. The High Field Results

The feature that distinguishes the high field method is that a force measurement is required in order to obtain

the desired quantity. In all experiments to date, a rec- tangular coil is placed on one side of a balance with the

upper end of the coil in zero magnetic field and the lower

end in a strong uniform field provided by an electro- or permanent magnet. One must map out the strong field using an NMR probe and thus obtain the average value

of 7p(high). The width of the coil must be measured

especially accurately in the region of high field gradients.

The force on the coil is measured by finding the mass

(and knowing the acceleration of gravity) that just counters the force change when the current in the rec- tangular coil is reversed.

The three high field measurements carried out since

the 1970 PMFC Conference are given in Table 2. All three techniques are similar so we will describe the most

precise determination, that of Kibble and Hunt [11, 12]. These workers developed a method to eliminate a major

source of systematic error. . The problem is that horizontal forces and torques acting on the movable coil can cause "fictitious" vertical forces and thus an error. They

eliminated these errors by introducing a flexure near the

middle of the long arm that was attached to the rec- tangular coil. Thus this long arm and coil formed a double

pendulum. They used an autocollimator to detect any

bending of the double pendulum, thus detecting the

unwanted torques and forces. They then adjusted their

coil position and balance to minimize these effects. This source of error was probably the cause of many of the

discrepancies in the earlier strong field measurements.

However, the latest measurements are in reasonable

agreement.

The Kibble and Hunt experiment had two other distin- guishing features worthy of mention. First, their tech- nique to shim the permanent magnet to produce a uniform field is simple and clever. They simply painted

magnetic paint in the appropriate places to accomplish

the final shimming. Second, the critical measurement of

the width of the coils was made by two methods. First,

they mechanically measured the wire position; then they

used an inductive technique similar to the one we will describe later in this paper to locate the current in the

wire. The two methods were in reasonable agreement.

At present, the major area of impact for the strong

field experiments is in providing a value of the ampere,

Ka- There exist significant discrepancies in the determination of the ampere, but this is discussed by B. Kibble

in another review paper in these proceedings. Until rea- sonable agreement about the value of Ka can be reached,

the high and low field results cannot be directly compared. Because the high field method has far fewer

discrepancies at the present time, most of this paper will concentrate on the low field experiments.

Table 2. High field 7' measurements since 1970.

Authors (year) and Laboratory 7;(high) - 26751.00 x lO"" s"' TkiLt, Uncertainty

(ppm)

Kibble, Hunt (79) NPL»

Chiao, Liu, Shen (80) NIM*^

Forkert, Schlesok (80) ASMW<

0.689(27)

0.572(95)

0.32(41)

1.0 3.5

16.0

3Refs. [11, 12]. bRefs. [6, 7]. <^Refs. [8, 9].

498
*

3. The Low Field Experiments

The ac Josephson effect has had a major impact on the

low field experiments. Before the laboratory unit of vol- tage could be maintained by the Josephson effect and re- lated to the constant 2e//i, the low field 7p experiment

seemed destined to play the role of monitoring the ampere, and thus the volt, since the ohm can be accurately

determined via the calculable capacitor. But at the same

time that 7p(low) lost this job, the Josephson effect created a new role for it in the area of fundamental constants. The uncertainty in 7], (low) soon became the most

important limitation in the determination of a without

the use of QED theory. The pertinent equation is

a-2 - [(c/4i2oc)/(ftLAB/n)] X

(|jLp /|jLs )(2e/h )lab /y'p ( 1ow)lab • (6)

Here c is the speed of light in vacuum; R a,, the Rydberg

constant for infinite mass; Vli^p^^lH, the ratio of the lab- oratory as-maintained ohm to the absolute or SI (International System) ohm; (x^ /|xg , the magnetic moment of the proton in units of the Bohr magneton (again, the

prime indicates protons in a spherical sample of pure H2O

at 25 °C); (2e//i)LAB, the ratio of twice the elementary

charge to the Planck constant measured in terms of the

LAB as-maintained volt using the ac Josephson effect; and 7p(1ow)lab, the gyromagnetic ratio of the proton

measured in terms of the LAB as-maintained ampere by

the low field method.

With the motivation provided by Eq. (6), a number of laboratories set out to accurately measure 7p(low). Note

that it is 7p(low) in as-maintained units that is of greatest interest,. not the absolute value of 7^ given in Eq. (3). The results reported by the various laboratories are summarized in Table 1 and can best be described as being in complete disagreement with each other, falling

roughly in two sets. But there is no common parameter

that seems to distinguish the two sets, other than the

value of 7p(low).

The Chinese group of Chiao, Liu, and Shen from the National Institute of Metrology (NIM) has reported a value of 7p(low) [6] which is also reported in these proceedings by

Prof. Wang [7]. In the NIM experiment, the dimensions of

a precision single layer Helmholtz coil were measured by

comparison with end standards. The end standards were

in turn measured via an interferometer. The impressive

part of the NIM work is the precision obtained in the NMR

part of the experiment, 0.07 ppm, considering the low

field, about 2 x 10"^ T, produced by the Helmholtz configuration. The final 0.8 ppm accuracy was primarily limited

by the dimensional measurements.

The VNIIM (USSR) group has reported a new value of 7p(low) in these proceedings [10]. They use a four seg- ment solenoid and mechanically locate the wires, using a

laser interferometer to measure the dimensions. They

use a two coil free induction decay method to measure

the NMR in an approach similar to that used by the

NPL.

P. Vigoureux and N. Dupuy at the NPL (UK) have

spent many years obtaining their results [3, 4]. In measuring the pitch of their solenoid they use a unique

technique to locate the wire position. Their solenoid is bifilarly wound and the pitch is measured by placing a

voltage between the two landings. A metal sphere that

is attached to one end of an interferometer is moved

along a generator of the solenoid, shorting out the double

helix as it goes. The pulse that results from this short

triggers the recording of the position of the sphere by

the laser interferometer. This information is recorded for different generators by rotating the solenoid. These data

are then combined to describe the average position of each turn of the solenoid.

Because of the large discrepancy among low field 7^ experiments and because our technique for measuring the

pitch of the solenoid differs significantly, we will describe the NBS measurement in greater detail than

those of the other laboratories. The reader will note that

this obviously represents a bias on the part of the authors, but we hope it will be informative.

While all other experiments locate the surface of the

wire at many positions and then calculate where the current will flow from these measurements, the NBS has

developed a method to locate the average position of an

entire current loop using the magnetic field produced by

that current [5]. Coils A and A' in Fig. 1 form a linear

differential transformer which locates the axial position

DIIViENSIONAL IVIEASUREIVIENT SYSTEiVI

LASER INTERFEROMETER

P"'lGURE 1. Solenoid dimensional measurement system used to determine the axial position and radial variations of the wires. The five coils A, A' , B, B', and C are attached to a silica tube T and can be pushed or pulled along the axis of the solenoid. Coils A and A' locate the axial position of the injected current and coils B, B', and C form a diameter-to-voltage transducer. Mirrors M and M' are part of a laser interferometer. The Pijrex tube is evacuated.

of the alternating current injected into selected turns of the solenoid. Coils A and A' are connected so that their

output voltages cancel when centered on the activated turns of the solenoid, and a servo system locks the coil assembly to the null point with a precision of better than

0.05 |jLm. A mirror (corner cube M') located in the center

of the coil assembly is part of a laser interferometer sys- tem, with the reference mirror (M) connected to the end

of the solenoid. Thus, with the aid of a computerautomated system, the relative position of successive turns can be recorded. We chose to activate 10 turns at one time, and then move the current injector to a successive 10 turns until information about all 1000 turns of the

solenoid was obtained. Coils B, B' , and C form a radiusto-voltage transducer which measures the variations in the radius of the injected current. The voltage induced in

coil C is inversely proportional to the radius of the ac- tivated turns of the solenoid. To detect small changes we

first "buck out" most of the voltage in coil C by using

the two coils B and B' . At the same time that we ai-e bucking out the voltage in coil C, we are also increasing

499
«

the sensitivity of this three-coil system to changes in the

radius R of the solenoid, because the voltage in coils B

and B' increases when the radius R increases. This

three-coil radius-to-voltage transducer is then calibrated

by having a few turns of wire on both ends of the

solenoid that are 25 iJim larger at one end and 25 yjm

smaller at the other end. With this system, the axial position and the radius variations of the turns of the

solenoid were measured to an accuracy of 0.05 |xm. No

turn deviated from that expected of a perfect solenoid by more than 2 iJim. Figure 2 shows the pitch variation (axial position) and

the variations in the radius as a function of the wire position. These data are one measurement set, where the

central wires from +230 mm to -230 mm were measured

four times and the outer wires measured twice. The total height of the error bars represents twice the rms scatter

of the measurements recorded during this measurement

set, and it takes about 4 hours to complete such a set. The field at the center is then calculated using the ap1 1

i 0

cno

1 0

=»COa

< 1

1 1 1 1 1

^il / \ I / \

I I I Ij /* \

I I y

1 ^ Vi

1 1 1 1

r I

I r I

II X

I

I

' A/ . WEIGHTING FUNCTION

1 1 1 1 I 1 1 r 1

1 1 1 1

I I

1 1 1

1

WEIGHTING FUNCTION

\

1 1 1 1 \

/ ^

^

/ 1 1 1 1 -500 -400 -300 -200 -100 0 100 200 300 400 50u

WIRE POSITION (mm)

Figure 2. One measurement set of the axial position (pitch

variation) and radial variation as a function of distance from

the center unre. For the upper plot a pitch of 1.000087 mm/turn

has been subtracted from the measured positions. The total height of the error bars is tvnce the rrns value of the yneasure- ments taken on each unre during this measurement set. The

structure is reproduced from^ day to day within the error bars. The curves are weighting functions which must be multiplied

by the measured variations to calculate the field at the center of the solenoid.

propriate weighting functions which are also shown in Fig. 2. Our final value of y'p was calculated from 13 sets

of pitch variation and 22 radius variation measurements,

all very similar to Fig. 2. The statistical scatter was

small compared to other systematic uncertainties, such

as the measurement of the mean diameter. From these

diameter and radial variations, we calculated the gra- dient in the magnetic field along the axis of the solenoid

to be 0.2 ppm/cm. Using the NMR data we measured the

residual gradient after compensation and found good

agreement with that predicted from the dimensional

measurements.

One source of systematic error is very important and

its elimination is of interest. If the injected current is a sinewave, then an error we call an end effect is intro- duced. Because this current induces a voltage in the

other windings of the solenoid and because there is some

capacitance across these turns, some current flows in these turns thereby producing an error signal in the

detector coils [13]. This effect varies inversely with the

square of the frequency. We eliminate it by employing

the special wave form for the injected current shown in the upper trace of Fig. 3. The output of the detector coil

is just the derivative of the injected current. The second

trace of Fig. 3 results when the probe is +2 mm off null, and the bottom trace when it is -2 mm off null. In the next to last trace we are locked to the null. Using a

lock-in amplifier with a flat frequency response, we can

look at the integral of the output voltage from time A to time B minus the integral from B to C. This integral is proportional to the mutual inductance or total flux change in the output coil. The transient signals, however, integrate to zero. The effectiveness of this method

is demonstrated by either showing that the null position

is independent of frequency or that it is independent of the phase of the lock-in. We find an error of less than

0.04 |xm for a phase shift of up to 90°.

Figure 3. Injection current and resulting signal. Upper trace: current injected into a portion of the solenoid. Second trace: voltage observed across the pickup coils when the probe is dis- placed +2 mm from the center of the activated loop. Third

trace: probe is centered. Bottom trace: probe is displaced -2 mm. Letters indicate various instants of time along the oscillo- scope trace.

Our solenoid is 1 m long and 0.280 m in diameter. The

diameter was compared to a quartz end standard that was measured with a laser interferometer. The 0.8 jxm

(2.9 ppm) uncertainty in the diameter measurement pro- duced a 0.25 ppm error in y'p. With the high accuracy of

500
the radius variation and pitch measurements, the diameter measurement became the limiting factor in the experiment. The basic problem in determining the diameter is that even if we make a perfect measurement of the posi- tion of the surface of the copper w^ire, it is still difficult to know where the current flows in the wire. Because of our concern over a possible systematic error in measuring the diameter, a second approach was developed. The

technique is to find a coil geometry which produces a magnetic field that is nearly independent of the average

diameter. While a longer solenoid is one solution, we

were able to find a suitable configuration for the existing

solenoid in which current could be taken out of selected

turns. Using a computer, we searched for a configuration

where the sum of the diameter-variation weighting function is small and the second, fourth, and sixth-order gradients are compensated. The final configuration (see Fig.

4) uses five 1 A current sources, one of which is used to pass current through the entire solenoid in the normal

fashion. The other four are connected to selected turns on the opposite side from the main return lead so that

the net current in these selected turns is zero. This five- current system produces a 0.8 mT field (uniform to 0.1 ppm over the H2O sample), instead of the 1.2 mT field of the conventional solenoid, but this 0.8 mT field is 8.6 times less sensitive to the average diameter of the

solenoid.

The 1 A current placed in the solenoid is measured by

comparing the voltage across a 1 Q precision resistor in series with the solenoid, with a saturated standard cell calibrated in terms of the United States legal volt, which

in turn is maintained via the Josephson effect. A cable is used to transfer this current to the NBS nonmagnetic facility, thus providing us with the closest possible tie to the

Josephson voltage standard [14]. The observed scatter in the NMR measurements (0.02 ppm) reflects the precision we obtained in making the voltage comparison. At the

nonmagnetic facility a rubidium magnetometer system is used to reduce Earth's magnetic field fluctuations.

The two values of y'p which we obtained from the two

different current configurations agreed within their

respective uncertainties and were combined together in a least-squares fit to give our final value and 0.21 ppm uncertainty. The final accuracy of this determination was

primarily limited by the diameter measurement either

directly in the conventional configuration or indirectly by

the uncertainty for the corrections due to the compensation currents which reduced the diameter sensitivity in the five-current configuration.

4. Comparison of Results

The major purpose of a new proton gyromagnetic ratio measurement is to help test the validity of the convenCOMPENSATED NBS SOLENOID

VARIATIONS OF MAGNETIC FIELD UNIFORMITY

I 1 AMPERE

'

10 cm

Figure 4. The two ciirreiit configKrations. The top solenoid is compensated with two 0.1 A current sources for maximum uniform field. The field is uniform to better than 0.1 ppm over a center 7.5 cm sphere. The bottom configuration has four one ampere compensation currents so that the resulting field is independent of the mean diameter of the solenoid. The field is uniform to 1.0 ppm over the light shaded region and 0.1 ppm over thedark region. The If cm H^0 sample just fits within theO.l ppm region. All compensation

currents are injected on the opposite side of the solenoid. Leakage between current sources must be carefully controlled.

501
tional QED theory. The important role of 7p in testing

QED comes from the need for a value of the fine- structure constant, a, which is essentially independent of QED (see Eq. 6). The importance of Eq. (6) was first demonstrated in the late 1960's when the value of a

derived from it was used to resolve the discrepancy

between the theoretical and experimental values for the

ground-state hyperfine splitting in hydrogen. Subsequently, the Josephson-effect measurement of 2e/h advanced quickly, and by the early 1970' s, the primary limitation for obtaining a more accurate value of a from Eq.

(6) was the uncertainty associated with the experimental

determination of the proton gyromagnetic ratio. At that

time, the most accurate measurement of 7p(low) had an

uncertainty of two ppm and was obtained m our labora- tory [1]. Figure 5 shows a comparison of some values of a which

are derived from various atomic physics experiments plus

the appropriate theories. (These values were kindly pro- vided by B. N. Taylor and were calculated in preparation

for a new least-squares adjustment of the constants.)

Each of these QED quantities was discussed by various

authors at this conference. The curve plotted is simply a

visual guide which points out discrepancies and is called an ideogram [15]. We have included two theoretical

corrections for the electron anomalous magnetic moment

reported at PMFC-II, one by Levine and Roskies and the

other by Kinoshita and Lindquist. The uncertainty in

is now limited by the uncertainty in the eighth order

correction terms that Kinoshita and Lindquist reported.

Larger, faster computers are all that are needed to reduce this uncertainty significantly. The difference

(a„ - a^' )/a = (0.22 ± 0.13) ppm represents a ree '/mNBS markable success for atomic and solid state physics and

De V i at i on ppm /d i V

)

Figure 5. Values of the fine-structure constant derived from

QED theory and various experimental determinations. The

horizontal axis is the deviation from the mean of the measurements. (The values have been provided by B. N. Taylor.) At the top we have the value obtained mthout QED from the NBS

determination of y\, using Eq. (6). Then in descending order we have values from the anomalous moment of the electron; the muonium. hyperfine splitting; the hydrogen hyperfine splitting; the hydrogen fine structure; and the helium fine stnicture measurements. The curve is provided as an aid to the reader.

It is a composite of Gaussian lines with the value and width of each Gaussiayi corresponding to each measurement. The area of each Gaussian is inversely proportional to the uncertainty.

QED theory.

Two recently reported values of Vpdow) sharply

disagree with this pleasing situation [4, 10]. This dis- agreement is apparent in Fig. 6 which shows values of a

obtained from the various values of 7p(low) and for comparison, the value aa^. Also included in the figure are two values of a which were derived from a new effect described by von Klitz'ing and others in these proceedings, called the quantized Hall resistance, R^. The per- tinent equation used to obtain a from Rh is:

a/^^ = [2/(jjLoC) ] (HLAB/ft) {Rh)i.kb (7) Equations (6) and (7) can be combined to eliminate the

quantity Olab /

X (2e (low) Ilab} (8)

1/ni pha

Wtd. Hve.

137.035998 (64)

Eq.6 VNIIM

Eq.S NIM

Eq.G NPL Tp Eq.6 NBS Tp Electron g-2

Eq . 7 Be n -NBS R„

Eq . 8 Be 1 1 -NBS R^.Tp

De V i at 1 on ( 1 ppm /d i V )

Figure 6. Values of the fine- structure constayit from various non QED sources. The four uppermost values use Eq. (6) and

the various values of y], from Table 1. For comparison, the value from the anomalous moment of the electron of Fig. 5 is included. The last two values are based on a yiew Bell Laboratories ayid NBS value of the Hall resistance and Eqs. (7) and

(8), respectively. The three NBS values are not independent.

Equation (8) is obviously not independent of Eq. (7), but

represents an alternate path to obtain a from the Hall

resistance, and the cube root dependence helps reduce

the uncertainty in a.V ^ Of course, all three results could be combined together to obtain a least-squares

value for a, but for now we choose to show the results from Eqs. (6), (7), and (8) separately. (The reader should

keep in mind that the three values are not independent.)

The preliminary value and uncertainty shown for R^j were obtained in a collaborative effort between Bell Laboratories and NBS [16]. This value is in agreement with

less precise values reported by the German and Japanese

groups [17, 18]. The two discrepant values of a are in disagreement with the value obtained from a variety of

different physical phenomena; however, these two meas- urements should not be ignored because if they are

correct, the implications for physics would indeed be

dramatic. For example, if a derived from the NPL value

of 7p(low) is correct, then oLa^ is in error by 36 standard

deviations.

502
5. Experiments and Ideas in Progress

The need for new y'p determinations is obvious. For the

high field experiments the major motivation is to resolve

the inconsistencies in the absolute ampere measurements.

We know of one experiment in progress at ETL in Japan,

in which the field of 2.8 T is produced by a superconducting magnet. Because of the extra strong field, they can

use thin film techniques to deposit the conductor on fused

silica to produce the movable rectangular coil and still have a force of sufficient magnitude. They are presently

Hmited by magnetic field inhomogeneities.

In the low field experiments, two new ideas are dis- cussed in these proceedings [19, 20]: one by K. Weyand

of the PTB (FRG); the other by G. Greene of Yale

University. The PTB experiment is under construction

and nearly ready for experimental testing. Greene's pa- per discusses a new idea that employs the Ramsey

separated oscillating field technique.

Continued progress in low field experiments at VNIIM

(USSR), ASMW (DDR), and NIM (China) is expected. At

ETL (Japan), a new non-magnetic facility is being con- structed and a new precision solenoid is ready to be

measured. At NBS we plan to extend our accuracy to a few parts in 10^. A new 0.3 m diameter, 2.4 m-long precision solenoid has been constructed and will be used to reduce the major sources of uncertainty in our last ex- periment. We also plan to incorporate a new technique

using Ampere's law for an independent method of measuring the solenoid's dimensions [21]. The quantized Hall effect has added a new dimension to the electrical standards and fundamental constants field. Figure 7 is a geometric representation of the significance

of this new technique. The tetrahedron is a way of summarizing our options for the present and future measure- QED

Hfs Qe Mhfs

Figure 7. This pyramid is simply a visual representation of

the over-determination of the electrical quantities by the four

measured constants. The base of the pyramid is formed by the

three maintained electrical quantities: ohm, volt, and ampere.

The three lines that rise to form the apex represent the three paths for obtaining a from Eqs. (6), (7), and (8). The degree of

closure in this pyramid contains a great deal of physics and is indicative of the quality with'which one maintains the electri- cal standards. The quantity ( -Kq) means "unthout" the absolute ohm experiment, K^.

ments of y'p and other electrical quantities. Currently, we maintain the electrical units of resistance, voltage,

and current in laboratory units by two of these four

electrical quantities, namely the calculable capacitor and

2e/h. From that triangular base we now have three

paths to obtain a. The -R^ path (meaning "without"

Hall resistance) is formed from the well known equation,

Eq. (6). "Without" {2e/h, y'p) is the new Hall equation,

Eq. (7); and we can also obtain a "wdthout" the calcul- able capacitor experiment Kn = CIlab/^, using Eq. (8). The degree to which one obtains closure on this tetrahedron contains a great deal of physics (i.e., validity

of the Josephson effect and the quantized Hall effect) and

verifies the quality of our electrical units. Beyond the internal self-consistency of this tetra- hedron, the values obtained for a from various QED ex- periments provide additional physics and constraints on

the validity of the tetrahedron itself. Presently, the NBS

group is showing closure at the 0.2 ppm level in a, and

the possibility of extending this to a few parts in 10^ in the near future seems realistic. 7p(low), 2e/h, Rh, and the absolute ohm experiments

are now joined, through their relation to a, in a common

effort to achieve the highest possible accuracy. The reali- zation of this goal by all these experiments will provide

not only an exacting test of QED, but also a crucial in- sight into the validity of our maintenance of electrical standards.

References

[1] p. T. Olsen and R. L. Driscoll, in Atomic Masses and Fundamental Constants i, Ed. by J. H. Sanders and A. H. Wapstra (Plenum Publ. Corp., New York, 1972), p. 471.

[2] P. T. Olsen and E. R. Williams, in Atomic Masses and Fundamental Coyistants 5, Ed. by J. H. Sanders and A. H. Wapstra (Plenum Publ. Corp., New York, 1976), p. 538.

[3] P. Vigoureux and N. Dupuy, NPL Report DES44, (1978).

[4] P. Vigoureux and N. Dupuy, NPL Report DES59, (1980).

[5] E. R. Williams and P. T. Olsen, Phys. Rev. Lett. 42, 1575

(1979) . [6] W. Chiao, R. Liu, and P. Shen, IEEE Trans. Instrum. Meas. IM-29, 238 (1980).

[7] Z. Wang, these proceedings.

[8] W. Schlesok, IEEE Trans. Instrum. Meas. IM-29, 248

(1980) . [9] J. Forkert and W. Schlesok, Metrlogischen Abhandlungen

des ASMW, Heft 3 (1981).

[10] Y. V. Tarbeyev, these proceedings.

[11] B. P. Kibble, in Atomic Masses and Fundamental Constants 5, Ed. by J. H. Sanders and A. H. Wapstra (Ple- num Publ. Corp., New York, 1976), p. 545.

[12] B. P. Kibble and G. J. Hunt, Metrologia 15, 5 (1979).

[18] E. R. Williams and P. T. Olsen, IEEE Trans. Instrum. Meas. IM-21, 376 (1972).

[14] E. R. Williams, P. T. Olsen, and B. F. Field, IEEE Trans. Instrum. Meas. IM-23, 299 (1974).

[15] A. H. Rosenfeld and P. R. Stevens, in Proceedings of the Fifth Biennial Internatioyial CODATA Conference, Ed. by B. Dreyfus (Pergamon Pi'ess, New York, 1976), p. 19. [16] D. C. Tsui, A. C. Gossard, B. F. Field, M. E. Cage, and R.

F. Dziuba, Phys. Rev. Lett. 48, 3 (1982). Information in Fig. 6 has been revised to reflect the data which became

available after the conference.

[17] E. Braun, P. Gutmann, G. Hein, F. Melchert, P. Wamecke,

S. Q. Xue, and K. v. Klitzing, these proceedings.

[18] C. Yamanouchi et al., these proceedings.

[19] K. Weyand, these proceedings.

[20] G. L. Greene, these proceedings; and Metrologia 17, 83 (1981).

[21] E. R. Williams and P. T. Olsen, IEEE Trans. Instrum. Meas.

IM-27, 467 (1978).

503

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

The Development of Precision l\/leasurement

and Fundamental Constants in China

Wang Zhu-xl

Peking University, Beijing, P.R.C.

In China there is some e.xperimental work on precision measurement and fundamental constants, such as cesium and hydrogen frequency standards, iodine and methane stabilized lasers, the determination of the proton gyromagnetic ratio yp and the gravitational acceleration g, the realization of the SI

electrical resistance unit, and so on. Key words: cesium and hydrogen frequency standards; electrical resistance; gravitational acceleration; iodine and methane stabilized lasers; proton gyromagnetic ratio.

1. Introduction

Great advances have been made in precision measure- ment and fundamental constants since PMFC-I in 1970. Although this work has only been done for a short time

in China, there has been some good results. I am very

pleased to give a brief review of these results to the specialists participating in this conference.

In China, the research work which has had good

results includes the cesium beam frequency standard, the

hydrogen maser as a frequency standard, iodine and

methane stabilized He-Ne lasers as wavelength standards, the determination of the proton gyromagnetic

ratio y'p, the absolute measurement of the gravitational

acceleration g, and the realization of the SI electrical resistance unit.

2. The Cesium Beam Frequency Standard

Two primary cesium beam frequency standards, CsII

and CsIII, have been constructed at the National Insti- tute of Metrology (NIM), Beijing, China.

The cesium beam apparatus is shown in Fig. 1. The

beam tube is made of stainless steel; the total length is

Figure 1. NIM cesiiDii beam frequency standard.

about 5.3 m and the diameter 290 mm. The active region

of the cavity is about 3.7 m and the line v^ddth is about

42-46 Hz with a 150 °C oven temperature.

In December 1980, CsIII was operated for 50 h to carry out a frequency comparison, using HP5061A cesium

oscillators v^dth a high performance tube as stable refer- ence standards. The resulting frequency stability data is given in Table 1. There are many factors which affect frequency uncertainty, such as pulling of near lines, cavity phase difference, spectral purity, second order

Doppler effect, and so on. The RSS of the total uncertainty is 4.5 X 10"^^.

Table 1. Frequency stability data of CsIII.

Is 10s 100s 1000s

1.8 X

10"ii 4.2 X 10-12 1.3 X 10-12 4.0 X 10-13

Ih 2h 3h 5h

2.8 X 10-1^ 2.5 X 10-13 2.3 X 10-13 2.4 X 10-13

3. Hydrogen Maser

Hydrogen maser experiments have been done since

1973 at the Shanghai Bureau of Metrology (SBM). A photograph of the SBM hydrogen maser is shown in Fig. 2. The main accuracy limitation of a hydrogen maser is the uncertainty in the determination of the wall shift. When the hydrogen atoms collide with the coating of the

storage bulb, an additive phase shift is introduced in the

hydrogen atom wavefunction, so that the maser fre- quency is offset from the free oscillator frequency of the

hydrogen atom. The frequency offset value is given by an

empirical formula as follows:

A/u. = -^[1 + cc{t,)it - td] ,

where k is a wall shift coefficient, D the bulb diameter, a

the temperature coefficient in "C'l, and ti and t the

505
Figure 3. NIM iodine stabilized He-Ne laser.

Figure 2. SBM hydrogen maser.

reference temperature and temperature of the bulb,

respectively. The parameters k and a are obtained from

our experiments on the hydrogen maser wall shift. In these experiments, we used five different size quartz

storage bulbs in turn, which were all coated in the same way with a locally made F4 coating. The results of the

experiments are [1] km °C) = (-293 ± 17) mHz • cm

and

a(40 °C) = (-17 ± 2) X 10"=^ "C'l . The frequency stabilities for different sampling times

are 4 x lO-^^/is, 6 x lO-^VlOs, 9 x IQ-i^lOOs and 2.5 x

10"^'* /day, respectively.

The value of the unperturbed hyperfine frequency of the ground state of the hydrogen atom /o, which we obtained with respect to TAI, is /o = 1420 405 751.768 ± 0.002 Hz . This value is in close agreement with the average value

of /o published by other authors after 1970. The total uncertainty is 2.5 mHz.

4. Iodine and Methane Stabilized He-Ne

Lasers

Iodine and methane stabilized He-Ne lasers have been

constructed at NIM. The third harmonic lock servo control circuit was designed by NIM and Peking University.

The He-Ne laser tubes and absorption cells are all made

in China. An iodine stabilized laser is shown in Fig. 3. The frequency stability of the iodine and methane sta- bilized lasers is about 2 x 10""/ vT"and 3 x lO'^^/V^T

respectively. We have carried out an international inter- comparison of these two stabilized NIM lasers with similar BIPM lasers at BIPM in April 1980. The frequency

stability of the NIM and BIPM iodine stabilized lasers NIM2 and BIPM2 relative to each other was 6 x IQ-^^

from 900 s to 2700 s. The difference in frequency

between NIM2 and BIPM2 was 13.8 kHz and between

NIMCH4I and BICH46, 0.56 kHz. These differences

correspond to relative differences of 2.9 x 10~" and 6.3 X 10^^, respectively [2]. The comparison of iodine stabilized lasers was done 13 times from April 2 to 17, 1980. The standard deviation was 4.5 kHz, corresponding to 0.95 x 10"". Various fre- quency shifts of NIM2 were also measured at the BIPM.

The pressure shift and modulation shift were found to be

-(9 ~ 11) kHz/Pa and -6.6 kHz/MHzpp, respectively, and

the power shift -0.05 kHz/jjiW, which is very small. This

is good for frequency reproducibility. The success of the

intercomparison of NIM and BIPM iodine stabilized lasers is significant because the NIM2 He-Ne laser tube

and iodine cell are different from those of BIPM2 as well

as from those of lasers built by other laboratories.

5. The Absolute Measurement of the

Gravitational Acceleration

A three point, freely-falling body system for determining the gravitational acceleration has been built at NIM.

The experimental work began in 1970. This stationary

laser gravimeter has been used for the absolute measure- ment of gravity since 1975. The standard deviation of g

measured with the apparatus is 100 fxGal. Since the great

earthquake in Tangshan on July 28, 1976, we have con- structed a new, precise transportable gravimeter for earthquake forecast investigations and other scientific re- search. The transportable gravimeter is shown in Fig. 4. The object with the corner-cube falls in a high vacuum

chamber. The dropping distance is determined by a Michelson interferometer. The He-Ne laser light source

is stabilized by locking to the Lamb-dip. The time base is obtained from a highly stabilized rubidium frequency

standard which is compared with the NIM cesium fre- quency standard.

Some absolute gravity stations have been established

in China from 1979 to 1981, for example, at NIM, Xiangshan in Beijing, and some other stations in Yunnan

Province in southwest China.

We have also determined the gravitational acceleration

at Sevres A3, BIPM. The value found was 980 925 914 ±

16 ixGal. The difference between this value and that of A. Sakuma is only 6 jjiGal.

506
Figure 4. NIM transportable gravimeter.

6. The Determination of thie Gyromagnetic

Ratio of the Proton 7^

The gyromagnetic ratio of the proton y'p, was determined at NIM in 1977 by both the low and high field methods using NMR [3]. The average of the experimental

results from 44 groups of 7p(high) measurements ob- tained in 1977 was

7; (high) - 2.675 163 7(95) x 10^ Anim-77 • s • kg"! , with a total RSS uncertainty of 3.5 ppm.

The apparatus for the low field determination of y'p is shown in Fig. 5. The work was done at the NIM nonmagnetic laboratory in a suburb of Beijing. In this experiment, the earth's residual magnetic field was compensated automatically and reduced to less than 50 x 10"^" T;

its variable component could be kept to less than 5 x

2Q-iorp_ rpj^g average value of 7^ (low) obtained was

7'p(low) = 2.675 132 6(22) x 10«s-i • T^l^,

with a total RSS uncertainty of 0.8 ppm.

1

7'.

Figure 5. NIM apparatus for measuring the gyromagnetic ratio of the proton in a low magnetic field.

From the results for -yp(high) and 7p(low), the value of K, which is called the ampere conversion factor, and the

value of 7p(abs), may be obtained:

K = I - (5.81 ± 1.80) ppm , and

7;(abs) = 2.675 148 1(48) x 10^ s-^ • T-^ . This value of K implies that the ampere maintained at NIM is smaller by 5.81 ppm than the absolute or SI

ampere.

7. The Absolute Determination of the Ohm

An absolute determination of the ohm by a cross- capacitor was carried out at NIM in 1978. The uncertainty of the cross-capacitor and the 1 fi absolute determination are ± 0.35 ppm and ± 0.48 ppm, respectively.

The cross-capacitor is shown in Fig. 6 and is different

from those of other laboratories. The null electrode is a fused quartz tube with its surface covered with a composite metal film of chromium-aluminum-chromium. Two

turns of fine lines are made by photo-etching of the film to provide an insulating air gap vdth the distance

between the two line centers determining the capacitance

value. (The nominal value is 0.5 pF.) The advantage of

this construction is that it keeps the capacitance constant, so that it is a convenient standard for operation

and measurement.

Figure 6. NIM cross-capacitor.

The ohm standard has been measured via the cross- capacitor periodically since 1978 with the result %iM = HabsCI - 0.36 X 10-*^) ± 0.48 fxH . (1) The international resistance intercomparison between

NIM and BIPM [4] allows Eq. (1) to be expressed as

^61-78.9 = f^ABs( 1 - 0.91 X lO"*') ± 0.48 (xTl . (2) Equation (2) is in good agreement with the results of other countries. In addition, the NIM cross-capacitor was

compared to those of other countries by NIM's participation in a cycle of the international intercomparison of 10 pF fused quartz capacitors organized by BIPM in August

1980. The value obtained at NIM was only 0.2 ppm larger

than that given by BIPM.

507
A cross-capacitor with a movable guard rod is currently under investigation at NIM. A new method for compensating the end effect has been proposed and a further improvement in the precision of the NIM cross- capacitor can be expected.

References

[1] Y. M. Cheng ef al., IEEE Trans. Instrum. Meas. IM-29, 316

(1980).

[2] BIPM Report, 1980 (unpublished).

[3] W. Chiao, R. Liu, and P. Shen, IEEE Trans. Instrum. Meas. IM-29, 238 (1980).

[4] Cross Capacitor Group, NIM, Acta Metrological Sinica (in Chinese) 1, 16 (1980).

508
Precision Measurevient and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

The 7p-Experiment at PTB

Kurt Weyand

Physikalisch-Technische Bundesanstalt, D 3300 Braunschweig, Bundesallee 100, F.R.G.

This paper will give a brief summary of the mechanical apparatus and the electronic set-up which

have been constructed to determine the gyromagnetic ratio of the proton by observing free precession

in a low magnetic field. A novel scheme is used to determine the coil constant of a multilayer field coil system consisting of four winding packages by measuring a quantity proportional to the magnetic flux density at a number of points on the axis of the set of coils carrying a stabilized current. Key words: coil constant; fundamental constants; nuclear magnetic resonance; precision measurement;

proton gyromagnetic ratio; SQUID magnetometry.

1 . Introduction

In order to determine the gyromagnetic coefficient of

the proton (^p) by the low-field method, the precession

frequency of protons is observed in a magnetic field of known flux density, produced by an electric current in a

coil of known dimensions. That means the experiment

has to be performed in a magnetically neutral surrounding, to avoid errors in determining the actual magnetic

field strength during the measurement of the precession

frequency. Therefore a nonmagnetic building has been

erected about 300 m away from other laboratories. Before starting the construction in 1978 the available ter- rain was magnetically surveyed. A place was found

where the spatial gradient in the earth's magnetic field shows a maximum value of about 0.25 nT/m [1]. Alternating magnetic fields at that place were found to be of the

order of 4 nT peak-to-peak at 50 Hz and 0.5 nT at 150 Hz

respectively. In order to compensate the earth's magnetic field, a

large tridimensional Braunbeck-coil system was con- structed consisting of four octagonal coils in each direction

[2, 3]. By careful adjustment of the currents through

these coils the main field in a spherical volume, 80 cm in diameter, at the center of the system has been reduced to values of not more than 1 nT. Variations of the earth'

s

magnetic field are observed by means of a triaxial flux- gate magnetometer giving a sensitivity of 100 mV/nT up

to 0.3 Hz. The sensor is placed halfway between the

nonmagnetic building and an observation station. (The ob- servation station, which contains all the electronic equipment, is about 80 m from the nonmagnetic building.) The

variations are compensated by means of additional currents through the coils, resulting in an instability of the residual field of 0. 1 nT in the day and 0.01 nT at night, as measured by means of a SQUID-magnetometer.

There are two main problems in doing the y^-

experiment: to obtain a relaxation time as close as possible to the natural relaxation time for protons in water

and to determine the coil constant, i.e., the value of flux density produced by a geometrical arrangement of turns

carrying a current /.

2. Detection of the Precession Signal

If a NMR experiment using water as the source of pro- tons and done in the free precession mode does not give

the natural relaxation time, there may be a gradient in the magnetic field over the probe volume. With a field

coil arrangement, which should provide an adequate field homogeneity (AB =3 nT/cm), a gradient may nevertheless be caused by extraneous materials near the probe,

such as those used for the probe supports or for coil forms. Impurities in the water may cause additional re- laxation. Whatever the source of the relaxation, the

determination of the precision frequency becomes more

uncertain.

Relaxation due to impurities in the sample itself may

be identified by performing a spin echo experiment [4]. In order to reduce extraneous material near the probe as much as possible the free precession experiment is done

using a single coil with about 400 turns for polarizing and

energizing the nuclear moments as well as for detecting

the precession signal. The probe itself is a spherical sample 4 cm in diameter, filled in an artificial N9-atmosphere

with triple distilled water by means of a special distilling

apparatus [5].

Figure 1 shows the block diagram of the signal detect- ing circuit. Fig. 2 indicates the time sequence of the

measuring procedure in the upper part and shows the ob- served signal trace in the lower part. On the starting

pulse, the control unit initiates the polarizing current source and afterwards the burst oscillator for the rf-90°- pulse. To avoid spikes the polarizing current is not switched abruptly but is slightly smoothed. Some mil- liseconds after disabling the burst oscillator the signal

amplifying path is enabled and the precession signal is fed through a preamplifier acting as a line driver to a high gain tuned amplifier. Its output voltage is mixed

with the frequency /j. of a synthesizer by means of a phase sensitive detector PSD. That frequency is set some

hertz lower than the expected signal frequency /o, thus

the PSD is acting as a narrow band filter. Its output signal, consisting of a frequency /o - /r- is fed through a low-frequency bandpass filter to a transient recorder for observing the signal and to a computing counter for measuring the difference frequency.

As may be gathered from Fig. 2, the relaxation time

T2 amounts to 2.2 s for the observed precession signal,

which shows a signal to noise ratio better than 50 for a 8-Hz-bandwidth of the PSD.

509
70ni distance

tuned

amplifier burst

oszillator

pliase

sensitive detector

^0-

band pass 2Hz<^-=10Hz

frequency

synthesizer

procedure

control

unit

0.5mT = B^ 16 mT

Bp^5.0mT

transient computing

recorder counter

Figure 1. Block diagram of the precessioyi signal detection

circuit.

polarization enable

polarization current

90°- pulse

precession signal

frequency-measurement. n10s

3. Determination of the Coil Constant

Theoretical Background

Up to now the dimensions of singlelayer coils have

been measured by mechanical and optical methods [6, 7], sometimes in combination with electromagnetic indicator

systems [8], and from the results of those measurements

the flux density at the center of the field coil has been

evaluated.

Novel schemes have been investigated and one of them

will be used, in order to find the coil constant of a system

of coils, i.e., as it shall be understood here, the value of

flux density at the center of the system as a function of the current through the coils [9, 10]. By means of a SQUID-magnetometer a quantity proportional to the

magnetic flux density will be measured in arbitrary units

at a number of points along the axis of the coil system

when a current is maintained in it. If the flux profile

along the axis of any coil system is calculated in terms of

its as yet unknown dimensions, the comparison of the

magnetometer readings with the calculated profile should

in principle yield a set of effective dimensions, from

which the coil constant may be evaluated again. Thus in general the coil constant of any arrangement of turns may be determined in this way leading to a more or less complicated set of unknown values. But in practice

measurements by means of two flux gate sensors con- nected in a differential bridge and thus rendering a resolution of 1 nT have shown, that it will be nearly impossible to record the flux profile of a long solenoid [11]. This

is due to the scatter of the angle between magnetic axis

and sensor axis while one of the sensor heads is dis- placed. For that reason a quite different coil arrangement has been investigated, which may itself be shifted

precisely whereas the magnetometer probe is fixed. Assuming a single circular turn with radius a as shown

in Fig. 3—then the field along the j-axis is given by the

Biot-Savart law according to ^[1 + (x/a)2]-3^.

2a

Br = (1)

Start

Thus the only unknown geometric dimension a may be

evaluated by measuring the field B o in the center and B^

at any other point on the axis of that turn at the distance X and by forming the field ratio V^- = B^/Bq as given by

a = x[l/V,)^ - 1]-^ (2) where all B values may be measured in arbitrary units.

510
If the current path differs from a circular path, it can be assumed that there exists an effective circular path

and using the method above an equivalent radius r* can be obtained. Supposing a polygonal current path, where n is the number of sides and a is now the radius of the

circumscribed circle, the field on the axis follows the

equation

Rp = }^ 2L g cosTT/n sin-rr/w

(3) By using the ratio of the field values at the positions x =

0 and X = a, produced by the polygon, the radius r* of the effective circular current loop amounts to

= a

2(1 + cos^TT/n)

cos^ TT M

1

-1/2

(4)

producing a field at the center point B*o = fx//2r*. The

fractional difference from the real field is drawn as a

function of n, as line (1) in Fig. 4, according to Bo - BR

V2"(l + cos^tt/w)

Bi

cos"^ Tiln

1/2 TT cos IT In n sin7T/n

1. (5)

Bp- Bp

(l)U\\'\)l^\

r* a / a / /

. \(5) 2 turn \ \ Vs in distance 2 A / b-a

a

0\ 1

%

1

1

10^

r-a

a

If the field value at x = a/2, instead of that at x = a, is taken to perform the field ratio the difference in the center field becomes about six times higher, and is given

by line (2) of Fig. 4 according to

5(1 + 4 cos^ TT /n )

5g

ac3

8 cos^ tt/w

1/2 T[_ cos IT /w n simr/n - 1.(6)

In order to simplify the measuring procedure for determining the field ratio function it is desirable to make use

of two current loops connected in series and axially dis- placed by ±A; then by measuring additionally the flux profile when both the loops are counter-connected the center point x - 0 may be found more precisely. For two

loops the field function must be changed containing now

two position dependent terms {x + A )/a and (x - A )/a. Thus a system of two nonlinear equations has to be

solved using two field ratios at different positions to evaluate numerically the effective geometric values r* and A*. Assuming again both loops as polygonal and us- ing the field values at x = a/2 and x = a related to that

at X = 0 in the case a /A =4, the error in evaluating the center field is given by line (3) in Fig. 4. For more clarity, the geometric proportions from which the presented

field differences result are also shown in that figure.

Looking to the righthand ordinate one may find the

differences of the effective radius r* and that of the in- scribed circle b = a costt/w from that of the cir- cumscribed circle a, both related to a. In order to achieve a center field strength which is high enough to do the 7p-experiment, winding packages

must be used instead of two singular loops. This is indi- cated in Fig. 3. Then a current NI flows through the

cross section ilh, where A'^ is the number of turns, and

produces a field on the axis according to the much more

sophisticated equation

B.r =

8lh

{x + I )ln a + h

+ (x - 1 )ln

a - h

1 +

1 + 1 +

X + I a + h

1/2

a + h

1 +

1 +

1 +

X + / a - h

X — I a - h

1/2

1 +

X - I a + h

vz (7)

For two packages displaced by ±A, as before, x must be

replaced by x + A and x - A leading again to two

equivalent position dependent terms. A set of about 360

equations of this type, each representing a field ratio at a different position x, is used to evaluate a set of at least

five effective geometrical values, which are able to describe the field function in the observed axis interval

Figure 4. Deviations of the center field, when polygonal

current paths are described by effective circular loops, and the

effective radius of such a loop, both as a function of the number of polygon sides.

4. The Coil System

A one-pair-winding coil system does not yield sufficient homogeneity over the sample volume, so the new coil system consists of two pairs of nominally identical coils

511
q

iasstibe r -Dewar anlirolate s ystem

^j<^^Fn\______ compressed Ngjr— air

compressed

air Laserbeam

Figure 5. Pneumatic field coil displacing apparatus.

as shown by Fig. 5 and in a more detailed manner by

Fig. 7. All the coils are of the same nominal radius

(160.85 mm) and are wound between flanges on a cylinder

of fused silica. The flanges, which are made of glass

fiber, are glued carefully on the cylinder to ensure coaxiality. The number of turns (1295 and 2916), the length of the windings (13.6 mm and 31.4 mm) and the mean dis- tances of the coils (79.5 mm and 307.8 mm) are so chosen,

that the coefficients up to the seventh order of the magnetic scalar potential expressed in zonal harmonics vanish. Because of the symmetrical arrangement of the coils the even terms vanish in any case, thus the inhomogeneity over a sphere of the NMR-sample size at the center should be less than 10"^ theoretically, while the

coil constant amounts to 17.5 mT/A.

It has turned out to be difficult to wind the multilayer

coils in a regular fashion by centering one turn on another by means of a fitting nylon thread wound

between adjacent turns—with increasing number of layers the winding began more and more to "float." Therefore the coils have been wound in the usual way but

the crossing lines from one layer to the next have been

staggered by a distance corresponding to the circumfer- ence of the cylinder divided by the number of layers.

This technique has yielded stable coils showing variations

of outer diameter at different generators not larger than

0.2 mm.

5. Determination of the Coil Constant

Hardware

In order to measure the flux profiles due to the coils a pneumatic coil carrying apparatus has been constructed

as shown in Fig. 5. The shifting device is made of glass

fiber and consists of a piston 600 mm in diameter and

550 mm in length inside of a cyhndrical tube of 1200 mm

length. By means of a conical aluminum block mounted

under the bottom of the piston the center of gravity is lowered. The small air-gap of about 0. 1 mm between pis- ton and tube enables a shift procedure having an angular

scatter of not more than 0.001°. A two-nozzle-system

prevents the piston from rotating around its axis. Thus the coils carrying a current equal to that which is to be used for the measurement of the precession fre- quency are moved in the axial direction past a SQUIDprobe. The magnetometer is a somewhat modified commercial type using a one turn, 6 mm diameter flux transformer coil and thus giving a resolution of 0.003 nT.

It is fixed inside of a special type of superinsulated

Dewar, which is made of glass fiber, and which rests on a

triaxially adjustable slab, allowing the probe to be positioned on the magnetic axis of the field coils. During the

motion the flux linking the probe is recorded, as well as the readings of a laser interferometer, which gives the

relative displacement of coil form and probe.

Three different measurements are necessary to determine the coil constant of the whole system. First the center of the inner pair is found by recording 'the flux

profile when both the windings are counter connected, as mentioned before. Then the flux profile of these windings

connected in series will be recorded and from that record

the set of dimensional values will be found just

representing the coil constant of the inner pair. Finally

the total coil constant of the four packages is found by

energizing the outer pair instead of the inner one and by

comparing the flux produced at the center point with that

due to the inner pair. Figure 6 gives the block diagram

to data acquisition systsm

BCD-memory

Laser

Interferometer

digital comparator Ax

- BCD-memory BCD-memory

1

1

digital output J-Oo

magnetometer

analog output

-=<t'o

clock 1

Sample & Hold

preset counter zn

digital comparator A<f

data registration controlled by

AO

A4>steps lAx f—

isteps 1 — AO steps - -

il5h

Act)=f(AI) 1 1 A4)^f(Ax) iAO=(AI)^si

1 x = 2R

thermal reset

Figure 6. Block diagram of flux profile recording circuit and

time sequence of such a procedure.

512
of the flux profile measuring circuit and the lapse of time

of one of the field measuring procedures. The magnetometer readings are split in two parts; one part represents

by means of a digital flux counter a value in terms of an

integer multiple of the flux quantum ^o, while the other

part is an analog signal proportional to the fractional part

of (j)o. This signal is observed by a fast acting sample and

hold digital voltmeter, from which a command for data

registration is initiated by comparing its output voltage

with a predetermined value, for example, when a preset

number of flux quanta h4>o is encountered. Thus a record

of correlated magnetometer and interferometer readings

will be obtained in nearly constant steps of n cj^o. In the

range close to the center point the recording mode is modified; here correlated readings are recorded in equidistant cc-steps.

6. Energizing the Coils

The flux profile measuring procedure takes more than

1 hour, meanwhile the current through the coils should

not change. A block diagram of the complete field coil en- ergizing set-up is given in Fig. 7. The left side of that

figure shows the circuit components of the regulated

main current source, essentially a digital to analog con- verter as a digitally presetable reference voltage source, the chopper stabilized control amplifier Vi, and the regulating unit Ti connected to the feedback resistor R y. By

means of the other components in this branch—the counter circuits, the comparator, and the oscillator—

linearly rising or decaying current with selectable time

constant may be generated. This is desirable for the beginning of a profile measuring run in order to avoid high

flux slewing rates to the magnetometer and for comparing the coil constant of the outer pair of coils to that of

the inner one. When the main current I q has reached the

desired value, a superposed control loop is closed, thus

providing a current stability in the order of 10"' for currents in the range 20 mA < I o < 100 mA. The loop

consists of a high gain' null detector, which compares the

voltage drop at the standard resistor R^' with the Zener

diode reference voltage Ur, and a proportional-integraldifferential (PID) regulating stage followed by the volt- age to current converter (V-z and T2) biased to about

30 |jlA. Instead of the shown superposed control loop

based on voltage comparison a more sophisticated one us- ing a SQUID-gradiometer system may be applied, result- ing in a stability almost two orders of magnitude higher

[12]; but this stability is not needed given the present

state of the experiment. By means of two other current

sources, each of them connected to one of the outer windings, additional currents up to 100 [xA may be adjusted

in order to optimize the field homogeneity at the center

of the field. These current sources have to be floating be- cause of the voltage drop at the windings due to the main

current 1 0, as indicated by the voltages Ufi and Ufz7. Systematic Errors

There are some sources of error limiting the accuracy

in determining a coil constant by the presented method.

By measuring mechanically the outer diameter of the

four winding packages at 18 different generators, differ- ences of about 0.2 mm have been found. Using this value

in constructing a polygonal current path would result in a polygon with 90 sides, as indicated by the dashed line in Fig. 4, and that would cause, theoretically, a worst case

relative error of nearly 10 . This error is small compared with those assigned to the magnetometry. Due to the area of the flux transformer loop the measured field values have to be corrected by considering the flux density over the loop area as a function of displacement x. This correction function may be derived from the vector

potential expressed in terms of the flux density and its derivatives along the axis of the two current carrying

loops:

B

1

2 1 +

(22/2)' a + i/fp (1 + yir^

-1

(1 + yf) -a/2 (1 + yi)

-3/2 (8)

main current source homogenizing current sources

Figure 7. CurTent sources and field coil system.

513
^10

Figure 8. Error influx measurement due to the size of the flux transformer loop (lefthand ordinate), and resulting error in

field ratio assuming an uricertainty in the loop radius of A p/p = ±0.001 (righthand ordinate), both as a function of dis- placement X.

with iji = {x + A )/a and yz - ~ A )/a [3]. Higher or- der terms of the evaluation need not be taken into ac- count, because they are correlated to higher order terms

of the ratio of radii of the sensor loop and of the field coil loop p/a. Equation (8) is illustrated in Fig. 8 with respect

to the dimensions of the coils used in our experiment. As- suming an error in diameter of the sensor coil not larger

than ±10"^, the uncertainty in field ratio depending on

position is given by the hatched area in Fig. 8, where it is to be neted that the right hand ordinate is to be used.

By examination of the correction function an adequate

alignment of the SQUID-probe to the axis of the field coil

is suggested by energizing the field coils by an ac-current and by looking for a minimum value of flux near the center and for a maximum value beyond the points of in- flexion. Thus the error due to misalignment of the axis

should be not larger than a few parts in 10^. An error of the same order will occur due to the nonlinearity of the

magnetometer.

Nevertheless, by means of some additional experiments it should be possible to determine the coil constant with a total relative uncertainty of less than 10"^.

The author would like to thank Prof. Dr. Capptuller for many helpful discussions. Prof. Dr. Cammenga and his co-workers for preparing the NMR-samples, Dr. Theile

and co-workers for doing the magnetic measurement in order to find the best place for the non-magnetic building, Dr. Vigoureux for the examination of the coil con- stant measuring method [14], and last but not least, the

co-workers of the project for their assistance.

References

[1] B. Theile et al., Magnetische Vermessung im Slidgelande der PTB, private communication.

W. Braunbeck, Z. Phys. 88, 399 (1934). A. L. Bloom and D. J. Innes, Appl. Phys. 36, 2560 (1965). K. Weyand, PTB-Annual Report 1978, 3.2.24, p. 155. H. J. Petrick and H. K. Cammenga, Ber. Bungenges. Physik. Chem. 8, 1105 (1977). P. Vigoureux and N. Dupuy, NPL Report DES 49 (1980). W. Chiao et al, IEEE Trans. Instrum. Meas. IM-29, 238

(1980). E. R. Williams and P. T. Olsen, Phys. Rev. Lett. 42, 1575

(1979). K. Weyand, PTB-Mitt. 85, 206 (1975). K. Weyand, PTB-Annual Report 1978, 3.2.24, p. 155.

[11] K. Weyand and E. Simon, PTB-Annual Report 1977, 3.2.23,

p. 146. K. Weyand, IEEE Trans. Instrum. Meas. IM-29, 324 (1980). W. Smythe, Static and Dynamic Electricity (McGraw-Hill,

New York, 1950). P. Vigoureux, PTB-report-E-16, 1980.

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9] [10]

[12]

[13]

[14]

514
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

A New Method for the Determination of the Proton Gyromagnetic Ratio

G. L. Greene

Gibbs Laboratory, Physics Department, Yale University, New Haven, CT 06520

A method for the determination of 7p is proposed which involves the separated oscillatory field magnetic resonance technique with flowing water. An association between the volume integral of a magnetic field in a cylinder (determined by a resonance frequency) and the line integral along its axis (determined by Ampere's law) is made. An accuracy of a few parts in 10^ may be attainable with this technique.

Key words: flowing water NMR; fundamental constants; magnetic resonance; precision measurements;

proton gyromagnetic ratio; Ramsey technique.

1 . Introduction

The principal interest in the measurement of the pro- ton gyromagnetic ratio (-/,) lies in its application as a very sensitive test of QED. The importance of y'p arises

in its role in the determination of the fine structure constant through the relation [1]: a-2 = [(c /4i2 (fVnLAB) (iJLp /|xb) (2e /h )] /y'

p

(1)

vi^here 2e/h and 7p are measured in terms of as- maintained electrical units and fi/fiLAB is the ratio of the

S.I. to as-maintained ohm. Such a determination of a is not dependent on QED and can therefore be compared

Math the results of - 2 experiments to provide a sensitive test of the theoretical predictions of QED. It shoud

be noted that the uncertainty in a as determined by

Eq. (1) is dominated by the uncertainty in yL Thus, a more accurate determination of 7p leads directly to an improved knovi^ledge of a. The major limitation to the determination of y'p has

been the accurate absolute determination of a magnetic

field [2]. It has been suggested [3] that a field determination technique based directly on Ampere's lav^^ may al- low a significant improvement in the accuracy of the field determination. This suggestion has been incorporated

into an experimental proposal which involves a movable

SQUID magnetometer or pickup loop [3] to take a pointwise average of a magnetic field along a path through a

solenoid. This technique, while extremely promising, has

the disadvantage that it requires a pointwise integration.

Such pointwise integrations are difficult to obtain to high

accuracy and can lead to unforeseen systematic errors. An alternative technique involving the Faraday effect [4] has problems related to variations in the index of refraction.

It has recently been demonstrated that Ramsey's

method of separated oscillatory field magnetic resonance, using flowing water, can be employed as a sensitive field averaging magnetometer [5-7]. This technique can be

applied to the measurement of 7p in such a way that

there is no need for a highly accurate pointwise integration [8]. With such a technique, the determination of 7p is reduced to a frequency measurement, a measurement of a current, and a measurement of a single length. An ac- curacy in the determination of y'p of one part in 10^ may

be possible.

2. Experimental Technique

A schematic outline of the apparatus can be seen in Fig. 1. To simplify the explanation of the technique the

follovdng assumptions are made (the vaHdity of these as- sumptions wdll be discussed later):

a) Only longitudinal fields (parallel to the axis of the

water tube) will be considered;

b) Demagnetization and chemical shifts are knovm;

c) The tube is aligned on the solenoid axis; d) The tube is uniform along its length;

e) Field inhomogeneity shifts, Bloch-Siegert effect,

etc., are negligible.

It is further assumed that, as suggested by Williams

and Olsen [3, 4], the windings of the solenoid are ar- ranged in such a way that the field falls off rapidly out- side the solenoid. In particular it is assumed that the

field at, and beyond, the separated Ramsey coils is insignificantly small (or at least accurately calculable).

Laser Interferometer nd llor measurement of AL)

Proton pre -polarizing

1

Figure 1. Schematic outline of yp apparatus. The second Ramsey coil can be shifted and its change in position accurately determined with the laser interferometer.

515
The resonance frequency w in the Ramsey arrangement

will be given by [5, 6] (with assumptions b-e)

L

r \B\dL (2) where L represents the Ramsey coil separation. Using

the assumption that the tube and the field are parallel, it is possible to express Eq. (2) as

CO = ^ B • de. (3) By Ampere's Law it is known that for any path through

the solenoid:

(4)

(5)

(6)

(7)

r B dt = iM)N I . Now using the assumption that the field at and beyond

the Ramsey coils (i.e., for Z < 0 and X > L) is negligible, Eqs. (3) and (4) can be combined to yield

l'piM)N I

It should be noted that in general it will not be possible

to determine L to high accuracy. The experiment is repeated with the coil separation re- duced by an amount AL. This change in coil separation can be measured very accurately by means of the laser interferometer. The resonance frequency oj' for this case

vdll be given by _ y'^i^N I " - L -ALCombining Eqs. (5) and (6) and solving for y'p one obtains

, _ AL oo' o) ~

lM)NI{oj' - w)

Note that all the quantities on the right of Eq. (7) can ul- timately be determined to very high accuracy (perhaps a few parts in 10^). This allows a very sensitive measure- ment of 7p to be made by (in principle) a determination

of:

1) The two frequencies co, co' 2) The distance AL,

3) The current /.

This promises to be a greatly simplified method for the accurate determination of "Yp.

3. Systematic Effects and Possible Sources

of Error

Several assumptions were required to lead to the

result of Eq. (7). These are dealt with in turn. All of the

effects, as well as other sources of systematic error, are

discussed in detail elsewhere [8].

3.1 Effects of Transverse Fields

Transverse fields may arise from external sources or from the solenoid. In the case of external fields, the

spurious effect may be examined by a systematic variation of the current /. Such a variation of / can also be

used to account for stray longitudinal fields. Transverse

fields from the precision solenoid may either be calcu- lated or examined experimentally by changes in tube di- ameter. If the field on the axis of the solenoid is reason- ably well known, the difference between the volume in- tegral of I B I and the line integral of B can be calculated

sufficiently well so that no significant spurious effect

results [8]. For a reasonably sized apparatus, the field on

axis need be known to —1% in order for this shift (—10"^)

to be known to approximately one part in 10^.

3.2 Diamagnetic Effects

The proton gyromagnetic ratio 7p is defined as the

gyromagnetic ratio for protons in a spherical sample of water (at a temperature of 25 °C). In the procedure here, a measurement of the gyromagnetic ratio for protons in a

cylindrical sample of H2O aligned vdth the magnetic field

is made. In this geometry, the field seen by the protons

wiW be altered from that experienced by protons in a

spherical sample (assuming the same field) due to the di- amagnetism of water. This shift will be given by (1 + i-uK/S) = -3.010(4) x

10"^ where k is the volumetric susceptibility of water (at 25 °C). This implies an uncertainty in the final measure- ment due to the correction for the bulk diamagnetism of water of about 4 parts in 10^. There is a variation in this correction with temperature

(due primarily to bulk expansion in water) of approximately -2 X 10"^ per °C. This imphes that the water

temperature must be kept to within ±0.5 °C for a meas- urement to a part in 10^.

3.3 Tube Alignment

It is vitally important that the axis of the tube be parallel

with the magnetic field 'of the solenoid. Otherwise, one

will not have equality between Eqs. (2) and (3). If the tube

is "canted" by an angle 0, one will expect the J | B | d£ to

differ from j B d( by approximately (1 - cose).To insure

accuracy to 1 part in 10^, ahgnment must be accurate to

10"'* radians. This may be difficult, but it is certainly possible. It should be noted that / B • d€ is a lower bound

for / I B \dL Thus any adjustment which minimized w is correct. Sagging or deformation of the tube must be controlled to a like degree, or calculated.

3.4 Tube Uniformity

If the tube's cross-sectional area varies along its length, then the field average taken by the protons will not be a uniformly "weighted" average. This problem

poses very serious constraints on the tube uniformity.

The effect of variations in tube cross section can be ac- counted for in a variety of ways which involve shifting

the tube within the coils. While it is possible to correct

for the systematic shifts due to such nonuniformity, this effect will probably pose a major problem in the successful application of this technique.

3.5 Other Systematic Effects

There are a number of small effects which can give rise to shifts and errors in any "beam" type resonance ex- periment. Among these are coil phase errors, BlochSiegert effect, and effects due to field inhomogeneities.

None of these effects is expected to give a significant

contribution to the error of a final measurement. However, it will be necessary to employ a series of trim coils to eliminate effects due to field inhomogeneities [8]. In addition to those mentioned in the above assumptions, attention must be paid to a variety of other effects which need to be well understood. These include details of fluid flow and mixing, field averaging, field inhomogeneity, T2 relaxation, determination of current /, deter516
mination of AL, line center determination, and signal to noise ratio. It appears [8] that the errors which would be

associated with these effects can be kept at the level of a few parts in 10^.

4. Conclusion

It is possible that a significant improvement over the current uncertainty of 0.21 ppm in -yp will be very diffi- cult without the application of a new technique. The

heart of such a new technique may well lie in the direct

application of Ampere's law to the determination of the

magnitude of a magnetic field as suggested by Williams

and Olsen [3, 4]. It is suggested that this approach may

be greatly simplified by the use of the Ramsey separated

oscillatory field magnetic resonance method using flowing

water. An ultimate accuracy of a few parts in l(r may be

attainable.

I wish to thank Dr. E. R. Williams, Dr. W. D. Phillips

and Dr. B. N. Taylor of the Center for Absolute Physical

Quantities, National Bureau of Standards, for several

suggestions and comments.

This work was supported in part by NSF contract PHY

78-25655.

References

[1] B. N. Taylor, W. H. Parker, and D. N. Langenberg, Rev. Mod. Phys. 41, 375 (1969).

[2] E. R. Williams and P. T. Olsen, Phys. Rev. Lett. 42, 1575

(1979).

[3] E. R. Williams and P. T. Olsen, IEEE Trans. Instrum. Meas.

IM-27, 467 (1978).

[4] E. R. Williams, W. D. Phillips, and P. T. Olsen, unpublished

(1978).

[5] G. L. Greene, Ph.D. Thesis, Harvard University (1977).

[6] J. M. Pendlebury, K. Smith, P. Unsworth, G. L. Greene, and W. Mampe, Rev. Sci. Instrum. 50, 535 (1979).

[7] G. L. Greene, et al., Phys. Rev. D 20, 2139 (1979).

[8] G. L. Greene, Research Proposal to NRG (1979), unpublished;

and G. L. Greene, Metrologia 18, 93 (1982).

517

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

The Quantized Hall Resistance in Two-Dimensional Systems

K. von Klltzing, H. Obloh, and G. Ebert

Physik-Department, Technische Universitat Miinchen, D-8046 Garching, F.R.G.

and

J. Knecht and K. Ploog

Max-Planck-lnstitut fur Festkorperforschung, D-7000 Stuttgart, F.R.G.

After the Josephson effect, a second solid state quantum effect—the quantized Hall resistance

seems to be useful to determine fundamental constants. We will demonstrate that electrons at the in- terface between two semiconductors or at a semi-conductor-insulator interface may form at low temperatures a degenerate two-dimensional electron gas with discrete energy levels (Landau levels), if a strong magnetic field is applied perpendicular to the interface. Under experimental conditions where

an integer number i of Landau levels is fully occupied with electrons, the value of the Hall resistance Rfj (ratio between Hall voltage and current through the sample) becomes quantized with values which

depend only on fundamental constants: Rf^ = h/e^i (h = Planck constant, e = elementary charge).

Our experiments on (100) silicon MOSFETs (Metal-Oxide-Semiconductor Field-Effect-Transistors) and

GaAs-Al^Gai_^.As heterostructures show that the measured value of the quantized Hall resistance is independent of device parameters. The analysis of high precision measurements on different MOSFETs

yields a value of h/e^ = (25812.79 ± 0.04) Cl, which is about 1 ppm smaller than the recommended

value. Calculations show that in our experiments corrections to the value of the quantized Hall resis- tance, which can be further reduced by increasing the magnetic field or decreasing the temperature,

should be smaller than 10"^. Key words: fine structure constant; Hall effect; magneto-quantum-transport; MOSFET; resistance standard; two-dimensional electron gas.

1. Introduction

Recent measurements of the Hall resistance Rh (Hall

voltage Uh divided by the current / through the sample) on a tv\^o-dimensional electron gas have shown that at lovi^ temperatures {T < 4 K) and strong magnetic fields (B > lOT) this resistance is quantized in units of

/i/e^i ===25813/1 ft (/i = Planck constant, e - elementary

charge, i - 1, 2, 3, ...) [1]. Since the present uncertainty in the realization of the SI unit Ohm is less than

10~^ [2], measurements of the quantized Hall resistance

can, in principle, be used to determine directly in SI

units with high accuracy the ratio h/e'^ and therefore the

fme-structure constant a:

(1)

with the permeability of vacuum po = 4it x 10'^ H/m and

the velocity of light c = 2.99792458 x 10"^ m/s. On the

other hand, the quantized Hall resistance can be used to develop a resistance standard based on fundamental constants, if the value of h/e^ is known.

The new quantum effect is based on the fact that the

energy levels of a two-dimensional electron gas (2DEG)

in a strong transverse magnetic field B becomes fully

quantized viath the same number of states A'^ = eB /h (degeneracy factor) for each level. Under experimental conditions, where an integer number of energy levels is occupied, we vidll demonstrate that the Hall resistance

depends only on fundamental constants and not on the

magnetic field, device geometry, or other parameters.

Such a result is not expected because the Hall resistance

Rh of a three-dimensional degenerate electron gas

depends on inaccurate parameters like the thickness d of the sample, the strength of the magnetic field B, and the

carrier concentration n.

Rf B

eucd

(2)

However, for a 2DEG the parameter d is incorporated in the two-dimensional carrier density Wj„„ and it will be

shown that the ratio becomes quantized in a strong

magnetic field in integer multiples of e /h. This leads to a quantization of the Hall resistance with resistance values

of

Rh —

h

2

• e I

i = 1, 2, (3)

In this paper we will, after an introduction to the tran- sport properties of a 2DEG in a strong magnetic field, summarize our experimental results obtained on two dif- ferent systems of a 2DE(j—the electrons in the inversion

layer of a (100) sihcon MOSFET (Metal-Oxide-Semiconductor Field-Effect-Transistor) and the electrons at the interface of a GaAs-Ala;Gai-j,-As heterostructure.

519
2. The Quasi Two-Dimensional Electron

Gas

A quasi two-dimensional electron gas (2DEG) is formed, if electrons can move freely within a plane {x-y

plane) but are trapped in the 2-direction. Such a 2DEG

can be realized at the surface of a semiconductor, if a suf- ficiently large electrostatic field is applied normal to the interface, which causes a drop in electron potential

towards the surface. If the width of this potential well is small compared to the wavelengths of the carriers, the

energy levels of the electrons are grouped in so-called

electric subbands Ej corresponding to quantized levels

for motion in the z-direction, the direction perpendicular

to the surface, and a continuum for motion in the plane

parallel to the surface [3]. Experimentally the existence

of a 2DEG was demonstrated by Fang and Howard [4]. They measured the channel conductance of a silicon MOSFET in strong magnetic fields at helium temperatures

and found that the observed quantum oscillations ori- ginate from a 2DEG. Figure 1 illustrates the situation

when such a 2DEG (inversion layer) is formed on a p- type semiconductor. The surface electric field is built up by a positive voltage (gate voltage Vy) at the metal

side of the metal-oxide-semiconductor capacitor. The

downwards bending of the conduction band originates

i

i

conduction band Ec

SiOFigure 1. Formation of a two-dimensional electron inversion

layer at the surface of a p-type semiconductor.

mainly from fixed negatively charged acceptors in the depletion layer close to the surface. At gate voltages above

a certain threshold voltage V,/, the depletion layer ddepi is thick enough to bend the conduction band near the sur- face below the Fermi level Ep forming a narrow nearly

triangular potential well in which the 2DEG is bound.

The concentration w,„„ per unit area of electrons in this potential well can be calculated from the capacitance of the surface capacitor Cox

(4) The energies of the electric subbands Ej (j =

0, 1, 2, . . . ) can be estimated by calculating the

energy eigenvalues of an electron in a triangular potential with an infinite barrier for ^ < 0 and a constant

electric field Eg = eNg l^o e.s for 2 ^0 (A^^ - concentration per unit area of positive charges at the gate, Co =

static dielectric constant of the semiconductor). The solu- tions are approximately [3]:

E, =

2m,

2/Z

ij = 0. 1, 2,

4

. . ) (5) where m, is the effective mass for the motion perpendicular to the surface. For silicon with a (100) surface orientation, two of the six valleys of the conduction band have

an mz - 0.916 rrio and four valleys have an

niz = 0.19 nio. The lowest electric subband Eg belongs

to the two valleys with the higher effective mass and

an effective mass for the motion parallel to the surface of

nix = niy - 0. 19 nio ; the other four valleys form a series of electric subbands E' j ij =0, 1,2, . . . ) starting at higher energies. For a more quantitative calculation of

the energies of the electric subbands it is necessary to solve the Schrbdinger equation for the actual potential V': which changes with the distribution of the electrons

in the inversion layer:

dx ^

2m, dz ^

2m

E^\l (6)

In the effective-mass approximation, the electronic

wavefunction for the subband can be characterized by

an envelope function

v|; = ^j(z) exp (i/c^aj + ikyy) (7) and the motion of the electron can be separated into a part parallel to the interface and a part perpendicular to

it with an energy

Eikj^kyj) 2mx

fkl

2w,, + E^ (8)

If only the lowest electric subband Eo is occupied, the

electrons form a strictly two-dimensional electron gas

with a constant density of states per unit area D{E) =2-77 ^[m^y Ih}

The effective potential Y 'z in Eq. (6) contains not only

the electrostatic potential due to the electrons in the

inversion layer and the charges in the depletion layer

(Hartree potential) but also many-body effects [5-7] (exchange and correlation potential). Different authors

have calculated numerically the energies of the electric subbands.

Experimentally, the separation between electric subbands can be measured by analyzing the resonance

absorption of electromagnetic waves with a polarization

of the electric field perpendicular to the interface [8]. Actually the resonance absorption energy is not identical with the difference between the one-electron energies

E \ - Eo, because the interband excitation induces a

polarization in the inversion layer and the electron-hole

interaction leads to an exciton shift. However, the total correction is relatively small because the exciton shift cancels to a certain extent the polarization shift [9]. Figure 2 shows measurements of the resonance absorption energy for transitions between the electric subbands

Eo and i as a function of the electron concentration n;„„ in the surface channel of a (100) siHcon MOSFET [10]. The results are plotted for two different substrate bias

voltages VsB ~ the externally applied voltage between

520
O^i , . . 0 2^6 810^2 »-n|ny/cm-2

Figure 2. The resonant positions of iyitersuhband transitions El — Eq as a function of the inversion carrier concentration

'^inv fo^ ^ (100) silicon MOSFET. The results are plotted for

two different substrate bias voltages Vgg [10]. The dotted line corresponds to the Fermi energy Ep relative to the lowest sub- band Bathe surface channel and the p-type substrate. Such a substrate bias voltage changes the thickness of the depletion

layer

ddeyi - V2eoesi(V; - VssVeCA^A - No) , (10a)

and therefore the charges in the depletion layer,

\Qd I = V2€„es,e(iVA - Nd){Vj_ - Vsb) (10b)

The built-in voltage Vi for silicon at low temperatures is about 1.1 V corresponding to an energy gap of 1.1 eV.

Since the contribution of the depletion charge to the electric field at the surface is proportional to Qd , the surface

electric field can be varied without changing the electron

concentration in the surface channel. Therefore the

energy distance between the electric subbands becomes

larger with increasing negative substrate bias voltage as shown in Fig. 2. The lowest subband separation at a

fixed Uinv is obtained at Vsb - + 1.06 V corresponding to a vanishingly small depletion charge. The Fermi energy

Ef - Eo (dotted line in Fig. 2) is for surface carrier con- centrations less than n;,,,, = 6 x 10^^ cm"^ and is always

smaller than the energy separation between the electric subbands. Therefore only the lowest electric subband Eo

is occupied at low temperatures (electric quantum limit).

The question, whether the higher electric subband E i or

a new subband series E o becomes occupied at extremely

high surface carrier concentrations, has not been solved

unambiguously [11, 12]. At least at inversion carrier con- centrations below 2 X 10^^ cm"^ or for measurements

with a negative substrate bias voltage, the condition for the electric quantum limit and therefore for a strictly

two-dimensional electron gas is fulfilled. A two-dimensional electron gas can not only be formed

at the semiconductor-insulator interface but also at the

interface between two semiconductors [13]. Figure 3

illustrates the situation for a GaAs-Al^-Gai-j. As heterostructure. In this case the narrow potential well for the

2DEG originates from the difference in the energy gaps

of the two semiconductors and the depletion layer on the

GaAs-side. The electron concentration in the interface

2DEG

Alj^Ga^.y As / GaAs

Figure 3. Schematic energy level diaxjram of a GaAsAl^Gaj^jAs heterostmcture. The p-type GaAs is weakly doped

compared to the doping level of the n-type Al^Ga,_^s.

channel is mainly determined by the concentration of ion- ized donors in the depletion layer of the highly doped

Alj.Gai-j-As. Without interface states, the slope of the

conduction and valence band at the interface is the same

for both semiconductors. For the 2DEG in GaAs only one

series of electric subbands Ej is expected, because the

conduction band of GaAs has only one minimum in the

first Brillouin zone with an isotropic mass of - niy = = 0.07 fUo. Contrary to the MOS system, the carrier concentration

of the 2DEG in a heterostructure cannot be varied over a wide range. Small changes in n,>„, up to hni,a, = 4 x 10^^ cm^^ are observed, if the device is illuminated with

intrinsic light [14] or if a voltage between a Schottky

contact at the surface of the heteroj unction and the inter- face channel is applied [15]. Since leakage currents across the Schottky barrier may falsify experimental results, our high-precision measurements are performed on devices without a Schottky contact and without intrinsic illumination. GaAs-MOSFETs are not available up to now

because of a large number of interface states located

energetically within the energy gap. Top views and

cross-sections of typical silicon MOSFETs and GaAsAl^Gai-a;As heterostructures used in our experiments

are shown in Fig. 4. For our MOSFETs a Si02 layer is thermally grown with a thickness of typically O.ljjim -

l|xm on a single crystal of p-Si. A metal (usually Al) on

the top of the oxide forms the gate. Highly doped n regions at the semiconductor surface are used as ohmic

contacts to the inversion layer. The depletion layer

between the contacts or the n-type inversion layer

and the p-type substrate separates electrically the sur- face channel from the bulk. The arrangement of the n contacts shown in Fig. 4 allows measuring in the

presence of a magnetic field either the conductivity comFlGURE 4. Typical geometries and cross-sections of devices used

in our experiments. From left to right: (a) long-silicon MOS-device for Hall effect measurements

with a typical length of 0.5 mm;

(b) circular MOS-device for conductivity measurements with a diameter of about 100 ixjn; (c) cross-section and top view of a GoAs-AIqj Gao^-^s hetero- structure used in our experiments.

521
ponent (Txx (circular device) or the resistivity components

Pa:j- and (long device).

For the GaAs-Al^Gai-xAs heterostructures, the dif- ferent layers are grown using MBE (molecular beam epitaxy) techniques [16]. Basically the heterostructure consists of a silicon doped Ala^Gai-;^ As-layer {x ^ 0.3,

Nsi = 3 X 10^'^ cm"^) on top of an undoped GaAs-layer

as shown in Fig. 4. The resulting carrier density of the

2DEG is about 5 x 10" cm'^. The Hall geometry of the

device is obtained by a mesa-etching process. Indium

metal was alloyed into the Al^Gai-^: As-layer to make

electrical contacts to the 2DEG for Hall effect and resis- tivity measurements.

3. Quantum Transport of a 2DEG in Strong

Magnetic Fields

3.1 Density of States

A strong magnetic field B with a component normal

to the surface causes the inversion layer electrons to move in cyclotron orbits parallel to the interface.

As a consequence of the orbital quantization the energy

levels of a 2DEG in a strong magnetic field can be writ- ten schematically in the form [siHcon(100)-inversion

layer]:

En,s,v ^ Eo + in + |-)^a), + sg^i-BB + vEy . (11)

where s = ± 1/2 (spin quantum number), g is the Lande

5^-factor [17], f = ± 1/2 (valley quantum number), Ey is the valley splitting [18, 19], and %Wc = t.ieBzNm m, ) is the energy difference between Landau levels with quantum numbers n and n + 1. For a 2DEG in GaAs the last term in Eq. (11) is zero. The Landau splitting 'fewc depends on B^ , the magnetic field component perpendicular to the x-y plane of the 2DEG, whereas the spin splitting depends on the total magnetic field B. This allows one to change their energies independently by changing

the orientation of the magnetic field relative to the interface. The wavefunction of a 2DEG in a strong magnetic field may be written in a form where one of the center coordi- nates of the cyclotron motion is a good quantum number

[20]. The degeneracy of each level En,s.v is given by the

total number of center coordinates of the system. Since

only one center exists in a region of area -nd^ [21] {( =

^iueB is the radius of the ground Landau orbit) the degeneracy factor per unit area is given by

N = ^ . (12a)

h

This degeneracy factor for each energy level £'„,s,y can

also be calculated from the number of states (for 5 = 0)

in the energy range ftwc , since the condition of total state conservation leads to a condensation of the states within

this energy range on one Landau level N = D{E)%oic = ^ . (12b)

ri

This degeneracy factor (number of states per unit area

for each energy level) plays an important role in the

analysis of the quantized Hall resistance because the sur- face carrier concentration for fully occupied Landau levels becomes quantized in units of eB Ih.

I

-

'I

! 3.2 Magneto-Conductivity

The modification of the density of states in a strong

m.agnetic field leads to a drastic change in the conductivity. Ando has calculated the level broadening and the transverse conductivity q^x of a 2DEG in strong mag- ) netic fields [20, 22] and finds that the density of states of i each level En,s,v is not a delta function, but has an elliptic

form in the self-consistent Born approximation (SCBA) ) with energy gaps between the levels. Additional tails at the edges of the density of states appear in a higher ap- , proximation [23]. The conductivity a^x at low tempera- I tures is mainly determined by the square of the density

of states at the Fermi energy and has peak values (if one

level is half filled) of ct^.T'' = {e^/T^1i)in + 1/2), if the

scatterers are of short-range. In the case that the

scatterers are of finite range, this peak value is reduced

[20].

' The characteristic features of the conductivity can be

understood by a simple diffusion of the center coordinate

of the cyclotron orbit, where the conductivity is proportional to the product of the diffusion constant and the

density of states at the Fermi energy. Without scattering, the electrons cannot diffuse in the direction of the

electric field (they are drifting perpendicular to the electric and magnetic field). Therefore an observed vanishing i' conductivity (Jxx in strong magnetic fields corresponds to j a vanishingly small scattering rate. Figure 5 shows a typical result for the conductivity Uxx (measured with a circular MOS device) as a function of i the gate voltage at two different magnetic field values of

j B. The oscillations are related to the variation of the

density of states at the Fermi energy. At gate voltages

where cr^x drops to zero, the density of states D{Ef) is zero. The valley splitting is resolved only for the Landau

levels up to n =1, whereas the spin splitting is visible for Landau levels up to n = 3. The period in the gate

voltage is constant within 1% and increases linearly with

increasing magnetic field within the experimental accu- racy of 0.5%. This result is compatible with Eqs. (4) and

(12), which means that the inversion carrier concentration increases proportionally to the gate voltage Vg, and

the degeneracy of each level increases proportional to the

magnetic field.

c

^ 310-^ /b=o B = 10T B = 18T

X 2-10"^

110"^

n = 2 n=2 \ ^

\\

i-^ 5 10 15 2 D 25 30 35 Vg/V

Figure 5. The conductivity ct,,.,. as a function of gate voltage Vg

yneasured with a circular silicon MOSFET at different mag- ' netic field values B(T = 1.5 K). ^

i At gate voltages where a^x drops to zero, an energy \ level En,s,v is fully occupied. This means that exactly at | these gate voltages the inversion carrier concentration is |,| a multiple of the degeneracy A'^ of a Landau level

Tiinv = iN ^ (i = 1, 2, 3, . . . ) . (13)

522
At these gate voltages a Hall resistance

Rh =9ry - = -|t (1 = 1, 2, 3, . . . )(14)

is expected. For high-precision measurements of this quantized resistance an accurate adjustment of the gate

voltage Vg corresponding to fully occupied Landau levels

is necessary. Such an adjustment of Vg with an uncertainty of less than 10"^ seems to be impossible due to instabilities in both the gate voltage and the threshold voltage. In addition, the a^x curve close to aTM" is strongly asymmetric

(Fig. 6, see also Ref. [24]) and the position of the

minimum is not well defined. Moreover, at very high

magnetic fields (Fig. 5, B = 18 T) u^x becomes immeasurably small (aj-a- < 10"^ Ct~^) within a certain reFlGURE 6. Temperature dependence of cr^x (measured at B =

74 T) close to the gate voltage of the fully occupied Landau

level n = 1.

gion AVp, usually referred to as the localized region.

Therefore, the gate voltage position of the minimum in

(Jxx cannot be determined. Kawaji and Wakabayashi in- vestigated this localization for silicon MOSFETs in detail and they found that for Landau quantum numbers

3= 1, approximately a fraction l/(2n-i-l) of a Landau

level is localized [25]. Different theories (localization due

to lattice imperfections [26-28], electron-electron interaction [29, 30], or simply an exchange of electrons between

the inversion channel and an electron reservoir in the

Si02- or the Al^Gai-^ As-layer [31], are able to explain

the observed "localization" of electrons. Up to now, it is not clear at which gate voltage Vg

within the localized region all states of the energetically

lower Ijang Landau levels are occupied. The results in Fig. 6 (r = 0.4 K) seem to indicate that in the localized

region states of the higher energetic Landau level be- come occupied (in agreement wdth calculations for a twodimensional disordered electron system in strong magnetic fields [28]), but measurements by other authors [32] or on other samples do not confirm this result. The uncertainty in the location of the gate voltage for which Eqs. (13) and (14) are fulfilled, seems to prevent a highly accurate adjustment of = ieB /h and therefore

an accurate determination of the quantized Hall resis- tance. Fortunately, Hall effect measurements show that

the Hall resistance remains extremely constant in the

gate voltage region where localization is observed and

has a value corresponding to a complete occupation of an

integer number of energy levels [33].

3.3 Hall Conductivity a

Experimentally, the Hall conductivity Uxy cannot be

determined from measurements with a circular device,

because the azimuthal component of the current

i(t> = (^xyEsD cannot be measured {Esd = source-drain

field). However, for a long device both the current along

the long axis (source-drain current) and the electric field perpendicular to the current (Hall field) can be measured. If the current has only one component (j = jx),

the potential drop along the current direction is proportional to the component pxx of the resistivity tensor, and

the Hall field is directly proportional to pxy [34]. It should be noted that in two-dimensional systems pxy is given directly by the ratio of the measured quantities Uh

(Hall voltage) and Ix (source-drain current) and is (apart

from small corrections) independent of the dimensions of the sample. Therefore, the experimentally determined

quantities Rh = Uh/Ix (Hall resistance) and Rx (resistivity of the sample measured between potential probes)

are identical with pxy and pxx , Rx = 0. The corrections

(Rh - PxyVpxy and (Rx - pxxVpxx for Rx 0, which will be calculated in Section 3.4, are unimportant for the

following discussion and we can set Rh = ?xy and

Rx Pxx • The corresponding components of the conductivity ten- sor (Jxx and (Jxy can be calculated from the equations (p^j- — Pyy> Pxy ~ ~ Pyx)'-

O-xx = Pxx/(plx + p4) (15)

O-x!/ = - PxyKplx + Piy) (16) In the limit of fully occupied Landau levels {(Jxx = 0) eqs.

(15) and (16) reduce to p^:; - 0 and pxy = - VfJxy

3.3.1 Measurements on Silicon MOSFETs

A typical result for Rx ~ Pxx and Rh ~ p^.,, as a function of the gate voltage at a constant magnetic field is shown in Fig. 7. The oscillations in PxxiVg) are mainly

determined by the oscillations in (JxxiVg), because Eq.

(15) reduces to (Jxx = Pxx/Pxy in strong magnetic fields

iPxy > Pxx)- Similar to the measurements on circular device (Fig. 5) "locahzed" regions (Rx = 0) around Vg = 5 V, lOV, and

21 V are visible. In these regions a constant Hall resis- tance is measured corresponding to a value of an integer

number of fully occupied Landau levels. Fig. 8 shows

high-precision measurements of Rx and Rh close to gate

voltages where the lowest Landau level w = 0 is occupied. Within the experimental uncertainty of 3 ppm for these measurements, the Rh -curve is flat in the gate vol- tage region 11.65 F < Vg < 12.15 V and has a value

which agrees with the recommended value [35] h/ie- = (6453.204 ± 0.005) H. This experimental result

is inconsistent wdth the idea that the Hall I'esistances Rh

should decrease with increasing carrier concentration and

therefore w^th increasing gate voltage Vg [Eq. (4)] Rh ~ Pxy

' 7f7

^

• (17)

ni„v (Vg - Vth)

The unexpected result that Rh remains constant

523
B=18.9 T T^l.SK

(b)

Figure 7. Gate voltage dependence of the measured Hall resis- tance Rfj (~ p^^) and resistivity (~ p^^) at B = 18.9 T for a long silicon MOS device (length to width ratio L/W = 8, dis- tance between potential probes A = L/3).

6^53,5-

6453,0

6452,5

6452,0

B =13,5T

T - 1,5K

Q R

30

25

20

11,50 11,75 12,00 12,25 V

^V.

Figure 8. High-resolution measurements of the Hall resistance Rfj and the resistivity R-^ for a silicon MOSFET at gate vol- tages close to the occupation of the n = 0 Landau level (B =

13.5 T,T = 1.5 K).

{^Rh'Rh < 3 X 10"^), even if the gate voltage is changed by AV^/F^ = 5%, can be explained in different

vi^ays: (a) the assumption that the carrier concentration in the inversion layer changes linearly with the gate

voltage [Eq. (4)] is not correct. A change in the

gate voltage in the region of the plateaus leads to the occupation of states not connected with the

inversion layer (e.g., states in the depletion layer,

in the Si02 etc.). This process seems to be important in GaAs-Al,.Gai_,,.As heterostructures [31]

and will be discussed later.

the carrier concentration in the inversion layer

changes proportionally to the gate voltage, but in the region of p^x ~ ^xx ~ 0 localized states in the

tails of the Landau levels are occupied. Prange

[36] as well as Aoki [37] calculated the contribution of localized states to the p.^^ component and

found that localized states do not contribute to cfxx but contribute to p,r^ in such a way that a gate

voltage independent value corresponding to a fully

occupied Landau level is expected. A summary of the theoretical calculations is shown in Fig. 9. This result agi'ees very well with the experimental data but is inconsistent with a former publication [38] which stated that the Hall conductivity is determined by the number of mobile carriers

below the Fermi energy Ep.

X

XO

XD

I

Figure 9. The density of states D, dc conductivity a^^ , and the calculated Hall conductivity u^y [37] as a function of the nor- malized inversion carrier concentration /A^.

Up to now it was not possible to decide which one of these two explanations applies to our devices. All theoretical calculations published after the discovery of the quantized Hall resistance [36, 37, 39, 40] show that

independent of the origin of the plateaus, the correct value p^.^ = h/eH is expected as long as the condition

Pxx = (^xx =0 is fulfilled. Since ctxx is never exactly zero, the most important

correction to the Hall resistance seems to be the finite scattering time t (a„. i= 0). The value of the conductivity

(Jxx (or resistivity p,.,r) at gate voltages where

corresponds to completely filled Landau levels can be in- creased by increasing the temperature or decreasing the

magnetic field. Figure 10 shows measurements on the same sample as shown in Fig. 8, but at a lower magnetic

field and a higher temperature. The plateau in Rh disap- pears, but the value of Rh and Vy = V„ip^'^) remains

unchanged (uncertainty ± 0.05 fl), even ii the resistivity

is increased by more than a factor of 100. If we assume

that relative to the measurements shown in Fig. 8 minima in p^^. are observed at gate voltages corresponding to rtinv — iN (which is not correct if an overlap

between adjacent Landau levels vdth different lineshapes

is present), and the correction' to the Hall resistivity Ap^j,

is proportional to p."^'", we can estimate Ap^-,, < 0.1 p"yTM. Therefore, the correction to Rh at the minimum of R^c in Fig. 8 should be smaller than 10"^. This correction is still smaller if, as assumed by Laughlin [39], ARh changes

524
9,00 9,05

Figure 10. Measurements of the Hall resistance and the

resistivity R^. for the same sample as used iyi Fig. 8 but for a

lower magnetic field (B = 11.5 T) and a higher temperature T

= 1.9K.

with the square of the resistivity Rx Our high-precision

measurements on two devices at a fixed gate voltage in the middle of the plateau between the Landau levels n = 0 and n - 1 yield a value p^.^^ = h/4e^ = (6453.198

± 0.009) n [33]. This value should be corrected by about

-0.6 ppm due to the finite input resistance of our voltmeter [46]. The comparison between Fig. 8 and Fig. 10 demonstrates that a strong magnetic field is necessary for high-precision measurements of the quantized Hall resis- tance. Figure 11 shows a series of measurements at dif- ferent magnetic fields B. With increasing B the plateau

shifts to higher gate voltages and the width AF^ of the

"localized region" increases drastically. AVg seems to be proportional to the effective energy gap /S.E between

two Landau levels [34], but a quantitative analysis is not

possible vdthout a clear definition of the width AVg . The

depth of the minimum on the left side of the plateau in

Pxy depends on the length to width ratio of the device and

originates mainly from the shorting of the Hall voltage at the source and drain contact. This correction will be dis- cussed in Section 3.4.

3.3.2 Measurements on GaAs-AljjGa.| ^^As Heterostructures

After the observation of the quantized Hall resistance on silicon MOSFETs different groups started to measure

this effect on GaAs-Al^-Gai-j As heterostructures [41, 42,

15], although earlier measurements on this system did not show steps in the Hall resistance [43]. But recent measurements on high-quality samples demonstrate that

this system too is suitable for the observation of magneto-quantum effects discussed in Section 3.3.1 for

silicon. Figure 12 shows an experimental result of the

resistivity p^.^ {B ) and the Hall resistance p^y (B) ef a GaAs-AUGai-a;As heterostructure with a carrier concentration of Uinv ~ 5 X 10^^ cm""^. Since the carrier concentration is fixed, the population of the Landau levels is changed by varying the magnetic field and therefore the

degeneracy - eB /h of each level. The Hall resistance

is not a linearly increasing function as expected from the

simple theory of the Hall effect, but shows a step-like

behavior. At magnetic field values where p^.^. goes to zero (fully occupied Landau levels), p^.,^ becomes

magnetic-field independent with values equal to h/e~i (i = number of filled levels). The origin of the plateaus in

Pxy can be explained, if an exchange of electrons (tunneling) between the 2DEG and the donors in the

Al^Gai-j-As is assumed. Baraff [31] calculated under this assumption self-consistently the ratio Ui^jB as a function

of the magnetic field and found that this ratio remains

constant within certain magnetic field regions. This

behavior originates from the fact that the Fermi energy

Ef (strictly speaking the energy bands relative to Ep)

cannot jump from one Landau level to the next lower

one, if the magnetic field is raised, but moves monotonically through the energy gap between two Landau levels.

Figure 11. Measurements of the Hall resistance Rfj and the

resistivity R^ as a function of the gate voltage Vg at different magnetic field values B. The curves are plotted for gate vol- tages close to the fully occupied Landau level n = 0 (aspect ratio of the MOS device L/W = 1, distance between potential

probes A. = 0.6 L).

Figure 12. p^y and p^x measurements on a GclAs-AIqjGag -As

heterostructure as a function of the transverse magnetic field. The inversion carrier concentration is about 5 x lO^^cm^'. The magnetic field region where a "contact" resistance higher than 1 MQ, is observed, is characterized by the dotted part of

the RffiB )-curve.

525
because a jump would be connected with an increase in the width of the Al.^Gai_.rAs depletion layer (= step-like

increase in n,„,.). Therefore, within a certain magnetic

field range an integer number i of Landau levels remains

completely filled which leads to a constant Hall resis- tance Rh = h/eH as a function of the magnetic field. This means that the plateaus in p^^ and p.„, which look

very similar to the corresponding measurements on sili- con MOSFETs, can be explained without assuming a lo- calization of carriers in the tails of Landau levels. The dotted part of the curve in Fig. 12 characterizes

the region where the relatively high "contact" resis- tance of the sample used in this experiment leads to an

error in our potential measurements. Even with an input

resistance of the volt-meter of 10^^ fi the uncertainty in our measurements is larger than 10"^ and therefore

highly accurate measurements are not possible. The

"contact" resistance oscillates with the magnetic field B

and has maxima at R,r{B) = R""". This result demonstrates that systematic errors in high-precision measure- ments due to "contact" resistances are possible. For

most of our silicon MOSFETs the contact resistance is negligibly small (< 100 ft).

3.4 Corrections Due to Sample Geometry

and Problems in Accurate

Hall Resistance Measurements

Generally the measured Hall resistance RjI^p is always

smaller than the theoretical value i?^^'"' = p^.y expected

for an infinitely long sample:

R^"^ = G Rit" 1 (18)

The factor G, which depends on the Hall angle 0 (tan© =

^x^/^xx) and the aspect ratio L/W, has been calculated

usmg the conformal mapping published by Lippmann and

Kuhrt [44]. This result agrees with recent calculations of

Girvin and Rendell [45]. Figure 13 shows results for the

correction 1 - G for different values of L/W. In these

calculations the position i of the Hall electrodes is exactly in the middle between source and drain (f/L = 1/2).

1-G

Hall electrodes

Drain

Figure 13. Calculations of the correction term G in Hall resis- tance measurements due to th^ finite length to width ratio LIW

of the device (Hall electrodes are assumed to he exactly in the middle between source and drain).

If the position of the Hall electrodes in asymetric {ilL <

1/2), the correction 1-G becomes much larger. Fig. 14 shows results for a sample with an aspect ratio L/W = 5 and ilL values ranging from ilL = 1/4 to ilL - 1/13. For a strongly asymmetric position of the Hall elec- trodes (|- « -|), the correction 1-G' can be found from

Fig. 13 using an aspect ratio 2llW:

1 1 - G( It

W

1

L

(19)

This correction is especially important outside the pla- teaus as may be seen in Fig. 15. The reduction of the

Hall resistance relative to the value originates from

the shorting of the Hall voltage at the source and drain

Hall electrodes 1-G

Figure 14. Calcnlatioris of the correction term G in Hall resis- tance measurements for a device ivith an aspect ratio L/W = 5

a7id differeyit positions C/L for the Hall electrodes.

Figure 15. Comparison between the measured quantities Rfj

and and the corresponding resistivity components p^y '^'^

Pxx < respectively. The silicon MOS device has a length to vndth

ratio L/W = 1 and the positions of the Hall electrodes (potential probes) are i/L = 0.2 (see Fig. U).

526
contact. However, for a Hall angle of 90° the influence of

the contacts on pTM and p^a- measurements is zero. For

typical minimal values of pi-"'Vpxy = 10"^ and aspect ratios of L/W > 3 the correction should be smaller than

10~^. This correction is unimportant because, at present,

the uncertainty in the value of the quantized Hall resis- tance is determined by experimental uncertainties and we would like to summarize some problems which arise in measurements of the quantized Hall resistance.

(a) The Hall voltage is relatively small because the

upper limit for the current through the sample

(typically 10 |jlA for a width of the device of

200 ixm) is limited by heating effects of the elec- tron gas. Since the current lines for a long device

in strong magnetic fields converge at corners of

the device, the upper limit for the current is not a

simple function of the width of the sample. Moreover, the inhomogeneous heating of the electron

gas close to the source and drain contacts leads to an inhomogeneous conductivity within the surface

channel. Under this condition the correction G

(Fig. 13, Fig. 14) which can be calculated only if the conductivity is spatially constant, is inaccurate.

(b) MOS devices are extremely sensitive to high fre- quency pick-up. Rectifying metal-semiconductor

contacts may lead to dc voltages even without an

applied source-drain voltage. The pick-up of spikes may lead to an injection of charges in the

oxide or the substrate connected with a shift in the threshold voltage Vth (relaxation time up to some hours).

(c) In order to eliminate the influence of the ther- moelectric power, the difference in the Hall vol- tage for different current directions is usually

measured. However, a change in the current

direction leads automatically to a change of the potential within the surface layer relative to the

source. This means that the gate voltage and the

substrate bias voltage relative to the surface channel changes because these voltages are applied relative to the source contact. For MOS devices with

a thick oxide and weakly doped substrate material, the influence of the voltage drop across the

device becomes less important.

In spite of the problems discussed above we believe

that measurement of the quantized Hall resistance using

cryogenic current comparators [46, 47] should allow one

to determine h/e^ with an uncertainty of about 10~'.

4. Conclusion

Measurements of the Hall resistance Rh of a twodimensional electron gas, realized' with a silicon MOS

field effect transistor or a GaAs-Al^-Gai-xAs heterostructure, demonstrate that under the condition of fully occu- pied Landau leVels Rh is (within our uncertainty of less than 10"^) independent of external parameters. We believe that the value of Rh is given by the equation

Rh = h/eH (i = number of fully occupied energy levels) as expected for an ideal two-dimensional electron gas,

since all known corrections (finite scattering rate, in- teraction between Landau levels, shorting of the Hall

voltage at the contacts) should change with magnetic

field, temperature or sample geometry, which is not ob- served. The silicon MOSFETs have the advantage that

the channel conductivity can be changed and becomes

zero for negative gate voltages, which allows determining the leakage current of the experimental set-up. On

the other hand, GaAs-Al^Gai-^As heterostructures can be used for measurements of the quantized Hall resis- tance at magnetic fields as low as 4 T and at temperatures of r = 4.2 K, which is much more pleasant than a magnetic field of 14 T and a temperature of 1.5 K neces- sary for silicon MOSFETs. If we can establish that the

value of the quantized Hall resistance measured on

GaAs-Alj-Gai-j-As heterostructures is independent of device parameters (no leakage currents, no contact resis- tances), this system seems to be (at present) the most

useful one for the practical application as a standard

resistor or for the determination of h le ^.

For our experiments on silicon MOSFETs we used a

large number of different devices obtained from different sources. We would like to thank M. Pepper (Cavendish

Laboratory, Cambridge and Plessey Company), G. Dorda

(Siemens Forschungslaboratorien, Miinchen), D. Tsui

(Bell Laboratories, Murray Hill), and R. J. Wagner (Naval Research Laboratory, Washington) for providing us with samples. The high-precision measurements were not

possible without the help of the PTB, Braunschweig, and we would like to thank E. Braun, V. Kose and F. Melchert for their assistance. This work was supported by

the Deutsche Forschungsgemeinschaft (SFB 128) and the

Bundesministerium fiir Forschung und Technologie of the

Federal Republic of Germany.

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Hail Effect in Siiicon IVIOS inversion Layers for h/e Determination

C. Yamanouchi, K. Yoshihiro, J. Kinoshita, and K. Inagaki

Device Fundamentals Section, Electrotechnical Laboratory

Sal<uramura, Niiharigun, Ibaraki 305 Japan

and

J. Moriyama, S. Baba, and 8. Kawaji

Department of Physics, Gakushuin University,

Mejiro, Toshimaku, Tokyo 171 Japan

and

K. Murakami and T. Igarashi

Electrical Standards Section, Electrotechnical Laboratory

Sakuramura, Niiharigun, Ibaraki 305 Japan

and

T. Endo, M. Koyanagi, and A. Nakamura

Quantum Metrology Section, Electrotechnical Laboratory

Sakuramura, Niiharigun, Ibaraki 305 Japan

The quantized Hall conductivity has been measured in n-channel inversion layers of silicon metal- oxide-semiconductor field effect transistor devices at temperatures below 1.4 K using magnetic fields up to 15 T with improved accuracy and resolution. This work aimed to reexamine a new method for h/e^ determination proposed by von Klitzing, Dorda, and Pepper. The result, which corresponds to

/i/4e^, and its one standard deviation uncertainty are -I/ct^-^ = (6453. 1969±0. 0046) Oetl or

(6453. 1969 ±0.0056) n^BS- This result agrees with that obtained by other methods to 1 ppm. Key words: electric quantum limit; inversion layer; precision measurement; quantized Hall conductivity; silicon MOSFET; strong magnetic field.

1

1

1. Introduction

A new method for h le ^ determination has been pro- posed by von Klitzing, Dorda, and Pepper [1]. The

method consists of precise measurements of the Hall emf

and the source to drain current flowing in an n-channcl

inversion layer on a silicon (100) surface at low temperatures under a strong magnetic field normal to the inter-

\ face. Appropriate surface carrier concentrations for the

\ measurements are attained by adjusting the gate volt- age. At these concentrations, the Fermi level lies halfjj way between the highest spin state of the Mh Landau

(j state and the lowest spin state of the (A^+l)th state. The

|

' electron system is effectively two-dimensional, degen-

; erate, and fully quantized in orbital motion,

j: The new method is based on the fact that plateau reij gions occur in the Hall conductivity a,,.,^ versus gate voltage curves and the values of cr,,.,^ at the plateaus are given by

a,.y = - ie-lh (1) when a)(.T » 1 and interactions between different Landau states are absent, and when the Fermi level lies near

the tail of Landau states, i.e., a^- = 0 [1,2] as predicted

theoretically by Ando, Matsumoto, and Uemura [3]. Their theory has shown that the plateau region of a^-i^ ar- ises from immobile carriers, i.e., immobile holes in the

upper edge of the lower Landau states and immobile

electrons in the lower edge of the higher Landau states. Here, i is the number of Landau levels below the Fermi

level, = eB hii^ the cyclotron frequency, and - the zero field relaxation time.

The experimental results [4] show that the value of the

Hall resistance, the ratio of Hall emf to channel current,

529
and its one standard deviation uncertainty are

-l/crj.i,(z = 4) = 6453. 198 ±0.009 O, in surprising agree- ment with the corresponding value h/Ae- -

6453. 204±0. 005 fi based on the recommended value for the fine-structure constant, 1/a = 137.03604±0.00011.

This fact strongly suggests that the simple relationship

Eq. (1) holds almost exactly in the real experiment.

The experimental conditions available are such that

ojcT — 15, fL
c — 9 meV, and the broadening of a Landau

subband T - 46(B/|x)i'2 = 2-3 meV [5,6,7], when the

zero field mobility value |jl — 10'* cm^/volt-sec and 5 = 15 T are used. It has been shown in theoretical studies [8,9]

that (Tj-y should be exactly an integral multiple of - e'^/h at T = 0 when the Fermi energy lies in the localized re- gime and the magnetic field is strong enough. However,

for finite magnetic field strengths, no theoretical basis to evaluate the level of precision has been obtained as yet.

The object of the present experiment is to test the

vahdity of Eq. (1) making similar measurements to Ref.

[1] by use of different measuring systems with improved

accuracy and resolution, in the hope of finding a many

body effect which is supposed to quench the effect of in- teractions involved in the two-dimensional disordered

system and/or establishing confidence in the use of the

quantized Hall resistance to determine h/e"^.

2. Experimental Procedure

Measurements were made on n-channel (100) inversion

layers of silicon MOSFET devices at temperatures below

1.4 K with magnetic fields up to 15 T. The magnetic

fields were supplied by an Intermagnetics General Corp.

150M superconducting magnet at the ISSP, University of Tokyo.

The samples used were obtained from A. Yagi of the

Research Center, Sony Corporation. The shape and

probe arrangement of the samples are shown in Fig. 1. Table 1 lists the samples, their oxide thickness and di- mensions, and the maximum value of the Hall mobility

|jL/y, the peak value in the Hall mobility versus gate volt- age curves.

V I X T

f~DW W

J,—jr —f I

Figure 1. Shape of samples.

Table 1. Sample geometry and Hall mobility.

Sample

Oxide

thickness (nm)

(|i.m)

w

(|xm)

(ji^ msx.

(cm2/V-s)

ififi L\j\J lo,OvU 72H57-1-13 200 16.00 1600 11,000 62H53-17-1-L1 200 600 100 10,300 62H53-17-1-L2 200 600 100 10,300 44-4-11-L 800 500 100 11,850

The channel current source consists of eleven 1.35 V

mercury batteries and a series of step-variable resistors

(380 kfi at maximum current) with two selector switches

and a current reversing switch. These switches and emf

selector switches are mounted on acryhc plates and

operated via teflon rods to minimize the electrical leak- age and heat flow from the operator to the contact

points. The channel current source and a Yokogawa Electric Works 2781 standard 10 kfi resistor were immersed

in an oil bath to minimize the effect of changes in room

temperature. The leads inside the Dewar are Formvarinsulated, 0.16 mm diameter copper wire, and are doubly

enclosed in vacuum tight cupronickel cylinders, which ex- tend from 30 cm above the top flange of the Dewar down

to just above the sample holder, in order to keep a fixed

temperature distribution along the leads to minimize

changes in thermal emfs, and also to protect against possible moisture condensation during liquid He transfer. The leakage resistance of the whole circuit was more

than 1012 ^

The Hall emf Vh and the potential drop Vgtd across a standard 10 kfi resistor are respectively balanced against

a Guildline Instruments Model 9930 Direct Current Comparator Potentiometer to obtain a null reading on a

Guildline Instruments Model 9460A Nanovolt Amplifier

without feedback and a strip chart recorder. The current

sensitivity of the amplifier (galvanometer) and recorder

system is (3 ~ 5) x 10"^^ A, and the voltage sensitivity of the system is 20 ~ 50 nV for the present measurements.

For each measurement of Vu and Vgtd, eight pairs of readings were taken with opposite directions of the channel current so as to eliminate the thermal emfs, their uni- form drifts, and long term drifts of the channel current. A typical run takes about 30 minutes to complete 16

readings. Another typical run consists of 24 readings in which both measurements of Vh and Vsd are included, as

is shown in Table 2. Here, Vsd is the potential drop

across the source and drain electrodes. We refer in this paper to Rh = Rstd '^Vn/'^stdi ^sd =

R.td SVsd/SVsw, and Rpp^ = R,td ^VppJlV.td =

(d/W) p,,..,., where Vpp^ is the potential drop between two

probes P and P', d the distance between P and P' , (d/W) = 4 for the samples used, R^td is the resistance value of the standard 10 kfi resistor, and S denotes a summation

over the corresponding readings in a run.

530
Table 2. Example of a typical ruyi. Date: 23 March 1981

.

Sample: 72H57-1-13. B = U.71 T. T = 14 K.

Position Channel . Gate „ „ of^f current

, f"^,,, voltage ^std res. ^He

selector ^oiai-Uy (xlOnV) (am) ,y.* (°C) (Torr) switch t> y

23.9 2.8 Vg

1 std + 9888253 5:05.0 2 SD + 6381046 08.5

3 H + 6381046 10.0 4 H — 6381008 12.0 5 SD — 6381006 15.0 6 std — 9888231 17.5

7 H — 6381009 5:19.0 8 SD — 6381001 21.5

9 std 9888229 25.0

10 std + 9888237 27.5

11 bU + 6381035 29.5

12 H + 6381036 31.5

13 std 9888225 5:34.0 14 SD 6381001 36.5

15 H 6381001 38.0 16 H + 6381035 40.5

17 SD + 6381033 43.5

18 std + 9888226 45.0

19 H + 6381032 5:48.0 20 SD + 6381037 50.5 21 std + 9888231 52.5 22 std 9888224 54.5 23 SD 6381001 56.5 24 H

Vg

6381000 58.5

23.9 2.3

3. Experimental Results and Assignment of

Uncertainties

It has been found during the course of the experiments

that the Hall resistance and the source-to-drain resist- ance RsD are independent of the channel current in the

range up to 10 |jlA at gate voltages where Rpp- {Vq) <

0.01 ft and the lowest four Landau subbands {N = 0) would be fully occupied, within the accuracy of the

present work. When the current exceeds —15 |jlA, Rppincreases abruptly and the Hall resistance begins to devi- ate from a constant value. The effect of geometrical configuration and electrodes was also checked. For "wide" samples with L/W =0.1

and L/W = 1.0, both Rh and Rsd were measured.

When CT.„. = 0, the relationship Rsd = Rh - ~ ^o-^y

should hold if the spread resistance of the source and

drain electrodes is not present. In this case Rsd can be

measured in place of Rh in a bridge configuration, and

the samples could be connected in series or parallel (or

both) to get various, probably very stable resistance

values. Figure 2 shows the result of measurements for a sample with L/W = 1.0 obtained at 5 = 14.21 T, T = 1.4 K, and with a channel current of Isd = 9.46 |xA. No

72H57-1 -13

6A52-5I 1 1 1 I I I ui i_

KO 1/;-2 IAS K-8 GATE VOLTAGE ( V )

Figure 2. Hall resistance and source to drain resista^ice Rsd of a "wide" sample as a function of gate voltage. At gate

voltages between li.3 V and U.6_y, the_avemge values of and Rsd O-^e almost eqxial, i.e., (Rsd " Rh^Rh = -0.15±0.36

ppni, ivhich is within the random uncertainty of the present

yrwasurement.

531
difference is found within the resolution of the present

work between these Rh and Rgo data in the gate voltage

region between 14.3 V and 14.6 V. For "long" samples with L/W = 6, Rh and Rpf were

measured. A result obtained at B = 14.21 T, T = 1.4 K, and with Isd = 9.46 |jlA is shown in Fig. 3. In the

gate voltage region between 14.4 V and 14.6 V, Rh

passes through a plateau. Little differences are found

again between the Rh data of the "long" sample at the

plateau and those of the "wide" sample.

In order to ascertain whether the measurement result

is independent of the method of measurement, a high in- put impedance (>10''^ fi) digital voltmeter, Datron Electronics Ltd. Model 1071, was also used instead of the

Guildline potentiometer and galvanometer system.

Results of Rpp' measurements obtained by use of the dif72-17H53-19-L1

6453-5

64530

cr

6A525

B = 1A,21T T=1 A K

Is[):i9A6 pA

h/Ae2 \

11 Isd X

p

f-H' R

-+ + + + 4 L

10

0-5 0-

1A0 lA-2 1AA 1A6 GATE VOLTAGE ( V )

1A8

Figure 3. Hall resistance and tmyisverse resistance Rpiof a "long" sample as a function of gate voltage. At gate volt- ages between U4 V and U.6 V, Rp,,- passes through a )niHimum f = .5 x 10''' fl) and R^ passes through a plateau of

value Rh = 6i53:202i ±0.0008 il. This result was not taken

into account in the final result because this measurement was made without the oil bath.

ferent methods are shown in Fig. 4. It is noted that the

direct reading of potentials with a high impedance digital

voltmeter could cause large and qualitative errors in the

RpP' measurements, supposedly due to a loading effect caused by transient imbalance in the feedback loop of the

digital voltmeter. These errors might produce some diffi- culties in measuring Rh values or in choosing the gate

voltage value for Rh measurements.

10^

10"'

72-17H 53-19-Ll

10"'

10'-

1A-8 GATE

150 15-2 VOLTAGE ( V

15 A

Figure 4. Values of Rpp- obtained in different ways are plotted

against gate voltage.

The results of the present measurements are summarized in Table 3, in which average values of Rh and Rsd

with their respective one standard deviation uncertainties are listed. The final value of Rh obtained at B

one standard deviation uncertainty are

14.71 T and its

or

Rh = (6453.1969±0.0046) Oetl (0.72 ppm),

Rh = (6453.1969+0.0056) Habs (0.88 ppm).

Table 4 lists the sources of uncertainty associated with

the experimental system when it is operated at Isd —

9.5 |xA and Vh — 60 mV, and their one standard deviation estimates.

Table 3. Result of precision measurement.

Sample or Rsd

("etl)

o"h -1 (^^ETL)

Rpp'

iCl) B

(T)

72-17H53-19-L1 =6453.1966 ±18 0.0015 14.71 62H53-17-1-L1 =6453.1969 ±23 0.0005 14.71 62H53-17-1-L2 =6453. 1982 ±26 0.0056 14.71 72H57-1-13 /J/y =6453.1958 ±15 14.71 =6453.1948 ±18 14.71 44-4-11-L «/y =6453. 1974 ±31 10.50

=6453.1969 ±10flETL Fluctuation of Mean Values ±20 Random Uncertainty

±41 Systematic Uncertainty

Estimated from Table 4 =6453. 1969 ±0.0046 Hetl Final Result

532
Table 4. Sources of Uncertainty.

Sources of Uncertainty

One Standard Deviation

Uncertainty in ppm

Random Uncertainty of around the Mean of Each Sample

Fluctuation of the Mean Values of Each Sample

(Possible sample dependent uncertainty) Root Sum Square

0.31 0.15

0.34

Sources of Systematic Uncertainty

(a) Thermal emfs variation Less than 50 nV/2 hr and almost linear function of time. Measuring sequence can cancel the effect if linear. A single run takes ~ 30 min.

(b) Variation of measuring current

(1) Potentiometer current (from spec.

)

(2) Mercury battery current Caused by emf change due to variation of oil bath temperature, less than 0. rC/2 hr and almost linear in time. Measuring sequence can cancel the effect if linear.

(c) Errors in potentiometer

(1) Linearity of dials (from spec.)

(2) Power coefficient of emf generating resistors -0.1 ± 0.05 ppm for comparing 10 kO and 6.5 kfl in 0.2 V range.

(d) Insulation Normally greater than 5 x 10^^ O.

(e) Standard resistor value

(1) Temperature uncertainty

(2) Aging and effect of transportation -1.5 ppm/year ±0.6 ppm. Compared to the same type

resistor kept at rest at 20°C.

(3) Calibration including Qeti/^abs

Root Sum Square e.xcept e-3 Root Sum Square

0.1

0.1

0.05

0.05

0.05

0.1

0.6

0.5

0.64

0.81 RSS (Random + Systematic e.xcept e-3) RSS (Random + Systematic)

0.72 0.88

4. Discussion

4.1 Self Test of the Relation -(T-yH\) = h/ie^

As is seen in Table 3 and Table 4, the difference in Rh

values corresponding to h/Ae^ between the measured

samples is of no statistical significance within the resolution of the present work. However the uncertainty associated with the standard resistor value makes it difficult to test the relation up to the order of, for example, 0.1 ppm or higher in precision, and to check whether many

body effects determine the precision of the relation. A

relief to this situation would be obtained by measuring

a^^HS) and/or aj.~H2) as well as (r.ij^i'i) on the same sample. If the relation a-^c'yi^.) = a,,ryH4) x 2 = a7/(8) x 4 is verified to hold experimentally up to the highest precision, the many body effect on the unit of quantization of Hall conductivity in Si-MOS systems would be ruled out,

at least up to the order of the precision, since many body

effects depend primarily on the carrier concentration.

A preliminary measurement with a magnetic field of

14.7 T on sample 72-17H53-19-L1 reveals that

2a-\8)/(T~M) = 0.9999984 ±0.0000010 at a channel

current of 2.46 |jlA [10]. However, higher magnetic fields should be used to get higher precision since the plateau

region of Rh was narrow and the highest limiting current was low at B = 14.7 T. This kind of measurement will provide a method of self-test of the quantized Hall con- ductivity phenomenon independent of the calibration ac- curacy of the standard resistor.

4.2 Experiment With Different Materials

In the case of e/h determination via the ac Josephson

effect, an experiment with different materials was essential to establish confidence in the use of the effect. Use of different materials is aimed at changing carrier concentrations and/or screening properties. In the case of MOSFET devices, these changes are realized by simply ad- justing the gate voltage and magnetic field. The carrier

concentration can be varied at least a factor of four and even a qualitative difference in screening properties is expected to occur by changing the magnetic field: the

carriers in the lowest Landau subband, which are con- sidered to be localized, become delocalized at magnetic

fields above 25 T [11], for example. This fact suggests

that the role of material dependence measurements

would be less important to establish confidence in the

h determination by means of the quantized Hall con- ductivity in two dimensional systems.

533
The authors would like to express their sincere thanks to Professor S. Tanuma and Dr. I. Oguro of the ISSP,

University of Tokyo, for extending to us the use of their

high magnetic field facilities, and to Dr. A. Yagi of the

Sony Corporation for providing us with samples. In particular, the authors wish to acknowledge the stimulating

discussions with Professor T. Kinoshita of Cornell

University, Professor T. Ando of Tsukuba University,

Professor Y. Uemura of the University of Tokyo, Professor N. Miura of the ISSP, and Dr. S. lijima of ETL.

References

[1] K. V. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett.

45, 494 (1980).

[2] S. Kawaji and J. Wakabayashi, in Physics in High Magnetic

Fields, Ed. by S. Chikazumi and N. Miura, Proc. Oji International Seminar, Hakone, 1980 (Springer, Berlin,

1981), p. 284.

[3] T. Ando, Y. Matsumoto, and Y. Uemura, .J. Phys. Soc. Jpn.

.39, 279 (1975).

[4] E. Braun, E. Staben, and K. v. Klitzing, PTB-Mitteilungen

90, 350 (1980).

[5] T. Ando and Y. Uemui-a, J. Phvs. Soc. Jpn. 36, 959 (1974).

[6] J. Wakabayashi and S. Kawaji, J. Phys. Soc. Jpn. 48, 333

(1980).

[7] S. Kawaji, Surf. Sci. 73, 46 (1978).

[8] H. Aoki and T. Ando, Solid State Commun. 38, 1079 (1981).

[9] R. E. Prange. Phys. Rev. B 23, 4802 (1981).

[10] K. Yoshihiro, J. Kinoshita, K. Inagaki, C. Yamanouchi, J. Moriyama, and S. Kawaji, Surf. Sci. 113, 16 (1982).

[11] N. Miura, Y. Iwasa, T. Itakura, and G. Kido, J. Phys. Soc. Jpn. .36, 1228 (1982).

534
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Cryogenic Method for the Determination of the FineStructure Constant by the Quantized Hall Resistance

E. Braun, P. Gutmann, G. Hein, F. Melchert, P. Warnecke, and 8. Q. Xue*

Physikalisch-Technische Bundesanstalt, 3300 Braunschweig, F.R.G.

and

K. V. Klitzing

Physik Department der Technischen Universitat Munchen, 8046 Garching bei Miinchen, F.R.G.

Recently a determination of the fine-structure constant with a relative uncertainty of 1.3 parts in

lO*"' has been carried out. Since then attempts have been made to construct an e.xperimental setup based on a potentiometric method using modern cryogenic measurement techniques, which should lead

to a significant decrease of the uncertainty.

Key words: cryogenic current comparator; cryogenics; fine-structure constant; flux transformer; fundamental constants; Hall resistance; SQUID.

1. Introduction

As shown by v. Klitzing et al. [1] the Hall voltage Um

of a MOSFET in high magnetic fields and at low temperatures shows steps at which U// is given by

Uh = ~ \ l (1) a = 1, 2, 3. . .), where h is the Planck constant, e the elementary charge,

and / the source drain-current. The ratio of the Hall volt- age to the current is the so called Hall resistance Rh'- Rh - - \ . (2)

I e^

It has to be pointed out that the Hall resistance at the

i steps is independent of magnetic field, temperature and

j any geometric dimension of the sample. Therefore, it is

• possible to determine the ratio of two fundamental conI stants h/e"^, by measuring the Hall resistance.

Moreover, by use of

where \x.o is the permeability of vacuum and c the speed

of light in vacuum—presently known with a relative uncertainty of 4 X 10"^—^it is possible to determine the

fine- structure constant a with about the same uncerii tainty as hle^. It has already been shown [2] that with a

l' rather simple experimental setup it is possible to deter-

;j mine h/e"^ and thus a with a relative uncertainty of about

\ 1 part in 10". Since the fine-structure constant is of great

I importance to quantum electrodynamics and for the reali-

' zation of electrical units, great efforts are undertaken to

) *0n leave from National Institute of Metrology, Peking, People's

j Republic of China.

decrease its uncertainty. Up to now the uncertainty of other experimental methods for the determination of a is one part in 10^ while the currently recommended CODATA value which is based on the 1973 least-squares ad- justment of the fundamental constants has a relative uncertainty of 8 X 10"^ [3]. One further aspect of interest in the quantized Hall

resistance is the fact the h /e^ has the dimensions of a re- sistance. One may hope to decrease the experimental uncertainty to a value which allows the use of this new

method for the maintenance and the reproducibility of the SI unit of resistance, the ohm, similiar to the use of the Josephson effect in connection with the volt. In the following we shall at first give a short discussion

of the initially used experimental setup and then describe

the new method.

2. Method of Voltage Comparison

The MOSFET and a standard resistance of 10 kH were

connected in series and fed by a current of 5 |xA to 10 |jlA as shown in Fig. 1. The Hall voltage at the MOSFET Uh and the voltage drop at the standard resistance

Un were measured by means of a high impedance digital

voltmeter (DVM). The Hall resistance is then given by

Uh Rh = (4) UN

In the result given in Ref. [2] a systematic error had

been corrected with the wrong sign. The correct results based on the same experimentally obtained data of Ref.

[2] is -\ = (6453.194 ± 0.009) H (5)

and = 137.03584 ± 0.00018. (6)

535
Z3 €>

Uo

B=15T

T=1.5K

DVM

Figure 1. Experimental setup for the method of voltage comparison. Rh is the quantized Hall resistance, Rx the standard

resistance. S is a two-way selector switch. The uncertainty of this method is limited for various reasons:

(a) the resolution of the voltmeter;

(b) possible deviation from linearity of the voltmeter

(for a current of 10 |jlA, Uh ~ 64 mV and Un ~

100 mV); and

(c) drifts of thermal emf s in between different readings.

The potentiometric method described in the foUow^ing

should overcome these difficulties and should lead to a smaller uncertainty.

3. Cryogenic Potentiometric Method

3.1. Principle

The Hall voltage Uh at the MOSFET is balanced by an

equal voltage U;^ across a standard resistance Rnc- The

MOSFET and the standard resistance are in two separate

circuits and the ratio of the two currents has then to be

known with high accuracy. In order to decrease thermal

emf's and thermal noise, most of the components of the

two circuits are at liquid helium temperature. The

standard resistance for instance wiW be a cryoresistor

made of a copper-phosphor alloy.

The setup is in principle shown in Fig. 2. The ratio of the two currents 1 1 and /2 in the two circuits is kept con- stant by a cryogenic current comparator (CCC) in con- nection with a SQUID (SQl) [4]. Each of the currents passes through the CCC via a coil with A''i and A''2 turns,

respectively, and wound in opposition.

Any deviation of the condition

Nih - N2I2 - 0 (7)

is coarsely controlled through feedback by master (MCS)

and slave (SCS) current sources. The fine regulation is automatically done by the SQUID.

3.2 Balancing Condition

Since the two-voltage drops Uh and U^ cannot be ex- pected to be equal at the beginning of the measurement,

there has to be the possibility to change 1 2 into / '2 in or- der to get

Uh =0 U'n = Rnc 1 2- (8)

T=4.2K

Figure 2. Principle of the cryogenic potentiometric method. Rh is the quantized Hall resistance,

Rf^(^ the cryogenic standard resistance,

CCC the cryogenic current comparator,

MCS the master current source, SCS the slave current source, SCFT the supercond^icting flux transformer,

SQl and SQ2 are SQUIDS.

If for instance Rmc is smaller than Rh, 1 2 has to be in- creased for achieving a balance. In order to keep the au- tomatic feedback by the CCC and SQl still working, a

fraction /s of / 2 h = r\r2. (9)

is fed through an additional coil of the CCC with N2, turns and opposite sense of winding in comparison with

A''2 as shown in Fig. 3.

'

1 ^13

c c c 1^3

Figure 3. Scheme of the additional circuit to achieve the bal- ance at SQ2 by adjustment of tlie resistance R^ and R^.

536
Now, as the switch S is closed, the condition

Nih - N2I2 + Nsh = 0 (10)

is satisfied by means of the CCC. As seen from Fig. 3, r\ is given by

-n- r (11)

R 4 The ratio R^toRi has now to be adjusted until the bal- ance Rh 1 1 - Rnc 1 2 (12)

is achieved. As balance indicator a further SQUID (SQ2)

in connection vdth a newly developed superconducting

flux transformer (SCFT) with a 1200 to 1 turns ratio will be used [5]. Together vdth Eqs. (9) and (10) the un- known Hall resistance is now given by ^^ = w,^^'~ S^- ^^^^

The same setup with either different windings or even

another CCC can be used for the comparison of any two

other resistances, for instance to compare R^: with a known standard resistance.

By assuming thermal noise from a resistance of about

10 kO in the balancing circuit the current resolution of our system is given by 2 x lO"^''^ A at 1-Hz-bandwith [5]. Measuring at a current of 10 |jlA means that a determination of the fine-structure constant to a few parts in 10*^ should be possible if R^c is known with a corresponding

uncertainty.

We would like to thank Dr. Pepper from Plessey Co. and Dr. Dorda from Siemens A.G., who provided us with

samples and all our colleagues who assisted us. Prof. Kose stimulated the work with fruitful discussions.

References

[1] K. V. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett.

45, 494 (1980).

[2] E. Braun, E. Staben, and K. v. Klitzing, PTB-Mitteilungen

90, 350 (1980).

[3] E. R. Cohen and B. N. Taylor, J. Phys. Chem. Ref. Data 2, 663 (1973).

[4] S. Q. Xue, P. Gutman, and V. Kose, to be published.

[5] S. Q. Xue, P. Gutman, and V. Kose, Rev. Sci. Instrum. .52, 1901

(1981).

537

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Status of the NBS-NRL Determination of the Fine-Structure Constant

Using the Quantized IHall Resistance Effect

M. E. Cage, R. F. Dziuba, and B. F. Field

Electrical Measurements and Standards Division, National Bureau of Standards, Washington, DC 20234

and

C. F. Lavine* and R. J. Wagner

Electronics Technology Division, Naval Research Laboratory, Washington, DC 20375

Measurements of quantized Hall steps have been made on (100) Si MOSFET devices using a potentiometric method. In this approach the quantized Hall resistance at a step is compared to a nominally

equal room temperatui'e reference resistor. The standard deviation of a single observation is 2 parts in

10' at 10 |jlA source-drain current. The measurement system is described, along with a report on some

of the problems encountered. Key words: fine-structure constant; fundamental constants; inversion layer electrons; precision measurement; quantized Hall resistance; silicon MOSFET; superconducting magnet.

1 . Introduction

The paper of von Klitzing et al. [1] on precision Hall

resistance measurements of metal-oxide-semiconductor

field-effect transistors (MOSFETs) suggests that this method can be used to determine the fine-structure constant and to establish an absolute resistance standard.

When a positive voltage is applied to the gate electrode

of an n-channel MOSFET, a thin layer of electrons (called

the inversion layer) is induced beneath the oxide. The

layer approximates a two-dimensional electron gas. A

magnetic field 5 > 10 T applied perpendicular to the

inversion layer constrains the electron motion in the

layer to quantized orbits vtath discrete energy levels (Landau levels). At sufficiently low temperatures, T <

2 K, the Fermi level can be adjusted so that a given

Landau level is filled while higher levels are empty, so

that an inversion layer electron cannot be scattered if the

spacing between Landau levels is sufficiently large.

Under these ideal conditions, the Hall resistance (the ratio between the Hall voltage and source-to-drain current

Isd) is given to high accuracy by

Rh — ho 2ai

25, 813

a,

where po is the permeability of vacuum (exactly equal to 4 77 X 10"''' H/m), c is the speed of light in vacuum, a is the fine-structure constant, and i is an integer.

Since resistances can be determined in SI units to within a few hundredths of a ppm (part per million) by

means of the calculable capacitor, and since the present

uncertainty in the speed of light is only 0.004 ppm, meas- urements of the quantized Hall resistance of MOSFET

devices can, in principle, be used to determine the fine- structure constant. Conversely, the effect can be

"inverted" and used to establish a known resistance

*Present address: Department of Physics, Saint John's University, Collegeville, MN 56321.

standard. The ultimate goal is to determine the fine- structure constant to an accuracy of a few parts in 10* and to develop an absolute, highly reproducible resis- tance standard based on fundamental constants of nature.

2. Experimental Procedure

Measurements have been made on Si MOSFET devices

using the 15 T superconducting magnet at the Naval

Research Laboratory's High Magnetic Field Facility. Our

measurement approach is indicated in Fig. 1. A constant current is applied to the MOSFET, cooled to 1.5 K, and

to a room temperature reference resistor. Reference

resistors have been constructed with values nominally

equal to that of the Hall resistance, i.e., 6,453.20 0 for

i = 4, and 12,906.40(1 for i - 2. Therefore Vh ~ Vr.

The potentiometer voltage is made almost equal to the

voltage drop across Vh or Vr, and an electronic detector

amplifies the difference voltage signal. The isolated

detector output is either displayed on an X-Y recorder or

is digitized for further processing by a computer. X-Y re- corder tracings of a Hall step are shown in Fig. 2. The

T = 1.5 K

STD. RESISTOR

6453. 20n

__l

-< «• 1 D I--

POTENTIOMETER

Figure 1. A simplified schematic of the measurement circuit used to compare the Hall Resistance of a MOSFET to a nominally-equal room temperature reference resistor via a po- tentiometer adjusted to vnthin a few ppm of Vjj . The entire system is battery-operated, including the electronic detector. D.

539
>

£

Q.

Q. O

>

<

J 1 1 L Vq ( 200 mV/div.)

Figure 2. High sensitivity X-Y recordings of an i = i Hall step for both current polarities. The dashed line shows the shift in gate voltage for the step upon reversing the current direction instead of interchanging the source-drain coymections. This is our only "perfect" step; all other steps have structure on one or

both sides. The arrow indicates the gate voltage position at

ivhich the voltage drop along the channel, V^, is a minimum.

The Hall voltages are displaced vertically by an arbitrary

amount for clarity.

current source, potentiometer, and electronic detector

are all battery-operated.

The potentiometer does not require calibration in this arrangement; it need only remain sufficiently stable during the measurement period. Thermally induced emfs and

linear drifts in the current source and potentiometer are

cancelled by reversing the current through the MOSFET

and reference resistor. A series of reversals in the order + h, H h is made for each of two measurements of

V[{ which bracket in time one measurement of Vfj. It re- quires 5 min to complete this sequence in order to obtain

a single data point, and typically between 10 and 20 data

points are obtained in a single run. The detected voltage signal has short-term peak-topeak variations of 2 ppm for a 10 |xA source-drain cur- rent, and is the same for both Vh and Vn. The standard

deviation for a data point is typically 2 parts in 10^ at 10

|xA. A low standard deviation of a single observation is required to study current-dependency effects, the struc- ture of the Hall steps, and the flatness of the steps.

3. Problems Encountered

3.1 Systematic Errors

The measurement system is checked for sources of sys- tematic error by replacing the MOSFET at the end of the

probe and the reference resistor with two calibrated

10 kft resistors that differ by about 10 ppm. The two

resistors are compared using the measurement sequence

discussed in the last section. The two resistors are then

interchanged and the sequence repeated. Systematic er- rors are immediately obtained by noting the deviation

from the mean value of the two measurement sets. For

example, rectification of external high frequency signals

by the constant current source (with resonant frequencies at 490 kHz, 4 MHz, and 6 MHz) caused a 2 ppm

systematic error. After introducing a radio-frequency in- terference filter at the current source output, the sys- tematic error dropped to 0.2 ppm. Measured leakage

resistances in the probe and the circuit account for this residual error. Obviously, the sources of error depend on

the particular measurement system; but it is vital that a comparable method for testing systematic errors be in- corporated into the measurement procedure.

3.2 Current Reversal in the IVIOSFET

If the polarity of Vh is changed by reversing the current direction then the negative polarity Hall step oc- curs at lower gate voltages. This effect is shown by the

dashed Hne of Fig. 2, in which the step is shifted by

0.13 V when Igo = 15.5 pi A and Vh = 100 mV. The

reason for the shift can be explained by studying the

equal-potential lines and electron flow patterns for the

two current directions as shown in Figs. 3 and 4 for IsD = ±5 |jlA. The inversion layer electron potential dis- tribution and the gate voltage are referenced to circuit ground at the source. The average electron potential is 4-16 mV for +5 |xA and -16 mV for -5 |jlA. Therefore, the gate voltage required to just fill the i = 4 quantum level shifts by 32 mV for Isd = ±5 (jlA, or by

100 mV for Isd - ±15.5 |jlA. The observed shift, however, was 130 mV for Isd — ±15.5 |jlA because the potential distribution also changes relative to the grounded

substrate, and this change depends upon the acceptor

atom doping concentration.

EQUAL- POTENTIAL LINES AND ELECTRON PATHS

Rh = 6,453.20

0 ®

32 &

®

32 ®

32 32

32 mV

32 mV

Figure 3. The resulting electron flow pattern and the potential distribution for Isd = +5 M-^ when the gate voltage is adjusted

so that the inversion layer electrons completely fill the four

N = 0 Layidau quantum states, but none of the N = 1 states. This shift is unacceptably large for high currents, but

it can be corrected by either changing the gate voltage

the appropriate amount on current reversal, or effecting current reversal by interchanging all the connections to the source and the drain. We are using the latter method

as demonstrated in Fig. 2. (In the example shown in Fig.

3, the 32 mV and 0 mV values are all interchanged on

reversal.)

Although this m.ethod has the advantage that the Hall

steps "line up" vdth respect to gate voltage when the current is reversed, it suffers from a lack of cancellation

of errors that result from leakage currents between the

540
2 ppm when Igo = 7.5 [xA) (V,^ ~ 50 mV). Because of

this error, we have been unable to obtain accurate

results with MOSFETs. We are correcting the problem

by replacing the source, drain and potential leads with

twisted pairs of teflon-coated wire, and the gate and substrate leads with coaxial cable.

4. Conclusions

We have developed a measurement system of sufficient stability and precision to enable us to study, in detail,

the structure and flatness of Hall steps. After correcting

the leakage resistance problems discussed in the last section, it is expected that measurements of (Rh)si and a

can then be made with a 1 a total uncertainty of less than

0.2 ppm.

The authors wish to acknowledge the encouragement of B. D. McCombe, NRL, B. N. Taylor, NBS, P. J. Stiles,

Brown University, the efforts of the staffs of the

Research Device Facility and the High Magnetic Field

Facility at NRL, and the financial support of the Office of Naval Research.

References

[1] K. V. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett.

45, 494 (1980).

I

1

EQUAL- POTENTIAL LINES AND ELECTRON PATHS

Rh = 6,453.20 X2

IsD = -5^A

® ®

T 0 T 0

-32 -32 ®-32

-32

-32 mV

-32 mV

-32

Figure 4. The electron flow pattern and potential distribution when the four N = 0 Landau quantum states are just filled and Igp = -5 \xA.

gate and the source or drain. In the present apparatus, a

3.3 X 10^^ n leakage resistance exists between the

formvar-insulated gate, source, and drain leads. This

results in a systematic error of 1 ppm for each 10 V applied to the gate when Isd = 15 (xA {Vh ~ 100 mV) and

541
11 I i
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Use of a Cryogenic Current Comparator to Determine the Quantized Hall

Resistance in a Silicon MOSFET

A. Hartland

Division of Electrical Science, National Physical Laboratory, Teddington, Middlesex, U.K. A measurement system based on a cryogenic current comparator bridge is described. The system

has the capability of being able to determine the ratio of a 6.453 kfl and 100 H resistors, at 295 K, with an uncertainty (Ict) of 2 parts in 10' when the larger resistor is energized with a current of 10 (jlA. The suggested application of this technique to the determination of the quantized Hall resistance in sili- con MOSFETs is described and sources of possible error are assessed. Key words: cryogenic current comparator; quantized Hall resistance; resistance ratio bridge; silicon MOSFET.

1. Introduction

It has been demonstrated [1] that the Hall resistance,

Rii, measured in silicon MOSFETs in high magnetic

fields, (>14tesla) and low temperatures (< 1.5 K) for particular values of the voltage applied to the gate elec- trode can have the quantized values h /ie'^, where i is an

integer, h is Planck's constant, and e is the charge on the

electron, e.g., when i = 4 then Ra = 6.45320 kO. Ru

has been measured [2] with a claimed uncertainty of

1.4 ppm using standard digital voltmeter (DVM) tech- niques. In order to assess whether i^H is invariant to geometric, material, or other physical variables associated with 2-dimensional electron systems in semiconductors, and, consequently would be suitable as a resistance

standard, it is essential for the measurement system to have the capability of relating i?H to the national standard of resistance (the ohm) with an uncertainty smaller

than 1 part in 10". This uncertainity is comparable to that

achievable [3] using a calculable capacitor in conjunction

with a combination of ac and-dc ratio bridges.

The basis of the system which has been developed is a cryogenic current comparator [4] used in a circuit to control the ratio of two dc currents, the smaller one of which

flows through the MOSFET and the larger through a standard resisitor maintained at a temperature of 295 K. The Hall potential is nulled against the potential differ- ence developed across the standard resistor using a bandwidth limited flux transformer coupled to a SQUID

(Superconducting Quantum Interference Device) as a current detector. The metrological problems are pre- sented in the next section, followed by a description of the measurement system and its application to the determination of a 6.453 kfi resisitor at 295 K substituted for the MOSFET, and finally, an estimate is given of the limiting uncertainties to be expected for a determination of Rn2. Metrological Considerations

In order to examine the conditions required for a high

precision measurement of i^H, a preliminary study has

been undertaken of the characteristics of silicon MOSFETs at a temperature of 1.45 K and in a magnetic field of 7.5 tesla. These devices which are nominally equivalent to some of those employed in the original determination of Ru [1] have conduction channel dimensions

400 |jLm long by 50 (jim wide, and mobilities, \x, ranging

from 0.8 - 2.0 T'^{m^V~h~^^ determined in the absence

of a magnetic field at the particular value of that gate

voltage, Vg, corresponding to the Fermi level lying

between the first and second Landau levels. The Hall

voltage, Vii, was measured with a Datron 1075 averaging

DVM and the source-drain current, /so, likewise determined by allovdng it to flow through a standard resistor

in series with the device. Despite the relatively high mobility of the electrons in these MOSFETs it was apparent

that for Bz = 7.5 T plateaus in the Ve versus Vg plots do

not exist and it is not possible to measure a unique value

of Ru- The value of i?H (i - 4), obtained when R-^x (—Vxx/IsD, where V^x is the voltage measured between

electrodes separated by 133 |xm along the conducting

channel) is a minimum, was within 5 parts in 10"* of h /4e , vdth a repeatability of 3 x 10"^ for /so = 1 p-A. This

lack of agreement between /2h and h/Ae^ is expected,

since the maximum value of the ratio {(Txy/ ctxx), where

(Txy and (Txx are the Hall and transverse conductivities

respectively, is 4.7 x 10^, which, based on the available

theroretical and experimental evidence is an estimated

factor of 100 less that that necessary to ensure Rn =

h/ie^ without systematic corrections.

There are two distinct relationships:

dxij — ie^/h (1) and

Rn = l''o".v(/ . (2) both of which only become exact in the limit (Jxy/^xx ~^ ^• The degree to which Eq. (1) is obeyed depends on the to- tality of quantization of the i- D electron system, with all mobile electrons confined to an integral number of Landau levels. For a given mobility, as is increased and T

reduced, the width of the energy gap [5, 6] between Landau levels increases, thermal excitation of electrons from

occupied to unoccupied levels is reduced, and Eq. (1) is expected to become more exact. One of the main objectives for the technique described in this paper is to ex- amine the conditions for the breakdowTi of Eq. (1). The classical theory of the Hall effect for electrons in homogeneous, isotropic media in low values of relates the X and y components of the electric fields, Ex.y and

543
current densities, J^.y to CTjj. and a^.^ as follows

(3)

(4)

However, when is large there is considerable electric

field distortion in the vicinity of the electrodes where the currents are introduced and it is necessary to solve the

total field distribution by the techniques of conformal

mapping. This has been done [7] and the results, as quoted by Kawaji [8], have been used to derive expressions which relate V^d, the voltage measured between

the source and drain electrodes, and Va, to /sd and /h,

the Hall current, by

^SD - •^A'A' ^SD + ZxY^H (5)

(6)

where the impedance parameters Zxx and Zxy are given

by

yZxX = CTxj. g i(Txy/<Jxx)

(7)

(8)

The quantities h and g are both functions of (ary/a^j-), and

L/W and P/L, where L and W are the length and width,

respectively, of the conducting channel, and P is the potential electrode contact length. If the device is allowed

to supply Hall current to an external resistance, Re , then

and

/h = -Vu/R,

Rh = (Vh/Ssd) = h/l/{R, + dxxg). (9) Consequently, the Hall resistance is a complex function

of the device geometry, (j^x, and R^.- However, in the

limit (cTxy/crxx) —> h 1 and g (^xy/^xx, and from

Eq. (9) R, l/(l//2e + CT.xy I

i.e., i?H = l/o'.r;/ when /h - 0. However, if Hall currents

flow then l/oxy can still be derived from measurements of (^h/^sd) versus R^.. This has been done for devices at 5z = 7.5 T and T = 1.45 K and it was found necessary to include an additional device dependent resistance, 2rp,

ranging in value from 1.7-3.0 kfl in series with the Hall

potential electrodes before agreement with Eq. (9) was

achieved. Clearly, Uxx 0 is an essential pre-requisite

for an accurate determination of h/ie'^ and from the avail- able experimental evidence this occurs for Si MOSFETs

when B, > 14 T and T < 1.7 K.

In order to determine Vu by a null technique the

equivalent circuit shown in Fig. 1 is appropriate. Direct currents /sd and Msd are supplied to the MOSFET and a standard resistor, R^, respectively, such that =

{h/ie'^VRs. Additional emf's Tpl^ due to a possible leak- age current, /b, flowing through r^, and, Vth, caused by

thermal emf's are included since tnese can cause errors

in the intercomparison of and Vr^. The current detector is a bandwidth limited dc flux transformer coupled to a SQUID. The noise sources of emf are also indicated and

it has been assumed that the principal noise generator,

<v,fi >^'^, is due to the conversion of shot noise, gen- erated by electron injection into the conducting channel,

into voltage noise by the Hall effect. The equivalent

noise resistance, r„, has been calculated so that the total circuit noise is equivalent to the Johnson noise generated

in r„ at a temperature of 295 K. When the bridge is balanced the minimum current to be observed for a 1 part in 10" resolution in Vh is A/. 10-" iIsD/(rxyR), (10)

where R - (l/cTxy + 2rp + Rs)- The current noise

<i n >/Vh7 flowing in the primary winding, inductance

L, for R » tl is <i,?>i^ = {2/R ){2kT r„/TT)^ (11)

for a bandwidth (rL/4L ). If G is the current transformation ratio of the flux transformer then for a given

bandvddth the optimum value of G is G <iS>''' = <il>1/2 (12)

where is the intrinsic SQUID equivalent current

noise /VHz at its input coil. Also the current signal /noise

ratio is A/min/<^i >

1/2

11;

10-^(1 SD /(Txy)i'n /8 kT r^r' (13) a 1 Hz bandwidth. Since for the model chosen Vn depends mainly on /sd, A/niin/<? n >^ is proportional to /sd- Unfortunately, /sd cannot be increased arbitrarily

since electron heating causes the effective temperature

1 I

Figure 1. Equivalent circuit for the null detection ofVfj.

544
to rise resulting in a decrease in the value of (a,.,y/a„)

with a consequent reduction in accuracy in the determination of h/ie^. In the currently available silicon devices

in magnetic fields 5z = 14 T serious electron heating oc- curs when /sD ^ 10 |JlA.

3. Cryogenic Measurement System

An outline circuit diagram of the measurement system

is shown in Fig. 2. The primary current, /sd, is gen- erated by an integrated circuit current source (LM334Z)

in a temperature controlled oven and has a short term

(ten minutes) stability of better than 5 parts in 10'' at 10 |xA. NI^n is generated by the operational amplifier

with MOSFET inputs (CA3140) driving a booster amplifier and operating as a voltage follower. The ratio of the

two currents is controlled by a cryogenic current comparator in conjunction with SQUID 2 (SHE, System 330)

sensing the unbalance signal in the comparator, which is then fed back to stabilize the ratio. The cryogenic

current comparator [9, 10, 11] was constructed by making

a "rope" consisting of 67 twisted strands of insulated,

copper clad, single core, niobium-titanium superconducting wire (Niomax S, IMI Ltd), having an overall diameter of 0.33 mm. The rope was wound onto an insulating

former and sixty-six windings of 109 turns and one winding of 58 turns each had their ends separately brought

out and labelled. The windings were then covered by

21/2 overlapping turns of lead shielding (0.5 mm thick)

with PTFE "plumbers" tape used as insulation between

the turns. A sense winding of 6 turns of Nb-Ti wire was wrapped around the outer-most layer of the shielding and

connected to the input coil of SQUID 2. The whole as- sembly was then surrounded with lead sheeting. The

quality of the comparator was determined by passing

10 mA through the windings joined successively in series

opposition in a binary build-up arrangement. Figure 3 shows the result of doing this. The most serious error occurred for the thirty-two winding set (position number

8) which indicated a maximum error of about 5 parts in

10^^ in the final ratio. The short winding of 58 turns could not be tested, but it is assumed to have coupling errors no worse than any of the other 66 major windings.

IsD=10^A

Sixty-four of the 109 turn windings were then joined in series with the 58 turn winding, producing a comparator

with an effective ratio N of 64.5321101. Since the inter- connections between windings had non-superconducting

joints it is important that there be no leakage between

windings greater than 100 k H if the effective ratio of the

comparator is not to be degraded below the intrinsic 5 parts in 10^°. At liquid nitrogen temperature interwinding resistance exceeded 10 Mil. The SQUID 2 output was fed back to the 100 O standard resistor and the N

winding set of the comparator via an isolation amplifier

which effectively decoupled the bridge circuit from the

earth of the SQUID system. SQUID 2, which has an in- trinsic equivalent input current noise of about 10 pA/VHz

was operated in the internal feedback mode and the out- put gain and bandwidth were set at x 100 and 100 Hz,

respectively. The stability of the control circuit, which

had a loop-gain of approximately lO*', is shown in Fig. 4. This demonstrates that for a current /sd = 10 |xA the ratio of the two currents drifts by about 2 parts in 10^ in a period of ten minutes. The drift is probably caused by

changes in the temperature of the liquid helium bath giving rise to changes in the SQUID' s zero. The, input bias

current, /b, of the CA3140 is about 2 pA for a 5 volt supply voltage and it is assumed that this changes by A/b =

0.2 pA as the input voltage is increased from 0 to 0.2 volt

(a sufficient voltage to supply NI^d to the standard resistor). Consequently, the change in the error voltage,

TpA/b, as the comparator is energized is expected to be

no worse than 0.5 nV or 1 part in 10^ of Vy{. A known and

variable fraction of NI^d can be diverted through a single

109 turn winding of the comparator to provide a calibra- tion signal.

The flux transformer coupled to SQUID 1 was con- structed in a similar manner to the current comparator

with approximately 7000 turns of Nb-Ti wire wound on a nylon former which was then almost completely sur- rounded (leaving a 1 mm gap) by lead shielding. Six turns of Nb-Ti wire wound on the shielding acted as a secondary winding which was connected to the input of SQUID 1. An 8 ft resistor (phosphor bronze wire) was

connected in parallel with the primary winding (induc- tance 0.25 H) resulting in an input bandwidth of 8 Hz.

Current

0 volts

Figure 2. Outline circuit diagram of the cryogenic measurement system connected to the MOSFET.

545
The current amplification factor, G, of the transformerSQUID input coil combination was measured to be 32.5,

which is more than adequate to satisfy Eq. (12). thus en- suring that the detection system was not detector-noise

limited.

Zero ' cur rent

2

3

5

7 (^-*vsA|

8 >rw I

Cal.

current

MO m A

Zero

current

Figure 3. Determination of errors in the cryogenic current comparator ratio. The numbers indicate smtch positions for

connecting the various iviyidings in series opposition in a

binary build-up manner, e.g., position 1 connects a single

winding IW^ in oppositon to IW.^; position k connects iWj +

IW^ in opposition to tivo windings, 2W, in series; and position 8 conyiects IW^ + IW., + 2W + + 8W + 16W in opposition

to 32W.

10 minutes

10"7 Current

ratio

uncertainty

ISD =

Figure 4. Output of SQUID 2 showing drift in the nmintained

current ratio, N.

Figure 5(a) shows an overall view of the cryogenic ap- paratus, and Fig. 5(b) is an enlarged view of the current comparator and the flux transformer. Great care was

taken to use shielded, insulated twisted pairs of wires

conveying currents and potential to and from the cryostat. It is particularly important to avoid stray leakage

paths for /so and N/sdVM 1

1 W««A>WWV»

1

1

1

Figure 5(a). Overall view of cryogenic apparatus. The Dural

block at the top is an electromagnetic and thermal shield for

the potential leads, and the niobium, can surrounding the comparator and flux tra nsformer is a superconducting shield.

4. Measurement Procedure

Initially, the measurement system has been used to relate the value of a 6.45 kCl resistor to a 100 Cl standard

resistor. These resistors together with two others of 10 n and 1 fi, all of a standard ac/dc construction [12], are maintained at a constant temperature of 295 K in an

air enclosure [13] with a long term temperature stability

better than 1 mK. Eventually, the intention is to determine i^H in terms of the 1 D. standard using the bridge

described here and, also, a similar bridge having a 10:1

ratio.

546
Figure 5(b). Enlarged view of the comparator (on the left), flux transformer, and SQUID sensors.

The measurement proceeds by closing the switch in the

potential circuit and monitoring the drift on a recorder

connected to the output of SQUID 1. /so is then in- creased from zero to about 10 (xA after ensuring the

WsD tracks with it by adjustment of the variable resis- tance R. If the resistance ratio is not equal to A'^ then a

deflection is observed. A calibration signal (1 part in 10^)

is injected into the potential circuit and the further deflection monitored. Finally both calibration signal and /so

are returned to zero. The sequence of events is shown in Fig. 6. The measurement is repeated four times and the

la standard error of the mean determined from the chart

recordings. The repeatability of the resistance ratio (la)

is ±2 X 10"^ for a current /so = 10 |jlA. The equivalent current noise observed with a post detection bandwidth

of 1 Hz is in excess of the anticipated circuit noise,

Eq. (11), probabily due to instabihty in the current ratio control circuit and /or fluctuating thermal emf's arising in the potential leads from the liquid helium bath to room

temperature. For the eventual measurement of /2h there

are two such connection paths, but, unless the thermal

emf's have a significant second-order time vaHation,

they are eliminated by the measurement procedure.

However, it will probably be essential that themal emf's

are kept less than 100 nV. The following is a provisional

Cal.

signal

1 in 10^

lOjuA in e-^kii

0-6A mA in lOOn

Zero current

1 minute

Figure 6. Recorder output of SQUID 1 showing a measure- ment cycle ivith the null detector circuit complete.

estimate of the major error contributions in the determination of jRh in terms of a 1 H standard.

Comparator ratio 5 X 10-10 Current ratio (during measurement) 1 X 10-8

Emf A/brp due to bias current 8 X 10-8

Thermal emf 1 X 10-8

Measurement uncertainty 2 X 10-^ Determination of 1 ohm: 100 ohms 2 X 10-8 RSS Total (la) 2.2 X 10-^

5. Conclusions

Preliminary investigations of the circuit parameters in a silicon MOSFET with attached Hall potential probes

have been undertaken to assess the problems of determining Riii- h/ie^) with an accuracy of the order 1 part

in lOMn terms of the national standard of resistance. A

current ratio bridge, operating at liquid helium temperatures, has been constructed and used to determine the

resistance ratio of two resistors of 6.453 kft and 100 D. situated in a 295 K temperature regulated enclosure ^^^th an uncertainty of 2.2 parts in 10'. A major limitation to- wards reducing this uncertainty is created by the restric- tion that the source-drain current must be less than

about 10 fjiA. It may be possible to alleviate this by us- ing devices with larger conducting channel areas or by

connecting a number of them in parallel. If this approach

is satisfactory, then the limiting uncertainty in the meas- urement will arise from the change in the emf, ?'p A /b, in547
duced in the potential circuit as /sd is increased from

zero. This could be eliminated, but with some additional

experimental complexity, by the complete isolation and

individual stabilization of the currents /sd and NI^dThanks are expressed to Dr. M. Pepper of the University of Cambridge for providing the MOSFET devices

and to Mr. P. Scrimshaw for technical assistance.

References

[1] K. von Klitzing, G. Dorda, and M. Pepper. Phys. Rev. Lett.

45, 494 (1980).

[2] E. Braun, E. Staben, and K. von Klitzing, PTB-Mitteilungen

90, 350 (1980).

[3] R. D. Cutkoskv, IEEE Trans. Instrum. Meas. IM-23, 305

(1974).

[4] I. K. Harvey, Rev. Sci. Instrum. 43, 1626 (1972).

[5] R. J. Nicholas, R. A. Stradling, and R. J. Tidey, Solid State Commun. 23, 341 (1977).

[6] Th. Englert and K. von Klitzing, Surf. Sci. 73, 70 (1978).

[7] R. F. Wick, J. Appl. Phys. 25, 741 (1954).

[8] S. Kawaji, Surf. Sci. 73, 46 (1978).

[9] D. B. Sullivan and R. F. Dziuba, IEEE Trans. Instrum. Meas. IM-23, 256 (1974).

[10] A. Hartland, in Euromeas 77 (lEE Conference Publication

152, 1977), p. 10. [11] F. Delahaye, IEEE Ti-ans. Instrum. Meas. IM-27, 426

(1978).

[12] F. J. Wilkins and M. J. Swan, Proc. lEE 117, 841 (1970).

[13] J. J. Hill, S. Harkness, and J. J. Denton, IEEE Trans. In- strum. Meas. IM-25, 501 (1976).

548
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

A Resistance Standard Using the Quantized Hall

Resistance of GaAs-AI^Ga^ -x'^^ Heterostructures

A. C. Gossard and D. C. Tsui

Bell Laboratories, Murray Hill, NJ 07974

We have observed quantization of the Hall resistance of the two-dimensional electron gas in GaAsAl.,.Gai_ ,As heterostructures. The quantized Hall resistance is given by p,„ = h/[2 e-{N + 1)] where

N is the quantum number of the Landau level immediately below Ef. Our results show that the quantized as given by this relation does not depend on temperature T, magnetic field B, and the elec- tron mobility of the sample, which in this experiment is varied from ^ = 1.8 m^/V • s to 6.9 m-/V s. In other words, B, T, and (x must suffice to reach the quantum regime, which is evidenced by the vanishing of the parallel resistance p_rj(<0.1 Q/n), but the value of quantized p-^^ is independent of them.

In our samples, this quantum regime is reached for S > 3 T at T = 1.2 K and for 5 > 4 T at T =

4.2 K. This modestly low magnetic field requirement makes this system a practical primarj' resistance standard. Key words: fine-structure constant; GaAs heterostructures; Hall effect; quantized Hall resistance; resistance standard.

I 1. Introduction

I It has been known for several years that at sufficiently

high magnetic field (B ~ 15 T) and low temperature

,| (r ~ 1.5 K), the gate voltage (Vg) dependence of the

j transport coefficients of the inversion layer in (100) SiI MOSFETS (metal-oxide-semiconductor field effect-transistors) shows anomalous behavior [1]. More specifically,

the magneto-resistance (p„) vanishes and the Hall resisjl tance (p,,.,,) goes through a plateau in a finite range of Vg,

I which corresponds to the situation that the Fermi level (Ef) lies between Landau levels. In a quantizing magnetic field, B, the energy spectrum

of a two-dimensional, 2D, electron gas is a series of discrete Landau levels, each having a degeneracy p =

yeB/h. (Here, h/e is the flux quantum and 7 stands for the spin- and valley-degeneracies.) Scattering removes

the orbital degeneracy and broadens each level into a band of states. The electrical transport properties of the

system are determined primarily by the position of the

Fermi level Ef in relation to these "Landau subbands''

[2]. When a "subband" is completely filled, the presence

of an energy gap between the filled and the empty Landau levels inhibits scattering and causes pj.,. to vanish at T - C. The off-diagonal Hall resistance then becomes

Pxy = Bine = hlie'^, where n is the electron density and

i is the number of filled Landau levels [3]. Recently, von Klitzing et dl. [4] made accurate meas- urements in the Si-inversion layer and showed that at the

plateau

h

P^ry = = a~^fXoC/2i (1) where i is the number of filled Landau levels, a the fine structure constant, |jlo the vacuum permeability, and c the velocity of light. The width of the Hall resistance pla- teau is apparently the result of localized states between

the Landau levels. They may be due to impurities or surface imperfections.

Von Klitzing et al. demonstrated the use of this quantization of the Hall resistance for a high-accuracy determination of a and noted the possible use of it as a resistance standard. However, to reach this quantization

regime, in the case of Si inversion layers, requires B >

15 T and r < 2 K. Such high B and low T requirements

will clearly limit its use as a standard resistor. However, 5 ~ 8.5 T at 4.2 K or -10 T at 2 K can be ob- tained cheaply with NbTi superconducting magnets.

Since this quantization is characteristic of 2D carriers, it should be possible to lower this requirement to 5 < 10 T

and T ~ 4.2 K, by using the 2D electrons in GaAsAlj. Gai-j- As heterostructures, whose cyclotron mass is approximately three times lighter (m* = 0.068 mo [5]) than that of electrons in the (100) Si inversion layer. We

recently prepared modulation-doped GaAs-Al^. Gai-^ As

heterojunctions in which the scattering rate of the 2D

electrons is comparable to that in Si-MOSFETs. We

measured p„ and p,-^ vs. B and observed quantization of the Hall resistance at J" = 4.2 K and at B as low as 4.2 T. These observations demonstrate the use of GaAsAljGai-jAs heterostructures as a practical primarj^

resistance standard [6].

2. Sample Preparation

The GaAs-Alj- Gai-.,. As heterostructures were prepared

by using molecular-beam-epitaxy techniques on Cr-doped

semi-insulating GaAs substrates of (100) orientation

which show no observable conductivity at 4.2 K. Each

sample consists of a 1 fxm thick undoped GaAs layer, a 0.07 |jLm thick n-type Si-doped Al^. Gai-jAs layer, and a top cap layer of 0.02 fa.m thick undoped GaAs. The inter- face between the top GaAs and the Alj-Gai-^As is de- pleted of free carriers by the surface potential, and the

2D electrons, which result from ionized donors in the

Alj-Gai-jAs layer, are confined to the interface between

the, Al.r Gai-.,; As and the GaAs underneath it. The elec- trons are confined by the potential well formed by the

conduction band discontinuity at the interface (lEc ~

300 meV) and the band bending in the GaAs [5]. Consequently, the electronic motion perpendicular to the inter- face is quantized and the electrons, still free to move

parallel to the interface, constitute a 21) system.

549
The specimens were made into standard "Hall

bridges' ', each having six side-arms symmetrically placed

to facilitate measurements of magnetoresistance and the

Hall effect [7]. The width of the bridges was either

0.1 mm or 0.38 mm and the separation between two

neighboring side-arms was 0.3 mm for the narrow ones and 1.0 mm for the wide ones. Electrical contacts were made by alloying indium into the epilayers at 400 °C in hydrogen atmosphere. The quality of the contacts and

the homogeneity of the specimens were checked by comparing the I-V characteristics measured across various

combinations of the voltage and current contacts.

3. Measurements and Results

Figure 1 shows the magnetic field dependence of p.„- and pxy for a sample with a real electron density of 4.2 x lO^/cm^ and low-field mobility of 7.9 x 10^ cnr/Y s at 4.2 K. The data were obtained by measuring the voltage

between appropriate side-arms with a high input impedance voltmeter and a current source of 10 |xA. The

0 2 4 6 8 10 B (T )

Figure 1. p^^ and p^^ vs. B taken from GaAs-Al^Ga^-j-As

heterostructure at U.2 K. B is perpe^idicular to the surface.

magnetic field, which was generated by a superconducting solenoid at 4.2 K and calibrated to better than 3%,

was applied perpendicular to the plane of the GaAsAlj. Gai_.,. As interface. The Shubnikov-deHaas oscilla- tions at B < 4 T are periodic in 1/5 and the electron

density was deduced from the period [7]. The 2D nature

of the electrons is confirmed by the fact that this period

depends only on the perpendicular component of B and no

oscillatory effect was observed with B applied parallel to the surface. The structure at B ~ 6 T depends strongly on T and it develops into two well resolved peaks at T =

1.2 K. We attribute it to spin-splitting of the = 1 Landau level. It should be noted that the effective g- factor of electrons in GaAs is g* = 0.522 [8] and the

splitting at B - 6 T is AE" =0.18 meV. In view of

kT ~ 0.36 meV at 4.2 K and /i/t = 0.25 meV, its observation is not expected and exchange enhancement of the

splitting of spin-up and spin-down states is appreciable

[9, 10]. The most striking feature in Fig, 1 is the vanishing of

px.r and the constant value of p^y (at —12 kft and ~6 kfi)

in a finite range of magnetic field (A 5) about B = 8.4 T

and 4.2 T. At these values of B, Ef lies between the

N ^ 0 and = 1 and between the N = 1 and N - 2 Landau levels respectively. As T decreases, AS of these

two regions increases and at T —1.2 K a new region be- comes apparent at B ~ 2.7 T, corresponding to having

Ef between the N = 2 and N = S Landau levels. These

data resemble those observed in p„. and p,,.y vs. Vg in Si inversion layers and manifest the quantization of Hall

resistance of 2D electrical carriers. In this case, the carrier density is fixed and filling of the Landau levels is accomplished by varying B, which varies the orbital de- generacy of the Landau level. Because of a smaller cyclo- tron mass in GaAs, the quantization is observable at much lower B and higher T in a sample having an elec- tron scattering time (t ~ 2.7 x 10 s) comparable to that in high mobility Si inversion layers.

Initial measurements were made by connecting a cali- brated resistor in series vdth the sample and comparing

the Hall voltage vdth the voltage developed across it us- ing a null detection scheme. The calibrated resistor was a decade resistor made by General Radio Co. (model no. 1433F), which in the range of resistance of interest here, was accurate to 0.01%. Within this accuracy, the Hall

voltage was strictly a linear function of current to / =

10 |jlA, the range checked in this experiment. The Hall

resistance at 4.2 T was (6453.4 ± 0.7) ft and at the

8.4 T plateau was (12907.0 ± 1.3) O. These values of p,ry in the observed plateau regions

agreed within the accuracy of the decade resistor with

that given by Eq. (1) [but with i = 2(A'' + 1)], using ac- cepted values for a, and c. Measurements on three

additional samples showed that, in addition to being in- dependent of B and T, the quantized p^-y is independent

of mobility, which varied from jjl = 1.8 m^/V • s to 6.9 m^/V • s for the samples used. In other words, B, T, and |x must suffice to reach the quantum regime, which is evidenced by the vanishing of p„. (<0.1 fi), but the value

of the quantized p^y is independent of B, T, or [x of the

sample. Clearly, pathological problems with electrical contacts to the 2D electron layer and inhomogeneities

which destroy the 2D character of the sample would

affect py:y. Our experience indicates that such problems

also manifest as differences in voltage drop across equivalent pairs of side-arms, lack of linearity and symmetry

in I-V, and irregular structure in p.^.^ vs. B. None of these were detected in these samples.

More recently, a preliminary measurement of p.,,, to a much higher precision has been made on one of our modulation-doped GaAs-Alx Gai-j. As heterostructures in col- laboration with M. E. Cage, R. F. Dzuiba, and B. F. Field [11]. This more recent measurement yielded a value

of p,,y = 6453.2057(10)fiNBS at the i = 4 plateau which

would yield a value of the fine structure constant a"^ =

137.035965(12). The precision obtained in these measure- ments seems encouraging for further development of these structures for resistance standards and for meas- urement of a.

4. Conclusion

In summary, we have observed the quantization of the

Hall resistance of GaAs-Al.v Gai-.^ As heterostructures at

550
4.2 K and at B as low as 4.2 T. The B and T requirement

is sufficiently simple to warrant its practical use as a primary resistance standard. Multi-layer or superlattice structures which have m identical 2D electron layers

electrically connected in parallel might also be used for resistance standards. The quantized Hall resistance

would then be given by p^j, = h/eHm, and the value of the resistance could be tailored by varying the number of

layers. It should also be noted that the quantum regime

might be reached with even lower B by using narrow gap

semiconductors such as InSb, InAs, PbTe, or

Hg,jCdi-_^. Te if junctions of suitable quality could be

formed in such materials.

We thank S. J. Allen, A. Y. Cho, P. M. Platzman, J. M. Rowell, and H. L. Stbrmer for discussions, and G. Kaminsky and W. Wiegmann for their most valuable

technical help.

References

[1] S. Kawaji and J. Wakabayashi, Surf. Sci. 58, 2,38 (1976).

[2] T. Ando and Y. Uemura, J. Phys. Soc. Jpn. 36, 959 (1974).

[3] T. Ando, Y. Matsumoto, and Y. Uemura, J. Phys. Soc. .Jpn. 39, 279 (1975).

[4] K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett.

45, 494 (1980).

[5] H. L. Stbrmer, R. Dingle, A. C. Gossard, W. Wiegmann,

and M. D. Sturge, Solid State Commun. 29, 705 (1979).

[6] D. C. Tsui and A. C. Gossard, Appl. Phys. Lett. 38, 550

(1981).

[7] D. C. Tsui and R. A. Logan, Appl. Phys. Lett. 35, 99 (1979).

[8] W. Duncan and E. E. Schneider, Phys. Lett. 7, 23 (1963).

[9] T. Ando and Y. Uemura, J. Phys. Soc. Jpn. 37, 1044 (1974).

[10] D. C. Tsui, H. L. Stbrmer, A. C. Gossard, and W. Wieg- mann, Phys. Rev. B 21, 1589 (1980).

[11] D. C. Tsui, A. C. Gossard, B. F. Field, M. E. Cage, and R.

F. Dziuba, Phys. Rev. Lett. 48, 3 (1982).

551

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

A Quantitative Theory for the Determination of hle^ from the Hall Effect in

Two-Dimensional Conductors

L. Bliek and G. Hein

Physikalisch-Technische Bundesanstalt, D 3300 Braunschweig, F.R.G.

In experiments on two-dimensional conductors, steps proportional to e'^lh have been observed in the quotient of the electric current and the Hall voltage. They can be explained on the basis of the

vi^ell-established theory of the Shubnikov-De Haas effect, provided the dependence of the scattering

lifetime on the density of states is consistently taken into account. Results of numerical calculations of the Hall voltage and of the electrical resistance agree closely with published experimental data. Key words: fine-structure constant; hle'^ determination; quantitative theory of resistance and Hall ef- fect; quantized Hall resistance; two-dimensional conductors.

1. Introduction

Both in the Hall voltage vs. gate voltage dependence of

Si MOS-transistors [1,2] and in the Hall voltage as a

function of magnetic induction [3] for GaAs-Al^ Gai-j. As

heterostructures, steps have been observed experimentally in high magnetic fields and at \ow temperatures. At

the step positions, Rh, i.e., the quotient of the Hall volt- age and the current, is given, within the experimental

accuracy, by

Rh - h/{e^j) (1) vdth j = 2, 3, 4, 5 Whenever Rh assumes such a constant value, the samples have no measurable electrical resistance.

If the steps are to be used for a more precise determination of h/e^ than currently available, theory has to prove that Eq. (1) is sufficiently accurate or even exactly

correct, and not just true within the present experimental accuracy. Previous calculations, assuming a uniform,

energy-independent scattering time had predicted steps

in the number of electrons—plotted against the Fermi

energy—corresponding to Eq. (1) with small, but signifi- cant, deviations [4]. Somewhat later Ando et al. [5], making different assumptions about the electron scattering

process, arrived at the conclusion that, to the accuracy of the approximations they made, the Hall conductivity

should assume values corresponding exactly to Eq. (1). In the follovdng, it is shown that the well-established

theory of the Shubnikov-De Haas effect, describing the

quantum effects in the high-field magnetoresistance,

predicts the exact validity of Eq. (1) for T = 0, provided

the dependence of the scattering lifetime on the density

of states is consistently accounted for. With the help of

this theory, the influence of a finite temperature can be

calculated. Moreover, it gives a good quantitative

description of the experimental resistance curves.

2. Fundamental Equations for the

Resistance and for the Hall Voltage

Resistance theory provides relations between electric

fields and current densities in a conductor. If a magnetic

induction B is present, directed in the z-direction, per- pendicular to a two-dimensional conductor, there will be

two electric fields corresponding to a current density i in

the direction of x: E^^ and the Hall field The resis- tivity tensor is then given by

. (2)

(3)

(4)

(5)

(6)

9XX ?l)X Pxy Pyy

9xx - E,/i

9xy = Ey/i

and

For the conductivity ct, on the contrary,

i/Er (Jr and

i/Ey (J:cy Hence, theoretical relationships between E and ; are resistivity formulae rather than conductivity formulae, a fact which is sometimes overlooked.

Keeping this in mind, we can obtain the follovdng two

relations from the literature [6]:

k I X Dk{E)f{E)D,iE)[l - fiE)] dE (7)

p^y = B/ie no). (8)

Here f {E ) is the Fermi function, A^j is the transition

probability from the quantum state k to the quantum

state I, where k and I stand for the complete set of quantum numbers designating a magnetic energy level, DiiE)

is the density of states associated wdth the level /, and no

is the electron density. The density of states is concentrated near the quantum energies En,s.v -

{n + 1/2) -heB/m + s As + v A,,, where n = 0, 1,2 ,3 . . . , s - -1-1,-1, V = +1,-1, mis the cyclotron effective mass. As is the spin splitting, and A^, is an additional

splitting, observed for Si, but absent in GaAsAl.rGai-xAs. Equation (7) holds for elastic scattering. If the scattering is treated in the Born approximation, one

finds A„,s,„, n\s\v' = n + 1/2. There are correction terms

to Eq. (8),' but they vanish in the regions of interest be553
cause there the resistivity vanishes. The missing factor

in Eq. (7) is found from the condition that p^x should

reach its zero magnetic field value as 5 —» 0.

3. Hall Voltage and Resistance in the

Absence of Scattering-Induced Broadening

of the Quantum States

Without any broadening of the quantum states, one has

Dn.s.v (E ) ~ h(E - E„^s,v) (9) For B ^ 0, the total density of states has to reach its zero magnetic field value Dq{E) - STrm/h^ which leads

to

and hence

D{E) = ieB/h) ^ h{E - En,s,v) (10)

no = (eB/h) X HEn,s,v). dD

Inserting Eq. (11) into Eq. (8), in the limit T 0, leads

to Eq. (1), j being equal to the number of states n,s,v, with En^s.i' below the Fermi energy ^, i.e., RniO shows

steps as given by Eq. (1). Experimentally, one does not measure RuiO, but

rather RniB) or RuiUG), where Uq is the gate voltage

applied to the MOS transistor and is proportional to the

total number of induced electrons. If this number were

equal to no, Re.(Ug) would be a simple hyperbola just as RuiB) should be a straight hne, given by Eq. (8). The

experimental observation of steps in both curves shows

that, apart from the no mobile electrons considered up to now, there are additional electrons in bound states, giving no contribution to /2h- Therefore, Eq. (11) has to be

replaced by

n = no + nb = (e5//i) ^ f(En,s,v)

n,s,i' + J^^ D^iE)fiE)dE , (12)

where n is the total electron density, n^ is the density of

electrons in bound states, and D^{E) is the density of bound states. This equation provides for steps in no{yi) at constant B, as observed experimentally with Si MOStransistors, and in no(S) at constant w, as observed in the GaAs experiments.

It should be noted that in all the experiments ^ remains

almost constant, apart from small oscillations around ^o, the zero magnetic field value. This is obvious for the

GaAs experiments, but it is also true for the Si MOS

work because, on changing the gate voltage, the number

of electrons is changed, not by changing ^, but by changing the position of the conduction band edge relative to I,. Consequently, only the density of state^ D^iE) near

has an influence on the width of the steps and the

number of states actually involved is small. Moreover, it should be noted that on passing several steps, the same

bound states are occupied by electrons and emptied again

several times.

For non-zero temperatures, the steps in no are not completely flat. If i

is at a distance 8 from the central

position between two adjacent energy levels, which are A

apart (the influence, even of next-nearest neighbors, is completely negligible), no deviates from the ideal value

h/{je^), by an amount [2h/{je^)]exp{-A./2kT) sink

(b/kT). For the Si MOS experiments, e.g., a distance of

0.1 times the observed width of the j = 4 step

corresponds to a relative change of 10"^.

4. Influence of an Energy-Independent

Scattering time

While Eq. (12) closely fits the observed Hall effect data,

the underlying Eq. (10), for the density of states, inserted

into Eq. (7), does not fit the experimental resistance data

at all. Rather than almost 8-shaped peaks, one observes

broad maxima. According to the uncertainty principle, a 8-shaped density of states corresponds to an infinite life- time. Therefore, since the electrons are scattered on the

average after a scattering time t, broadening is not unexpected. In fact, the energy levels should show Lorentzian

broadening [7], and Eq. (10) has to be replaced by

DiE) = ^X Mtt t) h „% {E - En,s,vf + ^^/T^ + D^,{E). (13)

The broadening parameter t is normally assumed not to depend on energy [6]. Figures 1 and 2 show examples of the results one then obtains [9]. Compared with the ex- perimental curves, the calculated peaks in the density of

states, and the corresponding ones in the resistance, are too narrow, and the values at the minima are far too

high. The steps in the electron density, and the

corresponding ones in the Hall voltage, appear at the

correct positions, but they are not nearly as flat as the

experimental ones. The explanation for the deviations of the calculated

results from the experimental ones, lies in the long tails

Figure 1. Density of states at the Fermi energy in multiples of Dq{Q, as a function of the electron density in multiples of 2eB Ih, for a spin-splitting of 0.2 ^lu), = 0, and several dif- ferent constant values for the broadening parameter A = 1/u)t.

554
a = 0.03

B = 0 OS

n = o.ic

a - 0.50 016 . zzo:i 2a.cs.i-t

Figure 2. Electron density in multiples of 2eB /h, as a function

of the Fermi energy in multiples of flai. Parameters as in Fig.l.

of Lorentzian functions for constant t. Such a constant

value of T is not consistent with Eq. (7) which implies

that t"^ is proportional to the number of states an elec- tron can be scattered into. Therefore, the assumption of

a constant broadening parameter is only justified if D (iQ

is almost constant. This is true for metals, where it has

successfully been used to explain details of the De HaasVan Alphen effect, but it is not so for the present case and for others where discrepancies have previously been

explained using ~ D{Q [8].

5. Hall Effect and Resistance with

Consistent Assumptions for the Energy

Dependence of the Scattering Lifetime

A consistent calculation of the resistance and the Hall

voltage has to start from the following equation for the

dependence of %h on D{E ) which underlies Eq. (7): h _ 2[D{E) - D^jEm - i{E)] t

DoiE) - D^iE) TO ' ^

^

where DoiE ) and tq are the zero magnetic field values.

Inserting Eq. (14) into Eq. (13), one finds that

D{E) - D\jiE ) — 0 is a possible solution. Neglecting

contributions of other states, one finds for the density of

states belonging to the energy level En,s,v at i : Dn,sAO ^

for (^-^;„.,„)2:

2-n-To ^ I

{i-En,s,vl_ for a- En,s,.)^ ^ „ V 2 77 V ^2^(2 77To) ^TTTo

(15)

which is essentially the result of Ando and Uemura [10]. Neglecting the contributions of other states is, however, not justified under the conditions prevailing in the

experiments. There the states clearly overlap, and for such a situation Eq. (15) is not applicable. Instead,

D {E ) has to be calculated numerically, starting from

Eq. (13) and Eq. (14), and then has to be inserted into

Eq. (7) and, after multiplication with f {E) and integration to give no, into Eq. (8). Results, obtained by such

numerical calculations are shown in Figs. 3 and 4. The

calculated curves agree closely with the experimental

ones. Only the amplitudes of the various resistance maxima relative to one another show considerable deviations

from experiment. Here, the results might have been improved by allowing for different transition probabilities

for different types of scattering processes. The average

amplitude of the calculated maxima differs by only 2.2%

from the experimental value.

15-

10-

5--

30- mV

25--

20-2,0

•1.0

1 Potential 2 Hall Voltage

BLH 26. 3.81 A1=.05B A2 = .015 S1-.224 S2 = .100 ZB = 1.50 T =.010

Figure 3. Potential drop in the direction of current floiv, Up,

and Hall voltage, Ujj , for a Si MOS transistor mth a ratio of

active letigth to width of 2.6, for B = 18 T and a current of

1 \lA, as functions of the gate voltage, Uq. The Hall voltage was calculated for T = 1.5 K. Because of the small influence of temperature near 1.5 K, the resistance curve ivas calculated for T = 0. With this exception, the parameters have the experimental values [1].

555
16i

0

I ' 0 5 10 15 20 25 V 30 ^

1 Fermi Energy 2 Hall Voltage

BLH 26. 3.81 Al=.055 A2 = .015 Sl=.224 S2 = .100 ZB = 1.50 T =.010

Figure 4. Hall voltage, Ufj, and separation of the Fermi en- ergy L, from the co7iduction band edge, in units of % w, as functions of the gate voltage Uq , for a Si MOS transistor. Parameters as in Fig. 3.

By fitting the calculated curves to the experimental Si MOS data, the following results were obtained for the

parameters:

(1) The two level splittings were found to be equal to 0.224 ^ to and O.lO^w.

(2) (X = e To/m was found to depend on n in the manner shown in Fig. 5.

(3) Z)b(0 was obtained as 7.16 x lO^^ /eV cm^ which means that, on changing Uq, at most 2% of the

electrons go into or come out of bound states.

(4) rfox/^r, i.e., the thickness of the oxide layer, divided

by its relative permittivity, was found to be 26.0

nm, or, with er = 3.78 [11], dox = 98 nm.

(5) UG depends linearly on n, as it should, but there is an offset, such that [/g is not zero for no = 0. The

corresponding number of bound electrons is four

10 15 20 V ' 25

-i

1—I

^

1

1

I ^

3 4 5/1/ m--" 6

Figure 5. Mobility, p., as a function of the gate voltage, Vq,

and the density of conduction electrons, N, for a Si MOS

transistor. Parameters as for Figs. 3 audi.

times as large as the number corresponding to the

step width.

The authors have the pleasure to acknowledge helpful

discussions with Dr. K. von Klitzing and Dr. H. Kbhler,

University of Wurzburg, Wiirzburg, Germany.

References

[1] K. V. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett.

45, 494 (1980).

[2] E. Braun, E. Staben, and K. v. Klitzing, PTB-Mitteilungen

90, 350 (1981).

[3] D. C. Tsui and A. C. Gossard, Appl. Phys. Lett. 38, 550

(1981).

[4] L. Bliek, results presented at the 12th International Confer- ence on the Physics of Semiconductors, Stuttgart, 1974.

[5] T. Ando, Y. Matsumoto, and Y. Uemura, J. Phys. See. Jpn. 39, 279 (1975).

[6] See for example L. M. Roth and P. Argyres, in Semiconductors and Semimetals, Vol. 1, Ed. by R. K. Willardson and

A. C. Beer (Academic Press, New York, 1966), p. 159.

[7] R. B. Dingle, Proc. R. Soc. London, Ser. A: 211, 517 (1952).

[8] L. M. Bliek, G. Landwehr, and M. v. Ortenberg, in Proceeditigs of the 9th International Conference on the Physics of Semiconductors, Moscow, Ed. by S. M. Rybkin, Acad. Sci. USSR (Nauka, Leningrad, 1968), p. 710.

[9] L. M. Bliek, in Proceedings of the 12th International Confer- ence on the Physics of Semiconductors, Stuttgart, Ed. by

M. H. Pilkuhn (Teubner, Stuttgart), p. 729.

[10] T. Ando and Y. Uemura, J. Phys. Soc. Jpn. 36, 959 (1974).

[11] Several different values for are given in the literature. The present one was measured at low temperatures by P. Thoma, PTB (private communication).

556
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Quantum Hall Effect: Role of Inversion Layer

Geometry and Random Impurity Potential

R. W. Rendeir

Electron Physics Group, National Bureau of Standards, Washington, DC 20234

and

S. M. Girvin

Surface Science Division, National Bureau of Standards, Washington, DC 20234

The recently discovered quantum Hall effect offers the potential for a new precision determination

of the fine-structure constant and establishment of a quantum standard of resistance. We present here

a discussion of possible sources of error associated with finite sample size. In addition, we present a preliminary report on an investigation of the density of states between Landau levels in the presence of a model impurity potential which is based on a non-perturbative variational calculation using recently

developed field theoretic techniques.

Key words: fine-structure constant; impurity scattering; measurement errors; quantum Hall effect; re- sistance standard.

1. Introduction

The recently discovered quantum Hall effect [1] has at- tracted a great deal of interest because of the possibility

of obtaining an improved value of the fine-structure con- stant and development of a quantum standard of resist- ance. It is well know^n that the Hall coefficient of a two

dimensional electron gas (e.g., a MOSFET or heterojunction inversion layer) oscillates with magnetic field be- cause of quantization of the electronic states into Landau

levels. At low temperatures and high magnetic fields, the

carrier scattering rate between Landau levels becomes

extremely low. When the P'ermi level is adjusted to fill i levels, current flow becomes nearly lossless and the Hall

resistance goes through a plateau assuming the universal

value

R = h/eH , (1) apparently to extreme accuracy. Two types of questions are of great current interest. The first is: What are the possible sources of experimental error in a high precision measurement of h/e~ using a quantum Hall effect device? We present here a discussion of errors introduced by end effects in MOSFET devices of finite length. Within a simple model of the inversion layer we formulate expressions for the current and

voltage distributions and for the (very small) deviation of the Hall resistance from its ideal value. Limitations of the model and other possible sources of error will also be

briefly discussed.

The second question of interest is: What are the fundamental physical hmitations of Eq. (1)? One of the most important topics is the effect of disorder and the conse- quent localization of some of the carriers [2^]. As a

small step toward developing a quantitative understanding of the role of impurity scattering, we have investi-

*NRC/NBS Postdoctoral Fellow

gated the density of states between Landau levels in the

presence of a random Gaussian white noise potential. We

present here preliminary results of a non-perturbative

variational calculation of the density of states using re- cently developed field theoretic techniques.

2. End Effect Errors

We have analyzed [5] a standard model [6, 7] in which

the inversion layer system is assumed to consist of a sharply defined (ideally two dimensional) rectangle

within which the Hall coefficient is constant. The source and drain electrodes are attached to the top and bottom

of the rectangle and the Hall probes are taken to be point contacts at arbitrary locations on the sides of the device.

It is assumed that no current is drawn from the Hall

probes.

In order to calculate the current and voltage distributions within the sample it is necessary to determine what

the appropriate boundary conditions are. Because the source and drain are heavily doped metallic regions, they

have a very small Hall coefficient and will short out any

electric field parallel to the boundary at the top and bot- tom of the device. The assumption that the Hall field is completely shorted out at the ends slightly overestimates

the error, since the Hall- coefficient in the electrodes,

though small, is non-zero and has the same sign as in the

inversion layer. The boundary condition on the sides is taken to be no current flow out the sides. We may take

advantage of the assumption of a uniform value of the

Hall coefficient to note that the current makes a fixed angle 8 with respect to the electric field. Thus since the current at the edges is necessarily parallel to the edges,

the electric field must ihake an angle 8 with respect to the sides. Assuming steady state conditions one now has a well defined electrostatics problem which can be solved employing the techniques of conformal mapping [5-7]. Our

method of using a direct conformal mapping is consider557
ably simpler than previous approaches and the results are more readily evaluated in the limit 8 tt/2 appropriate to the quantum Hall regime. Since the Hall angle is so close to -it/2 it is convenient to define the quantity e via

8 - 1^ (1 - €) . (2)

Because of the sudden change in the boundary conditions

at the corners the current is forced to enter the device

almost entirely from one corner and to exit from the

opposite corner [8]. In fact within this simple model having sharply defined corners, the electric field and current

density both have power law singularities of the form

fiy) ~ {T/yf-^ , (3) where y is the distance above the corner and T is the

length of the device. The equipotential lines for this situ- ation are shown qualitatively in Fig. 1.

EQUAL- POTENTIALLINES AND ELECTRON PATHS

Rh= 6,453.20 11

IsD = +5/iA

Figure 1. Qualitative picture of equipotential lines in the inversion layer. The numbers on the sides indicate the approximate potential distribution when the source drain current is 5 pA and the Fermi level is at the i = 4 Hall step.

By integrating the calculated electric field [5] on a path

connecting the two Hall probes we have determined the

ratio of the actual Hall voltage to the ideal Hall voltage

for an infinitely long device. These results are displayed

graphically in Fig. 2. In order to understand this figure

it is useful to refer to Fig. 3 which shows the geometry

of the devices used in the Naval Research Laboratory

(NRL)—National Bureau of Standards (NBS) collaboration. Vxx refers to the voltage measured between two

probes on the same side and is a measure of the carrier

scattering rate and the resistivity pxx- refers to the

Hall voltage measured across the device. In the NRL

geometry the probes are located 1/5 of the way along the

edge from the ends. Returning now to Fig. 2 one sees a dashed line from which the quantity e may be determined

after Vxx/^Haii has been measured. Given e one may then

determine the error from the quantity

AT ~ ^Hall) rA\ A = logio ^7 , (4)

Figure 2. Graph of error in measured Hall voltage due to the shorting effect. See text for a detailed discussion.

Figure 3. Geometry of NRL MOSFET shovnng the location of

the four potential probes.

where Voc is the ideal Hall voltage for an infinitely long

device. The sohd Hne labelled x = 1/5 in Fig. 2 gives this error for Hall probes located as in the NRL geometry.

The line labelled x - 1/2 is for the case of Hall probes

located at the midpoints of the sides. These curves were

calculated using the aspect ratio of 6.5 appropriate to the

NRL geometry. The smallest attainable value of e is not known but preliminary measurements [1, 9] give an upper limit of 5 x 10^'' and it is likely that e is much

smaller than this. From Fig. 2 one sees that the error due to the shorting effect is less than 3 x 10"^" for Hall

probes at the center of a device with aspect ratio 6.5. Within the present simple model it appears that the

shorting effect can be neglected as an important source

of error. It is clear however that this simple model has

several limitations. It will certainly fail in the singular

field regions in the corners but this should not have a serious effect in sufficiently long devices. Also we have

implicitly assumed that the electrons scatter in the inver558
sion region so that it is meaningful to speak of a fixed

Hall angle. It is quite possible that the scattering rate is so low that this is not valid. Boundary scattering effects

in this problem are generally poorly understood at the

moment.

3. Random Impurity Potential

In the presence of a random impurity potential the

sharp Landau levels will be broadened into a distribution

of energies. This broadening has been subjected to per- turbative analysis [10-14] but even self-consistent per- turbation theory is ill-suited to situations involving local- ized states [15]. We present here preliminary results of a non-perturbative variational calculation of the density of

states in a two dimensional white noise potential in the

presence of a magnetic field. No assumption of a smoothly varying potential is made as in Ref. [15]. Haughton and Schafer (HS) [17] have developed a vari- ational scheme for calculating the density of localized states at asymptotically deep energies. We have modified

their method to take into account the magnetic field and

use this to find the density of states in the tails of the

Landau levels. The probability distribution for the potential is taken to be

P{V) = exp (-S(y)) , where the action S is given by

1 2a^

(5)

(6)

and cr is a measure of the disorder. If one is interested in the density of states at some energy SI then one would

like to choose V to minimize S subject to the constraint

that the Schroedinger equation with potential V have a

solution vdth eigenvalue O. HS show that the wave function associated vdth this extremal potential satisfies a non-linear Schroedinger (instanton) equation. In the

presence of a magnetic field B with vector potential

X r

the HS result is readily modified to yield

(7)

(8)

This is the equation for a two dimensional harmonic oscil- lator with a self-interaction. We work here with energy

units (^a)c/2) where Wc is the cyclotron frequency and use

a length scale P"^ = (2^/ma)(.)^. The minimum action

solutions of this non-linear equation are believed to be

rotationally symmetric and so we have dropped an angular momentum term from (8). Follovidng HS we obtain for the extremal action.

where K = —

S = KSo

and

So = fdRR \^(R)1'^

0

-I

(9)

(10)

Evaluation of the fluctuations about the extremal solution is a non-trivial exercise which is rendered more diffi- cult by the presence of the magnetic field. Hence, as a

first crude approximation we neglect these fluctuations

and using Eqs. (5) and (9) approximate the density of

states by

The procedure used for evaluation of this equation is as

follows. The ± sign in Eq. (8) was chosen according to whether a self-attractive or repulsive interaction was

required to pull a state out of the nearest Landau level and Eq. (8) was then solved numerically. The results are shown in Fig. 4 which displays a graph of So versus ft. The curve centered on ft = -hi is for zero node solutions

Figure 4. Graph of the extremal action appearing in the variational theory of the density of states. The dashed line is a per- turbation theory result. The solid lines are based on numerical

solution of Eq. (8).

to Eq. (8) whose primary parentage is the lowest Landau

level ( at ft = 1). The curve centered on ft = 3 is for one node solutions to Eq. (8) whose primary parentage is the

second Landau level (at ft - 3). The dashed line is the

action corresponding to a lowest order perturbation

theory solution of Eq. (8):

So = (ft - I? (12)

p(ft) = exp (-/fSo) (11)

We see that the exact solution matches this closely for ft close to one but deviates further away. In the limit

ft 00, the exact S„ becomes linear in ft and independent of the magnetic field. This occurs because the size of the localized state becomes much smaller than the Landau radius.

The approximate density of states given by Eq. (11)

with = 4 for the lowest Landau level is shovra in Fig.

5. The dashed line is a simple Gaussian obtained from

Eq. (12) while the solid line uses the numerical result for the action. One sees that the numerical result is skewed

toward lower energies relative to the Gaussian. This is a

result of the Landau levels repelling each other under

the influence of the perturbing potential. The numerical

result crosses over from a Gaussian near the peak to a slower exponential falloff in the low energy tail. This is consistent vdth the known result at low energies in the

absence of a magnetic field [16]. We now need to consider the range of energies in which this variational approach is valid. We have seen

that the variational approximation agrees with the per- turbation result close to the peak. However, both the

perturbation theory and the variational calculation assume rotational symmetry which becomes a poor

559
Figure 5. Approximate density of states for the lowest Landau

level calculated in the variational theory. The dashed line is a Gaussian approximation based on lowest order perturbation theory. The solid line is the numerical result. Note the asymmetry.

approximation close to the peak due to the large number

of degenerate levels. In the absence of a magnetic field, one needs to be well outside the band of unperturbed

states in order for the variational approximation to be

valid. This would suggest that the present result is valid

only in the tails of the density of states—the region in- accessible to ordinary perturbation theory.

It is possible in principle to experimentally determine

the density of states by measuring the MOSFET dif- ferential capacitance. More detailed calculations of the

density of states are underway and will be presented

elsewhere [18]. Another interesting and important quantity is the location of each of the mobihty edges. It is beyond the present capability of theory to reliably esti- mate the positions of the mobility edges but the asymmetry we have observed in the density of states suggests

that the mobility edges will not be symmetrically placed

about the nominal Landau level position.

4. Summary

We have briefly reported on investigations into the

inversion layer voltage distribution in the quantum Hall

regime and on the density of states between Landau levels in the presence of a model impurity potential. The

quantum Hall phenomenon presents a great variety of new physics puzzles which challenge our understanding

of basic ideas of transport. Further study of the many

questions of current interest should prove fruitful for some time to come.

References

[1] K. V. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett.

45, 494 (1980).

[2] R. E. Prange, Phys. Rev. B 23, 4802 (1981).

[3] R. B. Laughlin, Phys. Rev. B 23, 5632 (1981).

[4] H. Aoki and T. Ando, Solid State Commun. 38, 1079 (1981).

[5] R. W. Rendell and S. M. Girvin, Phys. Rev. B 23, 6610

(1981).

[6] R. F. Wick, J. Appl. Phys. 25, 741 (1954).

[7] H. J. Lippman and F. Kuhrt, Z. Naturforsch. Tell A: 13, 462

(1958); 13, 474 (1958).

[8] S. Kawaji, Surf. Sci. 73, 46 (1978).

[9] M. E. Cage, R. F. Dziuba, B. F. Field, R. J. Wagner, and

C. F. Lavine, private communication.

[10] T. Ando and Y. Uemura, J. Phys. Soc. Jpn. 36, 959 (1974).

[11] T. Ando, J. Phys. Soc. Jpn. 38, 989 (1975).

[12] C. S. Ting, S. C. Ying, and J. J. Quinn, Phys. Rev. B 16, 5394 (1977).

[13] S. Fujita and M. Prassad, J. Phys. Chem. Solids 38, 1351

(1977).

[14] S. Das Sarma, Solid State Commun. 36, 357 (1980).

[15] John L. Cardy, J. Phys. C 11, 1321 (1978).

[16] Masaru Tsukada, J. Phys. Soc. Jpn. 41, 1466 (1976).

[17] A. Houghton and L. Schafer, J. Phys. A 12, 1309 (1979).

[18] R. W. Rendell and S. M. Girvin, unpublished.

560
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984). GRAVITY AND RELATIVITY

Experiments Relating to the Newtonian Gravitational Constant

H. de Boer

Physikallsch-Technische Bundesanstalt (PTB) D-3300 Braunschweig, F.R.G.

The Newtonian gravitational constant G is one of the natural constants, the exact knowledge of which might give us important answers to cosmological questions and contribute to the support of a theory of gravitation. In a survey, the problems are set forth which are related to the gravitational constant, appertaining measurements are described, and the uncertainties of measurements obtained so

far are compiled.

Key words: dependence of G on time; dependence of Newton's law on R; determinations of the gravitational constant; Newtonian gravitational constant.

1. Introduction

In all general lectures, v^^e are accustomed to the sig- nificant progress achieved in the respective field of metrology being discussed, a reduction of the measuring

uncertainty, and a theoretical interpretation of the problems in the field. With regard to the Newtonian gravitational constant G it is not, however, possible to report

any pioneering success, as the progress made in this special field is slow and the problems facing us are far from

being solved experimentally. The interest of astronomers

and physicists in the exact knowledge of G, and in the

questions bound up with this, has grown with the initial findings concerning the composition and interaction of the mass distribution in the cosmos. New gravitational

theories have been developed, and questions are being

raised in support of the various theories which are in- creasingly difficult to answer.

In this review paper we summarize the experiments

carried out to determine the Nev^d^onian gravitational

constant, and also briefly deal with some experiments

which are known to us under the following key-words:

inertial mass; active and passive gravitational mass; in- dependence of the gravitational constant; and time

dependence of the gravitational constant. At the end we

report on the experiment carried out in the PTB to determine G. In its simplest form, the law of gravity formulated by

Nev^rton in 1687 states that two spherical bodies with mass mi and m2 separated from each other by a distance

R attract each other with the force

G

mim2

where G is the Newtonian gravitational constant. To give an impression of the magnitude of the forces

occurring. Fig. 1 shows the forces as a function of the

distance between two spherical bodies of either platinum,

lead, and glass of a fixed radius (unbroken lines), and as

a function of the distance with a specified mass (broken

distance

Figure 1. Forces as a function of distance between spherical

bodies of platinum, lead, and glass at a fixed radium (unbroken

lines) and as a function of distance with specified mass (bro- ken lines).

lines). The forces which occur are vdthin the 10"^ N

range. More or less the same force of attraction would be pro- duced by a voltage of about 10 V between the spheres

separated from each other by a minimum distance. As

can be seen from this simple example, contact voltages

possibly occurring between the attracting masses at the

relative measuring uncertainty to be aimed at of AG/G = 10"*^ must be less than 10"^ V.

If one of the interacting bodies is the Earth and the

other a 1 kg weight at the Earth's surface, a force of at- traction of 10 N is encountered. The most precise bal- ances for the comparison of forces due to weight have a relative uncertainty of AF/F = Am/m ^ 10"^. Thus,

when using such a balance to compare two 1 kg weights

and when bringing a 1 kg weight closer from the top or

561
from the bottom to one of the weights placed on the

beams of the balance, the balance would not show any

reaction as a result of the gravitational effect produced.

The task of an experimenter who wishes to measure G

consists in the measurement of very small forces to be

carried out, as we shall see below, in the presence of a great many disturbing forces.

2. G Experiments and Results

Experiments hitherto carried out can be roughly

grouped as follows:

(1) measurements in which the Earth and mountains or

parts of the Earth's crust are involved as reaction masses (so-called large-scale experiments);

(2) measurements with Jolly beam balances; and

(3) measurements with torsion balances.

2.1 Large-Scale Experiments

The measurements within the scope of the first group

aimed primarily at determining the mean density of the

Earth. The possibility of calculating the value of the

gravitational constant from these measurements is only a secondary result. Three methods served to determine the

Earth's mean density (Fig. 2):

(1) pendulum measurements at sea level and, subsequently, on an elevated plane the geometry and density of which are measured;

(2) determination of the deviation of the direction of the

acceleration due to gravity from the radial direction

of the Earth at two sides of a mountain the geometry

and density of which must again be known; and

(3) determination of the mean density from measure- ments of the acceleration due to gravity, for example

with pendulums, at various depths of a mine. To cal- culate G, the mass of the spherical shells is related to the whole mass of the Earth.

Measurements according to the first two methods were

carried out for the first time in 1738 by Bouguer and de

la Condamine on a suitably shaped mountain ' in Peru. Measurement results obtained according to the third

method were published in 1828 by Airy and his collabora- tors who had carried out measurements in a mine 400 m

in depth. The G value which would at present be the

most precise and which was published by Stacey, Tuck

and others [57] who had carried out measurements in an

Australian mine according to the last method is G =

(6.712 ± 0.037) X 10-11 N-m^-kg-^

Stacey and Tuck furnish another G value of (6.730 ±

0.025) X 10-11 N-m^-kg-^ which they calculated from

comparable measurements by McCulloh in a mine 700 m

in depth. Both values deviate from the 1973 CODATA

value: G = (6.672 ± 0.004) x lO'H N-m^-kg-^ [13], and

is a possible hint that the 1/J?^ dependence of Newton's

law of force is not obeyed for short distances, as is sup- ported by other authors. We will return to this point

later. Figure 3 shows a summary list of measured values ob- tained in the large-scale experiments. We see that the

spread of the measured values is very large and deviates

from the CODATA value, the uncertainty range of which

has been entered as well. Hence it follows that measure- ments of this kind are not suitable for the precision

determination of G.

10'" Nm^ kg"^

738 Bouguer u. dele Condamine

1881 MendenhQil 892 Preston

78 1774 Moskelyne u Hutton 743 1821 Corlini 856 1842 Soigey u. Puissant 70 1854 James u, Clark 50 „ 1855 Pectiman

1828 Airy 1883 von Sterneck 1885 von Sterneck 1980 Stacey, Tuck, al Stacey, Tuck, aL(Mc Culloh)

Figure 2. Methods and results of "large-scale measurements'

ofG.

1980

year

1880

1930

1780

1730

Stocey, Tuck et al

Preston von Sterneck Mendenholl o

Pectiman Jomes u, Clork Saigey u Puissont

Airy

Carlini

Mackelyne u Hutton

Bouguer u de le Condamine _J \ _1 \ L_

(6 712=o.o37).io-"Nm^kg-^ (6 730' 0,025) 10" Nm^kg-^

Codoto 1973 [13]

5 00 6,00 7

Value of Gravitational Constant G

10""Nm^kg"^ 900

Figure 3. Compilation of G values attained by "large-scale

measiirements."

2.2 Measurements with Jolly Beam Balances

In the introduction it was said that it is not at present

possible for the most precise balances to reveal the

gravitational force between two 1 kg weights separated

by a minimum distance. This can naturally be remedied

by using greater masses and countermasses. Figure 4 shows a schematic drawing of the measuring arrange- ment used by Richarz and Krigar-Menzel [48, 49]. These

authors used a balance with a 23 cm long beam on the

ends of which pans were mounted at a vertical distance

of 200 cm. Lead weights of 0.67 kg were weighed in the

562
-230 - A

• 0 57 kg lead mass

1900

yeor

1890

1885

1880

1875

1870

[131 Codata 1973 ~ti

Poynting

Wilsing

Richorz , Krigar-Menzel

Jolly^s- J I I I I I L I I I I I I I I

0.67 kg lead mass

Figure 4. Jolly beam balance arrangement used by Richarz

and Krigar-Menzel (schematic). The beam length is 230 mm

and the pan separation about 2000 mm (2m).

pans A,D and C,B without the lead mass of about

100 000 kg in order to be able to correct for the differ- ence of the acceleration due to gravity between the

planes AC and BD. Subsequently, the same measure- ment was repeated with the lead block. These measure- ments allowed a G value of (6.683 ± 0.011) x 10""

N-m^-kg"^ to be determined. Figure 5 shows a summary

list of the best-known measurements of this kind with

the estimated uncertainties. The relative uncertainties of the measurements are within the 10"^ to 10"^^ range. The

Hmit of this method is probably almost reached with the

result obtained by Richarz and Krigar-Menzel. These au- thors indicate the following disturbance influences:

(1) air convection as a result of temperature gradients;

(2) non-homogeneity of the density of the sets of

weights;

6.40 6.50 6 60

Value of Gravitational Constant G

10'"Nm^kg^

Figure 5. Compilation of G values attained by Jolly beam bal- ance measurements.

(3) influences by the knife-edge bearing of the balance

beams; and

(4) disturbances of the ground.

Table 1 gives a summary about the most important bal- ance measurements. After the measurements by

Krigar-Menzel, and Wilsing, i.e., after 1850, no further G

measurements were carried out by the weighing method.

2.3 Measurements with Torsion Balances

The most successful laboratory experiments to determine the gravitational constant have been carried out with torsion balances. Figure 6 shows the fundamental

design of this balance type. Cavendish [12] was the first scientific researcher to use this measuring arrangement

in 1758 for the determination of G. The arrangement con- Table 1. Determinations of the gravitational constant by m^ans ofJolly beam balances

(from, L. Stegena and M. Sagitov [58])

Experiment:

author, year of publication,

reference, and

place of experiment

Beam length

and height dif- ference between

masses m and

M

(cm)

Attracted masses M:

material,

shape, weight

Attracting masses M:

material,

shape, weight

Constant of gravitation

G and rms

error

(10-11 N-m2-kg-2)

Jolly, 1873

[31, 32], Munich,

Germany

L = 60 h = 2100.5 Mercury in a sphere-shaped

glass vessel; 5 kg each

Lead sphere; 5775.2 kg6.447 ±0.11

Poynting, 1891

[43, 44], Birmingham,

England

L = 123.26 Lead-antimony

alloy, sphereshaped;

21.6 kg each

Lead-antimony

alloy, sphereshaped; 153.4 and

76.5 kg

6.70 ± 0.04

Richarz,

Krigar-Menzel,

1888, [48, 49], Spandau citadel

L = 23.32 Lead,

sphere-shaped;

0.67 kg each

Lead,

parallel-piped 211.1 X 211.1 X

209.5 cm; 100536.8 kg

6.683 ± 0.011

Wilsing, 1889

[62, 63, 64] Potsdam, Germany

L = 100 Copper and

lead-antimony

alloy; 540 g each

Lead circular cylinders; 325 kg each6.594 ±0.15

563
V ..for"

""^•^ position

..near

deflection torsion period

stotic detlection method dynamic method

Figure 6. Principles of torsion beam balance measurements.

sists of a balance beam with weights fixed to the end.

The balance beam is suspended from a torsion wire so that the direction of the wire and the direction of the

gravitational acceleration are the same. The outer reaction mass arrangement is such that the reciprocal force of

attraction lies in the plane perpendicular to the acceleration due to gravity. When G is determined from the mass

attraction with the aid of the angle of deflection of the

torsion balance, as is shown on the left of Fig. 6, this method is referred to as the static method. In this case, the torsional moment of the wire must be known for the

calculation of G; it is generally calculated on the basis of a measurement of the period of oscillation with the mo- ment of inertia being known. The measuring method is referred to as dynamic when the torque and, thus, G are determined on the basis of the mass attraction by two

measurements of the period of oscillation of the torsion

balance, the great masses being arranged first in the

"near" position and then in the "far" position. This

method is shown on the right of Fig. 6. It was used for the first time in 1896 by Ebtvbs [17, 18]. Figure 7 and Table 2 show a summary of the results of the most significant measurements with torsion balances

since the first measurements carried out by Cavendish.

As can be seen from Figure 7, the G values spread over

a wide range until 1920. After this date, the spread de- creases. If we wish to know the most precise G values,

980

yeor

940

900

850

820

o static dynamic ^ in compensation o in resonance q

C2>

•CodotQ 1973 1131

Sogitov et ol Karagioz et ol Focy u Pontikis Rose. Porker. Lowry et al

Heyl u Ctirzanowski Zohrodnicez

Heyl

Burgess

Eoetvoes

Luttier.Towler et ol Renner

Broun Boys

Cornu u Boille

Reicti Boily

Covendisti J L

6.60 B64 6 68 10" Nn

Value of Grovitotionol Constant G

576

we will have to consider the values obtained after 1920. Figure 8 shows the values obtained after 1920 together

with the simple standard deviations stated by the authors. The reports do not state in every case exactly how

the calculation was made, so that a direct comparison of the values stated requires caution. In most cases, the au- thors state measured values for series of measurements,

obtained successively when making measurements under

varying conditions, as, for example, using different materials for the masses or different torsion wires.

1980

year

1960

1950

1940

1930

1920

Codata -t-1973 (131

dynamic * : in compensation o in resanonce

Sogitov et al Korogioz et al Focy u Pontikis Rose, Porker, Lowry et al.

Luttier, lowler et ol Renner

Heyl u Chrzonowski

Zotiradnicez

Heyl

J I L

Figure 7. Compilation of G values attained by torsion beam

balance measurements.

5 65 6 67

Value of Gravilotional Constont G

Figure 8. Compilation of G values and appropriate one stand- ard deviation uncertainties attained by torsion beam balance measurements after 1920.

2.3.1 Measurements by Heyl (1930) and Heyl and Chrzanowski

(1942)

Heyl and, at a later date, Heyl and Chrzanowski

[24, 26] used the dynamic method. Heyl was the first to use large, cylindrical masses since these could be

manufactured and dimensioned with a higher precision.

He had, however, to pay for this advantage with a great

deal of mathematical calculation at a time when pro- grammable computers were not yet available. The large

66 kg masses were made of tool steel. The small masses

Heyl used were spherical bodies weighing about 50 g and

made of gold, platinum, and optical glass. The results ob- tained with the different materials differed from one

another. Heyl found that the gold masses increased in size during the test because mercury from the pressure

measuring system combined with the gold. Heyl obtained

the following result from his 1930 measurements:

G = (6.670 ± 0.005) x 10^" N-m^-kg-^

In 1942, Heyl and Chrzanowski reported on another

measurement. They intended to improve Heyl's measuring apparatus to allow a better G value to be obtained.

They carried out the following improvements:

(1) Extension of the balance beam from 20 cm to 80 cm

in length. It subsequently failed because the deflec- tion of the beam exceeded the permissible value. An

increase of the beam section resulted in the mass in- creasing to such an extent that it could not be handled by the torsion wire.

(2) Heyl and Chrzanowski made an attempt to use a disc-shaped balance beam offering the advantage of rotational symmetry with respect to the gravitational

forces. In this case too, difficulties arose concerning

the carrying capacity of the torsion wire.

(3) Furthermore, they attempted to replace the torsion

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wire by two wires or a bifilar suspension, to achieve

a higher carrying capacity and at the same time, a prolonged period of oscillation of the torsion balance.

The attempts to achieve a multiple filament suspension failed, as they did not succeed in loading the fila- ments uniformly. The 1942 tests used hard-drawn

tungsten wire 0.030 mm in diameter and annealed

tungsten wire 0.035 mm in diameter; thus the modifications in comparison with the 1930 tests were not very significant.

The results obtained with the different wires varied

slightly. The overall result was in agreement with the

results obtained in 1930. These workers held the view

that with their final result of G - (6.673 ± 0.005) x 10"" N-m^-kg-^

,

they had reached the metrological limits of their

apparatus.

2.3.2 Measurements by Rose, Beams and Others (1969) and by

Luther and Others

Chronologically, the G measurements by Rose [52],

Beams [3], and others in 1969 were the next to be carried

out; they were continued by Luther [38, 39] and others at a later date. Figure 9 is a schematic representation of the measuring arrangement. On an air-supported rotary table, two

mirror

small mass system

counterweight

tungsten spheres

rotary table

precision bearing

rotor

Hg

Hg slip rings

Qir tight chamber

quartz fiber

window

tracking optcol lever

incident line of light

light source

photo diodes

reflected line of light

stotor

copper probes

Figure 9. Measurement arrangement used by R. D. Rose et

al. [52].

tungsten spheres are fixed as large masses, which to- gether with a torsion balance can be rotated about the

balance axis. When placing the large masses at an ap- propriate angle to the small masses, the torsion balance

is deflected. This deflection is measured with a spot of

light by means of photodiodes. From this a signal is formed which controls a motor driving the rotary table. The rotary movement of the table is such that the deflec- tion of the torsion balance is just compensated for by the

forces of inertia. This interaction of the forces of inertia

and gravitation results in an ever-accelerating rotation

of the table, from which G can be determined. What

is remarkable and new about this arrangement is the

following:

(1) The measurement of the gravitational forces is not only based on geometric measurements but also on a frequency measurement (or, to be more precise, on

the measurement of a frequency change). There is no

deflection or oscillation of the torsion balance, so this method can be referred to as a compensation method.

The measurement can be carried out by integrating over a prolonged period of time.

(2) By the rotation of the table, influences of gradients of the gravitational field of surrounding masses are averaged out to first order.

(3) During the measurement, the coordinates of the reaction masses remain fixed and can therefore be es- timated with a smaller uncertainty.

2.3.3 Measurements by Facy and Pontikis

In 1972, Facy and Pontikis [19, 20, 21] published the fi- nal results of their G measurements, using the resonance method (Figure 10). The large masses fixed on a motordriven rotary table circle the small masses attached to

optical system

vacuum pump

ilament of plotinum

brass ball

reduction gear

^1^^

motor

electrical speed control

spot registration

Figure 10. Measurement arrangement used by Facy and Pontikis [19].

the torsion balance. At certain rotational frequencies of the large masses, there is a resonant excitation of the oscillation of the torsion balance. The movements of the

torsion balance are recorded. The measurements are evaluated using a computer, the respective geometric

data and the oscillation values actually measured being

used in the theoretical differential equation of oscillation from which the G value best suited for the measurements

is calculated. As other authors had previously done, Facy

and Pontikis also used different materials in their tests,

i.e., silver, copper, bronze, lead, and brass. Unfortunately, the documentation of the results is obscure and

incomplete; it is not clear, for example, which material

combinations were selected for the large and small masses to obtain the results published in 1972. The au- thors announced a final, detailed report which, however,

to my knowledge was not published.

2.3.4 Measurements by Renner Another G measurement was reported on in 1973 by

Ya. Renner, Budapest [46, 47]. He applied the dynamic

measuring method in which the period of oscillation in the near and in the far position of the large masses was

determined in the customary way. The large masses con566
sisted of four identical hollow cylinders mounted on a

table which could be rotated about the axis of the torsion

balance. Two cylinders arranged opposite each other were filled with mercury.

2.3.5 Measurements by Karagioz and Others

A preliminary measurement by Karagioz [33] and oth- ers in 1976 was the next to be published. The authors

used the dynamical method. They did not, however, measure in the near and in the far position of the large masses but used only one large mass which was shifted in the direction of the torsion beam.

The mounting for the large mass was provided with

boreholes 4 mm in diameter and each at a distance of 12 mm, into which the spherical mass was placed. The

large masses were made of steel, copper, brass, and

duraluminium and the small masses of oxygen-free

copper. Only with a mass of steel did it prove possible to carry out measurements with a small deviation of the

measured values. The unroundness of the spherical masses resulted in a positional uncertainty of the mass

centers of about 50 \xm. Karagioz states that the mass

center for a relative uncertainty of smaller than 10"^ must be positioned to better than 3 iJim.

2.3.6 Measurements by Sagitov and Others (1977)

In 1977 a measurement was carried out by M. Sagitov

[53, 54, 58] and others. Sagitov too uses the dynamic

method and like Karagioz, he shifts the large masses in the direction of the balance beam. As a reason for this arrangement, he states that it allows the geometries to be measured more easily and precisely. Sagitov uses

cylindrical masses made of steel and copper. His device

for detecting the oscillations of the torsion balance is so arranged that for periods of oscillation in the range of

approximately 2000 s to 2300 s, he measures the instan- taneous angular deflection every 20 seconds with a rela- tive uncertainty in the time measurement of 1.5 x 10"^ and a resolution of 10"^ rad. He thus obtains a reliable picture of the angle vs. time behavior of his oscillation. This knowledge enables him to recognize the nonlinear

influences upon the oscillation by comparison of the ex- perimental values with the respective differential equation of the oscillation and to take them into account in the evaluation.

Measurements for the determination of G are decidedly

long-term measurements, some of the individual series of

measurement taking several months. Between the measurements, the measuring apparatus is usually recon- structed or modified so that in many cases, a stationary

behaviour of the influencing parameters cannot be as- sumed. An indication of the validity of this objection is given by the measured values themselves: In the reports

of many authors, it is noticeable that the mean values

and the scatter of various series of measurements are not

consistent.

3.1 Inconstancy of the Torsional Moment of

Suspension

In the torsion balance experiments hitherto successfully carried out, the balance beam with the masses

fastened to it is held by a torsion wire. The torsion wire

has thus to fulfil two functions:

(1) It must carry the weight of the torsion balance.

(2) It must supply the torsional moment.

To obtain a very sensitive torsional balance, it is desir- able to select a very small wire diameter and a very long

wire. However, since it must support the weight of the

beam, a lower limit is imposed on the wire diameter by

the yield and breaking point of the wire material. In the

usual laboratory arrangement, the length of the torsion

wire is limited to a maximum of 1 m by the temperature

constancy required for the measuring arrangement. In the selection of the wire material, other aspects play an

important role, for example, the dependency of the

torque upon the magnetic field and temperature. To date,

materials such as tungsten, platinum, platinum-iridium,

steel, quartz, and carbon have been used. Almost all au- thors report difficulties encountered with the various

wire materials. They report a null drift of the torsion

balance, slow changes of the torsional moments, and erratic modifications when relieving or re-loading the wire.

Burgess in 1902 [10, 11] and Tangl in 1926 [59] at- tempted to find a way out by freeing the wire from the

load by using the buoyancy of a body in mercury or water

(Fig. 11). Burgess described great difficulties with the

mercury surface. With time, impurities in the mercury

result in a layer forming on the surface making the meas3. Influences on the Measurement

Uncertainty

Looking at the measurements which are at present the

best, one notices that the relative uncertainties hitherto

achieved are in the range of some 10"'*. When considering

the measurements more closely, one discovers that the

uncertainties represent the deviations of the measured

values in a given experiment. No publication states an

estimate of the possible systematic error due to disturb- ances of the measurements by influencing parameters.

This certainly cannot be explained by the fact that the

authors did not think of these but that most of the

parameters still exerted an influence upon the measure- ment although their influence had been eliminated to the

greatest possible extent. The influence of the disturb- ances consists of the interaction of all of the disturbing

parameters so that in the end, the overall influence can be described by statistical data at best. Describing the measuring uncertainty by simple statistical means, however, is allowed only if stationary processes are involved which are independent of time.

Scheme of Tongl's device for increasing Scheme of the Burgess setup

torsion balance sensitivity

Figure 11. Measurement arrangeynent ivith buoyancy bearings

[58].

567
uring conditions undefinable. Because of the complex

frictional conditions of the body floating in the liquid, this method can be used only for static measurements.

Another attempt to achieve a suspension free from fric- tion and torque using a magnetic suspension also encountered difficulties. It was not possible to eliminate the

influences of torque in the operation of the suspension.

3.2 Disturbances of the Ground

All measurements suffer from disturbances and tilting

of the ground, which vary locally and manifest themselves in disturbing oscillations of the torsion pendulum.

The frequency range of ground disturbances extends to

far below 1 Hz so that a decoupling of the measuring ar- rangement by mechanical filtering is not possible. Owing

to nonlinearities of the oscillating system, oscillations of

the balance beam about other axes of inertia can be cou- pled to the oscillation of interest. One could resort to an

attenuation by using suitable gaseous atmospheres, but

in dynamic measurements this possibility is precluded by

the fact that the frictional behavior and the adsorption

of gas on mobile components would result in systematic

errors. The influence of ground disturbances can be re- duced by using sufficiently long averaging times for the

measurements (long compared with the period of the

disturbances).

3.3 Influence of Temperature

As in almost all precision measurements, ternperature exerts a considerable influence upon the measuring

result. Most critical is the variation of geometrical di- mensions vdth temperature. For G measurements in particular, tolerances vdthin the range of 10"^ and 10"^ m

must be complied with and conserved over a prolonged

period of time; this requires a temperature constancy over this time of 10"^ to 10"^ K. Keeping the temperature stable is not the only requirement; there must not be any temperature gradients, either. In measurements

with sensitive torsion balances, between the upper and

the lower part of the measuring arrangement, for example, temperature gradients of 10"^ to 10"^ K produce

convection currents of the filling gas and, as a result, in- terfering torques at the level of the torques in the ex- periment. It might be considered that such difficulties can be avoided only by operating the torsion balance at a reduced pressure.

3.4 Inhomogeneity of the Masses

As is shown in Table 2, the various workers carried out measurements with widely ranging masses. Heyl used

the largest masses, 66 kg, and Facy and Pontikis the

smallest, 1.5 kg. The advantage of a greater torque on

the torsion balance produced by the larger mass, and

thus, an improved signal-to-noise ratio between the effec- tive signal and spurious signals due to spurious forces on

the torsion balance is, however, cancelled out by the

phenomenon that the inhomogeneity (density variation)

generally increases with the mass. This results in an uncontrollable systematic error in the measurement result.

It is very difficult to obtain information on the density

distribution of the masses by experiment—information

that could be used later as a correction in the evaluation.

The selection of the masses also calls for compromises.

Theoretically, it would be advantageous, for example, to manufacture the masses from materials with a high density such as platinum or tungsten. In practice, it has

been shown that knowledge of how to process these materials in order to achieve homogeneous density is insuffi- cient, so the theoretical advantage of the high density cannot be fully utilized for practical purposes.

3.5 Gradients in the Gravitational Field, Changes of

the Surrounding Field

In laboratory measurements, it is fundamentally impossible to block off the influences exerted by surrounding

masses. In the measuring arrangement area, the gravitational field always shows gradients which in a dynamic

measurement result in a disturbance of the torsional oscillation difficult to correct by calculation. As already

mentioned, G measurements are long-term measurements

with measuring periods sometimes covering several

months. Figure 12 clearly shows to what proximity an

object coming from infinity can be brought to the measuring arrangement with the influence upon the measure- ment result remaining smaller than the relative value of

10"^. Figure 13 shows by what distance the position of an object can be changed with respect to the torsion bal- ance so that the influence remains smaller than 10"^.

3.6 Magnetic and Electrical Influences

When setting up a measuring arrangement, it should

also be ensured that all the materials used are as free as possible from ferromagnetic impurities which can be

Figure 12. hifluence of masses approaching a torsion balance from infinity.

568
Figure 13. Influence of masses changing their distance relative

to the torsion balance.

caused by unclean tools during production. Particularly

in the case of large arrangements, it is difficult to achieve a complete shielding of the magnetic field of the

earth. In addition to the magnetic disturbances, electrical disturbances are to be anticipated. Here, electrostatic

charges of electrically insulated components and contact

potentials can build up if different materials are used to

set the arrangement up.

4. Perspectives

In the foregoing, a number of disturbing factors were

given which must be taken into consideration when

designing a measuring assembly. It is not possible for each disturbing factor to be dealt with independently in order to find a way to eliminate it. This task becomes a problem of optimizing several parameters simultaneously, with the further difficulty that analytical

knowledge of the influences of the parameters is very

inexact. When the question is asked: "What are the possible Hmits of an attainable measuring uncertainty?' ' , it

is difficult to find an answer. On the basis of his results which he pubhshed together with Chrzanowski in 1942

and which he had obtained with a relative uncertainty of

4.7 X 10"^, Heyl took stock, saying: "The conclusion

may be drawn that the limit of the possibihty of the tor- sion balance has been reached". According to estimates

which take into account the well-known disturbing influ- ences, it should be possible to attain relative uncertainties in the range of 10"^. To achieve this, it wiW however

be necessary to carry out several tests with different

test assemblies to improve our knowledge of the

disturbances.

The next question that could be put is: "Are there any

suggestions for G determinations which are based on

completely new measuring methods and which promise

substantial progresses with respect to the expected

reduction of the measuring uncertainty?". All experts

are obviously in agreement that the laboratory experiments which promise to be successful can be carried out only with torsion balances, whereby a number of good, new ideas would certainly result in a further reduction of the uncertainty in determining G.

I do not know of any new laboratory experiments.

Some suggestions have been made by several authors

[4, 5, 6, 14, 60]; David Berman, Robert Forward, and

John Vinti suggest that G experiments should be carried out in space, that is, on board a satellite in a force-free environment. They suggest two possible measuring

methods (Figure 14):

Satellite System Experimental Schematic Tunneled Sptiere Experimental Schematic

Figure 14. Proposals for G experiments on board forcefree satellites [iJ.

(1) In the force-free space of a satellite, a small test mass circles around a large mass. G can be determined on the basis of the known masses and the orbit

data.

(2) In a tunnel inside a large mass, a small mass oscil- lates around the center of the large mass. The determination of G is made possible by knowledge of the masses and the oscillation period.

The authors believe that such tests are feasible. Great

demands are made, however, on the symmetry of the

satellite's mass distribution and internal equipment as well as on a force-free orbit.

5. Other Issues Related to G and Newton's

Law

Up to now, we have dealt vdth laboratory experiments

for the determination of G. There are still some other in- teresting problems which are directly related to the

Nev^1:onian equation and thus to G. Let us take a look at these problems which can be defined as follows:

(1) active and passive gravitational mass, inertial mass;

(2) possible deviations from Newton' s l/R - law; and

(3) the time dependence of G.

569
5.1 Active and Passive Gravitational l\/lass, Inertial IVIass

10 ^. The result is not as convincing as the comparison

between gravitational and inertial mass.

Fundamentally and in the strictest sense, one may

identify three types of mass:

(1) the inertial mass which appears in Newton's second

law as the factor of proportionality between the force on and the acceleration of a body;

(2) the passive gravitational mass which is the mass of a body on which a gravitational field acts; and

(3) the active gravitational mass which is the factor of proportionality for the gravitational field emanating

from a body.

It is assumed that metrologically, the three properties of a body just described can be defined by the same quantity. In order to prove this, however, it must be shown

experimentally that these quantities are in agreement.

In the well-known Eotvbs [18] experiment, gravitational and inertial masses of bodies made from different materials are compared with one another by comparing the

gravity acting upon them in the earth's gravitational

field with the centrifugal force produced by the earth's

rotation with the aid of a torsion balance. Eotvbs found

that the inertial and the gravitational masses are in agreement viith a relative uncertainty of 10"^. In an ex- periment carried out at a later date by Roll, Krotkov,

and Dicke [51] who effected measurements by comparing

the influence of the gravitational force of the sun and the

centrifugal forces due to the earth's revolution around

the sun, an agreement between inertial and gravitational mass could be indicated with a relative uncertainty of 10-". Tests directly comparing passive and active gravitational mass are rare; I know only of an experiment by

Kreuzer [34, 23] (Figure 15). In his experiment, he compares the influence of different materials which have,

to motor

suspension fiber

optical lever

electrodes torsion beam electrodes

Figure 15. Set-up of Kreuzer to measure the equivalence of active and passive gravitational mass [3^].

however, the same density, on a torsion balance. A

Teflon cylinder immersed in a liquid of more or less the same density is moved periodically to and fro and the sig- nal caused by the gravitational force is recorded. By

altering the temperature, the density of the liquid can be

adjusted to that of Teflon so that—with the measuring

signal disappearing and the density of the material

remaining the same—proportionality between passive

and active gravitational mass can be assumed wdthin the

scope of the attained relative measuring uncertainty of

5.2 R-Dependence of the Gravitational Constant

Another problem which has stimulated experimenters

in the last few years to carry out relevant experiments is the presumed deviation of Newton's law from the IIR^

form. From the consideration of a general scalar tensor theory of gravitation. Wagoner [61], Fujii [22], and

O'Hanlon [24] derived a gravitational potential which is extended by a Yukawa term: V{R) = --^(1 + ae-^),

where G« is the Newtonian gravitational constant at in- finite separation, and a and |x are constants. We may

thus write

FiR) = GiR) ^^^fvdth

G(R) = Goll + ail + ]xRe->^)y.

Fujii surmises that a < 1/3, and 10 m < fx"^ <

1000 m. If this is correct, G (R ) would have the value of Go = (4/3)Gx over the short range (i.e., for small separations as used in the laboratory). In this case, all cosmic mass determinations resulting from assuming Gcc = Go

would have to be corrected accordingly. In the event,

however, of [x"^ being smaller than 0.01 m, the G values

measured in the laboratory would be in agreement with

the value Goo. Long' s statement for G {R ) over the short range is somewhat different [35, 36, 37]: GiR) = G{Ro)[l + XlniR/Ro)]

,

vdth \ - 0.002, Ro = I cm, Rl - upper limit for validity, and R < RlMeasurements carried out by Panov and Frontov [42] in 1979 on a torsion balance in which masses of different

weight acted upon the torsion balance from a distance

of between 0.4 and 10 m did not show any deviation

from the 1/R^ form within the range of the measuring

uncertainty.

Hsou-Tao Yu [29,30] and others carried out GiR)

measurements at a distance of 2 m to 15 m from an oil tank 54 m in diameter and 15 m in height, which was

sometimes filled and at other times empty. For these

measurements they used a Worden gravimeter. The

result they obtained was G(15 m) = God ± 0.18). In 1980, Hirakawa [27] and others made measurements

in the range of 2 m < R <4.2mby measuring the

dynamic gravitational field of a rotating steel bar using a mass quadropole antenna. They did not find any deviation within their measuring uncertainty.

Spero and others [56] made an attempt to demonstrate

a deviation from the 1/R^ law by a null method. If the

1/R^ law is valid, there must be no gravitational field within an infinitely long hollow cyHnder, due to the walls

of the cylinder. If this law is deviated from, it must be

possible for a gravitational field to be detected which

depends upon a lateral shift. In terms of Long's GiR)

representation, the limit obtained for X. was ±7 x 10"^. This result is at variance with that obtained by Long.

However, Long points out that for physical reasons null methods of this kind are perhaps not suitable for finding

the deviation from the 1/R^ law.

570
Another experiment which can provide significant in- formation because of its very wide R range is that of Sta- cey and others who carried out gravitational acceleration

measurements at different depths of a mine 950 m deep.

On the basis of exact density measurements of the

respective spherical shells, they state a value of G =

6.712 ± 0.037 X 10-11 N-m^-kg-^ which is higher than

the laboratory values and tends in the direction of the

\IR'^ dependence found by Long.

All these measurements show that the measuring uncertainty hitherto achieved does not suffice to give clear answers to the questions posed. It does, however, seem

possible to further reduce the measuring uncertainties,

so that an answer can be found in the foreseeable future

to the physically relevant question of a deviation from

the l/i?^ force law.

5.3 Time Dependence of the Gravitational Constant

In theories of cosmic development, it is presumed that

G is not constant wdth time. The relative change GIG is in the range of some lO^^Vyear or, accordingly, some

10-i^/s.

The proof of changes of G with time of this order re- quires the availability of processes or measuring devices

which are G-dependent and which are stable with respect

to all other influences over the necessary observation

period, this stability being greater than the anticipated ef- fect produced by G. The attempts to demonstrate a time

dependence of G which are the most promising up to now,

are based on exact observations of the orbits of planets of

the solar system and of the moon [1, 41, 55]. The present

state of this field and the measuring uncertainties hitherto

attained are discussed by R. C. Vessot, T. C. van Flandern, and R. C. Ritter in these proceedings.

As yet, no successful laboratory tests can be reported. V. B. Braginskii and V. L. Ginzburg [8] have submitted

proposals for a new measurement. Provided that the

change in gravity behaves in the same way as the change

of the gravitational constant with time, they suggest that

experiments should be made with special pendulums

especially designed for long-term stability. The change of G with time is to be demonstrated in these experiments

by means of the resulting change of the oscillation period

with time.

Another possibility of determining changes of G within

the scope of laboratory tests results from the assumed

relationship between a change of the gravitational constant and the formation of new matter which can be

detected by sensitive measuring devices. The state of these tests will be reported on by R. C. Ritter and G. T. GilHes [50].

6. PTB Experiment to Determine the

Gravitational Constant

In conclusion, I vidsh to report briefly on the principle on which the determination of G is based in experiments

carried out in the PTB. Figure 16 shows the most important design features. We have selected a static measuring method with a torsion balance. The balance beam

with the masses fixed to it is supported by a mercury

buoyancy bearing whose special design produces a centering effect in the direction of the torsional axis. This design allows the wire suspension to be dispensed

with. The torque produced by the gravitational forces is compensated by an electrostatic quadrant electrometer.

The masses are cylindrical in shape and are made, at

Figure 16. PTB set-up ferr the determination of the gravitational constant (schematic).

present, of glass. The countermasses are supported by

prisms and can be shifted in an axial direction. References

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571
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and J. W. Beams, Atomic Masses and Fundamental Constants U, Ed. by J. H. Sanders and A. H. Wapstra (Ple- num Press, London, 1972), p. 521.

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and J. W. Beams, in Atomic Masses and Fundamental

Constants 5, Ed. by J. H. Sanders and A. H. Wapstra

(Plenum Press, New York, 1976), p. 592.

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D. Boulanger and M. U. Sagitov (Nauka Press, Moscow,

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572
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Redetermination of the Newtonian Gravitational Constant "G"

Gabriel G. Luther

Center for Absolute Physical Quantities, National Bureau of Standards, Washington, DC 20234

and

William R. Towler

Department of Nuclear Engineering and Engineering Physics, University of Virginia, Charlottesville, VA 22901

The universal Newtonian gravitational constant, G, has been redetermined at the National Bureau

of Standards using the time-of-swing method, in which the period of a torsion pendulum in the form of a dumbbell weighing approximately 5 g is modified by the presence of two 10.5 kg tungsten balls. The

difference in the squared frequencies with and without the balls is proportional to G. In this experi- ment the difference was approximately 3% . The apparatus used, albeit with extensive modification, is the same as used in the previous redetermination done at the National Bureau of Standards, using the method of constant angular acceleration. The apparatus is described, improvements are discussed, data

is presented and evaluated. The resulting value of G is: 6.6726 ± 0.0005 x 10"^^ m^ • s"^ • kg"\ which is in agreement with the generally accepted CODATA value but with significantly smaller uncertainty.

Key words: gravitation; Newtonian gravitational constant; universal gravitational constant.

1. Introduction

The presently accepted value of G, defined by

Nev^i:on's equation

F ^ G MiM2/R^

and listed by Cohen and Taylor [1], was determined

mainly from measurements made by Paul R. Heyl and

Peter Chrzanowski in 1942 [2]. Of the several attempts to improve this measurement made during the past 39 years

[6, 7, 8], none seems significantly to have increased the

precision of the internationally accepted value of G =

6.6720 ± 0.0041 x lO'^^ m^ • s"^ • kg-\

This work is an extension of the work done at NBS [4] using Beams' s method of constant angular acceleration;

however, it has been reconfigured to use the time-ofswing method of C. V. Boys [3] and Heyl [2]. The salient modification is the addition of an autocollimator incorporating a diode array, suggested by R. D. Deslattes,

which allows for the continuous monitoring of the posi- tion of the small mass system. This additional information results in a more precise determination of the

frequency, amplitude, noise, drift, and linearity of the

torsion constant of the small mass system.

2. Theory

For a torsion balance with moment of inertia / and an- gular frequency oif, supported by a fiber with torsion constant Kf , the angular frequency squared is given by:

co/ = Kf/I or Kf = 7 0)^2 _ (1)

If the large masses are placed near this torsion pendulum, a small "fictitious" torsion constant Kg due to the

gravitational attraction is added to the torsion constant

of the fiber and the frequency of the oscillations becomes

0)/+, = (Kf + KgVI or {Kf + Kg) = li^l+g. (2)

Subtracting Eq. (2) from Eq. (1) yields

Kg = - co/) = 7A(co2)^

where Kg = U/d 0^ evaluated at 6 = 0 and U = gravitational potential of the small mass system in the gravitational field of the large masses calculated by integrating

U -GM,f ^

over the volume of the small masses. The detailed calculations of the potential J7 of a cylinder in the gravitational field of a spherical mass, are given in reference [9]. The expression for Kg is proportional to G, and is a function of the magnitude of the masses and the geometry of the apparatus. Thus, G can be expressed as G = Moj^)I/kg where kg = Kg/G . Note that A{(x/) is determined by measuring the fre- quency of the torsion pendulum and that I and kg are

calculated from the measurements of the appropriate di- mensions, angles, masses, and densities of the apparatus.

The above is true of an undamped harmonic oscillator. However, the frequency of a harmonic oscillator is changed when damped. This frequency change is second

order in the damping constant for small values of the

damping constant. Because the damping constant in our

case is approximately lO"**, the frequency change result- ing from this damping is negligible.

Evidence of anharmonicity was sought by looking for the second and third harmonics of the natural frequency.

None was found.

3. Apparatus

The apparatus is shown in Fig. 1 and consists of the

following:

573
MAGNETIC DAMPER

125 fJ.

VACUUM WALL

12/j. FIBER

AL TABLE

TUNGSTEN BALL

RETICON AUTOCOLLIMATOR

5gm TUNGSTEN

DUMBBELL

Figure l. Diagram of the apparatus ivith insert showing detail of the damper.

3.1 The Large Masses

The large masses are the same tungsten spheres used

by Beams at the University of Virginia and in our earlier

measurements [10]. They were fabricated from sintered

tungsten by the Y-12 Plant of the Union Carbide Corporation at Oak Ridge, TN. They are 10.165072 and

10.165108 cm in diameter, and their masses are 10.489980

and 10.490250 kg, respectively.

3.2 The Small Mass System

This system consists of two tungsten discs mounted in a dumbbell configuration. The discs are 2.5472 mm thick

by 7.1660 mm in diameter mounted on the ends of a

1.0347 mm diameter centerless-ground tungsten rod

28.5472 mm long. The mass of the entire assembly is ap- proximately 7 g.

3.3 Suspension

The small mass system is supported by an approximately 12 |xm diameter quartz fiber 40 cm long. The fiber was plated wdth chromium and gold to make it conductive. The period without the large masses in place is ap- proximately 6 minutes. The change in the period due to the large masses is a few percent.

3.4 Damper

The damper is shown in the insert in Fig. 1. It consists

of a small circular aluminum disc suspended in a strong

magnetic field approximately 10 cm from the top of the vacuum chamber by a 125 (xm diameter quartz fiber. The

spindle through the center of the disc serves as the sup- port for the much smaller 12 |xm diameter fiber which

holds the small masses. The larger top fiber is quite flex- ible with respect to motions of the small masses in the

pendulous mode which are introduced by seismic disturbances. These vibrations are damped in a few seconds.

The torsion constant of the larger top fiber is much

greater (10^) than that of the smaller fiber, i.e., the top

fiber is rigid, compared to the bottom fiber, with regard

to torsion motions: therefore, the damping of the torsion

mode is negligible. The tranquilizing effect of the damping is evident in Fig. 2.

TIME

Figure 2. Plot of angular position vs time without and with damping.

3.5 Detector

The detector consists of a 170 mm focal length autocol- limator with a 25 mm aperture in which the light through

a 15 ixm wide slit is collimated, reflected twice from the

mirror attached to the small mass system, and refocussed on a 1024 element diode array. The light-sensitive ele- ments are spaced 16 ixm on centers, which gives a range

of the autocollimator of 2 x 10^^ radians. The width of the refocussed image of the slit covers 4 to 5 elements of

the array.

3.6 Computer

Periodically (every 20 seconds as determined by the

internal clock of the computer and checked by a fre- quency counter), the computer turns the lamp of the au- tocollimator on for about 300 milliseconds, interrogates

the array, and calculates the center of the image group to 1/8 of the spacing between elements. This, the angular

position of the small mass system, is then stored.

574
3.7 Vacuum System

The small mass system, the fiber, and the damper are

enclosed in an evacuated brass cylinder which is maintained at a pressure of a few times 10"^ Torr by an ion

appendage pump. The windows in the chamber are coated

with a conductive coat of tin oxide. A rotating vacuum

seal through the top of the brass chamber allows positioning of the small mass system relative to the line joining the centers of the large masses.

3.8 Isolation Room

The apparatus is enclosed in an acoustically isolated,

thermally controlled cube about 2.5 m on a side. This is mounted on a reinforced concrete slab of about 5000 kg.

It is located in a large basement room of the Physics

Building at the National Bureau of Standards, Gaithersburg, MD, where the temperature is held constant to within approximately 2 °C. The inner room temperature

is controlled to vdthin approximately 0.1 °C.

3.9 Metrology

The large masses are positioned by being placed on

three small pads located in wells machined into a monolithic aluminum plate. The separation of the two large masses on the aluminum plate was measured in situ by

the 3-D measuring machine of the Dimensional Metrology

Group of the National Bureau of Standards, and determined to within 0.3 |xm. The consistency of the separation, after removing and replacing the large masses, was

found to be better than 0.05 jxm rms. The dimensions of the small mass system were determined initially by the Dimensional Metrology Group and

finally by the Gauge Block Group at the National Bureau

of Standards. The density of the tungsten in the small mass system was determined by the Length and Mass

Measurements and Standards Division of NBS.

4. Data Collection, Reduction, and Results

The strategy adopted involved taking data only on weekends continuously from Friday afternoon until Monday morning, alternating between measurements with

the large masses in place and with them removed. Usually, each of these conditions would extend from 6 to 12

hours. The frequencies for all the individual sets (6 to 12 hours) were determined using the algorithm of J. J. Snyder [11]. These frequencies are squared. The midpoint time of the run was assigned to each run. Each set

of runs was separated into two groups (masses on, masses off) and a linear least squares fit was made of (jT vs time for each group with the constraint that the slopes

of the fits be the same. The implication of this procedure

is that any underlying drift (slope of the fits) affects the runs in the same way (Fig. 3), i.e., is independent of whether the masses are on or off. As a check on the linearity of the system, runs were made using different starting amplitudes of the oscilla- tions. The smaller oscillations had a higher percentage

noise level (as may be expected). However, no difference

in the frequencies using different amplitudes was

detected beyond the normal scatter. The calculation of G was made from the average of the

A(a)2) value listed in Table 1. The final assignment of the uncertainty in this, determination of G is arrived at by the conventional method of the square root of the sum of the squares of the uncer0.15171

0.15170 -

0.15169

0.14673 -

0.14672

200 360

Figure 3. Plot of the least squares fits of the ii? vs time con- strained to equal slopes. (Note break in the scale for ii?.) The

unit of time is hours; of ui , radians per 20 seconds.

Date (1980)

Table 1. Average value of A (os^)

A(co2) Orientation

Aug 29 -Sept 2 Sept 5 -Sept 8 Sept 12 -Sept 15 Sept 19-Sept 22 Sept 22-Sept 26

Oct 3-Oct 4 Oct 8-Oct 10

0.00497866 ne-sw

0.00497837 ne-sw

0.00497816 nw-se

0.00497729 nw-se

let up to air, autocollimator modified realigned 0.00497773 ne-sw

0.00497788 ne-sw

Average: 0.00497801 ± 0.00000020 (40 ppm)

tainties listed in Table 2, and therefore represents a Ict uncertainty.

The resulting value is G = 6.6726 ± 0.0005 x lO^^^ m^ • s" .3 . .-2 . kg-l

5. Discussion

The measurement described here yields significantly

smaller error bars than previous attempts using this method. A legitimate question is: why is this so? This

experiment differs from Heyl's in that it has high vac- uum, high Q, high density of the large masses, high data

volume, and high damping of the pendulous modes. Some

of these factors are interrelated.

575
Table 2. Error budget

Source of uncertainty Uncertainty (ppm)

Position of the large masses 10 Mass of the large masses 1 Length of the small mass 22

Thickness of the small mass 36 Density of the small mass 6 Moment of inertia of mirror 23 A 40 RSS Total 64

Our choice of a quartz fiber which yielded the high Q

came as a result of long trial and error. Experience with

tungsten fibers indicates that it is difficult to achieve Q'

s

greater than about 100. Also, the tungsten seems to have

unpredictable changes in the ambient position of the

small mass system and a drift which gets smaller in time,

but never disappears entirely. Graphite, on the other

hand, has low internal friction (the Q may be several

hundred) but its torsion constant, even after being in vacuum for several months, changes by as much as 1%

per day, and the ambient position seems to be temperature dependent with a coefficient of about 0.1 rad/°C for the fibers we used (6 [x.m diam., 40 cm long). The choice

of a quartz fiber for its obvious advantages dictated high vacuum in order to exploit effectively the low internal

friction of the quartz. This in turn required the installa- tion of the damper.

In any G experiment of this type, the sensitivity is proportional to the density of the material from which

the large masses are formed. Tungsten, which has a density of 19, was used in this experiment; earlier experimenters used materials with densities of less than 10. Heyl did not have the advantage of the sophisticated

computer and data acquisition equipment which was

available to us. Under the circumstances, it is surprising

that Heyl did so well or that we did not do better.

6. Future Improvements

The four largest components of the error budget are the uncertainty in the length and thickness of the discs of the small mass system, the uncertainty of the moment of

inertia of the mirrors, and the uncertainty in the meas- urement of the frequency of the small mass system's

oscillation. We are now constructing a new small mass

system that will be more amenable to precise metrology

than the present one. It is made from tungsten slices lapped flat and parallel to gauge block tolerances (1/10

fringe). These tungsten end pieces will be wrung onto a

piece of pyrex, flat and parallel to the same tolerances. The dimensions of the pieces will be determined before

assembly and then again after being wrung together in order to determine the quality and thickness of the

wringing film. From this sandwich, a cylinder will be

formed with approximately the same dimensions as the

existing one. A small mass system made in this way will have 90% the sensitivity of the present one with the same mass, yet it should be much more rigid and maintain its dimensions through the metrology.

The mirror and stem of the small mass system will be

made much lighter (20% as heavy), thus their contribution to the error will be proportionally less. Finally, the largest contribution to the overall error is the uncertainty in the measurement of the frequency of the oscillation of the small mass system. This uncertainty

will be the most difficult to reduce. In order to minimize

this error, fibers which produce different periods will be

tried. Experience has shown that the present small mass

system, and consequently the new one, may be

suspended on fibers of such small diameter so as to pro- duce periods of up to twenty minutes. The frequency

change due to the large masses in that case will be about

20%.

It is difficult to imagine doing a measurement of this type without the personnel and facilities of an institution

such as the National Bureau of Standards. The following

list is by no means a complete list of all the assistance

given to us. Bruce Borchardt measured the separation of the large masses and did the initial metrology on the small masses; Bob Crosson transposed data; Rich Davis and Randall

Schoonover performed density measurements; R. D. Deslattes suggested the use of the diode array and participated in fruitful discussions; Jim Filliben and Stephen

Leigh provided time series analysis, and general computational assistance; P. Thomas Olsen checked the magnetic susceptibility of the large masses and the material

of the small masses; J. J. Snyder supplied the algorithm

for computing the frequencies; Clyde Tucker performed

metrology on the small mass system.

References

[I] E. R. Cohen and B. N. Taylor, J. Phys. Chem. Ref. Data 2, 663 (1973).

[2] P. R. Heyl and P. Chrzanowski, J. Res. Natl. Bur. Stand. 29,

1 (1942).

[3] C. V. Boys, Phil. Trans. R. Soc. London, Ser. A.: 1, 1 (1895).

[4] G. G. Luther et al., in Atomic Masses and Fundamental

Constants-5, Ed. by J. H. Sanders and A. H. Wapstra

(Plenum, New York, 1976), p. 592.

[5] R. D. Rose et al, Phys. Rev. Lett. 23, 592 (1964).

[6] W. A. Koldewyn, A New Method for Measuring the Newtonian Gravitational Constant, thesis, Wesleyan

University, 1976.

[7] C. Pontikis, C. R. Acad. Sci. Ser. B 274, 437 (1972).

[8] 0. V. Karagioz et al, Izv. Akad. Nauk SSSR, Fiz. Zemli

1976 (5), 106 [Izv. Acad. Sci. USSR, Phys. Solid Earth 12, 351 (1976)].

[9] W. R. Towler et al, in Precision Measurement and Fundamental Constants, Ed. by D. N. Langenberg and B. N.

Taylor, Natl. Bur. Stand. (U.S.), Spec. Publ. 343 (Aug.

1971), p. 485.

[10] J. H. Nash et al, "High-Density Tungsten Spheres," Union

Carbide Corporation Y-12 Plant Document Y-1654.

[II] J. J. Snyder, Appl. Opt. 19, 1223 (1980).

576
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

The Design of a Beam Balance for a Determination of G

C. C. Speake and A. J. F. Metherell

Cavendish Laboratory, Madingley Road, Cambridge CBS OHE, England, U.K.

We are currently engaged in the design and construction of a beam balance which will be used for a precision determination of G. The balance is designed to give a value of G to about one part in 10'^. In

place of knife edges the balance employs a cross-strip pivot for its fulcrum and flexure pivots are used

to suspend the test masses from the ends of its arms. The balance is servo-controlled so that it maintains its position in the horizontal plane to an angle of about 10"^ rad. The attracting masses are in the form of rectangular slabs with rectangular holes at their centers. This shape ensures that the test masses suspended from the arms of the balance lie in regions of space where the gravitational field due

to the slabs is highly uniform. A precise knowledge of the center-of-mass to center-of-m,ass distances is therefore unnecessary for a precision determination of G with this device. Key words: beam balance; cross-strip pivot; Newtonian constant of gravitation; precision measurement.

1 . Introduction

Apart from the very earliest measurements of G, which

depended on finding the difference between the astro- nomical and geodetic latitudes on the north and south

sides of a suitably surveyed mountain, all other measure- ments of the constant of gravitation have involved the use of either the torsion balance or the beam balance.

The torsion balance has been used either in the form of a

deflection instrument, as, for example, in the work of Cavendish [1], Boys [2], and Rose et al. [3], or in the

form of a torsion pendulum, as exemplified by the experiments of Heyl [4] and Zahradnicek [5]. This versatility

has meant that the torsion balance has played a dominant

role in the measurement of G, whereas the use of the

beam balance has been reported only three times in the

literature [6], [7], [8], and these experiments were con- ducted over a period of less than twenty years towards

the end of the 19th century. There are, however, no

physical reasons for supposing that the beam balance is intrinsically less accurate than the torsion balance in detecting and measuring the small forces encountered in a determination of G. It is clear, given all the advantages

of modern technology, that a great deal of scope exists to up-date and improve the beam balance method. We are

currently involved in the construction of a beam balance

designed for a precision measurement of G; the purpose

of this paper is to describe the salient features of our method and to report on the progress so far made.

2. The Principles of the Beam Balance

Method

The balance and its associated masses are shovm

schematically in Fig. 1. The arm lengths t'l and 62 (Fig. 2) are nominally equal vdth t'l = C2 — 0.3 m and the

test masses m\ and m2 suspended from the ends of the arms are also nominally equal wdth mi = m-z — 1 kg.

The attracting masses are in the form of a pair of rec- tangular parallelepiped slabs with linear dimensions a —

0.05 m, b — 2.0 m, c — 1.0 m (Fig. 1) and separated by

a distance d = 1.0 m as measured from their geometric

centers. Each slab has a rectangular hole cut in it with dimensions a ' = a, b' - 0.175 b and c

' = 0.7 c thus ensuring

4/^

Figure 1. Schematic arrangement of balance and attracting masses.

that mi and m2 sit in regions of space where the z components of the force possess extrema in the x, y, and 2 directions which are coincident [9]. The abandonment of the traditional spherical shape for these masses in favor

of the more complicated mass distributions chosen here means that the center-of-mass to center-of-mass dis- tances between the test masses and attracting masses

are no longer quantities which have to be determined

with great precision. The slabs produce gravitational

fields in the regions of space near m 1 and m 0 which are highly uniform and hence these test masses do not have

to be positioned vdth any great accuracy. Indeed our cal- culations show [9] that for a positioning error in mi or m2 as large as 0.1 mm G can be determined to within a few parts in 10^ provided that all other sources of error are negligible.

577
There is no need to remove the slabs from the vicinity

of the balance in order to reduce the gravitational torque

acting on it to zero, as is required for calibration purposes, since this can be achieved by the simple expedient

of moving the slabs vertically through an appropriate dis- tance. This arrangement therefore avoids problems asso- ciated with floor tilt since the mass distribution on the

floor remains unaltered as the slabs are raised.

The effect of the rectangular slabs is to produce a force

Fgi which acts upwards on mi and a force Fg2 which acts downwards on W2 and the resultant gravitational couple on the balance is cancelled by applying a voltage to an

electrostatic transducer of the cylindrical capacitor type

described by Durand [10]. This transducer produces the

force Fe i shown in Fig. 2. The transducer voltage is used

to servo-control the balance and maintain it in a hori- zontal position throughout the measurements. This ar- rangement facilitates error analysis because the voltage

readings can be read out continuously during each ex- perimental run. When the gravitational couple is balanced by the elec- trostatic couple we have

Fei ^1 - Fgi^I + Fg2 ^2

or Fe\ = Fg\ + /^g2-^ = G {ximi -I- X2'W2-^},

where G is the constant of gravitation and xi and X2 are geometrical factors which depend on the positions and

shapes of the attracting masses used in the experiment.

The electrostatic force Fe\ can be determined from the

geometry of the transducer, its capacitance, and the

apphed voltage, leaving only the ratio Ixliz to be determined. The factors xi and X2 are not sensitive to the positions of the test masses mi and m2 so that their positions

do not form a part of the precision measurements re- quired to determine G. The ratio 61/^2 does however have to be determined to a high precision and the way this measurement is achieved is as follows: The rectangular slabs are raised

into the position of zero torque (see above) and a second

transducer is used to apply a force Fe2 to the mass m2

which is then balanced by the first transducer. The result

is that the ratio Fe\IFe2 immediately gives i\lt2. Again

servo-control is employed so that the ratio i\li2 is monitored viithout altering the horizontal position of the

balance.

The quantities which have to be measured with high

precision are (a) the masses mi and m2, (b) the masses of the rectangular slabs, (c) the dimensions of these slabs,

(d) the dimensions and capacitances of the electrostatic

transducers, and (e) the voltages applied to these trans- ducers. To determine G to one part in 10^ requires all these quantities to be determined to within a few parts in 10^ and none of the above represents any special difficul- ties in this respect. The quantities which do not need to be known with any real precision are the positions of the

test masses relative to the rectangular slabs. This should

represent a real and marked improvement on all other

determinations of G so far reported in the literature

since all have required a precise knowledge of the

center-of-mass to center-of-mass distances between the

test masses and the attracting masses involved.

3. The Mechanical Design of the Balance

The overall length of the beam balance is restricted to about 0.6 m as a compact balance is a necessity when one

is faced with the problem of controlling the temperature

of its environment accurately and eliminating as far as possible undesirable thermal effects caused by temperature gradients. To this end the balance arm is con- structed from carbon fiber reinforced plastic which has a very low thermal expansion coefficient (-0.5 x 10"^ K"^

in the direction of its length). This material has the added advantages of low density, extremely high rigidity

and meets the essential requirement of being non- magnetic. The balance, together with its test masses, is to be operated inside a high-vacuum chamber to avoid

problems arising from atmospheric convection currents and Brownian noise. In addition to insulating the balance

as far as possible from thermal noise it may be necessary

to float the apparatus because mechanical noise vdll manifest itself in motions of the horizontal reference plane

defined by the fixed capacitor plates (Fig. 2). Noise at the pivot does not however couple to the rotation of the

beam and hence sophisticated anti-vibration mountings

are not required.

To avoid any problems arising from mechanical hysteresis, the arms of the balance will be allowed to rotate by only a very small amount (very much less than 10~^

rad.) This will be achieved by the use of servo-control to maintain the arms of the balance in the horizontal

position.

The balance employs a cross-strip pivot (Fig. 2) instead

of the conventional knife-edge for its fulcrum. The upper

plate A (Fig. 2) of the pivot is held rigidly to the laboratory bench and the balance arms are attached to the

lower plate B. The crossed-leaf springs of the pivot hold

A and B together but allow plate B to rotate about the

horizontal axis passing through the point P marked in Fig. 2. Detailed studies of this device have been published by

Haringx [11]. As far as we have been able to determine

its mechanical properties make it ideally suited for our purposes and some of its more important properties are

as follows: Firstly, its operation depends only on the

elastic deformation of the cross-springs, thus completely

eliminating the problem of sticking which ultimately limits the sensitivity of a knife-edge fulcrum. Secondly, pro- vided the angular rotation of the beam is small (as in our

case) the axis of rotation is particularly well defined [12].

A

< 1 D><1 7—

/

>

LJ (_l

1

/

/

B

"G2

Figure 2. The beam balance.

578
Thirdly, when a tension is applied to the pivot, the re- storing couple per unit angle of rotation decreases in pro- portion to the applied load until it reaches zero and the

beam becomes unstable. By choosing appropriate values

for the masses mi and m^, the natural period of the device can therefore be made very large and the balance

very sensitive.

The cross-strip pivot, which at present employs Be-Cu

strips for its elastic members, does however suffer from

the follovdng problehi: It is obviously desirable to per- form all measurements at room temperature and in order

to measure forces substantially smaller than those

corresponding to thermal motion at room temperature it appears that the balance should have a long relaxation

time [13]. Measurements made with angles of rotation of

about 10"^ rad indicate that the damping due to internal

friction in the Be-Cu strips is too large for our purposes.

However for the very small oscillations the beam is designed to cope vdth (—10"^ rad), the damping parameter might be very much smaller [14] than the value

obtained at an amplitude of 10"^ rad. We are currently

extending our measurements to these very much smaller

amplitudes and if the damping still proves to be too large our intention is to replace the Be-Cu strips with elastic members fabricated from fused siHca. The test masses are suspended from the rigid arms of

the balance by means of flexure pivots (Fig. 2). The axis

of rotation of this pivot is well defined provided that the

tension of its elastic member is kept constant [15]. In our

case mi and ma are fixed and the tension is therefore

kept constant to an extremely high level. We are confi- dent that with fixed loading and with proper temperature

control of the balance's environment the arm lengths £i and ^2 will remain constant during any given experimental run to a very high degree of precision as is re- quired for a measurement of G by this method.

4. Electronic Stabilization of the Balance

The electronic control of the balance is indicated

schematically in Fig. 3. A differential capacitance trans- ducer of the type described by Jones and Richards [16] is used to detect the motion of the beam. The off-set cur- rent produced by an imbalance between two small sensor

capacitors flows from the center tap of the secondary

windings of the bridge transformer (hand wound, ferrite pot core) through another transformer which acts as an

input-impedance matcher for a low-noise amplifier. The

beam is earthed. The oscillator produces approximately

10 V peak to peak at a frequency of 20 kHz. The control

system exerts precisely zero force on the beam when it is exactly balanced. The polarity of the dc voltage output of the phase-sensitive detector depends on the sense of dis- placement of the beam relative to the horizontal plane.

The micro-computer returns voltages to the electrostatic

transducer which applies the force Fe\ (Fig. 2) in such a way as to critically damp the motion and return the beam

to its horizontal position. It is very likely that the elec- tronic system is more than adequate for our requirements and we hope to restrict the motion of the beam, to angles less than about 10"^ rad. 5. Summary

With the device described in this paper it is hoped to determine G to an accuracy of about 1 part in 10^. The

Figure 3. The electronic control system.

measurements wA\ ultimately involve us in a comparison

vdth the kilogram, the meter, and the volt, and our aim

is to design a beam balance which will yield a value for G

whose accuracy is limited to the accuracy vdth which

these three comparisons can be made. Our design studies

indicate that the problems associated with achieving this aim, although considerable, are not insurmountable.

The authors are indebted to Professor A. H. Cook,

FRS for many useful discussions and for his continued in- terest and encouragement.

References

[1] H. Cavendish, Phil. Trans. R. Soc. London 18, 469 (1798).

[2] C. V. Boys, Phil Trans. R. Soc. London, Ser. A: 186, 1 (1895).

[3] R. D. Rose et al. Phys. Rev. Lett. 23, 655 (1969).

[4] P. Heyl, J. Res. Natl Bur. Std. (U.S.) 5, 1243 (1930).

[5] V. J. Zahradnicek, Phys. Z., 34, 126 (1933).

[6] J. H. Poynting, Philos. Trans. R. Soc. Ser A: 182, 565

(1891).

[7] Ph. Von Jolly, Abh. Bayer. Akad. Wiss. (1898).

[8] F. Richarz and 0. Krigar-Menzel, Anhang zu den

Abhandlung (1898).

[9] A. J. F. Metherell et al., these proceedings.

[10] E. Durand, Electrostatique (Masson et Cie, Paris, 1966), Vol. 2, p. 227.

[11] J. A. Haringx, Appl. Sol. Res. Sect. A: 1, 313 (1947).

[12] R. V. Jones, J. Sci. Instrum. 193, 39, (1962).

[18] V. B. Braginsky and A. B. Manukin, Measurement of Weak

Forces in Physics Experiments (University of Chicago

Press, 1977).

[14] R. V. Jones, private communication 1981. tl5] F. S. Eastman, Bulletin, University of Washington, Engineering Experiment Station Series Bulletin No. 86 4, (1933).

[16] R. V. Jones and J. C. S. Richards, J. Phys. E 6, 589, (1973).

579

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Optimizing the Shape of the Attracting Mass in Precision l\/leasurements of G

A. J. F. Metherell, C. C. Speake, and Y. T. Chen

Cavendish Laboratory, Madingley Road, Cambridge CBS OHE, England, U.K. and

J. E. Faller

Joint Institute for Laboratory Astrophysics, University of Colorado, Boulder, CO 80309

It is now recognized that the sphere is not necessarily the best shape to use for an attracting mass in a precision measurement of G. Two shapes which minimize the effect of errors in the center-of-mass to center-of-mass distance between the attracting mass and the test mass are (a) the right-circular hollow

cylinder and (b) a pair of rectangular parallelepiped slabs with rectangular holes at their centers. The

Idnematic factors influencing the design of these mass distributions are considered and the optimum

design parameters that should be employed with attracting masses of these shapes are determined.

Key words: attracting masses in gravitation; constant of gravitation, G; gravitation; gravitational at- traction; Newtonian constant of gravitation; Newtonian gravitation; optimum mass geometries.

1. Introduction

Since the first precision measurement of G was re- ported by Cavendish [1] in 1798 a further thirty or so determinations have appeared in the literature. In nearly

all these cases a spherical shape was chosen for the at- tracting mass and only during the last decade has it been

recognized [2, 3, 4] that this shape is not necessarily the

most suitable for a precision measurement of G. The major drawback of the sphere is this: If the center-of-mass

distance between the attracting and attracted masses is uncertain by an amount Ar, then to the first order in small quantities, the error AF^ in the measured force F^

between these masses is given by IaF^ /FJ = 2A r/r. The

ultimate precision to which G can be determined with a

spherical attracting mass is therefore limited by the ac- curacy Ar to which its center of mass can be located. Recently it has been recognized [2] that the field along

the axis of a hollow cylinder (Fig. la) possesses extrema

(Fig. lb) in the regions of space close to the end faces of the cylinder and that the quantity IaF^/F^I for z — Zm

vanishes to the first order of small quantities. Clearly

this property is not unique to the cylindrical shape; al- most any solid with a hole cut through it will possess

small regions of space in which the gravitational intensity

is highly uniform. Once the general shape of the attract- ing mass has been decided on, it is obviously desirable to optimize its design to ensure that the regions of uniform

intensity are as large as possible, which means that the

extrema should be as flat as possible. In this paper we report on an investigation aimed at finding the optimum design parameters for attracting masses of two quite different shapes; these being (a) the

right-circular hollow cylinder and (b) a pair of parallel

rectangular parallelepiped slabs with rectangular holes at their centers as illustrated by Fig. 3. The former has

been used by Long [4] in a test of the inverse square law

and the latter are to be used in a beam-balance method

[5] for determining G. Although our investigations have

Figure la. The right circular hollow cylinder.

o z

Figure lb. The variation of the axial component of the gravitational field with the axial coordinate z.

included a study of the effects of density inhomogeneities

and machining errors on the ultimate precision to which

G can be determined when attracting masses of the above

shapes are employed, in the interests of brevity the

results of these studies will not be included in this paper

but instead vdll be published elsewhere [6]. Here we ^^^ll concern ourselves exclusively v^ith the kinematic factors which determine the optimum design parameters; this means that throughout this paper it is assumed that the

581
materials from which the attracting masses are made are

of constant density, and furthermore that these masses

do not suffer from any machining errors.

2. The Right-Circular Hollow Cylinder

2.1 The Gravftationa! Field of a Hollow Cylinder

A closed analytical expression for the gravitational

field of a hollow cylinder does not appear to have been

published anywhere in the literature. Heyl [7] has published numerical values for the coefficients of the first eleven terms of a zonal harmonic expansion of the potential function of a solid cylinder. The resulting formula for the gravitational field is of open form and cumbersome to say the least. In principle the problem of finding the potential function for either a hollow or sohd cylinder in closed form is simple enough; in practice however the cal- culation is lengthy and tedious and a full ' treatment of

this problem is to be published elsewhere [8]. However,

since the general solution to this problem is one of some

importance we summarize the results of the calculation

by quoting in the appendix at the end of this paper the

relevant expressions for the field of a hollow cylinder.

2.2 The Paraxial Approximation

The expressions for the field intensities are very much

simplified if the point {z,y) lies close to the axis of the

cylinder. In this case the axial component of the intensity can be obtained by first noting that at the point iz,y)

.2tt

xr

y,2 _|. y2 2ry cos 0

)

3/2 dQ dr dx

where p is the density of the cylinder and r, 9, and x are

the coordinates indicated in Fig. la. On expanding the

denominator of the integral and neglecting terms involving powers of y higher than the second we obtain after integration

F, = 2ttGp

+ rf

A

1

B - C + D

1

1 A31 53

where A, 5, C, and D are the distances indicated in Fig. 2, Q being the axial point with the same value of z as the point P. This equation is a good approximation to

Fj, when y is small and is of course exact for all axial points {y = 0).

2£

2.3 The Optimum Shape for the Cylinder

A detailed analysis [6] based on the paraxial approximation shows that in the region of the point {Zm, 0), which lies on the axis of the cylinder and is the point

where IF^I is a maximum in the z direction, the force If^I

is a minimum in the radial direction. The point z^ is therefore a stationary point in both the axial and the radial directions and hence the field is constant to the first order of small quantities for small displacements in any

direction about z,„. Furthermore, the volume of space over which the field is effectively constant increases in size [6] if either the length of the cylinder is increased or

if the inner and outer radii are allowed to approach one

another. The optimum design for an attracting mass of

this shape is therefore one for which the length of the

cylinder is made as long as possible and the wall thick- ness as thin as possible, compatible of course with the re- quirement that sufficient mass of material be present in the cylinder to produce the required gravitational attraction. Finally it is worth noting as the length of the

cylinder is increased the point z,„ moves inwards towards

the face of the cylinder until in the semi-infinite limit it lies in the plane of the face.

3. Rectangular Slabs

We are currently involved in the design and construction [5] of a beam balance which will be used for a precision measurement of G. The arrangement of masses used

with this device is illustrated schematically in Fig. 3, where Ai and A2 are attracting masses in the form of a

pair of parallel rectangular parallelepiped slabs with

holes cut in them as shown in the figure. The gravitational couple of the slabs on the balance arms is zero whereas a couple is exerted on the test masses m

(Fig. 3). The balance is servo-controlled and its position

Figure 2. A cross-section of a hollow cylinder shov/ing the dis- tances A, B, C, and D of the paraxial approximation.

Figure 3a. A schematic diagram of the beam balance and its associated masses.

582
Om

Figure 3b. A cross-section in the plane x = 0 containing the balance.

where x, y, z, and d are defined in Fig. 3a, and it being

understood that

ai = ai = a if m = 1 or 3, or a' if m = 2 or 4 as = a2 = b if m = 1 or 3, or 6 ' if m = 2 or 4 = az = c if m = 1 or 3, or c ' if m = 2 or 4, where a, b and c are also defined in Fig. 3a.

in the horizontal plane is maintained by applying an equal

and opposite electrostatic couple (see Ref. [5] for further

details).

3.1 The Gravitational Field of the Rectangular Slabs

The potential function of a homogeneous rectangular

parallelepiped can be found in Ref. [9] and it is a simple

matter to use the expression given there to determine

the field at any point {x, y, z) in the vicinity of the slabs

illustrated in Fig. 4. The component of force in the

Cartesian coordinate direction x„ is given by "

= ^i-ir^\F,jm

m =1

where

iFx^)m^Gp

+ (xi+i - (-l)-'a;+i)8i,„)

n 2 2 2

l=n-\ i=l i=\ k =1

k=\ i=l ;=1

X tan

-1 (a^n+i - (-iya„+i)(a;„+2 - (-lFa„+2)

In this expression 8„,; is the Kronecker delta and

e^r.s = [Up - (-l)9ap)2 + (xp+i - (-l)'-ap+i)2 + (a^p+2 - (-lrap+2)']^^

it being understood that

X4 = Xi =x + d if m= 1 or 2, or a: -d if m = 3 or 4

^5 = ^2 =

Xg = X3 = z.

Figure 4. The variation of \Fz\ vnth z, y, and x. The broken

curve represents the variation obtained with solid slabs whereas the full curves represent those obtained with the slabs of Fig. 3a.

3.2 The Optimum Shape for the Rectangular Slabs

Insensitivity to position of the test masses can be

achieved provided that stationary points exist for the z component {F^) of the gravitational field (the x and y components produce zero torque)^ and provided that

these points can be made coincident vdth the positions of the masses m (Fig. 3) which lie in the plane x = 0. The

variation of IFJ as x, y, and z are varied is illustrated schematically in Fig. 4. If solid slabs (that is slabs without rectangular holes in them) are used then IF^I possesses two stationary points (S in Fig. 3b) in the

plane x - 0 and these points have extrema in all three

coordinate directions. For slabs with rectangular holes of suitable dimensions there are four stationary points (situated at m and m' in Fig. 3b) in the plane .r - 0 and these

also possess extrema in all three coordinate directions.

Slabs which possess this property are the ones most suit- able for the beam-balance experiment.

detailed analysis shows that when, as in our case, the test masses

are pivoted, the torque due to the y components is not identically equal to zero but is nevertheless negligible; not only are the torques due to the y

components acting in opposition and therefore subtractive but they also vanish to the first order of small quantities.

583
To determine the optimum shape of the rectangular

slabs the following procedure was adopted: Firstly the

dimensions of the slabs were chosen arbitrarily in the sense that a, b, c, and d (Fig. 3) were fixed by having re- gard only to the overall size of the apparatus. Secondly

the dependence of IFJ on the three coordinate directions was investigated (by numerical methods) as a function of

the dimensions a', b' , c' of the rectangular hole until the

optimum shape was determined. We have found that for slabs with dimension a = 0.025, c = 0.5, d = 0.5, rela- tive to b =1.0, the rectangular hole should have a' - a, b - 0.1756 and c' - 0.356. With these dimensions the

stationary points occur at x = 0, y = ±0.27396, z =

±0.66776, these being the positions at which the test masses should be placed if maximum insensitivity to positioning errors is to be achieved.

The work on the hollow cylinder described above was

done in 1973/4 while one of us (AJFM) was a Visiting

Fellow at the Joint Institute for Laboratory Astrophysics

and he wishes to record his deep appreciation to the Fel- lows of JILA for allowing his participation in this work

to be made possible.

Appendix: The Gravitational Field of a

Hollow Cylinder

The coordinate system chosen for the problem is illus- trated in Fig. 2, the radial coordinate being denoted by y

and the axial coordinate by z. The inner and outer radii of the cyhnder are rj and r2 respectively, its density is p and its length is 21' . The origin of the coordinate system

is chosen so that the planes z = ±C are coincident with

the end faces of the cylinder. We denote the binomial

coefficients by "C,- and define the following terms; C, =

(t^i^ " y^)^^, where i = 1 or 2, unless y ^ r, in which

case C, =0,

r„ =

yn,m

and

Inl

{2(2n - m))\

22''m!(2w - m)\{2(n - m))\

Observing the convention we take 0! = O" = 1. The components of the field intensity in the axial and radial directions at the point {z,y ) are then given by

Fy ^ Fy^"^ + Fy^"^ + Fy^'^ + Fy^^^

,

where Fj*"' = Fy^"-^ = 0 ^mless both z - f < Ci and z +

i > - C 1 in which case

F,(«' = 2ttGp X (-1)' n S i =1 « =1 n

2n m =0

p=0

F<»' = 2.GpX(-lVn Xjl^X i=l n =1 'i m=l

i-iryn.m X 2.p^Cj,y 2p [{Z +

p2(»-p)+ljC2

2{n - p) + 1

the limits and ^2 being given by

ii = -t if z - t > - Ci or - - 2 - Ci if 2 - e ^ - Ci;

^2 = K if z + t < Ci or & = - 2 + Ci if z + C > Ci

and where Fj"'^ = F^^"*' = 0 unless either z + t > C 2 or

z - ( < - C 2 in which case

i=l n=l m=0

i-iry^niz + if + y^) 2 ]^2

\ n -m -1

p=0

F/' = 27tGp i (-DV; i 2Tnn^--'X

1=1 71=1 TO =0

(-ir?/2'«[mi/-i/(„+„-i,+ 172

- in +m - ^)yl{n+m)+v2\l^

n -TO -1

p=0

where

9 +

2

(ti - 1)

in which

(2 + 0

yH2q - 1)((2 + if + 1/2)9-1^2

^ = 2(^ - 1) y\2q - 1)

'

the limits being

ii^-i if 2 - e >C2orif 2 - e <-C2

or

^l = -2 -C2 if C2^2 -e >-C2

and

i2^i if 2 6 > C2 or if 2 + e < - C2

or C2--2 +C2if C2&2 +t ^-Cz

and where Fi"^^ - Fj"^' - 0 unless both z - I < C2 cmd

z + t > C 1 in which case

P^ic) = p^(a)^i = 2 only) + F,<'" (i = 1 only) - 2ttG p[2 +

Q^f

and

Fy(c) = ^ 2 only) + Fy^^Hi = 1 only)

the limits being

ii^-t if z - t >Ci or ^1 = -2 +Ci if 2 - e ^Ci

and

& = e if z + t < C2 or ^2 = - 2 + C2 if 2 + e ^ C2

584
and where F}^^ = F/^ = 0 unless both z - I < - C\

and z + t >-C2in which case Fj-^^ and Fy^'^^ are given

by the equations above for Fj-'^ * and Fy^'^ ^ except that the

limits are = if z - I > - C2 or ^1 = - z - C2 if z - f ^ - C2

and

L2=t if 2+6 <-CiorC2=-2-Ciif2+e ^-Cj.

References

[1] H. Cavendish, Phil. Trans. R. Soc. London 88, 469 (1798).

[2] M. Hulett, Senior Thesis, Wesleyan Univ., Conn. (1969).

[3] D. R. Long and D. Ogden, Phys. Rev. D 10, 1677 (1974).

[4] D. R. Long, Nature 260, 417 (1976).

[5] C. C. Speake and A. J. F. Metherell, these proceedings.

[6] A. J. F. Metherell et al. to be published in Metrologia.

[7] P. Heyl, J. Res. Natl. Bur. Stand. 5, 1243 (1930).

[8] A. J. F. Metherell et al. to be published in Phil. Trans. R.

Soc. London.

[9] W. D. Macmillan, The Theory of the Potential (Dover Publications Inc., New York, 1958).

585

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

i

Vacuum Polarization and Recent l\/leasurements of the Gravitational Constant

as a Function of Mass Separation

Daniel R. Long

Department of Physics, Eastern Washington University, Cheney, WA 99004

We point out that the coulomb inverse square law has long been known to fail at small charge separations due to a vacuum polarization effect. We point out that gravitational inverse square law data should be analyzed for a vacuum polarization effect and present the QED result for the mass density shift of the vacuum due to an inducing mass. We point out that care must be used in analyzing the data because the

point mass situation is very different from the case where one of the masses is quite extensive. We

analyze the current data for the value of the vacuum polarization parameter X. Much of that data has er- rors too large to give definite results although it tends to agree with my earlier result of X. = 0.002. Of the more precise results, Stacey and Tuck are in agreement with my value while Spero et al. disagree. It is pointed out that the vacuum polarization analysis of the Spero et al. result is ambiguous.

Key words: gravitation; gravitational inverse square law; inverse square law failure; non-Newtonian

gravitation; vacuum polarization.

1 . Introduction

It is surprising that the gravitational inverse square

law w^as not experimentally investigated many years ago

since the coulomb inverse square lav^^ has long been

knovi^n to fail at small charge separations. The coulomb

inverse square law fails by a very general process known

as vacuum polarization. By this process, a source of force

polarizes attracted charges out of the vacuum and builds

up a charge density around itself. Just as a negative elec- tron accumulates a cloud of positron density around it, a mass can attract a cloud of positive mass density around

it. This mass density cloud around a mass causes the in- verse square law to fail because an approaching mass

sees less attracting mass as it penetrates the cloud. The idea of a mass density in the vacuum near a mass

seems strange; however, if the particles in such a cloud

do not interact except gravitationally such a mass density shift of the vacuum would be very difficult to detect. Indeed, there could be a large non-interacting mass density of the entire vacuum and it would be undetectable

since it would pull gravitationally equally in all directions. Hence, the mass density of the vacuum we are speaking of might be regarded as a small shift in an al- ready large mass density of the vacuum.

2. Theory

The mass density shift of the vacuum 8 p (r) has been

calculated from quantum field theory by Uehling [1] and

by Schwinger [2] and is given by

1.410 + log

R*

|r - r I

- r'\« R'

V2p(r')rfr' .(1)

Here, p(r') is the external, zeroth order mass density

distribution, R' is the Compton wavelength of the fer- mions which couple to the graviton, and X is the interaction strength.

When Eq. (1) is applied to a small, compact, mass it yields a mass density shift of the vacuum which drops off

inversely as the cube of the distance away from the mass. When the force, F, between two such masses, Mi

and M2, is calculated from Eq. (1), the result is

F{R)

GiR)

G{R )MiM2

R^

Go\l + Mog R_

R*

(2)

(3)

where R is the mass separation. Go the gravitational con- stant at /2 = R*, and X is the same fermion-graviton cou- pling constant as used in Eq. (1). It is also a result of

quantum field theory that the induced mass density of

the vacuum drops exponentially to zero for R > R *. Hence, the inverse square law holds perfectly at large mass separation, a result in accord vdth astronomical

observations.

It is important to understand clearly how to apply

Eq. (3) to gravitational data in order to test that data for vacuum polarization effects. By differentiating Eq. (3), we find that

dG {R)

(4)

X - (5)

Go dilogR) '

and so it is attractive to assume that X is simply the di mensionless logarithmic derivative given by

d (log GiR))

dilogR)

•

For the purpose of evaluating X from G iR ) data taken at two different mass separations R2 and Ri, R2 > Ri, we

can use the following expression as long as GiR) only

changes by a small percentage:

GiR2)-GiRy)

(1/2)[G(«2) + G(/2i)]log

R 2 (6)

It must be greatly emphasized that Eqs. (i), (5), and

( 6) only apply ivhen the attracting masses are small ayid

compact compared to the distance between them. When

one of the masses is very large and extensive, Eq. (1) must be used directly.

587
3. Comparison with Experiment

There now exist a number of sets of data involving

compact masses from which values of \ can be inferred.

Unfortunately most of these experiments quote errors too large to unambiguously support the claim that X. differs from the Newtonian value of zero. On the other

hand, the general trend of the data suggests that \ —

0.002 and hence that more careful experiments should be

done.

In 1974, I published [3] an analysis of the historic measurements of G at various mass separations. A least squares fit to that data gives a historic value for A, X (Historic) = +2.11 x 10"^ ± 0.70 x 10-^

where the error is one standard deviation. In 1976, I reported [4] a measurement which compared

the value of the gravitational constant at a mass separation 29.9 cm with the value at a mass separation of 4.48 cm. The possible systematic errors in that experiment were carefully assessed. The resulting value for A, was

\(Long) = +1.95 X 10-=^ ± 0.4 X 10-3 where the quoted error contains both the random and

systematic errors. Also in 1976, a Russian group, 0. V. Karagioz et al.

[5], happened to measure G at a number of different mass

separations. While they did not note the fact, their best

and most complete data yielded a X value of X (Karagioz) = +2.015 x 10"^ ± 0.61 x 10-^

where the given error is one standard deviation. In 1979, Panov and Frontov reported an experiment to

explicitly check out my result for X. Their experiment

measured the growing excitation of a torsion balance as the attracting masses were removed back and forth at the fundamental period of the torsion pendulum. In the

part of the experiment where they compared the value of the gravitational constant at a mass separation of

295.8 cm with that at 42.06 cm, they found the value of X X (Panov, 1) - +1.54 x 10"^ ± 3.07 x lO'^

They also used a 600 kg mass to compare the value of G

at a mass separation of 984.0 cm with that at 42.06 cm

with the resulting value of X X(Panov, 2) = -0.63 x 10'=^ ±4.12 x 10"^

It is evident that the errors quoted for both of these X values are too large to really check my result. However,

considering the problems of floor tilt and the precision lo- cation of a 600 kg mass, it seem clear that their first value, X(Panov,l) is to be preferred. That value is in good agreement with my result. In 1980, Hirakawa et al. [7] used a 1400 kg aluminum

gravitational antenna to measure the near wave amplitude due to a rotating bar mass at various mass separations. We can infer the following values of X from their

work.

X(Hirakawa, 1) = +77.4 x 10 ± 172 x 10

X(Hirakawa, 2) = +12.9 x 10"^ ± 93 x lO'^

X(Hirakawa, 3) = + 1.1 x 10"=^ ± 75 x 10'=^

X(Hirakawa, 4) = + 3.9 x lO"'^ ± 49 x 10"^

As can be seen, their stated errors are far too large to

really test my value of X; however, their two most accu- rate data tend to agree with my result. In 1979, Yu et al. [8] measured the force due to gravity near a large oil tank when it was full and when it was

empty. In this way, they obtained a ratio of their value

of the gravitational constant at a mass separation of some

30 m to the known value found by experiments at a mass

separation of 13 cm. The fact that their attracting mass was so large compared to the mass separation makes this case a borderline case which probably should be treated

using Eq. (1). We will, however, treat it as a point mass

situation and calculate X using Eq. (6). Their resulting

values for X are:

X(Yu, 1) X(Yu, 2)

X(Yu, 3) X(Yu, 4)

+ 1.80 X 10-3 ± 21.8 x 10-3 + 3.54 X 10-3 ± 34.7 x 10-3 + 15.2 X 10-3 ± 41.3 X 10-3 + 52.1 X 10-3 ± 60.6 X 10-3

Again we find that the stated errors are too large to test

for my value of X, but their best values tend to agree

with my results. The most unequivocal support for my value of X comes from the geophysical work of Stacey and his collaborators. Stacey et al. [9] measured the force due to the

earth's gravity at different points along a vertical mine

shaft about 1 km deep. By precisely measuring the density of the surrounding rock they inferred a value for the

gravitational constant at a range of about 0.5 km of G{R ^ 0.5 km) - 6.712 x 10-^

± 0.025 X 10-« iZIiecm! .

Furthermore, Stacey and Tuck [10] have examined all of the geophysical measurements of G available in the

literature. All of that data also gave large values for G

and the best value, that of McCulloh, was nearly identical

to the one measured by Stacey et al. A value of X can be inferred from the Stacey and Tuck

result by comparing that value of G with the authoritative result of Heyl and Chrzanowski [11] measured at a mass separation of 13 cm. However, in this case the at- tracting mass is immense and completely surrounds the

gravimeter attracted mass. In this case Eq. (1) must be

used to find X. Using Eq. (1) we find that the vacuum polarization mass density shift of the vacuum within the

1 km thick rock slab is very small and negative (tending

to reduce the observed G). The vastly dominant extra

force arises from the attraction by the rock slab of the in- duced mass cloud coupled to the gravimeter mass. The

compton wavelength of the fermions which couples to the

graviton, R enters into the calculation as a free parameter. For a value of * = 349 km the results of Stacey

and Tuck are in complete agreement with my work:

X(Stacey) = +2.00 x 10-3 ± 1.2 x 10-3.

It should be emphasized that the effects of the parameter

R* are small (-30%), and that the Stacey and Tuck

result essentially agi'ees with my work irrespective of the precise value of R*.

The paper by Spero et al. [12] presents the only result which, at first glance, appears to explicitly disagree with

my value for X. That group investigated the gravitational field inside of a long, uniform, cylinder which

should vanish according to Newtonian gravitation. They

found results which were consistent with Newtonian

588
gravitation to high accuracy. This null experiment involved an attracting mass which

is very extensive and even surrounds the attracted mass. Hence, we must go directly to Eq. (1) in order to deduce

a value for \. Unfortunately, it appears that Eq. (1) is too simple to follow such a complex topology. To illus- trate this point we shall consider a null experiment using

a spherical shell of mass rather than a cylinder. I have

explicitly integrated Eq. (1) for the case of a solid sphere. Outside of the sphere we obtain an inverse cubed

distance dependence of the induced, positive, mass density. Inside of the sphere we find the surprising result that the induced mass density of the vacuum is large and

negative. Now if we cut a hole in the center of the sphere we really do not change the inducing gravitational fields very much so we might expect that Eq. (1) would give us

a negative induced mass density at a point inside such a

hole. Such a negative induced mass density would tend to explain the Spero et al. result. Contrary to these physical expectations, Eq. (1) gives

a positive induced mass density in the hole in the sphere.

Physical intuition is offended even further by a detailed

study of how Eq. (1) gets this positive mass density in the hole. It turns out that for points in the hole Eq. (1) acts just as if it were at a point outside of the sphere.

This is very troubling because we would generally expect

the polarization of the vacuum to depend on the nearby

gravitational fields and the fields inside a hole in a sphere are very different from the field outside of a sphere. The fact that Eq. (1) "thinks" those two physical situations are identical tends to suggest that it is not

a sufficiently good approximation to deal with the spherical shell case or the cylinder used by Spero et al. At present, I am working to try to resolve this paradoxical behavior of Eq. (1) by expressing the induced mass density shift in terms of the nearby gravitational

fields, but this work is not finished. For the present we

shall use the point mass Eq. (6), to calculate a value of \

for the Spero et al. measurement and include a question

mark with the number to indicate the above ambiguities.

Thus,

\(Spero) = +0.01? x 10"^ ± 0.97? x IQ-^

The foregoing discussion presents all of the data re- lated to X that I am aware of. These results are summarized in Table 1, and tend, I think to indicate a non-zero

value of A.. References

[I] A. E. Uehling, Phys. Rev. 48, 55 (1935). See also D. R. Long,

II Nuovo Cimento 55B, 252 (1980). Equation (1) has been

modified for the gravitational case by the substitution o =

-37; \/4.

[2] J. Schwinger, Phys. Rev. 75, 651 (1949).

[3] D. R. Long, Phys. Rev. D 9, 850 (1974).

[4] D. R. Long, Nature 260, 417 (1976).

[5] 0. V. Karagioz, V. P. Isamaylov, N. I. Agafonov, E. G. Kocheryon, and Yu. A. Tarakanov, Izv. Adad. Nauk. SSR,

Fiz. Zemli 12 (5), 351-354 (1976) [Izv. Acad. Sci. USSR,

Phys. Solid Earth 12 (5), 106-111 (1976)].

[6] V. I. Panov and V. N. Frontov, Zh. Exsp. Teor. Fiz. 77, 1701 (1979) [Sov. Phys. JETP 50, 852 (1979)].

[7] H. Hirakawa, J. Tsubono, and K. Oide, Nature (London)

283, 184 (1980).

[8] H.-T. Yu, W.-T. Ni, C.-C. Hu, F.-H. Liu, C.-H. Yang, and W.-N. Liu, Phys. Rev. D 20, 1813 (1979).

[9] F. D. Stacey, G. J. Tuck, A. R. Maher, and D. Morris, Phys. Rev. D 23, 1683 (1981).

[10] F. D. Stacey and G. J. Tuck, Nature (London) 292, 230

(1981).

[II] P. Heyl and P. Chrzanowski, J. Res. Natl. Bur. Stand. 29, 1 (1942).

[12] R. Spero, J. K. Hoskins, R. Newman, J. Pellam, and J. Schultz, Phys. Rev. Lett. 44, 1645 (1980).

Table 1. Experimental Values for X.

Source X(10 R2

X.(Historic) + 2.11 0.70 80 cm 6.3 cm

XCLong) + 1.95 0.40 29.9 cm 4.48 cm

\(Karagioz) + 2.02 0.61 16.5 cm 5.6 cm

X(Panov,l) + 1.54 3.07 295.8 cm 42.06 cm

\(Panov,2) - 0.63 + 4.12 984.0 cm 42.06 cm

\(Hirakawa,l) -H77.4 172. 2.5 m 2.2 m

\(Hirakawa,2) -t-12.9 93. 3.0 m 2.2 m

\(Hirakawa,3) + 1.1 75. 3.5 m 2.2 m

X(Hirakawa,4) + 3.9 49. 4.2 m 2.2 m

X(Yu,l) + 1.80 21.8 30.47 m 13.0 cm

X(Yu,2) + 3.54 34.7 32.35 m 13.0 cm

X(Yu,3) + 15.2 41.3 34.35 m 13.0 cm

X(Yu,4) + 52.1 -+- 60.6 36.30 cm 13.0 cm

X(Stacey) + 2.00 + 1.2 1 km 13.0 cm

X(Spero) + 0.01? 0.07? -30 cm ~4 cm

589

Precision Measurement, and Ftindamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Tests of the Gravitational Inverse Square Law Using Torsion Balances

J. K. Hoskins, R. Newman, J. Schultz, and R. Spero*

Department of Physics, University of California at Irvine, Irvine, CA 92717

We describe experiments at U.C. Irvine to test the inverse square distance dependence of the

gravitational force at laboratory distances. One experiment has tested a distance range of approximately 2 to 5 cm, using a test mass suspended from a torsion balance to probe the gravitational field in- side a mass tube. Results of this experiment support an inverse square law: Assuming a force law devi- ating from inverse square by a factor [1 + elnr] with r expressed in cm, we find e = (1 ± 7) x 10~^. A

second experiment, now in progress, tests a distance range from 5 cm to 20 m. We discuss the feasibil- ity of an experiment to test the inverse square law at distances less than 1 mm. Methods are discussed

for reducing the effect of seismic noise on a torsion balance by damping pendulum oscillation modes of the balance. Key words: gravitation; inverse square law; Newtonian gravity; torsion balance.

1. introduction

The possibility of a deviation from inverse square

behavior of the gravitational force on a laboratory dis- tance scale has received considerable recent attention.

Several authors have pointed out that a suprisingly large

anomaly could be consistent wdth existing empirical evi- dence. An experiment indicating a failure of the inverse

square law over a distance range 4.5-30 cm has in fact been reported by Long [1]. Several recent theoretical

ideas suggest forces which could manifest themselves as a failure of the inverse square law of gravitation; of particular interest are the possibility of a long-range component of the strong interaction arising from two gluon

exchange [2], and the suggestion arising from supergravity unification theories [3] of a possible laboratory scale deviation from inverse square behavior.

We are conducting experiments to test the inverse

square law over distance ranges 2-5 cm and 5 cm-20 m,

and have considered an experiment to probe a range < 1 mm. These experiments are based on the use of the

torsion balance, an ancient instrument which has proven

admirably suited for this application, and shows promise

for vast further improvement in capability.

2. The 2-5 cm Experiment

This experiment has been completed, and results have

been published [4]; we reproduce some of the details here. The method is essentially a null experiment; a tor- sion balance (Fig. 1) is used to measure the change in the

force acting on a test mass suspended inside a long hol- low cylinder as the cylinder is moved laterally so that the

test mass hangs alternately near opposite walls. For an

infinitely long perfect cylinder and exact inverse square

force law the gravitational field due to the cylinder vanishes everywhere inside it, just as inside a spherical mass shell. For our finite cylinder of length L = 60 cm

and inside diameter D - 6 cm there exists a small net "end effect" force on a test mass located near an inside

wall, smaller than the nearly balanced opposing forces

due to near and far wall by a factor (D/L r = 10"^. Thus,

*Present address: Department of Physics, California Institute of Technology, Pasadena, CA 91125.

tungsten

torsion wire

electrostatic force plates

(rods)

vacuum enclosure thermal and

magnetic shielding

316 S. S. cylinder

test mass

chart recorder

l-^ computer

transport

mechanisms

» cylinder

0» motion

cart

Figure 1. Schematic cf the experimental apparatus for the 2—5 cm inverse square law test.

to compare the gravitational force at two distances

corresponding to the distances from the near and far walls in our cylinder, to a level of 1 part/10^, we need measure the end effect force to only 1 part/10^. Further- more, the end effect force is nearly a linear function of

test mass radial position in the cylinder, so that we need measure only the relative motion of the cylinder with. respect to the test mass, while the absolute position of the cylinder relative to the test mass need only be knowTi

to a few millimeters. These features of the finite cyhnder

591
"null" experiment greatly simplify metrology tasks. The

homogeneity and geometry of the cylinder itself must be

known with a precision on the order of 1 part/10'', but by

averaging data taken at a set of equally spaced azimuthal

orientations of the cylinder results are obtained

equivalent to those for an azimuthally symmetric and

homogeneous cylinder, leaving only axial cyhnder mass

variations to be precisely measured.

The cylinder, of mass 10.4 kg and wall thickness 1 cm, was made of high purity double vacuum melted 316 stain- less steel. Its associated magnetic field was measured to be less than 5 nT. The test mass was a 20 gm 4.4 cm long

rod made of high purity copper, hanging 83 cm below the

end of a torsion balance boom of total length 60 cm, made

of OFHC copper. All components were machined with

care to avoid imbedding ferromagnetic fragments. The

balance was suspended by a 32 cm long, 75 (xm diameter

tungsten wire in a vacuum of 2 x 10"' Torr, maintained

by an ion pump. The cylinder was external to the vacuum

system, which was surrounded in the region of the test mass by magnetic and thermal shielding.

Calibration of the balance to the level required for the

"null" test was accomphshed by applying a known gravitational force, using a 133 g copper ring of radius R =

12.1 cm, located so that the test mass lay on the ring axis

at the position z = R /V^where the force due to the ring

is a maximum and hence insensitive to the exact location

of the test mass. The ring mass was chosen to produce a

force on the test mass approximately equal to the end ef- fect force it calibrates. At intervals the ring was moved

by remote control to a corresponding position on the far side of the test mass, producing a calculable change in torque on the balance which served to calibrate it. In addition, the ring was used to enhance the null aspect of the experiment: whenever the cylinder was moved, the

ring was simultaneously moved to its opposite position to largely cancel the change in force on the test mass due to the cyhnder.

The experiment ran unattended during night hours,

under the control of a computer. Data was taken with

the test mass hanging 1.8 cm from the center of one wall. At intervals of 17 minutes the ring was moved, and the

cylinder translated by 3.43 cm on a screw-driven cart us- ing a remote magnetically shielded motor. From a total of 132 measurements of the torque change

AT on the balance associated with a roundtrip cylinderring cycle, at 8 cylinder azimuthal positions, we determined the experimental value AT - (4.49 ± 0.03) |xdyn- cm. (For comparison, the torque associated with only the

portion of the cylinder to one side of the test mass is ap- proximately 2000 fxdyn cm.) The corresponding calcu- lated value for AT, based on a Newtonian force law, is AT = (4.47 ± 0.14) iJidyn cm, in excellent agreement

with our measured value. The uncertainty in this calcu- lated value is dominated by approximately equal contri- butions from: (a) uncertainty in the balance arm mass

distribution, (b) limits on possible magnetic couplings,

based on runs without shielding, and (c) uncertainty in corrections for nonuniformity in cylinder geometry and

density. Deviations from perfect geometry were measured with 10"^ m resolution using a commercial differential gaging instrument. Variations in cylinder mass dis- tribution were determined by computer-controlled axial scans at various azimuths of the transmission of -y-rays through the cylinder wall. Analysing our data assuming the parameterization sug- gested by Long for a force law anomaly:

G = G{r) = Go[l + elnr]

we find: e = (1 ± 7) x 10 ^ for r measured in cm, compared to the value required to fit Long's data:

€Long = (200 ± 40) X 10-5

Another possible form for an anomaly involves a Yukawa potential, taking for the potential energy of masses M and yn separated by r:

Assuming this form we have determined the range in (a,

X.) parameter space which is excluded by our data. We

find our result to be inconsistent with Long's for all values of \. (See Fig. 3.) Long has argued [5] that the effect he finds may not be

observable in a null experiment such as ours. Non-null

tests of the inverse square law may be desirable.

3. The 5 cm-20 m Experiment

This experiment, now in progress, is not of a null type.

A torsion balance, consisting of a 60 cm long copper bar

suspended at its midpoint by a tungsten fiber, acts as a gradiometer to determine the field gradient produced by

a distant mass. The change in torque on the balance

which results when 3 tons of water are pumped between

two distant water tanks is compared with the torque

change produced by moving a one gram mass from a position 5 cm to one side of the balance bar to a corresponding position on the other side. The balance operates in a vacuum enclosure in a thermally insulated steel-lined pit at a depth of 2 m, in an open field near the UCI campus.

The balance arm has a mass of 500 gm and moment of in- ertia 157,000 g • cm^. The suspension fiber has a diameter of 90 |xm and torsion constant 49 dyn- cm/rad. The

undamped period of the torsion oscillation is 350 seconds.

Our hope is to achieve a sensitivity of 1 part/10^ in the

comparison of G(5 cm) and G(20 m), which would enable a determination of Long' s parameter e with an uncertainty

of 2 X 10"^. We consider in the following sections the

contributions to the uncertainty in this comparison.

3.1 Metrology

Geometric and mass metrology for this experiment are, we believe, comfortably under control at the 1 part/10^

level. Several features of the experiment simplify metrology tasks. The water tanks are suspended from single

points, so that a precise determination of the positions of their centers of mass in the horizontal plane reduces sim- \ ply to accurately surveying the suspension points. One of the tanks hangs from a precision load cell which determines the mass of water transferred between tanks to a

precision of 5 x 10 The arrangement of the 1 g "near

mass" is such that the torque change on the balance,

when the near mass is transferred from one side to the

other, is sensitive only in second order to uncertainties in

all three axes of the exact position of the balance arm

relative to the near mass (see Fig. 2): (1) for vertical dis- placements the torque does not change to first order due

to symmetry, (2) the near mass is located at the position

along the length of the bar at which the torque it pro- duces is a maximum; hence the torque is insensitive to small uncertainties in near mass position along the bar

axis, (3) the near mass is transported between opposite

sides of the balance arm by the rotating disk indicated in i Fig. 2, in such a way that the distance between the two , positions of the near mass is precisely determined by the

592
Figure 2. Schematic of torsion balance for the 5 cm—20 m test, showing near mass transport mechanism and damping plate

arrangement. The balance operates imthin a vacuum enclosure, to which the near mass is external.

spacer bar indicated in the figure. The resulting torque

change is sensitive to the exact position of the balance arm only in second order (if the bar is not quite centered

between the two near mass positions the increased

torque for one mass position is compensated to first order by a decreased torque in the other position). The

result of these three considerations is that the balance arm position within its vacuum enclosure need be known

only to about 1 mm in any direction for 1/10^ precision in

this experiment. A window in the vacuum enclosure

makes this determination readily feasible.

3.2 Nonthermal Noise

Noise in the torque signal is at present the main limita- tion in this experiment. Expressed in terms of the sensitivity of the balance as a gradiometer, the present effec- tive noise level under quiet conditions, at our

0.75 cycle/hour operating frequency, is about lE/VHz,

where IE = 1 Ebtvbs unit = 10"^ s"^. Since the total change in field gradient when 3 tons of water are pumped

between the two tanks is about 0.1 E, we can expect to compare G(20m) with G(5cm) with a reasonable integration time (10^ s) only to about 1%. To reach our

1 part/10^ goal will require a reduction of the present

noise level. A feature of the torsion balance used for this experiment is the use of electronic cooling not only of the tor- sion oscillation mode but also of pendulum oscillation modes. Pendulum motion of a torsion balance, excited by

microseismic noise, couples by several subtle nonUnear

mechanisms into torques. Because the mechanisms are

nonlinear the resulting slowly varying part of the tor- sion noise signal is proportional to the square of the pen- dulum amplitude, and hence to the Q of the pendulum

modes. Since the Q is on the order of 10^ or more, very

large reductions in the torsion signal noise may be ex- pected from damping the pendulum oscillation modes.

The torsion balance used for our 2-5 cm experiment

(Fig. 1) used a magnet below a copper disk mounted on

the torsion balance axis to produce eddy current damping

of pendulum modes, reducing their Q by a factor of about

1000; this greatly reduced the observed noise in the tor- sion mode. The balance used in our 5 cm-20 m experiment (Fig. 2) uses two orthogonal pairs of fixed electrostatic plates, flanking a copper cylinder mounted on the

balance axis, to sense pendulum motion of the balance

and to apply appropriate damping forces. The elec- tromechanical servo loop is similar to that applied by

Block and Moore [6] to a gravimeter, and the analysis is similar. In our case two such servo loops are used, one

for each orthogonal axis of pendulum motion. Also, in our

case the pendulum motion of the balance in each axis has

two normal modes, corresponding to swinging motion

vdth top and bottom points of the rigid balance structure moving in phase and 180° out of phase. This complicates

the analysis of the servo loop, but the resulting transfer

function for the mechanical part of the system is simply

expressed as a sum of two second order pole terms. This damping system successfully reduces the Q of the

pendulum modes of our balance from over 10^ to on the

order of 10", and greatly reduces the low frequency noise

in the torsion signal. However the noise is still higher

than desired, as indicated at the beginning of this section. What the dominant source of remaining noise may

be is not clear at this point. It may be in part a result of high Q vertical spring oscillations in the balance, which

are undamped by the present system, but were damped

by the eddy current system of the 2-5 cm experiment.

Gravitational noise from moving squirrels and rabbits in the vicinity of the balance may be a significant contribution. A number of other possible noise sources remain to be explored, and we still hope to reduce the noise to a

level which will allow a 1 part/10^ experiment.

3.3 Systematic Error We have taken pains to minimize the danger of magnetic couplings, through the use of high purity materials,

careful machining procedures, and magnetic shielding of the balance housing. Care was also taken to minimize

thermal effects by shielding the balance housing with

spaced concentric metal thermal shields. Our biggest con- cern is the possibility that ground tilt produced by the

transfer of water between the distant tanks will shift the

position of the hanging balance relative to the various

surrounding electrostatic plates, with a consequent

correlated change in torque on the balance. To determine

if this is a problem we have taken two steps: (1) sensitive

tiltmeters are mounted on the torsion balance base to determine the actual tilt generated when water is trans- ported, and (2) the torsion balance base is mounted on

three piezoelectric feet, w'hich may be used to generate a

large (3 x 10"^ rad) artificial tilt to determine the sensitivity of the torque signal to tilt. Preliminary measure- ments indicate that the tilt associated with water trans- port is less than 10"' rad, assuring no problem for a 1%

sensitivity experiment. A more precise measurement of

the correlated tilts is possible, and will be necessary.

3.4 Experimenter Bias

Being concerned about the many subtle biases which

can creep into an experiment of this type, such as the

temptation to look with less intensity for sources of sys593
2 1 \ I

I I I I I I r

I I I I I I I I I I -3-2-10 I 2 3 4 5 6

log X(cm)

Figure 3. Potential and present limits on the strength (a) and

range (\) of a Yukawa-type component of the gravitational

force, for torsion balance experiments discussed in this paper. The curve marked "Long" is the locus of (a, \) values consistent imth D. Long's experimental results. Long's data re- quire a < 0. The other curves are: (A) the 2 a limit on allowed

(a, k) values provided by our 2—5 cm experiment, assuming a < 0, (B) the target sensitivity of our 5 cm-20 m experiment,

and (C) the sensitivity which might be expected for an experiment with 0.05 cm mass separation.

tematic error when the answer begins to look "right", we have protected ourselves by arranging that the value

of the nominally 1 g "near mass" is known to us only to 10%, while the exact value is guarded by a colleague,

unassociated with the experiment, who is instructed to reveal the exact value of the mass only when the book is closed on other aspects of the experiment.

4. Possible Inverse Square Law Tests for

Mass Separations Less Than 1 mm

In view of the various theoretical motivations for test- ing the inverse square law [2, 3] it would be desirable to extend such tests to as small a mass separation as possible. In particular, the possible long range component of the strong interaction which could be associated with two

gluon exchange between hadrons would have a van der

Waals type distance dependence [2], increasing as a high

power of the inverse mass separation. To search for such

a force one should explore the minimum feasible mass

separations. A good technique for doing this with a tor- sion balance might be to suspend a thin-walled metalized

quartz tube of perhaps 2 cm diameter from the end of a

torsion balance in such a way that it hangs within a similar stationary tube of slightly larger diameter which can be moved laterally in the fashion of our 2-5 cm experiment. In effect such an experiment measures the force as a function of distance between closely spaced mass

plates. The possible advantages of using quartz tubes lie in the rigidity of a tube, high degree of homogeneity

achievable with fused quartz, and the ability to repeat

the experiment with a set of different azimuthal positions

of the stationary tube relative to the suspended tube to reveal possible effects of mass non-uniformity. Suitable

precision quartz tubes appear to be commercially avail- able. A thin foil shield between the two tubes would be

required to shield the electrostatic van der Waals forces, ; which become comparable to the gravitational force for j! mass separations of about 0.2 mm. | References '

[1] D. R. Long, Nature (London) 260, 417 (1976).

[2] G. Feinberg and J. Sucher, Phys. Rev. D 20, 1717 (1979).

[3] J. Scherk, Phys. Lett. 88B, 265 (1979).

[4] R. Spero et al., Phys. Rev. Lett. 44, 1645 (1980).

[5] D. R. Long, Nuovo Cimento B 55B, 252 (1980), and these ; proceedings. | [6] B. Block and R. D. Moore, J. Geophys. Res. 71, 4361 (1966).

|

594
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Measurement of Gravitational Forces at Separations Around 10 l\/leters*

Hsou-Tao Yu and Wei-Tou Ni

Department of Physics, National Tsing Hua University, IHsinchu, China (Taiwan)

and

Chin-Cheng Hu, Fa-Hslang Liu, Chia-Hsiang Yang, Wu-Nan Liu

Exploration and Exploitation Research Center, and Taiwan Petroleum Exploration Division, Chinese

Petroleum Corporation, Miaoli, China (Taiwan)

We used a Worden gravimeter to measure the gravitational forces at fixed positions when a large

oil tank was full and when it was empty. The results are reported here. Key words: gravitational constant; inverse square law; oil tank experiment; tests of Newtonian

gravity.

Since the last International Conference on Precision

Measurement and Fundamental Constants, there has

been considerable interest in the empirical foundation of the gravitational inverse square lavi? [1, 2, 3]. In other

words, the problem is whether the gravitational "constant" G(r) = Fgrav^^/Afm is independent of the separation of the two point masses m and M. Wagoner [4], Fujii [5], and O'Hanlon [6] have argued for the variation

of G(r) and proposed the a-|x model with G(r) = Gc [ 1

-I- a(l + \i.r)e''^'^\ where Gc, a and \x. are constants. Previous laboratory experiments had separations ranging from 5 cm to 1 m. Above 1 m there has been no direct

measurement. In 1978, we used a Worden gravimeter to measure gravitational forces at fixed positions when a

large oil tank was full and when it was empty. At a fixed

point, the readings of the Worden gravimeter varied with

time due to drift and tidal effect. To subtract these

effects, we selected a base point as reference. For simplicity in analysis, readings were repeated between the

base point and observation points within 1 hour. Since

this time lapse is short compared with the period of

tides, we can use linear interpolation methods to obtain

(fl' ~ Phase) at equal time. The difference in g is due to the environment. We repeated the observations seven

times to reduce errors. From these differences when the

tank was full and when it was empty, we calculated = g - g - g z- The results are presented in Table 1 [7, 8]. According to the analysis of Mikkelsen and Nevraian

[3], G (r) is nearly constant in the ranges 5 cm < r < 1 m

and 10"* km < r < 3 x 10^ km. Therefore in the a-|jL

model, we assume the range of variation \x'^ to be 1 m <

< 10"* km. Now using the data in Table I, the viable

range of a and jjl is shown as the shaded area in Fig. 1 [7,

8]. Mikkelsen and Nevmian [3] used models of the earth

and sun to conclude that 0.6 < (1 + ot)"^ < 1.3. Using

these constraints and the possible values of a and |x in Fig. 1, .we calculate the possible G(r )-range as the

shaded area in Fig. 2 [7, 8]. This result, together with

the accurate 2-5 cm experiment of Spero et al. [9], are incompatible with Long's result [2] in the a-|jL model.

1.2

1,0 0.8 0.6

0.4

0.20

-0.2 -Q4

-0.6

-0.8 -1.0 J i ^ J I I I I I 1 I \

I I lOm 15m

*Supported in part by the National Science Council of China (Taiwan).

Figure 1. Experimental constraints on the parameters a and |j. in the a-|x model by Wagoner H], Fuji [5], and O'Hanlon [6]. The viable range is shown as the shaded area. References

[1] D. R. Long, Phys. Rev. D 9, 850 (1974).

[2] D. R. Long, Nature (London) 260, 417 (1976).

[3] D. R. Mikkelsen and M. J. Newman, Phys. Rev. D 16, 919

(1977).

[4] R. V. Wagoner, Phys. Rev. D 1, 3209 (1970).

[5] Y. Fujii, Nature Phys. Sci. 234, 5 (1971); Ann. Phys. (N.Y.)

69, 494 (1972).

[6] J. O'Hanlon, Phys. Rev. Lett. 29, 137(1972).

[7] H.-T. Yu, W.-T. Ni, C.-C. Hu, F.-H. Liu, C.-H. Yang, and W.-N. Liu, Chinese J. of Phys. 16, 201 (1978).

[8] H.-T. Yu, W.-T. Ni, C.-C. Hu, F.-H. Liu, C.-H. Yang, and W.-N. Liu, Phys. Rev. D 20, 1813 (1979).

[9] R. Spero, J. K. Hoskins, R. NewTnan, J. Pellam, and J. Schultz, Phys. Rev. Lett. 44, 1645 (1980).

595
Gin

1,4 1.2

1.0

0.8

0.6

0.4

0.20 J I I L J I I

I I I I I I 5m 10 m 15m

Figure 2. Constraints on G(r) in the a-|jL nwdel. The viable range is shoum as the shaded area.

Table I. Experimental Results

Dates of measurement:

Tank diameter: 54.86 m

Density of the oil: 0.885 gm/cm^

March 31, 1978

April 3, 1978

Full tank

Empty tank

Oil level: 13.473 m from

the ground

Oil level: 2.32 m from

the ground

Distance of measured position 2m 4m 6m 8m

from the rim of the tank Mean experimental value 0.122 ± 0.085 ± 0.065 ± 0.060 ±

\AgE\(mGa\) 0.0155 0.018 0.017 0.018

rms deviations in percent 13% 21.5% 26% 30%

Errors due to the uncertainty in 0.1% 0.3% 1% 3%

the effective height of the massspring system of the Worden

gravimeter

Newtonian prediction I A I 0.120 0.083 0.059 0.044

^gE //^gN 1-01 - 0.13 1.02 ± 0.22 1.09 ± 0.29 1.35 ± 0.40

596
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Non-Newtonian Gravity: Geophysical Evidence

Frank D. Stacey and Gary J. Tuck

Physics Department, University of Queensland, Brisbane 4067, Australia

Six independent geophysical data sets have been used to estimate the value of the Newtonian gravitational constant G, yielding results that are consistently higher than the accepted laboratorydetermined value G* by 0.5% to 1.5%. Possibilities of unrecognized systematic errors preclude a defi- nite conclusion, so that new large scale measurements of G are needed to clarify the matter. Two such

experiments, using masses distributed on quite different scales, both much larger than any laboratory measurement of G, are under development. One determines the gravitational attraction by a 3.5 km

layer of sea water by measurements of gravity in a bathyscaphe. The other makes use of frequent level changes of a hydroelectric pumped-storage lake, involving a 10 m layer of water. Key words: geophysical measurements; gravitational constant; inverse square law; non-Nevd;onian

gravity.

1. Introduction

Precise measurements of the Newtonian gravitational constant G have been made over a very limited range of

separations of the attracting masses. Now that the validity of the inverse square law is being seriously questioned, measurements on scales quite different from

those of conventional laboratory determinations are. needed and geophysics has an important role to play.

The development of geophysical experiments to determine G is partly a matter of retracing history, since the

first methods of estimating G were geophysical [1,2].

They were abandoned 100 years ago in the face of overwhelming evidence that laboratory measurements

were much more precise. However, not only do we now

have a new reason for reintroducing such large scale measurements, but geophysical techniques have advanced so dramatically since then that it may even be

possible to produce a geophysical estimate that is more

precise than the laboratory ones. The feature of the current geophysical approach that

gives it a particular advantage is that it uses very extensive plane layers of material as the attracting masses. In

the case of water layers these can be very uniform and

very precisely horizontal as well as effectively almost in- finite in extent, so that their gravitational fields at all close points are independent of position and the forces

exerted on test masses are independent of their shapes

and homogeneities. All the geophysical estimates of G

that we have reported recently [3] and both the experiments discussed here make use of this property.

Although we are suspicious of a deficiency in the "in- verse square law over a range extending to tens of meters at least, at the present stage our data are inadequate to suggest, onlheir own, the form of any required

generahzation. In making our calculations, we therefore assume validity of the inverse square law and formally

calculate values of G in terms of it. A deficiency is then

apj)arent as a disagreement with the conventional value

G obtained from measurements on much smaller, labo- ratory scales. To proceed further we must recognize that we are measuring gravitational forces due to widely dis- tributed masses and that we can determine directly not

the law of gravity itself but some integral of it. Suffi- ciently precise data may nevertheless suffice to specify a

form of generalization, but the present data are inadequate for this purpose.

2. Theoretical Basis of Geophysical

Determinations of G

Although in principle any geological body of assumed

known mass may be used to estimate G the only usable

method in practice is that due, in its original form, to Airy [4] who measured gravity in a coal mine, in which

extensive horizontal stratification is a reasonable as- sumption. Applied to the situation of a spherical, nonrotating Earth of radius r and mass M we can write the

surface gravity

^(0) = GM/r^. (1) The product GM is inseparable in all astronomical and

whole Earth geophysical observations, including rotational effects, precession and wobble and seismological

observations including the free mode periods. To identify

G independently of M it is necessary to make measure- ments at some depth z which is inside a known fraction

A.M of the Earth's mass so that

g{z) = G(M - AM)/(r - z)^ (2) and G is obtained by eliminating the unknown M from

Eq. (1) and Eq. (2). In the approximation z « r, g{z) -gm «g(0),

giz) - giO) ^ 2gi0)z/r - 4j:G jjpdz . (3) A more complete equation that retains second order

terms and includes the effects of the Earth' s rotation and

ellipticity is [5] g{z) ' giO) ^ U{z) - A-aGXiz), (4) where

U = 2g{0)z/r[l + Sz/2r - (3/2)^2(3 sin2(t)o - 1)] + 3 0)^2(1 - sin2(j)o) (5)

X = (c/a) [1 + 2z/r + (1 - e~/a-)/2]

X J

pdz - (2/r) / pzdz , (6)

597
J2 = 1.08264 X 10"^ is the mass ellipticity coefficient,

(1) = 7.2921 X 10"^ rad s"^ is the rotation rate, 4>o is geocentric latitude, and c/a - 0.996647 is the ratio of polar

to equatorial axes. The refinements of Eqs. (4), (5), and

(6) are necessary to our work.

It should be noted that, although we have presented

the theory of the internal variation of gravity within the

Earth in terms of a model Earth v^ath complete spherical or elliptical layers, this is not in fact a requirement. All that is necessary is that the uniform layering should ex- tend to a distance that is large compared with the depth

range of the measurements. This is not an entirely trivial requirement for measurements over a depth range that

may be several kilometers, but boundary corrections can

readily be applied with sufficient accuracy.

3. Summary of Mine and Borehole Data

Equations (3) or (4) are the basis of a gravitational

method of estimating the in situ densities of rock layers

penetrated by mines and boreholes (by assuming the con- ventional value of G). In a few cases independent meas- urements of density have been reported, permitting the

procedure to be reversed to calculate G. Including our own report of mine measurements undertaken specifically to determine G [5] we have found five data sets that ap- pear reliable and give sufficient details for us to obtain

values of G with standard deviations representing scatter

of the data. The results are reported elsewhere [3] and

summarized here (Table 1). The values in Table 1 are consistently higher than the

conventional laboratory estimate [10], G* - (6.672 ±

0.004 ) X 10"^^ m^ kg"^ s"^, although only in the case of the McCulloh data is the discrepancy sufficiently large,

relative to the formal standard deviation, to compel us to look carefully for possible systematic errors. We used 31

of McCulloh' s gravity values [7] to obtain the estimate in Table 1 (4 values obtained in the top 50 m were omitted

because weathering and fracturing of the rock in that

range made the density estimates questionable, but in- cluding them makes little difference). Of these, the three

subsets of 11, 10, 10 data points obtained over different

ranges clearly indicate consistency of the data. The two

possibilities for doubt are a systematic underestimate of sample densities and an anomalous regional gravity gra- dient due to some mass irregularity outside the range of

observations. The supposition that the densities are all 1% low appears improbable, but is not totally impossible.

We have attempted to allow for the possibility of a bias- ing mass anomaly by fitting a mass of arbitrary magnitude and depth to the data and found that including it further increased the estimate of G, although only marginally. Thus, if the high value of G is to be attributed to a

systematic error, it must be due to a large, deep mass

anomaly under Ohio, where McCulloh' s measurements were made, causing an anomalously low free air gravity

gradient over a broad area. Evidence for this is currently

being sought.

4. Coniparison of Sea-Floor and SeaSurface Gravity Surveys

Before the development of methods of measuring gravity at the sea surface, marine surveys were made by a la- borious procedure of lowering gravity meters to the sea

floor. Subsequent surface measurements over the same

area permit G to be estimated from the density and

depth of the water. It is important to make the max- imum use of such data, since they avoid one of the doubts

that beset the mine and borehole data: there is no way

that we can admit a 1% uncertainty in the density of sea water. Limited comparisons of surface and sea-floor gravity data have been made by the U.S. Geological Survey, with a reported hint of a systematic discrepancy

[11], but from our examination of the data they appear

inadequate to draw any conclusion.

By far the best data set that we have been able to dis- cover and probably the only one in existence that is really

useful for our purpose is held by the Exxon Company's

Exploration Department. With their collaboration we

have extracted from these records 703 pairs of sea-floor and corresponding sea-surface gravity values equally

spaced over an approximately square area of 1100 km^ in the Gulf of Mexico in water up to 700 m deep. The gravity

differences show considerable scatter, which has at least two causes. One is noise in the sea-surface data, which are

less accurate than the sea-floor data. The other is an ef- fect related to topography of the sea floor. The survey

data were corrected for sea-floor topography, but the

correction may be imperfect, or else the data are influ- enced by sub-bottom mass anomalies that are correlated

with topography. In this circumstance it is important that we have data uniformly spread over a substantial area, because the area mean of the gravity gradient averages

the topographic effects. The mean vertical gradient of gravity in the sea over this area is (2.2124 ± 0.0020) x

lO"*' s-2, which gives G = (6.796 ± 0.016 ) x

10"^^ m^ kg"^ s"^. As we point out [3], this estimate

depends crucially upon the precision of the baseline ties of the two surveys. But the trend of the variation of gravity

differences i^g) with depth (z) is independent of these

ties and if instead we seek a method of eliminating the to- pographic effect, we can use this trend. Although such an approach is necessarily rather qualitative, it does not ap- pear reasonable to make assumptions that reduce the esti- mate of G below about 6.73 x 10"^^ m^ kg~^ s~^, unless in

Table 1. Estimates of Gfrom mine and borehole data.

Data source Type of data Depth range (m) (G±a)( 10 *^m^kg ^ ^)

Whetton et al. [6] Mine 96-587 6.795 ± 0.021 McCulloh [7] Mine 57-685 6.7390 ± 0.0025 57-208 6.724 ± 0.014 223 -388 6.726 ± 0.012 418-685 6.747 ± 0.013 Hinze et al. [8] Borehole 3712-3962 6.81 ± 0.07 Stacey et al. [5] Mine 0-948 6.712 ± 0.037 Hussain et al. [9] Mine 251-590 6.705 ± 0.060

598
this case also we happen to have hit upon an extensive low

in the free air gradient of gravity.

5. Bathyscaphe Measurements of Gravity

In collaboration with staff of Texas A & M University we are planning to obtain a set of vertical gravity profiles in the deep sea, using the submersible Alvin, which

has a maximum depth capability of 4000 m. That a free- floating bathyscaphe may be steady enough to permit

such gravity measurements was demonstrated by Drake

and Delauze [12], although their data were obtained in a

sea-floor trench and the very anomalous gradient could

be explained as a very local effect if the topography is much steeper than their map indicates. We are taking

the additional precaution of using a stabilized platform

for the gravity meter. Corrections for any slight vertical

accelerations will be made from the doubly differentiated

record of a very sensitive pressure gauge. The initial series of measurements is planned for the Sigsbee Abyssal Plain in the Gulf of Mexico, in an. area selected as gravitationally and topographically featureless. In principle such measurements permit a very precise determination of G [13]. If the measurements are successful they vdll provide

the largest scale determination of G. Measurements at the surface vdll not be possible as the gravity meter will not be adequately stabilized against wave action, but the

available depth range below the effect of the waves is about 3500 m.

6. The Splityard Creek Gravity Experiment

Splityard Creek is a minor tributary of the Brisbane

River in S.W. Queensland, just upstream of the now

nearly complete Wivenhoe Dam across the river. A subsidiary dam across the creek will impound an additional

volume of water at a higher level than the main lake and

will be used as a hydroelectric pumped-storage reservoir. In the course of its operation the level Mill rise or fall by

as much as 10 m in a few hours, depending upon the

demand for electricity, remaining constant for a few

hours before changing again. The situation is ideal for a fundamental experiment on gravity. A specially con- structed electricity pylon near the middle of the lake will support an observing platform for our experiments

(Fig. 1). The principal experiment uses a vacuum balance to compare the weights of 10 kg masses, hanging (in evacuated tubes) at different levels in the lake. Considering

two masses, one above high water and the other below

low water, a measured rise in water level introduces an

approximately infinite plane layer of uniform density p between the masses, pulling one up and the other dovm.

For a layer of water of depth h = 10 m the gravity

difference between the masses that is caused by the water layer is = 4ttG p/i = 8.38 X lO"*' m s'^ (7) which is nearly 1 part in 10^ of ambient gravity and so produces an unbalance of this magnitude in the comparison of the two weights. Assuming an initial balance of two masses m (10 kg) the balance may be restored by an

additional mass A m, where

Am - m A.g/g —8.5 mg (8) If the design sensitivity is achieved we vdll be able to measure an unbalance of this magnitude to 3 parts in 10^. This sensitivity is based on some special design features

EVACUATED

BALANCE

AND MASSES

Figure 1. Schematic view of the Splityard Creek gravity exper- iment. A balance compares the weights of masses sitspended in evacuated tubes at different depths in a lake as the lake level is changed. The balance is supported by an observing platform on an electricity pylon which minimizes corrections for the ab- sence of water in the volume occupied by the material of the pylon.

that are indicated in Fig. 2. Our balance design is based on the principles of capaci- tance micrometry that have had extensive use in our

laboratory for more than 10 years [14]. Horizontal positioning of the balance arm is measured relative to a reference plane in which there are electrodes activated

by a ratio transformer bridge operating at 3 kHz. This

plane supports interconnected pools of mercury which are used to sense the adjustment of the plane to a perfectly

horizontal position, the capacitances to the two pools being equal at perfect balance. The upper faces of the two

electrodes operate a similar bridge vdth the balance arm, whose deparature from horizontal is therefore indirectly

referred to the mercury levels. The balance arm itself is supported via its knife edge on an insulated support con- nected to a bridge preamplifier which is mounted on the

reference plane. It is important to the bridge operation

that, at balance of the bridge, electrostatic forces are

also balanced and the balance arm is at earth potential,

although as a precaution 1/V^ switch is fitted to the

bridge drive source. By dividing all voltages by "V^ all electrostatic forces and any consequent unbalance will be

T ^

' MERCURY

REFERENCE PLANE WITH ELECTRODES

Figure 2. Diagrammatic section of balance components for the Splityard Creek gravity experiment. The position of the bal- ance arm is sensed by electrodes mounted in the refere^ice plane on which it is mounted and this in turn is adjusted to horizontal by capacitance sensing to interconnected mercury

pools.

599
halved and the balance reading appropriate to zero applied voltage is readily calculated. The unbalance of the

bridge output is used in a servo-circuit to apply restoring

forces via additional electrodes in the reference plane

(not shown). These are also used to apply critical damping. Thus the balance reading is of a restoring voltage

that must be calibrated in terms of precisely knovra masses added to one side or the other. Four suspended masses will be used, one above the

high water level and one below on each side of the bal- ance, the two masses down in the lake being at different

levels. An electromechanical system for interchanging

supports to the masses is built into the balance. In this way masses supported at different levels by the same bal- ance arm can be repeatedly interchanged and precisely

compared and dimensional stabihty of the balance arms

over several hours is not required. The ends of the balance arm support the suspended masses not by conventional

knife edges, but by diamond points, because this makes

the exact point of support unambiguous. (The necessary

parallelism of three knife edges cannot be achieved.)

Apart from the operation of the balance itself, the only

quantities that must be measured accurately are the

depth and density of the lake water. The shapes of the

suspended masses and the geometries of the lake perimeter and supporting members of the instrument tower

(where these represent absence of water) are important

to the attainment of a very precise result but are not

critical. Regional gravity inhomogeneities, tidal effects,

depression of the lake floor by the water load and similar

effects are either constant or affect all suspended masses

so nearly equally that they can be ignored. The effect of observer mass is negligible at 8 m, which presents no dif- ficulty as the balance is connected to its control console

via an umbilical cord longer than this. Details of corrections to be applied for the finite lake area (and irregular outline) are of some interest. The

gravity due to a large sheet of density p and thickness dz

at a point distant a from it, where a is much less than

the distance r to any point of the perimeter is dg = 2itG pdzil - a <r-^> + <r-^> + . . ) (9) where <r"^> is the azimuthally averaged reciprocal

shoreline distance. This binomial expansion shows that

there is a first order correction and third and higher

order corrections, but no second order terms. Since

a/r ~ 2 X 10"^, third order terms are insignificant.

Thus we can write the gravity difference between points

at distances a above and {z - a) below the layer

Mdg) ^ d{Ag) ^ i-nG pdzH - (z/2) <r-^>) , (10)

where z is now identified as the vertical separation of two masses whose weights are compared. Now we can in- tegrate over a finite layer whose outline varies over its total depth range, h, vdthout complicating the form of the correction equation: = 4TrG p/i(l - (z/2) <r-i>). (11)

For a reason that follows, the lake perimeter is being

carefully surveyed at all water levels of interest, but for the purpose of this correction we can estimate <r~^> for the layer between any two levels from measurements on two pairs of masses at different vertical separations z. This is achieved by using the four masses referred to. The survey is a valuable check anyway, but is being done

for another particular reason, which is to orient the bal- ance so that <r~^> is precisely the same at both ends of

it. That this geometry has been achieved will be checked

by turning the balance around between observing runs.

7. Conclusions

While the present geophysical evidence for a non- Newtonian gravitational effect lacks the observer control

necessary to eliminate possibilities of systematic error, the fact that all of the available data give G at long range

higher than G* is circumstantial evidence that the effect

is real. It is also of interest that the effect is of the same

sign as that reported by Long [15] on the basis of experiments at laboratory ranges. Since we are contemplating an effect of order 1% for measurements on the scale of hundreds of meters, it cannot be long before sufficiently

well controlled large scale experiments settle the matter

conclusively.

The two experiments we have described are complementary. The bathyscaphe experiment is the largest

scale G measurement that we have been able to devise

and will be vital first in demonstrating whether a breakdown in Newton's law at kilometer range is positively

demanded and, if so, then in deciding whether the upper

range limit of the breakdown is vdthin an observable

range. If a breakdown is indicated then the greater accu- racy of the lake experiment will be needed to indicate

clearly the form of the breakdown. This experiment has

all the advantages of a controlled laboratory experiment,

^<vith masses repeatedly introduced to and removed from

an othervvise immobile apparatus. Further, it has several

advantages, even over conventional G measurements,

arising primarily from the geometrical simplicity and

symmetry of the large mass (a layer of lake water).

Corrections for attraction to the balance itself, extrane- ous cross attractions to masses on the wrong side of the

balance, and errors arising from inhomogeneities of the

attracting and attracted masses are all designed out of the experiment. Thus the 'lake experiment should provide

a useful estimate of G quite apart from the search for non-Newtonian effects.

Our work is supported by the Australian Research

Grants Committee and by Mount Isa Mines Ltd. Important ideas of Mr. G. Bell are incorporated in our balance

design.

References

[1] J. H. Poynting, The Mean Density of the Earth (Griffin, London, 1894).

[2] K. E. Bullen, The Earth's Density (Chapman and Hall, London, 1975).

[3] F. D. Stacey and G. J. Tuck, Nature 292, 230 (1981).

[4] G. B. Airy, Philos. Trans. R. Soc. London 146, 297 and 343

(1856).

[5] F. D. Stacey, G. J. Tuck, S. C. Holding, A. R. Maher, and

D. Morris, Phys. Rev. D 23, 1683 (1981).

[6] J. T. Whetton, J. 0. Myers, and R. Smith, Geophysical Pros- pecting 5, 20 (1957).

[7] T. H. McCulloh, Geophysics 30, 1108 (1965).

[8] W. J. Hinze, J. W. Bradley, and A. R. Brown, J. Geophys.

Res. 83, 5864 (1978).

[9] A. Hussain, G. Walach, and F. Weber, Geophysical Pros- pecting 29, 407 (1981).

[10] E. R. Cohen and B. N. Taylor, J. Phys. Chem. Ref. Data 2, 663 (1973).

[11] M. J. Yellin, U.S. Dept. of Commerce, ESSA Operational Data Report C & GSDR-2 (1968).

[12] C. L. Drake and H. Delauze, Ann. Inst. Oceanog. (Paris) 46, 7 (1965).

[13] F. D. Stacey, Geophys. Res. Lett. 5, 377 (1978).

[14] F. D. Stacey, J. M. W. Rynn, E. C. Little, and C. Croskell,

J. Phys. E (J. Sci. Instrum.) 2, 945 (1969).

[15] D. R. Long, Nature 260, 417 (1976).

600
Precision Measurement and Fundamental Constants 11, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Experimental Test of a Spatial Variation of the Newtonian

Gravitational Constant at Large Distances*

H. A. Chan and H. J. Paik

Department of Physics and Astronomy, University of Maryland, College Park, MD 20742

The Poisson equation of Newtonian gravitational potential provides a source-independent null test of the Inverse Square Law. A convenient Laplacian detector consists of superconducting gravity gra- diometers in three orthogonal directions. Matching and stability of the cryogenic detector are achieved

by utilizing superconducting circuits. Since the Laplacian of the gravitational potential produced by an

arbitrary source is zero outside the source in the Inverse Square Law, this experiment becomes a source-independent null test for the constancy of the gravitational constant. This characteristic allows

a precision test of the Inverse Square Law at geological distances using natural objects like an ocean or the earth. We discuss experimental procedures and expected sensitivities of the null experiment for three different sources: a swinging pendulum, an ocean tide, and the earth itself. It appears that the

empirical limits in the Inverse Square Law could be improved by three to six orders of magnitude in the range between 1 m and 10^ km by this new null experiment.

Key words: gravitational constant; gravitational null experiment; inverse square law of gravitation; su- perconducting gravity gradiometer.

1. Introduction

The gravitational constant G is considered to be constant both with time and space in Newi:on's Universal

Law of Gravitation and Einstein' s General Theory of Relativity. It is possible, however, to make G a function of

time [1] or a function of mass separation [2] vdthin metric

theories of gravity. A spatial variation of G has also been

predicted on other theoretical grounds [3-6]. Most of these theories favor a functional dependence

G(i2) = G[l + a(l + fjLR)e~^], (1) which arises from a potential of the form

GM

R

(1 + ae-^)

(2)

Therefore, a spatial dependence of G, if detected, could

imply the existence of an additional short-range force

which is superposed over the Newtonian long-range force

rather than necessarily a failure of the Inverse Square

Law.

Various authors have pointed out [7-9] that the exist- ing data on absolute G and the product GM for the earth

and other celestial bodies cannot rule out the possibility

of a having a value as large as 1/3, if the range p."^ falls somewhere between 10 m and 10 km. Efforts have been

made recently to extend the Cavendish-type experiments

up to a mass separation of /? ~ 10 m [10-14]. Inference

from orbits of artificial satellites on the constancy of G

for R < 10^ km is difficult on account of the rapid fall- off of any non-Newtonian signal as p."^ is reduced below

the satellite altitude. As a result, the intermediate dis- tance range 10 m < R < 10 km has been left largely

untested to this date. It is highly desirable to be able to examine GiR) on this range and improve the overall ex- perimental limits in the constancy of G as a function of R.

Such an experiment will test the scale invariance of the

Universal Law of Gravitation and help settle issues

raised by opposing theories of gravity.

In this paper, we discuss a series of experiments which

could cover the desired geological scale and might improve the empirical limits on a at large distances by

several orders of magnitude. These experiments are based on the principle of a null experiment proposed by

Paik [15]. The "source-independent" nature of the ex- periments permits use of a large natural object like a mountain, an ocean tide, or the earth itself as a source to examine the Inverse Square Law at the characteristic

distances of these objects. In the following sections, the

principle of the gravitational null experiment is re- viewed, experimental procedures for a laboratory,

geological-scale, and earth-orbit experiment are dis- cussed, and the expected resolutions of these experiments are presented. In addition, the principle and

design of the gravitational null detector are briefly

described.

2. Principle of a Null Experiment

The "source independent" null experiment invokes a well-known theorem in gravitostatics: Gauss's law for Newtonian gravity. In differential form, it is embodied in a Poisson equation

V2<})(x) = -V • g(x) = 4t7G p(x) , (3) where g = - Vcf) is the gravitational field (force per unit

mass). Since the unique solution of Eq. (3) is the 1/R potential, which is characteristic of an inverse square force law, it is clear that the Poisson equation for <i> is equivalent to a constant G, G i=G{R), in the force equation

g(x) = - iGiR) ^ (4)

*Supported in part by NASA.

where R = I x - x'l. Notice that the Laplacian of the

Newtonian potential, V^ct)N, vanishes identically in va- cuum, independent of mass distribution in the rest of the

universe.

Hence, a mill experiment for the constancy of G can be

performed in the following way. First, construct a detec- tor sensitive to V^(}), which we call a gi'avitational "La601
placian detector." Second, using a large object of an ar- bitrary shape, modulate the source-detector separation

periodically. Third, average the V^cj) signal over many

periods synchronously with the periodic motion. A statistically significant departure of V^cj) from zero would

then constitute a violation of the Inverse Square Law at the particular distance chosen for the experiment.

Unlike the Cavendish experiment, which basically

determines G by comparing the integral in Eq. (4) with a measured force, the present experiment tests a differential equation which connects the field to a local quantity.

As a result, the new experiment is insensitive to the glo- bal mass distribution of the source. This permits large

geological objects of irregular shape to be used as a

source. The source-detector separation is modulated

periodically to overcome the 1/f noise in the Laplacian

detector and to discriminate against gravity produced by

stationary objects at other distances.

For a point source whose potential is given by Eq. (2), the field equation is modified by

(5)

Of course, the right-hand side of this equation becomes a volume integral for an extended source:

P(x')

(6)

where R = | x - x'l as before. If the Inverse Square

Law should fail at certain distances, the exact functional

form of the gravitational potential could be determined

by integrating the new field equation, Eq. (5).

3. Gravitational Laplacian Detector

The Laplacian of the gravitational potential is, by definition, a sum of three orthogonal gravity gradients (the trace of the gravity gradient tensor):

V2<|) = - V g = - i — (7)

where the x, (i = 1,2,3) form any Cartesian coordinates.

A three-axis in-line component gravity gradiometer*

could, therefore, be employed as a Laplacian detector. A

sensitive superconducting gravity gradiometer to be used

for this experiment has been designed and is under con- struction [17,18]. Here we give a brief description of the

instrument.

Figure 1 is a schematic circuit diagram of a single-axis

in-line component gravity gradiometer. The shaded rec- tangles with arrows represent superconducting proof masses that are confined to move along the direction in- dicated by the arrows. A persistent current /q is stored

with opposite polarity in each of the two loops formed by

pairs of flat superconducting sensing coils. "When each

proof mass is displaced with respect to the coils in response to a gravitational force or a platform vibration,

it causes a current to flow into and out of the persistent current loop due to the Meissner effect. The signals from

the two accelerometers are directly added and subtracted

by two SQUIDs (Superconducting Quantum Interference

Devices) to obtain a common-mode force gi and a force gradient Fu = -dg[/dX[. A Laplacian detector is composed of three such single-axis gradiometers assembled

along three orthogonal directions.

similar null experiment in which a spherical mass shell is used as a Laplacian detector has been discussed by Paik [15] and proposed by Mills

[16] independently.

Figure 1. A schematic circuit diagram of a single-axis super- conducting gravity gradiometer. A Laplacian detector is composed of three single-axis gradiometers repeated in three or- thoqonal directions.

In order to reject the common-mode forces to high precision and realize a very sensitive gravity gradiometer,

the instrument must be fabricated with great care in a number of areas. The suspension springs for each proof mass must be soft along a well-defined axis and be linear. The sensitive axes of the two accelerometers in each gra- diometer must be aligned. The sensitive axes of three

gradiometers must then be aligned orthogonal to each

other and the scale factors matched.

Figure 2 is a cut-away view oi the superconducting

proof mass and suspension. The suspension structure containing eight "folded cantilevers" is machined out of a single block of niobium (Nb). Plane surfaces are lapped

parallel to each other before cantilevers are cut with an

Figure 2. A 55° cut-away view of the niobium proof mass and

suspension of a superconducting accelerometer. Two halves of

the proof mass are threaded into the suspension structure to be supported by eight "folded cantilevers."

602
electric discharge machine. After the two halves of the

Nb proof mass are threaded into the suspension structure, two "pancake" -shaped Nb coils each wound in a

single layer are brought near the two outer surfaces of the proof mass to form an accelerometer. To make a Laplacian detector, six such accelerometers are mounted on

six faces of a titanium cube whose deviations from or- thogonality are controlled to a few arcseconds. Each pair

of accelerometers located on opposing faces of the cube

are coupled by the superconducting circuit shown in Fig.

1. The common-mode rejection error in each axis due to misalignment of the accelerometer axes becomes a second

order effect when the cube is turned into three angular

positions [19]. The uncompensated orthogonality error in the three axes gives rise to a coupHng of 10"^ to zeroth

order Newtonian gradients. The folded cantilever flexure

employed here is expected to give a highly linear spring

due to its pure bending motion, as well as low resonance

frequencies of the proof masses.

In order to find V^c|) by summing the outputs of three

single-axis gradiometers, their scale-factors must be

matched precisely. A support structure which enables'

common-mode balance and scale-factor match of a threeaxis gravity gradiometer is shown in Fig. 3. An "umCylinder

Rotation

Lifter

Support

Cylinder

around the vertical and comparing outputs of the three

gradiometers for the same gravity gradient signal. A set of three positioning pins is located in the support struc- ture for this purpose.

A single-axis portion of the Laplacian detector

described here has been constructed and is undergoing

experimental test. The three-axis support structure is in the final stages of fabrication. An initial laboratory test of the Inverse Square Law is being prepared. The

instrument-noise-limited sensitivity of the prototype Laplacian detector under construction is expected to be ap- proximately 2 X 10"^^ s"^Hz~^. For the earth-orbit ex- periment, the goal is to achieve a white noise level of 2 X 10"^^ s"^Hz~^ in a larger model of the Laplacian

detector. These sensitivities will be assumed to compute

a minimum detectable a as a function of in Section 5.

4. Experimental Procedures

4.1 Laboratory Experiment

In the laboratory scale ~ 10 m, a dynamic gravitational field could be produced conveniently by a moving

source while the detector is kept at rest. In our earlier

paper [20], we considered an experiment in which a ro- tating dumbbell is used to produce a periodic quadrupole

field as in the experiment of Hirakawa et al. [13]. Here we replace the rotating dumbbell by a swinging pendulum. The latter generates a periodic monopole field which falls off more slowly (~R~^) than the quadrupole

field {~R and allows a larger separation experiment.

Figure 4 shows an experimental configuration of the source and the detector. Typical dimensions are

I = 0.16 m, /i — 5 m, u —2 m, and R = 10 m. For the

/////////

Superconducting

Disc

Figure 3. A support structure which enables common -mode

balance and scale-factor match of a three-axis gravity gr adiometer in an "umbrella" suspension.

brella" orientation, in which the three sensitive axes make an equal angle tan " ^ \/2 with respect to the vertical, matches the gravity bias in three gradiometers. For

common-mode balance, the gradiometer assembly is driven vertically at a desired frequency by means of a su- perconducting magnet. The ratio of two supercurrents in each of the three gradiometers is then adjusted until the

differential output caused by the applied linear acceleration becomes sufficiently low. The scale-factor match

between the three orthogonal components is accomplished by rotating the device in precise 120° steps

Figure 4. Experimeyital configuration of the source and the Laplacian detector in a laboratory experiment.

pendulum mass, we are constructing a spherical lead ball of 1600 kg. This vdll be used to test various instruments

and procedures. Eventually the pendulum will carry a 10^ kg stack of lead bricks for the Inverse Square Law ex- periment. If direct transmission of vibration through the

building structure proves to be important, one could at- tempt to cancel the vibration 'using two identical pendula

swinging in opposite directions. The detector could be lo- cated at the midpoint to double the gi-avity signal. This

symmetric source configuration has an additional advantage of producing a uniform field in the middle thus making a distance error bR less important.

603
A completely source-independent null experiment

requires a true Laplacian detector [21] which has a more

sophisticated design. The three-axis gravity gradiometer

that we are constructing at present is only an approximate Laplacian detector which still has a nonvanishing

coupling to higher order spatial derivatives of <{). Hence,

errors in the source geometry do not drop out completely. This residual coupling to the higher order

Newtonian terms arises from the fact that a practical

gradiometer measures a difference over a finite baseline I rather than taking a derivative at a point. For a potential

given by Eq. (2), one can show [20] that the output of the

three-axis gravity gradiometer is

3

1=1 GMl

1 -

21 a{ixE)^e-^ - _c

5 + Y* +

3

I

4- + 0 R

,(8)

where R = (Z, F, Z ) and /e = I R I. Thus, the finite baseline term varies as (l/R f and becomes important for a relatively short distance experiment.

In the laboratory experiment, {l/R)^ — 3 x 10~^. Since the zeroth order Newtonian term is balanced out to

10"^, the second order term needs to be compensated by

computation to about 3% to bring down the errors to the same level. In order to model the Newtonian gravity

correctly, we drive the pendulum to swing in a predetermined plane and read the actual pendulum position using

a shaft encoder mounted at the pivot. The encoder output

is then used to trigger a signal averager following the

Laplacian detector. In this manner, errors associated

vdth erratic pendulum motions are prevented from enter- ing into the signal process. The signal is averaged over many days to improve the signal-to-noise ratio.

4.2 Geological-Scale Experiment

The source-independent nature of the Laplacian experiment permits precision tests of the gravitational force law at kilometer ranges using geological sources. Figure

5 illustrates experimental arrangements for two different sources: a mountain and an ocean tide. When a mountain

is used as the source, the detector could be transported

horizontally, modulating the horizontal distance to the

mountain d {t ). An ocean tide gives the advantage of not having to move the detector at all though the signal fre- quency of 2.3 X 10"^ Hz is well inside the 1/f noise of the

Laplacian detector. The inherent density homogeneity

and the well-defined surface profile of water make the

tide a very attractive source for a geological-scale gravity experiment [22].

LAPLACIAN

DETECTOR

Moving a sensitive Laplacian detector without

deteriorating its performance poses a major challenge in the mountain experiment. The problem could partially be overcome by modulating d(t) in a square-wave fashion

and averaging signals at the two end points while the

detector is at rest. Although the Laplacian detector

would still have to be moved gently so as to remain

within the dynamic range of the SQUID detection cir- cuits, dynamic error sources such as platform vibration

and jitter could be eliminated by this procedure.

For an ocean-tide experiment, the cryostat containing

the Laplacian detector could be fixed on a support tower

above the water level or be lowered to the basin. The

dynamic source of gravitation is the water level differ- ence between the ebb and flow of the tide, h ± u. This can be approximated as a plane sheet of mass with uni- form density and thickness 2u for ranges smaller than

the closest distance to the shore. It is well known that

such a geometry produces a zero Newtonian force gra- dient so that our experiment becomes a double null ex- periment: a null source and a null detector. Since the height modulation {2u ) that is produced by

an ocean tide is approximately 10 m, the resolution in a

deteriorates at > 10 m. It is tempting to move the

detector vertically on a flat plane^ to a larger aniplitude

and improve the resolution at larger distances. In such an experiment, however, one encounters a large modulation of the vertical Newtonian gravity of the earth. Hence it is necessary to improve the common-mode rejection in the vertical direction.

4.3 Earth-Orbit Experiment

The three-axis superconducting gravity gradiometer

described in this paper is under development for a gravity survey satellite of NASA [18]. After a successful test of the prototype model under construction, a larger, more sensitive model may be built and flown in a low al- titude earth orbit to take a high resolution gravity map

of the earth. Such a mission will give an excellent opportunity to test the Inverse Square Law at a distance

range of the order of the earth' s radius and improve the

experimental limit of the Law by several orders of magnitude at iJL-i = 10^ to 10^ km [23]. For the gravitational null experiment, the satellite could initially be launched in an elliptical orbit which

modulates the satellite altitude from /ii to /12 at approximately 1.5 X lO""* Hz as shown in Fig. 6. A full modulaMOUNTAI

^ LAPLACIAN # DETECTOR

l*'iGURE 5. E'xperimental arrangements for two different geological sources: a mountain and an ocean tide. For the mountain

experiment, the detector is moved horizontally to modulate

d(t). For the ocean tide experiment, the tide modulates a verti- cal dista^ice h {t) while the detector is kept at rest.

Figure 6. The Laplacian detector in an elliptical earth orbit.

^An elevator in a rocket launch tower or a tall urban tower could be used to transport the cryostat vertically.

604
tion of the non-Newtonian signal will be achieved if hi — 200 km and h2 > 6000 km. There are several ad- vantages of the earth-orbit experiment over the

geological-scale experiment performed on the earth' s surface. The quiet, zero-g nature of space provides an ideal environment for operation of a sensitive gravity gradiometer. The orbital motion of the satellite gives a continu- ous and full modulation of gravity signals at a reasonable

frequency. Irregularities of the source are partially aver- aged out by the spin of the earth. It is also important

that the satellite itself can be spun quietly in space to eliminate some of the important error terms. The sensitivity of the flight model of the Laplacian

detector is expected to be 10 ""^rE Hz "^'- at the signal fre- quency of 1.5 X 10"" Hz, where Te = 3 x 10"^ s"^ is the

vertical bias gradient of the earth' s gravity. If the signal

is integrated for t = 10^ s (12 days), the instrument

noise level vdll go down to approximately 10"^ Fe implying a resolution of one part in 10^ in the Inverse Square

Law at the earth' s radius Re = 6400 km. An orthogonality error in the Laplacian detector, v^ithout compensation, would prevent a resolution better than one part in

10^. Fortunately, there are ways in which one could eliminate this error to first order and thereby reduce the coupling to the Newtonian term to a level of 10"^'^. One in- teresting approach is spinning the satellite around one of the sensitive axes of the Laplacian detector. Coupling to cross component gravity gradients arising from angular

errors is then modulated at harmonics of the spin fre- quency and can be distinguished from the Laplacian sig- nal which remains at dc. Errors in the Laplacian itself can be shown to be of the second order in misalignment

angles for the particular choice of the spin axis. A more

detailed analysis of the earth orbit experiment will be

published separately [23].

5. Expected Resolution of the Null

Experiment

In this section we discuss the expected resolution of various versions of the gravitational null experiment.

The peak-to-peak amplitudes of non-Newtonian signals

for various sources can be shown to be

2GM a{\x.R)'^ e-^^

1 - {u/Rf

sinh (|xz<) + — cosh iix.u) R

pendulum.

AirG Pw cte sinh {\xu) , ocean tide, (9)

1 2t:G Pe ct (1 + e

IjJi -a

-2ix/eE)

1 + Hi/Re 1 + hz/RE

earth.

where p^, and pe are, respectively, the densities of ocean water and the earth, and the other parameters have been

defined in earlier sections. We have assumed that the

size of the pendulum mass is small compared to or and the earth is a sphere vdth a uniform density pE- The parameter values used for computation are M = W kg, i2 = 10 m and w = 2 m for the laboratory

experiment; h = 10 m, u - 5 m and Ou, - 1.03 x 10^ kg m ^ for the

Re = 6400 km.

m, u

geological-scale

hi = 200 km.

P«' experiment;

hi = 6400

and

km.

Pe = 2.7 X lO'^ kg m"-'^ (surface density) for lo.""' < 10'^ km and pe = 5.5 x 10'^ kg m"-'' (mean density) for s 10^ km. The instrument sensitivities assumed for the three experiments are 2 x 10"^^ s"^ Hz"^ (at 0.3 Hz), 2 x 10-^0 s-2 Hz-i^ (at 2.3 x IQ-^ Hz) and 2 x lO'^^ g-2 Hz'i^ (at 1.5 X 10"^ Hz), respectively, the latter two

coming from the 1/f noise of the SQUID. In addition, an

integration time of t = 10^ s (12 days) has been assumed

for all experiments.

Figure 7 shows the expected resolution lamini of the null experiment as well as limits set by other experiments as

a function of p,^\ The soHd curves represent upper limits

in a implied by previous experiments. Curves labeled

"Panov," "Hirakawa," and "Newman" have been plotted from published results in Refs. [12-14]. The "lunar

surface gravity" curve is the one obtained by Mikkelson

and Newman [9]. The Hmit set by "LAGEOS-lunar rang- ing" has been obtained by comparing two recent data

points in the measurements of the geocentric gravitational constant^: one determined from laser ranging on

near-earth satellites [24] and the other from laser rang- ing of the moon [25]. The regions lying above the solid curves are forbidden by existing data. The shaded area

labeled "Long" is the region to which a is limited by

Long's experiment [10]. Long's positive result consti- tutes an exception in a general trend which favors the In- verse Square Law and is in direct contradiction with

Nevraian's data [14]. The three dashed curves in Fig. 7 represent the resolutions expected from the three versions of the V^(j) experiment considered in Section 4 and

Eq. (9). Notice that a combination of the geological scale and the earth orbit experiment is capable of resolving a

to better than 10~^ in ten decades over the range from

|jL - 1 m to p." 10'^ km filling the gap between the

laboratory and the astronomical scale. In the range of = lo^ km, the Laplacian experiment is expected to resolve the Inverse Square Law to better than one part

in 10^, matching the best limit obtained from solar sys- tem observations for much larger distances.

10^ 10"'

I ^ 10^ 10^ 10^ 10^ 10^ 10^ 10^ 10^ io'° RANGE OF FORCE ^"'(m)

Figure 7. Expected resolution in a of the Laplacian experiment (dashed curves) and limits set by previous experiments

(solid curves) as a function of the range fx"'. The null experiment is capable of resolving the Inverse Square Law to better than 10'" in ten decades of range from 1 m to 10'° m.

^The values of the geocentric gravitational constant in the two measure- ments are GAfg = 398600.44 ± 0.02 Ws~2 at R = 12300 km (semimajor

axis for LAGEOS) and GM e = 398600.461 ± 0.026 km^ s'- at R = 384400 km (mean earth-moon distance).

605
The proposed Laplacian experiment could remove embarrassingly large uncertainties that exist in our present

knowledge of G as a function of mass separation. It is a

null experiment which is capable of suppressing errors associated with the source. In the ocean and in earth orbit, the Newtonian terms could be eliminated completely

by the additional null nature of the tide as source and by

spinning the satellite quietly. As a result, excellent sensitivities are expected for the geological-scale and the

earth-orbit experiment. The laboratory experiment is not completely free from source errors due to the proximity

of the source and the detector. However, it will still be a

useful first step in testing out the techniques of the gravitational null experiment.

In conclusion, a new concept has been developed to test one of the most fundamental hypotheses in the laws

of physics: the scale invariance of the Inverse Square

Law of Gravitation. The experimental examination of the

spatial variation G constitutes a new test of General Relativity. An elegant null detector for the gravitational In- verse Square Law is being constructed using principles

of superconductivity and an intricate mechanical design.

We plan to carry out a series of null experiments in the

coming years.

We gratefully acknowledge extensive contributions of Dr. Vol Moody and Mr. Frank Desrosier in the development of the superconducting gravity gradiometer and the

construction of related experimental apparatus. We have

also benefited from discussions with Dr. Peter Bender

who has pointed out to us recent laser ranging data.

References

[1] C. Brans and R. H. Dicke, Phys. Rev. 124, 925 (1961).

[2] R. V. Wagoner, Phys. Rev. D 1, 3209 (1970).

[3] E. Pechlaner and R. Sexl, Commun. Math. Phys. 2, 227

(1966).

[4] J. O'Hanlon, Phys. Rev. Lett. 29, 137 (1972).

[5] Y. Fugii, Phys. Rev. D 9, 974 (1974).

[6] D. R. Long, Nuovo Cimento B 55, 252 (1980).

[7] Y. Fugii, Nature (London) Phys. Sci. 234, 4 (1971).

[8] D. R. Long, Phys. Rev. D 9, 850 (1974).

[9] D. R. Mikkelson and M. J. Newman, Phys. Rev. D 16, 919

(1977).

[10] D. R. Long, Nature (London) 260, 417 (1976).

[11] H. -T. Wu, W. -T. Ni, C. -C. Hu, F. -H. Liu, C. -H. Yang,

and W. -N. Liu, Phys. Rev. D 19, 2320 (1979).

[12] V. L Panov and V. N. Frontov, Sov. Phys. JETP 77, 1702

(1979).

[13] H. Hirakawa, K. Tsubono, and K. Oide, Nature (London)

283, 184 (1980).

[14] R. Spero, J. K. Hoskins, R. Newman, J. Pellam, and J. Schultz, Phys. Rev. Lett. 44, 1645 (1980).

[15] H. J. Paik, Phys. Rev. D 19, 2320 (1979).

[16] A. P. Mills, Jr., Genl. Relativ. Grav. 11, 1 (1979).

[17] H. J. Paik, E. R. Mapoles, and K. Y. Wang, in Future

Trends in Superconductive Electronics, Ed. by B. S. Deaver et al. (Am. Inst. Phys. Conf. Proc. 44, New York,

1978), p. 166.

[18] H. J. Paik, J. Astronaut. Sci. 29, 1 (1981).

[19] H. A. Chan, unpublished analysis.

[20] H. J. Paik and H. A. Chan, in Proceedings of the Second

Marcel Grossman Conference on General Relativity, Ed. by R. Ruffmi (North Holland, Amsterdam, in press).

[21] H. A. Chan and H. J. Paik, in Proceedings of the Ninth

International Conference on General Relativity and Gravitation (Jena, East Germany, 1980, in press).

[22] F. D. Stacey, Geophys. Res. Lett. 5, 377 (1978).

[23] H. J. Paik, paper in preparation.

[24] F. J. Lerch, R. E. Laubscher, S. M. Klosko, D. E. Smith,

R. Kolenkiewicz, B. H. Putney, J. G. Marsh, and J. E. Brownd, Geophys. Res. Lett. 5, 1031 (1978).

[25] A. J. Ferrari, W. S. Sinclair, W. J. Sjogren, J. G. WiUiams,

and C. F. Yoder, J. Geophys. Res. 85, 3939 (1980).

606
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

The Measurement of G for Small Inter-Mass Spacings*

William C. Oelfke

Department of Physics, University of Central Florida, Orlando, FL 32816

An experimental project is currently under way at the University of Central Florida to measure

the Newtonian gi-avitational constant G for inter-mass spacings r in the range 0.3 cm > r > 3 cm. A

Cavendish balance for measuring the gi'avitational attraction between disk-shaped masses has been

constnacted. This balance functions as a fully automated electrobalance with the incorporation of a mi- crocomputer for the monitoring and recording of data and the manipulation of the balance. A complete

description of the design and operation of this Cavendish balance is presented.

Key words: G {r ); inverse square law; measurement of G; short-range gravitational forces; torsion bal- ance.

1. Introduction

A Cavendish balance has been constructed to measure

the gi-avitational attraction between thin disks in order

to provide a means of determining the gi-avitational constant G for inter-mass spacings in the range 0.3 cm < r <

3 cm. It was pointed out in 1974 by Long [1] that the more recent values of G have been measured at inter- mass spacings of 7 to 10 cm and that although these

measurements specify an accurate G they do not verify

the inverse square law for gi'avity. In addition, historical data and some recent observations [2] seem to indicate

that the gravitational forces do not strictly adhere to an

inverse square law. Theoretical arguments by Fujii [3] and O'Hanlon [4] followed by those of Scherk [5] tend to support these ob- servations by suggesting that a massive spin 1 gi'aviton

may exist, along with the massless spin 2 graviton, that

could give rise to an overall gi-avitational potential of U(r) = -{GM/r )il - ^e'""') .

The range, m "\ of this force is unknown, however it may be assumed not to have a value around 10 cm be- cause of the consistency of G measurements in this range. It may be ruled out, as well, in the range of orbital distances by astronomical observations. Experimental

searches [6, 7] are therefore presently under way for short range forces consistent with in the ranges

m < 10 cm and 10 m < m "-^ < 1 km.

2. Experimental Design

This experiment is designed to measure gi'avitational forces between masses whose separations can be made as small as 0.3 cm. A flat disk geometry was chosen in order

to provide a measurable gi'avitational force at these

inter-mass spacings. In addition the Newtonian gravitational forces between such closely spaced disks has a much smaller gradient than the comparable force

between spheres and as a result a Yukawa force component, if it existed, would become much more apparent.

This is illustrated in Fig. 1 where a pure Newtonian

gravitational force of attraction between the disks is compared with a hypothetical force with a Yukawa component of range m = 1 cm.

.2-

0-1 , , , , , , , ,

, disk separation z

Figure 1. The plot of the gravitational force of attraction

betiveen two thin disks as a function of disk separation z. The

Neivtonian force is plotted as a solid curve whereas the dashed

curve shows the effect of adding the repulsive Yukaica type

tennfor the case of = 1 cm.

The gi'avitational balance is a symmetrical torsion bal- ance of the Cavendish type with two equal suspended

masses, each a disk 16 cm in diameter and 0.06 cm thick. Gravitational forces are exerted on these suspended

disks by a sjTnmetrical pair of disks, 20 cm in diameter

and 0.3 cm thick. The balance, shown in Fig. 2, is 75 cm

high with a torsion fiber length of 30 cm and a loaded

torsion beam mass of 300 g. Each suspended disk has a mass of 100 g and each movable disk, a mass of 680 g. The suspended disks, torsion beam, and torsion fiber are

shielded electrostatically and the entire balance is en- closed in a vacuum bell jai' and evacuated to 10"* Torr by

''Supported by a Joseph H. De Frees grant of the Research Corporation an ion pump.

607
The torsion balance is operated as an electrobalance so that any gravitational force tending to pull the torsion

balance away from its equilibrium position is continuously balanced by electrostatic forces exerted by a pair

of precision guard-ring capacitors located at the left end

of the torsion beam as shown in Fig. 2. These precision

capacitors can exert clockwise and counter-clockwise

torques on the beam and provide a means for accurate measurement of gravitational torques. The voltages on

these capacitors are controlled through digital-to-analog converters by a computer. Subroutines have been

developed which provide for accurate measurement of the

gravitational force while adjusting for torsion fiber drift and contact potential differences in the capacitors.

Hi 1^ RF DOUBLEBALAllCED MIXER

LO

BEAM

Figure 3. The beam-position transducer system. Movement of

the torsion beam, changes the value of the electrical capacitance

of the 500 MHz resonator producing a relative phase shift between the RF and the LO inputs to the double-balanced

mixer.

Figure 4. The 500 MHz resonator showing the input and out- put coupliyig liyiks, the inductance loop, and capacitors (partially hidden behind the ground plane).

Figure 2. The Cavendish balance for the measurement of short-range gravitational attractions.

Beam position is sensed by a uhf resonant transducer

system producing a voltage proportional to the angular

displacement of the torsion beam from equilibrium. This

transducer system, shown schematically in Fig. 3, is sensitive to angular beam displacements of 0.1 s of arc and

provides a sufficiently large dynamic range to facilitate servo-locking the torsion beam to a stable null in an en- vironment of floor .noise and fiber drift common to all Cavendish balances.

The uhf transducer shown in Fig. 4 is a unique

500 MHz resonator that consists of an inductance ring

folded so that it is parallel to two semicircular, coplanar

capacitance plates. A large capacitance of about 20 pF is established between these plates and the right end of the

torsion beam which serves as a ground plane, sandwiched

between the inductor and the capacitors. The electrical Q

of this resonant circuit is 10^ and when the resonator is operated as a transmission filter within its passband a small shift in the beam groundplane changes the resonant frequency of the transducer and produces a large phase

shift in the output 500 MHz signal. The double-balanced

mixer provides a means of detecting this phase shift and

converting it to a voltage that can be read directly by an

analog-to-digital converter.

608
Finally the manipulation of the large disks, the control

of the electrobalance servo loop, correction of baseline

drift, and the taking, processing and recording of data

are all controlled by an LSI 11 microcomputer. The

overall function of this system is represented in block di- agram form in Fig. 5. A clock program establishes the

sequence in which the movable disks are positioned rela- tive to the suspended disks and the sequence in which

GRAVITATIONAL

BALAiCt

LSI - 11

MICROCa'-lPiJTER

<5>

A/D DISK-ARM

POSITIONING

D/A SUBROlfTINE

DISK-ARM

DRIVE MOTOR

LINEAR

BEAI'l-POSlTION TRANSDUCER A/D

r- D/A

SEQUENCING

PROGRAM

BEAM POSITim

SUB^ffffE

DATA

ANALYSIS

SUBROUTINE

D/A BEAM POSITia^l ftASURING

SUBROUTINE

DISK

FILE

Figure 5. A block diagram of the gravitational balarice system

and microcomputer control.

the various measurements are taken. The movable disks are first positioned in the centered, symmetrical position

and a series of measurements are made by the beam position measuring subroutine determining gap distances of

the two guard-ring capacitors, their contact potentials,

and force necessary to hold the beam at its electrical null position. This measurement sequence is repeated twice

during each measurement cycle as a baseline check and

to provide calibration of the gravitational force F{z).

The sequencing program next positions the movable disks

in a clockwise position a predetermined distance z away

from the suspended disks. The position z is read from a precision potentiometer and the gravitational force is read as a balancing voltage generated by the beam position servo subroutine. The disk arm is again centered for a baseUne check and finally positioned a distance z away

from the suspended disks and the force measurement is repeated. The measurement cycle is repeated a number

of times for the same z and then a new series is initiated for a new z value. In this way a data set is generated

from which F {z ) can be determined and G (r ) can be de- duced.

The final assembly of this balance system was completed just before this conference. Work is now under

way to begin the program of software testing and calibra- tion and by the end of 1981 we will have begun making

measurements of G. References

[1] D. R. Long, Phys. Rev. D, 9, 850 (1974).

[2] F. D. Stacey and G. J. Tuck, these proceedings.

[3] Y. Fuji!, Nature Phys. Sci. 234, 5 (1971).

[4] J. O'Hanlon, Phys. Rev. Lett. 43, 103 (1979).

[5] J. Scherk, Phys. Lett. 88B, 265 (1979).

[6] J. K. Hoskins, R. Newman, J. Schultz, and R. Spero, these proceedings.

[7] H. A. Chan and H. J. Paik, these proceedings.

609

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Tests of Gravitation and Relativity*

Robert F. C. Vessot

Smithsonian Astrophysical Observatory, Cambridge, MA 02138

The weak but all pervasive force of gravity governs the overall behavior of the universe. The

dramatic discoveries in astrophysics place gravitation in the forefront of interest and raise questions about the range of validity of Einstein's General Theory of Relativity and at what level it may prove to be a classical limit to a quantum theory. At present, after more than seventy years, the General Theory continues to be the most acceptable description of space-time despite continuing tests to chal- lenge it by searching for contradictions to predictions based on this theory.

The kinds of tests being made can be roughly classified in terms of the contradictions to theory they

seek to expose.

1. Tests to look for variations in the conventional "constants," for example, the constancy and iso- tropy of the velocity of light and the time dependence of the gravitational constant.

2. Tests to look for variations in the physics of the flat space-time of Lorentz inertial frames, which

include studies of the second-order Doppler effect, length contraction, and mass variation.

3. Tests to confirm that gravitation is described by a metric theory in which a Riemannian geometry

describes space-time such that everywhere, locally, the laws of special relativity apply.

4. Tests to evaluate the validity of theories proposed as alternatives to the General Theory. To as- sess these alternatives, theorists have devised a score-keeping system, which, in essence, is a theory of theories called the Parameterized-Post-Newtonian (PPN) metric theory. This provides a valuable means to test theories by clearly establishing their observable experimental consequences.

5. Tests from the anticipated detection of gravitational waves. The recent discoveries in astrophysics

of superdense matter from gi'avitational collapse and the strong probability of the existence of black

holes have kindled a high level of interest in the search for gravitational radiation which, when

discovered, will open a totally new method of astronomical observation. Determining the characteristics of this radiation, i.e., its velocity of propagation and polarization qualities, will provide further

critical tests of general relativity.

The chief purpose of this paper is to review the results of recent tests and assess the status of experimental gravitation, concentrating on the last three categories outlined above. The status of applicable

present technology and developments for future measurements will be discussed. Key words: experimental tests of gravitation; General Theory of Relativity tests; relativity.

1. Introduction

The physics of gravitation deals with the weakest of the four known particle interactions. Though by far the

weakest force, the effects of gravity grow relentlessly

with mass and pervade all space-time, and they literally

encompass the whole universe. The dramatic recent

discoveries in astronomy and astrophysics have

heightened our interest in gravitation and raised questions as to the validity of the Einstein General Theory of Relativity when it is expected to apply to extraordinarily

dense matter and extremely high gravitational fields. Questions have been raised about the range of validity of

this theory and at what level we will find it to be a classical limit to a quantum theory.

After well over half a century, the General Theory con- tinues to be the most acceptable description of space- time and to many questioning individuals appears to have

almost become dogmatically accepted as a basis for theoretical research. At present, theoretical work has far outdistanced experimentation which, ovdng to the ex- tremely small manifestations of the General Theory,

makes testing relativistic gravity very difficult.

*This work was supported by the National Aeronautics and Space

Administration and the Smithsonian Institution.

Historically, the early tests of light bending from

eclipses of the sun, measurements of the perihelion of mercury, and the redshifting of spectral lines at the sun's surface gave little more than qualitative confirmation to the new theory. These tests, however, were crucial to its early acceptance and established Einstein's

General Theory as one of the true masterpieces of human

achievement. However, gravitational theories must, in the long run, be guided by experimental facts and it is only through the experimental approach that ultimate ac- ceptance can be gained. In other branches of physics, experiment and theory

generally develop concurrently, each guiding the pro- gress of the other. Gravitation is an exception because of the lack of instrumentation sensitive enough to provide

clean, accurate, hard-core facts. We have been limited by the levels of mass and velocity available to do truly definitive experiments in the lab- oratory, as distinguished from astrophysical observations. In the case of such observations, we clearly cannot

use gravity theory as a tool for unraveling physical proc- esses and, at the same time, expect the observation to provide a clear cut test of the gravity theory. This ambiguous position, at best, can lead us to a confirmation of the self-consistency between the physics of the observed

effects and the theory of gravity we have employed. The

theoretical development of gravitational physics contin611
ues despite the fact that no soUd evidence has yet been

found to invaHdate the General Theory. The need for such evidence becomes more and more essential as star- tling astrophysical data accumulate, and calculation and

speculation continue in our efforts to understand the new

wonders that we observe.

The recent advances in technology give us considerable

optimism in performing more crucial tests of gravitation.

Atomic clocks, very long-baseline interferometry, inter- planetary radar, laser and transponder ranging to space

probes and planets, superconducting gyroscopes and

other low-temperature techniques, provide means for making new instruments of previously unimaginable sensitivity. The spectacular advances in space technology

have extended the dimensions of our laboratory to en- compass the entire solar system—and beyond, and allow us to use massive bodies like the sun and planets as test objects for laboratory experimentation. As Misener,

Thorne, and Wheeler [2] have said, "General Relativity

is no longer a theorist's Paradise and an experimentalist's Hell."

It is clear that a rational testing program should, to a large extent, be guided by theory. However, in the case

of gravitation it is more often governed by what is possi- ble technologically and feasible economically. Also, we

should not be blind to the possibilities that nature may

hold some surprises, and opportunities to perform seemingly redundant tests by using disparate techniques

should not be neglected.

There is little question that in the near future the best,

clean, tests of gravitation will involve space-borne in- strumentation, both to make use of the available massive

bodies in our solar system and to escape the limitations

placed on terrestrial instrumentation by vibration and

earth tidal effects and the limits of signal propagation

through the earth's troposphere and ionosphere. We can

also expect that the use of space techniques in astron- omy, over the whole electromagnetic spectrum from x

rays to very low frequencies, will lead to discoveries

posing new questions that will make even further

demands on gravity theory.

At present, the most exciting prospect for the future is the anticipated detection of gravitational radiation. This

would open up a totally new era of astronomy by providing data from processes completely different from the

electromagnetic generators we have always observed.

1.1 Criteria for a Valid Theory of Gravitation

In seeking some orderly way to summarize gravity ex- periments, we should first look at the underlying criteria for a viable theory. Thorne and Will [3] give criteria for the viability of a gravitational theory as follows:

1. It must be complete—it must be capable of analyzing from first principles the outcome of every experiment of interest. The theory must

incorporate and mesh with a complete set of electromagnetic and quantum mechanical laws which can be used to calculate the detailed behavior of bodies in a gravitational field. (Currently this does not extend to the ex- tremes of quantum gravity.)

2. It must be self consistent. A theory must

predict uniquely the outcome of every experi- ment.

3. It must be relativistic. If gravity is "turned

off the nongravitational laws of physics be- come those of special relativity.

4. It must have the correct Newtonian limit. In the limit of weak gravity fields and slow motion, we should see convergence to Newtonian

physics.

5. It must embody the weak equivalence

principle—otherwise known as the Universality of Free Fall, the experimental fact that all bodies fail in the same way, regardless of their composition.

6. It must embody the Universality of Gravitational Redshift. These criteria provide guidance in classifying the types

of tests we can perform. Criteria 1 and 2 will ultimately be

met only by accumulation of experience of every possible

kind. Criterion 3 requires that the nongravitational as- pects of physics be governed by the Lorentz group as, for example, manifested by the everyday high-precision veri- fication of four momentum conservation and other relativistic laws of kinematics in high-energy accelerators. These particle physics tests, however, are very strictly lo- cal and not global in their extent. Criterion 4 expresses

the facts accumulated by centuries of solar system observation and the works of early astronomers and mathematicians culminating in the monumental Principia of Newton

(1687) [4]. This system of physics is fundamental and gives

the limiting description under conditions of weak gravitational fields and slow motion. Criteria 5 and 6 are state- ments that can be summarized as the Einstein Equivalence Principle, which he first enunciated in 1907 as a generalization of the observed proportionality between gravitational and inertial mass. This principle asserts that

there is no way of distinguishing locally between a gravitational field and an oppositely directed acceleration. One

of the consequences of the Einstein Equivalence Principle

is the so-called gravitational redshift. The Einstein Equivalence Principle (or strong

equivalence principle as it is sometimes called) leads

directly to the concept of a space-time that can be thought

of as consisting of a manifold of freely falling Lorentz

frames in each of which apply the laws of special relativity. This postulates that space-time is endowed with a metric, that the trajectories or world-lines of falling test bodies are geodesies of that metric, and that nongravitational laws of physics in any free-falling frame reduce to those of the Special Theory of Relativity.

We are thus led to a four-dimensional picture of space- time where, locally, the coordinates in four-space are rectilineal and orthogonal. Over larger excursions in space- time there occurs a bending of the coordinates that depends entirely on the distribution of matter and energy. This concept of how a spacetime continuum is shaped by

the presence of mass and energy is Einstein's entirely original view of the structure of our universe.

Testing this concept of space-time presents a formidable

challenge to metrology since it affects all aspects of space

and time. In it we identify two "constants," the velocity

of light, c, and the universal constant of gravitation, G. The nature of these two quantities throughout the space and time of our universe is fundamental to the study of cosmology and to the development of gravitation and rela- tivity theory.

The following review of tests of gravitation and relativistic theories is intended to sketch an overview of the

theory and describe the trends of the testing activity

made possible by present technology. To those who wish

to read further about the status of theory and experiment

I recommend the recent (1979) book celebrating the cen- tenary of Einstein's birth published by the Cambridge

612
University Press, edited by S.W. Hawking and W. Israel

[5, 6], entitled General Relativity. An Einstein Centenary

Stirvey; and C. M. Will's new book, Theory and Experiment in Gravitational Physics, by the same publisher.

1 .2 Tests of Gravitation

The tests that have thus far been done can be classified

in terms of the contradictions and inconsistencies they

seek to expose.

1. Tests to look for variations in the conventional "constants," for example, the constancy and isotropy of the

velocity of light and the time dependence of the gravitational constant.

2. Tests to look for variations in the physics of the flat space-time of Lorentz inertial frames, which include

studies of the second-order Doppler effect, length contraction, and mass variation.

3. Tests to confirm that gravitation is described by a metric theory in which a four-dimensional Riemannian

geometry describes space-time such that everywhere, lo- cally, the laws of special relativity apply.

4. Tests to evaluate the validity of metric theories pro- posed as alternatives to the General Theory. To assess these alternatives, theorists have devised a score- keeping system, which, in essence, is a theory of theories

called the Parameterized-Post-Newtonian (PPN) metric

theory [1, 7]. This provides a valuable means to test theories by clearly establishing their observable experimental consequences.

5. Tests from the anticipated detection of gravitational waves. The recent discoveries of superdense matter from

gravitational collapse and the strong probability of the existence of black holes have kindled a high level of interest

in the search for gravitational radiation which, when

discovered, will open a totally new method of astronomical

observation. Determining the characteristics of this radiation, i.e., its velocity of propagation and polarization, will provide further critical tests of general relativity.

2. The Constancy of Constants

The question of the behavior of the gravitational constant, G, and the velocity of light is central to any theory

of gravity and relativity. Since the focal point of this conference is on this type of question and because of the

many papers given on these topics including the excellent

review by H. de Boer that precedes this paper, I feel that I can justifiably give only a very rough overview of the behavior of the velocity of light and the universal

gravitational constant.

Historically, behavior of the velocity of light has been

the origin of our modern concepts of relativity. The

Michaelson-Morely ether drift experiments and its suc- cessors provide the early experimental evidence for the

Special Theory of Relativity. Central to this evidence is the question of the independence of c on source velocity,

its constancy, and isotropy regardless of wavelength. Recent experimental evidence is strong that the velocity of

light is independent of the relative velocity of the source and observer. If we take as a hypothesis that

c' — c + kv, the criterion for independence is given by the limits that can be set on k.

Measurements of 7 rays from -rr" mesons at energies

greater than 6 GeV are reported by Alvager et al. [8] who find

k (-3 ± 13) X 10" From astronomical observational data taken from a binary x-ray pulsar, Brecher [9] concludes that

/c < 2 X 10"^. Another postulate related to the velocity

of light is that it is a hmit to the velocity of massive particles. Tests have been made of the relative velocity of 7

rays and high-energy electrons in the 10-20 GeV range. Here the prediction is that

-5 X 10"

where is the velocity of 7 rays and Ve is the velocity of the electrons. Guiragossian et al. [10] confirm that (1 - t'e/c) ~

±2 X 10"l Brown et al. [11] report (1 - vjc) ~ ( 1.2 ±

2.7 ) X 10"^ for 11-GeV electrons and visible light and

also measure the relative velocity of light at visible wavelengths and of 7 GeV 7 rays giving a result C7GeV - (1.8 ±6) X 10"^

A recent and very elegant measurement of the isotropy

of the velocity of light has been reported by Brillet and

Hall [12] who have used a sophisticated laser technique

to search for an ether drift. Their work and its various

interpretations will be discussed at this conference. Their

conclusions are that the measurement of the length of a

cavity resonator using electromagnetic signals is constant

to within a few parts in 10^*. By making various hypotheses about the motion of this laboratory experiment

in the universe we can arrive at very impressive limits on the validity of the Lorentz transform, which is fundamental to special relativity.

The question of the behavior of the gravitational con- stant has arisen from speculations relating to the expansion of the universe and models have been suggested that

would necessitate a variation of G with time and with position in the cosmos and that there are possible violations

of the inverse-square law of gravitational attraction.

Again, since there are many excellent papers addressing this question, I shall only report the most recent

results from lunar occultation data and from planetary

radar measurements.

From Lunar Occultation

G_

G

< 4 X 10""/yr Morrison [13]

= (_3_6 + 18) X 10""/yr Van Flandern [14, 15]

G

From Planetary Radar Measurements < 4 X 10-i°/yr Shapiro et al. [16] < 1.5 X 10"^°/yr Reasenberg and Shapiro [17]

3. Tests of Special Relativity

The second category of tests is related to searches for violations of special relativity and comprises a number of subgroups that investigate the parameter 7 = (1 —

u2/c2)-l/2.

613
3.1 g-2 Tests

These tests are based on the concept [18] that the fac- tor 7 = (P/mo) dp/clE apphes differently from the cyclo- tron frequency coc, of an electron (lepton) in magnetic

field B (where m, = eB/ymoc), and the g-2 precession

frequency = {geB/2moC) + (1 - y)(DcThe g-2 experiments determine the quantity (cos - cj^) (e5/moc)"\ which, in the conventional theory, equals

ig - 2)12 = a. With the assumption that there is a dif- ferent y for electron dynamics and electron kinematics,

the quantity a becomes g/2 - y/y. The ratio y/y is writ- ten as an expansion y/y = 1 + Ciiy - 1) -I- ... , which

makes y/y = 1 in the limit 7^1 (low-velocity Hmit).

The term Ci is used to define a figure of merit with one

or more values of 7, made at differing electron energies.

to provide combinations like Ci = (a*^' - a'^0/(7'

For a given resonance linewidth, the larger the range of

7, the more sensitive the test. A summary of the figure of merit parameter, C i, made

from various tests is shown in Table 1 from Cooper et al. [19].

The pioneering experiments of Ives and Stilwell [25] using fast-moving hydrogen atoms have been repeated by

Mandelberg and Witten [26] and confirm the predictions

to about 5%

Experiments that observe the shift in a resonance owing

to thermal motion of the source have been made by Pound

and Rebka [27] as part of the necessary corrections of the resonance frequency of the '^^Fe Mossbauer radiation used

to measure the gravitational redshift. Similar thermal motion effects occur in hydrogen masers [28]; however, these

results assumed the validity of special relativity and were

used as corrections to the frequency data and are not a convincing test of special relativity.

Far more sensitive tests have been reported in the last few years from muon dec,ay measurements made with the

CERN storage ring and reported by Bailey et al. [29]. This is the most recent of a long series of high-energy

particle experiments to test the extension of the decay

time involving many different particles.

Table 1. Tests of g-2.

Particle 7*^' ^(2) References

|JL~, [i.^ 12 29.2 1.4 ± 1.8 X io-« Bailey et al. [20,21]

e 1 1.2 ( -2.6 ± 1.8) X 10-" Van Dyck et al. [22] Wesley and Rich [23]

e 1 2.5 X lO^ ( -1.0 ± .8) X 10-10 Cooper et al. [19]

3.2 Mass Increase Tests, m mo7

Measurements of the mass increase for 385 MeV

(v/c —0.7) protons have been reported by Grove and Fox

[24]. They measured the e/m ratio of 385-MeV protons in a synchrocyclotron and confirm the relation

1/2 e _ e m. m. 0

1 ,2 By putting the cyclotron frequency and the particle's

radius of movement, r, in terms of velocity, v, we have

em.emo

1 - ^ e 1

Their results are -0.0006 ± 0.001 for the quantity

{e/nii - e/m-i) (6/^2)"^ for 385-MeV protons.

3.3 Tests of Second-Order Doppler Effects or Time

Dilation, t = T07 These tests generally involve the measurement of an

electromagnetic signal, either in emission or absorption,

of a particle that is in rapid motion with respect to the

observer. The relation between the shift in the resonance

frequency, or the extension of the decay time, and the

relative velocity is experimentally determined and compared with predictions from special relativity.

Table 2 from J. Bailey et al. [29] summarizes these

results using the parameter (Ti/71 - T2/72) i'^i/yz)'^ = F as a figure of merit. From this recent work we see that time

dilation is confirmed at the 10"^ level. A proposed high precision measurement of time dila- tion is reported in the work of Nachman, Rayman, and

Hall [39] who will make time-dilation measurements us- ing two-photon absorption resonance in a fast beam of metastable helium. They expect to obtain a 500-fold improvement over the CERN storage ring results discussed

above. Their work will be reported elsewhere in these

proceedings.

These tests clearly establish that the Special Theory of Relativity is on solid ground and it appears that this foundation of gravity theory is becoming more secure as further testing is done.

An interesting observation of the meshing of the Special Theory of Relativity with the Einstein Principle of Equivalence is discussed in the following section where

the hydrogen maser space-probe relativity experiment is described. Owing to the 10,000-km altitude and very

nearly free fall to the earth's surface, values of v/c of ap- proximately 10^^ were encountered. In accounting for the

total frequency variation during the flight, second-order

Doppler effects at the 10"^° level are included in the relativistic prediction. As will be seen, the resulting overall confirmation of the predictions made from general rela- tivity was at the 70 x 10"^ level, and can be interpreted

as a test of time dilation over very nonlocal conditions.

614
Table 2. Second-order Doppler effects or time dilation t = j^y. Particle-decay experiments, criterion F = Tti/^j - ^2h2>l{-^2h'2)

Particle 72 ^ x 10^ Reference

TT- 1.0 2.44 -2.5 ±0.9 Nordberg ei ai. [30] Ayres et al. [31,32]

Dunaitsev et al. [33] K- 1.0 3.38 0.9 ±0.3 Ott and Pritchard [34] 4.17 Lobkowicz [35] Kj> 1.63 15.2 0.5 ±0.6 Carithers et al. [36] 20.2 0.2 ±0.7 Skjeggstad et al. [37] Geweniger et al. [38]

ix+ 1.0 29.3 0.02 ± 0.09 Bailey eiai. [29]

4. Tests of the Metric Hypothesis

The Einstein Principle of Equivalence asserts that

there is no way, locally, of distinguishing between a

gravitational field and an oppositely directed acceleration. The traditional argument used to describe this principle for the behavior of hght rays is shown in Fig. 1, where we consider the behavior of the frequency of a sig- nal from a transmitter on the left-hand wall of the lab re- ceived at the right-hand wall a distance f away. If we

accelerate the lab to the right with a bulldozer, the sig- nal received will be at a lower frequency because the

right-hand wall received the signal at a time t = t/c after it was transmitted during which time the receiver

had a velocity Av = aj away from the transmitter. The

Doppler shift, A/, from this velocity is A/// = Av/c =

a t'/c, where a is the acceleration of the lab. According to the Einstein Principle of Equivalence, if we tip the lab on

end we should see a shift A/// = g t'/c where g is the lo- cal acceleration due to gravity. This shift is often referred

to as the gravitational redshift. The question is, does a

act in the same way as gl

A very elegant test of the equivalence principle in a lab (minus bulldozer) was done using the extremely narrow

TRANSMITTER RECEIVER

LABORATORY

At = —

Av = a At= a;

CONSTANT ACCELERATION = a

TRANSMITTER DOES

Af Av g;

f " C - c2

Af _

f WHEN WE REPLACE

a BY g FROM GRAVITY ? LABORATORY

RECEIVER

Mossbauer effect emission and absorption characteristics

of ^^Fe gamma rays at a tower within Harvard's Jeffer- son Physics Laboratory by Pound, Rebka, and Snider

[27, 40]. Their results have confirmed the hypothesis at the 1% level. Another, but decidedly nonlocal, test of the gravitational redshift was performed [41] by launching a hydrogen maser oscillator (or clock) aloft in a rocket to an alti- tude of 10,000 km and comparing, with microwave signals, the frequency of the maser with other masers on

earth. The first-order Doppler effects from the rapid

changes in the propagation paths were canceled by a transponder system that measured the go-return (twoway) Doppler cycles, dividing the number of these cycles

by two and subtracting them from the cycles in the one- way microwave link that connected the probe clock to earth clocks as shown in Fig. 2. Since the probe had a high relative velocity vdth respect to the earth station

and since the earth station (located in Florida) was in motion owing to the earth's rotation, other effects of rel- ativity must be considered. The predicted relationship is given in geocentric inertial frame as follows: A/ ^

- <^e _

I - - l2 f 2c

Here (J)s - <i)e is the Nevi^onian potential between the

spacecraft and earth station, rge is the vector joining the

HHERRm ISUWD. aORIEM

GROUND STATION

COPPLEB TRACKING DATA

Figure 1. Illustration of gravitational redshift. Figure 2. Microivave system used rocket-home redshift experiment.

in the 1976 NASA-SAO

615
spacecraft and earth station, v.s and v^, are the velocities of the spacecraft and earth stations, respectively, and

is the acceleration of the earth station owing to the

earth's rotation.

The agreement of the measured data was within 70 x

10"'' of the total effect predicted in the above equation.

There are various ways that this result can be inter- preted. For example, if we affirm the correctness of the

second term giving the second-order Doppler effect and assume that the third term correctly accounts for a Doppler shift resulting from the change of velocity of the

earth station during the flight time, we can ascribe all the error to the equivalence principle and confirm it to approximately 140 x 10^^. (Since the redshift and

second-order Doppler terms are very nearly the same in magnitude, the fractional error in the total effect, now

half as large, is doubled.)

Alternatively, if we place all the error in the secondorder Doppler effect and assume the equivalence principle and the earth station term are correct, we have con- firmation that the second-order frequency shift is correct within 140 x lO"*". This conclusion is vahd over space- time intervals that have considerable curvature, which is a different situation than obtained for the purely local tests described earlier. Yet another interpretation of this data is possible if we

ascribe all the error to a possible anisotropy of the velocity of light during the experiment. Since this experiment involved a substantial change of velocity and

depended on the cancellation of the Doppler effect in the

one-way path, by taking an average of the two-way

Doppler and subtracting it from the one-way frequency, a

residual Doppler signature would be present in the data

if the velocity of light in the downlink and uplink signals was different. Under the above assumptions, a limit of Ac/c of ±3 x

10"^ can be set for the isotropy of the velocity of light.

The success of the Doppler-canceling techniques used

in this experiment leads us naturally to think of other ex- periments in space with clocks, such as measuring the

redshift to second order in a close-approach solar mission, and the use of a clock as an accurate distancemeasuring tool. With the current stability of 6 x lO"^*" for 1-h measuring intervals available from existing hydrogen masers [42], we can think of this in terms of a change of Doppler velocity Av Ic - If at a level where

Ay is about 2 x 10"'^ cm s"^ when compared over hourly

intervals. This sensitivity offers the possibility of detecting gravitational waves of very low frequency by looking

for small anomalies in the motion of spacecraft at dis- tances of many astronomical units from earth.

4.1 Universality of Free Fall as Historically Recorded

Tests leading to information that confirms the metric

hypothesis begin in the early 15th century with the apo- cryphal experiment of Galileo who is said to have

dropped a cannon ball and bocce ball from the Leaning

Tower of Pisa. However, there are records that Galileo

did perform dropping tests and also tests on an inclined

plane to dilute gravity. The results are stated as follows

by Galileo: "The variation in speed in air between balls of lead, gold, copper, porphyry, and other heavy materials is so slight that in a fall of 100 cubits (say 46 meters)

a ball of gold would surely not outstrip one of copper by

as much as four fingers. Having observed this, I came to the conclusion that in a medium totally void of resistance, all bodies would fall at the same speed" (see Ref.

[2], p. 16). Later, Isaac Newton repeated these tests with pendulums made of different materials and, in Principia, states in Definition I, "The quantity of matter is the measure of the same arising from its density and bulk

conjointly" ... and later ... "It is this quantity that I mean hereafter everywhere under the name of body or mass, and the same is known by the weight of each body,

for it is proportional to the weight, as I have found by

experiments on pendulums, very accurately made, which

shall be shown hereafter" [4]. Here began a long series

of experiments looking for a discrepancy of the proportionality of inertial mass to gravitational mass. Table 3 shows the trend of some of the tests of the Weak

Equivalence Principle [43] given in terms of y\' = 2(aA -

aBVidA + cib), where a a is the acceleration of substance

A and ae is the acceleration of substance B. These tests of the universality of free fall, or Weak

Equivalence Principle, are very important and a substantial effort has been made to connect the weak principle to the Einstein Equivalence Principle. This was begun by

the late Leonard Schiff [50] and was continued by Light- man and Lee [51] in 1973 who derive proofs under

specific mathematical frameworks for. certain restricted

conditions. This research has led to an understanding of the electrostatic and magnetostatic self-energies and

their contributions to the gravitational and inertial mass

of bodies. The resulting formalism points out possible de- viations in the manner in which the electromagnetic laws

and equations of motion mesh with gravitational fields if these laws do not follow the metric hypothesis. These de: viations are described in terms of parameters Fi and Aj,

etc., for each order of the gravitational potential when

expanded as a power series. These parameters describe

the electrostatic and magnetostatic self-energy variation

with potential, and are zero for metric theories. By making high precision resonators or oscillators depending on

different physical processes, we can see how their fre- quencies are affected by gravitational potential by ex- pressing their frequency dependence on gravity potential as prescribed by the nonmetric hypothesis,

A/ // = (1 + a)Act)/c . Here, a is the measure of the non- metric behavior or lack of proportionality in the redshift

vs. gravitational potential relation for a particular clock:

a = -3Fo + Ao for hydrogen hyperfine structure

as in the hydrogen maser

a = --^ (3Fo + Ao) for an electromagnetic cavity

2 resonator (superconducting cavity

oscillator) a = -2Fo for a high Q, dielectric crystal.

According to this hypothesis we can interpret the data

from the various tests of the Weak Equivalence Principle

shown in Table 3 and relate the data to tests that measure other parameters. Table 4 from Will [43] shows the

limits of validity of the parameters.

Recent work by Will [43] has shown that the measure- ment of gravitational redshift is a valuable resource for gravity theory. The basis for this is the following conjecture: "Every nonmetric theory of gravity predicts a gravitational redshift .which depends on the nature of the

clock whose redshift is being measured." This can also be stated as, "Any complete, self-consistent and relativistic theory of gravity that embodies the universality

of gravitational redshift is necessarily a metric theory"

616
Table 3. Universality offree fall—Weak Equivalence Principle.

Method and substances Limit on t\ Reference

Till Pendulums or

n various

• substances \^ 1 „^ 1 A-3 Newton |4J

Pendulums of various substances 2 X 10"^ Bessel [44] Torsion balance using various substances

5 X 10-9 Eotvos et al. [45]

Torsion balance using aluminum and

gold

10-11 Dicke [46]

Torsion balance using aluminum and 10-12 Braginsky and Panov [47] platinum

Projected experiments in free fall in an earth-orbiting satellite are expected to reach limits of lO"'** Everitt [48] Worden [49]

Table 4. Lightman and Lee [51] connection between the weak and strong

Principles of Equivalence [^3].

Parameter Limit Experiments

To 4 X 10-10 Eotvos-Dicke-Braginsky (EDB)

Electrostatic energy Ao 6 X lO-*5 EDB magnetostatic energy

Ti 4 X 10-2 EDB electrostatic energy

Al 600 EDB magnetostatic energy

SPo - Ao 2 X 10-4 Vessot et al. [52]

To + Ao 10-5 Hughes [53] Drever [54] f

(ro - Ao) <2 X 10-2 From H vs. SCSO test results, Turneaure et al. [55]

(from Will [56,57]). An alternative statement, with redshift replacing universality of free fall, is the Schiff Conjecture [58] (p. 343), "Any complete and self-consistent

theory that obeys the weak equivalence principle must

obey the Einstein equivalence principle." The conjecture

in quotes is the classical analog of Schiff s original quantum mechanical conjecture, reformulated by Thorne,

Lee, and Lightman [59]. This conjecture has led to experiments with clocks depending on various physical principles. In 1978 a series

of experiments were conducted at Stanford University by

Turneaure, Mattison, and Vessot, to compare the fre- quency of a superconducting cavity stabilized oscillator (SCSO) and a hydrogen maser (H). The object was to see

if the variation in the gravitational potential of the sun as the earth rotated and moved in its eccentric orbit would

manifest any change in the relative frequency of these

oscillators as shown in Fig. 3. The gravitational potential

variation is

u = Ln = -3.2 X 10-13 COS 2'u{t - to)

+ 2.8 X 10-12 a - to) ,

(where t is measured in solar days and ^o is noon April

4). The Lightman and Lee parameters [51] for such a comparison give the ratio

/h// SCSO = A \ + u

(To - Ao)

While the results of these tests are still to be published

[55] it is now possible to state that the departure

(ttH - ascso) < 2 X 10-2.

TIME (DAYS)

Figure 3. Illustrating the variation of the sun's gravitational

potential at Stanford, Califoi-nia, during the month of April.

617
This question of how electrostatic and magnetostatic

self-energy in particles behave as contributors to the

"mass" has its gravitational analog called the "gravitational weak equivalence principle" and is also called the

"super strong equivalence principle." The manner in which gravitational self-energy adds to the mass of bodies was studied by Nordtvedt [60, 61]. The gravitational self-energy for a body of mass m and

characteristic size, a, is Gmya - Eg and the total en- ergy is mc~. The ratio of these, Gm/c^a, is about 10"^^

for laboratory-sized bodies and the Eotvos-DickeBraginsky tests cannot help distinguish the gravitational

energy contributions to mass. However, for more massive bodies such as the earth, moon, and other solar sys- tem bodies, there could be a more measurable effect. Nordtvedt defines a parameter, ti, which is a measure

of the possible departure from the equivalence of gravitational and inertial mass in the following expression

J ( nig ry, V \ — 1 , G jx - x') — = 1 + f] -y 1 +

rrii p(a;)d% mc

where p(.r ) is the mass density at position x, G is the

gravitational constant, and c is the velocity of light.

He proposed tests of this effect using the earth-moon

combination in the sun's gravity field. Under the conditions where the equivalence principle is violated owing to the possibility that the earth-moon gravitation self- energy divided by is not behaving like ordinary mass,

the earth-moon separation would show anomalous motion.

This test has been performed using lunar laser-ranging

techniques and the results are reported by Williams et al. [62], who found -q = 0.00 ± 0.03, and Shapiro et al. [63], who determine that -q = -0.001 ± 0.015. Since this test also involves other parameters that come from the

assumption of a particular metric -of space-time, more

will be said in the following section. Theoretical work continues in efforts to understand the

relation between the Weak Equivalence Principle and the

Einstein Equivalence Principle. Recent work by Ni [64]

suggests Ebtv'ds-type experiments testing the universality of free fall be done on polarized test bodies.

While the convincing experimental work and theoretical arguments have done much to bolster our confidence

that whatever theory is found to be correct must be a metric theory, there is still room for error at some level. Theoretical study and experiments continue actively to pursue this question further.

5. Tests of Metric Theories

The most recent and complete classification of metric

theories proposed as alternatives to the general theory of

relativity is given by Will in General Relativity, An Einsteiyi Centeyiary Suiuey [43] and in his forthcoming book

Theory and Experiment in Gravitational Physics [6]. The role of the metric in such theories is quoted here

from Will [43]. When the postulates of metric theories of gi-avity are examined, one notices a crucial feature: no matter how complex the theory, no matter what additional gravitational or cosmological fields it deals with, matter and nongravitational fields respond only to the metric g The role of other fields which a given

theory may contain can only be that of helping to generate the space time curvature associated with

the metric. Matter may create these fields, and the

fields plus the matter may generate the metric, but

they cannot directly act back on the matter. The

matter respoyids only to the metric (author's italics). From this point of view, the metric g becomes the primary theoretical entity, and all that distinguishes one metric thing from another is the particular way

in which matter generates the metric.

A theory of theories has been developed by Thorne,

Will, and Ni [65] that now comprises ten parameters that can be included in the metric to adapt it to the various

proposed theories. This is known as the Parameterized

Post-Newtonian Metric or PPN metric.

Table 5 (from Will, see Refs. [43, 6]) lists these parameters, their function, and their value in general relativity.

A complete discussion of the validity set by experiment

on various different metric theories by the limits imposed

on these metric parameters is outside the scope of this outline. Instead, I should like to concentrate on the ex- perimental limits that can be assigned to the metric

parameters associated with general relativity.

One of the earliest parameterizations of the General

Theory was done by Eddington [66]. Later work by

Robertson [67] and Schiff [68] described the solar system

with a spherical nonrotating sun and treated the planets

as idealized test bodies. This metric has the form

ds2 = 1 + 2a^

M + 23

r 12-1 K

r

dt''

1 + 2y

M

idx^ + dy^ + dz^)

with a* = 1, r = (x^ + + z'^)^^ is the radial dis- tance, and M - GMsan/c^ and will be used in the follow- ing discussions as the "mass" of the sun. This simple metric shows the behavior of the line ele- ment in space-time, ds, and includes terms in its temporal part accounting for gravitational redshift, 2M/r,

and the second-order redshift governed by the term p, which describes the nonlinearity in the superposition law

for gravity. The use of the parameter a* to describe the

validity of the redshift at first order in c is redundant;

if a* were other than exactly unity there would be a conflict with the idea of the metric itself and we would have

a contradiction. In tests of planetary motion or particle

motion any such departure from unity would be included

in the definition of mass, M. However, in tests where the

behavior of electromagnetic signals is observed (such as

light bending and time delay from ray paths going near

the sun), part of the observed deflection or delay is due

to the temporal 2M/r term and part is due to the spatial

2yM /r term where -y is the amount of space curvature from unit rest mass. Even though we may eventually

have to face a logical inconsistency, I believe we can

benefit by keeping a* in the notation to help us see what

is happening in the experimental results. This metric provides a framework to test experimentally

the warping of time and space using the sun as a massive

test body by observing the behavior of light rays as they

graze the sun and by observing the behavior of test bodies

falling in the gravity field of the sun. The three early tests prescribed by Einstein, namely the advance of the per- ihehon of the orbit of the planet mercury, the deflection of starlight by the sun, and the gravitational redshift, are ex- amples of this type of experiment.

618
Table 5. The parameterized Post-Newtonian metric [Jf3,6].

What it measures Value in Parameter relative to General Relativity General Relativity

1 How much space curvature is 1i produced by unit mass?

p How much non-linearity is there 1

111 LIlC &UUC1 UUolLlUil i<XW yjL gldVlLj. Are there preferred location effects? 0 «1

Are there preferred frame effects? 0 as

^1

1 h Is there violation of conservation of 0 k total momentum?

u J

5.1 Solar System Tests of Falling Bodies

For the planet mercury in its eccentric orbit, the prediction is that the advance of its perihelion is given by

Set) =

6ttM

a(l - e^)

2 + 7-3 + J9 2Ma(l - e^) Here, R sun is the mean solar radius, a is the semimajor

axis, e is the eccentricity of the orbit, M is the "mass"

of the sun, and J 2 is the quadrupole moment of the solar gravitational potential.

The complication of the sun's angular momentum and

consequent centrifugal flattening giving the term J 2 led

to a very stimulating and fascinating alternative theory

of gravitation where it was postulated that the measured

value of the perihelion advance of about 43 arc seconds

per century might, at least in part, be due to an appreciable value of J 2- This led to the Brans-Dicke [69] scalar- tensor theory and a resurgence of interest in gravitation

that triggered a considerable effort to measure optically

the roundness of the sun's disk in an effort to detect some measure of Jo [70,71]. The Brans-Dicke Theory in- troduces a variable parameter, co, and gives the value

(1 -I- cl))/(2 + o)) for the parameter 7 in Table 5. As

0) ^ 20 the value of 7 1, as predicted by general relativity. As wall be seen, the value of co is being pressed

higher and higher and the validity of this theory is now

highly questionable.

Measurements of the quantity {2. + 2y - |3)/3, the perihelion parameter, have been made by radar observations

of mercury and the inner planets by Shapiro and his col- leagues [63] who report a value of 1.003 ± 0.005 assuming the J 2 contribution is negligible.

The question of measuring the value of J 2 of the sun is of great interest to solar and stellar physicists. A space

experiment to make an unambiguous measurement of J

is being studied by the U.S. National Aeronautics and

Space Administration. This mission, now labeled "Starprobe," consists of a spacecraft that passes within four

solar radii of the sun' s center, which will compensate for

all nongravitational forces such as solar light and particle

flux pressure by using a drag compensating servo sys- tem. By accurately tracking the extremely eccentric orbit of this freely falling "test body," the value of ^2 and

the other 7 and p combinations will be separately identified through their time signatures in the tracking data.

In addition, if the spacecraft were to contain a clock and

a suitable microwave system, the direct measurement of

p as the second-order behavior of the gravitational potential could be measured directly rather than derived from

ballistics of falling particles as will be discussed later. The Nordtvedt parameter r\, discussed in the earlier

section on the equivalence principle, when expressed in terms of the PPN metric coefficients [43] is

Ti = 43 - 7

3

2 . 1 . 3 ~

3 Using the laser reflectors placed on the moon during the

Apollo 11 mission a program of laser range measure- ments was begun in August 1969 at the McDonald Observatory in Texas [72] to measure the earth-moon motion

as a pair of bodies in the field of the sun. Typically, dis- tance measurements at the 30-cm level of accuracy have

been made in the path from the laser to the lunar reflector. As stated earlier, the results are that iq = 0 within

about 2 parts in 10". This offers a strong confirmation of the validity of metric theories and also provides information for separating the quantities 3 and 7 when used vdth data from

other experiments as will be seen later.

5.2 Tests of the Behavior of Electromagnetic

Signals

The predicted deflection of light grazing the sun 80 is given by

g* + 7 4M 1 + cose

2 d 2

' where M is the "mass" of the sun and 0 is the angle

between the earth-sun line and the incoming direction,of the light. For a grazing ray, d = R^m = 1 solar radius

and e a 0. Under these conditions the prediction is ^

^ X 1.75 arcsec .

80

F.or metric theories (a* + 7)/2 = 1. In 1964, I. I. Shapiro [73] proposed another test in- volving the round-trip time delay of radio or radar sig- nals passing very close to the sun. This "fourth" test of relativity predicts that the time is given by

250 - 201n

r

(IS

619
where d is the distance of closest approach of the ray in solar radii, and r is the distance of the planet or satellite, in astronomical units.

5.3 Solar System Light-Deflection Measurements

Light-deflection measurements begin early in the post

World War I era with the celebrated test by Eddington

and his co-workers who, with 30% accuracy, confirmed

Einstein's prediction. More recent optical measurements

have been made but the accuracy has only been improved

to ^

^ ~ 0.95 ± 0.11

from the solar eclipse of 30 June 1973 [74]. Many of the

difficulties with such optical lunar eclipse measurements

appear to have been overcome by recent photoelectric

and astrometric techniques [75] and better results are expected.

5.4 Solar System Light-Bendinq Tests Using Radio

Interferometry Techniques

Tests of light bending have been made using long baseline and Very Long Baseline Interferometry (VLBI)

techniques observing microwave noise signals from

quasi-stellar radio sources passing near the sun. Figure 4 (from Will [43]) shows the way the data have

been evolving in interferometric tests made in the last decade. These are plotted in terms of the quantity

(a* + y)/2, which, for the general theory is equal to 1. The latest results for this quantity are given by

Sramek [84] who puts (a* + y)/2 = 1.007 ± 0.009. According to Fomalant and Sramek [83,85] this technique is currently limited by uncertainties and irregularities in the tropospheric refractive index. While it is likely that a

factor of two, or so, can be gained by modeling the lenslike behavior of the troposphere lying over the radio tele- scopes, the future improvements are more likely to occur using VLBI techniques with spaceborne radio telescopes.

5.5 Solar System Time-Delay Tests

Measurements of the time delay of signals passing close

to the sun have given the most sensitive tests of (a* + y)/2 thus far available. These measurements were

first done by radar pulses reflected from planets at the

far side of the sun (superior conjunction) and later from

transponders on spacecraft in flight such as Mariner 6

and 7 and spacecraft orbiting the planet or on the surface

of the planets as in the Mariner 9 Mars orbiter and Viking lander and orbiter. These results are shown in Fig. 5 (from Will [43]) with later additions.

The latest results are truly impressive. Reasenberg

and Shapiro [91] give (a* + y)/2 = 1.000 ± 0.0001 as the result of using a 24-parameter model of the earth and mars motion and position. They have made over 100

least-squares solutions with several strategies to search

for hidden biases. Their data are from the Viking Lander

on mars, a space system that continues to operate and

provide information. From further data, and better

understanding of the propagation through the interplanetary plasma, Shapiro and Reasenberg express optimism

that the final uncertainty could be at the 2 x 10"^ level. This would be just about the present limit of the accu- racy of a* made in the space-borne rocket experiment

described in the section on Tests of the Metric Hypothesis and would bring the uncertainty in the

equivalence principle into the conclusion for the timedelay tests. By combining the lunar laser results cited earlier with

the light deflection and retardation results and treating

these as independent data sets, Shapiro, Counselman,

and King [63] obtain the following value for the parameters p and y excluding the other PPN parameters, that

is, assuming general relativity is correct; p = 1.003 ± 0.005

and

7 = 1.008 ± 0.008,

with a correlation of 0.6.

g +y

2 0.92 0.96 .00 1.04 1.

1970

1971

1972

1973

1974

1975

T—I—

Sramek

Sramek 77

Riley 78

Weiler et al .^^ Counselman et al . Weiler et al . Fomalont and Sramek

Fomalont and Sramek1_ J I L

T—

J L

g +y

2 0.88 0.92 0.96 1.00 1.04 T—I—I—I—I—I—

Passive radar to Mercury and Venus

86 Shapiro

Shapiro et al . 16

Active radar:

Mariner 6 and 7 Anderson et al .

Anchored spacecraft (preliminary results):

87

Mariner 9 Anderson

89 Reasenberg and Shapiro

90 Viking Shapiro et al .

Reasenberg and Shapiro"'

I I I I L

91

"1—I—[—

J I I L

Figure 4. Radio-ivave-deflectmi experiments (from Will [ItS]). (The superscript number associated with each nmne is the ap- propriate reference yiumber.)

Figure 5. Time-delay measurements. (The superscript number

associated with each name is the appropriate reference number.)

620
5.6 A Future Gyroscope Test in Earth Orbit

Thus far, all the tests discussed here have been related

to the gravitational effects of static matter. There is now

in preparation a test to measure the effects of moving

matter, namely the rotating earth, using extraordinarily

precise gyroscopes, operating at cryogenic temperatures,

whose spin axes are sensed using superconducting tech- niques [48,92,93]. The spin axis direction will be measured relative to the direction of a distant star established

by a telescope that is part of the space-borne system.

There are two relativistic effects that can be sensed by

a gyro. For a spacecraft in a polar orbit and with the

spin axis of the gyro in the plane of the orbit a relativistic effect known as the "geodetic precession," fii, is predicted that depends on the space curvature parameter, 7, and is given by

Oi = (1/3) (1 -I- 27) X 7 arcsec year^i

The second, more interesting effect, is the LenseThirring or frame-dragging effect produced by the

earth's rotation (see Ref. [2], §§ 19.2 and 33.4 for a dis- cussion) and in this case, the gyroscope spin axis is aimed

normal to the orbital plane as shown in Fig. 6.

Gyro 2

Figure 6. Earth-orbiting gyro experiment [92]

The PPN prediction for the Lense-Thirring effect in an

earth-orbiting gyro is

fl2 = (1/8) (47 4- 4) X 0.05 arcsec year"!

and is proportional to the earth's angular momentum.

The expected accuracy of this test is at the 10"^ arcsec year^^ level. Another test proposed by Van Patten and Everitt [94]

would involve two drag-compensated counter-rotating

earth satellites in polar orbits. These, in fact, act as gyroscopes, each of whose axes are expected to rotate

fl2 = (1/8) (47 -I- 4) X 0.18 arcsec year"^ if placed in a

circular orbit at 800-km altitude. Van Patten and Everitt

estimate that 0-2 would be measured to about 1% after two and a half years of data. Apart from the very obvious importance of this type of

test for confirming gravitational theory, there is the very

interesting possibility of using the Lense-Thirring effect

to measure the angular momentum of the sun by its ef- fect in the close solar approach or "star probe" mission

mentioned earlier. The measurement of both the J2 of the sun and its angular momentum would be a boon to

solar and stellar science.

A current assessment of the status of the measurement

of relativistic frame dragging is given by Van Patten at

this conference, and I should like to highlight the importance of this mission by reminding you that relativity and

gravitation theory is an essential tool in making measure- ments of importance to many other branches of science. After seeing these remarkably sophisticated experimental tests we are tempted to ask "How does general

relativity theory shape up when compared to other possi- ble theories?" The answer is that it has passed every

test so far with no difficulty. The problem is that there is an ever growing list of theories that also pass these tests and that we must begin to look elsewhere to make more

definitive tests. We must look for a laboratory with

masses, densities, and velocities quite beyond any accessible to us in our solar system. This does not suggest

that the use of our advancing technology in our terrestrial and solar system laboratory is of no avail, but that we should develop more sophisticated and sensitive

detecting systems to search for other manifestations of gravitation and relativity in the form of gravitational ra- diation originating in other parts of our universe. We

must realize that the region of space-time that mankind

has traveled is extremely brief and encompasses a region

that is infinitesimally small in the scale of distance, density, mass, and velocity of our universe.

6. The Search for Gravitational Radiation

Our present capability to devise laboratory experiments to test gravitation theory is limited by the masses

available in our solar system. For experiments that seek

to extend the range of validity of the theory we must

look for naturally occurring astrophysical phenomena in- volving super dense matter and enormously higher masses. We are thus forced to change our posture to being observers, rather than cause-and-effect experimenters, and face the problem of unravelling from the

data the characteristics of the generating phenomenon

and the manifestation of the theory we are testing.

Gravitational radiation appears to be the key to understanding gravitation under these more extreme conditions and to pointing the way to post-Einstein theories of

relativity. The instruments for detecting the expected

extremely weak signals are at the cutting edge of today'

technology and the sophistication, elegance, and astonishing sensitivity of these devices will surely contribute

to other areas of science and technology.

An assessment of the possible astrophysical sources of gravitational radiation and the magnitude and characteristics of the signals that we might receive has been

made in a recent book Sources of Gravitational Radiation edited by L. L. Smarr [95]. There are several ways

in which radiation can be generated. A continuum background radiation akin to the 3 K electromagnetic radiation associated with the Big Bang origin of our universe

is possible. It is highly probable that rotating binary star systems generate continuous radiation and wind-dowTi as they lose energy. There is very strong evidence for this

in the ongoing observations made by J. H. Taylor and his associates [96] from a binary pulsar system. However,

detection of radiation from such systems is far below the

sensitivity available in any existing gravitational wave

detector. Higher level, low-frequency (10"'^ Hz), pulsed

gravitational radiation is believed to result from cata- clysmic events such as supernova explosions and gi'avita- tional collapse or collision of black holes. Such signals

that might arise from super massive black holes, such as are thought to exist as the cores of galaxies, are within

the range of detectibility of highly sensitive Doppler

detecting systems used in interplanetary tracking of spacecraft.

621
These closely reasoned speculations have provided a

basis, or hunting license, to search for gravitational radiation. The largest amplitudes of radiation are expected

from pulsed sources, a study of which is shown in Fig, 7

[97]. In this figure the amplitude, h, is a measure of the

frequency difference observed between a pair of proper

clocks, one at the crest, the other at the trough of a wave, this is also a measure of the fractional change in distance over that wave. The frequency of the main pulse component is related

to the size of the collapsing object. To the left, the region

10^ to 10" Hz describes the results of supernova core col- lapses involving 10 to 100 solar masses in the Virgo Cluster of galaxies, which are at a distance of approximately

15 megaparsecs (Mpc) from earth. A burst rate greater

than one per year is expected. The efficiency of the en- ergy conversion, t^, is shown as 1% (lower Hne) and 20%

(upper line). As the frequency goes lower (to the right)

the effect of more massive black hole formation involving

10^ - 10"* solar masses at distances to 500 Mpc in globular clusters is shown.

At the lowest frequencies —lO"'^ Hz, the expected

sources are supermassive (10^ - 10^ solar masses) black

holes in galactic nuclei and quasars at distances associ- ated with the early formation of galaxies. The rate of oc- currence of these events is difficult to estimate, however.

Smarr [97] estimates this to be at least one per year on

ground-based resonant and free moss ontennas

spacecraft

doppler tracking

o

LOG z/.GW

t t t

ontenno

bosellne

.c —

Figure 7. Inteyisity vs. frequency of gravitational radiatiori and estimated sensitivity of detectors.

the assumption that galaxies like our own are assumed to have a supermassive black hole at its core and that it is caused by a sudden collapse. Above the lines shown as "Black-Hole Upper Limits" is a line labeled fundamental

upper limit. This is a constraint based on the conversion

of all galaxies into gravitational radiation in a Hubble

time, which characterizes the age of our universe. In order to exist at all we must clearly be well below this

limit. The question is, at what rate of occurrence and at what level do gravitational signals exist? We should

remember that proof of nonexistence of such waves at levels below the solid lines would be of enormous interest

to theorists, as all gravitation theories predict some form

of waves. The detection of gravitational waves was pioneered by

Joseph Weber during the 1960s [98] using a large aluminum bar as a resonant detector to look for signals in the

kilohertz range. Since that time similar detectors have

been built at a dozen or more sites throughout Europe

and North America and some of these have attained sensitivities of 1 X lO"^*" in the parameter h. At Stanford

University a second generation of these bars using cryo- genic techniques [99, 100] has reached a sensitivity of

3 X 10^^* at temperatures below 2 K. A generalization of the types of gravitational wave

detectors is shown in Fig. 8 [101] from Douglass and Bra- ginsky [10]. First is the ideally-free mass antenna where

differences in distance measured between the two masses

is used to detect the gravitational disturbance. Here we

visualize a pair of vddely separated stations, such as in a deep space probe and on earth, and the measurement of changes in separation arising from a very low frequency

(—10"^ Hz) wave crossing the solar system by use of Doppler tracking techniques [97]. The tracking of distant spacecraft has already provided some upper limits to the magnitude of gravitational pulses

[102]. The current levels of sensitivity are shown in the

GRAVITATIONAL WAVE DETECTORS

(After Douglass and Braginsky, 1979)

• Free Mass Antenna

• Almost Free Mass Antenna //////

direction of gravitational wave

//////

© 0

Resonant Mass-Spring-Detector

Resonant Bar Detector D

Figure 8. Gravitational wave detectors [lOlj.

622
upper right corner of Fig. 7. Substantially improved sensitivity is possible and design changes are being implemented in deep space tracking systems to improve the

sensitivity for gravitational wave detection.

Present atomic clock and Doppler tracking technology,

if fully utilized, is at the stage where it would be possible

to detect gravitational disturbances in the region

described as "proposed mission." This involves a multilink [42] Doppler system with time correlated processing

of Doppler data taken both at the spacecraft and at the

earth station. Tracking from an earth orbiting station us- ing cryogenically stabilized oscillators could improve the

performance to the levels shown within the solid line below the "proposed mission" region. Work is in progress [103] to assess the possibilities of this system by computer simulation of Doppler data. Another means of detection is the use of almost free masses. These are antennas made of several carefully

suspended masses whose distances are measured using

highly sophisticated laser techniques that extend the

path distance by many reflections back and forth. Considerable effort is being made at several universities to develop this antenna and results are expected in the near

future. The resonant mass-spring and the closely related bar

detector, discussed earlier, complete the present picture.

Studies of these various forms lead us to project the sensitivity of the bar and almost free mass antennas to the

10~^^ level at frequencies as low as 1 Hz. The projected

sensitivity of free mass Doppler techniques could reach

as low as 10"^^ and, given the very precise use of highly

stabilized lasers in position-stabilized spacecraft, we can

envision reaching the 10"-^ level. The signals, once we get to see them, wiW provide a mixture of astrophysical information, depending on the manner in which they were generated, along with infor- mation relating to gravity theory. The latter is contained

in the nature of the signals themselves, their velocity,

polarization states, and the multipole characteristics,

i.e., monopole, dipole, quadrupole, etc., of the radiation

process. The effort to detect gravitational waves is now a major, worldwide effort involving over 100 scientists. Progress has been very encouraging and the many inventions and new ideas from this work have provided valuable new technology. I beHeve we can look forward to some

very exciting and beautiful results in the next decade.

7. Conclusions

The study of gravitation has been going on since the

early days of recorded history. In the modern era, gravitational experiments and accurate astronomical measure- ments have guided the thinking of early scientists and

philosophers and led to the monumentally important

Principia of Isaac Newton and later to the immensely

beautiful legacy left by Albert Einstein in his General

Theory of Relativity. Questions of the validity of Einstein's theory still exist, however, and the field of

theoretical gravitation is currently wide open in mathematical speculation and stands badly in need of ex- perimental evidence. It is clear that Einstein's Special

Theory is on solid ground. Ongoing high-energy physics

experiments continue to press the speed of particles ever

closer to the velocity of light and the behavior of particles and radiation under local conditions continues to be

well described by the Special Theory.

The metric description of gravitation is based on the

idea of a Riemannian geometry in which everywhere, lo- cally, the flat space of special relativity is embedded. It

is today the most valid theoretical description of physics

that can accommodate gravitation. The original version

of the metric theories, Einstein's General Theory, so far has beautifully withstood all experimental tests. However, we have only scratched the surface of this important topic; we have only been able to work in a very

small range of gravitational potential such that at best we have only explored Ac|)/c^ for values below 1 x 10"^. An important question is, what happens for A<^/c^ for far larger values, even going to values, Acj)/c^ 1? The

highly probable existence of black holes and the beguiling ideas of mathematical singularities imply physical

conditions far beyond any we have ever tested and seri- ously challenge us to define some range of validity where

the now-classical Einstein Theory could be considered as a limiting case of a Post-Einstein Theory. Whether it will be possible to see evidence for this in tests done in the next decades within our solar system seems doubtful. It

is likely that any such evidence will have to come from

observations of astrophysical phenomena and the role of the experimenter will become that of an observer equipped

with the highest levels of available technology.

The future is very bright for this type of scientist- technologist-observer. I believe that we are at a threshold

where new information will be forthcoming from large

telescopes in space operating at all frequencies in the elec- tromagnetic spectrum. Very high angular resolution will be possible vdth pairs of space telescopes using very long

baseline interferometry techniques. The enormous in- crease in angular resolution and light gathering ability

available from space will vastly extend our view of the

universe. Also, I am convinced that the discovery of gravitational waves is not speculation, it will happen and will have an impact comparable to Galileo's use of the astro- nomical telescope in place of the naked eye. The likelihood

that the results of these measurements and observations

vdll pose far more new questions than answers to old ones

is a powerful and constructive manifestation of our evolution and our search for knowledge. There are few, more

worthwhile goals that our civilization can pursue.

I should like to thank Dr. B. N. Taylor for his recom- mendations and guidance in my attempt to organize this paper. The assistance and suggestions of Dr. Alan Lightman, in reviewing the manuscript, have been of great

value; I am grateful for his patience and willingness to

help. Also, I should like to thank Mrs. Anne Omundsen

and Mrs. Gerda Schrauwen of the SAO Editorial Department for their excellent advice and cooperation in putting

together this paper. The National Aeronautics and Space

Administration (NASA) and the Smithsonian Institution

have both provided support for this publication.

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Is the Gravitational Constant Changing?

T. C. Van Flandern

U. 8. Naval Observatory, Washington, DC 20390

Forty-four years after Dirac's original conjecture, experiments have become accurate enough to test for a time variation of the Universal Gravitational Constant. Present results utilize the lunar orbit, and do give a non-zero result. Interpreting the observed quantities with the Canuto-Hsieh scalar covariant cosmology, G/G = (-6.4 ± 2.2) x 10^^' per year. The implications for relativity and cosmol- ogy are briefly discussed. Key words: cosmology; gravitation; laws of physics;, occultations; universe.

1. Introduction

Can a fundamental constant of physics change with

time? No such variation has ever previously been

detected, lending confidence to the vddely-held belief

that the fundamental constants really are constant. However, the 1937 conjecture of P.A.M. Dirac [1] cannot be

ignored in this connection, not only because experimental

tests have only recently been possible, but also because

it provides a theoretical prediction that the Universal

Gravitational Constant G alone among the fundamental

constants of physics can vary.

Dirac's conjecture is a byproduct of the so-called Large

Numbers Hypothesis. In essence, if the fundamental constants are all expressed as dimensionless ratios to rid them of man-made dimensions, for example by using

units of length, mass, and time from the hydrogen atom,

the constants tend to all lie within a few orders of magnitude of unity, with the notable exception of the gravitational constant. This last stands out from the others, its reciprocal being forty orders of magnitude greater. It seems that the age of the universe is also forty orders of

magnitude greater than the light time for the Bohr ra- dius of the hydrogen atom. This coincidence already

mildly suggests a connection, unless nature is being

somewhat perverse. Additionally, however, Dirac

pointed out the formidable difficulties in explaining such

large numbers in terms of purely mathematical and

geometric concepts, as we hope to do someday with, e.g., the fine-structure constant. This reasoning also implies a time variation for these large numbers. In particular, if the coincidence with the age of the universe is not merely

fortuitous, G would have to vary inversely with time to maintain its relation to the age of the universe.

2. The Experimental Evidence

Tests for a time variation of G have only recently be- come possible due to the availability of such devices as atomic clocks and retroreflectors on the Moon, and the

capability of sending radar beams to the planets. The

principle behind such experiments is to detect the ac- celeration in orbital motion of astronomical bodies which

would occur if G changes. The precise relationship

between variations in G and changes in a body's angular

velocity (n) is theory dependent because G always ap- pears as a factor of mass (M) in orbital motion, and dif- ferent theories predict different behaviors for the product GM with time. In particular, no existing theory

predicts simple adiabatic changes in G with no accom- panying changes in M. In general, we have

G/G=fin/n),

where a dot denotes a time derivative, and / is a con- stant suppHed by the theory. For the Dirac theories, /

must be either +1 or -1, depending on the form of mass

change. For a so-called "primitive" theory in which mass does not change, / = 1/2. No existing theory known

to this author has / numerically greater than 2. Measuring changes in G therefore can be accomplished

by measuring changes in the angular velocity of astro- nomical bodies. This process of necessity involves the use

of an accurate clock. It will become apparent on reflec- tion that no astronomical body can serve as a clock for this purpose, since the clock must not disguise the effect sought. This means that Ephemeris Time and Dynamical

Time, both of which involve the orbital motion of bodies, cannot be used. On the other hand, Atomic Time is a

fully suitable time scale. For this reason, we cannot util- ize measurements of the orbital acceleration of, for example, the Moon over the last few centuries, when only

dynamical clocks are available, except as a control (for

example, to determine the angular acceleration from

other causes, such as tidal friction). Only if the Moon's

acceleration is different when measured vdth atomic and

dynamical clocks can we impute the cause to a changing

G.

3. Using the Moon

The Moon would provide an ideal celestial body for the

necessary measurements if it were not for accelerations

from other causes, such as tidal friction. However, for the reasons just detailed, it is possible to separate the ti- dal accelerations (using Dynamical Time) and the total accelerations (using Atomic Time) to determine whether

there is any non-zero contribution from changes in G. Experiments which measure the tidal lunar acceleration [2] include classical telescopic observations, transits of Mercury across the Sun's disk, ancient eclipse observations,

and ocean tide models. The mean of these individual

determinations is

?^ tidal = (-28.8 ± 1.5)"/cy2

(cy = century).

Determinations of the total lunar acceleration (using

625
Atomic Time, available since 1955) have heretofore relied

exclusively on timings of occupations of stars at the

Moon's limb. PreHminary results using this method [3] have been affected by systematic errors due, in part, to the star catalogs and lunar ephemeris in use for the analysis. (See Ref. [2] for more discussion of current determinations of the lunar acceleration.) These problems

have all been addressed and corrected insofar as possible;

and the lapse of additional time has made the matter

somewhat academic anyway, since the effects of acceleration eventually become larger than systematic errors of any given size. The lunar occultation results now give

wtotai = (-21.4 ± 2.6)"/cy2.

Additionally, results for this same parameter have re- cently become available from lunar laser ranging to the

retroreflectors left on the Moon by the astronauts. The mean errors of the laser experiment have also been ra- pidly decreasing, and are now for the first time better

than those from the occultation results. It is therefore

gratifying that the two experiments are in reasonable ac- cord, with the laser ranging results giving w total = (-23.6 ± 1.5)'7cy2

in the latest estimate [4]. Combining the occultation and

laser results (and including a few additional estimates by

other experimenters using the same laser data), we get

Wtotai = (-23.2 ± 1.2)' 7cy2.

4. The Rate of Change of G

Comparing the tidal and total acceleration determinations, we note a discrepancy which has at least formal

statistical significance. Taking the difference and divid- ing by the lunar angular velocity w - 1.7333xl0^^"/cy

gives for this discrepancy

ii/n = (+3.2 ± 1.1) X 10-"/yr.

The most complete and internally consistent cosmological framework for variable G to date is that of Canuto

and Hsieh [5], since it has addressed and resolved many

supposed conflicts between variations in G and physical,

geophysical, or astrophysical constraints. An important

feature of this scalar-covariant theory is that it retains

full consistency vdth the theory of general relativity,

since G does not change with respect to Dynamical Time

(only with respect to Atomic Time). Best agreement with

most physical constraints is secured if the parameter /,

relating n/n to G/G, has the value -2. In this case, we

arrive at the result G/G = (-6.4 ± 2.2) X 10-"/yr;

or equivalently, in the more usual choice of units in cos- mology, GIG = (65 ± 22) km/(s-Mpc) (Mpc = raegaparsec). This value can now be compared directly with vari- ous determinations of the Hubble rate of expansion of the

universe, with which it should be in agreement according

to Dirac's Large Numbers Hypothesis. Recent determinations of the Hubble constant have ranged from 50 to somewhat over 100 in these units, which is clearly consistent vdth the hypothesis in view of the errors in both

determinations.

The only other experiment with even preliminary

results to date is planetary radar ranging [6]. These au- thors stress that the errors in their results at this stage

make them consistent with a zero value for G/G. The

averaged determination from the individual results for each of the inner planets is n/n = (-f-12.4 ± 6.6) x lO'^Vyr,

which is in even better agreement with the above results than it is with zero.

5. Cosmologicai Implications

The simplest and most literal meaning of the experimental measurements of a different angular acceleration

of the Moon with respect to Atomic Time and Dynamical

Time is that the number of atomic seconds in a dynamical

interval (such as a revolution of the Earth about the Sun)

is becoming fewer. Presumably, if the result has any generality to it, this means that atomic phenomena are slowing down with respect to dynamical phenomena. If the

speed of light is constant (as assumed in the Large

Numbers Hypothesis and the Canuto-Hsieh theory, but an assumption nonetheless), then we can make the same

deduction about units of length: as time progresses, fewer and fewer hydrogen atoms will fit into the

distance from the Sun to the Earth, or into any orbit

whose dimensions are determined by the strength of gravitation.

The first deduction we can make about this result is that we cannot tell from existing data whether the

changes are occurring on the atomic level or the dynamical level. Moreover experiments which purport to show

that various atomic constants are invariant with respect ; to time do not rule out change occurring at the atomic

level, since no properties of atoms, measured in atomic

units, would be expected to vary. Indeed, we must take

care to reinterpret all previous experimental results in the light of this new (although tentative) finding. Assumptions such as the constancy of the velocity of light,

already made above, may be true only in one set of units

(atomic or dynamical), but not the other. It is precisely

such confusion which has led to some earlier erroneous

deductions about implications of variations in G, which

the scalar covariant theory of Canuto and Hsieh corrects

[5].The next important implication is that, even though

these variations of scale which we call a decrease in G

are related to the Hubble age of the universe, they are not by themselves a cause of or an explanation for the

observed expansion of the universe unless the speed of

light also varies, at least in dynamical units. The reason

is that the redshifts of distant galaxies, which are used

to measure the rate of expansion of the universe, can

easily be expressed as a fraction of the velocity of light,

c. For example, if the universe had constant linear di- mensions in both dynamical and atomic units, the in- crease in redshift with distance (or equivalently, with

lookback time) would imply an increase in c at past

epochs, or that c was decreasing as time moves forward.

If on the other hand c remains constant in dynamical

units, then so do redshift measurements; and the Hubble

rate of expansion of the universe in km/(s-Mpc) can be i non-zero only if the dimension of megaparsecs (a dynamical unit of length) varies with atomic time. For this rea- son we deduce that, as long as c is taken constant, the

Hubble age of the universe is its age in Dynamical Time.

If we let t measure time in dynamical units and t measure time in atomic units, then let and t\ be the respective measures on both scales for the present epoch. We j will vdsh to impose the constraint that <i = ti for con- | venience. Finally let to be the epoch of the "Big Bang"

or start of the expansion of the universe on the dynam626
ical time scale. Then the Large Numbers Hypothesis and

the experimental results apparently tell us that

dj/dt = 1/it - to),

from which we can easily derive

(t - to)/{ti - ^o)=e''"^\

As t goes to infinity, so does t; but note that as t ap- proaches to , T approaches minus infinity. The singularity

at the time of origin of expansion therefore occurs in atomic processes, but not necessarily in dynamical processes. More rigorously, it occurs in the ratio of atomic to dynamical time interval. This must certainly have pro- found implications for our current understanding of the

origin and evolution of the universe, although it is not immediately clear what all of the ramifications may be. It seems intuitively to represent a curious wedding of the

Big Bang and Steady State theories of cosmology which

contradicts neither.

It may be worth mentioning one additional implication

of varying G, insofar as it relates to Mach's Principle of

Inertia and the question of the influence of the distant

matter in the universe on local dynamical systems.

Nevd;on' s law of universal gravitation may be written as F - GMm/d\

where F is the force between two masses, M and m, at a distance d apart. We can also equate F to kma, where a

is the resultant acceleration of the body of mass m in response to the force, and is a constant of proportionality. It is customary to equate these two expressions for F, cancelling m in the process, and obtain an expression

for the acceleration of m which is independent of m itself: a = {G/k)M/d'K

However the cancellation of m disguises an important as- sumption. In the law of gravitation, m is the gravitational mass of the body, whereas in F = kma, m is its inertial mass. The ratio of gravitational to inertial mass

for a body has been shown to be constant to parts in 10^^ over a range of different chemical compositions; but the

time dependence of this ratio is not readily measurable.

It may be seen that observed variations in G may equally

well be ascribed to variations in k, or equivalently, to time variations in the ratio of gravitational to interial masses for all bodies. This may be thought of as a change

in the local property of inertia, perhaps caused by the ex- pansion of the distant matter in the universe, or perhaps

due entirely to changes at the atomic level. In either case

it permits the incorporation of Mach's Principle into cosmological theories in a new and interesting way.

6. Conclusion

The tentative conclusion from experimental evidence now available from the lunar occupation and lunar laser ranging experiments is that the universal gravitational constant G seems to be decreasing with time at a rate whose exact value is theory-dependent, but which is in qualitative agreement with the Hubble rate of expansion

of the universe, as predicted by Dirac's Large Numbers

Hypothesis. In accordance with the internally consistent

scalar covariant theory of Canuto and Hsieh, which also retains full consistency with General Relativity, the ob- served rate would be (-6.4 ± 2.2) x 10"^^ per year. Very

preliminary results from planetary radar ranging, which

has the advantage that the Moon's orbit is not involved,

in no way disagree with this measurement. The implications of this result for our understanding of the origin

and ultimate fate of the universe are profound, although

not yet fully elaborated.

References

[1] p. A. M. Dirac, Proc. R. Soc. London Ser. A: 165, 199 (1938).

[2] T. C. Van Flandern, Astrophys. J. 248, 813 (1981).

[3] T. C. Van Flandern, Mon. Not. R. Astron. Soc. 170, 333

(1975).

[4] J. 0. Dickey, J. G. Williams, and C. F. Yoder, EOS Trans. Am. Geophys. Union 61, 939 (1980).

[5] V. Canuto and S. H. Hsieh, Astrophys. J. 224, 303 (1978).

[6] R. D. Reasenberg and L I. Shapiro, in On the Measurement

of Cosmological Variations of the Gravitational Constant, Ed. by L. Halpern (Univ. of Florida Presses, Gainesville,

FL, 1978), p. 71.

627

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Experiments on Variation of the Gravitational Constant Using Precision

Rotations

G. T. Gillies and R. C. Ritter

Department of Physics, University of Virginia, Charlottesville, VA 22901

The classical rotor for gravitational studies, the earth, has a varying decay time of ~3 x 10^ years and a rotational roughness ~3 x 10"^ per day. A more rigid, highly protected, laboratory rotor might

be made more suitable for gravitational measurements such as tests for the temporal variation of Newton's G and for cosmological matter creation. In this paper past tests for variation of G with other parameters are reviewed and the first room-temperature tests of precision rotors for such '

' laboratory

cosmology" experiments are discussed. The design of two such experiments at Virginia and the early

results are presented for one protected rotor aimed at ultimately testing for matter creation at rates below -rnhn — 10^^*^ year"^ Rotor oscillations are observed in the rotating reference frame, having energy dissipation rates -10"^'* W, thus giving some evidence that "corotation protection" against gas drag is effective. Key words: gravitational constant variations; magnetic suspensions; matter creation; Newtonian gravitational constant G; precision rotations.

1. Introduction and Background

Much of our quantitative knowledge of the gravitational force has come from measurements of the

Newtonian constant of gravitation, G. Because of gravity's unique weakness and long range, however, highly

precise results are difficult to achieve in experiments of

this type. Nevertheless, several rather conclusive tests have been performed in the last century, and the conclusions drawn from these efforts are usually summarized as

follows: In weak fields, the gravitational force between

t\^o masses is almost perfectly described by Newton's

Law, and the force is independent of the physical and

chemical state of either the masses in question or the in- tervening medium (if any).

From other types of . experimental observations and

from certain types of astronomical data we can further

conclude that in strong fields, Einstein's General Relativity more exactly explains gravitational and mechanical

phenomenon. But is there more?

There have been about 200 measurements of G or

equivalently, of the mean density of the earth. For a thorough review of them, see Refs. [1] and [2]. While

most of these efforts were aimed solely at establishing

and then improving the absolute value of G, some were

studies of a more fundamental nature. For instance, while most workers used solid spheres or

cylinders as the fixed attracting masses in their balances,

Cremieu [3] used 20 kg of liquid mercury instead. The

good agreement between his result and the others rules out a dependence of G on the physical state of the masses. In a far more comprehensive experiment,

Ebtvbs [4] demonstrated that the gravitational force was

independent of both the physical state of the masses as well as their chemical structure down to the 10"^ G level. More recent experiments confirm this at even higher levels of accuracy (see Refs. [5] and [6]). A different, but contemporary, series of experiments was devoted to a search for variations in G with respect

to the direction of the test mass crystalline axis. Since so many properties of crystals depend on the direction of observation (e.g., refractive index, thermal conductivity,

etc.), it seemed reasonable to search for similar types of

gravitational anisotropies. Mackenzie [7] and Poynting

and Gray [8] were the first to do this. Although their ex- periments differed in design, the outcome of each was a

null result at the level of one part in 900 and one part in 16,000, respectively. Somewhat later, Stratton and then

Heyl [9] made new sets of measurements. Heyl found

that he could put a limit of 10^^ G on the gravitational

anisotropy of crystals, a number he arrived at by carefully weighing samples from each of the five nonisometric

crystal groups. Along similar lines, Louis Bauer of the Carnegie Insti- tution in Washington weighed both magnetized and un- magnetized steel samples in order to see if the magnetic

energy density (or the lack of it) produced a variation in the gravitational attraction between the steel and the

earth. Heyl points out [10] that Bauer found no such

variation.

Shaw and Davy searched for a variation in G with

respect to temperature. After a series of modifications

to their Cavendish-type apparatus [11], they were able to eliminate some troublesome systematic effects and finally

conclude that ifG = God + 0(7") where T is the temperature of the attracting masses in kelvins, then a is less than 2 x 10"^ kelvin.

Another classic series of experiments began in 1897

with Austin and Thwing's research on "gravitational

permeability" [12]. They used a Boys-type balance which

had been designed such that intervening masses made of various substances could be placed between the large and

small masses of the balance. Measurements of G were

then made with and without these intervening masses. Their data showed no variations in G as a result of screening of the gravitational force by the intervening masses, at least down to the 10"^ G level. This result,

although null, created a great deal of interest, and during the following 30 years, no fewer than ten other ex- perimental searches for gravitational screening were car- ried out. All of these reported null results, but the levels of sensitivity were steadily increasing. Then in 1920, Majorana [13] claimed to see a screening of the gravitational force at the 10"^*^ G level. Russell [14] disputed this, and

629
subsequent work by Majorana [15] with an altered

Ruprecht balance confirmed that his initial results were

in error and that gravitational screening was in fact undetectable even at the 10"^^ G level. Reviews of these

results are given by Majorana [16] and Heyl [17]. More recently, a number of workers have tested the exactness of the inverse square law by searching for a

variation in G with distance. While most of these experiments have produced null results, some maintain that the

question is still open [18]. For a review of this topic, see

Ref. [19]. In spite of all this careful work, one crucial test has not yet been attempted in the laboratory, and that is a test of the temporal constancy of G; and relatedly, of the temporal constancy of masses. Why is it crucial to search for a nonzero G/G? The

answer, in part, is that the existence of such an effect would be our first indication that General Relativity is incomplete in its ability to totally describe the gravitational force (since its field equations do not allow for a time varying G). From a Machian point of view one might

even expect, naively, that the observed recession of the

galaxies might give rise to a weakening of the gravitational constant locally. Einstein did, in fact, attempt to incorporate Mach's principle into his theory, but failed to do so. It would seem then that this would be a particularly vulnerable point in the theory, but hopefully one

subject to experimental clarification. Further, Dirac noticed that certain comparisons of the

Eddington numbers ie'^/Gniem^, yn^c^/e^H q, etc.) called for G to vary inversely as the age of the epoch, and for the total number of particles (say baryons) to vary

directly as the square of the epoch [20]. He called this his large numbers hypothesis (LNH). He later used Milne's

hypothesis to develop a two metric theory in which long

range mechanical phenomena are described by the "Einstein Metric" and short range, atomic phenomena are described by the "Atomic Metric." In so doing he was

able to link his LNH, in which G varies when measured

in atomic time, to standard General Relativity in which G

is constant. Recently, Canuto et al. [21] have succeeded

in formalizing these concepts into a scale covariant

theory of gravity from which the LNH emerges as a gauge condition. Dirac has recently changed his views on

the existence of matter creation [22], but this only makes

the overall question of its existence more intriguing. In summary. Sir Fred Hoyle has posed a question

which brings all these issues into focus. Our interpretation of Mach's principle and our knowledge of the fundamental constants is inseparable from an overall understanding of the nature of the universe. In turn, our understanding of the universe evolves simultaneously

Math our assessments of ever more precise experiments.

He asks therefore [23], "Does the universe affect the

laws we discover in the terrestrial laboratory, or is the

universe no more than an exercise in the application of

present laws. . .?" This question repeats a point made

earlier, in that it links the expansion of the universe to the local constancy of G, and also, for that matter, to the

constancy of masses. Most theories calling for a nonzero value of G/G

predict, in fact, that G/G = i/o - 5 x 10^^^ year"^ (see

for instance Ref. [20]). Rates of matter creation are predicted by the same theories to be m/m —

10"^° year"\ where Hq here and above is the Hubble

constant.

Clearly, measurements of effects this small will be difficult, and it was reasonable to expect that the first tests

would be astronomical. Reasonburg et al. [24] have used

radar ranging to the inner planets to obtain a value of G/G = 5± 10 X 10-1 Vear-^ y^^ Flandern [19] has

timed lunar occultations with an atomic clock and reports

a value of G/G = (-6.2 ± 2.0) x 10"" year"!. Finally,

J. G. Williams [25] has used lunar laser ranging to establish limits of |G/G | ^ 3.0 x lO'^^ year"^ (For a re- view of other articles reporting values of G/G ^ 0, see

Ref. [19].) The relatively large uncertainties in these

measurements and the fact that some of these results have changed algebraic sign over the last decade indi- cates that no concensus on a final value of G/G, as determined astronomically, can be expected for some time.

Some workers are, however, also starting to look at ro- tating bodies—as opposed to revolying bodies—as potential sources of data for evaluating G/G and m/m.

For instance, Paul Muller estimated the fractional de- crease in the spin rate of the earth, to/w, by analyzing ancient and modern eclipse records. After subtracting off components of w/co due to lunar tidal interactions and

various geophysical effects he found a residual change in the earth spin rate that he claimed was consistent with

the value of G/G predicted by several cosmologies [26]. At the same time. Beams and Ritter [27] began

developing a laboratory experiment for G/G which was

basically an altered version of Beams' rotation-based

measurement of the absolute value of G [28]. The isola- tion from gravity gradients provided by rotating the sys- tem at constant speed will most likely be essential to any

earth bound measurement of G/G. This and other details of that experiment are discussed below.

Ritter has pointed out that a freely spinning rotor with

a speed decay time constant greater than or equal to the

age of the universe could be used to test matter creation

cosmologies. The principles of such an experiment can be

understood by appealing to the conservation of angular

momentum: If matter is being created, the rotor moment

of inertia must change, but then its angular speed must

change too in such a way that the rotor angular momentum remains constant. But such a change in angular

speed could only be seen if the rotor were sufficiently

smooth and free of friction. Symbolically, we have a free rotor

(o = ajoe-"^, (1) where co is the rotor angular speed as a function of time,

t, and T* is the angular speed decay time constant. Since

most theories predict

m/m = 2Ho = 10''^^ year"i , (2) we must have a rotor with at most

or

T* > lO^'' years = 3 x 10^^ s . (4)

The earth as a rotor was an obvious candidate for study since it has presumably been spinning since its creation. Important obstacles to employing the earth for

this purpose are its large and nonconstant rotational en- ergy dissipation and fluctuations in the length of its day.

The rotation of the earth has been studied for many

years, and it has been found to have a broad range of speed fluctuations leading to length of day variations

(l.o.d.). The three types of variations which have been

measured are [29, 30]: The secular drift or spindown, the

630
medium range (diurnal to annual) variations, and the

short term-irregular variations. The spindown results from the unavoidable dissipation of the earth's rotational

kinetic energy via tidal mechanisms. The medium range

variations arise from the Chandler wobble and other nontidal geophysical mechanisms. Finally, the irregular

variations appear to be uncorrected with the others, yet

associated in an overall way with fluctuations in the an- gular momentum of the atmosphere. The irregular variations can change the l.o.d. by as much as 10"'^s [32]. The

spindown of the earth gives rise to a speed drift of

Wg/cDe = 25 X 10""/year'°[29]. This means the earth an- gular speed decay time constant is te = 4 x T^^ years

which, coupled with the suspected large variatio.. of this rate, would be much too small to measure rh /vi for the

earth at the 10"^° level. In reality, the l.o.d. variations

which average out to 10"^ day~^ from all sources [31]

might put a limit on the usefulness of this technique even

if TE were much larger. Clearly we have to look else- where for a rotor which is both less dissipative and

smoother.

Ritter et al. [33] proposed a laboratory experiment in which a rotor would be spun in a "protected" mode in order to achieve a sufficiently long speed decay time constant. The protection consists of a vacuum chamber

which would corotate around the rotor, superconducting

magnetic shields, and other necessary features. Beams

had built a double magnetic suspension where one rotor was freely (magnetically) suspended from inside a sec- ond, outer rotor itself freely suspended. This system

showed promise of virtually eliminating the otherwise

limiting rotor friction caused by the residual atmosphere

[34]. Reducing the friction increases the decay time constant. This can be seen by examining the differential equation for a free rotor whose moment of inertia is /:

/oj Kw = 0 . (5) Here k is the rotor damping coefficient. The solution is Eq. (1) where

T* = ^ . (6) K Both Beams [35] and Fremery [36] had achieved decay

times of —10^^ s with small (1 mm) steel spheres spinning

at high speeds (20 kHz) in vacuum. It became clear that

if the rotor moment of inertia could be increased by 10^ and the friction decreased by 10^ or more (with the corotation scheme) that decay time constants of 10^^ s could

be achieved and the m/m predictions tested at that level. The goals of the research reported here were to bring

the room temperature version of the matter creation ex- periment into existence and to demonstrate the effective- ness of the corotation scheme in reducing the rotor en- ergy dissipation. In the process of accomplishing these

goals many of the techniques which will be necessary for successful completion of the G experiment have been

developed and some of them carefully refined. This is a

significant step forward in the creation of the more diffi- cult G experiment.

2. Instrumentation

Since much of the instrumentation is common to both of these experiments, and since a G experiment has been

thoroughly described before [27], only a thorough

description of that used in the m experiment is given

here. In principle, the inner and outer rotors of the G

experiment are designed to maximize the gravitational

attraction between them, while the inner and outer ro- tors of the m experiment are designed to minimize the

attraction. In both experiments, all other forms of inter- rotor couplings including electrostatic, magnetic,

mechanical and gaseous couplings (among others) must be

minimized, unless they are to be used as part of a feed- back system.

2.1 Instrumentation for the m Experiment

In its present form, the experiment has required the

development and subsequent incorporation of five separate systems: A stable magnetic suspension for the

inner rotor, a highly constant rotary drive for the outer

rotor, a sensitive rotational period measurement system

for both rotors, a vacuum system to pump out the outer

rotor while it rotates, and a vibration isolation system

which can support most of the above. For specific details of the construction, many of which are necessarily omitted below, consult Ref. [37]. The design of the magnetic suspension system was dic- tated by the properties of the rotor to be suspended. In

this case a 250 g disk of glass ceramic (Zerodur) was

chosen for its thermal and temporal stability. The disk is coaxially connected to a 16 cm by 1 cm diameter Zerodur

rod at the other end of which the ferromagnetic support

sphere is fastened. The support sphere has only a few

percent of the total rotor moment of inertia to minimize

eddy current losses. The 16 cm rod keeps the disk out of the strongest part of the support field thereby minimizing Faraday disk effects. A mirror used for rotational

period measurement and counterweights are fixed to the

disk. The total moment of inertia of this assembly is 1100 g-cm^.

The position sensor for the suspension is a pulsed light

beam which is parametrically sensed by a split cathode

photodiode and the ensuing phase locked loop. A 200 W

amplifier drives the support coil at the output end of the

feedback loop, but a permanent magnet is used to pro- vide most of the lifting force for the rotor in order to reduce ohmic losses in the coil. There is sufficient com- mon mode rejection in this system to permit stable operation of the suspension while overhead room lights are on, and temperature drift of the photodetector is no

longer problematic. We have kept rotors in suspension

continuously for periods up to 0.1 year and could do so

indefinitely if necessary. Detailed discussion of this and

other magnetic suspensions developed in our laboratory

is given in Refs. [38] and [39]. It is important to note

that the pole piece used in the support coil does not ro- tate in the present system like it did in Ref. [37]. The need for an almost totally constant speed drive for the outer rotor is twofold: (1) as discussed above, in or- der to measure m/m to 10"^°/year, the decay time of the

inner rotor must be =10^" years. Any speed drift in the outer rotor at this level or above will cause the relative

positions of the inner and outer rotors to shift. Such

shifts would change the size of the restoring force acting on the inner rotor in such a way that the m signal might

be masked. (2) Large short term speed fluctuations, if rectified by measurement system nonlinearities, might

also introduce false m signals.

To overcome these problems, a new type of rotary

drive system had to be developed which could ensure the

long term accuracy ultimately needed, yet simultaneously

deliver driving torque to the load without generating ap- preciable speed fluctuations. After several attempts to develop such a drive, J. W. Beams proposed the design

presently in use called the magnetic averaging synchro- nous motor. It is a 50 pole synchronous motor whose ar- mature is a thin strip of slightly retentive stainless steel.

631
This strip is epoxied around the periphery of a circular

aluminum disk that is the motor rotor. Temporary poles are established in the armature strip

by flashing the drive coils at dc, and the motor is then

spun by hand and allowed to coast until the free period is within —1% of the desired driven period. At that point,

the drive circuitry is started, and within a few cycles,

the motor locks in either on the fundamental or (if desired) one of the odd subharmonics of the electrical driving signal. Each phase of this drive signal partially

rewrites the original induced poles, and eventually the

armature pole strengths and their respective positions

around the armature strip assume highly uniform values. This averaging effect smooths the drive. While the diffi- cult subrotation measurements of the motor speed fluc- tuations have not yet been completed, it is known that on

a period to period basis the fluctuations are drivingwaveform dependent, but can be made as small as 2 ppm

or less. Additional measurements have shown that these

fluctuations average down by a factor of ten for every

decade increase in averaging time, and it is very likely

that this process will only be limited by drift in the fre- quency standard used as the time base for this motor

(currently a cesium beam clock with Av/v = 10"^^). The speed range of this motor is 0.1 to 10 rad/s, and

the electrical drive signal is produced by a commercially

available frequency synthesizer (Hewlett-Packard

3325 A) synchronized to the atomic clock. The motor drives directly a smooth air bearing whose

undriven spindown quality factor is —3000.

A 30 L/s ion pump is attached to the lower part of the

motor bearing assembly. It is used to evacuate the 30 cm

long by 10 cm diameter glass outer cylinder which is mounted on top of the bearing. They are connected by a 30 cm pumping sleeve which passes through the center of the motor bearing assembly, along its rotational axis. The pump is powered by two 12 V rechargeable batteries

whose voltage is stepped up to —6000 V by a dc-dc converter. The batteries, converter, and associated metering

are all attached to the pump and the entire system is free

to rotate. A thin walled, cylindrical aluminum shroud is placed around the entire assembly and also corotates with it. The shroud streamlines the load driven by the

motor and therefore reduces the size of the load fluctuations. This is important since the rotary drive system is both softly coupled and (presently) nonfedback, and large

load fluctuations would result in unacceptable speed

variations.

Operating pressure in the outer cylinder is typically

Hmited to —10"^ mm Hg by the small diameter (—3 cm) of the long pumping sleeve. While this Hmits the battery

lifetime to about 4 days of corotation operation, the

chamber pressure during this time is well below the

molecular flow limit. This reduces the chances that col- lective gas motion, as might be expected in the viscous

regime, vidll limit the inner cylinder spindown decay time

constant. The measurements of both the inner and outer cylinder

periods were made with separate Jones recording optical

levers [40] modified by Towler [41] and later by (billies [37] for use as timers rather than angular displacement

transducers. The levers themselves are accurate to at least 0.1 |xs, and their output pulses are used to trigger

digital counters and more recently a stand alone computer

system used for making the actual timing measurements.

The magnetic suspension system, motor/bearing plus vacuum system load, and the Jones levers are all vibration isolated on a two stage passive system of our own

design. An especially quiet room in the basement of the

University of Virginia (U. Va.) Physics building was lo- cated and renovated for this work. The floor was sawed

to separate the working surface from the rest of the

building, and a 4000 kg stone was placed on Weber isola- tors in the center of that area. On this, the 250 kg sur- face plate which supports the experiment was set. It rests on pneumatically interconnected air bladders which

isolate against floor noise to =3 Hz with a damping time

of one second. Floor noise is unmeasurable (below

0.02 |jLm) at this station. An active isolation system

sensed and driven by piezocrystals is currently under

development and will be installed when necessary.

All control and sensing electronics are in an adjacent room, and the laboratory room containing the experiment

is temperature controlled to within 1 °C by a dedicated

air conditioner. Because a 1 Hz pendulum mode is easily

excited in the rotor while in suspension, no one is al- lowed in the laboratory room while an experiment is in progress.

2.2 Instrumentation for the G Experiment

While this experiment is still in the design phase, one

detailed proposal for construction has been formulated

and presented in Refs. [19], [27], and [42] by workers at the U. Va.

In principle, the experiment is a rotating Cavendish

balance. The angular position between the large and

small masses is held constant by a servo system. Barring

all other torquing mechanisms, a change in the gravitational interaction between the large and small masses would appear as a change in the restoring torque neces- sary to keep the system balanced.

The experiment, as mentioned above, is rotated to keep the effects of anomalous gravity gradients on the

balance very small. Such an experiment would have to be cooled to liquid

helium temperatures in order to keep thermal noise ef- fects small, and synchronous detection techniques would

be necessary to remove the effects of 1// noise. The small mass system would most likely be magnetically suspended in order to maximize sensitivity, and the

rotational angular speed of the whole system would have

to be better than Aw/oj = 10~^^. Subfringe laser interferometry or SQUID position

sensing show promise for use in angle sensing, but the

overall complexity of such an apparatus would make its construction both expensive and somewhat inelegant.

Further experience at the National Bureau of Standards

[43] with a rotating torsion balance has shown the practical difficulty of this combination, albeit with a fiber as the suspension method.

Two other schemes for measuring G were conceived at U. Va., but have not yet been studied in detail [44]. One

would employ two superconducting spheres as masses on

opposite ends of a "superconducting spring." This high

Q system would be driven and observed to see if the

strength of the drive signal varied presumably as a result of a change in G. A second system would synchronously

detect the response of a weakly coupled sphere floating

in a soft "ball trap" to a driving mass that rotated near

the trapped sphere and gravitationally induced some motion in it. In or.der for either of these systems to be

workable, it would be necessary to put them in drag-free

artificial satellites.

632
3. Results and Discussion

A number of spindown and spinup tests were run in or- der to study the intercylinder gas coupling. Rotor speeds were typically 20 radians per second or less during spindown, and decay time constants of =10'' s were observed

as expected at the levels of vacuum used. The spinup

tests were mostly done at atmospheric pressure, and the

rotor was accelerated by the viscous coupling and asymptotically approached the angular speed of the driven

outer cylinder. At relatively high angular speeds, 4 rad/s

and above, interrotor mode locking was observed.

Specifically, the inner rotor was observed to be smoothly

spinning up from rest and then suddenly it would lock in on the angular frequency of rotation of the outer cylinder

(within just a few cycles). While it is suspected that

asymmetric flow in the interrotor atmosphere may have

caused this, the question has not been important enough

to pursue to a conclusion. Details of the experiments and

samples of the data are available in Ref. [37]. The first corotation runs with the inner cylinder in vac- uum were troubled by the rotation of the magnetic

suspension support coil pole piece which was epoxied into

the center of the upper vacuum plate covering the outer

cylinder. The residual vertical and lateral misalignment

in the pole piece put an unexpectedly large torque on the

inner cyhnder and caused it to almost exactly follow the

motions of the outer cylinder. The strength of this "cranking effect" made it clear that it would be essential

to have the pole piece position fixed in the laboratory

frame.

The suspension was redesigned to permit this, and the

data immediately reflected the improvement. For the

first time, true corotation runs were conducted with an

inner cylinder now free enough to be of real use in studying the now corotating atmosphere's effects on the total rotor damping.

The first runs showed that when the motor angular

speed was within =0.5% of the rotor angular speed, the

rotor would be caught in the potential formed by whatever interrotor coupling existed. As the rotors corotated,

the inner one would oscillate about its preferred position

viith an amplitude of about 30 degrees or less and periods

ranging from 10^ s to 3 x 10^ s. Ten period averages of the data showed that the oscillations were sometimes

purely sinusoidal but often irregular and resembling a

sinusoid with 3rd harmonic distortion. Attempts to in- vestigate the dependence of the oscillation waveform on

initial conditions of oscillation are in progress. The na- ture of the coupling itself is also being studied. Dipole

moments of known amount are applied to the outer cylinder via a battery and cylindrical capacitor plates and the

subsequent effects on the rotor motion are observed. A

precision electrometer sensitive to 60 electrons /s of current flow is also being used to study the rotor charge

distribution both during and after the startup of the ion

pump. It is anticipated that coating of the rotor surfaces

with a conducting film will be necessary in order to reduce electrostatic coupling effects. Data covering a twenty-four hour period showing the

rotor undergoing a decaying oscillation while rotating is displayed in Fig. 1. The most notable feature of this data

is, in fact, the decay envelope bounding the data. It represents a power loss of =9 x 10"^'^ W, a remarkably

small amount of energy given the fact that the rotor has

0.3 mJ of rotational kinetic energy. Given that gas drag

is relative-velocity dependent, this result implies considerable effectiveness for the corotation protection method.

Figure 2 shows an equally interesting phenomenon.

£ 3. 108

AMPLITUDE. 30

DECAY r* = 2 X 10= sec

MOTOR

PERIOD_

3 11000

1000 1500 2000 2500 3000 CONSECUTIVE 10 PERIOD AVERAGES

Figure 1. Oscillations of inner rotor about synchronous position. The outer cylinder rotates at a constant rate with a period

of 3.11000 s. The roughness of the decay envelope is du£ to the spacing of the points from the 10 period averaging.

3.10

3,08 aO 3.06 u

. 3-04 Q

o

cr

uj 3.02

Q.

CCo 3,00 o

2.98

2.96

ROTOR: I = 1100 cgs E = 2445 erg

T* « 5 X 10^ sec P = 0.01 erg/sec

VERTICAL OSCILLATIONS

SEQUENTIAL 10 - PERIOD AVERAGES

Figure 2. Anomalous rotor slomng in non-corotation mode.

The return to the extended decay baseline shows the cotiservative nature of the temporary rnode. With the outer cylinder still, the inner rotor was allowed

to undergo spindown. At a period length of =2.980 s, it began to oscillate vertically, as if the suspension became

marginally unstable. The rotor suddenly lost rotational

kinetic energy, but as the oscillation stopped, it seemed to regain all it had lost to the vibratory mode when its period

reached =2.985 s; it returned to its original decay slope as

if nothing happened. This mysterious activity might result from some type of rotational /vertical mode coupling,

although it is difficult to understand the conservative na- ture of it in light of the electronic damping provided by

the magnetic suspension feedback circuitry.

Inevitably, the rotor decay time when suspended by

active magnetic suspensions will be eddy current limited. And, it is difficult to believe that vertical-rotational

mode coupling can be made sufficiently low that an active

suspension can be tolerated. For this reason it will probably be essential to do the final version of the m experiment with a passive, lossless superconducting bearing,

such as that built by Harding at the Jet Propulsion Laboratory [45]. Such a system, in simplified form, is currently under construction by Brown at the State University of New York at Binghamton [46]. No other problem seems to be so critical to the final outcome of the experiment as that of bearing design.

Speed fluctuations in the motor, while still only partially

633
characterized, might be corrected by suitable feedback,

providing that the long term accuracy is not lost in the

process. Higher levels of vacuum can be achieved by us- ing an air bearing (or, perhaps, a magnetic bearing) with

a wider throat—^this is a hardware problem only. The

nature of the couplings between the rotor and its en- vironment will have to be studied individually, and judicious application of superconducting shielding and other

protection mechanisms vvill be necessary. Ultimately,

spherical rotors may be necessary, since they can be

built freer of moment of inertia asymmetries than either

cylindrical rotors or rotors of a hybrid geometry. Finally,

if necessary, the experiment could be fedback and the

inner rotor served to a null relative position with the outer rotor at the bottom of their potential well in order

to obtain an unambiguous w. signal.

4. Conclusions

The variation of G with many factors has been laboratory tested at various levels of precision in the past and

no nonconstancy has been found. Now its variation with

time and the related matter creation question are becoming of greater interest to gravitation and cosmology. Astronomical tests have become inconclusive and will probably always be challengeable under the " liberahzation"

of physical theory needed to encompass time varying G

cosmologies. Laboratory tests are controllable and can

assault these challenges.

, Laboratory tests of matter creation are underway, and

G tests are being planned. These rely on the prospect

that laboratory rotors can be made much smoother and of longer decay time than the earth in its rotations. These

efforts to overcome the classical barriers of bearing and

gas-drag frictions use new methods, including magnetic

suspension and corotation of the gas atmosphere.

Beams and Fremerey obtained rotational decay times

of 10^** s for tiny steel spheres; and if the rotor moment of

inertia could be increased by 10^ while simultaneously de- creasing the bearing and gas drag by 10^, a cosmologically significant decay time of 10^^ s would result. While it is relatively easy, although not trivial, to scale up the rotor moment of inertia, it is a more difficult problem to reduce the bearing and gas friction. In fact,

ferromagnetic suspension bearings seem, at this time, to have insurmountable limits which may dictate the use of the more nearly lossless superconducting suspension sys- tems. Corotation of the gas, in principle, should eliminate

it as a source of drag. Since this is accomplished by coro- tating an outer cylinder around the inner cylinder, however, new problems such as interrotor couplings are

introduced. The electrical aspects of such couplings are only now beginning to be investigated, but other studies

discussed have yielded interesting information. Measurements of the interrotor oscillation amplitude show energy

losses in this mode of only lO"^"* W. When studied in con- junction vdth the rotor secular decay rate, we can con- clude that corotation can be effective in reducing the gas

drag component of the overall rotor damping.

A host of other technology has been developed and

much of it will be applicable to other areas of fundamental measurements as well. We gratefully acknowledge support by the U.S. National Bureau of Standards under Precision Measurement

Grant G8-9025, the National Science Foundation under

grants PHY78-3208 and PHY80-07948, and the University of Virginia. We also thank George Jones for his work

on the experiment and for taking the data presented

here.

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(1922).

[12] L. W. Austin and C. B. Thwing, Phys. Rev. 5, 294 (1897).

[13] Q. Majorana, Atti della Reale Accademia dei Lincei 28

(1919).

[14] H. Russel, Astrophys. J. 54, 334 (1921).

[15] Q. Majorana, C. R. Acad. Sci. 173, 478 (1921).

[16] Q. Majorana, Philos. Mag. 39, 488 (1920).

[17] P. R. Heyl, Science 55, 349 (1922). - :

'

[18] D. R. Long, Nuovo Cimento B 62, 130 (1981).

[19] R. C. Ritter, in The Proceedings of the Second Marcel

Grossman Conference on General Relativity, Ed. by R.

Ruffini (North Holland, Amsterdam, in press).

[20] P. A. M. Dirac, Nature (London) 139, 323 (1937).

[21] V. Canuto et al, Phys. Rev. D 16, 1643 (1977).

[22] P. A. M. Dirac, Proc. R. Soc. London, Ser. A: 365, 19 (1979).

[23] F. Hoyle, From Stonehenge to Modem Cosmology (W. H. Freeman, San Francisco, 1972).

[24] R. D. Reasonburg, in Proceedings of the Second Marcel

Grossman Conference on General Relativity, Ed. by R.

Ruffini (North Holland, Amsterdam, in press).

[25] J. G. Williams, W. S. Sinclair, and C. F. Yoder, Geophys.

Res. Lett. 5, 943 (1978).

[26] P. M. Muller, in Proceedings of the Workshop Meeting on the Measurement of Cosmological Variations ofthe Gravitational Constant, Ed. by L. Halpern (University of Florida, Gainesville, 1978).

[27] R. C. Ritter and J. W. Beams, in Proceedings of the Workshop Meeting on the Measurement of Cosmological

Variations of the Gravitational Constant, Ed. by L. Hal- pern (University of Florida, Gainesville, 1978).

[28] R. D. Rose et al., Phys. Rev. Lett. 23, 655 (1969).

[29] R. H. Dicke, in The Earth-Moon System, Ed. by B. G. Marsden and A. G. W. Cameron (Plenum Press, New

York, 1966).

[30] W. H. Munk and G. J. F. MacDonald, The Rotation of the Earth, (Cambridge University Press, London, 1975).

[31] G. P. Pil'nik, Sov. Astron. 20, 608 (1976).

[32] R. Hide et al., Nature (London) 286, 114 (1980).

[33] R. C. Ritter et al. Nature (London) 271, 228 (1978).

[34] J. W. Beams, Rev. Sci. Instrum. 34, 1071 (1963).

[35] J. W. Beams et al., Rev. Sci. Instrum. 33, 151 (1962).

[36] J. K. Fremerey, Phys. Rev. Lett. 30, 753 (1973).

[37] G. T. Gillies, Ph. D. Dissertation, U. Va. (University Microfilms, Ann Arbor, 1980).

[38] W. S. Cheung et al. Precis. Eng. 2, 183 (1980).

[39] G. R. Jones, Jr., M. Sc. Thesis, University of Virginia, 1980, unpublished.

[40] R. V. Jones and J. C. S. Richards, J. Sci. Instrum. 36, 90

(1959).

[41] W. R. Tovvler, Thesis, University of Virginia, 1967, unpublished.

[42] R. C. Ritter et al., in Atomic Masses and Fundamental

Constayits 5, Ed. by J. H. Sanders and A. H. Wapstra

(Plenum Press, New York, 1976), p. 629.

[43] G. Luther, private communication, 1981.

[44] R. C. Ritter and G. T; Gillies, unpublished, 1980.

[45] J. T. Harding, Technical Report Number 32-897 (Jet Propulsion Laboratory, Pasadena, California, 1966).

[46] Y. J. Brown, private communication, 1981.

634
Precision Measurement and Fundamental Constants 11, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Interpreting Dirac's Large Numbers Hypothesis

W. Davidson

University of Otago, Dunedin, New Zealand

Dirac's large numbers hypothesis (LNH) and its implications are reviewed. Recent work shows

that if the LNH is correct then not only does the Newtonian gravitational constant decrease on an

atomic time scale (G t~^), as inferred by Dirac, but the electrical force between two charged particles becomes stronger relative to their mutual gravitational force regardless of which of Dirac's two

space-time scales, atomic or gravitational, is used. This is made explicit by introducing a Coulomb constant 7 analogous to G. The derivation of the two metrics directly from the LNH is demonstrated. Some observational implications of the LNH are listed. Key words: Dirac cosmology; gravitational and electromagnetic constants; metric reciprocity; observations.

1. Introduction

As is well known, Dirac has suggested [1,2] that the

large dimensionless numbers of physics and astronomy

are related by equations in which the coefficients are of

order unity. This conjecture vidll be referred to as the

"large numbers hypothesis" (LNH). Three such numbers

are given special consideration by Dirac [1]. One is the

ratio of the electric to the gravitational force between

the electron and proton: e^/Gmg mp. Another is the ratio of the Hubble time T of the expanding universe to a

characteristic atomic time: T/{e^/me c^). The third is: M/mp, the number of baryons in the observable

universe. Dirac finds that the first two numbers are ap- proximately equal and the third is roughly the square of

this. McCrea and I have shovra [3] that if the LNH is as- sumed to be true then not only does the Newtonian gravitational constant decrease on an atomic time scale, as inferred by Dirac [1], but the electrical force between

two charged particles becomes stronger relative to their

mutual gravitational force, regardless of which of Dirac's

two space-time scales, atomic or gravitational, is used.

For this purpose we introduced a Coulomb constant y, to play the same role for the electrical force that G does for the gravitational force. Thus if we write appropriately

X

Y

N

_ ye'

G nip m„

7 e ^/rrie c

^ M

then very roughlyX ~ Y - W\ N ~ (lO^V.

(1)

(2)

(3)

(4)

It will be appropriate to write Y ~ t, since F is a measure of T in units of atomic time, and should vary

v^ath the age of the universe. In view of Eq. (4) the LNH

then requires that

X ~ Y - t N ~ t^.

(5)

(6)

It is then with remarkable self-consistency that we may

derive from Eqs. (5) and (6) the two space-time metrics,

atomic A and gravitational E, which have been suggested

by Dirac as providing separate scales for atomic and

gravitational phenomena [1]. This derivation direct from

the LNH will be given in the next Section.

2. Derivation of Dirac's A and E IVIetrics

from the LNH

It vdll be assumed, as in [1], that atomic masses and

charges are constant regardless of the space-time scale used. In the present treatment this means that any time

variations required by the LNH are to be borne by G and

7. We first seek a space-time metric A in which the

result Y ~ t of Eq. (5) stems from the assumption that

Ta « t (7) while 7 is constant. Suppose the metric has the

Robertson-Walker form - cHt^ - Rl{t){dr^ + rH^ + r^siu^Qd^^) (8)

in which, following Dirac's interpretation of the LNH

[1], the spatial curvature is assumed to be zero. Then as

in Ref. [3] the result Eq. (6) leads directly to

1/3 (9)

where to is an arbitrary constant. The result X ~ t of Eq. (5) then requires that in this metric G behaves as Ga ^' t-' (10)

since it is inherently assumed that

7^ — constant . (11)

Suppose alternatively, that the relation Z ~ f of Eq. (5) is realized because 7 varies while G is constant. We therefore seek a metric E having the properties that, on the E scale.

Ge - constant

7£ =c i .

(12)

(13)

635
Let the metric be of the form

ds i = c^di^ - Ri{T)idr^ + r^d<^ + sin^ Q d <^^) , (U)

giving the matter the same comoving coordinates r, 0, ^

as in the A metric. Let rhoriz be the limiting r value of observed matter (seen by light leaving the matter at

T = 0 and arriving at the observer at the present epoch

To). Then

rhom - c C -~T^ = F(tJ. (15)

The number A'^ of baryons observed up to the horizon at epoch To is proportional to rhoriz- Hence by Eqs. (6) and

(15) we obtain for general t F(t) 0^ and so, on differentiating,

1 a t -1/3 dt di (16)

It should now be noted that the relation F ~ ^ of Eq. (5) combined with Eq. (13) means that

But according to Eq. (14)

Te =

Re(t)

dREi-rVdT

'

Equations (16), (17) and (18) then give

REiv =

as well as T o<: t^, and hence we write

(17)

(18)

(19)

(20)

We choose t,-j in t time to correspond to t„ in t time.

Hence

To = jto. (21)

The A and E metrics are therefore

2^ dSi = c^dt^ (dr^ + r^d@^ + singed <^^) , (22) -

dsp - c^dT^ ~

4/3{dr^ + r^dQ^ + r^sm^ed<\>^) (23)

which evidently stand in the conformal relation

2

tn ds^A (24)

In the A metric Eqs. (10) and (11) hold; in the E metric

Eqs. (12) and (13) hold, and in particular ^ by

Eq. (20). There is therefore a simple reciprocity between

the metrics with regard to electromagnetic and gravitational phenomena. But evidently the electrical force between two charged particles becomes stronger relative

to their mutual gravitational force, regardless of which of Dirac's two space-time scales is used [3]. If the LNH were the correct explanation of the large

number coincidences, rather than say an anthropic principle [4], then according to the above interpretation elec- tromagnetic and gravitational phenomena can be

described in terms of conventional theory only if the

appropriate space-time scales, A and E respectively, are

used. It is therefore of the greatest importance to test such a radical proposition observationally.

3. Observational Interpretation of Dirac

Cosmology

3.1 Observational Metric in Dirac Cosmology

The most refined methods of observation in astronomy

and cosmology are undoubtedly those based on atomic and

electromagnetic principles. For example, within the solar system the best measure of distance is now made by radar

methods and atomic clocks. The distance to the nearer

stars is determined by Euclidean geometry and the

method of parallax. The distance to the nearer galaxies is the "luminosity distance" of their identifiable variable

stars whose absolute magnitude is of known calibration; alternatively the brightest stars in galaxies of statistically

known intrinsic brightness are used. The average absolute

magnitude of, say, elliptical galaxies is then known and so the Hubble law between luminosity distance and redshift

is established yielding the present value of Hubble' s con- stant and its reciprocal T. All these criteria are based

upon atomic or electromagnetic phenomena. Therefore the

theoretical basis of observations in Dirac cosmology

should presumably be the A metric Eq. (22). An exception

to this would have to be anticipated where mass values

have to be assigned using the gravitational properties of mass, and then the E metric would have to involved. In particular, the Hubble time derived from the ap- parent magnitude-redshift data must be identified as

in Dirac cosmology. On the other hand the metric E

given in Eq. (23) provides a theoretical Hubble time TeSince Einstein's gravitational field equations hold in this (Einstein-de Sitter) metric, it is Te which must be in- serted into the Einstein equation relating the present mass density to the present Hubble constant. A summary of predictions, observable in principle, for Dirac cosmology is given below. Full details will be published elsewhere. Where appropriate, comparison is made

with the predictions of general relativity (GR) for the

case flo = 1 (when the present mass density is equal to the so-called critical density 3/8 tt GT^).

3.2 Predicted Features of Dirac Cosmology

(a) In Dirac cosmology the atomic age of the universe

is only 1/3 of the observed Hubble time [1], compared

with 2/3 X r in GR (Ho = 1). If the theory is correct then the estimated ages of globular star clusters [5, 6] are too high.

(b) In Dirac cosmology the deceleration parameter ob- tained by the apparent magnitude-redshift test should

give iqo)A = 2; by the cosmic mass density method it should give {q„)E - 0.5. In GR (fto = D the prediction is Qo -0.5 by both methods.

(c) The present cosmic mass density in Dirac cosmology is the same on scales A and E and is equal to

3/8 TT Ge Ti where = Ta /4, Ta being the observed

Hubble time. For a given observed value of the Hubble

time the present mass density predicted for Dirac

cosmology is therefore 16 times the critical density in GR. This may be a powerful method of ruhng out Dirac

cosmology if the so-called "missing mass" is not found.

On the other hand the cluster evidence [7] for missing mass may support the Dirac theory.

(d) The epochs in atomic time for the cross-over from

radiation dominance to matter dominance, recombination

636
of hydrogen and formation of galaxies are all much earlier and at higher material densities in Dirac cosmology

compared with canonical GR. These circumstances to- gether with the result that G « i on the A scale seem

to assist the formation of galaxies in Dirac cosmology

(see also Ref. [8]). (e) The total mass density (after the cross-over) varies

as t^^ on the A scale compared with t"^ in canonical

GR (flo = 1). The radiation density varies as t~'^ compared with t"^ in GR.

(f) The redshift z in Dirac cosmology varies with

atomic time such that 1 + z o: t in contrast with 1 -I- z a t"^ in canonical GR (Oq = 1). Hence all cosmic events specified by redshift or temperature occur at a much earlier fractional age of the universe in Dirac

cosmology compared with GR.

(g) Conventional Maxwellian electromagnetic theory

can be applied consistently in A-scale Dirac cosmology,

whereas on the E gauge electromagnetic quantities

behave quite differently. In particular Planck's constant

varies with epoch such that He ^ t^, in addition to the

variation of the Coulomb constant yi^ t^). The latter re- lation means that the electrical space permittivity to ^

t"^ and the magnetic permeability (jl^ v. t"^. In addition

the adiabatic relation for radiation density, viz. p^/?" =

constant, does not hold on the E scale. Therefore if the

LNH is correct the results of combining Einstein's equations with the conventional mechanics of electromagnetic

theory would be invalid. The canonical GR big-bang

theory would therefore fail.

References

[1] p. A. M. Dirac, Proc. R. Soc. London Ser. A: .365, 19 (1979).

[2] P. A. M. Dirac, Proc. R. Soc. London Ser. A: 165, 199 (1938).

[3] W. Davidson and W. H. McCrea, Proc. R. Soc. London Ser. A: 374, 447 (1981).

[4] B. Carter, in Confrontation of Cosmological Theories and

Data: Proceedings, Ed. by M. S. Longair (Kluwer, Boston,

1974), p. 291.

[5] M. Schwarzschild, Q. J. R. Astron. Soc. 11, 12 (1970).

[6] P. Demarque and R. D. McClure, in The Evolution of Galaxies and Stellar Populations, Ed. by B. M. Tinsley and R.

B. Larson (Yale Univ. Obs., 1977), p. 199.

[7] G. 0. Abell, in Ann. Rev. Astron. Astrophys. 3, Ed. by L. Goldberg (Annual Reviews Inc., Palo Alto, 1965), p. 1. [8] R. H. Dicke, Rev. Mod. Phys. 34, 110 (1962).

637

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

New Laboratory Test of the Equivalence Principle

P. T. Keyser, J. E. Faller,* and K. H. McLagan

Joint Institute for Laboratory Astrophysics, National Bureau of Standards and University of Colorado,

Boulder, CO 80309

A test of the principle of equivalence using a large fluid (surrogate) fiber Ebtvbs apparatus is presently being undertaken at the Joint Institute for Laboratory Astrophysics. Preliminary measure- ments using a 0.25 m diameter fluid system were sufficiently encouraging that we have embarked on

the construction of a five times larger (1.27 m diameter) system employing approximately 500 kg of lead and 500 kg of copper as the test masses. The first experimental results on the equivalence of gi'av- itational and inertial mass with this new large apparatus are expected in 1983 or 1984. Key words: equivalence principle; general relativity; gravitation.

1. Introduction

The ratio of the inertial mass of an object to its passive

gravitational mass is postulated to be independent of the

material in metric theories of gravitation such as general

relativity [1]. In fact, in the last century, the equivalence

of inertial mass (m ) and gravitational mass {M ) for two

materials (A and B) as measured by the parameter t|, where

Ti(A, B) - K M_

m A m B

1 K + K

2 m A m B_

(1)

was known to 2 x 10"^ [2], and was measured to be less than 3 x 10"^ by 1922 [3]. Two more recent measure- ments have lowered the upper limit on ii to 3 x 10"" and

0.9 X 10"^^ respectively [4, 5]. Nevertheless, a fundamental motivation exists for checking this hypothesis to the highest possible accuracy: the principle of equivalence

is basic to theories of gravity.

At the Joint Institute for Laboratory Astrophysics

(JILA), we have developed a new type of torsionpendulum apparatus for the purpose of improving the

accuracy of the Ebtvbs experiment, which tests the

equivalence of gravitational and inertial mass. All of the

experiments mentioned above were done with a traditional torsion balance; but, despite a significant amount

of effort, none of these experiments reached the sensitivity that would be expected if they were limited by the

Brownian motion of the torsion balance. In all cases, other sources of noise dominated the fundamental ther- mal noise. The torsion fiber itself is a source of noise,

while the classical torsion balance is extremely sensitive

to seismic noise. Also, the sensitivity of a torsion balance

is limited since the fiber both supports and torques the masses. Any attempt to increase the sensitivity of tEe balance by increasing the size of the masses also requires an increase in the diameter of the fiber, and this then

places more severe requirements on the position detector

since the torsional stiffness of a fiber increases as the

fourth power of its diameter.

The approach we have taken at JILA to deal with

these limitations of the traditional fiber involves the util- ization of a new type of (surrogate) fiber, in which the

functions of support and torque are separated. We do

this by using a fluid (water at 3.98 °C, its maximum density point) to provide the support function of a fiber, and

an electrode array to provide the centering and restoring

functions. A 0.25 m diameter float using 3 kg test masses

of copper and tungsten was successfully tested and used

by Reiser and Faller [6]. While their result (t] = 0.2 ±

4.0 X 10"^^) was an order of magnitude worse than what

one would expect if the motion of the float were predominantly determined by Brownian motion at periods of 24 h, a power spectrum analysis showed that down to fre- quencies of 10 cycles /day the observed motion was in fact limited by the Brownian motion of the float.

2. Description and Analysis of the

Apparatus

We are now constructing a 1.27 m diameter float, shown schematically in Fig. 1. Note there are seven balls

in a hexagonal pattern on the lid of the float (see also Fig. 2), and three balls in a line on the underside of the

lid of the float container. These latter three are electri- cally insulated from the container. The central balls function as centering electrodes—an ac potential of about

1 kV is applied across them to center the float. The outer

balls function as torque electrodes. (The float can be

zeroed in any of six positions, differing by 60°.) Varying

*Staff Member,

Standards. Quantum Physics Division, National Bureau of Figure 1. Schematic of large float showing internal aiTange- ment of masses.

639
the ac potential applied to the outer balls serves to vary

the torsion constant. As seen in Fig. 1, we are using 24

cylindrical masses; 12 of copper and 12 of lead. The total mass of each 12-piece test mass is about 490 kg. A breakdown of equivalence (i.e., a non-zero t]) would result in a

slight difference in the acceleration of the two sides of the float toward the sun, and this would evidence itself as a 24-hour-period variation in the position of the float. The sensitivity of this type of apparatus can be ex- pected to improve with size since most of the noise terms

couple in through area (surface) effects, and therefore

their relative importance decreases as one scales up the

apparatus. Further, as one increases the size, D, of the

apparatus, the fundamental thermal-noise-limited sensitivity improves, scaling as Z)*^. To see this, note that the

noise torque in a bandwidth A/ due to Brownian motion

is given by: W = (2/c6r 6A/)i^ (2) where is Boltzmann's constant, T is the absolute temperature, and b is the damping coefficient in the equation

of motion for the float with an external torque, namely:

IB + he + KQ = Textemal • (3) Here I is the moment of inertia of the float, and K is the

torsion constant. The amplitude of the 24-hour-period

signal torque is given by:

Tsignai = y\M a D cosh (4) where M is the mass of one of the two test masses, a is the acceleration toward the sun (0.62 cm/sec^), D is the

effective moment arm of the test masses, and 8 is the

declination of the sun. The coefficient of 0 in Eq. (3) will depend on some geometric factor times the viscosity v of the fluid. Straightforward dimensional analysis

(remembering that the dimensions of v are M/LT) shows

that h dIP {D being a characteristic linear size of the

apparatus). Equating Tgig^ai and Tnoise (i-e., S/N = 1), not- ing that M ^ D^, and solving for r\, we see that

1 {2hTv^f)'^

D

5/2 a cos 8

(5)

That is, the kbT limit on r\ improves as D'^'^, or the sensitivity of a given apparatus improves as the five-halves

power of its size. For our new apparatus, the calculated

Ict, one-day Brownian motion limit on the equivalence of gravitational and inertial mass is 7 x 10"^^.

3. Discussion

In both the Princeton [4] and the Russian experiments

[5], seismic disturbances turned out to be a primary noise

limit for their systems; and to this type of noise, the fluid approach seems to be remarkably insensitive. We have

yet to detect any effect attributable to seismic disturb- ances on the position of fluid pendulums. One explanation

for this can be found by noting that at least for low fre- quencies, time-varying accelerations do not—as they do

with a normal fiber—stretch the system. Also, the fact that the zero of the system is established by a macroscopic electrode array rather than a microscopic fiber under stress probably accounts to some extent for the ob- served insensitivity of the zero to vibrations (and for that

matter, to changes in temperature).

Of the various noise sources identified in the earlier

experiment of Reiser and Faller [6], only the effect of the

residual mass-quadrupole moments can be expected to be

substantially worse in this new larger apparatus. This effect can be reduced by measuring the quadrupole mo- ments of the float and systematically tuning them out by mass tuning (as was done on the small float); or one can remove the effect of residual mass-quadrupole moments

on the data by directly measuring the changes in the gra- dients of the external gravitational field and then

correcting the float's response for these changing gradients. In the case of the large float, we expect to employ both methods.

A number of practical problems have been encountered

as a result of scaling up the smaller system to the large

one, which is presently under construction, and which is probably near the practical size limit for scaling. As one example, the float as received from the fabricator was

not as round on top as we had hoped. However, by using

a hydrauHc jack to deform the top just past the elastic

limit, we were able to improve its as-received roundness

of ±0.035 cm to ±0.010 cm, resulting in a fractional out- of-roundness of ^R/R = 1.6 x lO"'*. Also, due to the

vastly increased size of the apparatus, price and ease of fabrication became much more critical factors in choosing

the materials (e.g., not only were gold and silver out of the question, but even tungsten was). It appears that

lead and copper are the most practical choices. Initially we had planned to use Pb and Zn, however, the Zn cast- ings were not as nonmagnetic as a sample had led us to expect, and they were returned to the manufacturer. We

have subsequently switched to Cu. The paramagnetism

problem associated with CuO is avoided by using OFHC

(oxygen-free, high conductivity) copper, and by goldplating the masses to prevent any oxidation vdth time.

At least a part of the non-kbT performance at long

periods of the Keiser-Faller experiment has now been

traced to their use of an external auto-coUimator to monitor the float's angular position. This introduced not only

the stability of the auto-collimator, but also the stability of the stand on which it was located as well as the

mechanical integrity of the intermediate linkages

between the auto-collimator stand and the container lid whose electrode array determined the "zero" of the

fiber. A new sensing system (see Fig. 2) which we have

devised for this experiment promises to have a much

higher mechanical integrity in that the sensing optics are a part of the same container lid on which the zero- determining electrode array is located.

Differential Preamp

- 2"Aluminum

Centering Electrode Ball 2"Aluminum

Torque

Electrode Bolls

-0001 0 0,001 8 (radians

Float Position Output

for LEO Current of 10 mA

Figure 2. Schematic of new angle-measumig system.

In brief, this new sensing system consists of an in- frared light-emitting diode (whose output light is focused

into a line image by a cylindrical glass "lens" attached to the top of the float) and a split photodiode detector. As

the float turns, the cylindrical lens moves with respect to

640
the LED-photodiode pair, causing the hne image to move

on the split photodiode. This results in a change in the

output of the two halves of this device, giving rise, when

amplified, to a rotation-dependent voltage as shown on

the right half of Fig. 2. With two such detection

systems—one on each side—and proper summing of their

outputs, the detector is insensitive to sideways motion of the float without the sacrifice of any of its sensitivity to rotation.

4. Conclusion

This new large apparatus is now in the final stages of construction. We are awaiting delivery of the (encompassing) magnetic shield before installing this apparatus

in the "Spectroscopy Lab" in the sub-basement of JILA

to begin a prolonged (possibly a year-long) shake-down of the full apparatus. We plan to put a 0.2 to 0.5-cm layer of

oil on the surface of the water, which we have experimentally determined lowers the evaporation rate by a factor of about 1000. This should help avoid local evaporative cooling of the surface of the water (which could

lead to thermal gradients) and prevent the subsequent

recondensation of water on the float (which could produce

undesired quadrupole mass moments). We also plan to make the initial tests without cooling; and, by establishing a positive upward temperature gradient, we will try

to achieve sufficient temperature stability to inhibit convection. Cooling, while straightforward and serving to reduce the linear expansion coefficient, a, water—and

therefore the tendency to convect [7]—by more than an

order of magnitude (at 20 °C, a ~ 7 x \0-^V°C, while at 4 °C, a ~ 1.5 X 10"^/°C), extracts a certain price in experimental inconvenience. This is particularly true in a system of this size, and we would rather not have to pay

this price unless it proves absolutely necessary. However, even without cooling, the scale of this experiment

is such that the first improved results on the equivalence

principle using this large float should not be expected until 1983 or 1984.

This work is supported in part by the National Science

Foundation, Grant No. PHY79 -04928, and in part by the

National Bureau of Standards as part of its research pro- gram on improved measurement techniques for applications to basic standards.

References

[1] R. J. Adler, M. J. Bazin, and M. Schiffer, Introduction to General Relativity (McGraw-Hill, New York, 1975), p. 4.

[2] F. W. Bessel, Poggendorffs Annalen der Physik 25, 401

(1832).

[3] R. von Eb'tvbs, D. Pekar, and E. Fekete, Ann. Phvs.

(Leipzig) 68, 11 (1922).

[4] P. G. Roll, R. Krotkov, and R. H. Dicke, Ann. Phys. (N.Y.)

26, 442 (1964).

[5] V. B. Braginsky and V. I. Panov, Sov. Phys. JETP .34, 463

(1972).

[6] G. M. Keiser and J. E. Faller, "Ebtv'ds Experiment with a Fluid Fiber," in Proceedings of the Second Marcel Gross-

'inan Meeting on Recent Developments 07i General Relativity, Ed. by R. Ruffini (North-Holland, 1982), p. 969.

[7] C. Normand, Y. Pomeau, and M. G. Velarde, Rev. Mod.

Phys. 49, 581 (1977).

641

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

What Test Masses Are Best for an Eotvos Experiment?

D. F. Bartlett, J. Shepard, and C. D. Zafiratos

Department of Physics, University of Colorado, Boulder, CO 80309

No element lighter than aluminum has been used as a comparison mass in the recent precise tests of the equivalence of gravitational and inertial mass. We discuss how the substitution of solid hydrogen

for aluminum as the hght mass could increase the sensitivity of an Ebtvbs experiment by a factor between 10 and 100. Key words: Ebtvbs experiment; equivalence principle; gravitational mass; inertial mass; test mass.

1. Introduction

In an Ebtvbs experiment, the inertial-and gravitational masses of two test bodies of different materials are compared. The results of this comparison are characterized

by the coefficient r\:

v^^here M is the gravitational mass and rn the inertial mass of substances A and B. Since Galileo, the observed

universality of the acceleration of differing masses in a

gravitational field has been used to set increasingly

stringent limits on the departure of -x] from 0. Since these

tests form the basis of the equivalence principle, it is important to know how different A and B must be for the

principle to be effectively tested. Ebtvbs himself used magnalium and snakewood among

a variety of test masses for the comparison of inertial and gravitational mass [1]. As experiments have become more precise, the choice of materials has narrowed [2, 3] until in the most recent published experiment only two

materials were used [4]. Aluminum and platinum were

chosen partly for their convenient physical properties.

They also differ in nuclear characteristics such as ratio of neutrons to protons or of binding energy to nuclear mass. These differences in nuclear properties should make the

experiment a sensitive test of the equivalence principle.

However even aluminum is a heavy nucleus in that the

distribution of its nucleons is well described by the

standard two parameter Fermi function used to describe

all nuclei heavier than carbon. Can one increase the sensitivity of an Ebtvbs experiment by choosing an element

lighter than aluminum for the light test mass?

The answer is clearly yes if hydrogen is the choice.

With no neutrons and no nuclear binding, hydrogen is clearly unique. If the strong or electromagnetic interactions contribute slightly differently to the inertial and

gravitational mass of a nucleus, hydrogen is about 10

times as good a test mass as aluminum [5]. If the anomaly is in the weak interactions, the benefit is a factor of 100 [6]. More convenient choices for the light mass may include

lithium and beryllium. In the absence of any model for a breakdown of the equivalence principle it is impossible to predict which complex nucleus would be the best choice. We vdll suggest that lithium is a better choice than

aluminum, though not nearly as good as hydrogen.

2. Strong and Electromagnetic Interactions

Since most of an atom's mass is in the nucleus, it is most likely that a positive effect in an Ebtvbs experiment

would come from differences in the nuclei of the two test masses. The most obvious criterion is the neutron to pro- ton ratio. As is well known, the ratio N^/Z remains close to one for all elements between helium and calcium after which the ratio increases gradually, reaching 1.53 at

lead. As shown in Fig. 1, a reasonable approximation is N/Z =0, For A = 1 N/Z =1, For 2 ^ A < 40

N/Z = 0.1 + 0.7 logioA, for 40 < A ^ 208,

Where

A = N + Z .

H D He Li C Mg Co Ni Y Sn To Pb

M / ^ / 2 /

/ 0.50 -

/

II

I

I

0 1.0 2.0 log A

Figure l. Plot of N/Z vs logioA. Semilogarithmic plot emphasizes light nuclei.

If neutrons and protons contribute differently to inertial and gravitational mass, hydrogen is clearly a unique

choice for the light test mass. We may quantify this uniqueness by defining a figure of merit,

FMnp = UN - Z )/A )heavy - ((A^ ' Z )/A )ught

643
For platinum as the heavy mass and aluminum as the

light mass FMnp is 0.16. If hydrogen replaces aluminum

as the Hght mass, FMpp increases to 1.2. Alternatively, the test masses may differ in the

amount of binding of the nucleons in the nucleus. A plot of binding energy per nucleon {BE IA ) vs log A is given

in Fig. 2 (A). The uniqueness of hydrogen is also evident

from this curve. A convenient figure of merit is FMb£ = {{A - M)/A )heavy - ((A - M )M )iight , where A is A'' + Z and M is the atomic weight. For platinum as the heavy mass and either aluminum or hydrogen

as the light mass, FMs^ is 0.4 x 10"^ and 8.5 x 10"^,

respectively.

H D He Li C Mg Ca Ni Y Sn Ta Pb

^ ^ ^ ^^ ^ \ \

c

A / 1 \ /

/ / \ V /

^ 1/ \ /

A ~

/A ^

/-"^

// /

/ / .

1 0 1.0 2.0 log A

Figure 2. Plot of (a) BE /A, (b) T, and (c) \V \ = BE /A + T

vs log A. The solid dots are from- measurements of Ref. 8, the isotopes used being as naturally occurring except for ''Li and

Pb; the crosses are from calculations of Ref. 7. In either case the errors represent fitting uncertainties. Of course these comparisons are not to be taken too

literally—particularly since the mechanism for a violation of equivalence is not specified. However, the comparison

does indicate that the use of hydrogen as a test mass

should increase the sensitivity of the Eotvbs experiment

to either the strong or electromagnetic interactions by

roughly a factor of 10. Conceivably a positive effect in an Eotvbs experiment

could come from the motion of the nucleons in the

nucleus. This could happen, for instance, if the usual rel- ativistic increase in the mass of a moving proton or neutron were slightly different for inertiaPand gravitational masses.

It is thus convenient to separate the binding energy into

its kinetic and potential parts,

= I? I - T

A

In an earlier paper, we used Hulthen wave functions to calculate the rms velocity and hence the kinetic energy of the nucleons in deuterium and helium [7]. Some time ago, Moniz and coworkers made careful measurements of the

motion of nucleons in nine complex nuclei [8]. Using

500 MeV electrons as a probe, they determined that the

distribution of nucleon velocities was well described as that of a Fermi gas having zero temperature and a Fermi

momentum kp ranging from 165 MeV/c for ^Li to 265 MeV/c_for "°^Pb. We have used the non-relativistic

equations = (3/5) A: ^ and T - k'^/2m, where m is the mass of nucleon, to determine T for the complex nuclei

shown in Fig. 2 (B).

It is evident from this figure that although hydrogen is

still the best choice its superiority is not as dramatic as

in the previous discussion. Lithium is a close second. Using a figure of merit,

FM^ = (r/A)pt - (m)ught

we find FM^ = 4 x 10"^ 14 x 10~^ and 24 x 10"=^ for Al,

*^Li, and H, respectively.

Finallyj_ Fig. 2 (C) showing the variation of mean_ potential I V I with A is obtained from the addition | V | =

T + BE. In a separate calculation, we have determined V

directly for nuclei heavier than carbon. We made numerical integrations using as inputs the standard two- parameter Fermi functions for optical model parameters

[9] and for nuclear charge distributions [10]. Our results,

which are not shown, confirm the trend oX Fig. 2 (C),

although with somewhat higher values for V that those

obtained by addition.

3. Weak Interactions

The present accuracy of the Eotvbs experiment is al- ready sufficient to limit any anomaly in the interaction of gravity with the strong or electromagnetic interactions

to about t]/FMbe = 10-12/0.0004 ^ 10"^ Cognizant of

this severe limit, Haugan and Will have recently looked

at the sensitivity of the Eotvbs experiment to the weak

interactions [6]. They find that the contribution of the

weak interactions to the observed inertial mass of the nucleus is Ew/mi = 2.2 x IQ-s iNZ/A^)F{N, Z), where

F(A'^,Z) is a slowly varying function about equal to one. After taking weak neutral currents into consideration,

they find that

FMw(Al) = {EWmi)pt - (Ew/MOm = -0.8 x lO"!"

for sin^ 0w = 0.2, where Sw is the Weinberg angle.

By contrast,

FMw(H) = (EWmOpt - 0.7 x 10-^

about 100 times larger. In Fig. 3 we have used the explicit form of F{N,Z )

H D He Li Li C MgAI Ca Ni Y Sn TajPb

Figure 3. Fraction of nuclear energtj assignable to weak in- teractions {Evf/mj) vs log A. All data from calculation of Ref. 6. The errors represent the unkyioivn contribution from

nuclear structure to Eyy. For nuclei lighter than aluminum

this contribution becomes hard to estimate.
given in Eqs. (2) and (3) of Haugan and Will's article to

plot E^/ nil foi" various nuclei. The superiority of hydrogen is evident.

4. Conclusion

The virtues of hydrogen for an Edtvds experiment were first mentioned by Lee and Yang in 1955 [5]. At

that time the theoretical advantages of hydrogen were

outweighed by the obvious experimental difficulties in- volved in actually using hydrogen as a test mass. Within

the last decade, however, cryogenic techniques have

been increasingly used in precise gravitational measurements. Such techniques may well make solid hydrogen a

feasible test mass. Meanwhile, lithium differs sufficiently

in nuclear properties from heavier nuclides to make it an

attractive second choice.

References

[1] R. V. Ebtvbs, D. Pekar, and E. Fekete, Ann. Phys

(Leibzig) 68, 11 (1922).

[2] P. G. Roll, R. Krotkov, and R. H. Dicke, Ann. Phvs. (N Y ) 26, 442 (1964).

[3] G. M. Keiser and J. E. Faller, Nuovo Cimento (to be

published).

[4] V. B. Braginsky and V. I. Panov, Zh. Eksp. Teor. Fiz. 61, 873 (1971) [Engl. Trans.: Sov. Phys. JETP .34, 463 (1972)].

[5] T. D. Lee and C. N. Yang, Phys. Rev. 98, 1501 (1955).

[6] M. P. Haugan and C. M. Will, Phys. Rev. Lett. 37, 1 (1976)

[7] D. F. Bartlett, J. Shepard, C. Zafiratos, and B. F. L. Ward

Phys. Rev. D 20, 578 (1979).

[8] E. J. Moniz, I. Sick, R. R. Whitney, J. R. Ficenec, R. D. Kephart, and W. P. Trower, Phys. Rev. Lett. 26, 445

(1971).

[9] C. M. Percy and F. G. Percy, At. Data Nucl. Data Tables

17, 1 (1976).

[10] DeJager et al., At. Data Nucl. Data Tables 14, 479 (1974).

645

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Equivalence Principles and Precision Experiments*

Wei-Tou Ni

Department of Physics, National Tsing Hua University, Hsinchu, China (Taiwan)

We use a general framework—the x-9 framework—^to study the empirical foundations of the Einstein Equivalence Principle (EEP), and to analyze the theoretical significance of various precision ex- periments and observations such as timing observations of pulsar signal propagation through galactic

gravitational field, the Hughes-Drever experiments, the Eotvbs-Dicke-Braginsky experiments, and the Vessot-Levine experiment. These experiments constrain various linear combinations of the components

of X to be metric to various degrees of accuracy, leaving only one component out of 21 not constrained. The theory with this component different from zero can be reformulated as a torsion theory. To test EEP and this theory, experiments on polarized test bodies are suggested.

Key words: equivalence principles; gravitation; pulsar signals; redshifts; symmetry violations; test- body experiments; torsion; unification of fundamental interactions.

1. Introduction

In 1907, Einstein [1] proposed the complete physical

equivalence of a homogeneous gravitational field to a uni- formly accelerated reference system and derived clock

and energy redshifts in a gravitational field from this equivalence. When applied to a spacetime region where

inhomogeneities of the gravitational field can be

neglected, this equivalence dictates the behavior of matter in a gravitational field. The postulate of this equivalence is called the Einstein Equivalence Principle

(EEP). EEP is the cornerstone of the gravitational cou- pling of matter and non-gravitational fields in general

relativity and metric theories of gravity.

The renaissance of general relativity research in the

last two decades together with the fundamental

discoveries and developments in particle physics leads to renewed interest in the microscopic origin of gravity.

The discovery of parity violation [2, 3] and CP violation

[4] in the weak interaction puts us into a symmetry broken world. The success of the Weinberg-Salam [5, 6] theory of unified electroweak interactions makes spon- taneous symmetry breaking a promising way to generate

"fundamental" constants and to unify interactions. The

incorporation of spontaneous symmetry breaking in gravity to generate Newton's gravitational constant Gn has

been considered by various authors [7]. Considering the

role of spins and different gauge groups, various torsion

theories and gauge-type theories of gravity have been

proposed. Many of these theories violate EEP in one way

or another at certain levels. With our present knowledge, a unification scale can be

drawn in Table 1. Electromagnetism and the weak interaction would be unified as the electroweak interaction at the

W-boson energy (—100 Gev). According to the grand unification schemes [8], the electroweak and strong interacTable 1. Energy and distance scale of possible unification offundamental interactions. The distances

in the scale are the reduced Compton wavelengths of the correspondiyig energies.

Electromagnetism

Weak Interaction

Strong Interaction QCD

2 X 10-1*5 100 Gev 2 X 10-29 cm

10^5 Gev

Electroweak

(W-bosons, desert or more

etc.) structures

desert or more structures

Planck distance

1.6 X 10-33

Planck mass

1.2 X 10 19 Gev

Grand Unification

{X, Y particles etc.)

Gravity

Quantum

Equivalence Principle desert or more structures Gravity

Einstein Equation

* Supported in part by the National Science Council

647
tions would be unified at an energy of 10^^ Gev. At present

accelerator energies (Vs ~ 62 Gev), Quantum Chromodynamics (QCD) looks promising in explaining the strong

interaction. From this energy and the W-boson energy to the grand unification energy, there is a gap of 13-orders of magnitude. The important question is whether in this wide

gap, there are more structures besides electroweak unifi- cation and QCD. In gravitation, the equivalence principle

and Einstein equation are only verified empirically at lower energies and larger distance^. Using dimensional ar- guments, quantum phenomena should be important in gravity at the Planck-mass energy i'hc^/G)^'^ — 1.223 x

10^^ Gev or Planck distance itG/c^)^^ - 1.616 x

10"^^ cm. Above this energy, it is also possible that the

grand unified interaction and quantum gravity can further

be unified. Here the important question is whether the

gravitational coupling of matter still obeys the Einstein

equivalence principle. To answer this question, ever more

precise measurements in ever more diverse situations are

desired. This is a paramount challenge to the field of precision measurement.

As an illustration of why the naive point of view of no more structures might not be right, we quote the follow- ing historical example. Einstein's efforts of unifying elec- tromagnetism and gravity were not successful. Now we

understand why. Before electromagnetism and gravity can be unified, one has to take into account the weak in- teraction and, possibly, the strong interaction. Similar

things could happen again and there would be more

structures. Both the fundamental role of EEP in general relativity

and its possible violation at a certain level demand a close scrutiny of its empirical foundations. For this purpose, we

use a general framework—the xS framework [9, 10]—to study the empirical foundations of EEP, to analyze the

theoretical significance of various experiments and observations, and to propose new experiments.

2. The x-fif Framework and Three Theorems

Since here we are only concerned with the behavior of matter in a gravitational field, we treat the gravitational

field as an external field. Most experiments concern elec- tromagnetism. Therefore to be more specific, we con- sider electromagnetically interacting systems in this article. Generalizations to strong and weak interactions wiW

be presented elsewhere. In previous work we have used

a general framework—the x-9 framework [9, 10] to study

Schiffs conjecture and theoretical relations of various

equivalence principles. The X'9 framework is rather

comprehensive [10] and we use it in sections 3-6 to study

the empirical foundations of EEP.

This framework can be summarized in the following in- teraction Lagrangian density

2/ m/ S(x - X/)

(1)

where x'''^'' - X*^''' - ^x'^'^' is a tensor density of the

gravitational fields (e.g., gr,,, c}), etc.), and j'', Fjj =

Ajj-Aij have the usual meaning. The gravitational constitutive tensor density x''^' dictates the behavior of

electromagnetism in a gravitational field and has 21 in- dependent components in general. For a metric theory

(when EEP holds), x''*^ is determined completely by the

metric g'^ and equals i-g )^H^g'''g^^ - ^g'^g'^^)-

In Refs. [9, 10], we have proved the follo^dng theorem.

Theorem I. —For a system whose Lagrangian density is given by Eq. (1), WEP I holds if and only if

1/2 (2)

where cj) is a scalar function of the gravitational field and

, ijk [' _

1, if (ijTc i) is an even permutation of (0123)

-1, if {ijk i) is an odd permutation of (0123), (3)

0, othervdse.

By WEP I we mean Galileo's weak equivalence principle

that macroscopic test bodies with the same initial conditions follow the same trajectories. If cj) =it o in (2), the

gravitational coupling to electromagnetism is not minimal

and EEP is violated. Hence WEP I does not imply EEP.

But WEP I does constrain the 21 degrees of freedom of x

to only one degree of freedom (4)). In the actual empirical situation, since Ebtv'ds-Dicke-Braginsky experiments

are performed on unpolarized bodies, they constrain only

2 degrees of freedom of x (cf- §4). Only when these ex- periments are performed on polarized bodies with various

electromagnetic energy configurations, can they constrain the other 18 degrees of freedom.

In the nonmetric theory with x'-''^' ^ 0) given by

Eq. (2) [11], there are anomalous torques on electromagnetic-energy-polarized bodies so that different test bodies will change their rotation state differently, like magnets in magnetic fields (cf. §5). Since the motion of a macroscopic test body is determined not only by its trajectory but also by its rotation state, the motion of polarized test bodies will not be the same. We, therefore,

propose the follovdng stronger weak equivalence principle (WEP II) to be tested by experiments, which states that in a gravitational field, the motion of a test body

with a given initial motion state is independent of its internal structure and composition (universality of freefall motion). Therefore, in this framework, the imposition

of WEP II guarantees that EEP is valid. In this non- metric theory the first-order gravitational redshift is metrical. These results can be stated as the following two

theorems:

Theorem II—For the Lagrangian of Eq. (1), WEP II implies EEP.

Theorem III—WEP I implies that the first-order gravitational redshift is metrical.

3. Pulsar Signal Propagation in the

Galactical Gravitational Field

If EEP is observed, photons vdth different polarizations as test particles shall follow identical trajectories in a gravitational field. In pulsar observations, the pulses

and micropulses with different polarizations are corre- lated in their general structure and timing [12]. Due to precise timing and rich polarization data, pulsar signal

propagation in the galactic gravitational field constitutes a high precision test of EEP.

Since our galactic Newtonian potential U is of the

order of 10^^, we use the weak field approximation in the X-g framework. The vacuum Maxwell equation in this framework is

ix'"''^ Akj),j = 0, (4) Neglecting x'^*^p iii ^ slowly varying field, Eq. (4) be648
comes

X'-''^^Ai.fj =0. (5) For weak field, we assume

where

with Ti'-^ the Minkowski metric and | x'^^'s I < < 1. Using the eikonal approximation, we look for a plane- wave solution propagating in the z-direction. Imposing

the radiation condition in the zeroth order and solving

the dispersion relation for co, we obtain

o)^ = k{l + ^[{Ki + K2) ± V(/^i - K2)'' + 4^2]} (8)

where

/Dioio ^ ^(1)2020 _ 2^(1)2023 - ^(1)1020 _ ^(1)1023 ,

2^(1)1013 + ^(1)1313. ^ ^(1)2323

_ ,,(1)1320

(9) + X'(1)1323

Photons with two different polarizations propagate with

different speeds v± - oiJk and would split in 4- dimensional spacetime. The conditions for no splitting (no

retardation) is (0+ = o)- , i.e. K, = K2, K ^ 0. (10)

Eq. (10) gives two constraints on the x*^''s. The condition for no splitting (no retardation) in all directions gives ten constraints on the x*^''s. With these

ten constraints, x can be written in the following form

(11)

where i7 = det(iyi/). In actual observations, the pulses and micropulses with

different polarizations are correlated in general structure and no retardation with respect to polarizations are ob- served. This means that conditions similar to Eq. (10) are satisfied to observational accuracy. For the Crab pulsar, micropulses with different polarizations are correlated in timing to within 10"'* sec. The distance of the

Crab pulsar is 2200 pc; therefore to within 10"^ sec/(2200 X 3.26 light yr.) = 5 x lO"^*^ accuracy two conditions similar to Eq. (10) are satisfied. Over 300 pulsars

in different directions are observed. Many of them have

polarization data. Combining all of them, Eq. (11) is satisfied to an accuracy of 10"^^ - 10"^^. Since IJ ~ 10"^,

x'^Vf/ (or xlU) agrees with that given by Eq. (11) to an

accuracy of 10~^ - lO"^'*. Detailed analysis will reveal

better results. Thus, to high accuracy, photons are propagating in the

metric field and two additional scalar fields and vj;. A change of to X.i/"^' does not affect X/'^' in Eq. (11). Thus we have eleven degrees of freedom in Eq. (11).

4. Hughes-Drever Experiments, EotvosDicke-Braginsky Experiments,

and Vessot-Levine Experiment

Since Eq. (11) is- verified empirically to high accuracy

from pulsar observations, we start from Eq. (11) in this

section.

In Eq. (1), ds is the line element determined from the

metric gr,^. From Eq. (11), the gravitational coupling to electromagnetism is determined by the metric and

two scalar fields ^ and If is not proportional to gij

then the hyperfine levels of the lithium atom will have

additional shifts. But this is not observed to high accu- racy in Hughes-Drever experiments [13, 14]. Therefore

Hij is proportional to Qij to a certain accuracy. Since a change of H''^ to kH''' does not affect x'^'^'^^ in Eq. (11), we

can define H n = S'li to remove this scale freedom.

In Hughes-Drever experiments Am/m ^ 0.5 x 10"^ or

Arn/rne,m. ^ 0.3 x 10"^^ where me,m. is the electromagnetic binding energy. Using Eq. (11) in Eq. (1), we have

three kinds of contributions to Am/m^.m. • These three

kinds are of the order of (i) (i?^.„ - S'^lv), (ii) (Hq^ -

fifO|x)v, and (iii) {Hqo - goo)v^ respectively. Here the

Greek indices |x, v denote space indices. Considering the

motion of laboratories from earth rotation, in the solar system and in our galaxy, we can set limits on various

components of (H,, - g,j) from Hughes-Drever experiments as follows:

I 10-12

10"' -

10-4 .

10" (12)

where U (—10 ^) is the galactical gravitational potential.

Ebtvbs-Dicke-Braginsky [15, 16, 17] experiments are performed on unpolarized test bodies. In essence, these

experiments show that unpolarized electric and magnetic

energies follow the same trajectories as other forms of energy to a certain accuracy. The constraints on Eq. (11) are

1 - ^\l\/U < 10" (13)

and

\Hoo- goo\/U < 10-° (14)

where U is the solar gravitational potential at the earth. In 1976, Vessot and Levine [18] used an atomic hydrogen maser clock in a space probe to test and confirm the

metric gravitational redshift to an accuracy of 1.4 x 10-^

[19]. The space probe attained an altitude of 10,000 km

above the earth's surface. With Eq. (13), the constraint on Eq. (11) is

I ^00 - 900 \/U ^ 1.4 x 10-^ (15)

Thus, we see that for the constraint on | Hoo ~ 9oo \/U,

Hughes-Drever experiments, Ebtvbs-Dicke-Braginsky

experiments, and the Vessot-Levine experiment compete

among each other in accuracy. The empirical constraints from the last section and this section can be summarized in Table 2.

5. A Nonmetric Theory of Gravity, Torsion

and Polarized Test-Body Experiments

From the results of the above tw'o sections, we know

that X is constrained as in Eq. (2) to various degi-ees of accuracy. Eq. (2) together with Eq. (1) is the nonmetric

theory we proposed earlier [9, 10, 11] as a counter example to Schiff s conjecture. In the following we discuss

this theory.

The interaction Lagrangian is = 2^M) 4. ^^NM) (16)

649
Table 2. Empirical foundations of the Einstein Equivalence Principle.

1/2 4) eijkl With the constraints below, x'-''^'' = {-g) to

various degrees of accuracy, i.e., EEP is verified to various degrees of accuracy

except for the freedom in ^.

Experiments Constraints Accuracy

Pulsar Signal

Propagation

^ 2 2

10-8 . 10-10

Hughes-Drever

Experiments

^OQ-^fl'OO

10-12

10-'' -10-8

10-4

Eotvbs-DickeBraginsky

Experiments

vl, ^ 1 10-9

10-5

Vessot-Levine

Redshift Experiment

f^oo ^ fl'oo 10-4

where i?}^' is the usual metric Lagrangian and

1 ^ 1 77. t:. ,ijk( = (-

16lT = (-T-)(t>wA,A;,,ce''^'' (mod. div.). (17)

The Maxwell equations are

F'\,, + e'''^^ Fk„, = -4T7i'. (18)

The Lorentz force law is the same as in metric theories

of gravity. Gauge invariance and charge conservation are guaranteed. The Maxwell equations (18) are also con- formally invariant.

This theory can be put into the form of a torsion

theory. Define a metric compatible affine connection as

Pjk ={jk} + ^'jk (19)

where {^^ } is the Christoffel symbol obtained from Qij and

aVfc - 2(t),[-e|i . (20)

The Maxwell equations (18) can be written as = -4^ii (21)

where | denotes covariant differentiation with respect to r> (e.g., V\j = V\j + r etc. ) and

Fik ^ Ak\i - A,\k = Ah,i - Aa + 2ft Af . (22)

The nonmetric part of the Lagrangian can be vn-itten in the form

(23)

To complete this theory as a gravitational theory, we

have to add a gravitational Lagrangian to it. For example, the gravitational Lagrangian £g could be

(24)

or

16tt 9

where ri is a parameter and (0(4)) is a function of (j). Defining the electromagnetic stress-energy as

(em)

1

ij 4tt 4 we have the following matter response equation:

i:k

4-7T

(25)

(26)

(27)

(28)

From Theorem I in §2, we know that test bodies follow

geodesies in this theory. Choose geodesic frames such

that Qij^ic = 0 along the geodesic of the test body con- sidered. Define the angular momentum tensor as = / (XiTko - XkT,o)dV . (29)

The rate of change of the three angular momenta for the

test body is - /^c|),, - /A^. or J = I X Vet) (30) dt where

1 16i J

e^^''^ Fnj Fki x^dV (31)

is the gravitational polarization vector. Eq. (30) is similar to the one for a magnet in a magnetic field. Thus, the

motion would be oscillatory about the axis. The order

of magnitude of the torque would be Ee.m. ^ fraction of polarization x x V<j) where d is the dimension of the polarized electromagnetic field in the body. To constrain

the remaining <}) freedom of x and to test this nonmetric

theory, experiments on' polarized test bodies with

large are suggested.

The acceleration measurements on polarized test bodies

would be tests of WEP L The importance of these exper- iments as tests of EEP has been emphasized by Morgan

650
and Peres [20]. From Theorem I, WEP I constrains

20 degrees of freedom of x- Ebtvbs-Dicke-Braginsky ex- periments constrain 2 degrees of freedom of x- The ac- celeration measurements on polarized test bodies would

constrain the other 18 degrees of freedom. This would

constitute a check on the constraints obtained by other

experiments in Table 2. According to Theorem II, the acceleration and torque

measurements of both polarized and unpolarized test bodies would be a complete set of tests of EEP.

Polarized test body experiments can also probe the role of spin in gravity. This has to be treated in an extended

phenomenological framework.

6. Discussions

In this paper, we have used a general framework to study the empirical foundations of EEP and to analyze

the theoretical significance of various precision experiments. We set empirical limits on x's. This is useful in finding out the viable theories of gravity. For example,

Hojman-Rosenbaum-Ryan-Shepley theory [21] violates

the limits in Table 2, and, hence, is nonviable [22]. In the context of cosmology, the x due to all cosmological matter should be used in the analysis of pulsar signal

propagation and Hughes-Drever experiments, x could

also have a spontaneous-symmetry-breaking piece. This

piece is also constrained by the analysis of §3 and §4. Detailed analysis will be presented elsewhere.

References

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(1908); English translations by H. M. Schwartz in Am. J. Phys., 45, 512, 811, 899 (1977).

[2] T. D. Lee and C. N. Yang, Phys. Rev. 104, 254 (1956).

[3] C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, and

R. P. Hudson, Phys. Rev., 105, 1413 (1957).

[4] J. H. Christenson, J. W. Cronin, V. L. Fitch, and R. Turlay,

Phys. Rev. Lett. 13, 138 (1964).

[5] S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967).

[6] A. Salam, in Elementary Particle Theory, Ed. by N. Svartholm

(Wiley, New York, 1968), p. .367. [7] Y. Fujii, Phys. Rev. D 9, 874 (1974); P. Minkowski, Phys.

Lett., 71B, 419 (1977); T. Matsuki, Prog. Theor. Phys. 59, 235 (1978); A. D. Linde, Pis'ma Zh. Eksp. Teor. Fiz., .30, 479 (1979) [JETP Lett., 30, 447 (1979)]; A. Zee, Phys. Rev. Lett., 42, 417 (1979); 44, 703 (1980).

[8] H. Georgi and S. L. Glashow; Phys. Rev. Lett. 33, 438

(1974). For a review see J. C. Pati's talk presented at the Einstein Centennial Symposium held in Jerusalem, March

1979. University of Maryland Preprint #80-006.

[9] W.-T. Ni, Bull. Am. Phys. Soc, 19, 655 (1974).

[10] W.-T. Ni, Phys. Rev. Lett. 38, 301 (1977).

[11] W.-T. Ni, A Nonmetric .Theory of Gravity, preprint, Montana State University, Bozeman, Montana, USA (1973). [121 See, e.g., F. G. Smith, Pulsars (Cambridge University

Press, Cambridge, England, 1977).

[13] V. M. Hughes, H. G. Robinson, and V. Beltran-Lopez, Phys. Rev. Letters 4, 342 (1960); V. Beltran-Lopez, H. G. Robinson, and V. W. Hughes, Bull. Am. Phys. Soc. 6, 424

(1961).

[14] R. W. P. Drever, Philos. Mag. 6, 683 (1961).

[15] R. V. Ebtvos, D. Pekar, and E. Fekete, Ann. Phys.

(Leipzig) 68, 11 (1922); also R. V. Ebtvos, Math. Naturwiss. Ber. Ungarn (Budapest) 8, 65 (1890).

[16] P. G. Roll, R. Krotkov, and R. H. Dicke, Ann. Phys. (N.Y.)

26; 442 (1964).

[17] V. B. Braginsky and V. L Panov, Zh. Eksp. Teor. Fiz., 61, 873 (1971) [Sov. Phys. JETP 34, 463 (1972)].

[18] R. F. C. Vessot, and M. W. Levine, Gen. Relativ. Gravit.

10, 181 (1979).

[19] R. F. C. Vessot, et al, Phys. Rev. Lett. 45, 2081 (1980).

[20] T. A. Morgan, and A. Peres, Phys. Rev. Lett., 9, 79 (1962); A. Peres, Phys. Rev. D 18, 2739 (1978).

[21] S. Hojman, M. Rosenbaum, M. P. Ryan, and L. C. Shepley,

Phys. Rev. D 17, 3141 (1978).

[22] W.-T. Ni, Phys. Rev. D 19, 2260 (1979).

651
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Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

An Inertial Clock to Test the Non-Metricity of Gravity

W. Stephen Cheung and Rogers C. Ritter

Department of Physics, University of Virginia, Charlottesville, VA 22901

A certain class of non-metric gravitation theories implies that clocks of different electromagnetic nature will interact with local gravity differently and result in slightly different gravitational redshift measurements. An ideal rotating body would have its moment of inertia, hence timekeeping, depend on

the rotor dimensions, so that it would act as an "electrostatic clock." In the past, precision high speed

rotations of small spheres have exhibited long decay times, ~10^" s, and high Q. Simple extrapolation

indicates that larger rotors should have much longer decay times. A double magnetic suspension of two rotors—an inner "proof rotor and an outer "shroud" rotor—^has been built and is under test as a method for reducing bearing and gas drags. This paper presents a description of a non-metricity test, analysis of the corotation scheme, and details of construction of the first, room-temperature inertial clock. Key words: corotation; gravity; inertial clock; non-metricity; precision rotor.

1. Introduction

Except for the earth, precision macroscopic rotors have

seldom been employed in experimental tests of gravitational theories. The Hughes-Drever experimental tests

[1, 2] for the anisotropy of space, in fact, are one of the

few microscopic rotor tests of the nature of gravity. One

macroscopic rotor, the Stanford Relativity Gyro [3], is an example of a precision rotation but its gyro rather than

pure rotor property is what is of interest. A second ro- tor used for gravitation, purported to test a gi^avitational

radiation theory [5], was later found to be untenable [6]. As precision measurement devices macroscopic rotors

suffer primarily from gas and bearing friction, in addition

to the usual disturbances which constitute the ordinary

hostility of our earthbound environment for sensitive

measurements. If these problems can be solved suffi- ciently for a rotor to have a cosmological spin-down time,

say 10^^ sec, and if such a rotor can be interrogated appropriately, a useful new type of clock would result [7]. (Actually, the earth is such a clock but its spindown rate, ~3 X 10^ yr, is too great and too variable, ~3 x

10~^/day, for modern application in gravitation experiments.) Many appUcations might result from a clock

whose stability came from a rotor of Q ~ 10^^; we dis- cuss a particular one here. A certain class of non-metric theories can be treated in the THejx formaHsm [8] to predict the non-uniqueness of

the gravitational redshift as measured by ideal clocks of different electromagnetic nature [9]. The hydrogen maser

and superconducting cavity stabilized oscillator (SCSO)

have been compared for this purpose [10], but thus far with inconclusive results. The way in which the different

electrical effects modify the gravitational redshift in the

THep, formaHsm, independent of any particular gravitational theory, is given [11] below.

Here, the test might be made by comparing three

clocks as they rotate around the earth's axis, moving a "vertical" distance h in the sun's potential from noon to midnight. In this case g would be the mean local field they experience from the sun, about 0.006 N/kg, so gh/c^ = 8x 10-^1

Two clocks could not in principle uniquely determine

the parameters To and Ao from daily timekeeping differ- ences in such an experiment. Furthermore, many ordi- nary local disturbances also have daily variations. Three

different clocks, on the other hand, cannot only determine To and Aq, but can give redundant information to test for these other effects. The fact that the sought-for

variation is so small further amplifies the desirability for redundant measurement.

In this paper we discuss the conception, design, and

construction of an inertial clock based on precision rota- tions with special attention to the unusual problems: the

bearing friction and gas drag.

Redshift in THe|j.

Group - Aspect of EM Examples Formalism

I Electrostatic PrincipaUines of H Z = (l-Zr^jgh/

interactions Oscillations of high Q crystals Rotor clock

II Hyperfme H-MAser Z = (1 -STq + Ao)gh / interactions

III Electrostatic + SCSO Z = (1 - fro - ^A^)gk/e^

light

propagation

653
2. The Rotational System

Beams [12], Fremery [4], and others have achieved

long spindown times—10^° sec = 300 yrs —using small

(~3 mm dia) spheres ferromagnetically suspended. Magnetic bearing friction of a special kind and residual gas

drag limited the decay time in these experiments.

The Keith Coriolis torque [13, 14] is an eddy-current

drag on a ferromagnetic rotor, and has been tested by

Fremery. In this the spin-spin coupling of a local rotor to the earth causes a continuous precession of the rotor in such a way that it experiences a hang-off angle (towards

the north in the Northern Hemisphere) of a few microradians. This results in a compensatory eddy current in the rotor with a minute component causing drag. This is, surprisingly, almost independent of angular velocity. We

have been unable, thus far, to make low-drag ferrite ro- tors which might be expected to alleviate this problem.

Superconducting rotors, on the other hand, should

ideally be free of such an effect. (In addition, the shield- ing and low residual flux attainable makes a low temperature experiment the likely candidate for the ultimate

inertial clock.) Holdeman [15] has estimated the drag

torque for the superconducting Stanford Relativity Gyro

experiment [3] at w = 1200 rad/s. He arrives at Aco/co =

1.8 X 10"^ as the worst case for a 6 month mission,

which would be equivalent to a decay time t* > 10^'' sec. Beams [12] has shown that in the free-molecule regime,

gas drag on a rotor accurately obeys expected gas

dynamic estimates. If an otherwise free rotor obeys the

equation.

/e + K0 = 0 , (1) where / is the moment of inertia and k = Kg + Kb, the

drag coefficient composed of gas and bearing components, we can write [7, 16] for a spherical rotor of ra- dius a,

(2)

here k is the concentration of gas molecules of mass |jl at temperature T. For rotors ~ 10 cm diameter this would imply that a pressure < 10"^^ Torr would be needed to achieve spindown times T* > 10^* sec, a virtual impossibility.

Our solution has been to develop an old idea of Beams

[17]. In this, the gas is moved along with the rotor by

corotation of an outer cylinder. If collective gas effects are eliminated by operation in the free molecule regime,

a huge reduction in gas drag could be expected, since the

drag torque is proportional to the angular velocity of the

rotor relative to its surrounding gas. By this same reasoning (it can be shown formally) the outer cylinder exerts virtually zero torque on the inner

through the gas medium as long as they rotate nearly

together.

Thus the corotation method should virtually decouple

the inner rotor from any gas drag. We have developed

two versions of this: (1) Corotation Method I in which the outer cylinder is driven by a constant speed motor locked

to a cesium-beam clock [18]; and (2) Corotation Method II in which the outer '

' shroud' ' cylinder is servoed to rotate with the inner "proof" rotor. It is this second method

which is used as the basis for the inertial clock. In order to give some feeling for the corotation properties, we analyze a freely rotating corotation system. In

this system it is useful [7] to define an angle of departure

of a fixed point on the inner rotor from uniform rotation

at angular velocity coq in the laboratory system:

A = 6 - wo^ . (3) With appropriate substitution into Eq. (1), plus the addition of a rotating restoring torque c A and a driving

torque T {t ), this yields , I A + olA + a^Q + = Tit) (4) where k0 of Eq. (1) is replaced by a^A -i- a^^Q to account

for its effect in the two different (laooratory and rotating) reference frames. (We have shown [19] that the sup- port pole piece, hence the bearing, cannot, under ordi- nary circumstances, rotate as it develops a "cranking

torque" which locks the rotor to the outer cylinder.)

For a free, undisturbed rotor, c = ^(O = 0, so that

/A + (ttg -h ab)A = -Ob Wo . (5) If we define fi = A and assume Ub coq to be constant, the

solution to Eq. (5) can be written

n =

OLg + ttb

COO (e' 1) , (6)

where t* = -^/(ctg + ab) is the decay time due to all drag

coefficients. From this we get the lag angle

-A =

ttg -I- Ob

coo T* (e

-t/T* + t/j* - 1) , (7)

if A = 0 at i = 0. For a measuring time t « t*, this is approximately

-A

/

cooi' (8)

Thus, the free, passive, corotation system develops a phase lag -A which is quadratic in time but has negligi- ble dependence on the gas friction. Analysis of the servo behavior which is important to the clock properties is beyond the intent of this paper; and these properties are presently under study.

3. Corotation and the Double Magnetic

Suspension

In another experiment [18], which employs Corotation

Method I, it is feasible to use an air bearing for the

outer, constantly driven cylinder. In our experience,

however, such a bearing is quite rough on the scale of disturbance considered here, and would constitute a con- siderable complication in the servo loop for the Method II experiment of the inertial clock. Consequently we have

chosen to use the double magnetic suspension of Beams

[17]. Beams had two types of double magnetic suspension.

The inertial clock uses a variation of the second type, in which servo control of the position of the lower (inner) rotor is by means of a coil from below, thus affording

greater independence of the two rotors. We have studied

several topologies of the combination, including those in which the lower field is biassed to "push" or to "pull"

the inner rotor. The latter, which Beams used, is more

stable, but is more vulnerable to the cranking effect mentioned above. Ultimately a diamagnetic suspension, at least for the lower rotor, would seem to be most desir- able in avoiding both the cranking effect and other inter- rotor vertical coupling. The reason for such attention to vertical coupling is the fact that our studies [161 have in654
dicated the immense difficulty of avoiding vertical-rotary

mode coupling, vi^hich would, in effect, "shunt" the

inner-shroud rotor servo loop. In order to reduce drifts which had limited the earlier

experiments of Beams and Fremery, the present rotors use low angular velocity, thus avoiding centrifugal

stress. Typically 0. 1 to 10 Hz are the rotation rates we

use. Inter-rotor coupling via magnetic flux drive is also reduced.

4. The Hybrid Double Magnetic Suspension

The transducers in our current double suspension are

different in nature, i.e., hybrid. A schematic description

of this scheme is in Fig. 1. The vertical position of the

upper rotor, Ri, is optically sensed while the lower rotor, R2, is sensed by a sensing coil. Optical sensing is not employed in the lower suspension to avoid another possible

type of coupling between the angular velocity of Ri and

the lower servo loop due to optical reflection and scatter- ing from the attached shroud.

REF TO

FROM VCO

VCO

Figure 1. Diagram of the hybrid double magnetic suspension.

The upper rotor, R,, consists of the cold rolled steel ball, the cylindrical magnet, and the shroud. The lower rotor, Ro, is typically made of ferrite and is suspended inside the shroud

with the main lifting force provided by the permanent magnet.

R, is optically sensed while Rj is sensed by a coil. The labels are: SD = split-photodiode; DA = difference amplifier; LA =

lock-in amplifier; LC = LC tank circuit; FfV = frequency to voltage converter; VCO = voltage controlled oscillator; A =

typical suspension amplifier-differentiator.

A parametric optical system is used [20]. In this the

motional information of Ri amplitude-modulates the carrier frequency, typically 10 kHz, at which the light source is pulsed. The light source is phase-locked vdth

the subsequent signal demodulator through the voltage

controlled oscillator (VCO). This parametric method of suspension has demonstrated significant noise reduction

and disturbance immunity over the traditional optical sensor (dc light source). The 10 kHz light source is focused to give a parallel beam which passes evenly

above and below the cold rolled steel ball (5/8" diameter)

forming a symmetrical image on a split photodetector.

With careful mechanical and electronics design, the elec- tronic preamplifiers at the detector stage are balanced

and hence further stability is achieved. Preliminary

measurements show that the temporal drift, for example,

of a rotor so suspended is less than 2 x 10"^ cm/°C. This

drift is about a hundred times better than a similar drift

measurement using a single photodetector and a dc light source. Other disturbances such as ambient light fluctua- tion and electromagnetic pickup have practically no effect on this balanced parametric method [20]. The coil sensor for the lower suspension operates on

the principle of frequency modulation. The control motions of R2 modulate the resonant frequency (as well as the amplitude) of the ~1 MHz tank circuit. These FM sig- nals are mixed with a stable 1 MHz standard and the subsequent difference frequency is then converted into volt- age. The amplitude changes are ignored by means of

digital circuitry. This FM sensor is discussed further by

Bernard etal. [21]. A ferrite rotor suspended by this means

is shown in Fig. 2.

Figure 2. A ferrite rotor in suspension. The magnet above provides the main lifting force. The vertical motion of the rotor is picked up by the sensing coil below it. The corrective magnetic

fields come from the bigger coil below the rotor.

5. Present Apparatus

The first version of the inertial clock experiment is at room temperature and is designed mainly to test a variety of basic corotation questions, including feedback.

It is shown in moderate detail in Fig. 3. The aluminum

assembly sits on a vibration isolation table. The current-controlled pulsing LED and its detecting

circuitry are housed separately in two aluminum tubings

which are mechanically attached to the main aluminum

chamber. The windows are coated with a thin-, conducting

coating of tin oxide and provide vacuum sealing.

The permanent magnet provides the main lifting force

[22] for the rotor Ri. Thus the coil has only small, corrective currents and low heating. The shroud attached to Ri

has four baffle-like plates arranged so that gas molecules

must follow multiple straight paths in order to get into or out of the interior of the shroud.

Ri is driven to follow R2 by means of two pairs of eddy-current drive coils. One pair has shoulders to act as "catchers" of Ri if it falls. The second pair has hollow

interiors to house a dc light source and a split photodetector, respectively. A zero-crossing signal results

655
PERMANENT

MAGNET

ALUMINUM

HOUSING

Figure 3. Cross-sectional view of the present apparatus. R,

and R. are doubly suspended in a hybrid fashion. Optical

schemes of measunng the angular periods of the tivo rotors are

also shoivn. SD, and SD, are split-photodiodes employed for

the upper suspension and for measuremeyit of the ayigular

period of R,, respectively.

whenever the hole drilled in Ri aligns with this optical

path. The zero-crossing signals provide measurement of

the angular period of Ri. A simple scheme is adopted for the present apparatus

to measure the angular period of R2. A flat, shiny polished surface is machined on the teflon ringbelt attached

to R2. A laser beam is aimed at and reflected from this surface when it coasts by and is then detected by a photodetector. Modest performance of both Ri and R2

angle-sensors is sufficient for present experiments.

The vacuum level required here is not stringent as long

as it is the molecular flow regime. At present, 10"^ Torr

is acquired with a diffusion pump.

An overall picture of the present apparatus is shown in Fig. 4.

Figure 4. Overall view of the present apparatus. Attached to the lower left of the aluminum, housing is the vacimm port. The

aluminum, ttibings to both sides of the housing house the pulsing light source and the detecting circuitry, respectively. Above

the lid, one can also see the permanent magnet, the coil, and

the vacuum gauges.

6. Period Counting and the Angular

Servomechanism

A challenging problem to an ultra high-Q rotor is the

long experimental time needed to observe any significant

period change.

It can be seen that if stability of 10"^^ is finally

achieved, and is to be used to measure diurnal variations

at this level, the interrogation problem of a slow rotor is indeed significant.

An optical method for these relatively large period

measurements suffers white phase noise, n^, as well as quantization noise, Wq, of any electronic counting system.

In ordinary averaging over time, t, Wp, and go down

as t~^. An interesting algorithm suggested by Snyder

[23], a sort of "statistical vernier," causes np and Wq to

fall by t A microcomputer has been chosen to coordi- nate the period measurements and the angular control. The Snyder algorithm can be implemented in the microcomputer. When a difference between the periods of Ri

and R2 is detected, the microcomputer gates open the

eddy-current drive to drive Ri to follow R2 with the ap- propriate corrective phase.

7. Known Future Directions

Ferromagnetic suspension has exhibited [16] a ten- dency to couple vertical variations into rotary motion via

parametric interactions. This might be anticipated, at some level, for any active suspension method. Hence, a

particular point of diamagnetic, superconducting suspension would be to eliminate an active control. (It would, in addition, eliminate the CorioHs torque losses.)

An eddy-current drive, used thus far for convenience,

introduces rotating magnetic flux in undesirable places,

giving rise to stray inter-rotor coupling. The small

torque needed to maintain (not start) the shroud rotor angular velocity against gas drag, is about 10 dyne-cm.

This could be supplied by very modest laser power in a photon drive [7, 24]. The attendant intrinsic problems,

discussed by Braginsky [25, 7], seem readily surmountable, although practical aspects have not been suffi- ciently tested. Observation of the rotors will most likely require the

Snyder algorithm. Even so, the subdivision of — 10^ of

these slow rotations in one day into > 10^° parts is a for- midable question, likely to take us into the technology of

laser spectroscopy.

Hosts of disturbances of the standard varieties which

plague many other precision measurements will likely cause the appearance of this experiment to change radically. The present apparatus wall serve primarily to in- troduce us into the new problems of Method II Corotation. If our experience with Method I is an example,

these vidll be many and interesting.

We gratefully acknowledge support by the U.S. National Bureau of Standards under grant G8-9025, the National Science Foundation under grants PHY78-3208 and

PHY 80-07948, and the University of Virginia.

References

[1] V. W. Hughes, H. G. Robinson, and V. Beltran-Lopez,

Phys. Rev. Lett. 4, 342 (1960).

[2] R. Drever, Philos. Mag. 6, 683 (1961).

[3] C. W. F. Everitt, in Proc. First Marcel Grossman Confer- ence on General Relativity, Trieste, 1975, Ed. by R.

Rufini (North Holland, Amsterdam, 1977), p. 545.

656
[4] J. K. Fremerey, Phys. Rev. Lett. 30, 753 (1973).

[5] J. C. Keith, J. Math. Phys. (N.Y.) 42, 248 (1973); J. C. Keith, Rev. Mex. Fiz. 12, 1 (1963); G. D. Birkoff, Proc. Nat. Acad. Sci. U.S.A. 30, No. 10, 324 (1944).

[6] C. M. Will, in Experimental Gravitation, Proc. Int'l School

of Physics "Enrico Fermi," course LVI, Ed. by B. Bertotti (Academic Press, New York, 1974), p. 1. [7] R. C. Ritter, in Proc. Second Marcel Grossman Conference on General Relativity, Trieste 1979, Ed. by R. Ruffini (North Holland, 1982), p. 1039.

[8] C. M. Will, Metrologia 13, 95 (1977).

[9] C. M. Will, Phys. Rev. D 10, 2330 (1974).

[10] S. R. Stein and J. P. Turneaure, Proc. IEEE 63, 1249 (1975);

S. R. Stein and J. P. Turneaure, in Fidure Trends i)i Superconductive Electronics, Ed. bv B. S. Deaver et al. (Am. Inst. Phys. Conf. Proc. No. 44, New York, 1978), p. 192.

[11] C. M. Will, private communication.

[12] J. W. Beams, D. M. Spitzer, and J. P. Wade, Rev. Sci. In- strum. 33, 151 (1962).

[13] J. C. Keith, J. Res. Natl. Bur. Stand. Sect. D: 67, 533

(1963).

[14] J. K. Fremerey, Rev. Sci. Instrum. 43, 1413 (1972).

[15] L. B. Holdeman and J. T. Holdeman, J. Appl. Phys. 47, 4936

(1976).

[16] G. R. Jones, Jr., "Decay Time Studies of Magnetically Sus- pended Precision Rotors," M. Sc. Thesis, University of Virginia, 1981.

[17] J. W. Beams, Rev. Sci. Instrum. 24, 1071 (1963).

[18] G. T. Gillies and R. C. Ritter, these proceedings; R. C. Ritter et al.. Nature (London) 271, 228 (1978).

[19] G. T. Gillies, "The Development of High Q, Precision Mechanical Rotors for Use in a Dynamic Measurement of Matter Creation," Ph.D. Dissertation, Univ. of Virginia (1980).

[20] W. S. Cheung et al., Precis. Eng. 2, 183 (1980).

[21] B. E. Bernard et al.. Rev. Sci. Instrum., to be published.

[22] W. S. Cheung et al.. Am. J. Phys., to be published.

[23] J. J. Snyder, preprint and private communication.

[24] J. W. Beams, Phys. Rev. 72, 987 (1947).

1^25] V. B. Braginsky and A. B. Manukin, Measurement of Weak

Forces in Physics Experiments, Ed. by D. H. Douglass

(Univ. of Chicago Press, Chicago, 1977).

657
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:
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Assessment of the Prospects for a Measurement of Relativistic Frame

Dragging by 1990*

Richard A. Van Patten**

Department of Aeronautics and Astronautics, Stanford University, Stanford, CA 94305

Relativistic frame dragging, i.e., the Schiff motional effect or the Lense-Thirring effect, has never been measured. Just two experiments proposed to date promise measurement accuracies of a few percent. The two are the gyro experiment and the twin satellite experiment. These two experiments,

although very different, have a common element, a "gyroscope" in a rotating gravity field. With the gyro experiment, a small cryogenic gyro is placed in orbit and its spin axis history is compared with a

star to 10~^ arc sec/yr accuracy. With the twin satellite experiment, two drag-free satellites are placed

in reverse polar orbits and used as orbit-sized gyros. The sum of the nodal motion is checked to 30 cm

against that predicted from earth oblateness using polar satellite-to-satellite Doppler data. This paper compares the errors and risks involved in carrying out each of these very demanding scientific space

missions. Key words: gyro experiment; relativistic frame dragging; twin satellite experiment.

1. Introduction for Gyro Experiment

In 1960, Leonard Schiff [1] predicted, using Einstein's

general theory of relativity, that a gyroscope in orbit

about the earth would experience a precession of its spin

axis relative to the "fixed stars." Two relativistic pre- cessions are predicted: a "geodetic" precession associated with the orbital motion of the gyro about the earth

and a "motional" precession due to the earth's rotation.

Fig. 1 shows the two effects for a gyro in a 520 km alti- tude polar orbit with its spin axis in the equatorial and

orbital planes. The geodetic precession is north approximately 6.9 arc sec/yr and the motional precession is east approximately 0.05[(1 + y)/2] arc sec/yr, where the pa- rametrized post-Newtonian (PPN) parameter y is near

unity.

The Stanford Relativity Gyroscope Experiment, as presently envisioned [3], will use four superconducting

electrostatically suspended spherical gyros carried in a drag-free earth satellite with spin axes nominally aligned

Figure 1. Relativistic precessions: Geodetic and motional.

*Supported by NASA Contract NAS8-32355.

**Associate Investigator, "A Gyro Test of General Relativity in a

Satellite' '

with Rigel. Each spin axis orientation with respect to the line of sight to Rigel is measured using a SQUID

magnetometer to compare the direction of the London

magnetic moment with the readout of a telescope pointed

nominally at Rigel, to ± 0.05 arc sec. The satellite is roll- ing about the line of sight to Rigel with a period of 10 minutes. By rolling about the line of sight the body fixed

drift errors are eliminated leaving only inertially fixed

quantities such as gyro drift and relativistic precessions.

(Quantities such as solar heating on the sunlit side of the

rolling satellite and the earth's magnetic field have the

potential of looking like inertially fixed drift errors and

thus must be greatly attenuated (by 10"^^ for the latter)

in the vicinity of the gyros and telescope. A readout ac- curacy of 10"^ arc sec or 2 percent of the motional effect over one year is to be attained for any 70 hr estimating

period. The uncertainty in the proper motion of Rigel turns out to be the dominant error source at about 4% of the motional effect [6].

2. Introduction for Twin Satellite

Experiment

In 1974, the author and C. W. F. Everitt proposed a new test of the general theory of relativity [4, 5]. Using

general relativity, Lense and Thirring predicted that the

plane of a body orbiting a rotating mass is dragged a small amount in the direction of rotation of the mass. For

a satellite orbit 800 km above the earth, this relativistic motion is an angular rate of 0.2 [(1 + y)/2] arc sec/yr or four times the Schiff motional effect for a gyro in polar

orbit at the same altitude. This amounts to 31 m total eastward dragging for two orbits in 2.5 years. At first it was thought that a single drag-free satellite in polar orbit might be sufficient for making the test. But

the orbit plane position history relative to the poles

would have to be known to' centimeters to reduce the

nonrelativistic orbit plane precession uncertainty to a

level below that predicted by general relativity.

The two counter-orbiting (see Fig. 2) satellite experiment has significant advantages. Were it possible to

659
,DOPPLER SIGNAL

DRAG FREE SATELLITES

(1) The first study assumed the proper motion of Rigel was known exactly.

(2) The second study using the latest astronometric

data assumed the north-south proper motion and

east-west proper motion are uncertain by 0.9 and

1.7 milliarc-sec/yr respectively [9]. (3) The third study assumed the error in Rigel'

proper motion is a factor of 10 larger than the uncertainties computed from the accepted astro- nometric data and that an off-polar orbit could be

used along with the likely eventuality that the

measurements will show the geodetic term to be

very nearly correct.

Table 1 summarizes the results for the three studies

showing the expected error in the motional (frame dragging) term for the best orbit inclination under each as- sumption about proper motion [6].

. POLE OF ECLIPTIC Q LOCUS OF POLE OF MOOm'S ORBIT

Figure 2. Experimental configuration.

Table 1. Expected error in motional term.

Proper Motion Orbit Month Expected Error in Uncertainty Inclination Launch Frame Dragging

(milliarc sec/yr) i (milliarc sec/yr)

place the two satellites in identical orbits (even if not polar) their planes would precess due to the non-relativistic

effect of the earth's oblateness, in opposite directions at the same rate; i.e., the sum of the non-relativistic nodal

precession rates would be zero. It is not possible to place

the satellites precisely so, but now the non-zero sum of the non-relativistic nodal precession rates is a multiple

(related to earth oblateness and known to 4 significant

figures) of the mean plane separation which is very accu- rately deduced (to 0.2 cm accuracy) from satellite-tosatellite Doppler ranging measurements as they pass at or near the poles [7]. Subtracting the mean orbit plane

precession due to earth oblateness (obtained by measuring mean plane separation) from the actual mean orbit

plane precession (measured from ground tracking) a measurement of the Lense-Thirring effect is obtained to an accuracy of 1 to 3 percent for a 2.5 year experiment

[8].

3. Errors in the Gyro Experiment

No uncertainty 85.5° Sept. 0.49(1.1%)

NS = 0.9, EW = 1.7 85.5° Sept. 1.79 (4.1%)

NS = 9.0* 70.5° Oct. 2.67(6.1%)

*Geodetic precession assumed equal to theoretical value.

Fig. 3 shows the expected error in the motional precession measurement with the accepted values of proper motion uncertainty for all inclinations from 67° to 90°. Both Table 1 and Fig. 3 are based on a factor of 10 improvement in gyro readout noise relative to the existing

laboratory readout using well established, planned design

changes. This noise has a 1// power spectral density.

Thus a higher roll rate would decrease the effective

noise. The roll rate is limited however by spacecraft mass properties in combination with the available pointing control authority to about 0.01 radians per second.

The three major error sources in the gyro experiment

are:

(1) Precessions due to suspension system torques act- ing on a rotor which is not perfectly spherical or torques due to the center of mass being offset from

the center of support.

(2) Noise and scale factor error associated with the

SQUID magnetometer measurement of the gyro

spin axis (the London moment vector) and noise in the measurement of roll angle using a side-looking

star monitor (star blipper).

(3) Uncertainty in the proper motion of the guide star, Rigel.

Using a realistic measurement model, a Kalman filter covariance analysis was performed by R. Vassar and J. V. Breakwell [6] to analyze the effect of these errors on

the determination from a single gyro of the geodetic and

motional effect for a 1-year mission. Different a priori

uncertainties in the proper motion of Rigel were assumed

in three studies as follows:

2.5

2.0

Z JUN - O AUG ^ OCT X OEC Z FEB < APR

70 75 80 85

INCLINATION (DEGREES) 90

Figure 3. Expected error in the motional -precession with typi- cal proper motion uncertainties.

660
4. Errors in the Twin Satellite Experiment

Table 2 shows the error budget for the twin satellite ex- periment. As an aid to understanding Table 2 refer first to Fig. 4 and then to Fig. 5. Fig. 4a shows the form of the

lateral polar plane separation (as deduced from the polar

slant range Doppler data by Kalman filter). Fig. 4b shows

AXIS SHIFT yi FROM COLLISION

( EXPANDED FOR ILLUSTRATION )

Figure 4a. Form of lateral polar data.

the form of the sum of the nodal motions due to earth oblateness as calculated from the data of Fig. 4a and as measured from ground tracking. The non-relativistic noFlGURE 5. Lateral and vertical polar perturbations over one

sidereal day, as calculated from two different geophysical models. SAO 1973 Model (a) Lateral (b) Vertical, NOAA 197h Model (c) Lateral (d) Vertical.

Table 2. Twin satellite experiment error budget.

Error source

Equivalent Nodal Shift

at Equator

in 2.5 years

(meters)

1. Error in measurement of mean orbit plane

separation at the poles. 0.05

2. Error in determination of nodal angle

history of satellite No. 1 (from ground

tracking).

Same as above for satellite No. 2. 0.11

0.11

3. Error due to uncertainty of the Earth's quadrupole mass moment. 0.09

4. Error in nodal position caused by orbital adjustments performed at the equator. 0.08

5. Error in nodal position caused by fluctua- tions in self-gravity or mass attraction of

satellite No. 1 on its proof mass. Same as above for satellite No. 2.

0.04 0.04 Composite without time base error (root-sum-square) 0.21

Time Base Assumption

Optical

UT l(BIH) Laser (LURE)

VLBI

(POLARIS)

Equivalent time base error 0.5 0.01 0.05 Equivalent time base error allowing for correlation present with errors 2 above. 1.0 0.02 0.1 Composite with time base error 1.02 0.211 0.233 Combined relativistic effect 30.8 30.8 30.8 Probable experimental error (1 a) 3.3% 0.69% 0.76%

661
dal plane motions are removed by subtracting these two

results leaving only the relativistic plane motions. The total relativistic nodal plane motion over 2.5 years is made

up of two components; 27.8 m due to the Lense-Thirring

effect and 3.0 m geodetic precession due to the motion

about the sun. The total relativistic nodal plane motion

for both satellites is thus 30.8 m in 2.5 years. The en- velope of geophysical perturbations shown in Fig. 4a, in addition to the solar (twice yearly) and lunar (twice

monthly) gravity gradient perturbations, are shown more

clearly in Fig. 5a, b, c and d. These meter per day

vertical and lateral relative perturbations result from

various combinations of earth gravity harmonics . Error No. 1 in Table 2 has been determined from the

results of a covariance analysis due to D. Schaechter and

J. V. Breakwell [7], see Fig. 6, using a model with about

200 parameters for the polar vertical and lateral perturbations. The Doppler range error used for the analysis for a

single polar encounter was 1.0 cm which is applicable for

10

Eo

1

0.1

0.01

ICONSTANT

1 CONSTANT 2

1 1 1 1 1 1 WKS

Figure 6. Covariance evolution of polar mean plane separation

parameters (note orbit adjustment at 13 weeks).

distances up to 2 km assuming an oven stabilized crystal

oscillator. Errors No. 2 through No. 5 are described in detail in Ref. [4]. The dominant error for this experiment

depends on the time base assumption. The time base error

relates directly to the uncertainty in earth rotation with

respect to the inertial frame. This is needed to relate tracking data to the inertial frame. The equivalent time

base errors shown in Table 2 reflect the accepted uncertainties available or attainable for each of the three sources indicated. The optical UT 1 time base is available

while the laser and VLBI time bases may be available at the time the experiment could fly. Thus the expected ac- curacy for the twin satellite experiment is one to three

percent depending on the time base assumption.

5. Risks Associated witii the Gyro

Experiment

The primary technical risk areas affecting our ability to successfully complete precision measurement space ex- periments such as the gyro are, of course, (1) accuracy

performance and (2) reliability. The accuracy of the gyro

experiment has received extensive attention in the

research reports associated with the Stanford Gyro Relativity Project [6, 10, 11]. The reUability needed for obtaining an adequate probability of a successful flight is less well established and is also strongly coupled to accu- racy since accuracy degradation is capable of destroying

the scientific value of the mission even with a flight completely free of hard failures. A hard failure is any condition which prevents obtainment of additional valid

science data from which recovery is not possible or feasible. The cryogenic gyro experiment is extremely complex

and demanding in terms of being at or near the state of

the art in a number of high technology areas. There are

significant risks relating both to accuracy degradation

and hard failures which will be listed. It is important to note that MTBF (mean time before failure) numbers re- lating to system reliability are not available at this time

since flight hardware and redundancy are not yet fully

defined. The objective of the discussion is to give the

reader an idea of some of the numbers and conditions re- quired for attaining the stated accuracy. The quartz block assembly which contains 4 quartz

gyros, 1 drag-free instrument and one quartz telescope

along with associated plumbing, electrical, magnetic

shielding and vacuum equipment operates inside a helium

dewar at liquid helium temperatures. The gyro assemblies require a vacuum of 10"^ Torr to limit torques and

spin down rates. The residual magnetic field in the vicin- ity of the gyros should be ~ 10"^ to limit the trapped flux pinned to the superconducting niobium coatings on the

gyro rotors at the time the coating is cooled below its critical temperature. Excessive trapped flux would sa- turate the SQUID magnetometer gyro readout. The

earth's magnetic field must be attenuated by 10"^^ at the

gyro as the spacecraft rolls. This is accomplished by

proper design of the lead bag supconducting shield sys- tem in conjunction with the gyro mounting cavities which

also serve as superconducting shields. Magnetic properties of all parts inside the lead bag must be rigidly controlled, with magnetic inclusions eliminated to a level consistent with the 10"^ gauss design requirement. Thermal modulation of the SQUID magnetometer at the roll frequency due to solar heating on the exterior of the

spacecraft as the 10-minute roll takes place should not exceed ten micro degrees K. This lateral heat flux re- quirement is accomplished with super insulation already

required by the 1-year dewar hold time needed. Thermal

heating through the telescope aperture from the sun and

earth albedo must be similarly limited. A combination of gold coated windows, sun shade, and shutter arrangement is being studied as a solution. The quartz gyro rotor must be spherical to 0.4 to 1.0 microinches (Ifxin. — 0.025 ixm) with material homogeneity of 3 parts in 10'^ or better. The thin or thick film

(4 or 100 (xin.) superconducting niobium coating on the

rotor must be mass balanced well enough to be consistent

vdth the above. The rotor coating should also be suffi- ciently robust to permit gyro rotor levitation at 1 g^ for ground tests with an electrostatic suspension system.

The rotor diameter may be decreased from 1.5 in. to say

0.75 in. to help attain the above. The gyro rotor will be

spun up to 170 Hz with a gas spin-up system which re- quires spin-up channels to be located within 300 to 500 fx in. of the rotor surface. The flight suspension sys- tem must keep the rotor centered vdthin ± 50 |jl in. while supporting against acceleration forces of ± 1 g during the launch tests or ± 5 x 10'^ g during flight. The

support electrodes are located 1500 |jl in. away from the

^In this context the symbol "g" refers to a unit of acceleration equal to that of gravity at the surface of the earth.

662
rotor. The suspension preload levels must be decreased

to a low multiple of the gravity gradient acceleration

from the earth which vdll be a twice per orbit sine function at about 5 x 10"^ g. The suspension system

bandwidth must decrease from 4000 rad/s at 1 g to 4 rad/s during normal flight to reduce sensor noise ac- celeration to acceptable levels. During any emergency,

such as micrometeor impacts or inadvertent handling or docking events, the suspension system will switch up as required in time to prevent the rotor from contacting the

spin-up channels since any contact would cause severe damage to the gyro. The 5 1/2-inch aperture telescope assembly is optically

contacted together and must produce image linearity and

smoothness (but not absolute null stability) of better than

1 milliarc sec over the central ±0.05 arc sec in pitch

and yaw. The attitude control system must point the

telescope at Rigel within ± 0.05 arc sec including a ±0.03 arc sec sinusoidal dither command signal with

1-minute period. This is required to measure and control

the telescope photomultiplier readout scale factor to within ± 1 percent of that of the gyro. One hundred

hours of smoothing time is required to reduce the scale factor mismatch by 1/e without permitting the SQUID

noise to disturb the corrected scale factor by more than 1 percent rms. The above process permits the ± 50 milliarc sec attitude motions to be substracted from gyro to Rigel difference angle data to the level of 0.5 milliarc sec or better. The drag-free instrument is located at the center of mass of the spacecraft to ± 1 mm and will be designed to maintain the proof mass drag free to ±10"^° g. This

should be compared wdth the level of 5 x 10~^^ g attained

by the TRIAD discos drag-free satellite launched in September 1972. Both the drag-free and attitude control systems will use the helium boil-off gas in conjunction with proportional differential thrusters [11] for attainment of

smooth, but adequately responsive, control. The attitude

control system may employ inner pointing actuators for

fine pointing of the quartz telescope. The spacecraft vdll be designed with nearly equal moments of inertia to minimize the gravity gradient torques, and minimum

cross products of inertia to reduce pointing control au- thority requirements in the presence of the 10-minute

period roll. During initial spin-up vdth helium gas, steps must be

taken to assure that the spin axes end up being within 10

to 30 arc sec of Rigel to assure functioning of the high

resolution science data instrumentation systems.

The risk areas viewed as most important by the author

are listed in order of perceived importance at this writing in Table 3.

6. Risks Associated witli the Twin Satellite

Experiment

This experiment has received much less attention than

the gyro and therefore less time has been applied to both

discovering and solving problems. However, during the

three year period, 1975-1978, during which the experiment was actively studied a reasonably clear picture was

formed of the major areas of risk associated with a flight

program involving this mission.

To give the reader an understanding of the risks in- volved in conducting the twin satellite experiment it is necessary first to understand the nature of the drag-free

orbits required. The orbit requirements for performing

the mission are very strict, with a constraint on each of

the six relative orbital elements [12]. (1) Semi-major axes (a). The semi-major axes must He

within 10 cm of each other, since otherwise the difference

in orbital periods will cause the satellite encounter point

to drift from the poles. There are several difficulties with

these lower latitude encounter points. One is that a small

component of the polar mean plane separation is lost. A

second problem is that at lower latitudes the range meas- urement picks up small components of unmodeled fluctua- tions whose dominant effect is at the equator. Some work

has been performed [13] in which the encounter point is purposely allowed to drift from the poles, so that additional geophysical information is obtained. However, the

modeling of the earth gravity harmonic fluctuations at arbitrary latitudes (not just ±90°) is much more difficult. The normal altitude of both satellites is 800 km.

(2,3) Eccentricity (^, t)). A relative eccentricity

between the two satellites produces relative altitude fluc- tuation at the poles whose magnitude is proportional to the relative eccentricity and whose period is equal to the

perigee regression rate produced by the earth's oblateness. The separation of the mean orbit planes is determined using a Doppler ranging device which measures

slant range at closest approach. The introduction of any

relative vertical separation into this range measurement

vdll somewhat degrade the information on mean plane

separation. Also, the fluctuation model is greatly simplified by requiring the eccentricity to be nearly zero. Therefore, the two components of the eccentricity vector - e cos 0) and r\ - e sin w) for each satellite should be

trimmed to produce relative altitude fluctuations no

greater than 300 m over the poles.

(4) Inclination (i). The orbital requirements for the in- clinations of the two satellites are the most severe. This

is because the uncertainties of the earth oblateness and

relative inclination produce an uncertainty in the mean

non-relativistic nodal precession rate, which directly af- fects the accuracy with which the Lense-Thirring nodal

precession rate can be determined. As such, the average

relative polar separation of the two orbit planes over the

2.5 year duration of the experiment must be less than 10 meters. This will be accomplished vdth the aid of small

velocity vector corrections, applied every three months

at the equator.

The mean position of the orbit planes must also be

close to the pole of the earth. If it is not, the orbit planes

wiW scissor slightly (i.e., the ascending nodes wiW not remain exactly 180° apart). Present ground tracking can determine in-track orbit position to about 10 m. Even

with the scissor angle precisely knovra a component of the in-track uncertainty appears in the minimum range

determination at the poles. To keep this range uncertainty small compared to the 1 cm Doppler ranging accuracy, it is required that the mean of the two satellite planes, averaged over the duration of the mission, be

within 70 m of the earth's pole.

Both of these requirements on the inclinations must be

viewed in light of the fact that periodic small changes in the satellite orbital inclination, applied every three

months at the equator, will be required to prevent the

possibiHty of a satellite collision [14] and in addition for keeping up with the moving pole of the earth due to the

precession of the equinoxes.

(5) Longitude of the node (Q). The relative longitude of

the nodes, as mentioned above, must be kept close to

180°. So that the plane scissoring will not introduce large

uncertainties in the polar slant range Doppler measure663
Table 3. Risks associated with the gyro experiment.

Description of Risk Area

Specific Defect

or System

Type of Failure (accuracy or hard)

(1) Ball to cavity contact or gyro performance

la Thin film ball coating damage due

to Ig suspension

Hard

(pre-launch)

lb Gyro contamination ^Accuracy /hard

Ic Cryogenic deposits on gyro rotorXiL LXL civ V / 1 IcU \A

Id Arcing due to la or or lb Hard

le Thick film ball coating thickness non-uniformity

Accuracy

(2) Pre-launch testing 2a Inadequate ground

testing due to avoidance of la

Accuracy if 1 to 3 gyros fail to function on orbit /hard if none

lUIlLLlUIl Oil UXUIL

(3) Dewar system

performance

3a Loss of low (10~^

gauss) magnetic

field

Accuracy /hard

3b Loss of high (10~^

Torr) vacuum

.1 LliX Cii\-\ 1 110,1. \A.

3c Loss of superconductivity

Accuracy /hard

(4) Electronics, assuming

all critical functions are available

initially on orbit.

4a

4b

Loss of suspension

Loss of data readout system

Accuracy

iri.\„\, LAI 0\-\

4c Loss of control systems

2 i. V- V_ HI CVV V

4d Loss of flight

computers

xiCC ui ci\.y

4e Loss of communications Accuracy

4f Loss of power system Accuracy

^Note: Failure during the flight means reduced mission life which constitutes accuracy

degradation. Note: The study of relative advantages of thick and thin films is incomplete and coupled

vvdth the rotor diameter study which is also in progress (the electrode force per unit area at Ig decreases directly as ball diameter decreases, but gyro drift error increases).

ment, the nodes must be within 0.7 km of each other for the duration of the mission.

(6) hi-track Encotinter Point fS). The in-track encounter point requirement is that the encounter point is to oc- cur within 2 km of the poles. This allows an enormously

simplified fluctuation model to be used without introducing uncertainties comparable to the Doppler accuracy.

In addition to the usual method of using ground-tracking for orbit determination, this experiment can provide

very accurate relative orbit information from the satellite-to-satellite Doppler data taken near the north and

south poles. This is particularly useful in this experiment where high precision orbital information is re- quired, and many of the orbit requirements are constraints on the relative orbital elements. Currently,

ground tracking could supply the following accuracies on

the relative state vector; 8 a, ± 10 m; a 8 4, ± 40 m; a 8 -q, ± 40 m; a 8 i, ± 40 m; a 8 9, ± 40 m where 8 a, 8 ^

8 "n, 8 i, and 8 8 are the differences between the two satellites' state vectors. An analysis due to D. B. Schaechter

[15] has shown that the relative orbital elements can, in

fact, be estimated very quickly and accurately from

ground tracking augmented by the satellite-to-satellite Doppler measurement. An example of how the uncertainties evolve is shown in Table 4. Reference [15] also presents a scheme for maneuvering

the orbits into their required finely tuned configuration.

The scheme makes the best use of the satellite-to- satellite tracking and also avoids any possible mishap

through a satellite collision. In addition, Ref. [15] gives

664
Table 4. Uncertainty in relative orbital elements

after measurements.

Relative Orbital Elements

No. of Satellite- to-Satellite Measurements 5a abi a6 T) ahi a8fl

0 100 80 80 40 40

10 0.39 7 7 28 28

20 0.14 5 5 18 18 30 0.08 4 4 9 9

40 0.05 3 3 8 8

Table 5. Risks associated with twin satellite experiment.

Description of Specific Defect Type of Failure Risk Area or System (accuracy or hard)

(1) Keeping ahead of the la Excessive delay in Accuracy

orbit tuning orbit determination

and tuning

(2) Electronics, assuming

all critical func- 2a Loss of Dopplersystems

^Accuracy

tions are available 2b Loss of drag-free Accuracy

initially on orbit or attitude control 2c Loss of flight

computer(s)

Accuracy

2d Loss of communications Accuracy

2e Loss of power system Accuracy

(3) Collision of

satellites 3a Collision damage Accuracy

^Note: Failure during the flight means reduced mission life which constitutes accuracy

degradation.

closed form solutions for the station keeping requirements in terms of the relative orbital elements. Both two

and three-impulse strategies are optimal, depending on

the state of the orbits (state space regions).

From the above discussion it should be evident that orbit control wiW be a key factor in conducting the twin satellite experiment flight operations. The risk areas viewed as most important by the author are listed in order of perceived importance at this writing in Table 5.

7. Conclusion

A precise check of frame dragging would be an event of the highest scientific importance. It would extend the ex- perimental verification of Einstein's general theory of

relativity into a new area by providing the first measure- ments on the gravitational effects of moving matter, i.e., the "gravimagnetic" forces which are analogous in Einstein's theory to electromagnetic forces in Maxwell's

theory. Agreement of the measured result with the prediction of Einstein's theory would allow application of that

theory with much greater confidence to astrophysical cal- culations on rotating massive objects such as neutron stars or black holes. Disagreement vdth the theory would compell a radical rethinking of our concepts about gravitation.

Several ideas have been advanced for measuring frame

dragging either by observations with the earth, the rotating sun, or rotating astrophysical objects. One difficulty

vdth any observations involving the astrophysical bodies,

including the sun, is the lack of accurate data or absence

of any data on the angular momentum of the source. Although observations may be suggested which might re- veal the existence of frame dragging they cannot give a quantitative check of the theory and their significance

would remain uncertain. A more desirable approach is to make the measuements using the nearby rotating massive

body whose angular momentum we do know (the earth)

and to apply the information so obtained to deduce further

information about astrophysical objects. Of experiments

using the sun, the earth, or a laboratory body as the

source, only the gyro and counter-orbiting polar satelHte

experiments appear to be feasible [16]. One interesting example of an astrophysical phenomenon which has been attributed to frame dragging is the

existence of double lobed radio sources at the center of certain galaxies and quasars. The radio jets are observed

to be collimated over ranges of hundreds of kiloparsecs.

The currently most popular theory of this effect, due to R. Blandford and R. Shajek [17], is that there is a rotat- ing charged supermassive black hole at the center of the

quasar and that gravitational energy is coupled by frame

dragging to the electromagnetic energy of the charged

black hole to yield the radio signals. Another possible ex- ample of frame dragging is a theoretical explanation by

665
Fang and Ruffini [18] of the modulation on the large red

and blue Doppler shift in the H„ hnes of Object SS433.

SS433 is observed to have periodic red and blue shifts modulated with a 164 day period. Fang and Ruffini have

shown that a possible explanation of these observations

is a rotating black hole vidth a compact object in a lov/ orbit somewhat above the event horizon, r < 10^ m. On

this theory the red and blue shift result from the radial

component of the orbital velocity and the 164 day modulation results from the Lense-Thirring precession of the

orbiting compact object which is the source of the He

radiation.

Because of the present situation with a NASA Phase B

study in progress on the gyro experiment by the Marshall Space Flight Center and Stanford University fol- lowing 17 years of laboratory research, it is the author's

opinion that the gyro experiment must be successfully

flown in the presently projected 1988-89 time frame if we are to have quantitative experimental frame dragging

data by 1990. The twin satellite experiment also has the

potential to yield accurate frame dragging data; however

the prospects for its implementation during the 1980s are

less favorable.

References

[1] L. I. Schiff, Proc. Nat. Acad. Sci., USA 46, 871 (1960); Phys. Rev. Lett. 4, 216 (1960).

[2] J. Lense and H. Thirring, Phys. Z. XIX. 156 (1918).

[3] C. W. F. Everitt, m Experimental Gravitation, Ed. by B. Bertotti (Academic Press, New York, 1974), pp. 331-360; J. A.

Lipa, W. M. Fairbank and C. W. F. Everitt, ibid. p. 361.

[4] R. A. Van Patten and C. W. F. Everitt, Celestial Mechanics

13, 429 (1976).

[5] R. A. Van Patten and C. W. F. Everitt, Phys. Rev. Lett.

36, 629 (1976).

[6] R. Vassar, J. V. Breakwell, C. W. F. Everitt, and R. A. Van Patten, J. Spacecr. Rockets, in press.

[7] D. Schaechter, J. V. Breakwell, R. A. Van Patten, and C. W. F. Everitt, J. Spacecr. Rockets 17, 474 (1977).

[8] Richard A. Van Patten, in Proceedings of the International School of General Relativity Effects in Physics and Astrophysics; Experiment and Theory (Third Course) Ed. by J. Ehlers (Max-Planck-Institut fur Physik und Astrophysik,

Munich, 1977), p. 141.

[9] J. T. Anderson and C. W. F. Everitt, "Limits on the Meas- urement of Proper Motion and the Implications for the

Relativity Gyroscope Experiment" (internal report, Stanford University Dept. of Physics, 1979).

[10] C. W. F. Everitt, Report to the National Aeronautics and

Space Administration under grant No. 05 -020-019 and

contract No. NAS8-32-355 (June 1980).

[11] C. W. F. Everitt, Daniel B. DeBra, and Richard A. Van Patten, in Advances in the Astronomical Sciences, Ed. hy E.

J. Bauman (Univelt, San Diego, CA, 1981) Vol. 45, p. 265.

[12] D. B. Schaechter, Ph.D. Thesis, Stanford University (1977).

[13] J. V. Breakwell, C. W. F. Everitt, D. B. Schaechter, and R. A. Van Patten, Acta Astronautica, to be published (1982).

[14] David Schaechter, John V. Breakwell, Richard A. Van Patten, and Francis W. Everitt, J. Astronaut. Sci. 24. 137

(1976).

[15] D. B. Schaechter, J. V. Breakwell, R. A. Van Patten, and

C. W. F. Everitt, in Preprints of the 7th Triennial World

Congress of the IFAC, Helsinki, Finland, Ed. by A. Niemi (Permagon, Oxford 1978) Vol. 1, p. 1197.

[16] I. I. Shapiro, et al., "Strategy for Space Research in Gravitational Physics in the 1980' s," prepared by the Committee on Gravitational Physics, Space Science Board, National Academy of Sciences (27 July 1980).

[17] R. D. Blandford and R. L. Snajek, Men. Not. R. Astron.

See. 179, 435-456 (1977).

[18] Fang Li Zhi and Remo Ruffini, Phys. Lett. 86B, 193 (1979).

666
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984). .

, .

The Status of the Velocity of Light in Special Relativity

Ernst Breitenberger

Department of Physics, Ohio University, Athens, OH 45701

The special theory of relativity is shown to be independent of the assumption that the velocity of hght, c, is a universal constant. No more than an intuitively obvious monotonicity postulate of velocity addition is needed to prove the existence of a universal, kinematic limit velocity cq, and the existence

of a universal dispersion relation oj^ = c^ik^ + |x^' for vacuum waves. The best empirical support for the signal postulate Cq = c stiil comes from uncorroborated observations of the Crab Nebula pulsar

which admit a photon rest frequency of up to 7 MHz, far above the limit inferred from the consistency

of quantum electrodynamics (QED). Existing theory-dependent arguments purporting to demonstrate

the constancy of c are shown to be inadequate. Further experimental work on the variability of c vdth

frequency and with source conditions is desirable. Key words: photon rest mass; second postulate; special relativity; vacuum dispersion; velocity of light.

The special theory of relativity is traditionally developed using the premise that the velocity of light is a universal constant, c, which seems to rule out any logical

argument involving the contrary, and worse, if believed

to be unavoidable, would make any contrary experimental result appear as a refutation of relativity.

The subject can also be developed without reference to

c, in a purely kinematic manner. This approach has many

advantages (see Ref. [1]). It has a rich history dating

back to 1910, but was never taken quite seriously, pre- sumably because it remained strewn vvith redundant

assumptions and inaccuracies of proof. However, a satis- factory kinematical derivation of the Lorentz transformation can be formulated [2]. In the following, I first outline

the main steps of the argument, and then show some conclusions of interest to this Conference. In essence [1], the

velocity of light becomes a legitimate object of investigation, both experimental and theoretical; it also emerges

that not enough is known about its dispersion which seems

intimately related to the validity of QED.

Note first that the traditional approach requires the

existence of a universal signalling velocity c in order to make sure that it is operationally feasible to compare

length and time intervals between two inertial systems

K, K' which are in relative motion. Note next that we

need not compare intervals because nowadays we can agree to use natural units furnished by atomic standards,

such as ^^Kr and ^^Cs. Of course, we still need to communicate the zeros of coordinate and time scales, but for that purpose we need only single marker pulses whose

velocity is not essential to the procedure; in fact, a particle burst could serve just as well as a light flash. The admissible transformations (ATs) between K and

K ' are restricted by the relativity postulate to be linear. A chain of well-known, elementary arguments [2] shows

that they have the form

X ' — y{x - ut)

t' - ex + yt , y' = y, z' = z ,

(1)

with yiu) > 0 and yi-u) - y{u), if the usual convention is made that K and K ' coincide dX t - t' - 0, and

that 0 ' moves in K with velocity u along the x-axis.

The traditional argument would next call upon the con- stancy of c and hence require the invariance of +

y"^ + z^ - c^t^ (from which the unknown coefficients y

and e follow easily). This is quite unnecessary, though.

First, the equivalence of K and K ' requires that the in- verse to the transformation of Eq. (1), X - (x' + ut ')/(y + u €) have the form x = yi-u)[x ' — (-u)t '], whence

y = l/{y + U€) (2) must hold. For convenience, set now

e = -M a-y (3) with a new function a {u ), then from Eq. (2) 7 = (1 - auY^ (4) follows; also, a{u) is seen to be even and therefore e {u ) is odd.

Second, the product of two ATs must be another AT; if we choose for convenience the second AT to lead from K

to di K" which moves at u ' against K ' along the x '-axis and is oriented such that K ' and K

'

' coincide at <' =

t" - 0, then it is necessary that

1

-u ' a' 7'

-u 7

7' 7 —u 7

-u ay 7

(5) 7 -u y

-u " a" y" y"

where 7' = y(u') etc., for transformations of the type

given in Eq. (1) as written in the notation of Eq. (3). Comparison of the matrix elements leads at once [2] to the triple conclusion a = const , (6) 7" = yy' (1 + au u') , (7) u" ^ (u + u ')/(! + au u'). (8) For the history of the results given in Eqs. (4) and (6), see Pauli [3]. Try here a negative and set a = —1/w'^; then

7 = (1 + tiW)-"^, (9) u" = {u + m')/(1 - uu'/w^). (10)

667
Evidently this AT has no limit velocity. Addition of two

small positive velocities according to the law of Eq. (10)

yields a resultant larger than their arithmetical sum, but if uu ' is made large enough the resultant becomes negative.

It is rather startling to recognize that such an aberrant velocity composition law is compatible with commonsense notions of kinematics although they have been shackled

under the severe restriction of the relativity principle. In order to exclude this absurdity, we need a specific as- sumption which can be conveniently formulated as a 7veak montonicity postulate of velocity addition: the resultant

of two parallel velocities never has the opposite direction. On the other hand, if a > 0 we obtain with a = I/cq

the usual 7 = (1 - u^/co)~^, and Cq plays the role of a purely kinematic limit velocity. However, the concomitant Einstein addition law

u" = {u + u')/{l + uu'/c'i) (11)

still admits complications which are best seen from its simple consequence [4] U" — U = M '(1 - M^/C(f)/(1 + UU'/C§).

If we again choose both u and u ' positive, the difference

u" - u will be negative if w > Cq (and only then). Thus,

adding on to the velocity will for a tachyon lead to a slowing down, as if cause and effect were interchanged

[4], but the phenomenon is purely kinematic in origin be- cause no dynamical (or "signalling") assumptions whatsoever have entered the argument. In order to exclude

tachyons we may accordingly adopt a strong monotonicity postulate of velocity addition: the resultant of two

parallel, positive velocities is never less than either. (The strong postulate evidently implies the weak one, which is the reason why the monotonicity formulation is here being put forward.)

Thus the price for the removal of c from the foundations of relativity is a kinematic axiom the need for which was never openly appreciated (to the detriment of our understanding of "causality" which ought to be shorn of such intrinsically kinematic elements). On the other side

there are great benefits [1]. In particular, it becomes le- gitimate to ask whether or not Cq = c holds ("signal pos- tulate" [1]). By an elementary argument [5], for any wave (harmonic or not, but purely periodic) capable of propagating

through vacuum, (k, ?' w/Cq) — k must be a four-vector. It follows that

K, K, = k~ - w^/c^ = const = -|x^. (12) The phase velocity is then

Cp = w//c = Cod + (x2/fc2)l'2^ (13)

and the group velocity Cg = dw/dk is found to satisfy

CpCg = C(f . (14)

The dispersion law of Eqs. (12) or (14) has long been

known for de Broglie and for Klein-Gordon waves, but no

one seems to have duly emphasized that it is universal; it is, in fact, a fundamental result of relativity with the same standing as, say, the addition law Eq. (11) or the

time dilation formula. It is purely kinematic in nature

(see the quoted argument [5]) and constitutes a restric- tion on all possible wave equations (whether tachyons are admitted or not). If the wave equation is linear and of the second order, the only possibility is the Klein-Gordon

equation - \\)tf/c§ = M-^vli, (15)

but there are many nonlinear and /or higher-order possi- bihties.

Once the kinematic origin of the constant |jl is clearly

understood, there is no harm in setting

fjL = 2tt/\c (16)

with A.C a length, and in furthermore linking up with the

established dynamical context by setting Kc = h/moCo (17) as for a Compton wavelength. As a pragmatic middle way

between logical accuracy and familiar context,

vo CoH./2Tr = Cq/Xc (18)

will here be called the "rest frequency" of the vacuum

wave. If |JL = 0, then w = CqA; and there is no dispersion.

If [x? > 0, we have a real vq to which a real rest mass may be thought to correspond. If [x? < 0, then vq is imaginary, while Eq. (13) shows that Cp < cq and hence

Eq. (14) that Cg > cq; this case may be thought to correspond to tachyons.

The signal postulate holds if and only if c is a constant

[1]. Given the unique dispersion law of Eq. (12), if rela- tivity is valid only two kinds of variability of c are possible: vacuum dispersion with one particular value of |x or

Vo; and vacuum dispersion with different values of jx which could only arise from different source processes, or

as we say traditionally, "dependence of c on the state of motion of the source." The best (but still uncorroborated) result on vacuum

dispersion is due to Warner and Nather [6]; for discussion, see Ref. [1]. They deduced from observations of the

Crab Nebula pulsar that between ~ 350 nm and ~ 550 nm

the velocity of light is constant to better than 5 parts in

lO^''; to be precise,

|ACg |/Cg < 4.8 X 10-1^ (19)

This astonishing figure certainly justifies the claim that

for commercial purposes c in the visible equals Co, yet it does not imply a satisfactorily low limit on vq. From Eqs. (13) and (14) it follows that cJco -

(1 + ^JL2//c2)-l/2 ^ 1 _ ^,2^2/8^^ hence

|Acg |/co ~ |jl2x |A\ |/4tt2. (20)

If in Eq. (20) we replace Co by Cg and combine it with the

definitions of Eqs. (17) or (18), we have [1]

vo = moc|//i ^ co[( |Acg|/Cg)/(\ |A\ |)]i^, (21)

and if we take Co = c and insert the experimental value,

Eq. (19), we find

Vo < 7 MHz (22)

(the value given in Ref. [1] is smaller by VIO because in Ref. [6] there is a misprint of one power of 10). This

result is a disappointment. A vq of some MHz would have

been seen long ago in resonance spectroscopy, or as a discrepancy vdth QED predictions which are always

based on vq = 0. For instance, the Lamb shift calcula- tions in H agree with experiment (at present, roughly)

within 10 kHz. A limit of vo < 10 kHz naturally suggests

itself; wherefore we still (in 1981) have the irksome situa- tion that QED provides comfort for standard Maxwellian

electrodynamics [1], instead of the latter' s being sup- ported by a direct experimental result on c. \ Evidently a photon rest frequency would show up

better at long than at short wavelengths [1]. To have a yardstick, suppose that at X ~ 300 m it were found that c remains constant to 1 ppm over a range of A A. = 1 m; i Eq. (21) then shows that vq < 17 kHz. That would be a

j result well worth the effort. It has never been clearly un- '

668
derstood that measurements of the variability of c were

just as important as knowledge of its numerical value [7] at some wavelength, for we never clearly recognized that

a putative "photon rest mass" was an intrinsic feature of

kinematics which should be checked out before relativity can safely be applied in electrodynamics. In addition to infrared or rf measurements, it may also be possible to exploit modern techniques in the visible; the feasibility of finding variations of c to 1 part in 10^'^ has already been

envisaged [8]. A gain of 10'' over the result of Eq. (19)

could improve the limit of Eq. (22) by 10^, depending on A \, and again, the effort would be worthwhile.

Claims have repeatedly been made on theoretical

grounds to the effect that '

' the photon rest mass must be

zero." They were almost invariably based on the assumption of harmonic waves and the Klein-Gordon equation,

Eq. (15), which holds for the vector potential components

in an electrodynamics due to Proca [9] (see Ref. [1] for brief comment). This generalization of Maxwell's theory

is not very plausible to start with, for it is not gaugeinvariant [10]. (Probably this is the reason for the occasional and otherwise unsupported statement that "gauge

invariance excludes a non-zero photon rest mass.") Still,

it is entirely reasonable to ask what finite values of the

Proca parameter |x would be compatible with existing

data. Through careful comparisons vdth known features

of Maxwellian large-scale fields it has been shown [11]

that the Proca must be effectively zero; in the present

terminology, a Proca vo could be at most a small fraction

of 1 Hz. Thus Proca' s electrodynamics is not a viable al- ternative to Maxwell's, and consequently cannot say anything about the photon rest frequency [1]. Indeed, there are (untold?) other alternatives to Maxwell's theory. Take the Born-Infeld one (even though it is not supported by evidence either, see Ref. [12]). It leads to a wave equation of the type

(1 - 4;f)i|;„ + 2^,^\,t^\,:,t - (1 + = 0 (23) for the vector potential components [13]. This has undistorted wave solutions f{x - t) and g(x + t), and

therefore is free of dispersion, yet its static solutions exhibit a typical (a"* + r'*)~^'^-dependence on distance [13],

much as a Yukawa dependence with e "*^''-terms is typical

for Proca fields [11]. Thus, a departure from the Maxwellian r "^-fields is here compatible with a zero photon

rest mass. Consider also the possibility of linear /owr^/i-order

equations. The obvious D^vJ; = 0 is dispersion-free. For

its static solutions vdth (V^)^ i|/ = 0 there is (amongst others) the well-known elementary result: if and A)2 are harmonic, then x-^\ + ^2 is biharmonic (see any treatise on elasticity). Once again, fields quite different from the

Maxwellian ones (and now with a departure from the

Coulomb r "^-character quite different from Yukawa's

e "'^'-factor) are here compatible with a zero photon rest mass. These (counter) examples sharply illuminate the

kinematic origin of a putative photon rest frequency

Vo 0: electrodynamical speculations about the character of static fields, unsupported by other evidence, do

not lead to unequivocal insights concerning vq, hence ob- servations of fields other than those of vacuum waves

cannot at present tell us anything definitive about vq. A similar reservation applies to the interpretation of

pulsar data subject to plasma dispersion in the interstellar medium [14]. It is not known how a non-zero vo would

couple with plasma dispersion; any hypothesis about that

is dynamical and thus leads back to the previous situation

inasmuch as it would require independent justification

before it can be directed at the question, "what is vq?"

A somewhat more complex objection must be raised

against the interpretation of a classic experiment often

named nowadays after Cavendish, but previously named

usually after Plimpton and Lawton [15] who obtained the

first high-accuracy result. The experiment is said to vali- date Coulomb's law within extremely stringent toler- ances, and hence to imply a stringent limit on vq. In this form, the claim is incorrect. First, the experiment does

not test Coulomb's law

E(r) = l/4TTeor2, (24) but the law of Gauss

Dir) = l/4iTr2, (25)

which leads to Eq. (24) only through the linearity hypothesis D - eoE. In fact, the experiment is an inverse

Faraday pail experiment which tests the efficacy of an

electrostatic shield by measuring the charge electrostati- cally induced on an electrometer placed inside when a known charge is put on the outside. Note that no meas- urement of a (ponderomotive) force such as Eq. (24) is in- volved, and that the induced charge would be' quite the same if the shield (or pail) were filled with a dielectric. There seems to be a universal lack of understanding

that D is being measured, not E. In order to arrive at Coulomb's law, Eq. (24), one may, of course, combine the

result on D vdth an independent estimate on the linearity of the vacuum; however, for the latter we can draw only

upon QED, as far as we can see at the present time.

Thus QED would again become involved in a fundamental, "macroscopic" problem. Furthermore, we again do

not know how, nor if, a non-zero vq would affect Coulomb's law over laboratory distances, because we do

not know what sensible dynamical hypothesis to make,

just as discussed above. (All this must on no account be

taken as a slight on the beautiful work done recently [16] to improve the Plimpton-Lavii:on result; indeed it is pleasant to know how accurately the law of Gauss is valid.)

Lastly, consider possible variations of c arising from

source processes. Ten years ago, the evidence in regard

of the traditional "second postulate" was weak [1], but

two results of considerable weight have been published

since. Brecher [17] deduced from the observations of selected x-ray pulsars which are revolving quite fast in close binary systems that less than 2 x 10"^ of the (nonuniform!) orbital speed adds to the velocity of the x-rays;

this is a very low limit on the traditional "ballistic" ef- fects. Brown et al. [18] measured the velocity of 7 GeV

bremsstrahlung bursts from SLAC to an accuracy of

close to 1 part in 10^ and found it to be consistent with

the accepted value of c; this shows that fairly strong ra- diative reactions do not alter c. From the present,

purely kinematic standpoint these observations are directed at the possibility that the \x in Eq. (12) is not a universal constant. Difficult as it is to imagine "several

photon rest masses," there seems to be no first principle

opposed to the idea. Thus it remains desirable to have

further results under circumstances where nuclear forces or great changes of source acceleration play prominent

roles, as pointed out in Ref. [1]. To leave nothing unsaid, it should be added that a non-null result on vq or |jl would raise ambiguities of in- terpretation: has a non-zero photon rest mass been

demonstrated, or is the dispersion relation, Eq. (12), violated, with dire consequences for relativity? Thus in the kinematic approach the velocity of light indirectly be- comes a test object for the validity of the basic result,

Eq. (12), enhancing the desirability of further work.

669
References

[1] E. Breitenberger, Nuovo Cimento IB, 1 (1971).

[2] E. Breitenberger, to be submitted to Ann. Phys. (N.Y.).

[3] W. Pauli, Relativitatstheorie (Teubner, Leipzig 1921), translated as Theory of Relativity (Pergamon, New York

1958); end of Sec. 4 in either version.

[4] A. Einstein, Ann. Phys. (Leipzig) 23, 371 (1907).

[5] For example, J. D. Jackson, Classical Electrodynamics, 2nd

ed. (Wiley, New York 1975), p. 508.

[6] B. Warner and R. E. Nather, Nature (London) 222, 158

(1969).

[7] See J. Terrien, Metrologia 10, 9 (1974); K. M. Baird, D. S. Smith, and B. G. Whitford, Optics Commun. 31, 367

(1979).

[8] A. M. Grassi Strini, G. Strini, and G. Tagliaferri, Phys. Rev. D 19, 2330(1979).

[9] A. Proca, J. Phys. Radium 8, 23 (1936), and references given

there.

[10] W. Pauli, Rev. Mod. Phys. 13, 203 (1941), Sec. n.2.

[11] General references: A. S. Goldhaber and M. M. Nieto, Rev. Mod. Phys. 43, 277 (1971); A. Barnes and J. D. Scargle,

Phys. Rev. Lett. 35, 1117 (1975); A. D. Dolgov and Ya. B. Zeldovich, Rev. Mod. Phys. 53, 1 (1981), Sec. XII.

[12] For example, J. Rafelski, W. Greiner, and L. P. Fulcher,

Nuovo Cimento 13B, 135 (1973).

[13] M. Born and L. Infeld, Proc. R. Soc. London, Ser. A 144, 425 (1934).

[14] For example, J. R. Rawls, Phys. Rev. D 5, 487 (1972).

[15] S. J. Plimpton and W. E. LawtoTi, Phys. Rev. 50, 1066

(1936).

[16] E. R. Williams, J. E. Faller, and H. A. Hill, Phys. Rev.

Lett. 26, 721 (1971).

[17] K. Brecher, Phys. Rev. Lett. 39, 1051 (1977); 39, 1236(E)

(1977).

[18] B. C. Brown, G. E. Masek, T. Maung, E. S. Miller, H. Ruderman, and W. Vernon, Phys. Rev. Lett. 30, 763

(1973).

670
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Relativistic Time Dilation: A Latter-Day Ives-Stillwell Experiment

Paul Nachman,* Marc D. Rayman, and J. L. Hall**

Joint Institute for Laboratory Astrophysics, National Bureau of Standards and University of Colorado,

Boulder, CO 80309

Two-photon spectroscopy on a fast atomic beam offers the possibility of a dramatically improved

test of the special relativistic time dilation. Our experimental arrangement will allow direct measure- ment of He atom resonance frequencies displaced from their rest-frame values by time dilation alone. The atomic beam velocity will be determined optically via the Doppler shift. The longitudinal interaction geometry will also permit us to test with increased precision for a possible vector anisotropy in the speed of light, a possibility raising renewed interest because of recent astrophysical discoveries. Further, we expect to refine experimental values for the energies of the Rydberg terms in He. Key words: helium spectrum; preferred frames; special relativity; time dilation; two photon spectros- copy. We are assembling a spectroscopic experiment for a precision test of the time dilation (second-order Doppler

shift) predicted by special relativity. The "moving

clocks" [1] to be used wiW be helium atoms in a fast beam

(v/c < 5 X 10"^). The atoms vdll be in metastable states 2s and 2s^S and will be probed by tvi^o-photon transi- tions to higher triplet and singlet S and D energy levels. The experiment, v^^hich is nearly ready for operation, is designed as show^n in Fig. 1. Helium ions, produced in a discharge source, are accelerated to form an essentially

monoenergetic beam (5 to 50 keV, with a <1 eV spread).

this two-photon transition, producing three peaks in the

402.7 nm fluorescence 2p^P ^ 5d^D [Fig. 3(a)] as the exciting wavelength is varied. The central resonance

results from excitation of the He(2s^S) atoms by two

counterpropagating photons and thus is shifted only by

relativistic time dilation. The two other resonances, dis- placed by ~2 nm (~2 THz) from the central one [Fig.

3(b)], correspond to excitation by two copropagating photons, and thus are shifted primarily by the first-order

Doppler effect. "Light shifts" of the resonances due to the ac Stark effect should be <10 kHz.

PMT

ION SOURCE

H.V. ACCELERATION

5 - 50 kV

(y3 < 5 10"') He(2^S) + Na

Figure 1. Block diagram offast-beam spectroscopy experiment.

The ion beam (vnth several ^xA flux) is focused through

and beyond a cell containing sodium vapor. Resonant

charge exchange with the Na produces a beam that in- cludes an appreciable fraction [2] of neutral He atoms in the metastable levels. In the —milliradian forward cone, the velocity dispersion of the beam is not increased. This

fast atom beam crossed, vdth small intersection angle, a standing wave light field from a stabilized cw ring dye

laser. A Fabry-Perot resonator (build-up cavity) is used

to increase the energy density of the light by approximately a factor 30 from that in the incident (-300 mW)

laser beam. Figure 2 illustrates the proposed optical

system.

The initial resonance used will be 5d^D <r- 2s^S. Light

at three discrete wavelengths near 587.2 nm vdll drive

*NAS/NRC Postdoctoral Research Associate, National Bureau of Standards.

**Staff Member, Quantum Physics Division, National Bureau of Standards.

The angle of intersection between the atomic and opti- cal beams is small (—20 milliradians) to provide long

interaction times, hence minimizing laser intensity re- quirements and transit-broadening of the resonances. This geometry also permits direct determination of the

atomic velocity by means of the first-order Doppler

shifts, an improvement over earlier work in this labora- tory wherein the particles' velocities were inferred from

the accelerating potential of the beam [3].

The time dilation shift of the 587.2 nm resonance for He atoms with 50 keV kinetic energy is ~6 GHz. The

line width of the resonance should be ~5 MHz, due to transit broadening. We expect to determine line center

to 25 kHz (one half percent of the line width), thus

achieving a measurement of the time dilation shift to a few parts per million. This accuracy level would

represent ~300-fold improvement over that achieved in an experiment at much higher velocities, with muons at the CERN storage ring [4].

671
To Experiment Electro- Optic

"^Acousto-Optic Crystal Modulator —r^r^

Nel

y \

Servo

Electronics

Doubly _

o , ' Frequency Balanced _ ^.

,, „ Synthesizer Mixer '

Figure 2. Dye laser stabilization and tuning/. A small portion of the output of a cw ring dye laser is sampled (after the optical isolator) to be used for diagnostics ( mode analyzer and Lambdameter)

and for laser frequency control. In the stabilization fast tuning system this latter sample is frequency-shifted urith an acousto-optic modulator (AOM) driven by a stable, hit controllable, rffre- quency source. The resulting beayn is phase modulated vxith an electro-optic crystal to produce an FM sideband spectrum for locking the dye laser to the stabilized Fabry-Perot, using a reflection- mode Pound/Drever optical stabilizer. Cfianges in the AOM drive frequency are directly imposed on

the laser by its frequency servo. For slower or larger scans, the cavity length is tuned via the fre- quency synthesizer which provides a programmable frequency offset between the HeNe local oscilla- tor laser and the 12-stabilized reference laser. The Fabry-Perot's transmission fringes are used to lock its leyigth to the resonance condition with the HeNe local oscillator's variable wavelength, thus ultimately mapping the synthesizer's frequency change into a proportional frequency change of the dye laser.

4 0 2.7 nm FLUORESCENCE

(a) 2 - PHOTON

LASER EXCITATION

( X = 58 7.2nm)

2"^ ORDER DOPPLER SHIFT ~6 GHz

(FREE OF isf ORDER DOPPLER EFFECT)

(b)

|St ORDER DOPPLER SHIFT =2.9 nm

-2.6 THz (§) /3 = 5 X 10-3

Figure 3. (a) Energy levels and transitions for initial two- photon experiment, (b) Amplitude of ^02. 7 nm fluorescence

[Fig. 2(a)] as function of exciting laser frequency. First-or- der-shifted peaks arise from absorption of two laser photons

from same propagation direction. Central resonance is visual "Doppler-free" two-photon peak arising from one photon from

each beam.

Important future accuracy improvements can be ob- tained by using much smaller beam crossing angles in conjunction with two-photon transitions to longer-lived,

higher excited states. Improved velocity definition can be accomplished with two-laser/two-photon techniques

related to saturation spectroscopy. Line vddths of 50 kHz seem realistic and would improve these expected

results by two orders of magnitude if the

wavelength/frequency metrology can be correspondingly

improved. One possibility for this would be the use of CO2 laser/dye laser sideband generation in a nonlinear

crystal, followed by heterodyne detection with the

second dye laser. The experimental setup will allow us to make several

other precision measurements. For example, recent years have seen renewed interest in the possibility of residual preferred frame effects. In particular, the ob- served anisotropics in the cosmic microwave background

[5] and in the distribution of "local" mass (out to the

Virgo Cluster of galaxies [6]) suggest that our galaxy has

a peculiar velocity (compared to the Hubble expansion)

with respect to the average mass of the universe. Using

the present apparatus to monitor the relative positions

(to ~10 kHz) of the first-order Doppler-shifted resonance and the central resonance as functions of sidereal time

will permit us to test at the sub-10^^ level for a possible vector anisotropy in the speed of light. Our beam is oriented east-west to obtain the maximum sidereal modulation of such a hypothetical one-way effect.

672
Another interesting precision measurement to be made

with our fast longitudinal beam is of the He electronic structure. Doppler-free two-photon spectroscopy of high-lying ("Rydberg") terms in He provides a way to refine substantially experimental knowledge of this structure. Results of such accurate measurements, in company with recent microwave-optical studies [7] of fine structure vdthin and among Rydberg terms, would

present a renewed challenge for theorists to improve our understanding of this basic three-body quantummechanical system.

Dye lasers tunable over the wavelength range

520-700 nm allow access by two-photon transitions from

the 2s metastable levels to all w^S and n^D levels with n ^ Q and to all n^S and n^D levels with w > 4, including the ionization limits of these series. Analysis of the experimental results, using quantum

defect ideas to model Rydberg series as they converge to the ionization Hmit, should allow us to determine absolute term energies with MHz accuracy (—0.00003 cm"^), one to two orders of magnitude improvement over

present knowledge. Similar two-photon measurements in a discharge afterglow have already been made by Giacobino [8]. Spectroscopy with a beam of He offers the advantage of minimizing pressure-broadening and pres- sure-induced shifts of the resonances compared to similar

spectroscopy with a low-pressure cell.

References

[1] H. E. Ives and G. R. Stillwell, J. Opt. Soc. Am. 28, 215

(1938).

[2] E. H. Pedersen, Phys. Rev. Lett. 42, 440 (1979); and C. Reynaud, J. Pommier, V. N. Tuan, and M. Barat, Phys. Rev. Lett. 43, 579 (1979).

[3] J. J. Snyder and J. L. Hall, in Laser Spectroscopy, Ed. by S. Haroche, J. C. Pebay-Peyroula, T. W. Hansch, and S. E. Harris (Springer-Verlag, Heidelberg, 1975), p. 6. [4] J. Bailey, K. Borer, F. Combley, H. Drumm, F. Krienen, F. Lange, E. Picasso, W. von Ruden, F. J. M. Farley, J. H.

Field, W. Flegel, and P. M. Hattersley, Nature (London)

268, 301 (1977).

[5] G. F. Smoot, M. V. Gorenstein, and R. A. Muller, Phys. Rev. Lett. 39, 898 (1977); S. P. Boughn, E. S. Cheng, and

D. T. Wilkinson, Astrophys. J. (Letters) 243, L113 (1981).

[6] M. Aaronson, J. Mould, J. Huchra, W. T. Sullivan HI, R. A. Schommer, and G. D. Bothun, Astrophys. J. 239, 12 (1980).

[7] J. W. Farley, K. B. MacAdam, and W. H. Wing, Phys. Rev. A 20, 1754 (1979).

[8] E. Giacobino, private communication (1981).

673

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

An Experiment to Measure Relative Variations in tlie One-Way Velocity of

Light

D. G. Torr and P. Kolen

Department of Physics, Utah State University, Logan, UT 84322

In this experiment two rubidium vapor frequency standards were placed approximately 500 m

apart and the phase of their signals compared as a function of time. The diurnal rotation of the earth was used to introduce a change in the direction of propagation of the signal, thereby providing a test of the assumption of isotropy of propagation of electromagnetic radiation. The relative phase difference between clocks was also compared for negligible separation of the clocks. The drift rate changed

detectably for the separated clocks, while the round-trip velocity remained constant to within 0.001% c. Typical variations observed in the one-way velocity imply a diurnal modulation of the order

of ± 0.1% to 1.0% c. The relative precision of the measurements amounted to 1 part in 5 x 10^^.

Key words: fundamental constants; precision measurement; velocity of light.

1 . Introduction

The assumption of the isotropic propagation of light in the special theory of relativity is traditional. The adoption of a convention involving anisotropic propagation

does not necessarily violate the principle of relativity.

For example, Winnie [1] formulated three synchrony free

principles which include the factual core of the observations which support special relativity. The equations are expressed in e-generalized Lorentz form, which requires no assumptions regarding the one-way velocity of light.

Formulated in this generalized way, the special theory of

relativity allows for the possibility of anisotropic propagation of electromagnetic radiation, which is not observable via round trip experiments, or any other similar

symmetrical arrangement of the propagation path. However, we have not been able to find any experiments re- ported in the literature which are free of the symmetries

which cancel the effects of possible anisotropics in space. Sources of possible anisotropy in the propagation of

light include anisqtropies in the distribution of matter in the universe at large, or possibly the existence of an

"absolute space" as advocated by Poincare [2], in his strong theory of relativity, which has at its base the

non-relativistic theory of Lorentz [3]. Giannoni [4] for ex- ample, has also formulated a spectrum of non-relativistic

theories which satisfy the essence of Winnie's [1] syn- chrony free principles, and Torr and Kolen [5] have

established that the special theory and the Poincare [2] formulation of the Lorentz [3] theory lie at the extreme

ends of a spectrum of theories, all of which satisfy the essence of Winnie's [1] principles as modified by Giannoni [4]. There is, therefore, significant theoretical justification

for the measurement of the one-way velocity of light,

provided that a suitable experiment can be devised for

this purpose. In this paper we describe the results of an experiment in which the rotation of the earth is used to introduce changes in the direction of propagation of elec- tromagnetic radiation with the purpose of detecting

changes in phase produced by variations in the one-way

velocity of light.

2. Historical Problems with the One-Way

Experiment

Historically there has been a misconception regarding

the feasibility of any experiment designed to measure the

one-way velocity of light. Generally it is believed that

the experiment is meaningless. The rationale behind this argument proceeds as follows. In order to make the

measurement by timing the one-way flight of a light

pulse it is necessary to synchronize two clocks separated

by some distance d. To do this it is apparently necessary

to make some assumptions about the propagation speed

as a function of direction, since motion affects the fre- quencies of clocks, i.e., they cannot be locally synchronized and then separated without some unknown change

in phase occurring. Thus a knowledge of the velocity of propagation is required to synchronize the clocks, which

introduces a circular argument which in turn renders the

concept of the experiment meaningless.

In the experiment reported in this paper we circumvent this problem by not requiring that the clocks be

synchronized. We look instead for variations in the rela- tive phases of signals generated by two clocks, which

should only occur if the one-way velocity of light is direc- tion dependent. It nevertheless transpires that even the

absolute velocity of propagation can be deduced from this experiment as described below.

3. Concept of the Experiment

The experiment that we conducted utilized two

Hewlett Packard model 5065A rubidium vapor frequency

standards to time the flight of electromagnetic signals across a distance of —500 m separating the clocks. The

experimental arrangement that yielded the best results

is schematically illustrated in Fig. 1. Each clock generates a -1.5 V rms 5 MHz sinewave. The signal from

clock A was used to trigger the start input of an HP

model 5370A Universal Time Interval Counter. The sig- nal from clock B was used to stop the counter. The theory behind the experiment argues that if perfect clock stability is assumed, at some arbitrary time

675
CLOCK INTERVAL CLOCK

, STOP INPUT A COUNTER B

START INPUT Figure l. Schematic illustration of the concept of the experi- ment.

the interval counter will measure a signal given by

Ml = M + (1)

where M represents some initial phase offset between

the clocks, d — 500 m, and c

^ is the velocity of propagation at that time and orientation of the experiment. The

experiment relies on the rotation of the earth to inter- change the positions of the two clocks, so that twelve

hours later we measureM2 = M + (2)

where c " potentially represents a different value for the

velocity of propagation of the signal. Hence subtraction

of the two measured intervals yields the quantity

5^ = Af2 - Ail = -4- - (3)

= d

(c+ - C-)

c c

(4)

Naturally if the special theory of relativity is correct c - c' - c and bt = 0. To a first order approximation

5;, dv

01 — —5- (5)

where Ai = —

c and V -

c - c

Hence v — —c

(6)

(7)

(8)

Since a roundtrip measurement yields M, c and c can be determined as a function of time. In reality, however, the simplicity of this concept is marred by errors of measurement which we discuss in section 5.

4. Experimental Arrangement

Figure 2 schematically illustrates the details of the ex- perimental arrangement that was used. The clocks were

placed 500 m apart in an east-west orientation. The

5 MHz sine wave signal was propagated via a nitrogen

filled coaxial cable maintained at constant pressure at ~2

psi above the ambient atmospheric pressure using a 2 stage regulator. Thermal control was achieved to within

± 1 K over a diurnal cycle by burying the cable at a depth of 5 feet below the surface, and the enclosures

housing the equipment 10 feet below the surface. Electrical isolation was achieved by enclosing the equipment in a Faraday cage. Figure 2 also illustrates the dc power

supply arrangement used to supply regulated power to within ± 10 mV. The power was introduced into the

Faraday cages via electromagnetic interference filters. The performance characteristics of the rubidium clocks were evaluated by running the clocks at a separation of

1 m. Limitations imposed in the measurement due to clock errors were determined, and requirements for a successful experiment established. These findings are discussed in section 6.

FftRADAY/MAONETIC SKP-D

REGULATOR

FUSATED MA 24V

BATTERY

a4V

EMI FILTER

24Y REMOTE

CLOCK DC

35V

1

5065A

RUBIDIUM SUPPLY FREQ STO

AC -DC INVER- TER POWER

SUPPLY

lav 96A 12V BATTERY

12V DC/EOV

AC INVERTER 9370A INTER- VAL COUNTER

24V_

MA 24V

BATTERY

S4V

EMI FILTER

24V 50«6A

RUBIOKJM FREQ STO

FljOtfTED LOCAL CLOCK DC SUPPLY

faRADAY/MAONETIC SHCLD

COMPUTER

CASSETTE

STORAGE IEEE-4B* BUS

IGAS FILLED COAXIAL CABLE

EXPER»«NT ELECTRONICS BLOCK DIAGRAM

Figure 2. Block diagram illustrating the details of the experimental arrayigement.

676
5. Evaluation of Errors of Measurement for

Rubidium Vapor Frequency Standards

According to manufacturer-supplied information there

are two primary sources of error which affect the

behavior of rubidium clocks. These are:

a) Settability—This represents the practical limit of setting both clock oscillators to the identical frequency.

This error accumulates with time.

b) Long Term Frequency Drift—This error is intro- duced by "long term" noise, and produces a non-linear

drift with time.

By observing the behavior of the clocks at zero separation (i.e. — 1 m) their relative drift rate could be measured and hence the clock errors functionally fitted and

accurately defined. According to the manufacturer's

specifications [6] the accumulated error of a rubidium

clock can be parameterized as follows:

Tit) = {l/2)at:- ± bt ± to (9) where

Tit) - total accumulated error with respect to a

perfect time base, a = frequency drift rate — 1 x 10"^^ per month,

f b = - 1) = settability = ± 2 x lO'^^ seconds/second.

O 12 3 4 5 6 DAYS

Figure 3. Typical relative drift rate observed for the Rubidium

Vapor Frequency Standards over a period of 7 days for zero separation.

fo - initial frequency of oscillator at i = 0,

fr — reference frequency, and

to — initial time error at i = 0.

To calculate the difference in relative accumulated er- ror between two rubidium clocks, Eq. (9) is modified to ^Tit) - l/2(ai - a2)<2 ± brt±to (10)

where

AT it) = relative accumulated error,

a 1,2 = frequency drift constants of the respective

clocks r r

/ 10 20 = initial frequencies of the respective clocks

at i - 0.

It can be seen from Eq. (10) that if Q j — a-z, AT it) can be approximated by

ATit) brt ± to (11) for a short enough time such that the quadratic term can be neglected. This point is important for using rubidium

clocks in this application which will be discussed in detail

in the data reduction section. In order to characterize the behavior of the clocks with

repsect to each other, we monitored their performance

for a period of ~ 8 weeks at zero separation. Figure 3 il- lustrates the typical relative drift rates observed. We

have plotted the relative accumulated clock error AT it) for 7 days. These results show a pattern that we have

confirmed is typical. The clocks tend to drift linearly for periods of time which amount to several days. These

linear drift periods are interspersed with non-linear drift periods were the slope of the drift curve changes rapidly, and then settles down to the linear pattern again.

Figure 4 represents a typical linear segment over one day for zero clock separation. Figure 5 shows the residuals, i.e., with the Hnear trend removed.

While the mean departures from linearity over many

days agree with the manufacturer's specifications, these

departures occur "suddenly" providing large variations over a few hours which are clearly identifiable. Over the

linear segments, the clock error is about a factor of 10 smaller than that specified by the manufacturer. Using

the manufacturer's specifications we calculate a mean

non-linear component of —15 ns per day. Typically, however, the observed departure from linearity over 24

hours for a "linear segment" seldom exceeds 1.5 ns, which allows one to make a measurement of relative precision of —1 part in 5 X 10^^.

O 2 4 6 8 lO 12 t4 [6 re 20 22 ZA LOCAL TIME (HOURS)

Figure 4. A typical linear drift segment over a 2i hour period for zero separation of the clocks.

677
CO

ill o

z

tu

o:u

Ul

u.o

9

7

5

3

I -

-I

-3

-5

-7 h -9

-I 1 I I -1 1 f—

Figure

trend

0 2 4 6 8 10 12 14 16 18 20 22 24 LOCAL TIME (HOURS)

5. Typical residual drift rate obtained ivhen the linear

is removed.

which do not exhibit the non-linear variations peculiar to rubidium vapor frequency standards.

If these variations are real and the experiment is indi- cating the presence of a dynamical "absolute space," or the effects of anisotropic distribution of matter in the

universe then the motion of the solar system in the

galaxy should be embedded in the observed variations.

This should be observable as a sine wave modulation of the signal over a 24 hour period. The component of the

velocity of the solar system in the ecliptic plane is -10'' m/s "^ which translates into an amplitude variation

of ~ ±0.5 ns. In the presence of variations which exceed

±3 ns for 30 percent of the time, significantly more than

a year of integration would be required to detect this signal at a signal-to-noise ratio of 3 to 1. We have therefore used

a sample of data restricted to variation amplitudes <3 ns. The time required to obtain a given signal-to-noise ratio (SNR) is given by

6. Rubidium Results

Several months of data have been obtained for the

clocks separated and for zero separation. The separated

clocks exhibited a behavior pattern which is significantly

different from that observed for zero separation. Very

large (— 10 ns) departures from linearity have been ob- served on several occasions for the separated clocks. Two

examples are given in Figure 6. However, one cannot be

absolutely sure that this behavior is not simply another

form of the non-linear pattern identified in Fig. 3 even

though the clocks have not exhibited this kind of behavior under zero separation conditions. Our results indicate that departures from linearity for the separated

clocks remain below threshold (— 1.5 ns) 30 percent of

the time. Amplitudes between 1 and 3 ns are most com- mon. However, not only does the amplitude of the signal

vary from day to day, but the phase does also. Since

there is no theory available which can account for these

variations, we believe that it is essential to repeat the

experiment with different clocks such as cesium beams

0 2 4 6 8 10 12 14 16 18 20 22 24 LOCAL TIME (HOURS)

or

SNR

i.e. N = (SNR faj

(12)

(13)

where ht - mean signal amplitude,

cTe = standard deviation of the data, and

N = sidereal day count.

If we let SNR = 3,

8< - 0.5 ns

CTe — 3 ns N - 324 sidereal days at an occurrence rate of 30 percent, which considerably exceeds the number of days of

data currently available. Hence the expected signal is not yet detectable with our present arrangement.

Figure 7 shows the result obtained integrating 23 days'

data for which the maximum amplitude never exceeded

3 ns. From these results we can place an upper limit on

bt of 0.5 ns, i.e. V <90 km/s -1

This fairly closely approximates the component in the

ecliptic plane of the velocity of the solar system in the

galaxy.

2.5

2.0

IS

CO5 1.0 UJu

Z 0.5

llJ K

Ui

U. 0.0

u.o

W -0.5

I

-'.0

-I.S

Figure 6. The asterisks show a typical diurnal variation ob- tained for zero separation when the linear drift component is removed. Results obtained for a typical day for the separated

clocks are shown as the triangles. The crosses and squares

represent the maximum diurnal variation observed, excluding

cases which exhibit sudden "quantum" changes in drift rate which are clearly due to clock errors.

-2.0 I I

I I I ! I I I I I 0 2 4 6 e 10 12 14 16 18 20 22 24 LOCAL TIME (HOURS)

Figure 7. The coherent sum of 23 days' data for the separated

clocks for the period February to -June, 1981. Summing was

carried out using half hour bins.

678
7. Conclusion

The main result that has emerged from this work is the

demonstration of the viability of a measurement of the

one-way velocity of light. Given perfect clocks the absolute one-way velocity could be accurately determined.

With rubidium vapor frequency standards integration over 3 years would provide an unambiguous detection of

the motion of the solar system in the galaxy, if such motion is indeed detectable by this technique.

The results we have obtained to date exhibit large

variations in c (0.1% to 1% c) for the separated clocks,

which are not observed (i.e., in the same form) for zero

separation. More observational time will be required to unambiguously establish whether the observed variations are indeed due to clock errors or not.

We believe that the ambiguity could be greatly reduced

by using cesium beam clocks with a high performance option which significantly improves the clock stability.

References

[1] J. A. Winnie, Phil. Sci. 37, 81 (1970); .37, 223 (1970).

[2] H. Poincare, C. R. Acad. Sci. 140, 1504 (1905).

[3] H. A. Lorentz, Proc. K. Ned. Akad. Wet. 6, 809 (1904).

[4] C. Giannoni, Found. Phys., 9, 427 (1979).

[5] D. Ton- and P. Kolen, Found. Phys. 12, 265 (1982).

[6] Hewlett Packard Company Report: Application Note 52-2, "Time Keeping and Frequency Calibration" (1976).

679

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

High-Energy Gamma Rays Might Be Faster Than Visible Light

Kunio Fujiwara

Institute of Physics, College of General Education, University of Tokyo, Komaba, Tokyo 153, Japan

Our possible confusion between the wave and particle pictures arising from the linearity of the de

Broglie relation might be the cause of the divergence in field theories. When we assume that every line segment in </;ree-dimensional space has a quantum structure analogous to the atomic chain of atomic

distance 2ilQ, the de Broglie relation is exponentially nonlinearized at sufficiently high momenta, and

the symmetry between the two pictures is broken. It has been shown that, under such situations, the

field theories automatically become finite without chariging their conventional form. The present

theory predicts that high-energy -/-rays would be faster than visible light, though the detectability of the effect depends on the value of the unknown constant I q. Key words: de Broglie relation; ether; light velocity; local field; quantization of space; regularization;

special relativity.

1. Introduction

It is well known that, when the divergent integrals are renormalized, quantum electrodynamics (QED) can

predict basic electromagnetic quantities with high accuracy. The agreement between the theory and the experiments is, in fact, excellent. What should we think is the

physical meaning of this success of renormalized QED,

then? In the author's opinion, the success implies that

the conventional form of QED is, despite its divergence

difficulties, already beautiful enough to be regarded as

established. Any complication of the form would no

longer be necessary upon the regularization of this theory. From this point of view, the author would show

here that there exists at least one way to regularize QED

as well as the other quantum field theories without any

change in their conventional form.

We shall begin our discussion by reexamining some of our fundamental concepts. According to our understanding of the wave-particle duality, the wave and particle

pictures are transformable into each other through the

linear relations

p = ^k and E - ftm . . (1) The equivalence between the wave and particle quantities assumed in these relations often implies a nearly

complete confusion between the wave and particle pic- tures. We know, on the other hand, two fundamental

theories, i.e., the special theory of relativity and the

quantum theory. In the former theory, relative motions

between solid frames are considered; we should therefore

understand that the former is implicitly based on the particle picture of matter, while the latter apparently emphasizes its wave properties.-'^ A divergence problem

arises, however, when we intend to unify these two

theories into relativistic quantum field theories. Many

^The opposite appears to be true for light. In the theory of relativity, light is treated as wave signals, while the quantum theory emphasizes its particle properties.

physicists have treated this problem mainly from technical points of view, and it is usual to see that the result remarkably complicates the form of field theories.

The above divergence could, however, be a warning

that our understanding of the wave-particle duality (cf. Eq. (1)) is inadequate at least at extremely high energies. In this connection it is worth noticing that we can assume, without conflicting with any past observations,

that our 3-dimensional space has an unknown quantum

structure characterized by a very short length, /q. If we

really assume so, since only waves should be sensitive to such a structure, the equality between the wave and particle pictures would be broken at sufficiently high energies. Thus we would have

p = mUk, lo) and E = (2) instead of Eq. (1). Here, ^ik, Iq) is the nonlinearization

factor whose functional form will be proposed later. In

spite of such a change in the de Broglie relation, however, the E-p relation for light, for instance, would

remain in the usual form

E = cp , (3) because any particle properties are thought to be essentially insensitive to the quantum structure of space. Then

substituting Eq. (2) into Eq. (3), we get a nonlinear co-A- relation for light

(0 = ckiik, lo) . (4) The set of relations Eqs. (3) and (4) suggests that a

perfect 4-dimensional description, such as in the conventional theory of relativity, would be exact only in the

particle picture, and that our possible confusion between

the wave and particle pictures should be carefully solved

whenever we want to discuss the high-energy behavior of

field theories. In a recent paper [1] the author studied

this problem by starting from the analogy of a linear

chain, the simplest example of a quantized medium, and

showed that the relativistic quantum field theories au- tomatically become finite without changing their conventional form. In the following, the basic ideas will be

briefly outlined with particular emphasis on the importance of the precise measurement of the velocity of highenergy 7-rays.

681
2. The Outline of the Theory

An atomic linear chain can be regarded as a continuous

string as long as the wavelength, 277 //c, of the elastic wave propagating on the chain is much longer than the

atomic distance a. The wave equation is then given by

{d^/dx^ - v--d-/dt^)uix, n = 0 , (5) which leads to a linear w-k relation, co = vk, similar to that for light waves in vacuum. When the wavelength be- comes comparable to or shorter than a, however, the ef- fect of the "quantum structure" of this medium becomes

important. In such cases, the w-k relation is known [2] to be modified to

oj = {2v/a) siniak/2) . (6) The differential equation leading to this nonlinear to-A: re- lation will be obtained if we replace d/dx in Eq. (5) by

(2i/a ) sin[ -i{a/2)d/dx] . (7) Returning to our problem, we first assume that the

above example is just suggesting how to quarftize 3- dimensional space. Imagine that our 3-dimensional space

is filled up with a special medium (we call it ether) in which any two points are interconnected by an "atomic

chain" having an atomic distance 2?7o iU) real). The as- sumption of the pure-imaginary atomic distance appears

to be essential for the successful regularization of field theories. The momentum operator in field equations

would then be modified to -i^V X V2)"/(2n + 1)!] , (8)

n =0 when described by the observers fixed to the ether. (A

similar, but 4-dimensional operator has been proposed by

Danos, Greiner, and Rafelski [3].) Needless to say, for

light and material waves whose wavelengths are long

compared with /q. these chains would behave as "continuous strings," and the operator, Eq. (8), would

thereby reduce to -ifiV. In such cases, the concept of ether as the medium of the waves would lose its importance, and we could be convinced of the absence of the

ether as we have been so far. When applied to a plane wave exp[/(k • x - wO], the

operator, Eq. (8), will give the nonlinearized de Broglie

relation

p = nk[{smhkl,,)/klo] ^ ftk i iklo) (9) as preannounced in Eq. (2). The energy operator ifld/dt

and the energy-frequency relation E - fm are, on the

other hand, kept as usual because they are independent

of 3-dimensional space quantization. The resulting w-k

and E-p relations for light in vacuum are shown in Figs,

la and lb, respectively; the former leads to the kl^- dependent phase and group velocities Cph = c ^ {klo) and Cg = c cosh/c/o • (10) Our next problem is to reformulate the special theory

of relativity. We start from the following two postulates:

(I) Every physical law (except that describing the

quantum structure of space itself) is the same in

all inertial frames.

(II) Light behaves isotropically, and the phase and

group velocities of each individual light wave are

invariant to all the inertial observers.

0"' l/lo k 0 p

Figure 1. The m-k (solid curve in (a)) and E-p (b) relations for

light in vacuum, ivheyi the quantized ether is introduced into three-dimeyisional space. Notice the nonlinearity of the former

relation. We may assume that we are now in the linear region

shown in (a).

Then, the first postulate requires that, in any inertial frame, the p-k relation should be of the form^

p = /ik[(sinh/cO/A:/] = /Tk ^ kl) . (11) We have already assumed that / = in the frame fixed

to the ether. In frames moving with respect to the ether,

however, we can no longer regard I as a constant^. Thus, so far as / is concerned, we must admit the speciality of the "ether frame." This fact does not, however,

put any essential restriction onto the covariant description of field theories, because I will appear in the

theories only in the form kl, and we can reasonably as- sume that

kl — invariant (12) to all observers. Since Hght velocity is found to depend on

kl, Eq. (12) accords also with the second postulate.

Next, to solve the possible confusion between the wave

and particle pictures, we define a vector in the "reduced

space" as X = x/iikl) , (13)

where x is the vector in the usual space. Since we find

-i^(a/aX)ct)k = pct^k and -i(d/dx)^i^ = kcjj^. , (14)

where c})k = exp[i(k x - a)^)], we can understand that

X is quantum-theoretically conjugate with the particle

quantity, p, while x is conjugate with the wave quantity,

k. The Lorentz transformations are to be generalized in the way that they leave x^ - c\Hkl and kl invariant.

Under such transformations (hereafter referred to as GLT), the following quantities in the particle picture

form 4-vectors: = (X, ict), ^ d/dX^,, and ^ (p, iE/c) . (15)

On the other hand, the quantities in the wave picture

such as x^ = (x, ict ), = d/dXfj, and A;^ = (k, iw/c) (16)

Note that in particle scattering, momentum conservation would hold but wave-vector conservation would no longer hold.

^According to Eq. (44) in Ref. [1], we find

/ = /gd + S cose )/(l - s2)l/2

for light waves, where B is the parameter analogous to (3 = v/c, and 9 is the angle between the wave vector k and the direction of motion of the ob- server with respect to the ether. Note that / is independent of A: = | k |. This formula will hold also for material waves when tk ^ {kl ) » mc is ful- filled in the ether frame. Needless to say, kl = 0 can be a good approximation if tk £,(kl) < mc.

682
can no longer form 4-vectors. Nevertheless we find that

k^^x^J, = inv., d*x = inv., and d'^k = inv., (17)

which are easily proved if the invariance of P^lX^^ , ^,{kl )

d'^X etc. is noticed, (cf. Appendix A in Ref. [1].) The in- variance of the quantities shown in Eq. (17) will play an

essential role in regularizing field theories without

changing their form.

The covariance under the GLT is now to be the guiding

principle in formulating field theories. The new field equations, for instance, are derived from the conventional ones by making the replacement 3^ —» Z)^. Nor will any formal change be required upon the reformulation of

local field operators though, in return for that, a drastic

change will be required in our concept of "localization"

itself as mentioned below.

Consider, as a simplified example, a classical local field fix^ ^ i27:)-^JaiP^)exp{ik^x^d*k (18)

defined by an observer 0 in his .r^. space. Due to the re- placement a{k^J a iP^J and also to the invariance of

k^x^^ and d^/c, this expression is covariant under the

GLT. Therefore, another observer 0' would also write it down in the form

f(x^') = i2iT)-^fa(P^')expiik^'x^')d'k' . (19)

To that observer, however, the field of Eq. (19) can no

longer be local because x^^' is now dependent on the kl- value^ of the Fourier component of the field. A typical

example of x is given [1] by

xi = [xi - B^ikDctVil - B-)^- , x'2^ X2 , etc. (20)

It follows then that a local field defined by 0 at a point

is accepted by 0' as a quasi-nonlocal field whose

Fourier components are distributed on a curve in his x space (see Fig. 2), suggesting that "localization" might

Figure 2. An illustration of the transformation properties of

the coordinate x ^, which suggests that "localization" might be a

relative concept. The new situation ivoiild be more easily understood if the kl-axis is introduced as shown in the upper half

of the figure. Recall the invariance of kl.

^Note that the transformation of x ^obeys = inv. and kl = inv.

be a relative concept. Such a change in "localization"

appears to have some resemblance to the well-known

change in "simultaneity" which was required upon the

establishment of the special theory of relativity. Equation (20) also suggests that the microcausality principle

should be reformulated as "The effect of the kl-component of a signal emitted at

x^j_ = 0 cannot reach those points for which -

c^^HkDt^ > 0." Our new field theories appear not to violate this require- ment. As expected from the replacement dp. D^, the

Fourier transforms of the propagators for free fields are

to be written in terms of the particle quantity, P^. On

the other hand, the wave quantity d^k should be used as the volume elements of the Feynman integrals; this is ob- vious from the expression of the field operators as illus- trated in Eqs. (18) and (19). If our confusion between the wave and particle pictures is solved in such a way, all the

Feynman integrals would become finite because we find

that p = tk £,ikl ) ~ "^"(exp kl )/2l for large k's.

3. On the Experimental Tests

The new theory is broad enough to include the conventional theory as its limiting case. The difference between

the two will be appreciable only at sufficiently high energies, and precise measurements of the velocities of highenergy 7-rays or massive particles would be most effec- tive for the test of the new theory. According to this theory, the group velocity of a light wave of photon en- ergy E would be given by

C (E) = c cosh[sinh-i(/o^ /he)]

|c[l +{loEfflcM] for E «-hc/lo ^ [loE/n for E » -hc/lo ' ^^^^

where Iq is the possible third natural constant to be

searched for. Equation (21) is also applicable to a particle

of rest mass m if p » nic. In Fig. 3, the relative in- crease in light velocity [C {E ) - c ]/c is plotted against E

for various values of /q. (The possible increase in light

velocity with increasing photon energy has been dis- cussed also by Pavlopoulos [4].) There are a few velocity

E (eV)

Figure 3. Possible relative increase in light velocity \c/c plot- ted against the photon energy Efor variotis values ofl^.

683
measurements at high energies [5, 6, 7]. Brown et al. [5] compared the velocities of 7 GeV photons and of 11 GeV

electrons with that of visible Hght, and obtained the rela- tive velocity differences, [c (GeV) - c (eV)]/c (eV) =

(1.8 ± 6) X 10-*^ and (c - v^Vc = (-1.3 ± 2.7) x 10-^

respectively. Since they found no effect exceeding their

smallest experimental uncertainty, 2.7 x 10"^, we may

conclude that Iq is shorter than 5 x 10"^^ cm. References

[1] K. Fujiwara, Foundations of Phys. 10, 309 (1980) and Contr. Paper No. 8 at 20th Int. Conf. High Energy Phys. (Madison, USA; July 1980).

[2] C. Kittel, Introduction to Solid State Physics, 2nd ed. (Wiley,

New York, 1956), p. 104.

[3] M. Danos, W. Greiner, and J. Rafelski, Phys. Rev. D 6, 3476

(1972) ; and Z. Phys. 258, 147 (1973).

[4] T. G. Pavlopoulos, Phys. Rev. 159, 1106 (1967); and Nuovo

Cimento B 60, 93 (1969).

[5] B. C. Brown, G. E. Masek, T. Maung, E. S. Miller, H. Ruderman, and W. Vernon, Phys. Rev. Lett. 30, 763

(1973) . [6] Z. G. T. Guiragossian, G. B. Rothbart, M. R. Yearian, R. A.

Gearhart, and J. J. Murray, Phys. Rev. Lett. 34, 335

(1975).

[7] T. Alvager, F. J. M. Farley, J. Kjellman, and I. Wallin,

Phys. Lett. 12, 260 (1964).

684
Precision Measurement and Fundamental Constants 11, B. N. Taylor and W. D. Phillips, Eds.

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Fiber Optic Ring as a Gravitational Wave Detector''

C. L. Mehta, D. Ranganathan, and G. Bose

Physics Department, Indian Institute of Technology, New Delhi-110016, India

We suggest the use of a Sagnac interferometer made of a fiber optic ring for detecting gravitational waves. Any distortion in the geometry of the ring on account of a gravitational wave is detect- able as a phase shift between the counter propagating optical signals. In order to discriminate against

angular velocity variations and any other local fluctuations a pair of orthogonal coils is to be used and

the system rotated about an axis lying symmetrically between the two coils. It is estimated that a stabihzed He-Ne laser used as the source and fiber coils of 1 m diameter with 100 turns rotating with an angular velocity of about 1000 rad/sec will provide adequate sensitivity.

Key words: fiber optic gyroscope; gravitational wave detector; Sagnac interferometer.

Soon after the advent of the laser, its use in rotation

sensing devices vi^as suggested and successfully demonstrated [1]. Optical gyroscopes using ring lasers have, more or less, been perfected and are soon expected to be

standard devices for navigational purposes. (IJonsiderable

interest has also been shovv^n in fiber optic gyroscopes

[2]. The basic idea involved in these rotation sensing de- vices is the Sagnac effect [3] first noi d ir ^913. Based

on this effect, Haughan [4] et al. proposed a method of testing general relativity using a ring laser. In the

present paper, we suggest, as a variant of this method,

the use of a fiber optic ring for detecting gravitational waves. We first give a brief description of the Sagnac in- terferometer consisting of a rotating fiber optic ring. We

then show how a pair of such fiber optic rings mounted

suitably may be used to detect small changes in length

caused by a gravitational wave.

Figure 1 illustrates the principle of a Sagnac interfer- ometer. A laser beam is split into two beams by a beam

divider and sent through a fiber optic coil of m turns and

radius R in counter propagating directions. We also in- troduce a modulator in one of the beams as indicated in the diagram. Effectively, it changes the wave length of

OPTICAL

L

DETECTOR

Figure 1. Schematic diagram of a Sagnac interferometer.

*Supported in part by the Depaitment of Electronics, Government of

India, New Delhi.

the clockvdse propagating beam from \ to \ - d\. The wave length of the counterclockvdse propagating beam

being X. If the coil is rotating with an angular velocity ft about its axis, one may readily see that the net phase

shift produced between the two counterpropagating

beams is to the first approximation given by

Ac}, = iTT^^^^ iu^imil. (1)

k'^ X c

(It is being assumed that dk « \ and ClR « c.) If the

modulation d X is so adjusted that there is no net phase

shift (A(|> = 0), then we find that

dx = 4r^ —n-^ — (2) 2ttR c P c where A is the area and P the perimeter of the coil cross-section. The measurement of d X thus enables us to measure ft, i.e., we may use the interferometer as a gyroscope. Alternatively, knowing ft we may determine the

ratio of A, the area enclosed by the coil, to P, its perimeter.

The main effect of a plane gravitational wave is to dis- tort the geometry in the directions transverse to its wave vector. Figure 2 illustrates the direction of "tidal

forces" due to a gravitational wave of e+ polarization

propagating in the z-direction [5]. Its effect on an object

of circular cross-section is shown in Fig. 3. The change

in length in the transverse direction is proportional to the strength of the gravitational wave. If h in suitable

units denotes the norm of a typical element of the perturbation metric, then [5]

Such a change is then detectable through a Sagnac inter- ferometer as described below.

Two Sagnac interferometers are mounted orthogonal to each other on a shaft whose axis is symmetrically placed

between the planes of the two coils as showTi in Fig. 4. The input and the output coupling can be made by bringing the fiber ends out along the rotation axis. The shaft

is rotated about its axis at an angular velocity ft. Each

coil is affected by the component of the angular velocity

normal to its plane. In absence of any gravitational wave, two modulating signals for a null phase shift in the

respective coil would be the same. Consider now a gi-avi685
Figure 2. Direction of "tidal forces" due to a gravitational wave of e+ polarization propagating along z-axis.

Z

Y

(al

Figure 3. Distortion of circular cross-section (a) gravitational wave of e+ polarization propagating

tational wave along oz, normal to O. When the line of in- tersection of the planes containing the two coils (line

KK' as shown in Fig. 4) coincides with oz, coil 1 gets

distorted into an ellipse of semi-major axis R and semiminor axis R - hR, whereas coil 2 gets distorted into an

ellipse of R + hR and R as semi-major and semi-minor

axes, respectively. In this position the difference of the

modulation signals in the two coils is maximum. On the

other hand, when KK ' is perpendicular to oz, both coils are affected symmetrically and hence the difference of

the modulation signals is zero. Assuming fi"^ to be much

less than the period of a typical gravitational wave, the

output modulation difference will be an ac signal of twice

the angular frequency of rotation of the shaft. Its amplitude is proportional to the strength of the gravitational wave. The system displays maximum sensitivity in the

plane normal to the axis of rotation and zero sensitivity for waves travelling along the axis. The advantage of mounting both coils on a common

Y

(b)

in the yz-plane, (b) in the xy-plane, due to a

along z-axis.

shaft is that any fluctuation in the angular velocity of the

system affects both coils equally and thus do not show up

in the output. Similarly, any variation on account of ther- mal changes which affect both coils equally can be ignored. In any case they will usually have a frequency dif- ferent from the modulation difference signal, and hence

could easily be eliminated. For further efficiency one may have to place the whole system in a temperature

controlled environment.

In the above analysis, we considered a fiber optic ring

of circular cross-section. However, it may readily be

modified to include other geometrical cross-sections such

as a triangular one etc. In fact, in place of a fiber optic

ring one may use a ring laser gyroscope directly with

minor modifications.

The probable estimate of the gravitational wave

strength is [6] dK ^ h - 10-12 _ 10-16 (4)

686
Figure 4. Schematic diagram of the suggested gravitational wave detector.

for waves with a period of 1 - 10^ sec. Let us use a pair

of fiber optic coils of 0.5 m radius and 1000 turns each ar- ranged as described above and the system is rotated at 10^ rad/sec. If a He-Ne laser is used then the minimum

detectable phase shift for such a system assuming shot

noise limited detection [2, 7] is 10"^ fringe. If we ignore

the modulation in Eq. (1), we find that

d (A <j)) = 5- . (5) \ c R

Taking rf(A(})) ~ 10"^ and using typical values for the

other parameters as given above, it is seen that the ap- paratus will be sensitive for measuring dR /R ~ 10^'^. Further improvement can be made by increasing the

number of turns or the angular velocity of rotation. Also one may use a laser gyroscope which operates at the

quantum limit [8]. It is estimated that a sensitivity adequate to measure h ~ 10"^^ could be obtained using a ring laser gyroscope. Fiber optic gyroscopes are less sensitive at present mainly on account of back scattering in the fiber.

References

[1] F. Aronowitz, in Laser Applications, Vol. 1, Ed. by M. Ross

(Academic Press, New York, 1971) p. 134.

[2] V. Vali and R. W. Shorthill, Appl. Opt. 15, 1099 (1976).

[3] G. Sagnac, C. R. Acad. Sci., 157, 708 (1913).

[41 M. P. Haugan, M. 0. Scully, and K. Just, Phys. Lett. 77A, 88

(1980).

[5] C. W. Misner, K. S. Thorne. and J. A. Wheeler. Gravitation,

(Freeman, San Francisco, 1973) p. 1212.

[6] D. H. Douglass and V. B. Braginsky, in General Relativity:

An Einstein Centenary Survey, Ed. by S. W. Hawking and

W. Israel, (Cambridge University Press, Cambridge, 1979)

[7] S. Ezekiel and S. R. Balsamo, Appl. Phys. Lett. 30, 478

(1977).

[8] T. A. Dorschner, H. A. Haus, M. Holz, I. W. Smith, and H.

Statz, IEEE, J. Quantum Electron. 16, 1376 (1980).

687

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

A Possible Laser Gravitational Wave Experiment in Space

J. E. Faller* and P. L. Bender*

Joint Institute for Laboratory Astrophysics, National Bureau of Standards and University of Colorado,

Boulder, CO 80309

An investigation has been started of possible designs for a laser gravitational wave experiment

with baseline lengths of roughly 10'' km or longer. The objectives of the experiment are to search for narrow-band signals with periods of seconds to hours, for pulses of gi'avitational waves, and for broad- band background radiation. One of the main goals is to detect signals from known rotating binary stars, such as Am CVn, WZ Sge, and i Boo. The corresponding gravitational wave periods are 8.76, 40.5, and

193 minutes. The expected strain amplitudes are roughly 0.4, 0.5, and 6 x 10"^\ respectively, which

correspond to equivalent accelerations of 50, 4, and 2 x 10"^^ g^ for a 10'' km baseline. The main uncertainty in the expected signal strengths comes from the uncertainty in the distances. In view of the ex- tremely tiny equivalent accelerations, care will be needed in designing the experiment in order to minimize spurious accelerations due to forces other than the gi'avitational attraction of the sun and

planetary bodies. The first approach considered [1] was to make laser heterodyne distance measurements over two

non-parallel baselines which remain equal in length to 10~^ over the duration of a single experiment. For a laser stability in a 1 cycle/h bandwidth of 5 x 10"''' or better, periodic changes in the difference

in length of the two baselines could be measured with roughly 10"^^ sensitivity in one day. Three

spacecraft equipped with laser transmitters and receivers are required. The laser beams would be sent both ways over each baseline, and the beat signals detected at each end. With 10 mW of transmitted

laser power in the visible from each satellite, the required diameter of the optical systems is 50 cm. One geometry considered was to have a central spacecraft at the LI (or L2) libration point of the earth-sun system, which is about 10*^ km from the earth, and the other two spacecraft 1.5 x lO'^ km

away at the L4 and L5 libration points. It is feasible to arrange the initial conditions so that the baselines from the central spacecraft to the other two stay equal to 10"*^ over moderately long times. However, the number of signal counts detected per second would be quite low, and the propulsion capability necessary to reach the L4 and L5 libration points in one year is substantial. A second geometry discussed involved putting two spacecraft about 1.4 x 10^ km apart in coplanar one year period circular orbits around the sun, which define a reference plane, and the third spacecraft

in a similar orbit which is inclined by about 0.3 degrees with respect to the reference plane [2]. With

proper phasing of the orbital positions, the third spacecraft will be near the highest point above the reference plane when the other spacecraft are equally far ahead of and behind it. The distances from

the third spacecraft to the other two would be 10'' km, and would stay equal to 10'*^ over roughly a

day. With this separation the signal to noise would be high enough so that the lasers in the first two

spacecraft could be phase locked to the signals arriving from the third spacecraft. However, thrusts would have to be applied to the third spacecraft about every day in order to reestablish the proper

geometry.

Another approach which we would like to suggest is to let the two baselines differ in length by up to perhaps one part in 10^, but measure the apparent changes in the length L of one baseline as well as the changes in the difference in length A L. The differences between the measured apparent changes in the baseline length L with periods of seconds to hours and those expected from celestial mechanics

would be used to determine the laser frequency variations. This additional information makes it possi- ble to correct for the laser frequency variations in determining whether changes in AL occur due to gravitational waves. The only restriction in interpreting the results is the implicit assumption that the change in L due to the gi'avitational waves is not orders of magnitude larger than the change in AL.

With the new approach, we are exploring a different geometrical arrangement with 10'' km spacecraft

separations which would allow the experiment to run for long periods of time without orbit corrections. The main spacecraft would be placed in a 1 year period circular orbit, which defines the reference

plane. The other two spacecraft would be placed in orbits with eccentricities e = 1/300 and inclinations with respect to the reference plane of very nearly V3e. By proper phasing, the other spacecraft will . appear to rotate about the main spacecraft with an annual period in a plane whose normal is tipped

downward by 30 degi'ees with respect to the direction toward the sun. The most difficult design problem for the proposed gravitational wave experiment probably will be

shielding out or compensating for perturbing forces on the spacecraft due to fluctuations in the solar wind pressure and other effects. The magnitude of the fluctuating accelerations at the frequencies of interest will be roughly 10"''^

g. A disturbance compensation system (DISCOS) which reduced perturbations from roughly 10"^ g to 5 x 10"'- g even at zero frequency was flown on the TRIAD satellite in 1972. Initial studies indicate that an improved DISCOS can be designed which would reduce spurious — accelerations at the frequencies of interest to the necessary level in a 1 cycle/h bandwidth of less than

10"'^ g. However, attention will have to be placed on avoiding the many sources of perturbations in such systems.

*Staff member. Quantum Physics Division, National Bureau of Standai-ds.

'in this context, the gravitational acceleration at the earth's surface g, is used as a unit of acceleration.

689
References

[1] p. L. Bender, R. W. P. Drever, J. E. Faller, and R. Weiss, unpublished manuscript, 1977; referred

to by R. Weiss in Sources of Gravitational Radiation, Ed. bv L. L. Smarr (Cambridge Univ.

Press, Cambridge, 1979), p. 26.

[2] R. Decker, J. L. Randall, P. L. Bender, and J. E. Faller, in Active Optical Devices and Applications, SPIE Vol. 220 (Society of Photo-Optical Instrumentation Engineers, Bellingham, Washington, 1980), p. 149.

690
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Amin, S. R 117 Anderson, J. T 359

Arditi, M 29 Baba, S 529

Baird, K. M 35

Bairnsfather, H 469 Barker, P. H 345

Bartlett, D. F 643 Basov, N. G 65 Becker, P 317, 321 Becker, W 99

Bell, G. A 445 Bender, P. L 689 Benjamin, D. J 469 Bergquist, J. C 93 Bertinetto, F 69

Blatt, R : 99

Bliek, L 553

Bollinger, J. J 141 Borchert, G. L 331 Bose, G 685 Bouchareine, P » 81 Boulanger, J.-S 21 Braginsky, V. B 109 Braun, E 535 Breeden, T I53 Breitenberger, E 667

Brillet, A 73 Brodsky, S. J 249 Burghardt, B 49 Cabrera, B 359 Cage, M. E 539 Caldwell, C. D 117 Camani, M 223 Cerez, P 73 Chan, H. A 601 Chan, K. K 127 Chen, Y. T 581 Cheung, W. S 653

Clothier, W. K 469 Coarasa, M.-J 453 Cohen, E. R 391 Colclough, A. R 263

Conti, R 207

Cordiale, P 69 Currey, M. F .. 469 Daams, H '.. .. 21 Daniel, H.-U 45 Davidson, W 635 De Boer, H 561 Dehmelt, H. G 215. 219 Delker, L 325

Deslattes, R. D 303 DiMauro, L 153 Dong, T 153 Dress, W. B 233 Dugan, G 325 Dunn, A. F 385 Dziuba, R. F 539 Ebert, G 519 Elnekave, N 465 Endo, T 489,529

Erickson, G. W 195 Fabre, C 169

Faller, J. E 405, 411, -581, 639, 689 Fang, Y 419 Fau, A 465

Author Index

Page

233, 5

Felch, S. B

Feldman, M

Field, B. F

Fontana, S

Frieze, W

Fujiwara, K

Gabrielse, G

Gahler, R

Gallop, J. C

Gao, J

Gillies, G. T

Girard, G

Girvin, S. M

Gossard, A. C

Goy, P

Graff, G

Gramlich, J. W

Greene, G. L

Gross, M

Gubin, M. A

Guildner, L. A

Guo, Y

Gutmann, P

Gygax, F. N

Hall, J. L 43 Hammond, J. A

Hansch, T. W

Hansen, P. G

Hara, K

Haroche, S

Hartland, A

Harvey, K. C

Haugan, M. P

Hein, G 535 Hellwig, H

Hinds, E. A

Hoeffgen, H

Hoskins, J. K

Howe, D. A

Hu, C

Huang, C

Huang, D

Hughes, V. W 149, Igarashi, T

Iliff, R. L

Inagaki, K

Itano, W. M

Ito, N

Iwasaki, S

Johnson, C. E

Johnson, W. H

Jonson, B

Kalinowsky, H

Kalus, J

Kawaji, S

Keyser, P. T

Kibble, B. P

Kinoshita, T 203. Klempt, E

Knecht, J

Kobayashi, Y

Kochsiek, M

Kolen, P

Kotz, U

Koyanagi, M 489, Kowalski, F. V

Kowalski, J

Page

359

153 539

69

149

681 219

257

365 419

629 453

557

549

169 353

357

515

16965 287

419

535 223

, 671 423

111 331 479

169 543

163

181 553

11 14949

59125 595

419 419

237

529

423

529

93

53

53 229

335

331 353

257

529

639

461 529

223

519

441 427

675

159 529

181 159

691
ivponou,

A

Kriiger, E 369 Kubota, T 479 Kugel, H. W 145 Kurosawa, T 53 Lahdenpera, K 493 Lakkaraju, H. S 103 Lavine, C. F 539 Lepage, G. P 185 Leventhal, M 145 Levine, M. J 201 Lewis, L. L 25

Li,

C 77

Li,D 419 Lichten, W 117 Lindquist, W. B 203 Littman, M. G 173 Liu, F 595

Liu,W 595 Long, D. R 587 Lovelock, M. J 345 Lu, D. C 325 Lundeen, S. R

127,

131, 141 Luther, G. G 573 Mampe, W

233,

257 Man-Pichot, C. N 73 Martin, J. E 291 McGrory, J. B 331 McLagan, K. H 639 Mehl, J. B 281 Mehta, C. L 685

Meisel,

G 49 Melchert, F 535

Metcalf, H.

J

153,

177 Metherell, A. J. F

577,

581

Miller, P.D 233 Moldover, M. R 281 Morinaga, A 53 Moriyama, J 529

Miiller, J.

W 375 Mungall, A. G 21 Murakami, K 529 Murnick, D. E 145 Murphy, T. J 357 Nachman, P 671 Nakamura, A

489,

529 Naylor, H 345

Nelson, E.

T 145 Neumann, R 159 Newman, D 207 Newman, R 591

Ni,W

595,

647

Nikitin, V.V 65 Nikulchin, A. V 65

Nistler,W 369

Niv,

Y 145 Noehte, S 159 Obloh, H 519

Oelfke, W.

C 607

Olsen, P.T

475,

497 Ono, A 299

Paik, H.

J 601 Panciera, R 449

Pasin,

W 449

Patel, C. K.N 145 Patterson, J. B 445 Pendlebury, J. M 233

Perrin, P 233

Petley, B.

W 365 Petrovskiy, V. N 65 Peuto, A 449

Phillips, W.D

173,

177, 475, 497

Pichanick, F. M.

J 149

Picotto, G.B 69

Pipkin, F.M

127,

131, 141 Ploog, K 519

Page

Powell, L.

J 357 Protscenko, E. D 65 Prowse, D. B 437 Quinn, T. J 291

Radcliffe, W.

J 365 Raimond, J. M 169 Ramsey, N. F 233 Ranganathan, D 685

Rasetti,M 449 Ravn, H 331 Rayman, M. D 671 Rebaglia, B. 1 69

Reilly, M.

L 287 Reinert, W. 49 Rendell, R. W 557

Rich,

A 207 Rinker, R. L

405,

411

Ritter, R.

C

629,

653 Robinson, H. G 229 Roskies, R. Z 201 Rowley, W. R. C 57 Roxburgh, L W 1 Riiegg, W 223

Sacconi,A 449

Safinya, K.A 127 Sakuma, A 397

Sakurai,T 53 Sands, R. W 423 Sauder, W. C 277 Schenck, A 223

Schilling,

H 223 Schuessler, H. A 103 Schult, 0. W. B 331 Schultz, J 591 Schulze, R 223 Schwinberg, P. B

215,

349 Seppa, H 493

Seyfried, P 313 Shen, N 77 Shepard, J 643 Shiner, D 123 Shiota, F 479

Siegert,H

317,

321 Sloggett, G. J 469 Smith, K 233

Smj'the, R.M 345 Sokolov, Y. L 135 Speake, C. C

577,

581 Spero, R 591 Speth, J 331 Stacey, F. D 597

Steiner,M 45 Suhr, H 159 Sun, Y 77 Sweetman, E 207 Tanaka. K ' 53 Tarbeyev, Y. V 483 Torr, D. G 675 Towler, W. R 573 Traut, J 353

Tsui, D.

C 549 Tuck, G. J 597 Tyruikov, D. A 65 Van Degi-ift, C. T 457 Van Dvck, R. S., Jr

215,

349 Van Flandern, T. C 625 Van Patten, R. A 659 Vessot, R. F. C 611 Vitrant, G 169 von Klitzing, K

519,

535 Vowinkel, B 49 Wagner, R. J 539 WalHn, P 493 Walls, F. L 25, 93 Walther, H 45 Wang, C 77

692
Page

Wang, Z 505 Warnecke, P 535 Weirauch, W 369 Werth, G 99 Weyand, K 509 White, R. E 345 Wieman, C 123 Williams, E. R 475, 497 Wineland, D. J 83, 93 Winkler, K 159 Wdger, W 383 Wolf, H 223 Wood, 0. R. II 145

Page

Wu, C. S 325 Wu, Y 77 Xue, S. Q 535 Yamanouchi, C 529 Yang, C 595 Yennie, D. R 185 Yoshihiro, K 529 Yu, H 595

Zafiratos, C. D 643 Zhang, G 419 Zhang, X 77 Zumberge, M. A 405

zu Putlitz, G , 159

693

Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Conference Registrants

Amin, S. R. Garvey, R. M.

Arditi, M. Garvin, D. Aspden, H. Giacomo, P. Baird, K. M. Gillies, G. T. Bankuti, L. Girard, G. Barker, P. H. Gladisch, M.

Bartlett, D. F. Gonella, L. Becker, P. Gossard, A. C.

Belecki, N. B. Goy, P.

Bell, G. A. Grabe, M.

Bender, P. L. Graff, G. Benjamin, S. Greene, G. L. Bennett, S. J. Gubin, M. N.

Bernard, B. Guien, M.

Bertinetto, F. Guildner, L. A.

Bliek, L. Guo, Y.

Bollinger, J. J. Hall. J. L. Borchert, G. L.- Hammond, J. A. Bouchareine, P. Hansch, T. W.

Bourdet, G. L. Hara, K. Braun, E. Hartland, A. Breeden, T. Harvey, K. C. Breitenberger, E. Haugan, M. P.

Brillet, A. Hazeltine, S. G. Brodsky, S. J. Hellwig, H. W.

Brown, B. L. Henins, A.

Cabrera, B. Hill, R. M.

Cage, M. E. Hinds, E. A. Chandra, K. Holdeman, L. B. Chen, Y. T. Holt, H. K. Cheung, W. S. Holt, R. A. Cohen, E. R. Hoskins, J. K.

Colclough, A. R. Hudson, R. P. CoUe, R. P. Hughes, V. W.

Conti, R. S. Itano, W. M.

Cooper, J. W. luculano, G. Costain, C. C. Jen, C. K.

Daniel, H.-U. Jimenez, J. J. Davis, R. S. Johnson, C. E. de Boer, H. Johnson, W. H. Dehmelt, H. G. Jones, G. R., Jr. Deslattes, R. D. Jones, 0. C. DeVoe, R. G. Jopson, R. Dong, T. Kantor, F. W.

Drever, R. W. P. Kessler, E. G., Jr. Dugan, G. F. Kessler, K. G. Dumas, J. Keyser, P. T. Dunn, A. F. Kibble, B. P. Dziuba, R. F. Kinoshita, T. Edwards, G. J. Kobayashi, M.

Egan, P. 0. Kobayashi, Y.

Eicke, W. G., Jr. Koch, W. F. Eisenhart, C. Kochsiek, M.

Erickson, G. W. Kolen, P. Evenson, K. M. Koyanagi, M.

Faller, J. E. Kowalski, F. V. Fau, A. Kowalski, J. Faustov, R. N. Kramer, B. Feldman, M. Kr^-netskiy, B. B.

Field, B. F. Ku, H. H. Frank, L. E. Kupper, W.

Frueholz, R. P. Layer, H. P. Fujiwara, K. Levine, M. J. Furukawa, G. T. Lewis, L. L. Fystrom, D. 0. Lewis, R. R.

Gabrielse, G. Leyh, C. H.

Gahler, R. Li, M. C.

695
Lichten, W.

Lightbody, J. W., Jr. Littman, M. G. Lockyer, T. N. Long, D. R. Lough, T. Lowke, J. J. Lundeen, S. R. Luther, G. G. Mangum, B. W.

Marinenko, G. Martin, J. E.

Martin, W. C. Mathiesen, 0. Mattison, E. Marx, E. Mehta, C. L. Meisel, G. 0.

Metcalf, H. J. Metherell, A. J. F.

Mills, A. P., Jr. Mohr, P. J. Moldover, M. R. Morimura, M.

Muijlwijk, R.

Muller, J. W.

Nachman, P. Nakamura, A. Nelson, E. T. Newman, D. E. Newman, R.

Ni, W.-T.

Nistler, W.

Oelfke, W. C. Olsen, P. T. Ono, A. Page, L. A.

Paik, H. J. Pakay, P. Petley, B. W.

Peuto, A. M.

Phillips, W. D.

Pipkin, F. M.

Powell, L. J. Prestage, J. D. Preston-Thomas, H. Prowse, D. B. Quinn, T. J. Ramsey, N. F. Rayman, M. D.

Reilly, M. L. Rich, A.

Rinker, R. L. Ritter, M. W.

Ritter, R. C. Robinson, H. G. Robiscoe, R. T. Rosasco, G. J. Roskies, R. Z. Rowley, W. R. C. Roxburgh, L W.

Rueger, L. J. Rumble, J. R., Jr. Sacconi, A. Sakuma, A. Sauder, W. C. Schuessler, H. A. Schumer, D. B. Schweitzer, W. G., Jr. Schwitz, W.

Seppa, H.

Series, G. W.

Seyfreid, P. Shen, N.

Shields, J. Q. Shiner, D. Shugart, H. A.

Siegert, H.

Silbar, M.

Sloggett, G. J. Snyder, J. J. Sokolov, Y. L. Speake, C. C. Stacey, F. D.

Steudel, A.

Stubbs, C. W.

Sweetman, E. Tanaka, K. Taylor, B. N. Thomsen, J. S. Torr, D. G. Towler, W. R. Twyford, L. C, Jr. Urnov, A. M. Van Baak, D. A. Van Degi-ift, C. T. Van Dierendonck, R. F. Van Dyck, R. S., Jr. Van Flandern, T. C. Van Patten, R. A. Verma, A. R.

Vessot, R. F. C. Volk, C. H. von Klitzing, K.

Wallin, P. T. Walls. F. L. Wang, Z. Weber, A. Weber, E. W.

Weirauch, W.

Werth, G. Weyand, K. White, J. A. Wieman, C. E.

Williams, E.R

Wineland, D. J. Witt, T. J. Wdger, W.

Woodle, K. Yamanouchi, C. Yennie, D. R. Zumberge, M. A.

696
NBS-n4A IREV. 2-8CI U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO. NBS SP 617

2. Performing Organ. Report No, 3. Publ ication Date August 1984

4. TITLE AND SUBTITLE

Precision Measurement and Fundamental Constants II

Proceedings of the Second International Conference, held at the National Bureau of

Standards, Gaithersburg, MD, June 8-12, 1981

5. AUTHOR(S)

B. N. Taylor and W. D. Phillips, Editors

6. PERFORMING ORGANIZATION (If jo/nt or other than NBS, see instructions)

NATIONAL. BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZiP)

Same as in item 6 above.

10. SUPPLEMENTARY NOTES

Library of Congress Catalog Card Number: 84-601083

Document describes a computer program; SF-185, FlPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual sumnnary of most si gnificant information. If document includes a si gnificant

bi bl io graphy or I iterature survey, mention it here)

This volume presents the Proceedings of the Second International Conference on Precision Measurement and Fundamental Constants, held at the National Bureau of

Standards in Gaithersburg, MD, from June 8-12, 1981. Like its 1970 predecessor,

the Conference provided an international forum for theoretical, experimental,

and applied scientists actively engaged in research on precision measurements

relating to the fundamental physical constants, and on the testing of related

fundamental theory. More specifically, the purpose of the Conference was to assess the present state of the precision measurement-fundamental constants

field, to examine basic limitations, and to explore the prospects for future

significant advances. The principal subjects covered were: frequency, wavelength,

spectroscopy, quantum electrodynamics, the gas constant, x-ray interf erometry

,

nuclidic masses, uncertainty assignment, gravitational acceleration, mass, electrical quantities, gravity, and relativity. These proceedings contain the vast

majority of both the invited review papers and the contributed current research

papers presented at the Conference. The new results reported at the Conference

were considered for inclusion in the 1983 least-squares adjustment of the constants

carried out under the auspices of the Task Group on Fundamental Constants of the

Committee on Data for Science and Technology (CODATA)

.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s) data analysis; experimental tests of theory; fundamental constants; least-squares

adjustments; precision measurements.

13. AVAILABILITY

[T] Unlimited

I I For Official Distribution. Do Not Release to NTIS

Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

646

15. Price

USCOMM-DC e043-P80

sir U.S. GOVERNMENT PRINTING OFFICE : 1984 0 - 450-414 : QL 3



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