Rational Bureau of Standard

ybrary, E-01 Admin. Bldfr

MAY 2 8

"1971

NBS TECHNICAL NOTE 399

U.S.

WENT

OF

ERCE

ional neau

lards

The Physical Basis

of Atomic Frequency Standards
NATIONAL BUREAU OF STANDARDS

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NATIONAL BUREAU OF STANDARDS • Lewis M. Branscomb, Director

NBS TECHNICAL NOTE 399

ISSUED APRIL 1971

Nat. Bur. Stand. (U.S.), Tech. Note 399, 54 pages (April 1971

)

CODEN: NBTNA

The Physical Basis

of Atomic Frequency Standards

A.S. Risley

Atomic Frequency and Time Standards Section

Time and Frequency Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide

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

(Order by SD Catalog No. C13.46:399), Price 60 cents

Table of Contents

THE PHYSICAL BASIS OF ATOMIC

FREQUENCY STANDARDS

A. S. Risley

Page

Abstract 1 Key Words 1

1 INTRODUCTION 1

2 ATOMIC ENERGY LEVELS 2

2. 1 Four Basic Ideas 3

2. 1. 1 The Electron 3

2. 1. 2 The Nucleus 4

2. 1. 3 The Quantization of the Energy of Atoms ... 5

2. 1. 4 The Quantization of the Electromagnetic Field . 6

2. 2 The Bohr Model of the Hydrogen Atom 7

2. 3 Quantum Mechanics 10

2.4 Spin- -Additional Energy Levels 13

3 THE INTERACTION BETWEEN ATOMS AND

ELECTROMAGNETIC RADIATION 14

3. 1 Maxwell's Theory of Electromagnetic Radiation ... 14

3. 2 The Dependence of Atomic Transitions upon

EM Radiation 16

3. 3 Transition Probability and Linewidth of Actual

Frequency Standards . 18

4 CHOOSING A SUITABLE ATOM 19

4. 1 The Energy Levels of a Many-Electron Atom 20

4. 2 The Hyperfine Interaction 25

4. 3 Criteria for Choosing an Atom 26

4. 4 The Need for State Selection 29

5 THE THREE MAJOR EXAMPLES OF ATOMIC

FREQUENCY STANDARDS 30

5. 1 General Considerations 30

5. 1. 1 Common Functions 30

5. 1. 2 Primary versus Secondary Standards 32

5. 2 The Hydrogen Maser 32

in
5. 3 The Cesium Beam 36

5.4 The Rubidium Gas Cell 39

6 SUMMARY 42

ACKNOWLEDGEMENTS 43

REFERENCES 44

BIBLIOGRAPHY 45

FIGURES . 46

IV
* THE PHYSICAL BASIS OF ATOMIC FREQUENCY STANDARDS

A. S. Risley

Atomic Frequency" and Time Standards Section

National Bureau of Standards

Boulder, Colorado 80302 USA

Abstract

A tutorial discussion of the physical basis of atomic

frequency standards is given. These principles are then

related to the conditions under which an atom can be used as

the working substance of a stable and accurate frequency

standard. The three primary examples of atomic frequency

standards- -the hydrogen maser, the cesium beam, and the

rubidium gas cell--are then discussed in terms of these principles and conditions. The functions of the fundamental parts

of each device become apparent through this development.

Key Words: Atomic frequency standards; Cesium beam;

Energy levels; Hydrogen maser; Hyperfine

interaction; Rubidium gas cell; Transition

probability.

1. INTRODUCTION

Many people have an interest in modern technological

developments. For some, these developments relate to their work and

for others, such as students, these devices offer evidence of the

realization of basic physical ideas. Atomic frequency standards are

an important example of such devices. Not only is their application of

significance economically but their design is based upon a foundation of

physical theory which is basic to both physics and chemistry- -the idea

*This paper was originally prepared for inclusion in an NBS monograph

whose subject is to be time and frequency. The monograph will be

prepared by members of the staff of the Time and Frequency Division

of NBS, Boulder.
of atoms and atomic structure. The purpose of this paper is to state

the physical basis of atomic frequency standards and, having done so,

to show how these principles determine the design of the specific

devices.

As the reader will observe, I have taken a historical approach in

discussing the atomic theory. In so doing, perhaps some of the excitement and beauty of physics will be apparent. If I have been successful

in this effort then it may be that some people who are already familiar

with the atomic theory will find this work stimulating. Whatever his

level of education, the reader should find the essentials of atomic

frequency standards given here.

2. ATOMIC ENERGY LEVELS

"The only existing things are the atoms and empty space; all else

is mere opinion" --Democritus (about 400 B. C. ). Today, the atomistic character of matter is taken for granted by

almost everyone who has been raised in a technological society. And

yet, for 2200 years (from the time of Democritus), a majority of

scientists, partly on theological grounds, actively rejected the idea.

As foreign as the atomistic concept is to the human senses, by the

early 1800' s the study of gases and of chemical composition had made

atoms a compelling idea. Atomic frequency devices are intimately

tied up with the ideas of atoms, atomic energy levels, and electromagnetic radiation that developed so rapidly from about 1800 to 1930.

A discussion of these ideas is essential to the understanding of these

devices. These ideas will be discussed in a historical manner with a secondary goal of suggesting the profound changes that took place in

physics during this period. The major goal will be to give sufficient

detail that the function of the various parts of each device will be

obvious.
2. 1, Four Basic Ideas

The knowledge upon which atomic frequency standards are based

developed from the study of the absorption and emission of electromagnetic radiation by matter. This study, which is a branch of

spectroscopy, has developed a very elaborate model of the energy level

structure of atoms and of their interaction with the electromagnetic

field in an attempt to account for the great number of spectroscopic

observations. It is an effort which has been remarkably successful.

2. 1. 1 The Electron

The atom, as a miniature solar system, is a model which is

familiar to many people. As familiar as this --the Bohr model--is,

100 years of ingenious experiment and profound change in theoretical

concept were the necessary preliminary to its birth.

At first it was not necessary to think of atoms as having structure,

of being made up of a nucleus and electrons. Progress was made in

understanding gases simply by assuming atoms to be spherical objects

with mass and a small but non-zero radius. Information about the

structure of atoms came from the electrochemical studies of Faraday

around 1834. These experiments showed that atoms had electrical

charge associated with them; and in fact that the charge comes only in

discrete amounts—it is not infinitely subdivisible. But neither the

mass nor the charge of individual atoms were separately known; rather

only the charge to mass ratio. These ratios depended on what substance

was used in the experiment, but for any given substance the ratio was

constant.

The electron itself, however, could not be isolated by the

electrolysis technique, and another 60 years passed before, in 1897,

J. J. Thomson conducted an experiment with electric discharges in
gases and was able to isolate the electron. This experiment,

sometimes called a cathode ray experiment, measured the deflection

of a stream of particles--electrons--the particles being deflected by

electric and magnetic fields. Nevertheless, this experiment was the

same as the electrolysis experiment, in the sense that it gave only the

ratio of charge to mass, e/m. Thomson realized that he had discovered a fundamental characteristic of all atoms because the

character of these cathode rays was independent of the gases and

metals he used in his experimental apparatus. Another thing Thomson

found was that the cathode rays were negatively charged.

The results of Faraday and of Thomson suggested a radically new

concept: atoms are normally electrically neutral; they are composed

of very light particles--electrons--which have a negative charge and a much heavier other part which has a compensating positive charge.

Thomson's work was followed by the Millikan oil-drop experiment in

the year 1910. Here, Millikan measured the absolute charge on the

electron, and, having e/m from Thompson's work, the mass, m, of

the electron became known.

2. 1. 2 The Nucleus

The positively charged "other part" of the atom was not yet

pictured as a very small, and therefore, very dense core--a nucleus.

During the period 190 9 to 1911, the work of Ernest Rutherford and his

colleagues evolved this profound idea. Their experiment was based on

letting alpha particles (the nuclei of helium atoms) coming out of

radioactive radium strike a thin metal film. A quotation from

Rutherford shows his excitement over the result [1]:

"... I had observed the scattering of ^-particles, and

Dr. Geiger in my laboratory had examined it in detail. He

found, in thin pieces of metal, that the scattering was usually
small, of the order of one degree. One day Geiger came to

me and said, "Don't you think that young Marsden, whom I am training in radioactive methods, ought to begin a small

research?" Now I had thought that, too, so I said, "Why not

let him see if any ^-particles can be scattered through a large angle?" I may tell you in confidence that I did not

believe that they would be, since we knew that the a-particle

was a very fast, massive particle, with a great deal of

[kinetic] energy, and you could show that if the scattering

was due to the accumulated effect of a number of small

scatterings, the chance of an ctf-particle' s being scattered

backward was very small. Then I remember two or three

days later Geiger coming to me in great excitement and

saying, "We have been able to get some of the ^-particles

coming backward. . . " It was quite the most incredible event

that has ever happened to me in my life. It was almost as incredible as if you fired a 15-inch shell at a piece of tissue

paper and it came back and hit you. On consideration, I realized that this scattering backward must be the result of a single collision, and when I made calculations I saw that

it was impossible to get anything of that order of magnitude

unless you took a system in which the greater part of the mass

of the atom was concentrated in a minute nucleus. It was then

that I had the idea of an atom with a minute massive center

carrying a charge.

2. 1. 3 The Quantization of the Energy of Atoms

Two more ideas were necessary ground work for the Bohr model.

These were (1) the quantization of the energy of atoms and (2) the

quantization of the energy of the electromagnetic (EM) field.

The idea of the quantization of the energy states of matter was

conceived by Max Planck around the year 1900. He introduced the idea

in his highly successful theory of black-body radiation.

Black-body radiation had been a subject of long-standing interest

in the field of thermodynamics as well as spectroscopy. The first, and

partially successful theoretical attacks upon this phenomena were

based upon conventional thermodynamic theory and the exceedingly
successful electromagnetic theory of James Clerk Maxwell. These

approaches were able to give good results for either the long or the

short wavelength ends of the emission spectrum, but no one model

satisfied the entire spectrum.

Planck based his theory on several assumptions: (1) Matter (as

far as thermal radiation is concerned) is made up of an exceedingly

large number of electrically charged oscillators. This collection contains oscillators at all possible frequencies. (2) Although all

frequencies are represented, any given oscillator has its own specific

frequency, say V, and it cannot radiate any other. (3) He further

assumed that the amounts of energy that a given oscillator could emit

were given by E = nhv, where V is the characteristic frequency of the

oscillator, n is an integer (1, 2, 3, . . . ), and h is a constant. The

quantity, h, is a fundamental constant of nature and is known as Planck's

constant. The third assumption is the third of the four basic foundation

stones of Bohr's theory.

So Planck's ideas succeeded in predicting black-body radiation,

but nobody, including Planck, was comfortable with them. To quote

Holton and Roller [2]:

"Planck himself was deeply disturbed by the radical

character of the hypotheses that he was forced to make, for

they set aside some of the most fundamental conceptions of

the 19th-century science. Later he wrote that he spent manyyears trying to save physics from the notion of discontinuous

energy levels, but that the quantum idea 'obstinately withstood

all attempts at fitting it, in a suitable form, into the framework of classical theory . '"

2. 1. 4 The Quantization of the Electromagnetic Field

In 190 5, Einstein proposed the following relation as an explanation

of the photoelectric effect: hV = P + E, where V is the frequency

k, max

of the electromagnetic radiation striking the surface from which
electrons are emitted, E, is the maximum kinetic energy of these

k, max

electrons, P is a property of the metal being studied, and h is a constant- -independent of the metal being studied and of the frequency V. Eventually, careful measurements showed that this constant, h, was

numerically equal to Planck's constant, and, with the passage of time,

the two constants have come to be accepted as identical--as representing the same immutable aspect of nature.

The idea behind the above equation was that light was quantized-

that its energy came in discrete amounts, YiV, that it had a particle-like

as well as a wave-like character. This new idea was no more

acceptable to the majority of scientists than was Planck's proposal that

the energy states of matter were quantized. One might have thought,

however, that Planck would have maintained a reserved silence--after

all, Einstein's idea and his own were so analogous and the two constants

appeared to be numerically equal. But he did not. In 1910 Planck

wrote that, if this new theory of light were accepted "the theory of

light would be thrown back by centuries --for the sake of a few still

rather dubious speculations." [3]

2. 2, The Bohr Model of the Hydrogen Atom

With these four ideas and with the boldness characteristic of these

ideas, Niels Bohr, in 1913, proposed his now famous model of the

hydrogen atom. Before discussing the Bohr model, however, it is

necessary to admit that it will not allow us to explain all the details of

atomic frequency standards. One reason is that the Bohr theory will

not correctly predict the intensities of the absorptions and emissions

that an atom can undergo. Even so, the Bohr model is of much more

than historical interest. For one thing, it does incorporate the

majority of the ideas which we need, and it does provide a mechanistic,

intuitive picture of the energy levels of an atom.
Bohr was faced with the problem of both acknowledging the idea of

the nuclear atom and of predicting its stability. Precisely what went

through Bohr's mind is something that is not known to us, but the

similarity between his problem and planetary motion is obvious: In

the simplest case there is the sun (nucleus), a very massive body, and

a planet such as the earth (electron). The two bodies attract one

another with a force which is inversely proportional to the square of

their distance of separation.

Bohr knew that the radiation from an atom was associated with its

electrons because in 1897 (just after the discovery of the electron by

Thomson) Zeeman and Lorentz showed that the radiation coming from

an atom was due to a charged particle whose charge-to-mass ratio was

the same as that for the electron. Maxwell's theory predicts that if an

electron is moving in a circular path with an angular frequency, ^2,

it will radiate electromagnetic energy at a frequency, V - —— . This

is because the electron is charged and because of the acceleration

experienced in circular motion. The difficulty with the planetary idea

is that, as the electron radiates away its energy, it slows down and

gradually spirals into the nucleus. Bohr dispensed with the problem

of a nuclear crash by assuming an arbitrary restriction.

Bohr's analysis was based on two equations. The first equates

the attractive force between the nucleus and the electron to the product

of the radial acceleration of the electron and its mass. The second

equation sets the angular momentum of the electron, mvr, equal to a constant, nh/(27T). Here, v is the magnitude of the velocity of the

electron upon its circular orbit, r is the radius of this orbit, m is the

mass of the electron, n is an integer (1, 2, 3, . . . ), and h is Planck's

constant. This second equation is completely arbitrary and is in

fundamental disagreement with Maxwell's electromagnetic theory
because it implies that an electron can travel around a circular path

without radiating.

Bohr solved these two equations for the "allowed" orbits, i. e. , those particular values of the radius which satisfy both conditions.

This results in

r = n

2 c (2. 1)

where c is a constant. The smallest value of the integer n is 1 (unity),

and when the electron is in the orbit corresponding to this radius it is

said to be in its ground state. (The definition of "ground state" has to

be modified for a many-electron atom. ) This result is one of the major

successes of Bohr's theory because if the constant is evaluated it is

found that r, for n = 1, is approximately equal to 0. 5 X 10 " m. This

is in good agreement with the size of the hydrogen molecule as obtained from the kinetic theory of gases.

The second important result of Bohr's analysis is the expression

of the total energy E of the atom in terms of the integer n. The result

is

1 em

where m and h are as defined above, e is the charge of the electron,

and £ is the dielectric constant of free-space. By taking the difference between the energies of two orbits, an equation is obtained

which predicts many important observations of optical spectroscopy.

The result is

E .E.-—22L .(-!_. -L). (2.2) n n Q ^2 , 2 ,2 2 8 c h n

1 n
When energy is absorbed or emitted the electron makes a corresponding

change in orbit- -this is commonly called a transition. Figure 1 shows

the energy level diagram for the hydrogen atom. The vertical lines

represent possible transitions. To understand the experimental results

of optical spectroscopy it was useful to group the various emissions.

These groups were given names according to those people who had

devoted the most study to them, Lyman, Balmer, etc. For the hydrogen atom, transitions to the n = 1 level involve radiation whose

wavelengths are of the order of 100 nm (1000 Angstroms). This

corresponds to a frequency of 3 X 10 15 hertz. (A hertz is equal to one

cycle per second. ) However, when the effects of the intrinsic magnetic

moments of the electron and the nucleus are included, it turns out that

additional levels are available whose spacings are at microwave

frequencies (somewhere between 10 9 and 10 hertz). Before discussing

these effects, I need to talk about an entirely different basis for

calculating the characteristics of an atom. This approach is essential

for obtaining the probabilities of transitions between the various

levels--a subject which is crucial to atomic frequency standards. This

theory is called quantum mechanics.

2. 3. Quantum Mechanics

It was not just its inability to account for all of the experimental

data that left something to be desired in the Bohr theory. It was the

fact that this theory- -just as the black-body theory of Planck and the

photoelectric theory of Einstein- -was an intellectually uncomfortable,

ad hoc, combination of the old ideas of classical physics and the new

idea of quantization. These conceptual difficulties were removed with

the advent of quantum mechanics.

10
This new theory was developed separately, and nearly

simultaneously, by Erwin Schrodinger and Werner Heisenberg about

1926. The work of these two men, at first sight, seemed quite

different; but it was soon shown that, despite their difference in

mathematical form, they were physically equivalent. A brief description of Schrodinger's theory is given here. Schrodinger's theory takes

the form of a partial differential equation. Specifically, it is an

equation involving the rate of change of a quantity, 0, with respect to

the variables of space and time. In its original form, this equation

was designed to predict the possible values of energy which could be

possessed by a material body. In its generalized form it is used to

calculate any measurable property of a particle. The quantity is

usually referred to as a wave function. The physical meaning of this

quantity was, at first, quite uncertain. It is now generally agreed,

however, that the square of the magnitude of ijj, at any given point in

space, represents the probability of finding the particle at that point

in space.

This probabilistic interpretation is perhaps the key feature of the

new physics --of quantum mechanics --which says that particles have a wave-like aspect. Einstein based his photoelectric theory on the idea

that light had a particle -like property. In 1924, Louis de Broglie

turned the cart around by suggesting that a particle has a wavelength

associated with it. In other words, a particle has some of the

characteristics of a wave. In particular he said that the wavelength,

X, of a particle should be given by X = where h is Planck's constant

mv

and the product, mv, is the momentum of the particle. To quote

de Broglie [4]:

"Determination of the stable motion of electrons in

the atom introduces integers; and up to this point the only

phenomena involving integers in physics were those of

11
interference and of normal modes of vibration. This fact

suggested to me the idea that electrons, too, could not be

regarded simply as corpuscles, but that periodicity must be

assigned to them. "

Schrodinger's equation brings together in a natural way the

localized (what we usually call the particle aspect) and wave-like

aspects of a particle. This is very clearly seen in the solution of

Schrodinger's equation as applied to the hydrogen atom. Some of the

results of the solution are: (1) The atom can have only certain discrete

values of energy, and these values are associated with an integer, n. This is qualitatively and quantitatively the same result as obtained

from the Bohr theory, but, (2) although for any given energy the

electron has a high probability of being found in a restricted region

about the nucleus, it has some chance of being found anywhere, and

certainly it is not to be thought of as being localized on orbits as in the

Bohr model. (3) In solving for the H atom it is found that two more

integers, I and m , occur as a natural result of the solution. Thus we

JO get three numbers, n, I, and m , each of which are integers. These

Xj integers are called quantum numbers, and they represent the fact that

here three properties of the atom are quantized. They are: The energy,

represented by n; the total orbital angular momentum, represented by

I; and the projection of the total angular momentum, along some

specified direction, represented by m . The importance of these

Xj additional quantum numbers will be discussed in Section 4. 1.

It is apparent that the theory of matter and electromagnetic energy

has gradually become more and more abstract, less and less intuitive.

Despite the fact that quantum mechanics is a profoundly different concept than that used by Bohr, he was prepared to accept the increasing

12
abstraction--he wrote [5]:

"To the physicists it will at first seem deplorable that

in atomic problems we have apparently met with such a limitation of our usual means of visualization. This regret

will, however, have to give way to thankfulness that mathematics in this field, too, presents us with the tools to prepare

the way for further progress. "

2.4. Spin- -Additional Energy Levels

The spectrum of the hydrogen atom, when looked at with sufficient

precision, is found to have some significant disagreements with the

Bohr model. These discrepancies take the form of shifts in energies

of the various levels. Looked at on an even finer scale, it is found

that nearly all of the lines split into two lines so that there are more

levels in the actual spectrum of hydrogen than Bohr predicted. For a time it appeared that if Bohr's theory were generalized to include both

elliptical orbits and Einstein's theory of special relativity (the

Bohr-Sommerfeld theory), the shifts (but not the extra lines) could be

accounted for. Later, however, it became clear that there were

inconsistencies in this approach. Then, in 1925, Uhlenbeck and

Goudsmit showed that the observed shifts in energy could be obtained

by assuming that electrons intrinsically possess angular momentum and

a magnetic moment. In 1924, W. Pauli had suggested that the

properties of intrinsic angular momentum and magnetic moment be

attributed to the nucleus of an atom as a possible means of explaining

some other of the details of atomic spectroscopy. It is now commonly

accepted that these two properties are as much a part of the intrinsic

character of electrons and nuclei as are those of mass and charge.

The two properties of intrinsic angular momentum and magnetic moment

are inseparable and are often referred to by the single name-spin.

13
The existence of the spin properties of the nucleus and electron

means that there are additional forces between these two particles for

which Bohr's theory did not account. Thus, in the more detailed

theory, the energy level structure is modified. For our purpose, the

result of all this is that there are additional energy levels, some of

which have a separation corresponding to microwave frequencies.

3. THE INTERACTION BETWEEN ATOMS AND

ELECTROMAGNETIC RADIATION

Section 2 discussed the idea that any given atom can only exist in

certain discrete states and that each state has a definite energy. If an

atom is not able to interact with other atoms it can only change its

state by absorbing or emitting electromagnetic (EM) energy. This

absorption (or emission) is the means whereby we know that the energy

level structure exists. In the case of atomic frequency standards the

EM radiation which causes these transitions is directly related to the

output of the standard. It is the purpose of this section to show that

the probability of making these desired transitions depends upon the

frequency and upon the intensity of the EM field and that the atom must

have a non-zero magnetic moment. In this paper I consider only

elemental atoms and not molecules. If the output frequency is to be

well defined, it is also necessary to have the atoms interact with the

EM field for a fairly long time.

3. 1. Maxwell's Theory of Electromagnetic Radiation

The earliest work in spectroscopy involved looking at the various

colors of light that were emitted and absorbed by heated solids, liquids,

and gases. The detailed work in this field had begun by the mid-18th

century. At this time, light was generally held to be corpuscular in

14
nature, i. e. , not a wave. By the first part of the 19th century,

however, the tables had been turned in favor of a wave theory of light.

This was due in major part to the work of Thomas Young and Augustin

Fresnel in studying the phenomena of diffraction, interference, and

polarization. The theoretical work of Maxwell and the experimental

work of Heinrich Hertz near the end of the 19th century seemed to give

the final, irrefutable evidence in favor of a wave theory as opposed to

a corpuscular theory of light.

By the mid- 19th century the most important facts of some two

centuries of study of electricity and of magnetism were embodied in

four equations. It was Maxwell's remarkable contribution to realize

that if one of these four relations (Ampere's law) were modified it would resolve a problem in understanding the conservation of charge

and it would allow the four equations to be combined to form wave

equations for both the electric and magnetic fields. That is to say,

these equations predicted EM radiation. It was only some five years

later, in 1888, that Hertz conducted experiments which verified

Maxwell's predictions including the prediction that the wave would

propagate with the speed of light. The obvious conclusion is that light

is an electromagnetic wave!

Maxwell's theory of the EM field says that the field's intensity can

vary continuously from zero to any arbitrary value. In Section 2. 1. 4,

however, Einstein's studies of the photoelectric effect were discussed,

and his conclusion was that the energy of a light wave is quantized.

(Actually, EM radiation has both particle-like and wave-like properties.

In most experimental situations, however, one aspect predominates

over the other. ) Although the particle -like aspect of radiation was

important to Bohr in his theory of the atom, for most of our purposes

we need only to make use of the fact that the radiation that causes

15
transitions is electromagnetic in nature and to treat its energy as continuously variable. (But see the discussion on particle detection

in 5. 1. 1. )

3. 2. The Dependence of Atomic Transitions

upon EM Radiation

The dependence of the transition probability upon the EM field was

first studied by Einstein. This work (which was a further examination

of black-body radiation) assumed that there were two mechanisms for

emission of radiation- -induced and spontaneous- -and, also, that

absorption, like induced emission, depended on the strength of the EM

field and the duration of the interaction. His calculations showed that

the probability per atom per unit of time for induced emission B , was

nn'

equal to that for absorption B , n'n

B__, = B . . (3.1)

This is a very important fact and will be discussed further in 4. 2. He

also found that the coefficient for spontaneous emission A , was

8771/ h

related to the coefficient B , by A , = s— B . . In the microwave

8w3 h

c region (say 10 hertz) g— is a very small quantity, and hence

c spontaneous emission--which causes noise--has, to date, not been a problem in atomic frequency standards. (Noise is any disturbance that

tends to obscure the desired signal. ) It is, however, an important

factor in the infrared and visible radiation regions, as, for example,

in state selection for the rubidium gas cell (see Section 5. 4).

A quantum mechanical (QM) analysis gives similar results. The

result that the induced emission and absorption probabilities (per atom)

are identical is obtained in both cases. In the QM case these

16
probabilities again depend on the intensity of the EM field and upon the

duration of interaction, but because it can be more detailed, the QM

analysis produces some new results which are of particular use in

atomic frequency standards.

In the Einstein calculation, no explicit dependence upon the

frequency V is obtained for the coefficient B ,, but the QM result

nn1 does have an explicit frequency dependence. In atomic frequency

standards the atoms are in a gaseous state at very low pressure when

interacting with the EM field. To a very good approximation, we need

only consider the interaction with a free atom, and, if the QM analysis

is done for this case, the following is obtained:

P , = sin2 sin2 ~. (3. 2)

nn 1 2

P , is the probability of making a transition from state n to n

1 after

nn' a time interval T if the atom was in state n at the beginning of the

interval. The quantity sin 9 is defined as -2b/a, where

a = [(£2 - £2) 2 + (2b) 2

] 1/s

. Here, = ~L (E - E ,), and b is the

o ohn n' product of the magnetic moment of the atom and the strength of the EM

field which is exciting it. The symbol Q, is defined as 2771^. One of the

things this result shows is that for P , to be non-zero, the atom must

nn 1 have a non-zero magnetic moment when in either state n or n'.

In figure 2, P .as given by (3. 2), is plotted versus (Cl - Cl )/(2b)

nn o

for several cases. The dashed curve shows that if the radiation

frequency is equal to the frequency difference associated with the two

277 levels, i. e. , ZTTV = CI = (E - E ,)-— , then- -after the time interval

o n n

1 h

T = Tr/ (2b) --the atom is sure to have made the transition between state

n and state n'. The dashed curve assumes that the EM field is

interacting with a single atom or a group of atoms all moving with the

same velocity.

17
In a real device the atoms have different velocities; the solid curve

shows the result of averaging eq (3. 2) over a Maxwelliam velocitydistribution. This curve applies to the specific case of the interaction

time T being equal to 1. 2777 (2b). This is the interaction time for which

the transition probability will be largest. The dotted curve gives the

transition probability for T much greater than 77/ (2b). The basic conclusion from eq (3. 2) and figure 2 is that for a given value of the

interaction time, the transition probability will be maximized for

£1 = Qi and that this maximum value depends upon both the magnetic

moment of the atom and the strength of the EM field.

3. 3, Transition Probability and Linewidth

of Actual Frequency Standards

Until now I have been emphasizing the maximum value of the

transition probability curve --its value when 0, = Cl . The extent to

which this curve spreads out is also important, and I will now discuss

this point.

The purpose of a frequency standard is to provide an output which

is as nearly a single frequency as is possible. There are certain

features of the real world that inevitably cause the output of an actual

standard to be less than perfect. The non-zero width of the transition

probability curve, as displayed in figure 2, is an important example of

one of these degrading features. In an active device- -a device which

generates its own EM field- -the meaning of this non-zero width is that

the device emits observable energy over a range of frequency. [The

linewidth of a resonance curve is usually defined to be the difference

(in frequency) between those two frequencies at which the intensity is

one-half its peak value. ] The intensity of the undesired energy falls

off rapidly as its frequency deviates from V by more than one linewidth.

18
A passive device--a device where the EM field is supplied by an

external source- -also has a non-zero linewidth.

We are now in a position to see two important requirements of a high-quality frequency standard. First, the peak value of the transition

probability must be high enough that the signal can be seen above the

noise, and, second, the interaction time must be long enough to make the

linewidth sufficiently narrow. In Section 5, the means by which these

two goals are achieved will be discussed.

In the case of atomic frequency standards, there is a useful

relationship between the linewidth, W, of the frequency response and

the interaction time 7. For example, in an ideal cesium beam

(mono-velocity beam) using "two-cavity" excitation (see item 4 of

Section 5. 3), W and T are related by

Wr = 1/2. (3. 3)

For other high quality atomic frequency standards, the Wr product is

also of the order of unity. Equation (3. 3) makes the importance of the

parameter r obvious: The linewidth of the atomic resonance improves

linearly with increasing t.

4. CHOOSING A SUITABLE ATOM

Sections 2 and 3 discussed two fundamental facts of physics,

atomic energy levels and the interaction of atoms with electromagnetic

fields. These facts are the basis for atomic frequency standards. We

need now to decide how to choose a suitable atom.

19
4. 1, The Energy Levels of a Many-Electron Atom

The purpose of this subsection is to describe the energy level

structure of atoms with more than one electron. In order to

understand this structure it is useful to consider a basic experimental

result of spectroscopy.

For simplicity's sake, consider the simplest of atoms- -hydrogen.

If a gas made up of hydrogen atoms is heated to a high temperature,

then it will emit EM energy. This energy is emitted at discrete frequencies. In the range of visible light, the standard means of

observing these radiations is to use an optical spectrograph. This

device causes the energy at the various frequencies to be split up

spatially so that if it is directed onto a strip of photographic film, the

developed film shows a group of lines known as the Balmer Series.

These are so labeled in figure 3. The response of the photographic

film is limited to visible light but if it could respond to higher and to

lower frequencies, additional groups of lines, which are also shown in

figure 3, would result. For any given group, it can be seen that the

lines become very closely spaced towards their high frequency (short

wavelength) end. The terminal frequency for each group is indicated

by dashed lines in the figure. The dashed line for a given group is

called the series limit for that group. The importance of the series

limit is that it tells us about a particular energy level which is common

to every line in the group. Now, the emission spectra of gases of any

other kind of atom are fundamentally the same as that of hydrogen.

This basic characteristic is often hard to see in the other atoms

because there are more energy levels involved and sometimes the

various series (groups) overlap. Nevertheless the spectra of any atom

is composed of an infinite number of these series (groups) of lines.

Because of this, the frequency of any line of any series in any atom can

20
be obtained from an equation of the same form as eq (2. 2). The

equation has the general form:

V = T

2

- T

r

(4. 1)

T and T n are called spectroscopic terms. T is the generalized

2 1 f f2 6

common term of a given series of lines.

As spectroscopists began to look more closely at the spectrum of

atomic hydrogen, they found that most of the lines of the spectrum,

which had previously appeared to be single, were actually double.

Before having discovered these double lines, they had found that to

classify all the lines in the spectrum unambiguously, they needed two

(not one) quantum numbers, n and I. Now, with double the number of

lines, four quantum numbers were necessary. The extra numbers that

are needed are s, the quantum number of the spin of the electron, and

j, the quantum number of the total angular momentum of the electron.

The spectra of all other atoms also require four quantum numbers to

characterize each line (energy level). (I should state here that I am

temporarily ignoring the extra complication of the interaction of the

atom with an externally applied dc field. ) At first these four numbers

were just an empirical scheme for specifying the levels, but in time

they were given meaning in terms of the structure of the atom. The

labels used are the principle number n- -which is analogous to the n of

the Bohr theory; the total orbital angular momentum number L; the

total electron spin number S and the total angular momentum number J.

In atoms with more than one electron, there is not only the

electrostatic force between an electron and the nucleus, and the force

between the orbital and intrinsic moments of an electron, but also the

electrostatic and magnetic forces between electrons. These forces

21
determine the energies and therefore the frequencies corresponding to

the possible transitions between states. An important reason for using

n, L, S, and J in labeling the levels is that each of these quantities is

usually important in characterizing the energies of the levels. The

quantity L. is formed by means of certain quantum mechanical rules

from the &

x s of the individual electrons. The quantity S is formed in a similar manner. The number J is then formed from L and S, again,

using the appropriate quantum mechanical rules. Now, you may ask,

"Why were all the l

l s combined to form L and s's to form S, why not

combine the individual £

l s and s's to form j's (the total angular momentum for individual electrons) or even some intermediate scheme?" The

method used here implies that the coupling of the individual orbital

momenta and the individual intrinsic momenta is very strong with

respect to the coupling of the orbital and intrinsic momenta of a given

electron. This is commonly called Russell-Saunders coupling and

usually applies to the ground state of any atom; this is the state of

interest to us. To gain insight into the quantum number n, and as a general aid in

understanding the energy level structure of an atom, the Pauli

Exclusion Principle is extremely useful.

One of the most prominent features of nature is the periodic

behavior of the elements. This periodicity expresses itself in the

emission (and absorption) spectra we have been discussing and in the

chemical behavior of the elements as first summarized in 1896 by

Dimitri Mendeleev. The periodicity is explainable by the Pauli

Principle. This principle is to be taken as a postulate, as fundamental

as the concepts of mass and charge. Wolfgang Pauli, in 1925, proposed this principle as a hypothesis based upon his observations of the

spectra of the alkali metals. The principle says: "In an atom there

22
never are two electrons having the same group of four quantum

numbers. " The numbers referred to are n, i, j, and m.. (The values

of m. are the projections of j along a dc magnetic field. ) In his book

on quantum mechanics, Persico [6] says: "It may be said that the

entire theoretical interpretation of spectra is based upon this principle

and constitutes an impressive confirmation thereof. " Using Pauli's

Principle as a theoretical guide, calculations show that, roughly

speaking, the electrons gather into localized groups called shells.

Figure 4 displays this result. These shells are analogous to the orbits

of the Bohr theory and are labeled by the quantum number n. To be

rigorous about the interpretation of figure 4 would require a much

deeper discussion of quantum mechanics than the purpose of this

article allows. The point, however, is that the Pauli Principle determines the number of electrons that can be located in each shell. The

letters K, L, and M are the traditional labels of the first three shells.

Another noteworthy thing observable in this figure is that the radius

of the K shell (and the L shell) decreases markedly as we go to atoms

with more electrons. The reason for this is that the heavier atoms

have a nucleus with a larger charge and therefore the attraction on the

electrons in the inner shells is larger.

At this point the discussion can be considerably simplified by

admitting that we are not interested in the most general energy states

of an atom that could possibly occur but rather in those that have to do

with an atom's emission electrons. In atoms other than the noble

gases, there are one or more electrons which are much more easily

caused to go into an excited state than are the rest. These electrons

are called emission electrons because the emission spectra of an atom

is usually due to them. These emission electrons lie outside the filled

shells and, to a first approximation, the rest of the atom (the nucleus

23
and non-emission electrons) appears as a point charge to these

electrons. Now, in the usual case there is more than one emission

electron and the energy of these electrons depends not only upon their

quantum number n (they are usually in the same shell), but upon the

"sum", L, of the i-'s of the emission electrons and also upon their S

and J values. The values that L, S, and J can assume are restricted

by the Pauli Principle.

To sum up, any given term (energy level) is labeled by the four

quantum numbers, n, L, S, and J. Figure 5 is an energy level

diagram for the single emission electron of potassium. This figure

makes use of the quantum numbers. The number n identifies the shell

(see the discussion of the Pauli Principle for the meaning of the word

shell). The capital letter that heads each column represents the total

orbital angular momentum L. The letter S (not the quantum number S)

means an L value of 0; P means an L of 1; D means an L of 2; and F

means an L of 3. The superscript is equal to 2S + 1. (I am now talking

of the quantum number S. ) For potassium, S = —, therefore, 2S + 1 = 2.

1 The subscript is the J value of the energy 11level. For S = — there are

only two possible values for J, L + — and L - — (except that for L = 0,

1 J can only be equal to — ). For small values of n there is an observable vertical separation

2 2

(an energy difference) between the P , and P . states. The amount

of separation betweel levels with different J value depends upon L as

2

well as upon n, and it can be seen that levels D_ ,_ _ ._ and levels

2 5/2, 3/2

F„ . _.„ are so close in energy that they can't be distinguished on

7/2,5/2 &y y &

this plot. Thus, they are not separately displayed. From figure 5, it

is not apparent that it is necessary to specify S. But in spectra where

S also changes, then the need for the extra quantum number becomes

apparent. Note that the ground electronic state of the emission electron

2

is 4 S, i where the 4 represents the fourth shell.

24
4. 2. The Hyperfine Interaction

The magnetic interaction, between the nuclear spin- -which is

symbolized by the letter I--and the spin of an electron is called a hyperfine interaction. It is this interaction which causes the desired

spacing between the pair of energy levels used in several important

frequency standards. Figure 6(a) shows how the ground electronic

2

state S 1 J of the hydrogen atom is split into two states by the hyperi. I L. fine interaction. One of these two states (the F = 1 state) is then

further split by the dc magnetic field, H. This figure also shows

the dependence of the energies of each level upon H.

The letter F labeling the ordinate in figure 6(a) represents the

quantum number of the total angular momentum of the entire atom.

This number is formed from J and I just as J is formed from L and S. That is, F = J + I and F . = J - l|. Every integer value in

max mm '

' between F and F . is also allowed. The nuclear spin has only max mm

one possible value, and, because we are interested only in the S 1

,

state, J has only the one value, 1/2. Thus, for the hydrogen atom,

which has an I value of 1/2, there are only two possibilities for F.

Here, F =1 and F . = 0, and no intermediate values are possible max mm

(see fig. 6a). I have restricted the atom to be in the S. , electronic

state for reasons to be discussed in Section 4. 3.

To deal with an external magnetic field we need yet another

quantum number m . This new number refers to the possible pro- F

jections of F along the direction of H. Quantum mechanics allows

F =m = -F. Therefore, for F = 1, the possible values of m are 1, F F

0, and -1. For the hydrogen atom, figure 7(a) shows that the two

energy levels between which we want transitions to take place are the

F = 1, m = 0, and F = 0, m = 0. This is so because these are the

i F

levels which have the least dependence on H for low values of the

25
field (see item (f) of Section 4. 3). [This transition is sometimes

symbolized by F, m =1,0 «— 0, 0. ] F

4. 3. Criteria for Choosing an Atom

I can now list those characteristics that are desired in an atom for

the purpose of a frequency standard and then, based on the Pauli

Principle and the spectroscopic information discussed above, decide

which atoms are most suitable.

The following are criteria for choosing a suitable atom:

(a) There should be a high probability of finding the

atom in the desired state.

(b) The transition frequency should be in a useful range.

(c) The magnetic moment should be large enough for a useful interaction with the EM field and also for the

purpose of state selection.

(d) The atom should be quite inert to interaction with

like atoms or with any container in which it is

stored.

(e) The atom should have a low ionization potential if

the transition is to be sensed by means of ionized

particle detection.

(f) The energy levels between which the desired

transition occurs should have as little dependence as

possible upon H.

As each of these items is discussed it will become apparent that

several of these requirements are mutually contradictory, and a compromise has to be reached.

Item (a) basically requires that the desired transition be due to the

ground electronic state of the emission electron and that the first

26
excited state of this electron should be significantly higher in energy.

Hydrogen, rubidium, and cesium are the three atoms that have seen

most use for frequency standards, and one reason is that the ground

electronic state in each case is S . . This type of state has, in

general, the highest possible separation from the first excited state.

The question of what is a useful range for the transition frequency--

item (b)--involves many things. The one aspect of this question which

properly belongs in this article was discussed above. This subject is

the hyperfine interaction. The reader should be reminded, however,

that a study of atomic frequency standards is incomplete if these other

topics are not considered. For example, in masers, the absolute

value of the transition frequency influences the ease of detecting the

maser output. In passive devices, such as the cesium and rubidium

machines, the spectral purity of the external microwave signal depends

upon the transition frequency. For a discussion of these and other

points, see reference 7. When all the considerations are taken into

account, it turns out that a transition frequency lying in the microwave

range is quite desirable. The hyperfine interaction often produces

energy level spacings in this region.

The importance of the magnetic moment of the atom- -item (c)--was

discussed in Section 3.2. It was pointed out there that if a frequency

standard is based upon a magnetic interaction, and the highest quality

devices are (I continue to restrict myself to devices based on atoms

and not molecules, but see the summary), then the net magnetic

moment must be non-zero. State selection, to be discussed in Section

4. 4, also requires that the net magnetic moment be non-zero. In the

most general case, the net moment is made up of contributions from

the orbital and spin magnetic moments of the electrons and from the

magnetic moment of the nucleus. The hyperfine interaction (which

27
determines the transition frequency) is crucially dependent upon the

nuclear moment, but, because the nuclear moment is about 1000 times

smaller than that due to the electrons, it can be ignored in calculating

the net magnetic moment. If the net magnetic moment is too large,

then the interaction of the atoms with themselves and with their container will result in broadening the linewidth, W, and in a frequency

shift. It turns out that the parameters involved in practical frequency

2

standards are such that an atom in the S . state (hydrogen and the

alkali metals, Li, Na, K, Rb, Cs, and Fr) will satisfy the magnetic

moment requirements. This is fortunate because an atom in this state

also best fits the requirement of item (a).

It can be seen that the desirability of being inert —item (d)--is in

conflict with item (c). The noble gases best fit the inert requirement,

but, because their ground state is S , they have zero net magnetic

moment. Atoms whose ground state is S , (such as H, Rb, and Cs)

form a reasonable compromise because they satisfy items (a), (b), and

(c) and yet have a very small magnetic moment.

But the alkali metals are surely not chemically inert- -they are

among the most chemically active of all of the elements --and a large

price is paid in the accuracy of the hydrogen maser (see Section 5. 2)

because of this.

Item (e) points up again the conflict between the various requirements. One way of sensing that the desired transition is occurring is

to collect the atoms which have made the transition. This is done by

ionizing those atoms and collecting the charged particles thus

generated. If an atom is to be ionized efficiently then its ionization

potential must be fairly low. But a low ionization potential guarantees

high chemical activity so, again, there is a conflict. The alkali metals

28
all have low ionization potentials, and cesium is particularly good in

this regard.

An external dc magnetic field--item (f)--is another source of

frequency shift. For example, even using the F = 1, m = and the

F

F = 0, m = energy level pair in hydrogen, several layers of

F

magnetic shielding are required if this shift is to be reduced to a tolerable level.

4. 4. The Need for State Selection

In order for any atomic frequency standard to work, it is

necessary to have a difference in population between the two energy

levels of interest. That is, there need to be more atoms in one energy

level than in the other. The reason for this is that the absorption

probability, B , . is equal to the induced emission probability, B , n'n nn1 (see Section 3. 2). Thus, if there are the same number of atoms in the

upper and lower states, then energy is absorbed by atoms in the lower

state as fast as it is emitted by those in the upper state. In a maser,

there must be more atoms in the upper than in the lower state. It is

this excess in the upper state that provides the energy for the emission

of EM energy. In the passive devices, a population imbalance is required because, if the upper and lower states are equally populated,

there is no net response to the externally applied radiation.

About I860, James Clerk Maxwell and Ludwig Boltzmann

considered the problem of the relative population of the energy levels

of a system with many possible levels. Equation (4. 2) gives the result

applicable to our case.

n

—j- = exp

n

1

(E - E ,) n n*

kT (4.2)

29
Here, n is the number of atoms in the upper energy state whose

energy is E , and n1 is the number in the lower state whose energy is

E .. The absolute temperature is given by T, and k is Boltzmann's

constant.

Eor hydrogen, the energy difference between the upper state

(F = 1, m = 0) and the lower state (F = 0, m = 0) is so small that F F

the ratio n/n1 is greater than 0. 99. That is to say, for all practical

purposes, there are the same number of atoms in each state. Under

these conditions the maser will not operate. The atoms in the lower

state must be removed from the beam of incoming atoms. (See Section

5.2.)

Any transition in the microwave region -will have n/n'«^ 1, and so

state selection will also be required in the cesium beam and rubidium

gas cell.

5. THE THREE MAJOR EXAMPLES OF

ATOMIC FREQUENCY STANDARDS

5. 1 8 General Considerations

The prime emphasis of this section will be upon the relationship

of the major parts of each device to the physical principles discussed

in the preceding sections. Some comment will be made on the performance characteristics of the devices, but a detailed discussion is

beyond the scope of this article.

5. 1. 1 Common Functions

There are three functions that each device must perform. First,

the atoms must be in the proper condition to emit EM energy (active

device) or to be responsive to an external source of EM energy (passive

device). Second, the atoms must be kept in interaction with the EM

30
field long enough that the transition linewidth is acceptably narrow.

(The EM field configuration must, of course, be such that an interaction with the atoms can occur. ) Third, some means of detecting

that transitions are occurring must be provided.

For the hydrogen, cesium, and rubidium devices, the transition

frequencies are in the microwave region, and, therefore, to place the

atoms in a proper condition, state selection is necessary. These ideas

are discussed in Section 4. 4.

The means by which the H, Cs, and Rb devices achieve long

interaction times are clearly evident in their mechanical construction,

and this construction has a strong effect on their performance characteristics.

Of the three devices, the hydrogen maser is the only active one,

and the fact of its oscillation is evidence that transitions are occurring.

If we define the word particle in a general sense, then particle

detection describes the means by which transitions are detected in

each of the other two devices. The collection of ionized atoms, as discussed in Section 4. 3 (item e), is the usual and obvious example of

particle detection. But, as mentioned in Section 2. 1. 4, electromagnetic energy has a particle-like as well as a wave-like character.

When the frequency of the EM radiation is high enough then the

radiation can be detected by means of a photocell. The radiation is

collected at the detector as discrete events. That is, instead of the

radiation appearing to be continuously there- -as, in fact, it appears

to be, at lower frequencies --it occurs in bursts. In this sense, it behaves as we normally think of particles as behaving. When the

particle-like behavior of the EM field is prominent, it is common to

refer to it as a photon field and to an individual event at the detector as

the detection of a photon. In the rubidium gas cell, photons are detected

as evidence for the occurrence of transitions.

31
5. 1. 2 Primary versus Secondary Standards

The ideal transition frequency of a frequency standard is that

frequency which its atoms would have if they were completely removed

from all surroundings including interactions with each other. This is

called the free atom frequency. But, in an actual device, the atoms

interact with each other, and with the rest of their environment, and

the resulting transition frequency is displaced from the free atom

frequency. From both experimental and theoretical studies, some

(and hopefully all) of the factors which cause significant disturbances

can be identified. Furthermore, by means of experiment and theory,

quantitative estimates of each disturbance can be obtained. But these

estimates are just that, estimates, and there is an uncertainty associated -with each of them. Those standards -whose net uncertainty is

the smallest—whose accuracy is highest —are often called primary

standards. Those other standards whose precision (i. e. , stability in

time) is also high but whose accuracy is inferior are often called

secondary standards.

5. 2. The Hydrogen Maser

Figure 6(b) is a simplified schematic of the hydrogen-atom maser.

It has six fundamental parts:

(1) a vacuum envelope,

(2) a source of hydrogen atoms,

(3) a state selecting magnet,

(4) a microwave cavity,

(5) a storage bulb, and

(6) a magnetic shield with an associated "C" field coil.

A detailed discussion of these six parts follows:

(1) In this device, as in all of the devices, the gaseous environment of the desired atoms (in this case, hydrogen) must be controlled.

32
If a hydrogen atom in the desired F = 1, rri = state collides with a F

foreign atom (such as an oxygen atom), then there is a good chance that

it will decay to the F = , m = state without having made the desired

contribution to the field in the cavity

(2) Hydrogen enters the source chamber in the form of H

molecules. By means of electrodes placed across the source, the

molecules are dissociated into atoms. A small diameter, elongated

hole at the top of the source collimates the atoms into a beam as they

escape from the source.

(3) When the atoms leave the source they are nearly equally

distributed among the 4 states shown in figure 6(a). The state selecting

magnet removes most of the F = 0, m = (and F = 1, m = -1) atoms F F

so that the beam entering the storage bulb is in a condition for a net

emission of energy. By considering figures 6(a) and 6(c), a simple

explanation of this type of state selection can be obtained.

Figure 6(c) shows an end view of a six-pole magnet. The beam of

atoms is directed roughly along the axis of the small hole in the center

of the magnet. The magnetic field intensity is zero on the axis of this

hole but is of the order of several thousand gauss at the pole tips.

From figure 6(a) it can be seen that for the states F = 1, m = -1 and F

F = 0, m = 0, the energy decreases with increasing H. Because of

the tendency for a system to take up the lowest possible energy configuration, the atoms in these states move towards the pole tips

thereby being defocused. By a similar argument, the other states are

focused. Only the F = 1, m = state is wanted in the storage bulb, F

since the other state can degrade the performance of the maser, but if

H is adjusted to a few times 10 ampere/meter (a few hundred microoersteds) then the effect of atoms in the wrong state can be kept to a tolerable level.

33
(4) Masers work on the principle of self- stimulated emission.

The purpose of the cavity is to obtain a higher field intensity- -and,

therefore, an increased transition probability- -from a small amount of

input power. The input power, which is supplied by an excess of atoms

in the F = 1, m = state, must be greater than the internal power F

losses. By internal losses I mean such things as escape of atoms

through the entrance of the storage bulb before they have made the

transition to the F = 0, m^ = state and decay to this state without F

radiating because of collisions with foreign atoms.

There is a minimum flux of atoms, in the correct state, needed in

order for any given maser to oscillate. Above this level a small

amount of energy is stored in the cavity in the form of EM energy in

the TE mode (the cavity is a circular cylinder). A small fraction

of this energy is coupled out and is the output of the maser. Its frequency is about 1420 MHz. (Actually, the usual output of the maser is

at a frequency of 5 MHz and is obtained from a quartz crystal oscillator which has been phase locked to the 1420 MHz signal. Thus, the

frequency stability of the 1420 MHz signal is partially transferred to

the 5 MHz signal. The details of this phase locking and the purpose of

using the quartz oscillator fall outside the scope of this article. (But

see the discussion of item b in Section 4. 3. )

(5) The storage bulb is usually a hollow, thin-walled, quartz

sphere about 0. 15 meters (about 6 inches) in diameter. It is coated on

its inside with polytetrafluoroethylene. There is a neck attached to the

sphere, and the incoming flux of atoms enters through a small hole

along the axis of this neck. On the average, the atoms are stored for

about one second within the bulb before they chance to escape out the

hole through which they entered.

34
This long storage time results in a very narrow linewidth for the

atomic resonance. It should be said in addition, that were it not for

this long storage time the maser would not oscillate- -the losses would

be too high. On the other hand, the bouncing of the atoms off the walls

perturbs the atoms with the result that (for a bulb 0. 15 meters in

diameter at room temperature) the frequency incurs a fractional shift

of about 2 parts in 10 . The fractional uncertainty in knowing this

12 shift (see Section 5. 1. 2) is about 2 parts in 10 and constitutes the

biggest single factor in the inaccuracy of the H maser [8],

(6) The intensity of the earth's magnetic field is typically about

40 ampere/meter. In addition, this field is spatially nonuniform, and

the nonuniformity would prevent the maser from oscillating. The magnetic shield (there are usually three layers of shielding) reduces the

intensity and improves the uniformity of H in the region of space

where the storage bulb is located. The "C" field coil produces a static

field opposing that due to the residual field of the earth. The "C" field

is typically adjusted to give a net H of a few times 10" " ampere/meter.

The magnitude of this field and, therefore, the resultant frequency

shift (see item f of Section 4. 3) can be measured quite accurately by

taking advantage of the other two F = 1 energy states. This is done by

equipping the cavity with a pair of Helmholtz coils which apply an audiofrequency field at right angles to H. (The microwave field, in the region

of the storage bulb, is parallel to H which is parallel to the axis of the

cavity. ) The audiofrequency field (whose frequency is typically a few

hundred hertz) causes transitions between the F = 1, m - level and

the other two F = 1 levels. These transitions are detected as a decrease in the output power of the maser, and the frequency

necessary to cause these transitions depends on the magnitude of H.

35
The accuracy of this measurement is sufficiently high that the error

in the maser frequency due to this source is much less than that due to

the wall shift discussed above.

The uncertainties in determining the perturbations to the free H

atom frequency are small enough that the H maser is used in several

of the world's laboratories as a primary standard.

5. 3. The Cesium Beam

Figure 7 shows a schematic diagrain of the cesium beam. It consists of eight major parts:

(1) a vacuum envelope,

(2) a source of cesium atoms,

(3) an "A" magnet,

(4) a microwave cavity,

(5) a "C" field region,

(6) a "B" magnet,

(7) a detector, and

(8) an external source of microwave power.

A detailed discussion of these eight parts follows:

(1) The vacuum envelope performs the same function here as it does in the H maser.

(2) Because, at room temperature, the rate of evaporation of

cesium is too low, the source is a chamber which is heated to about

373 kelvin (100 degrees centigrade). At this temperature, a sufficient

number of cesium atoms diffuse through a small, elongated hole and

are collimated as in the H maser.

(3) The purpose of the "A" magnet is to select certain atoms for

exposure to the EM field. The desired atoms are those which must

undergo a transition in the EM field in order that the "B" magnet later

36
will focus them onto the detector. The basic purpose of the "A" and

"B" magnets is to create a condition wherein it is possible to determine if the frequency of the microwave source is equal to the transition

frequency. In the cesium beam, just as in the H maser, the two states

between which transitions are to take place are essentially of equal

population. Thus, the cesium beam also requires state selection but

here it does not matter whether the atoms are primarily in the upper

state or primarily in the lower state because B , = B f (see Section

nn1 n'n

3. 2). (In fact, it turns out that if the state selecting magnet is of the

dipolar type, atoms in both states can be used. )

(4) The cavity serves four purposes. First, it enhances the

strength of the EM field for a given amount of input. This input is

supplied in the form of microwave energy from an external source.

Second, it results in a fairly long interaction time with the EM field.

This gives the narrow linewidth which is desired. Third, it defines a

definite region of interaction and fixes the direction of the EM field

with respect to the "C" field. Fourth, it allows the atoms to experience

only a standing- wave EM field (any running -wave component could, if

not transverse to the cesium atoms motion, cause a first-order Doppler

shift).

Combined with a fairly large value of the interaction time T, the

field enhancing effect of the cavity easily permits an optimum value of

the product bT to be obtained. (See Section 3. 2. ) Figure 7 shows a horseshoe- shaped cavity. The beam passes first through one end of the

horseshoe and then, after a time T, through the other. The reason for

using the horseshoe cavity is that the atoms behave somewhat as if they

were in a single, very large cavity. This increases the interaction time.

If this scheme is to work perfectly then the EM fields at either end of the

horseshoe must be exactly in phase -with one another. This condition is

difficult to achieve, and the failure to do it perfectly results in the

largest single inaccuracy in many cesium beams.

37
(5) The two states between which transitions are desired are the

F = 4, m = and the F = 3, m = 0. (To be specific, I am talking F F

about Csl33 whose nuclear moment is 7/2. There are actually nine

F = 4 states and seven F = 3 states, and they all have different energies

in the presence of a non-zero H. See Section 4. 3. ) These are the

two states with the least dependence on H. (See item f of Section

4. 3. ) But, just as in the H maser, the dependence is strong enough

that, in the EM field region, the atoms must be shielded from the

earth's field. Again, as in the H maser, a second function of the

shielding is to cause the residual field of the earth to be spatially

uniform in the region of interaction with the EM field. This is a particularly severe requirement here because the field must be uniform

throughout the entire path of the atoms between the two cavities. If it

is not, an error is introduced which is equivalent to that due to

inequality in phase between the two ends of the cavity.

(6) The "B" magnet can be identical to the "A" magnet. Its

function is to focus, onto the detector, those atoms which have made a transition and to de-focus those which have not.

(7) Those atoms which have made the desired transition impinge

on the collector and are ionized. The detector, in laboratory cesium

beams, is usually made of tungsten or a platinum -iridium alloy. When

the detector is heated to about 1200 kelvin, almost all of the atoms

hitting the collector are ionized.

(8) The external source of microwave energy in a cesium beam

is usually supplied by a quartz crystal oscillator whose frequency has

been multiplied up to the cesium transition frequency- -about 9192 MHz.

As in the H maser, the usual output frequency is not in the microwave

region but rather at 5 MHz. The quartz oscillator takes on the longterm stability of the cesium transition by being servoed to it. This is

38
done by automatically adjusting the frequency of the external source to,

in effect, maximize the number of atoms collected at the detector.

Those parameters which, in the cesium beam, alter the transition

frequency from its free atom value have been thoroughly studied [9].

The uncertainties in their determination are quite small, and the

cesium beam is used as a primary standard in many laboratories. The

high accuracy capability resulted (in October of 1964) in the declaration of the International Committee of Weights and Measures that

the physical measurement of time be based on the F, m_ = 4, <— 3, F

13 3 transition in Cs . The world's highest quality cesium beams--such

as NBS-III at the National Bureau of Standards in Boulder, Colorado;

the machine at the National Research Council of Canada; and the

machine at the Physikalisch-Technische Bundesanstalt in Germany--

have an evaluated accuracy (one sigma) of about 5 parts in 10 . This

is a greater proven accuracy than for any other type of device known

[9].

5. 4. The Rubidium Gas Cell

A schematic diagram of the rubidium gas cell is given in figure 8,

It has five major parts:

87

(1) a gas cell containing Rb gas,

87

(2) a Rb light source (and associated filter) for state selection,

(3) a microwave cavity,

(4) an external source of microwave energy, and

(5) a photocell detector.

A detailed discussion of these parts follows:

87 2

(1) Rb atoms at a partial pressure of about 10" newtons/meter

(about 10 Torr) are contained in an optically transparent cell. The

two levels of interest are the F = 2, m^ = and F = 1, m^ = states

r F

87

of the ground electronic state. (Rb has a nuclear moment of 3/2,

39
2 and the ground electronic state is S , ; therefore, the two possible

values of F are 2 and 1. ) The transition frequency is about 683 5 MHz.

87

In addition to Rb , the cell contains a buffer gas (which is usually

a mixture of neon and helium, at a partial pressure somewhere

between 1 X 10 and 2 X 10 newtons/meter ). This buffer gas performs for the rubidium cell the same function that the storage bulb

performs for the H maser and the horseshoe cavity performs for the

cesium beam: It increases, by several orders of magnitude, the

interaction time with the EM field, thereby resulting in a narrower

linewidth.

Unlike the H maser and the cesium beam, the atoms are used over

and over again. This is possible because state selection is done within

the gas cell itself.

(2). The method of state selection employed here is called optical

pumping. A simplified explanation of this technique can be obtained

from figure 9. This technique involves, in essence, three energy

levels. In figure 9, the levels labeled B and C are, respectively, the

F = 2, m_ = and the F = 1, m =0 levels between which we want

F F

transitions to occur. Level A is a level which is energetically far

removed from levels B and C. The frequency corresponding to the

spacing between level A and the other two levels is about 3.8x10 Hz.

This means that almost all of the atoms will be equally distributed

between states B and C, and hardly any will be in state A. (See

Section 4. 4. ) Before the light is turned on, almost all of the atoms are in levels

B and C. This is shown in part 1 of the figure. When the light is

turned on, many of the atoms absorb energy from the light beam and

are excited to level A. This is shown in part 2 of the figure. But, for

atoms in state A, it is very likely they will spontaneously decay to

40
either state B or C (see Section 3. 2). Decay to states B and C is about

equally probable. Part 3 of the figure shows the situation. Because

the spontaneous emission probability from state B to C is very low,

the net result of this pumping is that the number of atoms in state B is

built up at the expense of state C. Thus, the population imbalance required so that the atoms will be responsive to the external source has

been achieved. The reader should note that the rubidium cell is a passive device and that an excess population in state B is not

required- -an excess in state C would work just as well. The condition

is as it is because of the state selecting method.

(3) The purpose of the microwave cavity is field enhancement.

(4) The external microwave energy is again supplied by a quartz

oscillator and a frequency multiplier chain. If the external source is

at the proper frequency, some of the atoms in state B make transitions

to state C. This is detected at the photocell as a decrease in the

received light. By servoing the frequency of the quartz oscillator to

minimize the light at the photocell, the quartz oscillator is kept on

frequency.

(5) There is a great advantage to using a photocell to detect the

fact that transitions are occurring. The reason is that each photon at

the light frequency has a great deal of energy and a small change in the

number of photons arriving at the detector is quite noticeable.

The buffer gas used in the gas cell causes the transition linewidth

to be quite narrow because it increases the interaction time of the

atoms with the EM field. Unfortunately, it also produces a very large

shift in frequency, just as collisions with the Teflon coating of the

storage bulb produce a net frequency shift in the H maser. The uncertainty in the determination of the shift due to the buffer gas is quite

large, and, consequently, the rubidium cell serves only as a secondary

standard.

41
6. SUMMARY

The reason for the use of atoms for frequency standards is that

they have well defined energy states* In an atom which is useful as a frequency standard, the desired energy level pair is quite insensitive

to the atom's environment, and, hence, the energy difference between

this pair has the required high stability.

The energy level pair is used by observing transitions from one

level of the pair to the other. In an active type of device--a maser--

the atoms emit electromagnetic (EM) energy by making the transition

from the upper to the lower of the two states. In a passive device--such

as a cesium beam--an external source of EM energy causes transitions.

Devices are built where the transitions are from the lower to the upper

state or vice versa; in fact, devices are sometimes built where both

processes are used simultaneously. Be the device active or passive,

the central point is that it is the EM field which causes transitions.

There are three basic functions which any atomic frequency

standard must perform. First, the atoms must be put in a condition

in which they will emit EM energy (active device) or be responsive to

an external source of EM energy. The most commonly used transitions

are in the microwave region, and, for those transitions, state

selection is required. Second, the atoms must be kept in interaction

with the EM field long enough that the transition linewidth is acceptably

narrow. The means by which this long interaction time is achieved in

the H, Cs, and Rb devices has a detrimental side effect upon the

accuracy of each of the devices. Third, some means of detecting that

transitions are occurring must be provided. In the hydrogen maser,

its oscillation is evidence for the occurrence of transitions. Given the

broad definition of particles discussed in the text, the Cs and Rb

devices sense transitions by particle detection.

42
For a device to adequately perform these three functions, the type

of atom which is used must be carefully selected. A list of criteria

for choosing an atom was given, and it was seen that several of these

criteria tended to conflict. An atom whose electronic ground state is

2

S . is a good compromise in meeting these criteria, and the reasons

for this were given. All of the frequency standards described in this

article cause their atoms to be in a very rarified gaseous state when

interacting with the EM field. (The buffer gas in the Rb gas cell is at

a fairly high pressure. This gas, however, is not the working

substance; its purpose is the same as that of the Teflon coating in the

hydrogen maser. ) This is an important technique in achieving a stable

and accurate output frequency.

In this paper, I have confined the discussion to devices which use

atoms, rather than molecules, as their working substance. This was

done because these usually have been the more stable and accurate

devices. There is, however, at lease one case--the methane saturated

absorption cell—where the stability (and, perhaps, the accuracy

capability) is extremely high [10]. Even though molecules are used

rather than atoms, the physical basis is still a well defined energy

level pair and an appropriate interaction with the EM field.

The theory of atomic structure and of the interaction of an atom

with the EM field has become quite elaborate and abstract. Atomic

frequency standards provide concrete examples of a great many of

these ideas.

ACKNOWLEDGEMENTS

I want to thank Dr. Donald Halford, Dr. David Wait, and

Mr. Roger Beehler of the National Bureau of Standards, Boulder, for

many helpful criticisms of this paper.

43
REFERENCES

[1] Holton, G. , and Roller, D. H. D. , Foundations of Modern Physical

Science , p. 615 (Addison-Wesley Publishing Co. , Inc. , Reading,

Massachusetts, 1958).

[2] Ibid. , p. 581.

[3] Ibid. , p. 591.

[4] Oldenberg, O. , Introduction to Atomic Physics , p. 335 (McGrawHill Book Co. , Inc., New York, N. Y. , 1949).

[5] Holton, op. cit. , p. 651.

[6] Persico, E. , Fundamentals of Quantum Mechanics , translated by

Go M. Temmer, p. 454 (Prentice-Hall, Inc., New York, N. Y. , 1950).

[7] Cutler, L. S. , and Searle, C. L. , Some aspects of the theory and

measurement of frequency fluctuations in frequency standards,

Proc. IEEE 54, p. 136 (February 1966).

[8] Hellwig, H. , et al. , Measurement of the unperturbed hydrogen

hyperfine transition frequency, IEEE Trans, on Instrumentation

and Measurements IM- 19 , No. 4, pp. 200-209 (November 1970).

[9] Beehler, R. E. , Mockler, R. C, and Richardson, J. M. , Cesium beam atomic time and frequency standards, Metrologia 1,

p. 114 (July 1965).

[10] Barger, R. L. , and Hall, J. L. , Use of laser-saturated absorption

of methane for laser frequency stabilization, Proc. of the 23rd

Annual Symposium on Frequency Control, Fort Monmouth, N. J. , 6-8 May 1969, p. 306.

44
BIBLIOGRAPHY

1. Background

1. 1 Holton, G. , and Roller, D. H. D. , Foundations of Modern

Science (Addison-Wesley Publishing Co. , Inc. , Reading,

1.2 Oldenberg, O. , Introduction to Atomic Physics (McGrawHill Book Co. , Inc., New York, N. Y. , 1949).

1. 3 Persico, E. , Fundamentals of Quantum Mechanics , translated by G. M. Temmer (Prentice-Hall, Inc. , New

York, N. Y. , 1950).

1.4 Ramsey, N. F. , Molecular Beams (Oxford University Press,

Amen House, London E.C.4, 1956).

1.5 Herzberg, G. , Atomic Spectra and Atomic Structure , translated by J. W. T. Spinks (Dover Publications, New York,

N. Y. , 1944).

2. Devices

2. 1 Kleppner, D. , Goldenberg, H. , and Ramsey, N. , Theory of

the hydrogen maser, Phys. Rev. 126 , p. 603 (1962).

2.2 Beehler, R. E. , Mockler, R. C. , and Richardson, J. M. , Cesium beam atomic time and frequency standards,

Metrologia 1, p. 114 (July 1965).

2. 3 McCoubrey, A. O. , A survey of atomic frequency standards,

Proc. IEEE _54, No. 2, pp. 116-135 (February 1966).

2.4 Hellwig, H. , Areas of promise for the development of future

primary frequency standards, Metrologia 6, p. 118

(October 1970).

45
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46
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by Gerhard Herzberg, copyright by Dover Publications, 1944.)

48
2 * r ^*

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49
Figure 5 The energy level diagram for the emission electron of the

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for an explanation of the labeling of the levels. (This figure is a modification of Fig. 28 of Atomic Spectra and Atomic Structure by Gerhard Herzberg, copyright by Dover Publications,

1944.)

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scientific and technical activities. National Standard Reference Data Series.

NSRDS provides quantitive data on the physical

and chemical properties of materials, compiled from

the world's literature and critically evaluated. Product Standards. Provide requirements for sizes, types, quality and methods for testing various industrial products. These standards are developed cooperatively with interested Government and industry groups

and provide the basis for common understanding of product characteristics for both buyers and sellers. Their use is voluntary.

Technical Notes. This series consists of communications and reports (covering both other agency and

NBS-sponsored work) of limited or transitory interest. Federal Information Processing Standards Publications. This series is the official publication within

the Federal Government for information on standards

adopted and promulgated under the Public Law

89-306, and Bureau of the Budget Circular A-86

entitled, Standardization of Data Elements and Codes

in Data Systems.

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