UNITED STATES DEPARTMENT OF COMMERCE • Maurice H. Stans, Secretary

NATIONAL BUREAU OF STANDARDS • Lewis M. Branscomb, Director

Accurate Characterization of

The High-Pressure Environment

Proceedings of a Symposium held at the National Bureau of Standards, Gaithersburg, Md.,

October 14-18, 1968

Edited by

Edward C. Lloyd

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

National Bureau of Standards Special Publication 326

Nat. Bur. Stand. (U.S.), Spec. Publ. 326, 343 pages (March 1971)

CODEN: XNBSA

Issued March 1971

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 (Order by SD Catalog No. C 13.10:326). Price J4.50
Abstract

The volume contains 38 papers prepared for the Symposium on Accurate Characterization of the High-Pressure Environment held on October 14-18, 1968, at Gaithersburg, Maryland, under the sponsorship of the National Bureau of Standards and the Geophysical Laboratory of the Carnegie Institution of Washington. The papers are presented with the discussions that occurred during the sessions. The

book also includes reports of several informal committees of the conferees on choices of reference pressure materials and on other matters relevant to improved measurement and calibration. The

Symposium was intended to provide an authoritative survey of problems and techniques presently

in use or proposed for precise high-pressure measurement and for temperature measurement at high

pressure.

Key words: Accurate measurement; equation-of-state; fixed points; high pressure; high-pressure

equipment; instrumentation; pressure scale; shock wave technique; temperature.

Library of Congress Catalog Card No. 79-706140
Contents

Page Preface

Introduction and Summary ^ SESSION I

Chairman: P. H. Abelson

Hydrostatic Pressures of 50 kbar in a Piston-Cylinder Device: Measurement of Pressure and Characterization of the Pressure Medium 5 A. Jayaraman and R. G. Maines

Ultrasonic and Dilatometric Measurements at Very High Pressures 11 P. L. M. Heydemann and J. C. Houck

Characterization of the Bismuth i-ii and Barium i-ii Points Under Hydrostatic Pressure 25

R. J. Zeto, H. B. Vanfleet, E. Hryckowian, and C. D. Bosco SESSION II

Chairman: E. C. Lloyd

The Upper Bismuth Pressure Calibration Point 35

J. C. Haygarth, H. D. Luedemann, I. C. Getting, and G. C. Kennedy

Optical Interferometry at High Pressures 39

K. Vedam

Equipment for Generating Pressures up to 800 Kilobars 45

A^. Kawai

Manganin Resistance Gages as Accurate Instruments for High-Pressure Measurements 49

Y. A. Atanov and E. M. Ivanovo SESSION III

Chairman: F. P. Bundy

A Critical Review of the Effect of Pressure on Thermocouple emf's 53

R. E. Hanneman, H. M. Strong, and F. P. Bundy

The Effect of Pressure on the Thermal emf of the Platinum/Platinum 10 Percent Rhodium Thermocouple 63

P. M. Bell, J. L. England, and F. R. Boyd, Jr. Pressure Dependence of the Thermoelectric Power of Thermocouple Materials 67

P. J. Freud and P. N. La Mori

The Effect of Pressure on the E.M.F. of Thermocouples 77

/. C. Getting and G. C. Kennedy

Temperature Measurement by Thermal Noise at High Pressures 81 R. H. Wentorf, Jr. SESSION IV

Chairman: G. E. DuVALL

Equations of State for Sodium and Aluminum 91 D. J. Pastine and M. J. Carroll An Atomistic Theory of Shock Compression of a Perfect Crystalline Solid 105

D. Tsai

Effect of 2024 Aluminum Alloy Strength on High-Pressure Shock Measurements 125 M. van Thiel and A. S. Kusubov

Calculation of Equation of State From High-Pressure Sound Velocity Data 131

A. Holt and R. Grover

III
Shock Temperature Calculations for Silicone Fluid 137 M. Cowperthwaite and J. H. Blackburn

The Equation of State of Selected Materials for High-Pressure References 147 W. J. Carter, S. P. Marsh, J. N. Fritz, and R. G. McQueen

SESSION V

Chairman: C. W. Beckett

Effect of Pressure on the Lattice Parameters of Lead Chalcogenides and Nickel Arsenide-Type

Compounds 159

S. Minomura, H. Nagasaki, and I. Wakabayashi

The Compressibility and Thermal Expansion of LiF to 60 kbar and 600 °C as Determined by X-Ray Dif- fraction: Report of Progress 167

L. C. Carrison and C. B. Sclar

Ultrasonic and Static Equation of State for Cesium Halides 173

G. R. Barsch and Z. P. Chang

Calculation of the P-V Relation for Sodium Chloride up to 300 Kilobars at 25 °C 189

,/. S. Weaver, T. Takahashi, and W. A. Bassett The Hugoniot Equation of State of Sodium Chloride in the Sodium Chloride Structure 201

J. N. Fritz, S. P. Marsh, W. J. Carter, and R. G. McQueen

Consistency in the High-Temperature Equation of State of Solids 209

L. Thomsen and O. L. Anderson

SESSION VI

Chairman: F. R. BoYD, Jr.

The Solid-Liquid Phase Line in Cu 219

R. G. McQueen, W. J. Carter, J. N. Fritz, and S. P. Marsh

Shear Strength Effects on Phase Transition "Pressures" Determined From Shock-Compression

Experiments 229

0. E. Jones and R. A. Graham

Study of Phase Transitions in Insulators by the Dielectric Constant Technique 243

G. A. Samara and W. L. Chrisman

Experimental Determination of Curie Points of Ferromagnets up to 90 Kilobars — Possible Use for Calibration OF High Pressure 251

J. M. Leger, C. Susse, and B. Vodar

Fixed Points on the High-Pressure Scale Identified by Phase Transitions in Ammonium Fluoride 257

R. Kaneda, S. Yamamoto, and K. Nishibata

A Review of Resistance-Jump Phase Changes Useful for High-Pressure Calibration 263

F. P. Bundy

SESSION VII

Chairman: R. RoY

The Coesite-Stishovite Transition 273

S. Akimoto and Y. Syono

The Use of Solid-Solid Transitions at High Temperatures for High-Pressure Calibration 279

P. N. La Mori

The Use of Iron and Gold for Calibration of Higher Pressure and Temperature Points 283

H. M. Strong and F. P. Bundy

Transition Pressures of Bi 3-5, Sn, and Fe 291

M. Contre On Application of Mercury Melting Curve for Accurate High-Pressure Measurements 297

V. S. Bogdanov, Y. L. Levin, S. S. Sekoyan, and Y. L Shmin

High-Pressure Scale as Determined by X-Ray Diffraction Techniques up to Approximately 100 kbar 303

H. T. Hall

IV
APPENDIX

Temperature Measurement at High Pressures 309

Panel Discussion F. P. Bundy, Chairman

Fixed Points Near Room Temperature 313

Committee Report H. T. Hall, Chairman

Equation-of-State Standards 315

Committee Report O. L. Anderson, Chairman

Measurements of Pressure and Temperature in High-Pressure Apparatus 317

Committee Report F. R. Boyd, Jr., Chairman

Pressure Inhomogeneity: A Possible Source of Error in Using Internal Standards for Pressure Gages... 321

J. C. Jamieson and B. Olinger

Comparison of Four Proposed P-V Relations for NaCl 325

]. S. Weaver

Participants 331

V
VI
Symposium on

Accurate Characterization of the High Pressure Environment

Oct. 14-18, 1968

Photograph Key

1. John C. Bowen

2. M. van Thiel

3. George Jura

4. D. B. McWhan

5. R. Kaneda

6. C. Susse

7. M. Contre

8. G. Kennedy

9. H. Young

10. P. St. Pierre

11. Ikuo Kushiro

12. A. S. Kusubov

13. S. Akimoto

14. H. P. Bovenkerk

15. Ivan Getting

16. WiUis Mock, Jr. 17. U. O. Button

18. D. H. Chung

19. M. F. Rose

20. A. Jayaraman

21. R. S. Kirk

22. Tom Clark

23. Naoto Asami

24. I. L. Spain

25. O. B. Verbeke

26. Naoto Kawai

27. Daniel Decker

28. F. R. Boyd, Jr. 29.

30. Eugene Hryckowian

31. Harold D. Stromberg

32. J. Houck

33. W. A. Bassett

34. Francis Bundy

35. S. Minomura

36. Charles Beckett

37. Robert Zeto

38. Renato Bautista

39. Boris Vodar

40. David H. K. Mao

41. Alvin van Valkenburg

42. G. R. Barsch

43. P. H. Abelson

44. Charles Bosco

45. M. C. Krupka

46. Rene Cavailhe

47. D. L. Hamilton

48. A. Lacam

49. R. H. Wentorf

50. G. S. KeU

51. E. C. Lloyd

52. M. C. Gilbert

53. A. Taylor

54. Alan Webb

55. Leon Thomsen

56. Albert Holt

57. C. B. Sclar

58. H. M. Strong

59. Marilyn S. Werkema

60. D. J. Pastine

61. J. D. Barnett

62. R. E. Hanneman

63. J. R. Hastings

64. Leo Merrill

65. G. A. Samara

66. P. D. Calvert

67. Eric Lundblad

68. O. Jones

69. H. Tracy Hall

70. G. E. Duvall

71. M. Cowperthwaite

72. L. C. Towle

73. J. Haygarth

74. Stanley E. Babb

75. F. Seifert

76. G. S. James

77. Rustum Roy

78. Jac Paauwe

79. Elvin M. Compy

80. Eugene ShuU

81. Gene J. Scott

82. Ian MacGregor

83. D. P. Johnson

84.

85. Paul J. Freud

86. J. Jackson

87. James L. Cross

88. Y. A. Atanov

89. Z. P. Chang

90. Peter Bell

91. W. J. Carter

92. P. N. La Mori

93. A. Cezairliyan

94. E. V. Zolotykh

95. Donald H. Lindsley

96. J. L. England

97. C. E. Weir

98. Henry O. A. Meyer

99. K. Vedam

100. Peter L. M. Heydemann

VII

Preface

This Proceedings volume includes all the papers, for which manuscripts were received,

presented during the Symposium on Accurate Characterization of the High-Pressure Environment,

October 14 to 18, 1968. In addition, the Appendix includes one panel discussion and reports

of three informal committees, and two papers prepared subsequent to the meeting as an extension

of discussions during the meeting. There were several informal presentations during the symposium

for which no manuscripts were prepared, including opening remarks by the Session Chairmen and

a keynote discussion by the opening speaker.*

The Symposium was intended to provide an authoritative survey of problems and techniques

presently in use or proposed for precise high-pressure measurement and for temperature measure- ment at high pressure. It was hoped that a tentative consensus could be reached by the conferees on several important aspects of the measurement of high pressures and high temperatures, and a panel and three committees, whose reports appear in the Appendix, were established for this

purpose. Since the list of prospective participants represented the majority of the high-pressure

laboratories in the world, it was felt that the floor discussions would be of particular value. Thus,

time priority was given during the sessions to the discussion, and both a stenotype record and a magnetic tape record were made of all discussions. Edited versions of these discussions appear

with the papers. The Symposium organizers feel that these arrangements were successful, and it

is hoped that the added value of the Proceedings volume will compensate for the delay in publication which is due, in part, to the extensive editing and correspondence required with the conference

participants.

While no formal "referee system" has been used for the papers, it is felt that the discussion

accompanying the papers has served this purpose in many cases, at least on important points.

The organizing group included C. W. Beckett, E. C. Lloyd, and D. P. Johnson of the National

Bureau of Standards, and F. R. Boyd, Jr., and P. Bell of the Geophysical Laboratory, Carnegie

Institution of Washington. Dr. Beckett acted as General Chairman. Enthusiastic cooperation of

Dr. P. H. Abelson of the Carnegie Geophysical Laboratory and A. van Valkenburg of the National

Science Foundation gave important support to the effort. The staff of the NBS Office of Technical Information and Publications, particularly Mr. R. T.

Cook, made important contributions to the successful arrangements for the conduct of the Symposium. Mrs. Barbara Mayo-Wells has prepared the text material for the printers, with the assistance

of the Graphic Arts Section under Mr. C. F. Peters in preparing the illustrations and graphs. Mr.

Raymond Gates of that Section prepared the key to the group photograph of the participants.

Special thanks is due Miss Judy Eyler of the NBS Mechanics Division Office for her untiring

efforts before, during, and after the Symposium in efficiently handling many of the essential details

concerned with arrangements for participants and preparation of the Proceedings volume.

Edward C. Lloyd

February 15, 1971

*Dr. H. G. Drickamer, University of Illinois, opened the symposium with a discussion concerning the role of high pressure calibration in scientific research.

IX

Introduction and Summary*

The importance to science and technology of

precise measurements characterizing the highpressure environment is demonstrated by the

rapidly increasing activity in research, development, and industrial technology, involving such

measurements. It has already become apparent

that at an appropriate time it would be desirable

to reach agreement on a pressure scale somewhat

analogous to the temperature scale, over the range

from 5,000 to 200,000 bars, and possibly higher

pressures. Such a scale would be particularly helpful

in coordinating the results being reported by the

many laboratories now engaged in research involving the high pressure environment. Accuracy of

measurement of elevated temperatures at high pres- sures is also limited, and the need for development

of additional reference techniques for temperature

measurement up to 2000 K at high pressures should

be assessed.

This symposium was intended to provide an

authoritative survey of precise measurement problems and techniques now in use or proposed. It was hoped that the symposium would permit critical

evaluation of the need for efforts toward the development of improved measurement standards, and

identification of probable fruitful lines of investigation toward establishment of such standards. The

survey should provide the information needed to evaluate the feasibility of obtaining early agreement

on a provisional pressure scale.

The symposium was attended by about 140

participants from the United States and abroad.

Other countries represented included Canada,

England, France, Japan, South Africa, the Soviet

Union, Sweden, and West Germany. Thirty-eight

papers were presented, and four panel sessions

held, covering research at high pressures in static

systems and in shock wave experiments.

The tone of the meeting was effectively set by

the opening speakers, who emphasized the importance of improved accuracy in measurements

at high pressures and in calibrations relevant to such measurements. The point was made that in

current work, involving a wide range of phenomena

and many types of apparatus, a series of "interlocking" standards is needed.

About one-fourth of the papers dealt with matters

relevant to selection of reproducible and reversible

reference points on the pressure scale — so-called

"fixed points" — indicated by phase changes in

selected substances. The importance attached to establishing such agreed-on fixed points was in- dicated by these papers and the discussions throughout the week. On the final day these matters were

considered by an informal committee, and the foUow-

*Largely from a meeting report by the same authors appearing in Science 164,

860-2 (May 16, 1969).

ing outline of the report of this committee and of

the ensuing floor discussion summarizes the

informal consensus reached.

It was recommended that phase transitions and

accompanying pressures shown in table 1 be used

as pressure fixed points. The fixed points recom- mended represent equilibrium values.

Table 1.

Transition

Fixed-point pressure

(kilobars)

Present

estimated

uncertainty

(kilobars)

Mercury freezing point at 0 °C.. 7.569 0.002 Bismuth I to II transition at 25 °C. 25.50 0.06 Thallium I to II transition at 25 °C 36.7 0.3 Barium I to II transition at 25 °C 55 2 Bismuth III to V transition at 25 °C 77 3

Users are to consider the fixed points as exact.

The values of "present estimated uncertainty"

are given only to indicate the range within which

a value may be expected to shift as a result of

improved measurements in the future. The re- producibility of pressures based on these phase

changes niay be better or poorer than these uncertainties and, in any given case, depends strongly on technique. It is the responsibility of the ex- perimenter to establish reproducibility and hysteresis for his own apparatus and technique, and

the relationship between his experimental values

and the above equilibrium values.

In addition to the five points listed in table 1 covering the pressure scale up to 77 kbar, a consen-

.sus was reached that the cesium ll to III and III to IV

transitions on increasing pressure be taken as 42.5 kbar and 43.0 kbar, respectively, with a present

estimated uncertainty of 1 kbar, and that the

tin I to II transition be tentatively used as a fixed

point with an equilibrium transition value of 100 kbar

and a present estimated uncertainty of 6 kbar.

In addition to the fixed point of 7.569 kbar at the

freezing pressure of mercury at 0 °C, the committee

favored use of the mercury melting curve to establish other reference pressures up to 15 kbar, cor- responding to the freezing pressure of mercury at about 36.8 °C. It recommended that such reference

pressures be based on the Simon equation, adjusted

to agree with the value 7.569 kbar at 0 °C as foUows:

1
P = 38227

234.29,

1.1772

where T is the temperature in K on the International

Practical Temperature Scale (1948), and P is the

pressure in bars. Small adjustments in this equation

will be needed when the new temperature scale,

IPTS 1968, is used.

Several pressure scales derived from equations

of state of cubic solids were proposed at the meeting. Both metallic and nonmetaUic substances,

such as the cesium halides, were considered. Sodium can be treated most accurately from the

theoretical viewpoint, but its high chemical reactivity is inconvenient. Aluminum can be treated

by quantum mechanical methods if parameters are adjusted to fit some of the observed properties.

Both aluminum and copper have been investigated experimentally as standards in shockwave

measurements. These metals also could be used

as standards in x-ray measurements.

The new data on copper and aluminum shock

standards have been used to reevaluate the equation of state of several other metals that have been

used in determination of pressure based on x-ray

measurements of lattice constants.

Sodium chloride has been used most often as a reference material in recent applications of x-ray

methods for estimating pressure. An informal

committee on equation of state standards considered requirements in this application such as high

compressibility, low yield strength, chemical stability, availability of accurate data over a wide range

of pressure, and other properties. The committee

selected sodium chloride first, with copper and

aluminum as alternates.

Four evaluations of data relevant to the sodium

chloride scale now are avaiilable, including two

which were presented. In these evaluations dif- ferences in reported pressures are less than the

combined errors (about 4 percent) at pressures

from 25 to 300 kbar. When these sodium chloride

scales are combined with x-ray data on sodium

chloride media in which transitions of bismuth and

barium have been studied, the computed transition

pressures lie within the uncertainties indicated in

the table above for barium and for bismuth III to V. The committee recommended that a single sodium

chloride scale be adopted and that it be adjusted

to give values of pressure as close as possible to those selected for the fixed points. An average of

the two most recent sodium chloride scales would

very nearly fulfill these requirements. An exact

fit of selected fixed points appears unlikely without

arbitrary adjustments in the equation of state. Nevertheless, a provisional sodium chloride scale

consistent to within experimental error with the

fixed points selected for the region below 100 kbar

is attainable and, if accepted, could be very useful.

The committee suggested more comprehensive

theoretical and experimental investigations of the

equation of state of cubic sohds in order to improve

currently available scales. In particular, changes

in the vibrational energy states of sohds with volume

are required in order to obtain accurate theoretical

formulations of the equation of state and related

properties.

Temperature in high-pressure experiments is normally measured by thermocouples. However,

pressure affects the relation between electromotive

force and temperature for a thermocouple, provided

there is a thermal gradient within the pressurized

region. The reaUty of this effect is readily shown by

subjecting two dissimilar thermocouples (such

as Pt-Pt 10 percent Rh and chromel-alumel) to the

same conditions of high temperature and pressure

and monitoring the difference in apparent temperature. In various solid-media apparatus there are

differences on the order of 25 °C between the apparent temperature simultaneously read by chromelalumel and Pt-Pt 10 percent Rh couples at 1000 °C

and 40 kbar.

Quantitative evaluation of these effects is most

difficult. Direct determinations have been made

both in internally heated, sohd-media apparatus

and in externally heated, gas apparatus. The latter

results have not thus far been extended to sufficiently high pressures to permit critical comparison

with the soUd-media results. Indirect methods have

utilized a comparison of phase boundaries determined by experiment with boundaries calculated

from thermochemical data. Temperatures determined by thermocouples have also been compared

with those determined by the thermal noise

technique.

All investigators agree that the pressure effect on the chromel-alumel couple is less than ~ 5 °C

in the range up to 1000 °C and 40 kbar. Unfortunately, this couple is considerably less stable than

Pt-Pt 10 percent Rh in high-pressure cells due to chemical contamination and strain effects. AU

investigators also agree that the effect of pressure on the Pt-Pt 10 percent Rh couple is considerably

larger and that the magnitude of the effect increases

with both increasing temperature and increasing

pressure. Estimates of the pressure correction for

this couple range up to 35 °C at 1000 °C (for a cold

seal at 0 °C) and 40 kbar, but there are sizable

differences in the results of various investigations

in this range.

A conference concerned with the accuracy of

pressure and temperature measurement at high

pressures must necessarily be concerned with the

"nuts and bolts" of high-pressure experimentation.

A number of developments in technique were described which considerably extend the range, quality

or accuracy of high-pressure measurements.

Liquid cells have been constructed for use in

piston-cylinder and multi-anvil apparatus which

permit measurements under perfectly hydrostatic

conditions to be made at pressures up to 50 kbar

at room temperature. These cells are jacketed with

plastic or stainless steel and are filled with hquids

2
or mixtures of liquids such as pentane, isoamyl

alcohol, and methanol. Such cells have remarkable

mechanical stability and can be cycled over a wide

pressure range. Their use eliminates shearing

stresses which are inevitably present in systems

employing solid pressure media and which in some

instances affect the thermodynamic properties of

materials under investigation. An analogous technique for high temperatures is to use molten glass

as a pressure medium. Experience with molten

Pyrex in high-pressure cells suggests that thermocouples are more stable in such an environment

than with more commonly used ceramic insulation.

Until recently the pressure range of single-stage,

piston-cylinder apparatus was limited to approximately 50 kbar. This limitation was imposed by the

crushing strength of unsupported, carbide pistons.

However, such pistons wiU support stresses up

to 80 kbar if the ratio of the unsupported length of

the piston to its diameter is kept considerably

below 1. Failure along 45-degree shear planes is thus inhibited. A variation of this technique which

permits greater stroke is to segment the unsupported length of the piston with binding rings.

These binding rings slide back along the piston

as they make up on the face of the pressure vessel

during the compression stroke.

The possibility of using second-order phase

transitions as secondary calibration standards

has aroused considerable interest. Second-order

transitions proceed without volume discontinuities

and theoretically without hysteresis. The change

in Curie temperature with pressure is promising in

this regard although the effects thus far studied

are rather small. For example, the dTjdP slope of

the Curie temperature curve for nickel is 0.35 °C

per kilobar. Other materials, such as ferrites, may

show larger effects.

New developments in shockwave research now

permit a derivation of the fusion curve {P versus T)

of copper extending into the million-bar range. When

combined with accurate measurements of temperature and pressure in the static high-pressure

range (1100 to 1300 °C and 0 to 60 kbar) the fusion curves of several substances, such as copper,

silver, and others, could be used for in situ checking of calibrations of high-pressure, high-temperature apparatus. However, in a broader perspective,

accurately determined fusion curves may permit

solution of some rather fundamental questions in

geophysics and astrophysics involving extrapolation

of phase diagrams to very high pressure and high

temperature.

The kinetics of shockwave processes are being

investigated from several viewpoints. These in- clude lattice dynamic models of shocks in solids

and experimental investigation of nonequilibrium

processes at relatively low shock strengths in many

cases. The behavior of shocks in regions of phase

changes are especially interesting. The rate at which

such changes occur depends upon the types of

lattices involved and the orientation of the lattice

with respect to the direction of the shock. From the

viewpoint of pressure standards, additional studies

of the transformation of iron at about 126 kbar were

reported in the meeting.

The symposium was sponsored by the National

Bureau of Standards and the Geophysical Laboratory of the Carnegie Institution of Washington.

Expenses were covered by a grant from the National

Science Foundation.

Edward C. Lloyd

Charles W. Beckett

Institute for Basic Standards,

National Bureau of Standards,

Washington, D.C.

Francis R. Boyd, Jr.

Geophysical Laboratory,

Carnegie Institution of Washington,

Washington, D.C.

3

session I

Chairman: P. H. Abelson

Geophysical Laboratory

Carnegie Institution of Washington

Hydrostatic Pressures of 50 kbar in a Piston-Cylinder Device:

Measurement of Pressure and Characterization

of the Pressure Medium

A. Jayaraman and R. G. Maines

Bell Telephone Laboratories, Incorporated, Mountain Avenue, Murray Hill, New Jersey 07974

A method suited for making resistivity and other electrical measurements to 50 kbar hydrostatic pressure, using a conventional piston-cylinder device, is described. In this method a Teflon cell is used

to contain the pressure medium, which is a 1:1 mixture of n-pentane and isoamyl alcohol. After the

fluid mixture freezes (at about 46 kbar), further advance of the piston results in uniaxial stress. However, this can be readily relaxed by heating the medium to 70 °C. The resultant isotropic pressure dis- tribution stays when the pressure medium is cooled back to room temperature. The technique of probing the pressure distribution inside the cell, as well as pressure calibration procedure, is described.

1. Introduction

A technique for generating hydrostatic pressures

in excess of 30 kbar using a conventional pistoncylinder device was recently described [1].' In this, a Teflon cell contained the pressure medium, and

leads were brought out of the cell for electrical

measurements. The hydrostatic pressure limit of

about 38 kbar was dictated by the freezing of the

pressure medium at this pressure at room temperature. To evaluate friction correction, the Bi

I ^11 transition at 25.4 kbar was used and it was

shown that true pressure could be obtained from

the indicated pressure by multiplying the latter

with the ratio PtrBi I~*n/Pindicated the transition

point. In the present paper, efforts to extend the

hydrostatic pressure regime to 50 kbar, the methods

used to characterize the pressure medium, and the

measurement of pressure will be described.

2. Experimental Arrangement

Figure 1 is a schematic of the Teflon cell assembly

in position in the pressure chamber, while Figure 2

shows the Teflon cell and lead arrangement in

some detail. The materials used and the dimensions

of the cell are given in the figure. Consistent with

lower frictional correction and effective containment of the fluid, the optimum wall thicknesses

for the V2-in and 1-in diam Teflon cells are 50 and

' Figures in brackets indicate the literature references at the end of this paper.

Paper presented at the Symposium on Accurate Characterization

of the High-Pressure Environment, held at the National Bureau

of Standards, Gaithersburg, Md., October 14-18, 1968.

100 mils respectively. The Vs-in-thick Teflon disk

shown in the figure (see fig. 2) next to the cap serves

the purpose of an additional seal. In the bottom

side of the Teflon cell next to the piston, we use a

10-mil-thick steel shim as an additional seal and

this permits the use of a longer cell. We find that

leakage problems are virtually eliminated by slightly

offsetting the drilled holes in the Teflon disk with

respect to the holes drilled in the cap.

3. Characterization of the Pressure

Medium

The pressure medium could be conveniently

characterized by following the piezoresistance

effect of an n-type silicon plate. The piezoresistance

[2-5] effect relates the change 8p of the resistivity

tensor to the stress tensor. For a cubic substance,

the piezoresistance coefficient tt can be described

in terms of its 11, 12, and 44 components, referred

to the coordinate axes chosen along the cubic directions. In the case of hydrostatic stress — 8p/p = P{ttii + 2tti2) , where P is the pressure in

dynes/cm^ and the best value for the coefficient

(77-11 + 27712) is about 3.1X10-12 cm2 dyn-i for

n-siUcon [6]. For the uniaxial stress and the current

and field directions as used in our experiments, the

appropriate component is 7721 = 7112 which has the

value — 53.4XlO-'2 cm^ dyn~' for n-silicon. Be- cause of this large difference in piezoresistance for

the hydrostatic and uniaxial stresses, any deviation

from hydrostaticity shows readily in the resistivity versus pressure curve of a (100) oriented n-type

siUcon plate. It should be noted here that the pistoncylinder geometry favors a uniaxial stress distribution, after the pressure medium freezes.

5
\MILD HARD

STEEL STEEL

TEFLON

WC

^ — TOP END

LOADING PLATE

PRESSURE CHAMBER

PYROPHYLLITE

-FLUID

LEADS

ALUNDUM

CEMENT~~

t f t

THERMOCOUPLE

FLUID

V/ // // // // 'A

-SAMPLE

Figure 1. Teflon cell assembly inside the pressure chamber.

The diagram to the right is an enlarged view of the cell.

LEADS

PYROPHYLLITESTAINLESS STEEL ALUNDUM

CEMENT

INSULATION

(TAPE)"-^

PYROPHYLLITE

1950

TEFLON

|«-0.744-

- 0.994 > (C)

Figure 2. The dimensions of the parts in inches, for V2-m and

1-in diam (O.D.) Teflon cell assembly.

The connection of leads from inside the cell to the leads carried by the ceramic

tubing are shown to the right.

We used in our experiments a phosphorus-doped

(p = 4.4 ohm cm) silicon sample of square geometry

(160 mils to a side and 15 mils thick) cut parallel

to the (100) crystal plane. (The advantage of using

silicon is that there are no other conduction band

minima, excepting the ones located near the [100]

R/Ro

1.00

0.90

0.80

0.70

0,60

0 50

0.40

0.30

piezoresistance of (100) oriented n- silicon plate

o n- pentane + isoamyl alcohol

I.I mixture (I) A isoamyl alcohol (2)

0 20 30

P (kbar)

40 50

Figure 3. Resistivity ratio of an n-type, (100) oriented silicon plate, plotted as a function of pressure.

directions in k space. Further, the resistance re- sponse of 7i-silicon is almost linear with pressure.)

The plate was oriented inside the Teflon cell with

the plane perpendicular to the axis of the pistoncylinder geometry, so that when uniaxial stress

appeared, it was in the direction normal to the (100)

plane. Ohmic contacts were made at the four corners of the plate to which copper leads were

soldered. Two adjacent leads were used for current

while the opposite two served as potential leads.

Since the change of resistance Ap is small, both the

current and potential measurements have to be

very precise. A very convenient arrangement is to use a well stabihzed current source in conjunction

with a potentiometer, such as the Leeds and Northrup K3 for potential measurement. In figure 3 is shown the measured resistivity ratio plotted as a

6
function of pressure for a 1:1 mixture of n-pentane

and isoamyl alcohol (curve 1) and for pure isoamyl

alcohol (curve 2) as pressure medium.

The R/Ro versus pressure plot shows an abrupt

change of slope near 38 kbar, when isoamyl alcohol

is used as the pressure medium, and near 45 kbar

when the n-pentane + isoamyl alcohol (1:1 ratio)

mixture is used as the medium. This change of

slope is due to the appearance of uniaxial stress

perpendicular to the (100) plane of siUcon, after

the pressure medium freezes. Above the freezing

point, almost the whole of the applied stress appears

as uniaxial stress. We find that this uniaxial stress can be relaxed by heating the pressure medium to about 70 °C. Resistance measurements on n-silicon

obtained after heating and cooling back to room

temperature fall on the extension of the hydrostatic

data, showing thereby that the heating and cooling

procedure is an effective method to restore hydrostatic stress distribution, after the pressure medium

solidifies.

In describing the Teflon cell technique in earlier

publications, it was stated that the 1:1 mixture of

n-pentane and isoamyl alcohol solidifies at about

38 kbar. However, our present results show this to be incorrect. The reason for the discrepancy is due to the fact that n-pentane evaporates from the

Indicated Pressure (k bar)

Figure 4. Pressure calibration with the freezing point of Hg

at 22 °C {11.85 kbar), Bi I-U transition (25.4 kbar), the transition in Tl at 37 kbar, and Cs U—Ul at 42.5 kbar.

The ordinate gives the true pressure and the abscissa the appUed pressure.

mixture somewhat rapidly and after a length of

time, only isoamyl alcohol is left behind. This is what has apparently happened to the mixture used

in previous experiments.

4. Pressure Measurement

For pressure calibration we have used the freezing point of mercury [7] at 22 °C (11.85 kbar), the

Bi l-ll [8] transition at 22 °C (25.4 kbar), the transition in thallium [9] at 22 °C (37 kbar) and the

Cs II—III transition at 42.5 kbar. All the transitions

fall within the freezing pressure of n-pentaneisoamyl alcohol mixture. In figure 4 is plotted the

true pressure in kbar against the indicated pressure

(computed from the dial gauge) using the abovementioned three fixed points. The linear relationship

between the two pressures shows that the friction

correction is proportional to the pressure indicated;

^true~ (0-925)Pind- The multiplying constant is highly reproducible for the cell dimensions and

geometry shown in figure 2 and for n-pentaneisoamyl alcohol as pressure medium. Thus the

measurement of pressure is straightforward.

As a further check of the accuracy of our cahbration, we have compared the resistance data obtained

at Harvard by Kosicki and Paul on a sulfur-doped

1000

100

0.1

5 10 15 20 25

P(k bar)

Figure 5. Comparison of the resistivity of sulfur-doped n-GaSb

measured in Harvard University Bridgman-type hydrostatic

apparatus and in the piston-cylinder device (Teflon cell technique).

385-762 O - 71 - 2
n-GaSb in a Bridgman apparatus (calibrated using

a manganin gauge), with resistance measurements

on the same sample in our hydrostatic setup.

The results of these measurements are in very

good agreement (see fig. 5). We estimate that our

pressures are true to within one percent.

5. Conclusion

With the present technique it is possible to study

the transport properties of metals and semiconductors at least up to 50 kbar hydrostatic pressure, at room temperature. This extension of the hydrostatic

pressure range may be expected to yield valuable

information, particularly on the energy band struc- ture of semiconductors. Such usefulness has already

been demonstrated at least in two cases. Using the

technique of pressure generation detailed above,

the energy of the higher-lying conduction band

minima and the associated electron mobilities have

been determined in GaSb [10] and Ge [11, 12].

6. Acknowledgment

It is a pleasure to thank Dr. B. B. Kosicki for

helpful discussions and for the data on sulfurdoped GaSb, measurements on which were made in a Bridgman type hydrostatic apparatus at Harvard

University. We would also hke to thank Mrs. I. Kocserha for providing ohmic contacts on many

of the silicon slices.

7. References

[1] Jayaraman, A., Hutson, A. R., McFee, J. H., Coriell, A. S., and Maines, R. G., Rev. Sci. Instr. 38, 44 (1967).

[2] Smith, C. S., Phys. Rev. 94, 42 (1954).

[3] Morin, F. J., Geballe, T. H., and Herring, C, Phys. Rev. 105, 525 (1957).

[4] Herring, C, Bell System Tech. J. 34, 1 (1955).

[5] Keyes, R. W., in Solid State Physics 11, 187 (1960). (Seitz

and TurnbuU, Eds., Academic Press).

[6] Smith, A. C, Tech. Report H-P.-2 from the Division of Engineering and Applied Physics, Harvard University (1958), pp. 46-49.

[7] Bridgman, P. W., Proc. Am. Acad. Arts Sci. 47, 347 (1911).

[8] Heydemann, P. L., J. Appl. Phys. 38, 2640 (1967).

[9] Kennedy, G. C., and La Mori, P. N.. J. Geophys. Res. 67,

851 (1962).

[10] Kosicki. B.B.. Jayaraman, A., and Paul, W., Phys. Rev. 172,

764, (1968).

[11] Jayaraman, A., Kosicki, B. B., and Irvin, J. C, Phys. Rev. 171, 836 (1968).

[12] Jayaraman, A., and Kosicki, B. B., in Proc. 9th Intern. Conf. on Physics of Semiconductors (Vol. 1, p. 47, 1968).

DISCUSSION

R. E. Hanneman {General Electric Research and

Development Center, Schenectady, New York):

What are the upper Umits in pressure and temperature in using your technique?

G. C. Kennedy {Institute of Geophysics and Planetary Physics, University of California, Los Angeles,

California): There is an interesting extension of

this technique to higher temperatures. We have

been doing experiments for several years in which

the sample is immersed in molten glass. Glasses

are available that melt from 150 °C on up to quartz

1800 °C. An advantage is the self-seaUng of leaks.

When a leak occurs it is stopped by the glass

freezing off. Molten glasses work beautifully at 1200 °C to 1500 °C. We characteristicaUy run our experiments in molten pyrex, and it is a hydrostatic

medium under good conditions.

H. T. Hall {Brigham Young University, Provo,

Utah): I have a question about the calibration and

the effect of friction. I presume that the calibration

was made on rising pressure. How much friction

do you have, and where is it located?

This hydrostaticity-detecting device is interesting. Have you done experiments with Bridgman's

mixture of iso-pentane and ra-pentane in this device?

While your present mixture appears to be limited

to about 50 kilobars, I believe that there is evidence

that higher pressures can be reached with Bridgman's mixture.

S. E. Babb, Jr. {University of Oklahoma,

Norman, Oklahoma): On the matter of the pressuretransmitting liquid, Bridgman used impure pentane

to 30 kilobars. The purity greatly affects the degree

of subcooling, and I am suggesting that the pentane

becomes a subcooled liquid.

H. G. Drickamer {University of Illinois, Urbana,

Illinois): I doubt if necessarily the mixture is subcooled. A very long time ago we made some

viscosity measurements at one atmosphere, down

to the freezing point. We found that if you are careful

the mixtures can be supercoUed 10 to 15 degrees,

but the viscosity goes up enormously as the freezing point is approached. In general, you have some

kind of eutectic. If you are really subcooling, you

will get the kind of things you did with the siUcone.

I think you have a low melting-point eutectic of

some kind, and you are above the melting point on

that particular mixture.

R. H. Wentorf, Jr. {General Electric Research and

Development Center, Schenectady, New York):

I remember once talking with Bridgman on the use

of pentane. He said that when he used very pure

pentane in his equipment it broke. A mixture is

8
best, because you get the well-known depression

of freezing point with most organic mixtures, and

some kind of eutectic. Bridgman went back to the

conglomeration of different kinds of molecules in

order to get the low freezing point he wanted.

R. Zeto {U.S. Army Electronics Command, Fort

Monmouth, New Jersey): I would like to point out

two problems in the calibration procedure. In

your hydrostatic system you are calibrating on the

initiation of the transformation which is not the

real equihbrium point measured by Bridgman and

Heydemann. At the bismuth I-II point there is a V2 to 1 percent error involved. Also, if you are changing pressure in 1 kilobar increments there is an additional uncertainty due to overpressurization.

This can lead to an error of up to 4 percent for the

bismuth I-ll point.

We use an equivolume mixture of normal-pentane

and iso-pentane without indication of freezing to 50 to 60 kilobars. We find this pressure medium to be hydrostatic and can cycle any number of times.

J. D. Barnett {Brigham Young University, Prove,

Utah): I would like to know what your time scale is both on these piezoresistance effects and also on

the bismuth transitions. Let me suggest that on

your bismuth transition the time scale is dramatically influencing the hysteresis effect, as you

will see from Zeto's paper. A difference of the order

of several hundred bars, at least, can occur depending on how fast the pressure is changed.

M. Contra {Commissariat a I'Energie Atomique,

Paris): I am concerned about your use of Teflon.

Metals under pressure become elastic. Teflon becomes fragile under pressure. Has this caused

trouble in your apparatus?

AUTHORS' CLOSURE

In reply to Ur. Hall: All the measurements and

calibrations were done with increasing pressure. At

25.4 kilobars the friction was equivalent to 2 kilobars, and I think was mainly waU friction between

the piston and cylinder. We have not tried to relieve

this since we are able to correct for it. Concerning use of a mixture of iso-pentane and

n-pentane, we have tried this but we encountered

leakage problems.

In reply to Dr. Zeto: We make sure that the temperature rises minimally. Since the bismuth I-ll transition on the phase diagram has a fair slope, a

definitely lower pressure would exist at the calibration point if an appreciable temperature rise occurs.

In reply to Dr. Contre: I think Teflon can be used

at temperatures up to 200 °C or 300 °C quite easily.

We have repeatedly experimented in the range up

to 100 °C but have not done many experiments

above that. At higher temperatures, the problem of

leakage becomes more serious, particularly when

electrical leads are brought out. We have had no trouble with Teflon due to its becoming fragile. Our experience is that this material can be cycled many times to 50 kilobars, and it comes out in fairly good shape. There is some deformation. I think Teflon is probably one of the best

materials to use for this kind of experiment. It's better than any metal, especially when electrical

leads are to be brought out.

9

Ultrasonic and Dilatometric Measurements at Very High Pressures

Peter L, M. Heydemann and James C. Houck

National Bureau of Standards, Washington, D.C. 20234

A short-cylinder and piston assembly and its use for dilatometric and ultrasonic measurements

on solids and liquids at pressures up to 40 kbar are described. All necessary corrections for the evaluation of measurements are discussed and the systematic uncertainties are given. It is demonstrated

that ultrasonic methods can significantly reduce the uncertainties in the determination of density and

bulk modulus of both, solids and liquids.

Key words: Bulk modulus; density; dilatometric measurements; high pressure; liquids; solids; ultra- sonics.

1. Introduction

Simple piston and cylinder assemblies are widely

used to compress solids and liquids [1] ' and to study

their mechanical, electrical, magnetic, and optical

properties over a range of pressures of up to about

60 kbar. We have developed a particularly economical high-pressure cylinder and have also used ultrasonic methods extensively in connection with

high-pressure investigations on solids [2] and

liquids [3]. In the present report we investigate

the systematic uncertainties of such measurements.

Although we are dealing with our particular setup,

the uncertainties and corrections discussed are nevertheless representative of this general type of

equipment. The magnitude of the uncertainties is in many cases larger than was previously expected.

In investigating the estimated systematic uncertainties of ultrasonic measurements on compressed

materials, we beheve we can demonstrate the relatively greater accuracy that can be achieved — under

favorable conditions — with such methods.

2. Piston and Cylinder Assembly

2.1. For Ultrasonic Measurements in Solids

Figure 1 shows a piston and cylinder assembly

for ultrasonic and dilatometric measurements on

polycrystalhne materials and on polymers. The

cyUnder consists of tungsten carbide with 6 percent

cobalt binder. It has a bore of about 1.2 cm diameter

and is about 1 cm long. A steel ring, press-fitted

with 1.2 percent interference on the carbide

cyUnder, provides radial support. The normal stress

at the cyUnder-support ring interface is approximately 11 kbar. At the upper end the cylinder is closed with a tungsten carbide backup plate, and

at the lower end with the tungsten carbide piston.

The assembly is clamped against the top platen

' Figures in brackets indicate the literature references at the end of this paper.

Paper presented at the Symposium on,' Accurate Characterization

of the High-Pressure Environment, held at the National Bureau

of Standards, Gaithersburg, Md., October 14-18, 1968.

of the press (not shown in fig. 1) for axial support

of the cyhnder. A washer is used to transmit most

of the clamping force to the tungsten carbide cylinder. A clamping stress of 14 kbar at the faces of

the cylinder is sufficient to support the cylinder at internal pressures up to 40 kbars. No attempt has

been made so far to generate higher pressures.

A ram concentric with the one supplying clamping

force is used to press the piston into the sample.

A handle is provided as part of the piston stack

for rotation under load. Travel of the piston stack is monitored by two dial indicator micrometers

arranged diametrically on either side of the piston.

The travel of the piston stack is measured at a

point immediately below the lowest surface shown

in figure 1 relative to the surface on which the lower

face of the bridge plate rests. Rather large cor- rections have to be applied to obtain the advance

of the front end of the piston relative to the cylinder

from the dial indicator readings (see sec. 5).

The piston is fitted with about 5-/u,m diametral

clearance into the cylinder. Delta (anti-extrusion)

rings are used for very soft materials.

An ultrasonic transducer is cemented to the

backup plate as shown in figure 1. Several backup

plates with different lengths and with shear or

longitudinal transducers are usually at hand in

the laboratory. In cases where simultaneous

Transducer

Figure 1. Piston-cylinder assembly for solids.

11
measurement of shear and longitudinal wave

propagation times is imperative, a stepped piston

as shown in figure 1 is used. This carries a transducer at its larger rear surface. In all other cases,

the expensive stepped piston is replaced by two

tungsten carbide pieces of 2.54 cm and 1.27 cm

diameter. The transducers are accommodated in recesses of the adjoining parts. Electrical leads are brought out through radial grooves to BNC con- nectors mounted on the outside of the assembly.

Since the transducers are not subjected to mechanical stress from the high pressure on the sample,

a glyceryl phthalate resin (Glyptal 1202 varnish), or a cyanoacrylate cement (Eastman 910), are sufficient to make bonds.

Samples are often precompacted into the cylinder

in an auxiliary press and then machined inside the

cyUnder for plane and parallel surfaces. Alinement

of the piston stack is carefully checked prior to each measurement.

2.2. For Ultrasonic Measurements in

Liquids

Figure 2 illustrates the assembly used to contain

hquids in the cylinder. The cylinder is lined with

a polyethylene (PE) sleeve. The length of the liner

is slightly smaller than the length of the cylinder, so that the piston can enter and close the lower end

of the cylinder. Liquid is filled into the cylinder

with piston and cylinder in situ in the press. The

backup plate is then put in place and the assembly

is clamped.

If the bulk modulus of the liquid is smaller than

that of polyethylene, as is the case with most hquids

(see, for example, water in fig. 3), then the piston

will push against the sleeve as the contents of the

cylinder are compressed. This small excess of stress

in the piston-sleeve interface over the hydrostatic

pressure in the cylinder is sufficient to seal the

liquid, even at pressures as high as 40 kbar.

The extent to which this excess stress reduces

the internal hydrostatic pressure below the nominal

pressure computed from ram force and effective

area, was investigated at two pressures. The melting pressure of ice VI at about 22 °C was measured

as a function of the wall thickness of the sleeve.

There was no discernible variation of the nominal

pressure with sleeve wall thickness. Considering

the accuracy of pressure determination, to be discussed later, we assume that at the melting pressure

of ice VI of about 9200 bar at room temperature, the

internal pressure is reduced by less than one percent below the nominal pressure by polyethylene

sleeves with wall thickness as large as 0.25 cm. Two kinds of experiments were made at the Bi

I-II transition pressure. In the first experiment,

about 0.11 cm^ of polycrystalline bismuth was placed

inside the polyethylene sleeve. The rest of the volume was filled with isoamyl alcohol. The Bi I-II transition pressure was then determined for three

different sleeve wall thicknesses. The result is shown in figure 4. Plotted (X) for each of five ex- periments is the mean of the nominal pressures at which transition was observed under compression

30 kb

Pressure p

Figure 3. Bulk modulus Bt of water andj)olyethylene asJunction of pressure.

40 60 80 %

Sleeve wall thickness

T

1 1 .3 cnn

Figure 2. Piston-cylinder assembly for hauids.

Figure 4. Ram pressure necessary to reach the Bi I-II transition as function of the sleeve wall thickness.

12
and decompression against the volume of polyethylene in percent of the total volume contained in

the cylinder at zero pressure. With no polyethylene

present the curve should extrapolate to a ram pres- sure of 177.8 bars corresponding to an internal

pressure of about 25,660 bars, the Bi I II transition

pressure at 22 °C [4]. In the second experiment, an electrical lead was

introduced into the cylinder through the backup

plate, as shown schematically in figure 5. The

change of electrical resistance of a bismuth wire

connected to the electrical lead and to ground was

used to detect the transition. The results are also

plotted in figure 4. In three cases (#) isoamyl

alcohol was used as pressure transmitting liquid,

and in one case (+) methanol was used. The compression of both liquids to 25 kbar is quite similar

[5, 6]. At this pressure the volume of methanol is 67 percent and that of isoamyl alcohol is 70 percent

of their volumes at atmospheric pressure. Different curves in figure 4 would be expected for liquids of

widely different bulk modulus. For the most fre- quently used sleeve wall thickness of 0.15 cm, the

pressure correction amounts to only 1.1 percent at 25 kbar.

The pressure on liquids contained in these polyethylene sleeves can be cycled several times between zero pressure and pressures leading to compressions V/Vo of less than' 0.75. We have often

been amazed about the repeatability of these measurements, which usually is within the sensitivity of

both the dilatometric and the ultrasonic measurement. No such repeatability was found when, in the

beginning of this development, polytetrafluoroethylene was used as sleeve material. In this latter case very significant plastic deformation occurred even at moderate compressions.

3. Ultrasonic Measurements and

Typical Results

It is difficult to maintain parallelism of better

than 30 s between the front and rear face of the

samples in the assemblies of figures 1 and 2. Therefore, the ultrasonic measurement can not

be of the precision of which it is inherently

capable [7]. In most of our experiments, a wave

train of about one-half /is duration with a carrier

frequency of about 10 MHz was emitted from the

transducer. The reflections from various interfaces

in the acoustic path are picked up by the transducer, amplified and displayed on the screen of an

oscilloscope. Times between pulses or echoes are then determined using a calibrated sweep delay.

The echo pattern is often complex because of the

number of interfaces. Multiple reflections usually

further complicate the pattern. A typical echo

pattern is shown and interpreted in figure 6. Although we have not made any attempt to study

ultrasonic attenuation as a function of pressure,

the amphtude ratio of echoes and transmitted

pulses is often recorded during a measurement.

back-up plate

silver solder

joint \^

stieathed wire y steel button

sleeve

liquid —

seal

cylinder

i ~Bi wire

piston

1 1

Figure 5. Electrical feedthrough.

Distance Time

Figure 6. Ultrasonic echo pattern from isopentane (2-methyl

butane) at 20 kbar.

Sample echo sequences are marked with groups of arrows of equal length.

Significant changes of this ratio are observed at phase transitions in solids (fig. 7, compare also

[23]), or in the vicinity of the glass transition

of certain liquids. In the first case this change of the

signal amplitude is a facile way of detecting a transition. In the second case it indicates the increase

of the viscosity as the glass transition is approached.

Both with liquids and many solids, times between

pulses are usually measured to within ± 5 ns. These times are usually reproduced to within

better than 10 ns from one pressure cycle to the

next. When multiple echoes can be observed the

transit time t can often be determined to a few

nanoseconds.

13
0

Bismuth

relative amplitude of transmitted pulse

f = 30 MHz

I I

0 10 20

pressure

, f

< 30 kb

Figure 7a. Variation of amplitude of reflected pulses at the Bi I-II transition.

Mixtures of Aviation Instrument Oil with isopentane

Relative amplitude of first echo. f=l2 MHz

(numbers indicate oil/pentane ratio)

pressure

Figure 7b. Variation of amplitude of reflected pulses in the vicinity ofthe glass transition of mixtures ofaviation instrument

oil with isopentane.

Difficulties arise with materials showing large

anisotropy of the speed of sound along different

crystallographic axes, particularly when the grain

size is large compared to the wavelength. This leads

to high attenuation due to scattering and also to disintegration of individual echoes into a group of

smaller echoes. Figure 8 demonstrates this eff"ect with an example of the transmission of a 30 MHz

wave train through a bismuth sample of about 1 cm

length. Such samples are ground to a fine powder,

passed through a 325-mesh sieve, and precompressed into the cylinder. Nevertheless, after one

or two pressure cycles, very significant grain growth

is observed. Similar eff"ects have been observed

in other materials.

4. Determination of Pressure

The internal pressure p for an upstroke is calculated from

P = R{1 + firpr) ( 1 - /3cyl/?Pr) [Pr " (o + bpr) ] (1)

Change in transit tinne

kb

Figure 8. Ultrasonic pulse transmission through bismuth showing the effect of changing grain structure: separation of pulse

into two or more components.

Areas of black rectangles indicate width and amplitude of pulses. Hatched area shows

very large pulse amplitude in Bi 111.

where a bar above or below a symbol denotes rising or falling pressure respectively, and where

R is the ratio of ram area to piston area,

pr is the ram pressure,

1 + firPr is a correction factor for ram cylinder

expansion,

1 ~ jScyiPr/? is a correction factor for high-pressure

cylinder expansion, a is a parameter representing the reduction of internal pressure due to friction

extrapolated to zero ram pressure, and

6 is a parameter representing the change

of the internal pressure due to friction

per unit ram pressure.

Both /3cyi and R depend to some extent on the seal

in the high-pressure cylinder (eff"ective area) and

on the fraction of cylinder length exposed to high

pressure. The reduction of the internal pressure due

to friction is determined by including several small

loops in the upstroke or downstroke as illustrated

in figure 9. The friction data obtained from these

loops on the upstroke are fitted to the equation

reduction of internal pressure = /? (a + 6pr). (2)

On the downstroke a special provision has to be

made for correction in the range where the reversal

14
I n ram pressure

friction

Figure 9. Determination offriction.

of friction takes place (B to C in fig. 9). An exponential function was found to describe the friction

reversal very well.

P = Ril + l3rPr){l- iScylRpr) [pr+{a+bpr) — 2(a + bpr) exp [- F(pr, max - Pr)]] , (3)

where F is a factor chosen to fit the friction

reversal (it is determined with the help

of the OMNITAB FIT instruction [17]

and taken to be constant for a particular kind of experiment), and

Pr,max is the maximum ram pressure applied

during the run. The mass A/of the ram piston reduces the internal

pressure by MgR/A, where gis the acceleration due

to gravity and A is the area of the ram. A similar

correction must be applied for oil head in the line

connecting the ram and the gage measuring the ram

pressure.

Figure 10 illustrates how ultrasonic data taken

during one fuU pressure loop are reduced to a curve

of transit time versus internal pressure by applying

eqs (2) and (3) to the upstroke and downstroke ram

pressure.

Typical values for the constants in eqs (1) and (3)

taken from measurements with polyethylene sleeves

are:

R = 146.12

7.5X 10- ' bar-i

Pcyl — 5.0X10- ' bar-i

a 2.62 bar

h 0.035

F = .013

Pr, max 248 bar

A

I r 1 1 1 »

0 10 20 30 40 kb pressure

Figure 10. Dependence of time offlight of ultrasonic pulses on

pressure with and without correction for friction. Open circles upstroke, dots downstroke, both corrected for friction, triangles without

correction for friction.

In these runs no rotation was used to reUeve

friction. The friction at the highest ram pressure

therefore amounts to about 4.5 percent of the ram

pressure.

The systematic error in the internal pressure p is determined from the total differentials of eqs (1)

and (3). The terms of the total differential for p, the

uncertainties in individual constants or variables,

and their contributions to the total uncertainty are

listed in table 1. We believe that this list contains

all of the significant contributions to the systematic

uncertainty.

The uncertainty contribution of the ram-to-piston area ratio could be reduced by determining the

diameter of the high-pressure cylinder with higher

accuracy. The contribution of the ram cylinder ex- pansion is obviously negligible. The expansion of

the high-pressure cylinder makes by far the largest

contribution to the total uncertainty in the pressure

determination. We have described ultrasonic meas- urements of cylinder expansion in an earlier paper

[2], and are presently planning further experiments

to determine more accurately the expansion of cylinders under pressure. If necessary, the contribution of the ram pressure measurement can be re- duced by replacing the Bourdon gage, used by us, with a piston gage. The contribution of friction can

be reduced considerably by rotation of the piston.

In this computation of internal pressure we as- sume that friction on an upstroke equals that on a downstroke at a given internal pressure, if the loop

is reasonably small. The loop should be large enough

to develop friction fully in both directions of piston

travel. When liquids are pressurized in a polyethylene-lined cylinder, small loops close within

the precision of our measurement. This is not the case for many solids.

A solid under pressure in this apparatus is not

under hydrostatic pressure, but is under axial loading by the piston and radial loading by the reaction

15
Table 1. Total differential of p and contributions to estimated uncertainty of p.

Terms of total differential iicci idiiiiy oi iiic varidDic Contribution to total uncertainty

at 30 kbar

(1 +;8.p.) (1 -/3eyl/Jpr) [Pr" (a+ bPr)]dR 0.1 (ram area to piston area ratio) 20 bar

Rpr(\- /3ey,«Pr) [p, - (a + bpr)]dPr 1.5 X 10"' bar"' (ram cylinder exp.) 1 bar

R'Pr{l+ PrPr)[pr- {a + bpr)Wcyl 1 X 10"' bar"' (high-pressure 98 bar

cylinder exp.)

A (1 + fSrPr) (i — PcylAPrj U OjaPr 0.275 bar (ram pressure) oO D3,T

R(l + /3,.Pr)(l-j8cyi«Pr)a!a .2 bar ] 29 bar

\ (friction)

/? (1 + /3rPr) ( 1 - j8c_vl«Pr)Prrf6 .002 J 64 bar

Total uncertainty of p at p = 30 kbar 250 bar or 0.8 percent

force of the cylinder walls. This nonhydrostatic

stress distribution can result in axial stress components exceeding radial stress components by the

yield strength of the material on increasing pressure,

and the reverse on decreasing pressure. This causes

the size of the displacement versus pressure loops

to be dependent on the extreme pressure to which

the sample has been subjected [8, 9]. Therefore, we

do not expect a large downstroke to come close to points C (fig. 9) of small loops included in an upstroke, if solid material is compressed. With liquids

contained in polyethylene sleeves the downstroke

does, however, approach points C closely.

5. Evaluation of Dilatometric

Measurements

The advance of the piston into the cylinder cannot

be measured directly with mechanical means.

Instead, the travel of a piece in the lower part of

the piston stack is measured relative to the support

of the large bridge plate (fig. 1). Two diametrically

arranged dial indicator micrometers are read to 0.00001 in (0.25 fim) and their readings averaged.

A three-point measurement with gages spaced

120° would be preferable to the present setup, as it would permit detection of tilt of the piston stack

around any axis.

To arrive at the advance of the front end of the

piston relative to the cylinder, several corrections

have to be applied. The complete equation used

for an upstroke is

AL = L(p) — Lo = ho — h{p) + ^pr + r)pl

+ 7[1 -e-Pr] +/3ey,LoP (4)

where AL is the adjusted change in sample

length;

L{p) is the adjusted length of the sample

at pressure p;

Lo is the length of the sample at zero pressure;

ho is the dial gage reading at zero pressure;

h{p) is the dial gage reading at ram pres- sure p,-;

(f)Pr and rjpl are corrections for compression of

the piston stack and distortion of

other parts (the square term was

omitted from most equations in this

paper because its contribution is neghgible)

y[l — e~*^r] is a correction for initial slack in the

setup at low pressures, and

jScyi^-oP is a correction for the effect of cylinder

expansion which adjusts the length

to the length the sample would have

in a rigid cylinder, permitting the use

of L(p)/Lofor VIVo.

Typical values for the terms of eq (4) are:

Lo = 0.85 cm

4> = 1.5 X 10-4 cm bar '

V = 1.0 X 10-« cm bar^

7 = 6.7 X 10-3 cm

e — 0.005 bar- 1

16
h(p)-ho

Figure 11. Sample length and correction terms as function

of pressure for polyethylene and tellurium.

The dial gage reading h{p) increases as the piston

advances. Figure 11 illustrates the relative magnitude of the terms of eq (4) with data taken from a run

with polyethylene as sample material. The curve

for h{p) shows a large change at very low ram

pressures. This is mainly due to slack between

various parts of the piston stack, which is picked

up rapidly as the ram pressure increases. The fact

that this large initial change is not due to sample

compression only is illustrated in figure 12. Here

the cylinder was omitted, and the piston is pushing

directly against the tungsten carbide backup plate.

Even after all the slack is eliminated, the piston

stack is further compressed. This compression can be computed from the stress-dependent elastic

constants and the dimensions of the parts involved.

We prefer, however, to use the data shown in figure

12 to obtain the correction terms by fitting the

equation

h{p) = ho + y[l - e-'Pr] + (f>pr

+ -np-j- (5)

to the experimental data to determine ho, y, e, (p, and rj. Zero weight is assigned to ho, since a rea- sonable zero pressure dial indicator reading can- not be obtained either with, or without, sample

and cylinder. From the curve fitting operation, an extrapolated value for the zero pressure dial

indicator reading ho is obtained.

To solve eq (4), a value for ho has to be chosen

by extrapolating the h{p) versus pr curve to pr = 0. This is, however, often a very arbitrary value,

which may be off from the true value by as much as 0.005 in. A refined way of arriving at a value for

ho closer to the true value will therefore be discussed

below. With the sample length L known as a function of pressure, we can now proceed and determine

the density and bulk modulus. Assuming that a

solid sample was used we have

Figure 12. Elastic distortion of piston stack.

o\ 1 1 1 1 1— 0 2 4 6 8 10 kb

pressure

Figure 13. Bulk modulus and density ofpolyethylene. Influence of extrapolated zero pressure dial indicator reading.

where pip), po are the densities of the sample

material at pressure p and at p = 0, and

17
where Brip) is the isothermal bulk modulus.

Figure 13 shows the results for the data used in

figure 11. A quick glance at the p(p) curve does

not reveal any noteworthy effects. The Brip)

curve shows a break at the low-pressure end, while

the rest of the curve is more nearly a linear function

of pressure. If more runs than this one are compared, it appears that this break varies widely from

run to run. It is, in fact, the arbitrary choice of

ho which affects the lowest point of the Bt{p)

curve, since it appears as the difference of two

similar values h{p) and ho in the denominator

of eq (7). Experience shows that Bt is a hnear

function of pressure at least for compressions up

to F/Fo = 0.85. If we assume that for a limited

range

Bt{p)=Bo + BiP (8)

we can rewrite eq (4) and extrapolate ho from some

value of hip) taken at a higher pressure but from

within the range where eq (8) holds

, L/ \ Bo + Bip

ho = h{p) -^In (fypr Hi Do

-y[l-e"''']-/3ey,LoP, (9)

noting that

L{p)-Lo_

Lo

dp

Jo Bo + Bii

1 ^^^fl o + gip

Bx Bo

(10)

This extrapolation has been made for the data

in figure 11 and the extrapolated value for ho is indicated on the ordinate.

The extrapolated value for ho was then used to recompute L{p), p{p), and Brip). The dashed lines

in figure 13 indicate the improved values.

The computation of AL = L{p)—Lo for a downstroke is quite similar except that no correction

needs to be made for pickup of slack. Therefore,

we have for the downstroke

AL = L(p) —Lo=ho — h{p) +(f)pr + (3cy\Lop. (11)

If the sample is a fluid contained in a polyethylene

D^TT

sleeve, L{p) is determined as above. L{p)——,

4

with D diameter of the cylinder, is the total volume

of sample plus sleeve. The volume of the polyethylene to be subtracted from the total volume is

VpAp) = Vo,vE\l—^\n

L Oi.pE

Bq.PE + B\^pyJ)

Bo,PE

(12)

where Vp^{p) , Vo,pe are the volume of polyethylene

at pressure p and at p = 0, respectively,

and Bo,PE, fii,PE are the coefficients of a linear

bulk modulus equation analogous to eq (8).

D'^ — cF

Furthermore, Fo,pe = — ttLo, where c? is the in- 4

internal diameter of the polyethylene sleeve.

The volume of the fluid then is

^Fiuid(p) =—7-J^(p) 4 ttLoI ^0 1

1 ^p.pe + ^i.peP

Bl,PE Bo,PE

the density is

PFluid(p) =PFluid(o)

and the bulk modulus is

^^Fluid(o)

^r, Fluid (P) — ~ ^Fluid(o)

^Fluid(p)

Ap

Fluid

(13)

(14)

(15)

Again an extrapolation for ho will be necessary. It

is complicated somewhat by the presence of two

different materials in the cylinder. The expression to be used is

ho = h{p) —Lo

, Lo

+ d'[ 1

{D'^-cf) 1 Bx,

In ^0, PE'^ Bi, peP

PE B0, PE

_ 1 ^0. Fluid + ^1, FluldP \

^1, Fluid ^0, Fluid /

y[l-e-'Pr]-(/)p,-j8cyi^op. (16)

The systematic uncertainty of L{p) for a solid

sample is determined from the total differential of

eq (4). The significant terms of the differential and

their contributions to the systematic uncertainty

are shown in table 2 for 2 kbar and 30 kbar internal

pressure. We believe this to be a complete list of

all significant contributions. The total systematic

Uncertainty in the determination of L{p) is about

0.4 percent at 2 kbar and 1 percent at 30 kbar. As in

the determination of pressure, one of the largest contributions comes from cyUnder expansion. The

term prC?</) is related to the compression of the piston

stack. It could be reduced significantly if all measurements were referred to, say, p = 2000 bar, as is frequently done by other researchers [10] following

Bridgman's example [5] to avoid the difficulties

arising from the initial pickup of slack in the setup.

The large uncertainty in the determination of the

bulk modulus from dilatometric data is demonstrated in table 3 (compare also fig. 11). Listed are

the most significant terms only. A variation of all dial gage readings h{p) by the same amount in the

same direction does not have any influence on

Bt{p) , which depends on the difference between

two readings. Likewise, Br at some high pressure is not affected by a variation in ho.

18
Table 2. Significant terms of the total differential of L(p) and their contributions to the systematic

uncertainty of L(p)

Contribution to systematic

Terms of uncertainty

uilcciLdlJliy Lii iiic vdiiauic

at 2 kbar at 30 kbar

dho 0.00075 cm

] 0.0007 0.0007

[ (piston stack travel) dh .00025 cm J .0003 .0003

dy .00075 cm (slack) .0007 .0007

LopdPcyi 10"' bar' (cylinder exp.) .UUUo .UUzD

dLo 0.0013 cm (sample length) .0013 .0013

Prd<}) 1.5 X 10~^ cm bar"' (stack compr.) .0003 .0033

0.0036 cm 0.0088 cm

or 0.4 1 percent

percent

Table 3. Contributions to systematic

uncertainty o/Bx of tellurium at 35 kbar

Uncertainty of variable

Contribution to

total systematic

uncertainty

dPcy\ = 10"' bar"' (cylinder exp.) 17 kbar

dL = 0.008 cm (sample length) 0.5 kbar

d(f) = 1.5x10"^ cm bar"' (stack compr.) 141 kbar

Total systematic uncertainty 159 kbar

or 26 percent

of fi7-=590

kbar

These contributions to the total systematic uncertainty of Br were not determined

from differentials but from variation of input data of the computer program used to evaluate Bt. The reference value for Br was obtained from ultrasonic measurements

with appropriate corrections to convert from adiabatic to isothermal conditions.

Clip) _ L{p)

riip)"

ctip) _ L{p)

r,ip)

We must note, however, that L{p) as defined in

eq (4) is corrected for cyhnder expansion and that

for accurate evaluation t has to be corrected for

the same effect so that, with "cyi^ H")8cyiP?

Ci

L{p)

Ti(p)acyl'

Lip)

Tf(p)«cyl

(17)

The adiabatic bulk modulus Bs [24] is

«cyl \7iip) 3 T?(P

and since in a rigid cylinder or when suitable cor- rections are apphed

6. Evaluation of Ultrasonic

Measurements

The ultrasonic transit times r measured for

longitudinal waves (t;) and transverse waves

(tj) in soUds are readily converted into velocities

by division into the measured length Lip) of the

transit path

pip) = po

•

we have

Lip)

acyl

1 1 tKp) 3 Tfip)

(19)

(20)

The conversion of the adiabatic modulus Bs to the isothermal modulus Bt is made with the relations

19
Bt — ~ Vo

dp

dV (21)

where ap is the thermal coefficient of volume

expansion, ya is the Grlineisen constant,

7G

and

pCp

T is the temperature in K.

The problems encountered in the determination

of yc have been treated in several recent papers

[11, 12, 13]. If literature values for yc are not

available for the temperature and pressure range

in question, we can replace jg by ys at least for

simple solids from the modified Slater equation

[13]

13

1^ d In Ci _ 2 d \n Ct 3 a In (1/p) ~3 a In (1/p)

(22)

or we can determine yc as described below for

fluids from measurements along at least three isotherms. Often the term Tapy is small compared to unity and can be assumed to be constant.

Because of the large uncertainty in the determination of sample length [22] it may be advantageous

to eliminate L{p) in eq (20) and to compute Bs or Bt from ultrasonic data only. We note that the variation of L(p) with pressure for solids is relatively

small

L{p)=U 1

We can therefore replace Brip) under the integral

by Bsip) assuming apycT < 1 and, after combining

eqs (20) and (21), write for Brip)

BAp)

poLl

ItUp)

4 1

dp

JO acyl

1 4 1 3-2

t J

(24a)

Furthermore, we can assume that L{p) under the

integral in the correction term is constant and equal

to Lq. Whence it follows for Bt determined from

ultrasonic measurements alone:

The significant terms of the total differential of Bt

are listed in table 4, together with their contributions

to the systematic uncertainty of the bulk modulus

of tellurium at 35 kbar. The total systematic uncertainty is 9.3 kbar, which is about 1.6 percent of

the bulk modulus (^7 = 590 kbar). This result compares very favorably with the large uncertainty

shown in table 3 for the same material but from

isothermal measurements. Even if we allow for a

larger uncertainty of the term apycT, which was

not considered in table 3, the ultrasonic method is superior in those cases where ultrasonic measure- ments can be made.

We have not, for example, been able to make

ultrasonic measurements in polyethylene or in the

low-pressure form of polytetrafluorethylene because

of high attenuation. In the high-pressure polymorph

of polytetrafluorethylene, ultrasonic attenuation is quite low and ultrasonic measurements could be

made and evaluated, if the density were accurately

known at some pressure above the transition.

A word of caution must be said, however. The

solid samples in the piston and die assembly are subject to shear stresses, which may introduce

enough anisotropy to invalidate the ultrasonic

method [21]. This method should therefore only be

used for materials which have small shear strength,

small grain size, and a high degree of crystal

symmetry.

With liquids, no such problems exist, and it is therefore quite interesting to apply the ultrasonic

method to liquids contained in the polyethylene

sleeve described above.

Only longitudinal waves can be used. As in the

evaluation of eq (20) above, we choose not to use

the experimentally determined L{p) because of

its rather large uncertainty. Instead, bulk modulus,

speed of sound, and density are determined from

ultrasonic data alone.

The volume of the fluid Vf at pressure p is

Vfip) = ^fo exp — (l + apyoT)

i:

» dp

Bsip) (25)

where Vfo = d^TrLoj^ is the fluid volume at ambient

pressure.

If the adiabatic bulk modulus of the fluid

5 Up)

.it(p)

P(P) (26)

is not known as a function of pressure, a first approximation is made

Bt{p)-

1 4 1 tUp) s'tUp)

a^yi ( 1 + apycT)

poLl

«cyldp

1 4 1 rr(p) 3t?(p)

Bs.iip)

Lo

(27)

With this approximation we can now proceed to determine the approximate volume of the fluid

(24b) F/, 1 at pressure p. The volume of the polyethylene

sleeve is obtained from previous isothermal meas20
Table 4. Significant terms of the total differential of Bj determined from ultrasonic measurements, and

their contribution to the systematic uncertainty of Bx of tellurium at 35 kbar

Terms of

differential

Uncertainty of variable

Contribution to systematic uncertainty

2^»^»(r?(p) 3 r?(p))^^°

9 J i

3x?(p)

0.0013 cm (sample length)

5 X 10'" s (transit time of longt. mode)

10~* s (transit time of transv. mode)

2210 bar

5150 bar

1925 bar

Total systematic uncertainty 9285 bar or 1.6 percent of

^7X35 kbar) = 590 kbar.

urements of the bulk modulus fitted to a linear

function of pressure (see eq (12) and fig. 3).

From the sum of these volumes divided by the

cross-sectional area we obtain a first approximation

for the length of the fluid column

^ VPE{P) + Vf,l{p)

Z)27r/4 (28)

A first approximation for the density is obtained

from

PfAp)-Pfoy^- (29)

The first approximation length and density are now introduced into eqs (25, 26) and the computation

is repeated. This iteration is performed until the

change of the computed data from one cycle to the

next falls below a specified value. With water as a

test substance and pressures up to 12.5 kbar, four

iteration cycles were usually sufficient.

If measurements are made along three isotherms,

jG can be determined directly [14, 15, 16] by first computing the approximate volume along the

isotherms and then calculating

apc'-

Cp

with ocp^ji^^) and (^)^= -r(|^)^

(30)

An iteration process such as described above

but with two loops can be used to compute complete

and very accurate equation-of-state data over the

entire pressure-temperature plane.

We have made ukrasonic measurements on vari- ous liquids at pressures up to 40 kbar. Data obtained

for water along the 22 °C isotherm are used to illus- trate the merits of the method. The bulk modulus

data obtained from the iteration process using the

experimental transit time data [3] were fitted to a

linear function of pressure. The OMNITAB [17]

POLYFIT instruction was used for the least squares

fit. The result of the least squares fit for Bs and p

in bar is

5s = 23,222 + 6.677p.

The standard deviation of the fit was 743 bar. This

equation was integrated assuming constant apjGT

and converted to density. The density is plotted in

figure 14. For comparison, density data from the

works of P. W. Bridgman [18, 19] and L. H. Adams

[20] are entered. Our data are also in good agreement with the data by Vedam and Holton [16], who

used a fixed-path length interferometer. This

method provides greater accuracy but is Umited to much lower pressures.

Table 5 lists the contributions of various experimental data to the total uncertainty of density,

velocity, and bulk modulus of water at 10 kbar. It

is very interesting to note that the major uncertainties do not come from the transit time measurement, but from the pressure measurement and the

determination of the initial length of the polyethylene sleeve. This is even more obvious, when

the transit time t is determined with the pulse superposition method rather than from a simple delay

measurement. Pulse superposition will permit estimated uncertainties of less than one nanosecond

with a resultant contribution to the total uncertainty of the bulk modulus of less than 80 bar.

g cm'

1.3

density of water

at 22° C

integrated 6^= 23222 + 6.677 p

ultrasonic data

.Bridgman 1912 • Adoms 193!

5000 10000 bar

pressure

Figure 14. Density of water as function of pressure.

21
Table 5. Contributions of uncertainties in experimental data to the total uncertainty of the ultrasonically

determined equation of state of water {at 10 khar)

Experimental U ncertamty of Contribution to total uncertainty of

nil nntitv €xp€rifn€rit(il QUCLfitity

i-* c ii^iit y

rt 11 /If Tti fiHII 1 HQ

P 90 bar (rjim pressure) 0.00156 g/cm^ 5.5 m/s 478 bar

TLo 0.004 cm (sample length) .00186 g/cm^ 11.5 m/s 633 bar

Bt), I'K 1200 bar (bulk modulus of PE) .00017 g/cm^ 1.6 m/s % bar

0.0007 (Griineisen constant) .00021 g/m' 0.4 m/s 9 bar

Po 0.0005 g/cm' (density at p = 0) .00047 g/cm-' .3 m/s 52 bar

T 5 X 10-** s (transit time) .00064 g/cm^ 6.6 m/s 402 bar

Others .00025 g/cm3 4.5 m/s 149 bar

0.00516 g/cm^ 30.4 m/s 1819 bar

or 0.4 percent 1.1 percent 2 percent

P(io kbai = 1.235 g/cm^ C(,o kbar) = 2705 m/s fimo kbar) = 90350 bar.

7. Conclusions

The present evaluation of systematic uncertainties

of dilatometric measurements in simple piston

and cylinder assemblies demonstrates the large

influence of the deformation of the piston stack

(term </)pr) and of the expansion of the high-pressure

cylinder (term Lopftcyi) on the systematic uncertainties of density and bulk modulus derived

from such measurements. It also shows the relatively greater accuracy that can be achieved by

using ultrasonic methods under favorable conditions

to determine the mechanical properties of the

compressed materials. It appears that the development of improved dilatometric methods is imperative and that cylinder expansion and piston stack

distortion have to be known much better before

more sophisticated ultrasonic methods are employed.

8. References

[1] Jayaraman, A., Hutson, A. R., McFee, J. H., Coriell, A. S., Maines, R. G., ASME Annual Meeting 1966, Paper No. 66-WA/PT-lO.

[2] Heydemann, P., Houck, J. C, J. Res. NBS 71C, No. 1, 11

(1967).

[3

[4:

[5

[6: [7

[8:

[9

[10

[11

[12

[13 [14

[15

[16

[17

[18

[19

[20

[21 [22

[23

[24;

Heydemann, P., Houck, J. C., J. Appl. Phys. 40, No. 4, 1609

(1969). Heydemann, P., J. Appl. Phys. 38, No. 6 , 2640 (1967),

and J. Appl. Phys. 38, No. 8, 3424 (1967). Bridgman, P. W., Proc. AAAS 74, 399 (1942). Bridgman, P. W., Proc. AAAS 44, 38 (1913). McSkimin. H. J., Andreatch, P., JASA 34, No. 5, 609 (1962). Pistorius, C. W. F. T., Rapoport, E., Clark, J. B., Rev. Sci.

Instr. 38, No. 12, 1741 (1967). Tamayama, M., Eyring, H., Rev. Sci. Instr. 38, No. 8, 1009

(1967). Weir, C. E., J. Res. NBS 53, No. 4, 245 (1954). Bartels, R. A., Schuele, D. E., J. Phys. Chem. Solids 26,537

(1965). Anderson, O. L., J. Phys. Chem. Solids 27, 547 (1966). Pastine, D. J., Phys. Rev. 138, A767 (1965). Holton, G., J. Appl. Phys. 22, 1407 (1951). Holton, G., Hagelberg, M. P., Kao, S., Johnson, W. H., Jr., JASA 43, No. 1, 102 (1968). Vedam, R., Holton, G., JASA 43, No. 1, 108 (1968). Hilsenrath, J., et al., OMNITAB, A Computer Program for

Statistical and Numerical Analysis, NBS Handbook 101

(U.S. Government Printing Office, Washington, D.C.,

1966).

Bridgman, P. W., Proc. AAAS 48, 307 (1912). Bridgman, P. W., J. Chem. Phys. 3, 597 (1935). Adams, L. H., J. Am. Chem. Soc. 53,3769 (1931). Smith, R. T., Ultrasonics 1, 135 (1963). Montgomery, P. W., Montgomery, C, Wald, D. A., Rev. Sci.

Instr. 38, No. 8, 1073 (1967). Matsushima, S., Special Contributions, Geophysical

Institute, Kyoto University, No. 5, 117 (1965). Bergmann, L., Der Ultraschall (S. Hirzel Verlag, Stuttgart,

1954), p. 562.

DISCUSSION

P. M. Bell (Geophysical Laboratory, Carnegie

Institution of Washington, Washington, D.C.): There

are three ways to measure ultrasonic transit times

in an experiment of this type: a variable, cahbrated

delay for a comparison pulse, the pulse superposition or phase comparison techniques; and a direct

delay for a comparison pulse; the pulse superposimethod. But I think your sample length of about 1 cm is too short for that and I am surprised that you

get the stated accuracy. Have you used frequencies

above 10 MHz?

G. C. Kennedy (Institute of Geophysics and

Planetary Physics, University of California, Los

Angeles, California): The aim of your investigation

was to develop better compressibility measurements. We have been working on the evaluation

of precise shock wave data. These data are incredibly

22
good, far better than most of the people who made

them think they are. The difficulty, of course, lies

in the reduction of the Hugoniot to isothermal

conditions. Depending on who evaluates one and the

same curve you get different results. The problem

comes down to the variation of Griineisen gamma

with volume.

We have measured the specific volume of ten

compressible substances to three significant

figures. These include indium, calcium, potassium,

and so on. We find that our data agree to three

significant figures with data derived from shock

wave measurements, if the Dugdale-MacDonald

relation for the volume dependence of gamma is used.

For indium at 50 kbar our specific volume is 0.841, from shock waves we obtain 0.841. Bridgman's value is 0.802 or something like that.

Bridgman's values for gold and zinc agree both with our numbers and with those derived from shock

wave data.

Since with the Dugdale-MacDonald relation for

Griineisen gamma shock wave and static experiments yield the same results, we now have a way

to determine the variation of gamma by measuring

isothermal compressibihties and comparing these

with shock wave data. Our derived variation agrees

with Dugdale-MacDonald.

As a result tables of compressibilities of a large

number of standard substances are now being

recomputed at Livermore.

L. Thomsen (Columbia University, Palisades,

New York): In a paper by Orson Anderson and

myself we have derived yet another expression for

the Griineisen parameter as a function of density

which avoids all of the special assumptions that go

into the various curves which Dr. Kennedy just

drew. This paper will be presented on Wednesday

morning and we should then be able to pursue this

discussion considerably further.

D. J. Pastine (U.S. Naval Ordance Laboratory,

White Oak, Maryland): There is not any magic

formula that will give you the volume dependence of

the Griineisen parameter. Formulas like the Dugdale-MacDonald relation are essentially one-mode

approximations. There are, however, relations

which relate the Griineisen parameter to other

readily measurable thermodynamic quantities.

These will give you the first and second derivative

with respect to volume at low pressure.

AUTHORS' CLOSURE

In reply to Dr. Bell: We have in a few cases of

samples with low attenuation and the perfectly

alined endfaces used the more accurate phase comparison method. Usually we make a direct determination of pulse spacing with the help of a calibrated sweep delay. The position of the unmodulated rf-pulse is read at the peak of the pulse. This

reading is reproducible to within 10 ns. We also ex- pect it to be reproduced when we return to the same

pressure setting following a small loop. The time

between pulses measured in this way is equal to twice the transit time through the sample. However, in many cases five or more echos are observed

and the actual transit time can then be determined

to much better than 10 ns. Measurements are usually made at 10 to 12 MHz, but occasionally we go as high as 30 MHz.

In reply to Dr. Kennedy: If you have very accurate

isothermal data for the bulk modulus then we can

also use our ultrasonic data and, of course, thermal

expansivity data, to determine the Griineisen

parameter. We have tried this with our isothermal

data. But the accuracy of our isothermal data,

especially in the case of liquids, was not high enough

to yield really meaningful results.

The purpose of our paper was to emphasize how

much the accuracy of the determination of the

equation-of-state parameters can be improved by

using ultrasonic methods. This is particularly true

in those cases where the ratio of isothermal to adiabatic bulk modulus is close to unity. In all other cases we do have to rely on somebody's data for

the Griineisen parameter.

385-762 O - 71 - 3

23

Characterization of the Bismuth I-II and Barium I-II Points

Under Hydrostatic Pressure

R. J. Zeto, H. B. Vanfleet*, E. Hryckowian, and C. D. Bosco

Institute for Exploratory Research

U.S. Army Electronics Command, Fort Monmouth, N.J. 07703

Hydrostatic pressure experiments were made which have new and important relevance to high- pressure calibration in both liquid and sohd pressure systems for the bismuth I-II and barium I-II points on the high-pressure scale. The bismuth I—II transformations were characterized and the mechanism of the solid-state reactions was related to pressure cahbration. The strain hysteresis commonly

associated with the bismuth I-II point was shown to be invalid and was alternatively explained on the

basis of a thermally activated nucleation and growth mechanism which was demonstrated to govern

these transformations. With hydrostatic pressure, initiation of the bismuth i-ii transformation was

observed as a function of kinetic times as long as 15 hours while pressure and temperature were

maintained constant. The transformation behavior was a function also of the microstructure, purity,

and thickness of the sample. The bismuth I-II transformation was not necessarily sharp with respect

to kinetic time. As long as 42 hours were observed for completion of the transformation with very thin samples. Under hydrostatic pressure the bismuth I-II equilibrium point was not midway between the

initiation of the I-II and II—I transformations but was located appreciably closer to the former trans- formation. The bismuth I—II calibration point was correlated between liquid and solid pressure systems. For sobd systems it was shown that the "apparent" hysteresis was completely one-sided toward

the low-pressure side and that the equilibrium point coincided with the initiation of the I-II trans- formation which should be employed as the pressure calibration point. The bismuth I-II region of indifference was characterized with respect to nature of the sample and manner of measurement. An

experimental technique was indicated by which the center of the region of indifference was reproducible

within ±5 bars. The barium I-II point was measured at 56.27 kbar on the basis of a manganin gage

under hydrostatic pressure.

1. Introduction

The current high-pressure scale is based upon

fixed-point pressures of soUd-soUd phase transformations. Ahhough the bismuth I-II point at 25.380 kbars

at 25 °C is fundamental to the high-pressure scale

and is universally employed for high-pressure

calibration, very Uttle work has been done to characterize the nature of the I ^ II phase transformations upon which the fixed-point calibration

pressure is based. In view of experience and theory on the variable nature of solid-solid phase transformations, such a characterization and relation to the calibration pressure is warranted, both for

hydrostatic pressure systems which calibrate on the

equilibrium point and for solid pressure systems

which calibrate on the initiation of the transformations. This is particularly important for ultimate

accuracy of the bismuth I-II calibration point in the

former pressure systems in which pressures can be

measured with a sensitivity of 1 bar. The behavior

of the phase transformations can certainly be ex- pected to display an influence on the calibration

point in terms of this degree of sensitivity of pressure. More practically important at the present time

is the resolution in terms of phase transformation

theory of the several different calibration procedures

employed in solid pressure systems. During the last

several years in the high-pressure laboratory at Fort Monmouth, a unique hydrostatic pressure

capability was developed with a hexahedral press,

and hydrostatic pressure experiments were made

which have new and important relevance to highpressure calibration in both solid and liquid pressure

systems for both the bismuth l-II and barium I-II points on the high-pressure scale. In view of the

—t/3

*0n Sabbatical Leave from Brigham Young University, Provo, Utah. PRESSURE

Paper presented at the Symposium on Accurate Characterization

of the High-Pressure Environment, held at the National Bureau

ofStandards, Gaithersburg, Md., October 14- 18, 1968.

Figure 1. Schematic representation of the relative resistance of

bismuth for increasing and decreasing pressure in the vicinity

of the I-II calibration point.

25
large amount of information which resulted from

these experiments, the important results pertinent to pressure calibration and the goals of this conference

are presented in the manuscript in summary form.

More elaborate discussion and experimental detail

will be orally presented and will be submitted to

J. Appl. Phys. and to J. Phys. Chem. Solids in the near future.

To place the results of the present paper in context, it is helpful to review briefly the present

concept of the bismuth MI and II-I transformations.

A perusal of the literature reveals that the general

nature of these transformations, whether examined

by means of electrical resistance, volume change,

or x-ray measurements in either liquid or solid

pressure-transmitting systems, can be schematically

depicted as in figure 1 for the vicinity near the calibration point. With increasing pressure on a sample

composed completely of bismuth I, the I-II transformation initiates at a pressure corresponding to A.

After sufficient overpressurization to completely

convert all bismuth I to bismuth II, decreasing

pressure causes the II-I transformation to initiate at a pressure corresponding to B, which is substantially lower than A. Both transformations occur

isobaricaUy in liquid systems but may have to be

pressurized to completion in solid systems. Thus

the transformations to completion, C and D, exist as

illustrated for soUd systems but are vertical for

liquid systems. The pressure interval between

points A and B is commonly denoted as a hysteresis

for the transformations. In liquid systems this

pressure interval has been reported to have an

average value of about 0.71 kbar [1]', whereas in

solid systems the value is observed to be much

larger and is of the order of kbars due to the attendant and additional hysteresis of the pressuretransmitting medium, e.g., pyrophyllite or AgCl.

The transformations can be made to reverse over a much smaller pressure range near the equilibrium

pressure if the hysteresis is entered, either by

reducing pressure during the I-II transformation or by increasing pressure during the II-I transformation,

so that both phases are in intimate contact.

In his liquid system, Bridgman [2] measured a pressure range of about 60 to 100 bars within which

the transformation would not perceptibly run in

either the I-II or the II-I direction and denoted this

pressure range as the region of indifference. It is generally assumed, primarily because of a lack of

data, that the equilibrium pressure, E, is situated

halfway between the pressures at which the I-II and II-I transformations occur when measured

either at the hysteresis or at the region of indifference. Several investigators [3, 4], however, have

stated that such an occurrence might not be true

' Figures in bracicets indicate the literature references at the end of this paper.

with respect to the hysteresis, but in fact might

be located at a higher pressure, £".

2. Experimental

The basic apparatus was a hexahedral press with

2000-ton-capacity rams and a cubic pressure

chamber 2.25 inches on an edge. Hydrostatic pres- sures were generated within a stainless steel cell

which contained an equal-volume liquid mixture

of normal- and iso-pentane. The stainless steel cell

was situated within a pyrophyllite cube. As force

was applied to the pyrophyllite cube, the cell

experienced the uneven solid pressures within the

cubic volume, but the sample within the stainless

steel cell was exposed only to those forces transmitted by the liquid. The original hydrostatic pres- sure cell [5] was improved to increase its reliability

and usable high-pressure volume (Vie in diam X 1 in length) and to give 10 electrical leads for measure- ment from the hydrostatic pressure chamber. This

capability of volume and leads permitted fourterminal measurements, simultaneous comparison

of several samples in the same experiment, and

simultaneous measurement of manganin and

samples. For example, one experiment comprised a manganin coil, mercury (for calibration of the coil),

and four different bismuth samples. Four-terminal

measurements were always made on manganin

and also on the samples whenever possible. The

resistance of the manganin coil was measured with

a Leeds and Northrup Model 8069 G-2 Mueller

Bridge with a sensitivity of ±0.1 mil, which, for our coils, was equivalent to about ±0.5 bar. Quantitative experiments were conducted with a properly

seasoned and caUbrated (mercury L-I and bismuth

I-II points at 25 °C) manganin gage. Relative pres- sures from an uncalibrated manganin coil sufficed

for the examination of quahtative features of the

mechanism of the bismuth I-ll and II-I transformations. The sample parameters investigated were

purity (six nine's versus two nine's), microstructure

(single crystal versus polycrystal), and thickness

(0.040 in diam versus 0.013 in diam). Samples were

monitored with a sensitivity of at least 0.1 percent

either by means of a Keithley Model 150A microvoltammeter with the output recorded on a Varian

Model G-11 strip chart recorder, or directly by

means of a four-channel Model 240 Brush oscillograph-recorder. In order to avoid heating the

samples, the current level was adjusted such that

the power through the sample was kept at approximately 1.54 X 10"^ W. Temperature was measured

to 0.001 °C by means of a platinum resistance

thermometer immersed in a well of water which

made contact through a brass plug with the anvil

bearing on the pressure cell. Temperature corrections to 25 °C were applied for the resistance of

manganin and for the temperature coefficient of

50 bars/deg C for the bismuth MI point.

26
3. The Bismuth I-II Point Under Hydrostatic Pressure

3.1. The Apparent Hysteresis*

The hysteresis has important relevance to pres- sure calibration in solid pressure systems which

comprise the bulk of pressure apparatus in current

use. Two caUbration procedures are generally

employed in these systems: in one procedure only

the MI transformation pressure is measured and

the cahbration point is associated with point A in

figure 1, and in the other procedure both the I-II and II-I transformation pressures are measured and

the cahbration point is associated with point E in

figure 1. Obviously these are different calibrations,

and pressures reported on the basis of the one type

of cahbration will be different than those pressures

based on the other type of calibration. To resolve

this discrepancy it has been suggested [4] that the

former calibration method should add a pressure

corresponding to one-half of the hysteresis width,

i.e., A-E in figure 1. In any case, all of this type of

cahbration in sohd pressure systems depends upon

the hysteresis (both for bismuth and for the pressure

medium) exhibiting a fixed and reproducible value.

Two pubhcations [1, 4] associate a real hysteresis

with the bismuth I-II point, but each also implies the

possibihty of a thermally activated nucleation and

growth mechanism. Since a fixed strain mechanism

and a thermally activated mechanism are mutually

exclusive of one another, and in view of the obvious

importance of the hysteresis for solid pressure

calibration, it was considered desirable to conduct

hydrostatic pressure experiments to determine the

mechanism of the I-II and II-I transformations and

the basis of the hysteresis. Actually, results published in the literature appeared to indicate a

nucleation and growth type of mechanism. Also, our general experience with multiple cycles in

the hydrostatic system failed to reproduce a closed

loop with respect to resistance or to reproduce

either boundary with respect to pressure and cast

additional doubt on a fixed strain basis for the nature

of the hysteresis A-B as represented in figure 1. It was of particular interest to determine whether or not this pressure hysteresis was real, i.e., if the

initiation of the transformations beyond the equilibrium point was dependent upon a fixed strain energy

which would yield a fixed pressure interval /i-fi. The

alternative explanation of this pressure interval as being due to a thermally activated nucleation and

growth mechanism requires that the pressure interval ^-fi be time dependent.

Experiments made under hydrostatic pressure

showed that the hysteresis A-B of figure 1 was not a

real strain hysteresis but was time dependent

according to a thermally activated nucleation and

growth mechanism.

*See also "Interpretation of initiation pressure hysteresis phenomena for fixed-point pressure cahbration", Zeto and Vanfleet, J. Appi. Phys. (1970, in press) for later and more complete analysis.

By holding a sample composed completely of

bismuth I at constant pressure and temperature

below that represented by point A in figure 1, the

I-II transformation initiated as a function of kinetic

time. The observation of waiting periods prior to

initiation of the I-II and II-I transformations at constant temperature and pressure within the apparent

hysteresis has been noted numerous times for the

I-II transformation and also for point B and the II-l transformation. Furthermore the pressure coefficient of the waiting periods for the I-II transformation, for example, was consistent with a thermally

activated nucleation and growth mechanism; longer

waiting periods were observed for lower pressures

toward the equilibrium pressure. For the I II transformation, point A was located with a very slow

pressurization rate of 1 bar/min and then on dif- ferent cycles pressure was maintained constant at various lower values. Waiting periods as long as 15

hr at constant pressure and temperature were measured prior to initiation of the I-II transformation.

The waiting periods can be explained as a direct

consequence of the kinetic time for nucleation of the

new phase in a soUd state transformation. In the

vicinity of 150 to 200 bars over the equilibrium

pressure, the pressure coefficient of the waiting

periods appeared to be very large, about — 6 min/bar.

The waiting periods for the ll-I transformation did

not vary in a consistent manner with hydrostatic

pressure. The initiation pressure of this transformation appeared to be dependent on the pressure with

which the I-II transformation was overpressurized.

Hydrostatic pressure experiments in which

several different samples were simultaneously

examined and recorded showed very clearly that

the transformation behavior was a function of sample

purity, microstructure, and thickness. At constant

pressure and temperature, appreciably different

waiting periods for initiation of the l-Il transformation were observed for different samples. Also, the

times required for complete transformation to bismuth II varied not only from sample-to-sample but

also as a function of the holding pressure. Slower

transformation rates were observed for lower

holding pressure in accordance with a nucleation

and growth mechanism. Dramatically slow transformation rates were measured for very thin

samples. For example, a bismuth strip sample 0.0004

in thick required about 42 hr for complete transformation, whereas under the same conditions in

the same experiment a bismuth strip sample 0.011

in thick transformed completely in 1 hr. In the

literature, "sharpness" of the transformations is sometimes invoked as a criterion for "hydrostaticity" of pressure, but the role of kinetic time

in a solid-solid transformation must not be over- looked. Even though the transformations occur

isobarically in a hydrostatic pressure system, the

transformations may be very sluggish with respect

to kinetic time. The slow transformation rates are

further support for a thermally activated nucleation

and growth mechanism. The fastest initiation and

27
transformation rates were observed with thick

(0.040 in diam) polycrystaUine samples.

All evidence from these hydrostatic pressure ex- periments was consistent with a thermally activated

nucleation and growth mechanism. The hysteresis

which is generally associated with the bismuth I II point is not a real strain hysteresis but is actually an

apparent hysteresis due to a large activation energy

for nucleation and/or growth and the sluggish rates

of a solid-solid phase transformation. This apparent

hysteresis is variable with respect to kinetic time,

pressurization rate, and sample characteristics. The

assumption that the bismuth l-ll equilibrium point

is centered within the apparent hysteresis is also

invalid. The relationship of the equilibrium point

within the apparent hysteresis for liquid pressure

systems was determined and is illustrated in figure

2. The apparent hysteresis A-B was found with a slow and constant pressurization rate and was then

entered as described earlier for the measurement of

the region of indifference within which is the

equilibrium and calibration point. The region of

indifference was not at all midway between the

initiation pressures of the I-II and II-I transformations but was located appreciably toward the I-II side. The calibration point, E, was about 150 bars

below point A which corresponded to about Va

of the pressure interval A-B in the liquid system.

The transformations to completion, C and D, are represented as vertical lines since each transformation occurred isobaricaUy. The initiation

pressure of the II-I transformation, point B, was not

reproducible on different pressure cycles and after

completion of this transformation the initial bismuth

I resistance was not recovered. Additional evidence

of the variable nature of point B is shown in the

next section.

C I^I

PRESSURE

Figure 2. Schematic representation of the relative resistance of

bismuth for increasing and decreasing pressure in the vicinity

ofthe I-II calibration point under hydrostatic pressure.

Superficially these results have serious and detrimental implications for pressure calibration in

solid pressure systems since the "hysteresis" is not a stable transformation feature upon which to

fix a calibration. Consider, however, a nucleation

and growth mechanism in the context of general

experience with the kinetics of solid-solid phase

transformations. The observed waiting periods at constant pressure and temperature may be associated with induction times for the nucleation of the

new phase. It is known that scratching or grinding a

material wiU reduce and remove an induction

period, lower the activation energy for nucleation,

and facilitate the nucleation rate of the new phase.

Clearly, a scratching and grinding effect on the

bismuth sample can be expected from the solid

pressure medium and this would tend to eliminate

the waiting periods observed under hydrostatic

pressure. Theoretically, then, it would be expected

that initiation of the I-II transformation in a solid

pressure system should take place at a pressure

which corresponds, or at least is close to, the real

equilibrium point. Evidence that this correlation is valid is presented in the next section.

3.2. Correlation of the Calibration Point

Between Liquid and Solid

An experiment was designed to test the transformation hypothesis just mentioned in the previous

section. An experiment in the hydrostatic pressure

cell was conducted with two identical bismuth

samples: one sample was directly exposed to the

pentane-mixture pressure medium and the other

sample was completely embedded in an epoxy

solid (about 0.040 in wall thickness) which in turn

was directly exposed to the same pentane-mixture

pressure medium. These samples were simultaneously examined in the same experiment and

gave a direct comparison of the bismuth I ^ II transformations for a specimen subjected to a

hydrostatic pressure and one subjected to a solid

pressure-transmitting medium. The samples were

first pressure cycled with a slow, constant pressurization rate of about 15 bars/min in order to determine the apparent hysteresis for both samples. In

cycling the samples, pressure was taken well beyond

the bismuth II-III transformation in order to insure

the presence of only one or the other of phase I and

II at the initiation of the transformations. The true

equilibrium pressure was then measured with the

sample which was directly exposed to the hydrostatic pressure medium.

The correlation of the bismuth I-II equilibrium

point within the apparent hysteresis for liquid and

solid pressure-transmitting media is schematically

illustrated in figure 3. The equilibrium point, E,

measured with the sample in the liquid medium

corresponded within a few bars to the pressure. As,

at which the bismuth I-II transformation initiated

28
PRESSURE

Figure 3. Location and comparison of the equilibrium point

in solid and liquid pressure transmitting media for the bismuth

I ^ II transformations.

in the sample which was embedded in the solid

pressure transmitting medium. Verification of the

theory of the mechanism of the transformation was

obtained and these results provided striking cor- relation between liquid and solid pressure systems

for pressure calibration at the bismuth I-II point.

The hysteresis observed in solid pressure systems

is completely one-sided towards the II-I transformation. In solid pressure systems, calibration should

be conducted with a slow pumping rate and should

be fixed to the initiation of the bismuth I-II transformation.

This phenomenon was investigated in another

experiment in which an AgCl pressure-transmitting

medium was employed. The same effect was clearly

evident, although it was less pronounced than for

bismuth in epoxy. Comparison of the transformation curves from initiation to completion of the transformation, C, revealed a dependence on the nature

of the pressure transmitting media. This observation

was consistent with a thermally activated nucleation mechanism in which nucleation of the new

phase was accelerated by the pressure medium

which was more non-isotropic, i.e., from pentane to AgCl to epoxy. Initiation of the I-II transformation

occurred at the highest pressure,^/,, for the bismuth

sample in the liquid medium. The II-I transformation, D, was very fast in all cases but the initiation

of this transformation, B, occurred at a much lower

pressure when the II-I transformation was preceded

by the ll-lll transformations in the pressure cycle.

In this case the equilibrium point, E, was located

about the same pressure interval below Al as when

the II-III was not included in the pressure cycle, but

Ai,-E became only about one-sixth of the apparent

hysteresis Al-B.

3.3. The Region of Indifference Equilibrium

Point

The region of indifference comprises the equilibrium point and is employed for pressure calibration

in hydrostatic pressure systems. According to Bridgman [2], the region of indifference is measured

as the pressure range over which the transformations can be made to run in opposite directions when

starting from a pressure at which no transformation

is perceptible. The region of indifference is determined with about equal amounts of both phases I and II present in the sample. Nucleation is no

longer a consideration as the apparent hysteresis

and the transformations can be made to reverse over a pressure range less than 100 bars. Although

hydrostatic pressure systems comprise the minority

of pressure apparatus in current use, there are an

increasing number of laboratories utilizing these

systems in which small pressure changes of about 1 bar can be measured with a manganin pressure

gage. This precision cannot be utilized for pressure

calibration with a comparable accuracy, however,

since the equilibrium point has not been characterized with respect to the purity, microstructure, and

thickness of the sample, the width of the region of

indifference, the pressure direction of measurement,

and the variation of the equilibrium point with

respect to the relative proportions of bismuth I and

bismuth ll present in the sample. The pressure calibration point must be evaluated and considered in

the context of the solid-solid phase transformation

upon which it is based. General experience with

high-temperature transformations shows that transformations in the solid state are influenced by these

factors mentioned above. Characterization of the

I-II equilibrium point is therefore important for the

ultimate accuracy of hydrostatic pressure calibration, particularly in terms of the current techniques

which measure pressure with a sensitivity of 1 bar.

Hydrostatic pressure experiments were made to bear on such a characterization.

The technique which was experimentally employed to investigate the region of indifference was

as follows. The sample potential was recorded for 15-

min intervals at constant pressure. After the completion of an individual sample trace, pressure was

altered either up or down in increments of 5 to 8

bars and the sample potential was again monitored

for about 15 min at each pressure. The rate traces

of sample potential versus time were zero within the

region of indifference, positive for the II-I transformation, and negative for the I-II transformation.

The manganin resistance was measured at the

beginning and at the end of each trace and was

29
monitored on the recorder simultaneously with

the sample trace. Pressure was constant within 1 bar over the duration of each rate record of the

sample. The MI and II-I boundaries of the region of

indifference were noted as the pressure at which

the sample rate trace deviated from zero in the

negative and positive direction, respectively. The

equilibrium pressure was then taken at the center

of the region of indifference. By this procedure and

with the sample and manganin sensitivity employed,

20 bars was commonly measured for the width of

the region of indifference. This is the smallest

region reported to date.

With respect to the equilibrium point the question

arises as to the existence of a pressure region or a

single pressure value over which or at which the

transformation will reverse. Theoretically there

cannot be a single value of pressure for the equilibrium point. The bismuth I-II point is based on a

solid-solid phase transformation in which strain

energy is a thermodynamic variable. In the phase

rule sense, this extra degree of freedom means that

the transformation must occur over a range of

pressures at constant temperature instead of at a single pressure as normally expected for a pure

compound. While greater sensitivity than that employed in the present investigation might reveal

a region of indifference less than 20 bars, the region

of indifference will nevertheless not be zero. The

small value of 20 bars, however, is favorable for

utilization of the bismuth I-II transformation for

high-pressure caUbration. In terms of this discussion, the nature of the region of indifference is based

on the strain energy resulting from the volume

change between phases I and II and this is really the

strain hysteresis for the bismuth I-II point.

The nature of the region of indifference with

respect to pressure direction of measurement was

investigated with a polycrystalline bismuth sample

of six nine's purity and 0.040 in diam. Since the

determination of an equilibrium point should be

made by approach from each side of the equilibrium, there are two possible ways in which the

region of indifference may be measured: (1) I-II in

the up-pressure direction and II-I in the downpressure direction, and (2) I-II in the down-pressure

direction and the II-I in the up-pressure direction.

Four determinations were made by each method

with 45 to 55 percent bismuth I present in the sample

for each determination, and the results favored the

former procedure as employed by Bridgman. The

width of the region of indifference determined in

this manner was 20 ±6 bars, whereas a region of

indifference of 27 ±12 bars was determined by

the latter procedure. The average of each set of

four determinations gave the same absolute value

for the center of the region of indifference for each

method of measurement.

If the region of indifference is a strain hysteresis, as mentioned above, then this region could be a function of the relative proportions of bismuth I and

bismuth II present in the sample. Consequently,

measurement of the equilibrium point for highpressure calibration might Ukewise be a function of

the phase composition of the sample. To examine

this phenomenon the region of indifference was

measured at bismuth I proportions between about

20 and 80 percent with a polycrystalline sample of

six nine's purity and 0.040 in diam. In this percentage range the widths of the region of indifference,

about 20 bars in each case, overlapped. For bismuth

I proportions between about 20 and 60 percent, the

centers of the region of indifference were within

about 5 bars of each other; however there appeared

to be a definite bias, about 10 bars, towards lower

pressures for the measurements at higher proportions of bismuth I. This effect was also observed for

different samples with larger grain size (single

crystal) and less purity (two nine's). In terms of the

sensitivity of our pressure and sample measurements, this bias appears to be real, although it is difficult to make such a conclusion with certainty

for such small pressure values in view of possible

and unknown systematic errors of measurement.

Investigation of this point by other laboratories

seems desirable. In any case, it appears that highest

reproducibility (several bars) of the calibration point

is achieved when the pressure is adjusted so that

about 50 percent of phase I exists in the sample

when the region of indifference is measured.

As indicated above, the region of indifference was

investigated as a function of the purity (six nine's versus two nine's), microstructure (single crystal versus extruded polycrystal), and thickness (0.040 in

diam versus 0.013 in diam) of the sample. The center

of the region of indifference was reproducible within

±3 bars with respect to microstructure and thick- ness. Surprisingly, but favorable for pressure

calibration, there was very little difference between

six nine's and two nine's purity, the center of the

region of indifference being about 10 bars higher

for the less pure sample. The widths of the region of

indifference overlapped for all samples. The most

noticeable effect of these sample characteristics

was that the width of the region of indifference

became twice as large as a function of sample

thickness from 0.040 in diam to 0.013 in diam, even though the center of the region was the same in each

case. This width of the region was even 50 percent

larger (about 70 bars) when measured by method (2)

rather than by method (1) mentioned earlier. For

all samples except the latter, the width of the region

of indifference was about 20 bars.

The results described above have the following

relevance to hydrostatic pressure calibration with

the bismuth I-II point. First of all, the caUbration

point is reproducible at least within 50 bars regardless of pressure direction of measurement, proportion of bismuth I in the sample (between 20 and 80

percent), and sample characteristics (within the

sample variation investigated) of purity, microstructure, and thickness, except for the wide region

30
for very thin samples. With the exception noted,

all of these factors displayed their influence within

±25 bars of the calibration point. The influence

of these factors on the nature of the I-II transformations has been assumed to be negligent in previous

high-pressure calibration, but these factors have a

basis in phase transformation theory and in ex- perience with high-temperature transformations

and it is comforting to have experimental verification of the extent of their influence. These results

further indicate that the absolute reproducibility and

accuracy for pressure cahbration could be reduced

to about ±5 bars for the center of the region of

indiff'erence and to about ± 13 bars for the complete

width of the region of indifference by employing a pressure and sample sensitivity comparable (or

better) than that in this work and if the calibration

is conducted starting with a pure single crystal

sample about 0.040 in diam and by measuring the

region of indiff^erence after the manner of Bridgman

with about 40 to 60 percent bismuth I present in

the sample.

4. The Barium l-II Point Under

Hydrostatic Pressure

Most high-pressure measurements above 40 kbar

depend upon a calibration curve based on the barium

I-II point. The importance of the point for accurate

high-pressure measurements is clearly evident,

particularly for pressures greater than 60 kbar.

The barium l-ll point has been measured [6, 7, 4, 8]

with piston-cyUnder devices and rotating freepiston techniques and with tetrahedral apparatus

and x-ray techniques in conjunction with equationof-state. In view of the present uncertainty

associated with the cahbration pressure for this

transformation, the hydrostatic pressure cell

described earlier was employed to provide an independent measurement of the barium I-II point, this one with hydrostatic pressure and based on a calibrated manganin pressure gage.

Pressure is hydrostatic to the extent that the viscosity of the pressure medium permits a fast relaxation time for bulk stresses. The pressure-transmitting medium in the hydrostatic pressure cell was a

1 : 1 volume mixture of normal- and iso-pentane in

which pressures equalized by viscous flow through a prescribed capillary tube in times of the order of

seconds at 50 kbar and of minutes at 60 kbar [5]. An approximate value of 10^ poise at 60 kbars has

been measured for this pressure medium [9]. The

pressures generated at the barium I-II point were

hydrostatic within this context.

The relative resistance change, Ar/ro, of a properly seasoned manganin coil was calibrated

with pressure by means of a quadratic equation

P = ^(Ar/ro)+fi(Ar/ro)2.

whose coefficients A and B were determined by a

two-point calibration with the accepted pressures

and the measured relative resistance change of the

manganin coil for the mercury L-I [10] and bismuth

I-II [11] transformation at 25 °C. At the barium

point slow pressurization rates were employed and

adequate time for pressure equilibrium was ahowed.

The manganin relative resistance associated with

the region of indiff^erence of the barium l-II transformation was measured and the pressures corres- ponding to the width of the region of indifference

were calculated from extrapolation of the manganin

cahbration curve. Using this same experimental

and extrapolation technique, good agreement was

obtained for measurement of the thalhum II-III cahbration point.

The width of the barium I-II region of indifference

was 2.32 kbars according to manganin. This width

is appreciably larger than the region of indifference

measured with this same technique for bismuth

I-II and thaUium Il-lll, and is probably a reflection,

at least to some extent, of the decreasing degree of

"hydrostaticity" of the pressure medium. Nevertheless the pentane-mixture pressure medium is certainly more hydrostatic than the solid systems

employed for the barium I-II point, and the present

measurement of the region of indiff'erence is the

smallest reported for this transformation. According

to the usual procedure the equilibrium pressure was

determined from the center of the region of indifference and was 56.27 kbars. This value is 2.5 kbars

lower than the presently accepted value of 58.8

kbars employed for high-pressure calibration.

Based on this data, therefore, either the presently

accepted equilibrium pressure for the barium I-II transformation is high by about 2.5 kbars or the

curvature of the manganin curve changes quite

sharply between 37 and 59 kbars. Several other

recent measurements of this transformation have

indicated that this calibration point should be

lowered. Both of these measurements, 54.4 kbar

[4] and 55.0 kbar [8], are still lower than the value

obtained on the basis of manganin.

The results presented earlier for the comparison

of the bismuth I-II point in solid and liquid pressure

media have an interesting and potentially significant

implication for measurement and calibration at the barium I-ll point. If the mechanism of the barium

I-II transformation is one of thermally activated

nucleation and growth then, analogous to phase

transformation theory and verification in the case

of bismuth I-II, in solid pressure systems one would

expect the real equilibrium point to be closer to

initiation of the barium I-II transformation than

to the average of the I-II and II-l initiation pressures,

particular in view of the very wide pressure interval

measured between these transformations in the

solid systems.

5. Acknowledgment

Grateful acknowledgment is expressed to Dr.

31
H. H. Kedesdy for his encouragement during the course of this work, to Mr. G. E. Tomes for his able

assistance with the experiments, and to Mr. F. J. Becker for his expert machining of the materials

needed for the experiments.

6. References

[1] Davidson, T. E., and Lee, A. P., Trans. Met. Soc. AIME

2 30, 1035 (1964).

[2] Bridgman, P. W., Proc. Am. Acad. Arts Sci. 74, 1 (1940).

[3] Roy, R., 6th International Conference on the Reactivity of

Solids, J. Wiley & Sons, New York, N.Y. (Jan. 1969).

14J Jettery, R. N., Barnett, J. D., Vanfleet, H. B., and HaU, H. T.,

J. Appl. Phys. 37,3172 (1966).

[5] Barnett, J. D., and Bosco, C. D., Rev. Sci. Instr. 38, 957

(1967).

[6] Bridgman, P. W., Phys. Rev. 48 , 893 (1935).

[7] Vereshchagin, L. F., Zubova, E. V., Buimova, I. P., and

Burdina, K. P., Soviet Physics-Doklady 11, 585 (1%7)

(English translation), Dokl. Acad. Nauk, S.S.S.R. 169,

74 (1966), (original Russian).

[8] Haygarth, J. C, Getting, I. C, and Kennedy, G. C, J. Appl.

Phys. 38, 4557 (1967).

[9] Barnett, J. D.. and Bosco, C. D., J. Appl. Phys, 40, 3144

(1969).

[10] Boren, M. D., Babb, S. E. Jr., and Scott, G. J., Rev. Sci.

Instr. 36, 1046 (1965).

[11] Kennedy, G. C., and LaMori, P. N., J. Geophys. Res. 65,

741 (1960).

DISCUSSION

G. S. Kell {National Research Council of Canada,

Ottawa, Canada): We measured the phase equilibrium between ice-I and ice-III and got a similar sort of result in that as you increased the pressure the

transformation is rather sharp. When you lower the

pressure, you can lower the pressure by hundreds of

bars, without the transformation going back. We

explained this by saying that as you lower the pressure, the low-pressure phase gives mechanical

support to the high-pressure phase, which is then at a higher pressure than the hydrostatic pressure in

the system.

Now I ask: Could this sort of phenomen apply in

your case or, if not, is it possible that we have completely misdiagnosed our system? The phenomenon

is quite asymmetric that as you raise pressure, once

your transformation starts it runs to completion. As

you go from high to low, it can be undershot many

hundreds of bars and, if fact, perhaps you always

have high-pressure nuclei left.

J. D. Barnett {Brigham Young University, Provo,

Utah): You say you experience more consistency

going from I-II than II-I. Is this not just due to the

fact that generally you always run that transition

going I-II initially and, therefore, you have no nuclei

in the II phase at all? If you would cycle several

times in the region of transition, wouldn't you get as much variability going either way? If you were to go to 50 kilobar and come back, then would it be

reproducible? I wouldn't want to leave the impression that the II-I transition is inherently unreliable.

It looked as though, if you go to high enough pres- sure or low enough pressure, and get far enough

away from the transition so that no nuclei are present at all, it would be reproducible either way.

R. E. Hanneman {General Electric Research and

Development Center, Schenectady, New York):

Perhaps it would be instructive if we discussed the

nucleation expressions and maybe it would become

clearer as to why this hysteresis can occur on either

side, in principle. If one simply takes the normal

nucleation type expression, one obtains a pre-exponential term which I have shown to be essestiaUy

pressure-independent.' One obtains the pressure

dependence in the activation energy for the nucleation term. As you increase the super-pressure at fixed temperature, the critical nucleus size becomes

smaller.

The main point is that these are heterogeneously

nucleated transitions and not homogeneously nu- ated systems; in order to heterogeneously nucleate

a tetragonal phase versus an orthorhombic phase

the terms that go into the heterogeneous nucleation

barrier would be different going toward the orthorhombic phase versus toward the tetragonal phase.

For instance, we have an interfacial energy term

that goes into the nucleation expression. This term

can be much more pronounced in either direction,

depending on a specific heterogeneous nuclei

present, be it the cell walls or impurities in the

system. So I would submit we cannot assume the

hysteresis would always be on the low side. It could

be either way for transitions in different materials.

One other point related to this is if you use a homogeneous nucleation you end up with the largest

possible activation pressure or super-pressure

needed, so one might explore around and look for the

most heterogeneous nuclei that you can find to minimize the hysteresis. In other words, you are working with a purposely dirty system, if you will,

in terms of surface contact properties.

Another possibility would be working with a calibration type system where the free energy driving force versus super-pressure is very large and,

therefore, as you go up just a Httle bit in superpressure, it will enormously decrease the activation

of free energy needed to nucleate the process.

As you run a bismuth transition at a higher temperature, if this model is correct at all, then the

region should converge between hysteresis loops

' R. E. Hanneman, in Reactivity of Solids, John Wiley & Sons, Inc., New York,

edited by J. W. Mitchell et al., pp. 789-802 (1969).

32 4.
because your activation barrier for nucleation is dropping.

S. E. Babb, Jr. {University of Oklahoma, Norman,

Oklahoma): It seems to me, if you are interested

in measuring pressures better than say a tenth of a

kilobar, that you have got all kinds of problems with

this region of indifference. It is not clear where the

center is in terms of technique. For example if I measure a region of indifference of 70 bars, it is not clear that my 70 overlaps your 20.

I suggest if it is at all possible, when you are interested in pushing towards the bars, you should

avoid the solid-solid transition for calibration.

Therefore, in the 20-kilobar region, I would like to suggest, on the basis of Kennedy's measurements,

we go to melting and gallium.

R. Roy (Pennsylvania State University): I wish to expand on a point related to Dr. Hanneman's

remark. There is little question that the phases

formed either up or down pressure are in general

heterogeneously nucleated. However, what is seldom considered is, the nature of the phase nucleated. The way we write our equations misleads us

into thinking that each phase comes out in some

thermodynamically standard state as a single crystal. Experimentally, in most cases even a single

crystal wiU transform into an array of microscopic

crystals with all kinds of residual strains, enormous

grain boundary energies and possible orientation.

(Very often so-called semiconductor-metal transitions only measure the low resistivity of the grain

boundaries). Hence we actually observe the transition A —* B*. Conversely on the down pressure

cycle depending on how well annealed the sample is,

we certainly observe a reaction of the type B*—* A*

oxB-^A*.

The free energy differences between A and A*

and B and fi* may be sufficient to cause a displacement of a few degrees or bars of the equilibrium

curve. (In the case of MgO, the difference can be

equal to 10 percent of the total enthalpy, see Weber

and Roy, Am J. Sci. 263, 668 (1965). Even in atmospheric pressure work we do not know a transition

any better than ± 1 °C in the very best cases; perhaps to ±0.2°C in quartz. Hence I beheve that

we are up against the limitations of the physical

realities involved and will not be able to obtain

meaningful improvement of precision or accuracy

beyond the point now reached by these authors.

F. P. Bundy {General Electric Research and

Development Center, Schenectady, New York):

I have a question on the mechanics. You showed

very, very slow increase and decrease in pressure

with time. I wondered, if you are reading the pres- sure on the manganin gage in the fluid cell, whether

you observed any jumps? Most apparatuses where

you have solid gaskets, and so on, between the

flanges and punches will go by steps. Did you ob- serve that? The phenomenon I am referring to is stick-slip in the solid portion of the cell between the

punches and the gaskets, and the punch faces and

the pyrophyllite, as you increase the force on the

punches. Did you observe jumps in the cell pres- sure due to the stick-slip phenomenon? The pres- sure in the center of the cell doesn't necessarily

go smoothly.

AUTHORS' CLOSURE

In reply to Dr. Kell: We do not believe that the

large under-pressurization of the equilibrium

pressure required for initiation of the bismuth II-I transformation was due to mechanical support from

the low-pressure phase to the high-pressure phase,

primarily since the under-pressurization was completely eliminated when pressure was released

before the l-II transformation was completed such

that phase I was present in the sample. Also the

II-I transformation always proceeded to completion even after initiation instead of halting due to the

increase of pressure or mechanical support caused

by the volume change. Actually in our system we

measured a pressure increase of about 10 to 20 bars

concomitant with the transformation but completion

was attained nevertheless. Also it seems unlikely

to us that the formation of the first amounts of the

transformation could cause a higher pressure in the

matrix of the high-pressure phase equivalent to the

1.5 kbar under-pressurization observed in some

cases.

In reply to Dr. Barnett: Our data showed that the

initiation of both the I-II and II-l transformations

were a function of the microstructure of the sample

and we suggest that the II-I transformation is more

sensitive to this variable. We believe that this effect

is real, since we pressure-cycled the same sample in

the same experiment many times and for the I-II transformation only on the first pressure cycle

there was no possibility for phase II nuclei present

in phase I. In order to insure as best we could that

only one phase was present in the sample prior to the measurement of any initiation pressure, the

sample was generally allowed to sit overnight at a pressure several kilobars from the equilibrium

pressure. Under these conditions it seems unlikely

that residual nuclei of the old phase remained in

the sample although, of course, we cannot be

unequivocally certain. Perhaps the II-I transformation would be more reproducible by going to the

50 kbar or some other pressure but the sample

microstructure can also be expected to be a function

33
of the pressure treatment of the sample, particularly

in solid pressure media. Fortunately for pressure

calibration with the bismuth I<=^II point the results

that we presented showed that the initiation pres- sure of the II I transformation should not be included

in the calibration technique in any case. In reply to Dr. Hanneman: Our data were related

specifically to the bismuth I^II transformations and

we agree entirely that for other transformations the

location of the equilibrium pressure between the

initiation pressures is characteristic of the particular

material and may be toward either the high- or the

low-pressure side, it being less likely to be centered

due to the factors involved in the nucleation process

of each transformation. More detailed discussion

of the present results in terms of the thermodynamic

and kinetic aspects of phase changes will be found

in the separate publications of this work. We also

believe that nucleation of these transformations

is heterogeneous but regarding the use of impurities

to facilitate nucleation it should be mentioned that one must be careful that the purity and the calibrant

do not form a two-component system so that the

equilibrium pressure is altered. In this context we

showed that initiation of the I-II transformation

occurred first in two nine's purity bismuth versus

six nine's purity bismuth but the important point for

calibration was that the equilibrium pressures

measured for the same samples differed by less than

10 bars and were within the experimental uncertainty of the determinations. Also it should be

emphasized that the initiation of the bismuth I-II transformation is important for calibration only for

pressure systems in which the equilibrium pressure,

that is the true calibration point, cannot be measured. Our data showed that, in solid pressuretransmitting media, the initiation pressure of this

transformation with slow pressurization rates

corresponded quite closely to the equilibrium pres- sure and provides a good calibration. We have not

yet examined bismuth I <^ II above room temperature, but at higher temperatures we also would

expect to observe a smaller pressure interval between the initiation pressures with convergence of

the initiation pressures toward the region of

indifference.

In reply to Dr. Babb: We agree in principle that a

solid-liquid transformation rather than a solid-solid

transformation would be better for calibration, since

then there would be no region of indifference and

also initiation would always occur at the equilibrium

pressure. We have just commented on the favorable

coincidence of the initiation and equihbrium pres- sures for bismuth l-ll in sohd media and also we

extensively examined the width and reproducibility

of the region of indifference and found it to be quite

satisfactory for cahbration. We made 28 determinations of the region of indifference as a function of

the purity, microstructure, and thickness of the

sample, of the pressure direction of measurement,

and of the relative proportions of the two phases

present in the sample. The standard deviation of the

center of the region of indifference for these 28

determinations was ±0.008 kbar, which appears to be good evidence that the region of indifference

should be reproducible from laboratory to laboratory.

In reply to Dr. Roy: We agree that the region of

indifference of bismuth I «^ II is a good calibration

point even though it is a solid-solid transformation.

Although a solid-liquid transformation with zero

region of indifference is ultimately desirable, the

width of the region for bismuth l-li is smaller than

the present accuracy of absolute measurement and

also compares favorably with the precision of temperature measurements if one considers from P-T

equilibrium boundaries a rough equivalent of 50 to 100 bars per degree.

In reply to Dr. Bundy: For the slow pressurization

and depressurization rates of 1 to 25 bars/min that

were employed for detection of initiation of the

transformations and for the slower rates employed

for the region of indifference measurements, jumps

in the liquid cell pressure were only observed when

the hydraulic pressure behind the rams was in- creased externally to purposely change the cell

pressure. These jumps in the cell pressure were

probably not due to stickslip in the pyrophyllite

since there was no noticeable time delay between

the external hydraulic pumping and the response of

the manganin recording. After manganin started to change due to the external pumping the manganin

increased smoothly and leveled off at the new value

without discrete jumps in the recorded manganin

trace except in very infrequent cases. In these cases the jump in the trace was always less than the

increment in the pressure rate. The sensitivity of

the recording was such that pressure changes of 1 bar were clearly perceptible. Consequently for slow

pressure rates the external pressure changes

appeared to be accommodated in the pyrophyllite

part of the pressure chamber and gaskets by relaxation rather than by discrete stick-slip phenomenon.

34
session II

Chairman: E. C. Lloyd

National Bureau of Standards

Washington, D.C. 20234

The Upper Bismuth Pressure Calibration Point

J. C. Haygarth, H. D. Luedemann^, I. C. Getting, and G. C. Kennedy

Institute ofGeophysics and Planetary Physics, University ofCalifornia, Los Angeles, California 90024

The Bi ni-v and IV-V equilibrium boundaries were studied by monitoring electrical resistance

changes. Extrapolation of the III-V data to 25 °C yields an equilibrium transition pressure of 77.5 ± 1.0 kilobar (kbar). The extrapolation also agrees within experimental error with the pressure of 78.2 kbar

determined in a separate experiment at 22 °C. The use of the transition as a calibration point is dis- cussed, and a procedure for estimating the compression-stroke transition pressure from the equilibrium

transition pressure is proposed.

1. Introduction

In this paper, we describe a redetermination of

of portions of the bismth III-V and IV-V equihbrium

boundaries, with particular emphasis on a more

accurate estimation of the pressure of the bismuth

III-V transition at room temperature.

2. Experimental Procedure

End-loaded piston-cylinder apparatus as described

by Boyd and England [1] * was used, with modifications to enable it to reach higher pressures. The

pressure vessel used in this study had a core of

tungsten carbide cemented with 6 percent cobalt

(Carboloy 883). It was found that Carboloy 883

cores suffered a maximum of 0.1 percent permanent

enlargement of the bore diameter after exposure to the highest pressures, only one-tenth of that suffered

by Carboloy 55A. The pistons, diameter 1.270 cm,

were of Carboloy 999, which contains 3 percent

cobalt binder. The phase transformations were

detected by following the electrical resistance of the

sample. The setup is shown in figure 1. Silver

chloride was used as a pressure-transmitting

medium.

There was no visible evidence of reaction between

the bismuth and silver chloride. Examination of the

samples after the runs with an electron microprobe

failed to reveal the presence of any silver in the

bismuth.

' Publication #674, Institute of Geophysics and Planetary Physics, University of

California, Los Angeles.

2 Present address; Institut fur Physikalische Chemie und Elektrochemie, Universitat Karlsruhe, 75 Karlsruhe, Kaiserstrasse 12, W. Germany.

^ Figures in brackets indicate the literature references at the end of this paper.

Paper presented at the Symposium on Accurate Characterization

of the High-Pressure Environment, held at the National Bureau of

Standards, Gaithersburg, Md., October 14-18, 1968.

The resistance of the sample was followed by

passing a constant current through it and observing

the change of potential difference across it. The

potential drop across the sample was suitably biased

and displayed on one channel of a two channel

Honeywell "Electronik 16" recorder with a sensitivity of 1 mV. full scale deflection. Changes of

resistance of about 0.5 percent of that due to the

III-V transition were detectable.

The transition temperature at a given compression

was found by raising the temperature until the

transition was partially completed, then reducing

the temperature until the back reaction began to run. A dead span temperature interval in which the

UPPER END-LOAO PLATE

(CARBOLOr 883)

4-HOLE ALUMINA TUBE

MICA INSULATION

COPPER LEAD WIRES

TOOL STEEL SPACER

UPPER END -LOAD SHIM

STEEL SUPPORT

RINGS

STAINLESS

STEEL SEAL STEEL SLEEVE

TALC CORE (CARSOLOr

883) CARBON FURNACE

PVROPHYLLITE

SILVER CHLORIDE

PYROPHrLLITE LEAD FOIL, 0,0025 CM THICK

LOWER END -LOAD

SHIM

LOWER END- LOAD PLATE (STEEL) SLIDING BINDING

RING

CARBON

STEEL SUPPORT SLEEVES HARD STEEL SHIM STATIONARY

BINDING RING PISTON PUSHER

(CARBOLOY 883)

Figure 1, Apparatus and sample assembly.

35
reaction does not run detectably in either direction

with both phases present was found. This temperature interval was independent of the relative amounts

of the two phases present. It ranged in size from

about 4° at the lowest temperature to about 0.5° near the triple point for the III-V boundary, and was

0.5° for the IV-V boundary near the triple point and

less at higher temperatures. The transition temperature was taken as the mid-point of this interval.

This method of locating the boundary is referred

to later as the isobaric method.

The III-IV boundary was detected, and found to be

associated with large hysteresis, confirming the

findings of Klement et al. [2]. The hysteresis was so large that occasionally the metastable III-V boundary

was observed in the neighborhood of the III-IV-V triple point. The III-IV reaction was sc sluggish that no attempt was made to delineate accurately the

equilibrium boundary.

The III-V boundary was also studied at a number

of constant temperatures by cycling the pressure.

This is referred to later as the isothermal method.

The III-V boundary was also observed at room

temperature using a setup similar to that described

for the barium transition [3]. In this case the pres- sure was varied continuously at about 0.2 kbar per

second.

In aU experiments, the pressure of the oil activating the hydrauhc ram that drove the piston was

measured with a Heise Bourdon-type gage.

3. Results and Discussion

The data from the isobaric experiments were in

the form of thermocouple emfs and oil-pressure

values for measurements taken on compression and

decompression. They were plotted and resulted in

two similar curves whose separation was the doublevalued friction. It ranged from about 1.2 kbar at the

highest temperature to 1.6 kbar near the triple point

for the IV-V boundary, and from about 1.6 kbar near

the triple point to about 3.0 kbar at the lowest temperature for the III-V boundary. On the assumption

that the friction was symmetrical, the mean of compression and decompression values was taken at fixed values of thermocouple emf.

The results are plotted in figure 2. Circles represent points from the isobaric measurements, ex- cept that points on the metastable projection of the

III-V boundary are shown as squares. Triangles denote points derived from the isothermal experiments, and the point represented by a diamond is the result of the room temperature experiment.

The data for the isobaric experiments, in which

the equilibrium transition temperature was measured, were fitted to polynomials of degree two

through ten using a least-squares technique with the

U.C.L.A. IBM 360 computor. For both the III-V and IV-V boundaries the standard deviation was

essentially constant for all the polynomials of second

degree and above.

1 1 1 1

400

The Phase Diagram

of Bi

p 300 Liquid Y

\

Q.§ 200

100

m. \

\ll\

I \ \

1 1 W 1

nr

0 20 40 60

Press Kb

m O

o

o

*

isobaric dafo metastable M -Y boundory

isothermal data room temp, experiment

extrapolated values

1 1

1 1

1

1

250

200

° 150

Q.e

100

40 45 50 55 60 65 70 75

Pressure, kbar

Figure 2. Bismuth ni-v and iv-v equilibrium boundaries.

36
With P in kilobars and T in ° K, the second degree

equations were

P///-. =96.343-1.4431 X 10-3^-2.0774 X 10-"^

and

P,v-y = 54.192 + 1.0425 X lO'T- 2.3354 X \Q-*T\

The standard deviations were 0.08 kbar and 0.09

kbar respectively. Combination of these expressions

gives the III-IV-V triple point coordinates as 174.6°,

54.3 kbar, which compares well, expeciaUy as to temperature, with the result of Klement et al.

[2], 174°, 52.6 kbar.

Equations for the III-V boundary give the value for

the transition pressure at 25°, as 77.5 kbar; and predicted an abolute zero transition-pressure of 96.343

kbar. The extrapolated transition pressure at 22°

is 77.9 kbar, in good agreement with the measured

value of 78.2 kbar.

The pressure proposed for the lll-V equilibrium

boundary at 25° is 77.5 ±1.0 kbar, allowing for

absolute uncertainties in both pressure and temperature determinations. The standard deviation

of the data from the expression with which they were

extrapolated is less than one-tenth of the accuracy

estimation. Previous estimates of the III-V transition

pressure at room temperature have ranged from

88 to 90 kbar by Bridgman [4] and Vereshchagin

et al. [5] to 73.8 ±1.3 kbar by Jeffery et al. [6]. The consensus of recent opinion has favored a value near 80 kbar as reported by Klement et al. [2],

Stark and Jura [7], and Giardini and Samara [8]. Of the four [2, 4, 6, 7] papers in which the estimated pressure is assumed to be the equilibrium

pressure, only two [2, 4] describe the use of apparatus appropriate to measuring pressure directly.

The others [6, 7] describe techniques in which an

indirect method of calibration was used. The value

of 78 to 82 kbar suggested by Klement et al. [2] was

tentative and based on a long extrapolation of sparse

data obtained in double-stage piston-cylinder apparatus, where pressure determinations are less

certain than with single-stage apparatus. Considering this, the agreement with the present result is good. The fact that Bridgman [4] reports a value

about 10 kbar above the present value is consistent

with the finding that Bridgman overestimated the

pressures of a number of transitions frequently

used as calibration points [3, 9]. It is difficult to discuss the values reported by the other workers

because of the indirect pressure calibration procedures [6, 7, 8] and the fact that the equilibrium

pressure was not determined in all cases [5, 8]. On room temperature compression, Bi III persists

about 2 kilobars beyond the equilibrium transition

pressure into the Bi V field. Thus in normal calibration experiments the III-V transition may actually be

detected a few kilobars above the equilibrium

value reported above.

4. Acknowledgments

We wish to thank Dr. W. Klement for many helpful discussions, and Mr. L. Faus and Mr. J. Yamane

for their skiU and patience in machining the many

intricate parts required. We are also indebted to the Army Research Office, Durham, North Carolina

for partial financial support under contract

DA-31-124-ARO-D.

5. References

[1] Boyd, F. R., and England, J. L., J. Geophys. Res. 65, 741

(1960).

[2] Klement, W., Jr., Jayaraman, A., and Kennedy, G. C., Phys.

Rev. 131, 632 (1963).

[3] Haygarth, J. C., Getting, I. C, and Kennedy, G. C., J. Appl.

Phys. 38 , 4557 (1967).

[4] Bridgman, P. W., Proc. Am. Acad. Arts Sci. 74, 425 (1942).

[5] Vereshchagin, L. F., Zubova, E. V., Buimova, I. P., and

Burdina, K. P., Soviet Phys.-Docklady 11, 585 (1967).

[6] Jeffery, R. N., Barnett, J. D., Vanfleet, M. R., and Hall, H. T.,

J. Appl. Phys. 38, 3172 (1966).

[7] Stark, W.. and Jura, G., ASME paper 64-WA/PT-28 (1965).

[8] Giardini, A. A., and Samara, G. A., J. Phys. Chem. Solids 26,

1523 (1965).

[9] Kennedy, G. C., and La Mori, P. N., J. Geophys. Res. 67,851

(1962).

DISCUSSION

R. Zeto {U.S. Army Electronics Command, Fort

Monmouth, New Jersey): I would like to point out

that likely you have a systemic pressure error in the

isothermal IIl-v measurements due to the fact that,

in changing pressures and temperatures, only about one minute was allowed to observe whether a transformation occurred from one phase to the other. For

pressure calibration with sluggish transformations,

kinetic time at constant pressure should be allowed

for initiation of the transformation; otherwise the

observed initiation pressure is erroneously high.

depending on the pressure required to overcome the

kinetic barrier to nucleation.

R. E. Hanneman {General Electric Research and

Development Center, Schenectady, New York): The

nucleation theory predicts that as the temperature

rises the dead band or hysteresis region will become

narrower, and the fact that the dead band is down to a small pressure interval at the temperature of the

triple point indicates that in this particular case

nucleation time does not have too important an

effect.

37
R. Roy {Pennsylvania State University, University

Park, Pennsylvania): Referring to the bismuth

phase diagram, are you sure that the metastable

projections between III and IV are, in fact, between

the same phases? What is the evidence that the reverse transition is between the same phases as the

transition in the forward direction? In looking into

similar systems in detail we have found tremendous

complexity. So I am asking a general question as to whether we should all make the assumption that

when A goes to B, then B goes back to A, when in cases like germanium and sulfur we find there are many, many phases involved.

P. N. La Mori (Battelle Memorial Institute, Columbus, Ohio): I would like to raise a question conAUTHORS

We agree with Dr. Zeto that our uncertainty in

determining transition temperatures and pressures

could have been further reduced by our spending

more time on the measurements. However, in our

isobaric measurements we determined the transition temperatures with both phases present, and

we believe that this procedure allowed us to approximate the equilibrium temperatures with the precision quoted. In our experiments at constant temperature, the pressure was varied in increments of

V2 kbar every V2 minute or V2 kbar every

2V2 minutes. The variation of rate did not affect

the values of transition pressures obtained, and

the transition pressures determined in this way

agreed with those determined in the isobaric

experiments. The important point in this work

is that the transition pressure at 25° for the III-V

transition is shown to be near 77.5 kbar, rather than

89 kbar as previously accepted, and certainly not as low as 73 kbar.

In reply to Dr. Roy: The evidence we have that

the metastable III-V boundary is observed just

below the III-IV-V triple-point pressure is that on heating Bi III or cooling Bi V under these conditions we observed on occasion a single resistance

change whose magnitude was equal to that of the

III-V transition at pressures above the triple point,

rather than two distinct resistance changes corre- sponding to the III-IV (or IV-lIl) and IV-V (or v-iv)

transitions. The reason why the metastable projection of the III-V boundary may be observed at all

is that while the lll-v and V-III transitions occur

rapidly with little hysteresis under these conditions

while the III-IV and IV-III reactions are very sluggish

with large hysteresis, and the V-IV reaction is slower than the V-III reaction and shows some hyscerning the pressure correction you made for lateral

expansion or dila'tion of the piston under pressure.

I have made some measurements at 40 kilobars in a

similar type system using a 6 percent cobalt pres- sure vessel. Both by calculation from elastic constants as well as from measuring compressibilities

of known materials, I found a correction of approximately 1.4 percent at 40 kilobars. In your work

at 80 kilobars I believe the correction should be at least as large.

G. S. Kell (National Research Council of Canada,

Ottawa, Canada): You put weight in your argument

on the fact that your midpoint values with isothermal and isobaric changes were the same. Do

you have a proof for this?

CLOSURE

teresis. Thus, it is possible to heat Bi III into the

stability field of IV without IV forming, and as soon

as the metastable projection of the III-V boundary

is crossed, Bi V forms. Because of the sluggishness

associated with the V-IV reaction, Bi IV does not

form at once. Similarly, rapid cooling of Bi V will

result in formation of Bi ill just below the triple

point pressure; Bi IV does not have time to form before the metastable projection of the III-V boundary

is crossed.

In reply to Dr. La Mori: We feel that the discrepancy between his and our estimate of the dilation of

the piston under pressure may be due to the fact

that we used a grade of cemented tungsten carbide

containing only 3 percent cobalt. We estimated the

effective area of the piston by taking the diameter

of the piston after the experiment, and making a

correction for the elastic enlargement due to the

axial load, and for the thickness of the indium layer

between the pressure vessel wall and the piston.

The magnitude of the correction to the diameter of

the piston at 80 kbar was thus about + 0.6 percent

of the initial diameter. It is important to reaUze that

the effective diameter may be calculated by consideration of the piston alone, since it is under radial

constraint by the pressure vessel, and the two are thus separated only by the indium layer, whose

thickness may be measured.

In reply to Dr. Kell: We made the assumption that

the mechanical components of the total friction are symmetrical for compression and decompression.

The fact that the mean of the compression and

decompression values for the isothermal and isobaric determinations coincide would appear to indicate that the hysteresis for the transition itself

is also symmetrical.

38
Optical Interferometry at High Pressures*

K. Vedam

Materials Research Laboratory, The Pennsylvania State University, University Park, Pennsylvania 16802

After a brief review of the present status of optical interferometry at high pressures, the experimental technique of measuring the variation of refractive index with pressure to 7 and 14 kbars with

one- and two-stage optical pressure vessels respectively, is described along with some typical results obtained. The need for such measurements on liquids and gases which are used as fluid pressure media is emphasized.

1. Introduction

It is well known that optical interferometry is one

of the standard techniques by which one can make

precision measurements on the optical path length

and its variation with parameters hke temperature,

pressure, etc. This article describes a brief review

of the state of the art today in the field of optical

interferometry at high pressures.

A glance at the literature reveals that the optical

interferometric technique has been applied mainly

in the study of pressure dependence of the refractive index of gases [1],' Hquids [2, 3], and solids [4]. Since most gases and liquids solidify even at moderately low pressures of the order of 5 to 10 kbars, such

measurements were perforce Umited to this range.

Even in the case of solids the upper limit of pres- sures that can be safely employed is rather limited,

due to the inherent low strength of the optical

windows in the pressure vessel. By using a two-stage

pressure vessel with sapphire optical windows it is possible to increase the upper limit of pressure only

to about 15 kbars— or at best to 20 kbars. Even so, one can carry out precise, reproducible measure- ments which often yield linear relationship between

pressure and the change in refractive index. In view

of the reasons mentioned above this article will

deal mainly with the experimental procedure and

results obtained with solids. However, with sUght

modification the same techniques and procedures

are valid for liquids and gases as well.

2. Experimental Procedure

2.1. Single-Stage Optical Pressure Vessel

Until recently most of the experiments were

conducted with a single-stage optical pressure

vessel, which is described below in detail. In

principle, the experimental method of measuring

the variation of refractive index of solids with pres- sure involves the adaptation of Ramanchandran's

[5] interferometric technique to an optical high-

*Work supported by a grant from the National Science Foundation, Grant No. GK-1686-X.

' Figures in brackets indicate the literature references at the end of this paper. Paper presented at the Symposium on Accurate Characterization

of the High-Pressure Environment, held at the National Bureau

of Standards, Gaithersburg, Md., October 14-18, 1968.

pressure vessel, and is somewhat similar to the

techniques described by Cardona, Paul and Brooks

[6] and Waxier and Weir [4]. A schematic drawing

of the experimental arrangement is shown in

figure 1. The experimental sample to be studied

is suitably cut or cleaved, ground, and polished to be nearly optically parallel, so that one can easily

observe locaUzed interference fringes of the

Newtonian type produced by interference of Ught

reflected from the two surfaces of the crystal plate.

The final dimensions of the specimens are usually

about 0.8X0.8X0.3 cm. In the case of birefringent

crystal the edges of the specimen are usually made

parallel to the principal axes. In the case of cubic

crystals as well as with isotropic sohds, no such

crystallographic orientation is necessary. The sur- faces of the crystal were silvered by an aluminum

evaporation process so as to increase the reflectivity

and thus render the fringes sharp.

The optical pressure vessel is of conventional

design with a single-crystal alumina window whose

optically flat poUshed surface presses against a matched optically flat steel plug as shown in

figure 1. Seals at other closures have been effected

using Daniels' armored O-rings [7]. A combination

of Buna-N Tetraseal and a flat ring of copper have

also been found to be equally effective seals at these pressures. The c-axis of the alumina window

is parallel to the direction of propagation of hght.

The pressure-generating system employed for these

to

CAPS , , vy LIGHT SOURCE

Figure 1. Schematic diagram of the experimental arrangement,

for measurement of the variation of refractive index of sclids with pressure.

385-762 O - 71 - 4
measurements is a fairly conventional screw pump

in conjunction with a Harwood intensifies Sovasol, an optically transparent and colorless fluid marketed

by Mobil Oil Company, was used as the fluid pres- sure medium. The pressure was read on a 16-in-diam

100,000 psi, Heise gage which itself was calibrated

by a deadweight piston gage with an estimated

accuracy of 3 parts in 100,000.

The specimen, supported suitably in the bomb, is illuminated with parallel light from a sodium lamp

or a low-pressure mercury arc with appropriate

filters. In the case of birefringent crystals, a sheet

polarizer assembly is kept suitably oriented, between the alumina window and the specimen. This

allows the measurements to be made with appropriately polarized light without any interference

from the birefringence of the optical window. The

localized interference fringes are observed in the

telemicroscope for measurements in the visible

region of the spectrum.

As the specimen is subjected to hydrostatic pres- sure both the thickness and the refractive index of

the specimen change, with a consequent shift of the

fringes across a fiducial mark on the specimen. The

change in the refractive index An can be evaluated

from the well known interference formula

p\ — 2n-^t

where p is the number of fringes shifted, to the

initial thickness of the specimen. A? the change in

thickness of the specimen due to the applied pressure, and A. the wavelength of light employed. In

actual practice the values of An and Af at each

pressure (corresponding to some known fringe shift) are evaluated from a programmed calculation on an IBM 360 and the value of n corrected by this An

before it is used for the computation of An' for the

additional pressure producing the next fringe shift.

Since the pressure employed is fairly large, Af the

change in the thickness of the specimen, is evaluated with the help of the nonlinear theory of elasticity developed by Murnaghan [8], Birch [9], and

others [10]. According to this theory, if the deformation produced by the applied stress T,j does not

involve any rotation, as is the case in our present

measurements that employ hydrostatic pressures

only, then nj can be expressed in terms of rjij as (Barsch [11])

Tij— Cijkir}i;i + Dij, klmny]kl'nm)i, (2)

where

Dij, klmn =1/4 {Cinkl^jm + CJnkt^jm + Cimkl8j„

+ Cjynkl^in ~l~ C Umn 8jk ~l~ Cjlmn^ik ~l~ Cikmn 8jl ~l~ Cjkmn Si/ — 2Cijkl8mn ~ 2C ijmn 8kl + 2Cijklmn) , (3)

and Cijki are the second-order elastic constants,

Cijklmn are the Brugger's third-order elastic constants. In the case of cubic crystals and isotropic

solids under hydrostatic pressure P, eq (2) reduces

to

T = P = -{cu + 2c^2)r)

+ ^Cn + 2Ci2 —|cni — 3Cii2 — Ci23 ^

T?'^ (4)

where tj the Lagrangian strain is given by

litojz^tTjzA (5) ^ 2 tl

and the usual convention of replacing the index

pairs 11, 22, ... 12 by 1, 2 ... 6 has been

adopted for the suffixes of the elastic constants.

In the case of crystals of symmetry lower than

cubic system, eq (2) reduces to second-degree equations in the principal strains and hence the strain

components cannot be determined in a closed form.

However, starting from the approximate values

obtained by neglecting the higher order terms, we

can easily compute the true values by an iterative

procedure with the help of a computer. Thus,

whenever data on third-order elastic constants are

available the foregoing equations can be used to evaluate Af and hence An.

The third-order elastic constants are available

for only very few crystals. On the other hand,

Bridgman [12] has made extensive measurements

of the linear and volume compressibilities and their

variation with pressure of a number of solids and

fitted his data with quadratic equations of the type

^V^i = i^lLo) 1 =- a,P + b^P'^ (6)

and

^^^^{^VIVo)=-aP + bP-\ (7)

0

Using his values of a and b, the change in the

thickness of the sample and hence the value of

An can again be evaluated.

Typical Results and Discussions

Figure 2 shows the results obtained on the variation of the refractive indices of CaF2, BaF2, and

^-PbF2 with pressure to 7 kbars for X.5893 A. While

some crystals like CaF. [16], MgO [17], a-Al203

[18], diamond [19], etc., exhibit a perfectly linear

behavior in the pressure range investigated, some

others like BaF2 [16], the alkali halides [20], vitreous

silica [21], ct-quartz [22], etc., exhibit a nonlinear

variation of the refractive index with respect to

40
HYDROSTATIC PRESSURE (Kilobars)

Figure 2. Variation of the refractive indices of CaF2, BaF2, and /3-PbF2 with

hydrostatic pressure (T =22 °C)

4.0 -

30

LlJ >

!- 2.0 - <

q:

LlJ

1.0

<

I

X 5893 &

00

Ba F,

CoF,

PbF,

Sample •# I Sample #2

Sample # I Somple # 2 Sample # I Sample #2

Sample # 3

0.4 0.8 1.2 VOLUME STRAIN (%)

1.6

Figure 3. Variation of the refractive indices of CaF2, BaF2, and /3-PbF2 with

Lagrangian strain (T = 22 °C).

pressure, particularly at high pressures. But the

same results are found to show a perfectly linear

dependence between An and the volume strain

when the latter is computed, using the nonlinear

theory of elasticity, as can be seen from figure 3 for

the case of BaF2. In the case of potassium iodide

[20], even though the total volume strain is as high

as 9 percent, the linear relation between An and

AF/Fo is stiU found to be valid. A somewhat similar

linear relationship between the isothermal volume

strain and the melting temperature of various materials has been found by Kraut and Kennedy [23], even for strains as large as 60 percent in some cases. However, this does not mean that such a linear rela41
tion between the change in refractive index A.n and

the Lagrangian strain will be valid universally. On

the contrary, pronounced nonlinearity between An

and the strain is observed even at fairly low strain

levels in a and j8-ZnS, CdS, etc., for \5893 A indicating the influence of the nearness of the absorption edge in these cases. This aspect has been dealt

with elsewhere [24] and will not be described further

here.

During the course of these studies, some crystals

like /3-PbF2 [16J, rubidium halides [25], etc., ex- hibited phase transitions at high pressures and by

this interferometric technique it has been possible

to measure the variation of the refractive index of

these crystals with pressure up to the phase transition point, as in the case of the cubic form ^-PbF2

(fig. 2) which transforms irreversibly to the orthorhombic modification a-PbFo at 4.8 kbars at 23 °C.

2.2. Two-Stage Optical Pressure Vessel

A two-stage optical pressure vessel shown schematically in figure 4 has recently been constructed

and put into operation for pressures up to 14 kbars.

The principle of operation of this vessel is essentially

similar to the single-stage vessel described earlier

except for a few slight modifications. The two singlecrystal alumina optical windows are positioned in

such a way that at any time no one window experi- ences a pressure difference greater than 10 kbars.

The telemicroscope used to observe the localized

fringes in the single-stage vessel (fig. 1) has been

replaced by a vidicon camera attached to a TV

monitor. Such a procedure has two distinct advantages, namely (i) safety of the operating personnel

and (it) capability of carrying out these measure- ments over a wide spectral range from ultraviolet

to the near infrared, just by changing the vidicon

tube. Since sovasol, the fluid pressure medium used

in the single-stage vessel, solidifies around 8 kbars,

plexol 262, or n- or iso-pentane is used in the twoFlGURE 4. Schematic diagram of the experimental arrangement

with two-stage optical pressure vessel.

Stage vessel. For studies in the ultraviolet, n- or iso-pentane are preferable over plexol 262 in view

of their increased transparency at shorter

wavelengths.

Figure 5 shows a typical result obtained on a Type

I diamond with the two-stage vessel. Above 7 kbars

the pressure was read with a manganin cell in con- junction with a Carey Foster's Bridge. It is seen from figure 5 a perfectly linear behavior between

An and pressure and strain is observed in the case

of diamond for A,5893 A.

2.3. Optical Interferometry as an Interpolation Device in High-Pressure Techniques

From the above survey of the status of optical

interferometry at high pressures, it is seen that all the work so far has been limited to pressure systems

which employ gas or a liquid as the pressure medium

and further only to the pressure range where pure

hydrostatic pressures can be realized. This is understandable, for, under quasihydrostatic conditions normally attained at much higher pressures,

the experimental sample (which itself acts as the

interferometer), will be subjected to shearing

stresses as well, in addition to the hydrostatic

stress component. This invariably results in an

irreversible plastic deformation of the specimen.

If interferometric measurements are carried out

under such circumstances it is doubtful whether

any meaningful physical interpretation can be made

from the results so obtained. In other words, optical

interferometry can be used as an interpolation device for precision measurement of pressure only

in the region where it is possible to develop pure

hydrostatic pressures. One possible exception may

be the use of materials like Diamond as the interferometric sample and fairly soft materials like

AgCl or NaCl for the pressure medium, so that

VOLUME STRAIN (%)

1 I I I ^ 1 1 -i— O 5.0 lO.O IS.O PRESSURE (Kilobars)

Figure 5. Variation of the refractive index of Type I Diamond,

with pressure (T = 22 °C).

42
the permanent shear deformation under quasihydrostatic pressures can be minimized. Further, it would

be advantageous to use suitable crystallographic

orientation of the sample such that the stress-optic

birefringence introduced by the shear stresses is either zero or negligible. In any case, further

research work in this direction is necessary before

any definite statements can be made.

Having thus limited our range to the region of

pure hydrostatic pressures attainable in the laboratory it will be useful to consider the relative merits

of gas, liquids, and solids on which the interferometric measurements can be made to obtain an

accurate value of the pressure being used. As mentioned earlier, the use of solids for this purpose is not particularly advantageous, even though one can

use a material with a perfectly linear relationship

between pressure and the parameter measured. This

is due to the fact that the elastic constants of the

material and their variation with pressure must be

known rather precisely. However, at any particular

pressure, if one is interested in small deviations or

fluctuations of the pressure the interferometer can

be used with advantage, since then one is interested

only in a slight shift of the fringes which can be

measured with precision.

Here again the use of a liquid or gas would

definitely be more advantageous. For then one can

use Fabry-Perot etalons as the interferometers, with

their easy capability of measuring fringe shifts of the

order 0.01. Furthermore, since the compressibility

of these materials is much greater than that of solids,

the change in the refractive index for them is larger

by 2 to 3 orders of magnitude. Thus, even though

the elastic constants of the etalon-spacer material

may not be known precisely, the error introduced on

this account will be negligible.

However, experimental data on the variation of

the refractive index of gases [1] and liquids [3] are

at present limited to rather low pressures of only

1 to 3 kbars. Thus, this type of measurement, particularly on such gases and liquids which are cur- rently used as fluid pressure media, appears to be a

fruitful line of investigation.

3. Acknowledgments

The author would hke to express his sincere

thanks to Professor Rustum Roy for his kind interest

and encouragement, to Professor G. R. Barsch for

the many stimulating discussions, and to Messrs.

E. D. D. Schmidt, T. A. Davis, and J. L. Kirk for

carrying out some of the measurements reported.

4. References

[1] Michels, A., and Botzen, A., Physica 15 , 769 (1949) et seq.; de Groot, S. R.. Rev. Opt. 28, 627 (1949).

[2] Rosen, J. S., J. Opt. Soc. Am. 37, 932 (1947).

[3] Waxier, R. M., and Weir, C. E., J. Res. NBS 67A, 163 (1963);

Waxier, R. M., Weir, C. E., and Schamp, H. W., Jr., ibid. 68A, 4«9 (1964).

[4] Waxier, R. M., and Weir, C. E., ibid. 69A, 325 (1965);

Vedam, K., Schmidt, E. D. D., Myers, M. B., and Roy, R., ASME Publ. No. 66-WA/PT-3 (1966).

[5] Ramanchandran, G. N., Proc. Ind. Acad. Sci. 25A, 208

(1947).

[6] Cardona, M., Paul, W., and Brooks, H., J. Phys. Chem.

Solids 8, 204 (1959).

[7] Daniels, W. B., Rev. Sci. Instr. 28, 1058 (1957).

[8] Murnaghan, F. D., Finite Deformation of an Elastic Solid

(John Wiley & Sons, Inc. New York, 1951).

[9] Birch, F., Phys. Rev. 71, No. 11, 809 (1947).

[10] Hearmon, R. F. S., Acta Cryst. 6, 331 (1953); Nran'yai'

A. A., Sov. Phys. Solid State 6, 936 (1964); Barsch, G. R., Report No. 5., Contract No. Nonr-656(27) (1963),

unpublished.

[11] Barsch, G. R., ibid. Report No. 8 (1963), unpublished.

[12] Bridgman, P. W., The Physics of High Pressure (G. Bell and Sons, Ltd., London. 1958), p. 149.

[13] Anderson, O. L., J. Phys. Chem. Solids. 27, 547 (1966).

[14] Murnaghan, F. D., Proc. Nat. Acad. Sci. 30, 244 (1944).

[15] Thurston, R. N., J. Acous. Soc. Am. 41, 1093 (1967).

[16] Schmidt, E. D. D., and Vedam, K., J. Phys. Chem. Solids

27, 1563 (1966).

[17] Vedam, K., and Schmidt, E. D. D., Phys. Rev. 146, 548

(1966). [18] Davis, T. A., and Vedam, K., J. Appl. Phys. 38, 4555 (1967).

[19] Schmidt, E. D. D., Kirk, J. L., and Vedam, K., Am. Min.

53, 1404 (1968).

[20| Vedam, K., Schmidt, E. D. D., Kirk, J. L., and Schneider,

W. C. Mat. Res. Bull. 4, 573 (1969).

[21] Vedam, K., Schmidt, E. D. D., and Roy, R., J. Am. Ceram.

Soc. 49 , 531 (1966).

[22] Vedam, K., and Davis, T. A., J. Opt. Soc. Am. 57, 1140

(1967). [23] Kraut, E. A., and Kennedy, G. C, Phys. Rev. 151, 668

(1966).

[24] Davis, T. A., PhD. Thesis, Pennsylvania State University

(1968). [25] Vedam, K., and Schmidt, E. D. D.,J. Mat. Sci. 1,310(1966).

DISCUSSION

P. M. Bell {Geophysical Laboratory, Carnegie

Institution of Washington, Washington, D.C.):

Have you made measurements on the change in

index of refraction of glass as a function of pressure?

O. E. Jones {Sandia Laboratory Albuquerque,

New Mexico): I believe there has been some work

on change of index of refraction of glass under

shock loading, particularly by the Stanford Research

Institute people. Ahrens [J. Appl. Phys. 37, 4758

(1966)] did this on hquids. Then last year at the

Paris meeting on high dynamic pressures I think

there were some results reported on alpha quartz.

How do your results correlate with these made,

in general, at quite a bit higher pressure levels?

43
J. L. Cross {National Bureau of Standards, Washington, D.C.): In referring to the accuracy of the

pressure measurements, you mentioned the use

of a 100,000 psi Heise gage calibrated by a deadweight piston gage with an estimated accuracy of

3 parts in 100,000. I know of no piston gage capable

of this accuracy in this pressure range and, in any

event, I think that the dial gage could not be used

with this accuracy.

B. Vodar {Laboratoire des Hautes Pressions, Centre

National de la Recherche Scientifique, Bellevue,

France): The method of measuring pressure by

change in refractive index is presently being done

by Coulon at the University of Marseilles, at pressures up to about one kilobar or possibly higher.

Voice from the floor: Presumably there is a

relationship between refractive index and volume

change that comes about because the index of

refraction is related to the polarizability of the

system and the density. Thus, in selecting materials

for pressure calibration purposes, you would want

to pick materials in which the dimensions of the

atoms, which are being polarized, are small compared to the interatomic separation between atoms.

Presumably, the deviation for cadmium sulphide

occurred because the spacing between atoms

was of the same order of magnitude as the atomic

diameters.

AUTHOR'S CLOSURE

Yes, we have carried out measurements on vitre- ous silica. In this case the refractive index increases

linearly with pressure up to about 4 kbars, thereafter

it increases nonlinearly. It is interesting to note that

at high pressures the points fall above the extrapolated straight line. This is unlike the crystalline

materials where the points fall below the line. This

unique feature of glasses arises from the increase in

compressibility with pressure as the voids are

closed. We have made measurements on vitreous

silica only to 7 kbars. At higher pressures, perhaps

30 or 40 kbars, when all the voids are closed the

refractive index will increase at a slower rate with

pressure.

I have not compared my results with those observed from shock loading experiments referred to by Dr. Jones. Under shock loading conditions in

solids significant shear stresses are present, making

the conditions unhke those in my work. Further, the

shock experiments yield values under adiabatic

conditions whereas the hydrostatic measurements

yield the isothermal values. The difference between

these two can be significant, particularly in the case

of liquids.

In the choice of materials for pressure measure- ment by refractive index change, the departure from

linearity of many substances is an important problem. In many substances the refractive index de- creases with pressure due to the fact the polarizability of the ions or the atom is itself a function of

interatomic distances. In the case of gases, where

the atoms are widely separated, you would expect

the linearity to hold good. Unfortunately it does not.

*

44
Equipment for Generating Pressures up to 800 Kilobars

IVaoto Kawai

Department of Material Physics, Faculty of Engineering Science,

Osaka University, Toyonaka, Osaka, Japan

A type of high-pressure apparatus has been developed which extends considerably the range

of attainable very high static pressure. It consists basically of a sphere segmented into tapering trun- cated pistons. Hydrostatic pressures in the sample chamber in the centre of the sphere have reached

800,000 bars (11,600,000 psi). Pressures of over one million bars should be possible with the present

technology. Some new and interesting results from high-pressure studies are described.

1. Apparatus and Method

The apparatus consists basically of a metal solid

of revolution segmented into a number of tapering

sections. The preferred and usual form is a sphere,

but a solid cyhnder has also been used [1, 2].^ The

general array of pistons is illustrated in figure 1. The actual model for an eight-piston sphere is shown

in figure 2(a). The regions where four pistons meet

are flattened and covered by corner caps. The sections act as multipistions, directing an external

stress towards the center. The innermost apex of

each piston is truncated to form a polyhedral sample

chamber. The assembly of pistons acts as a pressure

intensifier, with the stress on an enclosed sample

being a function of the ratio of its surface area and

the outer area of the multipiston sphere.

An inter-piston separation, initially several millimeters, is necessary to allow inward motion of the

pistons. The pistons are separated by a distribution

of compressible spacers, usually pyrophyllite. A

sample holder is machined to surround the sample,

and fit in the central chamber (see fig. 1). It converts

the stress directed by the pistons to an approxi-

' Figures in brackets indicate the literature references at the end of this paper.

(a)

Figure 1.

Paper presented at the Symposium on Accurate Characterization of the High-Pressure Environment, held at the National

Bureau of Standards, Gaithersburg, Md., October 14-18, 1968.

mately hydrostatic pressure. Pyrophyllite, es- sentially AI2O3 • 4Si02 • H2O, is favoured for such

application because of its good compressibility but

high coefficient of friction and shear strength. Such

a sample holder preserves its shape and can hold the

sample firmly to a temperature of over 2500 °C.

The metal sphere is sealed in two thick hemispherical rubber shells (see fig. 3). The appropriate

wire leads, for measurement of temperature, electrical resistance, etc., are circuited out, and the

sphere is rested on a support cylinder. The whole

assembly is then placed in a high-pressure oil reservoir. In the experimental setup used, a confining oil pressure of 4 kilobars may be produced in

the reservoir chamber, using a uniaxial hydraulic

press (Mitsubishi Atomic Power Industries, rated

at 2,000 tons). With this external pressure acting on the jacketed sphere, the multipistons move

inward and exert a stress on the material in the

central sample chamber.

(a)

Tungsten Carbide

Figure 2.

45
(one of 4)

Figure 3.

Pyrophyllife Somple

Thermocouple

Graphite

Tube

Figure 4.

With increasing stress, the material in the gasket

structure between the pistons, and some of that

enclosing the sample, is extruded outward between

the pistons. As the pistons move closer together as well as inwards, this filler becomes thinner. It thus inhibits further outflow of the material from the

sample space, and supports the inner portion of

the pistons laterally.

The large radial pressure gradient means that

only the inner region of the pistons need be made of

the hardest, most expensive material. The flow

of the gasketing material acts to confine the pistons

laterally where the stress is most concentrated.

This discourages their destruction by compressive

failure and shear, and lateral ductile flow. No

appreciable plastic deformation has occurred on

the truncated piston inner faces, whereas this is frequently the case at a pressure of a few hundred

kilobars for apparatus with only two opposed

pistons.

The sample space may be equipped with instrumentation to monitor experimental proceedings.

Figure 4 shows a sample prepared for thermal treat- ment. It is confined in an octahedral pyrophyUite

block. A graphite heater and thermocouple are enclosed. The electric leads are taken out between

pistons, through a seal in the rubber shell, and

through the wall of the oil reservoir. A pressurecalibrating unit may similarly be imbedded in the

sample space. In conjunction with application of

pressure, the temperature of a sample may be

elevated to about 2,000 °C.

2. Calibration and Capability

There are many known indicators for calibrating

pressure. There are phase changes as evidenced by

an abrupt decrease in volume, sudden changes in

electrical resistance, and second-order transitions

such as resistance maxima. Some calibration points are listed in table 1. One aspect of using resistance

measurements for calibration is that there are two

Table 1. Some transitions for calibration of

pressure, based on changes in volume (AV) and

electrical resistance (AR) at 25 °C.

Conditions Substance

Transition pressure

(kilobars)

Abrupt AF KBr 18.0 KCl 20.2 AgCl 88

Abrupt AF and AR Bi 25.3, 26.8, 89

Tl 37

Ba 59

Abrupt AR Fe 133

Ba 144

Rb 193

GaAs 250

Resistance maximum Cs 42

Ca 375

Rb 425

CdS 465

46
causes of a change in resistance. There is the effect

of pressure on the specific resistance, and the

change due to possible aheration of the specimen

shape in the sample chamber.

In experiments here, a good reproducibility has

been obtained for calibration. The caUbration

is dependent on the particular apparatus used, be- cause of different relative sizes of sample chambers,

and on the arrangement of inter-piston spacers.

The values of extreme pressure are deduced by

extrapolation, since there is a lack of known refer- ence points at the very high experimental pressures

obtainable. The calibration is sufficiently uniform

to make this procedure of extrapolation acceptable.

The cahbration of pressures above about 500

kilobars, however, is presently not well known.

More known pressure-dependent changes at very

high pressure are needed. These could be discontinuous changes such as volume or resistance

changes, or continuous changes such as in resistivity or solubility of one component in another.

The maximum pressure attainable depends on the

radius of the sphere and the amount of truncation

at the centre. The smaller the central sample

spacer, the higher the possible pressure. It also

apparently depends on the number of pistons in

the assemblage. It may be noted that the eight pistons of the spheres described here are more than

in any of the other types of apparatus mentioned

earlier. However, in a sphere the more numerous

the pistons are, the smaller their apex angle be- comes. This gives less support to the highly stressed

piston apex regions, and allows failure at a lower

stress. This problem is counteracted by the pressure

of some gasketing material on the inner flanks, but

optimum results are not obtained by continuing

to increase the number of pistons.

The first sphere, constructed about three years

ago, had a diameter of 8 cm. The maximum pressure

obtained with it was about 200 kilobars. The highest

operating pressure used since then has been 800

kilobars, in the large-diameter, small-samplechamber sphere. This does not necessarily represent

the maximum pressure feasible with these particular

pieces of apparatus.

The extension of the range of pressure in the

future will most likely require pressure intensification by multi-stage apparatus. Now being

constructed is a design of multi-stage piston assemblages, with a central octahedral sample space.

This cascaded device is expected to allow a central

hydrostatic pressure in excess of 1 megabar (about

14,500,000 psi).

3. Conclusion

The apparatus described extends considerably the

range of possible very high static pressures. The

apparatus has significant advantages over other

existing devices. It opens the way for further re- search into the nature and properties of substances

under great pressures. For example, the deep in- teriors of the earth and other planetary bodies are pressure regimes. It is estimated that conditions in

the earth vary from a pressure of 10 kilobars and

temperature of 500 °C at a depth of 20 miles beneath

the surface to a pressure of 1,700 kilobars and temperature of about 3,000-4,000 °C at a depth of 1,800

miles. The latter is at the earth's solid mantle-liquid

core boundary. The presently available pressure of

800 kilobars corresponds to a depth of about 1,200

miles.

The new technology described here provides a means for probing further into the realm of superpressure science. It should allow investigation of

material, properties, and phenomena as yet

unknown.

4. Acknowledgment

Many thanks are due to the members of the working group who have been particularly involved in

development of the high-pressure apparatus and

research — Messrs. K. Hirooka, E. Itoh, and

S. Endoh.

5. References

[1] von Platen, Baltzar, A multiple piston, high pressure, high

temperature apparatus, Modern Very High Pressure Techniques, pp. 118-136, R. H. Wentorf, Editor (Butterworths, 1962).

[2] Kawai, Naoto, A static high pressure apparatus with tapering

multi-pistons forming a sphere. I, Proc. Japan Acad.

42,385-388(1966).

DISCUSSION

F. P. Bundy {General Electric Research and

Development Center, Schenectady, New York):

After the sample chamber has been pressurized

to, say, 400 or 500 kilobars, and then unloaded, do

the pistons remain as integral pieces or do they

fracture?

In estimating pressures in the 200 to 400 kilobar

region, I would suggest that you try the iron cobalt

alloys. I have more faith in the linearity of these

materials than in the Unearity of cadmium sulfide.

P. N. LaMori (Battelle Memorial Institute, Columbus, Ohio): What are the dimensions of your

apparatus?

F. R. Boyd, Jr. (Geophysical Laboratory, Carnegie

Institution of Washington, Washington, D.C.):

Would you tell us the grade of carbide that you used,

and whether any failures of the carbide have

occurred?

47
J. D. Barnett {Brigham Young University, Provo,

Utah): I note your use of the cadmium sulfide

transition for calibration of your apparatus in the

400 kilobar range. The reliabihty of a transition

reported by Drickamer in CdS is questionable. I have done considerable work on calibration in a tetrahedral apparatus up to about 100 kilobars,

and have found the relationship between true pres- sure and load to be very nonlinear, the curve becoming almost flat at the higher loads. We used

samples of about the size you used — around 4 mm

on an edge — and our equipment was capable of

applying a load of 600 tons force to each ram. The pressure was monitored continuously by an

x-ray beam on sodium chloride. We find that in

apparatus of this type most of the ram force is taken by the gaskets, with the pressure reached in

the sample being correspondingly limited.

G. Jura {University of California, Berkeley, California): Concerning your high calibration point,

there has been a reported bismuth transition at 415

to 425 kilobars. Did you look for that one?

AUTHOR'S CLOSURE

In reply to Dr. Bundy: When the pressure is re- lieved the pistons remain together, but they are

easily separated by hand.

In reply to Dr. La Mori: In our present apparatus

the outside diameter of the sphere is 250 mm, and

the diameter of the central sample chamber is 2 mm.

The area ratio is thus 15,000.

In reply to Dr. Boyd: The tungsten carbide used

for the piston contains 3 percent cobalt. We have

not experienced breakage, and we believe that this

is due to the lateral pressure of the gaskets exerted on the pistons. This is unlike the situation with two

flat pistons (Bridgman anvils) in which relatively

large deformations take place at 200 or 300 kilobars.

Our gaskets are initially about 2 mm in thickness,

and after pressure has been applied are found to be

0.1 to 0.2 mm in thickness.

Concerning Dr. Barnett's comment: His apparatus

had only four pistons while ours had eight, which I believe would reduce the load taken by the gaskets.

We hope to adapt the apparatus to use x-rays, and

to use sodium chloride as a reference material.

In reply to Dr. Jura: We could not find a bismuth

transition in the 400 kilobar region.

48
Manganin Resistance Gages as Accurate Instruments for High-Pressure

Measurements

Yu. A. Atanov and E. M. Ivanova

All-Union Research Institute for Physical

and Radiotechnical Measurements,

Moscow Region, Mendeleevo, U.S.S.R.

The rapid development of modern science and technology foretells the wide application of high

hydrostatic pressures from 20 to 40 kbar in the near future. The measurement of pressures in this range is usually carried out by manganin resistance gages. Many authors, however, report different metrological characteristics of individual manganin gages and present widely differing values of pressure coefficient, reproducibility, stability, nonlinearity, etc. There is no common basis for the selection of interpolation and extrapolation equations relating the change of resistance of a manganin gage

with the measured pressure.

In this work an attempt is made to investigate the scatter of metrological characteristics of a large

group of identical manganin gages by means of an absolute free piston gage, to find the best extrapolation equation and to evaluate possible errors of pressure measurements over 15 kbar.

1. Manganin gage

The spiral-wound manganin coil is mounted in

the groove of a ceramic holder which is fixed to the closure plug. Overall dimensions of the completed coil are 9.5 mm o.d. and 7 mm height. The

diameter of the wire is 0.08 mm. The nominal

coil resistance is 40 fl. Pieces of 0.12-mm copper

wire are welded to the ends of the manganin coil. Soldered copper joints improve the stabiUty of

the gage.

Seventy-two gages were manufactured. The same

batch of manganin wire was used. The completed

gages were seasoned at temperature + 140 °C for at least 80 hr with periodic dipping into the liquid

nitrogen bath (— 196 °C). Usually the high-voltage

pulse technique [1] > is used for seasoning of manganin gages in our laboratory. In this method, however,

it is difficult to control precisely the temperature of

seasoning, which influences the value of the pres- sure coefficient. In view of the fact that we tried

to minimize the scatter of gage characteristics, it was decided to use the low-temperature seasoning.

After seasoning the average decrease of the gage

resistance was equal to 0.28 percent of the initial

value. Any additional seasoning did not change

appreciably the gage resistance. This is of course

the evidence of satisfactory seasoning.

2. Calibration of the Manganin Gages

Against the Free Piston Gage

Twenty-four gages were taken from the group for

caUbration against the 15-kbar free piston gage

having the precision of 1 part in 1,000. Four gages

' Figures in brackets indicate tlie literature references at the end of this paper.

were being placed in the pressure vessel at the same

time. The temperature was maintained at 25 ±0.1

°C. The measurements were taken every 1 kbar. To

reduce the errors associated with the setting of the

temperature equilibrium in the pressure vessel the

readings were registered 3 to 4 min after every

application of pressure. Every gage was calibrated

three or four times.

3. Pressure Coefficient of Resistance

The pressure coefficient of resistance A; = -J-^^ Ro r

is one of the fundamental characteristics of manganin gages. (Here /?o is the resistance of a manganin

gage at atmospheric pressure. A/? the increment of

resistance at pressure P.) The pressure coefficients

of all gages decrease with pressure over 4 kbar

(fig. 1). Below 4 kbar the behavior of pressure coefficients is irregular. In most cases the values of

pressure coefficients increase with pressure in

this range, so there is the maximum in A:-versus-p curve. Since k values decrease practically linearly

with pressure, the calibration data were analyzed on the basis of the second-order expression

P = aAR + (3{AR)'^, where a and ft are constants

2,'hZ

2.1^0

2ie>

Paper presented at the Symposium on Accurate Characterization of the High-Pressure Environment, held at the National

Bureau of Standards, Gaithersburg, Md., October 14-18, 1968.

1

>

'—

1

— p

2 ^ 6 b to 12 il^.

Figure 1.

49
found by the least squares method. The expression

fits the calibration data with an accuracy better

than 0.1 percent. The pressure coefficient can be

written then

1 Ro {a + |3^R) oRo

my ko{l-y^R)

(1)

where ko - aRo

is the pressure coefficient of the

gage at P^O, and y — ~^ the slope of the A-versusAR plot. The scatter of metrological characteristics can be judged by the scatter of ko and y

values. For the gages calibrated the average pres- sure coefficient A:o= 2.41-10"^ bar~\ the standard

deviation cTko being 0.013-10"^ bar~^ At higher

pressures the scatter of k values diminishes. This

is the consequence of a correlation existing between

^0 and y values of individual gages. The corresponding values for y were:

7=0.00698, o-y= 0.00201.

The measured pressure can be expressed through

the pressure coefficient obtained in eq (1)

Ro k Ro ko{l-yAR) (2)

The substitution of ko and y into eq (2) gives the

approximate interpolation equation for any gage out

of the group investigated:

P =

1 Ro 2.41 • 10-6(1 -0.00698A/?)

The experimental pressures were compared with

pressures calculated from the approximate equation

for 4, 12, and 15 kbar. The mean difference is 19.6

bars (or 0.49 percent) for 4 kbar. The corresponding

values for 12 and 15 kbars are 40.2 bars (0.33 percent and 45.8 bars (0.31 percent).

The pressure seasoning of the gages was carried

out in apparatus of the piston-cylinder type, permitting generation of hydrostatic pressures up to 30 kbar in the 30 cm'' cell. A versus F plot is presented on figure 2. The average pressure coefficient

A'o showed no change after pressure seasoning and

the standard deviation cr/,„ decreased to 0.007 • lO"*"

bar~^ The new averaged equation for the group of

pressure-seasoned gages is

1 (3) Ro 2.41 • 10-«(l-0.00678Ai?)

The maximum difference between the experimental

K-ZOJ

pressures and calculated from eq (3) is less than 0.5

percent at pressures from 4 to 15 kbar. In other

words, eq (3) can be used for measurements of

pressures with an accuracy 0.5 percent with any

manganin gage of the group in the range from 4 to 15 kbar.

4. Intercomparison of

Manganin Gages

The intercomparison of manganin gages over 15

kbar was accomplished in the same 30 kbar hydrostatic apparatus. The measurements were carried

out at the room temperature without any temperature control. The readings of four to six gages were

registered at certain pressure in the cell. In all, 23

gages were intercompared. As a rule, after the

release of pressure the Ro values were higher than

initial ones but in a few hours they recovered

completely.

The treatment of data was as follows. The general

equation (3) was used to get the pressure values

from the readings of individual gages. The scatter

of pressure values thus obtained was then calculated

for every pressure at which the intercomparison

was carried out. The maximum deviation from the

mean value is less than 0.49 percent in the range

from 15 to 28 kbar. If the individual equations are used for each gage which were found from the

results of calibrations against the free piston gage,

the corresponding maximum deviation wiU be 0.12

percent.

5. Conclusions

The results obtained show that the scatter of

pressure coefficients of resistance does not increase

with pressure in the range from 15 to 28 kbar. The

manganin gages exhibit very good stability and

reproducibility in this range. It goes without saying

that the intercomparison of identical resistance

gages does not permit determination of the total error of pressure measurements over 15 kbar. The

0.12 percent deviation cited above can be associated

with the random error of measurements. Neverthe50
less the analysis of the pressure dependence of the

pressure coefficient k=f(P) in the range from 4 to 15 kbar makes it possible to claim rather low systematic error of pressure measurements over 30

kbar by means of extrapolation equations of the

type (3). For most practical needs, one can successfully use such extrapolation equations with the

constants determined from the calibration data.

Undoubtedly, for high-precision experiments one

needs to correct the constants of the extrapolation

equation on the basis of the mercury melting

pressure scale. At the moment such a comparison

is being prepared in our laboratory. We expect that

manganin gages calibrated against the mercury

melting pressure scale will have the total error less

than 1.5 percent in all the range of hydrostatic

pressures to 40 kbar.

Reference

[1] Alexeyev, K. A., Atanov, Y. A., and Burova, L. L. The

Proceedings of the Institutes of the Committee on Standards, 75 (i35). 44 (1964).

J. L. Cross (National Bureau of Standards,

Washington, D.C.): On the matter of the departure

from linearity of the pressure vs. resistance characteristic of manganin, we have observed a secondorder effect of opposite sign from that reported by

Adams and Goranson.' I believe that this is probably

due to our use of a strain-free coil configuration.

The elimination of residual strains in the wire may

have a large effect on the linearity observed, particularly at lower pressures. This may mean that

taking our zero at atmospheric pressure may not

be the most desirable way to use manganin gages.

It may be desirable, when possible and when the

accuracy required warrants it, to base the calibration upon points taken at elevated pressures only.

A matter that hasn't received much attention is the fact that the temperature coefficient of resistivity

of manganin is a function of pressure, so that the

temperature of the resistivity peak of manganin

increases as the pressure goes up, and in order to do

the best work with manganin, it may be necessary

to include the temperature and the resistivity

dependency on temperature in the calculations.

J. D. Barnett (Brigham Young University, Provo,

Utah): I notice that you season the manganin coils

by the Bridgman technique of cycling between

annealing temperature and liquid nitrogen. Is this

your standard procedure, or have you used other

seasoning techniques?

AUTHORS' CLOSURE

From the results of our measurements we felt that behavior of the manganin coils below four

kilobars was not representative of the behavior

at higher pressures. Accordingly, when the data

below four kilobars was not used all the curves of

pressure coefficient of resistivity vs pressure had

slopes of the same sign.

On the matter of temperature effects on the pres- sure coefficient of resistivity, we believe there

are too many other sources of error to justify strict

temperature control. If you work in the range from

say 20 to 40 degrees Celsius, you are on the flat portion of the temperature dependence of resistivity,

and the error due to the instability of temperature

is very low as compared with the other sources of

instability. Of course, the flat portion of the curve moves with the pressure, but we do not take this

into account.

In reply to Dr. Barnett: The objective of this

investigation was to obtain a large group of manganin gages with very similar characteristics. We

tried to obtain a very homogeneous group of manganin gages. That is why we used the Bridgman cycle.

But usually when we do the work in our lab with

standard 100-fl resistance gages, we use the very

fast technique of temperature seasoning. It is a sort of standardized procedure in our laboratory. It takes just a few milliseconds for the pulse from a

capacitor bank to flow through the gage. The

temperature is raised for a very small time up to 500 °C. After that, the residual stresses are eliminated and stability of zero resistance is very good.

We also tried to stabilize our manganin gages

subjecting them simultaneously to very high pres- sures and high temperatures, but results were not

satisfactory.

' L. H. Adams, R. W. Goranson, and R. E. Gibson, Construction and Properties of

the Manganin Resistance Pressure Gauge, Rev. Sci. Instr. 8, No. 7, 230-235 (July 1937).

51

session III

Chairman: F. P. BuNDY

General Electric Company

Schenectady, New York

A Critical Review of the Effect of Pressure on Thermocouple emrs

R. E. Hanneman, H. M. Strong, and F. P. Bundy

General Electric Research and Development Center, P. 0. Box 8, Schenectady, New York 12301

The effects of high pressure on the emrs of thermocouples are critically reviewed and best cur- rently available thermocouple corrections are presented. The important factors affecting these thermocouple corrections are discussed, including pressure-temperature gradients, contamination, and

deformation. A novel method for simultaneous and continuous in situ measurement of pressure and

temperature within a high-pressure cell through use of dual thermocouples is briefly described.

1. Introduction

As high-pressure research has progressed in the

past few years to include more quantitative experiments, the need for more accurate measurement of

temperature and pressure has become more critical.

Temperature is a particularly sensitive variable

because of its importance in numerous thermally

activated processes prevalent in materials. Since

thermocouples provide the primary means for in situ

temperature measurement in most high-pressure

experiments carried out at elevated temperatures,

the effect of pressure on the electromotive force

(emf) of thermocouples is often very important.

The central purpose of this paper is to review

critically our current knowledge of the effects of

pressure on the emf of thermoelectric materials

and to discuss the consequences of these effects on

the accurate in situ measurement of temperature

at high pressure. In addition, the concept of using

thermocouples to monitor continuously any local,

in situ pressure changes will be described.

2. Determination of Pressure Effects

on Thermoelectricity

The current theoretical understanding of thermoelectricity and the electronic properties of metals

and alloys is inadequate to permit any reliable

quantitative calculations of the effect of pressure on

thermal emfs. Thus, we must rely upon experimental determinations of the effect of pressure on

thermoelectric behavior.

2.1. Moderate Temperature-Hydrostatic

Pressure Range

The pressure effects on thermal emfs of metals

and alloys have been measured in hydrostatic systems by either single wire or composite thermocouple methods. Bridgman [1] ' carried out the first measurements of the effect of pressure on thermal

emfs of various materials using the single wire

method illustrated in figure la. Bridgman's results were accurate to within a few percent for the conditions studied (0 to 12 kbar and Tz-Ti^ 100 °C).

His work included some 20 metals and alloys but

his measurements did not include most of the

thermocouple materials commonly used.

Bell, England, and Boyd [2] have recently ex- tended the single wire hydrostatic method to meas-

' Figures in braclcets indicate the literature references at the end of this paper.

Paper presented at the Symposium on Accurate Characterization

of the High-Pressure Environment, held at the National Bureau

of Standards, Gaithersburg, Md., October 14-18, 1968.

I (b)

Figure 1. (a) Schematic diagram of Bridgman single-wire

method for determination of pressure dependence of thermal

emf. (b) Composite thermocouple methodfor measuring pressure

effects on thermocouples.

53
ure accurate corrections on Pt—Pt 10' percent Rh

and chromel-alumel thermocouples up to ~ 10

kbars and ~ 500 °C. The magnitude of the corrections obtained is in reasonably good agreement with

experimental data from non-hydrostatic experiments to be discussed later in this section. Since a

detailed paper by Bell et al. [2] on these results is included in this symposium, no further discussion of

their results needs to be included here.

The related composite thermocouple method for

measuring the pressure effects on thermocouples in a hydrostatic system is shown schematically in

figure 1(b). This method was first attempted by

Birch in 1939 over the limited pressure interval

0 to 4 kbars and for temperatures up to ~ 470 °C

[3]. Unfortunately, Birch failed to take into account

that only that part of the temperature gradient

lying within the pressurized zone can contribute to the pressure correction of thermocouples. Thus,

his deduced corrections were substantially smaller

than the true ones. It is perhaps fair to remark that

at least several other researchers have been caught

by the same trap in attempts to measure pressure

effects on thermocouples.

Bloch and Chaisse [4] have correctly used the

composite thermocouple method to determine the

effect of pressure on copper-constantan thermocouples over the range 0 to 5 kbars and — 196° C to 89° C. This is the most reliable data available for

low-temperature work and it is of particular interest for quantitative hydrostatic studies of magnetic

and crystaUographic transitions at low temperatures. Their copper-constantan results are in excellent agreement with Bridgman over the limited

mutual range of measurement. Figure 2 summarizes

some of the best current data available for thermoCu/CONST

Pt /Pt 10 RH Fe/CONST

0 Kb

5 Kb

10 Kb

10 Kb

200

"C

300 400 500

Figure 2. Compilation of some key pressure corrections for

thermocouples from hydrostatic systems.

couple corrections obtained from hydrostatic

systems.

In both of the above methods of measurement the

pressure gradient in the seal regions from Pj to Po is essentially isothermal. In addition, the internal

temperatures Tj and To are at the same temperature as the known external baths. As the internal

pressure is increased from the atmospheric pressure

Po to a higher value, Pj, the thermal emf output of a composite thermocouple (or single wire) will con- tinuously deviate from its corresponding value at Po for fixed values of Tj and To. Thus, accurate, absolute corrections for pressure effects on thermocouples can be obtained. The available physical

properties of seal and pressure vessel materials

have restricted measurements under the above

hydrostatic circumstances to pressures less than

about 15 kbars and temperatures of no more than

several hundred degrees.

2.2. Higher Pressure-Temperature Range

2.2.1. General Problems

Thermocouple corrections at much higher pres- sures and/or temperatures than discussed above

are often needed. Such corrections have been

achieved with much experimental difficulty using

non-hydrostatic systems but they are only approximate because of: (1) uncertainties in the absolute

pressure associated with non-hydrostatic systems

used, (2) lack of a precise knowledge of the temperature within the high-pressure region, (3) non- isothermal pressure seals and (4) other effects in- cluding plastic deformation. In general, the higher

the superimposed range of temperature and pres- sure the more difficult it is to obtain reliable,

approximate corrections.

These and other problems in achieving accurate

high pressure-high temperature measurement

using thermocouples have been discussed by

Hanneman and Strong [5, 6]. They show that

chemical contamination of thermocouples can

frequently occur within high-pressure cells at elevated temperatures and that this can strongly alter

the output of a thermocouple. Chromel-alumel

thermocouples are particularly susceptible to sizable high-temperature drift with time in pyrophyUite cells, while Pt/Pt 10 Rh thermocouples

appear to be relatively inert.

The magnitude of the effect of plastic deformation

(such as experienced within seal regions of nonhydrostatic cells) upon the thermal emf of Pt/Pt

10 Rh and chromel/alumel was found generally

to be within a few percent of the total pressure

correction at high pressures. As long as a high

resistance insulating tubing such as BN or alumina

surrounds thermocouple wires at high temperatures,

the electrical shunting effect on thermocouple

output should be negligible.

One of the most important modifications of

54
thermocouple corrections that can vary from one

type of cell to another is the presence of temperature

gradients within a pressure gradient region. The

total pressure-induced contribution to the emf of

a thermocouple is given by the following general

expression [6]:

dQ

BP P{T)dT (1)

where dQjdP is the pressure dependence of the

thermopower of the couple andP(T) is the pressuretemperature function along the thermocouple

path into the cell. One can independently express

P in terms of T in eq (1) without explicitly including

the distance variable, since both follow continuous

paths simultaneously. The quantities Tj and To

are the internal couple junction temperature and

the temperature where the pressure drops to atmospheric pressure, Pq, respectively. The quantity

dQIdP is, to a first approximation, a constant for

the most common thermocouple materials and

dEp is nearly proportional to a temperature shift.

Thus, the resultant decrease in temperature correction produced by a superimposed pressure-temperature gradient is approximately proportional to the

normalized area under a plot of the temperaturepressure path of that thermocouple. Figure 3 shows

schematically a diagram of a typical temperaturepressure path of a thermocouple (i.e.. To CD) in a

typical cell. The area ADTgTj corresponds to the

maximum theoretical correction if seals were isothermal. The ratio CDTjTqIADTjTq is the fraction

of the maximum correction applicable to the particular cell and Pj, Tj, and To of interest. Correction

modifications due to the superimposed pressuretemperature gradient effects sometimes exceed

10 percent of the total temperature correction. The

above types of gradient modifications apply to both

relative and absolute pressure-induced thermocouple corrections in any kind of apparatus.

Actual measurement of the approximate pressuretemperature gradient function along a thermocouple

path within any given cell assembly is a difficult

problem. To determine approximate temperature

gradients within a cell and its seal region, thermocouples are placed with their junctions in a series

of positions from the center of the cell sequentially

toward the seal. The raw mv values of all thermocouples (and hence the approximate temperatures)

along the path are determined from experiments

where both the cell pressure and internal temperature are varied over the desired range. Care must be

taken so that the extra thermocouples do not substantially alter the temperature gradients within

the cell. The pressure gradients within a cell can be similarly approximated by a series of localized

pressure calibrations as described earlier by

Hanneman and Strong [6]. In carrying out such

experiments, it should be borne in mind that the

local pressure within a non-hydrostatic cell at fixed

applied load can be dependent upon the temperature.

2.2.2. Methods of Determination and Results

Three major classes of measurement methods

have been utilized to evaluate absolute emf corrections in non-hydrostatic systems including: (1) a modified Bridgman method [7], (2) direct thermal

noise technique [8], and (3) indirect, self-consistent,

caUbration methods [5]. Modified Bridgman Method. A modified Bridgman

method was used in a key study by Bundy to obtain

thermocouple corrections aver a much wider

pressure range than previously studied (e.g., 0 to 72 kbars) [7]. Figure 4 illustrates schematically the

St(°c) PI/PR/

Cu /Const^

Mi/NiMo____-J

Chr/Alu

20 40

Figure 3. Schematic of the pressure-temperature profile along

a thermocouple path in typical and ideal high-pressure cells.

IDEAL ISOTHERMAL PRESSURE SEAL

ACTUAL CASE

to

Q_

TEMPERATURE

Figure 4. Schematic diagram of modified Bridgman methodfor

measuring effect of pressure on thermal emfs.

55

385-762 O - 71 - 5
HtAlING

COIL

i I UNDER

TEST

Figure 5. Summary of pressure corrections for thermocouples

for temperature gradient of 100 °C in non-hydrostatic range by

the modified Bridgman method.

Bundy apparatus. The maximum temperature

gradient that could be achieved by this apparatus

was approximately 100 °C because of the sizable

thermal conductivity of the system between the

hot and cold pistons. It was shown that errors due

to a small temperature gradient in the seal regions

of the cell did not exceed 5 percent of the thermocouple corrections obtained. Bundy's results for

three common types of thermocouples are shown in

figure 5. Recent similar measurements by I. Fujishiro, H. Mii, and S. Sakaida [9] have recently

extended the single wire alumel data of Bundy out

to 130 kbars for a 100 °C gradient with an opposed

anvil apparatus.

Getting [10] has attempted the very challenging

task of obtaining absolute thermocouple corrections

for Pt/Pt 10 Rh couples up to 40 kbars and 1000 °C

by the single wire modified Bridgman technique in a

piston cylinder apparatus. Although these experiments have met with the great difficulties inherent

in obtaining accurate corrections in this high-temperature range, some approximate corrections have

been deduced. While in common agreement with

nearly all workers over the range 0 to 100 °C up to 40 kbars these results do come into substantial

disagreement with other higher temperature meas- urements to be described later as well as extrapolations of the lower pressure high-temperature

measurements of Bell et al. [2]. Since aU of the methods described so far have

relied essentially upon external heat sources for

the known temperature gradients set up within the

high-pressure cell, the corresponding range of

experimental results have been mostly restricted

to rather limited temperature differentials within

the cells, especially at higher pressures. In order

to extend thermocouple corrections to the important

region of higher temperatures at high pressures,

new methods of measurement using internally

heated cells have been devised. These will be

described in the next two sections of this paper.

Thermal Noise Method. Studies using the thermal

noise method to obtain thermocouple corrections

at high pressure and temperature have recently

been initiated [8]. Briefly, this method is based on

the fundamental principle that a small pressure

independent thermal noise is present in a resistor

(or conductor). This may be given by:

{M:yiiRAf)=UT

where (A£')2 is the mean squared voltage fluc- tuations in a conductor of resistance R measured

over the frequency interval A/, and kT is the usual

thermal energy term [11]. Experiments using this

technique are very difficult, as they require ex- tremely low noise measurement circuitry and a low

noise ambient. Furthermore, d.c. heating is necessary in internally heated high-pressure cells to hold

noise to a tolerable level. Thus, salt type cells cannot

be used because of electrolysis effects. Since the

magnitude of the mean voltage fluctuations is of

the order of a microvolt per 1000 °K, measurements

must be carefully made.

The most notable results to date at high pressure

and temperature using the thermal noise method

are those of Wentorf [8]. He has determined approximate corrections for Pt/Pt 10 Rh thermocouple up

to 1000 °C and 40 kbars. The very limited data

available are in reasonable agreement with the

absolute corrections of Hanneman and Strong [6]

obtained by a totally different approach that is described in the next section. For example, at 40

kbars and 1000 °C the absolute correction to a Pt/Pt

10 Rh thermocouple determined by Wentorf is + 40 °C compared to a value of + 36 °C deduced previously by Hanneman and Strong.

Indirect Self-Consistent Methods. The most ex- tensive approximate determinations of absolute

type pressure corrections for thermocouples have

been obtained from indirect, self-consistent measurements. Until recent advances in low-noise electronics made the thermal noise technique possible,

only indirect type measurements could be used to obtain corrections in the important range of high

temperature at high pressure.

The first major effort to determine corrections in

this region was that of Hanneman and Strong [5],

who used the following indirect methods to deduce

approximate absolute couple corrections: (1) comparison of experimental and thermodynamically calculated results of the effect of pressure on nu- merous single-component phase transformations and

diamond growth from metal-carbon systems, and

(2) correlation of results of high-pressure-hightemperature diffusion data. The scheme upon which

the first class of indirect methods was carried out can be described most simply by referring to figure

6. This figure shows schematically three independent types of phase transformation common to vari- ous systems at high pressure. The solid curve A

with a very steep temperature-pressure transfor56
mation boundary is typical of phase changes with a

large ratio of volume change to entropy change, such

as graphite to diamond. Solid curve B shows a trans- formation with a relatively low slope (dT/dP), which

is typical of melting curves of metals with low volumes of fusion. Solid curve C depicts a transformation with a volume decrease on going to the higher

temperature phase. Typical examples of this trans- formation include the melting of many semiconductors and the y transformation in iron. The

dashed lines are representative of uncorrected, ex- perimental curves that are found for each of the

types of transformations. By designing a highpressure cell to measure simultaneously two or more phase transformations (of different types)

within the central isobaric pressure region, approximate thermocouple corrections to obtain a best fit of both sets of data can be deduced. Such data provide not only the approximate temperature corrections at pressure, but also the temperature coefficient of the internal cell pressure at fixed load.

A series of such experiments over various pressuretemperature ranges with a variety of different materials permits a reasonable self-consistent approximation of absolute thermocouple corrections of a given type of thermocouple. The most sensitive

types of combinations of transformations are A

versus C or B versus C, since the C transitions have

opposite slopes to the other two types. Numerous

experiments of the above types have been carried

out by Hanneman and Strong with self-consistent

results.

PRESSURE

Figure 6. Schematic diagram of various types of phase trans- formations used in indirect determinations of thermocouple

corrections.

Figure 7. Present approximate absolute pressure corrections for PtIPt 10 Rh thermocouples up to 1300 °C and 50 khar for gradient factor equal to unity.

57
5

40 kb

The second independent method they used to

test the approximate magnitude of the thermocouple

corrections was based on high-pressure diffusion

experiments and theory. This approach has two

indicators of the size of the correction. The first

is to find, by a best fit of data, the appropriate

temperature correction at a given pressure and

composition which gives a constant value of activation volume for diffusion, V*, over the temperature

range of experimental data while maintaining a linear log Dp versus l/T plot, where Dp is the dif- fusion coefficient at a pressure P and temperature

T. The second approach is based upon a comparison

of the value V* calculated from the experimental

atmospheric and high-pressure diffusion coefficients

at fixed temperature with that obtained from experiments in the hydrostatic range and from theory.

The approximate absolute thermocouple corrections

deduced from diffusion data in the iron-vanadium

and iron-nickel systems at 20 and 40 kbars are in

reasonably good agreement with those obtained up

to 1300 °C by the other methods described above

(i.e., within ~ 20 percent). Further details on the

diffusion analysis are treated in reference [5]. A

complete plot of the approximate absolute corrections for Pt/Pt 10 Rh couples deduced from the

best fit of all the indirect data of Hanneman and

Strong and the thermal noise results of Wentorf

adjusted for a pressure gradient factor of unity are presented in figure 7. Corresponding approximate absolute corrections for chromel/alumel

couples are shown in figure 8. The corrections shown

should be added to the uncorrected temperature.

For comparison the Getting-Kennedy corrections

[10] are shown as dashed lines in figures 7 and 8.

3. Effects of Pressure on the Relative

Differences Between Various Thermocouples

The relative correction differences due to pressure-induced effects on thermocouples can be

deduced by simply subtracting the known absolute

Figure 9. Approximate relative corrections between PtjPt 10 Rh

and chromeljalumel thermocouples up to 1000 °C at 20 and 40

kbar intervals. Gradient factor equal to unity.

corrections of two different types of thermocouples

at a given temperature and pressure. In the limited

pressure-temperature region where hydrostatic

measurements have been made and for small

thermal gradient regions (i.e., Ar ^ 100 °C) this

yields satisfactory relative corrections because the

absolute corrections are fairly reliable.

In the more uncertain high-pressure—hightemperature region, more accurate relative cor- rection measurements can be obtained by direct

experiments than can be deduced from absolute

data. Such experiments have been reported for a

direct comparison of chromel-alumel versus Pt/Pt

10 Rh couples by Hanneman and Strong [6] and

by Peters and Ryan [12].

In order to obtain direct relative data at high

temperature, two different types of thermocouples

are brought into an internally heated high-pressure

cell via equivalent pressure-temperature paths.

Their junctions are both placed at the same position

(i.e., pressure and temperature) and the wires are insulated throughout the cell. When the emf out- puts of the two different thermocouples are referred

to standard thermocouple tables, there is an apparent temperature discrepancy which increases

with pressure at a given temperature. These data,

together with a knowledge of the approximate cell

pressure and internal gradients, permit a good

approximation of relative corrections.

The best approximate relative corrections (for |

a pressure-temperature gradient correction factor

of unity) for Pt/Pt 10 Rh versus chrome!/alumel

couples are shown in figure 9. The estimated ac- curacy of these relative measurements is approximately ± 10 percent. Since these data are more

reliable than any absolute corrections at high temperatures, any absolute data obtained on both types

of couples must be in approximate agreement with

the data in figure 9 if it is correct. Limited additional

58
relative correction data has been published for

Pt/Pt 13 Rh and iron/constantan [5].

More extensive measurements of relative thermocouple corrections, especially at higher temperatures and pressures and for other types of couples,

are needed. These data will be particularly helpful

for the direct quantitative comparison of the

published results of various workers on the same

physical phenomena who have used different

thermocouples. In this regard, it is urged that

workers evaluate the pressure-temperature gradient

effects in their cells and beware of the other sources

of errors from thermocouple measurements described earlier in this paper.

4. Discussion and Conclusions of

Absolute Corrections

The present theoretical knowledge of thermoelectricity and electronic properties of metals is insufficient to predict quantitatively the pressure

dependences of thermocouples. We have shown

that quantitative measurements of corrections for

the pressure-induced deviations of thermocouples

are very difficult in the high-pressure-high-iemperature regime.

The most accurate corrections to date have been

obtained in the range —200 to several hundred

degrees centigrade and 0 to 15 kbars where hydrostatic cells can be used and deformation, contamination, and other problems can be minimized. A

summary of some of the key results is shown in

figure 2. Many of the results in this region are accurate to within a few percent.

Outside of the hydrostatic pressure range, the

next most reliable data to date have been obtained

from the modified Bridgman technique within a limited temperature range (0 to 100 °C). These

results are generally believed to have an accuracy

of better than 10 percent. Some of these by data

are shown in figure 5. The uncertainties in absolute corrections rise

substantially in the range up to 50 kbars and 1300 °C

where approximate corrections have been proposed.

Nearly all of the indirect and thermal noise data

[5, 6, 8] are compatible with the corrections given

in figures 7 and 8. In addition, the magnitude of the

corrections are compatible with extrapolations from

lower pressure data achieved from direct determinations [2, 7]. As shown in figure 7, the Getting-Kennedy corrections [10] determined by the modified Bridgman

approach do deviate substantially at higher temperatures from the other methods mentioned above.

Although these corrections are of the same sign and

magnitude as those from work in this laboratory,

they are smaller by nearly one-half at the highest

temperatures mutually examined. Possible inherent

errors in the Getting-Kennedy approach would tend

to make their corrections as lower limit values.

whereas the thermal noise method would tend to give upper limit values. Thus, the actual absolute

corrections probably lie somewhere in between the

solid and dashed curves in figures 7 and 8. Until more quantitative measurements can be

carried out, it appears that one could obtain reasonable but approximate absolute corrections by

averaging between the dashed and solid lines given

in figures 7 and 8, and applying liberal error limits

which span both sets of corrections. More accurate

and complete measurements are clearly needed

for better absolute corrections, but such tests will

be very difficult.

5. Simultaneous and Continuous in

situ Measurement of Pressure and

Temperature

One of the major problems in research and

applied technology in the area of high pressure has

been the lack of an accurate, reliable method for

continuous and simultaneous in situ measurement

of high pressures and temperatures. Since accurate

thermodynamic or kinetic studies under high

pressure require a means of independently evaluating the true pressure and temperature at the sample,

preliminary work in our laboratory was undertaken

to develop a reliable, convenient method to accomplish this requirement.

Pressure measurement in most previous highpressure investigations has relied completely on

precalibration methods, which use first-order phase

transformations of standard materials at room temperature to determine the relation between external

press load and internal cell pressure for a given

apparatus and cell geometry. In some cases approximate continuous measurements of pressure have

been possible by use of resistance measurement of

manganin or similar wire, but this is limited to < 250 °C and is sensitive to several variables. For

various reasons, the apparent pressure based upon

press load in a subsequent experiment often does

not indicate the true internal cell pressure. For

example, the pressure within a high-pressure cell

varies as a function of temperature and time due to thermal expansion, phase changes in the sample

holder or specimen, gasket leakage, and relaxation

of stresses in the apparatus. Concurrently, the

measurement of the true temperature in an internally-heated, high-pressure cell requires a knowledge

of the effect of pressure on the electromotive force

of thermocouples as described above.

By ulilizing two different thermocouples which

have differing, known pressure dependences, one

can uniquely monitor continuously both pressure

and temperature within any high-pressure cell. In

this method the junctions of two different thermocouples are placed together at the desired location in a high-pressure cell and both couples are sur- rounded by insulating material leading out of the

59
0 12 3 4 5 6 Au/Pt 40 Rh MILLIVOLTS

Figure 10. Direct mv versus mv comparison of two different thermocouples for in situ pressure determination within a high-pressure cell.

(Pt 40 Rh/Pt 10 Rh versus Au/Pt 40 Rh couples.)

1400

1200

1000 oo^ 800

r>

I—

<iaiUJz 600

400-

200 -i

60 Kb

_L

0 12 3 4 5 6 7 Au/Pt 40 Rh MILLIVOLTS

Figure 11. Pressure effect on calibration of Au/Pt 40 Rh

thermocouples for temperature determination.

cell to cold junctions. These leads are then con- nected to sensitive potentiometers or other devices

for accurately reading the emf output of both

thermocouples. Such measurements require choice

of thermocouple materials wrhich m\\ not be susceptible to chemical contamination drift. Figures 10

and 11 show the approximate pressure effects for

Pt 40 Rh/Pt 10 Rh and Au/Pt 40 Rh thermocouples

which have been used for simultaneous in situ

measurements of pressure and temperature. The

concurrent placement of both thermocouple emfs

on figure 10 fixes the pressure. Then, by reference to a more detailed graph similar to figure 11, the cor- responding temperature can be directly determined.

Such determinations have absolute accuracies only

as good as the absolute corrections available for

thermocouples. However, relative changes of pres- sure within a cell during a given experiment can

be seen to less than a kilobar under optimum

circumstances [13].

6. References

[1] Bridgman, P. W., Proc. Am. Acad. Arts Sci. 53, 269 (1918).

[2] Bell, P. M. England, J. L.. and Boyd, F. R., Carnegie Insti- tution: Ann. Kept, of Director Geophys. Lab., No. 1499,

pp. 545-7 (1967); also The effects of pressure on platinum/

platinum 10% rhodium and chromel-alumel thermocouples, this symposium.

[3] Birch, F. Rev. Sci. Instr. 10, 137 (1939).

[4] Bloch, D., and Chaisse. F., J. Appl. Phys. 38, 409 (1967).

[5] Hanneman, R. E., and Strong, H. M., J. Appl. Phys. 36,

523 (1965).

[6] Hanneman, R. E., and Strong, H. M. J. Appl. Phys. 37,

612 (1966).

[7] Bundy, F. P., J. Appl. Phys. 32 , 483 (1961).

[8] Wentorf, R. H., Temperature measurement by thermal

noise at high pressures, this symposium.

[9] Fujishiro, I., Mii, H., and Sakaida, S. BuU. Jap. Soc. Appl.

Phys. 37,621 (1968).

60
[10| Getting, I. The effect of pressure on the E.M.F. of the ther- mocouple, this symposium.

[11] Garrison, J. B., and Lawson, A. W., Rev. Sci. Instr. 20, 785

(1949).

[12] Peters, E. T., and Ryan, J. J., J. Appl. Phys. 3 7, 933 (1966).

[13] Hanneman, R. E., and Strong, H. M., New applications of duplex thermoelectric measurements in metals and alloys

under high pressure-high temperature conditions, G.E. R&DC report 65-RL-3913X (Mar. 1965).

DISCUSSION

D. B. McWhan (Bell Telephone Laboratories,

Murray Hill, New Jersey): Are there any data on

pressure corrections for thermocouples using some

of the newer materials, such as chromel versus

gold-iron, at low temperatures? This thermocouple

has a sensitivity of 15 fxY per degree at 4 K.

M. C. Krupka (Los Alamos Scientific Laboratory,

University of California, Los Alamos, New Mexico):

Do you have any information in the relative correction of an increased rhodium content couple — for

example a Pt-Pt 13 percent Rh couple?

I. Getting (Institute of Geophysics and Planetary

Physics, University of California, Los Angeles,

California): In some work I did on platinumrhodium alloys other than platinum 10 percent

rhodium, I found that the absolute correction

increases shghtly as the rhodium content increases.

Within the uncertainties of our measurements, the

correction could be taken to increase linearly with

the rhodium content.

G. C. Kennedy (Institute of Geophysics and

Planetary Physics, University of California, Los

Angeles, California): In applying two thermocouples

to measurement of the pressure by means of the

differences in pressure effect on thermo-emf,

wouldn't the most sensitive pressure-measuring

device be obtained by using a couple made of a metal of minimum compressibility versus a metal of

maximum compressibility?

I'd hke to comment on the indirect method of

measuring the pressure correction for a thermocouple. A number of years ago you published a suggestion as to the size of the correction, and

the basis of it was the difference between the

computed thermodynamic diffusion slope of a boundary and the measured slope. One of the

boundaries used was the germanium melting curves

that we published. We had also just done eight other

melting curves. For over half of our curves, a comparison of the thermodynamic computed slopes versus measured slopes would suggest a thermocouple correction in an opposite direction. My view

is that the uncertainties of the thermodynamic

parameters are such that from data of this sort

Uttle can be said about thermocouple corrections.

Finally, in very careful experiments with wellcharacterized materials and thermodynamic

parameters, I believe that useful, approximate

thermocouple corrections can be obtained by comparison of experiments and calculation. This is demonstrated by comparing our previous results

with those from "direct" methods reported at this

Conference.

H. Tracy Hall (Brigham Young University, Provo,

Utah): In some multi-anvil devices it is convenient

to make a thermocouple lead connection directly to an anvil rather than to bring the wire out. This is particularly true in a cubic press. Now, in a cubic

type of press with an internal heater containing a charge of good thermal conductivity — diamond

powder or something like that — leads have been

brought out to the anvil surface and then enfolded

in a metal, nickel in one instance, so that the contact

is between platinum-platinum rhodium wire and

nickel. Then the nickel is in contact with the tungsten-carbide (6 percent cobalt) anvil.

If you make a temperature calibration in the usual

way by controlling power input (say watts of electric

power) versus thermocouple temperature taken

from the tables, you will obtain a certain curve using

these nickel tabs. Now if the ends of the thermocouple wire are enfolded in, say, copper tabs, the

result will be different. And I suppose if you pick

some other metal it would still be different.

Because of the high symmetry of this system, it would appear that pressure gradients ought to be

pretty much the same in all cases, and the temperature at the anvil surfaces ought to be about the same. At 1500 degrees there is about a 40-deg C differ- ence between results using copper and using nickel.

The difference increases as you go to higher and

higher temperatures. At 400 or 500 degrees, there's

not much difference. At 1000 degrees, there may

be 20 deg C difference. Wherein would you say lies

the cause of this discrepancy? What effect is there

that gives you those differences? What would be

the basis for choosing a metal for the tabs so that

the temperature correction is very small?

A. Taylor (Westinghouse Research Laboratory,

Pittsburgh, Pennsylvania): I'm wondering whether

we are really looking at the problem from the right

point of view. Rather than saying what the temperature is at a particular pressure, should we not state

what the pressure is at a particular temperature?

If, from your curves, the temperature is approximately 40 degrees off at 1500 degrees, that is only a

relatively small percentage error in the temperature.

But at that temperature, a nominal pressure of about

60 kilobars in the tetrahedral anvil press can, in a matter of a few seconds, drop from let's say 60

kilobars down to only 40 kilobars, and you might

never know it from the outside conditions.

So the question reaUy is: How can we measure

the pressure at these high temperatures rather than

the temperature at the high pressures?

61
AUTHORS' CLOSURE

In reply to the question by McWhan: I believe that

data on copper-constantan from — 200 °C to room

temperature is significantly better than any other in

this temperature range. Thus, determination of the

relative correction between copper-constantan and

other couples would be worthwhile. No direct data on pressure corrections of chromel versus gold-iron

couples at cryogenic temperatures has been obtained to my knowledge.

In reply to Dr. Krupka: The effect of increased

rhodium content in Pt-PtRh couples is to increase

the correction shghtly on going from 10 to 13 percent

Rh. Some explicit results are given in our 1965

paper (ref. 5). In reply to Dr. Kennedy: Selection of metals of

high and low compressibility for pressure-measuring

thermocouples would not necessarily give the greatest sensitivity.

The thermopower, Q, of a material can be given

by:*

^ r a In ^ , a In \ 1

where k, T, and e have their usual meanings and A

and A. are the effective area of the Fermi surface and

the electron mean face path, respectively, evaluated at the Fermi energy cf. Although the pressure

dependences of A and X are not known sufficiently

weU in real metals and alloys to predict the effect of

pressure on thermocouples, it is instructive to estimate the order of magnitude of the effect.

Applying the free electron approximations to the

terms in A and A. and differentiating with respect to pressure, it foUows that:

—

dQ ttW )8 ~ ot —

dP e €f

where a is factor of the order of unity, /3 is the isothermal compressibility, and the other terms have

the previously stated meanings. Substituting in

typical values of 1,000 °K, ft of the order of

10"^- to 10"'^ cm^/dyne, and ~ few eV and using

equation (1) in the text, emf corrections of the order

of 0.1 to 1 mV are indicated at 50 kbar, for example,

for a given metal. Thus the total expected pressure

dependence of a thermocouple should be of this

magnitude or smaller, since it wiU arise from differ- ences in (jS/e/r) of the two materials used.

In the situation described by Dr. HaU, if the

pressures are really about the same in the two cases,

and if there is not a significant temperature gradient

through the metal, then one is forced to assume that

the difference between results with nickel tabs and

copper tabs is due to stress effect, deformation, or something of this sort. Chemical contamination

could be also rather important. A further key to the cause of the discrepancy might be obtained by

studying the effect of copper and of nickel tabs as a

function of time.

In choosing materials for the tabs, I would favor

materials which are as chemically inert as possible

and which keep the temperature-pressure gradients

as smaU as possible.

In reply to Dr. Taylor s question: I would submit

that for quantitative work, in situ measurement of

both pressure and temperature are important. That

is why I presented the dual thermocouple idea in

my paper. Thus an in situ pressure drop would be

immediately noticeable.

In addition to phase diagram definition under

pressure which seems to have been of most past

concern, it's clear to me that thermaUy activated

processes including nucleation, growth, diffusion,

and other kinetic processes are going to require

increasingly accurate in situ characterization of

both temperature and pressure. In these latter

processes an error of 40 °C in 1,000 °C to 1,500 °C

would generally lead to much greater errors than,

say, 10 kbars out of 50 kbars.

*J. M. Ziman, Electrons and Photons (Oxford Univ. Pitss, London, 196U).

62
The Effect of Pressure on the Thermal Emf of the Platinum/Platinum 10

Percent Rhodium Thermocouple

P. M. Bell, F. R. Boyd, Jr., and J. L. England

Geophysical Laboratory, Carnegie Institution of Washington, Washington, D.C. 20008

Temperature in high-pressure experiments is usually measured by thermocouples, which are themselves subjected to pressure. The effects of

pressure on the thermoelectric properties of these

thermocouples are significant in many experiments,

and they need to be quantitatively understood. This

problem has become acute with the development of

solid-media pressure apparatus capable of operating

at temperatures up to 2,000 °C and pressures of

100 kbar or more. Corrections for the effects of

pressure on thermal emf may be as much as 50

to 100 degrees under these extreme conditions.

Other papers in this volume discuss the evaluation

of these effects by experiments with solid pressure

cells in the pressure range above 10 kbar (Hanneman. Strong, and Bundy, p. 53; Getting and

Kennedy, p. 77). The purpose of the present

study has been to provide more extended data for

the region below 10 kbar, which can be explored

with an externally heated gas apparatus. Measurements that can be made with gas apparatus are limited in range, but they are inherently more

accurate than measurements with solid-media

apparatus. It is hoped that the data obtained in

the present study will provide a useful base for the

solid-media experimentation.

Early measurements with gas apparatus on the

effects of pressure on thermal emf by Bridgman

(1918) were remarkably precise but extended to only 100 °C and 12 kbar. Unfortunately Bridgman

did not study most of the alloys that are currently

Paper presented at the Symposium on Accurate Characterization

of the High-Pressure Environment, held at the National Bureau

ofStandards, Gaithersburg, Md., October 14^18, 1968.

used for thermocouples in high-temperature, highpressure experiments. He formed a thermocouple

by pressurizing part of a continuous homogeneous

wire. One pressure seal was heated, while the other

was cooled to 0 °C for reference. In such an experiment—now called a "single-wire experiment" — the

absolute pressure effect is measured directly for a

given element or alloy.

Birch (1939) has provided data on chromel/

alumel and platinum/platinuni 10 percent rhodium

thermocouples to 580 °C and 4 kbar. He subjected

these thermocouples to nitrogen pressure in steel

tubes immersed in a lead bath and compared their

emf readings with those from reference thermocouples at the same temperature but at atmospheric

pressure. As discussed by Bundy (1961), Birch did

not cool the pressure seal ii> his apparatus to 0 °C,

and hence he did not observe a maximum or "absolute" effect.

In the present study a platinum/ platinum 10

percent rhodium thermocouple was subjected to nitrogen pressure in a tube, one end of which was

heated by an electrical furnace while the other end

was cooled to 0 °C in an ice bath (fig. 1). The emf

of the pressurized thermocouple was read as a function of pressure relative to a second platinum/

platinum 10 percent rhodium thermocouple heated

to the same temperature but held at atmospheric

pressure. The data obtained are "absolute" in that

they are tied to a 0 °C reference. Most modern highpressure apparatus does not have pressure seals

at 0 °C, and practical applications of these data

will generally require correction for the temperature

of the pressure seal.

furnace

to Heise Bourdon-tube gauge

standard cone T -eal

'

to Manganin \ pressure gauge \

reference junctions

\ stirred ice batti (0°C)

copper leads to potentiometer -

null detector

standard tiigti pressure seal

ttiermocouple to be

tested under pressure

Figure 1. Gas apparatus used for the measurement of the effect of pressure on the thermal emf of the platinumjplatinum 10 percent

rhodium thermocouple.

The pressure medium used was nitrogen.

63
1. Apparatus

The apparatus is designed for the range 1000 °C

and 13 kbar, but most of the experiments reported

here were made in the range up to 5 kbar. The

pressure vessel (Rene 41) containing the pressurized

thermocouple was heated in an external electric

furnace constructed to have a thermal gradient less

than 17in at the hot spot and capable of precise

control. The monitor thermocouple at atmospheric

pressure was located in a small cavity in the outside

end of the pressure vessel. Leads from both thermocouples were brought into a stirred ice bath in which

the pressure seal was suspended. Two additional

thermocouples were employed, one in the ice bath

itself and another on the high-pressure side of the

cold pressure seal. Emf differences between

pressurized and unpressurized thermocouples were

measured with a potentiometer-nuU detector combination, which can be read to 1 X 10~^ V.

Readings were taken at constant temperature at intervals of pressure up to 10 kbar. The temperature

difference between the hot junctions of the pressurized thermocouple and the monitor thermocouple

was reduced to about 0.1 °C. The Ar observed at

sjDqo|!)| 'sjnssajd

Figure 2. Measurements of the change in emf (negative) of a

platinumlplatinum 10 percent rhodium thermocouple with

pressure. The cold junction and pressure seal were at 0 °C. This correction must be added to the observed emf of a pressurized couple to give the true temperature.

atmospheric pressure at the start of an experiment

drifts slowly during the experiment, apparently

owing to the large increase in conductivity with

pressure of the gas. Thus observed Ar readings

required a small correction for shift of the base.

Data were not accepted if the potential difference

of the pressurized and unpressurized cold-junction

thermocouples indicated a temperature difference

of greater than 0.01 °C.

2. Results

Pressure decreases the thermal emf of a platinum/

platinum 10 percent rhodium thermocouple, re- quiring a positive correction to the reading of the

thermocouple for some pressure above atmospheric.

This effect is not linear with pressure or temperature, but its magnitude increases with both pressure

and temperature.

Figure 2 shows the experimental data in a series

of isotherms ranging up to 509 °C, and these data

are plotted again in figure 3 as isobars. Some of the

points (fig. 2) were repeated in a series of heating

and cooling cycles and were found to be reproducible to better than a microvolt. The maximum effect

T

Temperature, °C

Figure 3. Measurements in fig. 2 plotted as isobars.

This correction must be added to the observed emf of a pressurized couple to give the true temperature.

.80

.60

.40

64
observed was 20 /xV at 509 °C and 3.5 kbar, cor- responding to an apparent AT" of 2 °C. Birch's

determination (op. cit., fig. 2) shows about 13 fx\

at this same pressure and temperature. The differ- ence between present resuhs and Birch's measure- ment is most likely due to the fact that the pressure

seal was cooled to 0 °C in our experiment but was

at some temperature above room temperature in

his determination. Present results agree in sign and

are roughly comparable in magnitude to results

extrapolated from solid-media experiments at much

higher pressures. Our results must be extended to higher pressures and temperatures, however, before

meaningful quantitative comparisons with the

sohd-media data can be made.

3. References

[Ij Birch, F., Thermoelectric measurement of high temperatures

in pressure apparatus. Rev. Sci. Instr. 10, 137-140(1939).

[2| Bridgman, P. W., Thermoelectromotive force, Peltier heat,

and Thompson heat under pressure. Proc. Am. Acad. 53,

267-386 (1918).

[3| Bundy, F. P., Effect of pressure on emf of thermocouples,

J. Appl. Phys. 32, 483-488 (1961).

[4] Getting, I. C, and G. C. Kennedy, The effect of pressure on

the emf of the thermocouple. This volume, p. 77.

[5] Hanneman, R. E., H. M. Strong, and F. P. Bundy, A critical review of the effect of pressure on thermocouple emfs. This volume, p. 53.

65

Pressure Dependence of the Thermoelectric Power of Thermocouple

Materials

Paul J. Freud and Phillip N. La Mori

Battelle Memorial Institute, Columbus Laboratories, 505 King Avenue, Columbus, Ohio 43201

Single-wire measurements of the pressure dependence of the thermoelectric power were made

hydrostatically to 8 kbar for chromel, alumel, copper, and constantan and in a piston cylinder apparatus

to 40 kbar for chromel, alumel, platinum, and platinum 10 percent rhodium. The temperature interval covered for the hydrostatic measurements was — 195°C to 290 °C and for the piston cylinder measure- ments it was 30 °C to 380 °C. A detailed discussion is given of the pressure-temperature distribution within the piston cylinder cell. Pressure emf values are presented with an uncertainty of ± 7 percent.

1. Introduction

Measurements of the pressure dependence of the

thermoelectric power (T.E.P.) have been made on a number of thermocouple materials using the singlewire technique. The objective of this work was to establish a pressure and temperature range in which

the pressure dependence of the T.E.P. could be

determined to an accuracy required for normal

thermocouple use. Therefore the main emphasis of

this study was to define the pressure and temperature conditions of the high-pressure ceU rather than

to push for the extremes of pressure-temperature

conditions. A number of high-pressure cell configurations were investigated and calibrated before we

arrived at a configuration using solid AgCl for the

main pressure column. The temperature distribution within the high-pressure cell was determined

carefully, since the calculation of the pressure dependence of the T.E.P. is crucially dependent upon

how the pressure and temperature are distributed

within the measuring cell. We believe that it is valuable to restate the con- cepts involved with the use of thermocouples under

high-pressure conditions so that a clear understanding of the important parameters is established. The

simplest way to look at thermal emf's is to imagine

the thermocouple circuit as divided up into small

segments each with a particular Ar across it. An

emf will be induced across the segment which is proportional to Ar. The proportionality constant is the

thermoelectric power (a) of the material being used.

The total thermal emf of the circuit is determined by

summing all the emfs from each segment. Thus,

V='2^'^Ti^j>a{T)dT. (1)

Figure la is a schematic diagram of a typical

thermocouple circuit used in a high-pressure

environment. The circuit consists of two dissimilar

wires which go from ambient pressure through

Paper presented at the Symposium on Accurate Characterization

of the High-Pressure Environment, held at the National Bureau

of Standards, Gaithersburg, Md., October 14-18, 1968.

pressure seals into the high-pressure and hightemperature environment. The seal is represented

by the crosshatched region, and in this region the

pressure increases from the ambient pressure to the high interior pressure. For this analysis it is assumed that the sealing region is isothermal. The

errors introduced when this assumption is not valid are discussed later.

The temperatures of significance in this circuit are the reference temperature To, the temperature

of the seal region, Ti, and the junction temperature,

T2.The T.E.P. of a given material is in general a function of temperature and pressure and can be

written as the sum of a pressure-independent term

and a pressure-dependent term:

aa = (Xa{T) + ^aaip,T). (2)

The subscript refers to the particular material in

question. The first term is the T.E.P. of the material

at zero pressure, and the second is the change in

the T.E.P. due to pressure. The second term is also a function of temperature. If we integrate the emf

around the circuit shown in figure la, the following

P = 0 i

; H 1 g ti pressure ; |\^

Pi:;:;

Wire a ., ^ Wire b •Vob •

To

High pressure . 1.

Wire o Wire 0

To

Figure 1. (a) Schematic diagram of thermocouple circuit for

high-pressure apparatus.

(b) Schematic diagram of high-pressure single-wire

circuit.

67
expression results:

r fTi rT2

Vab--<badT=\ a„iT)dT+\ [cxaiT)

+ A«„(p„ T)]dT+ K(r)

+ ^a,{pi,T)]dT+ \ a„{T)dT.

JT,

This reduces to the following when the pressureindependent and pressure-dependent terms are separated:

Va, = r [aaiT) - at{T)]dT

+ r' [Aaaipu T) - Amipi, T)]dT. (3)

jTi

The first term is just the emf, Vq, which would be

induced in the circuit at zero pressure (the value

tabulated for the temperature interval Tq^Tz),

and the second term is the additional emf induced

due to the wires being pressurized to p,. Note that

the temperature interval over which the pressuredependent term is integrated is the seal temperature

to junction temperature.

To measure the change in the T.E.P. as a function

of pressure, the circuit in figure lb can be employed.

This circuit is used for the so-called' single-wire

experiment, an accurate means of measuring the

T.E.P. pressure dependence. In this circuit, a single

wire is brought into a high-pressure vessel through an isothermal pressure seal at a temperature 7*1 and is brought out through a similar seal at a temperature T2. The induced voltage, using eqs

(1) and (2), is the following:

V{pi) = r aa{T)dT+ [(Xa{r)

JTo JTI

fTo

+ Aa„(p„ T)]dT + aa{T)dT

JTi = Aaaipu T)dT. (4)

jTi

The pressure correction term for a thermocouple

circuit, the second term of eq (3), can therefore be

determined by performing the single-wire experiment for each material of the thermocouple pair.

The problem of simultaneous pressure gradient

and temperature gradient which normally exists in a pressure seal and throughout a solid-media device

will modify this treatment. Equation (4) is still valid

but the variation of Aaip, T) has to be taken into

account, since the pressure is not constant over the

temperature interval within the cell. Hanneman and

Strong [1] ' have provided a means of approximating

the amount of uncertainty involved when the simultaneous gradients exist. They approximate Aq:(p, 7") where ^ is a constant and piT) is the internal pres- sure written in terms of the temperature. This is possible since T and p are both functions of position within the cell and therefore one can be written

as a function of the other. To determine the uncertainty associated with this problem, one has to determine the function piT). In other words the

pressure and temperature gradients within the highpressure cell have to be established. Equation (4)

reduces to V=A j%iT)dT, (5)

which is a constant times the area under the curve piT) versus T. The error introduced by assuming no

pressure gradients where there are temperature

gradients is just the difference in area between the

actual p ( T) versus T curve and the area calculated

if it is assumed that p = Pmax. from Ti to T2 and is zero everywhere else.

We have made a detailed analysis of the pressure

and temperature distribution in our piston cylinder

cell and have established the piT) relation so that

the area is determined to an accuracy of 2 percent.

2. Experimental Procedure

2.1. Hydrostatic Measurements

Single-wire measurements were carried out both

in a hydrostatic gas apparatus to 8 kbar, and in a

piston-cylinder solid-media apparatus to 40 kbar. A

schematic diagram of the hydrostatic apparatus is given in figure 2. The materials measured were

standard 36-gage teflon-coated thermocouple wire.

The wires were threaded through the 0.6-mm-diam.

bore of a stainless steel high-pressure tubing. One

end was brought out to ambient pressure through a frozen gas U-tube seal which is immersed in liquid

nitrogen. Argon, which is used as the pressure

medium, is solid in liquid nitrogen and therefore acts as the pressure seal around the wires coming out of

the U-tube. The temperature of this seal is therefore fixed at 78 °K.

The high-temperature seal is made by silver

soldering the thermocouple wire a small distance

into the high-pressure tubing. Since this region of

tubing is maintained strictly isothermal, no error in

the thermal emf is produced by shorting out this

section of the thermocouple wire with another material. The high-pressure tubing in this region is

• surrounded with a tight-fitting, thick-waUed copper

cylinder which insures that no temperature gradients will exist across the silver solder seal. The

thermocouple circuit is grounded at this point while

the rest of the circuit and the potentiometer are insulated from ground.

' Figures in brackets indicate the literature references at the end of this paper.

68
To H.P argon

Thermocouple leads

frozen gas

lead out

Pressure transducer

Copper cylinder

(15cm longx0.5cm I.D.x|.25 cm O.D.

)

Silver solder plug around

thermocouple wires

( I cm long )

Figure 2. Hydrostatic pressure apparatus for measuring single-wire high-pressure

thermal emf.

Cold thermocouple Sample leadTungsten carbide

piston

Ceramic insuloting tube -/

Hot thermocouple Sample lead

Figure 3. Piston cylinder high-pressure cell for measuring

single-wire high-pressure thermal emf.

Pressure was generated with a three-stage intensifier system and was measured with a strain gage

transducer which had been calibrated against a manganin coil. The maximum temperature was

limited to 300 °C due to the strength of the silver

solder plug. At the dry ice point the pressure was

limited to 5 kbar due to the freezing of argon. At all higher temperatures the maximum pressure used

was 8 kbar. A space was provided between the

low- and high-temperature ends for some slack

wire so that straining of the wires due to movement

in the cold seal would be minimized.

T-T,

Figure 4. Pressure temperature distribution in piston cylinder

high-pressure cell.

2.2. Piston Cylinder Measurements

The piston cylinder measurements were made in an apparatus of standard design and of dimensions

1.27 cm diam X 5.08 cm long. The cylinder material

was 94V4 percent WC 5^/4 percent cobalt binder

(Plansee 850). Figure 3 shows a schematic cross

section of the cell design for this experiment.

The design of the cell is obviously quite important

and the key to success or failure of the experiment.

The cell had to provide the conceptual physical

ideas discussed in the introduction with minimum

69
deviations from the ideal. Therefore, one requirement was to make the pressure seals as sharp as possible and/or as isothermal as possible. Additionally the pressure gradient over the temperature

gradient in the high-pressure cell was to be kept

to a minimum.

The reasons for these requirements are illustrated

in figure 4. If the pressure seals are infinitely sharp

and there is no pressure gradient in the highpressure cell, the dashed line represents the

p{T) versus T relation. This is the ideal or perfect

experiment. The pressure emf is proportional to the area under the curve. The actual experimental conditions are shown as

the solid line. The points on the graph have been

measured in our cell by use of thermocouples and

the bismuth transitions. This will be discussed in

detail below. The effects of the sUght temperature

gradient across the pressure seal have not been

indicated on the graph. The area under the dashed

curve is proportional to the value of the effect we

want to measure. We have chosen the experimental

pressure to be the pressure of the talc-AgCl interface (piston end) as this gives the best representation

of the true area. The pressure calibration was an important factor

influencing the final design of the cell shown in

figure 3. Large pressure gradients over the temperature gradients inside the high-pressure region would

cause large uncertainties in the results. We originally started with a cell using talc as the pressure

media, quite similar to that of Getting and Kennedy

[2]. This would correspond to replacing the AgCl

in our figure 3 with talc. A wire of bismuth 0.03 cm

diam X 0.6 cm long was placed axiaUy in the

furnace from the seal to the top of the furnace. The

bismuth 1 —* 2 transition started at 32 kbar force/ area on the compression stroke and took 7 kbar to complete. Using this design we found it impossible

to go beyond the end of the bismuth 2^3 transition

without danger of piston failure (44 kbar on the

piston). If the bismuth wire extended the length of

the cell the maximum pressure of the 1 —* 2 transition should remain the same but the gradient would

be extended from 7 kbar to 12 kbar. This would

seriously compromise the determination of area in a p(T) versus T plot and thus the final results.

An improvement was noted by replacing 0.64 cm

of the talc at the furnace end with AgCl. Here the

bismuth 1-^2 transition started at 28 kbar room

pressure but still took 7 kbar to complete. The

bismuth 2^3 transition was complete at 39 kbar on

the piston. Another improvement was made by going

to the configuration in figure 3 except that the BN

was 0.32 cm diameter. In this setup the 1-^2 transition started at 27.5 kbar and took 6 kbar to complete. For the first time the bismuth 2 resistance had

a flat portion. The bismuth 2-^3 transition was complete at 36 kbar on the piston.

The majority of the large AP from start to finish

of the transitions can be attributed to the AF of

50

I 40

o

ES 30

c

(rt in

<u CC£ 20

E

in CD

10

0

- 26 kb (-1-0.6 kb)

1/

,—26,8 kb{-0.2kb)

0 8 16 24 32 40

Force/Area - kb

BISMUTH CALIBRATION IN AgCl PORTION OF CELL

Figure 5. Pressure calibration for AgCl column in high- pressure cell.

transition. As the metal transforms it decreases in

volume. The shear strength of the talc is large

enough so that pressure must be added to the system to force the talc to follow the bismuth. Boyd

and England's [3] result using a 0.033 cm diam Bi

wire in 0.32 cm diam AgCl in talc confirm this. Here

the smallness of the of the small bismuth wire

and the weakness of the surrounding AgCl cause the

Af of the transition to be about 1.5 kbar.

Our setup used AgCl as the pressure media.

However, it was necessary to protect the thermocouple wires from the reactivity of AgCl, so they

were sheathed in BN. BN is stronger than AgCl

and therefore reacts to pressure in a similar manner

to talc. We quickly found that the diameter of the

BN was critical to the AP of transition. Figure 3

shows our final configuration with a 1 mm BN sleeve

and a 0.3 mm wire. Figure 5 shows the bismuth

pressure calibration for this cell with the Bi wire in

the 1 mm BN sleeve in the AgCl portion.

The AP of transition in this configuration is about

2 kbar and the bismuth 2^3 transition completes

at 31 kbar force/area. This represents a considerable

improvement over the all talc cell originally tried

(7 kbar and 44 kbar). In the actual experiment the

wire of thermoelectric material does not undergo a AV; therefore, the AP down the wire is less than

2 kbar. Bridgman has shown AgCl to have a shear

strength of 0.8 kbar at 25 kbar. This is the lower

limit AP to be expected along the AgCI with no BN.

Based on measurements made with other diameter

BN tubes and the pressure calibrations discussed

above, we find the AP along the wire in the AgCl to be 1.0 to 1.5 kbar. For figure 4 we have used the

value of 1.0 kbar. Using our measurements of pres- sure drop in the talc portions of the hot and cold end

70
^High pressure Low pressure —»~ ——\ H— Pressure seal

Distance from Pressure Seal, cm

Figure 6. Temperature distribution in the region of the hightemperature seal in the piston cylinder cell.

of the cell and the similar results of Boyd and

England [3], we estimate the AP to be 2 kbar in

the talc. Figure 6 shows the temperature calibration for

the furnace end of the high-pressure cell. These data

were taken in two runs of four thermocouples each;

the middle two thermocouples were common to both

runs. All the temperature caUbration data as well as the experimental data reported in this paper were

taken after the cell had been cycled to 40 kbar

pressure. It was necessary to do this in order for

the temperature caUbration to remain constant from

run to run and also during a run. This was also useful

in checking for failures in the experimental setup.

All the information is shown in figure 4. It is important to notice that the talc portions contribute

little to the measurement or its uncertainty and tend

to cancel out. The most important effect comes in

the AgCl portion of the cell. As long as the pressure

and temperature gradients are small or well-known

here, the uncertainty of the measurement is small.

If the pressure calibrations at room temperature are vahd for high temperature the uncertainty of the

method is less than 2 percent when the pressure is evaluated in the AgCl portion.

The location of the high-temperature thermocouple is shown in figure 3. Examination of figure 6

shows that this is the location of maximum temperature in the cell. A great deal of care was taken to keep the location of this thermocouple as well as the cold thermocouple in place. The ceramic tube

was epoxied to the stainless steel at the start of the

experiment. The seal design quickly locked the

ceramic tube in place as pressure was applied. AU

runs were inspected for movement of the thermocouple. This was not a problem at temperatures to 400 °C. We occasionally had movement before

using the epoxy but never after; those runs were

discarded. It will be necessary to modify the hightemperature seal for work at higher temperatures.

Behind the seal was a tungsten carbide bushing

whose I.D. was slightly larger than the ceramic

tubing. The high compressive strength of the WC

was used to keep the pressure on the wire at 1 atm.

Because the magnitude of the pressure emf's are

lO's to lOO's of microvolts, it is necessary to measure

the voltage to an accuracy of l/xV. Large currents

in the vicinity of the thermoelectric element could

induce voltages and limit the accuracy of the measurements. AC pickup would require filters, etc. We

have specially designed our high-temperature fur- nace to eliminate or reduce these problems.

The furnace shown in figure 3 is a graphite spiral

in the talc. This is accomplished by making a graphite screw which is threaded and cemented

into the talc and then has the center machined out. The resistance of our furnace was about 30 d. We

were able to reach 400 °C with only 3 A current.

This low current and coil geometry combine to eliminate problems with induced voltages in the

experiment. This is a much more desirable configuration than the solid carbon resistance tube

heater normally used in high-pressure experiments.

We have described in detail the high-pressure cell

and its calibration, since it is imperative to establish

accurately the pressure-temperature distribution

within the cell before any valid interpretation of

pressure emfs can be made.

2.3. Measurements

2.3.1. Hydrostatic Experiments

Measurements were taken of four thermocouple

materials: chromel, alumel, copper, and constantan.

The cold seal was maintained at 78 K and the hightemperature seal was maintained at a number of

fixed temperatures between 190 K and 560 K. With

the temperature interval between the seals fixed,

the pressure was varied between 0 and 8 kbar and

the induced emf was measured at each pressure.

In all cases, the induced emf was linear with pres- sure so only the slope of the emf versus p plot is presented. To insure that spurious readings generated in the frozen gas seal were not included in

the data presented, the emf generated over the

smallest temperature interval (i.e., 78 K to 190 K)

was subtracted from all the measurements at larger

temperature intervals. This eliminated any spurious

readings arising from the frozen gas seal, since they

would be independent of the temperature of the

high-temperature seal. The reproducibility of the

385-762 O - 71 - 6
100 200 300 400 500 600 700

Figure 7. Pressure emf per kilobar for temperature interval 190 K-^ T measured in hydrostatic apparatus.

data showed that if spurious emfs are generated,

they were consistent from run to run. The data are therefore presented in figure 7 in terms of the emf

per kilobar generated by temperature intervals of

190 K to r where T ranges to 560 K. Scatter in the

data introduced an error of ±0.25 )U,V/Kbar which is the total error of the measurement.

2.3.2. Piston-Cylinder Experiments

Measurements were made in the piston cylinder

apparatus on chromel, alumel, platinum, and

platinum-10 percent rhodium wires using the cell

described above. Attempts were made initially to

fix the pressure and measure the pressure emf as a function of temperature interval between the seals.

This method proved to be difficult since the pressure

and pressure distribution could not be maintained

at a constant value due to thermal expansion of the

cell upon heating and a subsequent redistribution of

friction forces within the cell. AU data reported

here were therefore taken as induced emf versus pressure at constant temperature intervals between

the hot and cold seal. The agreement between the

two sets of data was satisfactory, however.

The pressure on the axis of the cell undergoes an unusual cycle due to the presence of the highly

incompressible tungsten carbide column that makes

up the high-temperature seal. As the piston advances, the pressure at the top of this column

increases rapidly, compared with the pressure in

the rest of the cell. This continues until the pressure

medium cannot support the stress gradient generated. Further piston advances increase the pres- sure uniformly throughout the cell. On the pressure

release, the opposite effect occurs with the pressure

decreasing rapidly at the top of the seal column.

This pressure cycle wiU exhibit the opposite of the

40i

Figure 8. Pressure emf as a function ofpressure for alumel at a

constant temperature interval in piston cylinder apparatus.

Strain, percent

Figure 9. Effect of strain on the thermal emf of chromel thermocouple wire.

normal hysteresis cycle of a piston cylinder. Figure 8

shows the pressure emf for alumel at a fixed temperature interval between the seals as a function of

the piston pressure. The reverse hysteresis effect

is quite apparent.

Another effect was found when the chromel wire

was measured. At zero pressure a negative emf was

measured in the circuit with a fixed temperature

difference maintained between the seals. The effect

72
280

Figure 10. Chromel and alumel pressure emf as a function of

temperature interval at 20 kbar and 40 kbar.

of pressure was to decrease this negative emf

toward zero. It was found that chromel wire is quite

strain sensitive and that in preparation it had been

strained slightly. Tests were made using a strained

and unstrained chromel wire as a couple. Figure 9

shows that the resulting emf is a linear function of

permanent strain and is about linear in temperature

difference. Similar strain tests were made with the

alumel, platinum, and platinum-10 percent rhodium

wire, but no measurable strain-induced emf was

detected. Hanneman and Strong [4] have reported

that they found no measurable strain-induced emf

for all four materials. Our experiments show that

chromel is strain sensitive.

In solid-media, high-pressure apparatus, the

methods used to bring leads out of a high-pressure

cell generally result in a high degree of permanent

deformation of the lead wires. Therefore, caution

should be used when using chromel-alumel couples,

since an emf can be induced by the strain effects of

the chromel. However, the amount of temperature

correction is small, compared with the total emf of

the couple. For instance, if a gradient of 350 °C

exists across a 10 percent strained wire, the resulting error would be only 1.5 °C. To minimize this error, the high-pressure cell should be designed so

that regions of high strain are in regions of lowtemperature gradient.

Measurements were taken over a temperature

range 30 to 400 °C and a pressure range of 0 to 40

kbar. The pressure dependence of the emf was

obtained from the raw data, e.g., as shown in figure

8, using only the data where apparatus friction

effects were known. These would correspond to the

straight-line sections in figure 8. On the compression

T °C ' mox • ^

Figure 11. Platinum and platinum 10 percent rhodium pres- sure emf as a finction of temperature interval at 40 kbar.

curve this gives data to the highest pressures and

on the release curve gives data to the lowest pressure. The curves rarely went through the origin as they should. This was attributed to apparatus

friction and/or small spurious emfs in the circuit.

This offset from zero was always small except for

the chromel. The chromel offsets from 0 emf at 0

pressure could be accounted for by the above

mentioned strain effects.

The curves were first displaced so that they went

through the origin. The compression and decompression curves were then corrected for the compression and decompression values of the bismuth

transitions. These results were averaged to give the

final curve of emf versus pressure at a fixed temperature interval. During the experimental cycle

AT" drifted somewhat. Each individual experimental

73
point was corrected to a constant AT before the

above data reduction was made.

For all temperature intervals, the cold seal was

at a temperature above 30 °C. Therefore an additional correction to the final curves was made so

that they would aU correspond to temperature

intervals of 30 °C —^ T.

In general data were taken at Ar's of approximately 75 °C, 150 °C, 200 °C, 275 °C, 325 °C for each

material with about 20 pressure data points at each

temperature. We estimate the uncertainty of the

data in the piston cylinder experiments to be approximately 5 percent. This results from differences

in the compression and release data. Added to the

2 percent error in the area determination of piT)

versus T plots, we believe that the measurements

are good to ± 7 percent.

3. Results and Discussion

The hydrostatic pressure results are given in

figure 7 for the four materials investigated. The

induced emf for a fixed temperature interval was

found to be a linear function of pressure for all four

materials within the accuracy of the measurements

(0.25 /xV/kbar). For the copper and constantan

wires, the induced emf per kilobar was also a linear

function of temperature interval up to an interval

of 190 to 560 K. On the other hand, the induced emf

per kilobar for chromel and alumel was zero between 190 and 300 K, and then increased nonlinearly

above 300 K. Values of emf are considered positive

when the low-temperature seal side of the circuit in

figure lb is positive.

The piston cylinder results are given in figure 10

for chromel and alumel and in figure 11 for platinum

and platinum-10 percent rhodium. It was found that over the temperature intervals investigated alumel

and platinum had a pressure emf at constant

temperature intervals which was linear with pres- sure up to 40 kbar. A nonlinear pressure dependence

was found for PtlORh and chromel, both of which

became less sensitive to pressure at higher pressure.

Due to this nonlinear pressure dependence, the

results are also tabulated in table 1. The comparison between the piston cylinder

results and hydrostatic results on the chromel and

alumel wires indicates the two methods give the

same results within the experimental error. For a temperature interval of 30 to 300 °C the piston

cylinder results at 10 kbar show a pressure emf of

39 ±3 AtV for alumel and 43 ±3 /u-V for chromel,

while the hydrostatic results if extrapolated from

8 kbar to 10 kbar yield 36 ±3 /xV for alumel and

39 ±3 ixY for chromel. Also, it is seen that over

this temperature-pressure range the pressure emf

of chromel is equal to that of alumel within the

experimental error for both the hydrostatic experiment and the piston cylinder experiment. This

Table 1. Pressure emfifxV) for temperature interval

30°—> T of pressurized wire

T, °C 10 kbar 20 kbar 30 kbar 40 kbar

alumel

100 6 13 18 26

200 21 43 63 86

300 39 78 117 157

400 58 117 174 234

chromel

100 5 10 14 19

200 20 37 53 68

300 43 82 119 148

400 87 165 226 280

(chromel)-(alumel)

100 -1 -3 -4 -7

200 -1 -6 -10 -18

300 -f 4 + 4 + 2 -9

400 + 29 + 48 + 52 + 46

platinum

100 12 23 34 46

200 28 57 84 114

300 45 90 135 181 400 62 124 186 248

platinum 10 percent rhodium

100 5 11 16 20

200 14 27 38 50

300 25 48 68 90

400 35 72 105 130

(PtlORh)-(Pt)

100 -7 -12 -18 -26

200 -14 -30 -46 -64

300 -20 -42 -67 -91

400 -27 -52 -81 -118

shows that the two methods of measuring the

pressure emf give consistent results.

Other measurements are available for comparison

with our results, but we wiU only comment in general

about them.

Bundy's [5] single-wire experiment gave consistently higher emf values than ours for alumel,

platinum, and platinum 10 percent rhodium. His

chromel value was negative which was probably the

result of the strain effect described above.

Bloch and Chaisse [6] measured Cu-Constantan

hydrostaticaUy and their results agree with ours

74
within our experimental error of ± .25 /u,V/kbar.

Bridgman's [7] single-wire experiments on con- stantan and copper yielded pressure emfs which were the same as ours within our experimental

error. The comparative measurements of Hanneman and

Strong [4] on Pt-PtlORh versus chromel-alumel

couples at 40 kbar with a temperature interval of

0 to 400 °C are within 10 percent of our results.

Their absolute values [8] for Pt-PtlORh are higher

than our values by 30 percent which is outside of

their stated experimental limits. In addition their

chromel-alumel corrections are positive while ours

are negative.

Of the results presented at this conference, our

results are most easily compared with those of

Getting and Kennedy [2]. The agreement is within

experimental errors for Pt, PtlORh, and alumel. For

chromel, their results are lower than ours by a

significant amount (the order of 30 percent at 20 kbar and Ar= 0-400 °C). It appears that the

strain effect described above could be the cause of

this discrepancy. However, our hydrostatic results are in agreement with our piston cylinder data within an error limit of ±7 percent. Since no strain effects are present under the hydrostatic conditions, the

agreement of our two sets of data indicates our

piston-cylinder chromel data to be more accurate

than that of Getting and Kennedy.

4. Conclusions

We have only discussed the data, and the data

gathering procedure so far, but have not described

how to use the data. In the introduction, eq (3) shows

that as the result of the pressure dependence of the

thermoelectric power, an additional emf is generated

in a thermocouple circuit if the pressurized region is in a region of changing temperature. It was also

shown that single-wire experiments could measure

directly the amount of additional emf generated.

To utilize the single-wire results, one must know the

temperature of the thermocouple junction ih the

high-pressure cell and the temperature of the seal

where the pressure drops to zero. The internal temperature is measured by the uncorrected emf generated in the two-wire circuit. This temperature is the

uncorrected temperature, but for purposes of determining the amount of the correction, it is accurate

enough. The seal temperature must be determined

independently and in the case of certain belt and

piston cylinder cell designs, this temperature can

be considerably higher than ambient. Thus, having

established the temperature interval from seal to high-temperature region (it is the temperature

interval which is needed, not just the difference)

and having established the pressure, one can determine the amount of additional emf induced in the

thermocouple circuit from an isobaric plot of pres- sure emf vs. temperature interval, such as figures

10 and 11 or table 1. The total emf induced in the

thermocouple pair is just the difference between the

two pressure emfs. The sign convention here is important. The pressure emf is positive if the highpressure seal at the low-temperature side of the

single- wire circuit is positive. This is the case for all materials we measured. The pressure emf for the

wire of the pair which is negative (constantan,

alumel, platinum) is subtracted from the pressure

emf of the wire which is positive (copper, chromel,

platinum, 10 percent rhodium). The resulting emf

is subtracted from (or added to, if it is negative) the

measured emf of the circuit to arrive at the zero pressure emf which can be converted to temperature

from a standard table. By adopting this convention, no ambiguity wiU result when applying the pres- sure correction. By leaving the corrections in terms

of emf values instead of temperature values, one can easily apply the correction needed for his

particular high pressure cell with its own particular

seal temperature, reference temperature, and

thermocouple materials.

Although the data presented in this work are over

a limited temperature range, we have determined

a region of pressure and temperature where welldefined experimental conditions were established

and where pressure emfs were measured to an accuracy needed for precision thermocouple

applications.

5. References

[1] Hanneman, R. E., and Strong, H. M., J. Appl. Phys. 36, 523

(1965).

[2] Getting, I. C, and Kennedy, G. C, The Effect of Pressure on

the EMF of the Thermocouple, this volume.

[3] Boyd, F. R., and England, J. L., J. Geophys. Res. 65, 741

(1960).

[4] Hanneman, R. E., and Strong, H. M., J. Appl. Phys. 37, 612

(1966).

[5] Bundy, F. P.. J. Appl. Phys. 32, 483 (1961).

[6] Bloch, D., and Chaisse, F., J. Appl. Phys. 38,409 (1967).

[7] Bridgman, P. W., Proc. Am. Acad. Arts. Sci. 53, 269 (1918).

[8] Bundy, F. P., A Critical Review of the Effect of Pressure on Thermocouple EMF's, this volume.

75

The Effect of Pressure on the E.M.F. of Thermocouples

Ivan C. Getting and George C. Kennedy

Institute of Geophysics and Planetary Physics,

University of California, Los Angeles, California 90024

A schematic diagram of a thermocouple as in- stalled in a typical high-pressure apparatus is shown

in figure 1. There is, of course, no effect on the

reading of a thermocouple if the temperature of a thermocouple junction at high pressure is the same

as the temperature at which the elements emerge

from the high-pressure environment to the lowpressure environment. However, in many installations, the temperature at which the thermocouple

emerges from the high-pressure environment is vastly different than the temperature of the thermocouple junction. Thus substantial effects on the emf

of the thermocouples may be expected. Pressure

thus creates various thermocouple junctions between compressed and uncompressed segments of

the thermocouple elements.

Both Bridgman and Bundy have examined the

emf generated by a thermocouple of compressed and

uncompressed segments of the same wire. However,

these measurements have been taken to modest

limits of temperature and pressure. It was the aim of

the present investigation to carry the measurements

on to very high pressures and temperatures.

In our experiments a tungsten carbide bushing

with a small axial hole was used to hold pressure off

the test wires over part of their length so that the emf

of a compressed-uncompressed thermocouple could

be measured. A furnace was used to heat the junction. Compressed-uncompressed junction temperatures were measured with separate thermocouples.

Our measurements were limited by the pressures

and temperatures at which the tungsten carbide

sleeve failed. This experiment, which we call the

single-wire experiment, is shown schematically in

figure 2 and a detailed drawing of our experimental

arrangement is shown in figure 3. The emf of various

wires thus can be measured and the results combined algebraically to give the correction for a thermocouple made of any two elements at high

pressures with a junction at one temperature and

pressure seal at a different temperature.

Our experiments were conducted in an end-loaded

piston-cylinder apparatus to a maximum pressure of

30 kbars and a temperature of 1000°. A coaxial

graphite resistance heating element was used to heat

the junction of the compressed versus uncompressed wire under study. It is, of course, necessary

to show that the pressure drop across the junction

is isothermal and the zones of temperature drop

Paper presented at the Symposium on Accurate Characterization

of the High-Pressure Environment, held at the National Bureau

ofStandards, Gaithersburg, Md., October 14-18, 1968.

Sample Temp.,T

Pressure Seal Temp., Tj

Test Pressure, P

Ice Bafh

Figure 1. Schematic view of typical high-pressure cell with

temperature and pressure distributions.

Hot Seal Ternp., T

Cold Seal Ternp., Tg

Figure 2. Schematic view of single-wire experiment cell with

temperature and pressure distributions.

77
<2?

As" > ^<y ^^^^

,'9 q.

Figure 3. Detailed view of single-wire experiment pressure cell. inside the cell are isobaric. Thus, six thermocouples

not shown in figure 3, were located in two groups of

three spanning the seal regions. It was shown that

the temperature variation across the seals was less

than 3 percent of the hot seal temperature. Bi I-II transition experiments at room temperature led to an expected 3-kb maximum pressure difference

within the cell. In figure 4 we show experimental results at 10, 20,

and 30 kbars up to temparatures of 900°. These

results are the plots of the emf generated by singlewire experiment- on platinum minus the emf gen-

' erated by the single-wire experiments on platinum10 rhodium.

The temperature axis shown in figure 4 represents

the difference in temperature between the hot junction of a thermocouple and the temperature where

the thermocouple wires merge from the high-pres- sure apparatus. In our particular apparatus, when

the hot temperature junction is at 1000° the cool

seal, where the thermocouple emerges from the

high-pressure apparatus, is at approximately 100 °C.

The data are corrected for this rise in seal temperature and the results are presented in figure 5. In most high-pressure apparatus, the seal temperature rises as the sample temperature rises. If

the relation between these two temperatures is known, the correction for the effect of pressure in

the thermocouple may be determined as shown in

figure 6. In the example shown in figure 6, we assume

a sample temperature of 800°, 30 kbars, and a seal

temperature of 150°. The voltage correction appHcable is the difference between the value at 800° and

,1V 150

Pt SW emf minus Pt 10 Rh SW emf SW 7

" 8

" 1

1

„ 12

_L_ _1_ _L_ _L_ 100 200 300 400 500 600 700 800 900 1000 PRESSURIZED TEMPERATURE INTERVAL (°C)

Figure 4. Pt SW. emf minus Pt 10 Rh S.W. emf.

150°, here llO/xV. To obtain the correction in degrees, the voltage correction must be divided by the

relative Seeback coefficient at 800°, 10.8/>tV/°C. The

correction is to be added to the observed value. The

pressure seal temperature must be known for every

78
+ 200

150

100

+ 50

Pressure Corrections for Pt vs. PI 10 Rh Thermocouples

with Pressure Seal at 20 *C

-i I I I I L O 100 200 300 400 500 600 700 800 900 1000

TEMPERATURE (°C)

Figure 5. Pressure corrections for Pt vs. Pt 10 Rh thermocouples

with pressure seal at 20 °C.

200

Sample Correction Colculolion for Pt vs. PI 10 Rh Thermocouples.

5 100

Indicated Temperoture 800 "C

Voltoge

' Correction HIO^v

Pressure Seol Temperature 150 "C

o-j,

ij = 10,8 /iv/°C AT = + 10.2 °C

0 100 200 300 400 500 600 700 800 900 1000 TEMPERATURE (°C)

Figure 6. Sample correction calculation for Pt vs. Pt 10 Rh

thermocouples.

0 100 200 300 400 500 600 700 600 900 1000 PRESSURIZED TEMPERATURE INTERVAL (°C)

Figure 7. Alumel S. W. emfminus Chromel S. W. emf.

Pressure Corrections for Chromel -P vs. Alumel with Pressure Seal at 20 *C

^v

100

oc 0

100 200 300 400 500 600 800 900 ^ 10 kb

[Vlanufacturers Guaranteed

Calibration Toleronce

for Standard Grade 30 kb

temperature measurement to which these corrections are to be apphed. Data for Chromel and Alumel

thermocouple wires are shown in figure 7. Here, as before, the data presented are the algebraic differ- ences between the voltages observed in single wire

experiments on each of the two thermoelements.

Unfortunately, in some of the runs chemical con- tamination of the thermocouple elements took place.

These erratic readings are also presented. The solid curves are drawn through results believed to be from

runs where no contamination took place. A smooth

plot with the pressure correction for Chromel-P

versus Alumel thermocouples with the pressure

seal at 20° is shown in figure 8. The relative Seeback

coefficient for Chromel/Alumel is nearly constant

at about 40jLtV/°. Thus, for temperatures up to about

-400^

Figure 8. Pressure corrections for Chromel-P vs. Alumel with

pressure seal at 20 °C.

600°, the corrections for Chromel-Alumel thermocouples are less than 2°. For pressure seal temperatures other than 20° the corrections are calculated

by the same method demonstrated in figure 6. Because of some uncertainties in the pressure of

the experiment and uncertainties in the temperature

as well as a small temperature gradient to the pres- sure seal we estimate a maximum certainty in the

final corrections to be about ± 3° at the highest pres- sures and temperatures of the measurement. Proportionately the uncertainty is less at lower values.

79
In typical high-pressure experiments where cer- amic tubes surround the thermal elements, stress

distribution can make the corrections uncertain by

very much larger amounts. In a number of our runs,

grossly non-hydrostatic conditions were noted along

the thermocouple wire where bits of ceramic tubing

pressed against the wire or alternatively protected

the wire from pressure. Thus, if precise temperature measurements are made at high-pressure, care must be taken to insure pressure uniformity along

the thermocouple wires in the high-pressure environment. Chemical deterioration of the thermocouple

wires can often introduce error several magnitudes

greater than the error introduced by ignoring the

effect of pressure on the thermocouple.

As a final check on the accuracy of these measurements, we intercompared platinum-10 rhodium

thermocouples with Chromel-Alumel thermocouples

at high pressures and temperatures. We find the

difference in the reading of the two thermocouples

in our intercomparison experiment to agree with the

corrections we indicate for these thermocouple pairs

within the stated uncertainty of the measurements.

Note added in press

Subsequent work on this problem has led to a

slight revision of the data presented herein. A complete discussion of the experiment and the revised

data are presented in J. Appl. Phys. Oct. 1970.

80
Temperature Measurement by Thermal Noise at High Pressures

R. H. Wentorf, Jr.

General Electric Research and Development Center, P.O. Box 8, Schenectady, New York 12309

The temperature of a small carbon resistor (700 to 3,000 fl) in a high-pressure cell next to a Pt-PtlO

percent Rh thermocouple was measured by thermal noise. Runs were made at 40 and 50 kbar to temperatures of about 1,400 K. The results suggest that at the highest temperatures the thermocouple

indicated temperatures from 40 to 60 K too low, in fairly good agreement with estimates made by

other methods.

1. Introduction

Many interesting phenomena occur at high

temperatures and high pressures, e.g., 1,500 K and

50 kilobars (kbar). However, it is difficult to determine pressure and temperature accurately at these

conditions. [1]'

One can mention two ways to determine the

temperature inside a high-pressure chamber:

(1) One method is based on the equipartition of

energy. The thermal agitation of the current carriers in an isolated resistor generates at its terminals

a minute, fluctuating voltage known as thermal or "Johnson" noise [2, 3]. One may write:

E^- = 4:kTRB (1)

where E- is the average of the square of the fluctuating vohage due to thermal agitation,

k is Boltzmann's constant,

T is the absolute temperature,

R is the value of the resistance,

B is the bandwidth of the frequencies for which E

is measured; the frequencies are generally

below 10i'»Hz (optical).

Although it is simple in principle, the method

demands careful attention to experimental details

and it is not so convenient. Garrison and Lawson [4]

demonstrated that the method could be used at high pressures.

(2) One might measure temperature by means of

thermocouples. This common and simple method

has some limitations. Apart from the annoyances of

broken or corroded wires, one notes that metals

change their electronic characteristics with pressure, and so the emf of a thermocouple depends on

the pressure and temperature system to which it is exposed. A satisfactory theoretical treatment of this

problem has not yet appeared, but several workers

have made experimental measurements at moderate

pressures or temperatures [5, 6, 7, 8, 9]. Hanneman

' Figures in brackets indicate the literature references at the end of this paper. Paper presented at the Symposium on Accurate Characterization

of the High-Pressure Environment, held at the National Bureau

of Standards, Gaithersburg, Md., October 14-18, 1968.

and Strong [10] have estimated the eft'ects of pres- sure on common thermocouples for more severe

conditions.

One of the aims of the work reported in this paper

is to measure the same temperature simultaneously

by methods (1) and (2) and thereby help determine

the effect of pressure on thermoelectricity.

2. Experimental Considerations

2.1. General

The apparatus was intended to provide a stable

Pt-PtlO percent Rh thermocouple at essentially

the same pressure and temperature as a suitably

isolated resistor whose Johnson noise could be

amplified and measured. The temperature of the

probe resistor could then be estimated by using

eq (1), assuming that the observation period was

long enough to obtain a sufficiently precise value

of E\

Equation (1) could be used directly if one had an

ideal noiseless amplifier of known, stable gain. For

real, noisy amplifiers with somewhat unsteady gain,

it is better to compare the signal from the probe

resistor with that from a reference resistor which is at a known temperature and has the same resistance

as the probe.

The amplifier noise (the first stage contributes

the most) is simply added to the noise from the

probe re sistor; the sum appears in the output and

is called AE2 (amplified average E-). Thus a plot of

AE2 versus T is linear, as shown in figure 1 for vari- ous values of the probe resistor R. The intercept at T— 0 is the amplifier noise when the probe is at 0 K.

For an ideal but noisy amplifier the intercepts for

various /?'s would coincide and the slopes would be

proportional to R. The ampUfier used in this work

was not ideal, for reasons to be given later, and so the intercepts did not coincide at T = 0 and the

slopes were not quite proportional to R because the

eff"ective bandwidth B changed with R. However,

one still had the important linear relationship

between T and AE2 for a given R.

Thus the following experimental information was

necessary to determine the temperature of the

probe:

81
1 1 1 1 1 /R = 4732

"

ohms

- R=2605

R = I682

R=I2I8 R^69 - —" R=83l

R=0 -

1 1

1

1 1

100 200 300

T,

400 500 600

Figure 1. Plot of AE2 versus T for various probe resistances from 831 to 47.32 n.

R

nL. B.

REFERENCE R

SWITCH

BOX

PRE-AMP

POWER

AMP

LOAD

4 OHMS

WHEATSTONE

BRIDGE

POTENTIOMETER

(a) The resistance R of the probe;

(b) The amphfier output AE2, averaged over a

sufficiently long time, with the probe at the input;

(c) AE2, with reference resistor, resistance R, at input;

(d) The temperature of the reference resistor

(usually near 300 K);

(e) AE2 with shorted input {R =0) to monitor the

amplifier gain over periods of hours;

(f) A high probability that only a negligible

amount of external noise had been included in

AE2 during the measurements of (a) through (e)

above. External noise comes and goes, substantially

independently of the experimenter^s desires, and so one always seeks the minimum AE2 values out of a

large number of observations.

Figure 2 is a block diagram of the amplifier and

switching arrangements used in this work. One can

read about them in greater detail further along in

the paper.

2.2. The Value of i?

Consider eq (1) and let R = 1,000 O, T= 1,000 K,

and 5 = 5 X 10*. Then E'^ is about 2.8 X 10- 12 V^, and

we can expect to deal with signals in the range 1 to 2

/xV. Larger signals could be obtained from larger

resistances, but there are practical limits:

1. The shunt capacitance of the system, of the

order of 50 X IO-12 farad (F). 2. The finite input impedance of the amplifier.

3. Inside a high-pressure chamber the volume

which is approximately isothermal is limited and

subject to various stresses. Long, thick wires are impossible.

4. The resistor must be an electrically passive

element and tolerably isolated from its surroundings.

Available insulating materials at 1,500 K have

maximum resistivities of about 10'' ohm-cm and so can provide isolation of 10^— lO'' fl.

Figure 2. Block diagram of amplifier and detector.

Evidently many factors conspire to limit R to less

than 5,000 H. For the range 100 to 10^ Hz, it is diffi- cult to achieve amplifiers with less noise than that

of 500 ohms at 300 K. Thus suitable values of R

appear to lie between 500 and 5,000 fi. In this work

values of R from 600 to 3,000 O were used.

2.3. The Value of B

Several factors limit the frequency band:

1. At frequencies above about 200 kHz one has

problems with tuned circuits, stray capacitance, and

feedback.

2. Ubiquitous 60 Hz power line frequencies and

their harmonics make it wise to ignore frequencies

below about 1,000 Hz.

3. The Miller effect operates to reduce the input

impedance of the amplifier and/or limit the amplification of higher frequencies unless tuned circuits are used.

4. The time constants of the resistor used to measure AE2 demand a small resistance which in

turn demands a transformer to feed it; broad-band

transformers do not operate too well above 100 kHz.

Thus the value of B comes to slightly less than

10^ Hz. In this work the amplifiers used made the

effective range for B to lie between 2 kHz and about

80 kHz; the cutoffs at each end were not perfectly

sharp but were more than 6 dB/octave. The relative uncertainty in AE2 is given by {Bt)~^'^

where t is the observation time [4]. For 5 = 6 X 10^, an observation time of 5 sec corresponds to an

uncertainty of about 0.6 X 10"^, i.e., 1 degree out of

1,500, which is well below the uncertainties from

other causes.

82
3. Experimental Apparatus

3.1. The High-Pressure Sample

Figure 3 shows a sample arrangement which

proved satisfactory. (Several other kinds were tried.)

It fits in a "Belt" apparatus [11] with 19-mm-diam

piston faces.

The probe resistor, R, begins life as a strip of

Saran (polyvinyl chloride) about 0.013 mm thick,

0.13 mm wide, and 25 mm long. It is carbonized

in situ at about 40 kbar to yield a resistance in the

range 500 to 5,000 Cl, typically 1,000 H. The working

length of the resistor is about 7.5 mm, well inside the

walls of the heater tube, H. A rod of fused silica

about 7.5 mm long and 0.36 mm diam lies alongside

the Saran and fixes the working length. Two 0.25

mm platinum wires, a and b, butt against each end of

the silica rod and connect the ends of the resistor

to two 0.50-mm Formex-coated copper wires, A and

B, which pass through the compressible stone

gasket, F, to the outside. The Saran, silica, and parts

of wires a and b lie in a Lucalox (AI2O3) tube about

0.38 mm I.D., 1.5 mm O.D., and 19 mm long. This

tube is the most important electrical insulating

barrier between the resistor R and the heater tube

H. The Lucalox tube fits snugly inside a porcelain

tube 2.5 mm O.D. and 19 mm long. This tube shields

the Lucalox tube against mechanical intrusion of

material from the heater H as the sample is crushed

and sheared during compression and temperature

changes. The two tubes are labeled / in figure 3. The Pt-PtlO percent Rh thermocouple is made

from 0.25-mm wires c and d with the junction in the

heated zone at J. Wires c and d lie in tubes of

Lucalox and porcelain like the resistor, to help J be

at the same temperature as the resistor. The 0.5-mm

Pt and PtlO percent Rh wires C and D pass through

the gasket and connect c and to a 0 °C cold junction and a Rubicon potentiometer. The thermocouple

gave very little trouble.

The various tubes, etc. are carried at the midlength of a cylindrical sample holder PS, about 24

mm long and 19 mm diam, made of pyrophUite previously fired in air to about 750 °C to remove much of

the water, etc., in the pyrophyUite which would

attack the procelain and Lucalox at high temperatures. The heater tube H is pressed from a mixture

of graphite and hexagonal BN (1:2 by weight), so

that its hot resistance is about 0.15 fl. It is about 24

mm long, 9.2 mm I.D., and 10.5 mm O.D. Copper

disks 0.25 mm thick carry the heating current from

the pistons. Inside the heater is a rod, PC, of fired

pyrophyUite. After PC, H, and PS were assembled,

two parallel holes 2.5 mm in diameter and 0.45 mm

apart were bored through the assembly at the midlength to receive the various tubes, and then the

wires, etc., could be inserted and fixed.

The heater tube H was heated by direct current

from a bank of 12-V storage batteries via an adjustable series resistor. Electrolysis effects on the heater

and pyrophylhte were negligible.

Figure 3. High-pressure cell containing probe resistor {leads

A, B) and thermocouple (leads C, D).

The maximum furnace temperatures used were

about 1,450 K. At temperatures much above this,

the electrical isolation of the resistor was unsatisfactory, partly because of the enhanced conductivity

of the insulation at high temperatures and partly be- cause of the disruptions and intrusions from thermal

expansion forces.

Leads A and B were connected by low-capacity

coaxial cable to the switch box. Either A or B could

be connected to ground of the system. The top piston

(and hence the heater tube), the pressure chamber

cyliner, and the press frame were also connected to ground. (The pistons are ordinarily insulated from

the press frame by about 10^ fl.) Thus the probe

resistor R had good electrostatic shielding. The

resistance between lead A or B and the heater tube

could be measured during a run by temporarily

disconnecting the ground to the heater circuit. The

resistance between^ or and ground was measured

for every data point; any persistent differences indicated faulty isolation of R. (A portion of the voltage

drop along the heater t ube was then acting to bias

the wheatstone bridge.) AE2 was also measured for

either lead A or lead B grounded, and persistent

differences in these readings or wild, unsteady variations in AE2 also indicated faulty isolation of R.

Such data points were ignored.

The thermocouple leads C and D went to an ice

bath where they joined copper wires to a Rubicon

potentiometer. Usually lead D was grounded as a safeguard against feedback oscillation of the amplifier. The resistance between D and the heater tube

83
could be measured to insure that the thermocouple

was suitably isolated, but the low resistance of the

thermocouple wire reduces its isolation requirements below that of the probe resistor, and thermocouple isolation was not a problem.

3.2. Reference Resistor

The reference resistor was made of four rotary

adjustable carbon strip resistors with resistances of

10 KH, 5 Kfl, 1 Kn, and 500 fl. They were con- nected in series and any reference value between

0 and 16,500 could be selected with the aid of the

wheatstone bridge. The resistors were mounted in a small aluminum box and connected to the switchbox by a piece of low-capacity coaxial cable having

about the same length as that used to connect leads

A and B, and adjusted so that the shunt capacity across the reference closely matched that across

the probe. The reference temperature was read

from a calibrated thermometer next to the box.

For calibration purposes, such as constructing the

straight lines of figure 1, a number of metal film

resistors on alumina cores were placed in a cavity

in an aluminum block, together with a calibrated

Pt-PtlO percent Rh thermocouple, and heated over

a gas flame to about 525 K maximum. A cool switch

at the end of a stainless steel tube permitted one to select various resistances. This assembly replaced

the probe R for calibration experiments.

3.3. Pre-amplifier

Figure 4 gives a circuit diagram of the preamplifier. It was built inside a small aluminum box

and operated while buried in dry ice (195 K). Its dynamic input impedance, as determined by the

value of input series resistance needed to reduce

FROM

SWITCHBOK

«OUT

FET PREAMPLIFIER

ALL RESISTANCES IN OHMS

ALL CAPACITIES IN MICROFARADS

Figure 4. Preamplifier.

All resistances in ohms. All capacities in microfarads.

the output by half, went as follows:

kHz 5 10 20 50 60 80 100

kO, 220 220 260 210 120 75 52

Cooling the pre-amplifier from 300 K to 195 K

reduced the shorted input noise by 25 percent and

increased the gain by about 10 percent with negligible

effect on the bandwidth. The small input capacitor

and the dynamic input impedance caused the

amplifier gain spectrum to vary somewhat with input

(probe) resistance.

3.4. Power Amplifier

3.4.1. First Section

A schematic diagram of the first section of the

power amplifier is given in figure 5. A l-ixV signal at the pre-amplifier input appeared as a 0.1-V signal

at the output of the first section of the power

amplifier.

3.4.2. Second Section

A schematic diagram of the second (final) section

of the power amplifier is shown in figure 6. This

amplifier delivered a signal of about 1.5-W maximum

to a 4-ohm output load resistor, to provide the AE2

power in the detector described below. The output

transformer must be of high quality to provide good

frequency response.

3.5. Detector

If the fluctuating output voltage is impressed

across a resistor, then the power dissipated in the

resistor is proportional to or AE2. The power

appears a-; heat, and a suitable resistor can thus be

heated and its temperature measured by means of a thermocouple attached to the resistor.

A suitable resistor-thermocouple detector for this

work was made from 0.13-mm-diam chromel and

+ 22V

POWER AMPLIFIER

FIRST SECTION

Figure 5. Power amplifier, first section.

All resistances in ohms. All capacities in microfarads.

84
Figure 6. Power amplifier, final section.

All resistances in ohms. All capacities in microfarads.

alumel wires. Three pieces of chromel wire and one

of alumel were fused together at one end in a small

bead. A chromel-alumel pair was taken out as a thermocouple and the remaining chromel pair was

wound non-inductively into a small helix with the

bead inserted partway into the core of the helix so that the bead would be as hot as possible. The

chromel pair was about 50 mm long and had a resistance of about 4 ft. The ends of all four wires were soldered to terminals bolted through an aluminum plate and insulated from it by mica

washers. The aluminum plate formed part of a sealed can which was immersed in an ice bath.

Thus the resistor dissipated its heat to a sink at constant temperature and the thermocouple cold

junction was kept at 0 °C. The can was electrically

grounded.

The chromel wire load resistor was connected to the output of the power amplifier in series with a

100-)U,F capacitor. The potentiometer which measured the emf of the thermocouple was connected

in series with a 1-H, 60-ft choke. In this way the

potentiometer was isolated from the amplifier.

This resistor-thermocouple detector had a thermal

half-life of about 5 sec, a desirable value for both

accuracy and waiting time in the presence of

ambient noise, etc. Two were made and calibrated

with the direct current heating, using the average

of both current directions to cancel the effect of a small voltage drop across the thermocouple bead.

The maximum calibration temperature was about

350 °C, produced by about 1 W, and in use the

resistor temperature never exceeded about 230 °C,

except when one detector was decisively incinerated

by throwing the wrong switch.

3.6. Calibration of the Entire Electronic

Apparatus

As remarked in section 2, a plot of AE2 versus T

should be linear, and the slope of the Une should

depend upon R. Various practical considerations

made it necessary to work with an imperfect

amplifier and so the slope of the line was not

exactly proportional to R. However, the dependence

upon R was obtained in a series of experiments

done at 1 atm with hot resistors having resistances

in the range 831 to 4730 ft at temperatures up to about 525 K, as mentioned in section 3.2. When the

probe resistor and the reference resistor, both at one atmosphere, were at the same temperature, one

could determine the small effects, e.g., 0.1 percent

for 1600 ft, of the slightly greater shunt capacitances

for the probe — the noise from the probe was slightly

less than expected. The noise measurements from

the hot probes were used to construct the lines of

figure 1. The ratio of line slope to resistance decreased by about 2.0 percent upon going from 830

to 4,730 ft, in agreement with a simple circuit

analysis and the measured dynamic input impedance of the amphfier.

A second type of cahbraiion was performed as the

last step in a high-pressure run after the pressure on

85
the probe resistor had been reduced to about 2 kbar.

At this pressure the thermocouple error due to pres- sure is very small, and noise measurements could be

obtained on the actual resistor which had been used

at high pressure. The mechanical strains of decompression followed by heating limited the temperatures attainable with satisfactory isolation to about

1,000 K, and usually the probe resistance was much

higher than it had been at high pressures, but these

measurements permitted estimates of the effective

shunt capacitance of the probe resistor as well as a

partial check on the slope of the AE2 versus T line

and the performance of the amplifier and detector.

4. Experimental Procedure

4.1. Pressure Calibration

The pressures existing inside the high-pressure

cell of figure 3 were estimated from the oil pressure

driving the press ram pushing the pistons of the

Belt apparatus. The apparatus was calibrated at room temperature by reference to the resistance

transition in thallium at 36.7 ±0.3 kbar. It was also

calibrated at higher cell temperatures by comparing

the emf of a Fe-Pt, 10 percent Rh couple with that of a Pt-Pt, 10 percent Rh couple according to the

method of Strong et al. [12]. This method permits one

to detect the a — y transition in iron in the range 40

to 50 kbar at temperatures of about 900 K. The cold

and warm calibrations agreed well for 40 kbar, but

differed by a few kilobars at 50 kbar. There is much

evidence of several kinds which indicates that the

pressure inside a pyrophyllite cell changes in complicated ways in response to temperature changes,

and so the actual pressure in the hot cells is not

known to within 2 or 3 kbar. Further comments

on this problem appear in section 6.

4.2. Gathering Noise Data

After the sample shown in figure 3 had been as- sembled and placed in the high-pressure apparatus,

the pressure was increased to about 40 kbar and

then the sample was slowly heated by direct current

to reach about 1,350 K in 1 to 2 hr. Thereby the

Saran strip was pyrolyzed and its resistance fell to something less than 3,000 fl. After that the drift in

resistance was slow enough to permit noise temperature measurements.

The noise temperature measurements were made

in the following way. At some pressure and heating

power level, after probe resistance and the cell

thermocouple temperature became relatively steady,

one could roughly balance the potentiometer which

read the output resistor thermocouple temperature

(which was related to AE2), and then watch the

galvanometer pointer on the potentiometer as it swung in response to the thermal noise signal and

other electrical disturbances. In the absence of

disturbances, the galvanometer pointer asymptotically approached a rest position according to the

5-sec thermal half-life of the output resistor, so that

30 sec of freedom from electrical disturbance was

ample to establish a minimum emf value for the

output resistor temperature.

Usually the disturbances were caused by switches,

relays, commutating motors, ignition, etc., operating

in other parts of the building. Some periods of the

week were quieter than others, but disturbances were abundant even at midnight. The disturbances

usually added about 1 percent to the output thermocouple emf, but not in a systematic way. The loudspeaker on the output helped one identify periods

of intense disturbance during which measurement

attempts were futile.

Usually 5 to 15 min of observation sufficed to establish the minimum output thermocouple emf

between disturbances. Once this had been established, the Pt-Rh thermocouple emf and the probe

resistance could be determined in about 15 sees; these two changed relatively slowly. Next the

reference resistor was adjusted to the same resist- ance as the probe, and the output resistor thermocouple emf was determined for it. Fortunately this

circuit was not so sensitive to disturbances. Finally

the output resistor thermocouple emf was measured

with the preamplifier input shorted, to monitor the

amplifier behavior.

Following these steps, the probe resistor connections were reversed by means of the switchbox, e.g., lead A of the probe was grounded instead of lead B,

and the foregoing procedure was essentially

repeated. If all was well, the two sets of measure- ments were in substantial agreement, after allowing

for slight changes in cell heating power. If the probe

resistor was poorly isolated, the two sets of measure- ments did not usually agree very well and were

ignored.

From time to time the resistances between probe

and heater tube or between thermocouple and heater

tube could be determined as a check on the isolation, and the reference resistor temperature was

noted.

Thus one accumulated a number of values of

output resistor thermocouple emfs, Pt-Rh thermocouple emfs, and probe resistances for various cell

pressures and heating power levels.

The data were gathered first at a pressure of about

40 kbar and with falling temperature steps; then at 25 °C the pressure was increased to about 50 kbar

and the temperature was increased in steps to a maximum (usually limited by isolation failure) and

then decreased in steps (with improving isolation) to room temperature. Finally the pressure was reduced

to a few kilobars and the temperature was raised

and lowered as described in section 3.6.

86
4.3. Reduction of Data

The first step in obtaining useful information from

the data was to convert all good output resistor

thermocouple readings into AE2 values using the

relationship determined by the calibration w^ith

direct current heating of the output load resistor.

Next, one took into account the effe ct of probe

resistance on the slope of the line of AE2 versus

T. (See fig. 1 and section 2.6.)

One also made small corrections to allow for the

differences in shunt capacitance between probe and

reference resistor, on the basis of the room temperature high-pressure data and the low-pressure, hightemperature data, for both of which the pressure

corrections to thermocouples are negligible.

Finally one estimated the probe temperature using

the linearity of the relationship between AE2 and

T for each particular value of R observed. I.e., one knew two points on the straight line for a particular

R and determined the T corresponding to a given

AE2 on that line. The closer T is to the reference

resistor temperature, the more accurately it can be

fixed because the uncertainty in the slope of the

line affects it less.

5. Results

Three satisfactory high-pressure runs occurred.

The results are summarized in figure 7, for 40 kbar

and faUing T only, and in figure 8, for 50 kbar and

rising and falling T. In these two figures the abscissa

is the uncorrected temperature which one would

obtain from the emf of a Pt-Pt, 10 percent Rh thermocouple, according to standard reference tables,

without allowing for pressure effects, and the ordinate is the difference between the temperature estimated from thermal noise and the uncorrected Pt-Pt,

10 percent Rh temperature. The temperature estimated from thermal noise was always higher than

that indicated by the uncorrected thermocouple.

As noted in section 6.1.2 the effect of combined pres- sure and temperature gradients is to make the pres- sure corrections (the ordinates of figs. 7 and 8) from

5 to 10 percent too low.

6. Discussion

6.1. Errors

6.1.1. Fluctuation Errors

The scatter of the points in figures 7 and 8

indicates that the experimental conditions were not

as constant as one might have hoped in view of the

pains taken. The scatter is associated with the use

of high-pressure apparatus; at one atmosphere and

500 K the scatter in temperature estimation was

about 1 K. The most probable causes for this scatter

N

' PIRh 70 r

60 -

300 400 600 800 1000 1200 1400 1600

K by Pt-PIIO%Rh

Figure 7. Thermal noise estimated temperature minus Pt-Pt 10 percent Rh indicated temperature, versus Pt-Rh indicated

temperature at 40 kbar;fallingT only.

70 r

1600

K by Pt- PIIO%Rh

Figure 8. Thermal noise estimated temperature minus Pt-Pt 10

percent Rh indicated temperature, versus Pt-Rh indicated

temperature at 50 kbar.

The upper line is for increasing temperature; the lower line is for decreasing

temperature.

are the long-term (several hours) variations in

ambient electrical noise, the variations in cell

pressure resulting from temperature changes, and

slight variations in cell construction.

Past experience with high-pressure, hightemperature reaction vessels has shown that the

actual pressure in the cell depends on its pressure

and thermal history [13]. The data indicate that the

Pt-Rh thermocouple experienced a higher pressure

during a slight temperature increase than it did

during a temperature decrease, i.e., the thermocouple correction (ordinate of figs. 7 and 8) was

larger for a slight increase in temperature. The

points of figures 7 and 8 also indicate this same

effect on a broad scale in that the thermocouple

correction is generally larger for the ascending

temperature path than for the descending path, so

that the points fall on a rough loop instead of a line.

Even though as much care as possible was used

during the construction of a cell, variations of from

0.2 to 0.4 mm in the positions of various elements

were possible. Furthermore, one should not expect

a cell made of such a wide variety of materials to compress in a uniform manner; instead, erratic

motions are likely.

The emf of a thermocouple may be affected by

the amount of cold work it suffers. Variations of

385-762 O - 71 - 7
several degrees in the indicated temperature may be

produced by this effect [9, 14].

6.1.2. Systematic Errors

A likely cause for systematic error which can not

be detected by general examination of the data is the generation of spurious extra noise by current

flowing through the probe resistor due to imperfect

isolation.

Such current noise power is generally inversely

proportional to frequency and is not too important

above 100 kHz. For most carbon resistors one

expects about 1.5 /xV per applied volt for the range

1 kHz to 100 kHz. In the high-pressure cell, if the

isolation resistance is 100 times the probe resistance,

e.g., 100 kn, and the spurious driving voltage is 1 percent of the total of 10 V, then the voltage on the

probe available to produce current noise is about

10"'' V. The current noise would then produce about

1.5 X 10""* /xV of noise compared with the thermal

noise of about 1.7jU,V. But power, not voltage, is to be added or compared; on this basis the current

noise contribution would be less than 10"^ of the

thermal noise. Even if the probe resistor developed

100 times the usual current noise voltage, the contribution of current noise would be less than 10~* of

the total. The above considerations notwithstanding, when

the isolation resistance failed at high temperatures,

the noise from the probe suddenly increased and

became extremely erratic. It was not possible to measure the emf of the output resistor thermocouple

because the galvanometer in the potentiometer was

too restless.

Therefore it appears likely that extra current noise

from the probe resistor was certainly not important

below 1,000 K where the isolation resistance was

excellent; and was probably not important up to the

highest temperatures for which results are given in

this paper.

In an ideal measurement of the effect of pressure on a thermocouple, the temperature gradient occurs

at constant pressure and the pressure gradient occurs at constant temperature [6]. In the apparatus

used in this work these ideal conditions did not

occur. As figure 3 indicates, the thermocouple wires

C and D experience the bulk of the temperature fall along their radial portions. At the internal joint

where the wires begin to travel vertically in the

gasket, the temperature was at most 250 °C, judging

from the decomposition of traces of glue used in

assembly. In this radial portion the pressure on the

wires probably changes by less than 2 kbar, if other

types of experiments are used as a guide. In their

vertical travel through the gasket the wires en- counter the major pressure drop to one atmosphere

in a region having a temperature of about 70 to 100 °C when the hot junction is at about 1,400 K.

The wire temperature finally drops to about 25 °C

while the wires are at one atmosphere.

These considerations indicate that the ideal

correction to a Pt-Pt, 10 percent Rh thermocouple

for the effect of pressure would be somewhat larger

than indicated in figures 7 and 8, probably 5 to 10 ^{

percent larger, because only about 90 to 95 percent

of the temperature drop in the wires occurred at high pressure.

6.2. Comparison With Other Work

As mentioned in the Introduction, some work hasj

been done on the effect of pressure on thermocouples at more moderate conditions of pressure or temperature than attempted here.

The Work of Bundy [6] shows that at a pressure of ! 50 kbar and a temperature of 100 °C, the Pt-Pt 10

percent Rh couple should indicate a temperature

which is about 6 °C too cool. The present work

agrees satisfactorily with this previous finding.

The work of Bell, Boyd, and England [8] carried

out to temperatures of about 500 °C at pressures of^

5 kbar shows that a Pt-Pt, 10 percent Rh thermocouple at 500 °C and 5 kbar should indicate a temperature which is about 3 °C too cool.

Freud and Lamori [9] estimate from their work'

that at 40 kbar and 300 °C the Pt-Pt 10 percent Rh

thermocouple should read about 13 °C too low.

This work is in satisfactory agreement.

The work of Getting (1968) shows that a Pt-Pt,

10 percent Rh thermocouple at 600 °C and 30 kbar

should indicate a temperature which is about 13 °C

too cool. This is about two-thirds the correction

estimated from the present work.

Hanneman and Strong [10] and Bundy [14] con- sidered a number of experimental phenomena at high pressures and temperatures such as diffusion

in metals, melting curves of metals such as germanium, gold, iron, etc., and phase equilibria such as a — yFe and diamond-graphite to construct a self- consistent set of pressure corrections to thermocouples. Their estimates and the results of the

thermal noise work in this paper are in fairly good

agreement. E.g., for a pressure of 50 kbar and a temperature of 1 ,370 K, the foregoing authors estimate a

correction of 47 K; the average correction found by

the thermal noise method for the same nominal

conditions is about 60 K. At 40 kbar and 1,270 K the

foregoing authors estimate a correction of 36 K, and

40 K was found in the present work.

It appears that the present work is in satisfactory

agreement with results found by other or earlier

workers; the thermal noise temperature estimates

tend to be somewhat higher, in agreement with the

expectation that undetected extra noise may have

been added to the thermal noise signal from the

hot probe.

7. References

[1] Dachille, F., and Roy, R., Chapt. 9 of Modern Very High

Pressure Techniques, R. H. Wentorf, ed. (Butterworths,

Washington, 1962).

88
[21 Johnson, J. B., Phys. Rev. 32, 97-109 (1928).

[3] Bell, D. A., Electrical Noise (Van Nostrand, London, 1960).

[4J (iarrison, J. B., and Lawson, A. W., Rev. Sci. Instr. 20,

785-794 (1949).

[5] Bridgman, P. W., Proc. Am. Acad. Arts Sci. 53, 269 (1918). Or see his collected papers (Harvard, 1964).

[6] Bundy, F. P., J. Appl. Phys. 32,483-488(1961).

[7] (letting, I. C, and Kennedy, G. C., The effect of pressure on

the E.M.F. of the thermocouple, paper presented at this meeting.

[8] Bell, P. M., Boyd, F. R.. and England, J. L., The effect of pressure on the thermal emf of the platinum/platinum 10 percent rhodium thermocouple, paper presented at this meeting.

[9] Freud, P. J., and LaMori, P. N., The effect of pressure on

thermoelectric power of thermocouple materials, paper

presented at this meeting.

[10] Hanneman, R. 6. and Strong, H. M., J. Appl. Phys. 36, 523

(1965); Ibid., 37, 612 (1966).

[11] Hall, H. T., Rev. Sci. Instr. 31, 125 (1960).

[12] Strong, H. M., and Bundy, F. P., The use of iron and gold for calibration of higher pressure and temperature points,

Report No. 68-C-402, General Electric R&D Center,

Schenectady, N.Y. Paper presented at this meeting.

[13] Strong, H. M., Chapt. 5 in Modern Very High Pressure Techniques, R. H. Wentorf, ed. (Butterworths, Washing

ton, 1962).

[14] Hanneman, R. E., Strong, H. M., and Bundy, F. P., A critical review of the effect of pressure on thermocouple Emf's. Report No. 68-C-367, General Electric R&D Center,

Schenectady, N.Y. Paper presented at this meeting.

DISCUSSION

H. T. Hall {Brigham Young University, Provo,

Utah): Are your results on corrections that are to be

applied to platinum-platinum rhodium couples consistent with other results of your laboratory and with

results of other laboratories?

I. Getting {Institute of Geophysics and Planetary

Physics, University of California, Los Angeles,

California): May I suggest that if you used a dc

heater there is a likelihood of dc pickup on the

thermocouple. We tried the Bridgman single-wire

experiment with a dc furnace only once, due to unsatisfactory results from this cause.

Question from the floor: Does the insulating

material surrounding the resistor have any effect on the noise voltage?

M.Waxman (National Bureau ofStandards, Washington, D.C.): What is the ac loading on the noise

resistor?

Would you comment on the integrating time, and

describe the power fluctuations?

AUTHOR'S CLOSURE

On the question of agreement with results of

others, the corrections I found agree pretty well

with estimates of Hanneman and Strong, but are about 50 percent larger than Getting has found.

I think that my results may tend to be high because

of the extraneous noise from the environment, but

I don't think they are 50 percent too high.

I was aware of the problem of dc pickup, and the

data from one run was discarded because of this.

However, I monitored the resistance between the

thermocouple and the heater, and between the noise

resistor and the heater and, at 1400 °C, these resist- ances were usually of the order of 10^ to 10^ fi. The

key to this high resistance was the use of a very pure

shielded alumina insulator (Lucalox). I think that

this is one of the best materials available.

The insulating material has no effect on the noise

voltage because it is not carrying any current. While

the insulating material does comprise a resistor in

parallel with the noise resistor, they are both at the

same temperature, and the only effect is that the

resistance of the combination is slightly lower than

that of the noise resistor alone.

The ac loading of the noise resistor depends, of

course, on the preamplifier input impedance, and

was of the order of 50,000 ft. This affects the slope

of the calibration curves, and requires a separate

curve, done at low pressure, for each resistance

range.

On the matter of integrating time, the time constant of the circuit was such as to give an integrating

time of about 5 seconds. However, random fluctuations due to extraneous causes often required observations to be made over a period of 15 minutes or more in order to obtain three measurements, say,

that had the same minimum, so that the integrating

time might reaUy be of the order of 10 to 20 seconds.

89

session IV

Chairman: G. E. DuvALL

Washington State University

Seattle, Washington

Equations of State for Sodium and Aluminum

D. John Pastine and M. J. Carroll

U.S. Naval Ordnance Laboratory,

White Oak, Silver Spring, Maryland 20910

Equations of state are calculated for sodium and aluminum. In each case this is done by first calculating the 0 K isotherm and then adding the thermal contributions to the pressure. The calculations in both cases are theoretical and very few experimental data are used. Comparisons of theoretical prediction and experimental data are very favorable.

1. Introduction

It seems wise to initiate any equation-of-state

calculation with a general expression for the specific Helmholtz free energy, A. For, by this procedure, one can at least ascertain what it is that

he must know to go further. Accordingly, since this

article treats two elemental metals in the solid

state, we will begin by writing the general equilibrium expression

^=^0 + ^q.h.+^a.h. (1)

This equation is a legacy of statistical mechanics

[1] ' and it is very helpful because it makes an

orderly separation between three distinct contributions to A. In eq (1) Ao is the total specific Helmholtz energy of the solid at absolute zero of^the

temperature, T, excluding all vibrational contributions. Ao therefore has a definite value for every

fixed configuration of the nuclei of a solid and is a function only of the relative positions of these

nuclei. Under conditions of uniform pressure, P,

the relative positions of the nuclei can in principle be specified completely by the specific volume

V. Under these conditions therefore Ao is a function

of V only.

The quantity ^q.h. in eq (1) is the contribution to A arising from thermal vibrations in the quasiharmonic (q.h.) approximation. In this approximation, the normal vibration frequencies of the solid are assumed to be functions only of the mean nuclear

positions and therefore do not depend explicitly on

T. As in the case of these frequencies will be

functions only of v if the pressure is uniform, ^q.h.

' Figures in brackets indicate the literature references at the end of this paper.

Paper presented at the Symposium on Accurate Characterization of the High-Pressure Environment, held at the National

Bureau of Standards, Gaithersburg, Md., October 14-18, 1968.

has the general form

3n A^,^,^kT 2 {eil2T+ln [1-exp (-Oi/T)]} (2)

! = 1

where di — hfijk, h is the Planck constant, k is the

Boltzmann constant, /, is the ith normal mode fre- quency, and n is the number of atoms per gram.

The quantity Aa.\\. in eq (1) is the contribution

due to anharmonic effects not already included

(implicitly) in A^^ ^^ . It is therefore a correction to ^q.h. which in essence accounts for the fact that

the fi cannot really be independent of T at constant

volume. For most elemental metals at room temperature ^a.h. is a very small correction, but in the case of sodium, it is significant at low pressures and

therefore a crude (but effective) estimate will be

made in this case. In subsequent sections of this

article P, v, T equations of state are calculated for

solid sodium and aluminum. In each case it is done

by considering separately the important contributions of each term on the right side of eq (1).

2. The P, T Equation of State

of Sodium

2.1. The Evaluation oi Ao

Since in general P{v, T) =— (dAldv) t, then

according to eq (1) the contribution of to the pres- sure is Poiv) =— {dAoldv)T = — dAoldv where Po{v)

is a function of volume only. It is convenient to express volume in terms of the dimensionless parameter X — v/vo where vo is the specific volume at P = 0 and r=0 K. For sodium i;o = 0.9873 cm^/g.

With this definition, the pressure becomes P{x, T) ^ — po{dAldx)r where po=l/vo is the density at P = 0 and r=0 K. Likewise the expression for

91
Po becomes

Po(x)--po • (3)

From eq (3) it is clear that in the range of interest one need consider only those contributions to Ao which change appreciably with volume and therefore contribute significantly to the pressure. For

sodium this excludes a good deal. It excludes,

for example, the necessity of accounting for the

volume variation of the wave functions of the ten core electrons in each sodium atom [2]. This is because the core electrons are tightly bound relative to the valence electrons and are locahzed in a region of volume, Vc, which is small compared

to the volume per atom, Va, in the solid. The compression of core electrons is therefore unlikely to become an important factor until Va — Vc, which in

sodium corresponds to jc — 0.2 and a pressure of

well over 500 kbars at room temperature. Therefore,

for sodium, the calculation of Po{x) requires an

evaluation only of the interaction energy between

the singly charged atomic cores themselves, the

interaction energy between cores and valence

electrons, and finally the energy of the valence

electrons themselves. Two small contributions

to the core-core interactions can be established

immediately. These are the repulsive contribution,

A^, due to the mutual overlap of core wave functions

and the attractive contribution, A^, due to the Van

der Waals interaction. Reasonable semitheoretical

expressions for the pairwise contributions to A^ and A^ already exist in the literature [3-5],

and from these one finds

^f,'= (130.92 XIO^) exp [5.07 (1 - 2.0611x'/3)]

ergs/g. (4)

and

A^ = -{0.U87Xl0^)lx^

ergs/g. (5)

Equations (4) and (5) represent the results of

calculations in which only nearest neighbor interactions are included. The remaining contributions

to ^0 can be viewed as the sum of four conceptually

distinct energies. These are as follows:

(1) Ao'^\ the contribution (ground-state energy)

which one would have if all valence electrons were

in the ground state (i.e., not translating throughout

the lattice) and interacted with nothing save their

respective ion cores;

(2) Ai, a contribution (the Fermi energy) which

occurs because the exclusion principle prohibits

all the valence electrons from being in the same

energy state;

(3) AV\ a contribution (exchange energy)

which results from interactions between electrons

and degeneracies which occur because of their

indistinguishability;

(4) Ai"^'^, a contribution (correlation energy) : which arises because valence-electron motions are somewhat correlated because of the mutual Coulomb

interactions.

Of these four energies, A-d is presently the easiest

to evaluate. This is because the band structure of

sodium has already been calculated in considerable

detail. Bienenstock and Brooks (B-B), for example

j [6], have given sphericalized analytic relations con- | necting valence electron energy with wave vector '

in sodium for various values of the lattice parameter.

The latter calculations are very detailed and are

| based on the quantum defect method. Since the j Fermi surface of sodium is known to be very nearly

spherical [7], the energy versus wave vector relations given by B-B can be used to obtain the (mean)

Fermi energy for several values of the sodium lattice

parameter by simple integrations over wave vector

space. When this is done one finds in every case

that the mean Fermi energy of compressed sodium

remains within 3 percent of the free electron value

for electrons of equivalent number density. The

B-B calculations are particularly interesting be- cause they were performed in several ways. One

set of calculations, for example, includes the effects

j of core polarization and another does not. These j inclusions and exclusions were made so as to dis- cover their effect on theoretical shift of the Lande ' ^ factor for sodium.

The results are interesting in that the set of cal- | culations which provides the best agreement with

the pressure dependence of the experimentally ! measured ^-shift is also the set for which the mean ! Fermi energy is closest to the free electron value.

As a matter of fact, in this case the mean Fermi

energy remains within 0.7 percent of the free electron value over a range of the lattice parameter

which for sodium corresponds to 30 percent compression (i.e., 0.7 ^ ^ 1.0). The indications are, therefore, that for sodium A{^ is very close to the

free electron value, which is given by

A{ix) = (81.511 X 109) ergs/g.

(6)

The closeness of Af( to the free electron value is not

surprising. The free-electron-hke nature of the

metallic sodium valence electron has been emphasized by many authors for the past 35 years [8].

'

So far as the equation of state is concerned, it is a j fortunate fact, because it makes it sensible to calculate A()^- and A'i/'^^- in the same approximation,

especially since these terms are only small corrections to A^. Expressions for the free electron ex- change energy may readily be found in the literature.

A particularly good treatment of this subject has

been given by Gombas [9]. Estimates of the free

electron correlation energy may also be found.

Here, an analytic expression given by D. Pines [10]

is used. Great accuracy is not attached to the latter

expression (±20 percent) but, as it happens, for

the monovalent alkali metals the two energies

92
Jji^ and ^o""' tend to cancel one another and txs a

result their sum contributes very little to either Aq

or Fo. The possible inaccuracy of /4o"'''' is therefore

not a matter of great concern. Using the literature

mentioned above, one finds that for sodium ^

At^'ix) +A^r-'-{x) =109[41.2058M3- 79.887

+ 35.7213 In (3.933j»:>/3)-5.8515;c1/3] ergs/g. (7)

The constant term in (7) has been kept for the purpose of calculating the cohesive energy later on.

The ground-state energy A^-^- may be evaluated

in either a semi-theoretical or a purely theoretical

fashion. From the standpoint of accuracy, the semitheoretical approach is most desirable because with

this approach one can incorporate some of the

observed experimental behavior in the final expressions for pressure. This can be done by constructing a reasonable and adjustable analytic expression

for the ground-state pressure P^'^- and then manipulating the adjustable constants until satisfactory

agreement is obtained betw^een the theoretical and

experimental 0 K isotherms. The theoretical 0 K

isotherm Poo{x) is given by the sum

PooW=PoU)+PSU)

where Po{x) denotes the contribution of 0 K lattice

vibrations. In the range O^Poo{x) ^20 kbars, an

accurate experimental isotherm can be obtained

from the data of Beecroft and Swenson [11]. The

big advantage of the semi-theoretical approach is that it permits errors arising either from smaU omissions or from inaccuracies in the expressions (4)

through (7) to be absorbed in the adjustable constants of Pq'^'. This is particularly important at low

pressures, where the theoretical isotherms are given

by the small differences between relatively large

quantities. In this region small errors in .any of the

terms contributing to the 0 K isotherm (^o"" for

instance) are likely to show up in a conspicuous way.

At higher pressures this is no longer true. Consequently in the higher pressure region one should

expect good agreement between a purely theoretical

calculation and a good semi-empirical estimate of

the 0 K isotherm, provided of course that the error

in each contributing term of the pure theoretical

calculation is sufficiently small. It is therefore a good

procedure to check the results of an analytic, semiempirical calculation at high pressure by comparing

them with the prediction of pure theory. This is the

procedure which wiU be taken below.

As in the case with the other quantities which

contribute to Ao, it is easy to obtain a reasonable

analytic expression for Al'^-. The first such expression was derived for the alkaU-metals by Hellman

- The potential arising from the Coulomb interaction in the Hartree field is included

in eq (7).

and Kassatotschkin [12] (H-K) and in our units it has the form

^fi-^ U) =10«{-435.273/x'/'''+ (201.0432/;c) [1

-il + yx'l^)e-y-^'l']} ergs/g (8)

where and y are adjustable constants. The expression (8) is used in subsequent calculations because

it is simple and analytic and because its theoretical

basis is sufficiently sound. The development of

eq (8) is based primarily on an approximation due

to Wigner and Seitz [8] (W-S), according to which

one assumes that Af^- =nX{t\\e ground state energy

of one valence electron and one core contained in a spherical cell of volume Va). Corrections to this

approximation for sodium have been investigated

in considerable detail and have been found in general to be quite small [13—14]. In the H—K development it is further assumed that ground-state wave

function i//g s is equivalent to that of a free electron

(i.e., i//g.s. = l/fa. The energy in a single cell Ea

is then calculated with the expression

r R

where <p{R) is the effective valence electron coreinteraction, R is the distance from the center of

the core (which coincides with the center of the

spherical cell), and Rg is the cell radius. The adjustable constants appear in the expression for

ip{R)j for which (H-K) used the form

<p{R)^-{eVR)[l-Ae-'<'^]ergs. (9)

Here e is the electronic charge in e.s.u. and the

constants A and K must be determined from ex- periment. In eq (8), z= (^/A:^) X lOi^cm^ and

y—KR^ where R° is the value of Rg at v — Vo. As mentioned previously z and y can be evaluated

by forcing agreement (to within experimental error)

between the theoretical expressions for Poo{x)

and the experimental 0 K isotherm. To do this it is necessary to jump ahead to the approximate

expression for Pq derived in the next section, that

is,

PS(;c) - (0.5435)/;ci ««- kbar. (10)

With Ao{x) given analytically by the sum of eqs

(4), (5), (6), (7), and (8), and with given by eq (10),

the 0 K isotherm is easily derived from the relation

Poo{x) = — po dAoldx + P'^{x). When this is done

and when z and y are adjusted to optimize agreement with experiment, one finds z = 0.43258 and

y=6.71. The results of this adjustment are shown

graphically in figure 1, the complete analytic expression for the 0 K isotherm is given in Appendix A.

93
THEORETICAL Table 1. Comparison of values of A^-^- and P^-^ calculated

using the Prokofjew Potential for sodium with those calculated

using (8).

0.850 0.900 0.950 1 .000

Figure 1. The result offitting the theoretical 0 K isotherm to the experimental data.

As was mentioned earlier, there exist more purely

theoretical approaches to the calculation of ^§'*'. Of

these the most successful (for sodium) has been

that of Wigner and Seitz [8]. These authors merely

solved the Schrodinger equation for a single groundstate valence electron in the spherical W-S cell. An

eigenvalue of the energy per valence electron, Ea,

was assumed and the Schrodinger equation for

'I'g.s. was numerically integrated until the proper

boundary condition was met. In this case, the

boundary condition compatible with the translational symmetry of the lattice requires [8] that

jAg.s.C^s) =0. By solving the Schrodii;iger equation

in this way (W-S) were able to generate the curve

of Ea versus Rs or equivalently the relationship

between A^-^- and x. The valence electron-core

interaction used by (W-S) in the Schrodinger equation was one originally derived from spectroscopic

data by Prokofjew. Here the W-S procedure has

been repeated to obtain theoretical values of Aq-^-

and Pq'^'. These values are much more accurate

(than those of (W-S)), since they were obtained by a computerized Runge-Kutta integration technique.

In table 1 values of Af)-^- and Pt'^' calculated from

eq (8) are compared with the theoretical (W-S)

values as a function of x. It can be seen immediately that the agreement between the two sets of

values is quite good to about 40 percent compression. This agreement implies several important

things. First of all, it implies that the theory is sound

and that all the analytic terms which contribute to Poo(^) are represented quite accurately. Secondly,

it impHes that the error in the calculated 0 K pres- sures should amount to Uttle more than the dif- ference between the theoretical and semi-theoretical

values of Pq-^-.

At x = 0.6 this difference is only 0.23 kbars,

whereas the calculated pressure. Poo, at this compression is 94.2 kbars. Since the analytic version

of Pooix) is forced to reproduce the correct (experimental) pressures below 20 kbars and since

up to 94.2 kbars this same function agrees with the

theoretical values of Poo{x) everywhere to within

X

Prokofjew Equation (8)

pe.s.

' 0 pg.s.

' 0

1.0235

1.0133 0.9914

0.9701

0.9619

(lO^ergslg)

-347.10

-347.70

-349.02

-350.30

-350.80

(kbar)

-59.76

-60.19

-60.96

-61.66

-62.00

(lO^ergslg)

-347.71

-348.32

-349.64

-350.94

-351.45

{kbar)

-60.56

-60.86

-61.49

-62.09

-62.31

0.9215

0.8882

0.8569

0.7801

0.7374

-353.30

-355.40

-357.40

-362.40

-365.20

-63.34

-64.33

-65.14

-66.48

-66.46

-353.96

-356.06

-358.05

-363.02

-365.79

-63.37

-64.16

-64.80

-65.80

-65.78

0.6914

0.6441 0.6360

0.6009

0.5937

0.5367

-368.20

-371.2

-371.70

-373.80

-374.20

-376.90

-65.39

-62.80

-62.29

-60.27

-55.80

-40.65

-368.77

-371.78

-372.29

-374.44

-374.86

-378.05

-65.14

-63.41

-62.97

-60.50

-59.84

-52.50

0.8 kbars, it seems a fair conclusion that the analytic

0 K isotherm reproduces the true sodium isotherm

with an accuracy of 2 percent up to pressures of

95 kbars. For pressures beyond about 95 kbars the

two»^ calculations begin to deviate significantly and

one cannot be entirely sure which is the more

correct. Estimates of accuracy are therefore limited

to the region of f*oo(^) ^95 kbars. As a last test of the analytic 0 K isotherm, the theoretical bulk

modulus and cohesive energy at ;c = 1 are compared

with the experimental values. The experimental

cohesive energy of sodium is [16] 48.28X10^

ergs/g at 0 K and the experimental bulk modulus

is [11] 78.2 kbars. The theoretical values are 47.314

X 10» ergs/g and 79.0 kbars.

2.2. The Contribution of ^q.h.

At high temperatures {dilT<l for all i), the

expression (2) ior A^ has the limiting form

3n ^,.h.= A:r^ In (dilT)- (11)

This makes a contribution Pq.h. to the pressure of an amount

Pfi.hix, T)=po3nk iyoix)! = 3.293 (yo/x) (r/300) kbars (12)

94
3)1 where yo(-v) = — ^

yj, and yj, 1=1 fi \(^x It

=_£ (

Although the form of eq (12) is quite simple, the

theoretical evaluation of yo{x) (quasi-harmonic

Griineisen parameter) is full of obvious difficulties

and some rather drastic approximation must be

made. This isn't really a cause for concern because

it is already known [17] that for sodium near x=\, yo should be of the order of unity and moreover should diminish with decreasing x. This means

that at 40 percent compression (x = 0.6) and room

temperature (here r=300 K) the contribution of

h should be no more than five kbars. Since at this compression Po{x) is 92.8 kbars, then a 50 percent error in yo (which is quite unlikely)

would lead at the most to a 2.6 percent additional

error in the total pressure. In estimating yo(^),

therefore, one can afford to be a little crude. The

approximation which will be used for yo(it;) is the (apparently accurate) "two-mode" approximation [18-20]. In this scheme, one replaces the un- wieldy average over the 2>n logarithmic derivatives

of the normal mode frequencies,/;, with an average

over only two types of long wavelength acoustic

modes. Of these two types of modes, one is purely

longitudinal (p.l.) with a propagation velocity

Ci, given by

(13)

whfre C?i is the usual elastic constant, evaluated

along the 0 K isotherm. The other type is purely

transverse (p.t.) (having two independent directions

of polarization) with a propagation velocity, Ct,

given by

Ct —xClil (14)

where C^4 is the usual elastic shear constant evaluated [21] at 0 K. Since as a solid is isostatically

compressed, one expects the wavelength of a p.l. or p.t. mode to vary as x^'^, then the propagation

frequency of a p.l. mode,/;, is given by / °c Ci/x^^^

and that of a p.t. mode,/, is given by / ^ Ctlx^l^.

With these relations for frequency and the expressions (13) and (14) for velocity, one can calculate

Jo-

: ——(

^

/ \ dx

16 2C?iV dx

and

yo—T

x_/ dft_

ft \ dx

1 X (dC%,

6 2^4 \ dx

(15)

(16)

In the "two-mode" approximation, one assumes

\ 3n that the average ^ ^ y{) is given approximately by

on j^j^

7- {"y& + 2ray()}. In other words, the 3n values of on

yf) are replaced by n of the type y{) and 2n of the

type y{). It remains now to compute C?i and C% from

theory.

For the alkali metals,^ the theory of the elastic

shear constants C44 and C°= (Cn — Ci2)/2 has been

developed by Fuchs [22]. According to Fuchs, the

electronic contributions to 0%^ and C" are given, re- spectively, by 53.2/x'*'^ kbars and 7.146/x'*/^ kbars.

To the electronic term of C44 one should add the

contribution due to ion core-ion core repulsion [23]

(this contribution can be neglected [23] in the case oif C°). When this is done the expressions which result

for C54 and C are

^4 = {53.2^3 + ( 1654.6/^1/3 _ 312. 1/^2/3)

exp [5.07(1-2.0611x1/3)]} kbar (17)

Co = 7.146M3 kbar. (18)

With Co given by (18), C?i can be calculated immediately from the identity

dPn 4 -

(19)

With (15), (16), (17), (18), and (19) an analytic ex- pression for the theoretical yo(x) can be constructed

Table 2. Calculated values of yo, yQ, yj,', and

P" /(T/300) as a function of x for sodium.

X

y'a y'a (7/300)

(Dimensionless)(Dimensionless)(Dimensionless)

(kbar)

1.06 0.895 0.214 0.957 2.784

1.05 0.893 0.205 0.933 2.803

1.04 0.891 0.196 0.905 2.824

1.03 0.890 0.187 0.897 2.845

1.02 0.888 0.178 0.891 2.867

1.01 0.886 0.169 0.875 2.890

1.00 0.884 0.161 0.845 2.913

0.96 0.879 0.128 0.788 3.015

0.92 0.874 0.097 0.745 3.130

0.88 0.871 0.068 0.693 3.259

0.84 0.869 0.042 0.640 3.406

0.80 0.868 0.017 0.596 3.571

0.76 0.867 -0.006 0.538 3.758

0.72 0.868 -0.026 0.473 3.970

0.68 0.869 -0.043 0.391 4.210

0.64 0.871 -0.057 0.290 4.483

0.60 0.874 -0.066 0.148 4.796

95
in the "two-mode" approximation.

The resuhing values of yoix), its first and second

derivative (70 and 70), and the function Pq.h./(^/300) are hsted as functions of x in table 2. In the low-temperature limit eq (2) has the form

^q.h. = ^2e</2 = 2/i/i/2. (20)

i=l i=l

Using the Debye approximation [24] to estimate the

right side of (20), one finds

^q.h. - (9/8) nk doix) = (4.0651 X 106) ergs/g (21)

where Ooix) is the Debye temperature which for

sodium at x=l is [25] = 152 K. Now since the

Debye temperature has the definition: h fmaxl^

where /max is the highest normal mode frequency,

then one should expect that the quantity — {xlfmax)dfmaxlldx = - {xl do) ddoldx

behaves very roughly as yoix) which, as it happens,

is fairly constant (see table 2) with values remaining

near 0.86. As a rough approximation it will do to set — {xIdD) dOoldx^yoix) 0.86. After integrating

this expression one has

0z)(:t) = 9z)(l )/%»•«" =152^«-86 K. (22)

By substitution (22) into (21) and differentiating, one can obtain an estimate of which is the contribution of /4q h to the pressure at 0 K. This is

Pji(x) = (0.5435)/xi-«« kbar (23)

which is the expression already given in (10).

2.3. The Contribution of ^a.h.

- From the outset it should be clear that /4a. h. will

modify the isothermal pressures near room temperature only in a minor way. In effect it is a small cor- rection to a small correction.

It has already been said that in the quasi-harmonic

(q.h.) approximation the /; are assumed to be (for

all practical purposes) functions of x only and that,

in essence, /4a. h. could be viewed as accounting for

the temperature dependence of the If one finds

therefore that some of the /, vary appreciably with

temperature at constant volume (i.e., if ((9 In fi/d In T)x

is not small compared to unity), then a correction

should be made. In the case of the present calculation on sodium, consistency demands that whatever

corrections are made should be made only on the

frequencies (fi and ft) on the p.l. and p.t. modes

used in the approximation of Aq,h.- The necessity of

a correction in this case is easy to discover. Obviously, for the q.h. "two-mode'! approximation to be

perfectly correct it is necessary that and Cj be

given exactly by (13) and (14). In other words, it requires that the elastic constants Cn and C44 be

functions of volume only (i.e., Cn C^,; Ci4 = C^^).

This is because in the q.h. approximation it is assumed that the /, are independent of T, which

implies that for a given x they are essentially the

same at elevated temperature as they are on the

0 K isotherm. Consequently the temperature dependence of/; and ft will show up in the temperature

dependence of Cn and C44. More precisely one can

write

and

a In /A /> d In Cn

dlnT), ^V a In r

a In /A .^ d In C44

V a In r

For sodium under normal conditions the derivatives on the right side of (24) and (25) can be evaluated from the experimental data of Diederich and

Trivosonno [23]. From these data one finds

(a In CiJd In r).r=i =- 0.351 and {d In Cn/d In r)x=i = 0.0197. This tells us immediately that in the "twomode" approximation only the shear frequencies

ft will contribute appreciably t0/4a h - the contributions of /( being negligible by comparison. Since,

as has been said, ^a.h. will make only a small contribution, Pa.h., to isothermal pressures, only a crude

estimate of Pa.h. is necessary. The most obvious

way to make this estimate is simply to examine the

correction to yi which results from the temperature

dependence of C44. To do this one needs an expression for C44, which gives the temperature dependence of C44 at least to first order (i.e.,

C,, = CUl+gix)T],

where g{x) is a function of x only). It is not hard to find such expressions [1] but unfortunately in the case of C44 the function g{x) is not easily evaluated.

One can, however, estimate the temperature dependence of the shear constant C = ¥2(011 — C12). To

first order in T it can be shown that

C = CHl + ~gix)T) (26)

in which g{x) is given roughly by [26]

g{x)=-syl>ia (27)

where s is a positive constant.

Now since Fuchs [22] has pointed out that C%

and C" should be almost exactly proportional to each other, then it seems reasonable to expect that

the first-order temperature-dependent terms of C44

and C are also simply related. As a matter of fact,

from experimental data it is possible to deduce

that for sodium

96
g{x) ^ agix) +b (28)

where a and b are constant. Daniels [27], for ex- ample, has shown experimentally that for sodium

under normal conditions

dC44 dx dx )i

(29)

This implies directly that, for some range of isothermal compression, C44'^ C; or, using the theoretical expressions for C44 and C,

(i+g(;t)ro)-8C»(i+^(%)ro), (so)

where 8 is the proportionality constant and To = 300 K. But Fuchs has shown (and also the ex- perimental data) that C/C44 is very nearly constant [28]. Therefore, setting the constant e = C'V^44 ' one can rearrange terms in (30) to obtain

g{x)^heg{x) + {he-\)IT^ = a~g{x)+b. (31)

Using (31) for g{x) and (27) for g{x), C44 becomes

C4A = OU{\ + bT) - {sale)y'J. Setting 5a/e = /3 this

becomes simply

C,,= {l + bT)Cl^-liy'J- (32)

With eq (32) it is possible to obtain an expression

for the first order effect of temperature on the

derivative

-(a In /,/alnx)r=-l/6-(l/2) {d\nCi^ld\nx)r

Using (32) for C44 and noting that the term containing the temperature is small relative to [29] (1

-\-bT)C%i one can obtain (Note also, bT < 1)

Pr—Pq.h. + Pa.h. = 3.293(7o + A7)(r/300)/;t kbar. (35)

Since analytic expressions for 70 and C^^ have

already been given, it is necessary only to evaluate

/Lt. This can be done by demanding that the total

pressure (Po + Pt) be equal to zero at room temperature (300 K) and the observed value of volume

[30] {x= 1.043). In this way one obtains /x= 17.345

kbar.

The evaluation of jx completes the P, v, T equation

of state for sodium, but it is interesting to pursue the

matter further and find the expression for A^.h.

implied by eq (34). This is done in Appendix B, from

which one finds

^a.h. = -/^(2.1674 X 10«) (r/300)Hy;/C<'J ergs/g.

(36)

With eq (36) it is possible to calculate the total

specific energy E in the high-temperature limit

{T>dD) through the relation £ = - [a {AIT)ldT]jc.

This gives us E^Ao+ (3.25118 X 10") (r/300) [1

+ (2/3)At(r/300)7;/C«J ergs/g. (37)

With eq (37) one can make a more stringent test of

the accuracy of ^a.h. by calculating quantities more

sensitive to it than the pressure. For example, one can calculate the specific heat at constant pressure,

Cp, and compare it with the experimental [31] values.

One can use the identity

Cp — C(,+

(XX

Po "Mil (38)

din ft

d In X J T

d In CI,

d\n X {Tim)IC% (33)

where several constants have been lumped into jx. Viewed as a correction to yo the second term on

the right of eq (33), which we will call Ay', contributes an amount h to the pressure, which is given

by

Pa.h=poSnkmylx= 3.293(Ay/;c) (r/300) kbar

(34)

where Ay= (2/3) Ay'. The total pressure Pt due to thermd effects is therefore

where both Cv = {dEldT)x and (dEldx)T are given by

eq (37) and a^\lx{dxldT)p^(dPldT):rl{-x{dPldx)T\ can be obtained from the P, v, T equation of state

already derived.

In this manner Cp has been calculated (with small

quantum corrections [32]) along the P = 0 isobar.

The results are compared with experiment [31] in

figure 2; the agreement is excellent. The derivative

{dCpldP)T was also calculated under normal conditions and was found to have the value 1.657x10"*

cm^/g K. The experimental value is [33] 1.653 XlO~*

cm^/g K. The theoretical room temperature isotherm is compared with experimental data in table

3; values of Ay/(r/300) are also listed in this table.

The theoretical isothermal bulk modulus at 349 K

and normal pressure was found to be 64.3 kbars; the

experimental value [11] is 63.9 kbars.

The theoretical Hugoniot equation of state (which

is somewhat sensitive to yo + Ay) was also calculated. An outline [34] of the procedure for doing so

is given in Appendix C. The results are compared

97
12.6

12.4

12.2

12.0

U.8

11.6

U.4

11.2

11.0

1.57.

/ /

THEORETICAL

N.B.S. COMPILATION

31

_] I i_ 200 225 250 275 300

T (Kl

Figure 2. Comparison of theoretical and experimental Cp values. The Dulong-Petit value is 10.84 X 10+'^ ergs/g K.

with experiment [35] in table 4. The agreement is again excellent.

At this point it seems a fair conclusion that, for

T in the neighborhood of room temperature, the

theoretical isothermal equation of state for sodium

gives pressures accurate to within 3 percent at a given volume in the range of compression

(0.6 ^ ^ 1).

3. The P, V, T Equation of State of

Aluminum

3.1. The Evaluation of Aq

From the viewpoint of electronic structure,

aluminum is in many ways similar to sodium. Like

sodium, it has ten tightly bound core electrons

which will contribute little to Po{x) for sufficiently

small compressions. Also like sodium, it appears

that the mean Fermi energy of aluminum is close to the free electron value. The band structure calculations of B. Segall, for example [36], indicate that for

aluminum, /i{,= 0.976 X the free electron value at the normal volume. Calculations for smaller volumes

have not been carried out, but here it will be

assumed that A{ stays near the free electron value

for reasonably small compressions. We will therefore use the free electron approximation for

^eorr. +^ex.

g^^^^ 0.976 of the free electron value for

A-^. The appropriate expressions are

and

^^=109(734.35/a;2/3) ergs/g (39)

^eorr.+^ex. ^jQ9|-_j7o. 755 + 91.283 In (2.9849x'/3)

-59.162/a;1/3-11.447x'/3] ergs/g (40)

where again x=vlvo, and for aluminum [30]

Table 3. Theoretical and experimental 300 K isotherms and values of A7/(T/300) as a function of

X for sodium.

1.06

1.05 1.04

1.03

1.02

1.01

1.00 0.96

0.92 0.88

0.85

0.80

0.76

0.72 0.68

0.64 0.60

P(kbar) P(kbar)

Theoretical

-1.034

-0.426 + 0.206

0.883

1.591 2.323

3.081 6.518

10.637 15.569

19.92 28.671

37.368

47.991 61.061 77.280

97.604

Experimental "

AP(kbar) A->'/(r/300)

Approximate

experimental

error

1.51 ±0.1

2.19 ±0.1

2.92 ±0.14

6.16 ±0.20

10.07 ±0.20

14.98 ±0.30

19.50 ±0.30

(Dimension

less)

0.290

0.275

0.260

0.250

0.240

0.229

0.215 0.176

0.144

0.115 0.090

0.067

0.050

0.033 0.019

0.008

-0.003

" Reference [11].

Table 4. The theoretical and experimental Hugoniots of sodium

X n(kbar) P/,(kbar)

Theoretical Experimental "

0.7341 48.64 47.08

0.6762 71.06 70.47 0.6517 83.61 83.35

0.6404 89.67 90.54

0.6094 113.31 112.77

« Reference [35].

j;o = 0.36945 cm^g. These expressions are somewhat

different than eqs (6) and (7) for sodium. This is because, unlike sodium, aluminum has three valence

electrons per atom. Also unlike sodium the inclusion

of ^0 the expression for Aq'\?, unnecessary

[37], as they make very small contributions to Po(x)

for low compressions (0.8 ^ x). They have therefore

been neglected. The remaining problem is that of

evaluating Af^- (in the W-S scheme).

98
Unfortunately there exists as yet no Prokofjew

potential ifriR) for aluminum which will give the

interaction between a single valence electron and

the triply charged aluminum core. Without this

interaction one cannot integrate the Schrodinger

equation to obtain A^-'^-. The situation is not hopeless, however, because near the normal volume the

pressure Pq^' for aluminum should depend primarily on the value of ^pp{R) , near R — Rg and the integral

of (fi'iR) over the atomic volume Va. This is so be- cause of the free electron like behavior of the

valence electrons in the region between cores. For values of R greater than the approximate core

radius Re, ^Zs. "As s. should behave hke 1/d„. In the

region R < Rc set i//g.s. = Rg) where is the

wiggly part of the wave function in the region of high

kinetic energy. The energy per ground-state valence

electron Ea is given by

Ea^ + 4nTj'^^ iljiipp{R)R^dR

+ — <pp{R)R^dR

Va Jrc

iPtV^ijjiR^dR (41)

2,m jo

where m is the electron mass. The pressure contribution Pa arising from is Pa = — ( lI'lTrR'DdEaldRs

which according to (41) is given by

-4^RIPa=+^\ 2rjjnlj'i<ppiR)RhlR

\,f^^^^R'dR+^^,{Rs)Rl

J I I I L 0.600 0.700 0.800 0.900 1.000

Figure 3. Comparison between values of P''^- calculated using

Prokofjew potential with those calculated using (/)(R).

Table 5. Theoretical values of Pf^- (x) and Po (x)

for aluminum

X Pus.

' 0 P„

(kbar) (kbar)

0.9922 -1210.17 5.73 0.9852 -1218.85 11.93 0.9702 -1238.37 25.29

0.9606 -1251.05 34.38

0.9461 -1269.59 49.85

0.9311 -1289.70 66.44

0.9221 -1301.32 77.63 0.8941 -1338.12 115.75 0.8705 -1370.01 152.12 0.8389 -1412.04 209.70

0.8036 -1456.74 288.94

0.7950 -1467.47 306.86

0.7820 - 1468.24 360.82

(42)

where ^'i = d^JdRs and Va = dvaldRs. All the terms

in (42) depend on integrals of (pp{R) over the volume

except one which depends on ipp{Rs). Thus it can be expected that if (pp{R) is well known

outside of the core and approximately inside the core (so that the integral of (pp{R) over vc is reasonably accurate), then Pf^- can be well estimated.

This is easy to demonstrate. If instead of the

Prokofjew potential for sodium one substitutes

<^(/?) =— (e^//?) (1 + 10e~"'*) ergs and adjusts a

so as to reproduce one theoretical value of Pq^\ it is possible to closely reproduce the theoretical Pq^' curve for sodium in the range 0.6 ^ x < 1.0. Calculations of this type have been carried out using

0 = 4.3931X 10^ cm"^ The results are shown in

figure 3, where they are compared with values of

Pg.s. derived from the Prokofjew potential. This

procedure works because near Rs (when Rg > Rc)

,

<p{R) ~ — e-IR and so is the Prokofjew potential.

When a is adjusted to reproduce one theoretical P%'^' value (calculated from the Prokofjew potential), it forces the average value o{ (p{R) in the core region

to have nearly the same value as the average of the

Prokofjew potential itself. Aside from this, both

potentials have the same value near R = 0, namely — lle'^/R. This is the procedure which will be used

to find Po'^- for aluminum. In this case, however, a

will be determined by adjusting so as to force Poix) to zero at the observed value of volume

(fo = 0.369 cm^g). For aluminum, therefore, the

valence electron-core potential is taken as

(^(«)=-(e2/i?)(3+10e-^) ergs (43)

where the three accounts for the triply charged

aluminum core. To eq (43) one must add the potential contribution due to the presence (on the aver- age) of two other valence electrons in the volume

Va. For a first approximation one can assume a

99
charge density of —ejva per valence electron. In

this approximation one must add to eq (43) the potential [38]

= [m-R'] ergs. (44)

Using eqs (43) and (44), P^-^- was calculated for

aluminum in the manner already described. The

parameter a was found to have the value

q: = 4.3459 X 10"^ cm~'. The results are given in

table 5. The total pressure Po(^) is also given in

this table.

3.2. The Calculation of Pq h.

Unlike sodium aluminum is not well above the

Debye temperature do under normal conditions.

For aluminum [39] do =^ 395 K. For T < dp the

contribution of eq (2) to Pq.n. is

.3// 3/1

/'q.h. = (Polx) 2 y?ei + ipolx)^ 7?(A/;/2) (45)

(=1 /=!

where e,- = /i/i/ [exp {hfilkT) — !] is the mean

thermal energy of the ith mode. If one defines

3/) 3/! y{x, T) = ^yieil^ei,

i=t 1=1

then eq (45) may be rewritten as

3//

/'q.h. = ipolx)y{x, T)Et+ (polx) 2 yUhm) (46)

where Et is the thermal energy per gram

/ 3/1 \ yle., Et = ^

etj.

Barron [40] has pointed out that in most cases

for T > O.SOo. y{x, T) is practically independent

of T. Since the calculation here wiU be limited to T > 0.300 then the first term on the right in eq

(46) will be viewed as a product of Et and a function

of X only (i.e., {polx)y{x)Er). The second term on

the right of eq (46) is the contribution P^' of the 0 K

vibration to the pressure. In a sense this has already

been included in the calculation of Po{x) since the

latter function was forced to zero at the observed

value of volume at 0 K. At the observed value of

volume (i.e., x= I) it is the sum Po{x) + P^j which

should be zero. Thus, in effect, Po has been absorbed

into the calculation of Po, and for aluminum the

calculated Poix) should be approximately equal to Poo{x). Consequently the second term on the right

of eq (46) should not be included in the calculation

of /'q.h. for aluminum, and therefore when

O.SOd < T < 6d one should expect that

/'q.h. = {polx)y{x)ET. (47)

The evaluation of y{x) is somewhat simpler in

the case of aluminum. This is because for aluminum

one can expect Pa.h. — 0 (this point is discussed in

the next section) and therefore Pr — Pq.h.- This in

turn implies that y{x) can be obtained directly

from eq (C-7) (see Appendix C) if one sets Ay = 0

and F= 1. In other words one should expect that

for aluminum

P„ = {Po + ipoy{x)lx)[Ao(xi) -Aoix)

+ {ET)i]}l[l-yix)ixi-x)l2x] (48)

where Xi — Vi/vo, vx is the normal volume for aluminum [vi = 0.3740 cm^/g) and {EtIi is the specific

thermal energy under normal conditions [41]

[(£r), = 1.636 X 10^ ergs/g]. Since Pn can be

obtained from experimental data and since

Ao{xi) — Aq{x) — Po{x)dx [where Pq is Po Jx

given in table 5 or by equation (52)], then yix) can

be calculated directly from (48). Calculations of this

type have been carried out earlier by P. McKenna

[43] . It should be noted that the latter computations

were effected in such a way as to compensate for

the effects of yielding on P/, at low shock pressures.

The results of these calculations are given in table 6.

Table 6. Calculated values of yo(x) and Pq.h. for

aluminum.

X y Pq.h.

300 K

(kbar) 0.98 2.15 9.63

0.96 2.06 9.05 0.94 1.93 8.41 0.92 1.78 7.61 0.90 1.58 6.84

0.88 1.38 6.12

0.86 1.23 5.45

0.84 1.12 4.90

0.82 1.09 4.52

0.80 1.00 4.34

The latter data can be represented accurately by

an expansion of the type

y{x)^yo + a{l-x)+b{l-xy + c{l-x^) (49)

where yo = 2.302, a =-3.86, 6 = -56.13, and

c = 215.4. Equation (49) reproduces the data of

table 6 to within 10 percent for a given x. To complete the expression for Pt it remains to evaluate Et{x, T). For this the Debye approximation [24] wiU be used according to which

100
Et{x, T) = (2.771 X W)eD{TldDr \ {rV(e''

-l)}c?yergs/g. (50)

The Debye temperature do has been determined

experimentally for aluminum by Raimondi and

Jura [39] . These authors have found that

ep = 395 exp {- 1.78(0.9877 v-l)}K. (51)

By substituting eq (51) into eq (50), Et can be calculated as a function of x and T. Values of h calculated at 300 K using eqs (47), (49), (50), and

(51) are listed in table 6 as a function of x.

3.3. The Smallness of Pa.h.

As has been mentioned earlier, the relative small- ness of Pa.h. is indicated by the smallness of

{d In C44/3 In 3 )j- relative to unity. For sodium this

derivative is ——0.3, which leads to a contribution

of about 20 percent to Pt (the thermal pressure) at low pressures. For aluminum the experimental

data [43] indicate that {d In d^ld In T)^ --0.03.

This is a factor of 10 less than the value for sodium,

indicating that C44 depends far more on volume than

temperature. One should expect then that for aluminum Pg.h. is quite small andPr ~ P^.h.- Assuming Pa.h. — 0, the P, v, T equation of state

for aluminum above the Debye temperature is then given by P = Po(x) + Pq.h.. The theoretical

room temperature isotherm of aluminum is given

in table 7, where it is compared with two p-v

points taken from Bridgman's [44] data, which

have been corrected according to a suggestion of

Jamieson [45, 46].

For convenience we have fit (by the method of

least squares) the calculated values of Po{o(,) to an expression of the type

Pa{x)=K{ exp[2fe(l-;c'/3)]

-exp [6(l-^i/3)]}/^2/3 (52)

This expression eq (52) fits the calculated points in

table 5 quite well (to within 0.65 kbar) in the range

(0.8 ^ < 1.01) when 71 = 627.1 kbars and b

Table 7. Comparison of the theoretical 300 K

aluminum isotherm with corrected" Bridgman''

data.

X (Theoretical)'^ Corrected

experimental

P(kbar) P(kbar)

1.01245 0.00 0.00 1.010 2.55

I.IMJU 1 A 1 1 lU. 1 1

1 7 Q7

0.980 26.34

0.973 32.53 29.0 to 30.5 0.960 44.51 0.950 54.39

fiA An

/o.yo

0.920 87.69

00. y4 o/.D to 00.0 0.900 113.38

0.880 142.33 0.860 174.95

0.840 211.78

0.820 253.39

0.800 300.41

" References [45] and [46].

" Reference [44].

^ Using eq (52) for PoW. = 3.7721. Equation (52) should be recognized as the

result of assuming that ^o(.v) is given by a Morse

function.

In the range (0.9 ^ ^ 1.01) Pq^- is represented

accurately by the analytic relation

PI' = - 100 In [^ + + Ce-^'^] - 1057.8/;k4/3 kbar (53)

where ^ = 26.9302, P = - 16.4253, C = - 208.096,

and £) = 3.4857. In the range (0.8 ^ ;c ^ 0.9) one must add the correction — 4131.4 + 9378. 5x — 5322 x^ kbars to P^'- to obtain an accurate representation of the calculated values.

Appendix A. Analytic Form of

Since Po — — podAoldx and Ao{x) is the sum of

several terms, it follows that Po is the sum of terms,

each of which corresponds to one term in the expre?*

sion for Ao{x). Thus,

Po{x) = P|-^- + Pf+P^^''- + P^""- + P« + P^'.

(A-1)

le Sodium 0 K Isotherm i^^ (;»:)

Taking the appropriate derivatives of the terms in

Ao, one obtains (units are kbars)

pg s. = _ 146.8015M3 + 203.4154z[l - (1 + yx^'^

+ y2^2/3/3)g-j/x./3]/^2 (A-2)
where z = 0.43258 and y = 6.71;

P/-54.9861M3. (^_3)

Pf,'"-'-- + Pl-^- = 13.9118M3 - 12.0602/X+ 1.9756M3;

(A-1)

n = 461.89{exp[5.07(l -2.061Lri/3)]}/.r2/3;

(A-5)

and = - 0.2404/:t3- (A-6)

To obtain Poo(x) one must add to Pq{x) the vibrational pressure PjJ at 0 K

Pg = 0.5435/;c'««. (A-7)

Appendix B. The Calculation of ^a.h.

According to eq (34),

Pa.h. = -po(^r^) =2.1953(Ay/:t)(r/300) kbar

where

Ay= ju. ^7o - To

, d In C«,

In (r/300)/c«4

(B-2)

An expression for ^a.h. can be obtained quite simply by integrating eq (B-1) with respect to x. This

results in

^a.h. =- 10«(37.594) (jmWi,\C\, + q{T) ergs/g

(B-3)

where q{T) is a general function of T only. The

function q{T) introduces either a constant or a pure

function of T into the energy through the relation

E

dT

But general theoretical calculations of first order

anharmonic effects in solids indicate that there is no pure function of T or non-zero constant which

contributes to the anharmonic part of the energy [1]. Therefore for purposes of this development q{T) can be set equal to zero and for^a.h. one can use the

expression

^a.h. = - 10^(37.594) (r/300)2y;/C»4 ergs/g.

(B-i)

Appendix C. Calculation <

The shock Hugoniot is the locus of pressure

volume points obtained through the shock compression of a solid. For a sohd which is initially in

the normal state, the stress, (Xh, specific energy.

Eh, and compression, x, of the material behind a

steady-state plane shock wave are related to the

initial conditions through the exact conservation

relations [34]

(Th = PoUsUp/xi (C-1)

x^XiH- Up/Us) (C-2)

Eh = Ei + Xio-h ( 1 — x/xi ) /2po (C-3)

where Xi — Vilvo, vi is the normal specific volume,

Ei is the initial specific energy. Us is the velocity of

shock propagation, and Up is the velocity of matter

in the shocked state. When the yielding stress of

the solid is small compared to CTh, then cTh is (for all practical purposes) equivalent to a hydrostatic

pressure Pn- In this case eq (C—3) becomes

En ^Ei + XiPh ( 1 - xlxi ) /2po. (C-4)

the Theoretical Hugoniot

But Ph should be given by the sum Pq + Pt or in the case of sodium

Pa = Po(x) +3.293 (yo + A7)(n/300)/xkbar (C-5)

where Th is the temperature in the shocked state. With eq (37) one can substitute for {Th/300) with

the expression

109 X 3.25118(^/300) = (£„ -^o)/[l

+ (2/3))Lt(n/300)(7;/C:j]. {C-6)

By substituting eq (C-4) for Eh into eq (C-6) and

then substituting eq (C-6) into eq (C-5) and solving

for Ph, one can obtain the expression

Po + po(yo + Ay) [Aojxi) -Aojx) + (£r),]/(xF)

I - (yo + ^y){xi-x)|{2xF)

where P = [1 + (2/3)/A(n/300)(y;/CS4)] and

(Pr)i is the specific thermal energy under normal

conditions [i.e. Ei = Ao{xi) + (ET)i]. For sodium
the quantity F decreases rapidly with volume at constant temperature. For example at x = 0.8 and

(r/300) - 1, F - 1.003. As a consequence, little error results in setting F equal to unity in eq (C-7)

provided Tn is less than about 2000 K. Since Th is also contained in Ay in eq (C-7) an iteration is necessary to obtain Pu. One first sets Ay = 0 in eq (C-7)

and obtains values of Ph as a function of x. Then

Th is calculated as a function of shock compression

via eqs (C-4) and (37). These same values Tn can

be used to calculate Ay, which can in turn be substituted into eq (C-7) to recalculate Pi,, and so on. This is the procedure which has been used to calculate Ph for sodium.

References

[ll See for example Leibried, G., and Ludwig. L., Solid State Physics Advances in Research and Application, Vol. 12,

(Academic Press, New York, 1961).

2] Brooks, H., Nuovo Cimento Suppl. 7, 207 (1958). 3] Born, M., and Mayer, J. E., Z. Physik 75, 1 (1932). 4] Mayer, J. E., and HelmoUz, W., Z. Physik 75, 19 (1923). 5] Fuchs, K., Proc. Roy. Soc. (London) A153, 622 (1936). 6] Bienenstock, A., and Brooks, H., Phys. Rev. 136, A787

(1964). [7] See for example Schneider, T., and StoU. E., Phys. Kondens

Materie 6, 135 (1967). [8] Wigner, E., and Seitz, F., Phys. Rev. 43, 804 (1933). See

also ref. [2]. [9] Gombas, P., Die Statistische Theorie Des Atoms und Ihre Anwendungen (Springer-Verlag, Vienna, 1949).

[10] Pines, D., Nuovo Cimento Suppl. 7, 329 (1958). [11] Beecroft, R. I., and Swenson, C. A., J. Phys. Chem. Solids

18, 329 (1961). Because of a small calibration error the pressures reported in this article should be increased by

3 percent [C. A. Swenson (private communication)].

[12] Hellmann, H., and Kassatotschkin, W., Acta Physicochim.

URSS 5, 23 (1936). 13] Wigner, E., and Seitz, F., Phys. Rev. 46, 509 (1934). 14] See ref. [2], p. 203.

15] A corrected version of the Prokofjew potential is given in

ref. [8]. [16] To obtain the cohesive energy (at x=\) the ionization energy of the free atom valence electron in (ergs/g) must

be subtracted from Aq(\).

[17] See, for example, ref. [11].

[18] Bijl, P., and PuUan, H., Phil. Mag. 45, 290 (1955). [19] White, G. K., and Anderson, O. L., J. Appl. Phys. 37,

430 (1966). [20] Pastine, D. John, Phys. Rev. 138, No. 3A, A767 (1965). [21] The elastic constants must be evaluated at 0 K because in q.h. approximation /; and /( (and therefore Ci and Ct) are assumed to be independent of T. It automatically follows

that in this approximation the temperature dependence

(at constant volume) of Cu and C44 must not be included

in the calculation of the frequencies.

[22

[23

See ref. [5].

Diederich, M. E., and Trivisonno, J., J. Phys. Chem. Solids

27, 637 (1966). [24] See, for example, Mayer, J. E., and Mayers, M. G., Statistical Mechanics (John Wiley & Sons, Inc., New York and

London, 1940). Martin, D., Phys. Rev. 139, No. lA, A150 (1965). Pastine, D. John, J. Phys. Chem. Solids 28, 522 (1967). Daniels, W. B., Pbys. Rev. 119, 1246 (1960). This can be ascertained by relating the data in ref. [23] to the theoretical relations given here for Co and C-14. [29] This can also be determined from the data in ref. [23]

.

[30] The specific volumes at 0 K for both sodium and aluminum

can be calculated using the tables of Corrucini, R. J., and

Gnieweck, J. J., Natl. Bur. Std. (US) Monograph No. 29

(1961). [31] Corrucini, R. J., and Gnieweck, J. J., Natl. Bur. Std. (US)

Monograph No. 21 (1960). [32] Quantum corrections were made by multiplying Cr given

by eq (37) by the factor [1 - l/20(e«/7')2]. See for example ref. [24].

[33] Rodionov, K. P., Sov. Phys. Tech. Phys. 11, 955 (1957). [34] For more details on the physical properties of shock

Hugoniots see Rice, M. H., McQueen, R. G., and Marsh,

S. P., Solid State Physics Advances in Research and

Application, Vol. 6 (Academic Press, New York, 1958).

[35] Rice, M. H., J. Phys. Chem. Solids 26, 483 (1965). [36] Segall,B., Phys. Rev. 124,1797(1961).

[37] This is because of the smallness of the aluminum cores

relative to the separation between them.

[38] It should be noted that this potential assigns all of the energy of interaction to the valence electron in question.

This was done so that the resulting (after integrating the Schrodinger equation) valence electron energies could be

compared with Hartree-Foch calculations on the free atom. Since each valence electron should be assigned

only 1/3 of the interaction in eq (44), after each integration of the Schrodinger equation, an amount \.2e-IRs was subtracted from the energy (i.e., two thirds of the

total interaction between free electrons in a sphere of radius /?,,). [39] Raimondi, P. L., and Jura, G. J., Phys. Chem. Solids 28,

1419 (1967). [40] See for example Barron,T. H. K., Ann. Physics 1,77 (1957). [41] The specific thermal energy under normal conditions can

be obtained from the tables in American Institute of Physics Handbook (McGraw-Hill Book Co., Inc., New

York, Toronto, and London, 1957).

[42] McKenna, P., and Pastine D. John, J. Appl. Phys. 39,

6104 (1968). [43] Lazarus, D., Phys. Rev. 4, 545 (1949). [44] Bridgman, P. W., Proc. Am. Acad, of Arts and Sci. 76,55

(1948). [45] Jamieson, J. J., A discussion in High Pressure Measure- ment (Butterworths, Washington, 1963). [46] Jamieson gave the separate volumes of NaCl and Al (in a mixture of NaCl and Al) measured accurately at two

different pressure levels. Detailed calculations on the equation of state of NaCl (based on shock data) have been

done by J. Fritz of Los Alamos, who has kindly given us

the pressures corresponding to these two levels.

DISCUSSION

R. Grover {Lawrence Radiation Laboratory,

University of California, Livermore, California): For

sodium, there appears to be a lack of agreement

between shock data, Bridgman's compressibility

data, and equation of state values. I believe that

some additional experimental or theoretical work

would be desirable to resolve this.

L. Thomsen {Columbia University, Palisades,

New York): You indicated that the pressure correction Pa.h. (eq 34) arose from the temperature dependence of the eigenfrequencies. Now, the eigenfrequencies for all eigen vibrations are rigorously

defined as functions of volume only, so this Pg.h.

is "correction" in which you made a two-mode

103

385-762 O - 71 - 8
approximation for gamma. To make this correction

you introduced the velocities of sound. By introducing these velocities of sound you introduce a pseudotemperature dependence of the eigenfrequencies

which is then subtracted out again with your

Pa h

. Have I followed your logic correctly?

O. Anderson {Columbia University, Palisades,

New York): From a theoretical point of view, which

do you think we can rely most on in the long run: equations of state of aluminum, or of sodium

chloride? Would you find that the equation of state

of sodium has advantages theoretically over

either of these?

R. H. Wentorf, Jr. {General Electric Research

and Development Center, Schenectady, New York):

Could calculated isotherms be readily calculated

AUTHORS'

In response to Dr. Gravers comment: The lack of

agreement with Bridgman's data depends to some

extent upon which of Bridgman's data comparison

is made with. There is reasonably good agreement

with his 1945 data, as indicated in my paper [Phys.

Rev. 166, 703 (1968)]. There is a later set of

Bridgman's data, however, with which agreement

is poor.

Concerning Dr. Thomsens comments: It is, of

course, true that eigenfrequencies are rigorously

functions of volume only. One can't have eigenfrequencies if the modes of vibration aren't independent. As soon as you add anharmonic content to the vibrational spectrum, that is as soon as you make

the frequencies temperature-dependent, the normal

modes start interacting and are no longer independent. However, it turns out in certain cases that, to a reasonable degree of approximation, you can look

at these normal modes as having shifted in fre- quency due to anharmonic effect.

for sodium and aluminum at, say, 500 K and 1000 K,

for possible use of these materials as high-pressure,

high-temperature reference materials?

G. E. Duvall {Washington State University, Pullman, Washington): I'd hke to make a comment

which bears on this subject, remotely at least.

For several years R. M. Rosenberg* at the University of California has done some remarkable

work on the properties of lattices which have nonlinear forces connecting them. He has discovered

some quite remarkable things about normal mode

and eigenfrequencies. One of the things that astonished me was that there is not necessarily any cor- relation between the number of normal modes and

the number of degrees of freedom in a system, once the linear force assumption is violated.

CLOSURE

In response to O. Anderson: The question of

whether equations of state are to be preferred

for sodium, sodium chloride, or aluminum is a

difficult one. I accept the accuracy limitations

placed on the equation of state by Fritz and Decker.

Perhaps the choice depends upon the circumstances in any particular case. If a very compressible substance is desired, use sodium; if a

relatively incompressible material is needed,

aluminum might be chosen. In the case of sodium

chloride, distortion of the lattice probably occurs under pressure, and thus the actual pressures will

not be the same as those predicted by the equation

of state. This is not as likely to happen with sodium,

because the pressures depend almost entirely on

the volume.

In response to R. H. Wentorf: The isotherms at elevated temperatures are easily calculated. The

1968 paper referred to above describes calculation

of any isotherm.

*On non-linear vibrations of systems with many degrees of freedom. Advances

Appl. Mech. 9, 155-242 (1966).

104
All Atomistic Theory of Shock Compression of a Perfect Crystalline Solid

D. H. Tsai

National Bureau of Standards

Washington, D.C. 20234

Lattice dynamical calculations have been employed to study the process of compression of a

perfect, two-dimensional, fee lattice by a strong shock wave. The interaction energy between the'

lattice points is assumed to be the Morse-type potential function, and the interaction is assumed to extend to the fourth nearest neighbors. The formulation of the dynamical problem is described. The

details of the computation of the shock wave stress and density profiles, shock velocity, energy density

profile, local energy distribution, velocity distribution, and the components of the Grueneisen parameter are discussed. Of special interest is the thermal relaxation process behind the shock front. It is found that the relaxation time is not constant, and that the steady, relaxed region in the shock profile

trails farther behind the shock front with increasing time. The implications of these results on the

calculation of the high-pressure equation of state of a crystalline solid are examined.

Key words: Equation of state of solids: lattice dynamics; shock compression; shock wave; thermal

relaxation.

1. Introduction

The study of high-pressure phenomena in solids

has become important in recent years in connection

with geophysical investigations of the earth's interior, studies of the cratering phenomena on terrestrial and lunar surfaces from meteoritic impact or from nuclear explosion, research and development

in ballistics and high explosives, industrial manufacture of new high-pressure materials such as

synthetic diamonds, and other related problems of

interest to physics, geophysics, and technology. In

all of these studies, a central problem is the equation of state of solids under high pressure and high

temperature. In the laboratory, pressures up to several megabars, accompanied by temperatures

of several thousand K, can be obtained by compressing a material by means of a strong shock wave. Such a shock wave may be generated by detonating

high explosives placed next to the material, or by

impact of a high-velocity projectile on the material,

or by a combination of the two. These methods

have been employed extensively in the past 20

years to study the shock compression of a variety

of solids. From the dynamics of the shock wave

propagation, and with suitable assumptions, the

equation of state and other thermodynamic properties of these materials under high-pressure conditions have been obtained [1 , 2, 3].'

We would like to discuss briefly the basic assumptions which are usually made in these studies. For

the interpretation of the dynamics of the shock wave,

two assumptions are usually employed: First, the

compressed material behind the shock front is assumed to be in thermodynamic equilibrium. This

implies that the shock stress profile is also steady in

' Figures in brackets indicate the literature references at the end of this paper.

Paper presented at the Symposium on Accurate Characterization

of the High-Pressure Environment, held at the National Bureau

ofStandards, Gaithersburg, Md., October 14-18, 1968.

time. Second, the compressed material is assumed

to have yielded completely, so that the stresses may

be assumed to be hydrostatic. This is based on the

observation that the stresses behind the shock front are usually many times higher than the yield strength

of the material. These assumptions essentially re- duce the solid to a fluid, obeying the continuum

theory of fluid dynamics. With these assumptions,

and with the aid of the Hugoniot relationships of

conservation of mass and of momentum across the

(steady) shock front, one easily obtains the pres- sure and density behind the shock front from the

measured shock velocity and the particle velocity

behind the shock front. An additional assumption

giving the relationship between the Grueneisen

parameter and density is then employed, and this

directly allows the calculation of the PVT relationship for the material behind the shock front. The

equation of state obtained this way is the MieGrueneisen equation of state. A closer examination of these assumptions reveals

a number of questions. In the first place, the yield

strength referred to above is the static yield strength

of the material. Under dynamic conditions, with the

shock wave propagating at a supersonic velocity,

the yield strength of the material probably is much

higher, perhaps approaching the theoretical strength.

In any event, yielding is known to be a strain-rate

dependent phenomenon. If the material behind the

shock front yielded to a hydrostatic condition, then

it would be important to know the density and the

stress histories behind the shock front before one

could apply the Hugoniot relationships. The problem

of time-dependent yielding under the transient condition of shock compression is very complex. To our knowledge, this problem has not been exhaustively

studied either experimentally or theoretically in the

high-pressure regime.

In addition to the yielding problem which may

affect the stress profile behind the shock front, there

is a basic question in the continuum assumption.

A solid may be considered as a material made up of

discrete particles (atoms) bound together rather in a

105
regular pattern by attractive and repulsive interatomic forces acting at close range. It is the purpose

of lattice dynamics to study the properties of such

a solid on an atomistic scale, and to relate the microscopic properties to macroscopic properties. On

this subject there exists a vast and distinguished

body of literature. From a lattice viewpoint, the

continuum approach would be valid if the wavelength

of the shock wave were long compared with the

lattice spacing. For waves of shorter wavelength, the

discrete nature of the atoms causes a dispersive

effect. Now a shock wave has high- as well as lowfrequency components. Also, because of the an- harmonicity in the interatomic forces, there is coupling among waves of different frequencies:

for example, the wave front of a low-frequency compression wave has a tendency to steepen in the course of propagation and to generate high-frequency components. The effect should be especially

pronounced in waves of large amplitude. Thus as the

shock wave propagates into the soUd, the dispersive

effect should progressively cause the high-frequency

components to separate out from the low-frequency

components, so that the stress profile of the shock

wave perhaps should change continuously with time.

If this be the case, one would not be justified in

using the Hugoniot conditions to calculate the

stress and the density behind the shock front, without first investigating the process of approach to equilibrium.

A few authors [4, 5] also have studied the theoretical problem of the shock wave structure in a

solid from a continuum viewpoint. They show that

if a dissipative, viscous effect in the solid is postulated, then a steady shock wave proHle is obtained.

The thickness of the shock front in such a profile

increases with increasing viscosity. In our opinion,

these theoretical results are not satisfactory, be- cause the physical nature of the viscous effect is not

clear. As the viscous effect is reduced to zero, the

theoretical solution produces a discontinuous shock

front, but without the effect of dispersion.

The foregoing considerations have motivated us to study the process of shock compression of a solid

from a more detailed, lattice dynamical viewpoint.

In two earlier papers [6, 7], we have investigated the

propagation of a strong, one-dimensional shock wave

in a semi-infinite cubic lattice, with simple cubic,

fee or bcc configuration. Our main purpose was to study the role of the interatomic forces and the range

of their interaction on the dynamics of the shock

wave. For our purpose, we assumed different interaction energies ranging from the simple harmonic

type of potential between nearest neighbors to the

more complicated Morse-type potential with interactions extending to more distant neighbors. We

further restricted the motion of the lattice points to only one- and two-dimensional motions by assuming

certain symmetries of the relevant interatomic

forces. Finally, we assumed that quantum mechanical effects were unimportant at high temperatures,

and considered the lattice to be a simple system

obeving classical mechanics. Under these assumptions and some others Hsted in our earlier papers

[6, 7], we studied the dynamic response of the

lattice following an impact at the face of the semiinfinite lattice model, by solving numerically the

equations of motion of the lattice points. From the

solutions we then obtained the density and stress

profiles of the shock wave, as well as other properties such as the average energy density and the

Grueneisen parameter in the compressed lattice.

Our earher papers were rather brief, because of

editorial requirements. In this paper, we would hke

to present a more complete account of our ca' ulations, and to discuss additional results from our

investigations of the lattice energy, thermal stresses,

etc., with a view to examing in greater detail the

problems of studying the equation of state of a shock

compressed lattice by numerical computation, es- pecially the problems of temperature rise behind the

shock front, and of thermal relaxation.

It may be noted that our numerical approach to

this problem is basically the same as that employed

by Fermi, Pasta, and Ulam [8], Alder and Wainwright [9], Northcote and Potts [10], Rahman [11],

and others, in their investigations of the dynamic

behavior of lattices, sohds, and liquids. Recently,

Payton, Rich, and Visscher [12] have also studied

the thermal conductivity of a lattice by numerical

computation. The scope of this line of investigation

is therefore quite broad. Numerical solutions suffer

from rather well-known limitations. In our present

problem cost considerations further restrict us to studying lattices of small sizes, and over short

time intervals. From these solutions, it is not

easy to develop a broad understanding of the various

phenomena we wish to study. On the positive side,

it is clear that in the absence of a more powerful

mathematical theory which treats large anharmonic

and perhaps also anisotropic effects in a large system

of oscillators closely coupled in a complicated manner, we must continue to rely on the numerical solutions to provide insight to our problem. Fortunately,

in the approximation of classical mechanics, our problem is fairly well adapted to numerical computation, and the results are often not difficult to interpret on physical or intuitive grounds. Thus if the

validity of the lattice model could be established,

then the numerical results clearly would be useful

for the interpretation of experimental data, as well

as for the development of mathematical theories.

In section 2, we outline the formulation of the

mathematical problems of calculating the dynamics

of the lattice under shock compression. In section 3,

we discuss the numerical method of solution. In

sections 4 through 9, we discuss the calculation of

the properties of the lattice from the numerical solutions thus obtained. These include the density and

the stress components, shock velocity, energy

density and energy distribution, the problem of

thermal relaxation, velocity distribution and the

problem is fairly well adapted to numerical compuof the Grueneisen parameter. In section 10 we give a

106
summary of the results and discuss our view on the

problem of shock compression of solids in the light

of our results.

2. Formulation of the Numerical

Problem

In references [6 and 7] , we have described briefly

the formulation of the problem of the shock compression of a semi-infinite lattice constrained to one- and two-dimensional motions. In both cases, our problem was formulated as an initial-boundary value

problem in a cartesian coordinate system which was

stationary with respect to the uncompressed lattice.

In the present report, we give a more detailed, alter- native formulation of the same problem, in a car- tesian coordinate system stationary with respect

to the compressed lattice behind the shock front.

This resulted in a slight simplification in the calculation of some of the properties of the compressed

lattice. In every other respect, the two formulations

are identical, because the relative velocity of the

two coordinate systems, Up, is constant.

We limit ourselves here to the case of an fee,

monatomic lattice.^ The 100, 010, 001 directions are

alined in the x, y, z directions respectively, and the

face of the semi-infinite lattice is normal to the x

axis. The lattice is constrained to two-dimensional

motion, in the x and y directions, the motion in the

z direction being identically zero, because an exact

balance of the z component of the interatomic forces

would foUow from the assumed symmetry in the z direction. If there is also symmetry in the y direction, then the motion is reduced to the simple, one- dimensional case. We assumed that the interatomic forces were

central forces, and that the interaction energy (/) between a pair of isolated atoms was given by the

Morse potential function:

(/) = Z)|exp 2aro( 1

2 exp aro ( 1 (1)

where D is the dissociation energy of the atom pair, a is a constant related to the "stiffness" of the potential, r is the separation distance of the atom pair,

and To is the separation distance at the potential

minimum. If we define a "stiffness parameter" Ri

as that value of R (= r/r„) which makes 0 equal to zero, then aro = ln 2/(1 — In the present paper,

we assumed /?i = 0.7. Earlier, we have also set R\

equal to 0.8 and 0.9. In keeping with the treatment

of lattice vibrations of crystals in the literature (see,

for example, the review article by Slater [13]) we

assumed that there was no damping in the interactions between the lattice points.

Figure 1. A face-centered cubic lattice showing lattice point

OA and some of its neighbors.

The unit cell with dimension 2re in each direction, and with OA at a comer, is shown

in heavy outlines. \A and similar points in lattice plane A are the nearest neighbors

of OA in plane A. Similarly for 2A, iA in lattice plane 16, 3B in lattice plane B; and

2C, 3C, 4C in lattice plane C, \B' is in lattice plane B' which is on opposite side of OA

from plane B. There is also a plane C on opposite side of OA from plane C. The neighbors in B' and C except for 15', have been omitted in order to keep the figure simple. The unit area is formed by the perpendicular bisectors oi OA and \A in plane A, and

IS used in the calculation of stress across plane A.

To simplify our calculations, we assumed that the

interaction engeries were pairwise additive, so that

the total potential energy of an atom is given by

<I> = (2)

^ We have also studied the bcc lattice in two-dimensional motion under comparable

conditions, but we shaU not specifically refer to the bcc results except in figure 5.

where j = l, 2, . . . denotes the various neighbors

with which the atom interacts, and the factor 1/2

arises because each is shared between this atom

and its neighbor. For our fee lattice, we assumed

that the interactions extended to the fourth nearest

neighbors (fig. 1). These neighbors are in lattice

planes two lattice spacings away from the atom in

question, and according to the results of reference

[6], with the Morse potential and /?i = 0.7, neighbors

farther than the fourth nearest neighbors, lying in

planes three lattice spacings away, would have no

appreciable effect on the propagation of the shock

wave in the lattice.

We obtained the equilibrium configuration of the

lattice by equating d^jdr to zero, and solving for the

equiUbrium lattice spacing r — Ve based on interactions extending to the fourth nearest neighbors.

For Ri — 0.7, re was found to be 0.5313 ro. The cor- responding force constant in the x direction was

A = 10.5437Ao, where ko^ {d-(t)ldr-}r=ro = 2dW is the force constant for the iolated atom pair [6]. The

longitudinal velocity of sound was c=/v(A/m)^/^ = 1.7252 Co, where m is the mass of the atom, and

Co=ro(Ao/m)'/^ is the longitudinal volocity of sound in a chain of atoms spaced at ro apart, and interacting

with nearest neighbor interactions given by (1). We

assumed that these properties extended throughout

the semi-infinite lattice, and we ignored the surface

effect near the face of the lattice.

107
Equations of motion . Under dynamical conditions,

the force between an atom and its nth neighbor is given by

ifx)n=— {a^(f)/^r)„^

{fy)n = -{l3d<Pldr)„\

where fx and fy are the x and y components of the

interatomic forces, a and are the cosines of the

direction angles between r and the x and y axes,

respectively. The total force on the atom is given by

(3) summed over all the relevant neighbors,

^ _

"

. (4)

n

The equations of motion of the atom then become

Tnx — md'^xldT^ = ^x^

my=md'^yldT'- = ^y]

where x and y are respectively the coordinates of the

atom in question in the cartesian coordinate system.

Boundary conditions: For simplicity, we assumed

that the x-y lattice plane was divided into strips

running in the x direction, each strip containing rows (A'^= 10 to 50) of lattice points in the y direction.

For the boundary condition in the y direction, we

assumed that the solutions in two neighboring strips

were mirror images of one another about the common boundary, and that the solution in the entire

x-y plane was pieced together with these alternating strips of solutions. Actually, two sUghtly dif- ferent conditions were used along the y boundaries:

(a) In the majority of cases, we assumed that the

mirror planes contained the first and the Mb row of

lattice points, and the y components of the velocities

of these points, therefore, were always equal to zero. The lattice points in the other rows of two neighboring strips were symmetrically disposed about

the common boundary, (b) In some cases, we also

moved the mirror planes one-half of a lattice spacing outward, so that the Mb row of the lattice

points in one strip and the first row of the lattice

points in the adjacent strip were mirror images

of one another, and similarly for the (A'— l)th

row in the first strip and the second row in the

next strip, etc. This was done in order to investigate the effect of the boundary condition on the

details of the numerical solution, especially the

velocity distribution.

The boundary condition described above is equivalent to assuming that the boundaries in the y direction are rigid and perfectly reflecting. An incident

wave /„ impinging on the boundary is reflected as wave Ra of the same strength, Ra = — Ia- Similarly, an incident wave h impinging on this boundary from

the other side is reflected as Rb = — Ib- K Ia = ~Ib,

then Rb = Ia and Ra = Ib- Thus an alternative interpretation of the rigid boundary condition is that as an incident wave /„ reaches the boundary, it passes

through the boundary without change as Rb., but

l(, is always accompanied by h^ — Li approaching

the boundary symmetrically from the other side, and

passing through the boundary as /?&. This situation

is analogous to the "cyclic" boundary condition

which is often employed. In the cyclic case, we

assume that as 1(1 approaches the boundary of a

lattice strip, Ib = Li simultaneously approaches the

opposite boundary of the strip, /& and /« have the

same direction, and cross the boundaries at exactly

corresponding points.

Both the rigid and the cyclic boundary conditions

are physciaUy unreaUstic. Dean [14] has discussed

the effects of these boundaries, as well as the effects

of the "free" boundary, on the vibrational frequency

of a lattice. He showed that for a large lattice, the

effects of the different boundaries on the frequency

spectra were small, approaching zero as the number

of lattice points approached infinity. In several

cases, we have compared the solution of a lattice

with ^^=10 and that with A^=50. The solution in

the central strip (also 10 spacings wide) of the lattice

with A^=50 differed in minor details from the solution of the lattice with A'^= 10, but the time-averaged

dynamical behavior of the shock wave was not at all affected. With A'^= 10, the boundary conditions (a)

and (b) resulted in some difference in the velocity

distributions in the transverse direction. However,

with A'^=50, this difference became very small.

Thus our results were in agreement with Dean's

analysis, and the lattice with A'^= 10-50 appeared

to be wide enough for our purpose.

In the x direction, we assumed that the boundary

condition at the face of the semi-infinite lattice was

the same as that of (b) described above. As the shock

wave propagated into the lattice, we terminated our

calculations of the shock wave profile when the

shock wave no longer produced an appreciable

disturbance to the lattice points ahead of the shock

front. We have also tried to isolate a piece of the

shock compressed lattice, in order to study the problem of approach to equilibrium. This was done by

replacing the boundary condition at the shock front

with the boundary condition (b) inserted somewhere

in the shock compressed lattice, and studying the

time-development of the oscillations of the lattice

now isolated by boundary conditions (b) on all four sides.

Initial conditions: To simulate thermal motion in

the lattice before shock compression, we initially set the lattice in transverse motion by first displacing one or more of the interior lattice rows in the y

direction from their equilibrium positions, and then

allowing the lattice rows to oscillate freely as they

sought to reestablish equilibrium. With the assumption of central forces, it was necessary to have lat- tice motion in the y direction to provide a mechanism

for scattering longitudinal waves generated by the

108
—I w

' 0

X= TIME

/ / IMPACT

/ /

/ /

/ BEGINS

IMAGEAVERAGE SPACING OF

COMPRESSED LATTICE

AFTER COMPRESSION

"^SHOCK FRONT

BEFORE COMPRESSION

TRAJECTORY OF LATTICE POINT

EQUILIBRIUM LATTICE SPACING

SEMI-INFINITE LATTICE

ax/dT= - Up

Figure 2. Schematic diagram in the x-t plane showing the events near the beginning of

impact of a semi-infinite lattice with its image.

shock wave in the transverse direction. We limited

the "thermal" oscillations to the transverse direction purely for reasons of economy. In this way, it was necessary to calculate only the one-dimensional

motion of plane waves in the y direction. As the

shock wave moved in the x direction, the shock front

would encounter the uncompressed lattice at various

stages of transverse oscillations, so that the scatter- ing of the shock wave in the y direction might be ex- pected to become somewhat random, and thus simulating thermal scattering. To test this hypothesis, we

tried several initial configurations, with different

lattice rows displaced in different directions, but

with the initial lattice energy kept constant. Our

results again showed only minor differences in the

details of the solutions, but the dynamical behavior

of the shock wave was not affected.

In addition to the transverse "thermal" oscillations, we also set the lattice in motion in the —x

direction toward the origin of the coordinate system,

with the X component of the velocity x—^Up for all the lattice points. This was to simulate the conditions

of impact: We imagined that the mirror image of this

lattice was simultaneously (and symmetrically)

moving in the +x direction toward the origin from

the other side, and with a uniform velocity +Up, and

we considered that the impact started when the

faces of the two lattices were one equilibrium lattice

spacing apart from one another. At this point we

allowed the atoms in the two lattices to interact. As

the separation became smaller than the equilibrium

spacing, the repulsive potentials between the surface

layers would decelerate these layers, and cause them

to stop and rebound. Similarly, the interior layers

would progressively decelerate and compress. By

this process, we caused a shock wave to be generated at the interface, and the shock would propagate

into the lattice as long as the lattice continued to

move toward the origin. Figure 2 describes schematically in the x-t plane the events near the

beginning of the impact for a simplified, one-dimensional, semi-infinite lattice.

With these initial and boundary conditions, our problem was to calculate the position, velocity, and

acceleration of all the lattice points affected by the

shock wave, by solving the equations of motion at successive instants of time, Tp, Tq, etc. From these

solutions, we study the details of the shock wave

propagation and of the energy conversion, and

calculate the properties of the compressed lattice.

3. Numerical Computation

It is convenient to work with dimensionless

quantities. In table 1, we list the dimensionless

quantities used in our calculations. Most of the

symbols have been defined earher. In addition,

Ra = aro = ln2/(l-/?i), M^ = KRaRe, o-^-^ is the

normal stress in the x direction, p is the density, and

other symbols will be introduced as they are needed.

Numerical Method. In terms of the dimensionless

variables, the equations of motion become

X=d^Xldt^ = Fa:IMi

Y =d^Yldt^ = FylMA

We solved these equations numerically by the

"improved Euler-Cauchy method" [15]: Consider

two successive steps, p and q, in the t-X-Y co- ordinate system (fig. 3). For step p, at time tp, points

Ip, 2p, etc., are in an X-Y plane which contains the

line Xp passing through the equilibrium positions of

these points in the X direction. In this plane, there

are also other points which are the neighbors of Ip,

2p, etc., but we do not show them here, in order to

109
t = TlME

X

Figure 3. Schematic diagram of the numerical procedure of iteration.

keep the diagram simple. Point Ip is also in the

surface layer of the semi-infinite lattice, and 2p,

3p, etc. are in the successive interior lattice planes.

These planes are generally normal toXp. Notice that

for our fee lattice, point 2p is actually the projection

of the nearest neighbors of Ip and 3p onto the X-Y

plane: these neighbors are physically in the 2pth

lattice plane, but they lie above and below the X-Y

plane in the Z direction. With Ip and 3p, they form a

face of a primitive cell of the lattice. Similar considerations apply to points 4p, 6p, etc. In figure 3, w^e

also show points Ipi and 2pi which are the image

points of Ip and 2p, respectively, about the plane

X — 0. The interactions between Ip and its neighbors,

including those in planes Ipi and 2pi, determine the

boundary condition at the face of the lattice.

At step p, the position, velocity, and acceleration

(X, Y; X, Y; and X, Y) of points Ip, 2p, etc., are

Table 1. List of dimensionless quantities

Dimensional quantity Normalizing

factor Dimensionless quantity

Name Symbol

Equilibrium lattice spacing re To Re = relro

Distance between lattice r=(r2 + r§)i/2 re R^rlre={R} + Riyi^

points.

Lattice point position X, y re X^x/re, Y^ylre

Lattice point velocity Ujc = x = dxldr, etc. c Ujc = X = dXldt = Ujclc, etc. Time T Tele t - - TclVe

Velocity of sound c = re(A;/m)i/2 Co = ro{kQlm)^l^ cico

Potential energy

n D Ep = ^ID

Kinetic energy efc =^

(aj + up D E, = RaReM,{U^+Ul)

Force ^x =-^ iad<f)ldr)n, etc.

n 2aD Fjc^.^xl2aD, etc.

Force constants k ko = 2aW K^klko

Stress (see text) (Txx, (Tyy, etc. Sxx = a-xxlpoc^, etc.

110
known, either from previous calculations, or from

the prescribed initial conditions. Step q follows step

p at an interval Af later, tq = tp + At. At time tq, the

line Xp has moved to the new position Xq, and Ip,

2p, etc., to Iq, 2q, etc., respectively. Our problem is to determine the position, velocity, and acceleration

for points Iq, 2q, etc. As a first approximation, we assume

Xi^^ = Xnil-, yd):

-'y(i)=

nq 1ylip

X<H'^=X„p + AtX„p + i {Atyx„p

lip

rt= 1,2, (7)

The superscript (1) indicates the value of the first iterate. The subscripts np and nq for n=l,2, . . . correspond to the lattice points in figure 3. To obtain improved values, we proceed as follows:

Let us fix our attention on particle 1. From (7) we

have the coordinates X[^^ and F*,y of particle 1 and

those of its neighbors. Note that some of these

neighbors are in the image planes Iqi and 2qi, be- cause particle 1 is next to the boundary at ^^ = 0. If

particle 1 is also next to a transverse boundary, then

it also has neighbors in the image planes across the

transverse boundary. The coordinates of these

neighbors in the image plane are obtained from (7)

by reflection. From these new position coordinates,

we calculate new pairwise force components on

particle 1 from (3). We then obtain the new total

force components by summing the pairwise components over aU the relevant neighbors according

to (4). These are F'/i' and in dimensionless

form. The superscript (2) denotes the second iterate.

From (6) we have

V(2): ^ iq

y (2) : iq

^ XI, xlqlM^

y^qlM :} (8)

These are the improved values of the acceleration

components for point Iq. From (8), improved values

of velocity and position may be obtained:

A:i|'=j,p+iAi(;f,^+l(2,))

X\l^=X,p + iAt{X,p+X%^)

(9)

We apply equations 3, 4, 8, and 9 repeatedly to all the

lattice points, starting from Ig in the F direction to cover the width of A'^ rows, and then from 2q, . . ., 3g, . . ., etc., always using the newest available

values of Fj., Fy; X, Y; X, F; and X, Y. In this way,

we extend the calculation in the X direction until

we pass through the shock front and reach the

region undisturbed by the shock wave (see fig. 2). To test this condition, we require that

\X^^^— 1)1 < C, for two successive ra's

(9a)

Here C is a prescribed tolerance. This completes

the second iteration.

In the undisturbed region, there are of course the

free oscillations in the F direction set up initially to simulate the thermal energy in the lattice prior

to the arrival of the shock wave. The iterative procedure for computing these thermal oscillations is identical to the procedure just described for the

shock compressed region, except that we are con- cerned here only with the iteration of the motion in

the F direction, the motion in the X direction being

given by the uniform motion with a velocity equal

to —Up = — Up I c, as described earlier.

The third and higher iterates are obtained in the

same manner. The computation for step q is terminated when the maximum change in the values of

Xnq and Y„q between successive iterations is less

than the tolerance C:

n q iy(/0-

\^ nq

'^ nq -Y(knq

-1)|

1)1 <c

<c for aU «'s. (10)

Numerical Accuracy, Step Size, and Other Details

of Computation. We employed a double-precision

scheme for computing the positions of the lattice

points in an effort to improve the numerical accuracy. This was necessary because the interatomic

forces depended on the differences between the

position coordinates of an atom and those of its immediate neighbors, and when the position coordinates assume large values, small errors in these

coordinates would cause large uncertainties in the

computed forces. Our scheme consisted simply in

dividing each position coordinate into two parts: an integer part and a decimal part, and we performed the arithmetic operations separately for the

two parts, and then combined them for the purpose

of calculating the interatomic forces. In this way,

we were able to keep the same number of significant

figures (~ 7 places) in the decimal part of the position coordinates through the entire calculation. The

velocities and the accelerations of the lattice points

all were of comparable magnitudes, less than unity.

We used only the regular single-precision arithmetic

for these quantities in order to conserve storage

space in the computer.

The numerical accuracy was also affected by the

tolerance C (9a and 10) and by the step size At, their

choice being dictated by the accuracy requirement

as well as the economy of computation. We used a value of 10 ~^ for C and 0.2 to 0.25 for At. These were

found to be satisfactory in our earlier work [6, 7]. With C=10~^, 2 or 3 iterations were usually re111
quired for each step of Af. A value of — 0.2 corre- sponded to a step size of 10"'^ s, if we took the

period of a lattice oscillation at the highest frequency

to be 10~'^ s (wavelength=2/v), cf. Rahman [11].

We obtained a further check on the numerical

accuracy from our study of the problem of approach

to equilibrium of an isolated system of 25 X 25 lattice

points taken from a lattice that was previously compressed by a shock wave. For such a system, the

energy should remain constant in time. Our results

showed that after 860 steps, at Ai = 0.25, the energy

density had changed from 1.011424 to 1.010916, or

5 parts in 10^.

Apart from the above investigations, we have not

analyzed the problem of errors in our numerical

method of integration, although the need for such

an analysis clearly exists. In addition, we also should

examine other numerical methods which may be

more efficient than the improved Euler-Cauchy

method of integration. These problems have now

become important as our computation becomes

lengthier with the increased size and dimensionality

of the lattice.

The size of the lattice is determined by the size

of the computer, and even more importantly, by the

cost of computation. With a computer of 65 X 10'* words (each word = 36 bits), we could study 3,000

lattice points by our present method of computation.

In a two-dimensional array, our maximum lattice

size was adjustable from 300x10 (.A^-F dimensions)

to 60 X 50. The 300 X 10 lattice was used mainly to study the propagation of the shock wave in the X

direction, while the 60 X 50 lattice was used mainly

to study the boundary conditions and the problem

of approach to equilibrium. In both cases, we periodically recorded the computed values of position,

velocity and acceleration of the lattice point on a magnetic tape, for later analyses of the stresses,

energy density, velocity distribution, velocity autocorrelation, etc., of the compressed lattice. In the

shock wave calculations, we have thus far carried

the computation to a maximum time o{ t= 160 in the

300 X 10 lattice, with the shock front reaching about

205 lattice spacings from the face, but for most of

the runs, we stopped at i = 60. Earlier, in a one- dimensional lattice with harmonic as well as anharmonic forces, we had carried the computation to t= 1000 [6], in order to investigate the steadiness of

the shock stress profile. Finally, in the equilibrium

calculations, we have carried the computation to

^ = 250 in the isolated 25 X25 lattice mentioned

earlier, but without definitely ascertaining that

equilibrium had been established. The time required

for this last calculation was about 4 hr.

4. Density and Stress Calculations

In the fee lattice, if the unstrained equilibrium

lattice spacing is re, then the normal density po is

mjlrl, and the normal specific volume vo is 2r^ per

lattice point. Under compression from external

forces, the equilibrium lattice spacings of the deformed lattice become rej- and re,j {rpz = re in our two-dimensional case), so that the density p now is ml2rere.,rey, and the specific volume is 2rerexr,-i,. In

dimensionless terms, we define the relative density

p and relative volume V as follows

P'^plpQ^rj/rexrey (11)

V=vlvi) = rexrevlrl (i2)

These are geometrical quantities, and are easily

computed once the equilibrium positions of the

lattice points are determined. If the lattice points

oscillate, we can define the instantaneous local

density and specific volume, and their local spatial

and time averages. If the energy density in the

shock compressed region should be non-uniform,

we might expect the local average of the density and

specific volume to differ, also, from the values of

(11) and (12), since different parts of the lattice

would have different thermal expansions. However,

in the small lattice we studied, over relatively short

time intervals, we could not detect any difference

due to non-uniform energy density profile.

The calculation of the stress components in a

lattice with the lattice points at rest at their equilibrium positions has been discussed in detail by Love

[16]. The ideas are quite straightforward, and may

be summarized as follows:

Consider the unit area A (fig. 1) constructed from

the perpendicular bisectors between QA and its nearest neighbors \A. The plane of A is normal to the X axis. We wish to calculate the normal component of all the interatomic forces (up to the fourth

nearest neighbors, by assumption) which intersect

A. This is the usual definition of the normal stress over A. We denote this by ctj-xo with the subscript

o indicating that the lattice points are not oscillating.

For clarity, we imagine the center of QA (and those

of lA, 2A,^A) to be slightly to the right of A in the

X direction. Then the interactions between OA and

its neighbors to the right {\A, 2A, 4<A, IB', etc.) do

not intersect the plane of A, and need not be con- sidered in the calculation of ctxxo- The interactions

between OA and its neighbors to the left (IB, 3B,

2C, 3C, 4C, etc.) intersect A at its center, and the

normal components of the interatomic forces (eq 3)

all contribute to (Xxxo- The interaction between IB'

and IB I is like that between OA and 4C. This type of

interaction also intersects A at the center, and the

normal component therefore also contributes to

CTixo- The interaction between 15' and 1^2 is like

that between OA and 3C This type of interaction

intersects A at the midpoint of an edge of ^, and is shared between two adjacent areas having the edge

in common, therefore only one half of the normal

112
0.40 n

LATTICE POSITION

Figure 4. Shock stress profile in a one-dimensional fee lattice at t = 60, U,, = 0.1, other conditions as given for case (8) in figure 5.

component contributes to (Txxo- Finally, the interaction between IB' and IBa is like that between OA

and 2C. This type of interaction intersects A at a. corner, and is shared among four adjacent unit areas having the point at the corner in common,

therefore only one-fourth of the normal component

contributes to (Xxxo- The resultant aj-xo is the algebraic sum of all the components enumerated above.

In a similar way, cryyo and other stress components

may be calculated. We worked with dimensionless

stresses. These were obtained by dividing the stress

components by the factor poc '. When the lattice points are oscillatory, we can

calculate the instantaneous stress components ctxx

and cTyy in the same way, except now some of the

neighbors lA, 2A and 4-A may be to the left of the

plane of ^4, and their interactions with OA also have

to be considered. (Jxx and dyy now fluctuate in

time and in space, so we must also calculate the

appropriate time and space averages of the stress

components: For example, ctxx may be averaged

over an area in the plane of A to give the instantaneous normal stress in the neighborhood of OA.

This local stress averaged over time is the normal

stress that would be measured on a macroscopic

scale at point OA. If the lattice as a whole is not

moving, the local macroscopic stress obtained this

way would be the same over the entire lattice. The

difference between the macroscopic stress ctxx and

CTxxo is the thermal stress component in the x direction. Similarly for cTyy and CTyyo, etc. These may then

be used for calculating the components of the

Grueneisen parameter.

In reference [7], we also computed the macroscopic stress (Txx by equating the work done by (Txx

per unit time (in a coordinate system which was

stationary with respect to the uncompressed lattice)

to the gain in the energy of the lattice per unit time,

and solving for (Xxx- The stress obtained this way

was the stress at the interface exerted externally on the lattice, and we were able to verify that this

"external" stress was equal to the "internal" stress

computed by the method described above. This was

as it should be, since the interface did not have an

average acceleration in either coordinate system.

The disadvantage of the method of computing the

external stress is that the motion of the boundary

must have a component in the direction of the stress.

If the boundary does not move, as in the case of the

boundary in the y direction, then this method is not applicable. But the internal stress always can

be calculated by the method outlined above.

5. Shock Profile, Us versus t7p, and

Summary of Earlier Results

Figure 4 shows the shock wave stress profile,

Sxx, in a one-dimensional lattice at a certain instant

of time. This was taken from reference [7]. It is a

typical profile for a stable lattice. The stress profile

in a two-dimensional lattice was found to be very

nearly the same as in the one-dimensional lattice,

when they were similarly compressed by a shock

wave of the same strength (same Up). We arbitrarily

defined the position of the shock front as that point

in the leading portion CD (fig. 4) at which the particle

velocity was equal to zero in the coordinate system

of figure 2. From the shock front position at successive instants of time, we determined the propagation

velocity Us of the shock wave.

Figure 5 shows the computed relationship between Us and Up for a large number of cases described in the caption of the figure. This figure was

taken from reference [6], except that curve (8) for

the fee lattice, and curve (9) for the bcc lattice apply

to both one- and two-dimensional cases. The dashed

portion of curves (8) and (9) shows a change in the

slope of the Us versus Up curve for the fee and bcc

lattices with i?i = 0.7. At this point, C/p = 0.5, we

found that both lattices showed an interesting

instability.^ This resulted when a lattice point with

sufficient energy overcame the potential barrier of

the neighboring lattice plane, and jumped into the

neighboring cell. We observed that this jumping

seemed to occur first in the x direction of shock

propagation, and could take place also in the reverse

direction, thus causing the lattice to "heal" itself. In the case of the two-dimensional bcc lattice, we

also observed instabilities in the y direction, which

followed those in the x direction in certain places

^ We wish to point out that in reference [7], we did not notice the instability which

also occurred in the two-dimensional fee lattice, /ii— 0.7, at t/,, = 0.5. This was an

error which we now wish to correct.

113
/

0 0.2 0.4 0.6 0 8 1.0 Up

Figure 5. Computed Us versus Up.

The conditions for the various curves are given in the table below. All the curves

apply to the one-dimenslona! lattice, except (8) and (9) which also apply to the two- dimensional lattice. Experimental data for graphite, CA and CC, for shock wave

propagating in the A and C directions, are taken from Coleburn [17]. Experimental

data for quartz, QX and QZ, for shock wave propagating in the X. and Z directions, are taken from Wackerle [18],

Lattice Linear

chain

Square

lattice

sc sc sc fee bcc

Farthest neighbor taken

into account Isl 2d 3d 6th 7th 4th 3d

"(1)

(2)

(3)

Parabolic Morse R, = 0.7 (4) (5) (10)

(6) (7) (11)(8) •(9) Morse, R, = 0.8 (12) (13)

"/r = i(r-r„).

'•fj-=k(r-H) - {k'lro)(i -ioy\ k' = It tor case (2). ' See eq (l)-(4). ''Numbers in parentheses refer to similarly numbered curves in the figure. ^Curves apply to both one- and two-dimensional lattices. Others apply to one- dimensional lattice only.

in the highly agitated region immediately behind the

shock front. This caused a progressive disarrangement of the regular lattice structure, in a manner

suggestive of the phenomenon of melting. Presumably a similar situation could occur in the threedimensional lattice as well. In the one-dimensional

case, since there was no motion in the y direction,

instability in the y direction was of course not observed. We should note that these results are some- what inconclusive, because in our computational

program we did not take proper account of the

changing role of the more distant neighbors after

the instability first occurred. But up to this point, our results should be reasonably valid. The change

in the slope of curves (8) and (9) is especially interesting, and may have a bearing on the interpretation of

phase changes observed in shock wave experiments

[3]. Further studies are needed to clarify the various

details of our results. For this report, we shall not

refer to this problem again, but shall confine our- selves only to the stable fee lattice.

Figures 4 and 5 have been discussed in detail in

references [6, 7]. For the sake of completeness, we

give here a very brief summary of the main results:

(a) We found that in both the one- and two-dimensional cases, the lattice was stable under fairly high

compression, and that in the stable region, the

computed Us versus Up in the two cases was very

nearly identical, (b) The computed Us was very

sensitive to the assumed interatomic forces, and

the assumed range of interaction. With realistic

anharmonic forces. Us increased with increasing

Up approximately in a linear manner, in good agreement with experimental results (fig. 5). This suggests

the possibility of our using the shock wave data for

evaluating the repulsive part of the interatomic

forces and the range of their interaction, if the structure of the lattice is indeed the same as we have

postulated, (c) The typical shock wave stress profile

was made up of a steady shock front CD (fig. 4), a compressed region AB with high frequency, small

amplitude oscillations trailing far behind the

shock front, and a transition region joining the

shock front and the compressed region. The transition region resulted from the interactions of waves

of various frequencies generated at the shock front

by the anharmonicity of the interatomic forces, and

propagating in the dispersive lattice. With increasing time, the transition region also increased in

extent. Thus we found that the shock profile was

not steady in time, and that the stresses and specific

volume we computed for region AB were different

from those computed on the assumption that the

solid was a continuum with a steady stre'ss profile,

(d) The computed relationships between the specific

volume and the vibrational energy, and between

the specific volume and the Grueneisen parameter

were in qualitative agreement with published results

from other investigations on three-dimensional

solids.

The quantities we obtained in (d) above were

those averaged over the entire shock wave profile.

We would like to examine now in greater detail the

problem of energy calculations, with a view to studying the temperature of the shock compressed lattice.

Also, in (a) above, although the computed Ug versus Up was the same for the one- and two-dimensional

lattices, the details of the solutions were different.

For example, in the one-dimensional case, there were no oscillations in the transverse direction and

we would like to know how this would affect the

temperature of the lattice. There are two parts to the temperature problem: the calculation of the

local energy density and energy distribution in the

lattice; and the study of the relaxation process and

approach to thermodynamic equilibrium. Of these,

the second part is the more difficult, and our results so far are still not extensive enough to give us a

114
clear understanding on the existence of thermodynamic equilibrium. In the following discussion, our conclusions bearing on this part of the problem

are therefore necessarily somewhat tentative.

6. Energy Calculations

Two-dimensional lattice. The energy density em- of a lattice of N lattice points may be defined as

follows:

= + {^j+muJI2) (13)

j j

where $j is the potential energy of the jth lattice

point computed from eq (2), and e^j is the kinetic

energy of the jth lattice point, and is equal to mu'jl2,

where uj is the velocity of the jth lattice point which

an observer would see if he were attached to the

lattice. A'^ is associated with the region whose energy

density we wish to calculate. A point on the

boundary or at the shock front is counted as one

half of a lattice point. In dimensionless terms, we

have

Eav=eavlD, E„j=^jlD, E,j=ekjlD ^RaReM.Uf.

Figure 6 shows the local energy density in the

shock compressed region as a function of lattice

position at several instants of time. The positions

of the shock front at different times are marked by

arrows at the top of the figure. A'^ was taken to be 90

for each local region. The energy density computed

this way fluctuated somewhat in different regions,

especially near the shock front. If were still smaller, say N= 9, the local fluctuations would have

been larger. For this discussion, we consider only

the mean curves drawn through each set of computed points. These curves show that in the tail part of the shock profile (AB in fig. 4), the average

energy density remained very nearly constant in

time, and that in the front part of the profile {BC in

fig. 4), the energy density increased toward a maximum at the shock front. Beyond the shock front,

the lattice was not compressed, so that the energy

density was simply equal to the average "thermal"

energy given to the lattice as an initial condition.

This is indicated by a line marked £'(/,o = 0.037 on on the ordinate scale. As we have found earlier [7],

the energy density averaged over the entire shock

profile also remained constant in time. This is in- dicated by a line marked £,„ = 0.678 on the ordinate

scale. These results imply that the energy density

profiles at different times were actually geometrically

similar, i.e., we could reduce the different profiles to one by linear adjustments of the scale of the

abscissa. The mean curves in this figure, in fact,

have all been made geometrically similar this way.

Their fit with the computed local energy densities

is seen to be satisfactory.

Figure 7 shows a series of histograms of the local

energy distribution at various instants of time,

corresponding to the curves of figure 6. In each

histogram, successive abscissas give the range of

energy i^ + ek), and the ordinate gives the count

of the lattice points in each energy range. The values

of M gives the range of lattice positions covered by

SHOCK FRONT POSITION

118.9 143.2 167.5 191.9

t = IOOv t=l20A t = l40o t=l60o

0 40 80 120 160 200

LATTICE POSITION

Figure 6. Energy density profile in a two-dimensional fee lattice under shock compression.

Up = 0.2, other conditions as given for curve (8) in figure 5. Ep, is the lattice energy at zero temperature. £,„. is the energy

density averaged over the entire shock profile. is the thermal energy initially contained in the lattice.

115
t=IOO t= 120

n r t=l4C t = 160

"1 r

M = I8I-I9I.9

0 1.0 2.0 ENERGY ENERGY ENERGY ENERGY

Figure 7. Local energy distribution in the two-dimensionalfee lattiee corresponding to the curves in figure 6

each histogram, as along the horizontal axis of figure

6. M=l— 20 denotes the first 20 lattice columns, = 21 — 40 denotes the next 20 lattice columns, and

so on, until the shock front is reached. The count

in each block of 20 columns is 180 points. We shall

designate each block by the first value of M.

The histograms in figure 7 clearly show the time

and spatial development of the local energy distribution in the lattice under shock compression. For

example, consider the block Af=121 and its time

development from t = lOO to ?=160. At ?=100, the

shock has not yet arrived at this position, and the

energy distribution, not shown here, would have

been the distribution corresponding to the "thermal"

energy which the lattice initially contained, with

the average value at £'f/,o = 0.037, as in figure 6. At

t=120, the shock front has just passed this block,

and the energy distribution shows that a number of

lattice points have now acquired higher energies.

This distribution is typical for the part of the lattice near the shock front, cf., M= 141 at ^= 140, M= 161

al t= 160. As the shock front passed farther into the

lattice, the energy distribution of M=121 relaxed

toward the distributions of those blocks at the tail end of the shock profile. For these blocks (M=l,

21, 41, and 61), the energy distributions exhibited

fluctuations with respect to position and time. It is clear, however, that their spatial and time averages

were very nearly the same, and, indeed, the average

energy density for this region remained constant, as shown in figure 6. In addition to the energy density, we are also

interested in studying the vibrational energy, Cvib,

of the lattice, defined as follows:

evib = eav — ^o (14)

where €av is computed from eq (13) and <i>o is the

116
potential energy per lattice point in a lattice that is compressed to the same specific volume as in eq

(13), but with all the lattice points at rest, so that

the lattice is at zero temperature. In dimensionless

terms, we have E,-ii,— CrihlD, Eav, and Ei,o = ^olD.

If E„r is the energy density of the lattice in thermal

equilibrium, then Ei-n, computed this way is directly

related to the temperature T of the lattice: in a two-dimensional lattice, E,ib=2kTID, where k is the Boltzmann's constant.

We shall postpone the discussion of the question

of equilibrium until section 8. For the present, it is seen from figures 6 and 7 that in region BC, Env and

the energy distribution were both relaxing with

time, so that this region certainly was not in equilibrium. On the other hand, in region AB, both Eav and

the energy distribution and therefore the value of

£', (6 remained very nearly constant in time, so that

it is possible to define a steady-state temperature

based on Evu,. In figure 8, we have plotted the computed AEvib^ Efii, — E,ho as a function of the relative

volume V. This figure is similar to figure 2 of refer- ence [7], except that £",,,6 now is computed from Eav

for the local region AB (fig. 4) whereas in reference

[7], Evib was computed from Eav for the entire shock

profile. For comparison, we have also plotted

Pastine's results [20] on aluminum. Pastine employed a fee lattice model with nearest neighbor

interactions, and assumed a Morse potential function with /? 1 = 0.78, and uniform compression in

three dimensions. His method for computing temperature was also different from ours. We converted

his computed temperatures to ^vib, by setting

^vib = ^kATID. With these qualifications in mind,

we found our results in qualitative agreement with

Pastine's.

One-dimensional lattice. The results for the one- dimensional case were rather different. For a

linear lattice with harmonic interactions between

nearest neighbors, subject to boundary conditions

similar to those employed here, there is an exact

solution [19] given by the Bessel function of the

first kind. For such a solution, the lattice points at the tail end of the shock profile become nonoscillatory as t approaches infinity, and there is no

temperature rise in the compressed lattice. Our

numerical results for this case indeed agreed with

these conclusions, within the accuracy of our calculations. For example, for f/,j = 0.2, at f=100, we

found that the local energy was distributed about an

average value of £"01; = 0.02 over the entire shock

profile. Near the shock front, the width of the distribution was about ± 15 percent of Eav and this

width decreased steadily toward the tail of the shock

profile. Presumably, after a very long time, the distribution would shrink to a single line at — 0.02.

The value of Evtb which we calculated was effectively zero.

For a linear lattice with anharmonic interactions,

Eav no longer remained constant over the entire

shock profile but decreased from a maximum at the

2.0

1.5

AE vib

1.0 -

0.5

n 1 1

1

, -o- THIS CALCULATION

\ _ PASTINE'S RESULTS

1 ~ F0RAUJMINUMC201"

0.5 1.0

Figure 8. Computed AEvib versus V in region AB for the two- dimensional fee lattice under shock compression, under

conditions given for curve (8) in figure 5.

t = 100 t = 120

M = I2I-I43.5 1^ -rM=I0I-II9.2 M = I01-I20

'

I

' nM = 8I-I00

—I 1 1 M = 6l-80

n T—^ 1 1 1 M = 4|-60

1 T-^—I1 1 r- M=2l-40

n 1 1 M = I - 20

ENERGY

0 1.0 2.0 ENERGY

Figure 9. Local energy distribution in a one-dimensional fee

lattice under shock compression.

L',, = 0.2, other conditions are the same as those given in figure 6.

shock front toward the steady value at the rear of

the shock profile, as in the two-dimensional case

(fig. 6). In fact, as was found in reference [7], the

values of Eav averaged over the entire shock profile were nearly the same for the one- and twodimensional cases. However, the energy distributions for the two cases were very different: for

the anharmonic linear lattice, the energy distribution showed a tendency to become peaked at the local

value of E„v toward the tail of the shock profile,

similar to the harmonic linear chain, whereas in the

two-dimensional (anharmonic) lattice, no such

tendency was observed. This is shown in figure 9

which is to be compared with figure 7. It seems

likely that after a long time, the energy distribution

in the tail portion of the shock profile in figure 9

117
also would shrink to a single line at £"01, = 0.545.

This means that there would not be any oscillation

in that part of the lattice, and that the Evib calculated from eq (14) and hence the temperature rise

would be zero. Suppose we now interpose a rigid boundary in

the semi-infinite linear lattice to make it finite. Then

the shock wave, upon reaching this boundary, would

be reflected back and forth between the two boundaries and eventually the energies in the various

modes would be equilibrated [10]. The energy density of the whole lattice would be and there

would be a corresponding Emb and a temperature

rise. It seems rather curious that in the one- dimensional lattice there should be no temperature rise in the tail part of the shock profile until

after the shock wave has had a chance to be re- flected from the rigid boundary, whereas in a twodimensional lattice there should be a temperature

rise upon the first passage of the shock. We do not

have a simple physical explanation for this phenomenon. Apparently the transverse boundary condition

for the one-dimensional model (Uy = 0) has the

effect of making the "temperature" in the transverse

direction zero, and this also makes the "temperature" in the longitudinal direction zero behind a transient shock front. This suggests that the one- dimensional model is not realistic. The results of

the two-dimensional model seem more reasonable,

but we really should study the three-dimensional case in order to gain a deeper insight to this problem.

For the present, in the discussion of the temperature

rise behind the shock front, we limit ourselves to the

two-dimensional case only.

7. Thermal Relaxation Behind the

Shock Front

In the discussion of figure 4, we have noted our

earlier results which showed that the transition

region BC in the shock wave profile increased in

extent as the shock front propagated into the

lattice. We see from figures 6 and 7 that this

transition region corresponds to a region in which

the energy density relaxes from a maximum value

at the shock front to a steady value in region AB.

So the unsteadiness of the shock wave stress

profile is associated with the relaxation of the energy

density, and in our computation of the shock profile,

we have also obtained a description of the relaxation

process behind the shock front.

We may investigate the relaxation time behind

the shock front by studying the velocity Ub with

which point B moves behind the shock front. The

geometric simiUtude of the energy density profiles

(fig. 6) at diff^erent times indicates that region AB

is a constant fraction of the shock profile, so that

Ub is also a constant fraction of Us- The precise location of point B is difficult to determine from either the energy density profile or the stress profile. In a further attempt to dis0 .1 .2 .3 .4 .5 .6 Up

Figure 10. Velocity of propagation of point B, Ut,, plotted as Ub/Us versus Up.

tinguish region AB from BC, we also studied the

velocity distribution of the lattice points in these

regions. We found that in region AB the distribution

of the velocity components f/j and Uy was nearly

maxwellian, but in region BC, especially near the

shock front, the distribution was non-maxweUian.

Thus, by calculating the width of the distribution curve for the lattice, starting from the tail end of

the shock profile and taking successively greater

portions of the lattice toward the shock front, we

could identify region AB as one with approximately

constant width of the distribution curve, and region

BC as one in which the width increased toward a maximum near the shock front. We used this

method to locate point B, and computed the velocities of Ub at different values of Up. The results are plotted in figure 10.

Figure 10 shows that Ub/Us was approximately

equal to 0.58 at low values of Up, corresponding to low compression, and this value decreased to about

0.5 at Up = 0.4. If we consider AB to be a truly

steady region, with a steady temperature given in

figure 8, then these results give the propagation

velocity of point B at the front of this steady region.

The value of 0.58 is especially interesting: It is very

nearly equal to l/VS = 0.5774 which is the velocity

of second sound in a crystal at normal pressure and

low temperature [21] . We are naturally led to speculate that what we have computed here is the equivalent second sound in a crystal under high-stress,

high-temperature conditions. Alternatively, we could

consider BC as a region of thermal relaxation. Our

results in figure 10 show that at higher shock compression (higher Up and Us) the relaxation region is a

larger fraction of the shock profile.

8. Investigation of the Existence of

EquiHbrium in Region AB

Thus far our results show that in region AB the

local averages of the following quantities remained

uniform along the lattice and steady in time:

118
(1) Specific volume

(2) Stress components Sxx and Syy

(3) Energy density and Evib

(4) Energy distribution

(5) Approximately maxwellian velocity distribution.

However, even at ^=1000 (=5X lO"" s), the time

of our calculation was short on a macroscopic scale

(say 10"^ s or greater), so the question of the equilibrium properties of the lattice remains.

To study this problem, we investigated the time

development of the velocity distributions. According to the principle of equipartition of kinetic energy

in statistical mechanics (see, for example, Guggenheim [22]) the average kinetic energy in each classical degree of freedom is equal to AT/2 and the distributions of the velocity components are maxwellian. Hence the average widths (standard deviation) of the distribution curves should be identical.

Figure 11a shows the distributions of the velocity

components Ux and Uy when the lattice in region

AB was compressed more in the X direction than in

the Y direction. The velocity distribution in the X

direction was also narrower. Thus in this case, we

see that the "kinetic temperatures" have not been

equilibrated.

We could adjust the final unit cell configuration by

compressing the lattice statically first in the Y direction, and then dynamically by means of the shock

wave in the X direction. Figure lib shows that when

the final unit cell configuration was more nearly

square, the corresponding velocity distributions were more nearly equal.

Closer examination of figure lib shows that the

two velocity distributions were actually not identical,

and one wonders whether this was due to the slight

difference in the average spaciugs in the X and Y

directions (0.846 in X and 0.850 in Y). To study this

problem, we isolated a portion of the lattice in fig- ure lib, 25 X 25 lattice points in size, by introducing

the boundary condition (b) (see section 3) on all four

sides. We also adjusted the distances between opposite boundaries so as to make the average spacing

in both the X and Y directions equal to 0.847. We

then continued the computation with the conditions

of figure lib as the initial conditions. Our purpose

was to observe the time development of the velocity

distributions. Since the lattice was now identical in

X and Y directions, we would expect that at equilibrium we would not be able to see any differ- ence in the velocity distributions, or in any other

property of the lattice, in the X and Y directions. A

knowledge of the time required to reach equilibrium,

starting from the conditions of region AB behind the

shock front, is of considerable interest. Given this

knowledge we could then carry out similar calculations over a similar time interval for the case of

figure 11a with unequal strain and obtain a detailed

description of the equilibrium condition appropriate

to this case, assuming that the two cases required

similar times to reach equilibrium after shock

compression.

G(U)

.30

.20-

G(U)

.10 -

G(U)

-.10

VELOCITY, U^.Uy

Figure 11. Velocity distribution in three two-dimensional fee

lattices. The abscissa gives the range of the velocity components Ux and Uy. The ordinate

gives the value of G(t/), the fraction of lattice points having velocity in a given range.

(a) Region AB of shock profile, Uf = 0.2, i = 100. AB is 60 lattice spacings in X

direction and 10 lattice spacings in Y direction. /?.r = 0.833, Ry= 1.000.

(b) Region AB of shock profile. Up = 0.2, t = 35. AB is 25 lattice spacings in X direction and 50 lattice spacings in K direction. = 0.846, /fy = 0.850.

(c) Isolated lattice at / = 250. 25 lattice spacings in both X and K directions.

fi^ = «„ = 0.847.

The data for figure lib were obtained dX t — 35.

Starting at this point with the isolated lattice, as described above, we carried the computation to a maximum of ? = 250. The velocity distributions at

t = 250 are shown in figure 11c. Comparing figures

lib and 11c, we see that during this time interval,

the velocity distributions had not changed appreciably. The standard deviations for Ux were

119

385-762 O - 71 - 9
0.0245 at f = 35 and 0.0220 at t = 250, and those for

Uy were 0.0274 at f = 35 and 0.0279 at f = 250.

During this time interval, the velocity distribution

also fluctuated, and the averages of all the standard

deviations at the end of the interval, at f = 250,

were 0.0213 for Uj- and 0.0327 for Uy. Hence we

must conclude that no appreciable equilibration

had occurred during this time in the isolated lattice

of figure 11c.

We should point out that the isolated lattice of

figure 11c enclosed by reflecting boundaries on all four sides was not the same as the lattice of figure

lib which was in contact with region BC toward the

shock front. The equilibration times for these two

systems need not be the same, because in the closed

system, equilibration could take place only through

the exchange of energies among different modes that

were coupled by anharmonic effects, but in the open

system, in addition to the exchange of energies, it was also possible for the unequilibrated part to leave

the system.

Within the limits of our numerical data, we have

also calculated the velocity autocorrelation function

which may be defined as [10]

Ui{t + T)-Ui{r)dT

PiitJ) - , T > t

Jo {U;{r)ydr

where p, is the velocity autocorrelation function of

particle i, and Uiir) is the velocity of i at time r. We

shall not discuss the details of our results, but

briefly, for a point near the middle of the lattice, we

found that p; did not decrease nearly so rapidly with

t as in the cases studied by Rubin [23], by Northcote

and Pott [10], and by Rahman [11]. This was consistent with our conclusion of lack of equilibrium in

figure 11c, although it was also possible that we have

not made T large enough compared with t.

If in figure lib we did not achieve equihbrium in

region AB, we were not very far from it either. Presumably, for some case not too different from that of

figure lib, we could obtain very nearly identical

average velocity distributions in the X and Y directions. If we then followed this with additional calculations, as in figure 11c, for the isolated lattice,

we would not expect to see appreciable change in

the lattice even in long times, and we would con- clude that for this case, we might have reached

equilibrium. But then the question is why we should

be able to reach equihbrium rapidly only in this particular case. This is a curious question to which we

do not have an answer. Our basic difficulty at this

moment is twofold: We do not have an adequate

theoretical model of equilibrium for a small, anharmonic and anisotropic lattice, and we do not have

enough numerical results to permit us to construct

such a model. We also do not know how this possible lack of equilibrium affects shock wave calculations based on the assumption of equihbrium behind

the shock front [1-5]. Further study of these problems and further numerical computation are therefore needed in order to achieve progress in this

direction.

9. Grueneisen Parameter Calculations

and Summary of Results

The components of the Grueneisen parameter

(Grueneisen tensor) may be defined as follows [24]:

"Yxx — V ( o-jj — o-j-xo ) /evib Jyy^vi (Jyy — (Tyyo)lCvib

etc.

In dimensionless terms, we have

Jrx = 2RaReMiV{S^^-S^^o)IEvib

yyy=2RaReMiV{Syy-Syyo)IE,ib\ (15)

etc.

Here a. ' yyoi etc. are the stress components

when the lattice points are at their equilibrium positions and not oscillating, corresponding to the lattice at zero temperature.

The computed results for the steady region AB in our two-dimensional fee lattice are listed in table 2

which also summarizes the other results. The values

are averages over the time (t) period shown. For the

isolated lattice, the averages were taken from the

first step after i = 35 to t — 250, and these values are probably more rehable than the others because of

the comparatively long time of averaging. Here, the

unequal values of yxx and yyy for the lattice under

equal compression (R^ — R y — O.S'iT) indicate

again the lack of equihbration in the X and Y directions. The other values of the y components are less

reliable because of the shorter time average, and,

especially in the case of f/p = 0.1, because of the

small differences between Sxx and Sj-jco, Syy and

Syyo, and E„p and E,,,,- There seems to be a systematic decrease of Epo with increasing time. We do

not have sufficient data to say whether this trend

would continue monotonically, or become oscillatory as the lattice rebounds under shock excitation.

Despite the above considerations, the differences

between the y components for the lattice with equal

and unequal compression in the two directions are probably real. Specifically, at Up — OA and Rx Ry,

the specific volume of 0.712 was approximately the

same as the specific volume of 0.719 with Rx — Ry = 0.847, but the y components for these two cases

differed by a factor of almost 3 or 4. Thus we see that

the Grueneisen parameter depends not only on the

specific volume, but also on the details of the state of

strains (and of the stresses) in the crystal [25].

120
Table 2. Summary of results

Up t

Lattice spacing Specific

volume

y

St ress Eav Epo Eint) yxx yyy R, c c^xxo ^yyo u.

0.1

50

70

1.096

1.101

0.908

0.908

1.000

1.000

0.908

0.908

0.1103

0.1109

0.1099

0.1094

0.0500

0.0472

0.0471

0.0468

0.188

0.188

0.151

0.152

0.036

0.037

0.58 (0.7) (1.3)

100 1.217 0.833 1.000 0.833 0.243 0.230 0.111 0.106 0.619 0.549 0.070

0.2

120

140

1.216

1.219

0.833

0.834

1.000

1.000

0.833

0.834

0.241

0.245

0.230

0.229

0.113

0.112

0.106

0.106

0.618

0.617

0.546

0.544

0.072

0.073

0.58 5.2 2.3

160 1.221 0.834 1.000 0.834 0.246 0.228 0.112 0.105 0.619 0.541 0.078

0.3

50

60

1.311

1.317

0.766

0.767

1.000

1.000

0.766

0.767

0.395

0.396

0.369

0.367

0.196

0.194

0.182

0.181

1.306

1.303

1.177

1.168

0.129

0.135

0.55 5.1 2.5

0.4

50

60

1.416

1.423

0.711

0.713

1.000

1.000

0.711

0.713

0.567

0.570

0.512

0.508

0.293

0.293

0.267

0.264

2.222

2.223

1.945

1.922

0.277

0.301

0.50 4.6 2.1

0.5

50

60

1.489

1.470

0.660

0.660

1.000

1.000

0.660

0.660

0.747

0.734

0.673

0.673

0.408

0.406

0.369

0.369

3.255

3.284

2.909

2.908

0.346

0.376

3.9 2.2

0.2 35 1.316 0.846 0.850 0.719 0.404 0.398 0.394 0.394 0.990 0.966 0.023

Isolated

lattice 250 0.847 0.847 0.717 0.402 0.400 0.401 0.400 1.011 0.964 0.047 1.2 0.8

10. Conclusions

We have discussed in some detail the dynamical

response of a lattice under shock compression, and

the associated problems of calculating the properties of the lattice from the dynamical solutions. Our

results show that a perfect, two-dimensional lattice

(fee or bcc) is stable under fairly large one- dimensional strain, but under still larger compression, the lattice develops instabiUties which are suggestive of melting. In the stable regime, the shock

profile is made up of a relaxation region (BC in

fig. 4) and a steady region {AB in fig. 4). The relaxation time is not constant, but increases with time,

so that the steady region trails further behind the

shock front with increasing time. In the steady

region, the energy density, energy distribution, velocity distribution, stress components all remain

steady with time. The computed steady-state temperature for this region increases with increasing

compression (fig. 8). The propagation velocity of this

region decreases with increasing compression (fig.

10), and at low compression, this velocity is about

0.5 to 0.6 times the longitudinal velocity of sound, or approximately equal to the velocity of second sound

in a crystal under zero pressure. Our results do not

show that there is equilibrium in this steady region.

The computed Grueneisen components do not appear to be a function of the specific volume of the

lattice alone, but show a strong dependence on the

state of strains in the lattice. Finally, we found that

although the energy density profiles in a one- dimensional lattice may be nearly identical to those

in a two-dimensional lattice, the energy distributions

in region AB in the two lattices are different: In the

one-dimensional case, the energy distribution becomes narrower with increasing time, and presumably becomes a single line at very large times, so

that there is no vibrational energy or temperature

rise in a one-dimensional lattice; whereas in the twodimensional case, we do not observe a similar nar- rowing of the energy distribution or a decrease of

the vibrational energy with time. We therefore con- clude that the one-dimensional model is not a reaUstic model for studying the energies of the lattice

under shock compression.

The above conclusions apply to the twodimensional lattice model. All real solids are threedimensional, and probably imperfect, and at this

moment we are unable to say what modifications are necessary before we can apply our results to the

three-dimensional case. It is disturbing, nevertheless, that none of our results so far supports the

usual assumptions of a continuum solid employed

in the analysis of shock wave experiments. On the

other hand, our computed relationships between

Us and Up are in good agreement with experimental

data over a wide range of experimental conditions.

If this be taken as an indication of the "correct"

interatomic forces used in our calculations, then

the consequences also must be valid for the model

121
postulated. We must therefore extend our calculations and refine our model as we continue our study on an atomistic scale. For example, we should

extend our calculations to the three-dimensional

case and study its energy density and distribution.

If we included imperfections in the threedimensional model, we should be able to study the

problem of stress relaxation under shock compression. Also, by propagating the shock wave in an

arbitrary direction with respect to the crystalline

axes, we could study the effect of shear on the temperature of the lattice. The stability problem and the

melting phenomenon of a three-dimensional lattice are also important problems with some hope of

experimental verification. Finally, the problem of

equilibration is vital to the calculation of the

equation of state and other properties of the lattice.

These are very lengthy problems, but it seems clear

that they must aU be studied in comparable detail

if we are to complement more effectively the vast

amount of experimental work already in progress

in this area.

11. Acknowledgments

The author wishes to thank Dr. M. Klein, Chief

of the NBS Equation of State Section, for his interest

in this work and for his support. The author has

benefited from discussions with Dr. C. W. Beckett,

Mr. L. Harr, Dr. M. Klein, Dr. R. MacDonald,

Dr. R. Nossal, Dr. B. Robertson, and Dr. R. J. Rubin. Mr. J. Barnett provided invaluable assistance

in the computational work in the latter part of this

investigation.

12. References

[1] Rice, M. H., McQueen, R. G., and Walsh, J. M., Compression of solids by strong shock waves. Solid State Phys.

6, 1-63 (1958). [2] Deal, W. E., Jr., Dynamic high pressure techniques, in Modern Very High Pressure Techniques, ed. by R. H. Wentorf, Jr., pp. 200-227 (Butterworths, Washington,

D.C., 1962).

[3] Alder, B. J., Physics experiments with strong pressure

pulses, in Solids Under Pressure, ed. by W. Paul and

D. M. Warschauer, pp. 385-420, (McGraw-Hill Book Co.,

New York, N.Y.. 1963).

[4] Band, W., Studies in the theory of shock propagation in

[8

[9

[10

[11

[12

[13

[14

[15

[16

[17

[18

[19

[20

[21

[22

[23

[24

[25

solids, J. Geophys. Res. 65, 695-719 (1960). Bland, D. R., On shock structure in a soUd, J. Inst. Math.

Applications, London 1, 56-75 (1965). Tsai, D. H., and Beckett, C. W., Shock wave propagation in cubic lattices, J. Geophys. Res. 71, 2601-2608 (1966). Tsai, D. H., and Beckett, C. W., Shock wave propagation in a two-dimensional crystalline lattice, in The Behavior of Dense Media under High Dynamic Pressure, Symposium

H.D.P., International Union of Theoretical and Applied

Mechanics, Paris, September 1967, pp. 99-108 (Gordon

and Breach, New York, N.Y., 1968).

Ulam, S. M., A Collection of Mathematical Problems, pp. 109-113 (Interscience Publishers, Inc., New York, N.Y.,

1960).

Alder, B. J., and Wainright, T. E., Studies in molecular

dynamics. I. General method, J. Chem. Phys. 31,

459-466 (1959); II. Behavior of a small number of elastic spheres, J. Chem. Phys. 33, 1439-1451 (1960). Northcote, R. S., and Potts, R. B., Energy sharing and

equilibrium for nonlinear systems, J. Math. Phys. 5,

383-398 (1964). Rahman, A., Correlations in the motion of atoms in liquid argon, Phys. Rev. 136, A405-A411 (1964). Payton, D. N., Rich, M., and Visscher, W. M., Lattice ther- mal conductivity in disordered harmonic and anhar- monic crystal models, Phys. Rev. 160,706-711 (1967). Slater, J. C, Interaction of waves in crystals. Rev. Mod.

Phys. 30, 197-222 (1958). Dean, P., Atomic vibrations in solids, J. Inst. Math. Applications 3, 98-165 (1967). CoUatz, L., The Numerical Treatment of Differential Equations, 3d ed., pp. 54-57 (Springer-Verlag, Berlin, 1960).

Love, A. E. H., A Treatise on the Mathematical Theory of

Elasticity, 4th ed., pp. 616-627 (The University Press,

Cambridge, 1934).

Coleburn, N. L., Compressibility of pyrolytic graphite, J. Chem. Phys. 40, 71-77 (1964). Wackerle, J., Shock wave compression of quartz, J. AppL

Phys. 33,922-937 (1962). Bateman, H., Partial Differential Equations of Mathematical Physics, American ed.. pp. 228-236 (Dover Publications, New York, N.Y., 1944).

Pastine, D. J., An equation of state for face-centered cubic

metals. J. Appl. Phys. 35,3407-3414 (1964). Chester, M., Second sound in solids, Phys. Rev. 131,

2013-2015 (1963). Guggenheim, E. A., Boltzmann's Distribution Law, pp. 34-36 (Interscience Publishers, Inc., New York, N.Y.,

1955). Rubin, R. J., Momentum autocorrelation functions and

energy transport in harmonic crystals containing isotopic

defects, Phys. Rev. 131, 964-989 (1963). Pastine, D. J., Thermal expansion and structure of anisotropic monatomic solids, Phys. Rev. 148, 748-758 (1966). Anderson, O. L., Some remarks on the volume dependence

of the Grueneisen parameter, J. Geophys. Res. 73, 5187-

5194 (1968). See table 1, p. 5188 for acoustic transverse gamma and acoustic longitude gamma compared with

thermal gamma and other calculated ''gammas."

DISCUSSION

D. J. Pastine {U.S. Naval Ordnance Laboratory,

White Oak, Maryland): Referring to figure 8, I would like to make the point that the upper curve

ought to be reduced by one-third to compare with

yours, because you are dealing with a twodimensional calculation. This will put you in good

agreement on the low end. If you take into account

the more recent calculations, the upper curve will

be lowered even a little more on the high compression side, so I think the agreement then will

be very good between the two.

L. Thomsen (Columbia University, Palisades,

New York): You said there was not thermal equihbrium. Is it possible to define a local thermodynamic

equilibrium or an instantaneous thermodynamic

equilibrium?

M. van Thiel (Lawrence Radiation Laboratory,

University of California, Livermore, California): It

is difficult at this stage to compare wave profiles

from the calculation with experiment because our time resolution is much less than that which would

122
be required. It is possible, of course, that over this

short a time scale the system has not equilibrated.

In figure 4, showing the wave profile, it looked as though your phonon spectrum from an analysis of

the profile was peaked along certain frequencies.

We also mentioned earlier that the coupling between the various modes is slow. It is possible, if you can put in the proper zero-point motion for the

lattice, that the order that you start with will be

disturbed enough that coupling between the vari- ous modes becomes easier.

AUTHOR'S CLOSURE

It is true that if we reduced the ordinates of

Pastine's curve in figure 8 by one-third, our results

would be more directly comparable, and the agreement would improve. However, it is not easy to assess the significance of the improved agreement,

because of the differences between Pastine's model

and ours, including the unequal compression and the

apparently unequilibrated kinetic temperatures in

the X and Y directions in our model. Moreover,

Pastine's calculations did not consider the thermal

relaxation process behind the shock front. His

results therefore corresponded to

^\ib Eav ~" ^po £'tho

(in fig. 6). Indeed, as we showed in reference [7]

,

our A^vib based on £'av was in fair agreement with

Pastine's results. But there is a more serious

difficulty: Our one-dimensional results showed that

the oscillations in region AB tended to decrease to zero with increasing time, so that in this region the

oscillatory (thermal) energy was not one-third of

the three-dimensional value, but zero. This clearly

indicates that the one-dimensional model was not

suitable for studying the problem of temperature

rise under transient compression. Whether this

difficulty is completely removed in a twodimensional model is uncertain at this moment.

On the question of local equilibrium, we found

that the oscillations in the Y direction, introduced

into the lattice initially to simulate thermal vibrations, interacted rather strongly with the large

amphtude oscillations near the shock front, and

equilibrated rapidly along themselves, even in

region BC. The oscillations in the X direction

equihbrated more slowly, not until they reached

region AB, because of the combined effects of

dispersion, and of the moving shock front which

steadily lengthened the compressed region. In

region AB, the amplitude of the oscillations and

hence the anharmonic coupling between the X

and Y directions were both small. Our results

showed that if these oscillations were not equilibrated before they entered region AB, they would be

slow in coming into equihbrium on the time scale of our calculation.

In order to try to find at least a partial answer to the question raised by van Thiel, we displaced

different rows in our model by varying amounts

before the shock arrived, so that the shock was

going into a lattice which was vibrating in the

transverse direction. This gives the possibility of

some transfer of momentum in the transverse

direction. The model still has a rather high

symmetry, but what we have done probably simulates fairly well the zero-point vibrational energy

or even room temperature vibrational energy in the

solids studied experimentally.

123

Effect of 2024 Aluminum Alloy Strength on High-Pressure Shock

Measurements*

M. van Thiel and A. Kusubov

Lawrence Radiation Laboratory, University of California,

P.O. Box 808,

Livermore, California 94550

A shock wave technique is described by which wave profiles may be determined in metals. Piezoresistive manganin gages were used to determine these profiles to 130 kbars ini 2,024 aluminum alloy. The data are consistent with the assumption that the Hugoniot is the hydrostat. A maximum of 28 kbars is obtained for the maximum yield strength on the release wave following the strongest shock. The

derived loading-unloading curves in the pressure volume and pressure mass velocity planes indicate

that errors of about 2 percent are produced by use of purely hydrostatic release paths.

When we determine an equation of state from

shock wave data, we usually rely on the assumption

that the strength of the material is small. We say

then that, to a good approximation, the stress in

the shock direction is the same as that in the other

two directions. This is reasonable insofar as extrapolation of shock wave data to low pressures often

yields bulk moduli in good agreement with sonic

data. This assumption has, furthermore, been justified by the agreement of shock wave equations of

state with Bridgman anvil data. We say this, knowing that Professor Kennedy at the University of

California at Los Angeles, in cooperation with some

of our coworkers at Lawrence Radiation Laboratory

(LRL), have reopened the issue and have questioned

the term "reasonable agreement." This is, of course what should happen when, with growing confidence

in the validity of our measuring techniques, we can

look more critically at possible discrepancies.

Disagreement with the assumption of fluid behavior of the high-pressure Hugoniot has, however,

appeared earlier in the shock wave literature.

Barker et al. [1]* and Hartman [2] have shown that

strength, indeed, is a factor — at least up to' a pres- sure roughly equal to ten times the Hugoniot elastic

hmit. But Al'tshuler [3] demonstrated that, even at higher pressure, a sizable strength effect exists in

the release process that follows a shock wave. The

magnitude of this effect is described by Erkman and

Christensen [4], by Barker [5], by the authors [6],

and by some unpublished work of Wilkins at LRL.

It is evident from this work that strength in metals, even above 100 kbars, cannot be ignored. This

paper is intended to describe some further measure- ments taken by us (with manganin gauges) of the

strength of the 2024 aluminum aUoy at high pres-

* This work performed under the auspices of the U.S. Atomic Energy Commission.

' Figures in brackets indicate the literature references at the end of this paper.

Paper presented at the Symposium on Accurate Characterization

of the High-Pressure Environment, held at the National Bureau of

Standards, Gaithersburg, Md., October 14-18, 1968.

sures and the effect of this strength on the temperature and the shape of the release curve. The method used here consists of measuring a wave profile. To understand this, we will consider

loading and unloading curves in the pressure

volume plane. The sample is initially compressed

by a shock indicated by the dashed line in Figure la,

where this loading curve is drawn to indicate a two-wave process. This double-wave process occurs

for aluminum up to about 100 kbars, while above

this pressure the (first) elastic wave, moving with

a velocity corresponding to the Hugoniot elastic

2 3

P

ELASTIC-PLASTIC NOTE; NUMBERS CORRESPOND TO

RELEASE THOSE IN FIG. 3. SHOCK LOADING

(o) SHOCK AND RELEASE PROCESSES (b) PRESSURE VS . TIME AT DETECTOR

TIME —

METAL INTERFACES SHOCK CHARACTERISTICS

RELEASE CHARACTERISTICS

(c) POSITION TIME PLOTS

Figure 1. Shock and release processes.

125
CABLES

Al PLATE

INSULATION

SILVER RIBBON MANGANIN RIBBON

Figure 2. Target assembly.

limit (HEL), is overtaken by the plastic shock. When

the wave starts unloading, simple elastic-plastic

theory [7] indicates that initially the material unloads

along an elastic release curve until the difference in

the stresses in the x and y direction are sufficiently

large to cause dislocation flow and yielding. Thereafter, the release curve is essentially that of a hydrostatically compressed material, and the slope,

therefore, is given by the bulk modulus. In fact, this

release curve is a nonisentropic adiabat, with the

energy dependence

dE = -P^dV=^Y{V)dV-P{V)dV,

where Y is the von Mises yield strength. We also

assume that the material can be described by a cubic or isotropic stress tensor.

The wave shape that each part of the loading and

unloading curve implies is illustrated in figure lb.

Here we show the pressure as a function of time at a point in the sample. The velocity of the two loading

waves is different, since there is a break in the

loading curve at the yield point. The same is true of

the unloading curve. Each portion of the loading and

unloading curve is characterized by a velocity

dPxIdp^c'. Therefore, once we have determined

these velocities as a function of pressure, we can

construct the hysteresis curve in the f, V plane.

The shock and release wave loci are illustrated

in the time-space plane of figure Ic. A plate projectile impacts a target, sending shocks into the

target as well as into the projectile. We have drawn

these trajectories for the most general case, in

which elastic and plastic shocks run at different

speeds. A measurement of the velocity of these

shocks then yields the two loading curves. Similarly,

the detection of the unloading wave at an interface

jusec

Figure 3. Experimental manganin gauge record.

(1) Elastic precursor

(2) Plastic shock

(3) Start of semi-elastic release wave

(4) Point at which release wave velocity is comparable to (dPtdp)^''^

(5) End of release process

containing a pressure gauge yields the characteristics c+u, where u is the local mass velocity, and

c is the desired velocity in the mass coordinate

system. We can calculate u from the Riemann

equation

dP~\ dP) pc

and solve for c by iteration. A characteristics code

was written to make these calculations as well as to construct the P, p curves from the calculated c values.

The experimental assembly used by us is schematically represented in figure 2. Twelve-micronthick manganin ribbons spot-welded to silver

ribbons are placed between two 25-micron-thick

insulating sheets of Kapton.' A small amount of oil

is used to provide a hydrostatic environment for the

gauge and to hold the gauge in place on the insulation during assembly. The Kapton insulation is

' DuPont polyimide film.

126
sandwiched between two aluminum plates that

make up the target. Voltage and current leads

soldered to the silver ribbons are potted in epoxy

to strengthen the assembly. Pins are placed around

the gauge area to measure projectile velocity and

tilt. Shock velocities were not measured, since the

projectile velocity and the known shock properties

of the material yielded these to sufficient accuracy.

All the c values were, therefore, obtained relative to the calculated velocity of the shock wave.

Figure 3 shows the results of a measurement. This

is a voltage-time trace of the gauge output. The

numbers indicate the various shock and release

wave features mentioned earlier. A small disturb- ance produced by the elastic precursor is followed

by the main shock wave. We do not beUeve that the

gauge gives a good measure of the wave shape

during the first part of the compression process,

since initially it is not in intimate contact with its immediate environment. The behavior, therefore, is not reliable until any small gaps near the manganin

gauge element have been eliminated. These gaps

must certainly be closed when the release wave

reaches the interface between plates where the

gauge is located. We note here that the release

process does not start with a simple elastic release wave. It starts as a ramp along which the velocity

decreases rapidly, from a value that corresponds to a longitudinal modulus to one that corresponds to a bulk modulus. After that, the shape of the release

wave is close to the one expected from the usual

variation of modulus with pressure.

The gauge is used here in its most reliable mode;

namely, as an interpolation device. This is possible,

since the peak pressure can be determined from the

known shock Hugoniot [8] and the measured

projectile velocity. The pressure coefficients of

resistivity obtained on the gauges used are some- what lower than those determined by Keough [9] on

gauges potted in epoxy. He obtained coefficients near 0.29 percent/kbar, while the values measured

here were near 0.26 percent/kbar; i.e., closer to the

statically measured coefficients.

A loading-unloading hysteresis curve calculated

from these data is shown in figure 4. The pressure

scale on the left shows a peak pressure of 131 kbars.

At this shock pressure no elastic precursor exists,

and the shock process is described by a single

Rayleigh line. The large curvature of the release

wave at high pressure is a direct consequence of the

shape of the unloading curve in the region where a simple elastic release process was expected. While

the initial slope yields a reasonable Poisson ratio,

the slope rapidly decreases and goes over to a normal bulk decompression curve. It seems,

therefore, that incipient yielding occurs almost

immediately at the start of the release process.

Related to this observation is the question of the

location of the hydrostatic release curve along which

the stresses, both in the direction of the wave and

Figure 4. Experimental loading and release curve with esti- mated hydrostal.

30 -

1

THIS WORK

RESULTS OF ERKMAN AND

CHRISTENSEN FROM FREE SURFACE MEASUREMENTS

RESULTS FROM OTHER FREE SURFACE MEASUREMENTS

BY THE AUTHORS

100 200 300

P , kbar X

Figure 5. Summary of experimental dynamic strength of 2024

aluminum alloy at high shock pressures.

at light angles to it, are the same. The release curve must lie below this hydrostat before the direction of

the resolved shear stress is such that it can cause

dislocations to move. Since some yielding is observed at the very beginning of the release process,

this suggests that the Hugoniot point from which

the release starts is hydrostatic — or at least close

to it. While this may seem surprising, there is no

evidence to the contrary. In fact, it is known from

double shock experiments [1, 10] that the Hugoniot

does not lie on a shock yield surface. In these ex- periments, [1, 10] elastic compression waves

resulted from a second shock immediately following

127
the first. Following this line of reasoning, we have

drawn the hydrostatic release curve through the

Hugoniot point. The total energy change along this

hydrostat is equal to the change along the release curve plus a small residual energy at Px = 0, due to the small elastic compression stresses that remain

in the y and z directions (see Appendix for details

of this calculation). After complete release, the final

temperature is 51 °C — greater than the initial value.

This is 24 °C higher than the final temperature

obtained from an isentropic decompression. We will

come back to temperature effects later. In figure 5 we give a data summary of yield

strengths obtained in the work described here, as

well as three pertinent measurements taken

earlier [4, 5], using a free surface technique to detect

arrival and decay of the elastic wave. The data

obtained by the latter method were reduced,

assuming a simple elastic-plastic model. While not

entirely correct, this model yields reasonable yield

strengths. Turning first to the manganin gauge

results, we note that, starting at the Hugoniot

pressure (where, by our interpretation of incipient

yielding, the yield strength is zero), the yield

strength rapidly increases. This yield strength is 1.5 times the stress difference (APa-) between the

hydrostat and the release curve, consistent vfith the

von Mises' yield criterion

X(P;-p,)2=2r-,

and the assumption that

3 1/3 ^ Pi = f= hydrostatic pressure.

,1

Similar hardening of the sample during pressure

release was observed by Barker [5], who used an

interferometer method on a somewhat purer

aluminum alloy (6061). But, as the release process

approaches zero stress, the strength decreases,

which is the reverse of what is expected from

work-hardening.

From the other three measurements obtained with

a free surface technique, it would appear that there

is a gradual increase in the hysteresis or strength.

If the peak strength levels off at about 28 kbars and

then stays constant up to the melting point, we might

speculate at what shock pressure melting should

first be observed on the release curve. If the release

process were isentropic, melting would occur at the

foot (P = 0) of the release wave from a shock of

680 kbars or stronger. With a constant value of F,

the discrepancy from the isentropically calculated

temperatures is roughly proportional to pressure

above 300 kbars. This results in a reduction of the

shock stress required to initiate melting. Using

these data, a shock stress in excess of 520 kbars is sufficient to melt some portion of the sample on

release. If the yield strength increases above 300

kbars, this number should further decrease.

One reason for the selection of 2024 aluminum

alloy for the present study is its use as a standard to

140 1

' 1

'

1

' /I 1 1

1 11 11

120 HUGONIOT- > RELEASE CURVES -

100 V INTERFACE

>\ CONDITIONS:

\a.

80

60

40

20

-

-

' 1 ,\\ , \

EXPERIMENTAL

FLUID - \\ MODEL

\ , 1 , 1 , l\v

0

0 0.2 0.4 0.6 O.i) 1,0 1.2 1.4 1

PARTICLE VELOCITY, mm/Msec

Figure 6. Hugoniot and release curves in the stress-versus- particle velocity plane showing the difference in the interface condition between the duraluminum sample holder and a sample with dynamic impedance poUj for isentropic and

measured release curves.

2.00

r 1 1 1 j t 1 t 1 1 1 1 1 1 11 '

}

O

\ FLUID MODEL CALCULATION

- O FREE SURFACE APPROXIMATION O EXPERIMENTAL RESULTS

I I I I I 1I I I I . I ' 0 100 200

P , kbar

X Figure 7. The ratio of free surface-to-particle velocity.

determine Hugoniots of other materials. To note how

this is done, we turn to figure 6, where the data are reproduced in the stress-versus-mass velocity plane.

The Hugoniot along which pressure increases with

mass velocity behind the wave is represented by the

solid line. Here the pressure of the experimental

release curves decreases as the mass velocity

increases. Calculated isentropic release curves obtained with a fluid model [11] are given by the

dashed Hnes. These curves are used by many shock

wave experimentalists to obtain the mass velocity

behind a shock in a sample placed on an aluminum '

plate, through which this shock pressure was applied.

The dynamic impedance line of the sample, which

has a slope poUg in this plane, meets the measured

release curve at the correct pressure and mass i velocity of the interface between the sample and the

plate. In the example of figure 6, this is about 2 percent lower than would be indicated by the isentropic

release curves. Another, somewhat more approximate but quite

convenient, method for determining the particle

velocity behind a shock wave is the free surface

approximation, where one assumes that the velocity

achieved by the sample surface upon arrival of the

shock is twice as large as the particle velocity. Those

wh© have used this technique should turn to figure 7.

128
Here we show the ratio of the free surface velocity

to the particle velocity as a function of the shock

pressure. Again, a fluid model calculation [11] using

the Griineisen equation of state with the DugdaleMacDonald [12] formulation for {dPldE)v was used

to derive the solid curve. Note that the free surface

approximation becomes progressively worse at higher Px according to this model. The three experimental release curves measured here, however, yield a value less than two. One may expect, however,

that this approximation will hold at higher pressures

than expected prior to now, because of the compensating effects of shock heating and strength.

Barker [12] has observed deviations in 6061 aluminum of 5 percent at even lower pressures.

In summary, we note that the effective yield

strength at high pressure increases drastically along

the release curve, but that the Hugoniot appears to be hydrostatic, or nearly so. This strength increase causes small but noticeable errors in equation-ofstate points measured by using hydrostatic release curves for the 2024 aluminum standard. The free

surface approximation is similarly affected. It is also

noted that melting will occur on the release waves of

shocks weaker than those predicted from fluid

models.

Appendix

We consider a plate which has been shock-compressed in the x-direction and subsequently decompressed to Px = 0. We wish to calculate the yield

strength (the residual stress in the y and z directions) and the temperature.

The stress-strain condition after decompression

to Px — 0 are defined by

8P^ = 8F, = 0-C„6ei + 2C,28e2 (i)

8Py=^8P, = 8P2 = C^28e^ + iC,, + C^2)8€2, (2)

for an isotropic medium without shear. From eq (1),

(3)

since at the yield surface 8Py=Y {the yield strength)

we have from eqs (2) and (3)

therefore

8e2

(Cii + Ci2) 862; (4)

Cj^j + Ci 1C12 2Cj2

and similarly,

8ei =

•2C,2F

Cjj + CiiCi2 2Cj2

(5)

For small 8ei at the yield surface,

8V

8ei + 2862

Cn-Ci2

and

Y,

V'2 Cu-Ci-z

(6)

(7)

Therefore, by substituting eq (7) into eq (5), we get

1 8V Cn

862 = : (8) 2 V Cxi-Ci2*

The adiabatic energy change to the yield surface is

8EVY

(862 + 863) = me2.

Now, substituting eq (7) and (8),

V (8V\ 2 C„ (Q, + CnC,2 -2Q2)

8E = -4 V F (Cn-Ci2)2 (9)

If the plate was compressed from VqPoEo then

decompressed back to Vf, Ej, with Px — ^ and

Py=Y=Pz, we must first estimate the thermal

energy increase before we know the value of the

total elastic volume change,

8V_ Vo-Vj l(dV\

V V ^V\dE)p^^'

where the thermal energy change, ^E, is the

difference between the final and initial energy plus

the elastic energy

AE^Ef-{Eo + 8E).

Since 8E is usually small, the proper value of {8VIV)

and F may be obtained in a few iterations.

Note: The analysis given here was made by

estimating the elastic constants for duraluminum

from

Cn^povl and Ci2 = po

with vl — 0.63 cm//xsec and 1;^ = 0.535 cm//xsec.

Therefore, we have

Cii = 1.1054 Mbar,

Ci2 = 0.7971 Mbar,

8V

862 =^-1.94,

129
and

^ 10.8324 , 8V^^^

References

[1] Barker, L. M., Lundergan, C. D., and Hermann, W., J. Appl.

Phys. 35, 1203 (1964).

[2] Hartman, W. F., J. Appl. Phys. 35, 2090 (1964).

[3] Al'tshuler, L. V., at al., J. Exp. Theoret. Phys. (USSR) 38,

1061 (1960).

[4] Erkman, J. O. and Christensen, A. B., J. Appl. Phys. 38,

5395 (1967).

[5] Barker, L. M., High Dynamic Pressure Symposium

I.U.T.A.M., Sept. 11-15, Paris (1967).

[6] Kusubov, A. S. and van Thiel, M., J. Appl. Phys. 38, 893

(1969).

[7] Morland, L. W., PhU. Trans. Roy. Soc. (London) 251, 341

(1959).

[8] van Thiel, M., et al.. Compendium of Shock Wave Data,

UCRL-50108 (Clearing House for Federal Scientific and

Technical Information, Natl. Bur. of Std., U.S. Dept. of Commerce, Springfield, Virginia) (1966 plus 1967 suppl.).

[9] Keough, D. D., Stanford Res. Inst., Report No. DASA1414-1 (1964).

[10] Taylor, J., GMX-6, Los Alamos Scientific Laboratory,

private communication (1968).

[11] Rice, M. H., McQueen, R. G., and Walsh, J. M., Solid State Phys. 6, 59 (1958).

[12] Barker, L. M., private communication (1968).

DISCUSSION

O. E. Jones {Sandia Laboratory, Albuquerque, New

Mexico): If you assume the von Mises yield criterion

holds on unloading, then how can you explain the

fact that work hardening doesn't occur on loading

as well as unloading?

D. J. Pastine {U.S. Naval Ordnance Laboratory,

White Oak, Maryland): I can understand how you

calculate the Px part of the curve on the release

portion, and I understand how you calculated the

energy difference between the release curve and

what ought to be the hydrostat. But it isn't clear to me how you got the details of the release hydrostat.

G. E. Duvall {Washington State University, Pullman, Washington): I would like to comment on the

strain rate effects. I understand that the construction of this release curve was based on the calculation of propagation velocities point by point in the

profile that you showed. I assume that these propagation velocities were then calculated by assuming

that you knew the form of the adiabat at the rear face of the driver and the position of the gage, and

you could therefore calculate the time of propagation of each stress level from the rear face of the

driver to the gage. I would suggest that if there were

a strain rate effect in the aluminum, this could have

a significant effect on the shape, and you might very

well get a "smearing out" of what would otherwise

be a discontinuity in the release curve.

AUTHORS' CLOSURE

In reply to Jones: I can think in terms of dislocations and dislocation tangles being produced. Let's

consider the pressure-time plane. A shock front

always has some finite width. And within this width

there is some sort of stress-time or stress-strain

profile. We know there are regions of very high strain

rate and regions of lower strain rate. One would ex- pect that the production rate of dislocations would

certainly be different in these two regions.

The question is, whether dislocations will be

trapped or restrained from further motion in the

interval between their production and the stabilization of the stress at the top of the stress wave. If the

production rate is very high during some portion of

the loading process, some dislocations may be

mobile at the top. We know that aluminum in the

annealed state is a very soft material and has a strength of less than a kilobar. AU we need to do to explain this profile is to say that the strength is less

that a kilobar due to the presence of mobile dislocations. On the way down the process is considerably

slower. The dislocation production and trapping

rates are therefore more like those that occur in

normal work-hardening processes.

In answer to Pastine: We assume the Riemann

formulation for the actual release curve, which of course is strictly correct only if the state points on

the wave are a function of stress only. Strictly speaking, it should be redone in detail for the viscoelastic

model. The Riemann formulation used assumes that

the mass velocity of the medium plus the sound

velocity is a function of stress only. You need only

one wave profile because it gives you all the stress

states reached during the unloading process. So you

get sound velocity versus pressure. The hydrostatic

release curve is calculated using a Griineisen equation of state and a Dugdale-MacDonald form for y,

similar to the method originally used by the GMX-6

group. This approach actually assumes that the

Hugoniot state reached is hydrostatic.

On Duvall's comment: The actual deviations from

the hydrostatic isotropic derivation Riemann made

are not known at this time. We don't know how much

the trajectory of each stress point in the positiontime plane curves. All the reduction of velocity-time

measurements or position-time measurements were done with this approximation so far. I haven't seen any good viscoelastic model that is better than that.

130
Calculation of Equation of State from High-Pressure Sound Velocity Data*

Albert C. Holt and Richard Grover

Lawrence Radiation Laboratory, University of California, Livermore, California 94550

We describe a method for calculating the Mie-Griineisen equation-of-state parameters for simple

metals from shock Hugoniot data and a knowledge of the volume variation of longitudinal sound

velocity at high pressure. The theory is applied to the calculation of the equation of state of aluminum.

1. Introduction

Shock-wave experiments are useful for equationof-state measurements because the quantities P, V,

and E behind a plane shock front can be calculated

unambiguously from the velocity [1$ of the shock

wave and the velocity Up of the material behind the

wave [1] ^ Thus, by determining Us as a function of

Up from the results of a series of experiments at dif- ferent shock strengths, one obtains a line, the

Hugoniot, on the PVE equation-of-state surface of

the material. Measurement of the Us (Up) relationship has, for some time, been a routine matter at some of the nation's laboratories, and recent development of improved measuring techniques [2] in- sures that these experiments will continue to provide interesting and useful scientific information.

More recently, however, some shock-wave experimentahsts have turned their attention to the meas- urement of material properties behind the shock

front [3] and to observations of the evolution of the

shape of the shock wave itself [4-5]. The most

striking characteristics of this wave evolution are the

effects of material rigidity and a number of useful

measurements have resulted from this work; for

example, the determination of material strengths for

the case of one-dimensional strain. Some conse- quences of these particular measurements will be

discussed in a later paper of this Symposium, by

Jones and Graham [6]. Another material rigidity effect is the elastic re- lease wave which overtakes the shock front from

within the shocked material. This wave travels at the longitudinal sound speed for the shocked material, so that observation of the wave shape at suc- cessive points in the solid permits one to calculate

the longitudinal sound speed at a point on the

Hugoniot. The results from a series of experiments

yields the variation of the longitudinal sound velocity

along the Hugoniot line. As we have seen in the preceding paper, the shape of the elastic release wave

can be used to estimate the corrections to the

Hugoniot for the material strength of the shock compressed material.

*This work was performed under the auspices of the U.S. Atomic Energy Commission.

' Figures in brackets indicate the literature references at the end of this paper. Paper presented at the Symposium on Accurate Characterization

of the High-Pressure Environment, held at the National Bureau

ofStandards, Gaithersburg, Md., October 14—18, 1968.

The object of this paper is to utilize the highpressure sound velocity data for simple metals to estimate the thermal corrections to shock data

which are necessary to obtain isothermal compression curves. For this purpose, a thermal equation of

state is derived from a modification of Slater's

theory, and the theory is applied to recent release

wave velocity data on aluminum.

2. Theory

We now proceed to make some assumptions

about the nature of the solid and to show that these

assumptions are sufficient to permit one to calculate

the equation of state from Hugoniot data and the

longitudinal sound velocity data mentioned in the

introduction. We begin by assuming that the solid

is described by the Mie-Griineisen equation of

state [7],

P-P^{V)=^[E-E^{V)], (1)

where y is the Griineisen parameter. Vis the specific

volume, P is the pressure, and E is the specific

internal energy. The functions P^{V) and E^{V)

represent the pressure and specific internal energy

along the isentrope which crosses the Hugoniot at P = 0. (Here, and in the remainder of the paper,

"the Hugoniot" means that Hugoniot whose initial

point is at P = 0, T— 300 K.) Further, we assume that

y is given by the modified Slater formula [9],

^1 \ d\nCi 2 d\nCt ^ 3 3alnF 3alnF'

where Ci and Ct are the longitudinal and transverse

sound speeds in the isotropic continuum approximation, and y is regarded as a function of volume but

independent of temperature.

Equation (2) improves on previous theories [8] in

the following way: Instead of inferring the volume

dependence of all the normal mode frequencies

from the volume derivatives of the pressure, eq (2)

infers this volume dependence from the actual

volume dependence of the three acoustic modes in

the isotropic continuum approximation. Thus we

allow for different average thermal properties in the

longitudinal and transverse modes of the lattice

vibration of the solid. This improved theory is still

131
not adequate, however, for more complicated

atomic systems such as diatomic solids which contain distinct optical modes of vibration whose behavior is not closely related to that of the acoustic

modes. We refer to the metals which do not have

these distinct optical modes as "simple" metals.

Finally, in this calculation we assume that the

normal stress in the shock wave, given by the jump

condition [1]

a-x=UsUnVo (3)

(where cr,v is the stress normal to the wavefront and

Vo is the initial specific volume) is equal to the

pressure behind the shock front. Thus we neglect

the effect of stress anisotropy in calculating the

pressure behind the shock front. For a more detailed calculation, one would correct (T/m to P as suggested in the preceding paper.

To derive expressions for y, P^, and E^, we note

that the pressure Ph and the energy Eh along the

Hugoniot curve are known as functions of Ufj and

Us by means of the jump conditions.

and

Eh=UII2

Ph = UsUpVo'.

The third jump condition,

1

(4)

(5)

(6)

and the experimentally determined relationship

between Us and Up,

Us-C + S^Up + S2Ul, (7)

are sufficient to determine Eh and Ph as functions of

V along the Hugoniot. We could now, at least in

principle, substitute Eh and Ph for E and P in eq (1) to obtain an identity in V among the three unknowns

y{V),P'{V),and E'(V).

At this point, we allow eqs (6) and (7) to define the

transformation to a new dimensionless variable,

x—UpIC, which will be used in place of V. By substituting eq (7) into eqs (4), (5), and (6), we obtain

Eh, Ph, and V as functions of x, and on substituting

these quantities into eq (1), we obtain the equation

+ {S^-l)x + CS2X^

l + Six + CS2S^

X + S^X'^ + CS2X^

VoP'^

y (8)

which holds identically in x. The quantities P\ E\

and y in eq (8) are now functions of which we would

like to determine.

Because of the limited range of data available,

adequate solutions to eq (8) can be found by means

of a Taylor series expansion in x for P\ E% and

yix). If we write

VoP' ax + bx^ + dx^ + ex'^ + . (9)

then the unknown coefficients a, b, d, . . . may be

determined by substituting y(x), E^(x), and P*(.v)

into eq (8) and equating coefficients of equal powers

of X. The expression for E^ follows easily, since

dE'

dV (10)

we have

' = - P'{V')dV'. (11)

By changing the independent variable in eq (11) and

substituting P^ from eq (9), we obtain

E'IC' =

ax"" , 6-2S,

x-' 2 3

{d-2Sxb + %Sia)

+- —x^+ . . ., (12)

which is the desired expression for E^{x).

In order to determine y{x), given by eq (2), in

terms of a, b, d, e, . . ., we suppose that Ci{Up), the

longitudinal sound speed, has been measured at points lying on the Hugoniot, and we write

C,/C = a + )8x+Sx2 + (13)

where a, ji,b . . . are obtained by fitting eq (13) to the experimental data. We will use eq (13) and the

bulk modulus /3s, obtained by diff"erentiating eq (9) to obtain a transverse sound speed Ct. An error is introduced here, since /3s is the bulk modulus on the

isentrope, while Ci was measured on the Hugoniot.

Strictly speaking, Ci should be corrected for the

temperature difference between the two lines at constant volume (or constant x). In the appendix, we

calculate {dCildT)\- for aluminum and find that the

temperature correction from the Hugoniot to the

isentrope would be much smaller than the other uncertainties in this calculation, and consequently we

ignore this small difference.

The bulk modulus on the reference isentrope can be obtained from eq (9) by means of the

differentiation,

dP'

dV V{x)

dx dP'

dV dx (14)

and we obtain Ct by substituting eqs (13) and (14)

132
into the equation

1/2 (15)

Finally, we obtain an expansion for y from eq (2),

a A [\a A

-!(f-f)}- -- <-)

where A, B, D, . . . are expansion coefficients for

which are determined by eq (15). This expansion

is given by

CilC^=^{A+Bx + Dx^ + .}, (17)

A = a^-1,

fi-2a/3-(4Si-2),

and

D^2a8 + IB^-3CS2-3d-i5Si-l){Si-l).

The indicial equations for eq (8) require that

a = l, (18)

6 = Si, (19)

and

d = CS2-^\l+l3la +^y, (20)

so that the desired expressions for P^, E^, arid y are obtained by substituting eqs (18)-(20) into eqs (9), (12), and (16).

3. Calculation

We now apply the theory from the first part of the

paper to the problem of calculating the equation-ofstate parameters for 2024 aluminum. For this calculation, we use the following values for the Hugoniot

parameters of eq (7):

C = 0.5282 cm//is

Si = 1.4922

S2=- 0.7229 /Lts/cm.

These values represent a least-squares fit to 300

data points obtained at Livermore up to pressures

of about 800 kbar.

To obtain the coefficients for eq (13), we have

combined results from four sources. The highpressure sound velocity data came from the paper

of Kusubov and van Thiel [7], which preceded this one on the Symposium program, and from an earlier

paper by Erkman and Christensen [10]. These data

are shown in figure 1. Because of the large uncertainties in this high-pressure data, we have used

ultrasonic data [11, 12] to determine the intercept,

a, and the initial slope, )8. The curvature was

obtained from the high-pressure sound velocity data.

The three curves in figure 1 are fits to the data made

in the following way: For curve A we made a leastsquares fit to all of the high-pressure data by varying 8. This represents our best solution for the

problem. Curves B and C were made by omitting,

in the least-squares calculation, each of the last two

data points successively. We use these two curves

to test the sensitivity of the calculation to the data

and to determine how the experimental uncertainties

are reflected in the results of the calculations.

The most sensitive test of the theory is in the

calculated values of y. In table 1, we compare our

values of yo and d In y/d In V with values obtained

from ultrasonic and thermodynamic data. We also

include the values of yo and d In y/d In V predicted

from shock-v/ave data by means of the DugdaleMacDonald theory. Our value of yo is, like the

acoustic value, not in good agreement with the value

of yo obtained from thermodynamic data. On the

other hand, the value of 6 In y/d In F which we obtain

is within 20 percent of the value of d In yjd In V

obtained from acoustic data. Further, the acoustic

values of yo and d In y/d In V with values obtained

from curves B and C, so that one can say that there

is agreement within the uncertainty of the calculation. The large experimental uncertainties are

2.0

1.9

1.8

1.7

1.6

lT' 1-5

1.4

1.3

1.2

1.1

1 1 1

Tb . A

C.

1

A DUNEGAN

0 KUSUBOV AND VAN THIEL 5^ o ERKMAN AND CHRISTENSEN ~

C CURVES: C^/C ^cr + ^x +6

1 1 1

0.1 0.2

U(/C

0.3 0.4

Figure 1. Experimental values of the longitudinal sound

velocities are given as afunction ofshock particle velocity.

Both velocities are normalized to the quantity C, the Up = 0 intercept of the U,(Up)

relation. The parameters a and /3 for the curves were determined from ultrasonic data. The curvature, 6, is obtained by the method of least squares. Curves A, B, and C are used to test the sensitivity of the calculated quantities to the experimental results.

133
Table 1. Values of yo and 3 In y/ 3 In V as obtained

from thermodynamic, acoustic, and shock-wave data.

The Dugdale-MacDonald value is calculated from shock-wave

data by means of the Dugdale-MacDonald theory.

Thermodynamic Acoustic DugdaleMacDonald

This work

A B C

To

3 In 7

din r"

2.13 2.68

1.65

1.98

2.31

2.97

1.33

2.97

0.246

2.97

1.825

Table 2. Comparison ofthe reference adiabat from

this work with the same adiabat calculated by

means of the Dugda;e-MacDonald theory.

Pressure Energy

(Mbar) (X 103 -Mbar cm^/g)

This From DM This From DM

theory y theory y

1.0000 0 0 0 0

0.9534 0.0416 0.0416 0.33 0.33

.9127 .0878 0.884 1.25 1.27 .8766 .1377 .1397 2.68 2.74

.8441 .1902 .1952 4.51 4.68

.8147 .2441 .2546 6.65 7.05 .7877 .2984 .3176 9.01 9.81 .7628 .3520 .3844 11.52 12.95

reflected in the wide spread in dlnyjdlnV from

curves B and C.

Finally, we compare our reference adiabat with an adiabat calculated for the same Hugoniot data

using the Dugdale-MacDonald y. This is shown in

table 2. The good agreement is not surprising since

this adiabat must lie very close to the Hugoniot for

the range of pressure we consider.

4. Concluding Remarks

We have shown how high-pressure sound velocity

data can be used, at least in principle, to calculate

Mie-Griineisen equation-of-state parameters for

simple metals. Although the results of the calculation for the case of aluminum are somewhat disappointing, we feel that there are two strong

arguments in favor of pursuing this investigation

further.

First, the calculation is an extension of the ultrasonic definition of gamma, which agrees well with

the thermodynamic gamma for almost all metals

except aluminum. This aluminum anomaly is not,

at present, understood. Thus, the fact that highpressure sound velocity measurements are available

only tor aluminum is singularly unfortunate. The

calculation should be better on the average for other

metals.

Second, if the error bars on the sound velocity

data could be reduced and if the measurements

could be extended to higher pressures, then numerical solutions of these equations can be carried out

to yield the high-pressure behavior of y. We believe

that the high-pressure longitudinal sound velocity

measurements can be extremely valuable for obtaining more reUable values of the Griineisen y.

5. Acknowledgments

This work has benefitted from conversations of

the authors with Dr. E. B. Royce. We wish to thank

Forrest Rogers for performing some of the equationof-state calculations referred to in the text.

6. Appendix

The purpose of this appendix is to calculate

{dCildT)v in order to estimate the magnitude of the

correction of Ci from the Hugoniot to the reference

adiabat. To this end, we use the thermodynamic

identity.

(f)r(f),-(f).7-

where a is the coefficient of thermal expansion, and

Xt is the isothermal compressibility.

The values of {dCildP)T and {dCildT)p were

calculated from the single crystal data of Schmunk

and Smith [12] and that of Kamm and Alers [13] by

means of the Voigt-Reuss-Hill-Gilvarry [14] averaging scheme. The values for a and Xr were taken from

the tables given by Gschneidner [15]. Using the

values

= 1.821 X 10-3(cm//As) kbar-i

= - 0.747 X 10-''(cm/pts)K-i

a = 7.23x 10-5K-1

XT = 1.385 X 10-12 cm2/dyne,

we obtain

= 2.22X 10-5(cm/^s)K-i.

Since the offset of the Hugoniot from the reference

adiabat is on the order of hundreds of kelvins, we

estimate that the error introduced by neglecting this

difference is less than 1.0 percent. This is much

smaller than the uncertainties in the measured

values of Ci which we consider.

134
7. References

[1] Courant, R., and Friedricks, K. O., Supersonic Flow and

Shock Waves, Chapt. 3, p. 121 (Interscience, New York,

1948).

[2] For example, see: Barker, L. M., Butcher, B. M., and

Karnes, C. H., J. Appl. Phys. 37, 1989 (1966). For a review,

see: Doran, D. G., Measurement of Shock Pressures in Solids, Proc. High Pressure Measurements Symposium,

New York, November 1962, p. 59, ed. A. A. Giardini and

E. C. Lloyd (Butterworth, 1963).

[3] For example: Mitchell, A. C., and Keeler, R. N., Rev. Sci.

Instr. 39, 513 (1968). Also see: Duflf, R. E., et al., ShockWave Studies in Condensed Media, Proc. lUTAM Symposium on Behavior of Dense Media Under High Dynamic

Pressures, Paris, September 1967, UCRL-70247, Rev. I.

[4] Some early work: Minshall, S., J. Appl. Phys. 26, 463 (1955).

[5] For example: Fowles, G. R., J. Appl. Phys. 32, 1475 (1961);

Taylor, J. W., and Rice, M. H., J. Appl. Phys. 34, 364

(1963); Lundergan, C. D., and Herrmann, W., J. Appl.

Phys. 34, 2046 (1963).

[6] Jones, O. E., and Graham, R. A., Shear Strength Effects on Phase Transition "Pressures" Determined from ShockCompression Experiments.

[7] Kusubov, A. S., and van Thiel, M., Effect of 2024 Aluminum

Alloy Strength on High-Pressure Shock Measurements.

[8] For a brief discussion see: Rice, M. H., McQueen, R. G., and

Walsh, J. M., Solid State Phys. 6, 1, p. 41 (1958).

[9] Pastine, D. J., Phys. Rev. 138, A767 (1965).

[10] Erkman, J. O., and Christensen, A. B., J. Appl. Phys. 38,

5395 (1967).

[11] Compendium of Shock Wave Data (UCRL-50108), ed. M.

van Thiel, Vol. 1, Sec. Al, Art. 29-12, p. 1. [12] Schmunk, R. E., and Smith, C. S., J. Phys. Chem. Sohds 9,

100 (1959). [13] Kamm, G. N., and Alers, G. A., J. Appl. Phys. 35, 327 (1964).

[14] Anderson, O. L., J. Phys. Chem. Solids 24, 909 (1963).

[15] Gschneidner, K. A., Solid State Phys. 16, 275 (1964).

DISCUSSION

D. J. Pastine {U.S. Naval Ordnance Laboratory,

White Oak, Maryland) : I would like to point out that

in your model, Poisson's ratio is not independent of

volume — that is, this approximation is not made —

since a continuum is assumed in calculating the

transverse and longitudinal sound speeds separately.

I would also like to make another point in favor of

this approach. Your method and the one that I discussed earlier are independent methods, but the

values obtained by the two approaches for the

Gruneisen parameter are in good agreement.

L. Thomsen (Lamont Geological Observatory, Columbia University, Palisades, New York): One possible source of trouble with this formulation is in the

high-frequency part of the dispersion curve where

the 0) versus A curve bends over so that the acoustic

velocity isn't well defined. In that case your expression for gamma is uncertain to the extent that the

velocity is uncertain.

G. E. Duvall (Washington State University, Pullman, Washington): I inferred that van Thiel thinks

that the slope of the release curve right at the

Hugoniot point already indicates some plastic flow.

If that were the case, his velocity, as indicated by

the slope, would represent a smaller velocity than

the actual elastic longitudinal velocity; I wonder if this would seriously alter your results.

AUTHORS' CLOSURE

As Mr. Thomsen points out, the velocity is not well

defined at higher frequencies, and to some extent our agreement must be considered fortuitous. This

seems, at least on the surface, to be a little surprising, but we hope that a further study may turn up a reason for this surprising agreement.

Note added in proof: The question raised by Dr. Thomsen

may now be answered. Holt and Ross have calculated the in- dividual normal mode gammas for several physically realistic interparticle force laws and for a wide range of densities (see

Phys. Rev. Bl, 2700 (1970)). The calculated values for the trans- verse modes are all very nearly equal to the average gamma for

the transverse modes and conversely the individual mode gam- mas for the longitudinal modes are all very nearly equal to the average gamma for the longitudinal modes. Thus it appears

reasonable to assume that the average gamma for the material

is approximated by the weighted average of the longitudinal and

transverse acoustic gammas. Since these calculations do not

rely on the concept of an "acoustic velocity" one need not concern himself with the fact that an acoustic velocity is not defined near the boundaries of the Brillouin zone. In reply to Dr. Duvall: I believe that the velocity

right at the breaking point is the longitudinal sound

velocity. There is some disagreement on this between myself and Dr. van Thiel.

135

385-762 O - 71 - 10

Shock Temperature Calculations for Silicone Fluid

M. Cowperthwaite and J. H. Blackburn

Stanford Research Institute, Menlo Park, California 94025

The problem of calculating shock temperature indirectly from experimental data without assuming

thermodynamic properties is formulated and solved theoretically. In principle, the {e—p—v) and

(T—p — v) equations of state can be constructed in an overlapping domain of the (p — v) plane from a family of Hugoniot curves centered at points of known energy and temperature.

Experiments were performed in an attempt to construct these equations of state for silicone fluid 210. Shock and free surface velocities were measured to determine Hugoniot curves in the 300-kbar

regime, and energies and densities were measured from — 30 to + 260 °C along the atmospheric isobar

to determine the initial states of the shock wave experiments. In practice, it was necessary to assume a form for the (e—p — v) equation of state, since the differences in volumes between states on Hugoniot

curves at the same pressure above 40 kbar were found to be comparable with the experimental error in measuring the volumes of each of these states. The data were fitted to a Mie-Griineisen type (e—p —v)

equation of state with variable Cv and (dpldT)v, since Hugoniot points indicated a linear dependence

of energy on pressure along an isochore, and Cv varied along the atmospheric isobar. Shock temperatures on the 25 °C Hugoniot were calculated at points of intersection with isentropes and by integrating

with constant atmospheric pressure values of Cv The position of the 296 °C isentrope limits the temperature calculation with isentropes to values below 522 °C and 58 kbar, and the values around 50 kbar

are 8 percent lower than those calculated with the 25 °C value of Cj,, since C„ increases along the Hugoniot curve. Temperature calculations above 58 kbar assumed the 296 °C value of Cv Considerably more experimental work would be required over the entire pressure range to permit determining

equations of state of silicone 210 without making assumptions.

1. Introduction

The use of shock waves to study the high-pressure

environment in the kilobar regime is based on the

assumption that thermodynamic equihbrium is estabUshed behind the shock, where material behaves as a perfect fluid. The method is limited at the present time, however, since the mechanical

state variables can be measured in shock wave

experiments but the thermal state variables cannot.

Thus shock wave data are insufficient to determine

an equation of state, and shock wave studies provide an incomplete characterization of the high pressure

environment. In addition, shocked states are in- complete thermodynamic systems unless shock

temperature can be determined, and their characterization is an equation of state problem equivalent

to that of determining the temperature-pressurespecific volume {T—p — v) equation of state. In

practice, this problem has been solved by assuming

thermodynamic properties that allow the unknown

state variables to be calculated. More specifically,

the form of a complete equation of state is assumed

and shock wave data are used as boundary conditions to determine arbitrary functions that would

otherwise be undefined. In some cases [1]* the form

of the complete equation of state is an expUcit

assumption, in others [2-4] it is implicit in the as- sumptions that enable the state variables to be

*This work was supported by Physical Research Laboratory, Edgewood Arsenal, under Contract No. DA-18-035-AMC-122(A).

' Figures in brackets indicate the literature references at the end of this paper. Paper presented at the Symposium on Accurate Characterization

of the High-Pressure Environment; held at the National Bureau

of Standards, Gaithersburg, Md., October 14-18, 1968.

calculated. But the significance of such a characterization depends upon the correspondence between

the assumed and the actual thermodynamic

properties of the material.

The present paper describes an attempt to determine shock temperature indirectly from experimental data so as to provide a complete characterization of shocked states without assuming thermodynamic properties. The problem of calculating shock

temperature is formulated and solved theoretically.

The equations governing shocked states and the

identities of thermodynamics [5] are combined to define the experimental data required to construct

equations of state without assuming their form. Dow

Corning silicone 210 fluid was used as a test liquid.

2. Theory

We first pose the problem of shock temperature

and formidate a theoretical basis for its solution.

Let e, 5, U, and u denote specific energy, specific

entropy, shock velocity, and particle velocity, and

let subscript o denote the constant state of stationary

fluid in front of the shock. Then the RankineHugoniot jump [6] conditions relating shocked and

unshocked states,

vU=Vo{U-u) (1)

uU^Voip-po) (2)

puvo=U{e — eo + hu'^) (3)

express the balance of mass, momentum, and energy
across the shock discontinuity, and the inequality

s(e, v) > sico, Vo)

expresses the second law of thermodynamics for

the irreversible shock process.

Eliminating U and u from eq 3 gives the Hugoniot

equation [7]

e — Co = i(p + Po) {vo — v)

.

(4)

If an (e—p — v) equation of state satisfies the condition {d'^pldv'^)s > 0, then eq (4) with v < Vo defines

the locus of compressed states on the (e —p — v)

surface that can be reached from an initial condition

(co, Po, Vo) by single shocks. The (e—p — v) equation of state and eq (4) define this locus of shocked

states as a curve in the (p, v) plane, p = Ph(Po , Vo , v), which passes through the point (po, Vo) and is called

the Hugoniot curve centered at (po, Vo)- The eHmination of u from eqs (1) and (2) gives the equation of

the Rayleigh line.

p-po"^ (Ulvo)^(vo — v). (5)

Since a shocked state satisfies eqs (4) and (5), the

intersection of the Hugoniot curve centered on

(po, Vo) and the Rayleigh Une of slope — (U/vo)'^

passing through (po, Vo) defines the mechanical

thermodynamic state (p, v) behind a shock propagating at constant velocity U into a stationary state

(Po, Vo).

With the assumption of thermodynamic equilibrium behind a shock, the state variables of a nonreacting shocked fluid satisfy the following

thermodynamic identities:

de — Tds —pdv (6)

(7)

The objective of the present work is to use shock

wave and low-pressure data to characterize completely the high-pressure environment in the kilobar

regime without additional thermodynamic assumptions. Since shock temperature cannot be measured

directly with present-day techniques and cannot be

calculated from knowledge of the energy along a Hugoniot curve, it is necessary to construct the

(T—p — v) equation of state. Such a construction

must be based on the mechanical properties of

shocked states. At present the only feasible way to achieve this objective is to construct the (e —p — v)

equation of state first, and then use it with the

identities of thermodynamics to calculate the

(T —p — v) relationship. Hugoniot curves form the

basis of the experimental method of constructing

the (e —p — v) equation of state using shock wave

data; the relationship between the (T —p — v) and

(e —p — v) equations of state forms the basis for

calculating the temperature of shocked states.

2.1. The (e—p — v) Equation of State

An experimental Hugoniot curve ph(Po, Vo, v) is the locus of experimentally measured pressurevolume states produced by passing constant velocity

shocks of various strengths into an initial state

(po, Vo). The change in internal energy along a Hugoniot curve is given by eq (4). A family of

Hugoniot curves, each of which is centered on a curve along which the energy change is known, is therefore sufficient to determine the (e-p-v) equation

of state over the domain of the (p-v) plane covered

by the Hugoniots.

In the present work we choose to measure a

family of Hugoniots centered on the atmospheric

(p ~ 0) isobar, because the energy change can

easily be measured along this cross curve.

p(s, v)

^

(8)

For thermomechanical processes, a knowledge of

e, 5, T,p, and v provides a complete characterization

of a thermodynamic state. Thus, the (e — s — v)

equation of state is called complete because of the

identities (7) and (8) that define the (T — s — v) and

(p — s — v) equations of state, but all other equations

of state among these variables are incomplete. The

(e —p — v) equation of state is incomplete because

it cannot be used to calculate temperature and

entropy without additional data. Similarly, the

(T — p — v) equation of state is incomplete because

it cannot be used to calculate energy and entropy

without additional data. However, a knowledge of

any two incomplete equations of state provides a complete characterization because of the identities

of thermodynamics.

2.2. Calculation of the (T—p — v) From the

(e—p — v) Equation of State

Since the (T—p — v) and (e — p — v) equations of

state are both incomplete, it is necessary to estabHsh what additional data are required to calculate

temperature when the (e—p — v) relationship is known. It follows from thermodynamic identities

that the (e—p — v) and (T—p — v) equations of

state are related through the isentropes. The position of an isentrope in the (p — v) plane is determined by the (e —p — v) equation of state and the

isentropic condition de — —pdv obtained by setting

ds = Q in eq (6). The temperature along an isentrope

is given as

T=Ti exp (9)

138
by integrating the identity

*={i),.f-(il* <«»

subject to the isentropic condition ds = 0. Thus to calculate the (T—p — v) from the

(e—p — v) equation of state, it is necessary first to construct a family of isentropes in the (p — v) plane

and then to calculate temperature along them with

eq (9). It is important to note that temperature cannot be calculated with eq (9) unless the temperatures

Ti at particular points (p,, Vi) on the isentropes are known. Measurement of the temperature along any

curve which intersects the entire family of isentropes

permits a value of Ti to be assigned to each isentrope. Thus the (T—p — v) equation of state is determined in the domain of the (p — v) plane covered by

a family of isentropes. For a given (e—p — v) equation of state, there is a thermodynamically consistent

(T—p — v) equation of state for each assignment of

temperature along a nonisentropic curve. Measurements of temperature and energy along

the atmospheric isobar are sufficient to calculate the

(T^p —v) and (e—p — v) equations of state from

a family of Hugoniot curves centered on this isobar.

However, the (e—p — v) and (T—p — v) equations

of state will necessarily be specified over different

but overlapping domains of the (p — v) plane. The

family of Hugoniot curves defines the domain where

the (e—p — v) equation of state is known, but the

family of isentropes constructed from the (e —p — v)

relationship defines the subdomain where the

(T—p — v) equation of state is known.

3. Experiments

Dow Corning silicone 210 fluid (100 centistokes)

was chosen because of its good thermal stability

and its large coefficient of expansion. Static experiments were performed to measure the variation of

density and specific enthalpy h, along the atmospheric isobar. Shock wave experiments were performed to determine a family of Hugoniot curves centered on the atomspheric isobar.

3.1. Static Measurements

The variation of volume with temperature at at- mospheric pressure between — 30 °C and 150 °C was

measured with a density balance. A least squares

fit for the data, with Tin degrees Kelvin,

v-i = 1.2566+ 1.0577 X 10-37+2.604 X 10-^2

was used above 150 °C to calculate initial conditions

for the shock wave experiments. The standard deviation from the least squares fit is less than 0.1 percent

of the largest volume measured.

The variation of specific enthalpy with temperature at atmospheric pressure between —26 °C and

318 °C was determined with a drop calorimeter.

A least squares fit for the data with T in degrees

Kelvin and = 0 at T, = 298 K is

/i= i6960(i/r- 1/r, ) -o.o98i8(r- r,

)

+ 4.842 X 10-4(^2-72).

The standard deviation from the least squares fit is less than ^/4 percent of the largest enthalpy increment measured.

3.2. Shock Measurements

Four explosive shots were performed to obtain

high-pressure equation of state data using an impedance match technique [1]. Each shot assembly

contained both cold and hot liquid samples.

A cross section of an assembly showing the brass

cells containing the liquid samples is illustrated in

figure 1. Plane shocks were induced in the liquid

samples by the interactions produced by a brass

flier striking the cells. The driver system for the

flier plate was a P-80 plane-wave lens in contact

with a 4-in pad of high explosive. Shocked conditions

in the liquid were varied by varying the composition

of the explosive pad.

Direct measurement of shock velocity in the

liquid and indirect measurement of the shocked

condition in the brass at the cell-liquid interface

suffice to calculate the shocked state in the liquid.

The measurements were recorded on 70-mm Tri-X

film with a Beckmann & Whitley 770 camera writing

at a speed of 10 mm//as; object-to-image ratio was

2.6/1 and the slit overwrite time was 0.01 /as. Figure 2 shows a detailed drawing of a streak camera

view of the liquid cells. Changes of reflectivity of

Figure 1. Cross-section of shock experiment.

139
COLD CELL HOT CELL

END OF BRASS FREE SURFACE (STEEL SHIM START OF SHOCK IN BRASS (STEEL SHIMS)

END OF SHOCK IN BRASS AND START OF SHOCK

LIQUID AND BRASS FREE SURFACE

END OF BRASS FREE SURFACE (STEEL SHIM)

END OF SHOCK IN LIQUID (MIRRORED SURFACE)

END OF SHOCK IN BRASS AND START OF SHOCK

IN LIQUID

SECTION B -B

LIQUID

START OF SHOCK IN BRASS ( STEEL SHIMS)- SECTION A-A

|^\\SS^1 BRASS Y''/y ''^'

A

<5I-ASS COVER Figure 2. Hot and cold cells: Camera view of hot and cold cells showing slit projection; and cross-sectional views through hot and cold cells along slit lines.

the steel shims E and C, the brass surface DF, and

the mirrored surface H produce signals that depict

the series of events for the cold cell in the shock

experiment. A typical streak record is shown in

figure 3, where the letters identifying the signals

match the letters in figure 2. The figure shows (a)

the filter plate striking the cold cell at C and J;

(b) the time of arrival of the shock at the surface of

the brass at D, F, G, and /; (c) the time of arrival of

the free surface of the brass at E; (d) entrance of the

shock into the liquid at G and /; and (e) time of

arrival of the shock at the liquid-glass interface at H. The shock velocity through the liquid was calculated from times recorded at G, I, and H. The shock

velocity in the cold brass and its free surface velocity

determine conditions at the brass-liquid interface.

The shock velocity in the cold cell was calculated

using the time differences of location C and D

divided into a corrected thickness of the brass at location D. The corresponding free-surface velocity

was calculated from times at D, F, and E. The time

differences between locations were reduced by

0.022 /AS to account for the transit time through the

steel shim at location E. Similar calculations were made for the hot cell.

Pressure in the brass was calculated using eq (2)

with Po = 0 and with the particle velocity assumed

to be one-half free surface velocity. Comparison of

the calculated (p — u) points for hot and cold brass

Figure 3. Streak camera photographs: (a) Static: (b) Dynamic.

140
Table 1. Summary of shock-wave experiments

SRI

shot No.

Initial conditions Shock measurements Calculated shocked states

Initial temp.Specific

volume

Specific

energies

(ref.Oat 25 °C)

Shock

velocity ^ Particlevelocity r ressure

Specific

volume ^ Specific energy

(°C) (cm^lg) UIg) {mmIns) {mmlixs) (kbar) (cmVg) UIg)

12,326 -32 0.983 -92 '•6.72 + 0.02 3.18 ±0.02 '2\6±2 < 0.518 ±0.005 <^4952 ± 70

11,939 -20 .994 -71 6.91 ±0.07 3.46±0.03 240 ±5 0.497 ±0.009 5887 ± 84

12,228 -20 .994 -71 4.87 ±0.05 1.87 ±0.03 91±2 0.611 ±0.014 1684 ± 44

12,496 -10 1.004 -55 7.11 ±0.04 3.11±0.01 256 ±2 0.494 ±0.004 6474 ± 39

11,939 158 1.178 218 6.47 ±0.06 3.56 ±0.02 195 ±3 0.530 ±0.008 6548 ± 89

12,496 158 1.178 218 6.61 ±0.02 3.72 ±0.01 208 ±2 0.516 ±0.004 7225 ± 55

12,326 159 1.179 220 6. 16 ±0.04 3.28 ±0.03 171±3 0.550 ±0.008 5607 ± 160 256 1.299 399 4.08 ±0.02 1.94 ±0.01 61 ±1 0.679 ±0.008 2290 ± 30

12,326 279 1.330 445 6.09 ±0.07 3.31 ±0.01 ^ 152±2 ^0.606 ±0.009 ^5927 ± 40

12,496 296 1.350 481 6.39 ±0.05 3.91 ±0.02 M84±3 '0.526 ±0.010 -^8095 ± 74

"Errors in shock velocity calculated by summing estimated errors in length and time measurements.

* Errors computed with absolute values of estimated errors: see ref. [9].

*^ Points omitted from analysis.

with the {p — u) points of McQueen and Marsh [8]

for brass initially at 20 °C shows that only the 20 °C

Hugoniot curve need be considered for the impedance match calculations. The mirror-image

approximation was used for the brass isentropes.

Thus pressure and particle velocity at the brassliquid interface were calculated at the intersection

of the Rayleigh Hne for the liquid and the brass

rarefaction curve, assumed to be a reflection of the

Hugoniot curve through the measured free surface

velocity. The specific volume of the shocked Liquid

was then calculated using eq (1), and the corresponding change of internal energy was calculated using

the Hugoniot equation in the form e — eo = ia-.

The shock wave data are summarized in table 1. The precision of the data is controlled mainly by

the pressure uniformity and planarity of the

explosive driver system.

4. Construction of (e—p — r)

Equation of State

The limited number of experimental Hugoniot

points and the restricted range of data along the

atmospheric isobar prohibit the construction of an

equation of state solely from experimental data.

It is important, however, to use both shock wave and

static data to indicate the most appropriate form of

the {e — p — v) equation of state. A graphic fit of

Hugoniot data in the 200-kbar regime, without the

three points from shots 12,326 and 12,496 that

indicate crossing Hugoniot curves, suggests a linear

dependence of internal energy on pressure along

lines of constant volume-isochores; static data show

that the partial derivative {deldp)v varies along the

atmospheric isobar. Thus, the {e — p — v) data were

fitted to the form

e = pf{v)+g{v) (11)

with {deldp)v—f{v) > 0 everywhere in the region of

interest.

Additional properties of this model follow from

thermodynamic relationships. The relationship

between specific heat at constant pressure Cp and

specific heat at constant volume Cv is

Cp=C„[l + T{dvldT)plf{v)] (12)

and Cv is constant along an isentrope, since

The equation for a Hugoniot curve centered at (po = 0, Vo) is

p[f{v)-Hvo-v)]=g{vo)-g{v) (14)

the differential equation for an isentrope is

/dp\ _ pil + dfldv) + dgldv

\ dv) s f{v) (1^)

and the equation [9] obtained by formal integration

of eq (15) shows that the first derivative of g{v)

must be positive, i.e., dgjdv > 0. The rapid increase

of pressure along an isentrope indicates that the

{e—p — v) relationship wiU satisfy the mechanical

stability condition {dpldv)s<Qiif{v) satisfies the

condition {l + df/dv) > 0. Values of /(f) in the specific volume range

0.994 < v < 1.35 cm^/g along the atmospheric

isobar were calculated with the identity.

The values of Cp and (dvldT)p were determined

141
experimentally. The values of {dpldv)s were calculated from the sound velocity data of McSkimin [10],

which were extrapolated to cover the range of initial

temperatures used in the shock experiments. Values

of / (v) in the volume range 0.515 < i; < 0.55 cm^/g

were taken to be the slopes of (e —p) isochores

calculated from the Hugoniot curves. Values of

f{v) in the intermediate range were assumed to lie on a smooth curve because values of (deldp)^

calculated in the neighborhood of 0.54 cm^/g were

approximately equal to the value calculated at 0.994 cm^/g. Least squares fits of the data give the

following expressions for /(f):

/(t;) =-23.055 + 23. 134i; if ?; ^ 1.152 cm^g

/(i;) -60.502 -121.866?; if 0.9693 ^t;^ 1.152cm3/g

+ 62.916i;2

f{v) = 1.3822 + O.lOSi; if y ^ 0.9693 cm^/g

where the constants are given to a number of

decimal places for computation.

Since h='e = g{v) when p = 0, the measured

enthalpies at atmospheric pressure give values of

g{v) in the volume range 0.985 ^ f ^ 1.66 cm^/g.

A linear least squares fit for g{v) in this volume

range is given by the expression

16.107+ 15.517?;.

For values of volume less than 0.985 cm^/g, fits for

g{v) were generated by patching together the highpressure Hugoniot data and the atmospheric data

so as to satisfy the condition dgldv>0. The best

least squares fits for g{v) with a slight discontinuity

in the slope at i;== 1.01316 cm^/g are:

g{v) = 2408.116 + 7566.432?;

-7949.11t;2 + 2787.845j;3 is ?; ^ 1.0136 cm^/g

^(v) =- 16.107 + 15.517?; if?; ^ 1.0136 cm^/g

The fits for fiv) and g{v) define the (e—p — v)

equation of state and allow calculation of Hugoniots

centered on, and isentropes passing through points on the p = 0 isobar in the (p — v) plane. The discontinuous change in slope of the g{v) function at ?;= 1.01316 cm^/g is manifest in the shape of these curves in the neighborhood of this volume.

5. Calculations

5.1. Construction of Hugoniot Curves and

Isentropes

The Hugoniot curves centered on p = 0 at — 20 °C,

25 °C, 158.5 °C, and 256 °C were calculated directly

with eq (14). The isentropes passing through p — 0

at 25 °C, 158.5 °C, 256 °C, and 296 °C were con- structed by integrating eq (15) numerically with a

0.8 09

VOLUME, V—cm'/g

HUGONIOT

CALCULATED

CURVE

CENTER POINT

To

(°CI

25 1 037

ISENTROPES

CALCULATED

CURVES

THROUGH

To

(°C>

^0 (cm'/gl

25 296

1 037

1 36

VOLUME . V — cm' /g

Figure 4. Isentrope and three Hugoniots calculatedfor silicone Figure 5. Hugoniot and two isentropes calculated for silicone 210. 210.

142
Runge-Kutta technique. The —20 °C, 158.5 °C, and

256 °C Hugoniots and the 296 °C isentrope are shown

in figure 4. The 25 °C Hugoniot and the 25 °C and

296 °C isentropes are shown in figure 5.

5.2. Calculation of Temperature

Equation (9) was used to calculate the temperature along the isentropes passing through 25 °C

and 296 °C on the atmospheric isobar. The values of

temperature along these isentropes are listed in

table 2. Calculation of temperature at points where

isentropes intersect the 25 °C Hugoniot defines

values of shock temperature along this Hugoniot

curve. The point of intersection (T'c = 522.1 °C,

Pc = 58 kbar, i;c = 0.661 cm^/g) of the 296 °C isentrope and the 25 °C Hugoniot is the highest point on the 25 °C Hugoniot where shock temperature can

be calculated with the present data. The temperature on the 25 °C Hugoniot below 58 kbar can be

calculated with the isentropes lying to the left of the

296 °C isentrope. The temperature where the

256 °C isentrope intersects the Hugoniot is 456.7 °C,

and the temperature where the 158.5 °C isentrope

intersects the Hugoniot is 291.5 °C.

It is not possible to calculate temperature on the

25 °C Hugoniot above 58 kbar without making

further assumptions. The temperature along the

Hugoniot above 58 kbar was calculated with

constant C,, rather than by extrapolating the lowpressure data further. The equation for shock

temperature [2] above a point {Tc, Vc) on a Hugoniot

centered at {po = 0, Vo),

with

/=exp I dvlf{v)

was used to calculate temperature on the 25 °C

Hugoniot above (re = 522.1 °C, i;c = 0.661 cm^/g).

The integral term in eq (17) was evaluated numerically with a constant value of Ci,= 1.3735 X 10

kbar cm^/g °C equal to the constant value along the

296 °C isentrope. Since d, is constant along an

isentrope, the significance of temperatures calculated with eq (17) depends on the variation of Cv

with volume along the atmospheric isobar above

t;=1.35 cm^/g. In the case that Cv increases with

increasing volume above i;=1.35 cm^/g, it also

increases with the increasing pressure along the

Hugoniot curve, and the values of temperature

calculated with eq (17) under the assumption of

constant d, would be upper estimates for shock

temperature above 58 kbar. The values of temperature calculated with eq (17) are listed in table 3. For comparison the method of Walsh and Christian [2] was also used to calculate temperature along

the 25 °C Hugoniot curve with eq (17). The integral

Table 2. Calculated temperature along the 25 °C

and 296 °C isentropes

Volume 25 °C 296 °C

Isentrope Isentrope

(cm-'lg) ( (>) ( 1.35 296.0

1.29 300.4

I. AD >5U4.U

1 in O 1 A O

1. 10

1.09 328.5

1.05 339.7

Zo.U

i.Ul 90 A ooo.y A QAO U-VOV OO.U 371 A O / 1.0

.909 50.8 398.3

.849 64.3 426.1

.ouy 7Q A lO.O

7/1

0

fjo.U 7no . /uy OQ 1 /I OA 9

.649 113.7 528.5

.609 124.5 551.0

.549 141.3 585.9

.509 153.1 610.1

Table 3. Calculated temperatures on the 25 °C

Hugoniot above 58 kbar

Calculated with

Pressure Volume

C„= 1.9353 X 10-2 kbar cm^/g °C

f{v) = 1.3822 + 0.10798?;

C„= 1.3735 X 10-2

kbar cm^/g °C

1.7839

Temperature Temperature

{kbar) (cm^lg) (°C) (°C) 58.3 0.661 522.1 559.8

71.1 .641 541.4 698.8

85.7 .621 588.9 866.8

102.5 .601 640.3 1069.8

121.5 .581 725.4 1313.8

142.9 .561 839.3 1591.8

167.0 .541 986.4 1926.8

193.9 .521 1171.3 2316.8

223.8 .501 1397.3 2756.8

143
was evaluated under the assumption that Cy and

f{v) were constant along the Hugoniot curve and

that the values of these constants were the values

of Cv and f{v) evaluated at 25 °C. The calculated

temperatures where the 158.5 °C, 256 °C, and 296

°C isentropes intersect the 25 °C Hugoniot are 296.8 °C, 507.1 °C, and 561.5 °C. The values of temperature above 58 kbar calculated by this method

are also listed in table 3.

6. Summary and Conclusions

Because of the scarcity and inaccuracy of experimental data, it was necessary to assume a simple

form of the {e—p — v) equation of state to calculate

the thermodynamic properties of silicone fluid. The

particular form of the {e—p — v) relationship,

e—pf{v)-\-g{v) with f{v) and g{v) arbitrary functions of volume, was suggested from the shock wave

data and also from the variation of {deldp)v along

the atmospheric isobar. Values oi f{v) calculated

from the shock wave data and values of/(t;) calculated from atmospheric static data with the identity

(deldp) V —— Cpidv/dp) sidT/dv) p were used to calculate values of f{v) in the volume range not

covered by the experimental data. The values of

g{v) calculated from shock wave data, the values

of giv) measured along the atmospheric isobar, and

the condition dgjdv > 0 were used to calculate

values of g{v) in the volume range not covered by

experiment.

The Hugoniot curves centered at —20 °C, 25 °C,

158.5 °C, and 256 °C on the atmospheric isobar were

calculated directly with the expression

p\f{v) — h{vo — v)] = g{vo) -g{v)

obtained by combining the Hugoniot equation with

the {e —p — v) equation of state. The calculated

Hugoniot curves were consistent with experimental

Hugoniot points up to a pressure of 240 kbar but

then started to deviate from them. The isentropes

passing through 25 °C, 158.5 °C, 256 °C, and 296 °C on the atmospheric isobar were calculated by

numerically integrating the differential equation

for an isentrope with a Runge-Kutta method; the

temperature along these isentropes was calculated

with the equation T=Ti exp — I dvlf{v). Calculation of the temperature where the 158.5 °C, 256 °C,

and 296 °C isentropes intersect the 25 °C Hugoniot

curve defines values of shock temperature on this

Hugoniot curve. The temperature 7'c=522.1 °C at the point of intersection (pc=58 kbar, i;c=0.661

cm3/g) of the 296 °C isentrope and the 25 °C Hugoniot is the highest temperature on the 25 °C Hugoniot

that can be calculated from the data without additional assumptions. To put a possible ujsper estimate on shock temperature along the 25 °C Hugoniot

above 58 khar. the temperature was calculated by

integrating along the Hugoniot curve with constant

Cv. The method of Walsh and Christian was used to calculate temperature along the 25 °C Hugoniot

pliove 0 kbar.

It is obvious from our calculations of the

{e—p — v) equation of state that many more experimental Hugoniot {p — v) points are needed to con- I struct an {e—p — v) equation of state without first assuming its functional form. Indeed, to test the

feasibility of constructing an equation of state from [ experimental data it would be necessary to deter- ! mine, with accuracy, the positions of at least three

Hugoniot curves in the {p — v) plane. With welldefined Hugoniot curves it is possible to test the

vahdity of thermodynamic assumptions such as {dejdp) x)—f{v) , and if necessary to fit the data with

more complicated functions. However, with a hmited

amount of Hugoniot data and the positions of the

Hugoniot curves not well defined, it is not in general

possible to test or to determine any thermodynamic

properties conclusively. Specifically, it is not possible to test the validity of {de/dp) v=f{v) with our

data because the experimental Hugoniot points do

not lie on isochores in the {p — v) plane. Many more

data points obtained with explosive driver systems

producing an improvement in wave flatness and

pressure uniformity are required to achieve a more

definitive characterization of the (e-p-v) surface.

The temperature calculations in this paper use more thermodynamic data than previous methods

of calculating shock temperature. The shock wave

Hugoniot data and the atmospheric data for silicone

fluid span a larger domain of the {p — v) plane than

either the water data used by Rice and Walsh [3] or the metals data used by Walsh and Christian [2]. In

the domain of the {p — v) plane where it is possible

to calculate temperature without making assumptions about specific heat, the present method of

calculating temperature is considered to be better

than methods based on the assumption of a constant

specific heat. The comparison of temperatures on

the 25 °C Hugoniot curve below 58 kbar calculated

using the present method, with those calculated

using the Walsh and Christian model based on the

constancy of Cv and {deldp)v suggests that the

Walsh and Christian model gives an upper estimate

for shock temperature. The comparison of temperatures on this Hugoniot above 58 kbar calculated

using the Walsh and Christian method but with

diff"erent values of Ci, substantiates this suggestion.

However, the diff^erence between the temperatures

calculated above 58 kbar resulting from the diff"er- ence between the numerical values of the constants

emphasizes the problem of determining the temperature of shocked liquids. The fact that it is possible

with the Walsh and Christian model to calculate

different temperatures with a constant value of

{deldp),, once again reflects the independence of

the (e—p — v) and (T—p — v) equations of state. Because of the identity (de/dp) l{dpldT)v,

the (T—o — v) equation of state is not determined

144
by an {e —p — v) equation of state based on an

experimental Hugoniot curve and the assumption

of constant {del dp) r. The constancy of {deldp)v

provides no guarantee that Cv and {dpldT)v are also

constant. In fact different sets of values of C„ and

{dpldT)r that satisfy the condition of constant

{de/dp) v can be associated with different (T—p — v)

equations of state that wiU give different values of

shock temperature.

It is concluded that the values of shock temperature calculated along the isentropes are more

realistic than those that would be calculated with

models based on less thermodynamic information.

7. Acknowledgments

The authors are grateful to Physical Research

Laboratory, Edgewood Arsenal, for supporting this

work under Contract DA-18-035-AMC-122(A),

and thank Miss B. Loo for performing the numerical

integration and plotting the diagrams.

8. References

[1] Rice, M. H., McQueen, R. G., and Walsh, J. M., Compression

of solids by strong shock waves. Solid State Physics Vol. 6 (Academic Press, Inc., 1958).

[2] Walsh, J. M., and Christian, R. H., Phys. Rev. 97, 1544

(1955).

[3] Rice, M. H., and Walsh, J. M., J. Chem. Phys. 26, 824 (1957).

[4] Cowperthwaite, M., Am. J. Phys. 34, 1025 (1966).

[5] Gibbs, J. W., The Scientific Papers of J. WiUard Gibbs; Vol.

1, Thermodynamics (Dover Publications, New York, 1960).

[6] Zeldovich, Ya B., and Kompaneets, A. S., Theory of Detonation (Academic Press, New York & London, 1960), pp. 9-10.

[7] Courant, R., and Friedrichs, K. O., Supersonic Flow and

Shock Waves (Interscience Publishers, New York, 1948).

[8] McQueen, R. G., and Marsh, S. P., Equation of state for 19

metallic elements and shock wave measurements to 2 megabars, J. Appl. Phys. 31, 7, 1253-1269 (1960).

[9] Cowperthwaite, M., and Blackburn, J. H., Temperature of Shocked Liquids, Stanford Research Institute Special

Tech. Rept. No. 14, Contract DA-18-035-AMC-122(A)

(1967).

[10] McSkimin, H. J., J. Acoust. Soc. Am. 29, 1185 (1957).

DISCUSSION

A. Holt {University of California, Lawrence Radiation Laboratory, Livermore, California): In your

present work I assume that you have concluded that

the Hugoniots lie too close to one another at high

pressure to make this method valuable in obtaining

equations-of-state. If this is true, would you say that

this result would be found with other materials

besides silicone?

D. J. Pastine {U.S. Naval Ordnance Laboratory,

White Oak, Maryland): I would like to make the

point that you can calculate the rate of change with

temperature of the initial slope of the Ug — Up relation, which gives you an idea of how the Hugoniots

are going to change with temperature, and this turns

out to be of the order of thermal expansions — that

is, very small. So you could expect that for a 300

degree temperature change in metal the Hugoniot

curves wUl be right on top of each other.

J. W. Forbes {U.S. Naval Ordnance Laboratory,

White Oak, Maryland): I'm interested in your choice

of brass as a material to shock as the bottom of your

cells, because, in work by Nat Colburn and myself,

we have found that due to preheating, brass gives a

large anomalous effect as far as Ug — Up relationship

is concerned. We reported this a year ago at Bethlehem. It is also my understanding that McQueen and

Marsh discontinued use of brass because they

found it is too difficult to use as a standard. In our

data the separations in Hugoniots show a decrease

in shock velocity of perhaps up to 20 percent for the

same free surface velocity. We don't understand

exactly why.

G. E. Duvall {Washington State University, Pullman, Washington): Your curves were plotted in the

p — v plane, and aU tend to fall on one another. If you

plotted in the p — T plane, they would have been

nicely spread out. Or, better yet, a three-dimensional

plot would be useful.

AUTHORS' CLOSURE

Reply to A. Holt: It is unfortunately true that at the present time the method is limited by the accuracy of the Hugoniot measurements and the close

spacing of the Hugoniot curves which wiU be even more pronounced for heated solids. The way to obtain Hugoniot points away from the crystal density

Hugoniot curves of metals is to shock porous

samples of low density as in the equation of state

work on aluminum performed at Stanford Research

Institute by G. D. Anderson, A. L. Fahrenbruch,

and G. R. Fowles.

Reply to D. J. Pastine: I agree.

Reply to J. W. Forbes: I don't understand why

either. The brass was both heated and cooled in our experiments and corrections were made accordingly.

Reply to G. E. Duvall: I think that is an optimistic

way to look at the data.

145
I
The Equation of State of Selected Materials for High-Pressure References

W. J. Carter, S. P. Marsh, J. N. Fritz, and R. G. McQueen

University of California, Los Alamos Scientific Laboratory, Los Alamos, New Mexico 87544

1. Introduction

Establishing a standard for high-pressure dynamic

shock wave work requires more than precise meas- urements of Hugoniot data. Indeed, ahhough a P — V relation may in principle be determined (at

least under the Rankine-Hugoniot assumptions) in

terms of mass, length, and time with arbitrary accu- racy if sufficient care is taken, in order for the material to be useful as a standard it is also necessary to extend the equation of state into regions off the

Hugoniot. For example, if the standard is to be used

in other shock wave experiments it is necessary to calculate reflected shocks and isentropes in order

to perform the impedance-matching calculations.

If it is desired to transform the shock wave data to a form useful in static experiments, it is usually

necessary to calculate isotherms. These calculations

all assume knowledge of thermodynamic properties

of the material in regions where such data are un- obtainable at present. Although these calculations are not difficult, the necessity of extending the shock

wave results to regions off" the Hugoniot probably

results in the major uncertainty in establishing a dynamic pressure standard.

The Los Alamos Scientific Laboratory has long

been interested in establishing such a dynamic

pressure standard. To this end, we have devised

experimental techniques which measure directly

both the shock velocity and material velocity needed

to establish pressure-compression curves through

the Rankine-Hugoniot equations. Although these

techniques have been applied to a number of dif- ferent materials, the direct or primary standards are now considered to be deoxidized copper and 2024

aluminum alloy. Once these primary standards are

well established, it is possible in turn to measure

the equation of state of other materials relative to the standards through the impedance match method

If the Hugoniot data obtained in this way are suffi- ciently well-determined, these materials may in

turn be used as secondary standards, particularly if they have properties useful in static experimentation. A number of such materials are included in

this report.

Familiarity will be assumed with the RankineHugoniot equations and some of their consequences,

as well as many of the techniques of dynamic high-

*Work done under the auspices of the U.S. Atomic Energy Commission.

Paper presented at the Symposium on Accurate Characterization

of the High-Pressure Environment, held at the National Bureau

of Standards, Gaithersburg, Md., October 14-18, 1968.

pressure work. For a more detailed discussion of

these questions, see reference [1].'

2. Direct Measurements of the

Hugoniot Equation of State

2.1. Experimental Techniques

Perhaps the simplest yet potentially most accurate method of obtaining Hugoniot data is to measure the velocity of a shock front created by the

symmetrical collision of two plates moving at known

relative velocity. In the laboratory system, this can

be done by explosively driving one plate (the driver)

into a stationary one (the target); then the measure- ments of interest are the velocity of the driver at the time of impact (ud) and the corresponding shock

velocity in the target (ug). Provided the target and

driver are in the same thermodynamic state, symmetry requires that the particle or material velocity

(up) behind the shock wave be equal to one-half the

measured driver velocity. Varying the driver velocity by changing the driver thickness, the explosive

system, or the length of free run between the driver

and target then allows the Hugoniot locus in the

Us — Up plane to be determined. The RankineHugoniot equations centered at f = 0

P=PoUsUp (1)

polp=VIVo={us — Up)lus (2)

E-Eo=ll2PiVo-V) (3)

then serve to transform the experimental data to other thermodynamic quantities such as pressure

(P), density (p), or energy (E). The method has

the obvious advantage that it is free of the necessity

of calculating isentropes and the usual accompanying assumptions, and furthermore does not depend

on zero-pressure thermodynamic data not directly

obtainable in the experiment, such as thermal

expansions and specific heats. These limitations are all inherent in the previous method used to establish dynamic standards through the measure- ment of free surface velocities [2, 3]. The experimental set-up is shown in figure L A

sweeping image camera is used to record shockwave arrival times by the flash gap technique developed by Walsh and Christian [4]. The shock velocity

is determined in the usual way by measuring the

• Figures in brackets indicate the literature references at the end of this paper.

147
ACRYLIC FLASH

BLOCKS

GROOVE FOR MEASURING

SHOCK VELOCITY

SLIT PLATE

TARGET

GROOVE FOR MEASURING

DRIVER VELOCITYDRIVER

POLYETHiYL'ENE FILM

HIGH EXPLOSIVE

Figure 1. Design of the experimental assembly used to obtain

shock velocity-particle velocity data.

The upper part of the figure shows the arrangement of the plastic flash blocks on

the target plate which in effect give four independent sets of u.s- — Urf data points. The

left hand side of the assembly is used to measure the shock velocity and the other side the driver velocity. Each set of blocks is viewed through four or five slits. The poly- ethylene film placed behind the driver plate helps prevent driver plate breakup before

collision.

transit time of the shock wave through a known

thickness of material. The driver velocity is determined by a differential measurem_ent of the transit

time of the front surface of the driver through a shallow groove milled on the back surface of the

target plate. The resulting photographic trace is shown in figure 2. As simple and straightforward as the technique

appears to be, there are nevertheless a number of

potential sources of error. Perhaps the most serious

of these is the lack of hydrodynamic equilibrium in

the shock, which of course is a property of the material and not of the technique. This will be discussed

in more detail in section 2.3. Other sources of

difficulty are the following:

1. The driver plate may not be moving at constant velocity at the time of impact. Studies have

shown that a series of accelerations, caused by

shock reverberations, are produced when a plate is driven by high explosives [5]. Hence, the driver

velocity may not be constant as the driver moves

through the groove and the measured particle and

shock velocities will not be compatible. Whether or not an acceleration occurs as the driver is moving

through the groove depends on such factors as the

length of free run, thickness of the driver, and the

pressure level produced by the high explosive as- sembly. These factors must all be considered in

design of the shot assembly in order to minimize

Figure 2. An enlargement of a photographic record obtained

using the assembly shown in figure 1.

In this record, time increases downward; hence, the early traces on the left hand

side represent the shock wave arrival at the bottom of the narrow groove. The corre- sponding reference traces estabUsh the wave arrival at the top of the plate. Since the

flash blocks on the right hand side are at a lower level, the reference traces for the measurements arrive earlier. The offsets represent the difference in the driver-shock

transit times through the small gap machined in the bottom of the plate.

such errors. However, one can guarantee that the

initial shocks on both the shock and driver velocity

sides of the plate are the same by machining the

groove directly into the back of the target plates.

Also, by placing the groove at this position, small

pressure pulses present in the driver plate at col148
lision time are more likely to result in an increase in

the driver velocity than an increase in the shock

velocity. This is advantageous, since the Us — u,, Hugoniot is relatively insensitive to errors in the particle velocity, especially at low pressures.

2. Rarefactions from the rear surface and from

edges in the assembly may give deceptively low^ apparent velocities. Since a shock is propagated both

into the driver and into the target on coDision, it is possible that the rarefaction wave from the rear of

the driver may overtake the shock wave moving

through the target before the measurements have

been made. The catch-up ratio, r = target thickness/

driver thickness, can be computed using the longitudinal wave velocity at pressure; these calculations

show that values of r greater than 4 are suspect and

should be avoided. For a given desired pressure

level, this limits the thickness of the target plate and

hence the accuracy of the experiment. This hmitation, of course, is most stringent at high pressures.

The width and depth of the groove must also be so

tailored that sidewise rarefactions do not pinch off

the wave front; this again Umits the depth of the Ud

groove and hence the accuracy of the experiment.

This limitation is most stringent at low pressures.

3. The thermodynamic states of the driver and

target may not be identical. This is certainly always

the case, since the driver has been shocked while

being accelerated but the target has not. However, a

correction for this shock heating of the driver can be

applied to the data if the shock history of the driver

is known. This is done by recentering the known

Hugoniot of the driver to an initial state of higher

energy and lower density through the equation

previous section. However, this method can be

used to determine the Hugoniots of a few selected

materials to be used as primary standards. Two of

the most satisfactory materials serving as shock

wave standards are deoxidized annealed Cu and

2024 Ai alloy; these materials have therefore been

HUGONIOT DATA FROM SHOCK - PARTICLE VELOCITY EXPERIMENTS

10

6

4

c4

1

c

2024 ALUMINUM

92IT ALUMINUM

IRON COPPER

URANIUM - 3 WT PER CENT MOL

1 i

fBDENUM

0 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

UP KM/SEC

Figure 3. Hugoniot u^-u^ data for 2024 Al, 921-TAI, Fe, Cu,

and U-3 percent Mo alloy.

Each set of data points was obtained by independent symmetrical collision experi- ments. The curves were obtained by the method of least squares.

E = EH+{P-PH)lpy. (4)

This recentered Hugoniot is then used in an impedance match calculation to give the corrected

value of Up. At low pressures the correction in

particle velocity is negligible. Since almost all the

shock heating arises from the first strong shock

from the high explosive, an attempt was made to minimize experimentally the magnitude of this

pulse. A small air gap between the high explosive

and the driver appears to accomplish this while

not appreciably lowering the final driver velocity.

This has the added advantage of reducing the ten- sion waves due to the Taylor wave in the high explosive. Using thick layers of high explosive is also

helpful in the latter respect.

Other errors in these experiments are probably

random in nature, leading to scatter of about ±¥2

percent in shock velocity. These errors arise from

the high degree of precision required in machining

and assembly of the shots.

2.2. Hugoniot Data for 2024 Aluminum

and Copper

It is obviously impractical to obtain Hugoniot

data for all materials by the method outlined in the

4.0 UP KM/SEC

Figure 4. Hugoniot F — u,, data for thefive standards. ' The curves drawn through the data were calculated from the u^ — u,, fits shown in figure 3. In this plane the Hugoniots for the two Al alloys are indistinguishable. A substantial area of the P — plane is covered by these five materials, so that any unknown

material can be studied by using a standard of comparable shock impedance. This minimizes errors arising from long extensions of isentropes and reflected shocks away

from the Hugoniot.

149
Table I. Thermodynamic constants at 293 K

Material Density Thermal

expansion

Specific

heat

Debye

temperature

Sound velocities Griineisen

parameter

po a Cp e Cl Cb C(,(e.c.) C6(comp) y

2024 Aluminum

g/cm^

2.785

deg-'

21.8 X 10-6

ergslg deg K

0.879 X 10' deg K350

kmjs

6.39

km/s

3.15

km/s

5.25 [14]

kmjs kmjs

2.00

Copper 8.930 16.3 0.386 306 4.76 2.33 3.93 [14] 3.91 [15

3.92 [16' 3.95 [17

3.98 [8] 1.96

Silver 10.490 18.8 0.234 225 3.14

3.14

3.08

[15]

:i8]

[19]

3.19 [8] 2.38

Sodium 0.968 20.8 0.122 160 2.54 [10]

2.61 [25]

1.17

1.17

Molybdenum 10.206 4.9 0.251 314 6.45 3.47 5.04 [14] 5.13

5.12

[15]

20]

5.19 [8] 1.52

Palladium 11.991 11.8 0.238 220 4.57 2.06 3.90 [14] 3.80 [8] 2.26

Periclase 3.585 8.7 0.913 760 9.70 6.01 6.76 [14] 6.54

6.58

6.80 6.74

15

2122

23

6.82 [10] 1.32

Magnesium 1.740 23.9 1.025 237 5.74 3.15 4.44 [14] 4.46 [24] 4.45 [9] 1.43

Thermal expansion and specific heat data were obtained from refs. [11-13].

The sound velocities c;, C(, and Cb are longitudinal, transverse and resultant bulk velocities obtained by the pulse-echo technique on polycrystalline samples. The velocity ctifi.c.) is the bulk velocity calculated from measurement of elastic constants on single-crystal

samples, while Cb (comp) is the bulk velocity from measurements of isothermal compressibility, transformed by us to adiabatic compressibility. References to original sources are indicated in parentheses.

thoroughly studied. In order to check the resuhs

and extend them to regions of higher impedance,

similar studies have been performed on Fe and a high-density alloy of U and Mo. Finally, because of

possible rigidity effects in the 2024 Al alloy, another

aluminum alloy (921—T) with a very small elastic

wave was also studied; unfortunately this material

proved to be unusable because of large variations in

porosity. The results of these Hugoniot measure- ments are shown in figures 3 and 4 for all five of

these materials. Table 1 presents some of the important thermodynamic constants for Cu and 2024

Al as well as further selected materials. Table 2

lists the results of a hnear least squares fit of the

data in the form

Us — Co + SUp. (5)

Values of the bulk sound velocities for these materials from compressibihty, elastic constant, and

ultrasonic measurements are listed in table 1; comparison with the leading coefficient in eq (5) shows

excellent agreement for Cu but somewhat higher

values of the Hugoniot intercept for the 2024 Al

alloy. This is attributed to rigidity effects and is discussed in section 2.3.

In order to check the consistency of the measured

shock velocity-particle velocity loci, we have tested

each material against each of the others by the impedance match technique, using each in turn as a

Table 2. Coefficients of the Hugoniot Equation

Us = Co + SUp

Material

2024 Aluminum

Copper

Silver Sodium

Molybdenum....

Palladium

Periclase

Magnesium

Co

kmis

5.328 3.940

3.229

2.629

5.124

3.948

6.597

4.492

1.338

1.489

1.595

1.223

1.233 1.588

1.369

1.263

Standard to remeasure the Hugoniot of all the others.

The overall agreement is quite satisfactory and

shows that all five standards satisfy this necessary

condition. It should be remarked, in view of the wide

range of impedances of these materials, that this

cross-check is a stringent test of the individual

Hugoniot loci and also a test of the adequacy of the

values of the Griineisen parameter y listed in table 1 and the widely used assumption of constant py.

It is of interest to compare the results of these

measurements with the free surface velocity meas- urements originally used to obtain the Hugoniots of

150
FREE -SURFACE VELOCITY DATA

10

\^

y

X 202

0 202

COP

<i ALUMINUM

4 ALUMINUM ( PIN DATA)

0 2 4 6 8 10 UFS KM /SEC

Figure 5. Shock velocity-free surface velocity for 2024

Al and Cu.

The data were obtained by flash-gap techniques quite similar to those described

earlier. The solid curves were obtained by evaluating the Riemann integral along the isentropes calculated from the shock velocity-particle velocity Hugoniots.

the standards [3]. These measurements consisted of

shock velocity-free surface velocity pairs obtained on the same shot. To obtain a. Ug — Up Hugoniot from

these, the Riemann integral

rvw

Ur = Ufs-Up= [-{dPldV)sVl^dV (6)

Jv(P)

was evaluated indirectly by an iterative procedure

in v^hich calculated free surface velocities were

made to agree with experimental free surface

velocities by varying the particle velocities. The

isentropes were tied down along the zero pressure

isobar by requiring the internal energy E{V) at the

foot of the rarefaction release wave to agree with

that calculated from specific heat and thermal

expansion data. In figure 5 the experimental Ug — Up

data points are compared with the Us — U/s locus

calculated directly from eq (6) using the new shock

velocity-particle velocity Hugoniots of Cu and 2024

Al. The agreement for Cu is satisfactory in view

of the uncertainty in the zero pressure data and the

approximations used in calculating the isentropes.

2024 Al, on the other hand, contains regions where

the agreement is poor. These deviations are probably real and might reflect the effect of phase

changes (melting and vaporization) that are not

handled precisely in the calculated rarefaction

release isentropes.

2.3. Effect of Rigidity

In order for the Rankine-Hugoniot equations to apply across a shock front, it is necessary that

(a) (b)

X Up

Figure 6. The effect of rigidity on the Hugoniot curves.

In (a) are drawn possible curves for the various wave velocity loci as a function of

particle velocity. The upper curve represents a one dimensional Hugoniot and could be used to find the instantaneous longitudinal, elastic wave velocity. The lowest curve

represents the transverse wave velocity due to the shear modulus along the Hugoniot as

a function of particle velocity. It has been estabhshed through static experiments that the shear modulus increases with pressure so it has been drawn with a positive slope. However, since the temperature along the Hugoniot increases in an exponential manner,

this curve has been drawn with a downward curvature. If the material melts on the Hugoniot this curve will go to zero and the longitudinal and bulk velocity curves will coincide. The remaining figures correspond to the lower pressure region of (a) where rigidity effects will be most pronounced. The relationship of the experimental Hugoniot and

true release isentropes with the equilibrium Hugoniot is indicated in (b). Here HEL rep- resents the Hugoniot elastic limit on compression and REL the elastic limit on the re- lease wave. Figure (c) represents the type of pressure profile curve that would result from the collision of a finite driver plate with a target plate of the same material if its shock-wave equation of state were given by (b). Figure (d) is the P — transform of (b). In figures (b), (c) and (d) the relative amplitudes of the various waves are the same.

hydrodynamic equihbrium be obtained. Particularly

at lower pressures, it is always possible that

fingering effects of material rigidity can alter the

ideal Hugoniot behavior. The behavior of a plane

shock and rarefaction wave with rigidity present

is sketched in figure 6. The pressure and volume

are initially related by a one-dimensional equation

of state, since initially there can be compression

in one coordinate only. However, as the pressure

increases, the deviatoric shear stress increases

and the material begins to shear plastically, i.e., to relax to a three-dimensional equihbrium state. If the maximum resolved shear stress tolerable to the

material is small compared with the shock stress,

and if plastic flow can occur sufficiently rapidly

(that is, if the relaxation time is small compared

to the measurement interval), rigidity will have no pronounced effect on the Hugoniot and the material

should exhibit true hydrodynamic behavior. However, if the material does not relax completely

385-762 O - 71 - 11
from the 1-D to the 3-D equnibrium Hugoniot, then

all states on the experimental Hugoniot will be

slightly stiffer than the hydrostat. This means that

measured shock velocities will be higher than the

true equilibrium Hugoniot and the resulting

Hugoniot intercept in the Us — Up plane should lie above the measured bulk sound velocity. As seen

earlier, this is indeed the case for 2024 Al, where

the Hugoniot intercept is 2.7 percent too high.

Previous free surface velocity profiles measured

here on 2024 Al and Cu by the dc capacitor technique have shown that the Hugoniot elastic hmit,

or dynamic yield point, of 2024 Al is about 5.5 kbar

but is practically nonexistent for Cu [6]. An

assumed residual stress of about 2 kbar in the ex- permental 2024 Al Hugoniot, when subtracted to obtain the equiUbrium Hugoniot, allows the

Hugoniot intercept and bulk sound velocity to agree.

On the basis of the sound velocity, the small amplitude of the elastic precursor, and the very steep

wave profile observed even at low pressures, it can be assumed that the experimental Hugoniot

of Cu is representative of a true hydrostat.

If the material does not relax entirely to the equihbrium Hugoniot, an elastic precursor release wave

which is even larger than the shock precursor should

also be expected (see fig. 6). This should be manifested by a smaller free surface velocity than the

true hydrostat would predict. Ideally, of course, in our experiments the free-surface and driver velocities should be equal in the low-pressure regime.

Symmetrical-coUision cannon shots using 2024 Al,

however, show that free surface velocities are less

than the projectile velocity [6]. High-pressure

explosively driven experiments on 2024 Al, where

the free surface velocity and the driver velocity

were measured on the same shot, show a similar

effect. Even at driver velocities up to 3 km/s, a

deficit in free surface velocity of about 1 percent was

P

V V* CRYSTAL 0 V

Figure 7. Schematic to illustrate how porous materials can be used to measure the Gruneisen parameter.

The difference in internal energy between the two Hugoniots at is indicated by the crosshatched area. The corresponding pressure difference is also indicated. By using samples of low initial densities, very high temperatures can be reached in the shock.

POROUS 2024 ALUMINUM

• 0

0^Vl* a, 0

0

0

•

F

c 0 0.91 • o.ec

RflCTION

RYSTSL OENS OFITY

n 0.7C

X 0.5

!

.22 .24 .26 .28 .30 .32 .34 .36 VOL cm^/g

Figure 8. Experimental values for (dE/clP)i versus volume

determined from shock wave studies on porous 2024 AL

In this and the following figure each data point was computed from the shock-wave

data for each porous sample and the fit of crystal density Hugoniot data. The curves were computed from least square fits of the Ug — Up data for each density jiroup. Samples

with lower initial densities represent average values of {dEfdP)\ measured over larger temperature ranges at a given volume. Within the experimental accuracy there does

not appear to be any measurable variation of y with temperature. The general appear- ance of the data gives support to using the simple relationship py constant.

POROUS COPPER

.10

o^.06

Q.o•s. U

•o

.04

.02

L •

0

0 ifesL « q

I

0

•

I

• 0 1 1 \ 0 1

FRAC ION OF

1

CRYSTAL DENSITY

0 0.88

• 0.62

0.71

X 0.64 + 0.51

^ I .075 .080 .085 .090 .095 .100 .105 .110 .115 VOL cmVg

Figure 9. Experimental values for (dE/dP)v versus volume

determined from shock wave studies on porous Cu.

In addition to the experimental curves the values of (dEldP)v versus V obtained from the Dugdale-MacDonald relationship (solid line) and y constant (dashed line) are

also plotted. It can be seen that the data agree quite well with the Dugdale-MacDonald

calculation. At small volumes experimental data lie well above the y constant line, which is known to be a poor representation of the volume dependence of y because it predicts vanishing sound speeds at large compressions.

152
found. Hence, elastic-plastic behavior is important

in 2024 Al even at high pressures. Copper does not

show this behavior except at very low shock

pressures.

Rigidity must obviously play a role when 2024 Al

is used as a standard for materials of lower shock

impedance since the impedance match solution re- quires calculation of isentropes. It is not clear how a

residual deviatoric stress will affect the results

when a material of higher impedance, requiring calculations of reflected shocks, is used with such a standard. It is hoped that the increase in the shear

modulus due to the offset in pressure is compensated by the increase in temperature also en- countered in strong shock waves.

2.4. Experimental Studies on Porous Samples

In addition to a Hugoniot Us — Up locus, it is necessary to know the properties of the Griineisen parameter y over wide regions of the P — V plane in order

for a material to be useful as a shock-wave standard.

This is because the impedance match technique re- quires the calculation both of reflected shocks and

of release isentropes. Experimentally, this is ex- tremely difficult to do. One way of attempting this is to use porous samples of the material of interest in

equation-of-state shots using the impedance match

technique. An average value of the Griineisen ratio

is then determined by the pressure and energy off- set from the crystal-density Hugoniot at a given

volume through the relation (see eq (4))

POROUS 2024 ALUMINUM

12 I I

2 1 ] 1 1 I \ I 0 1 2 3 4 5 6 UP KM /SEC

Figure 10. Shock velocity-particle velocity data for porous 2024

Al. The curves were computed from the least square fit of the crystal density Hugoniot and a constant value of {dEldP)i , obtained from the y and p in table 1, for the average

initial density of each group of samples. The data were also corrected for deviations

in initial density. In this and in figure 12 it can be noted that the calculated curves

originate at Ug = Up =0. Thus the higher density curves have considerably more curvature in the low-pressure region. The low initial density curves become almost linear, since they are approaching the initial density po = pcrysT/(T + 2) which yields a vertical P —V asymptote at crystal density.

This is shown schematically in figure 7. The values

of (dEldP)v computed from the crystal density

Hugoniot and the experimental data points are shown in figures 8 and 9 for 2024 Al and Cu respectively. Also shown in these figures are the smoothed

results obtained by fitting the experimental data

points with a quadratic Ug — Up fit which is then used

to find the pressure and energy offsets from the

crystal Hugoniot. It is obvious that the method is a very sensitive one, and the precision requirements

to measure y accurately are quite severe. However,

there is no indication in these figures of a volume

dependence of {dEldP)v, so that the results are compatible with the assumption of constant py.

It is possible to compute the Hugoniot of a porous

material from the crystal density Hugoniot through

the equation

P=Ph[1 - (py) (^0- n/2]/[l - (py) {V* - V)/2]

which gives the pressure as a function of volume

for a porous material centered at the volume

Fq. Here Vo is the initial volume of the crystal

POROUS 2024 ALUMINUM

« i.00\

o 0.91 / FRACTION OF • 0.80 ) CHVSTAL DENSITY

.4 .5 .6 .7 .8 .9 1.0

V/VO

Figure 11. Pressure versus relative volume for porous 2024 Al.

In this and in figure 13 the curves are the P—I'/' o Iransforqis of the computed Ug—Up

fits. The scatter in the data due to variation in density is more pronounced in these

plots. Vt, in the figures refer to the average density of each group of samples. Although

the compression is large for these low initial density samples, the final densities are

not. The approach to the asymptotic density is quite apparent.

153
POROUS COPPER MOLYBDENUM, MAGNESIUM. PALLADIUM, SILVER, AND SODIUM

y/

^— X MOL/BDENU

0 MAGNESIUM • PALLADIUM

SILVER

a SODIUM

UP KM/SEC

Figure 12. Shock velocity-particle velocity data for porous Cu. Figure 14. Hugoniot u^ — u„ data for Mo, Mg, Pd, Ag, andNa.

Where possible, bulk sound speeds derived from both compressibility and ultra- sonic measurements have been included, although these have not been used in the

least square Bts of the Hugoniot data.

POROUS COPPER

1,8

1.4

q:^ ID

0 « l.00~\

0 0.88 1 FRACTION OF

0.82 ( rOVCTAi ncMCIXV 071 ( 0.64

• 0.5 1^ ]

\ \ V

0 \

\ 0

V'

\ >

<\ ^

K 0

.6 .7

V/VO

1.0

Figure 13. Pressure versus relative volume data for porous Cu.

density material, and Th is the pressure on the

crystal density Hugoniot at the volume V. This

Hugoniot can be transformed to the Ug — u,, plane

through the equations

us-^v*[Pi{v*-v)yi^

u,= [P{v*-v)yi\

(8)

(9)

2 .5

2.0

1.5

1.0

.5

MOLYBDENUM, MAGNESIUM, PALLADIUM, SILVER, AND SODIUM

* MOLYBDENUM

0 MAGNESIUM • PALLADIUM

SILVER SODIUM

.6 ,7

V/VO

1.0

Figure 15. Hugoniot P - V/Vo data for Mo, Mg, Pd, Ag, and

Na.

The curves span a very large region in this plane. Although there is some scatter in the data, it is evident that the compression curves are well determined.

In figures 10 to 13 the experimental Ug — u,, and

P — VjVn data are given for crystal density and

porous 2024 Al and Cu. The crystal data is the shock

velocity-particle velocity data fitted with the linear

relation listed in table 2. The other Hugoniots were

154
calculated from eq (7) assuming py constant and

using the average initial volume of each group of

data points as the Hugoniot centering point. The

agreement with the data is quite good, a further

indication that this simple volume dependence of

the Griineisen parameter is adequate. Much of the

scatter in the data probably arises from the diffi- culty of measuring initial densities accurately.

3. Indirect Measurement of the

Equation of State

Once the Hugoniot Ug—Up curves and the Griineisen parameters have been determined for the primary standards as outlined in section 2, it is possible to use these materials to investigate the

equation of state of other materials by use of the

impedance match technique [4]. Although a large

number of materials have been studied in this way,

we have selected only a few substances which are under particular consideration as static highpressure standards. These include five metals, Ag,

Na, Mg, Pd and Mo, and one oxide, MgO.

The Us — Up and P — V/Vo Hugoniot data for the five

metallic elements are presented in figures 14 and 15.

There is evidently considerable scatter in the data,

and the possibility of phase transformations along

the Hugoniots is certainly not precluded. However,

the volume changes at such transformations are undoubtedly quite small, and since velocity meas- urements are essentially differential P—V measurements, as seen by eqs (8) and (9), the resulting

compression curves should be quite accurate.

The situation with periclase is not so favorable.

The Hugoniot Ug — Up and P-VjVo data for crystal

MgO and three lower density ceramics are presented in figures 16 and 17. The linear least square

fit of the crystal data extrapolates at zero pressure to 6.6 km/s, a value considerably below the currently

accepted bulk sound speed of 6.73 km/s. The discrepancy is in the wrong direction to be explained by

rigidity effects. This discrepancy is larger than can

be explained by the rather large scatter in the ex- perimental data. A possible explanation is that MgO

transforms to the CsCl structure somewhere on the

Hugoniot below about 200 kbar. Since the Hugoniot

data appear to be characteristic of the same phase,

a straightforward linear Hugoniot fit, together with

the thermodynamic value of the Griineisen parameter, has been used to calculate the Hugoniots of

the porous ceramics by the method of section 2.4.

The results are gratifying, and again give additional

confidence to the assumption of a constant value for

the product (py). The experimental values of

{dEjdP)v for periclase, also calculated by the

methods of section 2.4, are plotted in figure 18.

Although attendant with the usual large scatter

found with this sort of experiment, it is obvious that

a constant value of (dE/dP) v is again consistent with

the data.

PERICLASE

10

9

6

5

4

/0 y

I

I y

/ 0

n

/o / X RHO 0

0 RHO 0

= 3.585

= 3. 33 • RHO 0 RHO 0

-2.97

= 2. 83

0 1.0 2.0 3.0 4.0

UP KM /SEC

Figure 16. Hugoniot Uj — Up data for MgO.

The crystal density material (p() = 3.585) has been fitted by the method of least squares to a linear u^ — Up relation. The resulting curve extrapolates considerably below

the bulk sound speed, indicating that the material has possibly transformed to a differ- ent phase at these pressures. The curves through the data points for the lower density

materials have been calculated from this fit, using a thermodynamic Griineisen parameter with (dEldP)v =0.2ll. The agreement is seen to be good.

^ PERICLASE

1.2

.4

.2

0

< RHOO=

0 RHO 0= 9 RHO 0 = RHO 0=

3.585

3.33 2.97

2.88

>\ \

\l

I I I I J I J —

1.60 .70 .80 .90 1.00

V/VO

Figure 17. Hugoniot P —V/Vo data for MgO.

The solid curves are transformations of the calculated u, — Up curves from figure 16.

155
a p - 2.842 PERICLASE A P^- 2,970

-

» Pa'- 3. 3 44

•a *

• A

- • A • A

g • A *

- A * A •

• •

•

1 1 1 1 1 1 1 1 0.19 0.20 0.21 0.22 0 23 0.24 0.25 0 26 0.27 0.28

V ( cm'/ g)

Figure 18. Experimental values for (3E/aP)v versus volume

determined from shock wave studies on porous MgO.

The data points were computed by the same method as used in figures 8 and 9.

4. Transformation of Hugoniot Data

to Isothermal Compression Data

As mentioned earlier, the extension of the Hugoniot data to regions off the Hugoniot represents the

major shortcoming in using dynamically established

standards in static experimental work. However, it is nevertheless possible to approximate the room

temperature isotherms from the Hugoniot data

under certain assumptions. For such calculations

it is most convenient to work through the general

energy-pressure equation (eq (4)). This equation

states that the pressure and energy at any particular

volume can be related to these quantities on the

] Hugoniot through some average value of the

Griineisen parameter y. In view of the experimental

evidence for porous Cu, 2024 Al, and MgO, as well

j as theoretical considerations, y may be taken to be \' independent of pressure and a function of volume ' alone; the relationship (dEI dP)v = constant appears to be satisfactory in the absence of a more

compelling choice. More complicated volume dependencies of y lead to only slightly differing re- sults for most materials. Calculation of isotherms

requires an arbitrary choice for the specific heat as

well, since data are unavailable in the region of in- terest. We have chosen to use one of the simplest

forms of the Debye theory, characterized by a single

Debye theta, 0,(^)» which is a function only of

volume [7]. This form for the specific heat is consistent with the assumption that the Griineisen

parameter depends only on the volume, since the

specific heat obtained in this manner is a function

only of entropy. In these calculations, &{V) has

been chosen so that the Debye theory will give the

correct room-temperature, zero-pressure value for

the specific heat. With this assumption, the thermal

energy, Et — E — E-zq^ (where £293 is the energy of

the solid at 293 K and the same volume) can be calculated. In turn, this energy and the corresponding

thermal pressure, Pt— Ery/V, can then be removed

from the Hugoniot curve to yield the room temperature isotherm.

The thermodynamic and Hugoniot data used to perform these calculations are listed in tables 1 and

2 respectively. The resulting P-p isotherms are tabulated in table 3, and the P-V/Vo isotherms are

156
Pressture (Mb) Calculated Density on 293 ''k Isotherm

2024

Copper Aliiminum Molybdenum Silver PsLlladium Sodium

2.785 8.930 10.206 10.490 11.991 ,968

2.803 8.963 10.225 10.539 12i024 1,037

2,821 8.995 10.244 10.587 12.056 1.096

2.838 9.027 10.263 10.635 12.088 1.147

2.855 9.059 10,282 10.681 12.119 1.193

2.872 9.090 10.301 10,726 12.150 1.235

2.888 9.120 10.319 10.771 12.181 1.274

2.90* 9.151 10.338 10.815 12.211 1.310

2,920 9.180 10.356 10.858 12.242 1.344

2,935 9.210 10.374 10.901 12.271 1.376

2,950 9.239 10.393 10.942 12.301 1.407

2.965 9.268 10.411 10.983 12.330 1.436

2.980 9.296 10.429 11.024 12,359 1.464

2.995 9.324 10.447 11.064 12.388 1.491

3.009 9.352 10.465 11.103 12.416 1.517

3.023 9.380 10.482 11.142 12.444

12.472

1 .541

3,037 9.407 10.500 11.180 1.565

3,051 9.434 10.518 11.217 12.500 1.588

3.064 9,460 10.535 11.255 12.527 1.611

3.078 9,487 10.553 11.291 12.554 1.632

3.0?l 9.513 10.570 11.327 12.581 1.653

3,104 9.538 10.587 11.363 12,608 1.674

3,117 9.564 10.605 11.398 12.634 1.693

3.130 9.589 10,622 11.433 12.660 1.713

3,142 9.614 10.639 11.468

11.502

12,686 1.731

3. 155 9.639 10.656 12,712 1.750

3.167 9,664 10.673 11.536 12.737 1.768

3.179 9,688 10.690 11.569 12,763 1.785

3.191 9.712 10.706 11.602 12.788 1.802

3.203 9,736 10.723 11.634 12.813 1.819

3.215 9.760 10.740 11.667 12.837 1.835

3.328 9,987 10.902 11.972 13.075 1.980

3.431 10,198 11.059 12.251 13.297 2.102

3.527 10.394 11.210 12.508 13.506 2.206

3.616 10.579 11.356 12.747 13.703 2.298

3.701 10.754 11.497 12.972 13.891

3.781 10.920 11.635 13.183 14,069

3.858 11.079 11,768 13.383 14.240

3.931 11.230 11,898 13.573 14.403

4.001 11.375 12,0:^5 13.754 14,561

4,068 11.514 12,149 13.927 14,712

4,132 11.648 12,270 14.093 14,858

4,195 11.777 12,388 14,252 14.999

4,255 11,902 12,504 14,405 15,135

4,313 12,022 12,617 14,552 15.268

4.369 12,139 12,728 14,694 15,396

4,424 12,252 12,837 14,831 15,520

4.477 12.362 12.944 14.963 15.642

4.528 12.469 13.050 15,091 15,759

4.578 12.573 13.153 15,216 15,874

12.674 13.255 15,336 15,986

12.773 13,355 15,453 16,095

12.869 13.453 15,567 16,202

12.962 13,550 15,677 16.306

13.054 13.645 15,785 16.407

13.143 13.739 15,890 16.507

13.231 13.832 15,992 16.604

13.316 13,924 16.091 16.699

13.399 14.014 16.792

13.4B1 14,103

14 . 1 "*0 14.277

14.363

14.447

14.531

14.613

14,695

14,776

16,884

16,973

17,061

17,147

17,232

17,315

17,396

17,476

17,555

Magnesium MkO

0.000

.005

.010

.015

.020

.025

.030

.035

.040

• 045

,050

,055

.060

.065

.07o

.075

,080

.085

.090

.095

.100

.105

.110

.115

.120

.125

.130

.135

.140

.145

.151)

.200

.250

.300

.350

.400

.450

.500

.550

.600

.650

.700

.750

.800

.850

.900

.950

1.000

1.050

1.100

1.150

1 .200

1.250

1.300

1.350

1.400

1,450

1.500

1.550

1.600

1 .650

1 .700

1.750

l.tloo

1 .850

1 .900

1 .950

2.00U

1.740

1.765

1.789

1.812

1.834

1.856

1.876

1.896

1.916

1.935

1.954

1.972

1.969

2.007

2.023

2.040

2.0S6

2.072

2.088

2.103

2.118

2.133

2.1*7

2.161

2.176

2.189

2.203

2.216

2.230

2.2*3

2.256

2.376

2.483

2.581

2.671

2.75*

2.832

2.904

2.973

3.038

3.100

3. 158

3.585

3.597

3.608

3.620

3.631

3.642

3.653

3.664

3.675

3. 685

3.696

3.707

3.717

3.727

3.737

3.748

3.758

3.768

3.777

3.787

3.797

3.806

3.816

3.825

3,835

3.844

3.853

3.863

3.872

3.881

3.890

3.976

4.057

4,133

4.205

4.273

4,339

4.401

4,461

4,519

4.575

4.629

4.681

4.731

4.780

4.828

4.874

*.9l9

4.963

5.005

5.047

5.088

Table 3. Calculated 293 K isotherms

157
30

_ 20

Ag Cu MgO Pd

- Mo - Ag

Cu

V *

"

\ •

" MgO

Pd

\ *

* Mo

• \ ^ \ * '

* \ X \ * V ^

COMPARISON OF BRIDGMAN ISOTHERMAL DATA WITH CALCULATED 293° K ISOTHERMS FROM SHOCK

WAVE DATA

1 1 .97 .98 .99 1.0

V / Vo

Figure 20. Comparison of Bridgman isothermal data with

calculated 293 K isotherms from shock wave data.

The agreement to 30 Icbar, with the exception of the data for MgO, is acceptable. The

Bridgman data are from reference [9]. The solid lines are the lower ends of the calculated 293 K isotherms from figure 19.

100 1 * 1*

I

90

Mg (1948) • \* \

• No (1942) \ 80

*Na(l948) •!* \a

70 •\ A la

60 A A n

50 kA

40

30

rnUPflRlc:nM ^ U IVI nM 1 DVJIN OF\Jr HRint^MAM Dr\ 1 UvjfVIMIN \\ ISOTHERMAL DATA WITH ^ CALCULATED 293°

K

20

ISOTHERMS FROM SHOCK

WAVE DATA

10

0

1 1 1 1

1 1 ^

.8 1.0

plotted in figure 19. These isotherms have been

calculated to pressures only slightly below the

maximum pressures reached on the Hugoniots,

and therefore represent small extensions of the

data. They should therefore be used with caution

in the very high pressure ranges.

The calculated isotherms are compared with the

work of Bridgman [8, 9, 10] in figures 20 and 21 for

most of the materials included here. The agreement,

at least to pressures of 30 kbar, is in general quite

good. Above about 50 kbar there is considerable

disagreement for both Na and Mg, with the 1948

Bridgman data being consistently too high in pressure. The most serious disagreement is with MgO.

In view of the difficulties with the sound speed and

the likelihood of a phase change below 200 kbar in

this material, the Bridgman isotherm is probably

more correct in this low-pressure region. It may be

remarked that if the Hugoniot intercept is made to agree with the ultrasonic value of 6.73 km/s, then

the Bridgman isotherm and our calculated shock

wave isotherm agree very nicely as well.

5. References

[1] McQueen, R. G., Laboratory techniques for very high pressures and the behavior of metals under dynamic

loading. Metallurigical Soc. Confs., Vol. 22, K. A. Gschneider, Jr., M. T. Hepworth, and N. A. D. Parlee,

eds. (i-iordon and Breach Science Publishers, New York,

1964).

[2] Rice, M. H., McQueen, R. G., and Walsh, J. M., Solid State Physics, Vol. 6, pp. 1^3 (Academic Press, New

York, 1958).

V/Vo

Figure 21. Comparison of Bridgman isothermal data with cal- culated 293 K isotherms from shock wave data.

The 1948 Bridgman data (ref. [!0|l would indicate consistently lower compressibilities than the earlier 1942 (ref. [llj) work. The solid lines are the calculated 293 K

isotherms from fiiiure 19.

[3

[4

[5

[6

[7

[8

[9

[10

[11

[12

[13

[14

[15

[16

[17

[18

[19

[20

[21

[22

[23

[24

[25

McQueen, R. G., and Marsh, S. P., J. Appl. Phys. 31,

1253 (1960). Walsh, J. M., and Christian, R. H., Phys. Rev. 97, 1544

(1955). Shreffler, R. G., and Deal, W. E., J. Appl. Phys. 24, 44

(1953). Taylor, J. W-. private communication.

Ziman, J. M., Electrons and Phonons, p. 56 (Claredon

rress, Oxford, 1960).

Bridgman, P. W., Proc. Am. Acad. Arts Sci. 77, 189 (1949). Bridgman, P. W., Proc. Am. Acad. Arts Sci. 76,55 (1948). Bridgman, P. W., Proc. Am. Acad. Arts Sci. 74,425(1942).

Hampel, C. A., ed., Rare Metals Handbook, 2d ed., pp. 689, 695 (Reinhold Publ. Corp., London, 1961).

Clark, S. P., ed.. Handbook of Physical Constants (The

Geological Society of America, New York, 1966). JANAF Thermochemical Tables (The Dow Chemical Company, Midland, Michigan, June 1966).

Marsh, S. P., unpublished data. Hearman, R. F. S., Phil. Mag. Suppl. Vol. 5, 323 (1956). Overton, W. C, Jr., and Gaffney, J., Phys. Rev. 98, 969

(1955). Lazarus, D., Phys. Rev. 76 , 545 (1949). Neighbors, J. R., and Alers, G. A., Phys. Rev. Ill, 707

(1958). Rohl, H., Ann. Phys. 16, 887 (1933). Bolef, D. L, and de Klerk, J., J. Appl. Phys. 33,2311 (1962). Chung, D. H., Swica, J. J., and Crandall, W. B., J. Am.

Ceram. Soc. 46, 452 (1963). Susse, C, J. Rech. Centre National de Rech. Sci. 54, 23

(1961). Schrieber, E., and Anderson, O. L., J. Geophys. Res. 73,

2837 (1968). Goens and Schmidt, Physik Z. 37, 385 (1936). Daniels, W. B., Phys. Rev. 119, 1246 (1960).

158
session V

Chairman: C. W. BECKETT

National Bureau of Standards

Washington, D.C. 20234

Eflfect of Pressure on the Lattice Parameters of Lead Chalcogenides and

Nickel Arsenide-Type Compounds

S. Minomura, H. Nagasaki, and I. Wakabayashi

Institute for Solid State Physics, University of Tokyo, Tokyo, Japan

1. Introduction

There has been considerable interest on a pressure

scale in lead chalcogenides and nickel arsenide-type

compounds. Lead chalcogenides exhibit the first- order transitions at 24,680 (PbS), 43,320 (PbSe),

and 41,200 (PbTe) kg/cm^ in the Bridgman's

P-V data [1],' accompanied by the extremely sharp

rise in electrical resistance [2]. Nickel arsenide-type

compounds exhibit the large negative pressure

dependence of magnetic transition temperatures in

the rate of - 12.3(MnAs) [3]. -3.3(MnSb) [3, 4].

-6(CrTe) [5], and -6(NiS) [6] degree/kbar, etc. Bassett, Takahashi, and Stock [7] have observed

the x-ray powder patterns of high-pressure polymorphs in lead chalcogenides, using diamond anvils,

which have a correspondence with the SnS-type

patterns, but are not yet identified completely at present. Lynch [8] has measured the lattice parameters of MnSb as function of pressure up to 150

kbar, using diamond x-ray cell and discussed the

negative pressure dependence of Curie temperature

in the relative changes of three shortest Mn-Mn

distances; c/2, a, and {c^l4- + a^)^'^.

In this experiment the lattice parameters have

been measured in two lead chalcogenides (PbS and

PbTe) up to 90 kbar and in four nickel arsenide-type

compounds (MnSb, MnTe, CrSb, and CrTe) up

to 170 kbar.

2. Experimentals

Table 1 summarizes the materials, sources, and

atmospheric lattice parameters. The materials were

diluted in 5 to 13 times of weight by amorphous

boron powder for reduction of x-ray absorption. The

x-ray powder photographs were taken at quasihydrostatic pressure and at room temperature, using

' Figures in bracicets indicate the literature references at the end of this paper.

Paper presented at the Symposium on Accurate Characterization

of the High-Pressure Environment, held at the National Bureau

of Standards, Gaithersburg, Md., October 14-18, 1968.

a Drickamer-type high pressure cell [9]. The x-ray

diffraction angles allowed about 20 degrees using

Mo target at the maximum pressure of 170 kbar.

Pressures were determined by markers of NaCl,

which was scaled by Bridgman's P-V data [10] and

Drickamer's x-ray data [11]. The lattice parameters

calculated by least square method.

3. Results and Discussions

3.1. Lead Chalcogenides

The normal forms of lead chalcogenides have

NaCl-type structure. The x-ray powder patterns

which have a correspondence with the SnS-type

patterns were observed in PbS above 25 kbar and

in PbTe above 45 kbar. Tables 2 and 3 show the

observed and calculated c?-spacings of the orthorhombic patterns in PbS at 45 kbar and in PbTe at

Table 1. Materials, sources, and atmospheric

lattice parameters

Materials Sources Structures a c cja

(A) (A)

PbS Kanto Chem.

Co.

NaCl 5.935

PbTe Dr. Y. Sato of

Electrical Communication Lab.

NaCl 6.439

MnSb Dr. T. Kaneko

of Tohoku

Univ.

hex. 4.13 5.78 1.40

MnTe Dr. S. Anzai

Keio Univ.

hex. 4.15 6.71 1.62

CrSb Dr. T. Kaneko

of Tohoku

Univ.

hex. 4.11 5.44 1.32

CrTe Dr. T. Kaneko

of Tohoku

Univ.

hex. 3.98 6.21 1.57

159
Table 2. Observed and calculated d spacings for

orthorhombic phase of PbS at 45 kbar and relative intensities of SnS-fype compounds

Orthorhombic unit cell

hkl ///i(SnS) ///i(GeS)

(A)

3.35 vs 120 3.39 10 OA80

3.18 s 021 3.17 15 40

2.84 vs 040 2.85 100 100

2.53 5 (boron ?) 2.30 s 131 2.32 15 50

2.00 s 150 2.01 10 10

1.87 w 230 1.88 50

1.79 w 151 1.78 5 50

Table 3. Observed and calculated d-spacings for

orthorhombic phase of PbTe at 80 kbar and relative intensities of SnS-type compounds

^obs. hkl ^^calc. ///i(SnS) ///.(GeS)

(A) (A)

3.69 m 120 3.68 10 80

3.38 s 021 3.37 15 40

2.99 5 040 2.97 100 100

2.435 131 2.43 15 50

2.31 m 210 2.30 1 30

2.15 w 141 2.14 10 40

2.01 m 230 2.02 50

1.91 w 151 1.91 5 50

80 kbar, where the smoothed data of lattice parameters were used for the calculation, and the relative

intensities of SnS and GeS. The other calculated

patterns, such as 101 and 111 lines etc., were not

detected for our instruments, because of their low

resolution. However, assuming the SnS-type structure as the high-pressure polymorph in lead chalcogenides throughout this paper, the diffraction

patterns were indexed to orthorhombic set of indices.

Figure 1 shows the unit cells of NaCl-type and

orthorhombic structures. The orthorhombic lattice

parameters were calculated from 120, 021, and 040

lines. Figures 2 and 3 are a plot of lattice parameters

versus pressure in PbS and PbTe. Tables 4 and 5

summarize the smoothed date of a/ao, b/bo, c/co,

VI Vo, and P in the NaCl-type and orthorhombic

structures. Figures 4 and 5 are a plot of VIVo versus

pressure. The crosses are Bridgman's data. The

P—V data suggest the first-order phase transition

from the NaCl-type to the orthorhombic structures

which accompanies the volume discontinuity of

about 3 percent in PbS and about 2 percent in PbTe.

The c axes are slightly more compressible than the a and b axes in the orthorhombic phase.

o Pb atom

\ % Calcogen atom

NaCl-type unit cell

Figure 1. NaCI-type and orthorhombic unit cells.

1.00

090

080

Figure 2. Lattice parameters versus P in PbS.

3.2. Nickel Arsenide-Type Compounds

Table 6 summarizes the pressure dependence

of magnetic transition temperatures and the volume

magnetostriction given by the thermal dilatation

data at the atmospheric pressure in MnSb, MnTe,

160
Table 4. Lattice parameters and compressibility

o/Pbs

p ajaa c/co VIVo

NaCl-type structure

hbar

0 1.000 1.000 1.000 1.000

10 0.992 0.992 0.992 0.973 20 0.983 0.983 0.983 0.950

Orthorhombic structure

30 1.014 0.966 0.919 0.902

40 1.009 .962 .914 .887 50 1.005 .960 .905 .873 60 1.000 .958 .902 .864 70 0.998 .956 .900 .858

*NaCI-type structure is also described in terms of orthorhombic axes.

Table 5. Lattice parameters and compressibility

o/PbTe

p a/oo bibo c/co viv,

NaCl-type structure

kbar

0 1.000 1.000 1.000 1.000

10 0.992 0.992 0.992 0.978

20 .986 .986 .986 .956 30 .978 .978 .978 .938 40 .972 .972 .972 .920

Orthorhombic structure

50 1.030 0.931 0.926 0.889

60 1.029 .927 .917 .874 70 1.028 .922 .902 .859 80 1.027 .919 .897 .844 90 1.026 .915 .888 .832

*NaCl-type structure is also described in terms of orthorhombic axes.

Table 6. The pressure dependence of magnetic

transition temperatures and the volume magnetostriction at the atmospheric pressure

Compounds dTc.NldP

Volume magnetostriction

^a Ac

(K) (KIbar) (percent) (percent)

MnSb 586 (Tc) -3.3[3,4] + 0.3 [12] 0

MnTe 307 (Tx) + 2.6[13] 0 -0.8[14]

CrSb 713 (r,v) + 1.2[15] -2.9[15]

CrTe 339 (Tr) -6[5] + 0.16[16] + 0.14[16]

CrSb, and CrTe. Figure 6 shows the unit cell of

nickel arsenide-type compounds. The hexagonal

lattice parameters were calculated from the following hnes: 101, 102, 110, 103, and in some cases

202, 004, 211, 212, 114, 213, in MnSb; 101, 102, 110,

201, and in some cases 202, 203 in MnTe; 110, 102,

101, and in some cases 103 in CrSb and CrTe.

1. 10

1.00

0,90

1

"—-I

Q /Qo

a/Qo , b /bo , c/Co

b/bo

/

1

C/Co

0 50 100

P . kbar

Figure 3. Lattice parameters versus P in PbTe.

080-

1 1 1 I 0 20 40 60 80 100

P. KBAR

Figure 4. V/Vo versus P in PbS.

p. KBAR

Figure 5. V/Vo versus P in PbTe.

Figures 7-10 are a plot of lattice parameters

versus pressure. MnSb exhibits the Curie temperature at 65 kbar decreases up to the room temperature. The small but measurable irregularity in the

161
NiAs-type unit cell O Ni atom

As atom

(Q) Interstices

Figure 6. NiAs-fype unit cell.

4.15

4,10

405

400

3.95

3.90 50 100 150

P, kbar

Figure 7(»). a versus P in MnSb.

580

5.70 -

5.60 -

C,A

550

5.40

530

520

50 100 P. kbar

Figure 7(b). c versus P in MnSb.

a-P curve is observed at about 65 kbar, as shown in

figure 7(a). The distortion in the a axis below 65

kbar from the dotted curve which is drawn by the

extrapolation from the curve above 65 kbar is con- sidered as the magnetostriction. CrSb exhibits the

sharp rise in c axis and the sharp drop in a axis

at about 80 kbar, which is not yet established, but

may suggest the first-order or second-order phase

transitions. The experiment on the pressure

dependence of Neel temperature in CrSb is in

progress.

Tables 7 to 10 summarize the smoothed data of

a/ao, c/co, (c/a)/(c/a)o, and V/Vo, and P. The lattice

parameters are expressed by the following function

of pressure:

4.15

4.10

4.05

a,

A

4.00

3.95

3.90 50 100 P, kbar

Figure 8(a). a versus P in MnTe.

50 100

P. kbar

150

Figure 8(b). c versus P in MnTe.

MsSb,- Aa/ao =1.32X10-^P + 4. 64 X 1 Q-^P^ (below

65 kbar),

-Aa/ao = 2.7 X 10"^ + 4.64 X lO-^F - 1.21

X 10-6P2 (above 65 kbar), - Ac/co = 9.42 X lO-^P - 2.50 X IQ-^P^;

MnTe,- Aa/ao = 5.52 X lO-^P - 1.18 X 10-^P\ - Acico = 10.6 X 10-*P - 2.79 X lO-^P'^;

CrSb,

Aa/ao = 3.9 X lO-^P - 1.7 X 10-«P%

Ac/co = 5.8 X 10-4P - 1.0 X 10-«P2.

CrTe,- Aa/ao = 1.6 X IQ-^ + 3.0 X lO'^P, - Ac/co = 1.4 X 10-3 + 8.0 X lO-^F.

162
4.10

4.00

4.10

4.00

Figure 9(a). a versus P in CrSb.

5 60-

5.20

Table 7.

50 100

P , kbar

Figure 9(b). c versus P in CrSb.

Smoothed data of a/ao, c/co, (c/a)/(c/a)o,

and V/Vo in MnSb

p a/ao c/co (c/a)/(c/a)o VIVo

kbar

0 1.000 1.000 imo 1.000 10 0.998 0.989 0.992 0.985

20 .995 .980 .984 .971 30 .992 .971 .979 .955 40 .987 .964 .976 .940

50 .982 .957 .975 .923 60 .976 .951 .975 .905 65 .972 .949 .976 .895 70 .970 .946 .975 .891 80 .968 .941 .972 .882

90 .966 .937 .970 .873 100 .964 .932 .967 .866 110 .962 .928 .965 .859 120 .960 .925 .964 .852 130 .958 .921 .962 .846

140 .957 .918 .959 .840 150 .955 .914 .957 .834 160 .954 .911 .956 .829 170 .953 .908 .954 .825

a,A

4.00f

390

3.80 50 100

P. kbar

Figure 10(a). a versus P in CrTe.

C.A

6 30r

620

6.10

6.00

5.90

5.80

5.70

5.60

550

0 50 100

P, kbar

Figure 10(b). c versus P in CrTe.

Figure 11 shows the smoothed data of c/a ratio versus pressure. Empirical exchange interaction curves relating magnetic transition temperatures

to interatomic distances have been frequently

discussed in nickel arsenide-type compounds.

How^ever, the pressure dependence of magnetic

transition temperatures may not be quantitatively

understood by the relative change in the neighboring

atomic distances (c/2 or a) and the bonding angles

of nearest neighboring atoms (c/a).

163
Table 8. Smoothed data of a/ao, c/co, (c/a)/(c/a)o,

andYIYoin MnTe

p a/ao c/co (c/a)/(c/a)o VIVo

kbar

0 1.000 1.000 1.000 1.000 10 0.996 0.988 0.994 0.977

20 .989 .978 .988 .956 30 .984 .969 .985 .938 40 .980 .962 .982 .924

50 .975 .955 .979 .907 60 .971 .947 .975 .893 70 .967 .942 .972 .881 80 .964 .935 .970 .868 90 .960 .929 .967 .856

100 .957 .922 .964 .844 110 .954 .917 .962 .835 120 .951 .913 .960 .826 130 .948 .909 .958 .816

0 50 100 150

P . kbar

Figure 11. (c/a)/(c/a)o versus P.

Table 9. Smoothed data of a/ao, c/co, (c/a)/(c/a)o,

and V/Yo in CrSb.

p a/ao c/co (c/a)/(c/a)o VIVo

kbar

0 1.000 1.000 1.000 1.000

10 0.996 0.994 0.998 0.986

20 .993 .989 .996 .976 30 .990 .983 .993 .964 40 .988 .977 .989 .954

50 .985 .972 .987 .944 60 .983 .968 .985 .935 70 .982 .965 .983 .930 80 .981 .961 .980 .925 90 .980 .959 .979 .921 100 .980 .958 .977 .920

Table 10. Smoothed data of a/ao, c/co, (c/a)/(c/a)o,

and V/Vo in CrTe.

p a/ao c/co (c/a)/(c/a)o VIVo

kbar

0 1.000 1.000 1.000 1.000 10 0.999 0.995 0.996 0.993

20 .995 .987 .992 .977 30 .992 .979 .987 .963 40 .989 .969 .980 .94a

50 .985 .963 .978 .934 60 .982 .953 .971 .919 70 .980 .947 .966 .910 80 .977 .939 .961 .897 90 .974 .931 .956 .883 100 .971 .923 .951 .870

According to the molecular magnetic field approximation [17], the internal energy E of magnetic

substances includes the elastic energy Egi and the

magnetic exchange interaction energy E^^

E = Eel + Eex' (1)

where

Ee,^i^Vyi2kV (2)

and

1 3k

£'ex = — 2

^^^^~'2~^*- ''^~ const - — VexSi S2 (3)

AFis the volume magnetostriction, k the compressibility, y the coefficient of molecular field, cr the

intrinsic magnetization per unit mass at absolute

zero temperature, Tc,n the Curie or Neel temperatures, and Jex the exchange integral between spin

Si and spin S2. The differentiation of eq (3) as

164
regards P,

1 dTc.x 1 SJe 2 daTa,x dP Je^ dP adP'

dEldV^O leads using eqs (1), (2), and (3),

AV^ 3^ dTc,N

Vo 2Fo dP

•

(4)

(5)

When the volume magnetostriction is only in a axis,

eq (5) is replaced by

^^2 —

Vo Co

' (6)

Figure 12 is a plot of the magnetostriction in a

axis of MnSb versus pressure. The pressure

dependence of Curie temperature reported by

Samara and Giardini is expressed by the function

of P with the critical pressure 150 kbar and the

constant a = 1.52.,

15

Obs. smoothed curve

Figure 12. Magnetostriction of a versus P in MnSb.

where

and

r,= r«|i+a^)(i-£'

Jex =Jo ( l + «p^ )'

(T—(To{ 1 (7)

Equation (7) leads the initial pressure dependence

of Jo and ctq,

if=^=iox,o-.b„-.

o-o dP Pc

6.7x10-6 bar-i

These values coincide within experimental errors with the data reported by Hirone et al. [4]. The

negative pressure dependence of magnetic transition

temperature is due to the negative change in in- trinsic magnetization, which may be caused by the

relative change in band structure or redistribution

of electrons between the spin orientations.

4. References

[1] Bridgman, P. W., Phys. Rev. 57, 237 (1940); Proc. Am.

Acad. Arts Sci. 74, 21 (1940); ibid. 76,55 (1948).

[2] Samara, G. A., and Drickamer, H. G., J. Chem. Phys. 37,

1159 (1962).

[3] Samara, G. A., and Giardini, A. A., Physics of Solids at High

Pressures, ed. C. T. Tomizuka and R. M. Emrick, Aca- demic Press (1965), p. 308.

[4] Hirone, T., Kaneko, T., and Kondo, K., ibid. p. 298.

[5] Grazhdankina, N. P., Gaidukov, L. G., Rodionov, K. P., Oleinik, M. I., and Shchipanov, V. A., Soviet Physics —

JETP 13,297(1961).

[6] Anzai, S., and Ozawa, K., J. Phys. Soc. Japan 24,271 (1968).

[7] Bassett, W. A., Takahashi, T., and Stock, P. W., Rev. Sci.

Instr. 38, 37 (1967); T. Takahashi, private communication.

[8] Lynch, R. W., J. Chem. Phys. 47, 5180 (1967).

[9] Perez-Albueme, E. A., Foresgren, K. F., and Drickamer,

H. G., Rev. Sci. Instr. 35, 29 (1964).

[10] Bridgman, P. W., Proc. Am. Acad. Arts Sci. 76, 1 (1945).

[11] Perez-Albueme, E. A., and Drickamer, H. G., J. Chem. Phys.

43, 1381 (1965).

[12] WiUis, B. T. M., and Rooksby, H. P., Proc. Phys. Soc. (London) 67, 290 (1954).

[13] Ozawa, K., Anzai, S., and Hamaguchi, Y., Phys. Letters 20, 132 (1966).

[14] Greenwald, S., Acta Cryst. 6, 396 (1953).

[15] Willis, B. T. M., Acta Cryst. 6, 425 (1953).

[16] Ido, H., Suzuki, K., and Kaneko, T., private communication.

[17] KitteL C, Introduction to Sohd Physics, John Wiley & Sons,

Inc. (1956), pp. 402-408.

DISCUSSION

W. A. Bassett {University of Rochester, Rochester,

New York): We have calculated the intensities for

the diffraction pattern produced by lead and sulfur

in the SnS structure using the SnS lattice parameters (see accompanying table). These intensities

differ significantly from those for tin and sulfur in

the SnS structure. We feel that these differences

should be taken into consideration in identifying the

structure and in determining the lattice parameters.

In table 2 you do not report a reflection having the

index (110), yet the calculated intensity is ///i = 49

and should be observable. Your reflection d— 2.84 A

is indexable as (111) and (130) as well as (040). Of

these, the (111) should contribute the most to the

165
intensity. The reflection d= 2.00 A may be indexable as a combination of (141), (002), and (150). Of these

the (141) would contribute the most to the intensity.

The reflection d = 1.87 A may be indexed as a combination of (211) and (230) of which (211) contributes

the most intensity.

Table of calculated intensities for PbS in the SnS

structure

Wavelength = 0.70926 A

a=4.34 A, 6= 11.20 A, c=3.99 A

Pb: A?=0. 1 150, y=Q. 1 180, z=0.2500

S: x=0.5220, >=0.1500, 2=0.7500

d hkl ///. 4.047 110 49

3.430 120 67

3.250 021 100

2.937 101 53

2.841 111 86

2.830 130 9

2.800 040 64

2.308 131 51 2.130 210 25

2.027 141 39

1.995 002 28

1.991 150 17

1.879 211 35

1.876 230 23

1.789 112 13

1.781 151 19

1.725 122 23

1.715 160 11 1.698 231 17

AUTHORS' CLOSURE

We have compared the intensities of the highpressure phase of PbS with those of GeS and SnS

obtained at atmospheric pressure. But this must be

done only occasionally as you suggested. We also

think that the intensities of the high-pressure phase

of PbS should be compared with the computed

values for SnS-type PbS in order to identify the

structure of the high-pressure phase of PbS. The

relative intensities at high pressures, however, have

been found to change to a considerable extent without phase transitions. This may be due perhaps to some strains etc. Therefore, the computed values at atmospheric pressure are thought to be not necessarily directly appUcable to the pressure

measurements.

We have used MoKa-radiation and d= 4.047 A

corresponds to 0 = 5.03°. This angle is the limit

where the direct beam disturbs x-ray patterns, so we

could not detect the diff"racted beam at this angle.

In our study, we have calculated the lattice parameters and the compressibilities of high-pressure

phases of PbS and PbTe assuming that their phases

have SnS-type structures, although the problem of

identification of these structures at high pressures

has not been solved yet.

166
The Compressibility and Thermal Expansion of LiF to 60 kbar and 600 °C

as Determined by X-Ray Diffraction: Report of Progress*

L. C. Garrison** and C. B. Sclar***

Battelle Memorial Institute, Columbus Laboratories, 505 King Avenue, Columbus, Ohio 43201

Measurements were made of the compressibility of LiF to 60 kbars over the temperature range

25 °C to 600 °C by high-pressure high-temperature x-ray powder diffraction methods. Pressures were

determined by using NaCl as an internal calibrant. These results also provide data on the thermal

expansion of LiF between 25 and 600 °C at pressures of 38 and 57 kbar. This paper demonstrates

the feasibility of determining the unit-cell volume of crystalline solids under concomitant high pressure and high temperature by in situ x-ray methods.

1. Introduction

An important area of crystal physics which has

been relatively unexplored because of the limitations of existing experimental apparatus is that of

the compressibihty and thermal expansion of crystaUine soUds at high temperature and high pressure,

respectively. The recent development of a highpressure, high-temperature x-ray powder diffraction

apparatus by Freud and Sclar [1, 2]^ provides a means of determining the unit-cell volume of crystaUine soUds at concomitant high pressure and high

temperature to at least 100 kbar and 1000 °C. This

paper presents the results obtained to date on a study which demonstrates the feasibility of this

experimental approach.

The authors respectfully acknowledge the interest

and support of Dr. J. N. Plendl and Mr. L. C. Mansur

of Air Force Cambridge Research Laboratories

during the course of this program.

2. Experimental Approach

2.1. High-Pressure High-Temperature X-Ray

Diffraction Apparatus

The apparatus used in this study is a modified

miniaturized "belt" apparatus which belongs to the

family of internally heated compressible-gasket

solid-media devices. The unique feature of this device is that the die-support ring assembly is fabricated in two parts which mate along a plane normal

to the piston axis. This split-die design permits entry

of filtered MoKa radiation into the high-pressure

volume and egress of both the diffracted rays and the

undeviated beam through suitable grooves and fanshaped slots ground in the mating surfaces. The

•Supported by Air Force Cambridge Research Laboratories under Contract No. Fl%28-68-C-{)123.

**Present address: General Electric Co., Worthington. Ohio 43085. ***Present address: Department of Geological Sciences, Lehigh University, Beth- lehem, Pa. 18018.

' Figures in brackets indicate the literature references at the end of this paper.

Paper presented at the Symposium on Accurate Characterization

of the High-Pressure Environment, held at the National Bureau of

Standards, Gaithersburg, Md., October 14-18, 1968.

high-pressure x-ray windows are either a beryllium

ring with a wedge-shaped cross-section or epoxy

resin stops at the bore of the die.

The high-pressure medium is either "amorphous"

boron or boron nitride, and the sample is in the form

of a thin cylinder which is coaxial with the pistons

and normal to the x-ray beam. The extrudable

gaskets between the pistons and the die are made

of pyrophyllite as they are in conventional devices

inasmuch as they are not part of the x-ray path.

High sample temperatures are attained by resistance

heating of carbon rods adjacent to the sample.

Temperatures are determined with a thermocouple

in contact with the sample. The apparatus is portable and is used in conjunction with a conventional x-ray source. The diffracted rays are recorded on the cyUndrical film of a Debye-Scherrer camera 114.6 mm in

diameter. In this study, lattice parameters were

determined with a precision of about ±0.2 percent.

The diffraction lines used for this purpose were as foUows: for NaCl (200), (220), (222), (400), (420),

(422); for LiF (111), (200), (220), (311), (222). A

photograph and a schematic drawing of the apparatus are shown in figures 1 and 2, respectively.

2.2. Calibration of High-Pressure Apparatus

The high-pressure apparatus was caHbrated by

means of the established fixed points on the room- temperature electrical-resistance absolute pressure

scale, viz., Bil^II transition at 25 kbar and Bi

III V transition at 75 kbar. However, in a miniaturized compressible-gasket apparatus like the

"belt," it is recognized that the internal sample

pressure may not be the same for successive ex- periments at a constant ram force. Furthermore, the

electrical-resistance scale is a room-temperature

scale, and it is not possible to predict the effect of

elevated temperature on the internal sample

pressure. Consequently, an internal pressure calibrant is needed if one is to know the sample pres- sure accurately at high pressure and high temperature. Such an internal calibrant is available in NaCl

through the use of the semi-theoretical equation of

167

385-762 O - 71 - 12
Figure 1. Photograph of high-pressure high-temperature x-ray powder diffraction apparatus.

The scale is in centimeters.

State for sodium chloride proposed by Decker [3]. As a means of exploring the reproducibility of

internal sample pressure as a function of applied

load and to obtain information on the magnitude of

the errors involved, a series of calibration experiments was carried out. The results are summarized

in figure 3. Each line in this figure represents for an

individual run the change in internal sample pressure, as revealed by the lattice parameter of sodium

chloride (Decker scale), for a given internal geometry

with increasing applied load. Although a fairly good

linear relationship exists between ao and applied

load, if reproducibility has been achieved all the

lines for a common geometry would have over- lapped. Obviously, this is not the case, and the

spread in the line positions shows the magnitude of

the nonreproducibility of internal sample pressure.

As a result of these experiments, it was concluded

that an internal calibrant, namely, sodium chloride,

would have to be used with each run. All the data

reported in this paper are based on pressures determined by the lattice parameter of sodium chloride as related to the Decker equation of state. A second type of calibration experiment was run

at elevated temperature to determine for a constant

applied load the variation in internal sample pres- sure as a function of thermal cycling. The results are presented in figure 4, in which the numbered

Figure 2. Schematic drawing of high-pressure high-temperature

x-ray powder diffraction apparatus.

/^^hydraulic ram, B = lungsten carbide pistons, C = steel die and high-pressure volume, Z) = water cooling tubes, £== rubber shim for positioning, F = Debye-Scherrer Camera, 6= x-ray film, //= electrical power lead, /= press positioning table, 7= ad- justing screws for table.

points represent sequential runs at which the lattice

parameter of NaCl was determined. These results

show that it is not possible to thermally cycle from

room temperature to an elevated temperature and

back at constant apphed load without significant

changes in the internal pressure. There appears to be a cumulative loss in internal pressure with an

increasing number of thermal cycles at constant

load providing that each successive run is carried

out at a higher temperature. In any event, it is clear that an internal pressure calibrant must be

used to determine sample pressure if the run is cycled thermally.

3. Experimental Results on Lithium

Fluoride

3.1. Compressibility

Data on the isothermal compressibility of LiF

were obtained to 60 kbar at temperatures of 25, 200,

400, and 600 °C. The lattice parameters (ao) used for

NaCl and LiF at 1 atm and 25 °C in calculating

Aa/ao were 5.6402 A and 4.0270 A, respectively.

X-ray diffraction patterns were obtained of LiF-NaCl

mixtures at high-pressure and elevated temperature

168
300

Temperature, C

Figure 4. Graph showing variation in lattice parameters of NaCl for an individual experiment in which the sample was thermally

cycled at a constant applied load of 4000 psig.

169
1

1

X

^X —

X

o

X —-^

X.

NoCI ^X

30 40

Pressure, kb

50 70

Figure 5. Aa/ao versus pressurefor LiF at 25 °C using NaCl to determine pressure.

\\X —

\ A Estim Dted pressu e \

\

\

\ ^

\

\

\

\ 6 00 C

AN

\ 400 C m \ « » \ •

\^ 200

< V 25 C

20 30 40

Pressure, kb

50

using Zr-filtered MoKa radiation (\ = 0.7107 A).

Figure 5 is a plot of the data for LiF and for admixed NaCl used an as internal caHbrant carried out

in duplicate experiments at 25 °C. These results

agree with earlier data published by Bridgman [4]

and by Pagannone and Drickamer [5] within 0.2

percent. Figure 6 shows Aa/ao versus pressure for

LiF at elevated temperatures; thermal expansion

data at room pressure are those given in [6]. The

numerical data on which these graphs are based

are given in table 1.

Table 1. Aa/ao versus pressurefor uiF

Pressure Temperature, °C

25 200 400 600

kbar

0

8.520

29

38

40

42

53

57

65

0.000

-0.0062

-0.0087

-0.0116

-0.0171

-0.0177

* + 0.0070 * + 0.0158

-0.0114 -0.0030

+ 0.0045

-0.0220

-0.0206

-0.0264

**- 0.0166 ** -0.0099 **- 0.0037

Figure 6. Aa/ao versus pressure for LiF at 25, 200, 400, and

600 °C.

*From reference [6]. , «, "Estimated pressure; unable to measure lattice parameter of NaCl accurately because lines were weak.

170
3.2. Thermal Expansion Under Pressure

Data on the thermal expansion of LiF at 38 and

57 kbar are presented graphically in figure 7. For

comparison, the thermal expansion of LiF at room

pressure, based on data in the literature, is also

shown. The results show that the coefficient of

thermal expansion of LiF is essentially the same at 38 kbar up to 600 °C as it is at room pressure. On

the other hand, the coefficient of thermal expansion

appears to show a sUght decrease at 57 kbar in

the same temperature range.

4. Conclusions

Despite the experimental difficulties and the

limited accuracy of the results relative to roompressure measurements, useful data on the compressibility and thermal expansion of crystalline

solids at high pressure and high temperature may

be obtained by in situ x-ray diffraction methods. An

internal calibrant, such as NaCl, is necessary,

however, jn order to determine the pressure

accurately.

5. References

[1] Freud, P. J., and Sclar, C. B., Apparatus for high-pressure

high-temperature x-ray powder diffraction studies to 100

kbar and 1000 °C (abstract), Abstracts for 1967, Geol. See. Am. Spec. Paper 115 (1968).

[2] Freud, P. J., and Sclar, C. B., Apparatus for high-pressure

I

high-temperature x-ray powder diffraction studies. Rev.

Sci. Instr. 40, 434-437 (1969).

[3] Decker, D. L., Equation of state of NaCl and its use as a pres- sure gauge in high-pressure research, J. Appl. Phys. 36,

157-161 (1965).

[4] Bridgman, P. W., Linear compressions to 30,000 kg/cm^,

including relatively incompressible substances, Proc. Am.

Acad. Arts Sci. 77, 189-234(1949).

/

/

/ —

/

/

/ - /

(2 = 1.25 X 10-" C

38 kilobors _-4 — a -1.21

)

57 kilobors 3=9 97 X 10"^ C"'

o/ > o D a = C

Ota from Ref(6)

ermol —

e >ipansion oefficient

Q 100 200 3 00 400 500 600

Temperature, C

Figure 7. Aa/ao versus temperature for LiF at room pressure. 38 kbar and 57 kbar.

[5] Pagannone, M., and Drickamer, H. G., Effect of high pressure on the compressibilities of Nal, LiF, and NaF, J. Chem.

Phys. 43, 2266-2268 (1965).

[6] Handbook of Physical Constants, ed. S. P. Clark, Geol. Soc. Am., Memoir 97 (1966) (unpublished data of H. A. McKinstry).

DISCUSSION

I

J. D. Barnett (Brigham Young University,

Provo, Utah): The vertical dispersion of your points

appears to me to be simply a result of not having the

sample in the appropriate spot within the highpressure volume. With a camera of only fivecentimeter radius it is necessary, in order to obtain an accurate calibration by use of sodium chloride,

to have the sample location fixed within about

I, five-thousandths of an inch. I question whether you

can do this repeatedly and maintain the position, at

' pressure in a sample that is several milHmeters

across, particularly in view of the motion due to the

distortion of the pyrophyllite. A displacement of

your sample five-thousandths of an inch towards the

film wiU account for the spread that you are getting.

In a study we have made of lithium fluoride,

using something like 50 x-ray patterns rather than

the 8 or 10 that you used, our data differ only slightly

from your average values. If you take the median of

your points, and assume that your sample was

misalined to fit the average of the scattered points,

all of your compression data on lithium fluoride are

shifted. The resultant lattice spacing of lithium

fluoride is thus inaccurate, since it shifts the curves on your graph in different directions in different

experiments. This is one of the disadvantages of the

small camera, since precise x-ray work requires

that sample-to-film distance be controlled very

accurately.

I believe that you could correct your data by

"calibrating" the x-ray patterns, using your measured lattice parameters at zero pressure as a reference in each run. That would, in effect, determine the position of the sample within the apparatus.

171
AUTHORS' CLOSURE

I believe that Dr. Bamett has a valid point on the

matter of the effect of position of the sample on the

indicated pressure. In checking position of samples

after completion of the runs, however, we find that

the shifts from center after pressurization are less

than one-tenth of a millimeter (0.004 in). I don't

know how much less, because it is very difficult to make this measurement accurately.

It is also important to note that the cylindrical

sample is only 0.3 mm in diameter. As I indicated

in my opening remarks, the data that we have presented are fragmentary and preliminary, and are only illustrative. There are a number of factors

that can be responsible for the scatter shown, and

the problem is one of resolving the magnitude of

contribution from the diff"erent factors.

172
Ultrasonic and Static Equation of State for Cesium Halides

G. R. Barsch

Materials Research Laboratory and Department ofPhysics,

The Pennsylvania State University, University Park, Pa. 16802

and

Z. P. Chang

Materials Research Laboratory, The Pennsylvania State University, University Park, Pa. 16802

1. Introduction

In spite of several promising attempts, the

problems of accurate pressure measurement and

of an accurate pressure scale above 30 kbar are not

yet completely and satisfactorily solved. The

present paper is concerned with the equation of

state of solids as one of several possible approaches

to this problem. If the equation of state, i.e., the

relation between pressure and volume, is known for a standard, it is possible to calculate the pressure

from the dimensional changes of a gage specimen

subjected to hydrostatic compression that can be

measured by x-ray or by optical methods.

At present, theoretical equations of state will

contain a high degree of uncertainty because of the

many simpUfying assumptions which are necessary, even for the simplest solids, to make the calculations

feasible. Therefore it seems more promising at present to utilize empirical equations of state for

the task of pressure calibration and measurement.

Naturally those empirical equations of state are

less useful for setting up a pressure scale which

presume the availability of a pressure scale, such as

direct volumetric data obtained from x-ray methods.

The remaining two methods utilizing shock wave

and ultrasonic data should be regarded as mutually

complementary.

While the former has the advantage of being

based on direct measurements at very high pressures, it suffers from uncertainties that arise in the

conversion from the shock-Hugoniot to pure isothermal conditions and from the fact that the pres- ence and effect of nonhydrostatic stress in shock

loading is not fully understood. On the other hand,

an equation of state based on ultrasonic data of the

elastic constants and their pressure derivatives is free from arbitrary assumptions, but is confined to comparatively low pressure. As has been shown by

Anderson [1],' however, the ultrasonic equation of

state is valid over a larger pressure range than that

*This work was supported by the U.S. Atomic Enerev Commission.

' Figures in brackets indicate the literature references at the end of this paper.

Paper presented at the Symposium on Accurate Characterization

of the High-Pressure Environment, held at the National Bureau of

Standards, Gaithersburg, Md., October 14-18, 1968.

required in the ultrasonic measurements. It is this

fact which makes it possible to use the ultrasonic

equation of state as a pressure standard for high

pressure, because it is based on measurements at low pressure in a range where the pressure scale is already established more reliably. It has been shown

by Anderson [1] that the pressure-versus-volume

compression curves of 14 materials calculated from

Murnaghan's equation of state [2] by using the bulk

modulus and its first pressure derivative measured

ultrasonicaUy below 10 kbar agree with directly

measured data up to 100 or 200 kbar, and in the case of very incompressible substances even up to

1 Mbar. With increasing pressure, increasing

discrepancies between the two sets of data occur which may be attributed to one or several of the

following causes.

First, Murnaghan's equation of state is based on

integration of a linear relation between bulk modulus

and pressure and contains only two empirical - parameters, the zero pressure values of the bulk

modulus and its first pressure derivative. One must

expect, however, that the pressure dependence of

the bulk modulus becomes nonlinear at sufficiently

high pressures and that more than two parameters

corresponding to the higher pressure derivatives of

the bulk of modulus are necessary for the characterization of the equation of state over a wider range of

pressure. The second possible cause of the discrepancy consists of experimental errors, statistical or systematic, of the ultrasonic or of the directly

measured compression data.

The improvements that can be obtained by using

a three-parameter equation of state have also been

investigated by Anderson [3] on the basis of an

equation of state proposed by Keane [4]. A very

good fit of the directly measured compression curves

of all five materials considered can be obtained in

this manner up to the highest pressures measured if two parameters — the initial bulk modulus and its

first pressure derivative — are determined from

ultrasonic data, and the third parameter— the

2 One must distinguish between empirical parameters (which can be determined

from experimental data other than high pressure measurements) and adjustable parameters (which are determined by fitting directly measured pressure vs volume

compression curves to some phenomenological or theoretical equation of state). In

this paper empirical parameters will be considered only.

173
limiting value of the bulk modulus at infinite

pressure — is adjusted to fit the empirical compression data. As emphasized by Anderson [3], the

usefulness of this approach consists of combining

the merits of the low-pressure ultrasonic data and

the shock wave data which are more reliable at high

pressure so that the resulting equation of state can

be used for predictions beyond the pressure range

of the shock wave data. The question that remains

unanswered in this approach is whether the third

parameter is compatible with the second pressure

derivative of the bulk modulus or whether it provides

only an approximate fit of the empirical data at high

pressure which may be sufficiently accurate,

though, for interpolating and extrapolating the data

at high pressure. Since for the five materials con- sidered by Anderson [3] the second pressure

derivatives of the bulk modulus have not yet been

determined ultrasonically this point must await

future clarification.

For the cesium halides CsCl, CsBr, and Csl the

second pressure derivatives of the elastic constants

have been measured recently by an ultrasonic

method [5], so that for these materials the second

pressure derivatives of the bulk modulus are available in addition to the bulk modulus and its first pressure derivative. This provides the opportunity

of investigating the consequences and improvements

which can be attained with a purely ultrasonic

three-parameter equation of state. For Csl the

previous measurements have been repeated with

increased accuracy and extended as a function of

temperature. On the basis of the resulting ultrasonic

equation of state, it is possible to use Csl as a pres- sure standard over a wide range of pressure and

temperature.

To this end a special mathematical representation

must be adopted for the equation of state. Possible

forms with three parameters are, for example, (1) a truncated Taylor expansion of the volume with

respect to pressure up to and including third order

terms in pressure, (2) a generalized form of Murnaghan's equation based on integration of a quadratic

expression for the pressure dependence of the bulk

modulus [46], (3) the second-order approximation of

Birch's equation of state [6-8], (4) Keane's equation

of state [3, 4], (5) a semi-theoretical equation of

state based on a special model for the interatomic

forces containing empirical parameters. All these

forms may be designed to agree in the bulk modulus

and its first two pressure derivatives, but they will

differ in the higher than second-order pressure

derivatives of the bulk modulus, or equivalently, in

the quality of approximation to the true equation

of state.

It is well known, for example, that the Taylor

expansion (1) is so slowly convergent that it is in- sufficient for the description of empirical compression curves even over a moderate range of pressure

[7, 1]. Therefore, this form need not be considered

in the present paper. Objections have also been

raised against the quadratic expression for the pres- sure dependence of the bulk modulus [7, 3], because

a negative second pressure derivative of the bulk

modulus which is demanded by shock wave data

implies that the bulk modulus decreases with

pressure after passing through a maximum, which is unlikely on physical grounds. In this paper additional evidence against the use of the corresponding

form (2) for the equation of state will be presented.

The quality of the remaining equations can be

verified by comparison of the compression curves

calculated from ultrasonic data with direct experimental data, in connection with a careful error

analysis. In addition, theoretical considerations on

the rate of convergence of any form based on a truncated Taylor expansion permit an evaluation.

It is therefore the purpose of this paper to compare

several representations of the three-parameter

ultrasonic equation of state with static experimental

data of Bridgman [9, 10] and of Perez-Albuerne and

Drickamer [11, 12] that were obtained with a freepiston apparatus and with a high-pressure x-ray

camera, respectively. A complete error analysis

will also be carried out. In addition, the convergence

of the equations of Murnaghan, Birch, and Keane

will be checked by comparison with a lattice

theoretical equation with empirical parameters

which are determined from the bulk modulus and

its first two pressure derivatives.

While the present paper was inspired by the

observation of Anderson [1, 3] that the ultrasonic

equation of state is valid over a considerably larger

pressure range than the ultrasonic measurements

themselves, it extends the previous work of Anderson insofar as (1) the second pressure derivative of

the bulk modulus is included from ultrasonic

measurements and a purely ultrasonic threeparameter equation of state is used, (2) several

alternative expressions for three-parameter equations of state are critically compared and finally

Birch's equation of state in second-order approximation is used, (3) the effect of the omission of higher

order terms is estimated, (4) a careful analysis of

the effect of the errors of all input data is performed,

and (5) the variation with temperature is taken into

account.

The credit and priority for having proposed a pressure standard on the basis of the equation of

state of an alkali halide, NaCl, belongs to Decker

[13, 14]. However, only one of his three empirical

parameters, namely the zero pres^sure value of the

bulk modulus, is obtained from ultrasonic measurements. The remaining parameters are determined

from the lattice constant and the thermal expansion

as a function of temperature. Decker's equation of

state does not agree with the ultrasonic value for

the first pressure derivative of the bulk modulus

and, strangely enough. Decker dismisses the

ultrasonic value for the pressure coefficient of the

bulk modulus of Bartels and Schuele [15] as inaccurate because it differs from his theoretical

174
; equation of state [14]. Instead, it is just as likely that

Decker's equation of state is in error because of the

uncertainty in the empirical parameters and/or

because of the special assumptions about the

interatomic forces which are implicit in this equation

of state. This point will be discussed below in

section 3.2.

Since the bulk modulus and its pressure derivatives completely characterize the equation of state for constant temperature, it is an undisputable

requirement that any theoretical equation of state must agree with these experimental data. In view

of our very limited knowledge on the interatomic

forces in solids, the most natural way in meeting

this requirement is to use a model potential with

adjustable parameters which are fitted to the

experimental values of the bulk modulus and its pressure derivatives. It will be shown below in

section 3.2 that in the pressure range of interest

the Birch equation of state [6-8], which is an

empirical equation of state based on a rapidly con- verging Taylor expansion of the interatomic potential, is within the limits of experimental error indistinguishable from such a semitheoretical

equation of state. For this reason the ultrasonic

equation of state proposed in this paper will be

based on Birch's equation of state. Objections have also been raised against the use

of NaCl as a pressure standard because a phase

transition to the CsCl structure occurs at high

pressure [16]. Although the magnitude of the transition pressure is stiU controversial and may be

as high as 300 kbar [17, 18], it seems more advantageous to use alternative materials for which no

high-pressure transition is known. Therefore the

three cesium halides, CsCl, CsBr, and Csl, were

selected. Of these materials, CsCl has the disadvantage that its use as a high-pressure standard

is restricted in temperature to the range below

470 °C because of the phase transition to the

NaCl structure [19-21]. In spite of this limitation

Table 1. Bulk modulus Bo, first pressure derivative

B^'', and second pressure derivative B^-^ of CsCl

and CsBr at 300 °K from ultrasonic measurements

(calculated from the experimental values underlying the results reported in ref. [5])

CsCl CsBr

fio = B''(kbar) 167.4 ±0.9

143.4 ±.8

" \ dp 't

5.98 ±.09

5.95 ±.08

-0.042

±.04

-0.050

±.004

a detailed discussion of CsCl will be presented in

this paper because, of the three cesium halides

considered, CsCl is the only one for which equation

of state data from direct measurements are available

above 100 kbar [11, 12] which allow a more critical

comparison of the ultrasonic and the directly

measured equation of state than Bridgman's compression data below 100 kbar [9, 10].

Another advantage of the three cesium halides

is that they are the most highly compressible of all alkali halides which made it possible to determine

the second pressure derivatives of the bulk modulus

from ultrasonic measurements below 10 kbar [5]

and to propose a purely ultrasonic three-parameter

equation of state. The high compressibility is also

advantageous since it corresponds to a high sensitivity of the pressure measurement with respect to the dimensional changes of the crystal.

2. Dependence of Bulk Modulus on

Pressure and Temperature

The pressure dependence of the elastic constants

of one or of several of the cesium halides CsCl,

CsBr, Csl has been measured by several authors

[22-24, 5], but only in one case was the precision

sufficiently high so that not only the first, but the

second pressure derivatives also could be determined [5]. These data will be used in the present

paper.

The isothermal bulk modulus and its first isothermal pressure derivative have been calculated

for CsCl and CsBr from revised data for the adiabatic bulk modulus and its isothermal pressure

derivative of reference [5] by using the thermal

data tabulated in reference [25] and the conversion

formulae of Overton [26]. The results are listed in

table 1 together with the isothermal second pressure

derivative of the adiabatic bulk modulus corresponding to the unproccessed ultrasonic data. These sec- ond pressure derivatives were not converted to the

purely isothermal quantities since this would re- quire the value of the temperature derivative of the

first pressure derivative of the adiabatic bulk modulus, which was not available for the data of reference

[5]. It will be shown below that for Csl the difference

between the intermediate isothermal-adiabatic and

the pure isothermal derivative is not much larger

than the experimental error. For the comparison

of the ultrasonic equation of state with direct experimental data it is therefore sufficient to use the un- converted intermediate second pressure derivatives

listed in table 1, the more so since in the pressure

range to be considered the contribution from this

term is small.

The experimental error shown in table 1 was

calculated from a careful statistical analysis of the

raw data underlying the values reported in reference

[5]. This was done by fitting the pressure dependence

of the bulk modulus to a parabola and taking into

175
account that the bulk modulus is a linear combination of two elastic constants, 5= (1/3) (cii+2ci2),

which were determined from four independent

series of measurements for the three elastic constants Cii, Ci2, and C44. The method used is described

in chapter 6 of ref. [27]. The errors shown in table 1 correspond to the standard error arising from the

statistical mean square deviation plus an estimate

for all other errors which are represented primarily

by the uncertainty in the pressure reading. The

pressure was measured by means of a manganin

cell in connection with a resistance calibration curve

provided by the Harwood Engineering Co., Walpole,

Mass., and stated to be as accurate as 10 The

resistance was measured with an ac bridge in con- nection with a Foxboro automatic recorder with an accuracy of 2 percent. This was the weakest

hnk in the chain of accuracy limiting factors.

Part of this error appears in the statistical error, and the remaining possible systematic error has

been estimated as 1 percent and is included in

the total error shown in table 1. Also included in

the error shown is the uncertainty arising in the

conversion from the adiabatic and intermediate

ultrasonic data to the purely isothermal data, which

was calculated from a liberal estimate of the uncertainty of the thermal data used in the conversion.

Very recently, for Csl we have repeated the ultrasonic measurements of the pressure dependence of

the elastic constants. The objective was, first, to

attain higher accuracy by measuring the resistance

change of the manganin pressure cell with a CareyFoster bridge. Moreover, the measurements were

carried out at four different temperatures up to 250 °C. It was possible to determine the temperature

derivative of the pressure coefficient of the bulk

modulus and to convert the second pressure derivative of the bulk modulus from the intermediate to the purely isothermal value. The ultrasonic values

of the bulk modulus and its derivatives with respect

to temperature and pressure are shown in table 2

together with the converted purely isothermal

values. The first and second temperature derivatives

of the adiabatic bulk modulus were taken from

reference [28]. Details of the measured elastic constant data and the conversion formulae not con- tained in Overton's paper [26] will be published

elsewhere.'^ The data of table 2 agree within experimental error with those of reference [5], but they

are more accurate than these previous values. The

errors shown in table 2 were calculated in the same

manner as those in table 1, and for the possible

systematic error of the pressure measurement a value of 0.5 percent has been assumed. For the

thermal quantities which are required for the con- version from the adiabatic to isothermal conditions,

smaller errors have also been assumed. With the

The formula for converting the isothermal second pressure derivative of the adiabatic bulk modulus to the isothermal pressure derivative of the isothermal bulk modulus

has been published recently by Yu [47].

Table 2. First and second derivatives of the

adiabatic and isothermal bulk modulus with

respect to temperature and pressure for Csl

at 25 °C

B (kbar) 126.2 ±0.2

118.9 ±0.5

(-^)^(kbarrC) — U.UOD / •'±.0034 —U.lUZo

3 ±.0060

(^,)^(kbar/rCn '-5.0X10-^ -1.3X10-^

(-) 5.70 5.93 \dp / T ±.06 3 ±.08

-0.061

±.004

-0.073

"±.008

1.4X10-3 2.6x10-3

' From reference [28].

- Estimated error: 5 percent.

^ From error of data listed in table 3. 'Estimated error: twice the value of A(,d'B^ldp')T.

Table 3. Density po, linear thermal expansion

coefficient a with temperature derivative (da/dT),

and specific heat Cp with temperature derivative

{d(.pjdT)p and estimated errors for Csl at 25 °C

poig cm-3) 4.525 + 0.3% Ref. [29]

a (10-M°C)-') 4.05 ± 2% Ref. [30]

(daldT)p (10-«(°C)-2) 6.6 ±3% Ref. [30]

Cp (Jg-M°C)->) 0.200 ±1% Ref. [31]

(dCplST),, (10-3Jg->(°C)-2) 0.043 ±3% Ref. [31]

increased accuracy of the pressure measurement,

the effect of these errors becomes now comparable

to the error of the ultrasonic data. For the pressure

coefficient of the bulk modulus, for example, the

error is increased by as much as 33 percent due to the conversion from the intermediate to the purely

isothermal value. Since this error turns out to be

one of the main accuracy-limiting factor for the

equation of state, the thermal data that were used

in addition to the data of table 2 for the conversion

are listed in table 3, together with their estimated

errors. Since the formula for the conversion of the

second pressure derivative is so lengthy and de176
pends on a large number of variables, no rigorous

error analysis has been carried out, and an estimate

will be used in this case. It is assumed that the error

of (d'-B''ldp'-)T is twice as large as the error of

Since the temperature dependence of the bulk

modulus is nonlinear, and since there is no theoretical evidence that the pressure coefficient of the

bulk modulus depends linearly on temperature, the

data of table 2 wiU not be used for calculating the

high-pressure equation of state as a function of temperature. Instead, the bulk modulus and its first two

pressure derivatives will be calculated from the

formula

(2.1)

with the superscript R referring to the reference

temperature tR. Further it is [6, 32]:

R(i) = ^

95o (2.2)

fiofi<?'=^- [1 + ^'']^'' +

^

(2.3)

where C and D are linear combinations of the thirdand fourth-order elastic constants, respectively.

Their room temperature values are listed in table 4

together with their first temperature derivatives.

Since at high temperatures (above the Debye temperature) the temperature dependence of the higher

order elastic constants is linear in the first-order

approximation of the quasiharmonic approximation

[33, 34], the experimental data taken at different

Table 4. Linear combinations C —Ciii~l"6cij2~l~2c]23

and D = Ciiii + 8cni2 + 6cn22 + 12cn23 of the intermediate and ofthe isothermal third- andfourthorder elastic constants, and their temperature

derivatives for Csl at 25 °C

Intermediate Isothermal

C (kbar) -6.13 X 1(F -6.32X103

(|),*b.r,^) 3.6 2.6

D (kbar) 1.02 X 10^ 1.03 X 10^

-11.0 -16.0

temperatures were interpolated linearly,

C = C, +

(^^J

it-t,) (2.4)

/dDV D= Dn + i^—j it-tn). (2.5)

For the temperature dependence of the high-pres- sure equation of state, the bulk modulus and its

first two pressure derivatives will be calculated in

section 4 as a function of temperature according to eqs (2.1) to (2.5) with the coefficients Bo, {dB/dT),

(d^Bldr), C, (dCldT), D, and {dD/dT) listed in

tables 2 and 4 for the isothermal case.

3. Evaluation of Several Three-Parameter Equations of State

3.1. Phenomenological Equations of State

A variety of mathematical expressions have been

proposed for the description of the equation of state

of solids at high pressure, i.e., for the relation between pressure and volume for specified thermodynamic conditions. Most of these are based on some truncated Taylor expansion and can therefore

be expressed in terms of an arbitrary number of

parameters, corresponding to the order of the approximation. Which one of these expressions for a

given number of parameters is best suited for an

accurate representation of experimental (or theoretical) data over a specified pressure range depends

on the rate of convergence of the underlying Taylor

expansion and can only be determined a posteriori,

i.e., by comparing this approximate phenomenological equation of state with experimental or rigorous

theoretical data. Such a comparison wiU be made in

sections 3.2 and 3.3 by using the equations presented subsequently and in section 3.2.

For convenience,, isothermal conditions wiU be

assumed and all quantities will be given in dimensionless form. All parameters occurring in the

explicit expressions for the equation of state are

therefore dimensionless quantities also, which

depend only on the temperature and the material

considered. Let

R=VIVo (3.1)

and

tt = pIBo (3.2)

denote the compression ratio of the volume V and

the pressure p in units of the bulk modulus Bo,

respectively, with the index 0 always referring to zero pressure.

The classical expression for the equation of

state consists of a truncated Taylor series of the

volume with respect to pressure (see, e.g., [1]):

177
N

^0 (3.3)

From the definition of the bulk modulus at arbitrary pressure,

and its Taylor expansion with respect to pressure

(written here for later reference in the Mh order

approximation),

Bm{p)=J^\b^^P^ (3.5)

i=o

the expansion coefficients ti can be expressed in

terms of the zero pressure values of the bulk

modulus and its pressure derivatives Bj,''. The first four expansion coefficients are [1]

to= 1,

fi = -l,

i2-(l/2)(B(i)+l),

«3=(l/6) [l + 3B(i) + 2(fi(i))2-Bo5(2)],

U={im) [l + 6fi(i)+ll(5(i))2 +

6(5<i')'

-2(2 + 35(i))fiofi[,2' + B?5ff'] • (3.6)

It has long been recognized that (3.3) is very

slowly convergent and therefore useful only for

small pressures, tt <^ 1 [1, 7, 9]. If this form with a given number of terms is used as an empirical equation, then the coefficients that are obtained by

fitting the experimental compression data depend

for larger pressures, e.g., tt<\, on the pressure

range of the experiments and must be extrapolated

to zero pressure [1, 7, 24].

These disadvantages are avoided in the equation

of state proposed by Birch [6-8]

,

7T,{R)=-R-^l^^bAR-"'-iy (3.7)

i=i

which is based on the Taylor expansion of the strain

energy with respect to the Eulerian strain

components.

Again the expansion coefficients can be expressed

in terms of the pressure derivatives of the bulk

modulus. The first four coefficients are

6, = 1,

62= (3/4) (5(0^) -4),

63=(l/24)[143 + 9(By)-7)fi(,') + 95o5[f)]

64 = (l/192)[-1888+1122B(,i>-288(B(o'))2

+ 27 (fi(i')=^ + 36 (3fi(i' - 8)Bo5o<2' + 21B%B^i^'\ . (3.8)

The advantage of (3.7) is that it is much more

rapidly convergent than (3.3), and that even in the

pressure range as large as tt ^ 0.5 the approximation with only two terms corresponding to N=2 is sufficiently accurate for many practical purposes

such as arise, e.g., in geophysics.

Another approach to the equation of state is due to Murnaghan and consists of integrating the

Taylor expansion of the bulk modulus (3.4) under

the assumption that the linear approximation in

(3.5) corresponding to A'^= 1 is sufficient [2]. The

resulting "first-order Murnaghan equation of state"

is [2, 11]

Rr^il + mn)-'"" (3.9a) or

7Ti = m-'iR-"'-l). (3.9b)

Correspondingly, one obtains by integrating (3.4)

in connection with (3.5) for A'^^2 the "second-order

Murnaghan equation of state" [46]:

R2 = {[2+ (n+l)m7r]/[2- (ra-l)ra7r]}-i/«"'

(3.10a)

or

7T2 = 2m-^[R-'""-l][(n-l)R-'""+(n+l)]-K

(3.10b)

The parameters occurring are m = Bi'\ [1 - (2SofiS,2))/(5(}) ryi\ (3.11)

For fi[,-' = 0 it is n=l, and the second-order

Murnaghan equation (3.10) reduces to the first- order form (3.9).

The three types of equations considered so far are based on different kinds of Taylor expansions

and can be used with any desired number of parameters. In this paper only the two- and three-parameter

forms of Birch's and Murnaghan's equations will

be considered.

Another three-parameter equation of state has

been proposed by Keane [4] and recently been discussed by Anderson [3]. This equation is not based

on any Taylor expansion, but on the assumption

that the bulk modulus is a monotonously increasing

function of pressure, and its pressure coefficient a monotonously decreasing function of pressure,

according to

178
(3.12)

fi'j' is the limiting value of the pressure coefficient

of the bulk modulus for infinite pressure. Kean's

equation of state is obtained by twofold integration

of (3.12) in connection with (3.4) [3, 4]:

7r,-=(fi|;vfiL'')^) [/?-"*^'-i]

+ [(fi(^Vfi<i>)-l]ln/?. (3.13)

By comparing the Taylor expansion of (3.13) with

(3.3) and (3.6) the third parameter, fl^' be

expressed in terms of the second pressure derivative of the bulk modulus as follows:

fiO) = fiy)+(5o5<2VfigO- (3.14)

The two assumptions of Keane (B{p) monotonously increasing with p, (dB/dp) monotonously

decreasing with p) lead to the following condition

for the bulk modulus and its first two pressure

derivatives:

-(fiy))2<5o5(f)<0. (3.15)

Both inequalities are fulfilled for the experimental

data of CsCl, CsBr, and Csl listed in tables 1 and

2 so that it is consistent to use, for these materials,

Keane's expression (3.13) for the representation

of the equation of state.

3.2. A Lattice Theoretical Equation of State

and Comparison With the Phenomenological Equations of Section 3.1

An ionic crystal, such as one of the alkali halides,

seems ideally suited for deriving a theoretical equation of state because the interatomic forces can be

described quite adequately and successfully in

terms of Coulomb and van der Waals attraction,

and in terms of a Born-Mayer type repulsive potential with empirical or adjustable parameters. In

view of the considerable success of the classical

theory of ionic crystals, it is not surprising that this

approach has been used also for the equation of

state of alkali haUdes [12-14, 35]. It should be

remembered, however, that these equations of

state are semi-empirical, and that their accuracy

is limited by the accuracy of the empirical quantities that were used for the determination of the

parameters in the repulsive potential. Since these

parameters are determined from experimental

data measured at low pressure one must expect

that the effect of their errors increases with in- creasing pressure. Thus from this point of view

there would be no advantage in using a lattice

"theoretical" equation of state instead of a phenomenological equation as discussed in section 3.1,

provided the phenomenological equation selected

converges sufficiently fast and differs in the pres- sure range considered from the lattice theoretical

equation with the same number of parameters by

an amount smaller than the effect of the experimental error of the input data.

Another limitation of the kind of lattice theoretical

equation of state considered is that it is based on

several simplifying assumptions and that their

effect on the results is not easy to determine and

would require a much higher theoretical effort. First, it has been noted that the inverse power law

for the van der Waals potential should be screened

at smaller interatomic separations due to the interpenetration of the electronic shells [36]. Also, the use of an exponential law for the repulsive interaction is only an approximation because the radial

dependence of the electronic wave functions cor- responds to a series of terms consisting of products

of polynomials and exponential functions and is reflected in the radial dependence of the cohesive

energy [37, 38]. More specific approximations of

dubious justification made in previous lattice

theoretical equations of state consist of the use of an

inverse power potential for the repulsive interaction

and neglect of second nearest neighbor effects [35], or of the evaluation of the repulsive parameters

for the second nearest neighbor interaction from the

electronic polarizabilities [13]. The equation of

state of Decker is further based on the DebyeGrueneisen approximation, and the Grueneisen

constant and its first pressure derivative are determined jointly with the first nearest neighbor repulsive parameters from empirical data of the thermal

expansion as a function of temperature [13].

In view of the considerable difficulties encountered in deriving a sufficiently reliable lattice

theoretical equation of state, the question arises

whether an empirical ultrasonic equation of state based on any of the phenomenological equations

discussed in section 3.1 agrees within the uncertainty limit caused by the experimental error of

the empirical parameters with the corresponding

lattice theoretical equation of state. In order to explore this possibility, a simple lattice theoretical

equation of state with four empirical parameters

was compared with the phenomenological equations

discussed in section 3.1. This equation of state is based on the expression

ae~ CD --

<2/V3)r = p. +^26 p^ (3.16)

for the lattice energy per ion pair where the symbols

have the usual meaning [39], and the effect of

thermal motion has been neglected. This expression

contains the usual Coulomb and van der Waals

terms, and first and second nearest neighbor repulsive interaction. In general, two exponential terms

179
are required for the second nearest neighbor interaction, but these two reduce to one if it is assumed

that the same constant pz describes both cationcation and anion-anion interaction. The van der

Waals constants C, D were taken from references

[40] and [41], respectively, and the repulsive constants Ai, A2, p\, p-i were determined from the zero pressure values of the lattice constant, and of the

bulk modulus and its first two pressure derivatives.

The equation of state based on (3.16) according to

_ d(f) _ I dcj)

was compared with (a) Murnaghan's first-order

equation (3.9); (b) Birch's equation (3.7) in first- order approximation; (c) Murnaghan's secondorder equation (3.10); (d) Birch's equation (3.10)

in second-order approximation; and (e) Keane's

equation (3.13). The same ultrasonic values for

the bulk modulus and its first two pressure derivatives were used for these phenomenological equations so that the difference Ap = p lattice ~Pphenbetween the lattice theoretical equation of state

based on (3.16) and the phenomenological equations

(a) to (d) arises from differences in the second and

higher pressure coefficients of the bulk modulus

(cases (a) and (b)), or from differences in the third

and higher order pressure derivatives of the bulk

modulus (cases (c) to (e)). This difference Ap is therefore an indication of the quality of the dif- ferent phenomenological equations in approximating

the lattice theoretical equation based on (3.16),

and a measure of the rate of convergence of the

various Taylor-expansions underlying the phenomenological equations (a) to (e). Although the exact

or true potential energy of the lattice will be given

only approximately by (3.16) and the thermal energy

has been neglected, one may expect that the dif- ference between the two values of Ap referred to

(3.16) and to the volume dependence of the true

free energy is a higher order quantity, so that for

ionic crystals the difference Ap based on (3.16)

may be considered as a fair estimate of the deviation

of the phenomenological equations from the true

equation of state. The difference |Ap| is plotted for Csl in figure 1 for the five cases (a) to (e) as a function of pressure.

The ultrasonic data of table 2 were used in connection with the room temperature value of the lattice

constant from reference [30]. For CsCl and CsBr

curves very similar to those of figure 1 were obtained. It is apparent that in the entire pressure

range up to 500 kbar the quality of the second-order

Birch equation in approximating the lattice theoretical equation is one order of magnitude better than

Keane's equation, and about two orders of magnitude better than the second order Murnaghan

equation. As all three equations contain the same

number of three empirical parameters, this drastic

100 200 300 400 500

Pressure p (kbar)

Figure 1. Pressure difference between phenomenological and

lattice theoretical equation of state versus pressure for Csl.

Figure 2. Comparison of compression data for CsCl, CsBr,

and Csl by Bridgman [9, 70] with ultrasonic Birch equation

of state in first- and second-order approximation.

superiority of Birch's equation is quite surprising.

The first-order Birch equation also can be seen to be superior to the first-order Murnaghan equation,

but the difference is not quite as pronounced. As

expected, the two three-parameter equations accord180
ing to Birch and to Keane are superior to any of the

two-parameter (first-order) equations of Murnaghan

and Birch, but interestingly enough the secondorder Murnaghan equation gives above 80 kbar

larger deviations than the first-order Murnaghan

equation and shows, in fact, the largest deviation

from the lattice theoretical equation of all cases

considered.

These results suggest the superiority of the

second-order Birch equation over all other phenomenological equations considered containing an equal or smaller number of parameters.

The absolute magnitude of the discrepancy

ranges for the second-order Birch equation from 65

bars (or 0.065 percent) at 100 kbar to 4.6 kbar (or

0.9 percent) at 500 kbar. It wiU be shown below in

section 4 that this discrepancy is considerably

smaller than the errors that may be expected from

the error of the ultrasonic input data and from the

measurement of the lattice constant. Thus in the

pressure range up to 500 kbar it seems sufficient

to use a second-order (three-parameter) Birch

equation instead of a lattice theoretical equation

with an equal number of empirical parameters.

3.3. Comparison of Ultrasonic Equations of

State With Piezometric Data

In this section phenomenological equations of

state discussed in section 3.1 will be calculated

from the ultrasonic data of tables 1 and 2, and compared with direct piezometric data of Bridgman

[9, 10] and Perez-Albueme and Drickamer [11, 12].

The purpose is first, to determine whether the ultrasonic and the piezometric data are compatible

within experimental error; second, to check whether

the contribution from the second pressure derivative of the bulk modulus is noticeable in the pres- sure range considered and improves the agreement

with the piezometric data; and third, to provide

further support for the above conclusions concerning the superiority of the Birch equation of state as compared with Murnaghan's equation.

From the comparison of the ultrasonic equation of

state (based on the first- and second-order Birch

equation) with Bridgman's piezometric data in

figure 2, one may make the following observations.

First, the effect of the second pressure derivative

of the bulk modulus is noticeable in the pressure

range below 50 kbar, but is within the limit of error

of the ultrasonic data, calculated from the errors shown in tables 1 and 2 by means of the Gaussian

error propagation law. Second, the agreement

between ultrasonic and piezometric data is improved by including the second pressure derivative

for Bridgman's data of 1940 [9] for CsCl and

CsBr, but worsened in aU other cases. Third, for

CsCl and CsBr Bridgman's data of 1940 [9] agree

well above about 15 kbar (within experimental

error) with the ultrasonic data, but below that presp (kbar)

Figure 3. Comparison of compression data for CsCl by Perez- Albuerne and Drickamer i2] with (a) ultrasonic Murnaghan equation of state in first- and second-order approximation, (b) ultrasonic Birch equation of state in first- and

second-order approximation, and (c) extrapolated compression

data by Bridgman [9, 70].

sure and in all other cases, diff"erences which are outside the range of error of the ultrasonic data

occur. One must conclude then that Bridgman's earlier

data of 1940 are more accurate than his later data

of 1945 [10], and that the results for the less compressible materials CsCl and CsBr are more

accurate than for the more compressible Csl.

In figures 3a and 3b the ultrasonic equation of

state as based on the first- and second-order approximations of the equations of Murnaghan and of

Birch, respectively, are for CsCl compared with the

data obtained by Perez-Albuerne and Drickamer

[11] with a high-pressure x-ray camera. The error of

the ultrasonic data is indicated by the broken curves

in figure 3a, and by a few selected error bars on fig- ure 3b. The x-ray data were fitted to a third-order

Birch equation (four parameters) and are plotted as the full curve. In figure 3b the individual data are

plotted also in order to show the large scatter present. The error was calculated from the standard

deviation according to the method described in ref- erence [27], section 6.25, and is shown as error bars.

It is apparent from figure 2a that the ultrasonic

data agree well with the x-ray data up to about 225

kbar in the first-order Murnaghan approximation,

but only up to about 150 kbar in the second-order

Murnaghan approximation. Thus the inclusion of the

second pressure derivative B'^^ actually deteriorates

the agreement between the two sets of data. Since

the second-order Murnaghan equation shows an in181
flection point near 150 kbar, which is unlikely on

physical grounds, one has to discard the secondorder Murnaghan equation as inappropriate. This

confirms the conclusion arrived at before on the

basis of figure 1, and substantiates similar expectations expressed earlier by Anderson [3]. Figure 3b shows that the difference between the

Birch equations in first- and second-order approximation is much smaller than that for the Murnaghan

equations, and that both the first- and the secondorder equations are in substantial agreement with

he x-ray data. Surprisingly enough, the first-order

ultrasonic Birch equation agrees within experimental error with the x-ray data, but the secondorder ultrasonic Birch equation deviates markedly

above about 400 kbar from the x-ray data. Since the

effect of a possible systematic error in the pressure

scale used in the ultrasonic measurements is in- cluded in the error of the ultrasonic data, and since

the error indicated for the x-ray data is based on the

mean square deviation only, one is compelled to suspect the presence of a systematic error in the

x-ray measurements which becomes excessively

large above about 400 kbar. This conjecture is substantiated by the fact that the least square fit of the

x-ray data to a third-order Birch equation exhibits an

inflection point near 300 kbar, which is unlikely on

physical grounds. Although any discussion of a possible source for such a possible error is purely

speculative, one may expect the pressure scale

used in the x-ray measurements to be a very likely cause. This pressure scale is based on the equation

of state of Ag and Mo that were obtained from the

conversion of shock wave data.

A feature worth noting is that the total error of

the ultrasonic data is smaller than the purely statistical error of the x-ray data in the entire pressure

range considered.

Since significant differences were found for the

cesium halides between the ultrasonic data and

Bridgman's volumetric data, it was deemed worthwhile to compare Bridgman's data with the x-ray

data. To this end Bridgman's data were fitted to a

third-order Birch equation and then extrapolated over the pressure range covered in the x-ray measurements. The results are compared in figure 3c

with the x-ray data. Also shown are the error limits

of Bridgman's data based on the mean square

deviation. It is apparent that Bridgman's data of

1945 [10] are much more precise, but of so low ac- curacy, that they differ from the x-ray data above

about 180 kbar. On the other hand, Bridgman's

data of 1940 [9] show a very large statistical error when fitted to a third-order Birch equation, so that

the agreement with the x-ray data within the joint

experimental error is extended to as high pressures

as about 380 kbar. Since both curves of Bridgman

fall quite close to each other, it is still safe to con- clude that the agreement between Bridgman's

data and the x-ray data is quite poor. The fair agreement found between the x-ray data and the ultrasonic data must therefore be taken as evidence in

favor of both these sets of data, and against the

reliability of Bridgman's data. The observation

that Bridgman's data are of rather limited accuracy

only has been made by several authors who also

discuss possible causes of error [42-44].

4. Ultrasonic Equation of State in SecondOrder Birch Equation

From the discussion of the preceding section

it seems justified to propose an ultrasonic equation

of state for use as a high-pressure standard. For

this purpose Csl has been selected, and an equation

of state calculated from the ultrasonic data of tables

2 and 4 on the basis of the second-order Birch equation. The results are expressed in terms of the lattice

parameter and are listed in table 5 and plotted in

figure 4. The absolute values of the lattice parameter

at zero pressure were taken from the thermal ex- pansion data of references [30],

a = 4.549+1.767Xl0-^i+1.528Xl0^^f2 (4j)

with a in angstrom units and t in degrees Celsius.

The relative compression a{T, p)la{T, o) is, however, independent of these data (except for the

thermodynamic conversion from the mixed ultrasonic to the pure isothermal data).

Since for practical applications the reliability

of these results is of crucial importance a detailed

error analysis has been carried out at room

temperature.

The pressure is according to (3.2), (3.7), and

(3.8) in second-order Birch approximation a function of the bulk modulus i5o, of its first two pressure

derivatives fi^' and fijf', and via the temperature

dependence of these quantities, a function of

temperature. If instead of the volume compression

Pressure p (kbor)

Pressure p (kbor) ,

Figure 4. Equation of state of Csl, calculated in second-' order Birch approximation from ultrasonic data of tables 2 and 3.

182
Table 5. Pressure according to ultrasonic equation of state for Csl as a function of the lattice constant a and temperature.

a{p, T)

a(0, 25)

inn °c ZUU 900 °P\j Ann °c i^OO OUU Li

1.030 4.691 0.54

1.025 4.668 0.30 1.26 1.020 4.645 1.23 2.12

1.015 4.622 1.10 2.29 3.13

1.010 4.600 0.98 2.36 3.50 4.31

1.005 4.577 0.95 2.43 3.77 4.89 5.66

1.000 4.554 0.00 1.11 2.60 4.05 5.36 6.45 7.21 0.995 4.531 1.52 1.87 2.96 4.43 5.85 7.14 8.21 8.97 0.990 4.508 3.58 3.92 4.99 6.44 7.84 9.12 10.19 10.96 0.985 4.486 5.83 6.17 7.22 8.65 10.05 11.33 12.41 13.20

0.980 4.463 3.29 8.63 9.67 11.09 12.48 13.77 14.88 15.71 0.975 4.440 10.98 11.31 12.35 13.77 15.17 16.48 17.62 18.52 0.970 4.417 13.91 14.25 15.29 16.70 18.12 19.47 20.7 21.6 0.965 4.395 17.11 17.45 18.49 19.92 21.4 22.8 24.0 25.1 0.960 4.372 20.6 20.9 22.0 23.4 24.9 26.4 27.8 29.0

0.955 4.349 24.4 24.8 25.8 27.3 28.9 30.4 31.9 33.2 0.950 4.326 28.5 28.9 30.0 31.5 33.1 34.8 36.4 37.9

0.945 4.304 33.0 33.4 34.6 36.1 37.8 39.6 41.4 43.1 0.940 4.281 37.9 38.3 39.5 41.1 43.0 44.9 46.9 48.8

0.935 4.258 43.3 43.7 44.9 46.6 48.6 50.7 52.9 55.0

0.930 4.235 49.0 49.5 50.8 52.6 54.7 57.0 59.5 61.9

0.925 4.212 55.3 55.8 57.3 59.1 61.4 64.0 66.7 69.4 0.920 4.190 62.1 62.7 64.2 66.2 68.7 71.5 74.5 77.6 0.915 4.167 69.5 70.2 71.8 73.9 76.7 79.8 83.1 86.5 0.910 4.144 77.6 78.3 80.1 82.4 85.4 88.8 92.5 96.3

0.905 4.121 86.3 87.1 89.1 91.5 94.8 98.6 103. 107. 0.900 4.099 95.7 96.7 98.9 101. 105. 109. 114. 119. 0.895 4.076 106. 107. 110. 112. 116. 121. 126. 131. 0.890 4.053 117. 118. 121. 124. 129. 134. 139. 145. 0.885 4.030 129. 131. 134. 137. 142. 148. 154. 160.

0.880 4.008 142. 144. 147. 151. 156. 163. 169. 176. 0.875 3.985 156. 158. 162. 166. 172. 179. 186. 194. 0.870 3.962 172. 174. 178. 182. 189. 197. 205. 213. 0.865 3.939 188. 191. 196. 200. 208. 216. 225. 234. 0.860 3.917 207. 210. 215. 220. 228. 237. 247. 256.

0.855 3.894 226. 230. 236. 241. 250. 260. 270. 280. 0.850 3.871 247. 251. 258. 264. 273. 285. 296. 307. 0.845 3.848 270. 275. 283. 289. 299. 311. 324. 335. 0.840 3.825 295. 301. 309. 316. 327. 341. 354. 366. 0.835 3.803 323. 329. 338. 345. 358. 372. 387. 399.

0.830 3.780 352. 359. 369. 377. 391. 407. 422. 435. 0.825 3.757 384. 392. 404. 412. 427. 444. 461. 474. 0.820 3.734 419. 427. 441. 449. 466. 485. 503. 516. 0.815 3.712 456. 466. 481. 490. 509. 529. 548. 561. 0.810 3.689 497. 508. 525. 535. 555. 577. 597. 610.

183

385-762 0-71-13
where the factor

100 200 300

Pressure p (kbar) 400 500

Figure 5. Sensitivity of equation of state of Csl with respect

to changes of the bulk modulus Bo, its first two pressure derivatives B*J* and Bjj^', temperature T, and lattice parameter a, at 25 °C in second-order Birch equation.

ratio R according to (3.1) the hnear compression

ratio

a{p, T)

(4.2)

a(o, T)

is used as independent variable according to = (4.3)

then the functional dependence of the pressure on

all variables is of the form

p=p{BoiT), fi(>»(r), Bi'KT), L) (4.4)

The pressure as calculated from the ultrasonic

equation of state will therefore be subject to the

errors ^B,^{T), AB{,'^{T), AB^'^{T) of the ultrasonically determined input data at a given temperature, which will be taken as 25 °C. These errors

limit, therefore, the accuracy of the pressure scale

based on the ultrasonic equation of state. In addition to these errors in the pressure scale, the actual

pressure measurement will be limited in accuracy

because of insufficient control and characterization

of the temperature, corresponding to an error AT,

and because of an error Aa in the measurement of

the lattice constants. The relative error in pressure

due to each of these errors considered separately is given by

_ d In p

d In X

(4.6)

Ap^

P

Ax

(4.5)

represents the sensitivity and x denotes any of the

five variables B^^, 5,*,", Bj^^K T, and a. The sensitivity

was calculated according to (4.6) from the data of

tables 2 to 4 for Csl and is plotted in figure 5 as a function of pressure for these five variables. The

sensitivity with respect to errors in the bulk modulus

Bo and the lattice constant decreases with pressure,

while that with respect to all other variables in- creases with pressure. At pressures above 65 kbar

the sensitivity increases in the sequence B^^\ T, Bo,

B[}\a.

Of greater practical interest is the actual uncertainty in pressure Ap which occurs for any given

error Ax of the five independent variables. These

errors were also calculated for Csl as a function of

pressure from the values for ABq, Afij," and AB^^

given in table 2 for the purely isothermal quantities,

and by assuming a temperature error of AT= 1 °C.

For the lattice constant an error of Aa/a= 10"^ was

assumed which corresponds to the best accuracy

attainable at present with high-pressure x-ray cameras [45]. The results are plotted in figure 6 which

contains also the total error according to the Gaussian error propagation law.

It is apparent that below 100 kbar the accuracy

limiting factor is the measurement of the lattice

constant, and between 100 and 150 kbar it is the

first pressure derivative of the bulk modulus, and

above 150 kbar it is the second pressure derivative

of the bulk modulus. The contributions from the

errors of all other variables are in the pressure

range up to 500 kbar much smaller than the larger of

the two contributions arising from the lattice constant and the first and second pressure derivatives of

the bulk modulus, and may therefore be ignored for

all practical purposes.

The total accuracy of the ultrasonic equation of

state if it is used in connection with x-ray measure- ments of the lattice constant at high pressure ranges

from 600 bars at 10 kbar (6 percent) over 3 kbar at 100 kbar (3 percent) to 33 kbar at 500 kbar (7 percent). In the pressure range from 25 kbar to 150 kbar

the total error in pressure is always smaller than

3.5 percent. In this pressure range the ultrasonic

equation of state of Csl should therefore be useful

as an auxiliary pressure standard.

A similar error analysis can be carried out for

temperatures above room temperature. Since the

ultrasonic measurements were carried out up to 250 °C one should expect that the accuracy of the

equation of state of Csl proposed here will be approximately the same. The values above 250 °C are calculated by extrapolation of the third- and fourthorder elastic constants and are based on a linear

approximation for their temperature dependence.

Above 250 °C the accuracy will therefore be lower

than at room temperature.

184
Strictly speaking, the accuracy plotted in figure 6

refers to the sample of Csl only, which was used for

the ultrasonic measurements. Since it is known

that the elastic constants and therefore the bulk

modulus depend somewhat on the concentration of

impurities, dislocations, and crystal imperfections, one may have to consider the possibility that an

additional uncertainty is introduced by this effect.

One should expect, however, that this effect is more pronounced for the bulk modulus and its second pressure derivative than for its first pressure

derivative, because the first pressure derivative is given by a ratio of third- and second-order elastic

constants, so that this effect may partly cancel.

Since the first pressure derivative is the accuracy

limiting factor below 150 kbar and since the accuracy of this factor amounted to as much as 1.3

percent for the ultrasonic data used here, one may

expect that this value should not be significantly

increased by the effect of crystal imperfections.

If the accuracy of the first and second pressure

derivatives of the bulk modulus could be increased

by factors of three and four, respectively, so that

the error would amount to about 0.4 and 2.7 percent,

respectively, then the uncertainty in the lattice

constant measurement would be the accuracy limiting factor up to 500 kbar. The total error at 500

kbar would amount to 6.8 kbar (1.4 percent) only.

This increase in accuracy seems quite feasible at present and would require more accurate pressure

measurement in the ultrasonic experiments, and

more reliable measurements of the thermal expansion coefficient, of the specific heat, and of the

temperature dependence of these quantities. It is through the temperature derivatives of the thermal

expansion coefficient and of the specific heat that an additional error is introduced in the conversion

from the ultrasonically measured isothermal

pressure derivative of the adiabatic bulk modulus

to the purely isothermal value. The potential benefit

of this increased accuracy seems to justify the

pursuit of this work as a future task.

5. Conclusions and Summary

A three-parameter equation of state based on

ultrasonically measured values of the bulk modulus

and its first two pressure derivatives has been

proposed for CsCl and CsBr at room temperature,

and for Csl between 0 °C and 600 °C. The phenomenological equation of state of Birch has been

used in the second-order approximation and has

been found to be superior to the equations of

Mumaghan and Keane. The ultrasonic equation of

state does not agree within the carefully established

limits of experimental error with the piezometric

data of Bridgman, but it does agree up to 400 kbar

with data obtained from high-pressure x-ray measurements by Perz-Albuerne and Drickamer.

Even at these high pressures the accuracy of the

Pressure p (kbar)

Figure 6. Total error of pressure of equation of state of Csl

and contributions from individual errors of material parameters (ABo = 0.5 kbar, AB<,'» = 0.08, AB<f' = 0.008 (kbar)-'), of

temperature (AT= 1 °C) and of lattice parameter (Aa/a= 10~^)

at 25 °C in second-order Birch equation.

ultrasonic equation of state is better than that of the

x-ray data. Although at present the total accuracy

of the ultrasonic equation of state is at high pressure

limited by the error of the first and second pressure

derivatives of the bulk modulus, it seems possible

to improve this accuracy and to present an ultrasonic equation of state for cesium halides with high

enough accuracy so that the predominant accuracy

limiting factor will be the error in the x-ray measure- ment of the lattice parameter.

6. Acknowledgments

The authors would like to express their sincere

appreciation to Prof. H. G. Drickamer for sending

a table of the numerical results of his equation of

state data for CsCl which were published in refer- ences [11] and [12] in graphical form. Thanks are

also extended to Mrs. J. Schiff for most of the computer programming required for the numerical

calculations.

Note added in proof:

The ultrasonic data reported in tables 1, 2, and 4 differ slightly

from the numerical values presented at the Symposium in 1968. This change is due to the fact that the earlier data were based on

an approximation used in the processing of the ultrasonic data

which was later found to be unjustified. While the resulting

changes in the room temperature data for the ultrasonic equation

185
of state shown in figures 2 and 3 are insignificant a more pro- nounced change accrued for the ultrasonic equation of state data

at high temperatures, which are reported in table 5 and figure 4. Whereas for the earlier data the isotherms were found to inter- sect at high pressures, the revised data lead to an equation of

state with isotherms which are quite evenly spaced apart, and

which do not exhibit the unusual behavior of negative thermal

expansion at high pressure.

7. References

[1] Anderson, O. L., J. Phys. Chem. Sol. 27, 547 (1966). [2] Murnaghan, F., Proc. Natl. Acad. Sci. 30, 244 (1944). [3] Anderson, O. L., Phys. Earth Planet. Interiors 1, 169 (1968). L4] Keane, A., Austral. J. Phys. 7, 3 (1954). [5] Chang, Z. P., and Barsch, G. R., Phys. Rev. Let. 19, 1381

(1967). 6] Birch, F., Phys. Rev. 71, 809 (1947). 7] Birch, F., J. Geophys. Res. 57, 227 (1952). 8] Knopoff, L., J. Geophys. Res. 68, 2929 (1963). 9] Bridgman, P. W., Proc. Am. Acad. 74, 21 (1940). [10] Bridgman, P. W., Proc. Am. Acad. 76, 1 (1945). [11] Perez-Albuerne, E. A., and Drickamer, H. G., J. Chem.

Phys. 43, 1381 (1965). [12] Drickamer, H. G., Lynch, R. W., Clendenen, R. L., and

Perez-Albuerne, E. A., Sol. State Phys. 19, 135 (1966). 13] Decker, D. L., J. Appl. Phys. 36, 157 (1965). 14] Decker. D. L., J. Appl. Phys. 37, 5012 (1966). 15] Bartels, R. A., and Schuele, D. E., J. Phys. Chem. Sol. 26,

537 (1965). [16] Larson, D. B., Keeler. R. N., Kusubov, A., and Hord, B. L.,

J. Phys. Chem. Sol. 27, 476 (1966). [17] Jamieson, J. C, in Physics of Solids under Pressure, C. T. Tomizuka and R. M. Emrick, eds. (Academic Press, Inc., New York, 1965) pp. 444-455.

[18] Bassett, W. A., Takahashi. T., Mao, H. K., and Weaver,

J. S., J. Appl. Phys. 39, 319-325 (1968). [19] Menary, J. W., Ubbelohde, A. R., Woodward, F. R. S. and I., Proc. Roy. Soc. A208, 158 (1951). [20] Hoodless, I. M., and Morrison, J. A., J. Phys. Chem. 66,

557 (1962). [21] Rao, K. J., Rao, G. V. S., and Rao, C. N. R., Trans. Faraday

Soc. 63, 1013 (1967).

[22] Koliwad, K. M., and Ruoflf, A.L., Bull. Am. Phys. Soc. 10,

1113 (1965). [23] Reddy, P. J., and Ruoff, A.L., in Physics of Solids at High

Pressures, C. T. Tomizuka and R. M. Emrick, eds. (Academic Press, New York, 1965), p. 510. [24] Chang, Z. P., Barsch, G. R., and Miller, D. L., Phys. Stat.

Sol. 23, 577 (1967). [25] Barsch, G. R., and Chang, Z. P., Phys. Stat. Sol. 19, 139

(1967). [26] Overton, W. C, Jr., J. Chem. Phys. 37, 116 (1962). [27] Smart, W. M., Combination of Observations (Cambridge

Univ. Press, Cambridge, 1958).

[28] Slage, O. D., and McKinstry, H. A., J. Appl. Phys. 38,

451 (1967). [29] Standard X-ray Diffraction Powder Patterns, NBS Circular

539, Vols. 1 to 7, National Bureau of Standards, Washington, D.C., 1953-1957.

[30] Johnson, J. W., Agron, P. A., and Bredig, M. A., J. Am.

Chem. Soc. 77, 2734 (1955). [31] Landolt-Bornstein, 6th Ed., Vol. 2, Part 4, Springer-Verlag,

Berlin (1961). [32] Barsch, G. R., and Chang, Z. P., J. Appl. Phys. 39, 3276

(1968). [33] Nranyan, A. A., Sov. Phys.-Sol. State 5, 129; 1361 (1964). [34] Ghate, P. B., Phys. Rev. 139, A1666 (1965).

[35]^ Arenstein, M., Hatcher, R. D., and Neuberger, J., Phys. Rev. 132, 73 (1963). [36] Pavlovski, M. N., Vashchenko, V. Ya., and Simakov, G. V., Sov. Phys.-Sol. State 7,972 (1965). [37] Lowdin, P. O., Adv. Phys. 5, 1 (1956). [38] Friiman, A., and Lowdin, P. O., J. Phys. Chem. Sol. 23, 75

(1962). [39] Tosi, M. P., Sol. State Phys. 16, 1 (1964). [40] Chandra, M., Proc. Phys. Soc. 91, 257 (1967). [41] Mayer, J. E., J. Chem. Phys. 1, 270 (1933). [42] Jeffery, R. N., Barnett, J. D., Vanfleet, H. B., and Hall, H. T., J. Appl. Phys. 37 , 3172 (1966). [43] Rotter, C. A., and Smith, C. S., J. Phys. Chem. Sol. 27,267

(1966). [45] McWhan, D. B., Accuracy of High-Pressure X-ray Diffrac- tion Measurements, 1964 Symp. on High-Pressure Technology, ed. E. C. Lloyd and A. H. Giardini, pubL by ASME

(1965). [46] Rose, M. F., Phys. Stat. Sol. 21, 235 (1967). [47] Yu, J. S., Appl. Sci. Res. 19, 220 (1968).

DISCUSSION

D. J. Pastine {U.S. Naval Ordnance Laboratory,

White Oak, Maryland): While it is apparently

true that the Murnaghan and Birch equations-ofstate may converge rapidly, this depends upon

the second and higher derivatives of bulk modulus

with respect to pressure. However, the values of the

second derivatives are not well known, and the

values of the higher order derivatives are not

known at all. The second derivative may also be

obtained from shock data, and this might be advantageous in cases where the values are difficult to obtain otherwise.

In obtaining your results from ultrasonic data, are you comparing isotherms with isotherms?

AUTHORS' CLOSURE

The reason we chose to work with the cesium

halides was because these substances are very

compressible, and the plot of the elastic constants versus pressure shows sufficient curvature to permit

the second derivatives to be readily determined at pressures below 10 kilobars. It would be more

difficult to pick up the second derivative ultrasonically below 10 kilobars with equal accuracy for less

compressible materials. But if we have a pressure

scale based on cesium halides, then we could use an iterative procedure based on ultrasonic measure- ments at higher pressures, say up to 50 kilobars,

and use the second pressure derivatives obtained

to calculate another equation-of-state for less

compressible materials up to one or two megabars.

AU we have shown here concerning convergence

186
is that the deviation of the second order Birch

equation from an exact lattice theoretical equation

based on a model potential, which includes first and second nearest neighbors, is small in the

pressure range of interest. We have not included

thermal motion, which was justified in our case because the pressure dependence arises mainly

from the cohesive contribution, and because the

thermal equation-of-state is based on so many other

assumptions.

On the matter of reducing our ultrasonic data,

I should have made the point that all these data

were first converted to isothermal conditions. For

the first pressure derivatives, for example, we must

know the pressure derivative of the Griineisen constant, which we have calculated from thermal data

by means of thermodynamic identities.

187

Calculation of the P-V Relation for Sodium Chloride up4:o 300 Kilobars at 25°C

J. Scott Weaver, Taro Takahashi, and William A. Bassett

Department of Geological Sciences and Space Science Center, University ofRochester, Rochester, N.Y. 14627

The P-V relation for the Bl phase of NaCl was calculated up to 300 kilobars at 25 °C by means of the Hildebrand and Mie-Griiniesen equations. The effect of (1) the uncertainty in the parameters used, (2) the functional form for the temperature-dependent part of internal energy, (3) the anharmonic contribution, and (4) the second nearest neighbor repulsion on the calculated P-V relation is discussed. It was

found that the calculated P-V relation for NaCl is sensitive to the value chosen for Bto, and that the

effect of (2), (3), and (4) on the calculated P-V relation is small. The precision for the present calculation has been estimated to be ±2.5 percent. The calculated P-V relation is in agreement within experimental uncertainties with the fixed points on the absolute pressure scale, with that determined by

Bridgman up to 100 kilobars, and that obtained by Fritz from shock wave data up to 260 kilobars. Simultaneous determinations of the effect of pressure on the volumes of NaCl and MgO were made by

x-ray diffraction on an intimate mixture of these two materials. It was found that the pressure values

calculated from the volume of NaCl using the calculated P-V relation are in agreement with those

calculated from the volume of MgO using the P-V relation calculated from the values of Bto and Bto with the Murnaghan equation. This agreement verifies not only the calculated P-V relation for NaCl,

but also establishes the validity of the internal standard method for pressure determination.

1. List of Notation

The subscript zero denoted quantities evaluated

at P = 0, V= Vo. For example: Vo; 8"^; etc. The other notations used and not defined in the

text are: A^{d In yid In V)t

Bs = —{dPld In V)s the adiabatic bulk modulus

B's^{dBsldP)T

B'^^{dB'sldP)T

Bt = - {dPId In V)t

B[={dBtldP)T

B';={dB'JdP)T

B* = -V{dP*ldV)T

Cr = dipole-dipole van der Waals coefficient.

Cd= specific heat at constant volume.

Dr— dipole-quadrupole van der Waals coefficient.

e= electron charge or Naperian base.

E*{V, 7") = temperature dependent part of the in- ternal energy.

P= pressure.

P*^- {dE*ldV)T+T{dPldT)v

PM^idre^Irt

Ps^Crlrl

Ps = iDrlrh'

Pr = brol^p

r= nearest neighbor distance

S= entropy

T= temperature (K)

U{V, T) — internal energy

V= molar volume

x=VIVo

a = {d In VldT)p isobaric thermal expansion coPaper presented at the Symposium on Accurate Characterization

of the High-Pressure Environment, held at the National Bureau

of Standards, Gaithersburg, Md., October 14-18, 1968.

efficient

Qir = Madelung constant

y=TaBtVIC„ thermodynamic Griineisen parameter

ds^-{aBs)-'{dBsldT)p

8t = -iocB,)-'{dBtldT)p

e= (d In aid In V)t

6= Debye characteristic temperature

p = interaction length in the repulsion energy

(f) = lattice energy

2. Introduction

In the internal standard method of pressure determination in high pressure x-ray diffraction studies,

the pressure is inferred from the molar volume of a standard substance mixed w^ith the material being

studied. Hence the pressure-volume (P — V) relation of the standard must be accurately known over

the pressure range of interest. However, due to the

lack of direct and accurate means of determining the

P—V relation at pressures above 60 kbar, desirable

experimental data are not available in the pressure

range. One experimental approach to the determination of the P — V relation is the study of the

propagation of a shock wave through a material.

Such studies yield values for both pressures and

volume along a non-adiabatic, non-isothermal path

known as the Hugoniot from which the required isothermal P — V relation may be calculated. In the

conversion of the Hugoniot to an isotherm, the re- sults depend critically on the assumptions made

about the form of the equation of state. Another approach involves calculation of the P — V relation

from a suitable approximation to the equation of

state whose parameters are fixed on the basis of the

properties of the material determined at low pressures. In such an approach, however, large errors

in a calculated P-V relation can result from small

189
errors in the values of the various low-pressure

parameters or from inadequacy in the form of the

equation of state used. Hence, neither of these approaches is free from objection.

This paper presents a discussion of the P — V

relation of the Bl phase of NaCl for pressures up to 300 kbar at 25 °C. The approach taken here is an

attempt to calculate the P — V relation from lowpressure properties, and then to compare the re- sulting f— V relation with the results of shock-wave

propagation studies. An additional test of validity of

the calculated P — V relation and the internal standard method is obtained from simultaneous determinations of the molar volumes of NaCl and MgO.

The results of a study of the sensitivity of the calculated P — V relation for NaCl to the form of the equation and to the value of the low-pressure properties

for NaCl are used to estimate the precision of the

P — V relation.

NaCl offers a number of advantages as a substance for use as an internal standard. Properties

contributing to its usefulness include:

(1) The small bulk modulus of NaCl (237.4 kbar

at 25 °C) yields good sensitivity since a given change

in pressure results in a relatively large change in

volume.

(2) The isometric structure of NaCl yields an

x-ray diffraction pattern containing relatively few

lines, thus minimizing interference with the sample

pattern. In addition, each NaCl reflection yields a value for the molar volume.

(3) The low yielding strength of NaCl aids in

producing a more hydrostatic pressure environment

than other sohd media having higher yielding

strength. In addition, NaCl has a low absorption coefficient for a wide range of the electromagnetic

radiations including x-rays, visible, and infrared.

(4) NaCl has been the subject of a great number

of theoretical and experimental studies, so that most

of the properties necessary for calculation of the

P — V relation are available from the literature.

3. Previous Studies

Figure 1 shows the results of calculations of the

P-V relation for NaCl at 25 °C byj)ecker (1965,

1966), who used the vibrational Mie-Griineisen equation, and by Perez-Albuerne and Drickamer (1965),

who used the thermal Hildebrand equation. These

are compared with the shock Hugoniot data obtained by Christian (1957) and with five points along

a 25 °C isotherm calculated by Fritz (1967) from

shock wave data. The discrepancy between the calculated P — V relations and the experimental data

appears to be due to the values of Bto chosen by

Decker and by Perez-Albuerne and Drickamer.

A family of isothermal P — F relations may be obtained from the expansion of Bt as a Tayler series

in P:

PRESSURE (kbar)

Figure 1. Comparison of the P-V relations for NaCl calculated

by Perez-Albuerne and Drickamer (1965) using the thermal

Hildebrand equation and by Decker {1966) using the vibra- tional Mie-Gruneisen equation, with the Hugoniot data ob- tained by Christian (1957) and with five points along a 25 °C

isotherm calculated from Hugoniot data by Fritz (1967).

The last of these has been superseded by more recent work (Fritz et al., this volume) and may be disregarded.

Bt^- (dPId In F)r-fio+5o^-P/2 B'^P^+

(1)

Neglecting P'^ and higher terms, eq (1) leads to the

Murnaghan equation (Anderson, 1965; Murnaghan,

1944):

P=iBtolBQ [(F/Fo)-«'o-l].

(2)

If the P'^ term is retained, eq (1) leads to an equation

of the form:

P={2Bt,jBt',) [(r/ro)-''%'-l] [1 + g

-d-g) (r/ro)-''«'o]-i (3)

where q= [\ -2BhBtll {Bt'^YY^ Rose (1967) has

derived a similar equation.

Another family oi P—V relations may be obtained from finite strain theory (Birch, 1938, 1952),

by noting that the free energy is a function of strain,

e, where 2e = y= [{VlV^y^l^ -\]:

P= (3/2)fiioy(l + y)5/2(l-^y+Tjy2+ )

(4)

where

f = - (3/4)(fi?;-4) and rj= (3/8) [fitofifp

+ {BtlY-^Btl + U?,|9^.

190
The values of Bto and Bt^ for NaCl used to evaluate these relations are given in table 2. The value of

Bt'o for NaCl is not known at present. However, the

values for Bso, Bs'q, and Bs'q for CsCl, CsBr, and Csl

determined by Chang and Barsch (1968) yield a relation, BsoBso~ — Bso for these three compounds.

Hence the effect of Bt'^ was investigated by calculating pressure values for the two cases: (1) fifj,' = 0

and (2) Btl = -Bt'JBh. The calculated P-V relations for NaCl are shown in figure 2. It is seen that

none of these P—V relations agrees with the shock

compression results of Fritz (1967). In particular,

pressure calculated from Mumaghan equation is far

greater than Fritz's values. This suggests that

Btl is a negative number and that the effect of

Btl can not be neglected for VIVq ^ 0.8. The Rose

equation (1967) appears to be in better agreement

with the shock data for Bt'^— — BtQlBto. However,

use of this equation with Bt'^ < 0 leads to a physically

unreasonable condition at high pressures. For ex- ample, '\i BIq — — Bt'JBto, Bt increases to a maximum

at P — Bto, and thereafter, it decreases and becomes

negative for P > fifo [1 + (1 + 2/fifo)'/2]. A negative

value of Bt violates the First Law of Thermodynamics, while a negative value of Bt' seems unlikely for a single-component system in the absence

of any phase transformation.

Four P — V relations calculated with the Birch

equation are shown in figure 2. The curve Birch 1 was calculated assuming that ^ = 17 = 0. The effect

of the ^ term is shown hy Birch 2, which was calculated with rj = 0 and ^ = -1.013 (or fi?o = 5.35).

The effect of rj and the value of BtQ is shown by

Birch 3a in which 17 = 2.65 and 5fo = 0, and by

Birch 3a in which t} = 0.64 and Btl^ — Bt'JBtn.

From figure 2 it can be seen that both the ^ and 17 terms exercise a significant effect on the calculated

pressure. Thus, eqs (2), (3), and (4) failed to describe

adequately the P—V relation of NaCl up to 300

kbar from the parameters determined at low pressures. On the other hand, with the refined experiPRESSURE (kbar)

Figure 2. Various P-V relations for NaCl calculated by means

of the Murnaghan, Rose, and Birch equations.

The values for the parameters used are: B(o = 237.4 kbar, BJ^ = 5.35, and B"^ = 0.00 or —0.0225 kbar"'. The equations shown are; Murnaghan^ eq (2); Rose^ eq (3); Birch 1, eq (4) with f=T) = 0; Birch 2, eq (4) with ^=-1.013, 7) = 0; Birch 3a, eq (4) with f = -1.013, i) = 2.648 (B';„=0); Birch 3b, eq (4) with ^ = -1.013, t) = 0.642 (B" = ~0.0225 kbar"'). Five points along a 25 isotherm calculated by Fritz (1967) from Hugoniot data are shown for comparison. These points have been superseded

by more recent work (Fritz et al., this volume) and may be disregarded. mental values for Bto. the Hildebrand and MieGruneisen equations appear to be capable of

accounting the compression of NaCl in the pressure

range of present interest.

4. Calculation of the P—V Relation

The internal energy of a solid may be written in

the form

U{V, T)^4>{V)^E*{V, T).

In the vibrational formulation of the equation of state.

Table 1. Definition of the models of solid used for the calculation of the P — V relation for NaCl.

E* is the energy due to the lattice vibrations; P* the pressure resulting from the lattice vibrations; and B* the contribution of the lattice vibrations to bulk modulus.

Model E* p* B* Remarks

I constant 0 0 Thermal formulation at 0° K.

II (9/4)R6l (9/4)7«6l/F (l + y-A}P* Vibrational formulation at 0° K.

III 6RTDe(eiT) + (m)Re yE*IV -(A-l)- (yE*IVy- (yVV) (TCv-E*) Vibrational Mie-Griineisen IV 6RTDe(eiT) yE*IV -(yE*IV)-{yVV){TCv-E*) Thermal Mie-Griineisen

V ^**^ ra„Bf„(y/y„)(F„/F) -(A-l)P* Vibrational Hildebrand

VI ^**^ ToaBto 0 Thermal Hildebrand

reiT De(dlT)=3(Tiey

j

[u^l {e" - l)]du Debye function. 6»=Debye temperature.

*For these two cases, E* should be a function of temperature alone. However, it is not necessary to know the functional form of E *, since it is not used in the calculation.

191
(f> is the potential energy due to the static lattice while

E* is the energy due to the lattice vibrations in- cluding the zero point vibrations. On the other hand,

in the thermal formulation, the energy due to the zero point vibrations is included in (/>, and hence

E* is taken to be the thermal energy. A relation for

the pressure is found by combination of the expressions for the internal energy with the thermodynamic

identity:

P^-{dUldV)T+TidPldT)v,

and then

P=-d4>ldV+P*,

where P* is the pressure resulting from the lattice

vibrations. For NaCl, P* is less than 10 kbar so that

the lattice energy (/> predominates at high pressure.

The isothermal bulk modulus then assumes the

form:

B, = V{d^(j)ldV^)+B*,

where fi* is a contribution of the lattice vibrations

to the bulk modulus.

Since the elastic constants of NaCl determined by

Slagle and McKinstry (1967) closely obey the Cauchy

relation (C12 — C44), a central-force model for </> is reasonable. For an ionic solid, the principal terms

in the attractive part of (f) result from the Coulomb

and van der Waals (dipole-dipole and dipole-quadrupole) interactions. Since a unique form for the repulsive part of (/) is not available, the Bom-Mayer

exponential form between the nearest neighbors

containing two adjustable parameters was adopted.

Thus, the adopted form for 4> in terms of the nearest

neighbor distance, r, becomes:

0 = - are^r - Crlr^ - Dr/r^ + be-'lp

and hence

P= - PmX-^I^ - Pex-^ - Psx-^'l^

+ PrX-^I^ exp (- + p*

and

Bt = - W)Pmx-'I^-3P6X-^- (11/3)P8a:-ii/3

+ {ll3)PRX-^IH2+roxlp) exp {-rox'l^lp)+B *

x=VIVo,

where a new set of parameters, Pm, Pq, Ps, Pr, with

the dimensions of pressure have been introduced for

convenience. The quantities characterizing the

repulsion {b and p; or Pr and ro/p) can be determined

by two boundary conditions: that the expressions for

P and Bt yield the correct values of 0 and Bto respectively when F= Fo (or% = I)

.

A number of approximate forms for the temperature dependent part of the internal energy E* are

possible. The P — V relation was calculated for six

different choices for the form of £'* in order to test the sensitivity of the final results to the choice of a model for the vibrational energy. The models se- lected are shown in table 1. Models I and II are ex- treme cases in which the thermal contribution to E*

is ignored. Hence these models are appropriate for

calculation of the 0 K isotherm only.

Models III and IV are based on the Mie-Griineisen

approximation to the vibrational and thermal formulations of the equation of state. The Mie-Griineisen

approximation may be derived if £'* has the functional form T • Q{djT), where 0 is a purely volumedependent characteristic temperature, and therefore, — d\n Old In V is equal to the thermodynamic

Griineisen parameter, y. In the present study, the

Debye functional form was adopted for £*, since the

specific heat values calculated for NaCl using this

form agree closely with experimental values. The

Griineisen parameter y was assumed to obey a power law in the volume of the form:

y=yoiVIVo)^

where the value of ^4 was obtained from the thermodynamicaUy exact relation derived by Bassett et al. (1967):

A={dln yid In V)t = 1+ {1 + y a - T)8s-Bs' + y+TidyldT)v.

The term T{dyldT)v is zero from the assumptions

inherent the Mie-Griineisen approximation. The

power law form for y leads to a relation for the

Debye temperature as a function of compression:

d=do-exp {yoA-'[l-{VIVo)^]}

where the subscript "0" denotes the values evaluated at zero pressure. Fumi and Tosi (1962) have

shown that the assumptions leading to the vibrational formulation of the Gruneisen approximation

are valid for temperatures somewhat less than the

Debye temperature or greater, and that the thermal

formulation is not appropriate in a temperature

region near the Debye temperature. Therefore,

Model IV is less satisfactory than Model III. The Hildebrand approximation, on which Models

V and VI are based, results from the assumption

that £* is a function of the temperature only, so that

P* = TidPldT)y = T a -Bt. This implies that Cv is a function of temperature only, and therefore Cv is constant in a isothermal condition, and the relationship below holds:

P* = P*{ylyo)l{VIVo),

192
Table 2. Physical parameters for NaCl used for the calculation of the P — V relationship at 25 °C

Parameters Values References

(A) Experimentally Determined Parameters

oo 5.6403 A

Bso 250.0 ± 1.8 kbar

- {dBsldT)p 0.109±0.007 kbar "C-'

5.27 ±0.25

(11.85±0.20)10-5°C-'

10. ±1

290 ±12 K

1.74756

(180 ±36)10-'"' erg cm"

(180 ±90) 10-'" ergcms

00

a-T

Cr

Dr

(B) Parameters Derived From the Values Listed Above

Vo 27.012 cm3 mole"'

Pm 106.214 kbar

Ps 15.95 ±3.19 kbar

Ps 2.64± 1.32 kbar

Cvo 0.4760 ±0.0018 kbar cm^ deg-' mole-'

To 1.597 ±0.030

Bto 237.4 ±1.7 kbar

5.35 ±0.30

3.67 ±0.25

1.20 ±0.36

(C) Equations Used for Deriving the Values in (B)

Sso

A

Cvo = 6R-3(Tieo)^

J [u''e"/(e"-l)2]dM

yo(l + yoOtoT) = aoBsoVolCvo

Sso = — (dBsldT)plaoBso

-4 = 1+ ( 1 + yoao?-) 8i„ - Bs^ + y„

Bt; = ( 1 + -yoaoT-) -' [Bs^, + yoaoT(Sto + A)]

Sto = Sso -I- ( 1 -I- yottoT') [yo + yooioT(eo + A)]

Clark (1967).

Slagleand McKinstry (1967), 250.8 ± 1.8 kbar; Haussuhl (1%0), 250.0

±1.3 kbar.

Slagle and McKinstry (1967), 0.103; Bartels and Schuele (1965), 0.108; Haussuhl (1960), 0.117 ± 007 kbar °C-'.

Bartels and Schuele (1965).

Slagle and McKinstry (1967), 11.48; Collins and White (1964), 11.85; Enck

cited in Decker (1965), 12.00 X 10-^ °C->.

Slagle and McKinstry (1967), 9.03; Collins and White (1964), 11.07.

Yates and Panter (1962).

Tosi (1964).

Mayer (1933).

Mayer (1933).

Anderson (1967).

Bassett et al. (1967).

Bassett et al. (1967).

Bassett et al. (1967).

193
where the subscript "0" indicates the quantities

evaluated at zero pressure. According to Fumi and

Tosi (1962), the vibrational Hildebrand approximation is appropriate for temperatures exceeding a temperature somewhat greater than the Debye temperature, whereas the thermal formulation is less

satisfactory at temperatures near the Debye temperature. Model V is the vibrational formulation of

the Hildebrand approximation, and Model VI is based on a simple case of the thermal Hildebrand

approximation, in which F* is assumed to be constant. In this latter case, the Gruneisen parameter

is proportional to volume, or equivalently, the bulk

modulus is a function of the volume only. This is the

model used by Perez-Albuerne and Drickamer

(1965), and should be inferior to Model V.

The values adopted for the various parameters of

NaCl are listed in table 2. The values shown in

table 2a were adopted from the literature, while the

values in table 2b were calculated from the values in

table 2a using the relations indicated. Since the isothermal bulk modulus at zero pressure, Bto, is used

as a boundary condition, the calculated P—F relation is quite sensitive to the values chosen. In the

present study, the value of Bto was calculated from

the value of adiabatic bulk modulus, Bso, determined

by Slagle and McKinstry (1967) by means of the

pulse superposition technique. Their value was

selected rather than a smaller value found by Bartels

and Schuele (1965) in view of the high precision ex- pected from the pulse superposition technique and

the agreement with the value obtained by Haussuhl

(1960). Since there was no compelling reason to select one value of — {dBsjdT) p over another, the

arithmetic mean of these values listed was adopted

with an uncertainty calculated by assuming that the

values are independent measurements of the same

quality. Similarly, an intermediate value was chosen

for cto. The P—V relations calculated using the six

models are compared in figure 3, which also shows

the results of Decker's revised calculation (1968) of

the Mie-Gruneisen equation and five points along a 25 °C isotherm calculated from shock data by Fritz

(1967). Models I and II are seen to be inconsistent

with the results of experiments and with other

models. Such a discrepancy is expected due to the

lack of a temperature-dependent term in the internal

energy. The other four models yield a range of pres- sures of only 4 kbar at 280 kbar, showing that the

calculated pressures are not sensitive to the exact

form chosen for the temperature-dependent part of

the internal energy. However, Models IV and VI,

which are the thermal formulations of the MieGriineisen and Hildebrand approximations respectively, are probably inferior to Models III and V,

the vibrational formulations, as already discussed in

the previous page. The close correspondence between pressures calculated on the basis of either of

Models III or V and Decker's 1968 recalculation

P-P-x (kbar)

" Fritz (1967)

o

IT ^ Decker

(1968)

' 1 -z

0 100 200 300

Pvtkbar)

Figure 3. Effect of the form of the temperature dependent part

of the internal energy on the P-V relation for NaCl.

Curves I—VI correspond to different choices for E* (see table 1). Decider's (1968) revised calculations are included to illustrate the effect of the second nearest neighbor

repulsion. The calculated P-V relations are compared with five points along a 25 °C

isotherm calculated by Fritz (1967) from Hugoniot data. These points have been super- seded by more recent work (Fritz et al., this volume) and may be disregarded. The

uncertainty in the pressure values for Model V (P\ ) resulting from the uncertainties

in the parameters used in the calculation is shown by curves + 6 and — S.

Table 3. The sensitivity of the pressure values

calculated with the Mie-Gruneisen equation

{Model VI) to the uncertainties in the parameters

at a compression of V/Vo = 0.650 (~ 300 kbar).

In order to show the direction of the change in the pressure

values, only the positive values for the designated uncertainties were considered. The sensitivity for the Hildebrand equation

(Model V) is similar to that for the Mie-Gruneisen equation

shown here.

Parameter

U)

Uncertainty

(8xlx)

Effect on P

(8F/P)

Sensitivity

(bPIP)l(bxlx)

(Percent) (Percent) Bto 0.6 + 1.1 + 1.9 y 1.9 -0.2 -0.1

A 34 + 1.8 + 0.05 e 4.1 -0.01 -0.003

Cr 20 -0.5 -0.03

Dr 50 -0.2 -0.004

indicates that the effect of the second-nearest neighbor interaction which was included in his calculation

is negligibly smaU.

The total uncertainty in the calculated P was estimated to be ±2.5 percent on the basis that the uncertainties in the individual parameters are independent and Gaussian. If the change in P re- sulting from the uncertainty in the ith parameter is bPi, then the total uncertainty in P can be calculated by:

[{bPiYyi\

194
As can be seen from table 3, the main sources of the

uncertainty in P are the uncertainties on Bto and

A. The total uncertainty thus calculated is shown

with two curves labelled + 8 and — 8 in figure 2, and

is sufficiently large, so that the two models are in- distinguishable. Hence it does not appear reasonable to use the shock experiment data for selecting one of these models over another.

The sensitivity of Models III and V to the uncertainties in the parameters used was examined by

recalculating the repulsion parameters and the

P — V relation with each parameter individually

veried to the uncertainty limit given in table 2. The change in pressure divided by the uncertainty

in the parameter provides a measure of the sensitivity of the P — V relation to the value adopted for

the parameter. The results are presented in table 3

for F/Fo = 0.650. The calculated pressure is in- sensitive to the values of all the parameters except

BtQ. The sensitivity of P to the BIq value is due to the use of this parameter as a boundary condition,

and hence, a change in this value is not compensated

by a change in the repulsion parameters.

5. Internal Consistency of Calculation

Thomsen and Anderson (1968) have pointed out

that once a functional form for the lattice energy (/> has been chosen, then the vibrational spectrum and

hence y are determined. Therefore, in principle,

4> and y cannot be chosen independently as was done

in the present study. However, in practice, this in- consistency in </) and y probably does not affect the

results of calculations significantly. Using the same

values for the parameters for NaCl as used in this

work, they have calculated yo= 1.526 and ^ = 0.40

by means of a relationship derived by them on the

basis of the lattice dynamic treatment given by

Leibfried and Ludwig (1961). Their value for 70 is consistent with the value of 1.597 ±0.030 obtained on the basis of the exact thermodynamic relation.

On the other hand, their value for^ differs from the

value of 1.20 ±0.36 which was obtained in the present work on the basis of the thermodynamically

exact treatment of the experimental data. The discrepancy between these two values for A may be

attributed to (1) failure of the fourth-order anharmonic theory used by Thomsen and Anderson (1968), or (2) failure of the lattice energy form adopted in the

present work. It is also possible that this discrepancy may disappear when an allowance is made for

the uncertainties in the values of the parameters

used, since the discrepancy corresponds to only

twice the uncertainty estimated iox A in the present

work. If the value for A obtained by Thomsen and

Anderson were adopted, it would yield calculated

pressure values about 3.5 percent smaller than the

results presented here.

The internal consistency of the present calculation can also be examined by a comparison of the

calculated value for Bt^ and the experimental

value. This test is valid, since the experimental

value for Bs'q was only used for calculating the value

for A and was not used as a boundary condition.

The vibrational formulation of the Hildebrand

approximation (Model V) yields a value of 4.90

for , whereas the experimental value obtained

Bartels and Schuele (1965) is 5.35 ±0.30, and that

by Chang (1965) 5. 18 ±0.09. Since the uncertainty

in the volume determination by the x-ray diffraction

technique is 0.3 percent, the error in the P-V

relation for NaCl resulting from the discrepancy in

the calculated and experimental Bt^ is not significant in comparison with the uncertainty in the

volume determination. However, more work is required before this discrepancy is resolved.

6. Comparison Between the P-V

Relations for NaCl and MgO

The calculated P-V relation for NaCl may be

verified by a comparison of the effect of pressure on the volume of NaCl with that of a material of

a known P-V relation. For this purpose, the effect

of pressure on the molar volumes of NaCl and MgO

was obtained by simultaneous determinations of

their volumes by means of the x-ray diffraction

method. An agreement in the pressures calculated

from the volumes of NaCl and MgO would not

only verify the calculated P—V relationship for

NaCl, but would also indicate the validity of the

internal standard method for the determination of

pressure by means of NaCl and the diamondanvil type x-ray camera.

6.1. Techniques

The measurements reported in this paper were

made by means of the diamond-anvil x-ray diffraction camera described by Bassett et al (1967). In

this device, an intimate mixture of powders of MgO

(ground synthetic single crystal prepared by J. Swica, Ritter-Pfalder Co.) and NaCl (Baker analytical reagent) having particle sizes less than 1 fxm

was placed between two flat, parallel faces of

diamond anvils. One anvil face is approximately

0.3 mm in diameter, and the other is 1.8 mm in

diameter. The diamond anvils mounted on the ends

of pistons are driven together by a simple screw

device. An x-ray beam of filtered molybdenum

radiation, 0.07 mm in diameter, is directed through one of the diamonds along the axis of compression

to impinge upon the portion of the sample at the

center of the area while the sample is under pressure. The x-rays diffracted by the sample emerge

through the other diamond and are recorded on a

cylindrical film, of which the center of curvature

coincides exactly with the sample as in the case of

Debye-Scherrer geometry. The x-ray diffraction

195
Table 4. The results of simultaneous determinations of the effect of pressure on the molar volume

o/MgO and NaCl at 25 °C

Pueo indicates the pressure values obtained on the basis of the P-V relation for MgO calculated from the Mumaghan equation using Bto=1599 kbar and Bt'„=4.52 (Anderson and

Andreatch, 1966). PNaci indicates the pressure values obtained on the basis of the P-V relation for NaCl according to Model V. The precision for the determination of V/Vo is ±0.3 percent.

Uncertainty in pressure represents that resulting from the uncertainty in VIVo.

Run No. MgO NaClPMgO ~ '^NaCI VIVc ^MgO VIVo

kbar kbar

9 UP 23 0.983 28 ± 5 0.911 28±1 0±5

TUP 28 .940 115± 6 .764 122 ±2 -7±6

9 UP 20 .928 142 ± 7 .735 154 ±3 -12±8

9 UP 22 .907 196 ± 7 .700 199 ±3 -3±8

9 UP 18 .902 211± 8 .692 211±3 0±9

12 UP 3 .895 232 ± 8 .679 232 ±3 0±9

9 UP 19 .888 251 ± 8 .671 246 ±4 5±9

9 UP 21 .887 254 ± 8 .676 238 ±4 16±9

12 UPS .884 265 ± 9 .662 262 ±4 3±10

12 UP 4 .883 266 ± 9 .664 259 ±4 7±10

Root Mean Square ±7.4

In the present investigation, the part of the diffraction pattern produced by the portion of the sample

at maximum pressure was used. In practice, the

outermost part of each broadened line, i.e., the part

of the line with the highest 26 hence the smallest

c?-value, was measured and then corrected for the

line breadth due to causes other than pressure

gradient. This correction consists of subtracting the

1-bar line breadth from the outer-edge to outer-edge

distance. Lattice parameters calculated from meas- urements made in this way show less scatter than

those based on measurements made from center

to center distances of the broadened lines. It is believed that shearing of the sample within a high

pressure gradient environment produces preferred

orientation within the sample leading to a scatter

of data.

A standard Norelco Comparator was used to measure the distance between the diffraction lines.

An analysis of the measurements shows that the

comparator yields results with a precision of 25 /Ltm. Therefore, a precision of ±0.1 percent has been

assigned to the lattice parameter determinations

based on the precision of the comparator technique

and on the agreement between the values derived

from the (200) and (220) diffraction lines. The re- sulting precision in V/Vo is thus ±0.3 percent. The

results of the present determinations are tabulated

in table 4.

6.2. Results of the Simultaneous Measurements of the Volumes of NaCl and MgO

patterns consist of the lines produced by NaCl and

MgO both compressed together in the diamondanvil cell. In addition, the film is marked by the

diffraction pattern of a piece of polycrystaUine NaCl

mounted on the outside surface for the diamond

facing the film. This provides a means of determining sample-to-film distance and of correcting

for changes in film and instrument dimensions.

These corrections are small compared with the

scatter of data points.

The diamond-anvil press operates on the Bridgman anvil principle. As a result, pressure in the

sample ranges from 1 bar at the edge to a maximum

near the center of the anvil faces. The material at the center of the anvU area constitutes the most

desirable sample for lattice parameter determinations by x-ray diffraction for three reasons. First,

the pressure is highest there. Second, the pressure

gradient is smallest, allowing a minimum of pres- sure range in the x-ray beam. Third, the shearing

is least where the pressure gradient is smallest. One

way to take advantage of these features of the

Bridgman anvil geometry is to direct a narrow x-ray

beam through the sample center normal to the

anvil faces. In practice, some degree of hne broadening due to pressure gradient within the x-ray beam

is inevitable.

A direct comparison between the effect of pres- sure on the volume of NaCl and of MgO by means

of the x-ray diffraction method using an intimate

mixture of the powdered sample of NaCl and MgO

should provide a test for (1) the reliability of the

calculated P—V relation for NaCl, and (2) the

adequacy of the NaCl internal standard method

for pressure determinations.

The P—V relation for MgO calculated from the

Murnaghan equation, eq (2), using the two parameters Bto = 1599 kbar and Bt^ = 4.52 determined by

Anderson and Andreatch (1966) for single-crystal

MgO at 23 °C was adopted for the present study.

The single-crystal values were chosen over the

values for polycrystaUine samples in view of the

fact that the elastic properties of polycrystaUine

MgO may be sensitive to the chemical additives

used for sample preparation as pointed out by

Schreiber and Anderson (1968). in comparison with

the P-V relation calculated with the Murnaghan

equation, the Rose equation, eq (3), with

BtoBtQ = — Bt'o

yields pressure values about 2 percent smaUer, and

the Birch equation, eq (4), yields pressure values

between the Murnaghan and Rose equations. Since

196

i

i
P-Pj(kbar)

» This work

• McWhan(867)

« Fritz (1967) 0

i^-^" ' • i^rr '—'—'—' r * 1 ^ «—H

1 ; X

•

e

0 o

e I.I... 1 0 100 200 300

Pi(kbor)

Figure 4. Comparison of pressures calculated from Model V

for NaCl with those from the Murnaghan equation for MgO

using the values ofVIWo obtained by simultaneous determinations of the volumes of Nad and MgO.

Open circles are experimental points obtained in this study, and the solid circles are the data obtained by McWhan (1966). Uncertainty in Model V is shown by + 8 and —S. The 22 °C isotherm calculated for NaCl from shock-wave data by Fritz et al. (this volume)— curves "Q" and "i" — has been added for completeness but is not discussed

in the text.

the uncertainty in the determination of V/Vo leads

to uncertainties ranging from ±5 kbar at V/Vo — l to ±9 kbar at VIVo = 0.87 (-310 kbar), there is no

compelling reason to choose one of these equations

over another.

The values for Pmbo Usted in table 4 were obtained from the experimental values for V/Vo using

the P—V relation for MgO. Thus, Pmso is the pres- sure to which the MgO grains in the NaCl-MgO

mixture are subjected. Similarly, the values for

^Naci were obtained from the V/Vq values using the

P-V relation for NaCl according to Model V.

Figure 4 compares the values for these two pressures. The results obtained by McWhan (1967) are

also shown in figure 4. The mean difference between /*Naci and Pivigo has been calculated to be

—0.3 kbar from the data of McWhan and the present

study, and the root mean square difference is ±6.4

kbar. In view of the uncertainty in the pressure

values resulting from that in the V/Vo measurements,

it is concluded that /^Naci is consistent with PmsoReliability of the internal standard method for

the determination of pressure in a high-pressure

x-ray diffraction device has been questioned, since

heterogeneous pressure distribution must exist in a mechanical mixture of two or more materials

having different bulk moduli and other properties.

If such a heterogeneous pressure distribution exists

in the sample, the pressure indicated by the NaCl

volume should not be the same' as the pressure ex- erted on a material mixed with NaCl. Since MgO has

about seven times as large zero pressure bulk modulus as NaCl and has a far greater shear strength

than NaCl, a comparison of the pressure values

obtained from the NaCl volume (PNaci) with those

P-Pi(kbar)

Pi(kbar)

Figure 5. Comparison of pressures calculated for Model V with

those from the volumetric determinations by Bridgman (1945),

and with those for three fixed points on the absolute pressure

scale.

The pressures corresponding to the phase transformations in Bi and Ba are Usted

in table 5. The values of Pi (pressure calculated from VIVo on the basis of Model V for NaCl) were calculated from the NaCl volumes at the pressure of each phase trans- formation measured by Jeffery et al. (1967). The cross-hatched portion of the error

flags indicates the uncertainty in the determination of the fixed points, while the clear portion indicates the uncertainty due to the uncertainty in the measurement of VlVo

for NaCl. Curve "Q" has been added to show the 22 °C isotherm for NaCl calculated from Hugoniot data (Fritz et al., this volume) which has become available since the

writing of the text.

from the MgO volume {Pu&o) should provide a test for the internal standard method for pressure determination. As shown in table 4, the root mean square

value for (PMgo~^Naci) is ±7.4 kbar, and is well

within the experimental uncertainty in the pressure

values obtained from the MgO volume. Therefore,

the internal standard method appears to be valid.

7. Comparison Between the NaCl

Pressure Scale and the Fixed Points

Jeffery et al. (1966) have determined the molar

volume of NaCl at pressures corresponding to several of the phase transformations which have

been used as fixed points in the 0 to 100 kbar pres- sure range. These data permit a direct comparison

between Model V NaCl scale and the fixed points.

Table 5 shows the mean value of VjVo obtained from

the experiments with increasing and decreasing

pressure, the corresponding pressure from Model

V, the literature value of the pressure at the phase

transformation, and differences in the two pressures.

The differences are also graphically shown in

figure 5. It can be seen that the difference is small

compared to experimental uncertainties in the case of Bi I-II and Ba I-II transformations, and is comparable with the experimental uncertainty in

the case of Bi III-V. Figure 5 also shows the volumetric data obtained by Bridgman (1945) which

are in agreement with Model V within experimental uncertainty.

197
In the case of Bi lU V, the discrepancy between

the pressure value obtained by Haygarth et al. (1968)

by means of the direct single-stage piston-cylinder

method and that calculated from the V/Vq data of

Jeffery et al. (1966) using the Model V F-F relation

for NaCl may be attributed to the following causes:

(1) The Model \ P-V relation for NaCl may be in error. However, this is not likely because Model V

is in good agreement with the experimental data of

Bridgman (1945) up to 100 kbar, and is consistent

with the Bto value obtained by the high-precision

ultrasonic method; (2) The V/Vq for NaCl at the

phase transformation pressures determined by

Jeffery et al. (1966) may be in error because of a heterogeneous pressure distribution in a mixture of

NaCl and sample. Because of such an inhomogeniety

in pressure distribution, the errors in the measure- ments may be non-Gaussian, and hence the arithmetic mean of the determinations for V/Vo of NaCl

made during increasing and decreasing pressure

might not represent a true value at the transformation; (3) The pressure value determined by Haygarth et al. (1968) may be in error. This may be

tested if the compression of NaCl was determined

up to 80 kbar at 25 °C by means of the same experimental apparatus as used for their study of Bi III-V.

If the compression values for NaCl thus obtained

agree with those of Bridgman (1945), which is in

turn in agreement with Model V of this study, the

discrepancy between the pressure values obtained

by Haygarth et al. and by the present study for Bi

III-V could be due to the cause (2). On the other

hand, if the compression values for NaCl are dif- ferent from Bridgman's in such a way that the value

of Haygarth et al. becomes in agreement with Jeffery

et al., then the P-V relation determined by Bridgman and that calculated in this study would be in error. Future studies would hopefully dissolve the

discrepancy between the existing results.

8. Summary and Conclusions

On the basis of the discussion presented above,

the following conclusions may be made:

(1) The thermal and vibrational formulations

of the Hildebrand and Mie-Griineisen approximations yield the pressure values to within ± 1 percent

for a given VI Vo value for NaCl up to SOO kbar at 25 °C. Hence in the harmonic approximation, the

P-V relations are insensitive to the specific form

chosen for the temperature dependent part of the

vibrational energy.

(2) Effect on the pressure values of the anharmonic part of the internal energy is probably small,

since the specific form chosen for the potential

energy yields the correct value for yo. Yet, complete

internal consistency is not found as shown by the

failure of the theory to yield correct values for A and

Table 5. Comparison between the pressure values

calculated from the P—V relation for NaCl

using the V/Vo values obtained by Jeffery et al. (1966) for the phase transformations m Bi and^a

Phase

transfor- mation

1 IVqPI av_.i VIV r i y a

^NaCl

ivioaei V P

Direct

methods

p P ' 'NaCl

kbar kbar kbar

Bi i-n 0.917 ±0.002 25.4 ±0.8 2 25.5 + 0.1 0.1 Ba i-ii 0.855 + 0.002 54.4+1.1 2 55.0±0.5 0.6 Bi ni-v 0.821 ±0.002 75.5±1.4 "77.5 ±1.0 2.0

= 78.2 9 7

' Jeffery et al. (1966).

2 Heydemann (1967).

' Haygarth et al. (1967).

"* Haygarth et al. (1968), extrapolated from high temperature.

^ Haygarth et al. (1968), 22 "C direct measurement.

BtQ. However, the effect of these terms on the

calculated pressure is small.

(3) The close agreement of the P-V relations

obtained in this study with Decker's (1968) revisea

calculation suggests that the effect of the second

nearest neighbor repulsion is small.

(4) The calculated pressures are insensitive

to the values of the various parameters used with an exception of Bto. Thus a pressure scale based

on the relations presented in this study wiU not

require significant revision as more precise values

for the parameters become available.

(5) The estimated precision of ±2.5 percent

in the calculated pressure is of the same order as

the uncertainty in pressure resulting from the

uncertainty in the measured value of VI Vo for NaCl.

(6) The P-V relation based on Models III-VI

is in agreement with the fixed points on the absolute

pressure scale; with Bridgman's volumetric studies on NaCl to 100 kbar; with the results derived from

shock-wave measurements to 260 kbar; and with

the P—V relation for MgO which are based on the

Murnaghan, Rose, or Birch equations with the

parameters determined by means of ultrasonic

methods.

(7) The internal standard method for determining pressure using NaCl appears to be vaUd as shown by the consistency of the pressure values

obtained from the compression VIVo values for

NaCl and MgO.

(8) A NaCl pressure scale based on Model V

has been adopted for the experimental studies at the University of Rochester. Model V (the vibrational Hildebrand equation) was selected because

the vibrational formulation may be expected to be

superior to the thermal formulation, and because,

in practice, the Hildebrand equation is easier for

calculation. The relation can be conveniently

198
expressed as:

P = 8Ax'>' - 106.2a;-'»'3 - 15.95x-^ - 2Mx-''i^

+ x--'l- • exp (14.050 -9.292x'/3) kbar at 25 °C.

On this pressure scale, the B1-B2 phase transformation in NaCl reported by Bassett et al. (1968) occurs at a pressure of

301 ±4 kbar (F/Fo = 0.643 ± 0.002)

with an additional uncertainty of ±8 kbar due to the

uncertainty in the pressure scale.

(9) Significant improvement in the precision of

the calculated P-V relation will require improvement in the values of Bto and A. Although the P-V

relation is insensitive to the latter, the large uncertainty in this parameter yields a large uncertainty

in the pressure. However, a reduction of this uncertainty may be difficult since A depends on the

difference between ds and Bs'

.

9. Acknowledgments

We are grateful to Leon Thomsen and Orson L. Anderson of the Lamont Geological Observatory,

Columbia University, for providing us with their

manuscript in advance to its publication; to Daniel

L. Decker of the Brigham Young University for providing the results of his revised calculations for the

P—V relation of NaCl; and to J. N. Fritz of the Los

Alamos Scientific Laboratories for his unpublished

data for the compression of NaCl. The present re- search program has been supported by the National

Science Foundation through grants GP-5304 and

GA-1430.

10. References

Anderson, O. L., The use of ultrasonic measurements under

modest pressure to estimate compression at high pressure,

J. Phys. Chem. Solids 27, 547-565 (1966). Anderson, O. L., Equation for thermal expansivity in planetary

interiors, J. Geophys. Res. 72,3661-3668 (1967). Anderson, O. L., and Andreatch, P. Jr., Pressure derivatives of

elastic constants of single crystal magnesium oxide at 23 deg.

and - 195.8 deg. C, J. Am. Ceram. See. 49, 404-409 (1966). Bartels, R. A., and Schuele, L). E., Pressure derivatives ot the

elastic constants of sodium chloride and potassium chloride at 295 deg. K and 195 deg. K., J. Phys. Chem. Solids 26, 537-550

(1965). Bassett, W. A., Takahashi, T., and Stook, P. W., X-ray diffraction and optical observations on crystalline solids up to 300 kbar. Rev. Sci. Instr. 38, 37-42 (1967). Bassett, W. A., Takahashi, T., Mao, K. K., and Weaver, J. S., Pressure-induced phase transformation in NaCl, J. Appl.

Phys. 39, 319-325 (1968). Birch, F., The effect of pressure upon the elastic parameters of

isotropic solids according to Mumaghan's theory of finite strain, J. Appl. Phys. 9, 279-288 (1938). Birch, F., Elasticity and constitution of the earth's interior,

J. Geophys.. Res. 57, 227-286 (1952).

Bridgman, P. W., The compression of twenty-one halogen compounds and eleven other simple substances to 100,000 kg/cm^,

Proc. Am. Acad. Arts Sci. 76, 1-24 (1945). Chang, Z. P., Third order elastic constants of sodium chloride

and potassium chloride single crystals, Phys. Rev. A140,

1788-1799(1965).

Chang, Z. P., and Barsch, G. R., Nonlinear pressure dependence

of elastic constants and fourth-order elastic constants of cesium halides, Phys. Rev. Letters 19, 1381-1382 (1967). Christian, R. H., The equation of state of alkali halides at high

pressure, Univ. Calif. Radiation Lab. Report 4900, 1-89 (1957). Collins, J. G., and White, G. K., in C. J. Gorter, ed.. Progress in Low Temperature Physics, Vol. 4, 450-479 (North Holland

Publishing, Amsterdam, 1964).

Decker, D. L., Equation of state of NaCl and its use as a pressure gauge in high-pressure research, J. Appl. Phys. 36, 157-161

(1965). Decker, D. L., Equation of state of sodium chloride, J. AppL

Phys. 37,5012-5014(1966).

Decker, D. L., Personal communication (1968).

Fritz, J., Personal communication (1967). See also: McQueen

(1967). Fumi, F. G., and Tosi, M. P., On the Mie-Gruneisen and Hildebrand approximations to the equation of state of cubic solids,

J. Phys. Chem. Solids 23, 395-404 (1962). Haussuhl, S., Thermo-elastische Konstanten der Alkalihalogenide vom NaCl-Typ, Zeit. Physik 159, 223-229 (1960). Haygarth, J. C, Getting, I. C, and Kennedy, G. C, Determination of the pressure of the Barium I-II transition with singlestage piston-cylinder apparatus, J. Appl. Phys. 38,4557-4564

(1967). Haygarth, J. C, Ludemann, H. D., Getting, I. C, and Kennedy,

G. C, The upper bismuth pressure calibration point, presented

at this Symposium.

Heydemann, P. L. M., The Bi I-II transition pressure measured

with a dead weight piston gauge, J. Appl. Phys. 38, 2640-

2644 (1967). Jeffery, R. N., BameU, J. D., Vanfleet, H. B., and Hall, H. T., Pressure calibration to 100 kbar based on the compression of NaCl, J. Appl. Phys. 37, 3172-3180 (1966). Leibfried, G., and Ludwig, W., Theory of anharmonic effects in crystals. Solid State Physics 12 (Academic Press, New

York, 1961).

Mayer, J. E., Van der Waals potential in the alkali halides, J. Chem. Phys. 1, 270 (1933). McQueen, R. G., Jamieson, J. C, and Marsh, S. P., Shock wave

compression and x-ray studies of titanium dioxide. Science

155, 1401-1404 (1967). McWhan, D. B., Linear compression of a-quartz to 150 kbar,

J. Appl. Phys. 38, 347-352 (1967). Mumaghan, F. D., The compressibility of media under extreme

pressures, Proc. Nat. Acad. Sci. 30, 244-247 (1944). Perez-Albueme, E. A., and Drickamer, H. G., Effect of high pres- sures on the compressibilities of seven crystals having the NaCl or CsCl structure, J. Chem, Phys. 43, 1381-1387 (1965). Robie, R. A., et al., in S. P. Clark, ed., Handbook of Physical

Constants, Geol. Soc. Am. Mem. 97, New York (1966). Rose, M.'F., Higher order elastic constants and the equation of

state of F. C. C. metals, Phvs. Stat. Solid 21, 235 (1967). Schreiber, E., and Anderson, O. L., Revised data on polycrystalline magnesium oxide, J. Geophys. Res. 73 , 2837-

2838 (1968). Slagle, O. D., and McKinstry, H. A., Temperature dependence

of the elastic constants of the alkali halides. I. NaCl, KCl,

and KBr, J. Appl. Phys. 38, 437-446 (1967). Thomsen, L., and Anderson, O. L., Consistency in the hightemperature equation of state of solids (in press) J. Geophys.

Res.

Tosi, M. P., Cohesion of ionic solids in the Born model, in F. Seitz and D. TurnbuU, eds.. Solid State Physics 16, 1-120

(Academic Press, New York, 1964).

Yates, B., and Panter, C. H., Thermal expansion of alkali halides

at low temperatures, Proc. Phys. Soc. (London) 80, 373-

382 (1962).

199

385-762 O - 71 - 14

The Hugoniot Ecfuation of State of Sodium Chloride in the Sodium Chloride

Structure *

J. N. Fritz, S. P. Marsh, W. J. Carter, and R. G. McQueen

University of California, Los Alamos Scientific Laboratory, Los Alamos, New Mexico 87544

The Hugoniot equation of state of NaCl has been obtained by measuring the shock velocity through

NaCl on copper and 2024 aluminum base plates. Shock velocities through the base plates and standard

impedance-matching were used to obtain the Hugoniot curves for both single-crystal (in various orientations) and pressed-powder samples. The smooth behavior of the resulting shock locus up to 230 kbar

indicates that NaCl exists in the sodium chloride structure up to this pressure. In the shock-particle

velocity plane the best linear fit to the data reported here is Us (km/s)= (3.528±.012) + (1.343±.009)up.

A quadratic fit, which gives a large weight to the measured bulk sound speed in NaCl, is Us=3.403

+ 1.5422 Up — 0.07345 Isotherms at 293 K, using different forms for the Griineisen parameter and

a simple Debye model for the specific heat, are calculated from the Hugoniots and are presented here. They should prove useful when NaCl is used as an internal standard in high-pressure x-ray devices.

1. Introduction

Since Jamieson [1] ^ first used NaCl as an internal

pressure standard in his high-pressure x-ray apparatus, it has been a serious candidate for a pressure

standard in high-pressure x-ray work. Decker [2]

has advocated the use of NaCl as a standard. His

proposal was based on a Born-Mayer equation of

state. Parameters were fixed by the initial density

and sound speed of NaCl at zero pressure and he

achieved a prediction of the curvature of the P-V

isotherm, which can be converted in terms of variables familiar to workers in the dynamic-pressure

field to a prediction of the slope of the shock velocity versus particle velocity Hugoniot. His results were

supported by the then existing shock-wave data. An

increase of 1.3 percent in the value of the sound

speed he used in his work would bring his isotherm

into agreement with the result of the present report.

Objections have been raised [3, 4, 5] to using NaCl

as a standard because of the possibility of the transition to the CsCl structure at a pressure as low as 20 kbar. Such a low-pressure transition would un- acceptably complicate the use of NaCl as a continu- ous pressure standard. Others [6, 7, 8, 9] have found

no evidence for a transition as low in pressure' as

this. From the work of Jamieson [9, 10] on the solution salts, NaxKi-xCl, one can conclude that the

Bl to B2 transition must be considerably above 130

kbar in the pure end member, NaCl. Finally, Bassett

et al. [11] have found and reversed the Bl to B2

transition in NaCl at approximately 300 kbar at room temperature. It would seem that whatever the

remaining phenomena are at 20 kbar, it does not

involve the Bl to B2 transition, and that NaCl will

remain in the Bl structure to high and useful

pressures.

*Work done under the auspices of the U.S. Atomic Energy Commission.

' Figures in brackets indicate the literature references at the end of this paper. Paper presented at the Symposium on Accurate Characterization

of the High-Pressure Environment, held at the National Bureau

of Standards, Gaithersburg, Md., October 14-18, 1968.

Perez-Albuerne and Drickamer [8] have compared

NaCl by x-ray techniques with Agand Mo. If the compression curves of Ag and Mo are known, this would

determine a NaCl isotherm. Using older shock

wave data [12], results were obtained which, because

of the individual scatter in a data point, were used

only to verify the calculation of an isotherm from

a particular form of a Born-Mayer treatment.

Their calculated isotherm is in agreement with

the isotherm we report as well as that of Bridgman

[6].Direct shock-particle velocity data on NaCl

could not be obtained because of the impracticaUty

of fabricating the assembhes required for our shockwave techniques. Hence we have employed the

shock-wave impedance match technique using

2024 aluminum base plates as the standard. In

this way, NaCl becomes a secondary standard,

dependent on the Hugoniot and extended equation

of state of 2024 aluminum. If the 2024 aluminum

Hugoniot is accepted as known, a shock-locus for

NaCl is obtained that is felt to be accurate to

1 percent in pressure. Subsequent modifications,

if any, of the primary standard may be easily translated to a new Hugoniot locus for NaCl. As always,

the largest uncertainty in an isothermal curve

derived from shock-wave data is related to the

choice of the Griineisen gamma used to reduce the

Hugoniot to an isotherm. Several functions of volume

have been used for y{V) and the resultant isotherms

reported.

2. Experimental Methods and Results

The impedance matching technique used for

obtaining the shock locus for NaCl has already been

adequately described [12]. Single crystals in the

(100), (110), and (111) orientations as well as some

pressed powder samples were shock-loaded on base

plates of 2024 aluminum, copper, and 921-T aluminum. The data taken on 921-T aluminum base

plates have been discarded for reasons described

201
Table 1. NaCl Hugoniot data in the Bl phase

Us Us Std. Up p P

kmjs kmjs km/s kbar glcm^

100 orientation

4.03 5.67 0.33 29 2.358 4.10 5.75 0.40 35 2.397

4.12 5.79 0.44 39 2.423

4.31 5.94 0.58 54 2.502

4.36 5.96 0.60 57 2.510

4.59 6.17 0.79 78 2.613

4.59 6.19 0.81 80 2.625

4.95 6.45 1.03 111 2.735

A on 4.89 6.45 1.04 110 2.748

4.99 4.96* 1.07 116 2.755

A f\f\ 4.99 6.50 1.09 117 2.766

A nn 4.99 6.54 1.12 121 2.789

5.01 6.55 1.13 122 2.793

5.51 6.92 1.44 172 2.932

5.43 6.91 1.44 170 2.949

5.47 6.92 1.45 171 2.942

5.47 6.92 1.45 171 2.941

5.69 5.41* 1.53 184 2.977

5.67 7.07 1.57 193 2.992

5.84 5.58* 1.70 215 3.052

7.36 1.82 234 3.121

5.96 7.37 1.84 237 3.129

6.12 7.47 1.92 254 3.151

6.12 7.47 1.92 254 3.152

O.lc) 7.51 1.95 259 3.166

6.15 7.51 1.95 259 3.167

6.13 7.52 1.96 260 3.185

6.20 7.54 1.97 264 3.170 111 Orientation

4.08 5.75 0.40 35 2.399

4.32 5.96 0.60 56 2.515

4.31 6.00 0.63 59 2.538

4.59 6.19 0.81 80 2.626

4.59 6.20 0.82 81 2.632

4.96 6.44 1.03 110 2.729

4.98 4.96* 1.07 115 2.757

5.01 6.50 1.08 117 2.760

4.99 6.50 1.08 117 2.765

4.99 6.53 1.11 119 2.780

5.03 6.55 1.13 123 2.787

5.02 6.56 1.13 123 2.795

5.45 6.92 1.45 171 2.947

5.47 6.92 1.45 171 2.944

5.57 5.41* 1.53 184 2.981

*Copper base plates. The rest are 2024 aluminum.

**This sample had a (110) orientation.

Up (Km/sec)

Figure !. The shock velocity-particle velocity Hugoniot o/NaCl.

The elastic wave velocities were obtained from data in reference (14). The larger symbols indicate multiple data points.

elsewhere [13J. In this report only those data whose

final shocked state is believed to remain in the Bl

structure are reported. The majority of the data

were taken on 2024 aluminum with a few points

taken on copper base plates as a consistency check.

The data necessary for impedance calculations for

the primary standards are:

2024 Aluminum: po = 2.785 g/cm^,

«s= 5.328

+ I.338ttp km/s

{dEldP)v =0.19 cm3/g

Copper: po = 8.93 g/cm^,

«s = 3.940

+ 1.489upkm/s I

0£/aP), =0.057 cm3/g I

A constant {dEldP)v was used to generate the re- lease isentropes necessary for the impedance match

calculations. The data are reported in table 1 and

illustrated in figure 1. In addition to the tabulated

data, data points showing the onset of the Bl to B2

(presumably) phase transition are included in the

'

figure for both (100) and (111) orientations.

'

IS Both linear and quadratic terms in Up were used •

to fit these data. In the absence of other information,

j there would be no point in going to an order higher

than linear to fit the data; however, consideration

must be given to the measured sound speeds in

NaCl. In fact, the Co coefficient in our quadratic

fit was forced to agree with the sound speed ob202
tained from Haussuhl's [14] measurement of the

elastic constants of single-crystal NaCl. The least

squares analysis of the data gives r.m.s. deviation

of Af/,s = ± 0.056 km/s or less over the range of the

data. Within the range of the data, the quadratic

and linear fits agree to within this Aus except for

the lowest four data points. The low shock pressures

for these four points ( < 40 kbar) were produced by

relatively complicated driver arrangements that

depend on impedance mismatches. It is not impossible that overtaking waves could influence these

measurements. The shock velocities at these pres- sures are notably below some of the single crystal

longitudinal velocities. Premature closure of flash

gaps could give rise to the sort of discrepancy we

see here, but since a low-pressure cannon shot

has indicated that the Hugoniot elastic limit of

NaCl is only about 2/3 of a kilobar, there is not

sufficient motion imparted to a shim to close any

flash gaps used here. Gap closure corrections are more critical in this pressure regime and conceivably

could cause errors in shock wave velocity, even of

the size to explain these low pressure shots. In

any event, forcing the Hugoniot through the sound

speed prevents these points from appreciably

influencing the fit to the data. The large bulk of the

data at higher pressure with its linearity fixes the

next two terms of the series. Some evidence that

these lower points influence the slope in the right

direction is afforded by Bartels and Schuele's

[15] measurement of (afis/SP) r = 5.27.

From the thermodynamic identities

So=[l+{dBsldP)s]l^

and

{dBs/dP}s= {dBsldP)T+y{TIBs){dBsldT)p

and Haussuhl's [14] value for {dBsldT)p of

-0.000117 Mbar/°K, we obtain 5o= 1.513. This is in

satisfactory agreement with our fitted slope of 1.542.

Although a linear fit represents the shock-wave

data adequately, emphasis should be placed on the

ultrasonic measurements in the low pressure region.

The quadratic fit

«s = 3.403+ 1.5422a,, -0.07345u| (km/s) (1)

represents the ultrasonic data in the low pressure

rfegion and the shock-wave data in the high-pressure

region. The number of significant figures quoted in

the above equation gives the exact form of the

Hugoniot used in a subsequent reduction to an

isotherm. The r.m.s. spread in Ug about the fitted

values is ±0.05 km/s. Our data for NaCl in the Bl

phase are in agreement with the data of Larson, et aL

[3], to within the quoted error. Unpublished data of

Hauver and Melani [16] are in agreement with ours

in the vicinity of Up=\ km/s and are 0.07 km/s

higher than ours near Up = 2 km/s. However, their

data were based on the old 2024 aluminum standard.

Table 2. Thermodynamic parameters for NaCl

Quantity Value Reference

M

N

ao

^ v\aT)p

Bs

58.443 g/mole

6.02252X1023

5.6393 A at 22 °C

1.195X10 •»/K at 20 °C

0.2505 Mbar at 22 °C

12.05 cal/mole-Kat 20 °C

[17]

[19]

[19]

[14]

[20, 21]

Derived Values at 293 K

p = 2.1645 g/cms = 0.2507 Mbar

Cp = 0.8627xl0' ergs/g-K

Co = 3.403 X10« cm/s

-y = ^c2/C^= 1.6044

y;8r= 0.0562

Ci; = 0.8168X10' ergs/g-K

Br =0.2374 Mbar

^7-= 4.213 Mbar-'

OP/ar), =0.0284 kbar/K

,. = 0.2880 cm'' ink = 0.8535 X 10' ergs/g - K

C,/3raA:= 0.9570

6»D/r= 0.9420

6lo = 276 K

Revision of the necessary impedance match calculations will bring the results into better agreement.

3. Reduction of the Hugoniot to a

293 K Isotherm

Details of this calculation have been discussed

earlier [13]. The auxiliary information needed for

this reduction is given in table 2. Original references

should be consulted for the accuracy of these numbers. The table gives the exact input used in our

calculations. The particular numerical code used

integrates the equation

ydV (V,-

dl H — — 1 H—f} 1 V)dPH + PHdV

2Cv (2)

for the temperature along the Hugoniot. A Debye

expression,

ET=inkTD,{e{V)IT) (3)

Jo e^ —

1

y^-d\n didhi V

(4)

(5)

203
Table 3. The (dEldP)v constant, 293 °K NaCl isotherm and its antecedent Hugoniot

p Hugoniot

Ph

Th Isotherm P

a/ao r Hugoniot

Ph

TI H Isotherm

p

alao

kbar K

125 2.793 618 2.828 .7654 .9147 glcm^

130 2.810 637 2.847

au zyo Z. 104*0 1 noon

.7603 .9127

5 2.206 302 2.208 .9803 .9934 135 2.827 657 2.865 .7555 .9108

10 2.245 311 2.248 .9629 .9875 140 2.843 678 2.883

15 2.281 319 2.286 .9469 .9820 145 2.859 699 2.901 .7461 .9070

20 2.314 328 2.321 .9326 .9770 150 2.875 721 2.918 .7418 .9052

zo OOO Z.OOO ,y I AO 155 2.891 743 2.935 .7375 .9035

30 2.377 345 2.386 .9072 .9680 160 2.906 765 2.952 .7332 .9017

35 2.406 355 2.417 .8955 .9639 165 9 091 788 . 790n t Z7W

40 2.434 365 2.446 .youi 170 2.936 811 2.986 .7249 .8983

45 2.461 375 2.473 .yooo 175 2.951 835 3.002 .7210 .8967

t^n ou OOO z.ouu 9 '^nn 180 2.965 859 3.019 .7170 .8950

55 2.511 398 2.526 .8569 .9498 185 2.980 884 3.035 .7132 .8934

60 2.535 410 2.552 .8482 .9466 190 2.994 909 8010

65 2.558 423 195 3.008 934 3.067 .7057 .8903

70 2.581 436 Z.oUO .ooZo .y4U/

200 3.022 960 3.082 .7023 .8889

z.oUZ 4oU 205 3.035 986 3.098 .6987 .8873

80 2.624 464 2.645 .8183 .9354 210 3.049 1013 3.113 .6953 .8859

85 2.644 479 Z.OOY O 1 1 ^ .olio 215 3.062 1040 ^ 190 6018

yu Z.0D4 AQA 220 3.075 1067 3.144 .6885 .8830

Z.0o4 OlU 9 71 n 7QQ7 Q97ft .VZ / O 225 3.089 1094 3.159 .6852 881 fi

lUU z. /Uo oz / 9 7Q1 Z. /ol . 1 yZD 09 ^d. 230 3.102 1123 3.174 .6819 .8802

105 2.722 544 2.751 .7868 .9232 235 3.114 1151 3.189 .6787 .8788 110 2.740 562 2.771 .7811 .9210 240 3.127 1180 3.204 .6756 .8774

115 2.758 580 2.790 .7758 .9189 245 3.140 1209 3.218 .6726 .8762 120 2.776 599 2.809 .7706 .9168 250 3.152 1238 3.233 .6695 .8748

is used for the thermal part of the energy.

With a y that depends only on volume, the appropriate fractional thermal energy and corresponding

thermal pressure may be subtracted from the

Hugoniot to yield the desired isotherm. The choice

of Debye theta at zero pressure and 293 K was

dictated by the value of Cv at these conditions, since

this is the important quantity in the integration of

(2). Since the numerical value for E{P=0, 7=273

K)-E(0, 0) predicted by this model (+1.76xl0«

ergs/g) agrees with the value obtained by fitting

C/.(r)(A£= 1.77X109, JANAF Tables [21]) we

have some evidence that we have an overall fit for

Cv{T) at the lower T region as well.

Probably the largest uncertainty in transforming

a Hugoniot to an isotherm comes from ignorance

about the way the Gruneisen gamma behaves at high pressures and temperatures. In the few cases where its high-pressure behavior has been measured, the assumption that {dEldP)v is constant has

been adequate to represent the data within the

experimental precision, but other forms for y{V)

are not excluded. We have used this assumption to calculate our "base room temperature isotherm."

The results, finely spaced for more convenient

usage, are given in table 3.

It is of interest to see what effect varying the parameters that went into the calculation will have on

the calculated isotherm. The equation

f-2 1 d In {d[PK{V)V''']ldV} (6)

3 2 d\nV

is a generalization of the Slater, DugdaleMacDonald, and Free Volume relations between

y{V) and Pk{V) which has been used by Grover

et al. [22] in their comparison of static and dynamic

high-pressure data on the alkali metals. Values of

f of 0, 1, and 2 correspond to the above theories, but

the value of t can be chosen to give the proper

thermodynamic value of y at the Hugoniot centering point. Further, if the expression for y{V) obtained from differences between the Hugoniot curve and the zero Kelvin curve is equated with the above

204
expression for y, the Hugoniot centering point is taken to be at zero pressure and temperature, and

a linear Ug-Up Hugoniot is used, an integrodififerential equation for Pk{V) is obtained. When

solved this gives not only Pk{V), but also a 7(17),

where 17= 1 — poV, that parametrically depends only on t and the slope, 5, of the Ug-Up Hugoniot. At zerc compression, this reduces to the relation

7o = 2s-(f + 2)/3 (7)

between yo, t, and s. A linear Ug-Up Hugoniot for

NaCl that passes through most of the shock-wave

data and the measured sound speed has s= 1.429.

With the correct yo, (7) yields 1.761. The y(i7)

obtained in this fashion will not significantly differ

from one where the precise experimental Hugoniot

is used as input and where the difference between

the temperature of the Hugoniot centering point and

zero temperature is taken into account. The resulting y(i7), as well as others, are shown in figure 2. This y(T7) may be accurately represented by a polynominal in 7) for 0 < tj < 0.5.

y= 1.6044-0.9955 rj+ 1.4961 rj^- 1.9284 rj^ (8)

This representation, along with the more precise

Hugoniot fit, was then used in the numerical code

that calculates the isotherm. The resultant curve,

labelled (1), is shown in figure 3. An ionic solid may

not be the best place to use such a theory for y(T7),

but it does offer an alternate 7(17) behavior to compare with the base isotherm.

Another y(i7) behavior is offered by Decker [2]. A term linear in the change in lattice parameter was

added to yo to give the best fit to high-temperature

(i.e., a slight increase in volume) thermal expansion

data. This should give the right initial slope of y(T7)

but as Decker says, it is uncertain whether this

volume dependence remains accurate to the large

volume changes obtained in the shock wave data.

This y(T7) is given by

y=1.6044 + 2.55 ((l-r,)'/3-i). (9)

Decker actually used the value yo=1.59. A poly1.8

1.6

1.4

1.2

1.0

0.8

1 1

V GRUNEISEN

1 1

GAMMAS

\. FOR NqCI

1

1

(4)\ \.

1 1 ^

0.0 0.1 0.2 0.3 0.4 0.5 V

Figure 2. The variousfunctions usedfor the Gruneisen parameter.

-0.04 VARIOUS NaCI

ISOTHERMS

100 200 300

P(Kb)

Figure 3. Pressure variations in the base isotherm due to various changes in the input data.

In this figure we show the fractional change in pressure from the base isotherm, SPIP = P(new parameters)/P (base) - I

,

at a given density. The actual ab^issa is the pressure of the base isotherm at this density.

205
Figure 4. NaCl isotherms.

The solid line shows the isotherm calculated from the quadratic fit to the Hugoniot in the Ug—Us plane. The

isotherm from the linear fit is parallel to Decker's isotherm and roughly maintains a constant pressure offset from

Decker's isotherm from 200 to 320 kbar.

nominal fit to (9) was used to calculate isotherm (2)

in figure 3. Isotherms (3) and (4) show the effect of varying

7o by plus and minus 10%. For these isotherms

(dEldP)v was kept constant. Isotherms (5) and (6)

show the effect of scaling Cy by plus and minus

10%. This was done by changing the value of 3nk in

the calculations. Isotherms (7) and (8) were obtained by adding and subtracting, respectively, 0.05

km/s to Co in the input Hugoniot.

4. Comparison With Other Isothermal

Data and Discussion

The "base isotherm" is shown in figure 4 along

with Decker's, that calculated by Perez-Albuerne

and Drickamer, and Bridgman's data. The agreement between our isotherm and that of PerezAlbuerne and Drickamer's is well within the

experimental error of both methods. The latter isotherm is slightly lower (1 to 2 kbar) than ours in the

pressure range from 40 to 100 kbar. Bridgman's

isotherm is stiU lower by again this pressure dif- ference in this region. The lower portion of our isotherm is chiefly determined by using Haussiihl's

sound speed. The initial slope of the Ug — Up Hugoniot could be changed from 1.542 to 1.512 to agree

with Bartels and Schuele, but this would decrease

the computed isotherm by only V4 kbar at 40 kbar.

At 60 kbar, the shock-wave measurements are

slightly above the fit used. Above this, where the

shock-wave measurements should be relatively free

of elastic-plastic flow effects because of the low

Hugoniot elastic limit of NaCl, the fit and the data

are consistent. Decker's isotherm falls below ours,

with the spread in pressure being about 5 kbar at 100 kbar, 12 kbar at 200 kbar, and remaining

approximately the same thereafter. As he noted,

Decker's isotherm is most sensitive to the value of

the bulk modulus used. A choice of bulk modulus

equivalent to the sound speed we have used would

bring the two isotherms into essential agreement.

It is perhaps worthwhile to emphasize at this point

that at and above 80 kbar, and to a lesser extent at 60 kbar, our isotherm does not particularly depend

on the precise value of the bulk modulus used, be- cause the shock-wave data then determines the

Hugoniot used to calculate the isotherms.

Jeffrey et al. [23] have given the pressures of

various phase transitions by determining a/ao

of NaCl used as an internal standard. Using their

values of a/ao and Decker's pressure scale they

obtain 24.8 ±0.8 kbar for the Bi l-ll transition,

53.3 ±1.2 kbar for the Ba l-ll transition, and

73.8 ±1.3 kbar for the Bi III-V transition. Using

our isotherm and their a/ao values, these numbers

become 25.8 ±8, 56.3 ±1.2, and 78.2 ±1.3. Bridgman's volume scale as reported by Jeffrey et al.,

has 25.4, 58.8, and 88 kbar for these transitions.

McWhan [24] also measured the Bi III-V transition

with NaCl as an internal standard. He took his

values on the increasing pressure cycle, whereas

206
Jeffrey et al. averaged increasing and decreasing

pressure readings. McWhan's F/Fo 0.816 ± .006

corresponds to 81.7 ±4.5 kbar on our isotherm.

The Birch-Murnaghan equation and the input

parameters as used by McWhan for his pressure

value is consistent with our isotherm up to a pres- sure of 100 kbar. Beyond that, the Birch-Murnaghan

value for the pressure increases more rapidly than

our isotherm by 6 kbar at 150 kbar and 13 kbar at 200 kbar.

The highest measured pressure on our Hugoniot

for the Bl phase is 264 kbar. The offset down to the

isotherm is about 16 kbar at this volume. Our isotherm above 250 kbar is then a consequence of an

extrapolation of the Hugoniot data. The downward

curvature of the quadratic Ug — Up fit is a relic of

trying to fit the ultrasonic data and shock data

smoothly together in the lower pressure region and

probably should not be there. Both the data alone

and calculations of a Ug — iLp curve from a BornMayer form for the interaction potential indicate

a linear behavior in this region. Accordingly we

have Listed in table 4 the isotherm above 200 kbar

that obtains from the best linear fit to our data.

In this calculation we have again used constant

{dEldP)v. From 200 to 230 kbar this is essentially

identical to the isotherm resulting from the quadratic

fit. Above 230 kbar it becomes stififer. A clear indication of a phase transition is ex- hibited by the higher pressure data points plotted

in figure 1. Although (111) and (100) oriented

crystals have indistinguishable Hugoniots at lower

pressures, they clearly separate in the region where

the phase transition occurs. Since a uniaxial compression of 50 percent in the (111) direction produces

the B2 structure from the Bl structure, one can

expect that a shock wave in this direction will see a lower energy barrier in the way of this transition.

Indeed, the (111) data are lower. If a shock wave is not complicated by relaxation effects and deviatoric

stresses, a phase transition with a sufficient AV

and an appropriate slope in the P-T plane appears

as a horizontal plateau in the Ug-Up plane. This is because the flash gaps measure the shock velocity

of the first wave in a two-wave structure, which

remains constant as we increase the driving

pressure, and the particle velocity, measured

indirectly by measuring the shock velocity of the

standard, is increasing with the driving pressure.

Time-dependent effects clearly influence the NaCl

data in this region, but the onset of a phase transition

is beyond question. At these pressures, even though

a considerable time may be required for a phase

transition to go to completion, any part of the AV

of the reaction manifests itself immediately as a shock velocity less than would be expected from a smooth continuation of the initial phase. A plateau

has been drawn somewhat arbitrarily through the

(111) data. Although the data scatter upwards from

this Une, the clustering toward the bottom is sufficiently sharp to give some credence to the

plateau as drawn.

Table 4. The NaCl isotherm above 200 kbar using

the linear Ug-Up fit

p Hugoniot

Ph

Th Isotherm P

kbar 1 3 glcni'

200 3.023 950 3.081 0.7025 0.8890

205 3.036 977 3.096 .6991 .8875 210 3.050 1004 3.111 .6958 .8861 215 3.063 1031 3.126 .6924 .8847 220 3.075 1059 3.140 .6893 8834

225 3.088 1087 3.154 .6863 .8821 230 3.100 1116 3.168 .6832 .8808

^ 1R9

94.0 1 1 74, 0. 1 yyj 677^ R789

245 1904 910 .y>f>74^ t 'i'O 8760

250 3.148 1234 3.223 .6716 .8757 255 3.160 1264 3.236 .6689 .8745

^ 1 71 ^ 94.0 .uuuz 265 0. 1 (JO ^ 969 66'? .\J\JOO 8799

270 3.194 1357 3 97=1 6600 871

1

275 3.205 1389 3.288 .6583 .8699 280 3.216 1421 3.300 .6559 .8689 285 3.227 1454 3.313 .6533 .8677 290 3.238 1487 3.325 .6510 .8667 295 3.248 1520 3.337 .6486 .8656

300 3.259 1553 3.349 .6463 .8646 305 3.269 1587 3.361 .6440 .8636 310 3.279 1621 3.373 .6417 .8625 315 3.289 1656 3.384 .6396 .8616 320 3.299 1690 3.396 .6374 .8606

The behavior of the (100) data is different; it tends to drift slowly upward in a manner which one

might expect from a reaction whose relaxation time

is comparable to the time it takes a shock wave to traverse the sample. The (111) plateau intersects

both the hnear and quadratic fits to the Bl phase

at Us = 5.94 and Up= 1.80 km/s. The pressure on

the Hugoniot is 231 kbar, the density is 3.105 g/cm^

and the temperature is between 1120 and 1130 K.

This number should provide an upper bound to the

actual transition pressure.

Bassett, et al. [11], have observed the Bl—B2

transition at a VIVq = 0.643 ± 0.002. Using our isotherm from the extrapolated linear Hugoniot, we

find the pressure of this transition as 307 ± 5 kbar.

The extrapolation of the quadratic fit would have

yielded 295 ± 5 kbar. This should set the upper

limit of pressure for the use of NaCl in the Bl

phase as an internal pressure standard at room

temperature.

5. Acknowledgments

The authors would like to thank G. E. Hauver and

A. Melani for the opportunity to look at their un207
published data and D. L. Decker for sending us his

tabulated pressure-volume isotherms.

6. References

12

3

[4

[5

6

78

[9

[10

[11

[12

[13

[14

[15

[16

[17

[18

[19

[20

[21

[22

[23

[24;

Jamieson, J. C, Science 139, 1291 (1963). Decker, D. L., J. Appl. Phys. 36, 157 (1965). Larson, D. B., Keeler, R. N., Kusubov. A., and Hord, B. L.,

J. Phys. Chem. Solids 27, 476 (1966). Evdokimova, V. V., and Vereshchagin, L. F., J. Exp. Theor.

Phys. 43, 1208 (1962); 16, 855 (1963). Pistorius, C. W. F. T., J. Phys. Chem. Solids 25, 1477

(1964). Bridgman. P. W., Proc. Am. Acad. Arts Sci. 76, 1 (1945). Johnson. Q. Science 153 , 419 (1966). Perez-Albuerne, E. A., and Drickamer, H. G., J. Chem.

Phys. 43, 1381 (1965). Jamieson, J. C, Physics of SoHds at High Pressures, pp. 444-459, C. T. Tomizuka and R. M. Emrick, eds. (Academic Press, New York, 1965).

Jamieson, J. C. private communciation.

Bassett. W. A., Takahashi. T., Mao, H. and Weaver, J. S.

J. Appl. Phys. 39, 319 (1968). Rice, M. H., McQueen, R. G., and Walsh, J. M., Solid State Phys. 6, 1 (1958). Carter. W. J., Marsh, S. P., Fritz, J. N., and McQueen, R. G.. The equation of state of selected materials for high- pressure references, NBS Symp. on the Accurate Characterization of the High Pressure Environment, Washington, 1968. Haussuhl, S., Zeit. Physik 159, 223 (1960). Bartels, R. A., and Schuele, D. E., J. Phys. Chem. Solids

26, 537 (1965). Hauver, G. E., and Melani, A., private communication.

Johnson, W. H., Jr., and Nier, A. O., Handbook of Physics,

E. U. Condon and H. Odishaw, eds., pp. 9-65, 2d ed. (McGraw-Hill, New York, 1967).

National Bureau of Standards Circular 539, 2, 41 (1965). Rubin, T., Johnston, H. L., and Ahman, H. W., J. Chem.

Phys. 65 , 65 (1961). Barron, T. H. K., Leadbetter, A. J., and Morrison, J. A., Proc. Roy. Soc. A2 79 , 62 (1964). JANAF Thermochemical Tables, The Dow Chemical Company, Midland, Michigan (June 1966).

Grover, R., Keeler, R. N., Rogers, F. J., and Kennedy, G. C,

On the compressibility of the alkali metals, J. Phys.

Chem. Solids (to be published).

Jeffery, R. N., Barnett, J. D., VanFleet, H. B., and Hall, H. T., J. Appl. Phys. 37 , 3172 (1966). McWhan, D. B., J. Appl. Phys. 38, 347 (1967). DISCUSSION

D. J. Pastine (U.S. Naval Ordnance Laboratory,

White Oak, Maryland): In combining the effects of

the possible deviation of the several parameters,

illustrated in figure 3, should all the effects be added

if all the deviations in the initial conditions were

made at the same time?

D. Decker (Brigham Young University, Provo,

Utah): In our work, I varied aU the parameters by

the majcimum amount, but found that the total error was less than the sum of the possible individual

errors. The value of B'^ in the new calculations that

we have made is 4.93, which differs a little from

yours. Also, there are at least five ultrasonic

measurements giving values for B'^ ranging between

6.05 and 3.31 so that presently it appears that

ultrasonic data cannot be used to refine the equation

of state.

D. B. McWhan {Bell Telephone Laboratories,

Murray Hill, New Jersey): I would say that the

results and outlook concerning calculation of the

NaCl isotherm is encouraging. A couple of years

ago in some work on quartz, I looked at the volume

of sodium chloride at the bismuth transition. At

that time the Decker equation gave values below

those of the present shock data, and the Birch

equation gave values above. Since then the starting

parameters for both theories seem to have come into

question, and when one modifies fio, as Decker has

done, then the values fall right on the shock data.

If one does the same thing with B'q with the Birch

equation, it wiU also agree with the shock data.

Thus there are three independent results that are in

agreement at least up to about 150 kilobars.

I would like to note that, as I mentioned, we determine the volume of sodium chloride at the bismuth

transition only on the upstroke, and obtained a value

of about 0.816. The Hall group got a value of 0.818.

I think this is remarkably good agreement for two

completely different pieces of equipment, in

independent experiments.

J. S. Weaver {Department of Geological Sciences,

University of Rochester, Rochester, New York): In our work, the deviations are given in our table 3,

of our paper given earlier in this Session, listing

the sensitivity of the calculated pressure values to these deviations. If one parameter is increased the

pressure may increase or decrease. Our 2V2 percent

is obtained by squaring the individual deviations,

summing them, and then taking the square root. This assumes that the parameter variations are independent and gaussian.

I think that the explanation of your better agreement with other values of Bq is related to taking

account of the second-nearest-neighbor effect.

AUTHORS' CLOSURE

In reply to Dr. Pastine: The effects of possible

deviation of the parameters involved in the calculation may be added but the total effect is small. We

should point out that, in contrast to Decker's

calculation, the only assumed parameter that we

are varying which has any significant effect on the

resulting isotherm is the Griineisen parameter.

Whether or not one decides to include the un- certainty in the specific heat, the only other bit of

information one needs to convert a Hugoniot to an

isotherm, makes no practical change in the final

result.

208
Consistency in the High-Temperature Equation of State of SoUds

Leon Thomsen and Orson L. Anderson

Lamont Geological Observatory ofColumbia University, Palisades, New York 10964

Four equations of state for solids are examined theoretically for mutual consistency in the hightemperature regime. It is found that neither the Birch-Murnaghan nor the Murnaghan equation can be

shown to be consistent with either the Mie-Griineisen or the Hildebrand equation. An expression is derived for the volume dependence of the Griineisen parameter y(p). A recent theoretical treatment of NaCl, proposed for use as a pressure standard, is neither internally consistent, nor is it sufficiently

precise for such a purpose.

1. Introduction

In current use there appear several different

equations of state for solids which differ in their

derivations, in the quantities which they exhibit explicitly, and in the quantities which they leave as experimental parameters. It is the purpose of this

paper to discuss together some of these equations,

and to show to what extent their separate derivations are mutually consistent. Such consistency becomes important when two of these equations are used together in interpreting experimental data. For

example, one method [cf. Takeuchi and Kanamori,

1966] of reducing shock-wave data employs the MieGriineisen equation to account for thermal effects

in transforming the Hugoniot into an isotherm,

which is then compared with the Birch-Murnaghan

equation. Clearly, if this calculation is to have meaning, assumptions made in any one part of it must not

be violated in another part. As another example,

when a high-pressure experiment is calibrated using

the equation of state of NaCl provided by Decker

[1965], the resulting data are constrained by the

assumptions of the Mie-Griineisen equation of state, as used by Decker. If these data are then compared

and extrapolated with the Birch-Murnaghan equation of state, the problem of consistency again must

be faced.

Many equations of state can be classified into one

of two broad classes: 'isothermal' equations which

are concerned primarily with effects of compression

at constant temperature, and 'thermal' equations

which treat strictly thermal effects more precisely.

In this paper, we first give brief accounts of the

derivations of some of these equations of state. Second, we develop a simple framework in which

they can all be considered together. Next, we show

how the same reasoning which was used in the

isothermal case leads to an expression for y{p).

Then we consider the restrictions imposed upon

the isothermal equations by the thermal equations.

Turning finally to recent work on NaCl, we find that

internal consistency in the Mie-Griineisen equation

'Lamont Geological Observatory Contribution No. 1563.

Paper presented at the Symposium on Accurate Characterization

of the High-Pressure Environment, held at the National Bureau of

Standards, Gaithersburg, Md., October 14-18, 1968.

generates in this special case a yip) from the

microscopic lattice potential energy.

2. Isothermal Equations of State

Prominent among this class is the BirchMurnaghan equation

Pip, T)=iKoiT) (y^/3-r/3) [1 -^(D (//^- 1)]

where (1)

y(p, D-— (la)

poU )

Here P is pressure, p density, T temperature, the

isothermal bulk modulus pidPldp)T, and the subscript 0 refers to zero pressure. ^ is related to experimental quantities by

HT)^-nK'oiT)-^] (2)

where K'o is {dKldP)T as P->0. The B-M equation

is derived [Murnaghan, 1937; Birch, 1938, 1947,

1952] from the theory of finite strain, in which it is proved that the strain is given by

e^^iy'l^-l) = eip,T) (3)

and it is assumed that a Taylor expansion, in the

strain, of the Helmholtz free energy may be

terminated at the cubic terms. It should be noted

that the uniqueness of this definition of strain has

been questioned by Knopoff [1963].

Leaving such questions aside, in the present case

of homogeneous isotropic compression, the free

energy is assumed to be

The B-M equation then follows from

where V— lip is the specific volume. We wiU return

209
to the discussion of this equation shortly.

Another important member of this class is the

equation of Murnaghan [1944]

Pip, T)

KoiT)

K'oiT)

(6)

where fir = Ko{T) . This follows from the assumption

K{P, T) = KoiT) + K!,iT)P

and is, experimentally, nearly indistinguishable

from (1) over a considerable range of compression.

This fact may be understood on the basis of the

uniqueness of the Taylor expansion. Any two

functions /i(y) and /2(y) that are constructed so as

to satisfy the same boundary conditions

J/o dy"

for n = 0, 1 ... A/ will have the same Taylor

expansion about yo, through terms in y'^', and will

differ only in the higher-order terms. The B-M and

the Murnaghan equations represent two such

functions, with yo=l, and T not entering the

problem. They are both four-parameter equations

for the free energy F, with one of these parameters

explicitly set to zero, leaving only po, Ko, and

apparent. Since the M + 1 = 4 term of the expansion in which they first begin to differ is

81

Ko

Po L KoK'o + K'o — IK'o +

143

it is clear that the B-M assumption on K" is

The Murnaghan equation, of course, assumes

K'o=0.

(8)

(9)

For a typical oxide, /(^o = 4, Ko = 1,000 kbar, and the

RHS of (8) is 3.9x10-3 kbar-i. This number is sufficiently close to zero to insure that the two

forms agree closely over a wide range of y.

It is of interest to note that the RHS of (8) is not

zero for any real Kd, and reaches its extremum at

K() = 3.5. It seems to be a remarkable coincidence

that solids should have values of Ko so close to this

extremum value. An examination of the derivation of

eq (8) shows that this number, 7/2, has its ultimate source in the fact that the number of dimensions in

physical space is 3. We find it very interesting that

typical values of K'^ are close to 3.5 and not some

other number, like 10. In any case, this situation

helps explain why it is that the series (4) is so quickly

convergent.

3. Thermal Equations of State

This class is most often motivated by considerations of the theory of lattice dynamics. In this theory,

the lattice potential energy is expanded in a Taylor

series about the equilibrium lattice configuration.

Most, if not all, observed phenomena are explainable in principle if the series is terminated at the

fourth-order terms:

(J) = $0 + <J)j + $2 + $3 + (10)

where

k terms

V is the hxh order coupling parameter,

1 ,1, ITl and <PjU, .

and q^i i?, the ith component of the displacement

from equilibrium of the atom in the /ath position of

the unit cell which is located by the lattice vector

m. The temperature is introduced with the assumption of thermal equilibrium, and the free energy can

then be written down. It is

F(p, r) = cDo(p) + ;^ [i^a)(k, \)

k. A + A:rin (l-e-«-(ka)/Ar)]

+ F;{T)+F,iT) (11)

Here, <I>o is the energy of assembling the (static)

lattice to its equilibrium configuration; aj(k, X) is a

vibrational eigenfrequency, a function of the wave

vector k and the polarization; F3 and F4 are complicated functions of temperature only, which are given explicitly by Leibfried and Ludwig [1961]. At

high temperature, the energy enters the classical

regime, and the equations simplify. We will content

ourselves here with this restriction to temperatures

greater then 0d, the Debye temperature.

F=$o + Ar^ In

h(o{Vi, \)

kf

(Dkk'k'

XX X

{o,0)'oi"Y

k k k ^ '

X

16A^5

2(1) — 0)" CO

OJ + fa) — (U OJ + faJ + OJ %k

8yV5^^(coa.') (12)

210
This equation is the high-temperature form of equations 10.2 and 10.4 of Leibfried and Ludwig [1961].

is the number of elementary cells; s is the number

kk'k" ^ kk'

of atoms in a cell, ^^^/y/ ^"d describe,

respectively, the third- and fourth-order derivatives

of the lattice potential energy, and are defined

precisely by Leibfried and Ludwig. The frequencies

0) depend on the lattice configuration through k;

insofar as this configuration can be described by

the single parameter p, we can say that to = (o{p).

However, the frequencies appearing in F3 and F4

of eqs (11) and (12) must be evaluated at a fixed

density because of the presence of

kk k"

kk'X"

4 1^1^' and cD^^, .

This situation can be briefly summarized by the

statement that the anharmonic high T approximation

is

F(p, T) = A{T) + B{p) + C{p)T + D{p)r^ (13)

In the harmonic approximation, O3 and <1>4 are zero, implying that B{p) is quadratic in the strain, C{p)

is constant, and D{p) is zero. In the fourth-order

anharmonic approximation (which we shall call

the anharmonic approximation) and $4 are nonzero, but the next terms (<I>5 and are zero, implying that B{p} is quartic, C(p) is quadratic,

and D{p) = Z) is constant.

The anharmonic equation of state is, from (5),

Pip, T) =

+

1 (14)

(14a)

where

and

7i =

d In o), d\n p

€i = ho)i

1 1 2

gftmilk-T— I

The alternative assumption that Fyih is a function

of T only yields the Hildebrand equation [Hildebrand, 1931]

P(p,T)=-^'+aKT (14c)

where a = —\j p{d pldT)i> is the thermal expansivity.

These two equations were discussed by Fumi and

Tosi [1962]. At high temperatures they converge

together, as the first assumption becomes irrelevant,

the second is automatically fulfilled, and the pres- sure becomes

P(p. T) =-^+^^3NskT dV V = B'{p)+C'ip)T (14d)

where 7(p) = (l/3A^s)S7i and C'{p)=aK. This

equation is equivalent to

daK\

dT J,

(15)

and this is a simple statement of the anharmonic

high T approximation.

4. A Coherent Framework

In the present context, it is clear that the dif- ference between the isothermal B — M equation (1)

and the thermal equations (13) is merely one of

viewpoint. We may expand F(p, 7") about a point

po, 6 corresponding to zero pressure and some finite

temperature 6 in many equivalent ways. Ut = T—6

Fip,T) = [Foo + Foot +Foo^+ . . .]+g{t)

+ [F^o*+n*t+n*2 + -1 2"

^2 ^3

I r r *** 1 17   _i_ F** * - I- 1 + 1*^00 +^00 t + '^oo 2 "6

+ . . . (16a)

are the Griineisen parameter and the energy, re- spectively, of the ith normal mode of vibration.

The further assumption that all the y, are equal

leads to the Mie-Griineisen equation [Griineisen,

1926].

Pip, T)

d<^0 Fyib

dV

^

V

where Fvib^^^,.

(14b)

= [Foo+FoV+nr|-+^o*o**|-+ • • .]

+ [Foo+FoV+^orY+n*o**f + • •]«

+ [Foo + Fo*oe + Fory + n*o**f

+ . . .]|+ . . .+git) (16b)

211
where the symbol ' means (dldt)^, the superscript * means {dld€)(, and the subscript oo indicates evaluation at P — e — O, t = 0. The function g{t) contains

the nonpolynomial terms in t that are necessary to obtain the correct specific heat. The viewpoint in

(16a) is clearly that of the B — M equation; the coefficient of e will vanish when the boundary conditions at ^ = 0 are invoked, and the part independent

of e is one to which our attention need not be explicitly directed in the derivation of an isothermal

equation of state. But the viewpoint in (16b) is not that of the an- harmonic theory. This is because the strains, as defined, are functions of T as well as p, and thus the

coefficients of the various powers of t are not of the

proper form. What is needed is a recognition of the

fact that the free energy depends on the total strain

from a fixed initial state. This strain, containing both

isobaric and isothermal compression is clearly

(x-'/3-l) (17)

where x = pipoo and poo is the density at /* = 0, f = 0. Then we can expand F in terms of e and t in an

equally valid and more instructive manner:

F{p,T) = [Foo + Foot + Fooj+ . . .]+g{t)

+ [Ft+Fttt + 00 00

+ [Ftt + Ftn +

00 00

+ [Fttt +

00

+ [Ftttt+

00 24 (18a)

= [Foo + Fte + Ftt^ + Fttt^ 00

00 24

00 2 " oo' 5

] + [Foo + Fte

g(t) =-3Nsk^{t + e) \n(l+^'j-i (19a)

The expansions in (18) were truncated in accordance

with the approximations of the fourth-order an- harmonic theory, as noted above.

It is clearly possible to derive (18) from (16) by

expanding e in terms of e and t. But the degree of

the expansion will be necessarily different for each

term. In other words, each y(p, T) appearing in

(1) must be handled differently. We may conclude

that the B — M equation, in the form of (1) and (4), is not consistent with the anharmonic theory, or with its results, equations 14. However, in the

modified form implied by equation 17, it clearly is consistent, except for the term in e"*. This term wiU

contribute to K at high pressure and to {dKldT)p at zero pressure.

The coefficients in (18) are defined experimentally

by

F00 = 0

Foo = 0

Ft = 0

00

Ftt — 9

00 Poo

Fttt=-27^ (^:;o-4) 00

Ftttt

00

Poo

= -27 ^00 {

Poo \

00^00 Koo + K,00

ll9b)

Ft:

00 Poo

aooKoo Ftt=-9 >00'

00 Poo

Foo [3 Nsk — Cyoo]

u

"^oo+3j

00

+ Ftt^ + 00 2.]f+[Foo + 1-

2

Some of the quantities needed in the next (sixthorder) approximation are

+ , +git)

(18b)

Fm= -27 [Coo- (8oo-3)(/<:^o-4)]

Poo

where the symbol • means (bldt)p, the superscript t means {dlde)t, and the subscript oo indicates evaluation at t = 0, e = 0. The anharmonic theory is in- voked to give the nonpolynomial term in t; from

(12) it is clearly

Ft=-

00 daK

Poo \ dT Jp

._2_oo—00 [/^^^—28oo+/3oo]

Poo

(19c)

212
f-ff = — 18 Q'no^""

00 Poo

( KL ~ ^) (Soo + ^

poo ~fj

This inequality is not demonstrable, in general,

except for very small values of e.

1 + o

( Son ^oo) - ^00 -

00*^00 \dT' / I'

49

18

where the derivatives are evaluated, of course, at the 00 point. 8 is the anharmonic parameter

° aK\dT)p

and

/3

1 /da

a- \dT/p

^ l d'K{P,T)

a dPdT

are introduced here to simplify the notation.

These coefficients are complicated for two

reasons: the pressure is a volume derivative of the

free energy rather than a strain derivative; and the

values of K and a are such that in the laboratory we

always measure {dldT)p instead of {dldT)p. It is of

interest to note that because of the sums in the last

three equations of (19c), knowledge of {d-K)ldT-)p

is of limited usefulness without similar knowledge of

K{daldT)p. Both these terms are needed if the

anharmonic theory is applied as in this context.

The fourth-order anharmonic high T approximation is that terms in Fttt, /'t,Ftt, etc., are small

00 00 00 compared to all other terms. For example.

00 2

< < FUe00 (20)

or

(/^oo~28oo + /3oo) aoot < < 1

is actually an approximate statement of eq (15),

evaluated at the oo point. Of course, this will be true

for small enough values of t, and despite the large uncertainties in {dajdT)p, it seems that this inequahty

will hold perhaps to i ~ 1 ,000°.

However, (20) is not sufficient to insure the adequacy of the anharmonic high T approximation. It must be shown that each term neglected is much

smaller than each term retained, requiring, for

example,

or

00 2

< <

00 2

(21a)

(/^oo~Soo + jSoo)a;oor | < < 3 00

i5 5^ ooo— 2 |e

(21b)

5. Thermal Equations From the Isothermal Viewpoint

The expansion in (18) gives new insight into the

quantities appearing in the thermal equations (14).

The equation of state which follows from (18) via

(5) is, in the anharmonic high T approximation,

P(p, r) =1X00 (;C^/3 _ ^5/3) [1 (^2/3 _ 1)

1 SVoo 00 , 3!

(22)

l + 3e Soo-K' +^

Comparison with (14d) gives an expression for the

Griineisen parameter, as a function of density:

y(e)=roo(l-2e) l + 3e SOO Kqq +—

(23)

where 700= {VaK)(tolSsR. This expression for 7

contains no assumptions beyond those already implicit in the Mie-Grlineisen equation (14d), except

that the strain as defined in (17) is the appropriate

measure of the displacements in (10). In particular,

all questions concerning the number and weighting

of mode gammas have been avoided; the answers

are contained within the measured values of aoo,

etc. Whenever some other expression for 7(p), for

example, that of Dugdale and MacDonald [1953], is used, it must converge to (23) in the high-temperature regime. To the extent that it does not so converge, one may conclude that the additional

assumptions it contains are not verified.

Different materials exhibit a large variety of behavior of 7(p). Some of the possibilities are illus- trated in figure 1 for several minerals of possible

importance in the earth. It is seen that for some materials 7 decreases rapidly, and goes through zero at compressions found in the lower mantle. This is remarkable behavior, and raises well-known problems with the thermal expansivity and the sign of the

temperature changes accompanying adiabatic compression. The present work adds to our belief that

such problems are real and must be seriously faced.

6. Thermal Restrictions on Isothermal

Equations

While the demonstration of (20) and (21) is suffi- cient to establish the adequacy of the anharmonic

high T approximation, it is for some purposes more

convenient to consider the exact statement of the

213
approximation

0. (15)

In this section, we consider (15) as a property of

thermal equations in the anharmonic high T approximation that must be possessed by isothermal

equations used in the same context. We discuss the

Table 1. Mineral parameters

Parameter Polycrystal

AI2O3

Single crystal

MgO

Polycrystal

Mg2SiO^

Soo *5.06 *3.54 t4.88

t3.99 $4.52 W.9

6oo ~ ^00 + 1 2.07 0.02 0.98

/3oo *2.24 *2.85 t3.49

7oo 1.27 1.46 0.77 e 1042 K 946 K 650 K

*Soga and Anderson [1966]. tSoga and Anderson [1967]. tAnderson and Liebermann [1967].

1.5

2_

0.0

- 1.0

•2.0

1 1

^

MgO

^Mg^SiO^

AI203

1 11 0.0 -.2 -.3

Figure 1. Grilneisen parameter as a function of strain, from

eq (27), using the mineral parameters of table 1.

conditions necessary for the inclusion of (15) in two

generalized isothermal forms of the equations of

state. Consider the form

M

P{p.T) = ^ci{T)y"i (24)

of which (1) is an example. The a here are experimental quantities fixed by the boundary conditions

at y= 1. The n, are independent of T, and the only

restriction on M is that it be finite. If such an equation is to meet the anharmonic high T condition

(15), it must be that

2 [ci+ {2(XoCi + aoCi)ni + Cialnf]y"i = 0. (25)

i Since the y"' are linearly independent, each

bracket must equal zero independently, giving M

algebraic equations which can be solved explicitly

for the n, in terms of a, ao, and their derivatives. It

is obvious that any rational m (e.g., ni — 5/3) can only

meet this condition approximately, at least in

macroscopic physics. This approximation is separate

from, and additional to, the approximation in (21).

But it is not possible, under present limitations, to state whether or not it is serious.

However, it is of interest to know whether the

condition (15) can be met exactly by any equation of

state. Consider the form

P{p,T)=K,{T)f{y) (26)

of which the Murnaghan equation (6) is an example

(provided K'^ is independent of T). The secondorder B-M eauation (with ^= 0) is another example.

Some of the properties of this form are discussed by

Birch [1968]. The only equation of the form (26)

which meets the anharmonic high T condition is

where

ni — /?2

(y'U — yn2) (27)

K' n,-^[l+ Vl-/?o(r)]

with

Ro{T) -

The Murnaghan equation (6) is a special case of

(27) corresponding to RoiT) =0. Therefore, one may

use the M-G equation in conjunction with the

Murnaghan equation only after demonstrating that

The demonstration is not generally possible.

Soga and Anderson [1967] measured Ks at zero pres214
sure and high temperature for the polycrystalHne

oxides MgO, AlzOy, and Mg2Si04. This can be con- verted from the adiabatic modulus to the isothermal

modulus using

K = Ksl{l + Tay).

It is then found that the curvature d'^KoldT^ is so small as to be unmeasurable, being equal to zero within a fourth of a standard deviation for all materials. However, if the standard deviation of

d^KoldT'^ from its mean is taken as an upper bound

for d^KoldT^, then the quantity Ro may, because of

the factor l/Koao, be quite large, and one therefore

cannot conclude that Much more precise

measurements of the temperature dependence of

Ko will be needed before it can be shown that

7. Internal Consistency in the Q-H

Approximation

In applying the Mie-Griineisen equation of state,

various assumptions are necessary, and it is important that these be made in a self-consistent

way. In particular, we wish to discuss here the relations between y and $0.

When the atomic interaction is unknown, then

both y and $0 may be chosen independently.

But where the interaction is known, both 7 and <I>o are thereby determined. For example. Decker

[1965] in determining a numerical equation of state

for NaCl assumes an interaction with Coulombic,

dipole-dipole, dipole-quadripole, and exponential

repulsion terms. Since all these are central forces,

he obtains an expression for $0 (which he denotes

by <I>) which involves only the lattice parameter

r. He then assumes

7 = yo + A. r— ro

ro

for the volume dependence of y, where

aK

70 =

pCi

1.59±.16

where S' indicates summation over aU lattice points

except h = 0, v = ix. is the mass of atom /u,. Where the interaction depends only on the distance RfJiv between atoms, the potential is

and (28) becomes

(29)

where V- is the Laplacian operator. Griineisen's

parameter is

1 1 „h 3

(30)

Where the interaction (^^^ is specified, it is easy to compute these functions. For the potential used by

Decker [1965]

0)-

+

6r» 6r'o

b /M++Mp- \ M+M- 1-2 - I e

-rip

p-+M+ \ r

2b. 1-2^ ] e-'-^l^~ (31)

and

y=^=

36w2

^ r^o ^ p' M

+

8C* , lOD* , 6b M++M

plM,

b-12

+M^ V r p)

2 + 2^-^'|e-'-v^/p+

pi.M2 + 2

rV2

p- rV2

rVl P

g-r^~>lp- (32)

and ro = R{P=0). A. is empirically chosen to be

2.55.

However, 7 is the logarithmic derivative of

0)^, and (o- is determined by the atomic interaction. From Leibfried and Ludwig [1961]

1 3sN k,x

^00

3^ h

^ hO

<t>vix (28)

Here

C* =30

Z)*=56

L \M- M+

d , d++

M_ M M- M

and the c's, the c?'s, and the S's are defined by Mayer

[1933]. Using the values for the constants found in

Decker [1965] and in his reference, Mayer [1933], 7

can be calculated numerically, and is shown in

215

385-762 O - 71 - 15
1.5

1.0

1,59 t 2.55Ar/ro (DECKER, 1965)

EQUATION (32)

J_

-.10 -.20

Ar/ To

Figure 2. Griineisen parameter for the NaCl model of Decker

[1965].

figure 2. Also shown is the linear relation of Decker,

and an arrow marking the maximum compression

considered by him.

It is seen that, in the low-compression range, y is nearly linear with Ar/ro, with an initial value of

1.46, and an average slope of 1.70. It is nearly equivalent to a power law ylyo={VIVo)'J with exponent

q ~ 0.4. Decker remarks that, for such behavior

of y, "the fit to the experimental thermal expansion

became quite poor."

Decker's idea was that the equation of state of

NaCl was sufficiently well understood theoretically

that salt might be used as a pressure gauge in ex- periments on other materials. On the basis of the

present remarks, it appears that the fourth-order

anharmonic approximation, as used by Decker,

is not sufficient to accurately describe the behavior

of NaCl. The application of the sixth-order anharmonic treatment, straightforward in this case, might

be expected to yield better agreement.

However, a feature of the central force assumption implies that even this will not be sufficient,

unless an atomic interaction which includes noncentral forces is used. Leibfried and Ludwig

show that, for central forces in cubic crystals, the

inequalities

and

Ci2 C44

dCvy\ ( dC44

dT

<

dT

(33a)

(33b)

are implied for the adiabatic elastic constants

Cij. If these are not satisfied by the data, then noncentral forces must be involved. Such is the case

for NaCl [Barsch and Chang, 1964]. It thus appears

that a theoretical basis for accurately describing

the properties of NaCl must include non-central

forces.

8. Summary

The conclusions reached here are of three sorts; those relating to mutual consistency, those relating

to internal consistency, and those relating to adequacy of approximation. Of the first sort:

1. Because, in eq (1), each yiT) must be treated

differently, we conclude that the Birch-Murnaghan

equation, in the form (1) and (4), is not consistent

with lattice dynamics in the anharmonic high T

approximation, nor with its results, the MieGriineisen and Hildebrand equations.

2. However, with the modification implied by

eq (17), the B-M equation becomes identical to the

anharmonic high T equations, provided only that

the term of (18) in e'* can be shown to be negligible.

The demonstration of this negligibility is beyond

present experimental techniques.

3. The Murnaghan equation is consistent with

the anharmonic high T equations, provided only

that Ro(T), eq (25), can be shown to be negligible

compared to unity. The demonstration of this

negligibility is beyond present experimental

techniques.

4. Where two equations cannot be definitely

proven consistent, they should not be used in the

same calculation.

216
Of the second sort:

5. The Griineisen parameter yie) is defined in

terms of boundary conditions by the expression in

(23). Any other assumption on y(p), such as that of

Slater [1939J or of Dugdale and MacDonald [1953],

must converge to this at high temperature.

6. Once an assumption has been made con- cerning the interaction between atoms in a crystal,

then the lattice dynamics is completely determined,

and only the degree of approximation remains to be

chosen. In particular, in the fourth-order approximation, when the interaction is assumed, (f>o, (o'f

,

and y are all determined.

Of the third sort:

7. The data of NaCl are such as to require non central forces for an adequate theoretical

explanation.

9. Acknowledgments

We appreciate helpful criticism from Francis

Birch, and access to the unreduced data of Naohiro

Soga. We thank the Journal of Geophysical Research for granting permission to reproduce substantial portions of Thomsen and Anderson [1969]

in this review article.

This research was supported by the National

Aeronautics and Space Administration under contract NSG-445, and by the National Science Foundation under grant GA-1208, and by the Air Force

Office of Scientific Research under contract

F44620-68-C-0079.

10. References

Alder, B. J., Is the mantle soluble in the core, J. Geophys.

Res., 71, No. 20, 4973 (1966). Anderson, D. L., and Toksiiz, M. N., Spherical waves on a spheroidal earth, J. Geophys. Res. 68, No. 11, 3483 (1963). Anderson, O. L., and Liebermann, R. C., Elastic constants of oxide compounds used to estimate the properties of the earth's

interior, in Proc. NATO Advanced Study Institute: on the

[ Application of Modem Physics to the Earth and Planetary

I Interiors, March 29-April 4, 1967, Newcastle upon Tyne,

' John Wiley & Sons, 1969.

Anderson, O. L., Schreiber, F., Liebermann, R. C., and Soga,

N., Some elastic constant data on minerals relevant to geo- physics. Rev. of Geophys., 6, No. 4, 491 (1968). Barsch, G. R. , and Chang, Z. P. , Pressure derivatives of the elastic constants for cubic symmetry — II. Numerical results, Phys.

Stat. Sol. 19, 139 (1967). Birch, F., The eftect of pressure upon the elastic parameters

of isotropic solids, J. Appl. Phys. 9, No. 4, 279 (1938). Birch, F., Finite elastic strain of cubic crystals, Phys.. Rev. 71,

809 (1947). Birch, F., Elasticity and constitution of the earth's interior,

J. Geophys. Res. 57, No. 2, 227 (1952). Birch, F., Thermal expansion at high pressures, J. Geophys.

Res. 73, No. 2, 817 (1968). Decker, D. L., Equation of state of NaCl, J. Appl. Phys. 36,

No. 1, 157 (1965). Dugdale, J. S., and MacDonald, D. K. C, The thermal expansion

of solids, Phys. Rev. 89, No. 4, 832 (1953). Fumi, F. G., and Tosi, M. P., On the Mie-Griineisen and Hilde- brand approximations, J. Phys. Chem. Solids 23 , 395 (1962). Griineisen, F., Zustand des festen Kiirpers, in Handbuch der

Physik 10, No. 22 (Springr-VeHag, Berlin, 1926). Also, in translation, NASA Republication RE 2-18-59W.

Hildebrand, J. H., Gitterenergien vom thermodynamischem

Standpunkt aus, Z. Physik 67, 127 (1931). Johnson, L. P., Array measurements of P velocities in the upper

mantle, J. Geophys. Res. 72, No. 24, 6309 (1967). Leibfried, G., and Ludwig, W., Theory of anharmonic effects in crystals, Solid State Physics 12 (Academic Press, New York,

1961).

McQueen, R. G., Marsh, S. P., and Fritz, J. N., The Hugoniot

equation of state of twelve rocks, J. Geophys. Res. 72, No. 20,

4999 (1967). MacDonald, G. J. F., Radioactivity of the earth, J. Geophys. Res. 69, No, 4, 2933 (1964). Mayer, J. E., Van der Waals potential in the alkali halides, J. Chem. Phys. 1, 270 (1933). Mumaghan, F. D., Finite deformations of an elastic solid. Am J. Math. 59, 235 (1937). Mumaghan, F. D., The compressibility of media under extreme

pressures, Proc. Natl. Acad. Sci. 30, 244 (1944). Schreiber, F., and Anderson, O. L., Pressure derivatives of the

sound velocities of polycrystalline forsterite, with 6% porosity,

J. Geophys. Res. 72, No. 2, 762 (1967). Slater, J. C, Introduction to Chemical Physics (McGraw-Hill,

New York, 1939). Soga, N., and Anderson, O. L., High-temperature elastic properties of polycrystalline MgO and AI2O3, J. Am. Ceram. Soc. 49, No. 7, 355 (1966). Soga, N., and Anderson, O. L., High-temperature elasticity and

expansivity of forsterite and steatite, J. Am. Ceram. Soc. 50,

No. 5, 239 (1967). Takeuchi, H., and Kanamori, H., Equations of state of matter from shock-wave experiments, J. Geophys. Res. 71, No. 16,

3985 (1966). Thomsen, L., and Anderson, O. L., On the high-temperature

equation of state of solids, J. Geophys. Res. 74, No. 4, 981

(1969).

217
I
session VI

Chairman: F. R. BoYD, Jr. Geophysical Laboratory

Carnegie Institution of Washington

The Solid-Liquid Phase Line in Cu*

R. G. McQueen, W. J. Carter, J. N. Frilz, and S. P. Marsh

University of California, Los Alamos Scientific Laboratory, Los Alamos, New Mexico 87544

1. Introduction

It is well known that the increase in temperature

due to shock loading is great enough to melt many

solids with the shock strengths presently attainable

in the laboratory. Using Hugoniot data and zero- pressure data it is possible to calculate with some

confidence the shock strength required to melt

a material on the pressure release wave. In a previous paper, [1],^ these calculations ignored details

of the true location of the melting phase line, since no information was available for this at high pressure, and in effect actually forced the phase line

to be located at a fixed volume in the P-V plane.

However, these calculations did conserve energy

around the path determined by the Hugoniot, the

pressure release isentrope and the zero pressure

isobar. Subsequent measurements of the residual

temperature of shock loaded Cu by J. Taylor [2]

gave substantial agreement with these calculations.

However gratifying these experiments were, the

question of where the melting phase line at high

pressure was located remained unanswered.

Aside from the basic interest in melting phenomena, the location of the phase line and the

effect that melting might have on materials such as Cu when used as a shock-wave standard indicated

that additional work should be done on this problem.

2. Experimental Program

Direct measurements to determine the location

of the melting phase line at high temperatures and

pressures appear to be either impossible or at best

quite difficult. If the thermodynamics of the transition were of the correct type, a small break might be

observed in the Hugoniot where melting f^rst begins.

However, as for most normally melting materials,

the discontinuities that exist in the temperaturevolume isobars appear only as a change in slope in

the energy-volume isobars. Since it is the latter

*Work done under the auspices of the U.S. Atomic Energy Commission.

' Figures in brackets indicate the literature references at the end of this paper. Paper presented at the Symposium on Accurate Characterization

of the High-Pressure Environment, held at the National Bureau

of Standards, Gdithersburg, Md., October 14-18, 1968.

variables that are observed in shock-wave studies,

it is not surprising that Hugoniot data, even in the

sensitive Ug — Up plots, go smoothly through regions

where melting is beheved to occur. An alternative

approach is to calculate the phase line from known

zero-pressure data and the equation of state loci of

the solid and liquid phases. Good equations of state

exist for many solids and for some liquids but not

for both phases of the same material. We have made

an effort to obtain the Hugoniot equation of state

for materials in the liquid phase that are normally

solid. This was done by either preheating samples in

electric furnaces or by preshocking the material

under investigation by a shock strong enough to cause melting. Both techniques not only entertain

additional experimental difficulties but also suffer

from unfavorable precision requirements. An al- ternative approach is to measure the Hugoniot of

a material at low initial density.

It is well known that shock-loading porous materials cause the internal energy at a given shock

pressure to be substantially greater than that obP

Figure 1. Pressure versus volume schematic to illustrate how

the use of porous materials increases the internal energy on

shock loading.

The area designated by the vertical Hnes represents the increase in internal energy due to shock loading at a low initial density.

219
LIQUIOUS , Q

SOLiOUS y ±y

/ +/ / V Po = / 0 " = 8.930

7.858 o7 / 7.323 0 y

" = 6.340

X " = 5.715 /+ p'

1 ^ 1 1

1

Up(km/sec)

Figure 2. Shock velocity-particle velocity data on Cu with

various initial densities.

The curve through the density 8.93 g/cm" data has been drawn on the basis of the melted phase calculations. The onset and completion of meUing in the crystal density Hugoniot are indicated. All other curves were obtained from the p(i= 7.323 g/cm'

liquid Hugoniot and a 70 = 2.6.

tained on nonporous materials. The greater the

porosity the greater will be the shock heating. This

effect is demonstrated schematically in figure 1. In the past, experiments using porous specimens were used to obtain the high-pressure Griineisen

parameter or to check values of the Griineisen function being used in equation-of-state calculations. If

the porosity is large enough and the extra energy

is sufficient to cause the high-pressure state to be

in the liquid phase, one can use these data in an

attempt to establish the Hugoniot of the material

centered in the liquid state at a relatively high tem8.2 -

7.6

(_) \ 7.0 o

E

S 6.4

Q.

5.8

5.2

I I

COPPER

^ 4 ^vov-cp-o—OD-o—OD-o—ca—o-OD—o—(D~o—on—

c

T^356»K

Ji_ I

T, =2850''K

0 400 800 1200 1600

T(°K)

2000 2400 2800

Figure 3. Specific heat as a function of temperature for Cu.

These data are from the compilation of ref. [4] and were obtained from the following original data sources:

Metals Handbook (American Society for Metals, Novelty, Ohio, 1961), 8th rev. ed.

Stull, D. R., and Sinke, G. C, Thermodynamic Properties of the Elements (American Chemical Society, Washington, 1956).

Stull, D. R., and Sinke, G. C, The Thermodynamic Properties of the Elements in Their Standard States (Dow Chemical Co., Midland, Mich., 1955). Smithsonian Physical Tables, ed. W. E. Forsythe (Smithsonian Institution, Washington, 1956), 9th rev. ed. Metals Reference Book, ed. C. J. Smithells (interscience. New York, 1955). Vol. II, 2d ed. Rare Metals Handbook, ed. C. A. Hampel (Reinhold, New York, 1961), 2d ed. Hultgren, R., Specific Heat of Metals and Alloys, AFOSR Tech. Note 4 (June, 1960). Handbook of Chemistry, ed. N. A. Lange (McGraw-Hill, New York, 1961), 10th ed. Handbook of Chemistry and Physics (Chemical Rubber Publ. Co., Cleveland, 1949), 31st ed. Materials in Design Engineering, 50(5)141, Mid-October Issues, 1959. Kubaschewski, O., and Evans, E. L., Metallurgical Thermochemistry (Pergamon, Oxford, 1958), 3d ed. Mechanical Properties of Metals and Alloys, National Bureau of Standards Circular C447 (Washington, 1943). Engineering Materials Handbook, ed. C. L. Mantell (McGraw-Hill, New York, 1958). Lucks, C. F., and Deem, H. W., Thermal Properties of Thirteen Metals, ASTM Special Tech. Publ. 227 (1958). Butts, Allison, Copper; The Metal and Its Alloys (Reinhold, New York, 1954). Fieldhouse, I. B., Hedge, J. C, Lang, J. I., and Waterman, T. E., Measurements of Thermal Properties 1, WADC Tech. Rept. 55-495 (1956). Fieldhouse, I. B., Hedge, J. C, Lang, J. I., and Waterman, T. E., Measurements of Thermal Properties II, WADC Tech. Rept. 55-495 (1956). Seibel, R. D., and Mason, G. L., Thermal Properties of High Temperature Materials, WADC Tech. Rept. 57-468 (1958). Lucks, C. F., and Deem, H. W., Thermal Conductivities, Heat Capacities, and Linear Thermal Expansion of Five Materials, WADC

Tech. Rept. 55-496 (1955). Makin, S. M., Standring, J., and Hunter, P. M., Determination of the Coefficients of Linear Thermal Expansion at T^C, R and DB(C)TT-45 (1953).

Bell, I. P., Fast Reactor-Physical Properties of Materials of Construction, Culcheth Laboratory FRDC/P-112 (Lancashire, England, 1955). Johnson, V. J., A Compilation of Thermophysical Properties of Cryogenic Materials, WADC Progress Rept. 6 (1958). Goldsmith, A., Waterman, T. E., and Hirschhorn, H. J., Thermophysical Properties of Solid Materials, Vol 1, WADC Tech. Rept. 58-376 (August, 1960).

220
22.0

20.0

U)o

"i" 18.0

16.0

14.0

1 1 1— 1 1 1

COPPER

1 1 1 1 1 1 1

1

a «

0 a

-

0

-

* a

0

Ci 1 1 1 1 ' 1

T„=I356'K

1 1 1 1 1 1 1^1

200 400 600 800

T (°K)

1000 1200 1400

Figure 4. Linear thermal expansion coefficients as a function of temperature.

See ref. [4] for identification of the symbols.

perature. For the present application we have ex- amined the shock-wave data on Cu measured in

this laboratory. These data were obtained using the

flash-gap impedance match technique with deoxidized Cu as a standard [3] and are summarized in

figure 2. In addition to the shock-wave data it is necessary to know the thermodynamic equation

of state along the P—Q isotherm. For this a compilation of specific heat data, figure 3, and thermal

expansion data, figure 4, were used to determine the

internal energy, E, and temperature, T, versus volume, V, given in figure 5 (see ref. [4]). Hugoniot data were obtained for Cu at average

initial densities of 8.93, 7.858, 7.323, 6.340, and

5.717 g/cm^. It is seen from figure 5 that the density

7.323 g/cm^ corresponds to a liquid density at 7"= 2370 K, which is more or less in the center of

the liquid phase region. It was this set of Hugoniot

data that was used to determine the liquid Hugoniot

equation of state centered at T= 2370 K.

These data were fitted with a smooth curve in

the Ug — Up plane. Selected points on this curve were then recentered at this density and an initial

internal energy corresponding to that volume as given in figure 5. The transformation is effected

through the Hugoniot energy equation centered at Po-0,

3000

2 600

2200

I 800 -

», I 400

I 000 -

600

200

E = Eo + P{Vo-V)l2

and the usual assumption that

py=-dPldE)r

(1)

(2)

is a function of volume only. At a given volume, V,

or density, p, the pressure increase, AP, due to

.110

Figure 5. Energy and temperature asfunctions ofvolumefor Cu.

These curves are required to establish initial conditions for the calculations were

derived from the data presented in figures 3 and 4.

the assumed increase in initial thermal energy,

Af'o, is given by

AP = pyAEol{l-py{Vo-V)l2). (3)

221
8.0

7.0

6.0

• ^=2.44

^=2.60

5.0 -

4.0

3.01 1 1.0 2,0 3.0 Up(knn/sec

)

Figure 6. Two u^ — Up transforms of the po= 7.323 glcm^ data

centered at T = 2370 °K using yn = 2.44 and ya = 2.6

In the present study py was assumed to be constant. This gives the P — V Hugoniot centered at the high-temperature point which can be transformed to the Ug — Up Hugoniot through the usual

Hugoniot equations

and

us=Vo[{P-Po)iVo-v)yi-' (4)

(5)

The Us — Up Hugoniots so determined were found

to be quite linear in the region covered by the

data. Two Hugoniots obtained by this technique

using two values for py are given in figure 6. The

Hugoniot fits obtained from these curves are listed

in table 1, where

Us= Co + SUp. (6)

Although both sets fit the data adequately, it is apparent that the zero pressure sound velocities

and slopes are very dependent upon the value of

yo used.

The value yo = 2.44 comes from the room-temperature thermodynamic 7 = 2.0 and th° assumption

that (py)29o K= (P7)237o K- This relationship has

been shown to be reasonably compatible with the

crystal density Hugoniot and the one centered at

3.5

3.4

E 3-3

3.2

3.1

3.0

O GITIS a MIKHAILOV

CALCULATED

MELTING

POINT

_L ±

1200 14 0 0 1600 1800 2 000 2 2 0 0 24 00

T CK)

Figure 7. Sound velocity versus temperature for liquid Cu.

The dots, O, are the data for Cu as reported by Gitis and Mikhailov [5]. The two +'s

at T= 2370 "K are the Hugoniot intercepts Usted in table 1. The two +"s at T= 1356 °K

were calculated from the equation of state hsted in table 1.

Table I.

Po To Co s

(glcm^) (kmis) 7.323 2.6 3.25 1.49 7.323 2.44 3.00 1.54

pn = 7.32. However, close examination shows that

when this approximation is used the calculated

Hugoniot for po = 7.32 does not give as precise

agreement with the porous shock data as might be

desired. This is most noticeable in the lower and

upper pressure regions. Of course, there is no reason

to believe that such a simple relationship (py^ constant ) should hold across a melting phase boundary.

This value of y does give a zero pressure intercept

in the Ug — u,, plane that is in good agreement with

the sound velocity at r=2370 K extrapolated from

the data presented by Gitas and Mikhailov [5], but

the calculated sound velocity using this y and the

Hugoniot fit is too low in the vicinity of the melting

point (see fig. 7). The sound velocity, c, at any pressure F, and

volume, V, are computed from the Hugoniot and

Griineisen function through the relationship

1/2 fi dv

• py ( V, -V)I2] + V^pyiP-Pn ) 12.

(7)

222
Equation (7) can be obtained by considering a small

isentropic expansion at the point {P, V) and equating

the ratio of the pressure and energy differences

with the Griineisen parameter. If eq (6) is used to define the Hugoniot then

Phanc

\y,-s{v,-v)Y

dP, „ [^0 + 5(^0-^)] = — Co

(8)

dv ^" [Vo-s{Vo-v)y

which when substituted in eq (7) gives

ipcy'-\cWo+{s-yo}(Vo-V)]l[Vo-s(Vo-V)f+(py)P.

(10)

Using the equation of state for y=2.6 in table 1 and evaluating eq (10) at the melting volume yields a value of c — 3.47 km/s, which is in substantial

agreement with the Russian data in the temperature

region. If one believes that the temperatures at 7.32 g/cm3 and 8.01 g/cm^ are 2370 K and 1356 K

respectively as read from the graph in figure 5,

then sound velocity/temperature gradients of 0.22

m/s K and 0.29 m/s K are obtained in this temperature range. These slopes are about a factor of two

less than the value quoted by Gitis and Mikhailov.

There is no y compatible with their slope of 0.46

m/s K. Some error is undoubtedly associated with

the data in figure 5 but the thermal expansion data

is certainly not so poor as to explain the discrepancy.

The sound velocity measurements probably entertain larger errors than are quoted in the paper.

3. Calculated Phase Line

The P, V, E equation of state given by the

Hugoniot and the Griineisen function can be ex- tended to a complete thermal equation of state

through use of the thermodynamic identity

dS = CvY+ Cv{pyydV (11)

where Cv is heat capacity at constant volume.

Thus the Gibbs free energy

G = E + PV-TS (12)

can be computed for both the solid and liquid

phases. The solid and liquid phases are coupled at the phase line through the requirement that

'solid'liquid- (13)

Since the difference in internal energies, AEi.-s, and

the densities of the two phases at the melting point

have been measused, it is necessary that the two

equations of state also satisfy these conditions.

This was done by slight adjustments of the values

of Cv used for each phase so that a correct value

for the average value of the thermal expansion was

obtained. The correct initial internal energies are thus determined which are also required to be

compatible with the data given in figure 5. Once

this has been done, the phase line at higher pres- sures is located by satisfying eq. (13).

The results of one calculation are given in figure

8. Because of the thought that ordinary melting is determined by simple order-disorder arrangements

which are reflected by a change in entropy, it was

decided to see how these phase boundaries compared with lines of constant entropy. Hence we have

plotted the two isentropes originating at the phase

boundaries at P= 0. It can be seen that the two sets

of curves are very similar, especially for the liquid

boundary. This tells us immediately that the equation of state currently being used in impedance

match solutions gives satisfactory solutions even in

the high-pressure melted regime. P- u, plots

demonstrating this are given in figure 9, where

release isentropes currently being used are compared with those calculated with the postulated

liquid equation of state. For assumed po"s lines

drawn on the graph there is less than one tenth

percent disagreement in the region of interest, weU

within present experimental accuracy.

The standard room temperature Hugoniot intersects the phase line at a pressure of 1.33 Mbar. On

the basis of equilibrium thermodynamics, we have

computed the Hugoniot locus through the mixed

phase region and into the liquid state. These results are also given in figure 8 and in the Us — Up plane

in figure 2. The points where the Hugoniot crosses

the mixed-phase region are indicated in the graph.

It is only by sighting down the curve that these can be detected. This indicates that it will probably be

very difficult to establish the melting point on the

Hugoniot by detecting kinks in the experimental

data.

These results are comforting in that it suggests

that hydrodynamic calculations used to study the

effects of underground nuclear explosions can give

meaningful results without a detailed description

of the liquid equation of state. Moreover, since the

very low initial-density porous Hugoniot data can

be calculated relatively well, it appears that a detailed description of the vapor phase is also not

required.

4. Comparison of the Phase Line

With the Kraut-Kennedy Relation

Two papers have appeared recently in the literature [6, 7] in which it is reported that, for a wide

range of materials, the melting temperature, Tm,

varies linearly with the room-temperature isothermal

223
2.0

1.5

1.0 -

HUGOMIOT

PHASE BOUNDARIES

ISENTROPE

.0 7 .1 3

Figure 8. Pressure-volume plots of the two calculated phase line boundaries, the isentropes originating

at the melting boundary at P = 0, and the crystal density Hugoniot.

It can be seen that the phase-line boundaries predict a smaller volume change at high pressure for the transition than is indicated by

volume separation of the two isentropes. In some calculations the volume change for the phase change approached zero, a situation that could be construed as the approach to a critical point. In all these calculations the results are suspect in the one Mbar region and above,

since the liquid Hugoniot used for the calculations has been extrapolated far beyond the original data.

Crystal Hugoniot

3 LO -

Figure 9. Pressure-particle velocity curves from 1.75 Mbar and 2.0 Mbar.

The solid curves were calculated from the crystal density Hugoniot and are currently being used in impedance-match solutions. The

dots are selected points on the release isentrope and were calculated from the liquid equation of state. It is apparent that the current solu- tions are adequate.

volume compression {AVjVo)RT— iVo—V)l^o, as

T,„^Tl[l + a{AVIVo)HT]. (14)

Although the physical significance of the phenomena

is obscure, their results are impressive and worth

further investigation, especially with respect to possible uniqueness of the room temperature

isotherm in predicting phase lines. We have already

noted the simultaneity of the phase line with the

boundary isentropes, and while the agreement is close, it is not exact. In general the volume change

224
for the transition decreases more rapidly than the

volume separation of the two isentropes. In the

temperature-volume plane this disagreement is considerably magnified. This can be seen in figure 10

where the temperature and compression, AK/Kp/.,

along the phase line is compared with the temperature vs. compression along the two isentropes.

The Kraut-Kennedy relationship is also compared in this plot. To obtain this plot we calculated the room temperature isotherm in the usual

manner from the crystal density Hugoniot. The

temperature-compression points were then tabulated from the isotherm and phase line at fixed

pressure points. It can be noted that although these

points are fairly linear they do bend downward at large compressions while the plotted isentropes

are concave upward. The Kraut-Kennedy curve

is more linear but suffers from the fact that the

a coefficient is not a priori predictable. Similar

P-T plots are given in figure 11.

5. Comparison of Shapes of the

Mehing Phase Lines with Compressibility Data

In an earher paper [1] we compared the pressure-temperature data along the melting phase line

with calculated pressure-temperature curves along

isentropes. For the few selected cases the slopes

of the isentropes, dPldT\s, were comparable but

smaller than those of the phase lines. In light of

the recent developments we have attempted to make a more extensive study of existing data.

By considering the total differentials of the phase

line and the isentropes at the P—0 melting point

the relationship

dp dP

dV PI. dV s dp dP 1 dp dP dV PI. dV s. pyCv

dT PI. dT s

(15)

between the slopes along the phase line (designated

by PL) and isentropes (designated by S) can be

obtained. Examination of eq (15) shows that if the

P—V slopes are the same then the initial P — T

slopes will also be identical. If the pressure-volume

derivatives are not the same, considerably larger

discrepancies will be found for the pressure-temperature derivatives. For this study it was decided

to compare both pressure-temperature and pressure-volume derivatives.

With dS = 0, eq (11) gives

av

Vo

Figure 10. Temperature versus AV/V,).

The dots rfprcsfiit points calculated from the room .temperature isotherm and phase

line after Kraut and Kennedy. The solid line is simply a linear fit of the low pressure

region. The other two solid lines represent the actual compression along the phase

boundaries as given by the calculations. The dashed hnes are similar quantities ob- tained from the isentropes plotted in figure 8. The initial agreement between the phase

line and isentrope along the liquid boundary is quite good but becomes rather poor at higher temperatures, a manifestation of the more rapid decrease in the size of the volume change on the actual phase line.

1 1 1

1 I r

1.2 -

1200 1400 1600 1800 2000 2200 2400 2600

T (°K)

dP

dV

dP

dT Tmipy). (16)

Equation (15) can then be written as

Figure 11. Pressure versus temperature.

The solid line represents the calculated phase Une and the dashed lines the pres- sure-temperature loci along the two Isentropes. As before the agreement between the phase line and the calculated liquid isentrope is good initially. The phase line crosses

the solid isentrope at about one Mbar. indicating that although the isentropes do not

give the exact location of the phase line they indicate, in this case, that melting does

occur at nearly constant entropy.

225
dp

dV PL

dP

dT

dP

dJ\

+ TCvipyy

PL dPldT\pL-{py)Cx (17)

to find the P — V slopes of the phase line in terms

of other thermodynamic quantities. The quantity

dPjdV\s was computed from eq (10) and the re- lationship

§\=-(»^y'- (18)

All quantities are computed at P = 0 at the melting

point. The product p7= constant, was determined

from room temperature quantities. Results of these

calculations are listed in table 2. From these data it can be seen that the initial

P—V slopes of the isentropes at the phase boundary

are in general smaller than the P-V slopes of the

phase lines by several percent. This results in the

dsentropic slope, dPldT\s, being larger than the

slope of the phase line. The agreement is better

when phase lines are compared with the roomtemperature isentropic compressions. The room

temperature isothermal compressions will be

slightly larger and the zero-Kelvin compressions

slightly smaller than the latter. It would be fortunate

if the agreement were better so that these results

could be extrapolated to other materials with

greater confidence. However, one must keep in

mind that many of the quantities listed in these

tables are of a very dubious quality, and perhaps

the agreement, both good and bad, is fortuitous.

As was noted in the Kraut-Kennedy paper, their

law is an approximation of the equation

Tm = TI, exp [aAF/Fo]. (19)

This of course is the equation for the temperature

along an isentrope. Equation (11) when integrated

along the isentrope gives

r=ro exp [- {py)^v] (20)

where the compression is taken along the isentrope.

From the table it can be seen that eq (20) will give

very nearly the correct initial slope of the phase line,

better in general than that obtained when using the

isothermal compression at the phase line.

Table 2.

Material

"o '^O s ^0

'a) ^P '^M dP/dT|

^ •^PL*

c /c PL' MP ^pl'^o

g/cm km/sec ergs/g°K °K g/cm km/sec kb/°K kb/°K km/sec

Li 0.53 4.65 1. 13 0.81 32. 80 X 10* 453 0.512 4.51 .28 . 300*' 4.32 0. 96 .93 Na 0. 968 2.63 1.22 1. 17 12.40 371 0.951 2.59 . 14 .120** 2.62 1.01 1.00 K 0.86 1.97 1. 18 1.23 7.53 337 0.851 1.96 .078 .078*' 1.96 1.00 1.00 Rb 1.53 1. 13 1.27 1.06 3. 35 312 1.51 1. 12 .057 .069'' 1. 11 0. 99 .98 Cs 1. 826 0. 70 1.57 0.54 2. 18 303 1.844 0.713 .058 .04o' .719 1.01 1.03 Cu 8.93 3.94 1.49 2.00 3.70 1356 8.368 3. 71 .40 .280*^ 3.87 1.04 .98 Ag 10.49 3.23 1.50 2.50 2.34 1234 9.658 3.00 .26 .210*^ 3. 12 1.04 .97 Au 19. 32 2.99 1.56 2. 97 1.30 1336 18.28 2.89 . 37 . 160*^ 3.53 1.22 1. 18 Mg 1.74 4.49 1.26 1.43 10. 00 923 1.640 4.28 .21 .148^ 4.42 1.03 .98 Zn 7. 14 3.01 1.58 2.45 3.69 693 6.98 2.95 .35 .225^ 3. 15 1.07 1.05 Cd 8.64 2.43 1.68 2. 32 2.30 594 8.419 2.36 .33 . 16o' 2.60 1. 10 1.07 Hg 13.54 1.49 2.05 1.96 1.39 234 14. 148 1.64 .09 .183'' 1.80 1. 10 1.21 Al 2.706 5. 17 1.47 2.0 8.63 933 2.540 4.88 .30 .17

^ 5.28 1.08 1.02

In 7.279 2.42 1.54 1.97 2.43 430 7. 190 2.39 .48

e .240 2.49 1.04 1.03

Tl 11. 85 1.86 1.52 2.25 1.30 576 11.66 1.84 . 30 . 160^ 1.95 1.06 1.05

Sn 7.287 2.61 1.49 2. 11 2.27 505 7. 176 2.57 .44 .30o' 2.63 1.02 1.01 Pb 11.34 2.05 1.46 2.77 1.30 601 10.984 2.02 .26 . 12o' 2.28 1. 13 1. 11 Nb 8.59 4.44 1.21 1.47 2.65 2693 8.047 4.25 .34 .200' 4.43 1.04 1.00 Ta 16.65 3.41 1.20 1.60 1.37 3270 15.39 3.26 .29 .200' 3.37 1.03 .99 Pd 11.99 3. 95 1.59 2.26 2.33 1823 11.244 3. 70 .35 .25^ 3.88 1.05 .98

Ft 21.42 3.60 1.54 1 2.40 1.27 2043 20. 174 3.42 .45 .240* 3.70 1.08 1.03

*CpL=V(-dP/dV

a. Rare Metals Handbook, 2nd Ed. , C. A. Hampel, ed. , Relnhold I>ubllshing Corp. , London (1961).

b. H. D. Luedemann and G. C. Kennedy, J. Geophys. Res. 73, 2795 (1968).

c. L. H. Cohen, W. Klement, Jr., and G. C. Kennedy, Phys. Rev. H5, 519 (1966).

d. A. v. Grosse, J. Inorg. Nuc. Chem. 27, 773 (1965).

e. A. Jayaraman, W. Klement, Jr. , R. C. Newton and G. C. Kennedy. J. PhyB. Chem. Solids 24, 7 (1963).

f. Metals Reference Book, Vol. 2, 2nd Ed. , C. J. Smithells, ed. , Intersclence Publishers Inc. , New York (1955).

226
6. References

[11 McQueen, R. G., and Marsh, S. P., J. Appl. Phys. 31, 1253

(1960).

[2] Tavlor. J. W., J. Appl. Phys. 34 , 2727 (1963).

[3] McQueen, R. G., Marsh. S. P., Carter. W. J.. Fritz, J. N., and

Taylor, J. W., High Velocity Impact Phenomena, ed. E. V. Cohen (Academic Press, Inc., New York), in press.

[4) Carter, W. J., Thermodynamic Properties of Seven Metals

at Zero Pressure, LAMS-2640 (1961).

[5] Gitis, M. B., and Mikhailov, I. G., Akusticheskii Zhurnal 11,

434 (1965); Soviet Physics-Acoustics 11, 372 (1966).

[6] Kraut, E. A., and Kennedy, (j. C, Phys. Rev. Letters 16,

608 (1966).

[7J Kraut. E. A., and Kennedy, G. C, Phys. Rev. 151, 668

(1966).

DISCUSSION

M. van Thiel (Lawrence Radiation Laboratory,

University of California, Livermore, California):

The graph of figure 10 indicates that the mixed

phase region increases with increasing temperature.

Would you comment on the significance of this?

Also, in figure 10, would you explain the meaning

of AF?

J. O. Erkman (U.S. Naval Ordnance Laboratory,

White Oak, Maryland): In experiments with porous

material, to what degree do you have a steady shock?

How much error is introduced from this source?

R. Grover {Lawrence Radiation Laboratory,

University of California, Livermore, California):

Is the value of A5 and at transition, and the

agreement between the isentrope and the melting

line, particularly sensitive to the assumption of

a constant value for pyl I have tried to estimate

the changes in gamma on melting, and find values

between, say, 10 percent and 20 percent.

F. R. Boyd, Jr. [Geophysical Laboratory, Carnegie

Institution of Washington, Washington, D.C.):

You mentioned the fact that there was no substance

for which there is a Hugoniot for both the solid

and the liquid. Since there is a Hugoniot for liquid

mercury, would it not be a practical and significant

experiment to freeze mercury and shock it?

AUTHORS' CLOSURE

In reply to van Thiel: It is difficult to judge the

size of the mixed phase region from figure 10.

From figure 8, however, it can be seen that the mixed

phase region decreases with increasing pressure,

or equivalently with increasing temperature. Of

course, if the phase line is indeed an isentrope,

then from the Clapeyron equation it must be this

that is causing the curvature of the phase hne

seen in figures 10 and 11.

In reply to Erkman: We assume that we have a steady shock front. This is a bothersome problem,

particularly at very low pressures, and we do not

know to what extent we fail to realize this condition, nor how much error is introduced. However, at sufficiently high pressures, far above that required

for void collapse, steady state flow should be

attained. Much more work remains to be done on

the shock properties of porous materials.

Replying to Grover: Isentropes are not extremely

sensitive to the functional form of y(,v) , or even to the value of y. However, the position of the phase

line is somewhat more sensitive to y{v), since y

is an important thermodynamic parameter distinguishing the two phases and is needed to calculate the locus AC = 0. Here, the functional form of

y is much more important, and the assumption of

constant py may not be valid. Ten percent is about

the increase in gamma from the solid to the liquid

which we predict by this model, in good agreement

with Grover's estimate.

Dr. Boyd's suggestion for shocking frozen mercury

could, I think, be quite easily done, and would be

a most interesting experiment.

227

Shear Strength Effects on Phase Transition "Pressures" Determined From

Shock-Compression Experiments*

O. E. Jones and R. A. Graham

Sandia Laboratories, Albuquerque, New Mexico 87115

Plane-wave shock compression experiments provide an independent, and often unique, method

for estabhshing pressures at which phase transitions occur in solids. The transition stress which is measured in a shock experiment consists of two components: a mean isotropic pressure, and a deviatoric shear stress which is related to the yield strength, i.e., the Hugoniot elastic limit (HEL), of the solid when subjected to shock compression. The effects of the nonzero shear strength of a solid on the measured shock transition stress are discussed, and methods outlined for calculating the isotropic pressure component needed for comparison with hydrostatic results. Comparison requires that the compression

to initiate the transition be independent of shear distortion. The shock propagation characteristics of a solid which are necessary for establishing a reUable equilibrium transition pressure are considered.

Existing HEL data are collected in an Appendix and permit the importance of shear strength to be

assessed for various materials. Shock compression data for phase transitions in Bi, Fe, Ge, CdS, and

InSb are analyzed and shear strength corrections applied to obtain transition pressures.

1. Introduction

Discovery in shock compression experiments of

the "130 kbar" polymorphic transition in iron by

Bancroft et al. [1] > in 1956, the observation of a shock-induced transition in bismuth [2], and the

subsequent identification of the iron transition in

static high-pressure experiments [3-8] have re- sulted in interest in shock compression as a method

for establishing transition pressures. However,

because solids have nonzero shear strengths, shock

compression is not isotropic; if this shear strength

is a substantial fraction of the transition stress,

corrections to the shock measurements are required

to obtain values comparable to hydrostatic values.

Although it was recognized early [1, 9] that the

finite shear strength of a solid gives rise to an elastic

wavefront, whose amplitude is termed the Hugoniot

elastic limit (HEL), shock transition stresses have

frequently been directly compared to hydrostatic

values and have not been corrected for shear

strength effects. More recent development of techniques capable of examining the elastic wavefront

in detail have yielded considerable HEL data which

can now be used to calculate isotropic pressures

from measured shock transition stresses, provided

that equihbrium conditions are achieved.

After accounting for the contribution of the

shear strength of the soUd to the shock transition

stress, the resulting value must be further corrected

to isothermal conditions for comparison with hydrostatic results. In most cases, the temperature cor- rection is smaller than that for the shear strength.

In this paper, primary attention will be focused on

shear strength effects.

*This work was supported by the U.S. Atomic Energy Commission.

' Figures in braclcets indicate the literature references at the end of this paper.

Paper presented at the Symposium on Accurate Characterization

of the High-Pressure Environment, held at the National Bureau

of Standards, Gaithersburg, Md., October 14-18, 1968.

The object of this paper is to survey effects of

the finite shear strength of a solid on the stress

component measured in a shock-compression ex- periment [10] and to discuss how, in an idealized

situation, this value may be corrected for comparison with hydrostatic values [11]. Deviations

from this idealized behavior are examined, and an

extensive collection of Hugoniot elastic limit refer- ences is reported and summarized in an Appendix.

Then, several representative phase transitions, for

which relatively complete data are available, are discussed and finite shear strength corrections are made. The compressions at which the shock and

hydrostatic transitions occur are evaluated to assess

the effects of shear on the transition pressures.

2. Stress Configuration Under ShockWave Compression

2.1. Uniaxial Strain [10]

Although all static very-high-pressure experiments involve some shear stress components, the

idealized experiment involves only isotropic pressure, regardless of the pressure magnitude. On the

other hand, plane- wave shock compression produces

a precisely defined state of uniaxial strain in which

shear stresses can be appreciable. Uniaxial strain

is produced by simultaneously applying a uniform

normal stress over the entire face of a homogeneous

specimen in the form of a flat plate whose thick- ness is much less than its lateral dimensions. For

a short period of time, before the arrival of unloading waves from the lateral edges, symmetry requires

that each macroscopic volume element deform in

the same manner as the adjacent volume element;

thus, there is no net lateral deformation and compression occurs only in the direction of shock

propagation.

This condition is illustrated in figure 1, where

the principal strains ey = ez in the plane of the plate

229
UNIFORMLY LOADED

PLATE COMPRESSED

UNCOMPRESSED

Figure 1. Uniaxial strain compression of a volume element

of an isotropic plate.

are maintained equal to zero by the principal lateral

stresses <Ty and cr^. (For an isotropic material the

lateral stresses are equal.) That uniaxial strain

compression intrinsically involves shear strain is immediately evident from figure 1. The angular,

change 6 in the original right angle between the face

diagonals of the cube is equal to tan~' y, where y

is the engineering shear strain. Because a soUd has nonzero shear strength and can resist shear deformation the lateral stresses Cy and cr^ will not be

equal to the stress ctj, and the stress state is anisotropic. In shock experiments, the stress component

(Tj- acting normal to the shock front is determined;

except by inference, the lateral stress components

are unknown.

Thus, because of this stress anisotropy, shear

stresses are generated on planes inclined to the

shock front. Their magnitude is limited by the shear

strength of the solid. When the magnitude of the

maximum shear stress exceeds some characteristic

limiting value, the solid loses its resistance to shear

deformation and yields in a manner which may be

either plastic or brittle in character. In general,

shear stresses are present in the yielded state above

the yield point; however, in contrast to the elastic

state, they undergo only small changes with in- creasing compression.

A representative stress-volume Hugoniot curve

is shown in figure 2 for a solid which exhibits both

a simple yield point and phase transition point.

The yield point stress, obtained under shock conditions of uniaxial strain, is commonly termed the

Hugoniot elastic limit (HEL). By "simple" we mean

that rate effects are negligible and stress equilibrium is achieved. In this case shock wave propagation is steady, and the HEL and the transition

stress have time-independent values. (Deviations

from simple behavior will be discussed later.)

In this case, the Hugoniot jump relations [10],

expressing conservation of mass, momentum, and

energy, describe points of the Hugoniot curve:

1

Vi ~U-Ui'

T, TRANSITION

H, HEL

1.0 V/V„

of

-TRANSITION WAVE

EUSTIC. WAVE

(a) STRESS - VOLUME HUGONIOT (b) STRESS - POSITION PROFILE

Figure 2. (a) Hugoniot stress-volume curve for a solid exhibiting

simple yield and transition points, (b) Corresponding multiple- wave stress-position profile for driving stress cri.

— (Txi = PiiU— Ui) (u — Ui), (2)

(3)

where tj is the compression, F( = l/p) is specific

volume, p is density, (Tj- is the stress component

acting normal to the shock front, U is laboratory

shock velocity, u is particle velocity, and the subscript i denotes the state of the material ahead of

the particular shock front under consideration.

2.2. Elastic-Plastic Behavior [1214]

For an isotropic, plastically yielding solid, further

insight into shock compression results from ex- pressing the stress o"x in terms of its spherical,

of hydrostatic, component P and its deviatoric, or shear, component r. By definition,

P = ^{(Tx + (Ty+(T;:)=(Tx—l{(rx — Cry), (4)

where, for an isotropic, homogeneous soUd, (Ty=(rz

by symmetry. The maximum resolved shear stress occurs at 45° to the plane of the shock front and is given by.

1 (ax — a-y). (5)

Combining eqs (4) and (5),

(Tx^P+-^ T,

cry = P-^r.

(6)

(7)

(1)

Equations (6) and (7) are true regardless of the

stress-strain relation for the solid.

230
For stresses below the HEL in figure 2, Hooke's

law applies, and, for uniaxial strain (ey=ei=0),

becomes,

for cTx ^ (r§. (8)

^-3 /A] T7' for o-x ^ (T§, (9)

where is the bulk modulus, /a is the shear modulus,

and a" is HEL value of the stress ctx- Equations (8)

and (9) relate ctx and (Tyi

a-y — CTz 1 (Tx for CTx ^ Ox, (10)

where v is Poisson's ratio. For most isotropic solids

V = 1/3; thus, for an isotropic material stressed

below the HEL, the lateral stresses (of. fig. 1) are about one-half the applied stress. The maximum

resolved shear stress is obtained from eqs (5) and

(10),

-fM7),{0Ta-x^CT^, (11) 2il-v)

while the mean pressure P is, from eqs (6) and (11),

5 jl + v) „ P^^. T CTx, for (Tx

3(1-1') = Krf

(12a)

(12b)

which for v= 1/3 is equal to about two-thirds of the

applied stress <Tx- If the bulk modulus K is understood to be dependent on compression 17, eq (12b)

is also applicable for 0"x > cr". Equation (11) shows that as ctx and tj increase,

the maximum resolved shear stress r increases.

Because the shear strength of a solid is finite, the

material will fail in shear at some critical value of

stress (Tx corresponding to the HEL. This loss of

shear resistance results in a sudden increase in the

compressibility of the material and is responsible

for the cusp at point H in figure 3. The stresses (Tx,

P, and T are shown schematically in figure 3. Transformation of the material to the new phase at point

T in figure 3 again results in an abrupt increase in

the compressibility and a cusp in the Hugoniot

curve at point T. The cusps at points H and T create concave downward regions in the Hugoniot curve,

and result in the propagation of the multiple wave

structure [10] shown schematically in figure 2. For elastic-perfectly plastic behavior the shear

stress above the HEL is idealized as having a constant value, independent of the stress amplitude, as shown in figure 3. This limiting shear stress t* is,

from eq (11),

2 (l-i^) (13)

P, OR

T

V T, TRANSITION POINT

O-^ , EXPERIMENTALLY DETERMINED HUGONIOT CURVE

IH. HUGONIOT ELASTIC

i\ LIMIT

1.0

Figure 3. Schematic diagram of the stress-volume behavior

of a simple, elastic-perfectly plastic solid subjected to shock

compression.

It is likely that t* will increase somewhat with

increasing compression as a result of work-hardening; however, for our purposes, we will ignore workhardening as a higher order effect because its influence on (Tx (cf. eq (6) and fig. 3) will be smajl

due to the dominance of the mean pressure P.

It has also been^ suggested that t* may increase

with increasing P. In the case of r* constant, the

Hugoniot curve, as shown in figure 3, is parallel

to the isotropic compression curve and offset above

it by an amount (4/3)t* If the HEL is measured,

then the mean pressure P can be calculated from

the measured values of ctx- Early measurements by Fowles [14] and Lundergan et al. [15] demonstrated that an elastic-perfectly

plastic model adequately described the shock compression of aluminum. More recent measurements

[16—22] with improved time resolution show detectable rate effects in aluminum and other metals;

but, it is observed that these effects do not cause

significant departures from the final stress-volume

states predicted by the elastic-perfectly plastic

model.

Summarizing, in a simple (rate-independent) isotropic material exhibiting elastic-perfectly plastic

behavior, a shock-induced phase transition occurs

in the presence of shear stresses which cause a difference between the measured shock transition

stress and the transition pressure determined hydro231

385-762 O - 71 - 16
Figure 4. Alternative procedure for determining the transition pressure for a simple solid which does not have a sharply

defined HEL.

Statically. Because of these shear stresses, a transition pressure calculated from the shock transition

stress is comparable to the hydrostatic transition

pressure only if it can be assumed that the shear

stress does not aid in initiating the transition. One

way of testing this assumption is to experimentally

determine whether a characteristic compression

7]T will initiate the transition, independent of shear

stress amplitude. If the assumption is justified, and

if temperature effects are negligible, then the

transition pressure which should be compared with

hydrostatic results is Pt, and not the shock transition stress crS. From eqs (6) and (13), and figure 3,

Pr = o-?-| ^Z'^';^ o-g-o-?-|^Vg, (14a)

3 (1 — f ) 3 Cr = K'nH+{o-%-a-^), (14b)

where Cg and Ci are the ambient pressure shear

wave and dilatational wave velocities, respectively,

and 17// is the value of the compression corresponding to cr". For 1^=1/3, the correction amounts to about one-third the value of the HEL. Only if the

stress crj at which the shock-induced phase transition occurs is several order of magnitude greater

than the HEL is it reasonable to neglect the correction in eq (14) and assume that crj is essentially an

isotropic pressure. In any event the HEL must be

measured in addition to the transition stress before

the effects of shear strength can be accurately

assessed.

For anisotropic solids, such as many single crystals, (Ty 7^ (Tz and the above development leading

to eq (14a) is not applicable. However, if the material

continues after yielding to withstand a limiting shear

stress T*, then the offset between ctj- and P is constant, and the transition pressure Pt may be calculated from eq (14b) and from statically measured

values of the bulk modulus K.

2.3. Deviations

It is reasonable to expect that all solids will exhibit significant nonequihbrium and strain-rate

phenomena during some time interval (or, over some

propagation distance) immediately following shock

loading with the result that the transition stress

and/or the HEL will not exhibit simple behavior.

In these cases shock wave propagation is nonsteady

and the Hugoniot equations (1)—(3) are not properly

applicable. However, in the absence of any alter- native theory, they are commonly used to reduce

nonsteady shock data and obtain some kind of

averaged value.

Experimentally there are two common ways of

testing for nonsteady behavior of both transition

waves and elastic waves. One method is to vary

the shock propagation distance by changing the

specimen thickness. In general, if the transition

and HEL values depend on propagation distance,

the values decrease with increasing distance and

appear to asymptotically approach an equilibrium

value. A second method is to vary the driving shock

stress appUed to the specimen and see if the jump

properties of the shock front are affected. In general,

smaller overstresses result in smaller deviations

from simple behavior. In establishing a transition

pressure from shock compression experiments,

nonsteady behavior must be minimized; thus, the

input driving stress should be only slightly above

the transition stress and shock measurements

should be performed only at propagation distances

large enough to permit equilibrium conditions to be approached [23].

In some cases, even though the elastic wavefront is steady, the HEL may not be sharply defined, as shown in figure 4. In this event the transition

pressure may be determined by an alternative

procedure which is still based on the elastic-plastic

concept of a constant offset between ax and P. As

shown in figure 4, a point A is chosen in the plastic

state so that the slope of the Hugoniot curve passing

through it defines the bulk modulus K. The transition pressure is then given by

PT = KA'nA+{<rl-(Ti), (15)

where the subscript A denotes the point A on the

Hugoniot curve. A complication for brittle solids is that there are some indications that they cannot support shear

stress following yielding (t* = 0), so that the final

232
shock state lies on a Hugoniot curve corresponding

to compression by an isotropic pressure and no HEL correction is required. As suggested in

figure 5, the HEL for a brittle solid is, in general,

dramatically large and can exceed a hundred kilobars. In addition the HEL is observed to increase

substantially with increasing driving stress. Such

behavior has been observed for quartz [24]. Whether

or not the high-pressure states lie on an isotropic

pressure curve can be deduced by comparing the

Hugoniot curve with the statically determined

hydrostat.

2.4. Shock Determinations of Transition

Pressures

Although deviations from simple behavior, such

as those discussed above, are of physical interest,

their effects are highly undesirable if a fixed point

transition pressure is to be obtained from shockcompression experiments. This is necessarily true

of rate effects because of the inadequacy of present

shock-wave theory for describing such phenomena.

Thus, the most reliable transition pressures will

be obtained for solids whose shock response most

nearly approximates the simple, elastic-perfectly

plastic behavior outlined earlier for both the HEL

and the transition stress. In general, a series of experimental measurements

must be performed to establish whether the material

will exhibit simple behavior, i.e., the HEL and the

transition stress have time-independent values, for

experimentally feasible specimen thickness (consistent with the uniaxial strain requirement) and

driving stresses. Provided this is possible, the Hugoniot curve in figure 3 can be established and the

transition pressure Pt calculated from eq (14) for

the compression rjr. However, it remains to be

demonstrated in each instance that the shock transition occurs at the same specific volume as the

hydrostatic transition.

A temperature correction to the shock transition

pressure Pt can be estimated [2] by assuming that

the temperature at which the transition begins to occur can be approximated by the isentropic

expression

0=6>o exp (Lorj), (16)

where Oo is the initial temperature and To is the

Griineisen parameter at zero pressure. The change

in the transition pressure due to this temperature

rise is APr= {dPldd)Ad, (17)

where did = 6 — 6o is the temperature rise, and dPjdO

is the slope of the pressure-temperature phase line. If necessary, dPjdO can be estimated from shock

compression data taken in the mixed-phase re^on

above the transition point £2, 25]. Because APr is usually much smaller than Pt, a more sophisticated

analysis is normally unwarranted.

Figure 5. Schematic diagram of the stress-volume behavior ofa

brittle solid subjected to shock compression.

3. Survey of Hugoniot Elastic Limit

Measurements

The considerations of the preceding sections

show that a measured shock transition stress should

not be directly compared with the corresponding

hydrostatic pressure without first evaluating shear

strength and temperature corrections. Consequently, shock compression studies should include

simultaneous measurement of the HEL, as well as the transition stress. Hugoniot elastic limit measurements often require

different techniques than those employed for higherstress shock measurements. This results from both

the requirement to detect smaller amplitude disturbances as well as the requirement for better

time-resolving capability to remove apparent ambiguities in observed behavior. These improved techniques are of recent development and far fewer

HEL measurements have been reported than have

strong shock measurements [26].

Table A of the Appendix is a survey of existing

HEL measurements. Except for geological materials

it is reasonably comprehensive, containing about

120 entries describing about 50 materials, and permits the importance of shear strength effects to be assessed for a variety of materials. The measure- ments are grouped according to the following

material classes: (i) metals, (ii) brittle single crystals, and (iii) polycrystalline ceramics. Where

appropriate, each entry describes the metallurgical condition of the material, a reported

value for the HEL, remarks which include the prop233
agation distance (specimen thickness) and comments describing significant departures from steady

behavior, the shock loading method employed, the

measurement technique used, and the information

source. Assignment of a unique HEL value to an

elastic wavefront is often not obvious (cf. fig. 4);

howrever, a representative value can generally be

chosen, and this is given in the table. Although the

table provides a reasonably complete summary of

experimental results, individual references should

be consulted for specific details. Private communications and abstracts have been used only for

recent measurements of particular interest.

From table A it is immediately apparent that many

materials exhibit time-dependent strain-rate effects

in as much as their Hugoniot elastic limits decrease

with increasing propagation distances. Stress re- laxation immediately behind the elastic wavefront

is further evidence of time-dependent behavior.

Only at the larger propagation distances can a steady value for the HEL be expected. For example.

for iron the HEL decreases by less than 10 percent

for propagation distances exceeding about 25 mm

[27].

For metals, it can be seen that the HEL seldom

exceeds a few tens of kilobars; in contrast, for

brittle single crystals and polycrystaUine ceramics

it commonly exceeds fifty kilobars and may be as high as several hundred. Most of the observed HEL

values are large enough to warrant corrections to the transition stress measurements. Also, it is evident from table A that HEL values are strongly

influenced by the structure-sensitive properties

of the materials and, hence, are best measured

directly for the material under consideration.

4. Comparison of Shock and Static

Transition Pressures

In this section, several representative static and

shock transitions, which have been relatively well

characterized, will be compared. Bismuth is of

Table 1. Comparison of several shock and static transitions

Material

condition

Bismuth 11

Fe polycrystallineCe'' [111]

crystal

CdS " InSb"

Cast Pressed

Crystal

a-axis c-axis a-axis c-axis [100] [111]

a'x, kbar 25.6 25.4 24.6 25.6 /129 136-142 28 32 20 17 Pt. kbar "25.7 "25.5 '24.3 <-25.5 "125 114-122 " 18 " 18 ^3 '9

P static, kbar " 25.48-25.50 " 118 ' 120-125 23 '23

' 133

177- shock, percent 6.1 6.0 5.8 6.1 /6.4 12-13 3.3 3.3 3.0 2.1

r)T static, percent ^6.4 j 6. 1-6.6 12.5 1' 4.8 ' 4.6-5.3

" Shock data from D. B. Larson, ref. [11].

" Calculated from eq (14a).

' Calculated from eq (14b) and the bulk modulus value given by A. A. (iiardini et al., ref. [30].

« P. L. M. Heydemann, ref. [29]. 'A. A. Giardini et al, ref. [30]. ^T. R. Loree et al.. ref. [23].

" Calculated from eq (14b) and the HEL data of J. W. Taylor et al., ref. [27] Pt has not been corrected for the effect of temperature,

estimated to about plus 1 kbar.

" W. Stark et al., ref. [43].

' A. S. Balchan et al., ref. [3].

J Values computed from the lattice parameter measurements of H. K. Mao et al., ref. [8], at the transition pressures given in refs.

[3] and [43].

* Shock data from R. A. Graham et al., ref. [34]. Pr has not been corrected for the effect of temperature, estimated to be about

plus 1 kbar.

' S. Minomura et al., ref. [35].

'"J. C. Jamieson, ref. [36].

" Shock data from J. D. Kennedy et al., ref. [37].

" Calculated from the observed -qr shock and the bulk modulus value given by G. A. Samara et al.. ref. [38]. Temperature corrections

are negligible. G. A. Samara et al., ref. [38]. "Shock data taken from refs. [40] and [41].

' Calculated from the observed rjr shock and the bulk modulus value given by R. E. Hanneman et al., ref. [42]. Temperature corrections are negligible.

' R. E. Hanneman et al., ref. [42].

234
oo

0

® MINSHALL, 1961

+ LOREE, ET. AL., 1966

0 2 4 6 8 10 12 14 16 18 20 22

'(1-V)

(1-21')

Figure 6. Effect of shear stress on the shock compression tit at which the iron transitions occurs.

interest because it is used in static calibrations and

differences between static and shock transition

pressures have been reported [2]. Iron will be con- sidered and a shear strength correction applied to the transition stress after analyzing experiments

which examine the effect of shear on the transition.

Germanium provides an interesting example of a phase transition in a solid with a large elastic limit.

Finally, cadmium sulfide and indium antimonide

will be considered since they apprently exhibit

phase transitions in the range of elastic compression. Values associated with these transitions are summarized in table 1. Temperature corrections calculated from eqs

(16) and (17) show about a 1 kbar correction for

all the shock transitions listed in table 1. Thus, only

in the case of the Bi transition is the temperature

correction comparable to the shear correction.

4.1. Bismuth

Duff and Minshall's shock measurements of the

Bi l-II transition [2] were reported shortly after

the discovery of the iron transition. This transition

is now well established statically [28-30] and

affords an excellent opportunity for a direct comparison of shock and static measurements. Duff and

MinshaU's measurements showed that the shock

transition stress was higher than the hydrostatic

value. They speculated that one source of the differ- ence might be a shear strength effect which their

detection method could not observe.

Recently, Larson [11] published an extensive

investigation of the Bi I-II transition under shockwave loading which appears to resolve the differ- ences between static and shock values. The

instrumentation which he employed had much

better time resolution and better low pressure

sensitivity than that employed in the earlier studies.

Some of his results are compared to the corresponding hydrostatic values in table 1. The shock compression required to initiate the transition stress

agrees with the static value [30] to within experimental error. A shear strength correction of —0.8

kbar was calculated [11] from the observed HEL

values for polycrystalline material. The high compressibility of Bi and the large slope of its phase

line combine to yield a substantial temperature

correction of + 0.9 kbar.

When these corrections are made, the calculated shock transition pressure Pt is in agreement

with the static value to within experimental error. Further, the shear stresses present during shock

compression do not significantly influence the transition. This latter observation is in agreement with

the conclusion reached by Heydemann [29] in

comparing his hydrostatic data to the static data

of Kennedy et al. [28]

.

4.2. Iron

The "130 kbar" transition in iron has been widely

used for pressure calibration. The measured HEL

values (cf. table A) are large enough to warrant a shear strength correction to the transition stress. However, before making this correction, we will

consider the evidence which indicates that the

shock transition is not affected by shear stress. MinshaU [31] and Loree et al. [23], have reported

experiments which permit the effect of shear on the

transition to be quantitatively evaluated. The HEL

of iron can be changed with heat treatment and/or

small additions of carbon. In this manner, MinshaU,

and Loree et al. varied the HEL from 7 to 19 kbar and

measured the characteristics of the shock transition.

These data are shown in figure 6 where the compression to initiate the transition, r}T, is plotted

for various HEL values. It is apparent that 177- is

235
unchanged even though the shear stress varies by a factor of three [32]. These data indicate that the

shock transition is not influenced by shear stress

and the shear strength correction to the observed

transition stress can be applied according to eq (14a). This conclusion does not preclude the

possibility that shear stresses might affect the

kinetics of the transition.

For annealed Armco iron, Loree et al. [23] re- ported the iron transition stress to be 129 kbar at a volume compression 177- of 6.4 percent for a propagation distance (sample thickness) of 25.4 mm and

minimum applied driving pressure. They did not

simultaneously measure the HEL; howrever, the

HEL of iron has been sufficiently characterized to permit a shear strength correction to be calculated.

Basing the shear strength correction on the HEL

value of 8.3 kbar measured by Taylor et al. [27]

for 25.4-mm-thick annealed Armco iron specimens,

the transition pressure Pj is calculated to be 125

kbar. Stress relaxation and other deviations from

steady behavior should not introduce more uncertainty than ±0.5 kbar.

The shock transition pressure and compression

values are in general agreement with static results,

although the range of static values given in the liter- ature is substantial. It vyrould be highly desirable

to have values for the hydrostatic compression at the transition. This would be of interest because of

the anomalously low values of the bulk modulus

reported for shock compression of iron below the

transition [27].

4.3. Germanium

Germanium has not been extensively investigated

under shock compression but we consider its transition because it is an example of a transition

in a material with a relatively large HEL. For shock

propagation in the [111] direction, the transition

stress o-j has a value of about 139 kbar, and has

been shown to correspond to a solid-solid transition

from the diamond to the white tin crystal structure [34]. The HEL is about 44 kbar. The static

transition occurs at about 120 kbar [35]; clearly

shear strength effects cannot be ignored if the two

values are to be compared. As can be seen from

table 1, the transition occurs at very nearly the same

compression for both shock and hydrostatic loading, which indicates it is insensitive to shear stress,

in agreement with static observations [36]. The

shock transition pressure Pt calculated from

eq (14b) is between 114 and 122 kbar, which

compares well with the static value of from 120

to 125 kbar. The effects of shock heating are estimated to affect the shock transition pressure by

less than 1 kbar, an amount smaller than the experimental uncertainty.

4.4. Gadmium Sulfide and Indium

Antimonide

The transitions in CdS and InSb occur at relatively low pressures and are especially interesting

since they apparently occur below the HEL. Be- cause of this, there is the possibility of an incorrect

identification of the transition wave as an elastic

wave or vice versa. Kennedy et al. [37] have con- sidered this question and base their identification

of the transition in CdS on the very large (~ 18

percent) volume change indicated between first and

second wave. We have computed the shock transition pressure from the observed shock compression,

T7r, and hydrostatic high-pressure bulk modulus

values [38]. The shock transition pressure is significantly lower than the hydrostaticaUy observed

transition pressure [38]. Conversely, the hydrostatic compression -rjr is larger than the shock compression [38]. Thus, the wurtzite-to-rocksalt transition in CdS, which is martensitic in character [39],

is influenced by shear.

Similarly, InSb appears to exhibit an even greater

sensitivity to shear. The shock values shown in

table 1 were obtained by Kennedy [40, 41] in experiments identical to those used to study CdS, and are

as yet unpublished. His identification of the transition wave is again based on the very large volume

change (10 to 14 percent) between the first and sec- ond wave values. The shock transition pressure calculated from the observed compression and the

static bulk modulus [42] is only about 60 percent of

that observed statically. In addition, the compression observed under shock is roughly one-half that

observed statically [42]. If the identification of the

transition is correct, these values indicate an enor- mous influence of shear on this transition.

5 . Summary

The transition stress measured in shock compression experiments consists of both pressure and

shear components. If the shear strength of a solid is very small, it is reasonable to assume, as has been

widely done, that the measured stress is essentially a transition pressure. However, for solids which

have a large HEL and a phase transition occurring

at relative low stress, such an assumption is not

justified and a shear strength correction is required

to determine the mean transition pressure. Precise

characterization of phase transitions observed in

shock experiments should include such corrections and, to do this studies of the HEL need to be

undertaken at the same time the phase transition is being studied. The corrections are valid only if it can be demonstrated that the transition is not

influenced by shear. Sensitivity of the transition to shear should be carefully evaluated before an observed shock transition is used as a static calibration point.

236
6. Acknowledgement

The authors express their appreciation to Drs.

G. A. Samara and R. W. Rohde of Sandia Laboratory

for their helpful discussions and critical comments.

7. References

[1] Bancroft, D., Peterson, E. L., and Minshall, S., J. Appl.

Phys. 27, 291 (1956). [2] Duff, R. E., and Minshall, F. S., Phys. Rev. 108, 1207

(1957). [3] Balchan, A. S., and Drickamer, H. G., Rev. Sci. Instr. 32,

308 (1961). [4] Jamieson, J. C, and Lav^'son, A. W., J. Appl. Phys. 33,

776 (1962). [5] Clendenen, R. L., and Drickamer, H. G., J. Phys. Chem.

Sohds25,865(1964).

[6] Bundy, F. P., J. Appl. Phys. 36, 616 (1965). [7] Takahashi, T., and Bassett, W. A., Science 145, 483

(1964). [8] Mao, Ho-Kwang, Bassett, W. A., and Takahashi, T.,

J. Appl. Phys. 38, 272 (1967). [9] MinshaU, S., J. Appl. Phys. 26, 463 (1955). [10] Previous summary discussions of shear strength effects and shock wave propagation have been given by Rice,

M. H., McQueen, R. G., and Walsh, J. M., in SoUd State Physics Vol. VI, ed. F. Seitz and D. TurnbuU (Academic

Press, New York, 1958). Duvall, G. E.. and Fowles, G. R., in High Pressure

Physics and Chemistry, Vol. 2, ed. R. S. Bradley

(Academic Press, New York, 1963). Skidmore. I. C., Appl. Mat. Research 4, 131 (1965). Doran, D. G., and Linde. R. K., in Solid State Physics

Vol. 19 ed. F. Seitz and D. TurnbuU (Academic

Press, New York. 1966).

Larson, D. B., J. Appl. Phys. 38, 1541 (1967). Wood, D. S., J. Appl. Mechanics 19, 521 (1952). Morland, L. W., Phil. Trans. Roy. Soc. (London) A251,

341 (1959). [14] Fowles, G. R., J. Appl. Phys. 32, 1475 (1961). [15] Lundergan, C. D., and Herrmann, W., J. Appl. Phys. 34,

2046 (1963). [16] Barker, L. M., Lundergan, C. D., and Herrmann, W., 35,

1203 (1964). [17] Butcher, B. M., and Canon, J. R., AIAA Journal 2, 2174

(1964). [18] Butcher, B. M., and Munson, D. E., in Proc. Fourth Symposium on Detonation, ACR-126, Office of Naval Research, U.S. Government Printing Office (1965). [19] Butcher, B. M., and Karnes, C. H., J. Appl. Phys. 37,

402 (1966).

[20] Munson, D. E., and Barker, L. M., J. Appl. Phys. 37,

1652 (1966). [21] Karnes, C. H., Proc. Symp. on the Mechanical Behavior

of Materials under Dynamic Loads, San Antonio, Texas,

Sept. 6-8 (1967), to be published.

[22] Graham, R. A., Anderson, D. H., and Holland, J. R.,

J. Appl. Phys. 38, 223 (1967). [23] Loree, T. R., Fowler, C. M., Zukas, E. G., and MinshaU,

F. S., J. Appl. Phys. 37, 1918 (1966). [24] Wackerle, J., J. Appl. Phys. 33, 922 (1962). [25] Graham, R. A., Jones, O. E., and HoUand, J. R., J. Phys.

Chem. Solids 27, 1519 (1966). [26] Van Thiel, M.. Kusubov, A. S., and MitcheU, A. C, Compendium of Shock Wave Data, Lawrence Radiation

Laboratory, University of California, UCRL 50108.

[27] Taylor, J. W., and Rice, M. H., J. Appl Phys. 34, 364

(1963). [28] Kennedy, G. C, and LaMori, P. N., J. Geophys. Research

67, 851 (1962). [29] Heydemann, P. L. M., J. AppL Phys. 38, 2640 (1967) and

errata 38, 3424 (1967). [30] Giardini, A. A., and Samara, G. A., J. Phys. Chem. Solids

26, 1523 (1965). [31] Minshall, F. S., in Response of Metals to High Velocity Deformation, ed. P. G. Shewmon and V. F. Zackay (Inter- science, New York, 1961).

[32] The insensitivity of the iron transition to shear should not

necessarily be extrapolated to other iron aUoys. Signifi- cant shock-induced martensite to austenite transformation has been observed in martensitic 30 percent Ni-70

percent Fe at shock stresses as low as 20 kbar [33]. Martensitic transformations are observed to be sensitive to shear in static experiments; thus, shock transition stresses observed in martensitic iron alloys require inde- pendent verification of the effects of shear.

[33] Rohde, R. W., Holland, J. R., and Graham, R. A., accepted

for pubhcation. Trans. Met. Soc. AIME.

[34] Graham, R. A., Jones, O. E., and HoUand, J. R., J. Phys.

Chem. Sohds 27, 1519 (1966). [35] Minomura, S., and Drickamer, H. G., J. Phys. Chem. Solids

23, 451 (1962). 36] Jamieson, J. C, Science 139, 762 (1963). 37] Kennedy, J. D., and Benedick, W. B., J. Phys. Chem. Solids

27, 125 (1966). [38] Samara, G. A., and Giardini, A. A., Phys. Rev. 140, A388

(1965). [39] CorU, J. A., Phys. Rev. 157, 623 (1967). [40] Kennedy, J. D., and Benedick, W. B., BuU. Am. Phys. Soc. 10, 1112 (1965). [41] Kennedy, J. D., private communication (1968). [42] Hanneman, R. E., Banus, M. D., and Gatos, H. C, J. Phys.

Chem. Solids 25,293 (1964). [43] Stark, W., and Jura, G., ASME Pubhcation 64-WA/PT-28

(1964).

237
8. Appendix— Table A. Summary of Hugoniot Elastic

Limit Measurements

Material Condition " (Theu kbar * Remarks Technique Reference

METALS

Iron Alloys

Armco iron annealed 7 25 mm E-1 Bancroft et aL (1956)

Armco iron AR, annealed 15-7 25-133 mm E-1, 14 Mmshall (1961)

Armco iron normalized 8-7 20-51 mm +

* E-5 ¥ -1/1 C\ \ Jones et aL (1962)

Armco iron annealed 11-7 6-51 mm+

* T? £ L-6 Taylor et al. (1963)

Armco iron . . 10 25-60 mm+

* T^ "7 E-7 T .1/1 O \ Ivanov et al. (1963)

Armco iron annealed 12-6 6-40' mm+* E-12 McQueen (1964)

Armco iron .. 11-8 4-18 mm + E-12 Tl . 1 / T A^ ^ \ reyre et aL (1965)

r errovac t, iron varied 12-6 12 mm* T? r L-5 Holland (1967)

r errovac L iron annealed 10 13 mm*, temp T? r L-5 D 1 J /I A^O \ nohde (1968a)

76-573 K

SAE 1018 1 - I normalized 13-10 20-51 mm + * E-5 Jones et aL (1962)

SAE 1018 annealed-cold 16-9 19 mm T^ r L-5 T i 1 / 1 C\C A \ Jones et aL (1964)

roUed

SAE 1018 varied 12 19 mm* E-5 T . 1 /T A*r A\ Jones et aL (1968)

SAE 1020 , 10-12 127 mm E-1 Minshall (1955)

SAE 1020 AR, annealed 15-11 50-127 mm E-1 Minshall (1961)

SAE 1020 AR 10 51-76 mm E-1 Costello (1957)

SAE 1040 11 127 mm E-1 Minshall (1955)

SAE 1040 AR, annealed 6-14 127 mm E-1 Minshall (1961)

SAE 1055 AR, annealed 13-15 125-127 mm E-1, 14 Minshall (1961)

SAL 1055 varied 8-15 25 mm E-14 Loree et al. (1966)

1% carbon-iron varied 12-17 25 mm E-14 Loree et al. (1966)

1.5% carbon-iron vaned 16-19 25 mm E-14 1 • 1 /T A^ ^ \ Loree et al. (1966)

2% carbon-iron vaned 15-21 25 mm E-14 T . 1 /T A^ ^ \ Loree et al. (1966)

SAE 4340 annealed 19-14 6-50 mm+* G-5 Butcher et aL (1968a) CAT? O >1 A SAL 4340 annealed 23-21 3-6 mm G-8 Graham et aL (1967b) CAT? A Af\ SAL 4340 RL-15 17 25 mm E-14 H/f* 1 11/1 (\£. 1 \ Minshall (1961)

SAL 4340 T3 T r RL-15 1 /— 19 12 mm G-2 X\ .1 .1/1 AZT A \ Butcher et aL (1964)

SAE 4340 RC-30 16 23 mm E-5 Jones et al (1962)

SAE 4340 RL-32 16-20 12 mm G-2 .1 .1/1 A/" A \ Butcher et aL (1964)

SAE 4340 RC-35 25 25 mm E-14 "KIS' 1 11/1 A/T 1 \ Minshall (1961)

£? 1 A f> A i\ SAE 4340 RC-40 20-18 20-51 mm E-5 Jones et al. (1962) OAT? A O A f\ SAL 4340 RC-50 22 23 mm E-5 Jones et aL (1962)

SAE 4340 RC-54 14-31 12 mm G-2 Butcher et aL (1964)

Hampden tool steel RC-20 14 l9 mm E-5 Jones et aL (1962)

Hampden tool steel RC-62 ~ 22 19 mm0 E-5 Jones et aL (1962)

Hampden tool steel RC-66 > 24 19 mm0 E-5 Jones et aL (1962)

SAL 347 stainless steel AR 6 127 mm E-1,14 Minshall (1961)

3% silicon-iron crystal ~ 7 13 mm* G-6 Taylor (1968)

3.34% sUicon-iron annealed 10-9 6-25 mm+* E-5 Mote (1968) Invar, 36% nickel-64% iron AR 13 wedge E-10 Curran (1961) Invar, 36% nickel-64% iron annealed 5 13 mm G-5 Graham et aL (1967a)

30% nickel-70% iron annealed 5-3 10-13 mm G-5 Graham et aL (1967a)

30% nickel-70% iron martensitic ~ 20(ramp) 13 mm0 G-5 Graham et aL (1967a)

Russian steel 3 AR 13-6 20-120 mm+* E-7 Ivanov et aL (1963)

Russian steel 30khGSA annealed 17 60 mm+*

" E-7 Ivanov et al. (1963)

Russian steel SOkhGSA hardened 18 60 mm+* E-7 Ivanov et aL (1963)

Austenitic manganese steel RB93 8 6 mm E, G-5 Champion (1968)

304 stainless steel RB77 6 6 mm G-5 Butcher (1968b)

Vibrac RC36 21 51-76 mm E-1 Costello (1957)

238
8. Appendix — Table A. Summary of Hugoniot Elastic

Limit Measurements — Continued

Material Condition " o-HEL, kbar * Remarks Technique Reference

Aluminum Alloys

2024 T-4 5 wedge E-10 Fowles (1961)

2024 annealed 1 wedge E-10 Fowles (1961)

2024 4 25 mm E-5 Jones et al. (1962)

2024 .. 6 E-6 McQueen (1964)

2024 T-4 13 mm G-6 Taylor (1968)

6061 T-6 6 13-25 G-1 Lundergan et al. (1963)

6061 T-6 5 6 mm G-5 Halpin et al. (1963)

6061 5-7 25 mm G-2 Barker et al (1964b)

6061 T-6 5 13 mm G-4 Barker (1967)

6061 T-6 6 13-64 mm G-2 Butcher et al. (1966)

1060 annealed 0.3-0.5 25 mm* G-4, 5 Barker et al. (1966)

1060 annealed 0.6-0.2 6-24 mm+* G-4 Karnes (1967)

Russian alloy D-1 annealed 4U-1ZU mm L-7 INovikov et al. (l\)oo)

Russian alloy D-16 annealed 3-2 30-80 mm+ E-7 Novikov et aL (1966)

temp 283-473 K

Russian alloy U-16 hardened 5 30 mm L-7 INovikov et al. (1966)

French alloy AU4G 8 E-12 Peyre et al. (1965)

Other Metals

Copper, crystal [001] 2 5mm* E-5 Mote (1968)

[Oil] 1 5mm* E-5 Mote (1968)

[111] 1 5 mm* (relax- E-5 Mote (1968)

ation sup- pressed by

prestrain)

Copper, polycrystal .. —0.5 (ramp) 5 mm© E-5 Mote (1968)

Copper annealed 0.6-0.4 20-30 mm E-7 Novikov et al. (1966)

temp. 283-473 K

Copper annealed ~ 0.4 (ramp) 13 mm0 G-6 Taylor (1968)

Copper annealed ~ 0.5 (ramp) 0 E-2,3 Munson et al. (1966)

Copper cold worked 0 G-6 Taylor (1968)

Lead annealed <0.2 E-2 Munson et al. (1966) Brass annealed 3-2 30-80 mm+ E-7 Novikov et aL (1966) Brass ~8 6 mm0 E-5 Benedick (1965)

Beryllium, crystal c-axis 40 * G-6 Taylor (1968)

Beryllium, crystal a- axis 4

* G-6 Taylor (1968)

Beryllium, sintered .. ~ 2 (ramp) 0 G-6 Taylor (1968)

Tantalum annealed G-6 Taylor (1968)

Niobium annealed G-6 Taylor (1968)

Thorium G-6 McQueen (1964)

Uranium 0 G-6 Taylor (1968)

DisIIlULIlf K^l y OtcU 3 1-3 mm Hj, VT o I Qi-crtn /I Qf\7\ i_<arson \iy\jt)

Bismuth, polycrystal cast ~ 2 (ramp) 2-13 mm0 E, G-5 Larson (1967)

DibiiiuLii, puiycrysiai cast ~ 4 (ramp) 2 mm0 Present work

Tungsten annealed 38 10 mm G-5 Rohde (1968b)

Antimony cast 17-2 5-49 mm+ E-1 Warnes (1967)

BRITTLE SINGLE CRYSTALS

Quartz (SiOi) JC-CUt 94-48 5-25 mm+* E-9 Wackerle (1962)

Quartz (SiO.) x-cut 66-55 6 mm* E-11 Fowles (1967)

Quartz (SiOi) y-cut 110-82 10 mm* E-9 Wackerle (1962)

Quartz (SiO^) y-cut 86-65 3^ mm* E-11 Fowles (1%7)

Quartz (Si02) z-cut 145-120 10 mm* E-9 Wackerle (1962)

239
8. Appendix — Table A. Summary of Hugoniot Elastic

Limit Measurements — Continued

iviatenai Condition " o-HEL. kbar ° nemarks 1 ecnnique Keierence

Quartz (SiO.) z-cut 148-100 3-6 mm* E-11 Fowles (1967)

Quartz (SiOi) z-cut 145-60 ... E-12 Peyre et al. (1965)

Quartz (SiOa) fused 98 (ramp) 10-13 mm 0 E-9 Wackerle (1962)

Sapphire (Al^On) 60° cut

1 OA 1 '7A 120-170 10-13 mm ti-y Brooks et aL (lyoo)

Sapphire (AliO:)) z-cut

1 OA OAA 120-200 10-13 mm Brooks et al. (iy66)

Sapphire (Al^Orj) JC-CUt 135-180 ll)-13 mm Brooks et ai. (iyb6)

Germanium [111] 44 8 mm G-? Graham et al. (iy6o)

Oermanmm [111] 41-35 6-12 mm* E-12 McQueen (1964)

Germanmm [100] 53-46 6-12 mm* E-12 McQueen (1964)

(jrermanium [100] 45 7 mm* E-5 Kennedy (1968)

(-rermanium [100] 47 6 mm* r* o G—

o

trraham (iyo7b)

Germanium [114] 6-12 mm* E-12 JVlcyueen (1964)

Sihcon crystal free surface velocities McQueen (iyo4)

50% higher than Ge values. Cadmium sulhde (Las) c-axis >32 (*) Kennedy et al. (1966)

Cadmium sulnde (Cdb) a-axis >28 (*)

17 C E-5 T/^ J A 1 /T A^£\ Kennedy et al. (1966)

indium antimonide (Inisb) [100] >20 ... E-5 Kennedy et aL (1965)

Indium antimonide (Insb) [111] >17 T7 C Kennedy et al. (1965)

Indium antimonide (InSb) [110] >20 E-5 Kennedy et al. (1965)

Titania (TiO,) [100] 70 6 mm E-13 Linde et al. (1968)

Titania (TiO.) [001] 100 6 mm E-13 Linde et al. (1968)

Sodium Chloride (NaCl) [100] -0.3 5-14 mm* E-5 Benedick (1968)

POLYCRYSTALLINE CERAMICS

Lucalox (Al.O:,, po = 3.98) 99-123

Alumina (AI2O3, Po = 3.92) 140

Alumina (AlAi, Po = 3.81) 67-100

Alumina (AI2O:!, po = 3.76) 58-72

Alumina (AI2O3, po = 3.72) 8.0 Magnesium

Oxidei(MgO,po = 3.58) 89-35

Boron Caibide (BX, p., = 2.50) 150

Barium titanate (BaTiOn) 25

Barium titanate (BaTiO;)) —30

Lead zirconate titanate — 40

(PZT 95/5)

Lead zirconate titanate 19 (PZT 52/48)

Manganese-zinc ferrite 23

Yttrium iron garnet (po = 5.07) > 60

Polycrystal quartz rocks 47-130

Titania (TiO.. po = 4.24) 75

3-13 mm E-11 Ahrens et al. (1968)

E-11 Gust et aL (1968)

6 mm E-11 Ahrens et aL (1968) 13 mm0 G-5,9 Present work

E-11 Gust et al. (1968)

4-10 mm E-11 Ahrens (1966)

E-11 Gust et aL (1968) 13 mm E-9 Reynolds et aL (1962)

3-13 mm+ E-10 Doran (1968)

4-13 mm+ E-10, 11 Doran (1968)

13 mm E-9 Reynolds et al. (1962)

14 mm0 E-5 Present work

8 mm G-5 Present work

3-13 mm+ E-11 Ahrens et al (1966)

6 mm E-13 Linde et aL (1968)

" AR denotes as-received.

'' When a range of sample thicknesses is given and + is noted in remarks, the larger HEL value corresponds to the smaller sample

thickness, and vice-versa. "Numbers refer to sample thickness. Symbols: + sample thickness effect observed; * stress relaxation observed; 0 poorly defined

elastic wavefront.

Letters denote method of loading: E explosive loading, and G gun impact. Numbers denote measurement techniques:

1. Pins, Minshall (1955).

2. Slanted resistance wire. Barker et al.

(1964a).

3. Interferometer, Barker et al. (1965).

4. Velocity interferometer. Barker (1967).

5. Quartz gauge, Graham et al. (1965).

6. Capacitor, Rice (1961).

7. Capacitor, Ivanov et al. (1963).

8. Solid dielectric capacitor, Graham

et al. (1967).

9. Optical knife-edge, Davis et al. (1961).

10. Optical lever-oblique shock, Fowle,

(1961).

1 11. Inclined mirror, Doran (1963). |

12. Optical lever, McQueen (1964).

13. Manganin wire, Keough et al. (1964).

14. Capacitor, Hughes et al. (1961).

240
References Cited in Table A

Ahrens, T. J., J. Appl. Phys. 37, 2532 (1966). Ahrens, T. J., Gust, W. H., and Royce, E. B., accepted for publication, J. Appl. Phys. (1968). Ahrens, T. J., and Duvall, G. E., J. Geophys. Research 71, 4349

(1966). Bancroft, D., Peterson, E. L., and Minshall, S., J. Appl. Phys. 27,

291 (1956). Barker, L. M., and HoUenbach, R. E., Rev. Sci. Instr. 35 , 742

(1964a).

Barker, L. M., Lundergan, C. D., and Herrmann, W., J. Appl.

Phys. 35, 1203 (1964b).

Barker, L. M., and HoUenbach, R. E., Rev. Sci. Instr. 36, 1617

(1965). Barker, L. M., Butcher, B. M., and Karnes, C. H., J. Appl. Phys.

37, 1989 (1966). Barker, L. M., Proc. lUTAM CoUoq. on the Behavior of Dense

Media under High Dynamic Pressures, Paris, France, Sept.

11-15, 1967. Benedick, W. B., Rev. Sci. Instr. 36, 1309 (1965). Benedick, W. B., private communication (1968). Butcher, B. M., and Canon, J. R., AIAA Journal 2, 2174 (1964). Butcher, B. M., and Karnes, C. H., J. Appl. Phys. 37, 402 (1966). Butcher, B. M., and Munson, D. E., in Dislocation Dynamics,

ed. A. R. Rosenfield, G. T. Hahn, A. L. Bement, Jr., and R. I. Jaffee (McGraw-Hill, New York, 1968a).

Butcher, B. M., private communication (1968b).

Brooks, W. P., and Graham, R. A., Bull. Am. Phys. Soc. II 11,

414 (1966). Champion, A. R., private communication (1968). Costello, E. L., in Proc. Conf. on The Properties of Materials at High Rates of Strain, The Institution of Mechanical Engineers,

London, 1957. Curran, D. R., J. Appl. Phys. 32, 1811 (1961). Davis, W. C, and Craig, B. G., Rev. Sci. Instr. 32, 579 (1961). Doran, D. G., in High Pressure Measurement, ed. A. A. Giardini

and E. C. Lloyd (Butterworth, London, 1963).

Doran, D. G., J. Appl. Phys. 39, 40 (1968). Fowles, G. R., J. Appl. Phys. 32, 1475 (1961). Fowles, R., J. Geophys. Research 72, 5729 (1967). Graham, R. A., Neilson, F. W.. and Benedick, W. B., J. Appl.

Phys. 36, 1775 (1965). Graham, R. A., Jones, O. E., and Holland, J. R., Phys. Chem.

Solids 27, 1519 (1966). Graham, R. A., Anderson, D. H., and Holland, J. R., J. Appl.

Phys. 38, 223 (1967a).

Graham, R. A., and Ingram, G. E., Proc. lUTAM CoUoq. on the Behavior of Dense Media under High Dynamic Pressures,

Paris, France, Sept. 11-15, 1967b.

Gust, W. H., and Royce, E. B., BuU. Am. Phys. Soc. II 13, 901

(1968).

Halpin, W. J., Jones, O. E., and Graham, R. A., in Symp. Dynamic

Behavior of Materials, ASTM Special Technical Publication No. 336, American Society for Testing and Materials, Phila- delphia, 1963. Holland, J. R., Acta Metallurgica 15,691 (1967). Ivanov. A. G., Novikov, S. A., and Sinitsyn, V. A., Soviet Physics,

SoHd State 5, 196 (1963). Jones, O. E., Neilson, F. W., and Benedick. W. B., J. Appl.

Phys. 33, 3224 (1962). Jones, O. E., and Holland, J. R., J. Appl. Phys. 35, 1771 (1964). Jones, O. E., and Holland, J. R., Acta Metallurgica 16, 1037

(1968). Karnes, C. H., in Proc. Symp. on the Mechanical Behavior of Materials under Dynamic Loads, San Antonio, Texas, 1967. Kennedy, J. D., and Benedick, W. B., Bull. Am. Phys. Soc. II 10,

1112 (1965). Kennedy, J. D., and Benedick, W. B., J. Phys. Chem. SoUds 27,

125 (1966). Kennedy, J. D., private communication (1968). Keough, D. D., Williams, R. F., and Bernstein, D., Am. Soc.

IV^ch. Engrs., P^er 64-WA/PT-5 (1964). Larson, D. B., J.' Appl. Phys. 38, 1541 (1967). Linde, R. K., and DeCarh, P. S., private communication (1968). Loree, T. R., Fowler, C. M., Zukas, E. G., and Minshall. F. S.,

J. Appl. Phys. 37, 1918 (1966). Lundergan, C. D., and Herrmann, W., J. Appl. Phys. 34, 2046

(1963). McQueen, R. G., in Metallurgy at High Pressures and High Temperatures, ed. K. A. Gschneidner, Jr., M. T. Hepworth, and

N. A. D. Parlee (Gordon and Breach, New York, 1964).

Minshall. S., J. Appl. Phys. 26, 463 (1955). Minshall, F. S., in Response of Metals to High Velocity Deformation, ed. P. G. Shewmon and V. F. Zackay (Interscience

PubHshers. New York. 1961. Mote, J. D., private communication (1968). Munson, D. E., and Barker, L. M., J. Appl. Phys. 37, 1652 (1966). Novikov, S. A., Sinitsyn, V. A., Ivanov, A. G., and Vasilyev, L. V., The Physics of Metals and Metallurgy 2 1 , 135 (1966). Peyre, C, Pujol, J., and Thouvenin, J., in Proc. Fourth Symp.

on Detonation, U.S. Naval Ordnance Laboratory, White Oak,

Maryland, Office of Naval Research, ACR-126 (U.S. Govern- ment Printing Office, 1965).

Reynolds, C. E., and Seay, G. E., J. Appl. Phys. 33, 2234 (1962). Rice, M. H., Rev. Sci. Instr. 32 , 449 (1961). Rohde, R. W., accepted for publication. Acta Metallurgica (1968a)

Rohde, R. W., private communication (1968b).

Taylor, J. W., and Rice, M. H., J. Appl. Phys. 34, 364 (1963). Taylor, J. W., in Dislocation Dynamics, ed. A. R. Rosenfield,

G. T. Hahn, A. L. Bement. Jr.. and R. I., Jaff'ee (McGraw-Hill,

New York, 1968).

Wackerle, J., J. Appl. Phys. 33 , 922 (1962). Warnes, R. H., J. Appl. Phys. 38 , 4629 (1967).

DISCUSSION

M. Contre (Commissariat a I'Energie Atomique,

Paris, France): For the static transition pressure

in iron you have given Drickamer's value of 133

kbar when, in fact, this value derives from

Bancroft's shock measurements. In this regard, I have recently obtained a value, in a belt-type

apparatus, of 150 kbar or more. The transition

occurred slowly — in four or five hours.

G. E. Duvall (Washington State University, Pullman, Washington): You have made a point of

comparing the strains that were obtained. It does

appear to me that there may be one circumstance

in which the measured strain is not representative

of the curve in transition: If an elastic precursor

takes the material to some Hugoniot elastic limit,

and before the arrival of the transition shock a re- laxation occurs, the state into which the transition

shock advances is not necessarily the state that

existed immediately behind the elastic precursor.

The results you give for cadmium sulfide and indium

antimonide are particularly interesting in this

regard.

D. I. Decker (Brigham Young University, Provo,

Utah): You say you saw no shear effects on the iron

transition, but even for the materials with lower

Hugoniot elastic limits the amount of shear might

still be considerable compared to that observed

in hydrostatic-type systems. For instance, you may

241
always have had a large amount of shear compared

to what Contre had.

F. R. Boyd, Jr. (Geophysical Laboratory, Carnegie

Institution of Washington, Washington, D.C.):

With regard to the results that you have quoted on

the cadmium sulfide and the indium antimonide,

isn't this the first time that an equilibrium boundary

has been shown to be displaced by shearing stress?

There has been an argument about this in geological

circles for 50 years. In general the minerals which

used to be thought of as "stress minerals,"

geologically, have turned out simply to be highpressure phases.

I would be interested in knowing in what kind of

high-pressure apparatus the static results on those

phases were obtained. It would be very interesting

to look at results obtained in two types of highpressure apparatus in which there are wide differ- ences in the degree of shear that is present.

G. A. Samara (Sandia Laboratories, Albuquerque,

New Mexico): The static measurements referred to on cadmium sulfide and indium antimonide were

obtained in the cubic apparatus at Fort Monmouth

three or four years ago. At Sandia one of our staff

members has been studying the transition of

cadmium sulfide under hydrostatic conditions. I believe his results show a transition at 23 ±0.1 kbar

for samples which have initial resistivities over a range of seven or eight decades. A range of values

up to about 30 kbar has been given in the literature,

and so I think this value is one of the lower ones

obtained.

C. W. Beckett {National Bureau of Standards,

Washington, D.C.): You mentioned a transition in

antimony at around 90 kbar. Is that transition wellbehaved?

AUTHORS' CLOSURE

With regard to Dr. Contre's comments, it is likely that Drickamer's value was strongly influenced by the early shock measurements. However

as noted in the paper our current estimate of the

shock transition pressure for iron is 125 kbar. I'm

unable to critically assess the significance of the

150 kbar value observed by Contre. If the implication is that previous quasi-hydrostatic values have

been low because of shear effects nucleating the

transition at lower pressures, then I am surprised

that the shock transition value for iron is unaffected

by shear changes (different Hugoniot elastic limits).

I agree with Dr. Duvall's point. In the paper we

have discussed how such complications resulting

from time-dependent phenomena can be identified.

Our point is that a material exhibiting such effects

would be a poor candidate for a pressure calibration

standard. With regard to cadmium sulfide and

indium antimonide, our characterization of these

materials is incomplete at present; however, their

behavior does appear to be complex.

Dr. Decker's point may well be true that shock

experiments may always involve much more shear

than exists in quasi-hydrostatic experiments.

However, my impression is that it is difficult to assess the shear stresses existing in many quasihydrostatic experiments. In the case of shock compression, shear stresses can be accurately evaluated.

In reply to Dr. Boyd, I don't know whether this

is the first report of shear stress displacement of an

equilibrium phase boundary. My guess is that it isn't because transformations that are martensitic

in nature might be expected to show such effects.

For example. Dr. J. Corll of Sandia Laboratories

has suggested from ultrasonic pressure measure- ments that the cadmium sulfide transition is of a martensite-type. I believe Dr. Samara can better

answer the apparatus question than I. The transition in antimony, asked about by Dr.

Beckett, shows a range of transition pressures of

about 25 kbar out of 100 kbar depending upon the

propagation distance and the driving stress. It is one in which the kinetics are definitely important,

and so would appear to be a poor choice as a refer- ence transition.

242
Study of Phase Transitions in Insulators by the Dielectric Constant

Technique*

G. A. Samara and W. L. Chrisman

Sandia Laboratories, Albuquerque, New Mexico 87115

This paper proposes the measurement of the dielectric constant and dielectric loss as a probe for detecting and studying the properties of pressure-induced phase transformations in insulators. The

measurement is very sensitive, relatively simple, and can be adapted for use in many types of high pressure apparatus. To illustrate the general usefulness of the technique, results on a variety of sub- stances (solids and liquids) obtained from measurements in three different pressure apparatus will be

presented and discussed. The substances investigated include a number of alkah and thallous halides,

strontium titanate. and water.

1. Introduction

The study of pressure-induced phase transitions

is of interest from the standpoint of lattice stability

and kinetics of phase transformations as well as for

determining the influence of crystal structure on

physical properties. In addition, the transition pres- sures in many substances have been used, or are

potentially useful, as "fixed points" for. pressure

cahbration purposes.

The detection and study of pressure-induced

phase transitions in insulators have been hmited to techniques — usually the direct measurement of

volume change — which are not well suited for many

of the high pressure apparatus now in use. In this

paper we suggest the measurement of the dielectric constant as a probe for studying such transitions. The measurement is very sensitive and relatively simple. Results on a variety of substances

(sohds and liquids) obtained from measurements

in both hydrostatic and quasi-hydrostatic (cubic

multianvil and Bridgman anvil) pressure apparatus

will be presented and discussed. Whalley and co- workers [1] ' have been using the same technique

for studying the pressure-temperature phase diagram of ice and have also suggested its use for the

study of other phase transitions.

Aside from the above considerations, the study

of the pressure (volume) dependence of the dielectric constant and dielectric loss is of interest be- cause these quantities enter into the theoretical

treatment of various physical properties, e.g., the

theories of electron-phonon interactions and infrared dispersion and dielectric relaxation. Combined

with temperature measurements, the pressure

results make it possible to separate the temperature

dependence of the dielectric constant into its volume-dependent and volume-independent con5 tributions. The latter contributions are determined

entirely by anharmonic lattice effects [2].

' Figures in brackets indicate the literature references at tiie end of this paper.

*This wori< was supported by the U.S. Atomic Energy Commission.

Paper presented at the Symposium on Accurate Characterization

of the High-Pressure Environment, held at the National Bureau of

Standards, Gaithersburg, Md., October 14—18, 1968.

2. Theoretical Considerations

The purpose of this section is to give a brief

discussion of dielectric properties. This should

serve to define the quantities and relationships to be used later.

When a dielectric (insulator) slab is placed in a

static applied electric field, it acquires a surface

charge. The polarization so induced arises from the

displacement of positive and negative charges in

the dielectric. For an isotropic, linear dielectric,

the polarization P is proportional and parallel to the applied field E. The electric flux density or

electric displacement D is defined by (cgs units)

D = E + 47rP = eE (1)

where e is the static dielectric constant. The dielectric susceptibility x is defined by P = xE, and from

eq (1) it is seen that

e = 1 -f 47rx. (2)

For isotropic dielectrics e and x are scalar quantities.

They are dependent on the molecular properties

of the dielectric and, at low fields (i.e., in the linear

range), are independent of the magnitude of the

macroscopic field E.

The polarizability a of an atom (or molecule) is defined by

fi = aF (3)

where fi is the electric dipole moment, and F is the

local, or effective, field seen by the atom. The

polarization P is defined as the net dipole moment

per unit volume and is given by

P = ^Nitii=^NiaiFi, (4)

1 i

where iV, is the number of dipoles per unit volume.

The local field F at a given lattice site i is generally written as

243
Fi — E + Ejnt. = E + ^7T^<f)ijPj, (5)

where E is the appHed field, and Ejnt. is the internal

field acting on the ion i due to the other ions j and

is expressed in terms of the polarization. is the

internal field coefficient, which is a dimensionless

quantity that depends on the arrangement of the

ions in the lattice. For diagonal cubic crystals, i.e.,

crystals in which all ions have cubic environments,

the Lorentz internal field is applicable and <j)ij = 1/3.

In this case the above eqs yield

P =

1 477.

E = ^E.

32*

From eq (6) it is easily shown that

47T

(6)

e — 1 _ 477

e + 2~ 3

(7)

which is the familiar Clausius-Mossotti equation.

The individual ionic polarizabilities are difficult to determine, and it has become customary to use eq (7) in its macroscopic form which can be written

as

(8)

e = e —le.

The loss angle is defined by

tan 8 = e7e'.

(13)

(14)

tan 8 is simply related to the quality or Q factor of

the dielectric by Q—lltan8. It is obtained rather

directly from experiment.

3. Measurement Technique

A real dielectric can be represented by an ideal

capacitance (loss-free) plus a parallel resistance

which expresses the sum total of all the dissipative

processes occurring in the dielectric when subjected

to an applied field. The static dielectric constant e is usually determined from capacitance measurements

at audio frequencies, which also yield values of the

dielectric loss in terms of tan S, Q, or the parallel

resistance R, depending on the method of measurement. The dielectric constant of the sample is measured relative to that of free space. For a

parallel-plane capacitor of large area to thickness

ratio (i.e., negligible fringe field), e is related to the measured capacitance C by

e^CdjeoA. (15)

Here a is the total polarizability of a macroscopic

element of the material of volume V. Equation (8) is applicable for cubic or isotropic substances [3]. For anisotropic, linear dielectrics, e, Xi ^nd a are tensors and eq (1) must be written as D = e-E, (9)

with components

D,= Je,.,£,-(^=1,2,3). (10)

Other quantities foUow similarly.

Up to now we have considered only the static case. When a dielectric is subjected to an alternating field, both D and P will vary periodically with

time. In general, however, D and P cannot follow

the electric field instantaneously. There will al- ways be relaxation effects and losses in the dielectric, and these cause a lag in phase between E and

the response of the material. Thus for example if E = E() cos (ot, (11)

we have

D = Do cos {(ot-8), (12)

where 8 is the loss angle which is independent of

Eo, but generally depends on frequency [4]. Under

these conditions the dielectric function is written

as a complex quantity

Here d is the separation between planes, A is the area of the capacitor and €0 = 8.85 X 10'^ farads/

meter is the dielectric constant (permittivity) of

free space. For capacitors of small area to thickness

ratio, corrections for fringe fields can be made, or a

guard-ring arrangement requiring three-terminal

measurements can be employed. However, when the

primary interest in the measurement is the detection

of a phase transformation, then the absolute ac- curacy is not important, and two-terminal measure- ments without regard to fringe field effects are adequate. At frequencies up to several hundred

kHz the measurements can be made using a variety

of commercially available capacitance and impedance bridges.

Electrode attachment to the surfaces of solid

samples is very important since even small air gaps can introduce serious errors. Vapor coated metal

electrodes are best, but carefully prepared conducting paint brushed or sprayed-on, or fired-on silver

electrodes are often satisfactory. For measurements

in certain apparatus such as Bridgman anvils, the

faces of the anvils themselves can serve as

electrodes.

In the present study a transformer ratio-arm type

capacitance bridge (General Radio Type 1615-A)

with three-terminal connections was used.

The use of shielded electrical leads and sample

holders eliminated ground capacitance effects.

244
6.8 1 1 p-i I

I

I

1 |—11

1 11 1

1 1

1

1

1

1

1

1

1 1 r

6.6 -

TEMP. = 296°K

PRESSURE, kbar

Figure 1. The dielectric constants of single crystal samples o/NaCl, KCl, and RbCl

as functions of hydrostatic pressure.

KCl and RbCl transform from the NaCI-type structure to the CsCl type structure in the pressure range covered.

Sample capacitance and dielectric loss tan 8 were

measured at frequencies between 1 and 100 kHz.

Measurements were made as functions of pressure

and/or temperature in three different pressure

apparatus: a hydrostatic piston-cylinder using a 50-50 mixture of n-pentane and iso-pentane as pressure fluid, a cubic multianvil, and a Bridgman

anvil apparatus. The anvils were shielded from each

other to eliminate stray capacitances.

4. Results and Discussion

In the following we present and discuss results on

a variety of substances to illustrate the usefulness of

the technique. It is readily seen from eqs (8) and (15)

that a discontinuity in e, and hence the measured

sample capacitance, can be expected at a first-order

phase transition.

4.1. Alkali Halides

The alkali halides, with the exception of cesium

chloride, bromide, and iodide which crystallize in

the CsCl-type structure, crystallize in the NaCl-type

structure at standard conditions. However, it is well

known [5] that several of them, particularly the

chlorides, bromides, and iodides of potassium and

rubidium, undergo transitions from the NaCl to the

CsCl structure at high pressure. We have studied

the pressure dependence of the dielectric properties of many of these compounds to 25 kbar with

emphasis on the behavior near phase transitions.

The measurements were made in the hydrostatic

apparatus.

Figure 1 shows the effect of pressure on the dielectric constant of single crystal samples of NaCl,

KCl and RbCl. The latter two crystals undergo the

NaCl—* CsCl transition in the pressure range

covered. For all three crystals the dielectric constant

e in the NaCl phase decreases monotonically with

increasing pressure. For RbCl and KCl a sudden discontinuous increase in e is observed at the start of

the transition. The onset of the transition is also

accompanied by a loud burst-like sound. It is quite

remarkable that the sound of the transformation in

the relatively small crystals used (about 1 cmX 0.07cm thick) could be heard outside the massive

steel apparatus. On further increasing the pressure,

e rises at a rate which increases with increasing

pressure and finally goes through a maximum which

corresponds to the end of the transition. In the

CsCl phase e again decreases monotonically with

increasing pressure. The reverse transformation

exhibits a large hysteresis, and the calculated value

245
of e at 1 bar often does not return to its exact original value, probably because of small changes in

sample geometry caused by the transition.

Our measurements on RbBr, Rbl, KBr, and KI

show features very similar to those for RbCl and KCl

shown in figure 1. The macroscopic Clausius-Mossotti relationship,

eq (8), is apphcable for the alkali halides in both

the NaCl and CsCl phases. Its application at the

transition shows that, although e for each of the

above K and Rb salts increases by 35 to 40 percent

at the NaCl CsCl transition, the total polarizability per molecule is not strongly dependent on

crystal structure. Thus, for example, the data on RbCl and KCl in figure 1 combined with volume

changes of 13.7 and 11.4 percent at their respective

transitions, [5] yield for the ratio (ctNaci/oJcsci) values

of 1.00 for RbCl and 0.96 for KCl. Comparable

values are obtained for the bromides and iodides.

These results indicate that changes in e at these

transitions are mostly due to changes in volume.

The question of whether or not NaCl undergoes

a pressure-induced transition around 20 kbar has

received considerable attention in the recent

literature [6]. The results in figure 1 along with

elastic constant measurements [6] show no evidence

of a transition under hydrostatic conditions up to 26 kbar at 23 °C. Recent measurements using a diamond anvil x-ray cell [7] indicate that the

NaCl CsCl transition in NaCl does not occur

until — 300 kbar at room temperature.

We have also investigated KCl in a 1.25cm (flat)

Bridgman anvil apparatus. The samples were - (0.020-0.040) cm thick X 0.64 cm diam., fused

powder discs enclosed in a pyrophyllite ring. Typical results, reported as capacitance versus hydraulic

ram pressure, are shown in figure 2. The onset of

the transformation is fairly well defined, but the

transition is spread over a wide range. This difficulty

45 -

43

KCi

T - 23°C SAMPLE THICKNESS = 0.023 cm

SAMPLE THICKNESS = 0.040 cm

- 30

29 £

28

arises , of course, because of the large pressure

gradients across the flats of the anvils.

4.2. Thallous Halides

We have studied the pressure and temperature

dependence of the dielectric properties of TlCl,

TlBr, and Til in some detail [2]. TlCl and TlBr have

the CsCl structure and do not exhibit any trans- formations. Tll, on the other hand, crystallizes under

normal conditions in a double-layered orthorhombic

structure {Dll — Cjncm) with four molecules per

unit cell. It transforms to the cubic CsCl structure on increasing temperature (170 °C at 1 bar) and/or

pressure (4.7 kbar at 25 °C) [8]. The dielectric constant of Tll increases by 35

percent at the transition, [2] and this is reflected by

a large discontinuous increase in the measured

capacitance of the sample. Figure 3 shows typical

results. The two isotherms are for two diflfercent

samples, and the differences in the values of the

capacitance are due to diff"erences in sample

geometries. The transformation is characterized

by a large hysteresis which decreases with increasing temperature. The transition temperature de- creases very rapidly with pressure with an initial

slope dTJdp =- 53 ± 3 °C/kbar [8]. In a number of isothermal pressure experiments

we followed the progress of the Tll transformation

as a function of time at constant pressure (the

pressure being that at which the first indication of

the start of the transformation was noted) by monitoring the change in sample capacitance. Figure 4

shows the results of one such experiment. The

g

<

400 800 RAM LOAD, psi

1200 1600

30

28

27

26

25

24

23

22

20

19

1

1 1

1 1 1 1

o» 24»C

A A 75°C

1

ii 1,

t

c

1

1 1 1 1 1 1

14

o

z

<

12 <

2 3 4 5 PRESSURE- kbar

Figure 2. Capacitance versus applied load for KCl samples

run in a Bridgman anvil apparatus.

The transformation is spread over a wide pressure range due to pressure gradients across the anvil faces.

Figure 3. Capacitance versus hydrostatic pressure isotherms

for two different Tll samples.

The 24 °C isotherm data are read against the left ordinate, and the 75 °C isotherm

data are read against the right ordinate.

246
I 34

<

1 1 1

100% CUBIC

1 1 1 1

—o /

j

PRES. = 4.8 kbor

-

- TEMP = 23°C

' 100% ORTHORHOMBIC

1 1 1 1 1

-

1

1 0 5 10 15 20 25 30 35 40

TIME - minutes

Figure 4. Progress of the orthorhombic —* cubic transformation

in Til as a function of time at constant pressure and temperature.

transformation proceeds as a reaction rate process

approaching completion asymptotically. The percent completion can be taken as the fractional

change in capacitance between the limiting values

of the two phases. Thus it is seen that the technique

can be used for studying the kinetics of phase

transitions.

At high temperatures, the dielectric constant and

the dielectric loss, tan 8, of the thallous halides as

well as those of many, other ionic crystals, [9] in- crease exponentially with increasing temperature

and decrease with increasing frequency. In this

temperature region polarization effects associated

with dipoles produced by impurities and lattice

defects predominate, and the dielectric loss is dominated by the conductivity of the sample [2]. The

relationship is

tan 8 = e7e' = 47ro-/e'aj, (16)

where

o-=o-o exp {-WlkT). (17)

Here o) is the angular frequency, cr is the d-c conductivity, (To is a constant, and ^ is the activation

Table 1. Values of the activation energy (for the

formation and motion of defects) of the thallous

halides at 1 bar obtained from dielectric loss,

tan 8, measurements at high temperatures

Present results are compared with literature values obtained from ionic-conductivity measurements.

TlCl ^(eV) TlBr Til (ortho.) Tll (cubic)

Samara " 0.84 ± 0.05 0.85 ± 0.04 0.62 ± 0.04 " - 0.50

Lehfeldt < .79 + 0.03 .80 ± 0.05 Others " .75 to 0.87 .75 to 0.84

° Ref. [2], * Evaluated at 3 kbar. W. Lehfeldt, Z. Physik 85, 717 (1933). ''Values quoted by Lehfeldt.

2 3 4

ao"^/T)-°K'^

Figure 5. The temperature dependence of the dielectric loss o/TlBr.

In the high-temperature region tan S is proportional to the conductivity.

energy for the formation and motion of defects. It is seen from eqs (16) and (17) that activation energy

can be determined from tan 8 versus temperature

measurements at constant frequency.

Figure 5 shows log tan 8 versus IjT plots for TlBr

at three frequencies. At high temperatures the re- sults exhibit the expected linear dependence with

the slopes very nearly independent of (o. Similar

results were obtained for TlCl and Tll. Table 1 summarizes for the three compounds the activation

energies calculated from the slopes of the linear portions of the curves at high temperature. For comparison we also list in table 1 values of ^ for TlCl

and TlBr obtained from ionic conductivity measurements. It is seen that the agreement is quite good.

We did not extend the high temperature measure- ments of tan 8 to high pressures, but it is a simple

matter to do so, and thus study the effect of pres- sure on the activation energy.

4.3. Strontium Titanate

At standard conditions, SrTiO.i crystallizes in the

ideal cubic perovskite structure and exhibits normal

dielectric behavior (i.e., it is not ferroelectric or antiferroelectric). Although it does not exhibit any phase

transitions in the pressure and temperature ranges

of the present measurements, the data on it ex- tended to 50 kbar [10] and serve to illustrate the

247

385-762 O - 71 - 17
T

' r

PRESSURE, kbar

Figure 6. The pressure dependence of the dielectric constant e and l/e for single

crystal SrTiO.i.

general usefulness of the dielectric constant technique. In addition, the results have interesting implications for pressure calibration purposes.

The measurements were made on single crystals

and were performed in a cubic multianvil pressure

apparatus [11]. The samples were suspended in a nylon-teflon container filled with a low viscosity

silicone pressure fluid — a technique similar to that

described by Jayaraman [12]. The fluid undoubtedly

solidifies at the high pressure end, but this had no

apparent influence on the results. The pressure

dependence of e is shown in figure 6. A reversible

monotonic decrease of e with increasing pressure

is observed over the whole range.

It is known [13] that the temperature dependence

of e of SrTiO.) at constant pressure obeys a CurieWeiss law, e = CI{T-Ti)). The results in figure 6

show that the pressure dependence of e at constant

temperature obeys a similar relationship expressed

in terms of pressure, i.e.,

€=C*l{P-Po). (18)

At 25 °C, C*= 12,600 kbar and P., = -40 kbar.

The linear dependence of l/e on pressure over a wide pressure range suggests that SrTiOs can be

used as an accurate pressure gauge. In fact, this

feature of the results was used in determining the

pressure calibration in the range 0 to 15 kbar in the

present experiment. It is recalled here that the usual

calibration curves (i.e., sample pressure versus load curves based on "fixed" pressure points) for multianvil, belt, and similar apparatus exhibit anomalous

behavior in this low pressure range. For example.

such curves do not extrapolate to zero pressure for zero applied load [11]. In the present experiment,

the l/e versus pressure data above 15 kbar fell on a

straight line which extrapolated back to the zero pressure l/e value. Pressures in the 0 to 15 kbar

range were then determined from the linear l/e versus pressure response.

4.4. Water

The measurements of the dielectric constant and

dielectric loss are well-suited for studying liquidsolid transformations under pressure. We have

chosen water to illustrate this point. Water exhibits a truly remarkable pressure-temperature phase diagram. In addition to the liquid-ice transformation, at least eight different phases of ice have been identified [1, 14].

The present experimental setup was quite simple.

An empty parallel-plate capacitor was immersed in

triply distilled water contained in a glass tube open

at the top. The assembly was then placed in a sample holder inside the pressure chamber of the hydrostatic pressure apparatus. A 50-50 mixture of n- pentane and t50-pentane was used as pressure

fluid. Water is heavier than the pentane mixture,

and the two fluids have a very low miscibility so

that, for the present purposes, it was not necessary to keep them isolated from each other.

Sample capacitance and tan 8 were measured at 100 kHz as functions of pressure at room temperature. The results are shown in figure 7. Although the

water used was triply distilled, the dielectric loss

was very high, probably due to dissolved CO2. Both

248
the capacitance and tan S increased initially with in- creasing pressure. The leveled-off region (especially

in tan 8) between = 8 and 10 kbar represents the

upper limit for the sensitivity of the measuring

apparatus. Large discontinuous decreases in capacitance and tan 8 were observed at the water-ice VI

transition. This is mostly due to the large decrease in

orientational polarizability on freezing. Sharp de- creases in capacitance and tan 8 were also observed

at the ice Vl-ice VII transition. A hysteresis of ~ 0.5 kbar was observed at this transition. The

hysteresis at the water-ice VI boundary was not

measured because the glass tube broke on lowering

the pressure at this transition. This difficulty could

be avoided by using a plastic or metallic liquid

container.

Whalley and co-workers [T\ have been studying

the water-ice phase diagram in some detail using

the dielectric constant technique. Some of their

measurements were made in a Bridgman anvil apparatus at pressure to over 100 kbar. The water

was contained inside a thin mica ring between the

anvils. In addition to the determination of phase

boundaries, their results yielded information about

the nature of the relaxation processes in the vari- ous phases of ice.

T 23° C O ( A tan 6

LIQUID ICE VI

12 16 PRESSURE, kbar

5. Summary

In this paper we have presented a brief account

of the experimental and theoretical considerations

concerning the use of measurement of the dielectric

constant and dielectric loss as a probe for studying

pressure- and temperature-induced phase transformations in insulators. The general usefulness of

the technique was demonstrated by results obtained

from measurements in three different pressure apparatus: a 30-kbar hydrostatic unit, a cubic multianvil, and a Bridgman anvil apparatus. Measurements on a number of alkali and thallous halides,

strontium titanate and water were presented and

discussed. The technique is one of the most sensitive means for studying phase transformation in

insulators and can be easily applied.

6. References

[1] WhaUey, E., Heath, J. B. R., and Davidson, D. W., J. Chem.

Phys. 48, 2362 (1968) and references therein.

[2] See, e.g.. Samara, G. A., Phys. Rev. 165, 959 (1968).

Figure 7. Changes in the capacitance and dielectric loss of a

triply distilled water sample as functions of hydrostatic pressure.

The liquid —> ice VI and the ice VI —> ice VII transitions are clearly exhibited.

L3J Friihlich, H., Theory of Dielectrics (Clarendon Press,

Oxford, England, 1949), Appendix 3, p. 169. [4] In an ideal dielectric (loss-free) the current leads the

voltage by a phase angle </> of exactly 90°. In a real dielectric v^ith losses </> < 90°. The loss angle 6 measures

the angle between the current through the dielectric and

that which would flow through an ideal dielectric, i.e., 8 + (t) = 90°. [5] Bridgman, P. W., Proc. Am. Acad. Arts Sci. 74,21 (1940). [6] See e.g., Corll, J. A., and Samara, G. A., Solid State Comm.

4 , 283 (1966) and references therein.

[7] Bassett, W. A., Takahashi, T., Mao, H., and Weaver, J. S.,

J. Appl. Phys. 39, 319 (1968). [8] Samara, G. A., Wahers, L. C, and Northrop, D. A., J. Phys.

Chem. Solids 28, 1875 (1967). [9] Rao, K. V.,andSmakula,A.,J. Appl. Phys. 36,3953(1965).

[10] Samara, G. A., and Giardini, A. A., Phys. Rev. 140, A954

(1966). [11] Samara, G. A., Henius, A., and Giardini, A. A., ASME

Trans., J. Basic Engr. 86, 729 (1964). [12] Jayaraman, A., Proceedings of this conference.

[13] See e.g.. Samara, G. A., Phys. Rev. 151,378 (1966) and

references therein. [14] Bridgman, P. W., J. Chem. Phys. 3, 597 (1935).

DISCUSSION

Y. A. Atanov {All-Union Research Institute for Physical and Radiotechnical Measurements, Moscow):

What is your estimation of the reproducibility of the

pressure coefficient of the dielectric constant of

SrTiOs? It is my understanding that the dielectric

loss of water increases very rapidly with pressure,

v/hen you raise the pressure. Could you explain the

mechanism for this dielectric loss and why it increases?

G. S. Kell [National Research Council of Canada,

Ottawa, Ontario, Canada): I would like to point

out that as far as the behavior of water is concerned,

the permanent dipole moment makes a major

contribution.

I would also like to mention that there are at least eleven or twelve solid phases established for

water. The paper by Whalley (your ref. 1) lists nine.

Two of these are metastable. In addition to that there

249
is, I think, a phase in which the X-ray structure was

determined by McFarlan some place in the region of

two or three kilobars. There is cubic ice in the lowpressure region. Thus there are eleven phases which

can be prepared in bulk, and in addition there is the

amorphous phase which can be prepared by evaporation in small quantities.

Question from the floor: In the transformation of

the alkali halides, I was interested in the noise you

heard. Was this a result of the fracturing of the

crystal or what? Are the crystals intact afterwards?

AUTHORS' CLOSURE

Reply to Atanov: We have not made a systematic

study of the reproducibility of the pressure coefficient of the dielectric constant of SrTiO.-?. Results

from four or five independent measurements by us and others yield values of [^(l/e)/^/*]? that fall in

the range 7.7 -8.0 X lO-^kbar. On individual

samples I think the reproducibility should be better

than ±1 percent. It should, of course, be remembered that e is a strong function of temperature, and

good temperature control is required.

The dielectric loss of the water used was quite

high even though the sample was triply distilled.

We did not look into the cause of this because we

selected water simply to demonstrate the sensitivity

of the technique. Water is a highly polar substance

and dipolar relaxation effects are important. The

presence of some carbon dioxide dissolved in the

water may contribute to the high loss. The increase

of the loss with pressure is of course associated with

the increase of sample conductivity, but we do not

know at present whether this increase is an intrinsic

property of the sample or is due to impurities.

Reply to Kell: The total macroscopic polarizability a in eq (8) can of course be written as the sum

of three contributions, i.e.,

a = OLo,) + air + arf.

These contributions exhibit different frequency

response and can, therefore, be separated. The

optical, or high-frequency contribution, ccop, arises

from displacements of the electronic charge relative to the nucleus of each atom. The infrared contribution (often referred to as ionic contribution), «,>, arises from the combined effects of the displacements of the atoms and the resulting distortion of the

electronic charge. The dipolar contribution, is present in substances possessing permanent dipole

moments (e.g., water), and it is operative in the dc to millimeter frequency range. Even though it is possible to effect a separation of a into the above contributions on the basis of frequency response, each

contribution remains a macroscopic quantity which

generally does not tell us much about the polarizabilities of the individual atoms or ions.

Reply to question from the floor: The sound one

hears during the transformation of the alkali halides

is characteristic of martensitic transformations and

is quite familiar to people working on steels and

other alloys exhibiting such diffusionless transformations. It is associated with the rapid slip that occurs

during the transformation.

These transitions involve large volume changes

(14% in the case of RbCl) and after a crystal has

gone through the transition twice (on the up-anddown cycle), one recovers a polycrystalline mass,

but tile sample holds its integrity.

250
Experimental Determination of Curie Points of Ferromagnets up to 90

Kilobars — Possible Use For Calibration of High Pressure

J. M. Leger, C. Susse, and B. Vodar

C.N.R.S. —Laboratoire des Hautes Pressions —Bellevue, France

Second-order phase transformations which occur without volume discontinuity and theoretically

without any hysteresis are suggested as secondary pressure gauges and for interpolation under high pressure in different temperature ranges. Results are presented concerning the shifts of Curie temperatures of some ferromagnets up to 90 kbar. The pressure effect is generally small but may be accurately

determined.

Direct measurement of pressures above 20 kbars

is only possible in devices of the piston cylinder

type for which, in principle, the relation p= applied

force/piston area, applies. However even in these

devices, when a solid pressure transmitting medium

surrounds the sample, the measure of the pressure

is made inaccurate because it is impossible to determine the influence of the frictional forces by

increasing and decreasing pressure experiments.

This arises from the fact that the friction which

comes from the transmitting medium is unsymmetrical [1].' It is then necessary, in these devices as

in apparatuses of the compressible gasket type, to make a pressure calibration. This is usually achieved

either in a continuous manner in situ by measuring

the crystalline parameter of a standard by x-ray

diffraction, or more simply and more frequently by

monitoring some polymorphic 1st order transitions

such as those occuring in bismuth, thallium, cesium,

barium, iron, tin, lead .... These transitions are generally observed at room temperature by the

resistance discontinuity method, which is the easiest

to handle. The direct measurements of the pressure

transitions of these elements are made by volumetric measurements in a piston cylinder device without

any pressure transmitting medium. This has only

been made for the first four elements mentioned

above.

The use of first-order transitions as fixed points,

however, offers the following inconveniencies:

(a) The sudden contraction of volume of the standard at the instant of the transition cannot be

compensated for instantaneously by flow of the

surrounding material since it has a non zero re- sistance to shear; this ''cavitation" effect, well

known to the users, results in a spread of the

transitions and makes their location, imprecise.

This effect can lead to an apparent shift of the

transitions when the standard cell contains several

standards or when a standard suffers successive

transitions (bismuth, barium).

' Figures in brackets indicate the literature references at the end of this paper.

Paper presented at the Symposium on Accurate Characterization

of the High-Pressure Environment, held at the National Bureau of

Standards, Gaithersburg, Md., October 14-18, 1968.

(b) There is another major disadvantage in using

first order transformations; these transitions always

exhibit some hysteresis range related to the nucleation processes of the different phases, the amplitude

of which depends on various parameters (shape,

purity of the sample . . .). This hysteresis is not

negligible as is clearly shown by the following values

[2], in kbars, obtained at room temperature by x-ray

diffraction using an internal standard.

Bi I -II 2.8 to 5.3 BaI-II2.6

Tin -III 1.6 BiIII-V3.8

This hysteresis must necessarily be taken into ac- count in calibrations made at increasing pressure

(belts . . .) making use of data obtained by a mean

process in reversible working devices (pistoncylinder).

1. Second-Order Transformations in

Solids as a Means of Characterizing

the High Pressure Environment

The inconveniences quoted above of using first order transformations are eliminated if instead,

second order ones are employed as fixed points for

calibration. By definition the latter proceed without

any volume discontinuity, and theoretically without

hysteresis.

Little work at high pressures has been done so far on these transformations, probably because the

two most popular methods, the volume and the electrical resistivity discontinuity methods are not wei.

suited for their observation. However exploration of

their respective capabilities as secondary pressure

gauges would be most valuable. At first sight these

capabilities are quite different according to the

considered type of transformation: — order-disorder transformations in binary alloys

(a typical example is AuCua) offer the disadvantage

of being very slow. — transitions to a superconducting state, without

magnetic field applied, which occur at very low

temperatures would be of limited use. — Curie points of ferromagnets or some ferroelectrics.

251
CZl Pyrophyllite O Talc E3 Acier

Graphite

Q Alumine

Figure 1. Measurement cell.

Neel points of antiferromagnets appear to be the

most promising.

In the next section, we present our first resuhs

concerning the effect of pressure on Curie temperatures of several ferromagnets. Their possible use as pressure gauges is discussed below.

2. Curie Points of Some Ferromagnets

as a Function of Pressure

2.1. Present Theoretical State

For every second-order phase change, the behaviour of the transition temperature as a function

of pressure can be described by the general relations established by Ehrenferst and which corre- spond to the Clapeyron equation for first order

transformations

:

''index 1 old phase

index 2 new phase

ct' isobaric thermal ex- pansion coefficient

X isothermal

compressibility

,Cp isobaric specific heat

dp aL—oi[ Cpi —Cpi

These relations do not allow calculation of the

pressure effect, because the necessary data (Cp,

a'. . .) have not been determined under pressure.

It is also impossible to compute this effect from the

Landau's theory in which the thermodynamic

30

20

10

Nicki 1

y

y

y

y

y

P - Kb . 40 60 ao 100

Figure 2. Shift ofthe Curie temperature ofnickel under pressure.

potential is expanded in powers of an order parameter, because this expansion involves coefficients

only known at ordinary pressure (magnetostriction

constant . . .). On the other hand, work based on

microscopic models is very scarce. The Heisenberg

model does not allow understanding of the pressure

effect in detail for the transition metals of the first period. A model with itinerant electrons has been

proposed in the case of nickel [4], but the result,

though giving the right sign for the variations, is

still approximative.

Thus so far it has been impossible to predict the

pressure effect on Curie temperatures from a theoretical point of view, so that they cannot be used

as primary pressure gauges. However for the reasons mentioned earlier they are well suited to be employed as secondary standards.

2.2. Experimental Method

To determine the shifts of the Curie temperatures

under the effect of pressure we have employed the

transformer method, formerly used by several

authors [5, 6, 7]: without magnetic field applied, one

notes the variations of the permeability fjL of the

sample as a function of temperature; in our experiments this sample has the shape of a torroid on which the primary and secondary windings are made

from a coaxial cable. The primary is supplied with an ac current of constant intensity; the secondary

voltage and the temperature are fed onto an XY

recorder and we directly obtain the iJi=f{T) curves

at different constant pressures. The inflexion points

of these curves determine the Curie points.

This transformer method is easy to handle under

pressure and allows us the determination of small

shifts of Curie points, even at rather high temperatures; it must be noticed that it does not always give

the exact value of the Curie temperature; generally

we get a value a Uttle too low, because the method

using a low magnetic field is not very conventional.

We performed all our experiments in profiled

piston devices [8] of the belt type, calibrated at

252
550

50 60 70

PRESSION ( Kilobars )

Figure 3. Curie point and a-y transformation of iron under pressure. room temperature according to the usual technique

by monitoring the variations of the electrical resistivity of bismuth and barium. The corresponding

transition pressures have been chosen equal to 25.4 kbar for Bi I -II, 58.6 kbar for Ba I -II, 82

kbar for Bi III-V.

The design of the measurement cell is shown in

figure 1; the sample is completely surrounded by

talc for moderate temperatures or boron nitride for

high temperatures. The temperature is measured

with a chromel-alumel thermocouple made of a "thermocoax" cable. The use of such couples seems

to be desirable in high pressure experiments

because the outer jacket, made of stainless steel,

protects the wires and prevents ruptures. In no

case was a correction made to take account of the

pressure effect on the emf of thermocouples. The

pressure and temperature ranges so covered are

15 to 90 kbar and 200 to 1,500 K.

2.3. Experimental Results

Most of the results we obtained [9, 10, 11] con- cerning the shifts of Curie points as a function of

pressure are shown in figures 2 to 9. The corre- sponding Curie temperatures measured at atmospheric pressure are given in table 1.

Table 1. Curie temperatures measured at ordinary

pressure {in K) by the transformer method

Fe 1044 30Ni -70Fe 334 30Ni -70Co 1209

36Ni -64Fe 486 45Ni -55Co 1132

53Ni -47Fe 788 60Ni -40Co 1022

Ni 627 64Ni -36Fe 863 75Ni -25Co 900

Co 1388 75Ni -25Fe 852 93Ni - 7Co 726

93Ni - 7Fe 708

Aecc)

60 Ml - 40 C 0 1^—

P( Kb )

Figure 4. Shift of the Curie temperature of an alloy 60 Ni-40

Co under pressure.

253
The experimental method and the features of the

transformations themselves allow accurate determination of the transitions for the following reasons:

(1) It is possible to heat or cool the high pressure

cell at any given speed; particularly one can proceed

very slowly so that the phenomena associated with

the kinetics of the transformations are eliminated.

In our case, as Curie points correspond to rapid

transitions, a low rate of heating simply permits a better measurement of the temperature of the

sample. Therefore the situation is more favorable

than when differential thermal analysis is used in a high pressure apparatus where thermal losses are

large and so require high rates of heating.

(2) The shape of the sample is such that the

temperature gradient present is negligible; no evidence of it was found at 1400 K during the study of

cobalt.

(3) The geometry of the sample is also such that

its internal pressure gradient is very small; further,

since it has been shown [12] that an uniaxial stress

has the same effect on Curie points as an isotropic

-50

-200

t e CO

\

\ 1 N V

36 Nl

A R

64 Fc

<

10 20 30 40 SO 60 P(Kb)

stress of the same intensity, divided by three, the

error introduced due to pressure gradient is very

small. The pressure gradient shows its effect on the

/A=/(r) curves by the appearance of a "tail" — for

instance, in the case of nickel, the transition width

is about 1.5 °C at p = 0 and about 3°C above

p = 25 kbar. It is possible to take account of this

phenomenon to define the Curie point, (fig. 10).

(4) The Curie temperature is not very sensitive

to nonmagnetic impurities, in contrast to the

magnetic permeability or electrical resistivity.

(5) The results for iron, nickel, and cobalt

confirm the absence of hysteresis effect at all pressures as predicted by Landau's theory. This

is displayed in the curves (fig. 10) obtained on cooling

and heating.

(6) In all cases the results were found to be

perfectly reproducible.

Figure 5. Shift ofthe Curie temperature of invar under pressure.

Figure 6. Shifts of the Curie temperature of iron nickel alloys

under pressure (the weight composition in nickel is indicated near each curve).

49 CC )

,

A5Ni-5SCc

75Ni-25

==^^60 Ni

Co

- 40 Co

Co

) Ni-7Co

"Tooin"

ON i - 100 Co

Figure 7. Shifts of the Curie temperatures of nickel-cobalt alloys under pressure.

254
T'C

—

1

0<AF

/ ) /

/

/

/

10 20 30 iO SO 60 70 M P(Kb)

Figure 8. Phase transformations under pressure for an iron

rhodium alloy.

3i5

335'

T°C

325

at 5

305 /

50re-i6Rh-iir

k

10 20 30 40 50 60 P(Kb)

Figure 9. Phase transformations under pressure for a ternary

alloy 50 Fe-46 Rh-4 Ir. ( D.T.A.; permeability method). The following remarks regarding the results

ought to be mentioned: — In most cases the shifts of the Curie temperature are small. Their use as pressure gauges would

therefore require a very careful control of the temperature. Per contra by a suitable choice of the metal

or alloy (table 1), it should be possible to pressure

calibrate a device at any given quasi-constant

temperature; this would be an advantage in certain

cases. — The sharpest transitions are observed for pure

metals. Unfortunately for iron and cobalt the shifts

of the Curie points are practically zero. In any case

the a — y transition in iron would preclude its use above 15 kbar. On the other hand, nickel would be

useable. For this metal dTddp is constant up to 60 kbar and equal to +0,357kbar.

Figure 10. Changes in permeability for nickel under various

pressures in the vicinity of Curie point.

60

50

40

30

20

10

P(ltb)

y

'

1

0

1

1 pressure

' J

Figure 11. Calibration curve of a belt type apparatus deter- mined from shifts of Curie temperatures. — Alloys seem to be less favourable because they

generally present a small range of hysteresis; this

is particularly noticeable for Ni-Fe alloys and

precludes their use. Co-Ni alloys offer greater

capabilities, the hysteresis being smaller and the

shifts of the Curie points larger. — The preceding results can be related to those of

McWhan [6] on rare earths metals and alloys. The

order points behaviour of these materials under pres- sure is rather complex due to the presence of first- order transitions and sometimes of metastable

states. Therefore one cannot think of using them as

255
pressure gauges no more than the iron-rhodium

based alloys we studied and which equally show a

first-order transition, (fig. 8-9).

In the course of our work we had the opportunity

of using the Curie point of nickel as a pressure gauge

to determine the calibration curve of one of our apparatuses in the range 0—25 kbar where there is no

convenient standard. The obtained curve (fig. 11)

effectively shows a shape characteristic of devices

with compressible gaskets.

3. Conclusion

Compared to first order phase transformations

the second order ones exhibit two major advantages

as possible pressure gauges: they occur without any

volume discontinuity and theoretically without

hysteresis.

In the particular case of Curie points of ferromagnets the shifts so far measured under pressure

are unfortunately small. However it is possible to determine them accurately because they are generally rapid and reproducible. Moreover it is possible, in principle, to fix the Curie temperature of

the standard at a given value by proper choice of the

alloy composition, which allows calibration in the

neighbourhood of a given temperature.

It would be necessary to follow up this research on

other ferromagnets for which pressure would possibly bring larger changes in the Curie points. More

generally the study of other types of second order

transformations, particularly in certain ferroelectrics

and antiferromagnets, should be developed under

very high pressure and would perhaps lead to the

discovery of new pressure gauges.

4. References

[1] Pistorius, C. W. F. T., Rapoport, E., Clark, J. B., Rev. Sci.

Inst. 38, 12, 1741 (1967).

[2] Jeffery, R. N., Barnett. J. D., Vanfleet, H. B., Hall, H. T.,

J. Appl. Phys. 37, 8, 3172 (1966).

[3] Belov, K. P., in Magnetic Transitions, Consultants Bureau

N.Y. (1961).

[4] Lang, N. D., Ehrenreich, H., J. Appl. Phys. 38, 1316 (1967).

[5] Patrick, L., Phys. Rev. 93, 3, 384 (1954).

[6] McWhan, D. B., Stevens, A. L., Phys. Rev. 139, 3A, A682

(1965).

[7] Samara, G. A., Giardini, A. A., Rev. Sci. Instr. 36, 1, 108

(1965).

[8] Susse, C, Epain, R., Vodar, B., C.R. Acad. Sci. Paris 258,

4513 (1964).

[9] Leger, J. M., Susse, C, Vodar, B., Solid State Com. 4, 503

(1966). [10] Leger, J. M., Susse, C, Vodar, B., Solid State Com. 5, 755

(1967). [11] Leger, J. M., Susse, C, Vodar, B., C.R. Acad. Sci. Paris

265, C, 17, 892 (1967).

[12] Belov, K. P., Kadomtseva, A. M., Vecm. Mock. Ynib. 2, 133

(1958).

DISCUSSION

G. A. Samara (Sandia Laboratory, Albuquerque,

New Mexico): In figure 10, 1 believe that the "tailing"

off of the curves at temperatures above the Curie

point is most likely due to the fact that although

the long-range order is lost there are still shortrange order effects. We have found this "tailing"

effect even under hydrostatic pressure.

O. E. Jones (Sandia Laboratory, Albuquerque,

New Mexico): We have made some shock compression studies on a ferromagnetic 70% iron-30% nickel

alloy. The shift of Curie temperature was comparable to that obtained in hydrostatic experiments,

showing that here the effect of shear is not particularly important.

D. B. IVIcWhan (Bell Telephone Laboratories,

Murray Hill, New Jersey): On a curve of initial

susceptibility versus temperature, the Curie point

is sometimes taken as the inflection point, and

sometimes as the intercept of the extrapolation to the background susceptibility. Defined either way,

it is not clear that your values of the pressure dependence are free of an uncertainty due to possible

changes in the shape of the curves with pressure.

I would also like to note that if the Curie temperature shift is plotted as a function of volume, much

of the curvature in the curves for nickel would be

eliminated.

AUTHORS'

On the results of our work on nickel, at the

beginning we felt that the curves for the shift of

the Curie temperature would be found to be linear;

but this was not so at the highest pressure range,

whatever may be the pressure scale and even if this shift is plotted as a function of the volume.

We think that a maximum most probably occurs at a higher pressure; this could be deduced from the

CLOSURE

curves obtained for alloys, particularly iron-nickel ones where we did observe that maximum for one of

the alloys we studied. More work on this matter is needed. Although comparison with data obtained

by shock techniques are useful, the experiments

should preferably be performed in conditions of

pressure not too far from hydrostatic ones.

256
Fixed Points on the High-Pressure Scale Identified by Phase

Transitions in Ammonium Fluoride

R. Kaneda, S. Yamamoto, and K. Nishibata

National Research Laboratory of Metrology, Japan

The I-II and II-III transitions in NHjF, as fixed points on the high-pressure scale, have been in- vestigated under purely hydrostatic conditions at the temperatures of 0, 25 and 50 °C. By means of manganin resistance gages, the transition pressures at 25 °C have been determined to be 3605 ±10

bar for the I-II and 11531 ±23 bar for the II-III. Their respective dependences on temperature are + 2.5 bar/°C and +20.7 bar/°C. The realization of both transition points is easy at and above room

temperature, but at low temperatures the ll-lIl transition is very sluggish.

1. Introduction 2. Experimental Method

In almost all high-pressure works, usually manganin resistance and other secondary gages are used

for measuring fluid pressures, and for gage-calibrations up to 20 kbar, fixed points identified by liquidsolid phase transitions in Hg and H2O have commonly been used. However, the accurate realization

of these fixed points would not be easy, because the

pressures depend strongly on temperature. The

realization of triple points in H2O is also complex

and much use of them may be doubtful. This re- quires a new fixed point.

In view of insensitivity to temperature and easi- ness in detection of transitions, it may be argued

that solid-solid phase transitions in NH4F, which

occur twice at pertinent pressure intervals with

associated large volume changes, offer convenient

fixed points.

Of the phase transitions in NH4F, the first study

was done by Stevenson [1] ' who discovered a highpressure modification NH4F ll at about 3.8 kbar.

Later, Swenson andTedeschi [2] reported the second

high-pressure phase NH4F ill formed at 11.5 kbar

at room temperature. Recently, Kuriakose and

WhaUey [3] investigated the phase diagram to 340 °C

and 20 kbar, and found a new phase NH4F IV at temperatures over 160 °C. All these investigations were made with simple piston-cylinder equipments,

in which NH4F samples were assumed to act as pressure-transmitting media and subjected to stress

fields with non-hydrostatic components. Only one

previous work carried out under purely hydrostatic

conditions was of Morosin and Schirber [4]. Except

some suggestion on the crystal structure, their x-ray

investigation of NH4F II gave no information on

behavior of the transition.

The purpose of our study is to evaluate the

feasibility of the NH4F I-II and II-III transitions for

the fixed points, as well as to give more light upon

their behaviors in purely hydrostatic conditions.

The experiments are based on the volume change

method, which is the usual method to locate polymorphic transitions under pressure. Briefly, the ex- periment consists in pressurizing an NH4F sample

hydrostatically to observe the phase transition, as detected by the discontinuity in pressure corre- sponding to the change in volume of the sample,

and in measuring the equilibrium pressure by means

of a manganin resistance gage calibrated against a

free-piston gage.

2.1. Apparatus

The arrangement of the apparatus excluding its auxilliary parts is shown in figure 1. The pressuretransmitting fluid used was pure white gasoline. The

pressure in the system is generated by two in- tensifiers; one is the priming for compressing gasoline to about 3 kbar, and the other for producing

steady pressures up to 20 kbar. For monitoring the

Figure 1. Arrangement of apparatus.

' Figures in brackets indicate the literature references at the end of this paper.

Paper presented at the Symposium on Accurate Characterization

of the High-Pressure Environment, held at the National Bureau of

Standards, Gaithersburg, Md., October 14-18, 1968.

A: Pressure vessel

B: Bath

C: Thermocouple

I,: 20kb Intensifier

I2: 5kb Intensifier Gi, G-i. G3: Manganin

resistance gages

S: Sample

V,, v.: Valves

257
Figure 2. Details of pressure vessel.

advance of the piston, the 20 kbar intensifier is provided with an electrical displacement transducer

connected to a multi-pen recorder. As the lengths of

connection lines composed by tubings of 2 mm i.d. are reduced to a minimum, no time effect in pres- sure transmission is present.

As shown in figure 2, the pressure vessel having a

cavity 15 mm diam by 140 mm long is of 18 percent

Ni maraging steel, with a pressure port at the lower

end, and the upper end a mashroom-type closure,

The NH4F sample of about 5 g was hermetically

enclosed, together with white gasoline of nearly a

half amount of the sample, in a pliant bag made of a polyethylene sheet of 0.05 mm thick. Care was

taken to flush out all air bubbles. The sample was

then set in a thin-walled stainless-steel container,

which was in turn placed in the pressure vessel. The

container, with a threaded cap holed so as to provide free access of the fluid, acts as a sample

holder preventing loss of the NH4F in the event of

breakage of the bag. In the experimental runs, the

bag was sometimes broken; nevertheless it kept the

sample free from the constraints exerted by the

walls of the container. Thus, such arrangement as above secured the sample all right, being subjected

to purely hydrostatic pressures.

The pressure vessel containing the sample was

thermostatically regulated by immersing it completely in a PID-controlled Hquid (methyl-alcohol)

bath. The bath, having a capacity of 128 liters, provided with heaters, refrigerators, and a motor-driven

stirrer, is able to keep the temperature of the pres- sure vessel constant to ± 0.01 °C at any desired temperature in the range from —30 to 60 °C. Two copperconstantan thermocouples calibrated against a standard platinum resistance thermometer were

used; one was for monitoring the bath temperature,

and the other for measuring the temperature of the

sample. The latter thermocouple was inserted in the

pressure vessel so as to attach to the upper side of

the mashroom-type closure about 100 mm from the

sample. From the subsidiary experiments carried

out at atmospheric pressure, the relation between

the temperature of the sample and that of the upper

side of the closure was obtained. By applying this

relation to the actual runs, the temperatures of the

sample were determined within the accuracy of

±0.05 °C.

All significant pressure measurements were made

with the manganin resistance gages calibrated

against the Laboratory's standard 15 kbar freepiston gage. The apparatus is provided with three

manganin resistance gages as shown in figure 1,

two of which are connected to the pressure vessel

for measuring the pressure on the NH4F sample at any time, and the remaining one, acting as a reference standard, connected to the high-pressure

side of the intensifier.

The gage Gi, for monitoring the pressure, was

connected to the preceding recorder, with which the

transition was located. For more precise determinations, the gages G2 and G3 were used and

connected to a Leeds and Northrup G-3 Mueller

bridge. There might be undesirable temperature

effects in the gage G2, as it was situated close to the

bath. On the other hand, the gage G3 was, some 2

m apart from the bath, always kept at room temperature, at which all gage-calibrations were carried out,

but it could not be used directly for the purpose of

determinations, since the complete equilibrium

state in the NH4F was attained with a valve Vi

closed. Therefore, during the experimental run,

occasional intercomparisons of these two gages were made, and the equilibrium pressure was determined

by the gage G2. Before every experimental run, the

gage G3, as the reference standard, was usually

calibrated with the free-piston gage.

Several manganin coils for the present work were

prepared and examined. However, in some coils, the

pressure-resistance relationship was not so consistent as had been expected. The inconsistency of p-r

relationship was almost entirely due to the lack of

reproducibility of resistance in the lower pressure

range up to about 3 kbar. In the higher pressure

range, the fractional resistance change with the

pressure variation was sufficiently consistent. With

a few exceptions the reproducibility or uncertainty

258
of the manganin resistance gages, which scarcely

depended on pressure, was found to be ±4 bar.

The free-piston gage is of the controUed-clearance

type and has the estimated accuracy of ±0.05 percent of the pressure to be measured. The total errors

in the pressure measurements near 4 kbar and 12

kbar may therefore be assumed to be ± 6 bar and

± 10 bar, respectively.

2.2. Observational Procedure

Prior to a final experimental run, with building up

pressure, the NH4F sample was exposed to a few

runs through the MI and II-III transitions. Through

these subsidiary runs, the sample resulted in a fine

grain size and subsequent transformations became

more smooth.

Firstly, the bath was set up at a constant temperature, and the pressure in the system was raised to the extent a little lower than needed to initiate a

transition. With keeping this state for awhile (until

temperature disturbances due to the compression

heat disappeared), the gage G2 was checked by

comparing it with the reference standard G3. Then

the pressure was increased step by step until the

transition began as indicated by the drop in pressure

due to the loss in volume of the NH4F; and with the

valve Vi closed, the pressure vessel was isolated

from the rest of the system. The sample in the pres- sure vessel was thus allowed to reach automatically

the exact equilibrium state and the equilibrium

pressure was determined by the gage G2. After

that, the valve was opened and the pressure, if necessary, was increased further in order to make all amount of the NH4F transform completely, and

the intercomparison of G2 and G3 was made again.

The pressure was now decreased for observing

the reverse transition. With releasing pressure below

the equilibrium pressure and the valve closed, the

system was allowed to make its pressure rise automatically to the equilibrium line from below, and the

equilibrium pressure reestablished was determined.

In an experimental run, repeated observations of

both the I-II and II-III equilibrium pressures attained

in this way were usually made several times. For

the purpose of locating the I-II and II-III transitions, pressure steps about 100 bar and 200 bar,

respectively, were adopted. The magnitudes of

these steps were, of course, much smaller as compared with those of the "carry back" pressure

ranges of the transitions, which depended on total

volume changes of the sample, compressibility of

the fluid, etc. The respective "carry back" pressure

ranges were roughly 800 bar and 600 bar.

2.3. Sample

The sample used was the special-grade-reagent

NH4F supplied from Wako Pure Chemical In- dustries, Ltd., Tokyo. To the sample, no special

refining treatment was made.

3. Experimental Results

The experiments were performed at 0, 25 and

50 °C. Further, a few experiments were attempted at — 25 °C and the I-II transition was surely observed at about 3.47 kbar, but the II-III transition could not

be detected in spite of raising pressure on the

sample up to 14 kbar.

The experimental data are summarized in table

1, which shows the equilibrium pressures attained

from both lower and higher directions. The equilibrium pressure is taken as that pressure at which

the rate of change in pressure is reduced to less

than 0.1 bar/min. For each experimental run, the

pressure value is the mean of 3 to 5 observations,

except the II-III at 0 °C, and the dispersion of the

observations is represented by the maximum deviation from the mean. The temperature value is the

mean of all readings during the run. The dispersion

of these readings, not shown in table 1, did not

exceed ±0.02 °C.

Table 1. Experimental results

Run Temp. Equilibrium pressure (bar)

No. (°C) n II I II III III II

1 -0.05 3541 ± 1 3537 +

1

2 0.03 3540 ± 1 3538 It 1 11382 10596 ±112

3 -0.01 3536 ± 1 3535 ± 1 11173±75 10756

4 25.11 3607 ± 1 3603 ± 1 11540 11526

5 25.10 3607 ± 1 3605 ± 1 11540± 2 11528± 3

6 25.00 3604 ± 1 3602 ± 1 11533± 3 11525+ 2

7 50.25 3662 ± 1 3661 ±1 12014 ± 2 12007 ± 3 8 49.92 3668 ± 1 3666 ± 1 12015 ± 2 12011+ 3

In the experimental run, the dispersion of observed values for the I-Il did not exceed ± 1 bar,

and that for the II-III, except at 0°C, was in the

limits of ±3 bar. There was no recognizable dependence of the equilibrium pressure on the fraction of the NH4F sample transformed. However, a

slight difference in the pressure value was present

between the different runs. The magnitude of the

difference, i.e., several bars, corresponded nearly to the uncertainty of the manganin resistance gages.

Therefore, it might be supposed that the difference

was mainly due to the characteristics of the gages

and not due to the nature of the transition. The

reproducibility of the I-II transition on all occasions

and that of the II-III at room temperature or higher

were fairly good. On the other hand, the II-III transition at 0 °C was so sluggish that only one or two observations were made in the run. The dispersion and the difference in the pressure value

amounted to 112 bar and 284 bar, respectively. This

proved the whimsical character of the II-III transition at low temperatures.

259
Table 2. Equilibrium and transition pressures

Temp.

CQ

Equilibrium pressure

(bar)

Transition pressure

(bar)

l-^ II II I MI

0

25

50

3539 ± 9 3606 ± 9 3665 + 11

3537 ± 9 3604 ± 8 3663 ± 9

3538 ± 10 3605 ± 10 3664 ± 12

11^ III III -> II II-III

0

25

50

11278 ±190

11536± 17 12013 ± 16

10676 ±203

11525+ 17 12008 ± 19

10977 ± 504

11531 ± 23

12011 ± 22

From the data shown in table 1, the optimum

transition pressures at just 0, 25 and 50 °C were

deduced. The results are given in table 2. The

transition pressure is the mean of the equilibrium

pressures. The uncertainty of the transition pressure

is evaluated by adding a half width of the "region

of indifference" to the maximum hmit of estimated

_uncertainty of the equilibrium pressure. The values

of 3605 ±10 bar and 11531 ±23 bar are ascribed

to the NH4F I-II and II-III transition pressures at 25 °C, respectively. The widths of the region of

indifference or the hystereses of the transitions at 25 °C are the order of 0.1 percent of the respective

transition pressures, while the II-III transition

at 0 °C has the region as great as some 600 bar, or about 5.5 percent. It is recognized that the

region of indifference becomes narrower at higher

temperature.

The dependence of transition pressure on temperature, dp/dT, calculated from table 2 is shown

in table 3. The dp/dT at 25 °C or the average values

of dpjdT in the range from 0 to 50 °C are +2.5

bar/°C for the I-II and +20.7 bar/°C for the II-III. Table 3 shows that both the dependences become

smaller at higher temperature.

The other characteristics of the transitions

observed are as follows.

At room temperature or higher, both the I-II and

II-III transitions run rapidly. Particularly, the transition velocity of the I-II was much greater than had

been anticipated; that is, once the transition was

initiated, the volume change followed so rapidly the

advance of the piston of the intensifier that the gage

Gi connected to the recorder did not show any

change in pressure. At the low temperature (0 °C),

the II-III transition was intolerably sluggish. Table 4

lists the observed equilibrium times which depend

on not only temperature but also the direction of

the transition.

Figure 3 presents the typical transition curves obtained from an experimental run at 25 °C. As these

unsymmetrical "pressure-time" curves indicate,

there is considerable difference in behavior between the directions of the transition. As to the I-II

Table 3. Dependences of transition pressures on

temperature

Temp, range dpIdT (bar/°C)

(°C)

I-II II-III

0-25 + 2.7 + 22.2 25-50 + 2.4 + 19.2

Average + 2.5 + 20.7

Table 4. Equilibrium times

Temp. Equilibrium time (min)

(°C)

I II 11^ I II III III II

0 25 15 60- 200-

25 15 10 25 90

50 10 5 15 50

Table 5. Volume and entropy changes at transitions

Transition AF/F„(%) AS(calK-' -mol-')

I-II -24.0 -0.53

II-III -9.5 -1.78

transition, the I—^11 reaction was inferior to the reverse II —> I in such characteristics as the velocity

of reaction, the width of the super (sub)-pressing

region or the overshoot pressure range, and perhaps

the reproducibility of the equilibrium pressure.

Oppositely, in the II-III transition, the II —* III reaction was far superior to the reverse III II in such

characteristics as mentioned above. The approximate magnitudes of the overshoot pressure ranges at 25 °C were 100 bar for the I^ II, 50 bar for the

11^ I, 200 bar for the II ill and 350 bar for the

III^ II, respectively.

Although the present work was not designed for

precise determinations of volume changes AV, the

total piston displacements observed on the recorder

allowed rough determinations of A F at the transitions. The results are shown in table 5. In conjunction with dpIdT, AV allow to calculate entropy

changes A.S{= AV-dp/dT) , which are also shown in

table 5. The initial volume of the NH4F sample

was measured by a Beckman air comparison pycnometer.

4. Discussion and Conclusions

The determinations of the transition pressures of

NH4F have been carried out under purely hydrostatic conditions, and the values of 3605 ±10 bar

for the I-II and 11531 ±23 bar for the II-III have

been obtained at 25 °C. Our values are not incon260
II -« III reaction

o

eg

Equilibrium line

JL

—> II reaction

oo

I —> II reacti

Equilibrium line \

II —» I reaction

_L J

1 2 3 4 5 min

Figure 3. Pressure-time carves at transitions.

sistent with previous workers', though seldom agreeing in detail. For the I-II transition, Stevenson's

value is 3.8 ±0.3 kbar and that of Swenson and

Tedeschi is 3.8 ±0.2 kbar, both at room temperature. Kuriakose and Whalley have reported the

value of 3648 bar at 22.5 °C. Their value, ex- trapolated to 25 °C, is 3653 bar. Morosin and

Schirber's value of 3.63 ±0.02 kbar, which was obtained at 23 ± 1 °C under purely hydrostatic conditions, is fairly close to our value. For the II-III transition, the value at 21.5 °C is 11.61 ±0.09 kbar

according to Kuriakose and Whalley. Their value,

extrapolated to 25 °C, is 11.67 ±0.09 kbar. Swenson

and Tedeschi's value of 11.5 kbar at room temperature is almost identical to our value, though the

uncertainty involved in their value is unknown.

The other data for the transitions, i.e., dp/dT,

and AS, given in this paper are fairly close to those of Kuriakose and Whalley [3], except the sign

of dldT{dpldT). The sign of dldT{dpldT) for the

II-III transition is positive according to them, while

that of ours is negative. This conflict may be caused

by the great uncertainty involved in our value of

the II-III transition pressure at 0 °C.

A principal virtue of the present experiments

lies in observing the true behaviors of the NH4F

transitions, which was secured by the satisfactory

arrangement of the sample having been subjected to purely hydrostatic pressure, complete freedom from

pressure leak and precise monitoring of pressure

and temperature. Through the experiments, it has

been confirmed that both the I-ll and II-III transitions

have the following characteristic natures which

mostly satisfy the necessary conditions for the fixed

points.

(a) The transition pressures have fairly good

reproducibilities, i.e., about ±0.1 percent, and the

widths of the region of indifference are narrow, i.e.,

about 0.1 percent of the respective transition

pressures.

(b) The dependences of the transition pressures on temperature are very small, i.e., the respective

dpIdT are the orders of 1/80 and 1/10 of that of the

freezing point of Hg.

(c) The transitions run rapidly. The approximate

equilibrium times are 10 min and 30 min, respectively.

(d) The transitions are easily detected by the

volume change method. The relative AF are roughly

24 percent and 9.5 percent, respectively.

However, the II-III transition is very sluggish and

whimsical at low temperatures, so that the transition points should be realized at and above room

temperature. To improve the accuracy of realization

of the transition points, it is desirable to select the

direction of transitions; the 11^ I and 11^ III reactions are more satisfactory.

Thus, it may be concluded that the solid-solid

phase transitions in NH4F offer most convenient

fixed points. Moreover, they may be conductive to realize an idea for a "self-calibrated" pressure gage

or a working device accompanied with fixed points.

As additional remarks, it was recognized that the

lack of reproducibility of manganin resistance

261
gages, which put some restrictions on the accuracy

of the present work, was mainly due to insufficient

consistency of pressure-resistance relationship in

the low pressure range. This means that the manganin resistance gage, even if it is used only in the

higher pressure region, should be checked once with

a fixed point in the lower pressure region like the

NH4F I-II transition point.

It is expected that further determinations by the

direct use of the free-piston gage will be carried out.

5. References

[1] Stevenson, R., J. Chem. Phys. 34 , 346 (1961).

[2] Swenson, C. A., and Tedeschi, J. R., J. Chem. Phys. 40,

1141 (1964).

[3] Kuriakose, A. K., and Whalley, E., J. Chem. Phys. 48,

2025 (1968).

[4] Morof in, B., and Schirber, J. E., J. Chem. Phys. 42, 1389

(196.5).

DISCUSSION

F. R. Boyd, Jr. {Geophysical Laboratory, Carnegie

Institution of Washington, Washington, D.C.): I am

impressed by the large volume changes of these two

transitions, the total change in going from phase I to phase III being something like 35 percent. I know of no other substance that shows that high a change.

It might be a useful substance in the supporting

stage of a two-stage device, in which alkali halides

have frequently been used to give a big volume

change in order to get a long stroke.

D. B. McWhan (Bell Telephone Laboratories,

Murray Hill, NewJersey): I think that Pauling showed

some time ago that ammonium fluoride can form a

solid solution with ice. It is one of few such materials, and for this reason there could be a problem

due to the water content.

AUTHORS' CLOSURE

The authors quite agree with Dr. Boyd's point of

view, but as yet they have made no attempt at using

ammonium fluoride as a supporting material for a two-stage device.

In reply to Dr. McWhan's comments, there is probably some difference in water content among

samples, as this substance is extremely hygroscopic.

The authors wish to point out that subsequent to the preseritation of the paper, additional experiments were carried out for a different sample of

ammonium fluoride (Ist-grade-reagent from Kanto

Chemical Co., Inc.) which had been stored without

a dessicaior for a few years. Of this sample, the

transition pressures obtained at 25 °C were 3606 bar

and 11533 bar for the l-II and ll-III, respectively.

These values compare with those of the sample

given in the paper and the close agreement might

be an indication that the water does not greatly

affect the transition pressures. However, further

work under more examined conditions would be

necessary in order to solve the problem due to the

water content.

262
A Review of Resistance-Jump Phase Changes Useful for High-Pressure

CaUbration

F. P. Bundy

General Electric Research and Development Center

Schenectady, New York 12301

1. Introduction

Because of the limited strength of the best construction materials, apparatus capable of developing

more than about 30 or 40 kbar has to be more complicated than a simple piston sliding into a simple

cylinder. The necessary gasketing and side-support

arrangements always absorb a sizable fraction of

the force applied to the pistonlike part, or parts, of an ultra-high-pressure apparatus, making it impossible to determine exactly how much of the applied

force is directed against the material enclosed in the

high-pressure chamber. Thus a direct force/area

determination of the pressure in the chamber is not

available. In such apparatus it is much more con- venient to establish the chamber pressure in respect

to the apphed force by inserting in the material

enclosed in the chamber small wires or ribbons of

substances which have, at known pressures, abrupt

phase changes manifested by electrical resistance

jumps. By observing three or four such resistancejump phenomena spread over the force range of

the apparatus, the chamber pressure can be calibrated. It is therefore important to have available a number of substances which exhibit such sharp

and reversible phase changes at reasonable intervals

up the pressure scale. It is also very desirable that

the true absolute value of the pressures of these

transitions be known in order that the experimental

data can be used for absolute thermodynamic

calculations, and so that workers using this type of

calibration in different laboratories in the world can

meaningfully compare their results.

The purpose of this article is to list the most

commonly used calibrating materials and to summarize the pertinent information about them. A discussion will be given of the absolute accuracy to which the various transition pressures are known,

and the precautions that must be used in applying

this method of pressure calibration to different kinds

of cell geometries in various apparatuses.

2. Phase Transformations Useful for

Calibration

Some of the phase transformations which have

been used most widely are listed in table 1. In the

table, A/?//?i is the ratio of the change of the resistance to the resistance at the low pressure.

Paper presented at the Symposium on Accurate Characterization

of the High-Pressure Environment, held at the National Bureau

ofStandards, Gaithersburg, Md., October 14—18, 1968.

The room-temperature resistance behavior versus pressure of most of these substances is shown

graphically in figure 1. Of all the transitions, the Bi ones are probably the most sharp and reversible.

The Bi (I-II) transition is remarkably large and

abrupt. Of the group, the {a, e) transitions in Fe

and the Fe-Co alloys are probably the most sluggish.

In apparatuses which are capable of only barely

reaching the Fe transition pressure, the transition is difficult to observe.

The transitions in the Fe-Co alloys were first observed as volume discontinuities in shock comTable 1. (refs. 16, 1, 20)

Substance Transition Pressure

kbar

Bi "low" (i-n) 25.4 -0.8

Tl (IMII) 37 -0.3

Cs (nin-iv) 42 Sharp cusp

formed by

rapid increase

followed by

abrupt drop

Ba "low" (I II) 58 + 0.2 Bi "high" (Hi-v) 83 -0.5

Sn (i-n) 100 + 0.05 Fe (a-e) 130 + 1.0 Ba "high" (Il-Liq?) ~ 140 + 0.3 Pb (MI) ~ 160 + 0.15 Rb (n-Liq?) ~ 190 + 1.5 FelSCo (a-e) -240 > + 0.1 Fe20Co (a-e) ~ 290 > + 0.1

u

-la

FOR Ba . Fe

•(

Bi 3-

>

Ba 7^

/

/Fe

Fe

5 —

Bi

CS n ^1

1-

XFDR 60

CSI

-40

P(kb)

Figure 1. R/Ro versus P, at 25 °C, for a number of calibrating

substances.

263

385-762 O - 71 - 18
FiGURl 2. P,T phase diagram for Bi.

pression experiments by Loree et al. [42].' They

appeared to be the (a-e) transition, as in pure Fe,

and the pressure of transition increased smoothly

upward as the Co content of the alloys increased.

Bundy [20] then tested the behavior of these alloys

in static pressure apparatus to determine whether

the transitions were accompanied by changes in

resistance. Resistance jumps were found, as in

Fe, at pressures which also increased smoothly

with Co content. In the tests pure Fe showed the

usual A/? transition, starting at 130 kbar. However,

with the alloys, the pressures at which the A/?

transitions occurred increased with the Co content

much more rapidly than for the AF transitions as reported in the shock-compression tests. For ex- ample, at 20 percent Co content the AF shockpressure transition was reported at 185 kbar, while

in the A/? tests in the static pressure apparatus the

transition came at 290 kbar. It is improbable that

the and AR events are different. It is also

improbable that a shock-pressure-induced transition

would initiate at a lower pressure than under

static pressure conditions. This discrepancy could

be examined effectively by x-ray diffraction experiments similar to those of Mao et al. [43].

The pressures at which some of the listed transitions occur are quite temperature-dependent. For

this reason it seems appropriate in this article to present what is known about theP,r phase diagrams

of these substances.

Figures in brackets indicate the literature references at the end of this paper.

P(kb)

Figures. P ,T phase diagram for T\.

The P,T phase diagram of Bi is presented in

figure 2. The data used in constructing this diagram

were collected and evaluated from references [12,

14, 25, 26, 28, 33, 40. 44, 48, 50]. The points of

greatest interest for calibration work are the I-II, IMII, and III-V transitions at 25 °C, room temperaure. It is important to note that the pressures of

all these transitions decrease as the temperature

increases. The III-V transition is especially sensitive, — 0.2 kbar/°C. Furthermore, although this is a sharp, reversible transition, there is still lack of

agreement on its pressure among those who have

tried to establish its absolute value by using approximations to free-piston gages [48], or by monitoring

its electrical resistance simultaneously with x-ray

diffraction of surrounding well-behaved crystalline

materials, like NaCl [33]. The observed "absolute"

values range from 89 kbar down to 74 kbar; 82-83

kbar is a good probable value to use until more

definitive absolute pressure methods become

available.

The P,T phase diagram for Tl, taken primarily

from [32], is shown in figure 3. The II-III transition,

which is manifested by a sharp drop in resistance,

is seen to be relatively insensitive to pressure. The

triple point for I-II-III occurs at the relatively low

temperature of 120 °C. In the triple-point region,

phase II has the highest resistivity and phase III the lowest.

Cesium has a very interesting P, T phase diagram,

as shown in figure 4 [27, 32(a), 37, 49]. According to

[37], the melting point of Cs I reaches a maximum

264
f 100-

30

P(kbor)

Figure 4. P,T phase diagram for Cs.

of 199 °C at 18 kbar, then decreases to 187 °C at the triple point with Cs II at 23.5 kbar. Again the

mehing point of the Cs II increases with pressure

to a maximum at about 27 kbar, 195 °C, then de- creases to another triple point at 42.5 kbar, 98 °C.

As reported first by Hall et al. [27], and later by

Jayaraman et al. [32(a)], a third phase exists over a pressure band only about 0.5 to 1.0 kbar wide.

At pressures above this a fourth, solid phase becomes the stable one. The melting temperature of

this fourth phase increases with pressure in the

more normal way. Phase I is known to be bcc,

while phases il and III are fee [27]. The crystal

structure of phase IV has not been established yet

from the complicated x-ray diffraction pattern. In

[27] it is shown that at the 42.2 kbar, 25 °C transition

the resistance versus pressure curve is more than

a cusp (as it is shown in fig. 1, herein). Instead, at 42.2 kbar the resistance jumps up to a relatively

high level, remains constant until 42.7 kbar is reached, whereupon it drops abruptly and then

continues to decrease with pressure while in the

fourth phase. In an ordinary pressure apparatus,

with only moderate pressure resolution, this double

phase transition is usually blurred into a single

resistance cusp centered at about 42.5 kbar. The

data given in [32(a)] indicate that the pressures of

the II-III-IV transition complex are nearly independent of the temperature, as is also the case for the

I-II transition.

A P,T phase diagram for Ba is given in figure 5,

based on data from references [1, 12, 23(a), 50, 30

and 47]. Ba I exhibits the unusual phenomenon of

having a maximum in its melting line (at about 15

kbar, 770 °C) [30]. The two room-temperature transitions at about 58 and 140 kbar are quite well

known. On the basis of the R versus P behavior of

Ba at low temperatures and very high pressures.

Stager and Drickamer [47] suggested that the 140

kbar transition is a melting phenomenon. Later, on the basis of x-ray diffraction, Drickamer [23(a)]

200 300

P(kbar)

Figure 5. P,T phase diagram /or Ba.

400

300

200

100 -I

-100

\\

\

\

\\

\

1/ 1

LIQ. /

1

n

Ibcl)

Sn

1

I *HITE TIN \v

GRAY 1 TIN 1

I

(DOUBLE bcl)

-IDi cubic) 1

\

RQfiMJL _ r -

1

1 1 1 1 1 ^ -20 20 40 60 80 100 120

P(kbl

Figure 6. P,T phase diagram for Sn.

reported that the high-pressure phase at room temperature is definitely crystalline, probably fee, so

it certainly cannot be liquid. There is a real opportunity here to explore experimentally the P,T

diagram above 80 kbar at temperatures well above

room value.

The true pressure values for the commonly used

Ba (l-ll) transition range from 53 kbar ([33], based

upon Decker's equation of state for NaCl), to 58.5

kbar ([50], free-piston gage). The true value of

the "high" Ba transition at around 140 kbar is still not firmly established. In Drickamer-type apparatus,

or high compression belt apparatus, it is definitely

higher than the Fe (a-e) transition, and distinctly

lower than the Pb transition at about 160 kbar.

The P,T diagram for Sn is relatively simple,

(fig. 6). The pressure values shown for the (l-Il) transition at 25 °C range from about 114 kbar [46],

265
2500

2000

1500

1 1 1 Fe

LIO.

rtfcc)

_ ' •

a (bcc) \ e (hep)

^

_R00_M 11 _ , 50 100

P(kb)

150 200

Figure 7. P,T phase diagram for Fe.

1000

800

200

Fl(;URE 8. P,T phase diagram for Pb.

to about 92 kbar ([4], referred to Decker's equation

of state for NaCl). A good probable value is about

100 kbar, in order to be in fair agreement with Bi

83 and Fe 130. At low (and negative) pressure and

low temperature is the region of stability of "gray

tin," which has diamond cubic structure. The transition from "white" to "gray" tin occurs only at quite low temperatures and is quite sluggish.

Data on the P,T phase diagram for Fe (fig. 7)

have been obtained by a number of workers. Refer200

Figure 9. P,T phase diagram for Rb.

ences [2, 21, 34, 38, 35, 19, 43] cover the reports

which relate most directly to this diagram. At room

temperature there is only the a to e transition, which

is accompanied by doubling of the resistance. In- crease in temperature decreases the pressure at which this transition runs. The a, e, y triple point

has been established with fair accuracy at 110 kbar,

500 °C [34, 38, 35, 19]. The a, y transition involves

about a 30 percent increase of resistance as well as a marked change in thermoelectric power. H. M.

Strong has used this a, y transition to calibrate

cells for pressure in the 50 kbar range at temperatures around 600 °C by monitoring the Fe/Pt thermocouple emf against that of Pt/PtRh. This type of

pressure calibration at higher temperatures will be

discussed by Strong in another paper in this

Symposium.

The, phase diagram for Pb is quite simple as far as it is known (fig. 8). The melting temperature has

been explored to at least 30 kbars. Over this range

the melting temperature rises smoothly about 200

°C. At room temperature there is a sharp transition

at about 160 kbar [1], which gives a sharp rise of over 15 percent in resistance. While it has not been

proved by direct x-ray diffraction, it is highly probable that the high-pressure phase of Pb is hep [41].

The boundary line between the fee and hep phases

would be expected to have a large dTjdP slope [41]

and consequently the pressure of this calibration

transition is probably not very temperature-sensitive.

Rubidium has a P,T phase diagram (fig. 9) that

is somewhat similar to those of Cs and Ba in that it has a maximum in the melting point and the possibility of a number of phases at very high pressures.

266
room temperature. The 80 and 190 kbar transitions

at 25 °C are well-established phenomena [15, 1], although the true values of the pressures may be

subject to revision, as is the case for the "high" Bi

and "high" Ba transitions.

3. Determination of Absolute Pressures of the Transitions

In 1942 [9], and later in 1948 [11], Bridgman re- ported his measurements on the volume compressibilities of a large number of substances in a two-stage piston-and-cylinder apparatus, up to pressures approaching 100 kbar. The apparatus

was made and used in such a manner that its action

approximated that of a free piston gage. During his

investigations he discovered a number of volume

discontinuities which corresponded to abrupt phase

changes in the test materials. The ones which relate

to the subject of the present paper were Bi at 24.9

kbar, Tl at 39 kbar, Cs at 44 kbar, Ba at 59 kbar, and

Bi at 89 kbar. In 1952 [12] Bridgman reported on

the resistance behavior of a large number of

materials, as measured in an opposed anvil apparatus, which has an internal pressure distribution

which is far from that of a perfect free-piston gage.

According to his estimates of the pressures in this

apparatus, there was a large abrupt resistance

drop in Bi at 24.9 kbar (the same pressure at which

the volume discontinuity occurred), but for Tl the

resistance discontinuity came at an estimated 44

kbar, Cs at 54 kbar, and Ba at 74 kbar. Bridgman

had expected any resistance discontinuities to coincide with volume discontinuities in a given

material, but in the cases of Tl, Cs, and Ba he con- cluded that the differences in "observed" pressures

of the and A/? events were too great to allow

them to be the same event. At the time, however,

he knew that the pressures in his piston-andcylinder apparatus were known to much greater

absolute accuracy than in the opposed anvil apparatus. Later, Boyd and England [6], Kennedy and LaMori [36], Hall et al. [27], and others demonstrated

that Bridgman's pressure estimates in the opposed

anvil apparatus, above about 25 kbar, were progressively too high, and that the and A.R events in

Tl, Cs, and Ba do indeed occur at the same pres- sures and are the same physical events. The question now has become one of establishing the actual

absolute pressures of the transitions, and of

determining the effects of such disturbing factors as

local pressure gradients in the specimen and the

surrounding solid material, the over-pressures or under-pressures required to start the transformations in the forward and backward directions, and

pressure gradients generated by local heating within

cells. There have been a number of significant efforts

in this direction. Vereshchagin et al. [50] devised

a rotatable free-piston arrangement in the center

of a larger piston of a multipiston apparatus. This

apparatus was reported to work as a free-piston

up to pressures approaching 100 kbar but no details

of the techniques were given. With this method they

established the following absolute pressures of

transition at 25 °C: Bi(l-Il), 25.4 kbar; Bi(lI-IIl)

26.9 kbar; Tl(ll-lll), 36.9 kbar; Ba(l-Il), 58.5 kbar;

and Bi(lII-V), 89.3 kbar. Quite recently Haygarth,

Getting, and Kennedy [26], using an improved piston-and-cylinder apparatus, have done extensive

measurements on the Ba(I-Il) transformation. They

have found the value 55.0 ±0.5 kbar at 22 °C, and

a transition pressure coefficient of 0.015 kbar/°C near room temperature. Samples from three dif- ferent sources, all of purity better than 99.5 percent,

gave essentially the same results.

Another approach taken by Decker [23], and

developed by Jeffrey et al. [33], has been to work

out a semi-empirical theoretical equation of state

for the simple, weU-behaved, ionic material NaCl,

and through this relate pressure and temperature

to lattice parameter. Then in a pressure apparatus

capable of providing simultaneous x-ray diffraction patterns of the resistance specimen and the

adjacent NaCl, determine the pressure at which

the transition of the resistance specimen goes.

Using an x-ray tetrahedral apparatus Jeffery et al.

[33] investigated the Bi, Tl, and Ba transitions.

They found some hysteresis between loading and

unloading which depended upon the specimen

material and its geometrical shape. Relative to the "NaCl pressure scale" of Decker they found

Bi(l-ll) at 24.8 ±0.8, Bi(li-iil) at 28.0 ±0.8, Tl(ll-lll)

at 34.6 ±2.1, Ba(l-ll) at 53.3 ±1.2, and Bi(lll-v) at 73.8 ±1.3 kbar. The Decker equation of state for

NaCl agrees closely with the compressibility of

NaCl at room temperature as found by Bridgman

[7, 10] to 100 kbar. However, Bridgman's measure- ments of the compressibility of Ba [9], made in

the same apparatus under essentially the same

conditions as NaCl, yielded the transition pressure

at 59 kbar. Thus the 1966 results of Jeffery et al. show either that there was a discrepancy between

Bridgman's values for NaCl versus Ba, or that

there are some very localized pressure differences

in the specimen region of the x-ray-resistance cell. The fact remains that among all the most sophisticated methods of establishing absolute pressures

of the transitions the scatter of values obtained is about 5 percent at 50 kbar and around 10 percent

at 100 kbar. As pointed out in 1967 by Bassett et al.

[5], when the most recent value of the initial compressibility of NaCl (by Slagle and McKinstry [45]

is inserted in Decker's equation it yields slightly

higher pressure; not enough, however, to match

the free-piston values of Vereshchagin et al. or of

Haygarth et al. Figure 10 is a P,T phase diagram

of NaCl based upon the most recent data.

At pressures higher than those that are attainable

by any kind of free-piston apparatus, absolute pres- sure calibration becomes dependent upon compari267
0 100 200 300

P(kb)

Figure 10. P,T phase diagram for NaCl.

son with shock compression data. In shock

compression the shock wave velocity and the

particle velocity are measured. By application of

conservation of momentum and of mass through the

shock front, one calculates the pressure and density

attained. One also can calculate to some degree of

accuracy the temperature reached during the

abiabatic compression of the material. From these

P, V, T conditions one can correct back to the P, V

values for room temperature. The lattice spacings

obtained from x-ray diffraction patterns of specimens observed under pressure in a static apparatus can then be compared to the shock-compression

data to calibrate the pressure. There are varied

opinions about the accuracy attainable in each step

of this measurement sequence. The situation is complicated even more in those compressions in

which a phase change occurs, because of the complexity of the wave velocities and additional temperature rise due to volume coUapse.

4. Some Causes of Discrepancies and

Anomalies

After one has had a number of years of experi- ence with ultra-high-pressure equipment of different

types, he becomes aware of various kinds of discrepancies and anomalies that can occur in pressure

calibration. It may be helpful to discuss a few

examples.

If the cell which goes into the high-pressure vessel

200 r

APPLIED FORCE

Figure 11. Cell pressure anomally in Drickamer apparatus

due to phase collapse o/BN disks.

incorporates a material which may undergo a volume-reducing phase change, the cell-pressure versus applied-force curve for the apparatus may

take disappointing turns toward cell-pressures less

than anticipated. A good example of this, in the

author's experience, was with a Drickamer-type

apparatus which had the usual pyrophyllite flared

gasket, but which contained BN disks in the center

of the cell. The BN was being used for its refractory

characteristics (see fig. 11a). When the BN disks

were used, the cell-pressure calibration showed a

linear pressure increase with applied force up

through the high Bi transition; but the (a-e) Fe

transition could be initiated only by loading the

apparatus to near its breaking point (see fig. lib).

By contrast, when pyrophyllite disks were used, the

Fe transition occurred at a loading corresponding to a linear cell-pressure increase with applied force.

With pyrophyllite disks the high Ba and the Pb

transitions came "on schedule" on the same straight

line, while with BN disks these transitions could

not be produced at all. It was found that this anomalous behavior was caused by the transformation,

at room temperature, of the graphitic type BN to the much more dense wurtzite and zincblende

forms [17].

A similar effect is obtained in a high-compression

belt apparatus, or a Drickamer-type apparatus,

when thin, hard, steel shims are used on the piston

faces to increase the compression ratio of the sample

zone relative to the gasket zone. In this case, the

rate of rise of cell pressure with applied force de- creases in the 130 kbar region as the steel in the

shims transforms to the e phase.

268
Frequently, high-pressure cells incorporate

materials which are refractory and chemically inert

in order to satisfactorily contain specimens under

high-temperature conditions. Materials like MgO,

AI2O3, ThOu', etc., are frequently used. These

materials also have very high compressional strength

and may shield the contained specimen from the

pressure generated in the more plastic parts of

the surrounding cell. In such cases resistancejump transitions may be long and drawn out with

respect to the applied force, indicating large and

variable pressure gradients. Later, when the core

of a cell of this type is heated relative to the re- fractory walls, the local pressure in the specimen

may rise higher than that ambient in the outer parts

of the cell. In such cells a pressure calibration car- ried out at room temperature should not be assumed

to apply to the hot-core operating condition. In a paper which comes later in this Symposium,

H. M. Strong will discuss the use of Fe for determining some of the higher P,T points, utilizing the

a, y and a, e transitions at higher temperatures.

At our laboratory we also have found the melting

point of Ge to be useful in establishing pressure

rises produced by heating the core of a cell at constant applied force to the apparatus [18]. Melting of diamond-cubic Ge causes an abrupt large

drop in its resistance. Because the liquid phase is more dense than the solid, the melting point de- creases with pressure. The dT/dP slope of the melting line can be calculated with considerable accuracy

by inserting in the Clausius-Clapeyron relationship,

dTldP = AVI^S, the experimental values of

and AS [29]. Assuming this calculated melting line

for Ge (which cross-compares quite well with

measurements made utilizing the calculated a — y

transition in Fe), P,T points in a given apparatus can be established by observing the melting of

Ge in the cell and apparatus of interest. The pres- sure rise with temperature varies considerably

with the geometry and materials of the pressure

cell, and hence no general formula for the magnitude

of the effect can be given a priori. In the author's

work with a "high-compression belt" apparatus

it was found that the pressure rise caused by heating

the specimen at the core of the cell was roughly

proportional to the product of the initial (cold)

pressure and the temperature rise. For example,

a Ar of 500 °C from an initial pressure of 100 kbar

would cause a AP of about 16 kbar.

The question of the initiation and the reversibility of phase transformations has been raised by

many workers in the field of high pressure, and

experimentally studied by some. It is important in

the pressure calibration problem because over- pressures or under-pressures required to start forward or reverse phase transitions add to the

overall "hysteresis" effect. It is well recognized

that in most pressure apparatuses there is considerable frictional and mechanical hysteresis in- herent in the moving parts of the mechanical system.

Also, the fact that most of the resistance-jump

calibration situations consist of a solid sensor sur- rounded by solid cell materials of finite shear and

compressional strength, means that pressure shielding of the sensor material is always a factor. Experimentally, it is very difficult to separate these

factors of apparatus hysteresis, pressure gradient

in the cell material surrounding the sensor, and

inherent phase-change hysteresis. In his detailed

study of the Bi (l-Il) transition as a primary pres- sure calibration point, and manganin as a secondary

gage, Bridgman [8] found that the "band of indifference" of the Bi (l-II-l) transition as determined

in a fluid environment, monitored with a manganin

gage, was about 0.06 to 0.10 kbar at 30 °C. In Bridgman's measurements, both phases were present

simultaneously, so this is essentially a measurement

of pressure-shielding effect (or shear strength)

within the Bi specimen itself. In 1964, Davidson

and Lee [22] reported results on Bi tested in a fluid

environment monitored with a manganin pressure

gage in which they observed the hysteresis of the

Bi (l-Il) transformation from completely one phase

to completely the other, thus involving the nucleation of the new phase. They found hysteresis of

0.55 to 0.90 kbar. Once started, the transition proceeded isobarically to completion. In this case the

observed hysteresis must be the overpressure

required for nucleation of the new phase. Jeflfery et al. [33], in connection with their x-ray diffraction

experiments of Bi specimens in NaCl enclosures,

discuss this nucleation hysteresis phenomenon quite

fully, as well as the effect of pressure gradients in

the material surrounding the transition specimen.

The hysteresis found by Jeffery et al. in their observations of the Bi (l-Il) transition ranged from

2.8 to 5.3 kbar, depending upon the geometry of

the Bi-NaCl arrangement. The excess hysteresis, over that found by Davidson and Lee, is probably

a result of the pressure gradients in the NaCl

environment. This illustrates the effect of the

geometry and kind of material which surrounds the

specimen of interest.

5. References

[1] Balchan, A. S., and Drickamer, H. G., Rev. Sci. Instr. 32,

308 (1961).

[2] Bancroft, D. , Peterson, E. L. , and Minshall, S. , J. Appl. Phys.

27, 557 (1956).

[3] Barnett, J. D., Benion, R. B., and Hall, H. T., Science 141,

1041 (1963).

[4] Barnett, J. D., Bean, V. E., and Hall, H. T., J. Appl Phys.

37,875 (1966).

[5] Bassett, W. A. , Takahashi, T. , Mao, H.-K. , and Weaver, J. S.

,

J. Appl. Phys. 39, 319 (1967).

[6] Boyd, F. R., and England, J. L., J. Geophys. Res. 65, 741

(1960).

[7] Bridgman, P. W., Proc. Am. Acad. Arts Sci. 74, 21 (1940).

[8] Bridgman, P. W., Proc. Am. Acad. Arts Sci. 74, 1 (1940).

[9] Bridgmi^i, P. W., Proc. Am. Acad. Arts Sci. 74,425 (1942).

[10] Bridgman, P. W., Proc. Am. Acad. Arts Sci. 76, 1 (1945).

[11] Bridgman, P. W., Proc. Am. Acad. Arts Sci. 76, 55 (1948).

[12] Bridgman, P. W., Proc. Am. Acad. Arts Sci. 81, 165 (1952).

[13] Bridgman, P. W., Proc. Am. Acad. Arts Sci. 84 , 63 (1955).

269
[14] Bundy, F. P., Phys. Rev. 110, 314 (1958).

[15] Bundy, F. P., Phys. Rev. 115, 274 (1959).

[16] Bundy, F. P., Trans. ASME, J. Engg. Indus. 83, 207 (1961).

[17] Bundy, F. P., and Wentorf, R. H., Jr., J. Chem. Phys. 38,

1144 (1963).

[18] Bundy, F. P., J. Chem. Phys. 41, 3809 (1964).

[19] Bundy, F. P., J. Appl. Phys. 36,616 (1965).

[20] Bundy, F. P., J. Appl. Phys. 38, 2446 (1967).

[21] Claussen, W. L., Rev. Sci. Instr. 31, 878 (1960).

[22] Davidson. T. E., and Lee, A. P., Trans. Met Soc. AIME

2 3 0, 1035 (1964).

[23] Decker, D. L., J. Appl. Phys. 36, 157 (1965); ibid. 37,

5012 (1966).

[23(a)] Drickamer, H. G., Solid State Phys. 17, 103 (1965).

[24] Dudley, J. D., and Hall, H. T., Phys. Rev. 118, 1211 (1960).

[25] Giardini, A. A., and Samara, G. A., ASME 1964 Symposium

on High Pressure Technology, ed. E. C. Lloyd and A. A. Giardini (June 1965).

[26] Haygarth, J. C., Getting, I. C., and Kennedy, G. C., J. Appl.

Phys. 38, 4557 (1967).

[27] HaU, H. T., MerriU, L., and Bamett, J. D., Science 146,

1297 (1964).

[28] Heydemann, P. L. M., J. Appl. Phys. 38, 2640 (1967).

[29] Horn, F. H., and Hall, R. N., Gen. Elec. Res. Lab., private

communication, 1964; also Bradshaw, S. E., Wembley

Laboratories, Wembley, Middlesex, England, private

communication, 1964.

[30] Jayaraman, A., Klement, W., Jr., and Kennedy, G. C., Phys. Rev. Letters 10, 387 (1963).

[31] Jayaraman, A., Klement, W., Jr., and Kennedy, G. C., Phys. Rev. 130, 540 (1963).

[32] Jayaraman, A., Klement, W., Jr., Newton, R. C., and

Kennedy, G. C., J. Phys. Chem. Solids 24, 7 (1963).

[32(a)] Jayaraman, A., Newton, R. C, and McDonough, J. M.,

Phys. Rev. 159, 527 (1967).

[33] Jeffery, R. N., Barnett, J. D., Vanfleet, H. B., and HaU, H. T.,

J. Appl. Phys. 37, 3172 (1966).

[34] Johnson, P. C, Stein, B. A., and Davis, R. S., J. Appl.

Phys. 33, 557 (1962).

[35] Kaufman L., Clougherty, E. V., and Weiss, R. J., Acta

Met. 11, 323 (1963).

[36] Kennedy G. C, and LaMori, P. N., Progress in Very High

Pressure Research, ed. F. P. Bundy, W. R. Hibbard, Jr., and H. M. Strong (John Wiley & Sons, New York, 1961),

p. 304.

[37] Kennedy, G. C, Jayaraman, A., and Newton, R. C, Phys.

Rev. 126, 1363 (1962).

[38] Kennedy, G. C, and Newton, R. C, Solids Under Pressure (McGraw-Hill Book Co. Inc., New York, 1962).

[39] Kennedy, G. C, and Newton, R. C, Solids Under Pressure,

(McGraw-Hill Book Co. Inc., New York, 1963) Chapt. 7,

p. 172.

[40] Klement, W., Jr., Jayaraman, A., and Kennedy, G. C,

Phys. Rev. 131, 632 (1963).

[41] Klement, W., Jr., J. Chem. Phys. 38, 298 (1963).

[42] Loree, T. R., Fowler, C. M., Zukas, E. G., and MinshaU,

F. S., J. Appl. Phys. 37, 1918 (1966).

[43] Mao, H. K., Bassett, W. A., and Takahashi, T., J. Appl.

Phys. 38, 272 (1967).

[44] PhiUips, D., and Jura. G., Trans. Met. Soc. AIME 239,

120 (1967).

[45] Slagle, O. D., and McKinstry, H. A., J. Appl. Phys. 38,

437 (1967).

[46] Stager, R. A., Balchan, A. S., and Drickamer, H. G., J. Chem. Phys. 37, 1154 (1962).

[47] Stager. R. A., and Drickamer, H. G., Phys. Rev. 131,2524

(1963).

[48] Stark, W., and Jura, G., ASME 1964 Symposium on High

Pressure Technology, ed. E. C. Lloyd and A. A. Giardini

(June 1965).

[49] Stishov, S. M., and Makarenko, L N., J. Theor. Exper. Phys.

54, 708 (1968).

[50] Vereshchagin, L. F., Zubova, E. V., Buimova, I. P., and

Burdina, K. P., Sov. Phys.-Dokl. 11, 585 (1967).

DISCUSSION

C. Susse (Laboratoire des Hautes Pressions, Centre

National de la Recherche Scientifique, Bellevue,

France): I think we have a partial answer to the

problem of the melting curve for barium at 144

kbar. Recent work performed by differential thermal

analysis seems to show that the melting curve

doesn't go down regularly with pressure. After the

first known maximum at about 15 kbar, there is a

triple point Ba I — Ba II — hquid at 500 °C — 67 kbar

followed by a maximum at 525 °C — 75 kbar and by

a cusp at 380 °C — 85 kbar, which is most probably

related to a phase transition in the solid. Above this

pressure the MP increases with pressure at least

up to 95 kbar.* We intend to do some additional

work in this area.

G. A. Samara {Sandia Laboratory, Albuquerque,

New Mexico): Although there seem to be some

problems in the use of the lead transition, I think it is one of the sharpest and most reproducible transitions above 100 kbar, and I would urge that lead not

yet be given up as a possible reference material.

•Bastide, J. P., Susse, C, Epain, R., C. R. Acad. Sci. Paris 267,857 (1968).

A. Taylor (Westinghouse Research Laboratories,

Pittsburgh, Pennsylvania): Do you find any appreciable difference in the transformation pressure of

zone-refined iron as against iron of thermocouple

purity?

D. Decker (Brigham Young University, Provo,

Utah): We have made one measurement only on

the iron triple point, which appears to be considerably lower than 110 kbar.

It appears that an extrapolation of the data that

Madame Susse gave earlier on barium would show

a transition well below 110 kbar, perhaps at 80 kbar,

around 550 °C.

J. Haygarth {Institute of Geophysics and Planetary

Physics, University of California, Los Angeles): I would like to comment on the phase diagram of

barium below 100 kbar. Our findings differ some- what from the results given earlier by Madame

Susse.

We have been fairly careful to obtain barium of

very low hydrogen content. It is apparent that it

is rather difficult to get the hydrogen out. We

270
had two samples, one which contained one atomic

percent hydrogen; and the other which contained

four percent hydrogen. On the one containing four

percent hydrogen, we obtained a second maximum

in the mehing curve around 47 or 48 kbar. When

we ran the experiment with the sample which originally contained one percent hydrogen, we did not

find this maximum.

Now there is a question about the purity of the

sample which initially contained one percent hydrogen because, unfortunately, the capsule in which

it had been stored was found to have become open

to the air, so the hydrogen content was in question.

But from this work I think that it is important to make sure that the barium, especially for melting

point purposes, is very pure.

I would also like to state that the slope of the

barium melting curve after the second maximum

seemed to be the same as the slope just before it, agreeing with the diagram Dr. Bundy showed.

AUTHOR'S CLOSURE

Replying to Samara: I have always found the

lead transition to be sharp and clean when there

is good electrical contact with the sample.

Replying to Dr. Taylor: We haven't made a systematic investigation of it, but we can say that

we have had to use the purest iron we could make

ourselves, usually "carbonyl iron" that has been

remelted in vacuum. This proves to be superior to thermocouple grade iron.

271

session VII

Chairman: R. RoY

The Pennsylvania State University

University Park, Pennsylvania

The Coesite-Stishovite Transition*

Syun-iti Akimoto and Yasuhiko Syono

Institute for Solid State Physics

University of Tokyo, Roppongi, Tokyo, Japan

The coesite-stishovite transition curve has been determined over the temperature range 550 to 1^00 °C in the pressure range 82.0 to 98.0 kbar by means of a tetrahedral-anvil type of high-pressure

apparatus. Amorphous anhydrous silica, a-quartz, coesite, and stishovite were used as starting materials. The transition curve was fitted by the linear equation P(kbar) = 69 + 0.024 TCC) using pressure

scale proposed by Jeffery et al. This determination was found to be in a reasonable agreement with

the previous data, if the common pressure scale is used. Experimental information for the coesite- stishovite transition was also compared with the stability relation derived from enthalpies and entropies

of coesite and shishovite. It was suggested that the pressure scale proposed by Jeffery et al. was underestimated considerably around 100 kbar.

1. Introduction

The rutile-type polymorph of Si02 now known as

stishovite was first synthesized by Stishov and

Popova [1961] ' at high pressures in excess of 100

kbar. Since stishovite is accepted to be a probable

major constituent of the earth's lower mantle, there

has been much interest in determining the properties and stability of stishovite. Wentorf [1962],

Sclar et al. [1962], Ringwood and Seabrook [1962],

Bendelyany and Verestshagin [1964] and Minomura

et al. [1964] have described the synthesis of sti- shovite using different types of high-pressure apparatus, but they provide no detailed data on its stability range. The first attempt to determine the

coesite-stishovite transition curve was made by

Stishov [1963]. He derived the standard entropy of

stishovite from the linear relationship between

entropy and density empirically found in the rutiletype oxides, and calculated the transition curve

passing through the equilibrium point 130 ±5 kbar,

1600 °C ± 100 °C experimentally determined.

Further data on the direct experimental determination of the transition curve in the temperature range

410 to 830 °C were provided by Ostrovsky [1965,

1967]. Ryabinin [1964] also reported a stability

diagram of SiOz including the coesite-stishovite

transition curve, which differs markedly from the

estimation by Stishov and Ostrovsky. It is one purpose of the present work to extend the determination of the coesite-stishovite curve to the wider

'Reprinted by permission from J. Geophys. Research 74, No. 6 (March 15, 1969). Copyright 1969 by the American Geophysical Union.

' Figures in bracicets indicate the literature references at the end of this paper.

Paper presented at the Symposium on Accurate Characterization

of the High-Pressure Environment, held at the National Bureau

of Standards, Gaithersburg, Md., October 14-18, 1968.

range of temperatures and to provide an improved

estimate of its position and slope, using several

kinds of starting materials including coesite and

stishovite.

Another aim of the present work is to examine the

possibility that the coesite-stishovite transition

could be used as a pressure calibration point at high temperatures. Most calibrations of highpressure apparatus utilizing solid pressure-media

are carried out on the basis of the phase transitions

in Bi, Tl, Ba, Sn, etc., which are known rather precisely at room temperature. This method is not

satisfactory for high-temperature work on account

of the change in the shear strength of the pressure

media with temperature. Boyd et al. [1967] recom- mended the quartz-coesite transition as a promising

calibration point in the pressure range 30 to 40 kbar

at high temperatures. Recent development of the

capabilities of the high-pressure apparatus requires

a higher calibration point at high temperatures. The

present information on the coesite-stishovite transition might contribute to pressure calibration around

100 kbar at high temperatures.

Very recently, thermodynamic properties of

coesite and stishovite were reported by Holm et al. [1967]. Experimental information for the coesitestishovite transformation obtained in this work will

be compared with the stability relation derived

thermodynamicaUy from their data. Some contributions to the establishment of an accurate pressure

scale may also be expected through such a comparison.

2. Experimental Procedure

All the high-pressure and high-temperature ex- periments were made using the tetrahedral-anvil

type of high-pressure apparatus described previously

273
[Akimoto et al., 1965]. Two different sizes of

cemented tungsten carbide anvils containing

5 percent of cobalt were used. For most runs from

83 to % kbar the 9-mm-edge anvils were used with

a 15-mm-edge tetrahedron of pyrophyllite sample

container. A combination of 7-mm-edge anvils and

a 13-mm-edge pyrophyllite tetrahedron was also

adopted for several auxiliary runs. The pyrophyllite

tetrahedron which served as the pressure-media

was fired at 700 °C for 30 min before use, and its surface was painted with thick suspension of

Q:Fe203 in ethanol to enhance the surface friction.

Pressure values in the tetrahedral press were calibrated at room temperature by measuring sharp

resistance change in the flat strip samples of Bi,

Ba, and Sn on the pressure-increase cycle. The

values of transition pressure, 26 kbar for Bi I-II, 55 kbar for Ba I-II, 76 kbar for Bi lll-v, and 92 kbar

for Sn I-II were adopted as fixed calibration points

following the proposal of Jeffery et al. [I960].

The transition pressure was confirmed to be re- producible within an accuracy of ±3 percent.

A graphite tubing, 4.0 mm long, 2.5 mm in outside

diameter, and 1.7 mm in inside diameter, was used

for heating the samples in the runs with the 15-mmedge pyrophyllite tetrahedron. Smaller tubing,

3.0 mm long, 2.2 mm in outside diameter, and

1.4 mm in inside diameter, was used for the runs

with the 13-mm-edge tetrahedron. The tubular

graphite furnace so prepared was placed diagonally

with the axis of the cylinder between opposite

edges of the tetrahedron. SiOa powder samples

were embedded in the center of the graphite furnace.

A pair of boron nitride disks was placed on both

ends of the sample to reduce the uncertainties

resulting from the temperature gradient within the

furnace.

Temperatures were measured with Pt/Pt-13%Rh

thermocouple, 0.2 mm in diameter, without any

correction for the effect of pressure on the emf of

the thermocouple. The hot junction was placed in

the center of the samples. Thermocouple leads

were brought out to the faces of the pair of anvils

not used for the graphite furnace contacts. The

run temperatures were corrected for the increase

of surface temperature of the tungsten carbide

anvils.

Determination of the coesite-stishovite transition curve was made by the usual quenching method.

Pressure was apphed to the sample first, then the

temperature was brought to the desired value and

held for the desired interval of time. Then the

sample was quenched isobarically by turning off

the heating power. After quenching, only the central part of the samples, less than 1 mm thick,

adjacent to the head of thermocouple, was used

for the phase-equilibrium experiments. Identification of phases present was made chiefly by the

x-ray diffraction method. In order to estabhsh the

transition curve accurately and to examine the

reversibility of the reactions, runs with different

kinds of starting materials were practised. Amorphous anhydrous silica, a-quartz, coesite, and

stishovite were used as the starting materials.

Mineralizers, such as H2O, were not used in the

present work.

3. Results

Results of experimental runs on which the location of the transition curve is based are given in

table 1 and figure 1. The percentages of the phases

present in the run products, shown in the last

column of table 1, are roughly estimated from the

peak intensity ratio of the x-ray diffraction chart.

When a-quartz was used as the starting material,

phases other than coesite and stishovite were not

observed in the run products at temperatures above

800 °C. However, metastable a-quartz was observed

in addition to coesite and stishovite in the run

product at 88.5 kbar and at 650 °C. Maximum con- version to stishovite in the present pressuretemperature-time condition was about 80 percent.

On the other hand, when amorphous anhydrous

silica was used as the starting material, complete

conversion to stishovite or coesite was achieved even at temperatures as low as 550 °C. Synthesis of

stishovite was also confirmed in the run in which

the starting material was coesite. Successful reverse

reactions from stishovite to coesite were demonstrated at two different temperatures, and using

single-phase stishovite obtained in the present work

1600

1400

1200

1000

800

600

400

Starting material

amorptious silica j o a-quartz j

ite /

/

/St

V coesite A stistiovitt

COESITE

STISHOVITE

70 80 90 100 110 120

Pressure . kb

Figure 1. Pressure-temperature stability fields for the coesite- stishovite equilibrium.

The dashed line designated by 0^ represents the transition curve determined by

Ostrovsky [1965, 1967]. The chain line designated by St represents the transition curve

estimated by Stishov [1963].

274
Table 1. Results of runs on the coesite-stishovite

transition

Run Temperature Pres- Time Phases present No. sure "

Starting Material, Amorphous Anhydrous Silica

"C kbar min

S29 550± 5* 83.0 100 Stishovite S25 600 ±10 82.0 65 Coesite

522 600 ± 5 91.0 60 Stishovite S31 630 ± 5 83.0 40 70% Coesite + 30%

Stishovite S27 700 ± 5 86.0 60 95% Stishovite + 5% Coesite

S26 700 ± 5 90.0 60 Stishovite S28 750 ± 5 85.0 45 Coesite

S30 830 ± 5 90.0 20 Stishovite

Starting Material, a-Quartz

SI 650 ±10 88.5 60 50% a-Quartz + 30% Coesite + 20% Stishovite S12 800 ± 5 87.0 60 95% Coesite + 5%

Stishovite S4 800 ± 5 93.5 60 70% Stishovite + 30%

Coesite

S6 900 ± 5 90.0 45 90% Coesite + 10%

Stishovite S14 900 ±10 94.5 45 60% Stishovite + 40% Coesite

Sll 900 ± 5 96.0 45 80% Stishovite + 20% Coesite

S13 1,000 ± 5 91.0 30 90% Coesite + 10% Stishovite S2 1,000 ±10 96.0 30 70% Stishovite + 30% Coesite

SIO 1,100±5 92.0 30 Coesite

S8 1,100 ±10 97.0 32 60% Stishovite + 4{)% Coesite

S35 1,180 ±20 98.0 15 80% Stishovite + 20% Coesite

S5 1,200 ±10 93.5 20 Coesite

S7 1,200 ±10 97.0 20 Coesite

Starting Material, Coesite

S23 1,015 ±10 93.5 60 60% Stishovite + 40% Coesite

Starting Material, Stishovite

S20 850 ± 5 90.0 70 Stishovite S21 950 ± 5 92.0 60 Stishovite S18 1,050 ±10 93.0 40 50% Coesite + 50% Stishovite S24 1,200 ±10 95.0 30 60% Coesite + 40% Stishovite

" Precision of pressure control is ±0.5 kbar.

* Precision of temperature control.

as the starting material. Summarizing all the experimental data, the boundary curve for coesite-stishovite transition was fitted by the linear relation

P(kbar) = 69 + 0.024 T{°C).

The unit cell dimensions of the stishovite obtained

at 97.0 kbar and at 1100 °C (S8 run in table 1) were

determined to be a = 4. 178 ±0.001 A and c = 2.665

±0.001 A. This determination is in quite a good

agreement with those values, a = 4. 1790 ±0.0004^4,

and c = 2.6649 ±0.0004 A, published by other in- vestigators [Stishov and Popova, 1961; Chao et al., 1962].

4. Discussion

In figure 1 the transition curve obtained in this

work is compared with those reported by Stishov

[1963] and Ostrovsky [1965]. Since pressure values

used by these investigators are based on Bridgman's

"volume-scale," in which Bi lll-V transition was

fixed as 89 kbar, their curves in figure 1 are represented after some corrections to the pressure scale

by Jeffery et al. [1966] were made. The transition curve determined in this work is found to be located

between Stishov's and Ostrovsky's curve and to have the slope dPjdT which is nearly consistent

with Ostrovsky's determination but is slightly

larger than Stishov's calculation.

Taking the accuracy of the pressure measure- ment (±3 — 5 kbar) in these investigations into

consideration, we cannot find any significant dif- ference between the curves determined with different types of high-pressure apparatus, i.e., opposed

anvil type and multiple anvil type, if the common

pressure scale is used. The slope of the coesitestishovite transition curve established in this work,

about 40 ~ 50°/kbar, indicates that the transition

is very sensitive to pressure. Any minor errors

in temperature measurement will have a small

effect on the results. These situations suggest that

the coesite-stishovite transition is usable as a

pressure-calibration point at high temperatures.

Since the pressure value in this work was calibrated

at room temperature by the conventional resistance

method, absolute value of pressure used at high

temperatures remains still uncertain. In case of

the quartz-coesite transition, the transition curve determined by Takahashi [1963] by means of the

tetrahedral press with the pressure calibration

method similar to this work was found to accord

well with the value of transition determined carefully by Boyd et al. [1967] with the piston-cylinder

type apparatus. Accordingly, if the uncertainty of

± 3 kbar was taken for granted, the coesite-stishovite

transition curve determined in this work using the

tetrahedral press may serve as a pressure-calibration point at high temperatures.

when information is available on the thermodynamic properties of coesite and stishovite, we

can estimate quite independently the pressuretemperature stability fields for the coesite-stishovite

equilibrium by means of thermodynamical methods

developed by Ahrens and Syono [1967]. Very re- cently, direct determination of the thermodynamic

functions of coesite and stishovite over a wide range

of temperature was reported by Holm et al. [1967].

Enthalpy changes of transformation at 298 K of

quartz to coesite and to stishovite are determined

to be 1.21 kcal/mole and 11.79 kcal/mole, respec275
Molar Volume, cm' /mole Pressure . P, kb

Figure 2. Pressure-volume curves for quartz, coesite, and

stishovite used in free energy calculation.

Ouartz: B-W, alter linear compression data by Bridgman [1949] and shock-wave

compression data by Waciterle [1962]. Coesite: Numerals attached to the curves means that the compressibility of coesite was assumed to be 100 percent, 80 percent, 70 percent of that of quartz. Stishovite: I, after Ida et al. [1967]; M, after the estimation by McQueen et al. [1953] using shock wave compression data of SiO - at higher pressures than 150 kbar by Wackerle [1962].

lively. Entropy changes at 298 K are also determined to be —0.23 and —3.24 eal/mole K for the

transformations of quartz to coesite and to stishovite.

The increase of free energy AGjgs zero pressure

and 298 K of a transformation is generally given by

AG«98-AH^'98-298AS«98 (1)

where AZ/^gg and AS^g are the change in enthalpy

and entropy of the transformation at zero pressure

and 298 K. The equilibrium pressure P between

low-pressure and high-pressure polymorph of a material is determined at 298 K by putting free

energy change AG^gg given by the following equation

zero

AG!;,, = j\vdp + AGI,, (2)

where AF is the volume change associated with the

transformation. Combining (1) and (2) we can

obtain

- j' AVdp = A//^'gg - 298AS^'9g. (3)

On the basis of the thermodynamic data just mentioned, the values of right-hand side of eq (3) are now known for the transformations from coesite

to stishovite as 11.48 kcal/mole. In order to obtain

the equihbrium pressure at 298 K, the value of the

left-hand side of eq (3) was numerically integrated.

In this calculation we adopted the shock wave

compression data of quartz by Wackerle [1962],

which was smoothly connected with the static compressibility data by Bridgman [1949]. Recent linear

compression data of a-quartz reported by McWhan

[1967] are in a reasonable agreement with the

present compression curve, at least for low-pressure

region. The pressure-volume relation for stishovite

Figure 3. The values of — I AVdp versus pressure for the

Jo coesite-stishovite transition.

determined by Ida et al. [1967] by means of highpressure x-ray analysis based on the compression

data of NaCl [Perez-Albuerne and Drickamer,

1965] was also used. Since no compressibility data

for coesite are available, we estimated the bulk

modulus of coesite to be 70 percent of that of quartz,

based on the logarithmic Hnear relationship between

bulk modulus and volume per ion pair reported by

Anderson and Nafe [1965]. We also tentatively calculated for comparison the compression curves of

coesite, assuming the compressibility of coesite

to be 80 percent and 100 percent of that of quartz.

In figure 2, volume versus pressure data of three

Si02 polymorphs used for the free-energy calculation were graphically shown. ^CP

In figure 3 the values of — I AVdp for transformaJo tion of coesite to stishovite were plotted versus

pressure. The equilibrium pressures between

coesite and stishovite are 85, 88, 95 kbar, corre- sponding to the assumption of 70 percent, 80 percent, 100 percent compressibility of quartz as that

of coesite. Since errors in the enthalpy measure- ments are reported as 0.15 and 0.3 kcal/mole for

the quartz-coesite and quartz-stishovite transformation, the uncertainties of the calculated equilibrium

pressure between coesite and stishovite were

estimated to be ±4 kbar.

The calculated equihbrium pressure at 298 K

between coesite and stishovite, 85 ± 4 kbar (assumption of 70 percent), is remarkably higher than the

experimentally determined value of 70 kbar based on the NaCl pressure scale by Jeffery et al. [1966].

The value is almost comparable to the equilibrium

pressure at 298 K around 82 kbar obtained by adopting Bridgman's volume-scale for the pressure values

of the present phase equilibrium experiments, but

is considerably smaller than the predicted value,

97.5 kbar, by Stishov [1963].

^Similar calculations were rarried out by Holm et al. (1967]. in which the compressibility of coesite was assumed to be 80 percent of that of quartz. Furthermore, the pressure-volume relation of quartz given by them shows quite a large offset at higher pressures than 90 kbar from Wackerle's compression data. For these reasons we

have done more detailed calculation.

276
On the other hand, the value of dP/dT can be

calculated by means of Clausius-Clapeyron equation

dP|dT^^S|^V, and is estimated to be 0.019 kbar/K

at 298 K using the observed entropy change by

Holm et al. [1967]. The slope determined experimentally at higher temperatures is 0.024 kbar/K

on the basis of the pressure scale by Jeffery et al. [1966], and 0.036 kbar/K on the basis of Bridgman's

volume-scale. When the effects of pressure and

temperature on the slope are ignored, the former

value is almost comparable to that of the calculated

one, but the latter is too much higher than the

calculated one. Although there exist some ambiguities on the

pressure-volume data of coesite and on temperature

measurement at high pressures, it may be safely

concluded from the above results that the pressure

scale proposed by Jeffery et al. [1966] is too low,

while Bridgman's volume-scale is too high to give

consistent explanation on the stability field of

coesite-stishovite equilibrium derived from thermodynamic data. In the present argument the effect of

the correction for the thermocouple emf at high

pressures on the phase boundary curve was completely ignored. Since the correction has generally

been accepted to be positive, the value of dPjdT

will be decreased, while the equilibrium pressure

at 298 K will be increased after these corrections

are made. Such procedure will reduce the inconsistency between independent conclusions which

were derived from comparing two thermodynamic

properties, enthalpy and entropy change during

phase transformation, with the experimentally

determined equilibrium pressure and value of

dPjdT, respectively.

5. References

Ahrens, T. J., and Syono, Y., Calculated mineral reactions in the earth's mantle, J. Geophys. Res. 72, 4181-4188 (1967). Akimoto, S., Fujisawa, H., and Katsura, T., The olivine-spinel

transition in Fe2Si04 and Ni2Si04, J. Geophys. Res. 70,

1969-1977 (1965). Anderson, O. L., and Nafe, J. E., The bulk modulus-volume

relationship for oxide compounds and related geophysical

problems, J. Geophys. Res. 70, 3951-3963 (1965). Bendelyany, N., and Verestshagin, L., The synthesis of dense

modifications of silica at presence of water under pressure 150X103 ks/cm2, Dokl. Akad. Sci. USSR 158, No. 4, 819

(1964) (In Russian). Soviet Physics Doklady 9, No. 10, 870-871

(1965) (English transL).

Boyd, F. R., Bell, P. M., England, J. L., and Gilbert, M. C., Pressure measurement in single-stage apparatus, Carnegie

Inst. Wash. Year Book 65,410-414 (1967). Bridgman, P. W., Linear compression to 30,000 kg/cm^, including

relatively incompressible substances, Proc. Am. Acad. Arts

Sci. 77, 187-234 (1949). Chao, E. C. T., Fahey, J. J., and Littler, J., Stishovite, SiOj,

a very high pressure new mineral from meteor crater, Arizona,

J. Geophys. Res. 67, 419-421 (1962). Holm, J. L., Kleppa, O. J., and Westrum, E. F., Jr., Thermodynamic properties of polymorphic transformations in silica. Thermal properties from 5 to 1070 K and pressure-temperature

stability fields for coesite and stishovite, Geochim. Cosmochim.

Acta 31, 2289-2307 (1967). Ida, Y., Syono, Y., and Akimoto, S., Effect of pressure on the

lattice parameters of stishovite. Earth Planet. Sci. Letters 3, 216-218 (1967). Jeffery, R. N., Barnett, J. D., Vanfleet, H. B., and Hall, H. T., Pressure calibration to 100 kbar based on the compression of NaClJ. Appl. Phys. 37, 3172-3180(1966).

McQueen, R. G., Fritz, J. N., and Marsh, S. P., On the equation

of state of stishovite, J. Geophys. Res. 68, 2319-2322 (1963). McWhan, D. B., Linear compression of a-quartz to 150 kbar

J. Appl. Phys. 38, 347-352 (1967). Minomura, S., Ito, K., and Okai, B., Pressure and temperature

measurements in Drickamer's resistance cell up to 161 kbar

and 4000 °C, ASME PubHcation 64-WA/PT-6 (1964). Ostrovsky, I. A., Experimental fixation of the position coesitestishovite equilibrium curve, Izv. Akad. Sci. USSR (ser. geol.) No. 10, 132-135 (1965) (In Russian).

Ostrovsky, I. A., On some sources of errors in phase-equilibria

investigations at ultra-high pressure: Phase diagram of

silica, Geol. J. 5, Pt. 2, 321-328 (1967). Perez- Albuerne, E. A., and Drickamer, H. G., Effect of pressures on the compressibilities of seven crystals having the NaCl

or CsCl structures, J. Chem. Phys. 43, 1381-1387 (1965). Ringwood, A. E., and Seabrook, M., Some high-pressure trans- formations in pyroxenes. Nature 196, 883-884 (1962). Ryabinin, Y. N., The influence of pressure on some properties

of the substance, Tectonophys. 1, 227-232 (1964). Sclar, C. B., Young, A. P., Garrison, L. C, and Schwartz, C. M.,

Synthesis and optical crystallography of stishovite, a very

high pressure polymorph of Si02, J. Geophys. Res. 67, 4049-

4054 (1962). Stishov, S. M., On the equilibrium line between coesite and rutile

like modification of siHca, Dokl. Akad. Sci. USSR 148, No. 5, 1186-1188 (1963) (In Russian).

Stishov, S. M., and Popova, S. V., A new modification of silica, Geokhimiya, No. 10, 837-839 (1961) (In Russian); Geochemistry, No. 10, 923-926 (1961) (English transl.). Takahashi, T., Discussion in High Pressure Measurement, edited

by A. A. Giardini and E. C. Lloyd, pp. 240-244, Butterworths,

Washington (1963). Wackerle, J., Shock-wave compression of quartz, J. Appl. Phys.

33, 922-937 (1962). Wentorf, R. H., Stishovite synthesis, J. Geophys. Res. 67, 3648

(1962).

DISCUSSION

W. A. Bassett [University of Rochester, Rochester,

New York): I have made some measurements on the

compression, both of coesite and stishovite on the

tetrahedral x-ray press at Brigham Young University,

and my coesite data when plotted on the diagram

(fig. 2) would be roughly 60 percent of that shown

for quartz. This would tend to bring the phase transition pressure even lower, and closer to Dr. Akimoto's

experimental pressure.

I also measured stishovite and found it to be

somewhat more compressible than indicated by

the curve of figure 2. The value that I got for the

bulk modulus would be approximately three megabars. In this work, I did not see the peculiar expansion along the C-axis which Dr. Akimoto reported

in an earlier paper. The C-axis seems to contract

very slightly up to a hundred kilobars, but along a

straight line.

277
F. R. Boyd, Jr. {Geophysical Laboratory, Carnegie

Institution of Washington, Washington, D.C.): Can

you tell us how fast the reaction goes, that is, how

long it will run?

J. D. Barnett (Brigham Young University, Provo,

Utah): I understand that you determined pressures

from a room-temperature calibration, with no cor- rection made due to increased temperature in the

cell.

I suggest that at a cell temperature of 500 °C to 600 °C the pressure will increase at least five or six

kilobars, perhaps even 10 kilobars at the higher

pressures. This has been our experience in x-ray

studies.

As a second point, the value you used for the tin

transition as reported in our paper by Jeffery et al. was not intended as a transition calibration point

at all. There is strong indication that this is some- where around 100 kbar rather than 92 kbar.

Thirdly, you used calibration points based on

Decker's NaCl theory before a recent correction of

the initial compressibility of NaCl in the theory.

This will increase the pressure by approximately

2 percent.

I think when you take these" three things into

consideration the agreement with other measure- ments is really not bad at all.

AUTHORS' CLOSURE

In reply to Dr. Bassett: I don't know why there is some discrepancy in the compression curves of

stishovite determined by Drickamer-type high pres- sure x-ray apparatus and tetrahedral-anvil type highpressure apparatus. But if we use the revised data

by Dr. Bassett for stishovite and coesite, the discrepancy between the experimentally determined

transition curve of coesite-stishovite and the thermodynamically determined curve wiU be reduced.

The reaction time is about 20 to 30 minutes at high temperatures above 1000 °C.

278
The Use of Solid-Solid Transitions at High Temperatures for High-Pressure

Calibration

Phillip N. La Mori

Battelle Memorial Institute, Columbus Laboratories, 505 King Avenue, Columbus, Ohio 43201

Transition pressures have been determined for bismuth l—*2 and bismuth 2 —» 3 to the mehing

point (183 °C). The transitions parameters of thallium 2—» 1, 1 —» 3 and 2-^3 have been measured to 40 kbar and 220 °C. The thallium 1—»2^3 triple point has been determined as 114±1°C and 37.4 ±0.4 kbar. The use of the determined transition parameters as high-pressure, high-temperature

fixed points is discussed.

1. Introduction

The purpose of this paper is to make some initial

attempts to provide pressure calibration points

above room temperature. This higher temperature

calibration is of extreme interest because most

high-pressure experiments also are made with

concomitant high temperature. At the present time

(almost) all high-pressure calibrations in solid media

devices are made at room temperature. Possible

changes in the pressure calibration with temperature are generally ignored. In the few cases where

it has been considered, the conclusions have not

been testable against experimental data.

The status of high-pressure calibrations has

been dealt with extensively in another paper,

Decker et al. [1],* and will not be reviewed here.

Additionally, the reader should remember that the

comments in this paper are directed toward solid

media devices, even though they are not necessarily

restricted to them.

Fixed points for pressure calibration are accurate

enough below 40 kbar to allow accurate determination of thermodynamic quantities. The calibration

in the 40 to 60 kbar range is less certain but probably accurate enough, when combined with the

lower pressure points, to again provide reasonable

determination of thermodynamic quantities. However, a critical assumption in using these pressure

calibrations is that they remain constant at higher

temperatures, i.e., the temperature-pressure regime

of interest in most experiments.

Decker's [1] review paper on high-pressure calibration discusses in detail criteria for selection

of materials for and methods of measurement of

fixed pressure points. It is assumed that the reader

is familiar with Decker's review, and only a few

additional comments will be necessary in this paper.

After applying all of Decker's "criteria," in order

for a fixed point to be of value, it must be measurable

' Figures in brackets indicate the literature references at the <•") "f •'^

Paper presented at the Symposium on Accurate Characterization

of the High-Pressure Environment, held at the National Bureau

ofStandards, Gaithersburg, Md., October 14-18, 1968.

in the many different apparatus and experimental

setups that are in use. Solid (because solids are

easier to handle than liquids) transitions which can

be measured by volume, resistance, and DTA techniques appear to be most easily adaptable to apparatus and experimental conditions. Thus, metals

which exhibit solid transitions appear to be the most

useful materials for study.

Thalhum and bismuth were chosen for this initial

study of high-pressure fixed points at high temperature. Both materials are used for fixed points

at room temperature, and the solid transitions in

these materials have been studied by volume, re- sistance, and DTA techniques. La Mori [2] has

shown that the volume and resistance transitions

do, in fact, occur at the same pressure.

2. Experimental

In order to attempt the measurements outlined

above, it is necessary to have an apparatus capable

of accurate pressure measurement at higher temperatures. Until recently, such an apparatus has

not been available. The development of the externally heated piston-cylinder apparatus. La Mori [3],

has permitted the determination of pressure to an accuracy of about one part per 1000 to 400 °C and

pressures of 30 kbar.

The technique used to measure the transition

pressure is the same as described by Kennedy and

La Mori [4]. The piston is inserted into the cylinder

and rotated by means of a lever arm attached to the tool stack. This has the effect of relieving friction somewhat in the manner of a free piston gage.

Pressure is calculated by force/area of the main

hydraulic ram relative to that of the area of the

high-pressure piston. Appropriate corrections are made for the weight of the ram and tool stack, and

dilation of the pressure vessel with pressure.

Figure 1 is a schematic of the X-Y plot of an actual

experiment. The sample is first cycled completely

through the transition. On the second upstroke, the

cycle is stopped just after the transition is entered

(point 1). The piston is rotated until the pressure

drops to a constant value (point 2). Pressure is increased and then released (point 3). The piston

279

385-762 O - 71 - 19
Table 1. Data for bismuth transitions

Temperature, °C

Pressure, kbar

Temperature, °C

Pressure, kbar

CompressionRelease Compression

Release

Bismuth i —» Bismuth II Bismuth 11 Bismuth III

23.3 25.39 22.9 27.02

Zo.o 25.25 22.6 OA -7/1 ZD. (4 OA O ZO.W 24.8

OA 1 25.28 24.7 0£. £iO 26.68

OO.Z Z'i.y 33.5 O/C -70 ZD. 10

34.7 24.72 32.2 26.46

oo. / 24.70 48.0 0£. A 26.46 All OA O 48.0 26.14

4/.0 24.14 73.2 25.76

AO C\ 48.0 24.28 73.0 25.58

AO ^ 4o.z 24.16 98.8 24.97

73.1 23.13 98.8 24.85

73.U 23.05 127.5 24.08

110.6 21.45 127.2 24.00

110.7 21.35 151.5 23.32

127.0 20.61 151.3 23.21 127.5 20.57 163.9 22.95

127.2 20.51 163.9 22.91

151.5 19.45 172.2 22.5

151.6 19.35

163.5 18.84

163.7 18.78

173.0 18.21 173.0 18.15

is rotated until the pressuire rises to a constant

value (point 4). The actual pressure of the transition

must lie somewhere between points 2 and 4. These

are the points that are reported in the Results section. For purposes of drawing a phase diagram, the

transition line is drawn through the mean of compression and release values.

The high temperature is obtained by placing an

external heater around the outside of the 11-in.

diameter by 2-in.-thick pressure plate. The 6-in.-

thick heater also heats 2-in. -thick aluminum plates on both sides of the pressure plate. These aluminum

heat sinks are efficient enough so that there is no measurable temperature gradient over the sample

at 500 °C. Heating of the pressure plate assembly

causes the binding rings to expand away from the

carbide pressure vessel. This and the decreased

strength of the carbide at higher temperatures

reduce the operating pressures of the device from

40 kbar at room temperature to 28 kbar at 500 °C.

Additionally, corrections for the increase in diamPressure

Figure 1. Schematic of piston rotation experiment.

T T

200

O 150

<u3o

0)CLE^ lOOl—

50

-KSN (II) -- K,J,aK(9)

15 20 25

Pressure , KB

30

Figure 2. Phase diagram of bismuth.

eter of the pressure vessel due to temperature and

expansion of the binding rings, as well as the de- crease in elastic constants, must be included in the

determination of the diameter of the pressure vessel.

This is discussed by La Mori [3].

3. Results

The results of this investigation on bismuth and

thallium are given in tables 1 and 2, and figures 2

and 3. Corrections have been applied to the raw

280
240

60 —

40 —

20 —

I (BCC)

IT (HCP)

Jayaraman et. al.

m (FCC)

I 10 20 30

PKb

40 50

Figure 3. Thallium phase diagram.

Table 2. Data for thallium transitions

Temperature,

-c

Pressure, kbar

Compression Release

Thallium II —> Thallium III

25.5 37.54 35.37

70.0 37.74 36.25

98.5 37.36 36.74

114.5 37.70 36.98

Thallium II—* Thallium I

137.8 32.05 30.89

161.0 26.14 25.74

184.2 19.80 19.37 208.0 12.36 11.67

120.0 36.40 35.51

Thalhum I—* Thallium III

120.0 38.63 37.90

148.5 40.49 40.25

172.0 42.61

data as described previously. These pressure cor- rections amount to about 400 bars at 40 kbar. The

accuracy of the pressure measurements is about 1 part in 100, while the precision is about 100 bars.

The latter arises from uncertainty in the measured

quantities; the former from uncertainties in the

pressure measurements of the hydraulic ram. Temperature measurements are accurate to 0.1 °C (see

ref. [3]).

3.1. Bismuth

The bismuth l—> bismuth 2 transition has been

used as a high-pressure standard since Bridgman

[5] first accurately measured it. He reported a value

of 25.155 kbar at 25 °C; examination of his data

shows an uncertainty of about 5 percent. Later

Kennedy and La Mori [4] reported a value of 25.410

kbar ±95 bars and using the same data they re- ported a "best value" of 25.380 kbar ±20 bars.

This best value was computed from the lowest

pressure found on the series of compression values

and the highest pressure found on the series of

release values. The value of 25.41 ±.10 kbar is considered by the authors to be most representative

of their work. Heydemann [6] reported a value of

25.499 kbar ±60 bars (corrected from 25.306 kbar,

Heydemann [7]).

The present work finds a value of 25.37 ±0.2

kbar with an uncertainty large enough to agree with

the previous determinations. The reason for the

larger uncertainty reported here than that found

by Kennedy and La Mori is that the pressure gage

was not calibrated as accurately in the present

experiments. Because of the presumed greater

accuracy of the true free piston gage, the data on

figure 1 are forced through Heydemann's value of

25.50 kbar at 25 °C.

The bismuth 2 bismuth 3 transition was previously determined by Kennedy and La Mori [4]

281
Table 3. Triple points in bismuth

Reference [111 1.1 1hlQ lis WrtTK VV KJl Jv

Bismuth 19.4 kbar 16.7 kbar 17.6 kbar

1-2-Liquid 184 °C 191 °C 183 °C

Bismuth 27.5 kbar 23.6 kbar 21.9 kbar*

2-4- Liquid 175 °C 191 °C 183 °C

*This does not take into account the bismuth 2-4 phase boundary, which will decrease the transition pressure slightly.

at 25 °C as 26.96 ±0.20 kbar. This value is thought

to be better than Bridgman's [5] value of 26.54 kbar

which has 5 percent uncertainty. The value found

here is 26.82 ±0.20 kbar.

The bismuth experiments always ended in catastrophic failure of the pressure vessel at 183 °C.

Since the bismuth filled the entire bore of the pres- sure vessel, this phenomenon is what one would

expect on melting. If one accepts this inference as evidence for melting, then the 183 °C is in good

agreement with Ponyatovskii [8] and not in agreement with Klement, Jayaraman, and Kennedy's

[91 value of 191 °C.

Bridgman [10] and Ponyatovskii [8] have previously examined the bismuth 1-2 and 2-3 transitions. They are in close agreement. Our data for

bismuth 1^2 are at a slightly higher pressure than

the others and in close agreement for bismuth 2-^3.

Kennedy and Newton [11] and Klement, Jayaraman,

and Kennedy [9] have inferred the transition

boundaries by drawing straight lines between inflections in their melting curve determinations

(DTA) and the values of Kennedy and La Mori [4]. The differences between these data, as well as

their differences with the present data, are believed

to be caused by the unfavorable pressure determination that is probable in their experiment, as compared to the present method. See figure 2. These experimental differences result in significant differences in the triple point determination.

The triple points inferred from this work are compared with other values in table 3. Even if our determination of the melting temperature of bismuth 2 is low, significant pressure differences will exist in

the triple point determinations.

3.2. Thallium

Of the four previously reported studies of the

solid transitions in thallium, the work of Jayaraman

et al. [12] and Adler and Margolin [13] is considered

to be vahd. The data of Bridgman [10] and Ponyatovskii [14] have several errors. The data are shown

on figure 3. Thallium 2 —» 3 transforms very slowly

at 25 °C. The data at 25 °C were not taken over as long a time as that of Kennedy and La Mori [4]. This

caused a wider region of uncertainty. The thallium

phase diagram therefore uses the previous value of

36.69 ±0.10 kbar at 25 °C as the best value.

The 1-2-3 triple point as determined by the

present experiments is 114±1°C and 37.4 ±0.4

kbar. This is in agreement with Adler and Margolin;

the value found by Jayaraman (115 °C, 39.5 kbar) is outside our experimental error. The major differ- ence in our data and Alder and Margolin is in the

1-2 and 1-3 transitions at pressures near the

triple point. These differences are outside our limit

of error.

4. Discussion

Solid transition pressures and temperatures have

been accurately determined for bismuth and thallium. These solid transitions should provide fixed

points for high-pressure experiments at pressures

up to 50 kbar and temperatures approaching 250

°C. A gap exists in the calibration at about 10 kbar.

This is in the range attainable by hydrostatic

methods where pressure measurement is easy and

can be done with great accuracy. Therefore, use of

the hydrostatic apparatus should be encouraged for

accurate data in this pressure range.

Further work of this type can be done to 500 °C

and 30 kbar. This is the present limit on externally

heated piston-type apparatus. Specially designed

cells with AgCl or other salt pressure media and

internally heated systems may permit determination

of less accurate pressure points to 1000 °C.

5. References

[1] Decker, D. L., Bassett, W. A., Merrill, L., Hall, H. T., and

Barnett, J. D., High pressure calibration: A critical review,

NBS National Standard Reference Data System Publication, in press.

[2] La Mori, Phillip N., Calibration experiments in a pistoncylinder apparatus, in High Pressure Measurement, ed. Giardini and Lloyd (Butterworths, Washington, 1963).

[3] La Mori, Phillip N., Compressibility of Rocks and Minerals

to 450 °C and 36 kb and Their Application to the Upper

Mantel, Ph. D. Thesis, Northwestern University (1967),

205 pp.

[4] Kennedy, George C, and La Mori, Phillip N., J. Geop. Res. 67, 851 (1962).

[5] Bridgman, P. W., Proc. Am. Acad. 74, 1 (1940).

[6] Heydemann, Peter L. M., J. Appl. Phys. 38, 3424 (1967).

[7] Heydemann, Peter L. M., J. Appl. Phys. 38, 2640 (1967).

[8] Ponyatovskii, E. G., Soviet Physics Crystallography (translation) 5, 147 (1960).

[9] Klement, W., Jr., Jayaraman, A., and Kennedy, G. C, Phys.

Rev. 131, No. 2, 632 (1963).

[10] Bridgman, P. W., Phys. Rev. 48, 893 (1935).

[11] Kennedy, George C, and Newton, Robert C, Phase Transition in Some Pure Metals in Solids Under Pressure, ed. Paul and Warsehauer (McGraw-Hill, New York, 1963).

[12] Jayaraman, A., Klement, W., Jr., Newton, R. C, and Kennedy, G. C, J. Phys. Chem. Sohds 24, 7 (1963).

[13] Adler, PhiUp N., and Margohn, Harold, Trans. Met. Soc. AIME 2 30, 1048 (1964).

[14] Ponyatovskii, E. G., Soviet Physics Crystallography (translation) 4, 237 (1959).

282
The Use of Iron and Gold for Calibration of Higher Pressure and

Temperature Points

H. M. Strong and F. P. Bundy

General Electric Research and Development Center

Schenectady, New York 12301

1. Introduction

The calibration of pressure cells at room temperature by reference to the phase transitions in

Bi, Tl, and Ba is now a fairly settled procedure. But

temperature excursions away from room temperature, especially to high temperatures, quickly ob- scure the familiar landmarks and both pressure and

temperature become uncertain. Because there are no sure reference points established in this region,

navigation on the P-T plane is something hke

groping in a fog. High temperature affects the pres- sure in a number of ways: (1) it tends to increase

the pressure due to restraint of thermal expansion

of the heated parts of the cell by the surrounding

cooler parts of the cell and pressure vessel; (2) it tends to decrease pressure when it causes phase

transformations to more dense structures to occur

in the sample holder or its contents; (3) it may de- crease pressure by accelerating gasket leakage; and

(4) it may decrease pressure by accelerating the

relaxation of tension in the high-pressure chamber.

It is usually impossible to estimate the net effect

of these pressure-modifying factors, nor is there a simple direct monitoring method for the pressure.

Temperature can be monitored by a thermocouple,

but unfortunately, high pressure changes the calibration of thermocouples in an incompletely known

manner. It is the purpose of this paper to describe

a few techniques for improving the accuracy of

pressure and temperature estimates in a static

high-pressure cell. One of the methods for obtaining more useful

high-pressure, high-temperature data is to associate

the high-pressure phenomenon under study with

some other familiar and easily observable phenomenon, recording both simultaneously. In this

way the experimenter can standardize his own

experiments, and he can offer data which make it easier for another experimenter to duplicate or compare results of similar type. The reference

phenomenon is Ukely to be one for which there is a good deal of accurate data which help very much

in more closely describing the pressure and temperature conditions used in his own experiments. In

this paper, the use of iron, gold, and germanium as

a pressure and temperature reference material is described.

Paper presented at the Symposium on Accurate Characterization

of the High-Pressure Environment, held at the National Bureau

of Standards, Gaithersburg, Md., October 14-18, 1968.

2. Iron as a Pressure-Sensing Element

Iron is a rather remarkable element for meas- urements of high pressure because it has several

features which vary with pressure in a known manner in different regions of the P—T plane. Also, it is easy for the experimenter to use iron in highpressure cells. Some of its most useful features are shown in the Fe phase diagram for the region below

910 °C in figure 1. When any one of the solid-solid

phase boundaries shown in figure 1 is crossed, the

Fe responds by conveniently changing its resistance

sharply, its thermoelectric properties, or its heat

content, anyone of which is rather easily detected.

Because the slopes (dTjdP) of the three phase lines are quite different, not all of these boundary lines are equally useful for pressure measurement. The

e, y transformation above 110 kbar, 500 °C has

such a small slope * (+1.87kbar) that it is not a

particularly sensitive pressure probe for that region.

Because its temperature of transformation is nearly

the same for a wide range of pressures above 110

kbar, it might better be used as a temperature ref- erence point for approximately relating temperature

in the cell to power input. The a, e transformation

boundary is so steep (~25°/kbar) that it is primarily

sensitive to pressure in the narrow range between

about 130 and 110 kbar. The a, e transformation is in fact one of the most valuable pressure reference

points now known for the higher pressure region

in static apparatuses. Certain alloys of Fe with Co

or with V have a, e type transitions at pressures in

the range 160 to 300 kbar. Details about these higher

transitions are given by Bundy [1] ^ and by Loree

et al. [2].

The a, 7 transformation is in a more popular

region of the high-pressure spectrum. It has a satisfactory pressure sensitivity of about — 4°/kbar

in the pressure range 20 to 60 kbar. Furthermore it

is quite versatile in signahng the occurrence of the

transformation by electrical resistance change

(Ai?//?a ^ 0.5), latent heat (A// = 217 cal/mol) or thermoelectric change (an easily identified sharp

maximum in emf output of an Fe/PtlORh junction versus temperature). In addition, Fe has one other

feature not shown in figure 1. It has a relatively

large thermoelectric pressure error effect which is

' This value is less than the +2.8°/kbar given in ref. (4). This new lower value is based on the latest, most accurate, values of the AFs published by Mao, Bassett, and

Takahashi. J. Appl. Phys. 38, 272 (1967), and the experimental slope of-2.3°/kbar

for the y, a line near the triple point.

^ Figures in brackets indicate the literature references at the end of this paper.

283
KAUFMAN, CLOUGHERTY,

WEISS, 1963 (CALC)

20 40 60 80 100 120 140 160 180 200

PRESSURE kb

Figure 1. Fe phase diagram.

Solid lines: experimental data; broken line: calculated a, y boundary, Kaufman el al.

useful for pressure probing in a limited range of

temperature from about 1000 to about 1500 °C in

the pressure region from 0 to at least 60 kbar,

possibly much higher. (The authors have explored

the effect in the region 30 to 60 kbar). This latter

effect will be described in more detail in a later

section.

The establishment of the Fe phase diagram illus- trated in figure 1 is the result of the efforts of many

people over a period of about 10 years. It involved

a variety of experimental techniques including highpressure x-ray probing, metallography, dynamic

shock techniques, and theoretical approaches. It

is not useful to describe details of all this work here

but the reader can find summaries and detailed

references to former work in articles by Kaufman

et al. [3] and Bundy [4]. The a, y transition, being

in a more accessible pressure region, has received

the most attention with established techniques and

is the most accurately known boundary. In figure 1,

the solid line was determined by experiment using

a chromel-alumel thermocouple for temperature

measurement. This thermocouple is known to have

quite a small pressure calibration error. The dashed

line is the calculated curve of Kaufman et al. The

difference between the two curves over the span 20

to 60 kbar is probably due to thermocouple pressure

effect. The portion in the upper pressure region

(>80 kbar) is in a far more difficult region experimentally and naturally is less accurately known than

the region below about 80 kbar.

The high-temperature phase boundaries of Fe,

(y, 8), (8, liquid) and (y, liquid) will not be

covered in this article because they are in an ex- perimentally difficult region and lack reliable techniques for accurate observations.

For pressure (or temperature) measurement in

the regions defined by the Fe diagram shown in

figure 1, it is a simple matter to install a small piece

of iron in the pressure cell adjacent to the sample

under investigation. A wire, disk, or thermoelectric

junction may be used, depending on the mode of

detection to be apphed. The electrical resistance

changes associated with the three phase boundaries

0 200 400 600 800

T CO, UNCORRECTED

0 200 400 600 800 T (°C), UNCORRECTED

0 500 1000 1500 2000 RAM OIL PRESSURE PSI

Figure 2. Resistance changes at Fe(a, y), (o, e), (e, y) boundaries.

are illustrated in figure 2. (The Fe probe should have

a very small volume relative to the cell so that its transformation with — will not change the cell

pressure significantly.) For experiments conducted

under conditions very close to a phase boundary line,

for example, close to the a, y transformation, the

probe will give absolute pressure readings. If the

temperature of the experiment is considerably

different from the phase line, especially the a, y

boundary, then absolute pressure readings cannot

be obtained because temperature has a significant

effect on cell pressure due to thermal expansion

and other effects mentioned previously. Even under

this circumstance, the iron probe is useful for

establishing relative pressures between experiments.

In this laboratory, the a, y transition at ~ 650 °C was used extensively for pressure cahbration and cross-comparison of experiments on diamond growth at ~ 1400 °C. The iron reading was

taken during the preheats for the diamond growth

runs to measure the pressure differences at the ~ 650 °C stage of different experiment runs. The

284
iron probe (thermocouple-grade Fe wire) was in the

form of a thermoelectric junction with Pt and

PtlORh. The three wires were connected at a common point, forming a "thermotriplet," and the

milhvolt output of Fe/PtlORh was plotted against

the output of the Pt/PtlORh junction on an X — Y

recorder. The a. y transition, which occurred at temperatures between 670 and 630 °C for these

experiments, was marked by a rather sharp maximum in the output of the Fe/PtlORh couple. (See

figure 3. In practice, the a, y transition is displayed on a larger scale for better resolution.) The temperature at the center of this maximum was obtained

from the Pt/PtlORh junction. To correlate this

temperature with the a, y solid curve of figure 1,

it was necessary to add a relative correction,

AT(C/A-Pt/PtlORh) of 18 to 20 deg to allow for the

relative pressure error between Pt/PtlORh and

chromel-alumel (C/A) under these conditions [5, 6].

The temperature so obtained was referred to the

a, y transition solid curve of figure 1 for a pressure

reading. This pressure corresponded to the cell

pressure at approximately 650 °C (40 to 50 kbar)

and was invariably several kilobars below the pres- sures needed for diamond growth, which were (50

to 60 kbar), at the temperatures required, 1300 to 1500 °C. Nevertheless, the iron probe measurement

was very useful for detecting the inevitable pressure

variations between experiments, which were not

due to varying the applied load, but which were

caused by differences in gasket behavior, cell

packing, relative humidity, etc.

While using this thermotriplet junction as an Fe

a, y pressure probe, it was discovered that it had

the properties needed to measure relative pressures

at ~ 1000 to 1500 °C by the thermocouple pressure

error method. This two-thermocouple pressure-probe

method is described in detail in the next section.

It is also described in an article of this same

Symposium by Hanneman et al.

In the work described here, the Fe probe was

always used for pressure calibration when the transitions were running in the forward direction a—* y,

a—^ e, and y. There was hysteresis in recycling

the transitions. Part of this hysteresis was due to pressure-press load behavior, part to reafction

kinetics in the Fe. On the second pass in the forward direction, the transitions sometimes occurred

at slightly lower press load. This difference can be

caused by an effect called "pressure ratcheting" —

a

pressure increase obtained by cycling the cell

temperature up and down. The difference could

also be caused by reaction kinetic differences in the

Fe probe. On the second cycle, some seeds of the

high-pressure modification may be on hand, for

example. These are inherent weaknesses in the

method. Nevertheless, using the probes in the forward transformations, quite consistent pressure

measuring results were obtained. For more detailed

discussions of hysteresis effects see Bundy [4].

Figure 3. Plot of Fe/PtlORh versus Pt/PtlORh millivolts at two pressures.

Fe is in bcc cubic form below about 650 °C, in fee cubic form above 650 °C.

3. The Two-Thermocouple Method

for Pressure Measurement

It is well known that thermocouples change calibration under pressure, but that no two thermocouples change calibration in the same way [5, 6, 7,

8]. It is possible therefore to use the difference in

temperature readings between two thermocouples

under pressure as a measure of pressure. For reliability of measurements and convenience, it is desirable to have the differences in thermocouple readings

to be as large as possible. But the requirement of

having thermoelectric materials which are chemically stable under high temperature in a highpressure environment limits the choice of materials

that can be used. For example, chromel-alumel

junctions were found to be unreliable above about

800 °C. The higher the temperature, the faster was

its decay from normal calibration. Pt and Ptl0%Rh

alloy has been found satisfactorily stable up to about

1700 °C or more for brief periods and stable at 1400 °C for hours. Pt-PtlORh is satisfactory for one

of a pair of the junctions for pressure measurement.

Other metals found to be stable enough for use at 1000 °C to at least 1300 °C were Au, Ir, and Fe.

Various combinations of Pt, PtlORh, Pt40Rh, Au,

Ir, did show calibration differences due to pressure,

but the differences were too small to be of practical

use. The combination Fe/PtlORh with Pt/PtlORh

proved to be stable for short periods up to about

1400 °C and for quite long periods at 1200 to 1300

°C. (The cleaner and drier the environment of the

thermocouples, the longer they were stable.) Also,

the pressure effect was large enough to be useful. A

large number of pressure observations have been

made using the thermotriplet Pt-PtlORh-Fe in

diamond growth experiments.

In using this triplet combination, an Z-F recorder

was connected to Fe/PtlORh and to Pt/PtlORh

terminals. The millivolt output of the one junction

plotted in terms of the millivolt output of the other

junction produced a family of curves in which each

curve was characteristic of the pressure. In a high285
Fe/Pt 10 Rh MV

Figure 4. yFe/PtlORh versus Pt/PtlORh millivolts at 11 to 14 mv Pt/PtlORh for three pressures. The cell pressure near the a, y transition (~ 650 °C) is lower than at ~ 14<K) °C by

about 6 to 7 kbar.

pressure isxperiment, it was then possible to read

from the X-Y plot both the relative pressure and the

temperature from this three-wire junction. The

types of X-Y plots obtained are illustrated in figure

3. The upper portions of the curves represent the

thermoelectric behaviors of yFe/PtlORh; the lower

portions show the characteristics of aFe/PtlORh.

From these plots it will be seen that this thermojunction has a limited range for pressure measure- ment which hes between ~ 1000 and 1500 °C at ~40 to 60 kbar. Below ~ 1000 °C the curves were

complicated by the pecuUar thermoelectric behavior

associated with the a, y transition. At this transition, the Fe/PtlORh passes through a peak output.

The position of this peak is also a measure of pres- sure because as described above, the a, y transition

shifts to lower temperatures with increase in

pressure.

While it is clear from figure 3 that the a, y

transition is related to the path of the curve above

1000 °C, it does not follow that they both indicate

the same pressure. The additional heating beyond

the a, y transition causes cell expansion and

possibly increase of pressure unless the compression pistons are pushed out correspondingly. Since

the piston or anvil friction is usually very high, the

piston motion, if any, does not normally fuUy

compensate for expansion. On the other hand there

may be phase transitions to denser forms within

the cell which cause loss of internal pressure also

not compensated by piston motion. The curves in

figure 3 were made under circumstances in which

there was a simple pressure increase due to re- strained expansion. Curves showing a sharp shift

to the right due to sudden pressure loss in a phase

transformation have been observed. For these reasons the a, y transition does not furnish an

absolute calibration for the curves above 1000 °C.

The curves in figures 3 and 4 were assigned approximate pressures by comparison with the

diamond-graphite equihbrium (di/g) and by other

studies, to be described below, on the effects of

heating on cell pressures. Figures 3 and 4 have

tables which show the characteristic pressures for

each curve, the pressures obtained by the a, y

transitions for the low-temperature portions of the curves shown only in figure 3, and the pressures

for the high-temperature portions after allowing for

cell expansion. The circled point on curve (2),

figure 4, marks an experimental point lying on the

di/g equilibrium. In finding the pressure corre- sponding to this point it was necessary to apply a thermocouple pressure correction, and the one used

was obtained from those published by Hanneman

and Strong [5, 6].

Ideally, the curves of figure 4 could be assigned

absolute pressures so that they could be universally

applicable. Unfortunately different apparatuses can

have different pressure and temperature gradients

along the thermocouple wires as they leave the

pressure cell. This affects the apparent temperature

error so that the curves of figures 3 and 4 will be

more or less characteristic of a particular apparatus.

But the phenomenon illustrated here is very useful

for the experimenter to compare pressures at high

temperature in a series of individual experiments.

The method also has a distinct advantage in that it provides a means by which pressure changes can

be followed in a continuous manner at high

temperature.

4. The Effect of Heating on Cell

Pressure

The effect of heating and thermal expansion on

the pressure in a rigidly confined material can be

quite remarkable. A simple example is that of

sodium chloride, sometimes used as a pressure

medium [5, 6, 9, 10]. Referring to the work of

Decker [11] on the equation of state of sodium

chloride, its cubical expansion coefficient at 50

kbar is about 105 X 10"®. Raising its temperature

by 1000° will therefore result in a volume expansion,

+ AF/Fo, of 0.105 at constant pressure. The compressibility at 1000 °C and 50 kbar is about

3.5 X 10~^ kb"'. Therefore to confine the sodium

chloride rigidly while heating at 1000 °C would

require a pressure increase of about 30 kbar.

In a practical pressure cell, confinement is not

strictly rigid because gaskets do leak, pistons move

some, and the pressure chamber does yield some.

Furthermore, there may be internal phase changes

in the sample or sample holder material to more

dense forms. AU of these effects influence the pres- sure in a direction opposite to that of thermal

expansion. It is therefore necessary to measure

directly the effects of heating on cell pressure in

particular situations.

In this work, several methods were used for observing the influence of cell temperature and internal phase transformations on cell pressures. One

of the most effective of these was to observe the

variation of the melting point of a tiny, independently heated germanium element while the internal

286
cell temperature was varied. The schematic experimental arrangement is illustrated in figure 5 along

with the Ge fusion curve [12]. The portion of the

fusion curve used in this work was that part between

about 30 and 60 kbar. The Ge element was heated

independently by a tiny strip of carbon in contact

with the Ge bar (dimensions: 1X1X5 mm). When

melting occurred, the resistance of the heating element decreased sharply and the thermocouple re- corded a latent heat arrest. (Liquid Ge is a metallic

conductor; solid cubic Ge is semiconducting.) When

the cell pressure increased, the Ge melting point

decreased.

A test on cell pressure variation with heating was

started with no heat applied to the central carbon

tube heater and the first observation on the melting

point of the Ge bar was made. Then with heat

increased in the core of the cell, frequent Ge melting-point checks were made. After maximum

temperature was reached, the variation of the melting

temperature with time was observed. In this way a number of observations were made on both sodium

chloride and pyrophyllite sample holders and in the

presence of internal phase changes. Also, in a few

experiments a second Ge element was placed inside

the carbon tube furnace. In other experiments, Bi

or Tl elements were located diametrically opposite

the Ge element for observing the 25 and 37 kbar

transitions in these elements. By these techniques, it was possible to gain an impression about the effects

of heating on cell pressure and about internal pres- sure gradients.

The results obtained in this study are not applicable to all apparatuses and experimental arrangements, but they do represent the kinds of things that can happen in particular circumstances. The results

did reveal sharp internal pressure gradients within

the cell at relatively low temperatures. As the cell

temperature increased so did the internal pressure

until the core temperature reached about 1,000 to 1200 °C. This behavior was observed in both

pyrophyllite and sodium chloride pressure media.

At stiU higher core temperatures, the Ge melting

point started to increase, indicating that the cell

pressure was decreasing, possibly due to internal

relaxations, gasket leakage, or phase transformations. For pyrophyllite the decrease in pressure

with time was quite rapid and large, falling eventually below the calibrated pressure in the roomtemperature cell. For sodium chloride cells the

pressure always remained weU above the roomtemperature calibrated value and quickly reached

an equilibrium value, except when the cell con- tained a sample which changed phase to a more

dense form. In the latter case the pressure decreased until the transformation was ended. The

difference in behavior of sodium chloride and pyrophyllite pressure media at high temperature is due to the fact that salt has a stable lattice while

pyrophyllite does not. Pyrophyllite transforms to coesite, kyannite, and other denser structures.

Figure 5. High-pressure cell for "belt" apparatus and

germanium phase diagram.

Pressure cell contains an element for internal pressure measurement by variation of Ge melting point.

BOTH AT CELL T OF ~ lAOOX

500 1000 1500 2000 2500 RAM OIL PRESSURE - PSI

Figure 6. Pressure calibration at different temperatures for a

Hall belt-type apparatus.

An illustration of the effect of temperature on

internal cell pressure is shown in figure 6. The

curves show cell pressure in terms of the applied

load. In this case the pressure medium was largely

sodium chloride. The room-temperature pressure

calibration corresponds to the bottom curve in the

figure. Pressures found from the a, y transition at approximately 650 °C are shown by the middle

curve. On the other hand, the internal pressure at temperatures near 1400 °C were substantially

higher. The actual pressures existing were determined by the melting temperature of Ge and by the

di/g equilibrium.

These results serve to emphasize the very great

difficulty in knowing at all times and at aU temperatures what the internal cell pressure and temperature really are. Not only do the pressure and

temperature influence the normal calibrations for

each other, but cell characteristics also influence

the apparent values as compared to the true absolute

values. If a fixed pressure-temperature reference

point could be found that is located well into the

region where both P and T are large, this reference

point would help very much to check both pressure

and temperature calibrations of various sorts of P

and T probes. In the next section an attempt to locate a possible reference point using the intersection of the di/g equilibrium with the Au fusion curve is described.

287
5. A Possible High-Pressure HighTemperature Reference Point

Simultaneous, accurate, absolute measurement

of high pressure and high temperature is not

presently possible and no reliable fixed points at these conditions have been established which could

be used as standard references. An approach to the problem of finding a convenient reference has

recently been made, using the intersection of two

phase-boundary lines which have radically different

slopes. The phase lines chosen for this experiment

were the melting line of gold and the diamondgraphite equilibrium Une. These were chosen be- cause they intersect at a point where both gold

melting and diamond growth could be observed

simultaneously. The gold fusion curve has already

been experimentally observed [13, 14, 15] and some

thermodynamic data were available [16, 17, 18] for

approximately correcting the experimental data to the "true" curve. The di-g equilibrium has been

thoroughly explored [19, 20, 21] and its location in

the P-T plane is well established. The experiment

consisted in finding the apparent P and T where the

two curves crossed and then comparing this P and

T with the "true" calculated P and T for the intersection of the two phase boundaries. The differences

AP and AT between the observed and calculated

intersections were then taken to be the errors in

experimental calibration of pressure and temperature. Thus it was possible to compare the assumptions made about corrections of thermocouple error and pressure measurement error with another

reference point determined by quite independent

data and assumptions.

An experimental arrangement for melting gold

and observing the di-g equilibrium is illustrated

in figure 7. Experiments were run at several pres- sures and temperatures near 53 kbar, 1,400 °C

(corrected values) so that the di-g equilibrium point

could be established in relation to the gold fusion

curve. The thermocouple used was Pt/PtlORh and

melting was observed by latent heat arrests. The fusion curve of Au was observed by three

experimenters using Pt/PtlORh thermocouples

[13, 15] and chromel-alumel thermocouples [14].

Each experimenter had a different apparatus and

method of pressure determination, taking into

account thermal expansion effects. It is remarkable

how well the three sets of data agree when unCARBON TUBE HEATER —

Pt

Ni- ALLOY DISK ~GRAPHITE AND SEED DIAMOND

corrected melting temperatures were plotted in

terms of corrected pressures, figure 8. The "true" calculated melting curves in figure 8

were plotted in two ways to give upper and lower

bounds to the possible location of the correct curve. The upper curve was a linear extrapolation of the

initial slope obtained from Clapeyron's equation.

The AS"^'- for gold at melting according to data

given by Hultgren et al. [16] is 2.21 cal/deg g-atom.

The AF"^' is 0.60 cm-Vg-atom [17, 18]. The slope

was calculated from this data to be 6.5 deg/kbar,

and extrapolated to 55 kbar, the melting point was

1,420 °C.

The lower calculated curve was plotted with the

help of the Kraut and Kennedy melting law [22]:

where r„,„ is the melting point at 0 kbar (1336 K)

and Tm is the melting point at pressure P. AF/Fo is the room-temperature isothermal compression, and

C is a constant to be determined from melting data

at relatively low pressure, in this case 10 kbar. At

10 kbar there is very little pressure or temperature

calibration error so that the experimental and

calculated melting temperatures (using Clapeyron

equation) agree very closely at 1,401 K. The value

of C obtained from the 10 kbar melting data and

Bridgman's compression data [23] was 8.94. The

extrapolation of the fusion curve beyond 10 kbar

then depended on the Bridgman isothermal compression data which were taken only as far as 30

kbar. There was very little curvature in the compression data and for purposes of obtaining the melting line to 55 kbar, the compression data were

extrapolated to this pressure. The melting point

at 55 kbar was thus found to be 1673 K.

1400

1300

1200

1100

1000,

di/g EQUIL.

EXP. FUSION CURVE: PRESSURE CORRECTED TEMPERATURE AS

READ.

+ COHEN, KLEMENT 8 KENNEDY (1966) » DECKER a VAN FLEET (1965) -^PRESENT WORK

20 30 40 50 PRESSURE - kb

60 70

Figure 7. Arrangement for simultaneously observing melting

point of gold and diamond-graphite equilibrium.

Figure 8. Diamond-graphite equilibrium line and gold fusion

curve.

Upper curve: extrapolation of initial slope of gold fusion curve, 6.57kbar. Lower

calculated curve: from Kraut and Kennedy fusion curve formula. Bottom curve: experimental data, Pt/PtlORh thermocouple temperatures not corrected for pressure

error; cell pressure was corrected for pressure increase by internal thermal expansion.

288
The true di-g, equilibrium and Au fusion curves

thus appear to intersect at 52.5 to 53 kbar at 1388 to 1410 °C (fitis. 8 and 9). The experimental intersection of the two curves appears to occur at 42.5 to 43 kbar at 1344 to 1352 °C (fig. 9). In the experimental determination, no allowance was made for pressure error

in the Pt/PtlORh thermocouple. Also, the pressures

used for the data points were those obtained from

room temperature pressure calibration by Bi, Tl,

Ba-transition points.

The experimental temperature and pressure for

the intersection of the two boundary lines lie below

their anticipated values. The pressure requires a correction of —h 10 kbar. The temperature readings were 36 to 66 degrees too low. The pressure

error was consistent with the studies on effects of

thermal expansion described above and the temperature error was consistent with the earlier work of

Bundy, Hanneman and Strong and the recent work

of Wentorf [24] on thermocouple pressure error. A

more accurate fix on the intersection point could

be obtained if the reliability of the Au fusion curve were to be improved.

6. Summary

Simultaneous accurate measurement of both

high pressure and high temperature is not presently

possible. However, a useful experimental procedure

is to use one type of phase change as a reference

against which other phase relations may be studied.

Iron was particularly useful for this purpose because

it exhibits several easily observed pressure-sensitive

solid-solid type phase transformations. It also

displays a large thermoelectric pressure-error

behavior, which when used with Pt and PtlORh

thermoelectric materials, provides both pressure

and temperature references.

, CALC. ^INTERSECTION

//I OBSERVED, UNCORRECTED

di/g EQUILIBRIUM

V-UNCORRECTED GOLD

FUSION CURVE

It was shown that cell heating can substantially

change the internal pressure in a high-pressure

sample and that pressure at high temperature can decay with time at a rate dependent on cell materials

used. The internal cell pressure changes were

detected by changes in melting point of a tiny Ge

element.

A high-pressure, high-temperature fixed reference

point could be the intersection of two well-established phase boundary lines. The effectiveness of

this approach to the problem of high-pressure, hightemperature measurement was illustrated using

the diamond-graphite equilibrium and the gold

fusion curve.

1000, 20 30 40 50 PRESSURE -kb

60 70 80

Figure 9. Apparent gold fusion curve and dijg equilibrium in relation to theoretical gold fusion curve and dijg equilibrium. No corrections for pressure or temperature errors applied to experimental data.

[1

[3

[4;

[5

[6

[7

[8

[9

[10

[11

[12

[13

[14

[15

[16

[17

[18

[19

7. References

Bundy, F. P., Fe-Co and Fe-V Alloys for pressure calibration in the 130 to 300 kbar region, J. Appl. Phys. 38,

2446-2449 (1967). Loree, T. R., Fowler, C. M., Zukas, E. G., and Minshall,

F. S., Dynamic polymorphism of some binary iron alloys,

J. Appl. Phys. 37, 1918-1927 (1966). Kaufman, L., Clougherty, E. V., and Weiss, R. J., The

lattice stability of metals — III. Iron, Acta Met. 11,

323-335 (1963). Bundy, F. P., Pressure-temperature phase diagram of iron

to 200 kbar, 900 °C, J. Appl. Phys. 36, 616-620 (1965). Hanneman, R. E., and Strong, H. M., Pressure dependence

of the emf of thermocouples to 1300 °C and 50 kb, J. Appl.

Phys. 36, 523-528 (1965). Hanneman, R. E., and Strong, H. M., Pressure dependence

of the emf of thermocouples, J. Appl. Phys. 37, 612-614

(1966). Bridgman, P. W., Effect of pressure on thermoelectric

properties, Chap. X, p. 295, in The Physics of High Pres- sure (G. Bell and Sons, Ltd., London, 1952). Also Proc. Am. Acad. Arts Sci. 53, 269 (1918). Bundy, F. P., Effect of pressure on emf of thermocouples,

J. App. Phys. 32, 483-488 (1961). Strong, H. M., Pressure distribution in reaction vessels (sodium chloride sample holder), U.S. Patent 3,030,662

(1962). Jeffery, R. N., Barnett, J. D., Vanfleet, H. B., and HaU, H. T., Pressure caUbration to 100 kb based on compression of NaCl (cell contained a sodium chloride insert), J. Appl.

Phys. 37, 3172-3180 (1966). Decker, D. L., Equation of state of NaCl and its use as a

pressure gauge in high-pressure research, J. Appl. Phys.

36, 157-161 (1964). Bundy, F. P., Phase diagrams of silicon and germanium

to 200 kbar, 1,000 °C, J. Chem. Phys. 41, 3809-3814

(1964). Cohen, L. H., Klement, W., Jr., and Kennedy, G. C, Melting of copper, silver and gold at high pressures, Phys. Rev. 145, 519-25 (1966). Decker, D. L., and Vanfleet, H. B., Melting and high temperature electrical resistance of gold under pressure, Phys. Rev. 138, A129-133 (1965). Strong, H. M., and Hanneman, R. E., unpublished work on

fusion curve of gold.

Hultgren, R., Orr, R. L., Anderson, P. D., and Kelley, K. K., Selected Values of Thermodynamic Properties of Metals

and Alloys, John Wiley & Sons, Inc., New York, N.Y.

(1963). Esser, H., Eilender, W., and Bungardt, K., Arch. Eisenhuttenwesen 12, 157 (1938). Krause, W., and Sauerwald, F., Z. Anorg. Allg. Chem.

181,347 (1929). Berman, R., and Simon, Sir Francis, On the graphitediamond equilibrium, Z. Elektrochem. 59 , 333-338

(1955).

289
[20] Berman, R., Thermal properties of diamond, Chapt. 14,

Page 371, Physical Properties of Diamond, Ed. R. Berman (Clarendon Press, Oxford, 1965).

[21] Bundy, F. P., Bovenkerk, H. P., Strong, H. M., and Wentorf,

R. H., Jr., Diamond-graphite equilibrium line from

growth and graphitization of diamond, J. Chem. Phys.

35, 383-391 (1961).

[22] Kraut, E. A., and Kennedy, G. C, New melting law at high

pressures, Phys. Rev. Letters 16,608 (1966). [23] Bridgman, P. W., Linear compressions to 30,000 kg/cmincluding relatively incompressible substances, Proc. Am. Acad. Arts Sci. 77, 189-234 (1949). [24] Wentorf, R. H., Jr., Temperature measurement by thermal

noise at high pressures, paper presented at this meeting.

DISCUSSION

P. M. Bell (Geophysical Laboratory, Carnegie In- stitution of Washington, Washington, D.C.): In the

test cell, does iron diffuse into the platinumplatinum 10 rhodium thermocouple and affect the

temperature measurement?

In determining the diamond-graphite curve, is there catalyst present and, if so, does it affect the

equilibrium point?

C. W. Beckett (National Bureau of Standards,

Washington, D.C.): I was very much interested in

your remarks about gold. As you know, the gold

point is one of the fixed points on the international

temperature scale and at one atmosphere is reproducible to about a hundredth of a degree. One

approach to this problem of thermodynamic temperatures, pressures, and so forth, would be to have

a comprehensive program on the phase diagram, so

to speak, which would include not only the meas- urements such as those you have mentioned but

also measurements of the heat of fusion and the

volume change at high pressures. This is much

easier said than done, of course. Before undertaking this, one has to have a carefully selected group of substances. The advantage

of gold in this case is that it does not react with the

carbon significantly, at least at one atmosphere, be- cause for many years the gold-point temperature

has been reproduced in graphite containers.

Another metal which behaves fairly well in these

cells is copper. Now, of course, the melting point of

copper is only about 20 degrees above the gold melting point at one atmosphere, but the slope of the curve with pressure is different.

Have you considered using the same techniqiie

with other metals such as copper? Platinum in

graphite would not be suitable because platinum

does react to graphite. However, the platinum sec- ondary temperature point is measured in thorium

dioxide containers, and it is reproducible to better

than a degree at one atmosphere. Thorium is hard to use in a pressure vessel, but one could use other

oxides. Beryllium might be better.

AUTHORS' CLOSURE

Response to Dr. Bell: Our platinum 10 rhodium

thermocouples did not show any tendency to change

calibration with time due to their being in contact

with an iron wire. The temperature-heating power

relations were stable for long periods. At the end of

a test, the freezing and melting temperature arrests

at the metal-carbon eutectic furnished a fixed check

point. We did not observe an apparent shift in this

check point as a result of the platinum-platinum

rhodium couple sharing a contact with iron, even

after several hours. The junctions were in a nearly

isothermal region, so that a considerable amount of

diffusion would have to occur before the effect

could be felt. On the matter of the catalyst, we have published

data (H. M. Strong and R. E. Hanneman, J. Chem.

Phys. 46, 3668 (1967)) showing the displacement of

the diamond graphite equilibrium due to the pres- ence of the catalyst. The catalyst must be present

in order for the reaction to run freely in either

direction. The displacement of the diamond graphite equilibrium is quite small; it amounts to about

12°, which is equivalent to about one third kilobar.

Allowance was made for this displacement.

Response to Dr. Beckett: Other metals have been

considered for use in locating phase boundary intersections. Copper would be suitable to use for

finding an intersection between its fusion curve and

the diamond-graphite equilibrium. This intersection

should occur about 70° below the intersection for

gold. If both intersections were located, there

would be three quantities to check for consistency:

the two intersections and the slope of the diamondgraphite equilibrium between the intersection

points. This slope should agree with the calculated

slope of R. Berman and Sir Francis Simon (Z.

Elektrochem. 59, 333 (1955)).

Another possibiHty is to find the intersections of

the well-known iron y equilibrium with the

fusion curves of Al and Pb. These intersection

points should occur at temperatures between 600

and 800 °C, pressures between 20 and 50 kbar,

where thermocouple calibrations have smaller pres- sure errors. Finding these points would provide

fairly accurate check points from which to reach

further into the high pressure and temperature

region.

290
Transition Pressures of Bi 3-5, Sn, and Fe

M. Centre

Commissariat d I'Energie Atomique, Centre d'Etudes de Bruyeres-le-Chdtel, Croupe Hautes Pressions,

29-33, Rue de la Federation — Paris

Making use of "X type anvil," the change in resistance of two reference metals in each run was

simultaneously recorded in order to compare their transition pressures. A linear extrapolation through

the well-known points below 60 kbar showed inconsistencies in the most commonly used high pressure

scales. The recording of the pistons displacements lead to an exponential extrapolation which gave

transition pressures of 78±2 kbar for Bi 3-5, of 102±4 kbar for Sn and of 140±15 kbar for Fe.

1. Introduction

The exact measurement, or even the simple

evaluation, of the pressure which is built up inside a high-pressure solid-medium apparatus has from

the beginning always been a problem. It was only in

piston-cylinder apparatus that a direct measurement

of the pressure was possible. Thus it became

feasible, making use of the phase transitions of a number of metals, to evaluate the pressure which

was built up in more intricate apparatus like "belt"

below 60 kbar.

Above 60 kbar there is much confusion taking

into account the various data that have been

published to date (table 1). Lately, during the same

year 1966, two teams, one from U.S.A. working

with Professor Hall [11],' the other from Soviet

Union working with Professor Vereschagin [13]

published in earnest that the high Bismuth transition

occurred at 76.5 ± 2 kbar for the first one, at 89.3

kbar± 1 percent for the second one. Research workers in the high-pressure field

cannot remain unconcerned by that state of the art,

if they wish to make more precise measurements.

The present work is an attempt to show the inconsistencies of the main pressure scales in the particular case of an apparatus called "X-type anvil", and

to suggest possible values for the transition pres- sures of bismuth 3-5, of tin and iron. The four

most widely used pressure scales, called A, B, C, D

(table 2), have been chosen so as to be compared.

Making the assumption that the transition pres- sures below 60 kbar are accurate, a linear extrapolation reveals a few inconsistencies. From the piston

displacement recording, an analytical expression

of the calibration curve is then derived. It is thus

possible to evaluate the higher transition pressures.

2. Experimental Procedure

The apparatus called "X-type anvil" has already

been described [31, 32, 35] and first presented at the

Eindhoven meeting of the European High Pressure

' Figures in brackets indicate the literature references at tlie end of tiiis paper.

Paper presented at the Symposium on Accurate Characterization

of the High-Pressure Environment, held at the National Bureau

of Standards, Gaithersburg, Md., October 14-18, 1968.

Research Group (1966). It consists of a die and of

pistons with a special shape as shown on figure 1.

It allows higher pressures to build without damage,

inside a volume which is identical to that of a "belt."

The cell body is made of pyrophyllite and the

gaskets are of a mixed type, that is to say they

consist of a pyrophyllite ring and of a Teflon ring

(fig. 1). The experiments have been carried out by recording simultaneously the resistance of two reference

metal samples at room temperature. This procedure

eliminates the lack of reproducibility of the calibration curve. The specimens were located side by

side, at 1 mm from each other, inside a Teflon

cylinder ((^) = 4 mm, h = S mm) which was in the

center of the cell (fig. 2). The metal samples were

wires 0.5 mm in diameter and 4 mm long. The

electrical connections were established between

the pistons and the chamber. The chemical analysis

of the samples is given in table 3.

Figure 1. Cross section of the cell inside the X-type anvil

0 4 mm

samples

h = Jmrn , teflon cylinder

glectric tab and connection

to the pistons

FuiURE 2. Samples assembling, schematic.

291
Table 1. Pressure transition data

Apparatus Year Authors Ref.

Pressure Transition in Kilobars

Bi

l-II

Bi

ii-iii

Tl

II-III Ba

II-III Yb BiIII-V Sn

MI

Fe a — e Ba

Piston-cylinder

Bridgman's anvil

Shock

Belt

Shock

Drickamer's anvil

Piston-cylinder

Drickamer's anvil

Piston-cyclinder

Tetrahedral press Bridgman's anvil

Bridgman's anvil Piston-cylinder

Tetrahedral press

Cubic press

Tetrahedral press

Piston gage

Dead-weight piston gage

Piston-cylinder

fl941 11942 1952

1956

1958

1960

1961

1962

1962

1963

1963 1964

1964

1965

1965

1965

1966

1966

1967

1967

Bridgman — in volume

change (vol) Bridgman — by resist- ance (res) Bancroft et al. (res) Bundy (res)

Boy and England (res) Balchan and Dric- kamer (res) Kennedy-La Mori (vol)

Balchan and Dric- kamer (res) Klement-JayaramanKennedy

Hall and Merrill (res) Stark and Jura

Stromberg et al Roux

Jeffery (vol)

(res) Giardina and Samara

(ind. vol) Jeffery (res) Barnett f— sheet Van Fleet [—wire

Hall Vereshchagin, Zubova,

et al.

Heydemann (vol)

Kennedy et al.

25

25.65

25.65

25.2

25.38 ±0.02

25.5 ±0.15

25.0 ±0.5

26.2 ±0.8

26.5 ±1.3

25.4 1%

[25.48

125.50

27.08

27.08

26.96 ±0.18

27.6 ±0.15

28.0 ±0.6

29.1 ±0.8

29.7 ±1.4

26.9 1%

±0.06

39.2/ 4045

45

37.1

36.7 ±0.1

36.8 ±0.6

35.6 ±1.3

58.8/ 60

78.4/ 80

78.4/ 80

59 ±1

59.6 ±1

54.5 ±1.5

35.4 54.6 ±2.1 ±0.9

36.9 58.5 1% 1%

58.8

58.8

39.5

38.1 ±1.3

38.2 ±1.5

88.2/ 90

122.5

90 ±2

-78

81 ±4

76.5 ±2

81-

82

75.7 ±1.3

89.3 1%

no data below

100 kbar

131 114

133 ±1.5%

113/ 115

99 ±4

107

92 ±3.5

92 ±3

according to grain size and purity

55.0 ±0.5

144

118±6

Table 2. Pressure scales

Ref. Year Authors

Pressure Transition in Kilobars

Bi

III

Bi

II-III

Tl

II-III BaII-III

Bi

VI-VIII Sn

1-2 Fe

Bridgman 25.4 26.8 36.7 58.5 89 113/ 133 A 1962 Kennedy-La-Mori 115 Balchan-Drickamer ±0.1 ±0.1 ±0.1 ±0.6 ±1

[ 1965 Stark 25.4 26.8 36.7 59 81 107 133 B Jura — Stromberg

1 1966 ±0.1 ±0.1 ±0.1 ± 1 ±4 ±4

Jeffrey 25.0 35.6 54.5 76.5 92 C 1965 Barnett

Hall D 1967 Vereshchagin & al 25.4 36.9 58.5 89.3 ±1 % ±1 % ±1 % ±1 %

292
Table 3. Chemical analysis of the samples

Samples Bi Tl Ba Sn Fe

\^Purity

TotalX 98.5% 99.99% 99.2% 99.9% 99.8%

Impuritie^

ppm ppm ppm ppm ppm

Li < 10 Na 100 < 10 < 25

Mg 10-50 < 10 400 25 < 50

Al < 50 < 10 300 < 50 < 25

Si Traces < 50 80 200 80

K < 10 < 10 < 10 Ca < 100 0.1% < 50 < 50

Ti < 100 < 100 Cr < 50 < 50

Mn < 25 250 230

Fe < 50 < 10 < 50 < 50

Co < 10 Ni < 100 < 100 350

Cu < 10 < 50 < 50

Zn < 200 < 100 As Traces

Sr 0.5%

Zr Traces Mo < 250 < 250 800

Ag < 10 < 10 Cd < 10 Sn 1,000-2,000 < 50 < 50

Pb 1-1.5% < 10 < 50 < 50

C 450 ± 50

At the same time that the resistance changes were recorded, the displacements of the pistons

toward each other were measured, makinji use of

four dial gages located at 90° angle around the highpressure apparatus (fig. 3). All the measurements

have been carried out during the first increase in

pressure run and the loads have been measured

with a strain-gage dynamometer.

3. Experimental Results

Fifty runs have been carried out under those conditions, among which the Fe transition was obtained 3

times, the Sn transition 10 times, and the Bi 3-5

transition 20 times. A typical resistance recording

is shown on figure 4. The displacement of the pistons, as measured during each run, has the shape

which is shown on figure 5. The lower part corre- sponds to the extrusion of the gaskets without much

increase in the pressure, whereas the upper part

corresponds to the real compression of the cell. Calibration curves, based on the B scale (table 2), are shown on figure 6 for two different die diameters. It

is to be noted that the lower part is approximately a

straight line which goes through the origin of

coordinates.

From those experimental results, two types of

extrapolations have been made to get the transition

pressures of Bi 3-5, Sn, and Fe.

The first one is a simple linear extrapolation. The

second one makes use of an analytical function. The

piston displacement recordings give a clue as to the

Figure 3. Photograph of the "X-type anvil" showing the four

dial gages used to record the piston displacements.

15

Arbitrary unit &• R

Figure 4. Typical behavior of the resistance of the metals used

to calibrate the pressure generated inside the cell.

293
50 100 150 200 250 300

Figure 5. Piston displacement curve.

Fl(;URE 6. Pressure calibration curves for two different die diameters.

kind of function which is to be chosen. When plotted on a semilogarithmic scale those recordings are

quite linear in their upper portion (fig. 7). With a slope k thus the piston displacement e can be

expressed as:

-Flk

where eo is a constant and F is the load.

When this exponential stage is reached, the ex- trusion is considered as achieved and the assump150

LOAD

(ton)

DISPLACEHENT (mm)

4 5 6 7 8 9 10

—I 1 1—r- 2 3

Fl(;URE 7. Piston displacement curve.

tion is made that the change in volume of the cell is proportional to the piston displacement. On the

other hand, the pressures which are generated in- side the cell are related to the overall compressibility of its components through a law, which can be

shown empirically to be close to an exponential with

a good approximation within the range of experimentation [28]. From these considerations an expression

of the calibration curve follows:

P = -A log (B+exp -Flk).

A, B, and C are constants which can be evaluated

knowing accurately at least three experimental

values. They will be chosen among the best known

transition points (Bi 1-2, Tl, Ba). It is obvious that

when the load F is increased indefinitely the pres- sure P must go to a limiting value, which is the case

with the above expression provided that B is positive. It also gives curves whose concavity is toward

increasing load as expected.

4. Discussion

4.1. Linear Extrapolation

As the true caHbration curve must go to an

asymptotic value when F goes to infinity, the linear

extrapolation gives excess pressures. The lowest

among those are gathered in table 4 together with

the corresponding values of the four chosen scales

A, B, C, and D.

294
Table 4. Linear Extrapolation {pressures in kbar)

S(3 ales A B C D Alined

values

Bi3^5

Nominal

values

89 81 76.5 89.3

« 78 ± 2 Extrapolated values

77.7 77.7 75.3 78

Snl^2

Nominal

values

115 107 92 (115)

« 104 ± 5 Extrapolated values

118 108 118 118

Fe a-^e

Nominal

values

133 133 118 (133)

« 170 ± 17 Extrapolated values

182 170 144 182

Table 5. Exponential extrapolation (pressures in

kbar)

The following remarks can be made:

(a) For the Bi 3-5 transition all these excess

values are below the corresponding values of the

scales. It is a striking case of inconsistency, at least

for the scales A, B, and D. This results from the

fact that the calibration curve of the present

apparatus is nearly linear up to about 100 kbar.

The scale C is the more coherent.

(b) For the Sn transition the linear extrapolation

does not reveal any inconsistency even if this extrapolation uses the scale value of the pressure transition of Bi 3-5.

(c) For the Fe transition the linear extrapolation gives a result which is very far off, even if the scale value of the pressure transition of Sn is used.

If a linear extrapolation is made through all the

range the pressure transitions are found to be

78 ± 2 kbar for Bi 3-5, 104 ± 5 kbar for Sn, and

170 ± 17 kbar for Fe, which is certainly very far

for the true value.

4.2. Exponential Extrapolation

The exponential extrapolation hopefully should

give a better estimate of the transition pressures.

Table 5 gives the calculated values.

The accuracy given next to each value takes into

account the uncertainty of the pressure transition of

Bi 1-2 (25.4±0.1 kbar) ofTl 2-3 (36.7±0.1 kbar)

and of Ba using the latest value given by Kennedy

[29] (55 ±0.5 kbar), which fit better than others.

For the present apparatus, it turns out that by

using the following transition pressures the calibraScales a-d B C

Sn Nominal values 115 107 92

1^2 Extrapolated values 110 101.8 97.7

Fe Nominal values 133 133 118

a—>e Extrapolated values 150 106

tion curve is nearly linear.

Bi Bi 1-2 Tl Ba 3-5 Sn

25.4 36.7 55 78 102

±0.1 ±0.1 ±0.5 ±2 ±4

The case of Fe is a little different because the

nucleation of the transition seems to depend greatly

upon the pressure gradients inside the cell, as shown

by several people [26]. The present apparatus gives

stresses which are of a less uniaxial character as a Drickamer or a Bridgman anvil, which might explain

the high value found: 140 ± 15 kbar.^ It thus appears

that Fe does not constitute a good reference metal.

In order to fill up the gap in the high-pressure

scale, it would be desirable to find another reference

element such as germanium [30, 34].

5. Conclusions

With a "Z-type anvil", inconsistencies in the highpressure scales which are currently used have been

revealed. A new pressure scale which fits better

this apparatus has been estabhshed, which would

locate the transition pressures of Bi 3 —5 at 78 ±2

kbar and of Sn at 102 ±4 kbar, and of Fe at 140 ±15

kbar. However, many authors have discussed the

influence of the apparatus shape, of sample shape

(wire or ribbon), of the pressure-transmitting

medium, of the pressure gradients on the nucleation

of the aUotropic transformations under high pressure. Thus the above conclusions may be valid only

for the apparatus which was used. Rather than cahbrating it would be better to evaluate the pressure

at all times by the continuous change in the property

of a material such as the lattice parameter with x-ray

diffraction. Provided the equation of state of the

material is theoretically known, an apparatus such

as a hexahedral press built in our laboratory [31]

should bring in the future interesting results.

^ Recent experiments performed on iron and barium samples lying side by side in the high-pressure cell cast no doubt on the fact that the high barium transition definitely occurs at a much lower load than the iron transition, in our apparatus.

295

385-762 O - 71 - 20
6. References

[1] Bridgman, P. W., Proc. Am. Acad. Arts Sci..74,425 (1942); Phys. Rev. 60, 351 (1941).

[2] Bridgman, P. W., Proc. Am. Acad. Arts Sci. 81, 165-251

(1952).

[3] Kennedy, G. C, and La Mori, P. N., J. Geophys. Res. 67,

851 (1962).

[4] Balchan, A. S., and Drickamer, H. G., Rev. Sci. Instr. 32,

308 (1961).

[5] Bancroft, D., Peterson, E. L., and Minshall, S., J. Appl.

Phys. 27, 291 (1956).

[61 Starit, W., and Jura, G., A.S.M.E. paper 64WA/PT-28

(presented at Winter Meeting, Nov./Dec. 1964).

[7] Bean, V. E., Master's Thesis, B.Y.U. (1964).

[8] Jeffery, R. N., Master's Thesis, B.Y.U. (1965).

[9] Stager, R. A., Balchan, A. S., and Drickamer, H. G., J. Chem.

Phys. 37, 1154 (1962).

[10] Klement, W., Jayaraman, A., and Kennedy, G. C., Phys. Rev. 131, 632 (1963).

[11] Jeffery, R. N., Barnett, J. D., Vanfleet, H. B., and Hall, H. T.,

J. Appl. Phys. 37, No. 8, 3172-3180 (1966).

[12] Lacam, A., Institut frangais des combustibles et de I'energie.

[13] Vereshchagin, L. E., Zubova, E. V., Bujmova, I. P., and

Burdina, Dokl. Akad. Nauk SSSR 169-174 (1966).

[14] Boy, F. R., and England, J. L., J. Geophys. Res. 65, 741-748

(1960).

[15] Dieter, G. E., A.S.M. 279-340(1960).

[16] Adler, P., and Margolin, H., Calibration experiments with a tetrahedral-type pressure apparatus, A.S.M.E. paper

62-WA-314(1963).

[17] Bundy, F. P., Phys. Rev. 110, 314 (1958); and from Modern

Very High Pressure Techniques, ed. — R. H. Wentorf

(1962).

[18] Hall, H. T., and Merrill, L., Inorg. Chem. 2,618 (1963).

[19] Giardini, A. A., and Samara, G. A., J. Phys. Chem. Solids

26, 1523 (1965).

[20] Heydemann, P. L. M., J. Appl. Phys. 38, No. 6, 2640-2644

(1967); Erratum, J. Appl. Phys. 38, No. 8, 3424 (1967).

[21] Roux, C, Appareil pour hautes pressions a chambre

cylindrique. Private communication — C.E. A. (1965).

[22] Johnson, D. P., and Heydemann, P. L., Rev. Sci. Instr. 38,

No. 8, 1294-1300 (1967).

[23] Decker, D. L., J. Appl. Phys. 36, No. 1 (1965).

[24] Stromberg, H. D., and Stephens, D. R., A.S.M.E. paper

64-WA/PT-13(1964).

[25] Young. A. P., Robbins, P. B., and Schwartz, C. M., A.S.M.E. paper 62-WA-257 (Oct. 1963).

[26] Giardini, A. A., and Lloyd, E. C., High Pressure Measurement, Butterworths Press (1963).

[27] Bradley, R. S., High Pressure Physics and Chemistry, Acad.

Press, London (1963).

[28] Contre, M., Compressibilite isotherme des milieux solides isotropes. Private communication (1963).

[29] Haygarth, J. C, Getting, L C, and Kennedy, G. C, J. Appl.

Phys. 38, No. 12, 4557-4564 (1967).

[30] Evdokimova, V. V., Sov. Phys. USPEKHI 9, No. 1, 54-72

(1966).

[31] Contre, M., French Patent, Dispositif pour engendrer de

tres hautes pressions au sein d'un solide. Commissariat

a I'Energie Atomique, France No. 1, 457, 690, 23/9/65-

26/9/66.

[32] Contre, M., Enclume annulaire X, Private communication,

C.E.A. (1967).

[33] Drickamer, H. G., and Minomura, S., J. Phys. Chem. Solids

23, 451 (1962).

[34] Contre, M., French Patent, Appareil pour engendrer de

tres hautes pressions statiques au sein d'un solide hexaedrique. Commissariat a I'Energie Atomique, France

No. 1, 395, 599, 6/12/63 - 8/3/65; U.S.A. No. 3, 261, 057.

[35] Contre, M., Enclume annulaire type MCX, Private communication, C.E.A. (1965).

[36] Schaufelberger, M., Diffractometrie de rayons X sous tres hautes pressions. Private communication, C.E.A.

(14/12/67).

[37] Schaufelberger, M., Compressibilite de I'hydrure de lithium,

Private communication, C.E.A. DISCUSSION

F. Bundy (General Electric Research and Development Center, Schenectady, New York): I would like

to respond to Dr. Contre's suggestion for using

germanium as a calibration material near 120 kbar.

Germanium does undergo a drop in resistance of

10^ to 10^, but my experience with it has been that

the resistance change extends over an appreciable

pressure range. For example, it may start falling at 110 kbar or even 105, and continue on up to a nominal 120 or 125 kbar. Thus, in my experience,

it's too fuzzy for use as a calibration point. I have

run germanium in the high-compression belt and in

the Drickamer-type apparatus, and in both cases obtained a six orders of magnitude drop in resistance, but the change is spread out too much for a cahbration point, in my opinion. Silicon is worse yet.

Iron, apparently, is not much better.

I beheve that you used the x-type anvil for the

barium transition. I would Uke to ask how that result

came out. I am interested because I have found the

high barium transition to be a very sharp and

satisfactory calibration point. The ones you showed

looked quite sharp, and perhaps would fall on the curve quite well.

R. Roy (Pennsylvania State University, University

Park, Pennsylvania): May I recommend strongly

against the use of the germanium transition. We

have shown (Bates, Dachille, and Roy, Science

147, 860 (1965) that Ge-I goes at stable equilibrium

to Ge-lll at' less that 15 kbar at room temperature.

Hence Ge-I is highly metastable in the 100 kbar

regime, when it goes to Ge-II. While the transition

may appear to be sharp, different samples will

likely show different transition pressures in a manner similar to CdS which undergoes an analogous metastable transition. (IMiller, Dachille, and

Roy, J. Appl. Phys. 37, 4913 (1966).

296
On Application of Mercury Melting Curve for Accurate High-Pressure

Measurements

V. S. Bogdanov, Yu. L. Levin, S. S. Sekoyan, and Yu. I. Shmin

All-Union Research Institute for Physical and Radiotechnical Measurements, Moscow Region, Mendeleevo,

U.S.S.R.

The successful solution of the problem of highpressure measurement up to several hundreds of

kilobars is based on the further extension of the

range and the improvement of accuracy of hydrostatic pressure measurements. The maximum

hydrostatic pressure achieved at present is limited

to 50 kbar, while the maximum pressures that can

be reliably measured by means of free piston gages

do not exceed 25 kbars.

The long-term experience of VNIIFTRI in the

research of the melting process under pressure

shows that the above-mentioned problem can now

be solved most successfully on the basis of the

mercury melting curve equation as it was suggested

by Zhokhovsky as far back as 1957 [1].' One of the

merits of this method lies in the fact that under

certain conditions the process of mercury melting

is an equilibrium one, and can be reproduced to a high degree of accuracy. Besides the relation between pressure and temperature (measured quantity) may be represented by the universal Simon

interpolation equation [2]

p = a (1)

where a and c are empirical constants and To the

temperature of the triple point. This equation was

derived and studied in a number of investigations

[3-7].

Some substances such as Rb, Cs, KNO3, and

KNO2 however have a maximum in the melting curve and the Simon equation fits experimental data on these substances only up to pressures about

0.5 A (where A is pressure of the maximum). Therefore, the Simon equation with parameters chosen to

fit experimental data in limited pressure range

cannot be extrapolated to any high-pressure value.

For the development of the precision method for

measuring high hydrostatic pressures up to 50 kbar

based on the mercury melting curve (MMC), it is necessary to solve the following two problems:

1. The proper choice of the reliable equation for

the MMC in the pressure range where free piston

gages may be used.

' Figures in brackets indicate the literature references at the end of this paper. Paper presented at the Symposium on Accurate Characterization

of the High-Pressure Environment, held at the National Bureau

ofStandards, Gaithersburg, Md., October 14-18, 1968.

2. Substantiation of extrapolation of chosen

equation to higher pressures.

To solve both problems one needs more detailed

information on the melting under pressure, and in

particular the data on the volume change along the

melting curve. The investigation of the volume

change during melting permits one to prove experimentally the equilibrium character and reproducibility of the melting process under pressure, and

reveal possible peculiarities on the melting curve.

1. The MMC Equation

The apparatus, procedure, and results of the first experiments on estabUshing the MMC equation are described in [8]. The experiments were carried out

up to 10 kbar, pressure being measured by means of

manganin gage calibrated against a free piston gage.

The later and more accurate experiments were

performed on the apparatus described in [9]. In

this case the measurement of P-T data was carried

out together with the measurements of the volume

change, the pressure up to 15 kbar being measured

by a free piston gage. In addition there were used

several piston-cylinder systems with different clear- ance developed in VNIIFTRI and Mendeleyev

Metrology Institute (VNIIM). All experiments were

carried out in accordance with the results of the

studies of free piston deformation errors [10].

All results are divided into five sets which characteristics are given in table 1. The analysis of the measurement errors was made

by the method suggested by Dr. Dolinsky (VNIIM)

and the results obtained for 281 points were processed by a computer. As a result the parameters of

the Simon equation were obtained:

39016

234.29

1.1772 1 (2)

Standard deviations of a and c are

o- (a) =20 (kg/cm2), o-(c) = 5 • lO'^.

The dispersion analysis makes it possible to estimate the validity of the relation chosen for its extrapolation to higher pressures. Table 2 summarizes

standard deviations of pressure measurements

cr(p), calculated from possible errors of temperature measurements dT in different temperature

intervals.

297
Table 1

Set Year

Pressure

range

Relative error of pressure measurements

Temperature measurement

error

Number of measure- ments

Pressure-measuring instrument

kbar percent °C

1 0 7—1 ^ W.oO — 0.25 in the range (-35-0°C)

±0.1 in the range (0-35°C)

Manganin gage calibrated against free piston gage.

2 1964 0.7-15 0.1 in the range

u. i oKDar

0.2 in the range

O lO AUai

the same 116 VNIIFTRI free piston gage with re-entrant cylinder average clearance /i — 6.6

QO iVOD zn U.UO ^7 The same, with A = 6.3 and /i = 9.6

4 1966 1.5-15 do ±0.05 34 The same, with h = 6.6 and h = llA

5 1967 7-15 do 0.01 in the range 0-20°C

0.05 in the range 20-30°C

50 VNIIM free piston gage with conventional cylinder.

Table 2

T, °C -13.56 11.29 36.80 59.94 83.76 107.31 130.64 153.70 ST°C ±0.05 ±0.05 ±0.05 ±0.05 ±0.05 ±0.1 ±0.75 ±1.0

P, kbar 5 10 15 20 25 30 35 40

crip) , bar 7 11 16 22 27 34 97 130

crip), percent 0.15 0.11 0.11 0.11 0.11 0.11 0.28 0.33

crip) does not change appreciably up to 40 kbar. At 40 kbar, crip) is equal to 0.33 percent providing that temperature 154 °C is measured with an accuracy ± 1 °C. (In practice this error can be reduced to ± 0.2 °C by allowing for pressure dependence of the thermocouple e.m.f.)

It should be noted that in parallel with the determination of the melting P-Tdata, precision measure- ments of the freezing pressure of mercury at 0 °C

were carried out, temperature measurement errors being about 0.002 °C. The results of the measure- ment sets were found to be as follows:

1965- 7719 kg/cm^

1966- 7721 kg/cm2,

1967- 7725 kg/cm^

(eq (2) gives 7725.2 kg/cm^).

The discrepancy in mercury point values turned

out to be unexpectedly large, and apparently cannot

be explained by the experimental errors. The cause

of this discrepancy is not yet established and the

experiments will be repeated soon with the best

possible accuracy and utmost care.

If similar measurements were also carried out in

other national laboratories, the comparison of data

would provide the most reliable value of this very

useful reference point. On the other hand, the

results of such measurements may be considered

as a mutual comparison of the national pres- sure scales at this point that in itself is of great

importance.

2. Studies of Extrapolation of Eq (2)

to 40 kbar

The estimation of accuracy and reliability of the

extrapolated P-T equation can be made by comparison of the data obtained from measurement of

physical phenomena of different nature. The following phenomena were chosen: the pressure dependence of electric resistance of pure metals and

alloys and pressure dependence of the NaCl lattice

parameter.

The validity of the equation can be also estimated

from theoretical considerations of this equation and

results of measurements of the volume change

along the melting curve.

2.1. The Volume Change Along the MMC

The analysis of P-T melting data for substances

having a maximum (rmax) in the melting curve shows

that parameters of the Simon equation selected at the low-pressure section of the curve fit the experimental data up to p(rn,ax)/2 = A/2. Thus, if it is proved that the maximum in the MMC lies far

beyond 40 kbar this wiU be the confirmation of

correctness of extrapolation, providing no triple

points exist all over pressure range.

298
It follows from the Clapeyron-Clausius equation

Tdp|dT=\|^V (3)

that the maximum in the melting curve T=f{p)

may exist only when the volume change becomes

zero (if latent heat of melting A. 7^ 0).

According to Bridgman [11] the pressure dependence of of mercury is convex to the pressure

axis but all other 29 substances exhibit concavity

to the pressure axis. Since the solution of the problem of extrapolation requires the knowledge of

exact behaviour AF=/(p) and AV=f,{T) , the

measurements of AF were repeated. These measure- ments were carried out on the same apparatus that

was used in studies of P-T melting parameters. For

this purpose the apparatus was equipped with the

special gadget. Its full description and the measure- ment procedure to obtain p-AV-T data are given

in [12].

P-AV-T data were also obtained for benzene,

nitrobenzene, and cesium up to 17 kbar [12-14].

The experiments indicated that melting of benzene,

nitrobenzene, and partly cesium takes place within

small finite pressure range (r= const.). This range

increases with pressure. In case of mercury melting

process takes place at a strictly constant pressure over the entire range studied.

At the same time the experiments indicated that

the volume change AV of mercury may vary considerably (up to 30 percent) for the same p and T

values. The investigations established the direct

relationship between the conditions of freezing and

AF values [13]. Obviously this factor may account

for the anomaly in the mercury AF dependence

which was observed by Bridgman [11].

It was shown that the mercury volume change AV

behaves in the manner similar to all other substances investigated. The pressure and temperature

dependence of AF can be accurately represented

by the empiric equations

AV= 0.01028 - 7.354 X 10-^ In (37.667 + p) (4)

and

Ar= 0.0253- 7.54 X 10-4 In (7/234.3). (5)

From eq (4) it foUows that the volume change becomes zero at about 1,000 kbar. We believe this can

be regarded as a confirmation of the fact that

extrapolation of eq (2) to 40 kbar is justifiable.

Eq (3) may be used for the estimation of accuracy

of extrapolation. From P-AV-T measurements it is possible to assume that providing there are no triple

points on the fusion curve, the function K/AV—fip)

together with its first and second derivatives is a smooth continuous function of pressure, and at the

maximum point (p = A) the following conditions are

realized:

dp

^0,

^A

X(A) ^0.

This means the function klAV=f(p) has a pole of

the first order at the point A. So, keeping the general

nature of eq (3), it can be written as

A"

dT A"-P" [a + (iP + y{P)P'r (7)

where n, a, and ft are integer parameters and y{p)

an unknown function having no peculiarities. To

estimate the accuracy of the extrapolation of the

Simon equation for mercury it is sufficient to approximate yip) to a constant {y{p)—yo)- Eq. (7)

then takes the form

(8)

The approximating capabilities of eq (8) were

studied for cesium in the range from 1 kbar to p — A [14, 15]. Deviations from the experimental data

did not exceed 1 percent. Therefore, we can consider 70 to be a good approximation of the 'y(p)

function for the whole pressure range studied with an accuracy not more than 1 percent. The yo for

mercury is found from experimental P-T data up to 15 kbar. Since A is estimated to be about 1,000 kbar,

it is possible to eliminate from eq (8) the parameters

relating to the maximum pressure without introduction of any essential errors. Applying the obtained equation to experimental

data up to 15 kbar, we find

70 = = -3.6 X 10-7 cmVkg. (9)

Using the well-known theorem of differential estimates [16] we can evaluate the deviation of the

Simon equation from the exact melting curve up to 40 kbar. The calculations give

(Ap/p) = 1 percent. (10)

2.2. Electric Resistance of Metals and Alloys

The simple empirical way to confirm the correct- ness of extrapolation is provided by comparison of

the MMC equation with the readings of some resist- ance gages which characteristics were carefully

studied for many years [20-22], and one gold resist- ance gage.

The equipment, procedure and measurement re- sults at 25 kbar are described in [17]. The comparison between the MMC equation and the readings of

five resistance gages were carried out in 60 points

evenly spaced in the range from 7 to 25 kbar. In

most cases the results agree within ±0.35 percent.

Ar(A)-0, (6) *The solution of eq (7)afy(P) =0 andP < < A is the Simon equation.

299
Similar experiments were performed with a gold

resistance gage up to 40 kbar [19]. The pressure

dependence of gold resistance was expressed by a semi-empiric equation taken from [23]. The parameters of the equation were determined from the

results of calibration against a free piston gage.

The comparison of the readings was performed at 6 points. The divergence of the data did not exceed

0.5 percent.

2.3. Equation of State for NaCl.

The MMC equation (2) was also compared with

experimental data based on the equation of state for NaCl [24]. There were no attempts to intercompare directly these methods. The indirect compari- son of the equation of state for NaCl with the MMC

equation can be carried out by using experimental

data up to 70 kbar obtained by Klement [25].

Klement's data were obtained at the quasihydrostatic apparatus that was apparently calibrated

using the phase transition points in Tl ll-lll ( — 37

kbar) and Ba I-II (~ 60 kbarj that were taken from

[26].

Jeffery and others [24] investigated the same

phase transitions by the X-ray technique on the

basis of the Decker's theoretical equation of state

for NaCl. They reported the values 34.6 kbar and

53.3 kbar. The comparison of extrapolated MMC

equation (2) with the data obtained by Klement [25]

shows the same discrepancies as those between the

pressure values of phase transitions determined by

the two methods mentioned above. This fact may

serve as an argument in favor of the MMC extrapolation equation (2). At the same time such large

discrepancies cannot be ignored, so the careful

analysis of the methods used for pressure determination is needed.

In our opinion the experimental error of the X-ray

diffraction technique cannot be the reason for such

large discrepancies. Since the pressure measure- ment errors in instruments of the piston-cylinder

type are difficult to analyse, we tried to evaluate the

merits of the equation suggested by Decker [27].

The Decker theoretical equation of state is the

only microscopic equation intended for use in highpressure measurements. To evaluate this equation

we suggest another equation of state for NaCl based on the same assumptions as those of Decker but

differing from the latter in the following principal

features:

(1) It does not take into account Van-der-Valls

interactions for they play a secondary role in comparison with both Coulomb interactions of spherically symmetric rigid ions and non-Coulomb

interactions of the nearest neighbours in the lattice

and can be compensated by a variation of the

non-Coulomb interaction parameters [27]. Following

Ghate [28] the potential energy of A' elementary

cells of crystal is presented in the form

o=yv eff \-6b e p+2e'

V2-

P (11)

where A is Madelung constant, Qetf—Q ' e, effective

ion charge, g-electron charge, r-distance between

the nearest neighbours, e, b, p unknown parameters

of the theory.

(2) The parameters e, b, and p and parameter A. which takes into account the pressure dependence

of the Gruneisen constant 7[27] are determined from

the latest experimental data on the effect of hydrostatic pressure on the isothermal modulus of bulk

compressibility Bt. Such a choice of parameters,

that is, the choice in terms of elastic and not thermal

characteristics of the crystal, is more reasonable

than that used in [27], since ion crystal models on which both equations of state are based, have limited possibilities to be used for discription of all thermal and mechanical properties of the crystal.

Here we determined parameters of the theory from

the equation of state (at po=l bar, 7*0 = 298 °K,

To = 2.8205 X 10"** cm [27]) and conditions of coincidence of the calculated values.

with the corresponding experimental values. The

equation so obtained has less unknown theory parameters and secondary interactions not covered by

the present analysis, are effectively accounted for by

a comparatively small number of theory parameters.

For determination of e, 6, p, and A. we selected the

following initial values:

BBs

dp r„ = 5.38 and Bto = 2.523 X 10" dyne/cm 2,

[28]

(here Bs is adiabatic modulus of bulk compressibility), which are most reliable as judged from the

latest literature references.

The values (dBrldp) 5M and 5to = 2.396x

10" dyne/cm^ we used in the calculations were derived from [28] using the data from [29]. The value

(dBTldT)„„ = - 0.159 X 10» dyne/cm-' was also taken

from [29]. The values of the Debye temperature and

Jo are the same as in [27].

By solving the system of four equations with four

unknown quantities (e, b, p, X) the following parameter values were obtained: e= 1.00288, 6=2.53982 X

10-*' ergs, p = 0.301719 A, \ = 6.25.

According to the data obtained ge//=e q is very

close to q, as it is usually assumed for the crystals of

NaCl type.

The value of the parameter A differs greatly from

the Decker value A = 2.55, but is in good agreement

300
with \ = 6 derived from the experimental data given

in [30].

The coefficient of thermal expansion a calculated

from the Gruneisen equation [31] was found to be

11.73 X 10"^ deg~' that is close to the experimental

value 12.09 X 10-^ deg-' [29].

The agreement of the calculated values of elastic

constants of the second and third order with experimental data carried out in [33-35] was also satisfactory. All this testifies to the applicability of our model to the description of the elastic properties

ofNaCl.

The equation of state obtained for NaCl was compared with the Decker equation up to 150 kbar. The

comparison showed that the data practically coincide up to 60 kbar. At 150 kbar the divergence of

data is about 5 percent. The pressures of the phase

transitions in Tl II-III and Ba I-II calculated on the

basis of our equation have coincided with those

determined from the Decker equation. Hence, the

pressure values of the phase transitions Tl II-Ill and Ba l-ll obtained experimentally by Jeffery and

others [24] seem to be more reasonable.

If appropriate corrections are introduced into the

Klement's data on the MMC, the agreement of his

corrected data wiU coincide within experimental

errors with those calculated from eq (2). Such an agreement is an additional confirmation to the cor- rectness of extrapolated eq (2) even over 40 kbar.

As to the theoretical analysis of the equation of

state for NaCl it shows that X-ray diffraction may

find wide usage in high-pressure measurements.

3. Conclusions

1. On the basis of the experimental data up to 15

kbar the MMC eq (2) is obtained. This equation is recommended for pressure measurement in the

range from 7 to 40 kbar.

2. The errors of extrapolated eq (2) are estimated

not to exceed 1 percent up to 40 kbar.

4. References

[1] Zhokhovsky, M. K., Izmeriternaya Tekhnika (Measuring

Technique), No. 2, 1 (1957).

[2] Simon, S. E., Glalzel, G., Z. fiir Anorg. Allgem. Chemie

178,309(1929).

[3] Domb. C. Phil. Mag. 42, 1316 (1951).

[4J deBoer,J.,Proc. Roy. Soc. A2 1 5, 4 (1954).

[5] Gilvarry,J. J.,Phys. Rev. 102,308(1956).

[6] Babb,S. E., J. Chem. Phys. 38,2743 (1963).

[7] Babb, S. E., Rev. Mod. Phys. 35,400 (1963).

[8] Zhokhovsky, M. K., Izmeritel'naya Tekhnika No. 5 (1955).

[9] Bogdanov, V. S., Izmeritel'naya Tekhnika, No. 8 (1967).

[10] Bakhvalova, V. V., Proc. Inst, of Comm. on Stds. 104,

No. 164, 21 (1969).

[11] Bridgeman, P. W., Proc. Am. Acad. 47,347 (1912).

[12] Zhokhovsky, M. K., and Bogdanov, V. S., J. Phys. Chem.

(USSR), 39,2520 (1965).

[13] Bogdanov, V.S., ibid. 40,2779(1966).

[14] Bogdanov, V. S. , JETR Letters 3 , 44 (1966).

[15] Kennedy, G. C, Jayaraman, A., and Newton, R. C, Phys.

Rev. 126,1363(1962).

[16] Kamke, E., Handbook of differential equations (Moscow,

1961).

[17] Zhokhovsky, M. K., and Razumikhin, V. N., Izmeritel'naya

Tekhnika, No, 4,43(1957).

[18] Zhokhovsky, M. K., et al., Izmeritel'naya Tekhnika, No. 11,

26 (1959).

[19] Bogdanov, V. S., and Miroshnikov. A. P., Proc. of the Insti- tutes of the Committee on Standards, 104 (164) (1968).

[20] Burova, L. L., and Zolotykh, E. V., ibid. 46 (106), 62-67

(1960).

[21] Alexeyev, K. A., and Burova, L. L., ibid. 75 (135), 1 (1969).

[22] Burova, L. L., ibid. 104 (164) (1968).

[23] Decker, D. L.. and Vanfleet, H. B., Phys. Rev. 138, 129

(1965).

[24] Jeffery, R. N., BarneU, J. D., Vanfleet, H. B., and Hall, H. T.,

J. Appl. Phys. 37, 3172 (1965).

[25] Klement, W., Jayaraman, A., and Kennedy, G. C, Phys.

Rev., 131, 1 (1963).

[26] Kennedy, G. C, and LaMori, P. N., in Physics at High Pressure, Solid State Physics, Vol. 1 1 (New York and London,

1960).

[27] Decker. D. L., J. Appl. Phys. 36, 157 (1965).

[28] Ghate, P. B., Phys. Rev. 139, No. 5A, A1666 (1965).

[29] Powell. B. E., and Skovo. M. J., J. Appl. Phys. 38, No. 1, 404(1967).

[30] Bartels, R. A., and Shule, D. E., J. Phys. Chem. SoHds 26,

537 (1965).

[31] Born, M., and Kun, H., DynamicalTheory of Crystal Lattices

(Clarendon Press, Oxford, 1954).

[32] Yates, B., and Panter, C. H., Proc. Phys. Soc. 80, 373 (1962).

[33] Chang, Z.-P., Phys. Rev. 140, No. 5A, A1788 (1965).

[34] Swartz, K. D.. J. Acoust. Soc. Am. 41, No. 4, Part 2, 1083

(1967).

[35] Gluyas, M., Brit. J. Appl. Phys. 18, 913 (1967).

DISCUSSION

J. L. Cross {National Bureau of Standards, Washington, D.C.): It is necessary, of course, to wait a

sufficient time to dissipate the latent heat. In the

actual initiating of the freezing you usually have to take the pressure above the freezing point to get a seed crystal started. But once the seed crystal is started and the system allowed to come to equilibrium, the reproducibility is excellent. In the free,

long duration runs that I made using two different

methods for observing the existence of two phases,

there was a spread of two-thirds of a bar. And for a period of 21 hours on one run where the two phases

were maintained in equilibrium, the total variation

in observed pressure at zero degrees, was about

three-tenths of a bar.

P. N. LaMori {Battelle Memorial Institute, Columbus, Ohio): I would like to comment on the value

of 60 kbars for the barium l-II transition, that you

quote from the paper by Kennedy and myself (ref.

26). The barium point data was included in the paper

as a value that we reached on one experiment just on entering the high pressure region. There was no

301
real attempt to determine that 60 kbars was the

maximum value. All the other values in the experiment were determined both on the compression and

the release strokes. So I think that you should not

make a comparison between this value and results

of experiments by Klement (ref. 25).

AUTHORS' CLOSURE

In agreement with comments by Mr. Cross we

want to say that in our experiments the equilibrium

pressure was reproducible within 0.5 bar. Obviously this variation is mainly due to the variations

of temperature and manganin gage readings. In our

opinion this high degree of reproducibility may be

effectively utilized in the establishment of the high

pressure scale. Such scale based on mercury melting curve would make all the measurements throughout

the high pressure laboratories comparable, consistent and uniform.

In reply to Dr. LaMori, in the paper by Klement

and co-workers [25] there is no direct indication on the actual values of fixed points used in calibrations of their apparatus. From the references

given in their paper however one can suggest that

Ba I-II point was taken equal to ~ 60 kbars. That is why we referred to the paper [26].

302
High-Pressure Scale as Determined by X-ray Diffraction Techniques up to

Approximately 100 kbar

H. Tracy Hall

Brigham Young University

Prom, Utah 84601

1. Post-Bridgman Pressure-Scale

Muddle

I suppose as much blame can be placed on me as on anyone for the pressure- scale muddle of the

emerging post-Bridgman era. No doubt the first Ba

transition observed outside of Bridgman's laboratory

was observed in February of 1954 in my Belt apparatus [1].' Due to secrecy, these results could not

be published until six years later. Meanwhile, I had

managed to circumvent the Belt secrecy by the

invention of the Tetrahedral Press [2]. The published

calibration for this apparatus (anvil load versus

pressure) used fixed points of 25, 44, 54, and 79

kbars (Bridgman's resistance points) for Bi, Tl, Cs,

and Ba transitions. (NOTE: Previous work expressed

in kg/cm- and atmospheres has been converted to kilobars in this paper). This precedent was followed

by others until Prof. George C. Kennedy, at the

June 1960 International High Pressure Conference

held at Bolton Landing, Lake George, New York,

conveyed the unsettling news that Bridgman's

"resistance fixed points" were too high and should

be lowered to Bridgman's "corresponding volume

fixed points."

Kennedy's insight ran ahead of the experimental

evidence. In hindsight one might possibly say

Bridgman recognized that certain of his resistance

points corresponded to lower volume points. However, a study of the crucial paper "The Resistance

of 72 Elements, AUoys and Compounds to 100,000

kg/cm-" [3] wiU impress the reader that Bridgman

was certain that the pressures given for his electrical

resistance transitions were correct.

2. Identity of Resistance and Volume

Transitions Established

In view of the Nobel Laureate's 50 years of

accomplishment in the field, it would have been

imprudent to challenge Bridgman's results without

some definitive experiments. A definitive experiment would be one in which the volume transition

and the electrical resistance transition were

simultaneously observed in the same sample specimen. The shapes of the resistance curves in the

' Figures in brackets indicate the literature references at the end of this paper.

Paper presented at the Symposium on Accurate Characterization of the High-Pressure Environment, held at the National

Bureau of Standards, Gaithersburg, Md., October 14-18, 1968.

regions of the fixed points (for example in Tl, Cs,

Ba, etc.) are very characteristic and if these were

found to occur simultaneously with the volume

transition, a one-to-one correspondence between

the transitions, i.e., an identity, would be established.

The first one-to-one correspondence to be established at pressures beyond 30 kbar was in the

instance of Ba. This work took place in the Brigham

Young University high-pressure laboratory at Provo.

A tetrahedral x-ray diffraction press was employed

for this purpose [4]. With this equipment, the

electrical resistance of Ba foil was monitored (at

room temperature) simultaneously with the monitoring of its Debye-Scherrer diffraction pattern as pressure was slowly increased [4, 5]. As this was

done, the characteristic electrical resistance pattern

described by Bridgman [6] with an abrupt increase

in electrical resistance at 70 kbar (Bridgman's

resistance value) was found.

A new x-ray diffraction pattern appeared simultaneously with the crossing of this resistance discontinuity. Compressions determined 3 kbar above

and 10 kbar below the discontinuity by x-ray

diffraction were each only 4 percent higher than

those determined from Bridgman's data [7]. (Bridgman determined his volume transition to occur

at 59 kbar). The x-ray determined change in compression at the transition was the same as found

by Bridgman. In addition, the overall Ba compressibility curve determined by x-ray matched a corresponding curve plotted from Bridgman's

compression data quite well if a value of 59 kbar

were selected for the Ba transition pressure. This

furnished compelling evidence that the 79 kbar

resistance and 59 kbar volume transitions were

one and the same and that the correct transition

pressure was about 59 kbar.

In addition to establishing the identity of the

resistance and volume transitions, the x-ray work

indicated the 59 kbar transition to be from the

normal body-centered cubic symmetry to a hexagonal close packed symmetry with cell parameters

of a = 3.90 A, c=6.15 A, and c/a= 1.58 at about 3

kbar above the transition pressure.

Even though the above research constituted the

first experimental evidence that the electrical

resistance transition pressures should be lowered,

most high-pressure researchers had already as- sumed the identity of Bridgman's 44. 54, and 79

kbar resistance transitions with his 40, 44, and 59

kbar volume transitions in Tl, Cs, and Ba respec303
lively. Also, they assumed the volume transitions

to give the approximately correct pressure values.

This decision may have been influenced by P. W.

Bridgman himself. It has been reported that at a Gordon Conference on High Pressure held shortly

after the Bolton Landing Conference, Bridgman

indicated that the pressures of discontinuities in

the resistance as given in his 1952 paper were not

suitable to use as fixed reference points for pressure

calibration. There are no written reports of Gordon

Conference proceedings and I did not attend that

particular conference; consequently my information on this point is only hearsay.

Irregardless, subsequent high-pressure x-ray

diffraction work has established resistance-volume

change correspondences for transitions in Tl,

Cs, Yb, Bi, and other metals that have been used as pressure fixed points. In all instances it has been

necessary to decrease the transition pressure of

Bridgman's electrical resistance transitions.

3 . Establishment of Continuous

Pressure Scale by X-ray

After the one-to-one correspondence between

the resistance and volume transitions had been

established, one could focus attention on the

problem of obtaining better values for the fixed

points. Of particular interest, in this connection,

are the Bi III-V and Ba I II transitions which Bridgman had established (by volume change) to occur

at 88 and 59 kbar respectively.

It would be very useful to have some type of pres- sure calibrant that would give continuous readings.

Fixed points might then be compared to this. Such

a calibrant might be based on the atomic separation

of the atoms of a simple substance, said separation

being determined by x-ray diffraction techniques.

A simple substance for which the compression can

be predicted with some degree of certainty is NaCl. Jefifery, Barnett, Vanfleet, and Hall— using

the previously mentioned tetrahedral x-ray dif- fraction press and an equation of state developed

by D. L. Decker— used this approach [8]. In this

work, Bi or Ba metal was imbedded in polycrystalline NaCl and the transitions were detected by the

electrical resistance change of the metal while

the lattice parameter of the adjacent NaCl was

simultaneously measured by x-ray diffraction.

The Ba I-II and Bi III-V transformations at room

temperature were assigned values of 53.3 ± 1.2

kbar and 73.8 ±1.3 kbar, respectively, corresponding to NaCl linear compression values of

Aa/a = 0.0510 and 0.0637.

These values were 9.4 and 16.2 percent smaller

than Bridgman's volume values and were, at first,

treated with reserve and skepticism. However,

recent personal communications from other

laboratories give similar values. They are listed

below together with the B.Y.U. values. AU x-ray

values have been corrected to Decker's improved

equation of state of NaCl [9].

Laboratory*

B. Y. U. Naval Res. Lab.^

Westinghouse

Nagoya U.'' U.C.L.A."

Signal Corps,

Ft. Monmouth ^

Method Used

X-ray tet. press X-ray tet. press X-ray tet. press "Bridgman Anvils"

Piston-cylinder, vol. Manganin gage

Ba I II, kbar

Bi III-V, kbar

54.4: 1.2 75.6:

75.0

1.3

76 ±4

55.0±0.5 77.5±1.0

56.3

The B.Y.U. , Naval Res. Lab., Westinghouse,

and Nagoya U. values all depend on Decker's

equation of state for NaCl. At this juncture, the

agreement among the above parties has been nar- rowed to about ±3 percent. There is a Russian

determination of the Ba and Bi transition in a free

piston gage, for which no experimental details are supplied, of 58.5 and 89.3 kbar ± 1 percent [10].

This appears to be the only holdout for Bridgman's

resistance values at the present time.

The work of Kennedy and his colleagues at U.C.L.A. and of Zeto and colleagues at Fort Monmouth is commendable in that they are using nonx-ray approaches to pressure-scale problems. Full

confidence in a pressure scale wiU only be achieved

when data from several different approaches are

in agreement. Kennedy is attempting to push the

pressure limit of piston-cylinder apparatus toward

100 kbar and measure volume changes, by piston

displacement. Vanfleet and Zeto are utilizing a manganin wire gage immersed in a true liquid

within a large pyrophyllite cubic pressure ceU.

The manganin gage is accurately calibrated to the

Hg and Bi I-ll transitions, fit by a quadratic equation,

and extrapolated to higher pressures.

4. Improved X-ray Diffraction

Presses

The tetrahedral x-ray diffraction press in use at B.Y.U. was designed years ago. Several improvements that could be made, particularly if the objective is to establish a pressure scale. Incidentally,

duplicates of the B.Y.U. press are installed at Commissariat a I'Energie Atomique, Paris, and also at the Iron and Steel Institute of Tohoku University,

Sendai, Japan. The design of these presses was completed before the invention of "anvil guide" [11].

Correct anvil position and sample location with

respect to the x-ray beam are, therefore, more

difficult to achieve than in presses equipped with

anvil guide.

I have designed a tetrahedral press with anvil

guide for use with x-ray diffraction. Two devices

*Courtesy of (a) Perry Ahlers, (b) A. Taylor, (c) H. Mii, (d) G. C. Kennedy, (e) H.

Vanfleet.

304
of this design have been buiU by McCartney Manufacturing. One is located at the Westinghouse

Monroeville, Pennsylvania, Laboratory; the other

at the U.S. Bureau of Mines Laboratory, Albany,

Oregon.

Years of experience at B.Y.U. with tetrahedral

and cubic presses show that tetrahedral presses can achieve the highest pressures with the least

anvil breakage. Pressures of 120 kbar on Decker's

volume scale have been obtained with V4 in (6.3 mm)

on edge triangular-faced tetrahedral anvils.

Because of its higher pressure capabilities, I would choose the tetrahedral press over the cubic

press in any new design for x-ray diffraction work.

A new design for pressure-scale work should be

built with increased ruggedness and with the highest

possible precision. The stretching of the tie-bar

frame under load introduces correction factors in

former tetrahedral x-ray presses because the

x-ray tube, goniometer track, etc., are attached to different parts of the frame. This could be eliminated

in new presses based on the "ram-in-tie-bar" type

equipped with the "sure-guide" mechanism [12].

In such a press, all the x-ray equipment could

be mechanically connected to the "sure-guide"

frame, which remains in fixed position with respect

to the exact center of the press under any load.

Such a press, specifically designed with pressure

scale research in mind, could be built for about

$50,000 (exclusive of x-ray diffraction equipment).

5. Improvement of Solid Pressure

Media

In addition to excellent prospects for the improvement of high-pressure diffraction apparatus to provide more precise and accurate measurements

relating to the pressure scale, there are good prospects for improving solid pressure-transmitting

media. It is known that the media surrounding a wire of a fixed-point caUbrant (such as Bi) influences

the pressure required for onset and completion of a transition and also influences the hysteresis

connected with the reverse transition. For example,

Davidson and Lee [13], using a liquid media, have

measured a hysteresis of from 0.55 to 0.90 kbar for

the Bi I-ll transition. On the other hand, Jeffery,

Barnett, Hall, and Vanfleet [8] measure an average

hysteresis of 5.3 kbar for a cylindrical 0.010-in-diam

wire and of 2.8 kbar for a 0.002-in-thick flat strip

surrounded by NaCl. Keep in mind that the pres- sures in the latter work were determined from

atomic spacings in the NaCl immediately adjacent

to the Bi wire or strip. Various explanations have

been offered as to why transition sluggishness and

hysteresis are always greater in solid media. Regardless of the explanations, the large hysteresis is undersirable. Ideally, both the NaCl and the Bi

wire should be immersed in a true hydrostatic

fluid in the x-ray work. For the most part, this is impractical. However, it should be possible to obtain

a closer experimental approximation to the ideal

situation than that of a Bi wire embedded in polycrystalline NaCl. For example, NaCl powder could

be mixed with an excess of powdered material

which possessed a lower coefficient of shear friction

than NaCl. This mixture could then serve as the

matrix for embedding the Bi or other wire. The

material admixed with the NaCl must, of course,

be sufficiently transparent to x-rays and meet other

requirements pertinent to the experiment at hand.

At 25 kbar, our shear friction measurements with

rotating Bridgman anvils show NaCl powder to have

a friction coefficient of 0.12. Boron nitride powder

and graphite powder have coefficients of 0.07

and 0.04 respectively. Mica sheet has a coefficient

of 0.07. In powdered form, mica would have an

even smaller coefficient. The coefficient for

powdered polyethylene is not known but should be

lower than that for NaCl. The friction coefficient

for NH4F has not been measured but extrusion

experiments at high pressure indicate it to be very

low. If a search were conducted, no doubt a great

many materials would be found with shear friction

coefficients weU below that of NaCl.

While solid pressure-transmitting substances

offer many practical advantages in all types of

high-pressure experiments, there still seems to be some unknown if not mysterious factors con- nected with their use. Additional research in this area is certainly needed.

6. References

[1| Hall. H. T., Rev. Sci. Instr. 31, 125-131 (1960).

[2| Hall. H. T., Ibid. 29, 267-275 (1958).

[3] Bridgman, P. W., Proc. Am. Acad. Arts Sci. 81, 167-251

(1952).

[4] ,BarneU. J. D., and Hall, H. T., Rev. Sci. Instr. 35, 175-182

(1964).

[5] Barnett, J. D., Bennion, Roy B., and Hall, H. T., Science

141,534-535(1963).

[6] Ref. [3], pp. 198 and 199.

[7] Bridgman, P. W., Phys. Rev. 60, 351-354 (1941).

[8] Jeffery, R. N., Barnett, J. D.. Vanfleet, H. B., and Hall, H. T.,

J. Appl. Phys. 37 , 3172-3180 (1966).

[9] Decker, D. L., Brigham Young University, Unpublished

(1968).

[10] Vereshchagin, L. F., Zuhova, E. V., Buimova, I. P.. and

Burdena, K. P., Dokl. Akad. Nauk SSSR 169, 74 (1966).

[11] Hall, H. T., Rev. Sci. Instr. 33, 1278-1280 (1962).

[12] Hall, H. T., Rev. Phys. Chem. Japan 37, 63-71 (1967).

[13] Davidson, T. E.. and Lee, A. P., Trans. Met. Soc. AIME

2 3 0, 1035 (1964).

305
DISCUSSION

A. Taylor {Westinghouse Research Laboratories.,

Pittsburgh, Pennsylvania): I would like to describe

some of our experience with the tetrahedral press

referred to by Dr. Hall.

The stretch in the press frame under load does not concern us, nor does the fact that the assembly

is "sloppy" at the joints, because a little initial

pressure will take the "slop" out of the system, and

the stretch can be accommodated by moving the

x-ray tube up and getting a stronger beam.

The most serious difficulties with the apparatus

are these: First, the sample moves off-center

under load because of the extrusion of the tetra- hedron, and that introduces errors in spacing which

are far more serious than those produced by the

stretching of the press. In measurements of pres- sure coefficients where one takes differences in

lattice parameters, the error that is introduced can

be very serious. Our procedure is to use a calibration material which is mixed with the sample. We

take the press up to full pressure, take the x-ray

diffraction pattern, and then bring the load down to a very small value — a couple of kilobars — and take

another diffraction pattern. From the lattice

parameter of the calibrating material — which is sodium chloride in this case — we are able to determine how much the sample has moved off center.

A second problem connected with x-ray dif- fraction measurements is associated with the large

amount of incoherent scatter coming from the tetra- hedron, apart from the sample. We use a mixture of

80 percent boron and 20 percent polystyrene.

If we attempt to get a stronger diffraction pattern

from the sample by making the diameter of the

collimator a little larger, it is found that an increase

in the background intensity occurs in relation to the line intensity, and a poorer line-to-background

ratio is obtained. It is thus necessary to make the

diameter of the collimator system quite small.

We find that a diameter of about 8 or 9 thousandths

of an inch is about right. But, unfortunately, when

the collimator is small and the sample is very thin,

there are very few crystals in the beam. Then the

line profile is spotty, and it's very difficult to find

the true maximum.

Until we can overcome these difficulties with the

tetrahedral press, our errors in pressure in, say,

the 70 to 80 kbar range might easily be plus or minus

two or more kilobars.

P. N. LaMori (Battelle Memorial Institute, Columbus, Ohio): In your paper you have attempted to give a historical account of the "pressure scale

muddle." As a participant I would like to offer a few additional comments.

You have suggested that Bridgman's results

should not have been challenged without definitive

experiments such as simultaneous measurement of

the volume and resistance transitions. This was

done as early as 1962 when I demonstrated by

simultaneous volume and resistance measurements

the bismuth 1^2, bismuth 2—*^ 3 and thallium

2—* 3 transitions. ["Calibration Experiments in a Piston-Cylinder Apparatus," High-Pressure

Measurement, edited by A. A. Giardini and E. C.

Lloyd, Butterworth's, 1963, p. 321.] In that paper

the barium transition was also clearly demonstrated

to be 58 ±2 kbar by a resistance measurement.

The additional proof that you say was necessary

of Kennedy's and my earlier conclusions was thus

available shortly after our original papers.

You have suggested that Kennedy's insight ran ahead of the experimental evidence. As a participant in the events leading to the discovery of the

error in Bridgman resistance pressure scale, I can only say that the insight was based on solid

experimental evidence. The experimental evidence

consisted of accurate measurements of the bismuth,

thallium and cesium transitions. It was only after

aU the experimental evidence was available that

Professor Kennedy, myself, and Professor D. T.

Griggs simultaneously and independently reached

the obvious conclusion about the pressure scale.

This work was presented at the Bolton Landing

Conference — G. C. Kennedy and P. N. LaMori,

"Some Fixed Points on the High Pressure Scale",

Progress in Very High Pressure Research, edited

by Bundy, Hibbard, and Strong, John Wiley, New

York, 1960 (314 p.).

An unpublished outcome of this work shows

clearly that there is a linear relationship between the

differences in volume and resistance measurements

above 25 kbar. This strongly suggested that the

resistance measurements were incorrect. See Figure

7, P. LaMori, "The Determination of Some Solid

Transition Pressures to 50,000 Bars", M. S. Thesis,

U.C.L.A., 1963. However, this was known at the

time of and presented at the Bolton Landing Conference (see discussion section of our paper).

Thus, the information Kennedy presented was

based on (1) solid experimental evidence, and (2) a demonstrable relationship between the volume

and resistance measurement.

Kennedy's insight came at a step earlier than this.

This was in the conception of the piston-rotation

experiment. Also important was his insistance in

doing the experiment in spite of much objection

from myself as well as others. The importance of

this has been clearly demonstrated.

306
Appendix

307

Panel discussion

Temperature Measurement at High Pressures

Chairman: F. P. Bundy

General Electric Research and Development Center

Schenectady, New York

P. Bell

Carnegie Institution of Washington

Washington, D.C.

P. J. Freud

Battelle Memorial Institute

Columbus, Ohio

I. Getting

University of California

Los Angeles, California

R. E. Hanneman

General Electric Research and Development Center

Schenectady, New York

G. C. Kennedy

University of California

Los Angeles, California

H. M. Strong

General Electric Research and Development Center

Schenectady, New York

R. H. Wentorf

General Electric Research and Development Center

Schenectady, New York

G. C. Kennedy: I think that there is a good deal

of confusion among the audience that doesn't

really exist for most experimenters in this field, in

view of the apparent disagreement between the GE

measurements and Getting's measurements.

Now most of the measurements are relative

measurements. Relative measurements in question

include measurements of the difference between

an alumel-chromel couple and a platinum-platinum/

rhodium couple. That experiment has been done

three times, by LaMori's group, by us, and by the

General Electric Research Laboratory, and all three results agree. Now, GE also has estimated

the absolute effect, but because our absolute effect

for platinum is positive and for alumel-chromel is negative the net absolute effect is the difference.

and our absolute effect is thus half the size estimated

by GE. But the three sets of measurements are

all in agreement involving both the difference between two thermocouples and measurements on

a single wire. So actually I think the situation is really very much under control. The only measure- ments which disagree are the noise experiments.

So I think that, compared to other problems in

high pressure, this effect of pressure on temperature

measurement is in very good shape.

R. E. Hanneman: The disagi-eement is not quite as bad as a factor of two, up to 30 kbar and 1,000 °C.

George Kennedy indicated earlier that he had little faith in use of phase transformation boundaries as

309
a basis for indicating temperature corrections. I think part of the problem is this question of what

the pressure really is at high temperature. Another

difficulty may lie in the point that we have not used

simply the Clausius Clapeyron equation, but have

integrated the heat capacity as a function of temperature. This does not give, of course, a linear curve,

but we have tried to bring this into the calibration.

Although at GE we may have been lucky and

chosen the "right" six or eight systems, we have

never had the system give results in the wrong

direction. There's been scatter of the order of 30

percent in experiments by using different pairs of

the different kinds of materials.

I. Getting: I think that even if we can come to some

agreement as to what the hydrostatic effect on calibration of thermocouples may be, this may not be

the major problem facing temperature measure- ment in high-pressure apparatus in general. For

roughly the first year that I worked on this problem, I was not able even to see the pressure effect. Chemical effects, particularly in chromel-alumel above 600

or 700 °C, were much larger. When I did the Bridgman experiment with a silver chloride pressure

medium, and heated to only 400 °C at 1 kbar nominal pressure, the voltages I observed for the chromel

lead itself were greater than any I have ever seen from the pressure effect. So chemical effects can be

very severe. As an example, take two identical thermocouples

and don't weld the junctions together, but just put

them side by side at the end of a Vs-in- or Ve-in-diam

ceramic tubing, and put them in the location where

you would normally put a thermocouple in any solid

pressure medium, high-temperature, high-pressure

apparatus.

Apply heat and look at the difference in the indicated temperatures. Sometimes, due to effects I don't know how to explain, we have gotten an honest

1 percent difference. That's 10 degrees at a thousand. After one has solved these problems and can

get two identical thermocouples to read together,

try two different composition ones, and you wiU find

another order of magnitude difference. When these

effects are all resolved, you are ready to talk about

whether you are experiencing a pressure effect or

not.

I think that if we are realistic about temperature

corrections in the region of a thousand degrees and

tens of kilobars, we will not make statements like

"temperatures were measured to plus or minus five

degrees." And at 1,500 degrees the situation is much worse.

P. Bell: The problem that I would like to draw at- tention to is the matter of contamination. While

comparisons against chromel-alumel are worthwhile, I think certainly in our apparatus and in

Professor Kennedy's there is boron nitride, and

there is carbon. Chromel-alumel behaves very badly

in a reducing atmosphere. Professor Birch mentioned this in his papers — and this is the reason why

he never used it again.

R. E. Hanneman: I think that this is an appropriate place to bring up the fact that we need to search

for thermocouple materials which are indeed more

stable, and we might start off with this new material,

platinel. We have done some work with some goldlike thermocouples, and these are not bad under

some conditions. So I think that instead of driving

so hard for the absolute corrections, perhaps we

should spend some of our time and effort making

relative measurements and looking for materials

that are good and stable.

Now, one other question that was inferred today is the matter of pressure variations that occur as the

temperature is changed. If you go up and down in

temperature you get these hysteresis effects which

Bob Wentorf very clearly showed. And these things

are all so large that they are a serious pitfall for a newcomer in the field.

I. Getting: I think the point about stress distributions on the wire being unknown is important. After

three years of experimentation, I claim only to have

some idea of the effective pressure on the wires to

within three kbar— that's 30 percent at 10 kbar.

As a further problem, in alumina ceramic tubing

in a standard piston-cylinder apparatus the effective stress on the wires is so uncertain that I will no longer say that the results give the corrections for

thermocouples.

I think we really would like a thermocouple, of

course, which is chemically stable and which has a small pressure coefficient. But if you look at the data

from Bridgman single-wire experiments to try to find a pair of materials which have the same emfs, the

corrections would be zero. As it turns out, if you

made a thermocouple out of platinum and alumel

you would have virtually no pressure dependence;

but this would be to little avail since the perform- ance would then be dominated by the chemical susceptibility of alumel and the mechanical weakness of

platinum.

R. H. Wentorf: We have been talking a lot about

thermocouples, but I'd like to point out that there is another good way to measure temperature, and that's

optically. If a person could get some well-defined

phase transformations which aren't too sensitive

to pressure, but run at a pretty high temperature

where they do emit, then one could do two-color

pyrometry on these samples and get, pretty well,

the absolute temperature. These materials could

then be run in a different configuration with thermocouples next to them without windows, and in

that way one could get another grip on the problem of thermocouple calibration and the general

temperature problem.

G. C. Kennedy: On the problem of chemical stability, there is a way of stabilizing the thermocouple

310
that works remarkably well. Platinum-rhodium at 1,500 °C in a proper environment doesn't drift. This

proper environment, we have found, is in molten

glass.

As an example, if an iron wire and a thermocouple

beside it are placed in an ordinary environment, in a matter of 3 or 4 minutes the thermocouple will

drift so fast that it may be useless. In the presence of

iron or graphite at 1,500 °C it drifts so fast that it really becomes useless.

If you surround the couple by molten Pyrex or any

molten high-silica glass, then it can sit for hours at 1,500 °C and it will not drift. The evidence that it does not drift is that one can repeatedly measure

the same value for the melting point of iron. The

pressure of molten glass seems to inhibit any migration of metallic substances to the platinum-rhodium

and also inhibits any migration of carbon to platinumrhodium.

H. M. Strong: I'll make just two short comments.

Following what George Kennedy has just said, we

have also found that platinum and platinum-10

percent rhodium in an environment of pure boron

nitride, or in clean dry salt, is also quite stable for

several hours at around 1,400 °C as indicated by the

fact that for the same power input one continues to get the same reading. Also the thermocouple output

at melting points of those materials remains the same.

I might amplify what Rod Hanneman said on pres- sure variation. We have tried to measure the pres- sure variation in cells as you change the temperature, using a tiny germanium element off to one side,

heated independently— and with a carbon tube in

the cell center to heat the cell proper. With this ar- rangement you can observe changes in pressure by

observing changes in melting point of the germanium. The pressure may not be known absolutely,

but the pressure change can be seen. By this method

we have observed in pyrophyllite quite distinct

changes of pressure upwards with heating amounting, at a 50-kbar level, to 10 to 20 kbars. But after a while the pyrophyllite tends to transform to a denser

phase, and then the pressure will drop, sometimes

quite far below the original pressure of 50 kbar.

Sodium chloride environments are quite a bit more

stable, but there is a distinct increase in pressure

with temperature.

F. P. Bundy: A comment along this line that I would like to make on Freud's work, and also

Getting's, is that in their exploration of the pres- sure distribution in their cell, along the temperature

gradient, measurements were made at room temperature using bismuth in most cases. I would suspect

from what I know of Strong's measurements under

these same conditions that during the heat-up of

the cell one would find that at the high temperatures

the pressure goes higher than was indicated on

the graphs.

P. J. Freud: The way we did the experiment was

to heat up at a low loading and then increase the

load.

I. Getting: One can chase this effective pressure

problem, without results. I tried to do it by following

piston displacement during cycling of piston load

at various pressures, and got a hysteresis loop for

something possibly corresponding to the average

pressure in the cell as determined by the piston

displacement. One can then, in temperature cycles,

relieve the load on the piston as the temperature

increases to try to correspond to that narrowing

frictional hysteresis loop, and so on and on. I think

the saving grace here is that even if you don't know

the stresses in the wires to better than three kilobars

it doesn't matter to more than a degree or so in the

correction.

D. Decker (Brigham Young University, Provo,

Utah): There is also another possible way of looking

at temperature effects— by doing a Mossbauer

experiment and looking at the Doppler shift. This

effect should be independent of pressure. And since

the isomer shift itself is a very slow- varying function

of pressure, it gives you an estimate of the temperature. We have tried this at 80 kbar and up to 400 °C.

However, we only used chromel-alumel thermocouples and found no effect. That is, measurements

on thermocouples directly and by the isomer shift

essentially agree.

385-762 O - 71 - 21 311

committee report

Fixed Points Near Room Temperature

Participants

L. Abbot

P. B. Alers

Y. A. Atanov

J. D. Barnett

F. BUNDY

R. Cavailhe

T. Clark

H. Tracy Hall, Chairman

M. CONTRE

J. Cross

J. Haygarth

J. HOUCK

A. Jayaraman

D. P. Johnson

R. Kaneda

N. Kawai

G. S. Kell

A. Lacam

G. J. Scott

C. E. Weir

M. S. Werkema

R. J. Zeto

In our committee discussion of fixed points, we

realized that several factors were of concern: Are

these points equilibrium points thermodynamically?

We realized that kinetics often enters into the establishment of these points. We also recognized that

the apparatus plays a part in deciding where an

individual person may find a fixed point in relation to his press loading or other factors.

We decided that the equilibrium value was the

important value, and that we should select the

fixed point data that were closest to equilibrium.

Then it would be up to the individual user to worry

about kinetics and apparatus factors in his own

situation.

In this connection, we advise that all highpressure workers describe their apparatus adequately. It is necessary to know what kind of device

is used, and its important dimensions. If it's a pistoncylinder apparatus, the diameter of the piston and

the length of the sample should be given. Materials

in the cell, such as pyrophyUite, boron nitride, or

silver chloride should be specified and their dimensions given. Give sufficiently detailed information

so that an estimate of the pressures can be made at a later date when fixed point values become more

accurate.

We decided that there were several ranges into

which the pressure scale could be advantageously

divided. The first range is called the Free Piston

Gage Range. It was the consensus of the group that

the free piston gage could be used up to pressures of

about 30 kbar. The first two fixed points that we

discussed — the mercury freezing pressure at 0 °C

and the bismuth I-ll transition — lie in the Free

Piston Gage Range. There may be other points that

should be included.

Our assignment was concerned with pressures

at room temperature. We decided that this meant

25 degrees Centigrade, and we have shown in the

accompanying list of "Pressure Fixed Points" our judgment of the best values of the pressure, in kilobars, of the fixed points at 25 degrees Centigrade.

Some measurements have been made above

room temperature, and we wish to include the

mercury point over a range of temperatures as a

fixed point, rather than just giving it as one point.

It recommended that such reference pressures be

based on the Simon equation, adjusted to agree

with the value 7.569 kbar at 0 °C as follows:

P = 38227

_V234.29/ J

where T is the temperature in K on the Inter- national Practical Temperature Scale (1948), and

P is the pressure in bars.

In the case of the bismuth I-II transition, it was

the consensus of the group that Heydemann's

value determined at the Bureau of Standards is the best value at this time and should be 25.50 ±0.06

kbar at 25 degrees Centigrade.

We called the next range, beginning at 30 kbar,

the Upper Force-Over-Area Range, with a present

upward limit of about 80 kbar. In the present

state-of-the-art this limit probably cannot be

exceeded in apparatus such as piston-cyhnder

equipment. In this range, we have selected the

thallium ll-lli transition to have a value of 36.7 ±0.3

kbar.

The value chosen for caesium (Cs Il-III = 42.5 ± 1 kbar) was the result of careful consideration. We

determined who had performed experiments where

the system was in a truly liquid environment, and

gave considerable weight to that. We considered

experiments that did not have a liquid environment

but in which an attempt was made to approach a

free piston condition. Here, one may have had

silver chloride for the environment. We included

x-ray diffraction data. We weighed these, one

against the other, and came up with a good con- sensus as to the best value. We then tried to put

some kind of an estimate on the plus or minus

value. The caesium ll-lll value appears to be

known only on an upstroke, so there could be a

313
question as to whether or not the caesiuin values

are equihbrium values.

The barium l-ll transition was placed at 55 ±2

kbar, and bismuth IIl-v at 77 ±3 kbar, at which

point we reached the limit of our upper force-over- area range.

Then we moved into a more difficult range — the

100 Kiiobar Range — in which tin, iron, barium, and

lead were considered. We understand that there

are no shock data on the transitions in tin, barium,

and lead.

There are problems, apparently, in trying to run

these transitions by shock techniques. The shock

data that have been obtained pertain only to iron,

and according to at least one member of our panel,

the iron transition is very poorly behaved. It was

the consensus of the group that iron probably ought

to be left out of the system, although there were a few who objected. Iron seems to be somewhat

irregular in behavior, and there is an uncertainty

as to the correct transition value.

It is the consensus of this group that the tin,

barium, and lead points are on much less firm

ground than the points in the Force-Over-Area

Range, which in turn are on less firm ground than

the points of the Free Piston Gage Range. We

feel that pressure is probably established up to the neighborhood of 80 kbar within plus or minus

3 kbar. In the 100 Kbar Range and above there is a great deal of uncertainty. How much is difficult to estimate. Points in this range have been chosen

from older work of Drickamer's and probably

should be revised downward on account of the

downward revision in the barium point, and the

iron point, t

PRESSURE FIXED POINTS

Free piston gage range <

Upper F/A range

Hg 7569 ± 2 bars at 0 °C

Bi III 25.50 ±0.06 kbar at 25 °C

Tl IMII 36.7 ±0.3 kbar

Cs n-ni 42.5 ± 1 (up only)

Cs III IV 43.0 ± 1 (up only)

Ba III 55 ±2

Bi III-V 77 ±3

Sn J.J J 100 ±6

Ba 140 (up only)

Pb 110 to 160 (up only)

100-kbar range

H. T. Hall

t Added in proof by the Editor: In the following table are Drickamer's revised values based upon a later publication (Rev.

Sci. Instr., Notes Section, Vol. 41, No. 11, 1970):

Approximate Location of Transitions, kilobars Old New

Bi 88 73-75

Sn 113-115 92-96

Fe 133 110-113

Ba 144 118-122

Eu 150-160 122-130

Pb 160 128-132

Rb 190 142-153

Cs*(max) 170-180 133-142

Ca*(max) 350-375 235-255

Rb*(max) 420-435 290-320

CdS*(max) 460 330-340

ZnS*(max) 550 410-420

*Maxinium in resistance-pressure curve.

314
committee report

Equation-of-State Standards

J. Jamieson

George E. Duvall

D. John Pastine

G. R. Barsch

W. A. Bassett

D. L. Decker

D. McWhan

Participants

O. L. Anderson, Chairman

W. J. Carter

C. Beckett

R. H. Wentorf, Jr.

K. Vedam

M. van Thiel

R. Grover

Max Swerdlow

J. S. Weaver

L. Thomsen

S. E. Babe, Jr. E. M. COMPY

P. J. Freud

P. L. M. Heydemann

M. Klein

D. H. K. Mao

The appointed committee included J. Jamieson,

George E. Duvall, D. John Pastine, G. R. Barsch,

W. A. Bassett, D. L. Decker, D. McWhan, W. J. Carter, C. Beckett, and O. L. Anderson. The

chairman made all persons attending into a committee of the whole.

The panel reached a consensus on the topics

to be discussed. They were: definition of equation

of state; basis for selecting a standard reference

material; the reference materials themselves;

properties an ideal standard should have; the theoretical problems facing the equation of state of the

standard materials; special new experiments desired on standard materials.

I. Equation of state defined. The consensus was that general definitions should be avoided,

and that the focus be made on the P,V,T relationship of the standard material. In particular, generalized stress and strain relationships seemed

only remotely possible. However, it was recom- mended that a feasibility study be made on the

prospect of determining shear constants as a function of pressure.

II. Basis of selecting standard materials. The

primary consideration is that the material be usable

as a standard in experiments. The secondary consideration is that the material be amenable to theoretical analysis. The experimental fields concerned

are: (1) x-rays under pressure; (2) shock waves; (3)

pressure vessels closed to x-rays. Experiments

specifically not considered in the selection of the

materials were low-temperature (liquid helium)

high-pressure experiments, and acoustics under

pressure. It was recommended that a feasibility be

made for a material whose equation of state is useful

for low-temperature experiments.

III. The materials selected. It was the consensus

that a material must suit various kinds of experimenters. It was recognized that no one material

would satisfy all experimentahsts, but, on the other

hand, that selecting many materials is equivalent to selecting none. The recommended materials were,

in order of preference:

1. NaCl

2. Cu

3. Al

It was agreed that sodium was a material to be

considered in the future. A minority viewpoint held

that something should be done to choose a particular

liquid as a calibrant.

These three materials were selected as the ones which were the best compromise to the ideal one which would have the following properties (in order

of importance): a useful value of bulk modulus,

chemical inertness, isotropic structure, the absence

of phase changes, low values of atomic number, low

yield points, and optical transparency.

IV. Theoretical problems to solve on recommended materials. Not necessarily in order of

priority, they were: (1) Fermi-Dirac approximations;

(2) quantum mechanical approaches, especially calculations in deformed lattices; (3) noncentral forces

in lattice dynamic theories; (4) sixth-order anharmonic approximations in lattice dynamic theories;

(5) non-equilibrium thermodynamics, especially on

lattice models; (6) behavior of the Griineisen parameter with volume; (7) uniqueness problems in the

strain definition of finite strain theories.

It was further recommended that the Rochester

group, the BYU group, and the Los Alamos group

compromise their differences in their individual

published tables of equations of state of NaCl, and

together issue a common table.

It was further recommended that the equations of

state of the three standard materials be cross corre- lated with each other, and with the recommended

fixed pressure points.

315
y. Some special experimental problems (not

presently well attended) which should be pursued. The panel urged that experiments be pursued

showing the shift of vibrational frequency with volume at all frequency ranges. These include the pres- sure dependence of Brillouin scattering, neutron

scattering, and Raman scattering. New experiments

involving secondary standards were encouraged.

These were: pressure shifts arising from NMR, the

Mossbauer effect, and various electronic transitions.

The panel further recommended that, since all experimental values on these three materials would

be of use to theoreticians working out equations of

state, whenever possible experiments on these

materials be performed. Of special import are those

experiments that are boundary conditions of phenomological equations of state.

O. L. Anderson

316
committee report

Measurements of Pressure and Temperature in

High-Pressure Apparatus

S. Akimoto

N. ASAMI

P. Bell

H. BOVENKERK

I. C. Getting

M. C. Gilbert

Participants

F. R. Boyd, Jr., Chairman

D. Hamilton

G. S. James

R. S. Kirk

I. D. MacGregor

H. O. A. Meyer

R. Roy

P. St. Pierre

F. Seifert

H. M. Strong

C. SUSSE

A. Taylor

H. S. Young

The measurement of pressure at high temperature includes all of the difficulties of measuring

pressure at 25 degrees with a few more thrown in. Probably the biggest additional difficulty arises from

the fact that at high temperatures it is difficult to get a truly hydrostatic condition. Most of the hightemperature cells that are used in high-pressure

apparatus are composed of materials like lavastone,

talc, and graphite, and these do not permit a very

close approximation to a hydrostatic condition.

There is also the problem that arises because most

of the effects that one would Uke to study are temperature-sensitive as well as pressure-sensitive.

At high temperatures, particularly above a thousand

degrees centigrade, making a good pressure meas- urement involves the difficulty of making a good

temperature measurement as well.

Now, in the last ten years or so there has been

accumulated quite a large body of data on melting curves and solid-solid transitions, some of which

have been most imaginatively and carefully determined. Various other effects have also been

measured as a function of pressure and temperature. Nevertheless we decided at the outset of our

session that we would not dignify any of these data

by characterizing them as calibration curves or

fixed points or anything of that sort. Thus having

"disposed" of the main problem, we gave our

attention to the question of which future work we

or others should undertake to provide better meas- urements and calibrations at high temperatures.

We divided the problem into two parts: the use

of phase transitions as possible calibration points;

and "other effects".

We considered the "other effects" first and the

discussion of these is summarized as follows:

One possible calibration scheme uses thermocouples including, in particular, the iron platinum10-rhodium couple that Dr. Strong has described.

There are problems with this device in that it's hard to get a really big effect that's very sensitive

to pressure.

We talked about in situ lattice parameter measurements, for example by x-raying a substance at both high temperature and high pressure. This is a good method, but it requires very specialized equipment, and the average high-pressure worker would

not be able to readily make such a measurement in

order to calibrate his device.

We talked about the effect of pressure on the

index of glass. This is rather an interesting phenomenon in that many glasses, when placed under

high pressure, have rather large increases in their

index of refraction. With certain glasses, particularly

SiO-i glasses, this effect is large enough so that it quite possibly would provide a useful means of

pressure calibration. Use of glass is also attractive

in that one could run a small piece of glass with an unknown, and by measuring the index after the

experiment, determine the pressure of the experiment. There are difficulties. It is a relatively lowtemperature phenomenon, and there also appears

to be a problem sometimes of reaching an

equilibrium state in the index of the glass.

We talked also about vacancies in TiO as a function of pressure, and this can be established by a

lattice parameter measurement after the event.

In other words, after exposure to high pressure one can, through measurement of the lattice parameter, get an estimate of the pressure.

We discussed magnetic effects, and the largest

of these Curie-point changes at present seems to be

the effect of pressure on the Curie point of nickel,

which Dr. Susse estimated to be about 0.35 °C

per kilobar. This is thus not quite as sensitive as we

would like to have.

•We talked about solution phenomena which are

really phase transformations of a specialized kind.

There are transition loops and solvus curves which

are extremely sensitive to pressure. In some cases

it is possible to measure the composition of, say,

phase A in equilibrium with phase B with an accuracy which would be equivalent to a change in

pressure of the order of a kilobar, so that there

317
25 50 75 100 125 150 Pressure , kb

Figure 1. Phase transitions which can be studied by the quenching method and which are potentially

useful for calibration at high temperatures and pressures.

Sillimanite-Kyanite: Richardson, Bell, and Cfilbert (1968); Quartz-Coesite: Boyd, et al. (1966); (iraphite-Diamond: Berman and Simon

(1955): Fayalite-Spinel-Liquid: Akimolo. Komada. and Kushiro (1967): B.Oa: Dachille and Roy (1959); Coesite-Stishovite: Akimoto and Syono (1960); ZnO: Bates, White, and Roy (1962); CozSiO^: Akimoto and Sato (1968). are possibilities here. And it was suggested that for

very high-temperature situations one might look

at solvii in oxide systems; at more moderate

temperatures, silicate systems; and at low temperatures there are salt systems that could conceivably

be used.

We then discussed univariant phase equilibria.

We gave attention to those which, through the

group's experience, were well-behaved. We felt that if we could establish some of these in a

sufficiently absolute way they would make useful

calibration points or curves. We divided the phase

transitions into two groups according to reaction

rate. There are those which require run times of

the order of ten minutes to ten hours to some days

in order to achieve equilibrium, and which can be

studied by the quenching method; i.e. bring the

material up to pressure and temperature, hold it, quench it, and then examine the results by optical,

x-ray or electron probe methods. Some of these,

for example quartz-coesite, can be done with high

precision. We made a P-T diagram of phase transitions that would fall in this category.

We did the same thing for a group of phase transitions which are essentially displacive transformations, very fast ones, in which one can detect the

transition either through a latent heat effect or,

let's say, by a change in electrical resistance.

The accompanying diagrams illustrate the pres- sure and temperature ranges in which some of

these transitions, both slow and fast, could be useful.

At the highest pressures there is a quenchable

transition in zinc oxide (fig. 1) which goes to an NaCl

structure, and there are data now on this in the range

above 100 kbar and up to about 500 degrees. At a

little lower pressure there is the coesite-stishovite

reaction. Then further down there are three forms of

Co2Si04 with a triple point at 930 °C and 68 kbar.

Still further down there's a transition in Fe2Si04

which goes from an olivine to a spinel form. This

transition intersects the melting curve at 1520 °C

and 62 kbar.

At lower pressures is the quartz-coesite transition

for which we probably have better agreement and

more determinations than for any other. I would say

that at a temperature, for example, of 1,400 degrees

the three most recent determinations of the quartzcoesite transition are within two kilobars of each

other at approximately 40 kbar.

In a lower temperature range there is a transition

in BaO.'i and at pressures below quartz-coesite there

is the kyanite-sillimanite transition.

These are all fairly well-behaved transitions, and

it is possible to bracket many of them very closely.

If we could solve some of the measurement diffi- culties I think almost any of these would provide

useful calibration points or calibration curves. Now, the more rapid transformations which can be observed through DTA or an electrical resist- ance change fall in two groups (fig. 2). Of course, all of the fixed points that we've been working with at room temperature can be extended to a greater or 318
1

2000

75

Pressure , kb

Figure 2. Rapid phase transitions which can be determined by DTA, resistance-jump, etc., and which are potentially useful for calibration.

Diopside: Williams and Kennedy (1%9); NaCl: Akella, Vaidya, and Kennedy (1969); a-fi Quartz: Cohen and Klement (1967):

Iron: Bundy (1965): Bismuth l-ll-lll and Thallium I-II-III: LaMori (1970); Bismuth III V: Haygarth, Ludemann. and Getting (1969).

lesser extent up the temperature scale. For example.

Dr. LaMori earlier showed us his results up to several hundred degrees on the bismuth I-II and bismuth ll-lll transitions, and on the thallium transitions. Also, we have high-temperature data on the

bismuth III-V transition, discussed earlier in this

meeting. A second group in a higher temperature

range includes the alpha-gamma iron transition

which has been determined up to quite high pressures. There is also the alpha-beta quartz transition

at lower pressures.

Quite a lot of work has been done on melting

curves. For the most part, the melting curves of

metals have slopes that are less than five degree per

kilobar. Thus they are not as sensitive as would be

desirable for pressure calibration at high temperature. Salts, however, exhibit much steeper melting

curves. Initial slopes on the order of 20 degrees per

kilobar are quite common in salts, and there would

be a number of these that could be considered. One

that is of current interest is sodium chloride, which

also is in a reasonably manageable temperature

range (fig. 2). At higher temperatures there are silicate melting

curves, which have initial slopes on the order of ten

degrees per kilobar. A number of these curves have

been determined and for the most part they lie in

the range of 1,500 to 2,200 degrees Centigrade.

Among substances of this type for which melting

curves have been determined are diopside, albite,

and fayalite.

There are basically three ways to improve meas- urements on these various equilibria to the point

where one could really have confidence in them as

Ccdibration points. One is by thermochemical calculation, another by the shock wave approach, and

a third I think by use of the piston-cylinder apparatus.

Much data has now been obtained with the piston-cylinder apparatus, particularly since the UCLA

group found it possible to extend simple pistoncylinder measurements out to 80 kbar. But there

still are problems in relating the load pressure to the actual pressure in a piston-cylinder device.

Even when one can quite accurately determine the

amount of the friction, there is still the possibility

that in a high-temperature cell, composed of concentric layers of such materials as talc, boron

nitride, and graphite, the actual load is borne disproportionately by these shells, with resulting

uncertainties in the pressure on the sample.

For this reason, even though the precision of some

of the measurements in this high-temperature, highpressure, field is now very high, we are uncertain

as to the absolute accuracy.

F. R. Boyd, Jr.

References

Akella, J., S. N. Vaidya, and G. C. Kennedy, Melting of sodium

chloride at pressures to 65 kbar, Phys. Rev. 185, No. 3, 1135-

1140, (1969). Akimoto, S., E. Komada, and I. Kushiro, Effect of pressure on

the melting of olivine and spinel polymorph of Fe2Si04, J. Geophys. Research 72 , 679-686 (1967).

319
Akimoto, S., and Y. Sato, High pressure transformation in Co-.SiOj olivine and some geophysical implications, Phys.

Earth and Planet Interiors 1 , 498-504 (1968). Akimoto, S., and Y. Syono, The coesite-stishovite transition, this volume, p. 273. Bates, C. H., W. B. White, and R. Roy, New high pressure polymorph of zinc oxide. Science, No. 3334, p. 993 (1962). Berman, R., and F. Simon, On the graphite-diamond equihbrium,Z. Elektrochem. 59,333-338(1955).

Boyd. F. R.,P. M. Bell, J. L. England, and M. C. Gilbert, Pressure measurement in single-stage apparatus, Carnegie Institution of Washington Year Book 65, 410-414 (1966). Bundy, F. P., Pressure-temperature phase diagram of iron to 200 kbar, 900 °C, J. Appl. Physics 36, No. 2, 616-620 (1965). Cohen, L. H., and W. Klement, Jr., High-low quartz inversion:

Determination to 35 kilobars, J. Geophys. Res. 72, 424.5-4251

(1967). Dachille, F., and R. Roy, A new high pressure form of BjO.i and

inferences on cation coordination from infrared spectroscopy,

J. Am. Ceram. Soc. 42, 78-80 (1959). Haygarth, J. C, H. D. Ludemann, I. C. Getting, and G. C. Kennedy, Determination of portions of the bismuth III-V and

iv-v equilibrium boundaries in single-stage piston-cylinder

apparatus, J. Phys. Chem. of Solids 30, 1417-1424(1969).

LaMori, P. N., The use of solid-solid transitions at high temperatures for high-pressure calibration, this volume, p. 279. Richardson, S. W., P. M. Bell, and M. C. Gilbert, KyanitesiUimanite equilibrium between 700° and 1500 °C, Am. J. Sci. 266,513-541 (1968). Williams, D. W., and G. C. Kennedy, Melting curve of diopside

to 50 kilobars, J. Geophys. Res. 74, 4359-4366 (1969).

320
Pressure Inhomogeneity: A Possible Source of Error in Using Internal

Standards for Pressure Gages*

John C. Jamieson and Bart Olinger

Department of the Geophysical Sciences

University of Chicago, Chicago, Illinois 60637

Studies have been made on two intimate mechanical mixtures of NaCl and Nb having different composition at high "pressures" using x-ray diffraction. A plot of K,,/F„ for Nb versus ViilV,, for NaCl

showed a different curve for each of the two mixtures. Our analysis and the consequences of this apparent violation of the criteria for an internal pressure gage are discussed.

It has been pointed out several times in the past

that the stress configuration in compressed solids is much more complicated than the homogeneous

hydrostatic case. In particular, stress differences in

materials due to encapsulation have been treated

by Bobrowsky [1] ' and Corll and Warren [2], while

pressure differences developed by flow and geometry have been described by Jamieson and Lawson [3], Lippincott and Duecker [4], and Jamieson

[5]. The encapsulation enhancement has been ex- perimentally verified by Corll [6]. The use of an

internal standard to measure pressure in highpressure experiments implies immediately that the

system under study will consist of two or more

phases chemically and mechanically different in

their properties. One is the material where properties are to be studied as a function of pressure; the

other serves as an internal pressure gage. In the case

of the present experiments the "gage" gives an

x-ray diffraction pattern from which its density at any pressure may be directly calculated, in principle. Then a comparison made with data obtained by

other means on the "gage's" compression gives the

pressure. While this application seems straightforward, we felt that it was not at aU obvious that in a system of stressed solids, stress would be suffi- ciently homogeneous for such a gage reading to be

representative of the pressure in the second phase.

This was previously stated quite clearly by Bobrowsky [1].- The following experiments were performed

to investigate this possibility.

Two mechanical mixtures of dried NaCl and Nb

were weighed and then agitated in a "Wig-L-Bug."

Each was uniform to the eye under the microscope.

One consisted of 89 percent NaCl by volume, the

other 81 percent. These were formed into pellets

and studied by in situ high pressure x-ray diffraction as described elsewhere [3, 7]. We chose NaCl since

it is frequently used as an internal standard in such

'(Editor's Note: This paper is an expanded version of the discussion presented by

Professor Jamieson as part of the discussion of the paper "Calculation of the P-V

Relation for Sodium Chloride up to 300 Kilobars at 25 °C," by Weaver, Takahashi, and Bassetl.l

' Figures in brackets indicate the literature references at the end of this paper.

- We wish to acknowledge receipt of a letter from G. C. Kennedy, shortly after start- ing the experimental portion of this research, in which independently the same point was emphasized.

experiments. The other material chosen was Nb

because of its small compressibility relative to NaCl, its simple bcc x-ray pattern, and its position

next to Mo on the Periodic Table. This latter

property guaranteed comparatively low absorption

and high scattering power in the Mo radiation, which

was used in this experiment. Absorption coefficients

for the two mixtures studied were 45 cm"' for the

19 percent mixture and 34 cm"' for the 11 percent.

For our sample diameter of 0.35 mm no serious

absorption effects were to be expected using the

empirical 2lfx rule [7] for optimum sample size.

Another reason for the choice of Nb was the fact

that its Poisson's ratio of 0.35 differed grossly from

the (average) one for NaCl of 0.21 as determined

from handbook values.

Table 1. Sample indexed pattern

*N(Nh) rfcalc(Nb) *Af(NaCl) t/ealc(NaCl)

A A A

2.558 4 2.576

2.292 2 2.293

1.824 8 1.822

1.618 4 1.622

1.491 12 1.487

1.325 6 1.324

1.283 16 1.288

1.146 8 1.147 20 1.152

1.059 24 1.057

1.023 10 1.026

0.9950 27 0.9915

.9374 12 0.9363

.8692 14 .8669 .8104 16 .8109 40 .8146

.7647 18 .7645

.7251 20 .7253 .6916 22 .6915

.6613 24 .6621 .6358 26 .6361 .5923 30 .5922 .5578 34 .5563

321
.90 .85 Vp/Vo-NaCI

Figure 1. Relative compression of mechanical mixtures of II percent NB-89 percent NaCl (dark circles) and 19 percent NB81 percent NaCl (light circles).

Pressures quoted are for Kp/Fo of NaCl from reference [10]. Cross is error derived from pattern given in table 1.

In order to illustrate the pattern quality obtained,

we display in table 1, an indexed pattern. From this

pattern we obtain a = 3.2435 ± 0.0016 A for Nb

and a = 5.1520 ± 0.0048 A for the NaCl unit cell

edge. The error figures quoted are standard deviations for this one pattern. All in all, two runs

consisting of 16 diffraction patterns at different

elevated pressures were taken of the 11 percent Nb

mixture. In one run, tapered face pistons were

used [7]; in the other, flat face pistons. Six different runs were made of the 19 percent Nb mixture.

Piston geometries were tapered face and flat-face as weU as one run in which a precompressed and

glued boron annulus was used with one flat piston

bearing on one tapered. A total of 29 exposures

were made of this composition. During each run up to its termination by sample extrusion or piston

collapse pressures were increased or decreased at

will. For each diffraction pattern the unit cell edges

for the co-existing NaCl and Nb were calculated

and the ratios Fp/Fo calculated in which Va is the

unit cell volume at zero pressure. All high-pressure

films had been calibrated from 1-bar photographs

using 1-bar values of the respective cell edges.

Under these circumstances one would expect that a plot of VplVo for Nb versus V^jV^, for NaCl would

be a smooth curve except for experimental scatter

if the two co-existing phases were truly at equal

pressures. We have found this not to be true on two counts. The first (and less well substantiated)

is illustrated in figure 1, a composite plot of all runs. The lines connecting the points of one 11

percent run indicate successive load changes; a movement to the right is increased load, to the left

is decreased. It seems apparent that there is no

unique curve along which the relative compressions

lie. The number of points in this run is too smaU

to make firm statements but there does seem to be a trend that repeated cycling leads to lower VplV^t for

NaCl at a constant VpjVa for Nb. Similiar effects can be seen in the other runs which are not

individually shown to avoid confusion. The second

example of aberrant behavior is also illustrated

in figure 1 where it can be seen immediately that

the points fall almost uniquely into two groups.

The dark 11 percent Nb form a distinct set from

the open circle 19 percent Nb. This is a strong

violation of an imperative criteria of an internal

standard, namely that the presence of the standard

or its amount should not influence the behavior

of the material whose state it measures. For this

work we wish to present the data in figure 1 as empirical facts to illustrate phenomena which need

to be taken into account in high-pressure research.

For the nonce, we offer only a few qualitative

comments on the possible underlying physical

explanations.

As we mentioned earlier, "pressure intensification" has been predicted both theoretically and

experimentally. The "Brobrowsky-Corll-Warren

effect" uses as a model a sphere of one material

embedded in one or more shells of a different material. In the one-sheU model, pressure varies as

c + d/r' in the shell. We now consider our mixture as a dissemination of Nb spheres of radius a in a matrix of NaCl and draw spherical surfaces of

radius b around each, such that these spherical

surfaces about each grain are in contact. We as- sume a uniform pressure pe in the interstitial

regions. Then [1, 2]

PcIPe

l + K

Kil- BJBo) (a/6)3 + 1 + KB.IBo

(1)

in which pc is the pressure at the core's exterior. Bo

its bulk modulus. Bi is the bulk modulus of the

jacket (NaCl) and K a function of its Poisson's ratio

given by

1 + z. (2)

Using the values given in Clark [8] for the elastic

constants of a quasiisotropic aggregate of NaCl

averaged by the Voigt method, 1^ = 0.24 and = 0.83

at 1 bar. From Bridgman's [8] value for the compression of Nb at 30 kbar, X in figure 1, we estimate

BJBo as 0.18. Numerically eq (1) becomes

PcIPe

2.69

{alby + 1.69

and for widely spread spheres of Nb in NaCl a pres- sure intensification of 1.6 is predicted. If we as- sume the particles of Nb are uniformly separated in

space, then their distances apart may be calculated

to be 3.37 a and 2.81 a for the 11 percent mix and 19

percent mix. respectively. Since b as defined previously is one-half this distance, we obtain

PclPe=lA2 and 1.31

322
from substitution in eq (3). This is qualitatively in

agreement with figure 1 where it is seen that a constant NaCl "pressure", i.e., F;,/Fo= constant,

the compression of Nb is indeed less for the 19

percent mix than the 11 percent. The coefficients

in eq (3) are known only at 1 bar. In addition, flow

may occur (this may explain the cycling effect

shown in figure one) and alter the conditions of its derivation. Hence, no further application of the

Brobrowsky theory will be made at this time.

There are further difficulties with the use of any

internal standard in which x-ray diffraction is used

to determine lattice spacings in a stressed polycrystalline aggregate. In order to obtain a given

diffraction peak, sufficient matter must be present

with its atoms arranged in a sufficiently regular

fashion to give coherent radiation in a given direction. In solid pressure systems we may expect

great variations in such coherency where small

regions may be sufficiently coherent to diffract

and transition regions with a gradually varying

interplanar spacing may not. Such an effect is well

known among metallurgical studies of stresses

"frozen in" metals (Barrett [9]). By intuition one would expect the least stressed regions to be most

coherent; hence, any lattice spacing determined

by x-ray diffraction is apt to be a maximum value

corresponding to lower pressure regions.

It is necessary to mention stiU another possible

explanation for the apparently different behavior

of the 11 percent and 19 percent Nb mixes as shown in figure 1 and that is the possibihty of

random error affecting the caUbration of each in- dividual run. Figure 1 is a composite of two 11 percent and five 19 percent runs. In each run a 1-bar

diffraction pattern has been used to calibrate the

camera for absorption, film shrinkage, and sample

position [7]. It might be argued that sheerly by

chance all measuring errors accumulated so that

both 11 percent runs gave systematically low values

for Nb compression while all five 19 percent gave

systematically high ones. While we cannot rigorously rule out this possibility, if it did occur then

many more data points and separate loadings must

be made in any compressibility study than had been

thought reasonable before. The points in any one

run are all calibrated from the same 1-bar pattern

and thus are independent of this possible random

error. Yet in figure 1 we find variations within one

run (and others nof illustrated) ascribed to cycling

and sample reworking which are well beyond any

measuring error and are comparable to the 11

percent to 19 percent difference.

In conclusion, we point to the second X at 100

kbar in figure 1 which corresponds to the Los

Alamos shock compression data for Nb from ref- erence [8]. This is a Hugoniot point rather than on the isotherm, but for Nb these are almost

identical. The NaCl pressure values in the graph

are taken from Fritz et al. [10]. This point hes

intermediate between the 11 percent and 19 percent

Nb compression in this pressure region. It is obvious

that much work remains to explain the rather

complicated phenomena which are probably oc- curring not only in the mechanical mixtures studied

here, but also in the two-phase system generated

when a substance undergoes a polymorphic transition in a high stress environment.

Acknowledgments

This research has been supported by NSF Grant

NSF GA 1270 and ARPA grant SD-89-Research

and by the Petroleum Research Fund of the Ameri- can Chemical Society. Grateful acknowledgment

is made to the donors of that fund.

References

[1] Bobrowsky, A., in High-Pressure Measurement, ed. A. A.

Giardini and E. C. Lloyd, Butterworth, Washington D.C.

(1963). [2] CorU, J. A., and Warren, W. E., J. Appl. Phys. 36, 3655

(1965). [3] Jamieson, J. C, and Lawson, A. W., J. Appl. Phys. 33, 776

(1962). [4] Lippincott, E. R., and Duecker, H. C, Science 144, 1119

(1964). [5] Jamieson, J. C, Bull. Geol. Soc. Am. 74, 1067 (1963). [6] CorU, J. A., J. Appl. Phys. 38, 2708 (1967). [7] Jamieson, J. C, in Metallurgy at High Pressures and High

Temperatures, ed. M. T. Hepworth, N. A. D. Parlee, and

K. Gschneidner, Gordon and Breach, New York (1963). [8] Clark, S. P., ed.. Handbook of Physical Constants, Geol. Soc. Am. Memoirs 97 (1966). [9] Barrett, C. S., Structure of Metals, 2d ed., McGraw-Hill,

New York (1952). [10] Fritz, J. N., Marsh, S. P., Carter, W. J., and McQueen,

R. G., The Hugoniot equation of state of sodium chloride

in. the sodium chloride structure, presented at this meeting.

323

Comparison of Four Proposed P-V Relations for NaCl

J. Scott Weaver

Department of Geological Sciences and Space Science Center,

The University of Rochester, Rochester, New York 14627

Some comparisons among four P-V relations for the Bl phase of NaCl discussed at this symposium

are presented. The two scales (Models HI and V) proposed by Weaver et al. (this volume) and a 1968

revision of Decker's P-V relation (Dss) are shown to agree closely. The three models yield zero-pressure thermal expansion curves in reasonable agreement with experiment, but give values for B',^ smaller

than the measured value. This appears to be due, in part, to the form used for the lattice energy. The

three models yield pressures about 2.5 percent smaller than those calculated from the pressure scale

{Fkh) based on Hugoniot data proposed by Fritz et al. (this volume). Comparison of Models III, v, and

Fkk in the — u,, plane suggests that part of this difference must result from either the form chosen

for the lattice energy in Models III and V or from the Hugoniot data upon which fti« is based. Omission

of part of the data used in Fm improves the agreement of this model with the sonic value of B'l^ and

partially removes the discrepancy between and Models III and V. Suggestions for additional studies which may be useful in choosing a P-V relation for NaCl for pressure calibration purposes are presented.

It is concluded that, since the differences among these scales are comparable to the experimental uncertainties involved in applying them, and since there is no compelling reason to choose one over the

others, any of these scales may be used for the present.

1. Introduction

Following the discussions at this symposium, it seemed desirable to expand the treatment of the

various proposed P-V relations for NaCl presented in

Weaver et al. (this volume). In addition, the new

pressure scale proposed at this symposium by Fritz

et al. (this volume) differs significantly from those

described in Weaver et al. In this supplementary

paper, four proposed P-V relations for NaCl are compared in an attempt to find criteria favoring one

of them, and some suggestions for additional studies

bearing on the problem are made.

For convenience, the models discussed here are referred to by abbreviations: III and V refer to Models III and V presented in Weaver et al. (this

volume); Fes refers to the 295 °K isotherm based on shock wave data reported by Fritz et al. at this

symposium. The discussion here is based on the

values given in tables 3 and 4 of their paper (Fritz

et al., this volume). In 1968, Daniel L. Decker

kindly provided a copy of computer output containing a revision of his published calculations (Decker,

1965) of the equation of state for NaCl. This revised

model will be referred to as Dhh. In general, the

notations of this paper foUow those in Weaver et al. (this volume). Additional notation is defined as introduced.

2. Thermal Expansion

The thermodynamic Gruneisen parameter for

III, V, and Dh8 (already assumed to be a function of

the volume only in both the Hildebrand and MieGruneisen approximations) was assumed to be of

the form: y^yox^. In ill andv, the value: /4 = 1.2±0.4

was calculated from the relation: ^4 = 1+ (l+-yar)Ss — Bg + y evaluated at P = 0. In Des, the value:

^ = 0.767 was determined to yield the best agreement (in a least square sense) between the observed

zero-pressure thermal expansion curve for NaCl

and that calculated from the model. In order to compare these two different approaches to the determination of the volume dependence of the Gruneisen

parameter, calculated zero-pressure thermal expansion curves for III, V, and Des were compared with

V/V,

1. 10

1.08

1.06

1.04

1.02

1,00

25

o • /

O 9 /

/

/

/

Model ni

Model Y

D68

McKinstry (cited in Clark, 1966) J , I 200 400

TEMPERATURE (°C)

600

*Paper and Discussion prepared subsequent to the Symposium at suggestion of Symposium organizers. Paper received in final form April 1970.

Figure 1. Comparison of the zero-pressure thermal expansion

for NaCl from models III, V, and Dhh with results from experi- ment.

J'V. is the volume of NaCl at 25 °C and f = 0. The agreement between the calculated and measured thermal expansion curves suggests that it is not essential to use the measured thermal expansion as a boundary condition on the models (as was done

forD,»).

325
experimental results obtained by McKinstry (cited

in Clark, 1966) as shown in figure 1. The points for

III and V were obtained by numerically solving the

P-V-T relation for V and T such that P = 0. The

points for Des were calculated by three-point inverse

Lagrangian interpolation in Decker's table to estimate the value of r/ro for which P = 0 at various

temperatures. The experimental curve was obtained by five-point Lagrangian interpolation using

McKinstry's values given in Clark (1966). As can be

seen in figure 1 the thermal expansion curves for

V and Das agree closely with the experimental

results, while III yields values for the expansion

about 5 percent larger at high temperatures. The

general agreement among the three models suggests that the thermal expansion is insensitive to the method used to determine A, and that it is not

essential that the model be forced to fit the thermal

expansion curve. However, the different values of

A chosen for III and V versus lead to different

values of {dPldT)y at high pressures. Hence, caution must be used in applying these models to pres- sure or temperature measurement under simultaneous conditions of high temperature and high

pressure.

3. Pressure Derivatives of Bt

The values of the pressure derivatives of Bt calculated for III, V, and Dsn may be compared with ex- perimental values to yield a test of the models since

only P and Bt are fixed by the boundary conditions

at V= Vt), while the derivatives are then determined

by the form of the model. The values for the pres- sure derivatives of Bt for Fan are largely determined

by the experimental Hugoniot for NaCl and, to a

lesser extent, by the procedure used to reduce the

Hugoniot to an isotherm. Values for the dimensionless parameters fi/,, and —BtoB"g for models III and

V were calculated by means of seven-point finitedifference formulas to yield the results shown in

table 1. Since Dhs and F^^» are presented in tabular

form, numerical derivatives are subject to large

errors due to rounding-off of the tabulated values.

This difficulty may be avoided by using an approximate P-V relation such as the Murnaghan or Birch

equation to convert the tabulated P-V relation to a more slowly varying function of V. For example, the

parameter a obtained from the Murnaghan equation:

P = Btoa~^{x~" — \) is an approximately hnear function of X for all of the models considered here as is shown in figure 2. A linear least squares fit to the

a —X relation for each model together with the

relations:

B'lg = lim a and Bt„B','

,. da

lim —

(which follow easily from the Murnaghan equation)

was used to compute the values shown in table 1.

6.0

5.5

5.0

4.5 -

4.0

o Model m

• Model V

« 068

> Fee (Q)

t Fee (L)

Battels a Schuele (1965) A Chang (1965)

> B

1.0 0.9 0.8 0.7 V/Vo

Figure 2. Murnaghan equation parameter a versus volume for

several models for NaCl.

a was calculated from the Murnaghan equation: /"= "

' [ (K/Kti) "' — I ]; with «,„=237.4 kbar for III, V, and and B,„= (0.00422)-' kbar for /)««. Q and /. are the quadratic and linear forms of f,in. The error fla^s ^how the uncertainty in a result- ing from rounding off of the tabulated values of P and V for Dm and Fun. It can be shown

that u = lll^ at VIV»~ I. Two measured values for for NaCl are shown for comparison.

Table 1. Values for and — BtoBj'o calculated

from models HI, V, Dhk, and Fan and values for

B^ determined by experiment for NaCl.

Source "BtoBl Method

Experiment 5. 18 ±0.09 Sonic (uniaxial loading);

Chang, 1965. Experiment 5.35 Sonic (hydrostatic pressure); Bartels and

Schuele, 1965.

III 4.916 5.964 Numerical differentiation. 4.902 6.217 Murnaghan equation.

V 4.894 7.169 Numerical differentiation. 4.8% 6.405 Murnaghan equation.

D„„ 4.858 6.416 Murnaghan equation,

Bto = 237.0 kbar.

Fbh 5.453 10.991 Murnaghan equation.

Bio =237.4 kbar.

The values obtained for III and V using this procedure differ somewhat from those calculated by

direct differentiation since the a — x relation exhibits a slight curvature. The difference is probably

not significant for the present purposes.

As can be seen from table 1, models III, V, and

Dan all yield values for B^o smaller than those determined experimentally. The value of the parameter — BtoB'f'o has not been measured for NaCl, although

326
the values: — Bs„B'^„ — 1.15,4, found for CsCl, CsBr,

and Csl (Chang and Barsch, 1967) and the value

-B,oB"„ \ .06B;„ for Csl (Barsch and Chang, this

volume) suggest that the values —BioB'lu—l.ZB',,,

calculated for III, V, and Ajh are not unreasonable.

The reason for the discrepancy between the values

for B',„ calculated from the models and that obtained

from experiment is not clear. The agreement among

the models shows that inclusion of the second

nearest neighbor interaction does not significantly

change the value calculated for B'to, while the effect

of the vibrational energy terms is too small to re- move the discrepancy without assuming unreasonably large values for the parameter^. (For example:

in model V, 5/o = 5.18 would require that /i = 3.42,

while B',0 = 5.35 would require A = 4.85. Either value

of A is incompatible with the observed values of

B'so and 8so.)

The value for B'to calculated from Fes is larger than

that found from experiment, and the value for — BioB'i'o — 2B!o is larger than that expected by analogy with the cesium halides. As will be discussed in

the next section, the value for B'lo can be brought

into agreement with that measured by Bartels and

Schuele (1965) if the eight lowest pressure data

points are omitted from the Hugoniot determined by

Fritz et al. (this volume). This procedure would also

reduce the value of — BtoB'/g.

4. Comparison With Shock-Wave Data

In comparing F^n with III, V, and Dun, the effect of

the Hugoniot to isotherm reduction used in Fan

may be eliminated by calculating the Hugoniot

corresponding to the assumptions already used in

III, V, or DtiH- For these models, the pressure, Ph,

along the Hugoniot is given by:

Us (km/sec)

Ph 7(1 —x) — X y y

TT (4) - (t>o) + X^' --<^0

y 0 yo

where (f) is the lattice energy, (I)' = d^ldV, and the

subscript "0" refers to quantities evaluated at V= V{). The corresponding shock (u,) and particle

{u,j) velocities are then determined by the RankineHugoniot relations: Up = po^''-P}l''{l—x)^^'-; and

Us= Up(l — x)"'. The calculated Uj — relations for

III and V are compared with data from experiment in

figure 3. The results for ill and V are in agreement

with the values obtained by Christian (1957) and by

Al'tshuler et al. (1961), but lie below those found by

Hauver and Melani (cited in van Thiel, 1966) and

those found by Fritz et al. (this volume) which form

the basis for Fun. (According to Fritz et al., the data

from Hauver and Melani are based on the old 2024

aluminum standard, and will be brought into better

agreement with the data from Fritz et al. when the

new standard is used.) The difference between the

6.0 -

5.0

4.0 -

34

Al'tshuler et al (1961)

Christian (1957)

Hauver a Melani (1966)

Fritz et al. ( 1968) — Models in a V ^ Bulk sound speed

1.0 Up (km/sec)

2.0

Fl<;URE 3. Shock versus particle velocity for NaCl calculated

from III and V compared with results from experiment.

The calculated u„ — u,> relations for III and V are indistinfiuishable on the scale

plotted. The bulk sound speed tor NaCl measured by Slagle and McKinstry (1967)

is shown by the filled trianj;le. The value determined by Haussuhl (1960) would be

plotted within the triangle. As noted in the text, the experimental results of Hauver

and Melani will be in belter agreement with those found by Fritz et al. when the 2024 aluminum standard is revised.

Ug — Up relation measured by Fritz et al. and that

calculated for III and V is most pronounced for

Up < 0.7 km/s and is somewhat reduced for larger

values of Up. Since the discrepancy between F^is and III or V persists when the u.s — Up relations cor- responding to these models are compared, the

procedure used by Fritz et al. does not appear to be

responsible for the difference.

The effect of the vibrational terms in the internal

energy for III, V, and Ajh on the Hugoniot pressure

was investigated by calculating y from the lattice

energy and the observed Ug — Up relation (Fritz et al.,

this volume) using the relations given above. It was

found that d In yld In V must have a parabolic dependence on In V if the models and the observed

Us — Up relation are to agree. For example, in the case of model v, the quantity d\nyld\n V is reduced

from 1.2 at VIVo = 1.0 to zero (or even slightly negative) at VIVo = 0-S8 and then increases rapidlyreaching the value 3.2 at F/Fo = 0.70. Although such

behavior of y cannot be excluded, it seems more

plausible to seek to resolve the discrepancy between

the calculated models and Fus by another means. The data obtained by Fritz et al. are compatible

with a linear Us — Up relation which yields a bulk

sound speed about 4 percent larger than that calcu327

r

ll

385-762 O - 71 - 22
'-PjCkb)

Figure 4. Comparison of pressures from several models for

NaCl with those calculated from V.

The curve 5 is an estimated upper bound on the uncertainty in the pressure cal- culated from V resulting from the uncertainties in the parameters used in the model. Curves Q and are the portions of Fkh derived from the quadratic and linear fits to the measured u^ — u,, relation for NaCl. Q' is derived from Q by omitting the 8 lowest pressure data points from the — u,, relation.

lated from sonic measurements on NaCl. Hence,

Fritz et al. chose to fit their data with a quadratic

«.s~«p relation constrained to pass through the

measured bulk sound speed. Since the lowest pres- sure data points appear to deviate from the fitted

Ug — Up relation, it is interesting to examine the

effect of these points on both the Hugoniot and on

Ftis. The omission of at least the four lowest data

points may be justified since Fritz et al. state that

these points were obtained by a different driver ar- rangement (using impedance mismatch to produce

low shock pressures) and may have been influenced

by overtaking waves. When all 43 data points listed

by Fritz et al. (this volume; table 1) were fitted, the

Ug — Uj, relation was found to be: u.s = 3.403

+ 1.5492ttp— 0.07833«^; while, when the eight lowest

pressure data points (those for which u,) < 0.70 km/s)

are omitted, the fitted relation becomes: Ms = 3.403

+ 1.5141u„—0.05738ap—where w., and u,, are in km/s,

and the bulk sound speed measured by Haussuhl

(1960) has been used to fix the a.s — Up relation at Up — 0. Most of the change results from the omission

of the four lowest points, while omitting more than

eight points was found to have little effect on the

fitted Us — Up relation.

One effect of omitting the lowest points is to reduce the values for Blf, and —BtoB"„ calculated

from the a, ~ Up relation. In particular, the value of

B'to is reduced from 5.47 to 5.33 when the eight

lowest points are excluded. The latter value is in

excellent agreement with the value of 5.35 measured by Bartels and Schuele (1965) but stiU larger

than Chang's (1965) value of 5.18. The value of — BioB'i'o will also be reduced, although the lack of

measured values for the second pressure and temperature derivatives of Bt for NaCl prevents calculations of the actual values. The omission of the

eight lowest data points was not found to have any

significant effect on the calculated temperature

along the Hugoniot, so that only the change in the

Hugoniot pressure need to considered in estimating

the effect of these points on Fhh. Reducing the pres- sure calculated from Fhh by an amount equal to the

difference in the Hugoniot pressures for the two

Ug — Up relations, yields a revised model {Q') which

is compared with the others in figure 4. It can be

seen that, although the discrepancy between Fan

and III, V, and is reduced, most of the difference

remains when Fm is revised.

5. Conclusions

The discussions in both Weaver et al. (this volume)

and the present work serve to show the close similarity among models III, V, and D^n. It seems that,

within the quasi-harmonic formulation of the equation of state for an ionic solid, the room-temperature

isotherm and the zero-pressure thermal expansion

curve for NaCl are insensitive to the form chosen for

the vibrational energy (Hildebrand or Mie-Gruneisen

approximation), the neglect or inclusion of the repulsion between second nearest neighbors, or the ex- ponent {A) chosen for the volume dependence of the

Gruneisen parameter. However, use of these models

at elevated temperatures will require that the

parameter A be determined. Since A for III and v

satisfies thermodynamic constraints at P — Q, and

since these models yield good approximations to the

zero-pressure thermal expansion curve for NaCl,

they should be useful at high temperatures.

Models III, V, and Dhh yield pressures about 2.5

percent smaller than those calculated from Fbk

(see fig. 4). This difference cannot be attributed to the procedure used to convert the Hugoniot to an

isotherm in Fob- Changing the form for the volume

dependence of the Gruneisen parameter in ill or v

so as to remove the difference leads to implausible

behavior of y and its derivative with volume. Hence,

either the form chosen for the lattice energy in ill,

V, and AiK or the Ug — Up data used in Fkh must be

responsible for much of the discrepancy between

these models. If the lattice energy is in error, V and

FijH can be brought into agreement by inclusion of a term of gaussian form, suggesting that covalent

bonding may be involved. However, it seems premature to include such a term without additional

information.

Model FfiH yields the value 5/o = 5.45, while in,

V, and Z)(iH give values near 4.90. If the lowest pres- sure Hugoniot data (eight points) are omitted, Fs8

yields fi/o = 5.33— in excellent agreement with the

value measured by Bartels and Schuele (1965), but

higher than that found by Chang (1965). Raising the

value of B't,, for III, V, and to either of the measured values probably will require modification of the

form used for the lattice energy, since unreasonable

328
values of A are required to accomplish this with the

vibrational energy terms. An additional measure- ment of fi/o for NaCl — either by sonic methods or perhaps, by means of low-pressure shock-wave techniques—would be most useful since a value near

that obtained by Bartels and Schuele would tend to support FiiH, whereas a value near Chang's would

suggest that the P-V relation for NaCl lies between

Ffi8 and III, V, or Dun.

Additional determinations of the Us — u,, relation

for NaCl at particle velocities less than 1.0 km/s

would be useful to determine whether, as suggested

here and in Fritz et al. (this volume), the low-pres- sure portion of Fan is influenced by errors in the

Hugoniot data. In addition, studies on porous

samples might help to determine the best choice

for the parameter A. Measurement of the volume

of NaCl by means of piston-cylinder techniques at pressures in the 50 to 100 kbar pressure range may

be able to distinguish between Fes and III, V, and D«8.

In conclusion, I believe that it is premature to attempt to choose one of these scales over the

others. Fortunately, the differences between them

are, at worst, comparable to the experimental uncertainties in the application of the internal standard

method of pressure determination in x-ray diffraction studies at pressures above 100 kbar. For the

present, it would be desirable to report the volume

for the internal standard together with the pressure

based on any of these pressure scales.

6. References

Al'tshuler, L. V., Kuleshova, L. V., and Pavlovskii, M. N., The

dynamic compressibility, equation of state, and electrical conductivity of sodium chloride at high pressures, Soviet Physics

(JET?) 12, 10-15(1961).

Barsch, G. R., and Chang, Z. P., Ultrasonic and static equation of

state for cesium halides. this volume.

Bartels, R. A., and Schuele, D. E., Pressure derivatives of the

elastic constants of sodium chloride and potassium chloride

at 295 deg. K and 195 deg. K, J. Phys. Chem. Solids 26,

537-550 (1965). Chang, Z. P., Third order elastic constants of sodium chloride

and potassium chloride single crystals, Phys. Rev. A140,

1788-1799(1965).

Chang, Z. P., and Barsch, G. R., Nonlinear pressure dependence

of elastic constants and fourth-order elastic constants of cesium

halides, Phys. Rev. Letters 19, 1381-1382 (1967). Christian, R. H., The equation of state of alkali halides at high

pressure, Univ. Calif. Radiation Lab. Report 4900 (1957). Clark, S. P., Ed., Handbook of physical constants, Geol. Soc. Am.

Memoir 97, p. 80(1966).

Decker, D. L., Equation of state of NaCl and its use as a pressure gauge in high-pressure research, J. Appl. Phys. 36, 157-161,

1965. Decker, D. L., personal communication (1968).

Fritz, J. N., et al.. The Hugoniot equation of sodium chloride in the sodium chloride structure, this volume.

Haussuhl, S., Thermo-elastische Konstanten der Alkalihalogenide vom NaCl-Typ, Z. Phys. 159, 223-229 (1960). Slagle, O. D., and McKinstry, H. A., Temperature dependence of the elastic constants of the alkali halides. I. NaCl, KCI, and

KBr, J. Appl. Phys. 38, 437-446 (1967). van Thiel, M., Compendium of Shock Wave Data, UCRL-50108

(Vol. 1)(1966).

Weaver, J. S., Takahashi, T., and Bassett, W. A., Calculation of

the P-V relation for the Bl phase of NaCl up to 300 kbar at 25 °C, this volume.

DISCUSSION

D. L. Decker {Brigham Young University, Provo,

Utah): The three equations, which are labeled III,

V, and Des, aU assumed a volume dependence of

the Gruneisen parameter to be of the form

y = y,{VIV„r. (1)

Weaver compared the various choices of A and the

models for these equations by comparing the volume

expansion versus temperature calculated from the

three equations. This is a very insensitive test for

the correct value of A. A far better approach is to compare the calculated coefficient of linear thermal

expansion versus temperature against the experimental results. (Enck and Dommel, 1965.) This is done in figure 1 where one observes that even at atmospheric pressure the value of ^ plays a definite

role in thermal expansion. A =0.1 gives the best

fit to the experimental thermal expansion. A — IA

gives the best fit of the calculated adiabatic bulk

modulus at atmospheric pressure to the measure- ments of Slagle and McKinstry (1967). In the

modification referred to by Weaver, I used A = 0.93

which gives a reasonable fit to these two experimental results simultaneously.

The direct value of calculated in Dks is 4.92,

which differs slightly from the interpolated value

I I I I I 0 200 400 600 800

T(°C)

FiCURE 1. Coefficient of linear thermal expansion versus tem- perature at atmospheric pressure. The solid curves are calcu- lated using various values for the parameter A. The darkened

area gives the range of the experimentally measured values.

329
TABLE 1. Comparison of calculated pressure and

the pressure measured by shock techniques " along

the Hugoniot and the 293 K isotherm, ao is the

lattice parameter at 25 °C and atmospheric pressure. Pressure in kbar.

/'(F^s)

Along the Hugoniot 293 K isotherm

alcia T(K) P(D„«) a/a<i P(D««)

25 0.9732 336 24.7 0.9721 24.6 ±1.1%

50 0.9546 386 49.0 0.9529 48.9 ±1.6%

75 0.9402 450 73.2 0.9387 73.0 ±1.9%

100 0.9284 527 97.6 0.9252 97.6 ±2.4%

150 0.9095 721 147.2 0.9050 146.6 ±2.9%

200 0.8945 960 198.3 0.8887 197.8 ±3.4%

250 0.8821 1,238 250.4 0.8746 252.2 ±3.7%

"Fritz, J. N., S. p. Marsh, W. J. Carter, and R. G. McQueen,

The Hugoniot equation of state of sodium chloride in the sodium

chloride structure, this volume.

* The uncertainty in D38 due to uncertainties in the experimental parameters used in the calculation. This does not include any

uncertainty due to the approximations in the theory.

given by Weaver, and is in excellent agreement

with III and v. There is also another experimental

measurement of which should be added to Weaver's table 1 and included in the discussion.

This is the value 4.98 ±0.20 measured by Drabble

and Strathen (1967) which is smaller than the other

measurements but agrees very well with the theoretical values in ill, V, and D^h. There is another

recent measurement yielding 5[^=3.2 (Gluyas,

1967) but this is hkely too low. The measurements of

B'^ indicate that the theories are credible but

these measurements are not precise enough to give any guidance as to how one might improve

the theory.

I agree that, to within the uncertainties involved,

the differences between F^h and III, V, or Dm are

not related to the procedure of converting from the

Hugoniot to the isotherm. This is borne out in

table 1 where the calculated and measured pres- sures are compared along the Hugoniot as well

as along the room temperature isotherm. The final

column in the table gives the estimated accuracy

of the calculated equation of state as limited by

the uncertainties in the input parameters. These

uncertainties include a variation of A ranging from

0.7 to 1.4. It is concluded that all these theoretical

equations of state lie within the uncertainties

given and give equally reliable approximations to the true pressure. Fr^, however, is definitely

different from Dgs but possibly not outside the

experimental error.

Bibliography

Drabble, J. R., and Strathen, R. E. B., The third-order elastic constants of potassium chloride, sodium chloride, and lithium

fluoride, Proc. Phys. Soc, 92, 1090 (1967). Enck, F. D., and Dommel, J. G., Behavior of the thermal expansion of NaCl at elevated temperatures, J. Appl. Phys. 36,

839 (1965). Gluyas, M., The second- and third-order elastic constants of sodium chloride at 295 °K, Brit. J. Appl. Phys. 18 , 913 (1967). Slagle, 0. D., and McKinstry, H. A., Temperature dependence

of the elastic constants of the alkali halides. I. NaCl, KCl,

and KBr J. Appl. Phys. 38, 437 (1967). Weaver, J. S., Comparison of four proposed P-V relations for NaCl, this volume.

J. N. Fritz {University of California, Los Alamos

New Mexico): We concur with the views and conclusions presented by Dr. Weaver. There now exists

further experimental justufication for casting out

the lowest pressure data points reported by us. New

experiments have indicated (at least for our four

lowest data points) the existence of a second compression wave in the shims that close the gap over

the NaCl sample. This wave arises from the rare- faction coming back through the shim and its interaction with the NaCl. The additional particle ve- locity imparted to the shim results in an early gap

closure and a higher apparent shock velocity in the

NaCl sample. This effect becomes progressively less

important at high pressures, and vanishes altogether above a certain critical pressure, and is absent for the bulk of the data we reported. The net

effect of all this is well represented in Weaver's

Q ' curve in his fig. 4.

330
Symposium Participants

L. Abbot, Harwood Engineering, Walpole, Massachusetts

P. H. Abelson, Carnegie Institution of Washington, Washington, D.C.

S. Akimoto, University of Tokyo, Tokyo, Japan

P. B. Alers, U.S. Naval Research Laboratory, Washington, D.C.

E. Ambler, National Bureau of Standards, Washington, D.C.

O. Anderson, Columbia University, Palisades, New York

N. ASAMI, Mitsubishi Atomic Power Ind., Inc., Saitama, Japan

Y. A. Atanov, AU-Union Research Institute for Physical and Radiotechnical Measurements,

Moscow

S. E. Babb, University of Oklahoma, Norman, Oklahoma

J. D. Barnett, Brigham Young University, Provo, Utah

G. R. Barsch, Pennsylvania State University, University Park, Pennsylvania

W. A. Bassett, University of Rochester, Rochester, New York

R. Bautista, Rice University, Houston, Texas

C. W. Beckett, National Bureau of Standards, Washington, D.C.

P. Bell, Carnegie Institution of Washington, Washington, D.C.

S. Block, National Bureau of Standards, Washington, D.C.

C. BOSCO, U.S. Army Electronics Command, Fort Monmouth, New Jersey

H. P. Bovenkerk, General Electric Company, Detroit, Michigan

J. C. BOWEN, Pressure Products Industry, Hatboro, Pennsylvania

F. R. Boyd, Jr., Carnegie Institution of Washington, Washington, D.C.

W. R. BOZMAN, National Bureau of Standards, Washington, D.C.

J. W. Brasch, University of Maryland, College Park, Maryland

F. BUNDY, General Electric Research and Development Center, Schenectady, New York

P. D. Calvert, Massachusetts Institute of Technology, Cambridge, Massachusetts

W. J. Carter, University of California, Los Alamos, New Mexico

R. Cavailhe, Commissariat a I'Energie Atomique, Paris, France

A. Cezairliyan, National Bureau of Standards, Washington, D.C.

Z. P. Chang, Pennsylvania State University, University Park, Pennsylvania

D. H. Chung, Massachusetts Institute of Technology, Cambridge, Massachusetts

T. Clark, General Electric Company, Detroit, Michigan

N. L. Coleburn, Naval Ordnance Laboratory, White Oak, Maryland

E. M. COMPY, U.S. Naval Research Laboratory, Washington, D.C.

M. Contre, Commissariat a I'Energie Atomique, Paris, France

M. COWPERTHWAITE, Stanford Research Institute, Menlo Park, California

J. L. Cross, National Bureau of Standards, Washington, D.C.

D. Decker, Brigham Young University, Provo, Utah

T. B. Douglas, National Bureau of Standards, Washington, D.C.

H. G. Drickamer, University of Illinois, Urbana, Illinois

G. E. DUVALL, Washington State University, Pullman, Washington

D. Edwards, Naval Ordnance Laboratory, White Oak, Maryland

J. L. England, Carnegie Institution of Washington, Washington, D.C.

J. O. Erkman, Naval Ordnance Laboratory, White Oak, Maryland

J. L. Feldman, Naval Research Laboratory, Washington, D.C.

J. Forbes, Naval Ordnance Laboratory, White Oak, Maryland

P. J. Freud, Battelle Memorial Institute, Columbus, Ohio

R. C. Frisch, National Bureau of Standards, Washington, D.C.

331
I. Getting, University of California, Los Angeles, California

M. C. Gilbert, Virginia Polytechnic Institute, Blacksburg, Virginia

R. Grover, Lawrence Radiation Laboratory, Livermore, California

W. E. Hake, American Instrument Co., Inc., Silver Spring, Maryland

H. T. Hall, Brigham Young University, Provo, Utah

D. L. Hamilton, University of Manchester, Manchester, England

R. E. Hanneman, General Electric Research and Development Center, Schenectady, New York

J. R. Hastings, National Bureau of Standards, Washington, D.C.

G. E. Hauver, Aberdeen Proving Ground, Aberdeen, Maryland

J. Haygarth, University of California, Los Angeles, California

P. L. M. Heydemann, National Bureau of Standards, Washington, D.C.

A. Holt, Lawrence Radiation Laboratory, Livermore, California

J. HOUCK, National Bureau of Standards, Washington, D.C.

R. Houidobre, Air Force Office of Scientific Research, Arlington, Virginia

E. Hryckowian, U.S. Army Electronics Command, Fort Monmouth, New Jersey

R.P. Hudson, National Bureau of Standards, Washington, D.C.

U. O. HUTTON, Brinklow, Maryland

J. Jackson, University of Maryland, College Park, Maryland

G. S. James, Adamant Research Laboratory, Johannesburg, South Africa

J. C. Jamieson, University of Chicago, Chicago, Illinois

A. Jayaraman, Bell Telephone Laboratories, Murray Hill, New Jersey

D. P. Johnson, National Bureau of Standards, Washington, D.C.

0. E. Jones, Sandia Laboratory, Albuquerque, New Mexico

G. Jura, University of California, Berkeley, California

R. Kaneda, National Research Laboratory of Metrology, Tokyo, Japan

N. Kawai, Osaka University, Toyonaka, Osaka, Japan

G. S. Kell, National Research Council of Canada, Ottawa, Canada

G. C. Kennedy, University of California, Los Angeles, California

R. S. Kirk, General Electric Research and Development Center, Schenectady, New York

M. C. Krupka, University of California, Los Alamos, New Mexico

I. KUSHIRO, Carnegie Institution of Washington, Washington, D.C.

A. S. KusUBOV, Lawrence Radiation Laboratory, Livermore, California

A. Lacam, Laboratoire de Bellevue, Bellevue, Seine et Oise, France

P. N. LaMori, Battelle Memorial Institute, Columbus, Ohio

Le Neindre, Laboratoire de BeUevue, Bellevue, Seine et Oise, France

D. H. LiNDSLEY, Carnegie Institution of Washington, Washington, D.C.

E. C. Lloyd, National Bureau of Standards, Washington, D.C.

E. Lundblad, Scandiament AB, Robertfors, Sweden

1. MacGregor, S. W. Center for Advanced Studies, Dallas, Texas

D. H. K. Mao, Carnegie Institution of Washington, Washington, D.C.

D. B. McWhan, Bell Telephone Laboratories, Murray Hill, New Jersey

R. Meister, U.S. Geological Survey, Washington, D.C.

L. Merrill, Brigham Young University, Provo, Utah

H. O. A. Meyer, Carnegie Institution of Washington, Washington, D.C.

S. Minomura, University of Tokyo, Tokyo, Japan

W. Mock, Jr., U.S. Naval Weapons Laboratory, Dahlgren, Virginia

R. L. Nuttall, National Bureau of Standards, Washington, D.C.

J. Paauwe, Naval Ordnance Laboratory, White Oak, Maryland

D. L. Pastine, Naval Ordnance Laboratory, White Oak, Maryland

L. Peselnick, U.S. Geological Survey, Washington, D.C.

E. J. Prosen, National Bureau of Standards, Washington, D.C.

S. V. Radcliffe, Case Western Reserve University, Cleveland, Ohio

332
C. R. RoBBlNS, National Bureau of Standards, Washington, D.C.

E. C. Robertson, U.S. Geological Survey, Washington, D.C.

M. F. Rose, Naval Weapons Laboratory, Dahlgren, Virginia

R. Roy, Pennsylvania State University, University Park, Pennsylvania

P. St. Pierre, General Electric Company, Detroit, Michigan

G. A. Samara, Sandia Laboratory, Albuquerque, New Mexico

K. H. SCHUMAKER, Air Force Office of Scientific Research, Arlington, Virginia

C. B. SciAR, Lehigh University, Bethlehem, Pennsylvania

G. J. ScOTT, University of Oklahoma, Norman, Oklahoma

F. Seifert, Institut fuer Mineralogie, Kiel, West Germany

J. V. Sengers, University of Maryland, College Park, Maryland

H. R. Shaw, U.S. Geological Survey, Washington, D.C.

E. Shull, Pennsylvania State University, University Park, Pennsylvania

E. F. Skelton, U.S. Naval Research Laboratory, Washington, D.C.

V. SOKOLOVSKIY, All-Union Research Institute for Physical and Radiotechnical Measurements,

Moscow

L L. Spain, University of Maryland, College Park, Maryland

H. D. Stromberg, Lawrence Radiation Laboratory, Livermore, California

H. M. Strong, General Electric Research and Development Center, Schenectady, New York

C. SUSSE, Laboratoire de Bellevue, Bellevue, Seine et Oise, France

M. SWERDLOW, Air Force Office of Scientific Research, Arlington, Virginia

T. Takahashi, University of Rochester, Rochester, New York

A. Taylor, Westinghouse Research Laboratories, Pittsburgh, Pennsylvania

L. Thomsen, Columbia University, Palisades, New York

L. C. TOWLE, Naval Research Laboratory, Washington, D.C.

G. Uriano, National Bureau of Standards, Washington, D.C.

M. van Thiel, Lawrence Radiation Laboratory, Livermore, California

A. VAN Valkenburg, National Science Foundation, Washington, D.C.

H. Vanfleet, Brigham Young University, Provo, Utah

K. Vedam, Pennsylvania State University, University Park, Pennsylvania

O. B. Verbeke, University of Maryland, College Park, Maryland

B. VODAR, Laboratoire de Bellevue, Bellevue, Seine et Oise, France

M. Waxman, National Bureau of Standards, Washington, D.C.

J. S. Weaver, University of Rochester, Rochester, New York

A. Webb, Naval Research Laboratory, Washington, D.C.

C. E. Weir, National Bureau of Standards, Washington, D.C.

R. H. Wentorf, Jr., General Electric Research and Development Center, Schenectady, New York

M. S. Werkema, Dow Chemical Company, Golden, Colorado

H. WOOLLEY, National Bureau of Standards, Washington, D.C.

H. S. Yoder, Jr., Carnegie Institution of Washington, Washington, D.C.

H. Young, E. I. duPont de Nemours and Co., Wilmington, Delaware

R. Zeto, U.S. Army Electronics Command, Fort Monmouth, New Jersey

E. V. ZoLOTYKH, All-Union Research Institute for Physical and Radiotechnical Measurements,

Moscow

U.S. GOVERNMENT PRINTING OFFICE ; 1971 OL-385-762

333
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