Equations of State for Solids at High Pressures and Temperatures

265 Pages·1971·15.996 MB·English

by  V. N. Zharkov, V. A. Kalinin (auth.)

V. N. Zharkov
Equations of State
for Solids at High
Pressures and
Temperatures
EQUATIONS OF STATE FOR SOLlOS AT HIGH PRESSURESAND TEMPERATURES
URA VNENIYA SOSTOYANIY A TV~RDYKH TEL PRI VYSOKIKH DA VLENIYAKH
I TEMPERA TURAKH
YPABHEHVIH COCTOHHVIH TBEP ~h!X TEJI nPvt Bh!COKVIX ~ABJIEHVIHX
VI TEMnEPATYPAX
EQUATIONS OF STATE FOR
SOLIDS AT HIGH PRESSURES
AND TEMPERATURES
V. N. Zharkov and V. A. Kalinin
Labaratory of Theoretical Physics
0. Yu. Shmidt Institute of Physics of the Earth
Academy of Seiences of the USSR
Tranalsted from Russian by
Albin Tybulewicz
Editor, Soviet Physics-Semiconductors
SPRINGER SCIENCE+BUSINESS MEDIA, LLC 1971
The original Russian text, published by Nauka Press in Moscow in 1968 for
the 0, Yu. Shmidt Institute of Physics of the Earth of the Academy of Seien•
ces of the USSR, has been corrected by the authors for the present edition.
The English translation is published under an agreement with Mezhdunarod·
naya Kniga, the Soviet book export agency.
B. H. JKap~;oe, B. A. Ka.~unua
'Y paBHeHBß COCTOßHHß TBep;tLIX TeJI
DpH BLICORHX ;taBJieHHßX H TeMnepaTypax
Library of Congress Catalog Card Number 77·120027
ISBN 978-1-4757-1519-4 ISBN 978-1-4757-1517-0 (eBook)
DOI 10.1007/978-1-4757-1517-0
© 1971 Springer Science+Business Media New Y ork
Originally published by Plenum Publishing Corporation, N ew York in 1971
All riehts reservPrl
No part of this publication may be reproduced in any
form without written permission from the publisher
PREFACE TO THE AMERICAN EDITION
We started our work on theoretical methods in the phys ics of high pressures (in connec
tion with geophysical applications) in 1956, and we immediately encountered many problems.
Naturally, we searched the published Iiterature for solutions to these problems but whenever
we failed to find a solution or when the solution did not satisfy us, we attempted to solve the
problern ourselves. We realized that other investigators working in the physics of high pres
sures would probably encounter the same problems and doubts. Therefore, we decided to write
this book in order to save our colleagues time and effort. Apart from the descriptions of ex
perimental methods, the book deals only with those problems which we encountered in our own
work.
Allproblems in high-pressure physics have, at present, only approximate solutions,
which are very rough. Therefore, it is not surprising that different investigators approach the
same problems in different ways. Our approach does not prejudge the issue and we are fully
aware that there are other points of view. Our aim was always to solve a glven problern on a
physical basis. For example, the concept of the Grüneisenparameter needs further develop
ment but it is based on reliable physical ideas. On the other hand, Simon's equation for the
melting curve has, in our opinion, no clear physical basis and is purely empirical. Equations
of this type are useful in systematic presentation of the experimental material but they are un
suitable for any major extrapolation.
A couple of years have passed since the manuscript of this book was sent to press.
Since then, additional experimental data have become available. A high proportion of these
data can be analyzed by the methods described in the present book. However, their incorpora
tion would not improve the book very greatly. One of the most pressing needs in high-pressure
physics is to improve the accuracy of the measurements. When this barrier is surmounted,
we shall then see which methods for the analysis of the experimental data should be refined
and which should be replaced.
We are pleased to hear that our book will be published in the home country of P. W.
Bridgman, and we are grateful to Plenum Publishing Corporation for arranging the translation.
Moscow, November 1969 V. N. Zharkov
V. A. Kalinin
V
PREFACE
The theoretical methods used in the high-pressure physics of solids are the subject of
the present book. The development of these methods has been stimulated primarily by geo
physics problems to which such methods have been applied in the first instance. Special atten
tion is paid in the book to the equation of state, i.e., to the dependence of the pressure P on the
volume V and the temperature T.
The equation of state is the basic relationship in high-pressure physics. It contains
valuable information on the properties of a medium and it allows us to carry out transforma
tions from one thermodynamic potential to another. Moreover, investigations of a certain prop
erty of a solid can be used to find the theoretical dependence of this property on the tempera
ture and the lattice constant. The variables normally employed are P and T. The equation of
state makes it possible to go over from the "theoretical" variables V and T to P and T.
The range of high pressures and temperatures considered in' the present book is deter
mined by the P-T conditions in the interior of the earth (P ::s 3.5 · 106 bar, T ::s 6000°K). * Un
fortunately, such pressures and temperatures are outside the present capabilities of theoretical
physics, i.e., practically none of the high-pressure and high-temperature problems can be
solved in a systematic manner by the methods of modern theoretical physics. Consequently,
we shall have to use semi-empirical methods. In other words, we shall first establish a func
tional dependence of a given quantity on the temperature and volume, and ~en we shall use the
experimental data to determine the numerical parameters in this dependence. This approach
is known as the method of potentials.
The method of potentials permits us to write the equation of state in the analytic form,
which has definite advantages in the derivation of the equation and in its applications. This
method is used to obtain equations of state for all those substances for which the necessary ex
perimental data are available.
A detailed list of the topics dealt with in this book is given in the contents.
*1 bar= 106 dyn/cm2 = 1.01972 kgf/cm2 = 0.986924 atm. The kilobar (1 kbar = 103 bar) and
the megabar (1 Mbar = 106 bar) are usually employed in high-pressure physics.
vii
CONTENTS
CHAPTER 1. Equation of State at T = O.
Method of Potentials • • • • . . • • . . . • . • • • • • . • • • • . . • • • . • 1
1.1. Introduction ............................................ ~~ 1
1.2. Forces in Solids ........................ 2
Cll • • • • • • • • • • • • • • • •
1.3. Interaction Potentials in Various Types of Solid • . • • • • • • • • • • • • • • . • • • 4
1.4. Gontribution of Zero-Point Vibrations to the Equation of State • • • • • . • . • • 8
1.5. Murnaghan- Birch Equation of State . • . • • . . • • • • • . . • • • • • • • • • • . • • 9
1.6. Other Types of Potential • • • • • . • • • • • • • • • • • . . • • . • • • • • • • . • . • • . 12
Literature Cited .•••. • • . • • • • • • • • • . • • • • . . . • . • • . . • • • • . • . • • • • • • 18
CHAPTER2. Equation of State at T ;o< O.
G r ü n e i s e n P a r a me t e r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1. Introduction 20
2.2. Free Energy of a Crystal Lattice in the Quasiharmonie Approximation •••• 20
2.3. Low Temperatures •.•••.•••..•••..•..•....•.•.•.••••..••• 21
2.4. High Temperatures ...................................... . 23
2.5. Debye and Einstein Approximations •.••••••••••.••••••••••••••. 24
2.6. Phonons ............................................. . 26
2. 7. Approximate Methods for the Determination of the Grüneisen Parameter ..• 28
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
CHAPTER3. High-Temperature Corrections to Equation of State.. 32
3 .1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2
3 .2. Azilia.I"monicity . . . . . . . . . . • . . . . . . . . . . . . . . . . .. . . . • . . . . . . . . • . 33
3.3. Thermally Excited Conduction Electrons in Metals • • • • • • • • • • • • • • • • • • 35
3.4. Phenomenological Theory of the Equation of State Aceurate to within T2• • • • 41
3 .5. Thermal Defects. • . • • • • • . • • • • • . . • • • • . • . • • • • . • • • • • . • • • • • • • 45
3.6. Electrons and Holes in Insulators and Semiconductors. • • • • . • • • • . . . . • • 60
3. 7. Melting Curves. • . . • • • . . • . • • • . • • . • • . • • • • • • . • • • • • • . • . • . • • • 73
Literature Cited ................................ ~~ . ~~ . . . . . . . . . 84
CHAPTER 4. E qua t i o n s o f S tat e D e du c e d f r o m E x p er im e n t a 1 Da t a • • 86
4.1. High-Pressure Techniques . • • • • • • • • . . • . . • . • . • • • • • • • • • . . • • • • • 86
4.2. Review of Experimental Data • • • • • • • . • • . • • • • • • • • • • • • • • • • • • • • • 90
4.3. Determination of Equations of State from Experimental Data • • • . • • . • • • • 91
Literature C ited • • . • • • • • • . • • • • . . • . • . • • . • . . . • . • . • • • . • • • • • . . • • 104
CHAPTER 5. Compression of Solids by Strong Shock Waves •••.••. 106
5.1. lrltroduction ......................................... ~~ . 106
5.2. Hydrodynamic Relationships • • • • • • • • . • • . • • • • • • • • • • • • . • • • • • • • • 107
5.3. Stability of Shock Waves. Two-Wave Structures . • • • • • • . • • • • • • . . • . • 123
ix
CONTENTS
X
5.4. Hugoniot Curve and Method of Potentials. • • • • • • • • • • • . • • • • . • • • • • • . 129
5.5. Experimental Methods for the Determination of Shock-Wave Parameters • • . 143
5.6. Determination of the Equation of State from Dynamic Data . . • . • • • • • • . • • 151
5. 7. Equations of State from Static and Dynamic Data . . • • • • . . • • • . . . • • • • • 165
Literature Cited • . • • • • • • . • • . . . . . • • • • • . . . • • • • • • • . • • • • . . • • • • • • 176
CHAPTER 6. Influence of Pressure on the Anharmonicity
and the Elastic Moduli, and Their Temperature
Dependences for Crystals with Pair Interactions..... 178
6.1. Introduction ....................................... 178
8 •• 111 •
6.2. General Relationships . • • • • • . • • • • . • • • • . • • • • • • • • • • • • . . . • • • • • 179
6.3. van der Waals Crystals . • • • • • • • . . • . • • . . . • • • • . • • • • • . . • . . • . . • 186
6.4. Ionic Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
Literature C ited . • . • • • • • • . • . • • . • • • • • • . • • • • • • • • • • • • • • . • • • . • • • 2 06
CHAPTER 7. Applications of Equations of State to the
Physics of the Earth's Interior ••.•..•.•••••••••.•• 207
7 .1. Interna! Struc ture of the E arth • . . • • • • . . • . • • . • • • • • • • • • • • . • . • . • 2 07
7.2. Determination of the Composition of Layer D in the Earth's Mantle . • • • • . • 217
7.3. Determination of the Composition of the Earth 's Core • • • • • • • • • • • • • • • 220
7.4. Distribution of the Electrical Conductivity in the Earth's Mantle • • • • • • • . • 225
Li teratt.Ire Ci ted . . . . . . . . . . . . . . . . • . • . . . . • . . . . . . . . . . . . . . . . . . . • 234
APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
A. Relationships between Thermodynamic Quantities • • • . • • • • • • • • • • • . • • • 236
B. Debye Approximation • • • • • • . • • • • . . • • • . • • • • • • . • • • • • . . • • • • • • • 241
B.1. Calculation of the Debye Function • • • • • • . • • • • • • • • • • • • • • • • • . • 241
B.2. Thermodynamic Functions in the Debye Approximation • • • • • • • • • • . • 242
c.
Ideal Gas . . . . . . . . . . . . . . . . . . . • . • . . . . . . . . 243
II • • • • • • • • e • • • • • •
C.1. Boltzmann Ideal Gas • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • . • 243
C.2. Ideal Gas with Constant Specific Heats • • • • • • • • • • . • • • • • • • • . • • • 245
C.3. Degenerate E lectron Gas at T = 0 • • • • • • • • . • • • • • . • • • • • • • . . • • 246
C.4. Degenerate Electron Gas at T ~ 0 • • . • • • • . • • • • • . • • . • • • • . . • . • 247
C.5. Photon Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 252
D. Calculation of Parameters in the Potential of a Solid . • . • • • • • • • • • • • • • • 254
D.l. Isotherm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
D.2. Hugoniot Curve • • • • • . • • • • • • . •. • • • • • • • • . • • • . • • • • • • • . . . • 256
D.3. Grüneisen Parameter • • • • • • • • • • • • • • • • . • • • • • . . • • • • • • . • . . 2 57
CHAPTER 1
EQUATION OF STATE AT T = 0.
METHOD OF POTENTIALS
1.1. Introduction
The solid state of matter differs from the gaseous and liquid states by a more compact
and symmetrical distribution of atoms. The distance between the atoms in a solid is of the
order of the atomic diameters and considerable forces act within a solid. At the very short in
teratomic distances found in solids, we must take into account not only the interaction of the
nucleus and electrons within each atom but also the interaction between the nuclei and electrons
of different atoms. Asolid is a very complex quantum-mechanical system and, therefore, no
systematic and consistent theoretical calculations can be carried out without some approxima
tions. It is usual to consider models whieh differ from the real system by certain restrictions
that are imposed on them. These restrictions can differ from problern to problern but they
should apply, whenever possible, to the secondary processes which do not alter basically the
main properties of the solid in question.
One such restriction is the adiabatic approximation, which allows us to consider separate
ly the motion of electrons and nuclei in a solid [1]. This approximation is based on the ob
servation that nuclei are much heavier than electrons and, consequently, the average velocity
of nuclei is much less than the average velocity of electrons. Thus, a solid can be divided into
two subsystems: electron and nuclear. When the electron subsystem is considered, it may be
assumed that the nuclei are at rest although they need not occupy crystallattice sites. If we
take R to denote the vectors R 1, R2, ••• , Rn, which describe the positions of n nuclei, we
can use the solution of Schrödinger's equation, corresponding to the motion of N electrons in
the field of n nuclei with the configuration R, to obtain a wave function which contains the vec
tors R as parameters. The energy eigenvalues also depend on the parameters R. In particu
lar, this method can be used to determine the energy of the gro'\lnd state of the electron subsys
tem, E0(R). When we consider the motion of the nuclei, we can assume that the state of the
electron subsystem at any moment is described by the same wave function in which the posi
tions of the nuclei at that moment are represented by R.
The adiabatic approximation is satisfied well by filled electron shells but it is inapplic
able to conduction electrons in metals. The separation of neighboring energy levels in the con
duction band of a meta! is less than the excitation due to the motion of nuclei and this motion
alters the wave functions of the conduction electrons.
We shall consider a crystalline solid at T = 0 and we shall ignore, for the time being, the
motion of the nuclei due to zero-point vibrations of the crystal. We shall assume that the nu
clei are at rest and are located at crystal lattice sites. Under these conditions, the equation of
state for a solid can be obtained if the model of the solid satisfies only two basic conditions:
the model must describe correctly the attractive forces which arise when nuclei move away
from one another, and the repulsive forces when nuclei approach one another.
1
2 EQUATION OF STATE AT T = 0. METROD OF POTENTIALS [CHAP. 1
The binding energy of the atoms in a crystal is of the same order of magnitude as the
binding energy of the valence electrons in an atom but it is less than the binding energy of the
inner-shell electrons. Hence, we may conclude that the forces in a solid are primarily due to
the interactions of the outer electrons and that the inner electron shells are practically unaf
fected by the various processes in a solid, This is also supported by the evidence that the x
ray spectra, corresponding to the transitions of electrons between deep energy levels, are
practically identical for free atoms and for atoms in a solid, Therefore, instead of consider
ing a system comprising all nuclei and electrons, we may consider, in approximation, two sub
systems: one cons isting of ions (formed from nuclei) and inner electrons, and the other con
sisting of outer electrons alone.
1.2. Forces in Solids
The forces in solids are basically electrostatic but their actual nature depends on the
distribution of electrons between the various structure elements forming a crystal. By struc
ture elements, we mean the particles into which atoms and molecules are converted when they
form a solid,
These structure elements include positive ions, each consisting of a nucleus and its as
sociated inner-shell electrons. These ions are formed, for example, from metal atoms which
lose their valence electrons quite easily, The energy required for the detachment of an elec
tron from an atom, and the consequent formation of a positive ion, is known as the ionization
energy. Another example of a structure element is a negative ion, formed by the capture of
excess electrons by an atom. Typical of the elements which form negative ions are halogens
whose atoms capture excess electrons and assume the electron structure of inert-gas atoms.
The energy evolved in the attachment of an electron to a neutral atom is known as the electron
affinity. In inert-gas crystals, the structure elements are the atoms themselves since their
electron structure is practically the same in the gaseous and crystalline states. Finally, the
free-electron gas in the metal and the valence electrons participating in the formation of co
valent bonds can also be regarded as structure elements,
The forces in a crystal can be regarded as acting between the structure elements. Let
us now consider these forces in detail.
Coulomb Forces
The Coulomb forces can act only between charged structure elements. Such elements
are positive ions, negative ions, and free electrons.
The Coulomb interaction of ions needs no further comment because the charge distribu
tion in an ion is approximately spherical. We may assume that each ion has a radius outside
which the electron density is negligibly small. Therefore, the interaction between ions is
equivalent to the interaction between point charges. The Coulomb interaction between positive
ions and a free-electron gas can be analyzed using the observation that the density of free (con
duction) electrons is almost constant everywhere except in the immediate neighborhood of nu
clei, The energy of this interaction is equal to the electrostatic energy of a system comprising
positive point charges (which represent ions) and a uniformly distributed negative charge (rep-
resenting the free-electron gas). ·
If r represents some characteristic dimension of the crystallattice (for example, the
distance between nearest neighbors), the dependence of the Coulomb energy on this dimension
is of the very simple type
c Ir, (1.1)
where c is a constant independent of r.