Table Of ContentQuantum Statistics of Charged
Particle Systems
Quantum Statistics
of Charged
Particle Systems
by
Wolf-Dietrich Kraeft
Sektion PhY8ikjElektronik, Ern8t-Mo ritz-Arndt-Univer8itat, Greifswald
German Democratic Republic
Dietrich Kremp
Sektion PhY8ik, Wilhelm-Pieck-Univer8itat, R08tock
German Democratic Republic '
Werner- Ebeling
Sektion PhY8ik, Humboldt-Univer8itat zu Berlin
German Democratic Republic
Gerd Ropke
Sektion PhY8ik, Wilhelm-Pieck-Univer8itat, R08tock
German Democratic Republic
PLENUM PRESS· NEW YORK AND LONDON
ISBN-13: 978-1-4612-9273-9 e-ISBN-13: 978-1-4613-2159-0
DOl: 10.1007/978-1-4613-2159-0
© Akademie-Verlag Berlin 1986
Softcover reprint of the hardcover 1s t edition 1986
Plenum Publishing Corporation, 233 Spring Street, New York, NY 10013, USA
All rights reserved
No part of this book may be reproduced, stored in a retrieval system, or transmitted,
in any form or by any means, electronic, mechanical, photocopying, microfilming, record
ing, or otherwise, without permission from the Publisher
Preface
The year 1985 represents a special anniversary for people
dealing with Ooulomb systems. 200 years ago, in 1785, Oharles
Auguste de Ooulomb (1736-1806) found "Ooulomb's law"
for the interaction force between charged particles. The authors
want to dedicate this book to the honour of the great pioneer
of electrophysics.
Recent statistical mechanics is mainly restricted to systems of neutral particles.
Except for a few monographs and survey articles (see, e. g., IOHIMARU, 1973, 1982;
KUDRIN, 1974; KLIMONTOVIOH, 1975; EBELING, KRAEFT and KREMP, 1976, 1979;
KALMAN and CARINI, 1978; BAUS and HANSEN, 1980; GILL, 1981, VELO and WIGHT
MAN, 1981; MATSUBARA, 1982) the extended material on charged particle systems,
which is now available thanks to the efforts of many workers in statistical mechanics,
is widely dispersed in many original articles.
It is the aim of this monograph to represent at least some part of the known results
on charged particle systems from a unified point of view. Here the method of Green's
functions turns out to be a powerful method especially to overcome the difficulties
connected with the statistical physics of charged particle systems; some of them are
. mentioned in the introduction. Here we can point, e.g., to the appearance of bound
states in a medium and their role as new entities.
The presentation begins (in the second Chapter) with the basic physical ideas and
a short survey of the exact results known for quantum Coulomb systems. The authors
- being no experts in the field of mathematical physics - be!ieve that the exact
results obtained by DYSON.and LENARD (1967, 1968), LEBOWITZ and LIEB (1969, 1972),
LIEB and THIRRING (1975) and many other authors (see THIRRING, 1980; VELO and
WIGHTMAN, 1981) are of fundamental importance and should be explained there
fore - without proofs - at the very beginning. The third Chapter is devoted to the
systematic representation of the main quantum statistical methods used in this book.
Chapter 4. covers the entire density-temperature plane by using the apparatus of
Green's functions. There especially the single- and two-particle properties as well as
the dielectric behaviour of charged particle systems are dealt with. Chapter 5. gives
some information about the classical case and is further devoted to the treatment
of nearly-classical (non-degenerate) plasmas. The results obtained so far are used in
the sixth Chapter in order to give thermodynamic functions in wide density-tempera
ture regions. Chapters 7. and 8. are devoted to transport and optical properties,
respectively.
Finally several restrictions have to be explained. Apart from a few remarks, the
material presented here is restricted to the three-dimensional case. Recently, the
interest in two-dimensional and in one-dimensional conductors is increasing rapidly
due to their unusual properties and to possible technological applications (GINTSBURG,
1971, 1981; ALASTUEY and JANOOVIOI, 1979, 1980, 1981; ALASTUEY, 1980, 1982;
VI Pre/ace
WILSON et al., 1980, 1981; AOKI and ANDO, 1981; WILLIAMS, 1982; GALLET et al.,
1982).
A comprehensive treatment of this quickly developing field is outside the scope
of this book, the reader is refered to the original work cited at several places through
out the volume. Such references to topics beyond the scope of this monograph are
added in smal1 print.
Other problems, also related to the Coulombic interaction, which we omitted from
this discussion, are treated in the same manner. We may mention here inhomogeneous
systems and band structure calculations (see, e.g., BRAUER, 1972, STOLZ, 1974),
properties of matter at extreme conditions (see, e.g., KrnzHNITs et aI., 1975), order
disorder transitions and delocalization effects (see, e.g., BONCH-BRUEVICH et al., 1981),
computer simulations (see, e.g., ZAMALIN, NORMAN, and FILINOV, 1977), superconduc
tivity (see, e.g., TINKHAM, 1975; LIFSHITS and PITAEVSKII, 1980), and far-from-equi
librium effects (see, e.g., EBELING and FEISTEL, 1982) .
The authors hope to contribute to the theory of Coulomb systems, a vastly growing
field of statistical physics, which is of high interest for science and technology. In
order to help graduate students and scientific workers to become familiar with this
modern area of physics the authors decided to include a part where the fundamentals
of the theory are presented in the manner of a textbook; this refers especially to parts
of Chapters 2. -4.
In conclusion, the authors would like to express their sincere thanks to Klaus
Kilimann, Hubertus Stolz, and Roland Zimmermann, who participated substantially
in the development of the results presented in this book. They contributed especially
to Chapters 4. and 6. -8. The authors would like to thank further J. Bliimlein,
F. E. Hohne, M. Luft, C.-V. Meister, T. Meyer, 1. Orgzall, R. Redmer, W. Richert,
T. Rother, M. Schlanges, T. Seifert, W. Stolzmann, and H. Wegener for invaluable
help and cooperation. Moreover, we express our gratitude for discussions and for pro
viding us with material to H. Bottger, V. E. Fortov, B. Jancovici, G. Kalman,
G. Kelbg, Yu. L. Klimontovich, K. Suchy, 1. T. Yakubov, and D. N. Zubarev.
H. Stolz and H. Bottger read the entire manuscript and made many useful com
ments and suggestions. Our thanks are moreover due to Lindsay Mann, Edinburgh,
who checked the English version of the manuscript. For technical assistance in the
preparation of the manuscript we are grateful to H. Bahlo, C. Berndt, R. Nareyka,
D. Rosengarten, and E. Wendt. Finally, our cordial thanks are due to Renate
Trautmann, Akademie-Verlag Berlin, for excellent editorial cooperation.
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1
2. Physical Concepts and Exact Results . . . . . . . . . . . . . . . 4
2.I. Basic Concepts for Coulomb Systems. . . . . . . . . . . . . . 4
2.2. Survey of Exact Quantum-Mechanical Results for Coulomb Systems 12
2.3. Survey of Exact Quantum-Statistical Results for Macroscopic Coulomb
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 16
3. Quantum Statistics of Many-Particle Systems . . 22
3.1. Elements of Quantum Statistics. . . . . '. . 22
3.1.I. Quantum Mechanics of Many-Particle Systems. 22
3.1.2. The Method of Second Quantization. . . . . . 24
3.1.3. Quantum Statistics. Density Operator. . . . . 26
3.1.4. Reduced Density Operators. Bogolyubov Hierarchy. 29
3.1.5. The Classical Limit, BBG:KY Hierarchy. . . . . . 31
3.1.6. Systems in Thermodynamical Equilibrium. . . . . 32
3.2. The Method of Green's Functions in Quantum Statistics. 35
3.2.1. Definition of Green's Functions . . . . . . . . . . . 35
3.2.2. General Properties of the Correlation Function and One-Particle Green's
Function . _ . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.3. Long Time Behaviour of Correlation Functions. . . .'. . . . . . . . . 43
3.2.4. Equation of Motion for the One-Particle Green's Function. Self Energy .. ' 45
3.2.5. Dynamical and Thermodynamical Information Contained in the Spectral
Function A(p, w) . . . . . . . . . . . . . . . . . 49
3.2.6. The Two-Particle Green's Function. . . . . . . . . . . . 52
3.2.7. Equation of Motion for Higher Order Green's Functions. . . 55
3.2.8. The Binary Collision Approximation (Ladder Approximation) 61
3.2.9. T-Matrix and Thermodynamic Properties in Binary Collision Approximation 66
3.3. Quantum Statistics of Charged Many-Particle Systems 71
3.3.1. Basic Equations. Screening . . . . . . . 71
3.3.2. Analytic Properties of Vs and 17. . . . . . 75
3.3.3. The "Random Phase Approximation'} RPA 77
4. Application of the Green's Function Technique to Coulomb Systems 80
4.1. Types of Different Approximations. . . . . . . . . . 80
4.1.1. Diagram Representation of 1: and 17. . . . . . . . . 80
4.1.2. The RPA and the VB-Approximation for the Self Energy. 82
4.1.3. Many-Particle Complexes and T-Matrices. . . . . . . 84
viii Content8
4.1.4. Cluster Formation and the Chemical Picture ............ 85
4.1.5. Cluster Decomposition of the Self Energy.. ............ 86
4.2. Dielectric Properties of Charged Particle Systems. Random Phase Approxi-
mation . . . . . . . . . . . . . . . . . . . . . . . . 87
4.2.1. Linear Response to External Perturbations. General Remarks 87
4.2.2. Properties of the RPA Dielectric Function . 94
4.2.3. Plasma Oscillations (Plasmons) 101
4.3. Single-Particle Excitations . . . HI
4.3.1. Quasi-Particle Concept . . . . . HI
4.3.2. Self Energy in VS-Approximation . 113
4.4. Two-Particle Properties in a Plasma H8
4.4.1. Bethe-Salpeter Equation for a Two-Particle Cluster. H8
4.4.2. Solution of the Bethe-Salpeter Equation. Effective Wave Equation and
Spectral Representations . . . . . . . . . . . . . . . . . . .. 121
4.4.3. Two-Particle States in the Dynamically Screened Ladder Approximation. 124
4.4.4. Two-Particle States in Surrounding Medium in First Born Approximation 126
4.4.5. Numerical Results and Discussion of the Two-Particle States. . . 130
4.5. Dielectric Function Including Bound States. . . . . . . . . . 137
4.5.1. Extended RPA Dielectric Function for a Partially Ionized Plasma. 137
4.5.2. Limiting Behaviour of the Extended RPA Dielectric Function. . 141
4.5.3. Self Energy and Vertex Corrections.to the Extended RPA Dielectric Func-
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
4.5.4. Local Field Effects and Enhancement of the Dielectric Function. . . . . 146
6. Equilibrium Properties in Classical and Quasiclassical Approximation. 150
5.1. The One-Component Plasma Model. . . . . 150
5.2. Many-Component Systems. Slater Sums. . . 154
5.2.1. Partition Functions and Effective Potentials. 154
5.2.2. Calculation of Slater Sums and Effective Potentials. 157
5.3. The Pair Distribution Function . 161
5.3.1. Basic Equations and Hierarchy . 161
5.3.2. Discussion of the Pair Distribution 163
5.4. Thermodynamic Functions . . . 165
5.4.1. Cluster Expansions of the Free Energy 165
5.4.2. Density Expansions of the Free Energy. 166
6. Quantum.Statistical Calculations of Equilibrium Properties . 170
6.1. Equation of State in the Screened Ladder Approximation 170
6.1.1. The Second Virial Coefficient . . . . . . . . . . . . 170
6.1.2. Evaluation of the Higher Order Contributions. . . . . 172
6.1.3. Evaluation of the Hartree-Fock and the Montroll-Ward Contributions. 176
6.2. Density and Chemical Potential in the. Screened Ladder Approximati~n . 184
6.2.1. Bound State and Quasiparticle Contributions. 184
6.2.2. The Mass Action Law. . . . . . . . . . . 188
6.3. One-Component Plasmas . . . . . . . . . . 190
6.3.1. Analytical Formulae for the Limiting Situations 190
6.3.2. Pade Interpolations between the Degenerate and the Nondegenerate Cases 193
6.3.3. Pade Approximations Including Higher Order Interaction Terms and Wigner
Crystallization ........... 199
6.4. Electron-Hole Plasmas . . . . . . . . 203
6.4.1. Analytical Results for the Plasma Model. 203
6.4.2. Pade Approximations . . . . . . . . 206
Contents ix
6.4.3. Ionization Equilibrium 208
6.5. Hydrogen Plasmas. . 210
6.5.1. The Two-Fluid Model 210
6.5.2. Basic Formulae for the Limiting Situations and Pade Approximations. 212
6.5.3. Ionization Equilibrium and Phase Diagram 216
6.6. Alkali Plasmas and Noble Gas Plasmas. . . . . 221
6.6.1. Pseudopotentials .... . . . . . . . . . . 221
6.6.2. The Chemical Potential of the Neutral Component. 222
6.6.3. The Chemical Potential of the Charged Component 224
6.6.4. Saha Equation and Ionization Equilibrium 225
7. Transport Properties. . . . . . . . . . . . . . . . . . . . . . 232
7.1. Linear Response Theory . . . . . . . . . . . . . . . . . . . 232
7.1.1. Many-Body Effects and Transport Properties in Non-Ideal Plasmas. 232
7.1.2. Transport Coefficients and Correlation Functions. . . . 235
7.1.3. Further Approaches . . . . . . . . . . . . . . . . 239
7.2. Evaluation of Collision Integrals Using Green's Functions 240
7.2.1. Green's Functions, Diagrams and Correlation Functions . 240
7.2.2. Evaluation of Correlation Functions in First Born Approximation 242
7.2.3. Results for a Hydrogen Plasma . . . . . . . . . 244
7.2.4. Inclusion of the Ionic Structure Factor. . . . . . . 249
7.2.5. Dynamically Screened Second Born Approximation. . 251
7.2.6. Statically Screened T-Matrix Approximation. Results. 254
7.3. Further Improvements of the Transport Theory . . 258
7.3.1. Self-Energy and Debye·Onsager Relaxation Effects. 258
7.3.2. Hopping Conductivity 259
7.3.3. Concluding Remarks . . . . . . . . . . . 261
S. Green's Function Approach to Optical Properties 264
8.1. General Formalism. . . . . . . . . . . . 264
8.1.1. Many-Body Theory of Absorption Spectra. . 264
8.1.2. Dielectric Function and Spectral Line Shape of Plasmas . 265
8.1.3. Doppler Broadening . . . . . . . . . . . 268
8.2. Evaluation of Line Shift and Broadening. . 268
8.2.l. Explicit Expressions for Shift and Broadening 268
8.2.2. Relation to the Impact Approximation. . . 272
8.2.3. Shift of Spectral Lines in Dense Hydrogen Plasmas 276
8.2.4. Estimation of the Shift and Broadening of Spectral Lines for an Argon
Plasma ............... . 279
8.3. Further Approaches and Concluding Remarks 281
9. References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
10. Subject Imlex . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
1. Introduction
Matter consists of charged particles, the negative electrons and the positive nuclei;
this fundamental discovery, due to Rutherford, belongs to the mile-stones in the
history of physics. The famous communication of Rutherford "Scattering of iX- and fJ
rays and the structure of atoms" was presented on April 7, 1911 to the Philosophical
Society in Manchester.
Using the simple picture that matter consists of nonrelativistic point charges, the
founders of quantum mechanics and quantum statistics developed the idea that
quantum physics is able now to calculate most pr.operties of particl,e systems from
first principles. Having in mind the properties of atoms and molecules DIRAC wrote
in 1929: "The underlying physical laws necessary for the mathematical theory of a
larger part of physics and the whole of, chemistry are thus completely known, and
the difficulty is only that the application of these laws leads to equations much too
complicated to be soluble". The optimistic program of the pioneers of quantum physics
was not easy to realize since enormous mathematical difficulties had to be overcome:
As a result of their great efforts, we now have a well-developed quantum chemistry
which is able to calculate even the properties of quite complicated molecules, al
though of course there are still many open problems (LUDWIG, 1980; PRIMAS, 1981).
Furthermore, we have at least the fundamentals of a quantum statistics of macro
scopic matter consisting of very many nuclei and electrons. Here we are concerned
with the latter subject only, the theory of macroscopic charged particle systems.
Of course, a theory of macroscopic matter based on first principles only is of funda
mental interest for our understanding of the properties of matter. In order to underline
the importance of the quantum statistics of Coulomb systems let us quote LEBOWITZ
(1980): "In some sense, all of statistical mechanics, which is the microscopic theory of
macroscopic matter, deals with Coulomb systems. The properties of the materials we
see and touch ar~ almost entirely determined by the nature of the Coulomb force a"
it manifests itself in the collective behaviour of interacting electrons and nuclei.
In most applications of statistical mechanics however this fact is not explicit at all.
One starts with an effective short range microscopic Hamiltonian appropriate to the
problem at hand." For example, in order to discuss the properties of a real gas we
describe the system as a collection of neutral molecules interacting via Lennard-Jonet
potentials. There are two reasons for considering explicitly systems with a starting
microscopic Hamiltonian with Coulomb interactions. The first is primarily a theoreti
cal one; we woUld like to understand how the Coulomb forces give rise to the effective
interactions, which appear in the usual statistical mechanics, and we would like tc
2 1. I ntroducti,on
have a theory which is based on first principles only. The second reason.is more
practical. There are many systems, e.g., plasmas, me~als, semiconductors, electrolytes,
molten salts, ionic crystals, etc., where bare Coulomb interactions playa very important
role. In this way the quantum statistics of Coulomb systems is able to predict the
thermodynamical, optical, and transport properties for systems of high practical im
portance, e.g., states of dense matter with very great energy density. Such states are
of great importance for many modern technical devices such as nuclear fusion plants,
gas-phase fission reactors, plasma-chemical devices, etc.
In this way it is obvious that the development of a consistent statistical mechanics
of systems of point charges is of principal and practical interest, and it is of special
importance to deal with the specific Coulomb difficulties, such as the instability
(unboundedness of the energy) of a classical system of point charges; the long range
character of the Coulomb forces; and the divergence of the atomic partition function.
One of the most elegant and powerful techniques to overcome such difficulties is
the method of quantumstatistical thermodynamic Green's functions (MARTIN and
SCHWINGER, 1959; BONCH-BRUEVICH and TYABLIKOW, 1961; KADANOFF and BAYM,
1962; ABRIKOSOV, GOR'KOV and DZYALOSHINSKII, 1962; FETTER and W ALECKA, 1971).
The main part of this book is based on this technique. The great advantage of such
an approach is that it covers, at least in principle, the entire region of density and
temperature. Other methods describe - as a rule - only a certain part of the
density temperature plane.
There is for instance, on the one hand, the well developed theory of highly degener
ate electron liquids (see, e.g., ABRIKOSOV, GOR'KOV and DZYALOSHINSKII, 1962; KA
DANOFF and BAYM, 1962; PINES, 1962; NOZIERES, 1966; GLICKSMAN, 1971; STOLZ,
1974; ZIMAN, 1974; LIFSHITS et aI., 1975; ABRIKOSOV, 1976; RICE et aI., 1977; HERR
MANN and PREPPERNAU, 1979). On the other hand, there exist theories of non-degen
erate plasmas (BALESCU, 1963; KLIMONTOVICH, 1964, 1975; SILIN, 1971; ECKER, 1972;
KADOMTSEV, 1976; GRYAZNOV et aI., 1980; KHRAPAK and YAKUBOV, 1981; ZHDANOV,
1982; EBELING et aI., 1983, 1984; GUNTHER and RADTKE, 1984).
A unified Green's functions approach covers, as already mentioned, the entire
density-temperature plane; of course, one still needs different approximation schemes
in different regions.
The program to be dealt with by the theory is the following: Starting from the
elementary units (electrons and nuclei) and their equation of motion (quantum
mechanics and Coulomb's interaction law), it is to obtain the properties of matter
under different external conditions (temperature and pressure). This includes the
derivation of plasma, gaseous, fluid or solid states under certain conditions and the
transition between them.
At the present stage of the theory such a wide program seems to be unrealistic,
of course; therefore we have to restrict ourselves here to certain aspects of the program.
Some of the material given in this book is new and is of interest for theoretical
and practical reasons.' Here we mention
(i) the quantum-statistical treatment of two-particle properties as, e.g., energy
levels and the life time of two-particle states in dependence on density and temperature
(ii) the foundation of the chemical picture (Le., bound states are treated as new
species) and its application in thermodynamics and transport phe~omena;
