com Progress in Mathematical Physics Volume 37 Editors-in-Chief Anne Boutet de Monvel, Université Paris VII Denis Diderot Gerald Kaiser, The Virginia Center for Signals and Waves Editorial Board D. Bao, University of Houston C. Berenstein, University of Maryland, College Park P. Blanchard, Universität Bielefeld A.
Fokas, Imperial College of Science, Technology and Medicine C. Tracy, University of California, Davis H. van den Berg, Wageningen University www. Uvarov Quantum-Statistical Models of Hot Dense Matter Methods for Computation Opacity and Equation of State Translated from the Russian by Andrei Iacob Birkhäuser Verlag Basel Boston Berlin www.com Authors: Arnold F.
Novikov Keldysh Institute of Applied Mathematics Miusskaya sq., 4 125047 Moscow Russia e-mail : arnold@kiam. ru e-mail : novikov@kiam.ru Originally published in Russian by Fizmatlit, Physics and Mathematics Publishers Company, Russian Academy of Sciences 2000 Mathematics Subject Classification 80-04, 81-08, 81V45, 82-08, 82D10 A CIP catalogue record for this book is available from the Library of Congress, Washington D., USA Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed biblio- graphic data is available in the Internet at <http://dnb. ISBN 3-7643-2183-0 Birkhäuser Verlag, Basel – Boston – Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, repro- duction on microfilms or in other ways, and storage in data banks.
For any kind of use whatsoever, permission from the copyright owner must be obtained. © 2005 Birkhäuser Verlag, P. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced of chlorine-free pulp. TCF ∞ Printed in Germany ISBN-10: 3-7643-2183-0 ISBN-13: 987-3-7643-2183-8 987654321 www.com Contents Preface xiii I Quantum-statistical self-consistent field models 1 1 The generalized Thomas-Fermi model 3 1.1 The Thomas-Fermi model for matter with given temperature and density .1 The Fermi-Dirac statistics for systems of interacting particles .2 Derivation of the Poisson-Fermi-Dirac equation for the atomic potential .3 Formulation of the boundary value problem .4 The Thomas-Fermi potential as a solution of the Poisson equation depending on only two variables .5 Basic properties of the Fermi-Dirac integrals .6 The uniform free-electron density model .7 The Thomas-Fermi model at temperature zero .2 Methods for the numerical integration of the Thomas-Fermi equation 16 1.1 The shooting method .2 Linearization of the equation and a difference scheme .3 Double-sweep method with iterations .3 The Thomas-Fermi model for mixtures .1 Setting up of the problem.
Thermodynamic equilibrium condition .2 Linearization of the system of equations .3 Iteration scheme and the double-sweep method .4 Discussion of computational results .com vi Contents 2 Electron wave functions in a given potential 29 2.1 Description of electron states in a spherical average atom cell .1 Classification of electron states within the average atom cell 30 2.2 Model of an atom with average occupation numbers .3 Derivation of the expression for the electron density by means of the semiclassical approximation for wave functions .4 Average degree of ionization .5 Corrections to the Thomas-Fermi model .2 Bound-state wave functions .1 Numerical methods for solving the Schrödinger equation .2 Hydrogen-like and semiclassical wave functions .3 Relativistic wave functions .3 Continuum wave functions .1 The Schrödinger equation .2 The Dirac equations. 61 3 Quantum-statistical self-consistent field models 65 3.1 Quantum-mechanical refinement of the generalized Thomas-Fermi model for bound electrons .1 The Hartree self-consistent field for an average atom .3 Analysis of computational results for iron .4 The relativistic Hartree model .2 The Hartree-Fock self-consistent field model for matter with given temperature and density .1 Variational principle based on the minimum condition for the grand thermodynamic potential .2 The self-consistent field equation in the Hartree-Fock approximation .3 The Hartree-Fock equations for a free ion .3 The modified Hartree-Fock-Slater model .1 Semiclassical approximation for the exchange interaction .2 The equations of the Hartree-Fock-Slater model .3 The equations of the Hartree-Fock-Slater model in the case when the semiclassical approximation is used for continuum electrons .4 The thermodynamic consistency condition .com Contents vii 4 The Hartree-Fock-Slater model for the average atom 107 4.1 The Hartree-Fock-Slater system of equations in a spherical cell .1 The Hartree-Fock-Slater field .2 Periodic boundary conditions in the average spherical cell approximation .3 The electron density and the atomic potential in the Hartree-Fock-Slater model with bands .4 The relativistic Hartree-Fock-Slater model .2 An iteration method for solving the Hartree-Fock-Slater system of equations .2 Computation of the band structure of the energy spectrum 118 4.4 The uniform-density approximation for free electrons in the case of a rarefied plasma .3 Solution of the Hartree-Fock-Slater system of equations for a mixture of elements .3 Examples of computations .4 Accounting for the individual states of ions .1 Density functional of the electron system with the individual states of ions accounted for .2 The Hartree-Fock-Slater equations of the ion method in the cell and plasma approximations .3 Wave functions and energy levels of ions in a plasma. 138 II Radiative and thermodynamical properties of high-temperature dense plasma 143 5 Interaction of radiation with matter 145 5.1 Radiative heat conductivity of plasma .1 The radiative transfer equation .2 The diffusion approximation .3 The Rosseland mean opacity .4 The Planck mean. Radiation of an optically thin layer .com viii Contents 5.2 Quantum-mechanical expressions for the effective photon absorption cross-sections .1 Absorption in spectral lines .5 The total absorption cross-section .3 Peculiarities of photon absorption in spectral lines .1 Probability distribution of excited ion states .2 Position of spectral lines .3 Atom wave functions and addition of momenta .4 Shape of spectral lines .2 Electron broadening in the impact approximation .3 The nondegenerate case .4 Accounting for degeneracy .5 Methods for calculating radiation and electron broadening .7 The Voigt profile .8 Line profiles of a hydrogen plasma in a strong magnetic field .5 Statistical method for line-group accounting .1 Shift and broadening parameters of spectral lines in plasma 220 5.2 Fluctuations of occupation numbers in a dense hot plasma .3 Statistical description of overlapping multiplets .4 Effective profile for a group of lines .5 Statistical description of the photoionization process .6 Computational results for Rosseland mean paths and spectral photon-absorption coefficients .1 Comparison of the statistical method with detailed computation .2 Dependence of the absorption coefficients on the element number, temperature and density of the plasma .3 Spectral absorption coefficients .4 Radiative and electron heat conductivity .5 Databases of atomic data and spectral photon absorption coefficients .com Contents ix 5.7 Absorption of photons in a plasma with nonequilibrium radiation field .1 Basic processes and relaxation times .2 Joint consideration of the processes of photon transport and level kinetics of electrons .3 Average-atom approximation .4 Rates of radiation and collision processes .5 Radiation properties of a plasma with nonequilibrium radiation field .6 Radiative heat conductivity of matter for large gradients of temperature and density.
280 6 The equation of state 285 6.1 Description of thermodynamics of matter based on quantum-statistical models .1 Formulas for the pressure, internal energy and entropy according to the Thomas-Fermi model .2 Quantum, exchange and oscillation corrections to the Thomas-Fermi model .2 The ionization equilibrium method .1 The Gibbs distribution for the atom cell .2 The Saha approximation .3 An iteration scheme for solving the system of equations of ionization equilibrium .3 Thermodynamic properties of matter in the Hartree-Fock-Slater model .1 Electron thermodynamic functions .2 Accounting for the thermal motion of ions in the charged hard-sphere approximation .3 Effective radius of the average ion .4 On methods for deriving wide-range equations of state .2 Cold compression curves .4 Comparison with the Saha model .5 Approximation of thermophysical-data tables .1 Construction of an approximating spline that preserves geometric properties of the initial function .com x Contents III A P P E N D I X Methods for solving the Schrödinger and Dirac equations 337 ANALYTIC METHODS 339 A.1 Quantum mechanical problems that can be solved analytically .1 Equations of hypergeometric type .2 Bound state wave functions and classical orthogonal polynomials .3 Solution of the Schrödinger equation in a central field .4 Radial part of the wave function in a Coulomb field .2 Solution of the Dirac equation for the Coulomb potential .1 The system of equations for the radial parts of the wave functions .2 Reduction of the system of equations for the radial functions to an equation of hypergeometric type .3 Equations of hypergeometric type for the bound states and their solution .4 Energy levels and radial functions .5 Connection with the nonrelativistic theory .3 The variational method and the method of the trial potential .1 Main features of the variational method .2 Calculation of hydrogen-like wave functions .3 Method of the trial potential for the Schrödinger and Dirac equations .4 The semiclassical approximation .1 Semiclassical approximation in the one-dimensional case .2 Application of the WKB method to an equation with singularity. Semiclassical approximation for a central field .3 The Bohr-Sommerfeld quantization rule .4 Using the semiclassical approximation to normalize the continuum wave functions .com Contents xi NUMERICAL METHODS 392 A.5 The phase method for calculating energy eigenvalues and wave functions .1 Equation for the phase and the connection with the semiclassical approximation .2 Construction of an iteration scheme for the calculation of eigenvalues .3 Difference schemes for calculating radial functions .4 The radial functions near zero and for large values of r .6 The phase method for the Dirac equation. 406 Bibliography 409 Index 427 www.com Preface In the processes studied in contemporary physics one encounters the most diverse conditions: temperatures ranging from absolute zero to those found in the cores of stars, and densities ranging from those of gases to densities tens of times larger than those of a solid body. Accordingly, the solution of many problems of modern physics requires an increasingly large volume of information about the properties of matter under various conditions, including extreme ones.
At the same time, there is a demand for an increasing accuracy of these data, due to the fact that the reliability and computational substantiation of many unique technological devices and physical installations depends on them. The relatively simple models ordinarily described in courses on theoretical physics are not applicable when we wish to describe the properties of matter in a sufficiently wide range of temperatures and densities. On the other hand, experi- ments aimed at generating data on properties of matter under extreme conditions usually face considerably technical difficulties and in a number of instances are exceedingly expensive. It is precisely for these reasons that it is important to de- velop and refine in a systematic manner quantum-statistical models and methods for calculating properties of matter, and to compare computational results with data acquired through observations and experiments.
At this time, the literature addressing these issues appears to be insufficient. If one is concerned with opacity, which determines the radiative heat conductivity of matter at high temperatures, then one can mention, for example, the books of D. Cowan [49], and also the relatively recently published book by D. There are also a number of papers and collections of short conference reports that analyze theoretical models in use and software packages [45, 205, 246, 240, 241].
Let us mention here one of the most perfected software programs, OPAL, and the astrophysical library of opacity coefficients (at the Livermore Na- tional Laboratory, USA) that is based on OPAL [92]. A large amount of work on improved models of matter and tables of thermophysical properties was carried out by the T4 group at the Los Alamos National Laboratory, USA. The results of www.com xiv Preface this work are systematized in the SESAME database [246].