MATHEMATICAL PHYSICS RECIPES WITH MAPLE
I Quantum-statistical self-consistent field models 1 1 The generalized Thomas-Fermi model 3 11 The Thomas-Fermi model for matter with given temperature and density 4 111 The Fermi-Dirac statistics for systems of interacting particles 4 112 Derivation of the Poisson-Fermi-Dirac equation for the atomic potential 7 113 Formulation of the boundary value problem 9 114 The Thomas-Fermi potential as a solution of the Poisson equation depending on only two variables 10 115 Basic properties of the Fermi-Dirac integrals 11 116 The uniform free-electron density model 13 117 The Thomas-Fermi model at temperature zero 15 12 Methods for the numerical integration of the Thomas-Fermi equation 16 121 The shootingmethod 16 122 Linearization of the equation and a difference scheme 19 123 Double-sweep method with iterations 20 13 The Thomas-Fermi model for mixtures 22 131 Setting up of the problem Thermodynamic equilibrium condition 22 132 Linearization of the systemof equations 23 133 Iteration scheme and the double-sweep method 24 134 Discussion of computational results 27 2 Electron wave functions in a given potential 29 21 Description of electron states in a spherical average atom cell 29 211 Classification of electron states within the average atom cell 30 212 Model of an atom with average occupation numbers 33 213 Derivation of the expression for the electron density by means of the semiclassical approximation for wave functions 35 214 Average degree of ionization 39 215 Corrections to the Thomas-Fermi model 41 22 Bound-state wave functions 42 221 Numerical methods for solving the Schr¨odinger equation 43 222 Hydrogen-like and semiclassical wave functions 43 223 Relativistic wave functions 50 23 Continuum wave functions 58 231 The Schr¨odinger equation 58 232 The Dirac equations 61 3 Quantum-statistical self-consistent field models 65 31 Quantum-mechanical refinement of the generalized Thomas-Fermi model for bound electrons 66 311 The Hartree self-consistent field for an average atom 66 312 Computational algorithm 68 313 Analysis of computational results for iron 72 314 The relativistic Hartreemodel 76 32 The Hartree-Fock self-consistent field model for matter with given temperature and density 80 321 Variational principle based on the minimum condition for the grand thermodynamic potential 80 322 The self-consistent field equation in the Hartree-Fock approximation 83 323 The Hartree-Fock equations for a free ion 86 33 Themodified Hartree-Fock-Slatermodel 92 331 Semiclassical approximation for the exchange interaction 92 332 The equations of the Hartree-Fock-Slatermodel 96 333 The equations of the Hartree-Fock-Slater model in the case when the semiclassical approximation is used for continuum electrons 99 334 The thermodynamic consistency condition 103 4 The Hartree-Fock-Slater model for the average atom 107 41 The Hartree-Fock-Slater system of equations in a spherical cell 107 411 The Hartree-Fock-Slater field 107 412 Periodic boundary conditions in the average spherical cell approximation 111 413 The electron density and the atomic potential in the Hartree-Fock-Slatermodel with bands 114 414 The relativistic Hartree-Fock-Slatermodel 115 42 An iteration method for solving the Hartree-Fock-Slater system of equations 117 421 Algorithm basics 117 422 Computation of the band structure of the energy spectrum 118 423 Computational results 120 424 The uniform-density approximation for free electrons in the case of a rarefied plasma 122 43 Solution of the Hartree-Fock-Slater system of equations for amixture of elements 123 431 Problem setting 123 432 Iteration scheme 125 433 Examples of computations 129 44 Accounting for the individual states of ions 131 441 Density functional of the electron system with the individual states of ions accounted for 132 442 The Hartree-Fock-Slater equations of the ion method in the cell and plasma approximations 134 443 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 51 Radiative heat conductivity of plasma 146 511 The radiative transfer equation 146 512 The diffusion approximation 150 513 The Rosseland mean opacity 154 514 The Planck mean Radiation of an optically thin layer 155 52 Quantum-mechanical expressions for the effective photon absorption cross-sections 156 521 Absorption in spectral lines 156 522 Photoionization 164 523 Inverse bremsstrahlung 168 524 Compton scattering 170 525 The total absorption cross-section 171 53 Peculiarities of photon absorption in spectral lines 172 531 Probability distribution of excited ion states 172 532 Position of spectral lines 174 533 Atom wave functions and addition of momenta 176 54 Shape of spectral lines 183 541 Doppler effect 184 542 Electron broadening in the impact approximation 185 543 The nondegenerate case 186 544 Accounting for degeneracy 194 545 Methods for calculating radiation and electron broadening 198 546 Ion broadening 205 547 The Voigt profile 213 548 Line profiles of a hydrogen plasma in a strong magnetic field 214 55 Statisticalmethod for line-group accounting 219 551 Shift and broadening parameters of spectral lines in plasma 220 552 Fluctuations of occupation numbers in a dense hot plasma 226 553 Statistical description of overlapping multiplets 228 554 Effective profile for a group of lines 238 555 Statistical description of the photoionization process 243 56 Computational results for Rosseland mean paths and spectral photon-absorption coefficients 245 561 Comparison of the statistical method with detailed computation 245 562 Dependence of the absorption coefficients on the element number, temperature and density of the plasma 250 563 Spectral absorption coefficients 259 564 Radiative and electron heat conductivity 265 565 Databases of atomic data and spectral photon absorption coefficients 266 57 Absorption of photons in a plasma with nonequilibrium radiation field 267 571 Basic processes and relaxation times 268 572 Joint consideration of the processes of photon transport and level kinetics of electrons 271 573 Average-atomapproximation 272 574 Rates of radiation and collision processes 274 575 Radiation properties of a plasma with nonequilibrium radiation field 277 576 Radiative heat conductivity of matter for large gradients of temperature and density 280 6 The equation of state 285 61 Description of thermodynamics of matter based on quantum-statistical models 286 611 Formulas for the pressure, internal energy and entropy according to the Thomas-Fermi model 286 612 Quantum, exchange and oscillation corrections to the Thomas-Fermi model 294 62 The ionization equilibrium method 300 621 The Gibbs distribution for the atomcell 300 622 The Saha approximation 301 623 An iteration scheme for solving the system of equations of ionization equilibrium 303 624 Coronal equilibrium 305 63 Thermodynamic properties of matter in the Hartree-Fock-Slatermodel 307 631 Electron thermodynamic functions 309 632 Accounting for the thermal motion of ions in the charged hard-sphere approximation 314 633 Effective radius of the average ion 317 634 On methods for deriving wide-range equations of state 318 64 Computational results 319 641 General description 319 642 Cold compression curves 322 643 Shock adiabats 324 644 Comparison with the Saha model 327 65 Approximation of thermophysical-data tables 330 651 Construction of an approximating spline that preserves geometric properties of the initial function 331 652 Numerical results 334