The theory of atomic structure and spectra

Cover......Page 1
Los Alamos Series in Basic and Applied Sciences......Page 2
Title page......Page 3
Copyright page......Page 4
Dedication......Page 5
Contents......Page 7
PREFACE......Page 15
CHAPTER 1. EXPERIMENTAL BACKGROUND AND BASIC CONCEPTS......Page 19
1-1. Line Spectra......Page 20
1-2. Characteristic Spectra of Atoms and Ions......Page 22
1-4. Wavelength and Wavenumber Units......Page 24
1-5. Energy Levels......Page 26
1-6. Units for Energy Levels......Page 27
1-7. Energy Level Diagrams......Page 28
1-8. The Ritz Combination Principle; Grotrian Diagrams......Page 34
*1-9. The Profiles of Spectrum Lines......Page 35
*1-10. Wavelength Range and Accuracy......Page 42
1-11. Empirical Spectrum Analysis......Page 43
1-12. The Role of Theory......Page 45
2-1. Introduction......Page 51
2-2. Angular-Momentum Operators......Page 52
2-3. Eigenvalues of Angular-Momentum Operators......Page 53
2-4. Step-up and Step-down Operators......Page 56
2-5. Orbital Angular Momentum......Page 57
2-6. The Addition Theorem of Spherical Harmonics......Page 62
2-7. Electron Spin......Page 67
2-8. Addition of Two Angular Momenta......Page 68
2-10. Coupled Wavefunctions......Page 71
2-11. Coupling Schemes; LS Coupling......Page 74
2-12. Notation for Angular-Momentum States......Page 76
2-13. Parity......Page 79
2-14. Parity and Angular-Momentum Selection Rules......Page 80
*2-15. Appendix......Page 82
3-1. The Schrödinger Equation......Page 85
3-2. Central-field Problems......Page 86
3-3. Analytical Solution of the Radial Equation......Page 88
3-4. Numerical Solution of the Radial Equation......Page 90
3-5. Electron Probability Density......Page 96
3-6. Energy Levels and Wavelengths......Page 97
3-7. Relativistic Corrections......Page 99
*3-8. Appendix: The Virial Theorem......Page 106
3-9. Appendix: The Spin-Orbit Interaction......Page 109
4-1. The Schrödinger Equation......Page 111
4-2. The Matrix Method......Page 112
4-3. The Central-Field Model......Page 115
4-4. Product Wavefunctions......Page 116
4-5. Antisymmetrization; Determinantal Functions......Page 117
4-6. Coupling of Antisymmetrized Wavefunctions......Page 120
4-7. Electron Configurations......Page 121
4-8. Equivalent Electrons; Closed Subshells......Page 124
4-9. Permitted LS Terms for Equivalent Electrons......Page 126
4-10. Configurations with Several Open Subshells......Page 127
4-11. Configuration-Average Energies......Page 130
4-12. Relative Energies of Configurations......Page 131
4-13. The Periodic System......Page 133
4-14. Variation of Ionization Energy with Z......Page 138
4-15. Level Structure under LS-Coupling Conditions......Page 140
4-16. Hund's Rule......Page 142
4-17. jj Coupling......Page 143
4-18. Pair Coupling......Page 146
4-19. Example: Si I 3pns......Page 149
4-21. Statistical Weights (Hydrogenic Atoms)......Page 152
4-22. Statistical Weights (Complex Atoms)......Page 154
4-23. Quantitative Calculation of Level Structures......Page 156
5-1. The 3-j Symbol......Page 160
5-2. The 6-j Symbol......Page 165
5-3. The 9-j Symbol......Page 168
*5-4. Graphical Methods......Page 170
6-1. Diagonal Matrix Elements of Symmetric Operators......Page 174
6-2. One-Electron and Total-Atom Binding Energies......Page 178
6-3. Ionization Energy and One-Electron Binding Energies......Page 186
6-4. Numerical Example......Page 189
7-1. The Variational Principle......Page 194
7-2. The Hartree-Fock Equations......Page 196
7-3. The Classical Potential Energy......Page 199
7-4. The Exchange Potential Energy......Page 201
7-5. Solution of the Hartree-Fock Equations......Page 202
*7-6. Complications and Instabilities......Page 203
7-7. Homogeneous-Equation (Local-Potential) Methods......Page 208
7-8. The Thomas-Fermi (TF) and Thomas-Fermi-Dirac (TFD) Methods......Page 209
*7-9. The Parametric Potential Method......Page 211
7-11. The Hartree-Fock-Slater Method (HFS)......Page 212
7-12. The Hartree-plus-Statistical-Exchange Method (HX)......Page 215
*7-13. The Hartree-Slater Method (HS)......Page 217
7-14. Relativistic Corrections......Page 218
7-15. Correlation Corrections......Page 220
*7-16. Appendix: The Thomas-Fermi and Thomas-Fermi-Dirac Atoms......Page 224
*7-17. Appendix: Small-r Solution for the HXR Method......Page 230
8-1. Computer Calculation of Radial Functions......Page 232
8-2. Comparison of Methods......Page 236
8-3. Accuracy of Computed Configuration-Average Energies......Page 237
8-4. Relativistic Effects......Page 241
8-5. Variation with the Principal Quantum Number......Page 242
8-6. Variation with Z of One-Electron Binding Energies......Page 247
8-7. Variation with Z of Coulomb and Spin-Orbit Integrals......Page 254
9-1. Coupling of Two Angular Momenta......Page 258
9-2. Recoupling of Three Angular Momenta......Page 263
9-3. Transformations Between Coupling Schemes......Page 266
9-4. Antisymmetrization Difficulties for Equivalent Electrons......Page 268
9-5. Coefficients of Fractional Parentage......Page 273
9-6. Coefficients of Fractional Grandparentage......Page 277
9-7. The Seniority Number......Page 279
9-8. Antisymmetrized Functions for an Arbitrary Configuration......Page 283
9-9. Single-Configuration Matrix Elements of Symmetric Operators......Page 285
9-10. Uncoupling of Spectator Subshells......Page 288
*9-11. Appendix: Vector-Coupling Coefficients......Page 289
10-1. Kinetic and Electron-Nuclear Energies......Page 294
10-2. Effects of Closed Subshells......Page 296
10-3. One-Electron Configurations......Page 297
10-4. Two-Electron Configurations; Spin-Orbit Matrix Elements......Page 298
10-5. Two-Electron Configurations; Coulomb Matrix Elements......Page 300
10-6. Intermediate Coupling......Page 306
10-7. Eigenvector Purities......Page 309
10-8. Examples: ps and p5s......Page 313
10-9. Pair Coupling......Page 316
11-1. Calculation of Complex Matrix Elements......Page 321
11-2. Racah Algebra......Page 322
11-3. Irreducible Tensor Operators......Page 323
11-4. The Wigner-Eckart Theorem......Page 325
11-5. Matrix Elements of the Product of Two Operators......Page 328
11-6. Tensor Product of Two Tensor Operators......Page 329
11-7. Uncoupling Formulae for Reduced Matrix Elements......Page 331
11-8. Scalar Product of Two Tensor Operators......Page 332
11-9. Unit Tensor Operators......Page 334
11-10. Double Tensor Operators......Page 337
11-11. Spin-Orbit Matrix Elements (One-Electron Configurations)......Page 339
11-12. Direct Coulomb Matrix Elements (Two-Electron Configurations)......Page 340
11-14. The Effective Operator for Exchange Matrix Elements......Page 341
11-15. Summary......Page 344
12-1. Introduction......Page 345
12-2. Coulomb Matrix Elements: Equivalent Electrons......Page 346
12-3. Non-Equivalent Electrons: Direct Integral......Page 349
12-4. Non-Equivalent Electrons: Exchange Integral......Page 351
12-5. Spin-Orbit Matrix Elements......Page 353
12-6. Example: $f^3sd^2$......Page 355
12-7. Level Structure: Spin-Orbit Effects under Near-CS Coupling Conditions......Page 357
12-8. Level Structure: Configurations $l^w$......Page 359
12-9. Level Structure: Configurations $l_1^{w_1}l_2......Page 364
*12-10. Level Structure: More-Complex Configurations......Page 374
*12-11. Appendix: Center-of-Gravity Relations......Page 375
13-1. General Remarks......Page 376
*13-2. Wavefunction Orthogonality Problems......Page 379
13-3. Two-Electron Configurations......Page 384
13-4. Single-Configuration-Like Interactions......Page 386
13-5. Rydberg-Series Interactions......Page 387
13-6. One-Electron Configurations......Page 389
13-7. Brillouin's Theorem......Page 390
*13-8. Arbitrary Configuration Interactions......Page 391
*13-9. Matrix Elements of Symmetric Operators......Page 393
*13-10. Coulomb Matrix Elements; General Case......Page 396
*13-11. One-Electron Matrix Elements; General Case......Page 408
14-1. The Einstein Transition Probabilities......Page 413
14-2. Electric Dipole Radiation (Classical)......Page 416
14-3. Electric Dipole Radiation (Quantum Mechanical)......Page 418
14-5. The Dipole Line Strength......Page 420
14-6. Oscillator Strengths......Page 422
14-7. Theoretical Calculation of Line Strengths......Page 423
14-8. Selection Rules and Relative Intensities for LS Coupling......Page 424
14-9. One-Electron Configurations; The Radial Dipole Integral......Page 428
14-10. Transitions Involving an Electron in Singly Occupied Subshells......Page 430
*14-11. Line Strengths for General Transition Arrays......Page 435
14-12. Selection-Rule Violations......Page 439
*14-13. Line Strength Sum Rules......Page 440
*14-14. Oscillator Strength Sum Rules......Page 442
14-15. Cancellation Effects......Page 450
14-16. Experimental Measurement of Oscillator Strengths......Page 452
*14-17. Systematics of Oscillator Strengths......Page 454
*14-18. Radiative Decay in Low-Density Plasmas......Page 458
15-1. Magnetic Dipole Radiation (M1)......Page 460
15-2. Electric Quadrupole Radiation (E2)......Page 463
*15-3. Interference Between M1 and E2 Radiation......Page 467
15-4. Examples of Forbidden Transitions......Page 468
16-1. Computer Programs......Page 474
16-2. Ab Initio Calculations......Page 479
16-3. Least-Squares Calculations......Page 483
16-4. Basic Least-Squares Method......Page 486
*16-5. Modifications of the Basic Method......Page 491
*16-7. Effective Operators......Page 495
*16-8. LS-Dependent Hartree-Fock Calculations......Page 499
*16-9. Highly Excited Configurations......Page 501
17-1. The Zeeman Effect......Page 503
17-2. Matrix Elements of the Magnetic-Energy Operator......Page 504
17-3. The Weak-Field Limit......Page 506
17-4. Weak-Field Zeeman Patterns......Page 510
*17-5. Strong Magnetic Fields; The Paschen-Back Effect......Page 512
*17-6. Intermediate Magnetic Fields......Page 515
*17-7. The Stark Effect......Page 516
*17-8. Isotope Shifts......Page 523
*17-9. Hyperfine Structure......Page 524
18-1. Introduction......Page 530
18-2. One-Electron Continuum Basis Functions......Page 533
18-3. Normalization of the Radial Function......Page 534
18-4. The Energy Dimensions of $P_{\varepsilon l}$......Page 539
18-5. Relation Between Radial Integrals for Discrete and Continuum States......Page 540
18-6. Photoionization......Page 541
18-7. Interaction of a Discrete State with a Single Continuum; Autoionization......Page 544
*18-8. Pseudo-Discrete Treatment of Continuum Problems......Page 553
*18-9. Ionization Equilibrium......Page 562
*18-10. Radiative Recombination......Page 565
*18-11. Dielectronic Recombination......Page 567
*18-12. Generalized Oscillator Strengths......Page 581
*18-13. Plane-Wave-Born Collision Strengths......Page 585
19-1. Binding Energies in Isoelectronic Sequences......Page 588
19-2. Energy-Level Structures......Page 591
19-3. Configuration Interactions......Page 596
19-4. Radial Multipole Integrals......Page 598
19-5. Spectra......Page 599
19-6. X-Ray Spectra......Page 601
19-7. Comparison of Highly Ionized and X-Ray Spectra......Page 602
*19-8. Autoionization and Dielectronic Recombination......Page 607
*19-10. Temperature and Density Diagnostics......Page 608
*19-11. Iron $K_\alpha$ Diagnostics......Page 611
20-1. Lanthanide Configurations......Page 616
20-2. Level Structure; General Remarks......Page 618
20-3. Low-Level Structures and Coupling Conditions......Page 621
20-4. Spectra......Page 622
20-5. Aids to Empirical Spectrum Analysis......Page 624
20-6. Ions......Page 628
*20-7. Actinides......Page 630
20-8. Transition Elements......Page 631
*21-1. Thermodynamic Functions of Atomic Gases......Page 634
*21-2. Opacities of Thick Plasmas......Page 635
*21-3. Energy Distribution of Levels of a Configuration......Page 636
*21-4. Wavelength and Oscillator-Strength Distributions Within a Transition Array......Page 643
Appendix A. Physical Constants, Units, and Conversion Factors......Page 650
Appendix B. Conversion Between Vacuum and Air Wavelengths......Page 652
Appendix C. 3-j Symbols......Page 653
Appendix D. 6-j Symbols......Page 664
Appendix E. One-Electron and Total-Atom Energies......Page 683
Appendix F. Basis Functions, Matrix Elements, and Racah Algebra......Page 686
Appendix G. Matrix Elements of Spherical Harmonics......Page 695
Appendix H. Coefficients of Fractional Parentage, $U^{(k)}$, $V^{(kl)}$......Page 697
Appendix I. Relative Line Strengths Within an LS Multiplet......Page 712
Bibliography......Page 720
Name Index......Page 727
Subject Index......Page 735