Polarization in Spectral Lines

CONTENTS......Page 12
1.1. The Polarization Ellipse......Page 20
1.3.  Polarization Tensor......Page 24
1.4. Quasi-monochromatic Wave......Page 27
1.5. Polarizers and Retarders......Page 30
1.6. Stokes Parameters......Page 34
1.7. Measurements of the Stokes Parameters......Page 37
1.8. Stokes Parameters and Polarization Tensor......Page 41
1.9. Properties of the Stokes Parameters......Page 44
1.10. Photons and Stokes Parameters......Page 46
2.1. Eigenvalues and Eigenvectors of Angular Momentum......Page 48
2.2. Coupling of Two Angular Momenta: Vector-Coupling Coefficients and 3-j Symbols......Page 51
2.3. Coupling of Three Angular Momenta: Racah Coefficients and 6-j Symbols......Page 58
2.4. Coupling of Four Angular Momenta: 9-j Symbols......Page 64
2.5. Rotations and Euler Angles......Page 68
2.6. Rotation Matrices......Page 72
2.7. Irreducible Spherical Tensors......Page 79
2.8. The Wigner-Eckart Theorem and its Consequences......Page 84
2.9. Properties of Reduced Matrix Elements......Page 87
3.1. Zeeman Effect......Page 92
3.2. Classical Theory of the Zeeman Effect......Page 101
3.3. Classification of Zeeman Patterns......Page 108
3.4. The Paschen-Back Effect......Page 116
3.5. Magnetic Field and Hyperfine Structure......Page 129
3.6. Atomic Level Polarization and Density Matrix......Page 134
3.7. Multipole Moments of the Density Matrix......Page 141
4.1. Quantization of the Harmonic Oscillator......Page 150
4.2. The Electromagnetic Field as a Superposition of Plane Waves......Page 153
4.3. Quantization of the Electromagnetic Field......Page 156
4.4. The Stokes Parameters in the Formalism of Second Quantization......Page 160
4.5. The Density Operator of the Radiation Field......Page 163
5.1.Propagation of Electromagnetic Waves in Anisotropic Media......Page 166
5.2. Transfer Equations for Polarized Radiation......Page 172
5.3. Application to Magnetic Lines......Page 174
5.4. The Voigt Function and the Associated Dispersion Proffile......Page 181
5.5. Symmetry Properties of the Transfer Equations for Polarized Radiation......Page 191
5.6. Geometrical Interpretation of the Transfer Equations for Polarized Radiation......Page 195
5.7. Resonance Scattering and the Hanle Effect......Page 198
5.8. The Scattering Phase Matrix in a Particular Case......Page 204
5.9. Some Illustrations of the Hanle Effect......Page 209
5.10. The Scattering Phase Matrix Expressed in Terms of Rotation Matrices......Page 213
5.11. Spherical Tensors for Polarimetry......Page 221
5.12. Further Properties of the Scattering Phase Matrix......Page 231
5.13. Understanding Scattering Experiments through Oscillator Models......Page 234
5.14. The Role of Collisions......Page 239
5.15. Some Properties of the Collisional Kernels......Page 249
5.16. Classification of the Physical Regimes......Page 251
CHAPTER 6. INTERACTION OF MATERIAL SYSTEMS WITH POLARIZED RADIATION (THE QUANTUM APPROACH)......Page 256
6.1. Equations of Motion......Page 257
6.2. The Interaction Hamiltonian......Page 259
6.3. The Dipole Approximation......Page 263
6.4. Approximate Equations of Motion......Page 268
6.5. Evolution Equations for the Atomic System......Page 271
6.6. Evolution Equations for the Radiation Field......Page 284
6.7. Evolution Equations for the Stokes Parameters......Page 289
6.8. Magnetic Dipole Transitions......Page 293
CHAPTER 7. STATISTICAL EQUILIBRIUM EQUATIONS AND RADIATIVE TRANSFER COEFFICIENTS FOR ATOMIC SYSTEMS......Page 296
7.1. The Multi-Level Atom in the Standard Representation......Page 297
7.2. The Multi-Level Atom in the Spherical Statistical Tensor  Representation......Page 303
7.3. Conjugation Properties of the Rates......Page 309
7.4. The No-Coherence Case......Page 310
7.5. The Multi-Term Atom in the Energy-Eigenvector Representation......Page 314
7.6. The Multi-Term Atom in the Spherical Statistical Tensor Representation......Page 320
7.7. Conjugation Properties of the Rates......Page 330
7.8. The Multi-Level Atom as a Special Case of the Multi-Term Atom......Page 331
7.9. The Multi-Level Atom with Hyperfine Structure......Page 334
7.10. The Principle of Spectroscopic Stability......Page 340
7.11. Selection Rules......Page 344
7.12. Changing the Reference System......Page 347
7.13. Collisional Rates......Page 352
8.1. Generalities......Page 368
8.2. Formal Solution of the Radiative Transfer Equations the Evolution Operator......Page 369
8.3. Analytical Expressions for the Evolution Operator......Page 372
8.4. Solution of the Radiative Transfer Equations by Diagonalization......Page 377
8.5. Evolution Operator for Purely Dichroic or Purely Dispersive Media......Page 381
8.6. Evolution Operator and Mueller Matrices......Page 383
8.7. Perturbative Solution of the Radiative Transfer Equations......Page 385
8.8. An Alternative Form of the Radiative Transfer Equations......Page 387
8.9. Solution of the Alternative Form of the Radiative Transfer Equations......Page 389
9.1. Transfer Equation for Polarized Radiation in a Magnetized Stellar Atmosphere......Page 394
9.2. Comparison with the Classical Theory......Page 400
9.3. Collisional and Doppler Broadening......Page 402
9.4. Different Forms of the Transfer Equation......Page 405
9.5. Generalities and Symmetry Properties of the Transfer Equation......Page 408
9.6. The Weak Field Approximation......Page 416
9.7. Formal Solution through the Evolution Operator......Page 428
9.8. The Unno-Rachkovsky Solution......Page 430
9.9. More General Analytical Solutions......Page 437
9.10. Solutions for Special Magnetic Field Orientations......Page 441
9.11. The Stepanov Solution......Page 445
9.12. The Intense Field Solution......Page 447
9.13. The Seares Formulae......Page 452
9.14. The Magnetic Field with Variable Azimuth......Page 454
9.15. Numerical Solutions......Page 456
9.16. Response Functions......Page 468
9.17. Contribution Functions......Page 475
9.18. Blends......Page 478
9.19. The Magnetic Intensification Mechanism......Page 479
9.20. Net Linear Polarization in Spectral Lines: the Differential Saturation Mechanisam......Page 485
9.21. Net Circular Polarization in Spectral Lines......Page 492
9.22. The Importance of Magneto-Optical Effects......Page 507
9.23. Transfer Equation for Fine-Structured and Hyperfine-Structured Lines......Page 510
9.24. Line Formation in Stochastic Media......Page 514
9.25. Isotropic, Microturbulent Magnetic Field......Page 523
CHAPTER 10. NON-EQUILIBRIUM ATOMIC PHYSICS......Page 528
10.1. The Two-Level Atom: Generalities......Page 529
10.2. The Two-Level Atom: Resonance Polarization......Page 532
10.3. The Two-Level Atom: the Hanle Effect......Page 539
10.4. The Two-Level Atom: Spectral Details of the Hanle Effect......Page 543
10.5. The Two-Level Atom: Resonance Polarization for Strong Maganetic Fields......Page 547
10.6. The Two-Level Atom: the Role of Collisions......Page 551
10.7. The Two-Level Atom: The Role of Lower-Level Polarization......Page 554
10.8. The Two-Level Atom: the Hanle Effect with Lower-Level Polarization......Page 564
10.9. The Two-Level Atom: the Role of Stimulated Emission......Page 570
10.10. Three-Level Atoms: the Stepwise Excitation......Page 576
10.11. Three-Level Atoms: the Raman Effect......Page 579
10.12. Three-Level Atoms: an Example Leading to Population Inversion......Page 582
10.13. The Weak Anisotropy Approximation......Page 586
10.14. The Two-Level Atom in the Weak Anisotropy Approximation......Page 589
10.15. The Two-Term Atom: Generalities......Page 599
10.16. The Two-Term Atom: Resonance Polarization......Page 603
10.17. The Two-Term Atom: Spectral Details of Resonance Polarization......Page 608
10.18. The Two-Term Atom: the Hanle Effect......Page 617
10.19. The Two-Term Atom: the Franken Effect......Page 624
10.20. The Two-Term Atom: the Alignment-to-Orientation Conversion Mechanisam......Page 626
10.21. The Two-Term Atom: the Role of Lower-Term Polarization......Page 628
10.22. The Two-Level Atom with Hyperfine Structure......Page 635
CHAPTER 11. ASTROPHYSICAL APPLICATIONS: SOLAR MAGNETOMETRY......Page 644
11.1. The Longitudinal Magnetograph......Page 645
11.2. The Vector Magnetograph......Page 648
11.3. The Unno-fit Technique......Page 653
11.4. The Bisector and the Center-of-Gravity Techniques......Page 658
11.5. Unresolved Fields......Page 663
11.6. Other Inversion Techniques......Page 673
11.7. Disambiguation......Page 680
CHAPTER 12. ASTROPHYSICAL APPLICATIONS: RADIATION ANISOTROPY IN STELLAR ATMOSPHERES......Page 682
12.1. The Milne-Eddington Model Atmosphere......Page 683
12.2. The Grey Atmosphere......Page 687
12.3. Outer Atmospheres......Page 692
12.4. Symmetry-Breaking Effects......Page 696
13.1. The Flat-Spectrum Approximation......Page 708
13.2. Velocity/Density-Matrix Correlations and the Approximation of Complete Redistribution on Velocities......Page 710
13.3. Resonance Polarization and the Hanle Effect in the Absence of Complete Redistribution on Velocities/Density-Matrix Corr......Page 717
13.4. Diagnostics of Magnetic Fields in Solar Prominences......Page 729
13.5. Diagnostics of Magnetic Fields from Coronal Forbidden Lines......Page 734
13.6. Resonance Polarization in the Presence of Velocity/Density-Matrix......Page 739
13.7. The Hanle Effect in the Presence of Velocity/Density-Matrix......Page 749
14.1. The Non-LTE Problem......Page 756
14.2. The Two-Level Atom: Non-LTE Theory for Weak Magnetic Fields (Hanle Effect Regime)......Page 761
14.3. The Two-Level Atom: Non-LTE Theory for Strong Magnetic Fields......Page 779
14.4. The Non-LTE Regime of Order 1.5......Page 786
14.5. The Non-LTE Problem for More Complicated Atomic Models......Page 788
14.6. Applications to Realistic Atmospheres: Polarization in the Continuum Spectrum......Page 793
14.7. Applications to Realistic Atmospheres: Approximate Results on the Polarization of the Fraunhofer Spectrum\r......Page 802
14.8. Alternative Methods for the Solution of the Non-LTE Problem......Page 806
A1. A Fortran Code for Computing 3-j, 6-j, and 9-j Symbols......Page 810
A2. Sample Evaluation of a Quantity Involving the Contraction of 3-j Cofficients......Page 813
A3. Momentum and Angular Momentum of the Electromagnetic Field......Page 815
A4. Multipole Components of Collisional Rates......Page 818
A5. Explicit Expression for the Exponential of the Propagation Matrix......Page 822
A6. Diagonalization of the Propagation Matrix......Page 828
A7. Formulae for the Calculation of the Evolution Operator......Page 831
A8. The Feautrier Method: Numerical Details......Page 832
A9. The Diagonal Element Lambda-Operator (DELO) Method: Numerical Details......Page 837
A10. Equivalent Width in the Presence of Depth-Dependent Line Shifts......Page 839
A11 Net Circular Polarization in Blends......Page 841
A12. Evolution Operator in Stochastic Media......Page 843
A13. Properties of the Generalized Profiles......Page 848
A14. Properties of the Symbol [W[sub(KK\'Q)](β[sub(l)]L[sub(l)]Sβ[sub(u)]L[sub(u)];B)][sub(fs)]......Page 854
A15. A Property of the Hopf Function......Page 857
A16. A Numerical Algorithm for the Solution of the Hopf Equation......Page 859
A17. Symmetry Properties of the Comoving-Frame Radiation Field Tensor for for a Cylindrically Symmetrical Atmosphere\r......Page 860
A18. Redistribution Matrix for a Maxwellian Distribution of Velocities......Page 862
A19. Properties of the Kernel Κ[sup(K)][sub(QQ\')](RB)......Page 863
A20. The Multipole Coupling Coefficients......Page 864
A21. The Calculation of a Double Integral......Page 869
A22. The Generalization of the √ε......Page 872
A23. The Generalized Multipole Coupling Coefficients......Page 878
A24. Reduced Matrix Elements for Photoionization Cross Sections......Page 882
LIST OF TABLES......Page 886
REFERENCE......Page 888
AUTHOR INDEX......Page 900
SUBJECT INDEX......Page 904