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Lectures on Atomic Physics Lectures on Atomic Physics Walter R. Johnson Department of Physics, University of Notre Dame Notre Dame, Indiana 46556, U.S.A. January 4, 2006 Contents Preface xi 1 Angular Momentum 1 1.1 Orbital Angular Momentum - Spherical Harmonics . 1 1.1.1 Quantum Mechanics of Angular Momentum . 2 1.1.2 Spherical Coordinates - Spherical Harmonics . 4 1.2 Spin Angular Momentum . 7 1.2.1 Spin 1=2 and Spinors . 7 1.2.2 In¯nitesimal Rotations of Vector Fields . 9 1.2.3 Spin 1 and Vectors . 10 1.3 Clebsch-Gordan Coe±cients . 11 1.3.1 Three-j symbols . 15 1.3.2 Irreducible Tensor Operators . 17 1.4 Graphical Representation - Basic rules . 19 1.5 Spinor and Vector Spherical Harmonics . 21 1.5.1 Spherical Spinors . 21 1.5.2 Vector Spherical Harmonics . 23 2 Central-Field SchrÄodinger Equation 25 2.1 Radial SchrÄodinger Equation . 25 2.2 Coulomb Wave Functions . 27 2.3 Numerical Solution to the Radial Equation . 31 2.3.1 Adams Method (adams) . 33 2.3.2 Starting the Outward Integration (outsch) . 36 2.3.3 Starting the Inward Integration (insch) . 38 2.3.4 Eigenvalue Problem (master) . 39 2.4 Quadrature Rules (rint) . 41 2.5 Potential Models . 44 2.5.1 Parametric Potentials . 45 2.5.2 Thomas-Fermi Potential . 46 2.6 Separation of Variables for Dirac Equation . 51 2.7 Radial Dirac Equation for a Coulomb Field . 52 2.8 Numerical Solution to Dirac Equation . 57 2.8.1 Outward and Inward Integrations (adams, outdir, indir) 57 i ii CONTENTS 2.8.2 Eigenvalue Problem for Dirac Equation (master) . 60 2.8.3 Examples using Parametric Potentials . 61 3 Self-Consistent Fields 63 3.1 Two-Electron Systems . 63 3.2 HF Equations for Closed-Shell Atoms . 69 3.3 Numerical Solution to the HF Equations . 79 3.3.1 Starting Approximation (hart) . 79 3.3.2 Re¯ning the Solution (nrhf) . 81 3.4 Atoms with One Valence Electron . 84 3.5 Dirac-Fock Equations . 88 4 Atomic Multiplets 97 4.1 Second-Quantization . 97 4.2 6-j Symbols . 101 4.3 Two-Electron Atoms . 104 4.4 Atoms with One or Two Valence Electrons . 108 4.5 Particle-Hole Excited States . 112 4.6 9-j Symbols . 115 4.7 Relativity and Fine Structure . 117 4.7.1 He-like ions . 117 4.7.2 Atoms with Two Valence Electrons . 121 4.7.3 Particle-Hole States . 122 5 Hyper¯ne Interaction & Isotope Shift 125 5.1 Hyper¯ne Structure . 125 5.2 Atoms with One Valence Electron . 130 5.2.1 Pauli Approximation . 132 5.3 Isotope Shift . 134 5.3.1 Normal and Speci¯c Mass Shifts . 135 5.4 Calculations of the SMS . 136 5.4.1 Angular Decomposition . 137 5.4.2 Application to One-Electron Atom . 139 5.5 Field Shift . 140 6 Radiative Transitions 143 6.1 Review of Classical Electromagnetism . 143 6.1.1 Electromagnetic Potentials . 144 6.1.2 Electromagnetic Plane Waves . 145 6.2 Quantized Electromagnetic Field . 146 6.2.1 Eigenstates of . 147 Ni 6.2.2 Interaction Hamiltonian . 148 6.2.3 Time-Dependent Perturbation Theory . 149 6.2.4 Transition Matrix Elements . 150 6.2.5 Gauge Invariance . 154 6.2.6 Electric Dipole Transitions . 155 CONTENTS iii 6.2.7 Magnetic Dipole and Electric Quadrupole Transitions . 161 6.2.8 Nonrelativistic Many-Body Amplitudes . 167 6.3 Theory of Multipole Transitions . 171 7 Introduction to MBPT 179 7.1 Closed-Shell Atoms . 181 7.1.1 Angular Momentum Reduction . 183 7.1.2 Example: 2nd-order Energy in Helium . 186 7.2 B-Spline Basis Sets . 187 7.2.1 Hartree-Fock Equation and B-splines . 190 7.2.2 B-spline Basis for the Dirac Equation . 191 7.2.3 Application: Helium Correlation Energy . 192 7.3 Atoms with One Valence Electron . 193 7.3.1 Second-Order Energy . 194 7.3.2 Angular Momentum Decomposition . 195 7.3.3 Quasi-Particle Equation and Brueckner-Orbitals . 196 7.3.4 Monovalent Negative Ions . 198 7.4 Relativistic Calculations . 199 7.4.1 Breit Interaction . 201 7.4.2 Angular Reduction of the Breit Interaction . 202 7.4.3 Coulomb-Breit Many-Electron Hamiltonian . 205 7.4.4 Closed-Shell Energies . 205 7.4.5 One Valence Electron . 206 7.5 CI Calculations . 209 7.5.1 Relativistic CI Calculations . 210 8 MBPT for Matrix Elements 213 8.1 Second-Order Corrections . 213 8.1.1 Angular Reduction . 215 8.2 Random-Phase Approximation . 216 8.2.1 Gauge Independence of RPA . 218 8.2.2 RPA for hyper¯ne constants . 219 8.3 Third-Order Matrix Elements . 220 8.4 Matrix Elements of Two-particle Operators . 223 8.4.1 Two-Particle Operators: Closed-Shell Atoms . 223 8.4.2 Two-Particle Operators: One Valence Electron Atoms . 224 8.5 CI Calculations for Two-Electron Atoms . 227 8.5.1 E1 Transitions in He . 228 A Exercises 231 A.1 Chapter 1 . 231 A.2 Chapter 2 . 233 A.3 Chapter 3 . 237 A.4 Chapter 4 . 237 A.5 Chapter 5 . 239 A.6 Chapter 6 . 240 iv CONTENTS A.7 Chapter 7 . 243 A.8 Chapter 8 . 244 List of Tables 1.1 C(l; 1=2; j; m m ; m ; m) . 13 ¡ s s 1.2 C(l; 1; j; m m ; m ; m) . 14 ¡ s s 2.1 Adams-Moulton integration coe±cients . 35 2.2 Weights ai = ci=d and error ek coe±cients for trapezoidal rule with endpoint corrections. 44 2.3 Comparison of n = 3 and n = 4 levels (a.u.) of sodium calculated using parametric potentials with experiment. 45 2.4 Parameters for the Tietz and Green potentials. 62 2.5 Energies obtained using the Tietz and Green potentials. 62 3.1 Coe±cients of the exchange Slater integrals in the nonrelativistic Hartree-Fock equations: ¤lallb . These coe±cients are symmetric with respect to any permutation of the three indices. 76 3.2 Energy eigenvalues for neon. The initial Coulomb energy eigen- values are reduced to give model potential values shown under U(r). These values are used as initial approximations to the HF eigenvalues shown under VHF. 80 3.3 HF eigenvalues ²nl , average values of r and 1=r for noble gas atoms. The negative of the experimental removal energies -Bexp from Bearden and Burr (1967, for inner shells) and Moore (1957, for outer shell) is also listed for comparison. 86 3.4 Energies of low-lying states of alkali-metal atoms as determined N 1 in a VHF¡ Hartree-Fock calculation. 88 3.5 Dirac-Fock eigenvalues (a.u.) for mercury, Z = 80. Etot = 19648:8585 a.u.. For the inner shells, we also list the exper- imen¡ tal binding energies from Bearden and Burr (1967) for com- parison. 94 3.6 Dirac-Fock eigenvalues ² of valence electrons in Cs (Z = 55) and theoretical ¯ne-structure intervals ¢ are compared with measured energies (Moore). ¢nl = ²nlj=l+1=2 ²nlj=l 1=2 . 95 ¡ ¡ 4.1 Energies of (1snl) singlet and triplet states of helium (a.u.). Com- parison of a model-potential calculation with experiment (Moore, 1957). 107 v vi LIST OF TABLES N 1 4.2 Comparison of VHF¡ energies of (3s2p) and (3p2p) particle-hole excited states of neon and neonlike ions with measurements. 114 4.3 First-order relativistic calculations of the (1s2s) and (1s2p) states of heliumlike neon (Z = 10), illustrating the ¯ne-structure of the 3P multiplet. 120 5.1 Nonrelativistic HF calculations of the magnetic dipole hyper¯ne constants a (MHz) for ground states of alkali-metal atoms com- pared with measurements. 134 5.2 Lowest-order matrix elements of the speci¯c-mass-shift operator T for valence states of Li and Na. 139 6.1 Reduced oscillator strengths for transitions in hydrogen . 159 1 6.2 Hartree-Fock calculations of transition rates Aif [s¡ ] and life- times ¿ [s] for levels in lithium. Numbers in parentheses represent powers of ten. 160 6.3 Wavelengths and oscillator strengths for transitions in heliumlike ions calculated in a central potential v0(1s; r). Wavelengths are determined using ¯rst-order energies. 170 7.1 Eigenvalues of the generalized eigenvalue problem for the B-spline approximation of the radial SchrÄodinger equation with l = 0 in a Coulomb potential with Z = 2. Cavity radius is R = 30 a.u. We use 40 splines with k = 7. 189 7.2 Comparison of the HF eigenvalues from numerical integration of the HF equations (HF) with those found by solving the HF eigenvalue equation in a cavity using B-splines (spline). Sodium, Z = 11, in a cavity of radius R = 40 a.u.. 192 7.3 Contributions to the second-order correlation energy for helium. 193 7.4 Hartree-Fock eigenvalues ²v with second-order energy corrections (2) Ev are compared with experimental binding energies for a few.
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