Lectures on Atomic Physics Walter R. Johnson Department of Physics, University of Notre Dame Notre Dame, Indiana 46556, U.S.A. January 14, 2002 Contents Preface ix 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 1.2SpinAngularMomentum...................... 7 1.2.1 Spin 1/2andSpinors .................... 7 1.2.2 Infinitesimal Rotations of Vector Fields .......... 9 1.2.3 Spin 1 and Vectors ...................... 10 1.3 Clebsch-Gordan Coefficients .................... 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 Schr¨odinger Equation 25 2.1 Radial Schr¨odinger 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 Potential Models ........................... 41 2.4.1 Parametric Potentials .................... 42 2.4.2 Thomas-Fermi Potential ................... 44 2.5 Separation of Variables for Dirac Equation ............. 48 2.6 Radial Dirac Equation for a Coulomb Field ............ 49 2.7 Numerical Solution to Dirac Equation ............... 53 2.7.1 Outward and Inward Integrations (adams, outdir, indir)54 2.7.2 Eigenvalue Problem for Dirac Equation (master) .... 57 i ii CONTENTS 2.7.3 Examples using Parametric Potentials ........... 58 3 Self-Consistent Fields 61 3.1Two-ElectronSystems........................ 61 3.2 HF Equations for Closed-Shell Atoms ............... 67 3.3 Numerical Solution to the HF Equations .............. 77 3.3.1 Starting Approximation (hart)............... 77 3.3.2 Refining the Solution (nrhf) ................ 79 3.4 Atoms with One Valence Electron . ............... 82 3.5 Dirac-Fock Equations ........................ 86 4 Atomic Multiplets 95 4.1 Second-Quantization ......................... 95 4.26-jSymbols.............................. 99 4.3 Two-Electron Atoms .........................102 4.4 Atoms with One or Two Valence Electrons ............106 4.5 Particle-Hole Excited States .....................110 4.69-jSymbols..............................113 4.7 Relativity and Fine Structure ....................115 4.7.1 He-like ions ..........................115 4.7.2 Atoms with Two Valence Electrons .............119 4.7.3 Particle-Hole States .....................120 4.8HyperfineStructure.........................121 4.8.1 Atoms with One Valence Electron .............126 5 Radiative Transitions 131 5.1 Review of Classical Electromagnetism ...............131 5.1.1 Electromagnetic Potentials . ...............132 5.1.2 Electromagnetic Plane Waves . ...............133 5.2 Quantized Electromagnetic Field . ...............134 5.2.1 Eigenstates of Ni .......................135 5.2.2 Interaction Hamiltonian ...................136 5.2.3 Time-Dependent Perturbation Theory ...........137 5.2.4 Transition Matrix Elements . ...............138 5.2.5 Gauge Invariance .......................142 5.2.6 Electric Dipole Transitions . ...............143 5.2.7 Magnetic Dipole and Electric Quadrupole Transitions . 149 5.2.8 Nonrelativistic Many-Body Amplitudes ..........154 5.3 Theory of Multipole Transitions ...................159 6Introduction to MBPT 167 6.1 Closed-Shell Atoms ..........................168 6.1.1 Angular Momentum Reduction ...............170 6.1.2 Example: 2nd-order Correlation Energy in Helium ....172 6.2B-SplineBasisSets..........................173 6.3 Hartree-Fock Equation and B-splines . ...............176 CONTENTS iii 6.4 Atoms with One Valence Electron .................178 6.4.1 Second-Order Energy ....................179 6.4.2 Angular Momentum Decomposition ............180 6.4.3 Quasi-Particle Equation and Brueckner-Orbitals .....181 6.4.4 Monovalent Negative Ions ..................183 6.5 CICalculations ............................186 A 189 A.1Problems...............................189 A.2 Examinations . ............................193 A.3AnswerstoProblems.........................195 A.4AnswerstoExams..........................209 iv CONTENTS List of Tables 1.1 C(l, 1/2,j; m − ms,ms,m)...................... 13 1.2 C(l, 1,j; m − ms,ms,m)....................... 14 2.1 Adams-Moulton integration coefficients .............. 35 2.2 Comparison of n =3andn = 4 levels (a.u.) of sodium calculated using parametric potentials with experiment. ........... 43 2.3 Parameters for the Tietz and Green potentials. .......... 58 2.4 Energies obtained using the Tietz and Green potentials. ..... 59 3.1 Coefficients of the exchange Slater integrals in the nonrelativistic Hartree-Fock equations: Λlallb . These coefficients are symmetric with respect to any permutation of the three indices. ..... 74 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..................... 78 3.3 HF eigenvalues nl , 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, foroutershell)isalsolistedforcomparison. ........... 84 3.4 Energies of low-lying states of alkali-metal atoms as determined N−1 in a VHF Hartree-Fock calculation. ................ 86 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- imental binding energies from Bearden and Burr (1967) fpr com- parison................................. 92 3.6 Dirac-Fock eigenvalues of valence electrons in Cs (Z = 55) and theoretical fine-structure intervals ∆ are compared with measured energies (Moore). ∆nl = nlj=l+1/2 − nlj=l−1/2 .......... 93 4.1 Energies of (1snl) singlet and triplet states of helium (a.u.). Com- parison of a model-potential calculation with experiment (Moore, 1957). .................................105 N−1 4.2 Comparison of VHF energies of (3s2p)and(3p2p) particle-hole excited states of neon and neonlike ions with measurements. 112 v vi LIST OF TABLES 4.3 First-order relativistic calculations of the (1s2s)and(1s2p) states of heliumlike neon (Z = 10), illustrating the fine-structure of the 3P multiplet. .............................118 4.4 Nonrelativistic HF calculations of the magnetic dipole hyperfine constants a (MHz) for ground states of alkali-metal atoms com- pared with measurements. ......................129 5.1 Reduced oscillator strengths for transitions in hydrogen .....147 −1 5.2 Hartree-Fock calculations of transition rates Aif [s ] and life- times τ [s] for levels in lithium. Numbers in parentheses represent powersoften..............................148 5.3 Wavelengths and oscillator strengths for transitions in heliumlike ions calculated in a central potential v0(1s, r). Wavelengths are determined using first-order energies. ...............158 6.1 Eigenvalues of the generalized eigenvalue problem for the B-spline approximation of the radial Schr¨odinger equation with l =0ina Coulomb potential with Z = 2. Cavity radius is R = 30 a.u. We use 40 splines with k =7.......................175 6.2 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 =40a.u...............178 6.3 Contributions to the second-order correlation energy for helium. 179 6.4 Hartree-Fock eigenvalues v with second-order energy corrections (2) Ev are compared with experimental binding energies for a few low-lying states in atoms with one valence electron. ........181 − 6.5 Expansion coefficients cn, n =1··20 of the 5s state of Pd in a basis of HF orbitals for neutral Pd confined to a cavity of radius R =50a.u................................184 6.6 Contribution δEl of (nlml) configurations to the CIenergy of the helium ground state. The dominant contributions are from the l =0nsms configurations. Contributions of configurations with l ≥ 7 are estimated by extrapolation. ...............187 List of Figures 1.1 Transformation from rectangular to spherical coordinates. .... 4 2.1 Hydrogenic Coulomb wave functions for states with n =1,2and 3. ................................... 30 2.2 The radial wave function for a Coulomb potential with Z =2is shown at several steps in the iteration procedure leading to the 4p eigenstate. ............................ 40 2.3 Electron interaction potentials from Eqs.(2.94) and (2.95) with parameters a =0.2683 and b =0.4072 chosen to fit the first four sodium energy levels. ........................ 43 2.4 Thomas-Fermi functions for the sodium ion, Z = 11, N = 10. Upper panel: the Thomas-Fermi function φ(r). Center panel: N(r), the number of electrons inside a sphere of radius r. Lower panel: U(r), the electron contribution to the potential. ..... 47 2.5 Radial Dirac Coulomb wave functions for the n = 2 states of hy- drogenlike helium, Z = 2. The solid lines represent the large com- ponents P2κ(r) and the dashed lines