A Primer on Atomic Theory Calculations (for X-ray Astrophysicists) F. Robicheaux Auburn University Mitch Pindzola and Stuart Loch I.Physical Effects II.Atomic Structure (wavelength, decay rate, …) III.Electron Scattering (excitation, ionization, …)

Physical Effects: mean field Electrons screen the charge of nucleus. Near nucleus V decreases faster than -kZe 2 /r Low see deeper potential and are more deeply bound 3s is more strongly bound than 3p which is more strongly bound than 3d actual

Physical Effects: correlation The main interaction between two electrons is through V(r 1,r 2 ) = k e 2 /|r 1 – r 2 | 2 electrons can exchange energy & angular momentum 2p 6 1 S mixes “strongly” with 2p 4 3d 2 1 S but “weakly” with 2p 2 3d 4 1 S 2s2p 6 2 S can decay into 2s 2 2p 4 Ed 2 S (auto-ionization)

Physical Effects: Relativity Spin-orbit interactionMass-velocity KE = p 2 /2m – p 4 /8m 3 c 2 + … Darwin term Dirac equation ~ spread electron over distance ~h/mc Quantum Electro-Dynamics effects Self energy, vacuum polarization, Breit interaction

Structure: Hartree/Dirac-Fock Approximate the wave function by single antisymmetrized wave function. Example: 1s 2 1 S  (1,2)=R 10 (r 1 )Y 00 (  1 ) R 10 (r 2 )Y 00 (  2 ) (  1  2 –  1  2 )/2 1/2 Equation for unknown function determined by variational principle. No correlation! Difficulties Equations are nonlinear Only E variational Advantages Well developed programs Fast Fix by better calcs

Structure: Perturbation Theory Corrections to wave function can be small. Example: 1s 2 1 S + 2s 2 1 S + 3s 2 1 S + … E a = +  b | | 2 /(E a,0 – E b,0 ) + … 0 th order states determined by “simple” H. Numerical calculation of matrix elements Difficulties Not available for most states Strong effects Advantages Well developed programs Can be very, very accurate Higher order correlations

Structure: MCHF/MCDF Approximate the wave function by superposition of antisymmetrized wave functions. Example: 1s 2 1 S + 2s 2 1 S  (1,2)=[C 1 R 10 (r 1 )R 10 (r 2 ) + C 2 R 20 (r 1 )R 20 (r 2 )] (L=0,S=0) Equation for unknown functions and coefficients determined by variational principle. Difficulties Equations are nonlinear Solve 1 state at a time Mainly for deep states Advantages Well developed programs Can be very accurate Fewest terms in sum

Structure: R-matrix Approximate the wave function by superposition of antisymmetrized wave functions. Example: 1s 2 1 S + 2s 2 1 S  (1,2)=[C 1 R 10 (r 1 )R 10 (r 2 ) + C 2 R 20 (r 1 )R 20 (r 2 )] (L=0,S=0) Functions found outside but coefficients determined by variational principle. Difficulties Many basis functions Small/large corrections treated same Advantages Well developed programs Can be very accurate Equations are linear

Structure: Mixed CI & perturbative Use configuration interaction method to include some effects. Use perturbation theory to include other effects. Examples: Non-relativistic CI – mass-velocity, S.O., Darwin Relativistic CI – Q.E.D. Difficulties May not be accurate enough Not full pert. potential Advantages Complicated interaction included

Structure: Transitions Radiative decay computed using transition matrix elements. Transition matrix elements are not variational. Electric dipole allowed transitions are typically strongest. Beware Spin changing transitions (2s2p 3 P 1  2s 2 1 S 0 ) Dipole forbidden transitions (3d  2s) Two electron transitions (2p3d 1 P  2s 2 1 S) Nearly degenerate states !!!!! 0 !!!!!

e - Scattering: Non-resonant Pert. Th. Direct transition of target from initial to final state Example: (1s 2 1 S) Ep 2 P  (1s2p 3 P) E s 2 P Transition amplitude approximated T f  i = Plane wave BornNo potential for continuum Distorted wave BornAvg potential for continuum Difficulties No resonances Strong coupling Which average potential? Advantages Fast Accurate for ions More accurate target states

e - Scattering: Resonant Pert. Th. Direct & indirect transitions of target Example:(1s 2 1 S) Ep 2 P 1s3s3p 2 P (1s2p 3 P) E s 2 P Transition amplitude approximated T f  i = +  n V (0) fn [E – E n + i  n /2] -1 V (0) ni What potential to use for bound and continuum states? Interference and interaction through continuum? Difficulties Strong coupling Which average potential? Inaccurate bound states Advantages Fast Fix by better calcs Easy averaging

e - Scattering: R-matrix Variational calculation for log-derivative at boundary Basis set expansion of Hamiltonian in small region R ij = ½  n y in y jn /(E – E n ) Analytic or numerical function take R  T Long range interaction through integration/perturbation Diagonalize matrix once for each LSJ  Difficulties Less accurate target Pseudo-resonances Fine energy mesh Advantages Accurate channel coupling Radiation damp. & relativity Pseudo-states for ionization

e - Scattering: Other close-coupling Special purpose close coupling methods can be very accurate for specific problems. Important for testing more heavily used methods & experiment. Convergent close coupling (CCC)-solve Lippman- Schwinger equation using basis set technique Time dependent close coupling (TDCC)-solve the time dependent Schrodinger equation (usually grid of points) Hyperspherical close coupling (HSCC)-solve for the time independent wave function using hyperspherical coords

e - Scattering Example: Excitation Excitation cross section directly used in computing the radiated power. Li in electron plasma n e = cm -3 dotted—PWB dashed—DWB solid—RMPS Li Li + Li 2+ Perturbation theory worse for neutral. DWB not that bad. Thermodynamics can help less accurate calcs.

e - Scattering Example: Ionization blue dashed—DWB green dot-dashed—CTMC red solid—RMPS Perturbation theory worse for higher n-states. CTMC does not quickly improve with n DWB does better for ionization of Li 2+ Average over

e - Scattering Example: DR of N 4+ all orders pert theor Upper 4 calcs use exptl 2s-2p j splittings Bottom graph: diagonalization+pert Low T might have problems Hard work for 2 active electrons Glans et al, PRA 64, (2001). Details of 1s 2 2p5l

Concluding Remarks “Must” use CI/CC or mixed methods (CI+pert) for neutrals and near neutrals. Scattering from “highly” excited atoms very difficult but errors may not be important. Typical weak transitions are less accurate than typical strong transitions. Photo-recombination can be abnormally sensitive at low temperatures if low lying resonances are present. Ionization in neutrals and near neutrals is difficult. AMO + plasma modeling needed for practical error est.