A Primer on Atomic Theory Calculations (for X-ray Astrophysicists)

F. RobicheauxAuburn 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 -kZe2/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(r1,r2) = k e2/|r1 – r2|

2 electrons can exchange energy & angular momentum

2p6 1S mixes “strongly” with 2p43d2 1S but “weakly” with 2p23d4 1S

2s2p6 2S can decay into 2s22p4Ed 2S (auto-ionization)

Physical Effects: RelativitySpin-orbit interaction Mass-velocity

KE = p2/2m – p4/8m3c2 + …

Darwin termDirac equation ~ spread electron over distance ~h/mc

Quantum Electro-Dynamics effectsSelf energy, vacuum polarization, Breit interaction

Structure: Hartree/Dirac-FockApproximate the wave function by single antisymmetrized wave function.

Example: 1s2 1S

(1,2)=R10(r1)Y00(1) R10(r2)Y00(2) (12 – 12)/21/2

Equation for unknown function determined by variational principle. No correlation!

Difficulties

Equations are nonlinearOnly E variational

Advantages

Well developed programsFastFix by better calcs

Structure: Perturbation TheoryCorrections to wave function can be small.

Example: 1s2 1S + 2s2 1S + 3s2 1S + …

Ea = <a|H|a> + b |<a|H|b>|2/(Ea,0 – Eb,0) + …

0th order states determined by “simple” H.Numerical calculation of matrix elements

Difficulties

Not available for most statesStrong effects

Advantages

Well developed programsCan be very, very accurateHigher order correlations

Structure: MCHF/MCDFApproximate the wave function by superposition of antisymmetrized wave functions.

Example: 1s2 1S + 2s2 1S

(1,2)=[C1R10(r1)R10(r2) + C2R20(r1)R20(r2)] (L=0,S=0)

Equation for unknown functions and coefficients determined by variational principle.

Difficulties

Equations are nonlinearSolve 1 state at a timeMainly for deep states

Advantages

Well developed programsCan be very accurateFewest terms in sum

Structure: R-matrixApproximate the wave function by superposition of antisymmetrized wave functions.

Example: 1s2 1S + 2s2 1S

(1,2)=[C1R10(r1)R10(r2) + C2R20(r1)R20(r2)] (L=0,S=0)

Functions found outside but coefficients determined by variational principle.

Difficulties

Many basis functionsSmall/large corrections treated same

Advantages

Well developed programsCan be very accurateEquations are linear

Structure: Mixed CI & perturbativeUse 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 enoughNot full pert. potential

Advantages

Complicated interaction included

Structure: TransitionsRadiative decay computed using transition matrix elements.

Transition matrix elements are not variational.Electric dipole allowed transitions are typically strongest.

Beware

Spin changing transitions (2s2p 3P1 2s2 1S0)Dipole forbidden transitions (3d 2s)Two electron transitions (2p3d 1P 2s2 1S)Nearly degenerate states

n*if dV || T initfininitfin TT

!!!!! 0 !!!!!

e- Scattering: Non-resonant Pert. Th.Direct transition of target from initial to final state

Example: (1s2 1S) Ep 2P (1s2p 3P) Es 2P

Transition amplitude approximated Tf i = <f

(0)|V|i(0)>

Plane wave Born No potential for continuumDistorted wave Born Avg potential for continuum

Difficulties

No resonancesStrong couplingWhich average potential?

Advantages

FastAccurate for ionsMore accurate target states

e- Scattering: Resonant Pert. Th.Direct & indirect transitions of target

Example:(1s2 1S) Ep 2P 1s3s3p 2P (1s2p 3P) Es 2P

Transition amplitude approximatedTf i = <i

(0)|V|f(0)> + n V(0)

fn [E – En + i n/2]-1 V(0)ni

What potential to use for bound and continuum states?Interference and interaction through continuum?

Difficulties

Strong couplingWhich average potential?Inaccurate bound states

Advantages

FastFix by better calcsEasy averaging

e- Scattering: R-matrixVariational calculation for log-derivative at boundary

Basis set expansion of Hamiltonian in small region

Rij = ½ n yin yjn /(E – En)

Analytic or numerical function take R TLong range interaction through integration/perturbationDiagonalize matrix once for each LSJ

Difficulties

Less accurate targetPseudo-resonancesFine energy mesh

Advantages

Accurate channel couplingRadiation damp. & relativityPseudo-states for ionization

e- Scattering: Other close-couplingSpecial 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: ExcitationExcitation cross section directly used in computing the radiated power.

Li in electron plasma ne = 1010 cm-3 dotted—PWBdashed—DWBsolid—RMPSLi

Li+

Li2+

Perturbation theory worse for neutral.

DWB not that bad.

Thermodynamics can help less accurate calcs.

e- Scattering Example: Ionization

blue dashed—DWBgreen dot-dashed—CTMCred solid—RMPS

Perturbation theory worse for higher n-states.

CTMC does not quickly improve with n

DWB does better for ionization of Li2+

Average over

e- Scattering Example: DR of N4+

all orders pert theor

Upper 4 calcs use exptl 2s-2pj splittings

Bottom graph: diagonalization+pert

Low T might have problems

Hard work for 2 active electrons

Glans et al, PRA 64,043609 (2001).

Details of 1s22p5l

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.