a primer on atomic theory calculations (for x-ray astrophysicists) f. robicheaux auburn university...
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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.