Computing Relativistic Atomic Form Factors

Slides from my Physics Honours Thesis Presentation in November 1998 on the computation of relativistic atomic form factors. The research involved using the Dirac equation to look the interaction between 1 and 2-electron atoms and high energy X-rays using both analytical and computational approaches.

Transcript

1. INVESTIGATION OF THEORETICAL APPROACHES FOR COMPUTING RELATIVISTIC ATOMIC FORM FACTORS Michael Papasimeon Supervisor : Dr. Christopher Chantler Optics Group, School of Physics The University of Melbourne 1
2. EXAMPLES OF USEFULNESS OF ATOMIC FORM FACTORS: f • Crystallography: Structure Factors F(hkl) = j fj e−Mj e2πi(hxj +kyj +lzj ) • Materials Science: Optical Properties of Materials (Refractive Index nr and Dielectric Constant ǫ) nr = n + ik = √ ǫ = 1 − δ − iβ = 1 − r0 2π λ2 j nj fj • Applications in X Ray Optics Including experimental work undertaken in School of Physics X Ray lab 2
3. THE ATOMIC FORM FACTOR f: WHAT IS IT? Photon-Atom interactions are described by QFT. f = f0 + f′ + if′′ f0 (q) f′(ω) f′′(ω) 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 ATOM PHOTON PHOTON 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 PHOTON ATOM PHOTON 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 PHOTON ATOM EJECTED ELECTRON PHOTON ATOM IN EXCITED STATE NORMAL ANOMALOUS ANOMALOUS 3
4. HOW DO WE CALCULATE f = f0 + f′ + if′′ ? • NORMAL FORM FACTOR: The scattering power of an atom relative to the scattering power of a free electron. f0 (q) = ρ(r)eiq·r dr ; q = |kf − ki | = 4π sin(θ/2) λ ◦ A −1 • IMAGINARY COMPONENT OF ANOMALOUS FORM FACTOR: Related to the total photoionisation cross section σ(ω). (r0 = e2/mc2) f′′(ω) = ω 4πcr0 σ(ω) • REAL COMPONENT OF ANOMALOUS FORM FACTOR: f′(ω) can be calculated from f′′(ω) using a Kramers-Kronig dispersion relation. 4
5. THEORETICAL LIMITATIONS AND ASSUMPTIONS • Isolated Atom • Electromagnetic Field: Classical. Electric Dipole, Electric Quadrupole, All Poles, RMP • Atomic Structure: Schr¨ odinger, Dirac • Perturbation Theory: 1st order relativistic, S-Matrix (QFT) • Numerical and Computational Issues: singularities, convergence 5
6. PROJECT AIM AND RESULTS AIM: Investigate the issues, assumption and limitations in atomic form factor theory by a critical analysis and study of hydrogenic atoms • New analytic result for relativistic normal form factor • New semi analytic results for first and second order photoionisation amplitudes. • New numerical results for f′′(ω) using S-matrix theory and relativistic perturbation theory. • Calculated bound-bound relativistic transition amplitudes for the first three excited states for hydrogenic atoms. • Angular dependent results 6
7. NORMAL FORM FACTOR f0 (q) FOR HYDROGENIC ATOMS ANGULAR DEPENDENT CONTRIBUTION • ANALYTIC NON RELATIVISTIC RESULT (has been done before) f0 (q) = 2Z a0 4 2Z a0 2 + q2 −2 • NEW ANALYTIC RELATIVISTIC RESULT f0 (q) = Γ(2γ1 ) 2iqΓ(2γ1 + 1) 2Z a0 2γ1 +1      2Z a0 + iq 2γ1 − 2Z a0 − iq 2γ1 2Z a0 2 + q2 2γ1      • γ1 = 1 − (αZ)2, α = fine structure constant, a0 = Bohr radius, Z = Atomic Number. For low Z, γ1 ≈ 1. 7
8. ATOMIC HYDROGEN 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 fo(q) Momentum Transfer q (1/A) Normal Form Factor for Atomic Hydrogen Non Relativistic Relativistic 0 5e-07 1e-06 1.5e-06 2e-06 2.5e-06 3e-06 3.5e-06 4e-06 4.5e-06 5e-06 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Difference in fo(q) Momentum Transfer q (1/A) Relative Difference: Relativistic - Non Relativistic Difference • Approximately 0.015% difference between reltativistic and non relativistic results. • Current experimental precision: 0.1% – 1% 8
9. HYDROGENIC URANIUM 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 10 20 30 40 50 60 70 80 90 100 fo(q) Momentum Transfer q (1/A) Hydrogenic Uranium Relativistic Non Relativistic 9
10. PHOTOIONISATION COORDINATE SYSTEM +X +Y +Z Φ Incident Photon k = (kx,ky,kz) Atom (in this case k = (0,0,1) ) Ejected Electron k’ = (kx’,ky’,kz’) Coordinate System for Photoionisation Θ k′ x = |k′| sin(Θ) cos(Φ) , k′ y = |k′| sin(Θ) sin(Φ) , k′ z = |k′| cos(Θ) 10
11. IMAGINARY ANOMALOUS ATOMIC FORM FACTOR f′′(ω) • APPROACHES TO CALCULATING f′′(ω): Standard Perturbation Theory, Relativistic Perturbation Theory, Relativistic S-Matrix Theory • RELATIVISTIC PHOTON ABSORPTION AND EMISSION OPERATORS (QFT) Ai = j ¯ α · ˆ ǫj eiki ·rj A† f = j ¯ α · ˆ ǫj e−ikf ·rj Sum over j electrons, ¯ α = Dirac alpha matrix, ˆ ǫj = photon polarisation, k = photon wavevector, k′ = ejected electron wave vector, rj = coordinate of j-th electron. • RELATIVISTIC PHOTOIONISATION AMPLITUDE: HYDROGEN A1 (k, k′) = ψc |Ai |ψ0 = ψc |eik·r ¯ αj |ψ0 11
12. ALL POLES AND ELECTRIC DIPOLE RESULTS A1 (k, k′)(x y ) = G0 Γ(γ1 + 2) √ 4π × 2π 0 π 0 1 i sin(θ)[ξ(k′ x − ik′ y ) ± iF0 sin(θ)eiφ] (1 2 σ1 − iµ(q, θ, φ))γ1 +2 dθ dφ AE1 1 (k, k′)j = A1 (0, k′)j µ(q, θ, φ) = qx sin φ cos θ + qy sin φ sin θ + qz cos φ 12
13. THE FORWARD SCATTERING DIRECTION A1 (k, k′)x = π √ π 2Z a0 3/2 1 − ǫ1 2Γ(2γ1 + 1) × Z a0 −(γ1 +2) Γ(γ1 + 2) 2 F1 γ1 + 2 2 , γ1 + 3 2 ; 1; − 2a0 Z 2 (k − k′)2 + 1 8 (k − k′)2 Z a0 −(γ1 +4) Γ(γ1 + 4)× 2 F1 γ1 + 4 2 , γ1 + 5 2 ; 1; − 2a0 Z 2 (k − k′)2 A1 (k, k′)y = −iA1 (k, k′)x ; |A1 (k, k′)y |2 = |A1 (k, k′)x |2 13
14. RELATIVISTIC S-MATRIX THEORY APPLIED TO ATOMIC FORM FACTOR CALCULATIONS ImA2 (ω) = r0 f′′(ω) = ω 4πc σT OT (ω) A2 = −r0 mc2 p m|A† f |p p|Ai |n En − Ep + ¯ hωf + i0+ + m|Ai |p p|A† f |n En − Ep − ¯ hωi + i0+ AR 2 (ω) = A2 (k, k′)j = −r0 mc2 ∞ 0 |A1 (k, k′)j |2 E0 − Ec + ¯ hω + i0+ dEc − r0 mc2 ∞ 0 |A1 (−k, k′)j |2 E0 − Ec − ¯ hω − i0+ dEc AE1 2 (k, k′)j = A2 (0, k′)j 14
15. NUMERICAL CALCULATIONS • Approximately 5000 lines of C++ • Approximately 2000 lines of Mathematica • Quadrature Methods – Simpson, Trapezoidal – Gauss-Legendre (10 point) – Converging Romberg • Intensive/Expensive Computation: Triple Integrals θ,φ, and Energy • Singularities, open interval and numerical Cauchy Principal value integrations • Parameters: Bound-Bound i0+ = iΓ 2 Continuum i0+ = small value. 15
16. RESULTS - ALL POLES AND ELECTRIC DIPOLE -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 10 20 30 40 50 60 70 80 90 100 log10 f’’ (e/atom) Energy (keV) f’’ : E1 to All Poles Comparison All Poles Dipole 16
17. COMPARISON: CHANTLER (BOUND H), KISSEL (ATOMIC H) 0 0.2 0.4 0.6 0.8 1 1.2 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 f’’ (e/atom) Energy (keV) Comparison with Chantler (Bound H, DHF), Kissel (H, S-Matrix) Kissel and Pratt Chantler Papasimeon 17
18. CONCLUSIONS AND FURTHER WORK • Summary of Results: Hydrogenic Atoms – New analytic result for relativistic normal form factor – New Semi analytic results for first and second order photoionisation amplitudes. – New Numerical results for f′′(ω) using S-matrix theory and relativistic perturbation theory. – Calculated bound-bound relativistic transition amplitudes for the first three excited states for hydrogenic atoms. – Angular dependent results • Further Work: Refine convergence, develop relativistic perturbation theory computation of f′(ω), XAFS (X Ray Anomalous Fine Structure) - multiple scattering processes off multiple atoms (eg: molecular hydrogen). 18
19. QUESTIONS? 19