Xray Scattering from WDM

Transcription

1 from WDM in the Approximation W. R. Johnson, Notre Dame Collaborators: Joe Nilsen & K. T. Cheng, LLNL Computational Challenges in WDM

2 Outline 1 2 Scattering by Free Electrons Inelastic Scattering by Bound Electrons 3 Hydrogen Beryllium Titanium Tin

3 Procedure Use the average-atom model 1 to describe plasma Input: atomic species (Z, A), density, temperature Output: ψ a (r), n b (r), n c (r), Z i, µ... Evaluate Thomson scattering 2 with input from A-A 1 Feynman, Metropolis & Teller (1949) 2 Chihara (2000), Gregori et al. (2003)

4 Model Divide plasma into neutral cells that include nucleus and Z electrons [ ] p 2 2 Z r + V ψ a (r) = ɛ a ψ a (r) V (r) = V Kohn-Sham (n(r), r) n(r) = n b (r) + n c (r) 4πr 2 n b (r) = nl Z = r<r WS n(r) d 3 r 2(2l+1) 1+exp[(ɛ nl µ)/k B T ] P nl(r) 2 Number of equations = N b + N l N ɛ 500 Equations are solved self-consistently

5 Model Divide plasma into neutral cells that include nucleus and Z electrons [ ] p 2 2 Z r + V ψ a (r) = ɛ a ψ a (r) V (r) = V Kohn-Sham (n(r), r) n(r) = n b (r) + n c (r) 4πr 2 n b (r) = nl Z = r<r WS n(r) d 3 r 2(2l+1) 1+exp[(ɛ nl µ)/k B T ] P nl(r) 2 Number of equations = N b + N l N ɛ 500 Equations are solved self-consistently

6 Model Divide plasma into neutral cells that include nucleus and Z electrons [ ] p 2 2 Z r + V ψ a (r) = ɛ a ψ a (r) V (r) = V Kohn-Sham (n(r), r) n(r) = n b (r) + n c (r) 4πr 2 n b (r) = nl Z = r<r WS n(r) d 3 r 2(2l+1) 1+exp[(ɛ nl µ)/k B T ] P nl(r) 2 Number of equations = N b + N l N ɛ 500 Equations are solved self-consistently

7 Model Divide plasma into neutral cells that include nucleus and Z electrons [ ] p 2 2 Z r + V ψ a (r) = ɛ a ψ a (r) V (r) = V Kohn-Sham (n(r), r) n(r) = n b (r) + n c (r) 4πr 2 n b (r) = nl Z = r<r WS n(r) d 3 r 2(2l+1) 1+exp[(ɛ nl µ)/k B T ] P nl(r) 2 Number of equations = N b + N l N ɛ 500 Equations are solved self-consistently

8 Model Divide plasma into neutral cells that include nucleus and Z electrons [ ] p 2 2 Z r + V ψ a (r) = ɛ a ψ a (r) V (r) = V Kohn-Sham (n(r), r) n(r) = n b (r) + n c (r) 4πr 2 n b (r) = nl Z = r<r WS n(r) d 3 r 2(2l+1) 1+exp[(ɛ nl µ)/k B T ] P nl(r) 2 Number of equations = N b + N l N ɛ 500 Equations are solved self-consistently

9 Model Divide plasma into neutral cells that include nucleus and Z electrons [ ] p 2 2 Z r + V ψ a (r) = ɛ a ψ a (r) V (r) = V Kohn-Sham (n(r), r) n(r) = n b (r) + n c (r) 4πr 2 n b (r) = nl Z = r<r WS n(r) d 3 r 2(2l+1) 1+exp[(ɛ nl µ)/k B T ] P nl(r) 2 Number of equations = N b + N l N ɛ 500 Equations are solved self-consistently

10 Model Divide plasma into neutral cells that include nucleus and Z electrons [ ] p 2 2 Z r + V ψ a (r) = ɛ a ψ a (r) V (r) = V Kohn-Sham (n(r), r) n(r) = n b (r) + n c (r) 4πr 2 n b (r) = nl Z = r<r WS n(r) d 3 r 2(2l+1) 1+exp[(ɛ nl µ)/k B T ] P nl(r) 2 Number of equations = N b + N l N ɛ 500 Equations are solved self-consistently

11 Model Divide plasma into neutral cells that include nucleus and Z electrons [ ] p 2 2 Z r + V ψ a (r) = ɛ a ψ a (r) V (r) = V Kohn-Sham (n(r), r) n(r) = n b (r) + n c (r) 4πr 2 n b (r) = nl Z = r<r WS n(r) d 3 r 2(2l+1) 1+exp[(ɛ nl µ)/k B T ] P nl(r) 2 Number of equations = N b + N l N ɛ 500 Equations are solved self-consistently

12 Example: Al metal T=10eV A = 27 ρ = 2.7 (gm/cc) R WS = 2.99 (au) State W(au) occ# 1s s p N b 9.97 N c 3.03 µ = (au) Z i = 2.32 n i = cm 3 n e = cm 3

13 Al metal T=10eV, continued n l V (r) V =0 n n n n n n n n n N c

14 Al metal T=10eV, continued Continuum Wave Functions P 0 (pr)/pr P 2 (pr)/pr P 5 (pr)/pr P 9 (pr)/pr j 0 (pr) j 2 (pr) j 5 (pr) j 9 (pr) r (au)

15 Al metal T=10eV, continued πr 2 n(r) 10 5 N b =10 N c = 3 R WS n c (r) Ω WS 10 R WS r(a.u.) r (a.u.)

16 Al metal T=10eV, continued πr 2 n(r) 10 5 N b =10 N c = 3 R WS n c (r) Ω WS 10 R WS r(a.u.) 1 N free = r (a.u.)

17 Wigner-Seitz Sphere in Electron-Ion Jellium A simplified picture that emerges is of a single neutral average atom floating in a uniform sea of Z i free electrons per cell balanced by an equal but opposite distributed positive ionic charge.

18 Thompson Scattering Scattering by Free Electrons Inelastic Scattering by Bound Electrons (ǫ 0 k 0) (p 0 E 0) (ǫ 1 k 1) (p 1 E 1) In nonrelativistic limit, this leads to dσ dω 1 dω = ɛ 0 ɛ 1 2 r Exchange of photons ω 1 ω 0 S(k, ω) with k = k 0 k 1, ω = ω 0 ω 1, where S(k, ω) is the dynamic structure function of the plasma.

19 Dynamic Structure Function Scattering by Free Electrons Inelastic Scattering by Bound Electrons The dynamic structure function S(k, ω) of a plasma can be decomposed into three parts: 3 3 Chihara (2000)

20 Dynamic Structure Function Scattering by Free Electrons Inelastic Scattering by Bound Electrons The dynamic structure function S(k, ω) of a plasma can be decomposed into three parts: 3 1 f (k) + q(k) 2 S ii (k) δ(ω) elastic scattering by ions 3 Chihara (2000)

21 Dynamic Structure Function Scattering by Free Electrons Inelastic Scattering by Bound Electrons The dynamic structure function S(k, ω) of a plasma can be decomposed into three parts: 3 1 f (k) + q(k) 2 S ii (k) δ(ω) elastic scattering by ions 2 S ee (k, ω) scattering by free electrons. 3 Chihara (2000)

22 Dynamic Structure Function Scattering by Free Electrons Inelastic Scattering by Bound Electrons The dynamic structure function S(k, ω) of a plasma can be decomposed into three parts: 3 1 f (k) + q(k) 2 S ii (k) δ(ω) elastic scattering by ions 2 S ee (k, ω) scattering by free electrons. 3 S B (k, ω) inelastic scattering by bound electrons. 3 Chihara (2000)

23 Scattering by Free Electrons Inelastic Scattering by Bound Electrons S ii (k, ω) = f (k) + q(k) 2 S ii (k) δ(ω)

24 Scattering by Free Electrons Inelastic Scattering by Bound Electrons S ii (k, ω) = f (k) + q(k) 2 S ii (k) δ(ω) f (k) + q(k) = 4π R WS 0 r 2 [n b (r) + n c (r)] j 0 (kr) dr

25 Scattering by Free Electrons Inelastic Scattering by Bound Electrons S ii (k, ω) = f (k) + q(k) 2 S ii (k) δ(ω) f (k) + q(k) = 4π R WS 0 r 2 [n b (r) + n c (r)] j 0 (kr) dr S ii (k) is obtained from the Fourier transform of V ii (R)

26 Scattering by Free Electrons Inelastic Scattering by Bound Electrons S ii (k, ω) = f (k) + q(k) 2 S ii (k) δ(ω) f (k) + q(k) = 4π R WS 0 r 2 [n b (r) + n c (r)] j 0 (kr) dr S ii (k) is obtained from the Fourier transform of V ii (R) Formulas by Yu.V. Arkhipov and A.E. Davletov (1998)

27 Scattering by Free Electrons Inelastic Scattering by Bound Electrons S ii (k, ω) = f (k) + q(k) 2 S ii (k) δ(ω) f (k) + q(k) = 4π R WS 0 r 2 [n b (r) + n c (r)] j 0 (kr) dr S ii (k) is obtained from the Fourier transform of V ii (R) Formulas by Yu.V. Arkhipov and A.E. Davletov (1998) Generalized by Gregori et al. (2006) to include T i T e

28 Scattering by Free Electrons Inelastic Scattering by Bound Electrons S ii (k, ω) = f (k) + q(k) 2 S ii (k) δ(ω) f (k) + q(k) = 4π R WS 0 r 2 [n b (r) + n c (r)] j 0 (kr) dr S ii (k) is obtained from the Fourier transform of V ii (R) Formulas by Yu.V. Arkhipov and A.E. Davletov (1998) Generalized by Gregori et al. (2006) to include T i T e S ii (k) T i /T e =1 T i /T e =0.5 T i /T e = 0.1 S ii (k,ω) (a.u.) Be T=20 (ev) θ=40 deg fwhm= 10eV k (a.u.) ω 1 (ev)

29 Scattering by Free Electrons Scattering by Free Electrons Inelastic Scattering by Bound Electrons 1 k 2 [ ] 1 S ee (k, ω) = I 1 exp( ω/k B T ) 4πn e ε(k, ω) Random-Phase Approximation for Dielectric function ε(k, ω): ε(k, ω) = πk 2 1 [ dη 1 0 p exp[(p 2 /2 µ)/k B T ] dp 1 k 2 2pkη + 2ω + iν + 1 k 2 + 2pkη 2ω iν ],

30 Scattering by Free Electrons Inelastic Scattering by Bound Electrons Dielectric Functions for Be metal T=10eV θ=30 deg θ=90 deg Re[ε] Im[ε] -Im[1/ε] ω (ev) Re[ε] 1 0 Im[ε] ω (ev) ε(k, ω) for ω 0 = 2690eV, with ω = ω 0 ω 1.

31 Example: S ee (k, ω) for Be metal Scattering by Free Electrons Inelastic Scattering by Bound Electrons 3 Be metal T=20eV 3 S ee (k.ω) (a.u.) 2 1 S ee S ii 40 deg 2 1 S ee S ii 90 deg ω 1 (ev) ω 1 (ev)

32 Scattering by Free Electrons Inelastic Scattering by Bound Electrons Inelastic Scattering from Bound Electrons Plane-Wave Final States S nl (k, ω) = p dωp (2π) 3 [ d 3 r e iq r ψ nlm (r) m 2 E p=ω+e nl ]

33 Example: Al 5eV Plane-Wave Final State Scattering by Free Electrons Inelastic Scattering by Bound Electrons S(k,ω) (a.u.) deg 150 deg L L 2 L L L 2 L ω (a.u.) ω (a.u.)

34 Example: Be 10eV Final State Scattering by Free Electrons Inelastic Scattering by Bound Electrons S nl (k, ω) = 0.4 S(k,ω) (a.u.) p dωp (2π) 3 d 3 r ψ p(r) e ik r ψ nlm (r) m 30 deg 150 deg 0.3 Plane-Wave Plane-Wave Aver-Atom * 10 Coulomb * Coulomb Ep=ω+E nl. Aver-Atom ω (a.u.) ω (a.u.)

35 Hydrogen Beryllium Titanium Tin : Hydrogen (high density n e = cm 3 ) Beryllium (light element with available experimental data) Titanium (intermediate atomic weight element) Tin (heavy metal with interesting bound-state features)

36 Hydrogen: T = 50eV Hydrogen Beryllium Titanium Tin n c (r) & n e Hydrogen T=50eV n c (r) R WS 0.1 n e = cm r (a.u.) S(k,ω) (a.u.) deg 40 deg 30 deg ω 1 (ev)

37 Hydrogen Beryllium Titanium Tin Beryllium: Comparison with Experiment S(k,ω) (a.u.) ω 1 (ev) Intensity Energy (ev) model for xray scattering by Be metal (T = 18 ev, n e = ) compared with measurement. 4 ω 0 = 2963 ev & θ = S. H. Glenzer & T. Doeppner, private communication

38 Hydrogen Beryllium Titanium Tin Titanium metal (Z=22) at T = 10 ev, ω 0 = 2960 ev S(k,ω) plasmon 3p 3s 30 deg N b =17.67 N c = ω 1 ω 0 (ev) S(k,ω) p 3s 150 deg ω 1 ω 0 (ev)

39 Hydrogen Beryllium Titanium Tin Tin (Z=50) at T = 10 ev, ω 0 = 2960 ev S(k,ω) (a.u.) deg deg S(k,ω) (a.u.) ω 1 ω 0 (ev) ω 1 ω 0 (ev)

40 Hydrogen Beryllium Titanium Tin Tin (Z=50) at T = 10 ev, ω 0 = 2960 ev S(k,ω) (a.u.) deg deg S(k,ω) (a.u.) ω 1 ω 0 (ev) ω 1 ω 0 (ev)

41 Hydrogen Beryllium Titanium Tin Tin (Z=50) at T = 10 ev, ω 0 = 2960 ev S(k,ω) (a.u.) deg deg S(k,ω) (a.u.) ω 1 ω 0 (ev) ω 1 ω 0 (ev)

42 Hydrogen Beryllium Titanium Tin Tin (Z=50) at T = 10 ev, ω 0 = 2960 ev S(k,ω) s 4p S(k,ω) 2.0 4d s+4p+4d ω 1 ω 0 (ev)

43 Summary References Summary: A-A model is used to study Xray scattering from WDM. Scattering from bound-states easily accommodated To be done: Improve the treatment of S ii (k) (hypernetted chains? or molecular dymamics?) Go beyond RPA and include correlation corrections to S ee (k, ω)

44 Summary References References Feynman, Metropolis & Teller, Phys. Rev. 75, 1561 (1949) S. H. Glenzer & R. Redmer, Rev. Mod. Phys. 81, 1625 (2009) G. Gregori et al., Phys. Rev. E 67, (2003) J. Chihara, J. Phys.: Condens. Matter 12, 231 (2000) W.R. Johnson et al., JQSRT, 99, 327 (2006) S. Sahoo et al., Phys. Rev. E 77, (2008)