Optical Properties of Plasmas Based on an Average-Atom Model

Transkript

1 Optical Properties of Plasmas Based on an Average-Atom Model Walter Johnson, Notre Dame University Claude Guet, CEA/DAM Ile de France George Bertsch, University of Washington Motivation for this work: Joseph Nilsen, LLNL Average Atom (NR version of Inferno ) Linear Response Kubo-Greenwood formula for σ(ω) Kramers-Kronig Dispersion Relation Dielectric Function ɛ(ω) Index of refraction n(ω) + iκ(ω) = ɛ(ω)

2 2 Introduction Free electron model used in plasma diagnostics: n free (ω) = 1 ω2 0 ω 2 1 ω2 0 2ω 2 < 1 where ω2 0 = 4π e2 m N free vol Recent experiments on Al plasmas find n > 1 at few ev temperatures LLNL comet laser facility 1 (14.7 nm Ni-like Pd laser) Advanced Photon Research Center JAERI 2 (13.9 nm Ni-like Ag laser) Reason: Effect of bound electrons on optical properties. 1 J. Filevich et al. Proceedings of the 9th International Conference on X-Ray Lasers, May (2004) 2 H. Tang et al., Appl. Phys. B78, 975 (2004)

3 3 Average-Atom Model QM version of generalized Thomas-Fermi model 3 Inside a neutral (Wigner-Seitz) cell: [ p 2 2m Z r + V ] u a (r) = ɛu a (r) (1) V = V dir (r) + V exc (r) for r R and V = 0 otherwise. V exc (ρ) is given in the local density approximation 2 V dir = 4πρ (2) 3 R. P. Feynman, N. Metropolis and E. Teller, Phys. Rev (1949)

4 4 Thermal Average Electron Density Contributions to the density are ρ b (r) = ρ c (r) = 1 4πr 2 l 1 4πr 2 l 2(2l + 1) n 2(2l + 1) 0 f(ɛ nl ) P nl (r) 2 (3) dɛ f(ɛ) P ɛl (r) 2 (4) where f(ɛ) = exp[(ɛ µ)/kt ] The chemical potential µ is chosen to insure electric neutrality: Z = r<r ρ(r) d 3 r R 0 4πr 2 ρ(r) dr. (5) Eqs. (1-5) are solved self-consistently for ρ, V, and µ.

5 5 Example Al: density 0.27 gm/cc, T = 5 ev, R = 6.44 a.u., µ = a.u. Bound States Continuum States State Energy n(l) l n(l) n 0 (l) n(l) 1s s p s p Nbound Nfree

6 ρ c (r) 10-2 ρ 0 R WS πr 2 ρ b (r) 4πr 2 ρ c (r) Z eff (r) R WS r (a.u.)

7 7 Linear Response and the Kubo-Greenwood Formula Consider an applied electric field: E(t) = F ẑ sin ωt A(t) = F ω ẑ cos ωt The time dependent Schrödinger equation becomes [ T 0 + V (n, r) ef ] ω v z cos ωt ψ i (r, t) = i t ψ i(r, t) The current density is J z (t) = 2e Ω f i ψ i (t) v z ψ i (t) i

8 8 Kubo-Greenwood Linearize ψ i (r, t) in F Evaluate the response current: J = J in sin(ωt) + J out cos(ωt ) Determine σ(ω): J in (t) = σ(ω) E z (t) Result: σ(ω) = 2πe2 ωω (f i f j ) j v z i 2 δ(ɛ j ɛ i ω), ij which is an average-atom version of the Kubo 4 -Greenwood 5 formula. 4 R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957) 5 D. A. Greenwood, Proc. Phys. Soc. London 715, 585 (1958)

9 9 Example: Al T=3eV & density= 0.27gm/cc s-3p bound-bound bound-free n=3 ε σ (a.u.) n=2 ε p-3s 2s-3p σ (a.u.) free-free Photon Energy (a.u.) total Photon Energy (a.u.)

10 10 Optical Properties For a conducting medium, the dielectric function is related to the complex conductivity by ɛ(ω) = 1 + 4πi σ(ω) ω We know Rσ(ω); we must evaluate Iσ(ω) From analytic properties of σ(ω) one infers the dispersion relation 6 I σ(ω 0 ) = 2ω 0 π 0 R σ(ω) dω. ω2 ω R. de L. Kronig and H. A. Kramers, Atti Congr. Intern. Fisici, 2, 545 (1927)

11 11 Application of Dispersion Relation Conductivity (a.u.) Re[σ(ω)] Im[σ(ω)] T=5eV ρ=0.27 gm/cc Photon Energy (a.u.) Photon Energy (a.u.)

12 12 Index of Refraction Rɛ(ω) = 1 4π Iσ(ω) Iɛ(ω) = 4π Rσ(ω) ω ω, n + iκ = ɛ n(ω) & κ(ω) T=5eV ρ=0.27gm/cc n(ω) n free (ω) κ(ω) κ free (ω) Photon Energy (a.u.)

13 13 Al: Comparison with Free Electron Model Plasma with ion density n ion = /cc (n-1)/(n free -1) T=3eV <Z>=1.38 2p-3d 2p-4d 2p-3s Pd x-ray Ag x-ray Photon Energy (ev)

14 14 Plasma with ion density n ion = /cc Al: Penetration Depth Penetration Depth (µm) T=3eV <Z>= Photon Energy (a.u.)

15 15 Conclusions Linear response theory applied to average atom model provides a straightforward method for obtaining the frequency-dependent conductivity. The dielectric function (and index of refraction) can be reconstructed with the aid of a dispersion relation, The model explains observed behavior of low temperature Al plasmas in the ev frequency range. Even away from bound-bound resonances, the free electron model may be misleading.