Low Frequency Plasma Conductivity in the Average-Atom Approximation

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Low Frequency Plasma Conductivity in the Average-Atom Approximation Walter Johnson & Michael Kuchiev Physical Review E 78, 026401 (2008) 1. Review of Average-Atom Linear Response Theory 2.

1 Low Frequency Plasma Conductivity in the Average-Atom Approximation Walter Johnson & Michael Kuchiev Physical Review E 78, (2008) 1. Review of Average-Atom Linear Response Theory 2. Demonstration of a low-frequency divergence in σ(ω) 3. Finite relaxation time and resolution of problem Collaborators: C. Guet, G. Bertsch, Jim Albritton, Joe Nilsen, K.T. Cheng Seminar LLNL August

2 Average-Atom Model of a Plasma Reminder Plasma composed of neutral spheres with Wigner-Seitz radius R = (3Ω 0 /4π) 1/3 floating in a jellium sea. p 2 2m r + V! u a (r) = ɛu a (r) V (r) = d 3 r ρ(r ) r r + V exc(ρ) 4πr 2 ρ(r) = X nl 2(2l + 1) 1 + exp[(ɛ nl µ)/kt ] P nl(r) 2 = r<r ρ(r) d 3 r R 0 4πr 2 ρ(r) dr Seminar LLNL August

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

4 Reminder 10-1 ρ c (r) 10-2 ρ 0 R WS πr 2 ρ b (r) 4πr 2 ρ c (r) eff (r) R WS r (a.u.) Seminar LLNL August

5 Reminder 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 Seminar LLNL August

6 Kubo-Greenwood Reminder 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 m 2 ωω (f i f j ) j ɛ p i 2 δ(ɛ j ɛ i ω), ij which is an average-atom version of the Kubo 1 -Greenwood 2 formula. 1 R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957) 2 D. A. Greenwood, Proc. Phys. Soc. London 715, 585 (1958) Seminar LLNL August

7 QM perturbation theory Infrared Catastrophe in Scattering H 1 = V (r) e mc (ɛ p) e iωt e T 21 = 2πi δ(e 2 E 1 ω) p 2 ɛ p p 1 mc X p 2 ɛ p n n V p 1 X p 2 V n n ɛ p p 1 + E n n E 1 E n n E 1 ω e = 2πi δ(e 2 E 1 ω) 1 mc ω ɛ q p 2 V p 1 + Seminar LLNL August

8 QM perturbation theory Infrared Catastrophe in Scattering H 1 = V (r) e mc (ɛ p) e iωt e T 21 = 2πi δ(e 2 E 1 ω) p 2 ɛ p p 1 mc X p 2 ɛ p n n V p 1 X p 2 V n n ɛ p p 1 + E n n E 1 E n n E 1 ω e = 2πi δ(e 2 E 1 ω) 1 mc ω ɛ q p 2 V p 1 + Seminar LLNL August

9 Low-Frequency Theorem (QED) QM perturbation theory p 2 ɛ p p 1 1 (ɛ q) V (q) ω Relation between scattering amplitude and potential (Born approximation) f el (θ) = m 2π V (q) p 2 ɛ p p 1 2π mω (ɛ q) f el(θ) Seminar LLNL August

10 QM perturbation theory Example of Dipole Matrix Elements 0.0 Dipole Matrix Element < f p i > Al metallic density T=10eV l f =1 - l i =2 l f =3 - l i = ω(a.u.) Seminar LLNL August

11 Low-Frequency Kubo-Greenwood Simplification f 1 f 2 ω f E σ(ω) 2πe2 m 2 Ω d 3 p 1 (2π) 3 d 3 p 2 (2π) 3 f «p 2 ɛ p p 1 2 δ(e 2 E 1 ω) E D p 2 ɛ p p 1 2E ave 1 (2π) 2 3 m 2 ω 2 q2 σ el (θ) = 2 (2π) 2 3 m 2 ω 2 p2 (1 cos θ) σ el (θ) Seminar LLNL August

12 Low-Frequency Kubo-Greenwood Simplification σ(ω) 2πe2 Ω 2 (2π) 2 3 m 4 ω 2 d 3 p 1 (2π) 3 d 3 p 2 (2π) 3 f «E where σ tr (p) = p 2 1 (1 cos θ) σ el(θ)δ(e 2 E 1 ω) e 2 σ(ω) = 2 3 ω 2 Ω d 3 p (2π) 3 (1 cos θ) σ el (θ) dω = 4π p 2 f «v 3 σ tr (p) E X sin 2 (δ l+1 (p) δ l (p)) l=0 Seminar LLNL August

13 Low-Frequency KG Formula Comparison 1 = σ tr(p) Λ p Ω τ p = Λ p v = Ω vσ tr (p) σ(ω) = 2e2 3 d 3 p (2π) 3 f «v 2 1 E ω 2 τ p (Low-Freq K-G formula) σ static = 2e2 3 d 3 p (2π) 3 f «v 2 τ p E (iman formula) Rule: K-G formula reduces to iman formula under replacement 1/(ω 2 τ p ) τ p Seminar LLNL August

14 Influence of Collisions where Effect: ψ(p, t) exp» i (p r Et) Γ p 2 t Γ p 2 = 1 τ p 1 ( E) 1 2 ( E) 2 + Γ 2 p /4 = 1 ω 2 + 1/τp 2 With this in mind, the Modified KG Formula becomes σ(ω) = 2e2 3 d 3 p (2π) 3 f «E v 2 τ p ω 2 τ 2 p + 1 (Modified KG Formula) Seminar LLNL August

15 Comparison of Conductivity Formulas 0.03 Conductivity (a.u.) K-G Formula K-G Low Freq Approx Modified K-G Aluminum Plasma T=5eV 0.27 gm/cc Photon Energy (a.u.) Seminar LLNL August

16 Conductivity Sum Rule 0 σ(ω)dω = πe2 d 3 p f «3 (2π) 3v2 E = e2 π dedω p 3 f «= e 2 π 3 (2π) 3 m E = e2 π d 3 p m (2π) f(e) = e2 π 3 2m dedω (2π) 3 pf(e) Seminar LLNL August

17 σ (a.u.) σ (a.u.) Summary of Modified KG Formula 3s-3p bound-bound 2p-3s 2s-3p free-free Photon Energy (a.u.) bound-free n=3 ε n=2 ε total Photon Energy (a.u.) Seminar LLNL August

18 Dispersion Relations By Cauchy s theorem, a function f(z) analytic in the upper half plane that falls off as 1/ z satisfies f(x 0 ) = 1 «iπ P.V. f(x) x x 0 Apply to Modified K-G formula for Re[σ(ω)] to find Im[σ(ω)] = 2e2 3 d 3 p (2π) 3 f «E v 2 ωτ 2 p ω 2 τ 2 p + 1 σ(ω) as an analytic function of ω is therefore σ(ω) = 2e2 3 d 3 p (2π) 3 f «E v 2 τ p 1 iωτ p Seminar LLNL August

19 Dielectric Function, Index of Refraction Alternatively, ɛ r (ω) = 1 + 4πi σ(ω) ω Re[ɛ r (ω)] = 1 4π Im[σ(ω)] ω Im[ɛ r (ω)] = 4π Re[σ(ω)] ω Index of Refraction: n(ω) + iκ(ω) = q ɛ r (ω) Reflection Coefficient: R(ω) = 1 n(ω) iκ(ω) 1 + n(ω) + iκ(ω) 2 Seminar LLNL August

20 Applications With Joe Nilsen Seminar LLNL August

21 Conclusions Linear response theory applied to the average-atom model of a plasma leads to a version of the Kubo-Grenwood formula for conductivity. The resulting formula has a second-order pole at ω = 0. Including finite relaxation time in the time dependence of scattering wave function replaces pole by Drude type energy denominator. Evaluating the relaxation time using phase shifts from the AA model lead to the iman formula at ω = 0. Seminar LLNL August

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