Low-Frequency Conductivity in the Average-Atom Approximation

Transcription

1 Low-Frequency Conductivity in the Average-Atom Approximation Walter Johnson, Notre Dame University Collaborators: Joe Nilsen, K.T. Cheng, Jim Albritton, Michael Kuchiev, C. Guet, G. Bertsch Related Contributions: Poster #30 - P. Renaudin et al. Talk (Friday AM) - R. Piron and T. Blenski Average-Atom & Static Conductivity (Ziman) Kubo-Greenwood Formula (Infrared Catastrophe) R Proper Static Limit & Conductivity Sum Rule Application to Plasma Optics RPHDM-08 1

2 Average Atom & Static Conductivity Average-Atom Model of a Plasma Plasma composed of neutral spheres with Wigner-Seitz radius R = (3Ω/4π) 1/3 floating in a jellium sea. ( p 2 2m Z r + V ) u a (r) = ɛu a (r) V (r) = d 3 r ρ(r ) r r + V exc(ρ) 4πr 2 ρ(r) = nl 2(2l + 1) 1 + exp[(ɛ nl µ)/kt ] P nl(r) 2 Z = r<r ρ(r) d 3 r R 0 4πr 2 ρ(r) dr RPHDM-08 2

3 Average Atom & Static Conductivity 10-1 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.) Al: density 0.27 gm/cc, T = 5 ev RPHDM-08 3

4 Average Atom & Static Conductivity Phase-Shifts & Scattering 4 Al: density 0.27 gm/cc, T = 5 ev Phase Shift δ l (E) Electron Energy E (a.u.) l=0 l=1 l=2 l=3 l=4 l=5 l=6 f(θ) = 2ip 1 ( l e 2iδ ) l 1 P l (cos θ), σ el (θ) = f(θ) 2 RPHDM-08 4

5 Static Conductivity Transport Cross Section and Conductivity Classical Drude Formula: σ = ne 2 τ/m, where n is free electron density and τ is mean time between collisions. Generally: τ p = Λ p v σ tr (p) = (relaxation time) Λ p = Ω σ tr (p) (mean free path) (1 cos θ) σ el (θ) dω (transport cross section) Conductivity obtained as a thermal average of Drude Formula: σ = 2e2 3 d 3 p (2π) 3 ( f ) v 2 τ p E (Ziman formula) RPHDM-08 5

6 Comparison with Experiment Example Resistivity (Ω-cm) Al T=20,000 K Ziman BSM D&K K&K Density (gm/cc) BSM: J. F. Benage, W. R. Shanahan, and M. S. Murillo, Phys. Rev. Letts. 83, 2953 (1999); Kunze, Phys. Rev. E49, 4448 (1994); K & K: I. Krisch and H.-J. Kunze, Phys. Rev. E58, 6557 (1998). D & K: A. W. DeSilva and H.-J. RPHDM-08 6

7 Linear Response and Conductivity Linear Response 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 RPHDM-08 7

8 Kubo-Greenwood Linear Response 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(e j E i ω), ij which is an average-atom version of the Kubo 1 -Greenwood 2 formula. (n.b. bound-bound, bound-free, free-free contributions) 1 R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957) 2 D. A. Greenwood, Proc. Phys. Soc. London 715, 585 (1958) RPHDM-08 8

9 Infrared Catastrophe Infrared Catastrophe Particle in a potential V (r): p 2 ɛ p p 1 = 1 (ɛ q) V (q) ω Relation between scattering amplitude and potential Result: (Low-frequency theorem QED) f(θ) = m 2π V (q) p 2 ɛ p p 1 = 2π mω (ɛ q) f(θ) RPHDM-08 9

10 Low-Frequency Kubo-Greenwood Infrared Catastrophe Free-free contribution: p 2 ɛ p p 1 2 ave σ(ω) 2πe2 d 3 p 1 d 3 p 2 m 2 Ω (2π) 3 (2π) 3 = 2e2 d 3 ( p f ) 3 (2π) 3 E 2 (2π) 2 3 m 2 ω 2 p2 (1 cos θ) σ el (θ) f 1 f 2 ω f E ( f ) p 2 ɛ p p 1 2 δ(e 2 E 1 ω) E v 2 1 ω 2 τ p (Low-Freq K-G formula) RPHDM-08 10

11 Proper Static Limit Influence of Collisions on Wave Function Effect 3 : ψ(p, t) exp 1 ω 1 2 ω 2 + 1/τp 2 [ i (p r Et) t ] τ p τ 2 p ω 2 τ 2 p + 1 With this in mind, the low-frequency K-G Formula becomes σ(ω) 2e2 3 d 3 p (2π) 3 ( f ) E v 2 τ p ω 2 τ 2 p + 1 (Modified K-G Formula) 3 M. Yu. Kuchiev and W. R. Johnson, Phys. Rev. E 78, (2008); Green s Functions for Solid State Physics, S. Doniach & E.H. Sondheimer, Imperial College Press (1998), Chap. 5 RPHDM-08 11

12 Proper Static Limit 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.) RPHDM-08 12

13 Conductivity Sum Rule Conductivity Sum Rule 0 σ(ω)dω = πe2 d 3 ( p f ) 3 (2π) 3v2 E = e2 π dedω p 3 ( f ) 3 (2π) 3 m E dedω = e 2 π (2π) pf(e) 3 = e2 π d 3 p m (2π) f(e) = e2 π 3 2m Z RPHDM-08 13

14 σ (a.u.) σ (a.u.) Summary of Modified K-G 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.) RPHDM-08 14

15 Dispersion Relations Application to Plasma Optics 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 RPHDM-08 15

16 Application to Plasma Optics Dielectric Function, Index of Refraction ɛ r (ω) = 1 + 4πi σ(ω) ω, n(ω) + iκ(ω) = ɛ r (ω) n(ω) & κ(ω) T=5eV ρ=0.27gm/cc n(ω) n free (ω) κ(ω) κ free (ω) Photon Energy (a.u.) RPHDM-08 16

17 Application to Plasma Optics Comparison with Free Electron Model Plasma with ion density n ion = /cc (With Joe Nilsen) (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) RPHDM-08 17

18 Conclusions Application to Plasma Optics R Kubo-Greenwood formula for σ(ω) applied to the average-atom model diverges as R 1/ω 2 at low frequencies. Including finite relaxation time leads to an approximation for the free-free contribution R to σ(ω) that is finite and reduces to the Ziman formula at ω = 0. The modified Kubo-Greenwood formula provides a simple and useful approximation for studies of the optical response of plasmas. RPHDM-08 18