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 (ω)
• Index of refraction n(ω) + iκ(ω) =
– ND –
p
(ω)
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Introduction
Free electron model used in plasma diagnostics:
nfree(ω) =
r
ω02
ω02
1− 2 ≈1− 2 <1
ω
2ω
where
ω02
e2 Nfree
= 4π
m vol
Recent experiments on Al plasmas find n > 1 at few eV temperatures
• LLNL comet laser facility1 (14.7 nm Ni-like Pd laser)
• Advanced Photon Research Center JAERI2 (13.9 nm Ni-like Ag laser)
Reason: Effect of bound electrons on optical properties.
1
2
J. Filevich et al. Proceedings of the 9th International Conference on X-Ray Lasers, May 23-28 (2004)
H. Tang et al., Appl. Phys. B78, 975 (2004)
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Average-Atom Model
QM version of generalized Thomas-Fermi model3
Inside a neutral (Wigner-Seitz) cell:
2
p
Z
− + V ua(r) = ua(r)
2m r
(1)
V = Vdir(r) + Vexc(r) for r ≤ R and V = 0 otherwise.
∇2Vdir = −4πρ
(2)
Vexc(ρ) is given in the local density approximation
3
R. P. Feynman, N. Metropolis and E. Teller, Phys. Rev. 75 1561 (1949)
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Thermal Average Electron Density
Contributions to the density are
ρb(r)
=
ρc(r)
=
X
1 X
2
2(2l
+
1)
f
(
)
P
(r)
nl
nl
4πr 2 l
n
Z
∞
1 X
2
2(2l
+
1)
d
f
()
P
(r)
l
4πr 2 l
0
where
f () =
(3)
(4)
1
1 + exp[( − µ)/kT ]
The chemical potential µ is chosen to insure electric neutrality:
Z=
Z
3
ρ(r) d r ≡
r<R
Z
R
2
4πr ρ(r) dr .
(5)
0
Eqs. (1-5) are solved self-consistently for ρ, V , and µ.
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Example
Al: density 0.27 gm/cc, T = 5 eV, R = 6.44 a.u., µ = −0.3823 a.u.
Bound States
State
Energy
1s
-55.189
2s
-3.980
2p
-2.610
3s
-0.259
3p
-0.054
Nbound
– ND –
n(l)
2.0000
2.0000
6.0000
0.6759
0.8300
11.5059
l
0
1
2
3
4
5
6
7
8
9
10
Nfree
Continuum States
n(l)
n0(l)
0.1090 0.1975
0.2149 0.3513
0.6031 0.3192
0.2892 0.2232
0.1514 0.1313
0.0735 0.0674
0.0326 0.0308
0.0132 0.0127
0.0049 0.0048
0.0017 0.0016
0.0005 0.0005
1.4941 1.3404
∆n(l)
-0.0885
-0.1364
0.2839
0.0660
0.0201
0.0061
0.0018
0.0005
0.0001
0.0001
0.0000
0.1537
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-1
10
ρc(r)
ρ0
-2
10
RWS
-3
10
0
2
4
8
6
15
2
4πr ρb(r)
10
2
4πr ρc(r)
Zeff(r)
5
0
– ND –
RWS
0
2
4
r (a.u.)
6
8
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Linear Response and the Kubo-Greenwood Formula
Consider an applied electric field:
E(t) = F ẑ sin ωt
A(t) =
F
ẑ cos ωt
ω
The time dependent Schrödinger equation becomes
∂
eF
T0 + V (n, r) −
vz cos ωt ψi(r, t) = i ψi(r, t)
ω
∂t
The current density is
2e X
Jz (t) =
fi hψi(t)|vz |ψi(t)i
Ω i
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Kubo-Greenwood
• Linearize ψi(r, t) in F
• Evaluate the response current: J = Jin sin(ωt) + Jout cos(ωt)
• Determine σ(ω): Jin(t) = σ(ω) Ez (t)
Result:
2πe2 X
σ(ω) =
(fi − fj ) |hj|vz |ii|2 δ(j − i − ω),
ωΩ ij
which is an average-atom version of the Kubo4-Greenwood5 formula.
4
5
R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957)
D. A. Greenwood, Proc. Phys. Soc. London 715, 585 (1958)
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Example: Al T=3eV & density= 0.27gm/cc
0.005
σ (a.u.)
0.004
0.005
bound-bound
0.003
0.003
0.002
0.002
2p-3s
0.001
0
0.01
0.1
2s-3p
1
10
0.030
σ (a.u.)
0.025
– ND –
n=2 ε
0.001
0
0.01
0.1
1
10
0.030
free-free
0.025
0.020
0.020
0.015
0.015
0.010
0.010
0.005
0.005
0.000
0.01
bound-free
n=3 ε
0.004
3s-3p
0.1
1
Photon Energy (a.u.)
10
0.000
0.01
total
0.1
1
Photon Energy (a.u.)
10
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Optical Properties
For a conducting medium, the dielectric function is related to the complex
conductivity by
σ(ω)
(ω) = 1 + 4πi
ω
We know <σ(ω); we must evaluate =σ(ω)
From analytic properties of σ(ω) one infers the dispersion relation6
Z ∞
2ω0
< σ(ω)
= σ(ω0) =
−
dω.
π 0 ω02 − ω 2
6
R. de L. Kronig and H. A. Kramers, Atti Congr. Intern. Fisici, 2, 545 (1927)
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Conductivity (a.u.)
Application of Dispersion Relation
0.030
0.003
0.020
0.002
0.010
0.001
0.000
0.000
-0.010
-0.001
-0.020
-0.030
0.001
– ND –
Re[σ(ω)]
Im[σ(ω)]
T=5eV
ρ=0.27 gm/cc
-0.002
0.01
0.1
Photon Energy (a.u.)
1
-0.003
1
2
3
Photon Energy (a.u.)
4
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Index of Refraction
=σ(ω)
<σ(ω)
<(ω) = 1 − 4π
=(ω) = 4π
,
ω
ω
√
n + iκ = .
1
10
n(ω) & κ(ω)
0
10
-1
10
-2
10
T=5eV
ρ=0.27gm/cc
n(ω)
nfree(ω)
κ(ω)
κfree(ω)
-3
10
0.01
– ND –
0.1
1
Photon Energy (a.u.)
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Al: Comparison with Free Electron Model
Plasma with ion density nion = 1020/cc
5
0
T=3eV
<Z>=1.38
-5
-10
– ND –
2p-3d
2p-4d
2p-3s
Pd x-ray
Ag x-ray
(n-1)/(nfree-1)
10
0
20
40
80
60
Photon Energy (eV)
100
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Al: Penetration Depth
Plasma with ion density nion = 1020/cc
Penetration Depth (µm)
3
10
2
10
T=3eV
<Z>=1.38
1
10
0
10
0
– ND –
20
40
80
60
Photon Energy (a.u.)
100
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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
80-90 eV frequency range.
• Even away from bound-bound resonances, the free electron model may be
misleading.
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