thomson scattering from wdm - nd.edujohnson/publications/rpwdm.pdf · average-atom thomson...
Post on 12-Mar-2019
225 Views
Preview:
TRANSCRIPT
Average-AtomThomson Scattering
Application to Aluminum
Thomson Scattering from WDMAverage-Atom Approximation
W. R. Johnson, Notre DameJ. Nilsen & K. T. Cheng, LLNL
The cross section for Thomson scattering of x-rays by warm densematter is studied within the framework of the average-atom model.
RPWDM-15
Average-AtomThomson Scattering
Application to Aluminum
Outline
1 Average-Atom
2 Thomson ScatteringElastic Scattering by IonsInelastic Scattering by Free ElectronsInelastic Scattering by Bound Electrons
3 Application to Aluminum
RPWDM-15
Average-AtomThomson Scattering
Application to Aluminum
Average-Atom Model
Divide plasma into WS cells with a nucleus and Z electrons[p2
2 −Zr + V
]ψa(r) = εa ψa(r)
V (r) = VKS(n(r), r)
n(r) = nb(r) + nc(r)
4πr2nb(r) =∑
nl2(2l+1)
1+exp[(εnl−µ)/kBT ] Pnl(r)2
Z =∫
r<RWSn(r) d3r
Number of equations = Nb + Nl × Nε ∼ 500Equations are solved self-consistently
RPWDM-15
Average-AtomThomson Scattering
Application to Aluminum
Average-Atom Model
Divide plasma into WS cells with a nucleus and Z electrons[p2
2 −Zr + V
]ψa(r) = εa ψa(r)
V (r) = VKS(n(r), r)
n(r) = nb(r) + nc(r)
4πr2nb(r) =∑
nl2(2l+1)
1+exp[(εnl−µ)/kBT ] Pnl(r)2
Z =∫
r<RWSn(r) d3r
Number of equations = Nb + Nl × Nε ∼ 500Equations are solved self-consistently
RPWDM-15
Average-AtomThomson Scattering
Application to Aluminum
Average-Atom Model
Divide plasma into WS cells with a nucleus and Z electrons[p2
2 −Zr + V
]ψa(r) = εa ψa(r)
V (r) = VKS(n(r), r)
n(r) = nb(r) + nc(r)
4πr2nb(r) =∑
nl2(2l+1)
1+exp[(εnl−µ)/kBT ] Pnl(r)2
Z =∫
r<RWSn(r) d3r
Number of equations = Nb + Nl × Nε ∼ 500Equations are solved self-consistently
RPWDM-15
Average-AtomThomson Scattering
Application to Aluminum
Average-Atom Model
Divide plasma into WS cells with a nucleus and Z electrons[p2
2 −Zr + V
]ψa(r) = εa ψa(r)
V (r) = VKS(n(r), r)
n(r) = nb(r) + nc(r)
4πr2nb(r) =∑
nl2(2l+1)
1+exp[(εnl−µ)/kBT ] Pnl(r)2
Z =∫
r<RWSn(r) d3r
Number of equations = Nb + Nl × Nε ∼ 500Equations are solved self-consistently
RPWDM-15
Average-AtomThomson Scattering
Application to Aluminum
Average-Atom Model
Divide plasma into WS cells with a nucleus and Z electrons[p2
2 −Zr + V
]ψa(r) = εa ψa(r)
V (r) = VKS(n(r), r)
n(r) = nb(r) + nc(r)
4πr2nb(r) =∑
nl2(2l+1)
1+exp[(εnl−µ)/kBT ] Pnl(r)2
Z =∫
r<RWSn(r) d3r
Number of equations = Nb + Nl × Nε ∼ 500Equations are solved self-consistently
RPWDM-15
Average-AtomThomson Scattering
Application to Aluminum
Average-Atom Model
Divide plasma into WS cells with a nucleus and Z electrons[p2
2 −Zr + V
]ψa(r) = εa ψa(r)
V (r) = VKS(n(r), r)
n(r) = nb(r) + nc(r)
4πr2nb(r) =∑
nl2(2l+1)
1+exp[(εnl−µ)/kBT ] Pnl(r)2
Z =∫
r<RWSn(r) d3r
Number of equations = Nb + Nl × Nε ∼ 500Equations are solved self-consistently
RPWDM-15
Average-AtomThomson Scattering
Application to Aluminum
Average-Atom Model
Divide plasma into WS cells with a nucleus and Z electrons[p2
2 −Zr + V
]ψa(r) = εa ψa(r)
V (r) = VKS(n(r), r)
n(r) = nb(r) + nc(r)
4πr2nb(r) =∑
nl2(2l+1)
1+exp[(εnl−µ)/kBT ] Pnl(r)2
Z =∫
r<RWSn(r) d3r
Number of equations = Nb + Nl × Nε ∼ 500Equations are solved self-consistently
RPWDM-15
Average-AtomThomson Scattering
Application to Aluminum
Average-Atom Model
Divide plasma into WS cells with a nucleus and Z electrons[p2
2 −Zr + V
]ψa(r) = εa ψa(r)
V (r) = VKS(n(r), r)
n(r) = nb(r) + nc(r)
4πr2nb(r) =∑
nl2(2l+1)
1+exp[(εnl−µ)/kBT ] Pnl(r)2
Z =∫
r<RWSn(r) d3r
Number of equations = Nb + Nl × Nε ∼ 500Equations are solved self-consistently
RPWDM-15
Average-AtomThomson Scattering
Application to Aluminum
Aluminum Metal
0 1 2 3 4r (a.u.)
10-1
100
101
102
Nc(r
)
0 1 2 3 4r (a.u.)
0
5
10
15
4πr2 n(
r) (
a.u.
)
Nc(r)
RWS
AluminumT=5eV density=2.7g/cc
RWS
RPWDM-15
Average-AtomThomson Scattering
Application to Aluminum
Aluminum Metal
0 1 2 3 4r (a.u.)
10-1
100
101
102
Nc(r
)
0 1 2 3 4r (a.u.)
0
5
10
15
4πr2 n(
r) (
a.u.
)
Zf
Nc(r)
RWS
AluminumT=5eV density=2.7g/cc
RWS
RPWDM-15
Average-AtomThomson Scattering
Application to Aluminum
Elastic Scattering by IonsInelastic Scattering by Free ElectronsInelastic Scattering by Bound Electrons
Thomson Scattering
(ǫ0 k0)
(p0 E0)
(ǫ1 k1)
(p1 E1)
+ Exchange of photons
In nonrelativistic limit, this leads to
dσdω1dΩ
= |ε0 · ε1|2 r20ω1
ω0S(k , ω)
with k = |k0 − k1|, ω = ω0 − ω1, where S(k , ω) is the dynamicstructure function of the plasma.
RPWDM-15
Average-AtomThomson Scattering
Application to Aluminum
Elastic Scattering by IonsInelastic Scattering by Free ElectronsInelastic Scattering by Bound Electrons
Elastic Scattering by Ions
Sii(k , ω) = |f (k) + q(k)|2Sii(k) δ(ω)
f (k) + q(k) = 4π∫ RWS
0 r2[nb(r) + nc(r)] j0(kr) drSii(k) is obtained from the Fourier transform of Vii(R)
Arkhipov and Davletov (1998) & Gregori et al. (2003)Modified by Gregori et al. (2006) to include Ti 6= Te
RPWDM-15
Average-AtomThomson Scattering
Application to Aluminum
Elastic Scattering by IonsInelastic Scattering by Free ElectronsInelastic Scattering by Bound Electrons
Elastic Scattering by Ions
Sii(k , ω) = |f (k) + q(k)|2Sii(k) δ(ω)
f (k) + q(k) = 4π∫ RWS
0 r2[nb(r) + nc(r)] j0(kr) dr
Sii(k) is obtained from the Fourier transform of Vii(R)
Arkhipov and Davletov (1998) & Gregori et al. (2003)Modified by Gregori et al. (2006) to include Ti 6= Te
RPWDM-15
Average-AtomThomson Scattering
Application to Aluminum
Elastic Scattering by IonsInelastic Scattering by Free ElectronsInelastic Scattering by Bound Electrons
Elastic Scattering by Ions
Sii(k , ω) = |f (k) + q(k)|2Sii(k) δ(ω)
f (k) + q(k) = 4π∫ RWS
0 r2[nb(r) + nc(r)] j0(kr) drSii(k) is obtained from the Fourier transform of Vii(R)
Arkhipov and Davletov (1998) & Gregori et al. (2003)Modified by Gregori et al. (2006) to include Ti 6= Te
RPWDM-15
Average-AtomThomson Scattering
Application to Aluminum
Elastic Scattering by IonsInelastic Scattering by Free ElectronsInelastic Scattering by Bound Electrons
Elastic Scattering by Ions
Sii(k , ω) = |f (k) + q(k)|2Sii(k) δ(ω)
f (k) + q(k) = 4π∫ RWS
0 r2[nb(r) + nc(r)] j0(kr) drSii(k) is obtained from the Fourier transform of Vii(R)
Arkhipov and Davletov (1998) & Gregori et al. (2003)
Modified by Gregori et al. (2006) to include Ti 6= Te
RPWDM-15
Average-AtomThomson Scattering
Application to Aluminum
Elastic Scattering by IonsInelastic Scattering by Free ElectronsInelastic Scattering by Bound Electrons
Elastic Scattering by Ions
Sii(k , ω) = |f (k) + q(k)|2Sii(k) δ(ω)
f (k) + q(k) = 4π∫ RWS
0 r2[nb(r) + nc(r)] j0(kr) drSii(k) is obtained from the Fourier transform of Vii(R)
Arkhipov and Davletov (1998) & Gregori et al. (2003)Modified by Gregori et al. (2006) to include Ti 6= Te
RPWDM-15
Average-AtomThomson Scattering
Application to Aluminum
Elastic Scattering by IonsInelastic Scattering by Free ElectronsInelastic Scattering by Bound Electrons
Elastic Scattering by Ions
0 1 2 3 4k (a.u.)
0.0
0.2
0.4
0.6
0.8
1.0
Sii(k
)
Ti/T
e=1
Ti/T
e=0.5
Ti/T
e=0.1
-20 -10 0 10 20ω0−ω1 (eV)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
S(k,
ω) (
1/eV
)
k=0.43
Al metalT= 5eV
ω0=3.1 keV
θ=30 deg
fwhm=3.1 eVk = 0.43 (a.u.)
RPWDM-15
Average-AtomThomson Scattering
Application to Aluminum
Elastic Scattering by IonsInelastic Scattering by Free ElectronsInelastic Scattering by Bound Electrons
Inelastic Scattering by Free Electrons
From Gregori et al. (2003):
See(k , ω) = − 11− exp(−ω/kBT )
k2
4π2neIm[
1ε(k , ω)
]
Random-phase approximation for dielectric function ε(k , ω):
ε(k , ω) = 1 +4πk2
∫ ∞0
p2
1 + exp[(p2/2− µ)/kBT ]dp
×∫ 1
−1dη[
1k2 − 2pkη + 2ω + iν
+1
k2 + 2pkη − 2ω − iν
].
RPWDM-15
Average-AtomThomson Scattering
Application to Aluminum
Elastic Scattering by IonsInelastic Scattering by Free ElectronsInelastic Scattering by Bound Electrons
Dielectric Function
0 10 20 30 40ω (eV)
0
2
4
6
ε(k,
ω)
Re[ε]Im[ε]-Im[1/ε]
-40 -20 0 20 40ω1−ω0 (eV)
0.00
0.04
0.08
0.12
0.16
S(k,
ω) (
1/eV
) See
Sii
Scattering from Al metal, kBT = 5 eV, ω0 = 3.1 keV, θ = 30
RPWDM-15
Average-AtomThomson Scattering
Application to Aluminum
Elastic Scattering by IonsInelastic Scattering by Free ElectronsInelastic Scattering by Bound Electrons
Inelastic Scattering by Bound Electrons
SB(k , ω) =∑
nl
Snl (k , ω)
Snl (k , ω) =onl
2l + 1
∑m
∫p dΩp
(2π)3
∣∣∣∣∫ d3r ψ†p(r)eik ·r ψnlm(r)
∣∣∣∣2Ep=ω+Enl
.
Plane-wave Approximation: ψp(r) ≈ eip·r
Snl (k , ω) =onl
πk
∫ p+k
|p−k|q dq |Knl (q)|2, where
Knl (q) =
∫ ∞0
dr r jl (qr)Pnl (r).
RPWDM-15
Average-AtomThomson Scattering
Application to Aluminum
Elastic Scattering by IonsInelastic Scattering by Free ElectronsInelastic Scattering by Bound Electrons
0 5 10 15 20ω (a.u.)
0
0.01
0.02
0.03
0.04
S1s
(k,ω
) (a
.u.)
0 20 40 60 80ω (a.u.)
0
0.01
0.02
0.03
0.04
0.05
0.06
θ=30 degω0=2960 eV ω0=18 keV
θ=150 degk=0.41 k=9.32
CoulombAve Atom
(Plane Wave)/10
Bound-state contribution to S(k , ω) for x-ray scattering from theK -shell of Be metal at kBT=18 eV.
RPWDM-15
Average-AtomThomson Scattering
Application to Aluminum
Elastic Scattering by IonsInelastic Scattering by Free ElectronsInelastic Scattering by Bound Electrons
Comparison with Sahoo et al. Phys. Rev. E 77, 046402 (2008)
-1000 -500 00
1
2
-1000 -500 0
Energy Shift (eV)
0
0.3
0.6
Te = 5 eV, θ = 1300
L - shell
M - shell
T e = 5 eV, θ = 1300
-1000 -500 0Energy Shift (eV)
0
0.1
0.2
S(k,
ω) ω
pl
Al metal T= 5 eV
LI
LII
ω0=9.3 keV
θ=130 degL
RPWDM-15
Average-AtomThomson Scattering
Application to Aluminum
Application to Aluminum
-150 -75 00
0.2
0.4
0.6
0.8
S(k,
ω) ω
pl
-80 -40 0 400
0.4
0.8
1.2
-500 -250 0
ω1−ω0 (eV)
0
0.2
0.4
0.6
-150 -75 00
0.2
0.4
0.6
ω0=9300 eV
ω0=9300 eV
ω0=3100 eV
ω0=3100 eV
θ=30 deg θ=30 deg
θ=130 deg θ=130 deg
RPWDM-15
ConclusionReferences
Summary:AA model is used to study x-ray scattering from WDM.Scattering from bound-states easily accommodated.Present results disagree with earlier AA results by Sahooet al. for Al.
To be done:Understand why Ti/Te 6= 1 is needed in Sii(k , ω) in someequilibrium cases but not in others?Go beyond RPA for See(k , ω) and include correlationeffects.
RPWDM-15
ConclusionReferences
References
G. Gregori et al., Phys. Rev. E 67, 026412 (2003)
S. H. Glenzer & R. Redmer, Rev. Mod. Phys. 81, 1625 (2009)
G. Gregori et al., Phys. Rev. E 74, 026402 (2006)
Y. Arkhipov & A. Davletov, Phys. Letts. A 227, 339 (1998)
Johnson, Nilsen & Cheng, Phys. Rev. E 86, 036410 (2012)
S. Sahoo et al., Phys. Rev. E 77, 046402 (2008)
RPWDM-15
top related