Average-Atom
Thomson Scattering
Applications
Xray Scattering from WDM
Thomson Scattering in the Average-Atom Approximation
W. R. Johnson, Notre Dame
Collaborators: Joe Nilsen & K. T. Cheng, LLNL
Computational Challenges in WDM
PLWS-4
Xray Scattering
Average-Atom
Thomson Scattering
Applications
Outline
1
Average-Atom
2
Thomson Scattering
Elastic Scattering by Ions
Scattering by Free Electrons
Inelastic Scattering by Bound Electrons
3
Applications
Hydrogen
Beryllium
Titanium
Tin
PLWS-4
Xray Scattering
Average-Atom
Thomson Scattering
Applications
Procedure
Use the average-atom model1 to describe plasma
Input: atomic species (Z, A), density, temperature
Output: ψa (r ), nb (r ), nc (r ), Zi , µ . . .
Evaluate Thomson scattering2 with input from A-A
Applications
1
2
Feynman, Metropolis & Teller (1949)
Chihara (2000), Gregori et al. (2003)
PLWS-4
Xray Scattering
Average-Atom
Thomson Scattering
Applications
Average-Atom Model
Divide plasma into neutral cells that include nucleus and Z electrons
h
p2
2
−
Z
r
i
+ V ψa (r ) = a ψa (r)
V (r ) = VKohn-Sham (n(r ), r )
n(r ) = nb (r ) + nc (r )
P
2(2l+1)
Pnl (r )2
4πr 2 nb (r ) = nl 1+exp[(
nl −µ)/kB T ]
R
Z = r <R n(r ) d 3 r
WS
Number of equations = Nb + Nl × N ∼ 500
Equations are solved self-consistently
PLWS-4
Xray Scattering
Average-Atom
Thomson Scattering
Applications
Average-Atom Model
Divide plasma into neutral cells that include nucleus and Z electrons
h
p2
2
−
Z
r
i
+ V ψa (r ) = a ψa (r)
V (r ) = VKohn-Sham (n(r ), r )
n(r ) = nb (r ) + nc (r )
P
2(2l+1)
Pnl (r )2
4πr 2 nb (r ) = nl 1+exp[(
nl −µ)/kB T ]
R
Z = r <R n(r ) d 3 r
WS
Number of equations = Nb + Nl × N ∼ 500
Equations are solved self-consistently
PLWS-4
Xray Scattering
Average-Atom
Thomson Scattering
Applications
Average-Atom Model
Divide plasma into neutral cells that include nucleus and Z electrons
h
p2
2
−
Z
r
i
+ V ψa (r ) = a ψa (r)
V (r ) = VKohn-Sham (n(r ), r )
n(r ) = nb (r ) + nc (r )
P
2(2l+1)
Pnl (r )2
4πr 2 nb (r ) = nl 1+exp[(
nl −µ)/kB T ]
R
Z = r <R n(r ) d 3 r
WS
Number of equations = Nb + Nl × N ∼ 500
Equations are solved self-consistently
PLWS-4
Xray Scattering
Average-Atom
Thomson Scattering
Applications
Average-Atom Model
Divide plasma into neutral cells that include nucleus and Z electrons
h
p2
2
−
Z
r
i
+ V ψa (r ) = a ψa (r)
V (r ) = VKohn-Sham (n(r ), r )
n(r ) = nb (r ) + nc (r )
P
2(2l+1)
Pnl (r )2
4πr 2 nb (r ) = nl 1+exp[(
nl −µ)/kB T ]
R
Z = r <R n(r ) d 3 r
WS
Number of equations = Nb + Nl × N ∼ 500
Equations are solved self-consistently
PLWS-4
Xray Scattering
Average-Atom
Thomson Scattering
Applications
Average-Atom Model
Divide plasma into neutral cells that include nucleus and Z electrons
h
p2
2
−
Z
r
i
+ V ψa (r ) = a ψa (r)
V (r ) = VKohn-Sham (n(r ), r )
n(r ) = nb (r ) + nc (r )
P
2(2l+1)
Pnl (r )2
4πr 2 nb (r ) = nl 1+exp[(
nl −µ)/kB T ]
R
Z = r <R n(r ) d 3 r
WS
Number of equations = Nb + Nl × N ∼ 500
Equations are solved self-consistently
PLWS-4
Xray Scattering
Average-Atom
Thomson Scattering
Applications
Average-Atom Model
Divide plasma into neutral cells that include nucleus and Z electrons
h
p2
2
−
Z
r
i
+ V ψa (r ) = a ψa (r)
V (r ) = VKohn-Sham (n(r ), r )
n(r ) = nb (r ) + nc (r )
P
2(2l+1)
Pnl (r )2
4πr 2 nb (r ) = nl 1+exp[(
nl −µ)/kB T ]
R
Z = r <R n(r ) d 3 r
WS
Number of equations = Nb + Nl × N ∼ 500
Equations are solved self-consistently
PLWS-4
Xray Scattering
Average-Atom
Thomson Scattering
Applications
Average-Atom Model
Divide plasma into neutral cells that include nucleus and Z electrons
h
p2
2
−
Z
r
i
+ V ψa (r ) = a ψa (r)
V (r ) = VKohn-Sham (n(r ), r )
n(r ) = nb (r ) + nc (r )
P
2(2l+1)
Pnl (r )2
4πr 2 nb (r ) = nl 1+exp[(
nl −µ)/kB T ]
R
Z = r <R n(r ) d 3 r
WS
Number of equations = Nb + Nl × N ∼ 500
Equations are solved self-consistently
PLWS-4
Xray Scattering
Average-Atom
Thomson Scattering
Applications
Average-Atom Model
Divide plasma into neutral cells that include nucleus and Z electrons
h
p2
2
−
Z
r
i
+ V ψa (r ) = a ψa (r)
V (r ) = VKohn-Sham (n(r ), r )
n(r ) = nb (r ) + nc (r )
P
2(2l+1)
Pnl (r )2
4πr 2 nb (r ) = nl 1+exp[(
nl −µ)/kB T ]
R
Z = r <R n(r ) d 3 r
WS
Number of equations = Nb + Nl × N ∼ 500
Equations are solved self-consistently
PLWS-4
Xray Scattering
Average-Atom
Thomson Scattering
Applications
Example: Al metal T=10eV
A = 27
ρ = 2.7 (gm/cc)
State
1s
2s
2p
Nb
Nc
W(au)
-54.591
-3.388
-2.019
µ = −0.0209 (au)
ni = 6.02 × 1022 cm−3
PLWS-4
RWS = 2.99 (au)
occ#
2.00
2.00
5.97
9.97
3.03
Zi = 2.32
ne = 1.40 × 1023 cm−3
Xray Scattering
Average-Atom
Thomson Scattering
Applications
Al metal T=10eV, continued
nl
n0
n1
n2
n3
n4
n5
n6
n7
n8
Nc
V (r )
0.630
1.132
0.859
0.285
0.089
0.024
0.006
0.001
0.000
3.026
PLWS-4
V =0
0.601
0.838
0.533
0.236
0.081
0.023
0.005
0.001
0.000
2.318
∆
0.029
0.294
0.326
0.049
0.008
0.001
0.000
0.000
0.000
0.708
Xray Scattering
Average-Atom
Thomson Scattering
Applications
Al metal T=10eV, continued
Continuum Wave Functions
P0(pr)/pr
1.5
P2(pr)/pr
P5(pr)/pr
P9(pr)/pr
1.0
j0(pr)
j2(pr)
0.5
j5(pr)
j9(pr)
0.0
-0.5
0.0
1.0
2.0
r (au)
PLWS-4
Xray Scattering
3.0
Average-Atom
Thomson Scattering
Applications
Al metal T=10eV, continued
15
nc(r) ΩWS
Nb=10
2
4πr n(r)
100
RWS
10
Nc= 3
5
RWS
10
1
0
0
1
2
r(a.u.)
3
4
PLWS-4
0
Xray Scattering
1
2
r (a.u.)
3
4
Average-Atom
Thomson Scattering
Applications
Al metal T=10eV, continued
15
nc(r) ΩWS
Nb=10
2
4πr n(r)
100
RWS
10
Nc= 3
5
RWS
10
Nfree = 2.3
1
0
0
1
2
r(a.u.)
3
4
PLWS-4
0
Xray Scattering
1
2
r (a.u.)
3
4
Average-Atom
Thomson Scattering
Applications
Wigner-Seitz Sphere in Electron-Ion Jellium
A simplified picture that emerges is of a single neutral average atom
floating in a uniform sea of Zi free electrons per cell balanced by an
equal but opposite distributed positive ionic charge.
PLWS-4
Xray Scattering
Average-Atom
Thomson Scattering
Applications
Elastic Scattering by Ions
Scattering by Free Electrons
Inelastic Scattering by Bound Electrons
Thompson Scattering
(ǫ0 k0 )
(ǫ1 k1 )
(p0 E0 )
(p1 E1 )
+
Exchange of photons
In nonrelativistic limit, this leads to
ω1
dσ
= |0 · 1 |2 r02
S(k , ω)
dω1 dΩ
ω0
with k = |k 0 − k 1 |, ω = ω0 − ω1 , where S(k , ω) is the dynamic
structure function of the plasma.
PLWS-4
Xray Scattering
Average-Atom
Thomson Scattering
Applications
Elastic Scattering by Ions
Scattering by Free Electrons
Inelastic Scattering by Bound Electrons
Dynamic Structure Function
The dynamic structure function S(k , ω) of a plasma can be
decomposed into three parts:3
3
Chihara (2000)
PLWS-4
Xray Scattering
Average-Atom
Thomson Scattering
Applications
Elastic Scattering by Ions
Scattering by Free Electrons
Inelastic Scattering by Bound Electrons
Dynamic Structure Function
The dynamic structure function S(k , ω) of a plasma can be
decomposed into three parts:3
1
3
|f (k ) + q(k )|2 Sii (k ) δ(ω) elastic scattering by ions
Chihara (2000)
PLWS-4
Xray Scattering
Average-Atom
Thomson Scattering
Applications
Elastic Scattering by Ions
Scattering by Free Electrons
Inelastic Scattering by Bound Electrons
Dynamic Structure Function
The dynamic structure function S(k , ω) of a plasma can be
decomposed into three parts:3
1
|f (k ) + q(k )|2 Sii (k ) δ(ω) elastic scattering by ions
2
See (k , ω) scattering by free electrons.
3
Chihara (2000)
PLWS-4
Xray Scattering
Average-Atom
Thomson Scattering
Applications
Elastic Scattering by Ions
Scattering by Free Electrons
Inelastic Scattering by Bound Electrons
Dynamic Structure Function
The dynamic structure function S(k , ω) of a plasma can be
decomposed into three parts:3
1
|f (k ) + q(k )|2 Sii (k ) δ(ω) elastic scattering by ions
2
See (k , ω) scattering by free electrons.
3
SB (k , ω) inelastic scattering by bound electrons.
3
Chihara (2000)
PLWS-4
Xray Scattering
Average-Atom
Thomson Scattering
Applications
Elastic Scattering by Ions
Scattering by Free Electrons
Inelastic Scattering by Bound Electrons
Elastic Scattering by Ions
Sii (k , ω) = |f (k ) + q(k )|2 Sii (k ) δ(ω)
PLWS-4
Xray Scattering
Average-Atom
Thomson Scattering
Applications
Elastic Scattering by Ions
Scattering by Free Electrons
Inelastic Scattering by Bound Electrons
Elastic Scattering by Ions
Sii (k , ω) = |f (k ) + q(k )|2 Sii (k ) δ(ω)
RR
f (k ) + q(k ) = 4π 0 WS r 2 [nb (r ) + nc (r )] j0 (kr ) dr
PLWS-4
Xray Scattering
Average-Atom
Thomson Scattering
Applications
Elastic Scattering by Ions
Scattering by Free Electrons
Inelastic Scattering by Bound Electrons
Elastic Scattering by Ions
Sii (k , ω) = |f (k ) + q(k )|2 Sii (k ) δ(ω)
RR
f (k ) + q(k ) = 4π 0 WS r 2 [nb (r ) + nc (r )] j0 (kr ) dr
Sii (k ) is obtained from the Fourier transform of Vii (R)
PLWS-4
Xray Scattering
Average-Atom
Thomson Scattering
Applications
Elastic Scattering by Ions
Scattering by Free Electrons
Inelastic Scattering by Bound Electrons
Elastic Scattering by Ions
Sii (k , ω) = |f (k ) + q(k )|2 Sii (k ) δ(ω)
RR
f (k ) + q(k ) = 4π 0 WS r 2 [nb (r ) + nc (r )] j0 (kr ) dr
Sii (k ) is obtained from the Fourier transform of Vii (R)
Formulas by Yu.V. Arkhipov and A.E. Davletov (1998)
PLWS-4
Xray Scattering
Average-Atom
Thomson Scattering
Applications
Elastic Scattering by Ions
Scattering by Free Electrons
Inelastic Scattering by Bound Electrons
Elastic Scattering by Ions
Sii (k , ω) = |f (k ) + q(k )|2 Sii (k ) δ(ω)
RR
f (k ) + q(k ) = 4π 0 WS r 2 [nb (r ) + nc (r )] j0 (kr ) dr
Sii (k ) is obtained from the Fourier transform of Vii (R)
Formulas by Yu.V. Arkhipov and A.E. Davletov (1998)
Generalized by Gregori et al. (2006) to include Ti 6= Te
PLWS-4
Xray Scattering
Average-Atom
Thomson Scattering
Applications
Elastic Scattering by Ions
Scattering by Free Electrons
Inelastic Scattering by Bound Electrons
Elastic Scattering by Ions
Sii (k , ω) = |f (k ) + q(k )|2 Sii (k ) δ(ω)
RR
f (k ) + q(k ) = 4π 0 WS r 2 [nb (r ) + nc (r )] j0 (kr ) dr
Sii (k ) is obtained from the Fourier transform of Vii (R)
Formulas by Yu.V. Arkhipov and A.E. Davletov (1998)
Generalized by Gregori et al. (2006) to include Ti 6= Te
1.0
2.5
Ti/Te=1
Ti/Te=0.5
2.0
Ti/Te= 0.1
0.6
Sii(k,ω) (a.u.)
Sii(k)
0.8
0.4
0.2
0.0
Be T=20 (eV)
θ=40 deg
1.5
fwhm= 10eV
1.0
0.5
0
1
2
k (a.u.)
3
PLWS-4
4
2900
2920
Xray Scattering
2940
2960
ω1 (eV)
2980
3000
Average-Atom
Thomson Scattering
Applications
Elastic Scattering by Ions
Scattering by Free Electrons
Inelastic Scattering by Bound Electrons
Scattering by Free Electrons
1
k2
1
See (k , ω) = −
=
1 − exp(−ω/kB T ) 4πne
ε(k , ω)
Random-Phase Approximation for Dielectric function ε(k , ω):
ε(k , ω) = 1 +
1
4
πk 2
Z
0
∞
p2
1+
exp[(p2 /2
− µ)/kB T ]
1
1
dη 2
+ 2
,
k − 2pk η + 2ω + iν
k + 2pk η − 2ω − iν
−1
Z
dp
PLWS-4
Xray Scattering
Average-Atom
Thomson Scattering
Applications
Elastic Scattering by Ions
Scattering by Free Electrons
Inelastic Scattering by Bound Electrons
Dielectric Functions for Be metal T=10eV
θ=90 deg
θ=30 deg
6
5
4
3
2
1
0
-1
Re[ε]
-Im[1/ε]
Im[ε]
0
10
20 30
ω (eV)
40
6
5
4
3
2
1
0
-1
Re[ε]
Im[ε]
0
10
20 30
ω (eV)
ε(k , ω) for ω0 = 2690eV, with ω = ω0 − ω1 .
PLWS-4
Xray Scattering
40
Average-Atom
Thomson Scattering
Applications
Elastic Scattering by Ions
Scattering by Free Electrons
Inelastic Scattering by Bound Electrons
Example: See (k , ω) for Be metal
Be metal T=20eV
3
See(k.ω) (a.u.)
3
40 deg
Sii
2
Sii
90 deg
2
1
1
See
2920 2960
ω1 (eV)
3000
PLWS-4
See
2920 2960
ω1 (eV)
Xray Scattering
3000
Average-Atom
Thomson Scattering
Applications
Elastic Scattering by Ions
Scattering by Free Electrons
Inelastic Scattering by Bound Electrons
Inelastic Scattering from Bound Electrons
Plane-Wave Final States
Z
Snl (k , ω) =
p dΩp
(2π)3
"
2
X Z
3
iq·r
d re
ψnlm (r)
Ep =ω+Enl
m
PLWS-4
Xray Scattering
#
Average-Atom
Thomson Scattering
Applications
Elastic Scattering by Ions
Scattering by Free Electrons
Inelastic Scattering by Bound Electrons
Example: Al 5eV
Plane-Wave Final State
S(k,ω) (a.u.)
1.2
1.2
30 deg
150 deg
L
0.8
L2
L2
L1
0.4
0.0
L
0.8
0
5
L1
0.4
10
15
ω (a.u.)
PLWS-4
20
0.0
0
Xray Scattering
5
10
15
ω (a.u.)
20
Average-Atom
Thomson Scattering
Applications
Elastic Scattering by Ions
Scattering by Free Electrons
Inelastic Scattering by Bound Electrons
Example: Be 10eV
Average-Atom Final State
Z
2
p dΩp X 3 †
ik ·r
d
r
ψ
(r)
e
ψ
(r)
.
nlm
p
3
(2π) m
Ep=ω+Enl
Z
Snl (k , ω) =
0.4
0.4
S(k,ω) (a.u.)
30 deg
0.3
Plane-Wave
0.2
Aver-Atom * 10
0
5
10
15
ω (a.u.)
PLWS-4
Plane-Wave
0.2
Coulomb
Aver-Atom
Coulomb * 10
0.1
0
150 deg
0.3
0.1
20
0
0
5
Xray Scattering
10
15
ω (a.u.)
20
Average-Atom
Thomson Scattering
Applications
Hydrogen
Beryllium
Titanium
Tin
Applications:
Hydrogen (high density ne = 1024 cm−3 )
Beryllium (light element with available experimental data)
Titanium (intermediate atomic weight element)
Tin (heavy metal with interesting bound-state features)
PLWS-4
Xray Scattering
Average-Atom
Thomson Scattering
Applications
Hydrogen
Beryllium
Titanium
Tin
Hydrogen: T = 50eV
0.6
nc(r) & ne
0.4
Hydrogen
T=50eV
S(k,ω) (a.u.)
0.5
RWS
0.3
nc(r)
0.2
0.1
24
20 deg
40 deg
30 deg
0.4
0.2
-3
ne = 10 cm
0.0
0.0
0.5
1.0
r (a.u.)
1.5
PLWS-4
4900
Xray Scattering
4950
5000
ω1 (eV)
5050
5100
Hydrogen
Beryllium
Titanium
Tin
Average-Atom
Thomson Scattering
Applications
Beryllium: Comparison with Experiment
1.0
3.0
0.8
Intensity
S(k,ω) (a.u.)
2.5
2.0
1.5
1.0
0.6
0.4
0.2
0.5
0.0
2880 2900 2920 2940 2960 2980 3000 3020
2880 2900 2920 2940 2960 2980 3000 3020
ω1 (eV)
Energy (eV)
Average-Atom model for xray scattering by Be metal (T = 18 eV,
ne = 1.8 × 1023 ) compared with measurement.4 ω0 = 2963 eV &
θ = 40◦ .
4
S. H. Glenzer & T. Doeppner, private communication
PLWS-4
Xray Scattering
Average-Atom
Thomson Scattering
Applications
Hydrogen
Beryllium
Titanium
Tin
Titanium metal (Z=22) at T = 10 eV, ω0 = 2960 eV
10
2.5
S(k,ω)
plasmon
1.5
3p
30 deg
8
Nb=17.67
Nc= 4.33
6
1.0
0.5
0.0
-100
150 deg
3p
S(k,ω)
2.0
4
3s
2
3s
0
-50
50
ω1−ω0 (eV)
100
PLWS-4
0
-100
Xray Scattering
0
-50
50
ω1−ω0 (eV)
100
Average-Atom
Thomson Scattering
Applications
Hydrogen
Beryllium
Titanium
Tin
Tin (Z=50) at T = 10 eV, ω0 = 2960 eV
2.5
4.0
S(k,ω) (a.u.)
S(k,ω) (a.u.)
2.0
5.0
30 deg
1.5
1.0
3.0
2.0
1.0
0.5
0.0
-100
150 deg
0
-50
ω1−ω0 (eV)
50
PLWS-4
0.0
-100
Xray Scattering
0
-50
ω1−ω0 (eV)
50
Average-Atom
Thomson Scattering
Applications
Hydrogen
Beryllium
Titanium
Tin
Tin (Z=50) at T = 10 eV, ω0 = 2960 eV
2.5
4.0
S(k,ω) (a.u.)
S(k,ω) (a.u.)
2.0
5.0
30 deg
1.5
1.0
3.0
2.0
1.0
0.5
0.0
-100
150 deg
0
-50
ω1−ω0 (eV)
50
PLWS-4
0.0
-100
Xray Scattering
0
-50
ω1−ω0 (eV)
50
Average-Atom
Thomson Scattering
Applications
Hydrogen
Beryllium
Titanium
Tin
Tin (Z=50) at T = 10 eV, ω0 = 2960 eV
2.5
4.0
S(k,ω) (a.u.)
S(k,ω) (a.u.)
2.0
5.0
30 deg
1.5
1.0
3.0
2.0
1.0
0.5
0.0
-100
150 deg
0
-50
ω1−ω0 (eV)
50
PLWS-4
0.0
-100
Xray Scattering
0
-50
ω1−ω0 (eV)
50
Average-Atom
Thomson Scattering
Applications
Hydrogen
Beryllium
Titanium
Tin
Tin (Z=50) at T = 10 eV, ω0 = 2960 eV
0.4
S(k,ω)
0.3
4p
5s
0.2
0.1
0.0
S(k,ω)
2.0
5s+4p+4d
4d
1.5
1.0
0.5
0.0
-100
-75
-50
-25
PLWS-4
-100
-75
ω1−ω0 (eV)
Xray Scattering
-50
-25
0
Summary
References
Summary:
A-A model is used to study Xray scattering from WDM.
Scattering from bound-states easily accommodated
To be done:
Improve the treatment of Sii (k ) (hypernetted chains? or
molecular dymamics?)
Go beyond RPA and include correlation corrections to
See (k , ω)
PLWS-4
Xray Scattering
Summary
References
References
Feynman, Metropolis & Teller, Phys. Rev. 75, 1561 (1949)
S. H. Glenzer & R. Redmer, Rev. Mod. Phys. 81, 1625 (2009)
G. Gregori et al., Phys. Rev. E 67, 026412 (2003)
J. Chihara, J. Phys.: Condens. Matter 12, 231 (2000)
W.R. Johnson et al., JQSRT, 99, 327 (2006)
S. Sahoo et al., Phys. Rev. E 77, 046402 (2008)
PLWS-4
Xray Scattering