thomson scattering from wdm - nd.edujohnson/publications/rpwdm.pdf · average-atom thomson...

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.

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Outline

1 Average-Atom

2 Thomson ScatteringElastic Scattering by IonsInelastic Scattering by Free ElectronsInelastic Scattering by Bound Electrons

3 Application to Aluminum

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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

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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

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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

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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.

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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

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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)


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f (k) + q(k) = 4π∫ RWS



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f (k) + q(k) = 4π∫ RWS


Arkhipov and Davletov (1998) & Gregori et al. (2003)

Modified by Gregori et al. (2006) to include Ti 6= Te

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f (k) + q(k) = 4π∫ RWS



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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.)

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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

].

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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

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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).

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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.

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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

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-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

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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.

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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)

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