Optical Properties of Plasmas Based on an Average-Atom Modeljohnson/Publications/aussem.pdf ·...

Optical Properties of Plasmas Based on an Average-AtomModel

Walter Johnson, Notre Dame University

Claude Guet, CEA/DAM Ile de France

George Bertsch, University of Washington





• Average Atom Model of a Plasma






• Linear Response ⇒ Kubo-Greenwood formula for σ(ω)







• Kramers-Kronig Dispersion Relation ⇒ Dielectric Function ε(ω)







• Kramers-Kronig Dispersion Relation ⇒ Dielectric Function ε(ω)

• Index of refraction n(ω) + iκ(ω) =√ε(ω)

Seminar: UNSW Mar. 30, 2005

2

Motivation

Free electron formula for index of refraction is used to determine electron densities.

n =

√1− ω

20

ω2≈ 1− ω2

0

2ω2< 1 where ω2

0 = 4πe2

m

Nfree

Ω

2

Motivation


n =

√1− ω

20

ω2≈ 1− ω2

0

2ω2< 1 where ω2

0 = 4πe2

m

Nfree

Ω

Problem: Recent experiments on Al plasmas find n > 1 at few eV temperatures

2

Motivation


n =

√1− ω

20

ω2≈ 1− ω2

0

2ω2< 1 where ω2

0 = 4πe2

m

Nfree

Ω


Reason: Effect of bound electrons on optical properties.

2

Motivation


n =

√1− ω

20

ω2≈ 1− ω2

0

2ω2< 1 where ω2

0 = 4πe2

m

Nfree

Ω


Reason: Effect of bound electrons on optical properties.

• LLNL comet laser facility1 (14.7 nm Ni-like Pd laser)

• Advanced Photon Research Center JAERI2 (13.9 nm Ni-like Ag laser)

1J. Filevich et al. Proceedings of the 9th International Conference on X-Ray Lasers, May 23-28 (2004)2H. Tang et al., Appl. Phys. B78, 975 (2004)


3

TANG et al. Diagnostics of laser-induced plasma with soft X-ray (13.9 nm) bi-mirror interference microscopy 977

FIGURE 2 a Interference fringes after removing self-emission of the Al plasma. The fringes evolve with the increasing of diagnosing delay (0 ∼ 5 ns). Onlythe fringes very close to the target surface cannot be resolved due to the over intense self-emission. b Interference fringe pattern c Electron density map forthe 1 ns-delay and for an irradiance of 1.4×1011 Wcm−2

a 2p-3d transition of Al2+ ions, the wavelength of which be-ing 13.883 nm [11]. This wavelength is very close to the X-raylaser one (13.887 nm) [12]. At 2 ∼ 3 nanoseconds after theheating pulse, the plasma temperature falls-off and the highlyionized plasma starts to recombine, leading to a reduced freeelectron density and a larger Al2+ ion density, which bothincrease the refractive index. Another possible mechanism in-volves the contribution of bound-free transitions in neutralatoms to the refraction index. Al2+ ions and neutral atomsexist at this period of plasma development. Their effect onthe refractive index, which can be neglected in the highlyionized, laser-heated plasma, should be considered in therecombining plasma. Otherwise, estimation of the electrondensity from the fringe shift may be underestimated. More de-tailed quantitative investigation of the effect of the Al2+ ionswill be performed by hydrodynamic simulation in our futurework.

3 Summary

We present the results of an X-ray interference mi-croscopy of a dense laser-induced plasma using a wavefrontdivision bi-mirror interferometer and a transient picosecondNi-like Ag X-ray laser (wavelength ∼ 13.9 nm). Interferencemicroscopy pattern of the Al (Z = 13) plasma have beenrecorded successfully. Electron densities are measured fromthe interference fringe shifts under the pumping intensity of∼ 1011 Wcm−2. For the aluminum plasma diagnosed 2 and 3nanoseconds after the heating pulse, opposite fringe shifts areobserved. Two possible explanations for this behavior are dis-

cussed. These results will contribute to the validation of the1D and 2D hydrodynamic codes and a better understanding ofthe physics of laser-produced plasma.

REFERENCES

1 D.T. Attwood, D.W. Sweeney, J.M. Auerbach, P.H.Y. Lee: Phys. Rev.Lett. 40, 184 (1978)

2 P.E. Young, P.R. Bolton: Phys. Rev. Lett. 77, 4556 (1996); P.E. Young,C.H. Still, D.E. Hinkel, W.L. Kruer, E.A. Williams, R.L. Berger,K.G. Estabrook: Phys. Rev. Lett. 81, 1425 (1998)

3 L.B. Da Silva, T.W. Barbee, Jr., R. Cauble, P. Celliers, D. Ciarlo,S. Libby, R.A. London, D. Matthews, S. Mrowka, J.C. Moreno, D. Ress,J.E. Trebes, A.S. Wan, F. Weber: Phys. Rev. Lett. 74, 3991 (1995)

4 J.J. Rocca, C.H. Moreno, M.C. Marconi, K. Kanizay: Opt. Lett. 24, 420(1999)

5 J. Filevich, K. Kanizay, M.C. Marconi, J.L.A. Chilla, J.J. Rocca: Opt.Lett. 25, 356 (2000)

6 A. Klisnick, J. Kuba, D. Ros, R. Smith, G. Jamelot, C. Chenais-Popovics, R. Keenan, S.J. Topping, C.L.S. Lewis, F. Strati, G.J. Tallents,D. Neely, R. Clarke, J. Collier, A.G. Macphee, F. Bortolotto, P.V. Nick-les, K.A. Janulewicz: Phys. Rev. A 65, 033 810 (2002)

7 T. Kawachi, M. Kado, M. Tanaka, A. Sasaki, N. Hasegawa, A. Kil-pio, S. Namba, K. Nagashima, P. Lu, K. Takahashi, H. Tang, R. Tai,M. Kishimoto, M. Koike, H. Daido, Y. Kato: Phys. Rev. A 66, 033 815(2002)

8 R.F. Smith, J. Dunn, J Nilsen, V.N. Shlyaptsev, S. Moon, J. Filevich,J.J. Rocca, M.C. Marconi, J.R. Hunter, T.W. Barbee: Phys. Rev. Lett. 89,065 004 (2002)

9 D. Joyeux, R. Mercier, D. Phalippou, M. Mullot, M. Lamare: J. Phys. IVFrance 11, Pr2, 511 (2001)

10 R. Mercier, M. Mullot, M. Lamare, G. Tissot: SPIE Proc. 3739, 155(1999)

11 NIST Atomic Spectra Database(http://physics.nist.gov/cgi-bin/AtData/main_asd)

12 D. Ros: Thèse de l’Universite Paris-Sud XI, Orsay (1998) (in French)


4

Average-Atom Model

QM version of a model proposed by Feynman, Metropolis, and Teller3

Inside a neutral Wigner-Seitz cell: Ω = A/(Avagadro No.× density)[p2

2m− Zr

+ V

]ψi(r) = εiψi(r) (1)

4

Average-Atom Model

QM version of a model proposed by Feynman, Metropolis, and Teller3

Inside a neutral Wigner-Seitz cell: Ω = A/(Avagadro No.× density)[p2

2m− Zr

+ V

]ψi(r) = εiψi(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

3R. P. Feynman, N. Metropolis and E. Teller, Phys. Rev. 75 1561 (1949)


5

Thermal Average Electron Density

Contributions to the density are

ρb(r) =1

4πr2

∑l

2(2l + 1)∑n

f(εnl)Pnl(r)2

(3)

ρc(r) =1

4πr2

∑l

2(2l + 1)

∫ ∞0

dε f(ε)Pεl(r)2

(4)

where

f(ε) =1

1 + exp[(ε− µ)/kT ]

5



ρb(r) =1

4πr2

∑l

2(2l + 1)∑n

f(εnl)Pnl(r)2

(3)

ρc(r) =1

4πr2

∑l

2(2l + 1)

∫ ∞0

dε f(ε)Pεl(r)2

(4)

where

f(ε) =1

1 + exp[(ε− µ)/kT ]

The chemical potential µ is chosen to insure electric neutrality:

Z =

∫r<R

ρ(r) d3r ≡

∫ R

0

4πr2ρ(r) dr . (5)

5



ρb(r) =1

4πr2

∑l

2(2l + 1)∑n

f(εnl)Pnl(r)2

(3)

ρc(r) =1

4πr2

∑l

2(2l + 1)

∫ ∞0

dε f(ε)Pεl(r)2

(4)

where

f(ε) =1

1 + exp[(ε− µ)/kT ]

The chemical potential µ is chosen to insure electric neutrality:

Z =

∫r<R

ρ(r) d3r ≡

∫ R

0

4πr2ρ(r) dr . (5)

Eqs. (1-5) are solved self-consistently for ρ, V , and µ.


6

Example

Al: density 0.27 gm/cc, T = 5 eV, R = 6.44 a.u., µ = −0.3823 a.u.

Bound States Continuum States

State Energy n(l) l n(l) n0(l) ∆n(l)

1s -55.189 2.0000 0 0.1090 0.1975 -0.0885

2s -3.980 2.0000 1 0.2149 0.3513 -0.1364

2p -2.610 6.0000 2 0.6031 0.3192 0.2839

3s -0.259 0.6759 3 0.2892 0.2232 0.0660

3p -0.054 0.8300 4 0.1514 0.1313 0.0201

5 0.0735 0.0674 0.0061

6 0.0326 0.0308 0.0018

7 0.0132 0.0127 0.0005

8 0.0049 0.0048 0.0001

9 0.0017 0.0016 0.0001

10 0.0005 0.0005 0.0000

Nbound 11.5059 Nfree 1.4941 1.3404 0.1537


7

0 2 4 6 8

10-3

10-2

10-1

ρc(r)ρ0

0 2 4 6 8r (a.u.)

0

5

10

15

4πr2ρb(r)

4πr2ρc(r)Zeff(r)

RWS

RWS


8

Pressure

P =13

[Txx + Tyy + Tzz]∣∣∣r=R

≈ (2mkT )5/2

6mπ2 I3/2(µ/kT ),

where Ik(x) is a “Fermi-Dirac” integral

Ik(x) =∫ ∞

0

ykdy

1 + e(y−x)

8

Pressure

P =13

[Txx + Tyy + Tzz]∣∣∣r=R

≈ (2mkT )5/2

6mπ2 I3/2(µ/kT ),

where Ik(x) is a “Fermi-Dirac” integral

Ik(x) =∫ ∞

0

ykdy

1 + e(y−x)

P = P (Ω, T ) provides an equation of state for the plasma

n.b. atomic unit of [P ]: 294.21 Mbar.


9

Kinetic and Potential Energies

Ekin =∫d3r

∑i

〈ψi|p2

2m|ψi〉f(εi)

=32P Ω− 1

2Epot

9

Kinetic and Potential Energies

Ekin =∫d3r

∑i

〈ψi|p2

2m|ψi〉f(εi)

=32P Ω− 1

2Epot

This is the generalized virial theorem.


10

Entropy

The entropy S of a collection of fermions is given by the expression

TS = −kT∑i

f(εi) ln f(εi) + [1− f(εi)] ln [1− f(εi)]

where f(εi) = 1/ [1 + exp (εi − µ)/kT ]

10

Entropy

The entropy S of a collection of fermions is given by the expression

TS = −kT∑i

f(εi) ln f(εi) + [1− f(εi)] ln [1− f(εi)]

where f(εi) = 1/ [1 + exp (εi − µ)/kT ] This can be manipulated to give

TS =53Ekin + Ee−n + 2Ee−e − µN

=52P Ω +

16Ee−n +

76Ee−e − µN


11

Other Thermodynamic Quantities

The internal energy U and the Helmholtz free energy F are given by

U = Ekin + Epot =3

2P Ω +

1

2Ee−n +

1

2Ee−e

F = U − TS = −P Ω +1

3Ee−n −

2

3Ee−e +Nµ

11



U = Ekin + Epot =3

2P Ω +

1

2Ee−n +

1

2Ee−e

F = U − TS = −P Ω +1

3Ee−n −

2

3Ee−e +Nµ

From thermodynamics, we know that

dU = dQ− PdΩ = TdS − PdΩ

dF = −SdT − PdΩ

11



U = Ekin + Epot =3

2P Ω +

1

2Ee−n +

1

2Ee−e

F = U − TS = −P Ω +1

3Ee−n −

2

3Ee−e +Nµ

From thermodynamics, we know that

dU = dQ− PdΩ = TdS − PdΩ

dF = −SdT − PdΩ

The Helmholtz free energy is a thermodynamic function of the “natural” variables of the problem,

Ω and T :

S = −∂F

∂T

∣∣∣∣Ω

P = −∂F

∂Ω

∣∣∣∣T


12

Application: Plasma Conductivity

The Ziman formula4 for the static conductivity of a many-particle system is

σ = −2e2

3

∫d3p

(2π)3v

2τ(p)

∂f

∂E,

where τ(p) is the mean collision time and where f(E) is the Fermi function.

τ(p) =Λ(p)

vΛ(p) =

Ω

σtr(p)

σtr(p) =4π

p2

∞∑l=0

(l + 1) sin2

(δl+1 − δl) τ(p) =Ω

v σtr(p)

σ = −Ω

3π2

∫ ∞0

dE

[v2

σtr(p)

]∂f

∂E.

4G. D. Mahan, Many-Particle Physics, Plenum, 2000


13

Phase Shifts

0 1 2 3 4 5Electron Momentum (a.u.)

-3

-2

-1

0

1

2

Phas

e Sh

ift (

rad)

l=0l=1l=2l=3l=4

T = 10 eV and metallic density


14

Resistivity of Aluminum

0 20 40 60 80 100T (eV)

0

50

100

150

200

250

Res

istiv

ity (µ

Ω−c

m)

Ziman FormulaMilchberg et al. (1988)

Metallic density


15

Linear Response and the Kubo-Greenwood Formula

Apply an electric field to the average atom:

E(t) = F z sinωt A(t) =F

ωz cosωt

15




ωz cosωt

The time dependent Schrodinger equation becomes[T0 + V (n, r)− eF

ωvz cosωt

]ψi(r, t) = i

∂

∂tψi(r, t)

15




ωz cosωt

The time dependent Schrodinger equation becomes[T0 + V (n, r)− eF

ωvz cosωt

]ψi(r, t) = i

∂

∂tψi(r, t)

The current density is

Jz(t) =2eΩ

∑i

fi 〈ψi(t)|vz|ψi(t)〉


16

Solution Ansatz

ψi(r, t) = ui(r)e−iεit + w+i (r)e−i(εi+ω)t + w−i (r)e−i(εi−ω)t

[T0 + V (n, r)]ui(r) = εiui(r)

[T0 + V (n, r)− (εi ± ω)]w±i (r) =eF

2ωvz ui(r)

16

Solution Ansatz

ψi(r, t) = ui(r)e−iεit + w+i (r)e−i(εi+ω)t + w−i (r)e−i(εi−ω)t

[T0 + V (n, r)]ui(r) = εiui(r)

[T0 + V (n, r)− (εi ± ω)]w±i (r) =eF

2ωvz ui(r)

The current density becomes

Jz(t) =2eΩ

∑i

fi 〈ψi(t)|vz|ψi(t)〉

=2eΩ

∑i

fi[(〈ui|vz|w+

i 〉+ 〈w−i |vz|ui〉)e−iωt + c.c.

]


17

Eigenvalue Expansion

w+i (r) =

∑j

Xji uj(r) w−i (r) =

∑j

Y ji uj(r)

Xji =

eF

2ω〈j|vz|i〉

εj − iη − εi − ωY ji =

eF

2ω〈j|vz|i〉

εj − iη − εi + ω

17

Eigenvalue Expansion

w+i (r) =

∑j

Xji uj(r) w−i (r) =

∑j

Y ji uj(r)

Xji =

eF

2ω〈j|vz|i〉

εj − iη − εi − ωY ji =

eF

2ω〈j|vz|i〉

εj − iη − εi + ω

The response current may be written

J =4eΩ

∑ij

fi

[<(〈i|vz|j〉Xj

i + 〈j|vz|i〉Y j?i)

cosωt

+ =(〈i|vz|j〉Xj

i + 〈j|vz|i〉Y j?i)

sinωt]


18

Kubo-Greenwood

• Linearize ψi(r, t) in F

18

Kubo-Greenwood


• Evaluate the response current: J = Jin sin(ωt) + Jout cos(ωt)

18

Kubo-Greenwood



• Determine σ(ω): Jin(t) = σ(ω)Ez(t)

18

Kubo-Greenwood



• Determine σ(ω): Jin(t) = σ(ω)Ez(t)

Result:

σ(ω) =2πe2

ωΩ

∑ij

(fi − fj) |〈j|vz|i〉|2 δ(εj − εi − ω),

which is an average-atom version of the Kubo5-Greenwood6 formula.

5 R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957)6D. A. Greenwood, Proc. Phys. Soc. London 715, 585 (1958)


19

Free-Free Contribution to Conductivity

0.01 0.1 1Photon Energy (a.u.)

0.000

0.005

0.010

0.015

0.020

0.025

0.030

σ ff(ω

) (a

.u.)

T=5eVρ=0.27 gm/cc

19

Free-Free Contribution to Conductivity


0.000

0.005

0.010

0.015

0.020

0.025

0.030

σ ff(ω

) (a

.u.)

T=5eVρ=0.27 gm/cc

Michael Kuchiev has prepared an elegant note on the low-frequency conductivity

explaining the origin of the infrared divergence and proposing a remedy.


20

Bound-Bound Contribution to Conductivity

0.01 0.1 1 10Photon Energy (a.u.)

0

0.001

0.002

0.003

0.004

σ bb(ω

) (a.

u.)

3s-3p

2p-3s 2s-3p

T=3eVρ=0.27 gm/cc


21

Bound-Free Contribution to Conductivity

0.01 0.1 1 10Photon Energy (a.u.)

0

0.001

0.002

0.003

0.004

0.005

σ bf(ω

) (a.

u.)

3-ε

2-ε

T=3eVρ=0.27 gm/cc


22

Optical Properties

For a conducting medium, the dielectric function is related to the complexconductivity by

ε(ω) = 1 + 4πiσ(ω)ω

We know <σ(ω); we must evaluate =σ(ω)

22

Optical Properties

For a conducting medium, the dielectric function is related to the complexconductivity by

ε(ω) = 1 + 4πiσ(ω)ω

We know <σ(ω); we must evaluate =σ(ω)

From analytic properties of σ(ω) one infers the dispersion relation7

=σ(ω0) =2ω0

π−∫ ∞

0

<σ(ω)ω2

0 − ω2dω.

7R. de L. Kronig and H. A. Kramers, Atti Congr. Intern. Fisici, 2, 545 (1927)


23

Application of Dispersion Relation

0.001 0.01 0.1 1Photon Energy (a.u.)

-0.030

-0.020

-0.010

0.000

0.010

0.020

0.030

Con

duct

ivity

(a.u

.)

Re[σ(ω)]Im[σ(ω)]

1 2 3 4Photon Energy (a.u.)

-0.003

-0.002

-0.001

0.000

0.001

0.002

0.003

T=5eVρ=0.27 gm/cc


24

Index of Refraction

<ε(ω) = 1− 4π=σ(ω)ω

=ε(ω) = 4π<σ(ω)ω

,

n+ iκ =√ε.


10-3

10-2

10-1

100

101

n(ω

) & κ

(ω) n

nfreeκκfree

T=5eVρ=0.27gm/cc


25

Al: Comparison with Free Electron Model

Plasma with ion density nion = 1020/cc

0 20 40 60 80 100Photon Energy (eV)

-10

-5

0

5

10(n

-1)/

(nfr

ee-1

) 2p-4d2p-3d

T=3eV<Z>=1.38

Pd x

-ray

2p-3s Ag

x-ra

ySeminar: UNSW Mar. 30, 2005

26

Al: Penetration Depth

Plasma with ion density nion = 1020/cc

0 20 40 60 80 100Photon Energy (a.u.)

100

101

102

103

Pene

trat

ion

Dep

th (

µm)

T=3eV<Z>=1.38


27

Conclusions

• Average atom model is a simple way to understand the electronic structure of aplasma. (next step - look at neighbors)

27

Conclusions


• Linear response theory applied to the average atom model. provides a straight-forward method for obtaining the frequency-dependent conductivity.

27

Conclusions



• The dielectric function (and index of refraction) can be reconstructed with theaid of a dispersion relation.

27

Conclusions




• The model explains “anomalous behavior” of low temperature Al plasmas in the80-90 eV frequency range.

27

Conclusions




• The model explains “anomalous behavior” of low temperature Al plasmas in the80-90 eV frequency range.

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