dynamical structure factor in warm dense matter and...

Dynamical Structure Factor in Warm Dense Matter andApplications to X-Ray Thomson Scattering

Carsten Fortmann

Lawrence Livermore National LaboratoryUniversity of California Los Angeles

IPAM, May 24, 2012

This work performed under the auspices of the U.S. Department of Energy

by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

C. Fortmann (LLNL & UCLA) IPAM 2012 1 / 32

Contributors

S. H. Glenzer, T. Doppner, A. L. Kritcher, O. N. Landen, T. Ma,A. Pak

R. Redmer, K.-U. Plagemann, P. Sperling, G. Ropke A. Wierling

Ch. Niemann

P. Davis, L. Fletcher, R. Falcone

H. J. Lee

SNL M. Desjarlais


Outline

Introduction: X-Ray diagnostics of WDM

Many-body theory of light-matter interaction

The Dynamical Structure Factor

Semi-analytic calculations of structure factors in WDM

Applications: X-ray scattering from warm dense plasmas

Ab-initio calculations for the dynamical structure factor

Summary and Conclusions


Studying matter under extreme conditions is relevant tounderstand astrophysical and laboratory plasmas

Plasma parameters:

Coupling parameter Γ =Ze2

4πε0 kBT

(4π n

3

)1/3

Degeneracy parameter θ =kBT

EF, EF =

~2

2m(3πn)2/3


Probing the bulk properties of solid density plasmasrequires intense x-ray sources

Critical wavelength vs. electron density

10-2

10-1

100

101

102

103

104

1017 1019 1021 1023 1025

100

101

102

103

104

105

0.1 1 10 100

probe wavelength (nm)

electron density (1/cm3)

1ω

2ω3ω

4ω

FLASH

LCLS

probe photon energy (eV

)

λc=2πc/ωpl

λc/100

λc/10

Va K-αZn K-α

XFEL Mn K-α

plasma frequency hωpl (eV)

Cl Ly-α

X-rays penetrate deepinto solid matter

volumetric heatingpossible, mitigate ne/Tgradients

probe bulk properties ofdense matter, discardsurface effects

Need accurate theoryfor the interaction ofcorrelated matter withx-rays


The QED Lagrangian governs the particle and fielddynamics

L(x) =∑c

ψc(x)(i~cγµ∂µ −mcc

2)ψc(x)−ε0c

2

4Fµν(x)Fµν(x)

+∑c

Zcejµc(x)Aµ(x)

Free fermions, Dirac equation, particle current jµc = ψc(x)γµψc(x)

Free gauge fields, Maxwell equations, fields Fµν = ∂µAν(x)−∂νAµ(x)

minimal particle-field coupling⇒ emission/absorption, collisions, radiative damping


The current-current correlation gives the linear response toexternal fields (EM, pressure, particles)

4-current dynamical structure factor Sµν(k)

Sµν(k) =

∫d4x ′ exp(ikλx

′λ)〈j†µ(x + x ′)jν(x)〉x k = (ω/c , ~k), x = (ct,~x)

Correlation functions are expressed via Green functions

〈j†µjν〉k =1

πImGjj(k)

Green functions are systematically evaluated using MB Feynmandiagrams

Gjj(k) =


Dyson-Schwinger Equations

Fermion propagator G = G(0) + G(0)Σ G

Self-energy: Σ =Γ0 Γ

QP shift+damping (lifetime)

Field propagator: W = V + VΠ

W

Polarization: Π =

G

G

Γ(0) Γ Dynamical screening

Vertex: = + K eff. coupling

Dyson, Phys. Rev. 75 (1949), Schwinger, PNAS 37 (1951)

Hedin, Phys. Rev. 139 (1965)


The spectral function describes damped quasiparticles in adense medium

A(p, ω) = −2ImG (~p, ω + i0+)

A(~p, ω): probability for particle withmomentum ~p at energy ~ω.

Free particle:AFP(~p, ω) = 2π δ(~ω − p2/2m)

Dense medium:A(~p, ω) = −2Γ(~p,ω)

[~ω−Ep−∆(~p,ω)]2+[Γ(~p,ω)]2

Σ(~p, ω) = ∆(~p, ω) + iΓ(~p, ω)


Example: GW0 spectral function

Spectral function for diluteplasma (ne = 1022/cc,T = 1keV)

]-1

B

momentum p [a

0 1 2 3 4 5 6 7 8 910 [Ry]eµ+ω

frequency 0 20 40 60 80 100

) [1

/Ry]

ωA

(p,

00.20.40.60.8

11.21.4

Narrow width, ideal dispersion

No quasiparticle satellites

Spectral function for denseplasma (ne = 1026/cc,T = 1keV)

]-1

B

momentum p [a

0 1 2 3 4 5 6 7 8 910 [Ry]eµ+ω

frequency -40 -20 0 20 40 60 80 100 120140

) [1

/Ry]

ωA

(p,

00.05

0.10.150.2

0.250.3

0.350.4

Strongly damped & shifted

Plasmaron satellitesFortmann, J.Phys.A (2008) & PRE (2009)


The spectral function yields important observables

Equation of state

na(µa,T ) =

∞∫−∞

dω

2π

∫d3p

(2π)3fa(ω)Aa(~p, ω)

Vorberger et al. PRE 69, 046407 (2004)

Band structure

Ep = ~2p2/2m + ReΣ(~p,Ep)

Optical properties

Πe(~k, ωµ)=∑zν ,~p

∞∫−∞

dω

2π

dω′

2πΓ0

Ae(~p + ~k)

ω − zν − ωµAe(~p, ω′)ω′ − zν

Γ(ωµ + zν , zν ;~p + ~k , ~p)

Fortmann et al., Contrib. Plasma Phys. 94 (2007)


Applications to X-ray processes in WDM

Bremsstrahlung

σbrems(~k, ω) ∝ ImΠee(~k , ω)

Important energy loss/heatingprocess in dense plasmas

Spectrum typically ∝ e−~ω/kBT

⇒ Temperature diagnostic forhot electrons

Direct measure of collisionalityin the plasma

Electron distribution functioncan be calculated bydeconvolution of spectrumfrom elementary cross-section

Inelastic x-ray scattering

σIXS(~k , ω) ∝ See00(~k, ω)

Important ne,T ,Zf , ρ/ρ0

diagnostic

Sensitive to many-bodycorrelations (collisons, localfield effects)

Spectrum gives informationabout free-free, bound-free,and bound-bound correlations


X-ray scattering is a versatile dense matter diagnostic tool

A diagnostic for complex systems

scattering from free andbound electrons

sensitive to static and dynamiccorrelations

e-e and e-i interaction

strongly non-ideal system

Key advantages of x-rayscattering

undercritical for solids andcompressed matter

deep penetration

probe bulk properties

measures both dynamic andstatic plasma properties

direct ne, Te measurement viafirst principle relations (plasmafrequency, detailled balance)

high spatial and temporalresolution thanks to modernx-ray sources (backlighters,fel’s)


Scattering spectrum reflects the correlation of densityfluctuations

Scattering

on free

electrons

S

*

Optical Laser

D

Plasma 0

v

(v/c)sin(/2)]

Non-collective Thomson Scattering (* < D)

0

Inte

ns

ity

Wavelength

Te

Boltzmann distribution

X-ray source Scattering

on free and

weakly

bound

electrons

Solid

density

Plasma

p = mv

S

p = h/c

p = h’/c

2(h/mc2)sin2(/2) (v/c)sin/2]

X-ray ‘Compton’ scattering

Inte

ns

ity

Wavelength

Rayleigh

peak

Compton

peak

Te

*

0

Energy

Fermi or

Boltzmann

distribution e

C

m

kE

2

22


XRTS resolves collective and single particle dynamics


Dynamical Structure Factor S(k , ω)

Scattered power Ps per frequency interval dω per solid angle dΩ

Ps(~R, ω)dΩdω =Pi r

20dΩ

2πA

∣∣∣∣~k f × (~k f × ~E 0i )

∣∣∣∣2 NS(~k, ω)dω

S(~k , ω) =1

2πN

∫dt eiωt〈ρe(~k , t)ρe(−~k , 0)〉

S : density-density correlation function ⇔ coherent superposition ofdensity fluctuationsS determined via

scattering experimentstheoretical calculationscomputer simulations (MD, QMD, WPMD, ...)

r0 = e2/4πε0mec2: classical electron radius

N: number of scattering centers; A: area of illuminated plasma∣∣∣∣~k f × (~k f × ~E0i )

∣∣∣∣2 : laser polarizationC. Fortmann (LLNL & UCLA) IPAM 2012 16 / 32

Chihara formalism for the dynamical structure factor in dense plasmas

S(k, ω) = |fI (k) + q(k)|2 Sii(k, ω) + Zf S0ee(k, ω) + ZcScore(k, ω)

f (k) q(k) 2S (k, ) Z S0 (k, ) Z S (k, ')S (k, ')d( , )Probe

Energy

Inte

ns

ity

E0

Ion feature[tightly bound/screening e-]

Ecompton

Electron feature[free or delocalized e-]

Bound-free[weakly bound]

Ebind

Schematic Scattering spectrum

f (k): ionic form factor, elasticscattering from bound states

q(k): “screening cloud”: freeelectrons following ionsadiabatically

Sii (k , ω): ion-ion structure factor

S0ee(k , ω): Electron OCP

structure factor (e.g. plasmons)

Score(k , ω) ∝ 〈~p|ei~k·~r |nlm〉:bound-free scattering

Chihara, J. Phys.: Cond. Mat. 12 (2000)


Extended Born-Mermin theory and local field effects

Electronic subsystem

e − e correlations important atfinite k

treat e correlations via LFC

→ χOCPee = χ

(0)e

1−V (1−Gee)χ(0)e

Ionic subsystem

static structure factor Sii (k)

e.g. HNC, DFT-MDWunsch et al. PRE (2009)

Schwarz et al., HEDP (2010)

Coupling of e and i subsystems

e − i collision frequency

νBorn(ω) ∝ Zf

ω

∫ ∞0

dq q6V 2S (q)Sii(q)

[χ(0)(q, ω)− χ(0)(q, 0)

]Mermin ansatz with correlated electrons and e − i collision frequency

χxM(k, ω) =−izν(ω)

χOCP(k , z)χOCP(k, 0)

χOCP(k , z)− iων(ω) χ

OCP(k , 0), z = ω + iν(ω)

Fortmann et al, PRE 81 (2010)


Mermin response function is derived from Linear Response Theory

relevant statistical operator for multi-component plasmaZubarev et al. Statistical Mechanics of Nonequilibrium Processes (1996)

ρrel = exp− Φ(t)− βH + β

∑c

µcNc + β∑c

δµc(k , ω)nckeiωt + c .c .

Lagrange multipliers β, µ, δµc(k, ω) from self-consistency relations

Mermin response functionMermin, PRB (1970) Ropke et al., Phys. Lett. A (1999)

χMc (k , ω) =

iz

ν

χ(0)c (k, z)χ

(0)c (k , 0)

χ(0)c (k , z)− iω

ν χ(0)c (k , 0)

, z = ω + iν(ω) .

χM conserves local density

ν(ω): dynamical collision frequency, describes e-i collisions


Calculation of the collision frequency

Force-force correlation function

ν(ω) =βΩ0

ε0ω2pl

⟨Jz0, J

z0

⟩ω+iη

∝ G4

Approximations

Born ap-proximation

e−

V scee

Dynamicalscreening

T-matrix


Collision frequency in Born approximation

Redmer et al., IEEE Trans. Plasma Sci (2005)

Fortmann et al., Laser Particle Beams (2009)

νBorn(ω) ∝ Zf

ω

∫ ∞0

dq q6V 2S (q)Sii(q)

[εRPAe (q, ω)− εRPA

e (q, 0)]

Be, Zf = 2, T = 13 eV

0.1 1 10frequency [ω

pl,e]

0

0.05

0.1

0.15

0.2

colli

sion

fre

quen

cy ν

(ω) [

Ry]

-200 -100 0frequency shift [eV]

0

0.01

0.02

stru

ctur

e fa

ctor

See0 (k

,ω)

[1/e

V]

Born-MerminRPA

ne=3x1023

cm-3

, θ=0.73

ne=3x1024

cm-3

, θ=0.15

ne=3x1022

cm-3

, θ=3.4

Collisions most importantnear θ = kBT

EF' 1

At high & low degeneracyLandau damping dominates.

Inclusion of dynamicalscreening and strongcollisions is straightforward.Thiele et al., J. Phys. A (2006)


Treatment of e-e correlations via local field corrections (LFC)

χxM(~k , ω) =−izν(ω)

χOCP(~k , z)χOCP(k , 0)

χOCP(~k , z)−(

iων(ω)

)χOCP(~k , 0)

, z = ω + iν(ω)

χOCP(~k , ω) =χ0e(~k , ω)

1− Vee(k)(1− Gee(~k , ω))χ0e(~k , ω)

The LFC is calculated using Pade interpolationbetween low and high ω limits Wierling et al. CPP 2011

0 1 2 3 4k [units of k

F]

0

0.5

1

1.5

2

2.5

3

Gee

(k)

HubbardBornSTLSPathak, VashishtaIchimaru, UtsumiFarid et al.Moroni et al.

rs=2

static LFC

0 2 4 6 8 10frequency ω [units of E

F/h_

]

-0.2

0

0.2

0.4

0.6

0.8

Re G(k,ω), k = 1 kF

Im G(k,ω), k = 1 kF

Re G(k,ω), k = 2 kF

Im G(k,ω), k = 2 kF

rs=2

dynamic LFC


1) XRTS in isochorically heated Be

Glenzer et al., PRL 98, 065002 (2007)

µ

!"#$%&!!

&'&

&!(!

(&

!)&*+,

-)./&0)+

1! !

23& !

!-)+3.4

5(. !

(&!)&*+,

-)+./&

1! !

6237!

-)8.4

&'9 &!!"%:!$;2&(!!

!1&!!&! !

<)=9! 7) !&9 &! '1&

(.&!!!& 6237(! !*

&! 1 ('&&

! (. !(&!&!


Collisions are crucial to fit the scattering spectrum

Plasmonscattering

Best fitne = 3 x 1023 cm-3

Te = 12 eV

with collisions

Inte

nsit

y

0

3Ion feature

[elastic scattering]

Detailed

balance

Energy (keV)

3.02.9

1

2

)a

Glenzer et al., PRL 98 (2007)

BMA: Te = 12 eV (consistent w/ independent measurement)RPA-fit: Te ' 30 eV.First experimental evidence for the importance of collisions in X-rayplasma scattering.


Two component fully ionized plasma model

Fluctuation-dissipation theorem

See(~k , ω) = − ~πne

Imχee(~k, ω)

Dyson equations for pair correlation function and screened potential

χcc ′(~k, ω) = χ0c(~k , ω)δcc ′ +

∑d=e,i

χ0c(~k , ω)V sc

cd (~k , ω)χdc ′(~k, ω)

V sccc ′(

~k, ω) = Vcc ′(~k) +

∑d=e,i

Vcd(~k)χ0d(~k , ω)V sc

dc ′(~k, ω)

Vcc ′(~k) =

qcqc ′

ε0k2(fully ionized, replace by eff. potential in partially ionized plasma)

χ0c(~k, ω) = =

1

Ω0

∑~p

f c~p+~k/2

− f c~p−~k/2

E c~p+~k/2

− E c~p−~k/2

− ~ω


RPA: Self-consistent mean field approximation

Terminate Dyson series after 1st iteration

χRPAcc ′ (~k , ω) = χ0

c(~k , ω)δcc ′ + χ0c(~k, ω)V sc

cc ′(~k , ω)χ0

c ′(~k, ω)

Solution to linearized Dyson equations

χRPAee (~k, ω) =

χ0e − χ0

eViiχ0i

1− Veeχ0e − Viiχ

0i

χRPAei (~k , ω) =

2χ0eVeiχ

0i


0i

χRPAie (~k, ω) = χRPA

ei (~k , ω) χRPAii (~k , ω) =

χ0i − χ0

i Veeχ0e


0i


Calculation of the DSF

Apply fluctuation dissipation theorem

Imχee = Imχ0ee + V 2

ei

[(Reχ0

ee

)2 −(Imχ0

ee

)2]Imχii+

+ 2Reχ0ee Imχ0

ee Reχii

χ0ee =

χ0e

1− Veeχ0e

Compare to “Chihara formula” (fully ionized case)J. Chihara, J. Phys. Cond. Mat. (2000)

See(k, ω) = Zf S0ee(k, ω) + |q(k)|2 Sii (k, ω)

Identified high frequency component Zf S0ee and screening cloud q(k)

in TCP-RPA model

TCP-RPA contains additional dynamic terms


TCP-RPA model largely agrees with Chihara formula

DSF calculations for fully ionized H, ne = 1022 cm−3,T = 1 eV

0.0001

0.001

0.01

0.1

1

10

100

0.001 0.01 0.1 1

See

0

TCP-RPAChihara

S(k

,ω)

ω (ωpl)

Dynamical structure factor for H,Zf = 1, α = 3

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.5 1 1.5 2

TCP-RPAChihara

low

fre

qu

en

cy s

ca

tte

r W

(k)

k (1/aB)

Integrated low frequency signal

Chihara DSF evaluated in RPA, q(k) =√Zf Sei (k)/Sii (k)

Additional dynamical features not important for integrated signal


Application 2: Angle and energy resolved XRTS on shockcompressed boron

5 and 8 keV photons are used to probeshock-compressed B at various k

X‐rayprobebeam(250J,10ps)

Borontarget

Goldshield(CHcoated)

Drivebeam(527nm,400J,10ns)

Probefoil(V,Cu)

K‐α

tovanHamosspectrometer

Experiments performed at theTitan Laser, LLNL

HELIOS simulations of thedrive-beam target interaction

Neumayer, Fortmann et al., PRL (2010)


We completely characterize the plasma via XRTS

Plasmon position gives electron densitysolid: w/ collisions, dashed: w/o coll.

0

0.2

0.4

0.6

0.8

1

1.2

4x1023 cm-3

2x1023 cm-3

6x1023 cm-3

sign

al [a

.u]

θ = 31o

α=1.7

ΔEpl

k=1.3 Å-1 compressed

uncompressed

source

(a)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

4850 4875 4900 4925 4950 4975

BMA+LFCRPA

sign

al [a

.u]

photon energy E [eV]

67o

62o

47o

α=0.83

α=0.90

α=1.2

(b)

k=2.0 Å-1

k=2.8 Å-1

k=2.6 Å-1

Non-collective spectrum sensitive toionization

g/cc g/cc

g/cc

k=4.3/A a=0.56

Best fit parameters:ne = 4× 1023 cm−3, Zfree = 2.3,⇒ ρ/ρ0 = 1.3. .


Plasmon dispersion and width reveal influence ofmany-particle correlations

Plasmon dispersion data supportstatic LFC model

10

20

30

40

50

60

70

80

90

1 2 3 4

RPA

BMA+LFC

BMA

plas

mon

shi

ft Δ

E plas

mon

[eV

]

wavenumber k [1/Å]

ωpl

2k2/2me

Collisional plasmon width mainly dueto e-i collisions

0

10

20

30

40

1 2 3 4

RPA

BMA+LFC

BMA

wid

th σ

S [eV

]

wavenumber k [1/Å]

Still unresolved discrepancies between experiment and theory.


Summary

Many-body field theory is applied to calculate radiation effects instrongly coupled, non-ideal plasmas.

Dyson-Schwinger Equations are solved self-consistently yielding thequasi-particle shift and damping

We developed a reliable Thomson scattering code as a basis forXRTS plasma diagnostics

Collisional plasmon width and dispersion are succesfully described viathe externded Born-Mermin approximation.

First attempts to apply ab-initio methods for DSF calculations arepromising


dynamical structure factor in warm dense matter and...

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