The dynamic conductivity and the plasmon profile of Aluminum in the ultra-fast-matter regime; an analysis of recent X-ray scattering data from the LCLS

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by  M. W. C. Dharma-wardana

The dynamic conductivity and the plasmon profile of Aluminum in the
ultra-fast-matter regime.
M.W.C. Dharma-wardana
National Research Council of Canada, Ottawa, Canada, K1A 0R6 ∗
(Dated: January 29, 2016)
We use an explicitly isochoric two-temperature theory to analyze recent X-ray laser scattering
6 data for Aluminum in the ultra-fast-matter (UFM) regime up to 6 eV. The observed surprisingly
1 low conductivities are explained by including strong electron-ion scattering effects using the phase
0 shiftscalculatedviatheneutral-pseudo-atommodel. TheapplicabilityoftheMerminmodeltoUFM
2 is questioned. The static and dynamic conductivity, complex collision frequency and the plasmon
line-shapeareevaluated within aBorn approximation andare ingood agreement with experiment.
n
a
PACSnumbers: 52.25.Os,52.35.Fp,52.50.Jm,78.70.Ck
J

7
2 INTRODUCTION (T = T ) at low T are partly attributed to the posi-
e i
tion of the scattering momentum 2k falling within the
] F
h Introduction - Short-pulsed X-ray photons, e.g., from second minimum in the ion-ion structure factor S(q). In
c the Linac Coherent Light Source (LCLS) have begun an isochoric UFM solid, the ions have no time to ad-
e
m to provide data in hitherto inaccessible regimes of mat- just to the rapidly heated electrons. The ions (and their
ter [1, 2]. Such information is of interest in under- bound electrons)remain frozenat their lattice sites, and
-
t standing normal matter under extreme conditions [3– at Ti. Hence S(q), and the bare electron-ion pseudopo-
a
t 6], as well as at new frontiers in high-energy-density tential W(q) remain essentially unchanged, even up to
.s matter, astrophysics, fusion physics etc. Such non- Te =6 eV. The thermal smearing of the Fermi sphere is
at equilibrium systems are also produced in semiconductor setbyf′(k,Te)=f(k,Te)(1−f(k,Te)),wheref(k,Te)is
m devices [7]. The theory involves complicated many-body theelectronFermifunction. It’soverlapwiththe ion-ion
- effects andthe quantum mechanics of finite-temperature S(q), and the electron-ion scattering cross section deter-
d non-equilibrium systems. Standard ab-initio methods mine the conductivity σ(0) as well as σ(ω).
n
are inapplicable or computationally prohibitive for this The new experiment provides the profile of the plas-
o
c ultra-fast matter (UFM) regime. Extensions of elemen- mon resonance. We present a simple theory of the mo-
[ tary plasma models or Thomas-Fermi models fail badly. mentumrelaxationandenergydephasingfrequencyν(ω)

Hence computationally simple realistic theories of these (alsoknownasthe‘collisionfrequency’),usingaBornap-
1
v systems are essential in the interpretation of experi- proximation constructed to match the ω →0 conductiv-
6 mentsonUFMwhichisasub-classofwarm-dense-matter ity obtained from the NPA phase shifts. The calculated
6 (WDM) [8]. Here we use a finite-T density-functional plasmon profile is in good accord with experiment.
5 theory (DFT) calculation of the electronic charge distri- The DFT-NPA model for isochoric UFM Aluminum
7
bution n(r) and the ion charge distribution ρ(r) around - An aluminum nucleus is placed in an electron subsys-
0
. an Al ion in the system as the basic ingredient of such a tem and an ion subsystem, within a large sphere (R ∼
1
theory. The neutral pseudoatom (NPA) model of Perrot 30 au.) where all particle correlations reach bulk val-
0
and Dharma-wardana [9, 10] is used in this study. ues as r → R. Hence this ‘neutral-pseudo-atom’ (NPA)
6
1 The LCLS results [1] of the plasmon feature and the is not an ‘average-atom cell-model’ similar to the IN-
: dynamic and static conductivities σ of Al up to 6 eV, FERNO model of Lieberman or its improvements [12].
v
i isochorically held at solid density dramatically improves The electron density in the bulk, viz., ne is 1.81×1023
X on the accuracy of the earlier UFM experiments [6, 11]. electrons/cm3, has an electron-sphere radius rs = 2.07
ar Surprisingly low static conductivities σ(0) of UFM alu- au. The free-electron pile up nf(r) and the scattering
minum are reported in Ref. [1], even at 0.2 eV. phase shifts δkl around the Al nucleus are calculated
We present two-temperature (2T) calculations for iso- via the Kohn-Sham equations, using a step-function to
choric Aluminum. Atomic units (a.u., |e|=~=m =1) mimic the ion-ion pair distribution function g(r). This
e
areused,andthetemperatureisinenergyunits. Theion is known to work well for Al [9]. The phase-shifts sat-
temperature T is the initial ‘room’ temperature, while isfy the Friedel sum rule, and the DFT uses a finite-T
i
onlythe electrontemperatureT israisedto6eVbythe exchange-correlation contribution [13]. All the results
e
50 femto-second X-ray pulse. We do not get the gradual in this study follow from the NPA output. The many-
decrease of σ with T found for equilibrium non-isochoric ion system is built up via the S(q) as a superposition of
aluminum. Instead, we reproduce the low static con- NPAs, using the S(q) derived within the theory.
ductivities reported in the experiment. The high con- At room temperature, this calculation yields an ion-
ductivities of the normal solid and the molten metal izationZ =3 andanionWigner-Seitz radiusr ≃ 2.99
ws
2
3 sis where T = T . The real part of the complex con-
(a) (b) i e
T=0.2 eV S(q), T=0.06eV ductivity σ(ω) = σ1 +iσ2, obtained via B-LB-GDW-M
T=6.0 eV ii
0.5 f(1-f), T=6 eV f(1-f), T=0.2 eV 2.5 is two orders of magnitude too large comparedto exper-
f(1-f), T=.2 eV f(1-f), T=6.0 ev
q iment, although the imaginary part σ2(ω) as well as the
M
r rrw=s=22.0.97930 a auu.. 2 perlaaslmmoondeplrsoofifleSa(qre),ipnominutc-ihonbeCttoeurloamccborpdo.teTnhteiaylsuaseswseevl-l
facto 0 s 1.5S(q)ii as pseudopotentials. Since the ω → 0 limit of the σ(ω)
m givesapoorσ(0),theyuseaZimanformulawithsuitable
or Aluminum 1 models of S(q) and pseudopotentials.
F-0.5 n=1.81x1023e/cm3
e
2k
F 0.5 In our approach, the ion-S(q,Ti) at Ti remains intact
2kF for all Te. We first calculate σ(0) using the electron
-1 0 phase shifts obtained from the NPA and obtain good
0 1 2 3 1 2 3 4
k/k q (1/au.) agreement with experiment. The calculation of σ(ω) via
F
the phase shifts is more demanding. Instead, since Al
FIG. 1: (Online color). (a) The pseudopotential form factor is a “simple metal”, an Ashcroft pseudopotential VA(rc)
M(q) at T = 0.2 eV and 6 eV, and the thermal-smearing specified only by the core radius r that reproduces the
c
′
functio′ns f (k,Te)=f(k)(1−f(k)). (b)The overlap of S(q) σ(0) could be found. This rc is consistent with the NPA
and f (k,Te). The ion S(q,Ti) with Ti = 0.06 eV. value. This is used in calculating σ(ω). There is no
low-frequency‘diffusionpole’intheexperimentalspectra
as expected from Mermin theory. Mermin assumes that
au. The r is held constant while T is increased, to
ws e theionsrespondperfectlytotheelectron-densityfluctua-
mimic the isochoric UFM, where as normal solid or liq-
tionsandmaintainlocalchargeneutrality. Thisholdsfor
uid Al expands (i.e, r increases) with temperature. A
ws timescales t much larger than the electron-ion tempera-
static electron response function χ(q,T ) is constructed,
e turerelaxationtimeτ whichismanypico-seconds[23]if
ei
with its local field correction (LFC) satisfying the com-
T 6=T ,orfortimescalessignificantlylargerthanphonon
i e
pressibility sumrule at each temperature. This defines a
timescales if T =T . Thus the Mermin model is largely
i e
fully local pseudopotential W(q) = n (q)/χ(q,T ), and
f e inappropriate for most UFM-WDM systems. Hence we
an ion-ion pair potential U (q) = Z2V −|W(q)|2χ(q).
ii q examineasimpleRPA-likemodelwheretheionsaremere
The pseudopotential W(q) = −ZV M , V = 4π/q2 is
q q q immobile scatterers during the 50 fs signal, and obtain
fitted to a Heine-Abarenkov form for convenience. The
good overallagreement with experiment.
form factor M = n (q)/n0(q) obtained from the NPA
q f f
is shown in Fig. 1(a) at T =0.2 and 6 eV. Here n0f(q) is The conductivity σ(ω) can be expressed via the force-
thelinear-responsechargepileup. Thisapproachiscapa- forcecorrelationfunctionasgiveninstandardtexts(e.g.,
ble of milli-volt accuracy and reproduces even the high- Ref. [24] sec. 4.6). If plane waves are used for the free
temperature phonons [14] discussed by, e.g., Recoules et electrons, the limit ω → 0 recovers the Ziman formula.
al[15](butphononsdonotformduringUFMtimescales). However,if the electron-ioninteractions arestrong,then
The resulting Uii(q) is used in the modified Hyper- theelectronresponseχ(q,ω)andthedynamicconductiv-
Netted-Chainequation(MHNC)yieldingtheS(q)atthe ity σ(ω) should be expressedvia the electroneigenstates
ion temperature Ti (which is the initial temperature of φ(r)α of the system [25, 26]. The NPA provides these,
the system at the arrival of the X-ray pulse). Since the with α=n,l forcore-states,and α=k,l;E =k2/2 for
kl
initial Al-crystal has an FCC structure, it is sufficient to continuumstates,with the mquantumnumber andspin
usethesphericallyaveragedS(q)takenasa‘frozenfluid’, summedover[25,26]. Thecorestatesgivebound-bound
say, at 0.06 eV. The latter is the lowest temperature at transitions, while the bound-continuum and continuum-
which the HNC could be converged, since the melting continuum transitions are also included. If numerical
point is ≃ 0.082 eV. The results are insensitive to the eigenstates φ (r) are not available, hydrogenic func-
α
use of an S(q) at 0.06 eV or, say, 0.082 eV. Our MHNC tions can be used within a many-body theory as in
procedure is accurate enough to closely reproduce the Ref. [26]. Such “Green-Kubo” formulae for σ(ω) usu-
experimental S(q) of normal liquid aluminum [16]. ally need heavy numerical codes. Our NPA approach
The complex conductivity σ(ω) - The Drude theory gives a simpler evaluation of comparable accuracy with
with a static ν(0) is known to be inadequate for σ(ω) ordersofmagnituderapidity. Thecorrectionsbeyondthe
except at small and high ω [19]. Sperling et al [1] have non-interactingresponsecanbeexpressedasarelaxation
usedaMerminmodel(diffusionpole) [21]augmentedby frequency ν(ω)=ν1+iν2 given in terms of a scattering
plasmamany-bodytheory[22]wherethey combinecom- cross section. The ν describes momentum relaxation as
ponents of Born (B), Lenard-Balescu (LB) and Gould- well as energy dephasing. This can be expressed via the
DeWitt (GDW)-Mermin (M) approaches in their analy- NPA phase shifts [17, 18]. The real part ν1(ω) may be
3
(a) 2 LCLS-expt., Ref.1
l=1 l=2 0.6 Theory-This work
10 Theory-Ref.1
0 l=3 s)
d) nit n=1.8 x 1023cm-3
δ(rakl LCiinrcelse:s : TT==60..02 eeVV b. u0.4 Te=0.06 eV, T= 6 eV.
-2 ar i e
S/m) pl=ha0se shifts (b) nsity ( ωp=15.8 eV
( -40 1 2 nte0.2
σ, k/k I
F
y
t
i
v 7980 eV.
ti 1 0
c -3 -2 -1 0 1
u Energy Shift: ω/ω
d
p
n T=0.06 eV
o i
C LCLS-expt. UFM
FIG. 3: (Online color) The UFM-aluminum plasmon line-
tnio Sperling et al. (Ti=Te) shape at Te=6 eV, from experiment and theory.
P
g psudopotential σ, UFM
n
itle phase-shift σ, UFM scattering cross section is evaluated using plane waves
M Normal Liquid (T=T)
i e (i.e., Born approximation), Eq. 3 reduces to the Ziman
Normal Solid (T=T)
i e formula with the weak pseudo-potential W(q)=ZVqMq
0.1 (shown in Fig. 1). The Heine-Abarenkov W(q) gives a
0.1 1 10
T (eV) higherestimate ofσ(0), while the phase-shiftcalculation
e agrees with LCLS.
WereplaceW(q)byanAshcroftpseudopotentialV (q)
A
FIG. 2: (Online color) (a) The static conductivity σ(0) of
chosen to reproduce the static conductivity σ(0), and use
Aluminum. LCLS experiment and the σ(0) from theory for
it to evaluate the relaxation frequency ν(ω) in the Born
UFM aluminum (Ti 6= Te). Some data for the normal solid
approximationto Eq. (1). Thus,
and normal liquid are also shown. Sperling et al(Ti = Te)
dataareaprivatecommunication. (b)TheNPAphaseshifts
1
δkl are shown for l=0-3, as a function of k/kF. ν(ω) = q4|V (q)|2S(q,T )∆(q,ω)dq (5)
6π2Z Z A i
{χ (q,ω,T )−χ (q,0,T )}
e e e e
given as: ∆(q,ω) = (6)
iω
ν1(ω) = ℑ q2Σ(~k,~q)S(q) f(~k)−f(~k+~q) (1) Eq.(5)isbasicallyHopfield’sexpression[19],whilemod-
3Z Xq~,~k iω(ω+ǫ~k−ǫ~k+q~) ern discussions are found in Refs. [20, 22]. The S(q) is
for the cold ions at T =0.06eV, as shownin Fig. 1(b).
i
2
The plasmon profile and ν(ω). An important result of
Σ(k,q) = (cid:12)k−2 (2l+1)eiδklsin(δkl)Pl(cosθ)(cid:12) (2) the LCLS-experiment is the plasmon profile from UFM-
(cid:12) X (cid:12)
(cid:12) l (cid:12) Aluminum. We discuss T = 6 eV in detail. Eq. (5)
(cid:12) (cid:12)
The static limi(cid:12)t of Eq. 1 gives: (cid:12) evaluates ν1(ω) and ν2(ω) using the VA(rc) pseudopo-
∞ tential. Obtaining ν1 via ℑ{χ(q,ω)} in Eq. 6 and ν2
ν(0) = 1 f(k)(1−f(k)k2dkF(k) (3) via Kramers-Kronig is computationally convenient. A
3πZTe Z0 direct estimate of ν2 is also available from Eqs. (5) and
2k (6). The response function χ(q,ω) uses an LFC derived
F(k) = q3Σ(q,k)S(q)dq; q =k(1−cosθ)1/2(4) from the finite-T xc-potential [13]. The transverse di-
Z
0
electric function ε(q → 0,ω + iν(ω)) provides the op-
The original numerical implementation (see appendix, tical scattering cross section S (q → 0,ω). This is
ee
Ref. [17]) has been improved, using up to 38 l-states if ∝ℑ{1/ε(ω−ν2+iν1)}nB(ω)wherenB(ω)isaBosefac-
needed,usinganenergycutoffofE +2T ,togetherwith tor at the electron temperature T . Instead of Mermin
F e e
asymptotic corrections. Typical δ from the NPA are theory we use the simple RPA-like transverse dielectric
kl
showninfig.2(b). Resultsforσ(0)forisochoricAl,from function in the q → 0 limit. The calculated scattered
Eq. 3 covering 0.2 eV to 10 eV are given in Fig. 2(a) intensity is shownin Fig. 3. The predicted profile differs
whileσ(0)upto100eVareinTable.1ofRef.[18]. Ifthe on the red wing of the experimental plasmon line shape.
4
1.5 with the LCLS data. A simple Born approximation to
Theory: ν(ω) complex conductivity
Theory: ν12(ω) Te=6 eV. the dynamic conductivity using a pseudopotential fitted
)/ν(0) 1 EExxpptt:: νν21((ωω)) ttohethpelatshmeoonreltiinceaslhσa(p0e)apnrodvtidheesdayngaomodicacpopnrdouxicmtiavtitioynotbo-
ω
ν( Expt: σ1(ω) 0.01 tained from the LCLS experiment. It is argued that the
ncy 0.5 ETxhpeot:r yσ:2 ( ωσ)1(ω) σ(0) Mtheermioninsfhoarmveinsointaipmperotporiraetsepofonrdu.ltAraffuaslltcmaalctutelartwiohneroef
que Theory: σ2(ω) ω)/ ν(ω) entirely from the phase shifts via Eq. (1) may re-
e (
fr 0 σ solve some of the shortcomings in the present theory.
n
o
axati ν2(K-K) 0.0001 The author thanks Heide Reinholz, Philipp Sperling
Rel-0.5 ν2(direct) and colleagues for their comments.
-1
00 11 22 33 1 2
Energy shift ω/ω Energy shift ω/ω
P p
∗
Email address: [email protected]
FIG. 4: (Online color) (a) Theory and experiment for the [1] P. Sperling et al., Phys.Rev. Lett.115, 115001 (2015)
momentum-relaxation frequency ν1 and ν2 versus theenergy [2] S. Glenzer and R. Redmer Rev. Mod. Phys. 81, 1625
shift ω/ωp. ν2(ω)calculated from ν1(ω)viaKramers-Kro¨nig, (2009)
and via a direct numerical procedure are shown. (b) σ1(ω) [3] Andrew Ng, Int.J. Quant.Chem. 112, 150 (2012)
and |σ2(ω)|from experiment and theory. [4] Z. Chen et al., Phys. Rev.Lett. 110, 135001 (2013)
[5] JieChenaetal.,Proc.Nat.Acad.Sci.,10818887(2011)
[6] H.M.Milchberg et al.Phys.Rev.Lett.61, 2364(1988).
SinceV (r )wasfittedtothephase-shiftσonlyatω =0, [7] M.W.C. Dharma-wardana,SolidStateCommunications,
A c
this is not surprising. 86, 83 (1993)
The relaxation frequency ν(ω) and the conductivity [8] F. R. Graziani et al. Lawrence Livermore National Lab-
oratory report, USA,LLNL-JRNL-469771 (2011)
σ(ω) can be extracted from the experimental S(q →
[9] F. Perrot and M.W.C. Dharma-wardana, Phys. Rev. E.
0,ω). We use the experimental ν1(ω),ν2(ω) of Sperling 52, 5352 (1995)
et al., to test our methods, even though they assumed
[10] M. W. C. Dharma-wardana, Contributions to Plasma
a Mermin form to extract the data, assuming that the Physics, 55, 85 (2015)
modeling differences fall within the error bars. The ex- [11] M.W.C.Dharma-wardanaandF.Perrot,Phys.Lett.A
perimentalνex andνex arecomparedwiththecalculated 163, 223 (1992)
1 2
ν1,ν2 in the figure 4, where the energy shift ω is ω1−ω0 [12] Wilson et al., JQSRT99, 658-679 (2006)
[13] F.PerrotandM.W.C.Dharma-wardana,Phys.Rev.B15
with ω0=7980 eV., and hence negative (for the plasmon 62, 16536 (2000) Erratum 67, 79901 (2003)
studiedhere). Thetheoreticalν1decaysveryslowlycom- [14] L. Harbour, M. W. C. Dharma-wardana, D. D.Klug, L.
paredtheν1ex. Weexpectthistobecorrectedwhenafull Lewis, Contr. Plasma. Phys. vol. 55, 144-151 (2015)
evaluation using phase shifts is used. [15] V. Recoules et al., Phys.Rev. Lett.96, 055503 (2006).
The Drude formula provides σ(ω) from ν(ω). Set- [16] M. W. C. Dharma-wardana and G. C. Aers, Phys. Rev.
ting ̟ = ω − ν2, d(ω) = ν1(ω)2 + ̟2 we use α = B. 28, 1701 (1983).
ωp2/(4π), σ1(ω) = αν1/d and σ2(ω) = α̟/d. We have [17] 3F6.,P2e3rr8o(t1a9n8d7)M. .W.C.Dharma-wardana,Phys.Rev.A
recalculated σ1,σ2 from our theoretical ν1,ν2 given in
[18] F. Perrot and M. W.C. Dharma-wardana, International
Fig. 4(a), and from the experimental ν1,ν2 at T=6 eV Journal of Thermophysics, 20,1299 (1999)
givenin Fig. 3 of the supplementary material of Ref. [1]. [19] Hopfield, Phys.Rev 139, A419 (1969)
The resulting σ1(ω),σ2(ω) are displayed in Fig. 4(b). [20] G. D. Mahan, Many particle Physics, Sec. 8.1, Plenum
Note that although σ2(ω) is expected to tend to zero Publishers New York(1981)
asω →0,thishappens onlyquiteclosetoω =0because [21] N. D. Mermin, Phys. Rev.B 1, 2362 (1970)
of the strong negativity seen in both experimental and [22] H. Reinholz, R. Redmer, G. Ro¨pke and A. Wierling,
Phys. Rev.E 62, 5648 (2000)
theoretical numbers for ν2 (see Fig. 4(a)). Although σ1 [23] M. W. C. Dharma-wardana, Phys. Rev. E, 64 035401
is close to the experiment for small-ω, it begins to differ
(2001)
significantly from experiment as ω increases. [24] G. F. Giuliani et al., Quantum Theory of the Electron
In conclusion, the static conductivity calculated us- Liquid., Sec. 4.6, Cambridge University Press (2005)
ing phase-shifted NPA electron eigenfunctions for two- [25] H. Ehrenreich et al., Phys. Rev.132, 1918 (1963)
temperature ultra-fast aluminum are in good agreement [26] M. W. C. Dharma-wrdana, Physica 92A, 59 (1978)