outstanding questions in electron–ion energy relaxation, lattice stability, and dielectric...

Outstanding Questions in Electron–IonEnergy Relaxation, Lattice Stability,and Dielectric Function of WarmDense Matter

ANDREW NG

Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, Canada

Received 29 April 2011; Revised 26 May 2011; accepted 31 May 2011Published online 11 April 2011 in Wiley Online Library (wileyonlinelibrary.com).DOI 10.1002/qua.23197

ABSTRACT: Our earlier experiments in shocked silicon and gold heated withfemtosecond-laser have revealed some unexpected findings including a reducedelectron–ion coupling constant, superheating, and enhancement of interband transitions.The original experiments and their interpretations are reviewed together withdiscussions of new perspectives to identify key questions that need to be addressed inunderstanding electron–ion energy relaxation, lattice stability, and dielectric functionunder nonequilibrium, warm dense matter conditions. VC 2011 Wiley Periodicals, Inc. Int JQuantum Chem 112: 150–160, 2012

Key words: warm dense matter; electron–ion energy relaxation; lattice stability;dielectric function

1. Introduction

W arm dense matter refers to states in whichthe electron temperature becomes compa-

rable with Fermi energy of the state and the iondensity is sufficiently high to render ion–ion Cou-lomb potential energy greater than ion kineticenergy [1]. Such states are many-body systemsdominated by effects of electronic excitation, elec-tron degeneracy, and strong ion–ion correlation.They cannot be described satisfactorily by conven-

tional condense matter or plasma theory. Yet theylie in the important regime of solid-plasma transi-tion that is central to broad disciplines rangingfrom material processing to inertial confinementfusion. Although the name of warm dense matterfirst appeared in an international workshop in2000 [2], studies of warm dense matter can befound much earlier in other areas of researchincluding high pressure shock waves and stronglycoupled plasmas.

Interest in warm dense matter has been rapidlygrowing. Although much progress has beenmade, many questions about fundamental proper-ties of warm dense matter remain open. In thisCorrespondence to: A. Ng; e-mail: [email protected]

International Journal of Quantum Chemistry, Vol 112, 150–160 (2012)VC 2011 Wiley Periodicals, Inc.

article, we will present three such examples thathave emerged from our earlier investigations onelectron–ion energy relaxation in shocked silicon,lattice stability of gold in ultrafast laser excitation,and dielectric function of gold under nonequili-brium high-energy density conditions. These aredescribed in Sections 2–4. For each case, we beginwith a review of the original observations andinterpretations. This is followed by a discussionof new perspectives. Our descriptions focus onthe salient features only. More detailed accountscan be found in original publications included inthe list of references. Outstanding questions arediscussed in Section 5.

2. Electron–Ion Energy Relaxation

2.1. ORIGINAL OBSERVATIONS ANDINTERPRETATIONS

Energy relaxation between electron and ion (orlattice) subsystems is of long standing interest incondensed matter and plasma physics. It is tied tofundamental understanding of electron–ion (orelectron–phonon) coupling. At the same timestates with different electron and ion (or lattice)temperatures are prevalent in laboratory warmdense matter. An experimental approach thatreadily lends itself to the study of such process isa single shock wave propagating in a solid. Im-

mediately behind the shock front, compression ofthe solid gives rise to heating of the ions. Electronheating then follows via energy transfer fromions. These processes are often treated with theuse of a two-temperature model [3] where thetime rate of change in temperatures of electronand ion are given by the coupled equations:

Ce@Teðx; tÞ

@t¼ g½Tiðx; tÞ � Teðx; tÞ�

þ @

@xjðne;TeÞ @

@xTeðx; tÞ ð1Þ

Ci@Tiðx; tÞ

@t¼ g½Teðx; tÞ � Tiðx; tÞ� (2)

Te (Ti) and Ce (Ci) are, respectively, the tempera-ture and specific heat capacity of the electron(ion) subsystem, ne is the electron density, j iselectron thermal conductivity, and g is the elec-tron–ion coupling coefficient that is generallytaken to be a constant.

Figure 1 shows an example of the calculatedtemperature gradients corresponding to a 6 Mbarshock wave in silicon for an assumed g value of1016 W/m3K. At such a high pressure, the bandgap of silicon is expected to be closed giving riseto metal-like conductivity behind the shock front.Thus, observation of optical emission from theshocked material becomes limited to that emanat-ing from a layer at the shock front with a thick-ness comparable with the photon mean-free-path.Accordingly, pyrometric measurements wouldyield emission intensity or electron temperaturesignificantly lower than that of an equilibratedshocked state as illustrated in Figure 1; their exactvalues would depend on the electron–ion cou-pling constant. This was the basis of our experi-mental investigation in shocked silicon [4, 5].

A schematic diagram of the first experiment [4]performed at the University of British Columbiais shown in Figure 2. Shock waves in silicon were

FIGURE 1. Results of two-temperature model calcula-tions of electron and ion temperature profiles in siliconcompressed by a 6 Mbar shock wave. Optical emissionmeasurements would be localized to a layer behind theshock front with a thickness limited to an optical depth.[Color figure can be viewed in the online issue, which isavailable at wileyonlinelibrary.com.]

FIGURE 2. Schematic diagram of experimental setupfor measurements of optical emission from shockcompressed silicon. [Color figure can be viewedin the online issue, which is available atwileyonlinelibrary.com.]

PROPERTIES OF WARM DENSE MATTER

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produced by laser-driven ablation. A 532-nm, 2.3-ns (FWHM) laser beam was focused onto thefront surface of an intrinsic, <100> silicon waferwith a focal spot diameter of �150 lm. Maximumirradiance on target was 5 � 1013 W/cm2. Siliconwas chosen for the study because of its importantpractical advantages. It is readily available ashigh-purity samples with optical-quality surfaces.With an indirect band gap of 1.2 eV at normalconditions, its absorption coefficient is less than104 cm�1 for wavelengths exceeding 500 nm. Thismakes it possible to observe, with the availabletemporal resolution, optical emissions from ashock wave while it is in flight inside the samplethus eliminating the usual difficulty where theshock front is visible only when it arrives at targetfree surface but is immediately obscured by therelease of the surface. Silicon also becomes metal-lic at pressures exceeding �120 Kbar.

In the experiment, the shock wave was charac-terized by shock speed determined from transittime measurement through targets of differentthicknesses. Emissions from the shock wave inflight in silicon were observed through target freesurface using f/1.4 optics, narrowband interfer-ence filters, and a streak camera. Temporal andspatial resolutions of the measurements were 20ps and 4 lm, respectively. Figure 3 shows meas-ured intensity of shock emission at 430 and 570nm for a shock speed of 20 km/s (correspondingto a shock pressure of �6 Mbar). For each wave-length, results from three separate shots were dis-played to show shot-to-shot reproducibility.

To calculate the intensity of shock emission, weconsidered an electromagnetic wave propagatingthrough the unperturbed region of the target to-ward the shock front at normal incidence. The

reflected and transmitted fields were obtainedfrom solutions of Maxwell’s equations. Dielectricfunction of the shocked silicon was derived froma conductivity model for dense plasma [6],whereas known dielectric function data [7] wereused for cold silicon. Electron thermal conductiv-ity was also obtained from the dense plasmamodel [6]. This yielded absorption of the EMwave by the shock wave from which an effectiveemission power was derived via Kirchhoff’s law.First, a baseline calculation was made assumingequal electron and ion temperatures in theshocked state. The calculated emission intensity isrepresented by the dot-dashed line in Figure 3. Itshowed peak intensity about an order of magni-tude higher than what was observed. To assessthe effect of conductivity model on emission in-tensity, calculation was also made using Spitzerconductivity that is known to be valid for idealplasma only. It led to even greater deviationsfrom observation as indicated by the dashed linesin Figure 3. A new set of calculations was thenmade assuming initial heating of ions only as theelectrons remained cold. Temperature equilibra-tion between ions and electrons behind the shockfront was treated with a two-temperature modeldescribed by Eqs. (1) and (2) using once again theLee–More electrical and thermal conductivity.These revealed strong dependence of emission in-tensity on electron–ion coupling constant (Fig. 3),with the value of 1016 W/m3K giving the best fitto experimental data. Similar behaviors were alsofound in measurements made at different shockspeeds. As shown in Figure 4, the observed peakemission intensity at a shock speed of 18 km/swas consistent with an electron–ion coupling con-stant of 1016 W/m3K, whereas an even lower

FIGURE 3. Intensity of optical emission from shocked silicon for shock speed of 20 km/s [4].

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152 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY DOI 10.1002/qua VOL. 112, NO. 1

value was suggested by the data at 15 km/s. Thisbecame the first experimental determination ofelectron–ion coupling constant in the warm densematter regime. At the time, corresponding elec-tron–ion coupling constant estimated from thewell known Spitzer–Brysk model for weaklycoupled plasmas [8] was about two orders ofmagnitude higher.

However, there is a weakness in this earlystudy. Our interpretation of an electron tempera-ture gradient behind the shock front was based onthe comparison of observed shock emission inten-sity to results of calculation that (i) assumed a con-stant value for electron–ion coupling coefficientand (ii) relied on the use of a dense plasma con-ductivity model [6]. Such limitations were elimi-nated in a second experiment conducted at theMax-Planck Institute for Quantum Optics based onsimultaneous measurements of emission intensityand reflectivity of a shock front in flight in silicon[5]. The use of these two measured quantities inKirchhoff’s law allowed direct determination ofthe brightness temperature of the shock front freefrom any opacity or transport models. For a shockspeed of 20.7 km/s, the measured electron temper-ature at the shock front was found to be 1.4 eV,which was substantially lower than the 3.45 eVHugoniot temperature predicted for the equili-brated shocked state, corroborating the findings ofthe experiment described above [4].

2.2. NEW PERSPECTIVES

Two quantum mechanical calculations ofenergy relaxation in warm dense matter havebeen reported. The first was the work of Dharma-wardana and Perrot [9] describing a two-tempera-ture aluminum plasma with a mass density of

2.374 g/cm3 and an ion temperature of 943 K forelectron temperatures ranging from 2 to 50 eV. Inthe Fermi Golden Rule approach, electrons andions were treated as two weakly interacting sub-systems, ignoring dynamical interaction betweenthem. Energy relaxation was calculated via inter-action of the normal modes of the subsystems. Toinclude dynamical effects in the coupling of theelectron and ion subsystems, a coupled-modeapproach was used in which purely electronic orionic normal modes no longer exists. Instead,electron density fluctuations mixed with ion den-sity fluctuations to form ion-acoustic modes gov-erned by both electron and ion temperatures.Energy transfer then occurred from hot to coolercoupled modes. For electron temperatures of 3–5eV, the electron–ion coupling constant calculatedfrom the Spitzer model was (2.0–2.4) � 1018

W/m3K. Fermi Golden Rule yields much lower val-ues of (6.1–6.6) � 1017 W/m3K. The slower energyexchange between the subsystems was attributedto the extremely weak overlap between spectralfunctions of the electron density fluctuation andthe ion density fluctuation. With coupled modeapproach, the electron–ion coupling constant wasfurther reduced to (5.5–6.4) � 1016 W/m3K, under-lining the importance of dynamical couplingeffects. Undoubtedly these are encouraging resultsin view of the 1016 W/m3K electron–ion couplingconstant derived from the shocked silicon experi-ments. However, we need to be mindful that thesecalculations refer to energy relaxation in aluminumplasma with hot electrons and cold ions, whereasour experiment is a study of shocked silicon withhot ions and cold electrons.

More recently, the problem of energy relaxa-tion in shocked silicon was treated in a quantumkinetic approach by Vorberger et al. [10]. The

FIGURE 4. Peak intensity of optical emission from silicon as a function shock speed [4].



silicon plasma studied was chosen to correspondto that produced by shock compression at a shockspeed of 20 km/s as in our earlier experiment [4].The calculations yielded a temperature equilibra-tion time between the hot ion and cold electronsubsystems of �0.4 ps using Fermi Golden Rulemethod and �1 ps using coupled-mode methodwith local field correction for ion–ion correlation.These were more than two orders of magnitudeshorter than that inferred from the experimentusing a two-temperature model analysis with aconstant electron–ion coupling coefficient as illus-trated in Figure 1 above. However, such a com-parison is likely erroneous. The quantity that wasmeasured in the experiment [4] was intensity ofshock emission and not electron–ion couplingcoefficient. The assumption of a constant couplingcoefficient throughout the thermal gradientsbehind the shock front is open to question. Infact, it was found in the quantum kinetic calcula-tions that for a two-temperature hydrogenplasma, the coupled-mode energy relaxation rateincreases substantially with time during the relax-ation process [10]. A more appropriate compari-son between experiment and theory should focusinstead on temperature of the shock front thatwas measured in the second experiment onshocked silicon [5]. As explained above, tempera-ture obtained from optical pyrometry would belocalized to a layer behind the shock front with athickness comparable with optical mean free path,which is typically 5–10 nm for a metal. For shockspeed of 20 km/s, this would correspond to statesin which temperature relaxation has occurred for0.25–0.5 ps. According to quantum kinetic calcula-tions using coupled-mode approach with localfield correction, the corresponding electron tem-peratures are �2.6 eV at 0.25 ps and �3 eV at 0.5ps [10]. These values still leave a discrepancy ofabout a factor of two when compared with themeasured value of 1.4 eV [5].

3. Lattice Stability


Lattice stability is also a focus of long standing,broad research due to its importance in under-standing the fundamental process of melting.Recently, a new phenomenon was found in semi-conductors excited by femtosecond lasers, namely,

nonthermal or athermal melting that occurs in atime scale of �100 fs [11–14]. This was attributedto formation of antibonding states when valenceelectrons were excited up to the conduction band[15]. The finding drew significant interest inunderstanding lattice stability in metals underultrafast excitation conditions, particularly athigh-energy densities characteristic of warmdense matter.

Our investigation began with an isochoric laserheating experiment in which free standing, solid-density, 30-nm-thick gold foils were excited by a150 fs, 400-nm pump laser at normal incidence[16]. The experiment was performed in the JupiterLaser Facility at Lawrence Livermore NationalLaboratory. A schematic diagram of the experi-mental setup is shown in Figure 5. Laser energywas first absorbed in the foil via skin-depth depo-sition. Ballistic electron transport then led to iso-thermal heating of the foil. Heating of the foil alsooccurred under isochoric condition because thetime scale for ion expansion was much longerthan the duration of pump laser pulse. Thesecombined to produce a uniform warm dense mat-ter state of gold defined by initial mass density ofthe solid (qo) and excitation energy density (De)imparted by the heating pulse. Excitation energydensity was determined from measurements ofpump laser reflection and transmission. The firstmeasurement made was reflectivity and transmis-sivity of a 150 fs, 800-nm probe laser at 45� angleof incidence (Fig. 5). As presented in Figure 6,this revealed a quasi-steady state of gold withnearly constant reflectivity and transmissivity thatpersisted for >4 ps at an excitation energy densityof 4 MJ/kg [16]. This spurred some intense debatebecause for the laser intensity used in our experi-ment, thermal melting of gold was expected tooccur in <1 ps when the lattice temperaturereached the normal melting point of 1337 K.

To achieve more accurate determination of thelifetime of quasi-steady dielectric state of gold,

FIGURE 5. Schematic diagram of experimental setupfor femtosecond-laser excitation of ultrathin gold foils.[Color figure can be viewed in the online issue, which isavailable at wileyonlinelibrary.com.]

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the reflectivity-transmissivity diagnostics wasreplaced with frequency domain interferometry.The latter utilized a double-pulse 800-nm laserprobe to monitor surface motion of the heated foil

with better than 1-nm sensitivity through meas-urements of change in phase shifts in the reflectedprobe pulses. The experiment [17] used a configu-ration similar to that illustrated in Figure 5 exceptwith the single pulse probe diagnostics replacedby the frequency domain interferometer setup.The results obtained at several excitation energydensities are presented in Figure 7. They all dis-play three distinct temporal features: (i) initialrapid increase of change in phase shift inresponse to laser heating, (ii) a quasi-steadydielectric state exhibiting nearly constant changein phase shift, and (iii) rapid change in phaseshift due to hydrodynamic expansion. The slopeof the last change in phase shift varies with exci-tation energy density, depending on the combinedeffect of increasing dielectric gradient scale lengthand hydrodynamic motion developed at foil sur-face. Figure 8 displays the measured quasi-steadystate lifetime as a function of excitation energydensity. This shows persistence of the state over aduration ranging from 2 to 20 ps.

To calculate the dependence of quasi-steadystate lifetime on excitation energy density, we firstapplied a two-temperature model [3] to the

FIGURE 6. Measured reflectivity and transmissivity at800 nm of gold at excitation energy density of 3.5 MJ/kg [17]. [Color figure can be viewed in the online issue,which is available at wileyonlinelibrary.com.]

FIGURE 7. Temporal evolution in change in phase shift of reflected 800 nm, 150-fs laser probe for different excita-tion energy densities [17]. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]



electron and phonon subsystems of gold. Theusual equation for lattice temperature wasreplaced with that for ion energy density to avoidexplicit treatment of melting and vaporizationtransitions. The coupled equations are givenbelow.

CeðTeÞ dTeðtÞdt

¼ �g TeðtÞ � elðtÞqAu

Cl

� �þ SðtÞ (3)

qAu

dðelðtÞÞdt

¼ g TeðtÞ � elðtÞ qAu

Cl

� �(4)

with elðtÞ ¼ ClTlðtÞqAu

, CeðTeÞ ¼ @UeðTeÞ@Te

, and SðtÞ ¼ DeqAu

sPffiffip

p

exp � t2

s2P

� �where Te and Ce are, respectively, electron tem-

perature and heat capacity, el, Tl, and Cl are,respectively, lattice energy density, temperature,and heat capacity, qAu is mass density of gold, Sis pump laser energy deposition term, De is meas-ured excitation energy density, and sp is pumppulse width. Figure 9 shows an example of resultsfrom the two-temperature mode for excitationenergy density of 4 MJ/kg.

Then, we made the hypothesis that disassem-bly of the solid occurred when its lattice energydensity reached a critical value eD, which is a con-stant independent of heating rate or electron tem-perature. The predicted lifetime of the quasi-steady state would be determined by the value ofeD chosen (Fig. 9). Accordingly, eD was used as afree parameter to yield the best-fit to the quasi-

steady state lifetime versus excitation energy den-sity data in Figure 8. Evidently, this ad hoc heat-ing-disassembly model appeared to be consistentwith experiment.

At the time these experiments were reportedthe phase of our observed quasi-steady state ofgold was not known. Measurements showed thatearly on rise of the laser heating pulse, both thereal and imaginary parts of the dielectric functionhad reached values equal to that for an equilib-rium liquid phase of gold [17]. This led to theconfusion that the quasi-steady state might be anonequilibrium liquid. On the other hand, Maze-vet and et al. found that results of their AC con-ductivity calculations would be consistent withour experimental data if the quasi-steady statewas assumed to be a solid with fcc lattice and iontemperature of 2100 K [18].


The first experimental evidence that theobserved quasi-steady state of gold was a none-quilibrium superheated solid phase was providedby the persistence of interband (5d-6s/p) contri-butions in broadband dielectric function measure-ments [19]. Details of the experiment aredescribed in the Section 4.1 below. This was fur-ther corroborated by the work of Ernstorfer et al.[20] in which ultrafast electron diffraction meas-urements were used to monitor temporal evolu-tion of lattice structure in gold heated with femot-second laser. The time scale for decay of the (220)diffraction peak was found to be comparable withthe quasi-steady state lifetime measured in ourexperiment.

FIGURE 8. Lifetime of quasi-steady dielectric state infemtosecond-laser heated gold [17]. The two best-fitcurves are results of the two-temperature heating andcritical disassembly model described in text, assumingdifferent values of the electron–ion coupling constant g.[Color figure can be viewed in the online issue, which isavailable at wileyonlinelibrary.com.]

FIGURE 9. Results of two-temperature model calcula-tion of electron temperature and lattice energy densityin gold at excitation energy density of 4.2 MJ/kg [17].[Color figure can be viewed in the online issue, which isavailable at wileyonlinelibrary.com.]

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An interesting development in theory has alsobeen put forth by Recoules et al. [21]. Using abinitio calculations, they showed that in gold witha cold lattice and an elevated electron tempera-ture, excitation of 5d electrons reduced screeningand made the effective electron–ion potentialmore attractive. This in turn increased the effec-tive ion–ion potential and modified the phononspectra, giving rise to the phenomenon of phononhardening. Heat capacity of ion calculated fromthe modified phonon spectra was fitted to aDebye function. The result showed an increase inDebye temperature yD with electron temperature.Following Debye–Lindemann’s theory, meltingtemperature was taken to scale as yD

2 and henceincreased rapidly with electron temperature, pro-ducing a nonequilibrium superheated solid phaseof gold. At an electron temperature of 6 eV, themelting temperature of gold with a cold latticewas predicted to increase from 1337 K to �4,700K.

When compared with our phenomenological,ad hoc heating-disassembly model, the phononhardening model with its ab initio approach isclearly appealing. However, we are unable to dis-cern the applicability of either model. In ourexperiment, there was no measurement of latticetemperature at the onset of target disassembly toshow that it was the same, independent of excita-tion energy density. On the other hand, there isno calculation of time scale of superheating in thephonon hardening model to allow comparisonwith experimental data.

4. Dielectric Function


The fundamental significance of dielectric func-tion is well known. Embedded in it are informa-tion on electron band structure and density ofstates. These are properties particularly central tounderstanding the behaviour of warm dense mat-ter state that encompasses phase transformationand electronic excitation, such as femtosecond-laser heated gold at high energy densities.

Our new investigation into the phase of thequasi-steady dielectric state in gold describedabove was the measurement of broadband dielec-tric function [19]. The imaginary part of dielectricfunction of gold at normal conditions is shown in

Figure 10. Both intraband and interband contribu-tions are discernable in the readily measurablevisible spectrum from 1.55 eV (800 nm) to 2.76 eV(450 nm). The portion below �2 eV is Drude-like,being attributed to intraband contributions from6s/p electrons while that above �2 eV arises frominterband transition of 5d electrons. Our experi-mental setup for this investigation was similar tothat illustrated in Figure 4 but with the 150-fs,800-nm probe laser pulse replaced by a 450–800nm supercontinuum source with a 600-fs tempo-ral chirp. This experiment was also performed inthe Jupiter Laser Facility at Lawrence LivermoreNational Laboratory. Values of dielectric functionwere again derived from solutions of Helmholtzequations for the measured probe reflectivity andtransmissivity.

Figure 11 shows results of our measurement.Also included in the figure are values for gold atnormal conditions. Relative to the peak of pumplaser pulse, time delay of the probe pulse variedfrom 1.4 ps at 1.55 eV to 2 ps at 2.6 eV. For an ex-citation energy density 17 MJ/kg, no data wereobtained above 2.38 eV corresponding to probedelay of 1.9 ps. This was attributed to disassem-bly of the solid, consistent with pervious observa-tions [17]. In contrary, at lower excitation energydensities the entire probe pulse fell within thequasi-steady state. The dielectric function data inFigure 9 revealed three important characteristics:(i) the persistence of 5d-6s/p interband contribu-tions, (ii) increasing red-shift in interband contri-butions with excitation energy density, and (iii)increasing enhancement in interband contribu-tions with excitation energy density.

The observed persistence of 5d-6s/p interbandcontributions in the dielectric function was taken

FIGURE 10. Imaginary part of dielectric function ofgold at normal conditions. [Color figure can be viewedin the online issue, which is available atwileyonlinelibrary.com.]



as evidence of electron band structure and thusalso evidence of the quasi-steady state remaininga fcc solid phase of gold. As mentioned above,this was later corroborated by ultrafast electrondiffraction measurements [20]. Red shifts in theinterband region of the dielectric function wereattributed to the usual modification of electrondistribution near the Fermi energy as electrontemperature increases.

However, enhancement in 5d-6s/p interbandcontributions to dielectric function of goldemerged as a clear anomaly. Electron–electronenergy relaxation was often thought to occur in�100-fs time scale, much shorter than the quasi-steady state lifetimes of interest here. Accord-ingly, electron distribution was expected to followa Fermi function. This left open the questionabout unknown modification in electron densityof states due to electronic excitation under none-quilibrium conditions.


As noted above, AC conductivity of gold hadbeen calculated by Mazevet et al. for excitationenergy density above 1 MJ/kg [18]. The heatedstate was taken to remain a fcc lattice with ions at2100 K and electrons at elevated temperatures.Interestingly, the results of their broadband AC

conductivity calculation for excitation energy den-sity from 1.6–20 MJ/kg showed a Drude-likeintraband region up to photon energy of �1.3 eVbut from 1.3 to 2.1 eV there appeared to be apeak structure that increased with excitationenergy density. While it may be tempting to com-pare this to the enhanced interband region in ourmeasured dielectric function [19], it should benoted that the observed feature was seen above 2eV (Fig. 11). Furthermore, the origin of the calcu-lated peak structure is not known.

The phonon hardening model of Recoules et al.[21] was also mentioned above. Their calculationshowed that in gold, excitation of 5d electronsreduced screening and made the effective elec-tron–ion potential more attractive. The 5d statesbecame more localized around the nucleus. Theirdensity of states shrunk in width and shifted to-ward lower energy as the Fermi level shifted to-ward higher energy. This indicates that electrondensity of states can be substantially modified ina nonequilibrium solid phase of gold with ele-vated electron temperature. However, the impactof this on the interband region of dielectric func-tion remains to be shown.

5. Conclusions and OutstandingQuestions

Based on pyrometric measurements of a single,planar strong shock in flight in silicon [4, 5], theexistence of different electron and ion tempera-ture gradients behind the shock front due to finiteelectron–ion energy relaxation rate appears tohave been well accepted. The inference of an elec-tron–ion coupling constant from a two-tempera-ture model using shock emission intensity datayielded the first assessment of its value in thewarm dense matter regime and pointed to theneed to examine energy relaxation beyond theSpitzer model. The coupled-mode calculations ofDharma-wardana and Perrot [9] clearly revealedthe important role of dynamical interactionbetween electron and ion subsystems in warmdense matter although the system treated wasaluminum plasma with hot electrons and coldions. The work of Vorberger et al. [10] includedcalculations of temperature relaxation in shockedsilicon specific to our experiments. They notedorders of magnitude discrepancy between theircalculated electron and ion temperature

FIGURE 11. Real and imaginary parts of broadbanddielectric function of gold [19]. [Color figure can beviewed in the online issue, which is available atwileyonlinelibrary.com.]

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equilibration time of �1 ps for the coupled-modeapproach and the �200 ps inferred from two-tem-perature model interpretation of our experiment.Such an indirect comparison between theory andexperiment is likely misleading. A better bench-mark would be electron temperature at shockfront that was directly measured in experiment.Assuming that the observed shock emissionswere emanating from a layer behind the shockfront with a thickness of 5–10 nm, the predictedelectron temperature would be 2.6–3 eV. This isstill substantially higher than our measured valueof 1.4 eV. Hence, electron–ion energy relaxation inwarm dense matter remains an outstanding ques-tion in spite of the progress made in coupled-mode calculations. We are now turning our focusto issues of nonequilibrium effects on electron–ioninteraction potential and formation of coupledmodes, both of which can greatly impact energyrelaxation rate in warm dense matter.

Our study of warm dense matter states pro-duced by isochoric and isothermal heating of freestanding, 30-nm-thick foils of gold with femtosec-ond laser have revealed the persistence of aquasi-steady, nonequilibrium, superheated solidphase [17, 19]. Such a solid phase has also beenfound in ultrafast electron diffraction experiment[20]. In an ad hoc model, heating of the electronand ion subsystems is described by a two-temper-ature method while subsequent disassembly ofthe solid is taken to occur at some intrinsic latticestability limit characterized by a critical latticeenergy density [17]. Interestingly, calculationsusing this model predicted lifetimes of the quasi-steady state consistent with measurement over awide range of excitation energy density and theassociated critical lattice energy density wasfound to be about 2.5 times higher than the latticeenergy at its normal equilibrium melting point.However, the mechanism effecting such a changein lattice stability limit was unknown. On theother hand, the much more sophisticated phonon-hardening model [21] describes the effect of 5delectron excitation on screening and hence elec-tron–electron and electron–ion interaction poten-tial. This in turn modifies the phonon spectrum,leading to increase in Debye temperature andthus rapid increase in melting temperature withelectron temperature. In the absence of other ex-perimental data for validating either model, ourunderstanding of lattice stability of nonequili-brium warm dense matter remains incomplete. Inthe case of the ad hoc model, the outstanding

question is what ion–ion potential could producecollapse of the lattice structure at the observedcritical lattice energy density. As for the phonon-hardening model, we need to address how thephonon modes are populated in ultrafastexcitation.

Our observation of enhancement in interbandcontributions to dielectric function of nonequili-brium, warm dense gold [19] is as yet an anomalywith no theoretical explanation. A tantalizing pos-sibility is the effect of excitation of 5d electrons onelectron density of states under warm dense mat-ter conditions, as hinted by the work of Recouleset al. [21]. The question is how the modified den-sity of states would affect the 5d-6s/p transitions.

Future progress in resolving our outstandingquestions would undoubtedly require substantialtheoretical effort focusing on fundamental under-standing of charge screening, interaction poten-tials, electron and phonon density of states in thewarm dense matter regime. However, the impor-tance of more measurements on well-definedstates in a wide range of independent experi-ments should be equally noted.

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NG


outstanding questions in electron–ion energy relaxation, lattice stability, and dielectric...

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