is there any evidence of a positive sound dispersion in the high frequency dynamics of noble gases?

Is there any evidence of a positive sound dispersion in the highfrequency dynamics of noble gases?

A. Cunsoloa,* , G. Pratesib, F. Rosicac, G. Ruoccoc, M. Sampolid, F. Settea, R. Verbenia,F. Barocchid, M. Krischa, C. Masciovecchioa, M. Nardonee

aEuropean Synchrotron Radiation Facility, B.P. 220 F-38043 Grenoble Cedex, FrancebInstitut Laue-Langevin, B.P. 156 F-38042 Grenoble Cedex, France

cUniversitadi L’Aquila and Istituto Nazionale di Fisica della Materia, I-67100, L’Aquila, ItalydUniversitadi Firenze and Istituto Nazionale di Fisica della Materia, I-50139, Firenze, ItalyeUniversitadi Roma III and Istituto Nazionale di Fisica della Materia, I-00100, Rome, Italy

Abstract

The dynamic structure factor,S�Q;v�; of neon has been studied by Inelastic X-ray Scattering and by Molecular Dynamicsimulations in the momentum transfer regionQ� 1–25 nm21 atT � 295 K andP� 3 kbar: At this density, comparable to thatof the liquid at ambient pressure, the shape ofS�Q;v� evolves from a Brillouion triplet towards a complex single lineshape,which precurs the shape expected for single particle behavior. The data, analyzed using the three modes model of the molecularhydrodynamics, show the absence of positive dispersion in the sound velocity and a minimum in the dispersion curve.Analogous measurements have been performed on a sample of supercritical4He at a density double of the liquid one. Theinelastic shifts obtained from the fits coincide with the 0-frequency values predicted by the viscoelastic theory. The analysis ofthe whole set of measurements leads to the conclusion that no relaxation processes as those expected in the viscoelastic behaviorare observable for these systems in the explored THz frequency range.q 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Noble gases

During the last few decades a large effort has beendevoted to the study of the dynamic structure factor,S�Q;v�; of dense fluids in both liquid and gaseous phases.One of the aims is the understanding of the transition fromthe hydrodynamics regime towards the so-called kinetic one[1], as a function of the density,r , and of the exchangedmomentum,Q. These two different regimes span two oppo-site limiting �Q;v� regions corresponding to times anddistances separated by a time and a lengthscale character-istic of the system. These are often chosen respectively asthe Enskog mean free path,lE and the Enskog mean collisiontime tE; the former is defined aslE � lB=g�r0�; wherelB isthe Boltzmann mean free path,l21

B � ��2p

prr20; andg�r0� is

the pair distribution function evaluated at the particle’sradius, r0; the latter is a measure of the mean collisionaltime and is defined astE � �4

��pp

rr20g�r0��21 [1]. For

excitations with wavelength 2p/Q much larger thanlE, andfrequency much higher thant21

E ; the fluid appears as acontinuum and a large number of particle collisions areprobed; in such quasi-macroscopic conditions the study ofdensity fluctuations can be approached using the Navier–Stokes theory to derive theS�Q;v�: At intermediate�Q;v�-values, the breakdown of the hydrodynamics theory isexpected to occur. When probed at extremely highQ andv , QlE q 1; vtE q 1 the dynamics is basically the freestreaming of a single particle between two successive colli-sions. In the two limit cases the dynamic structure factor hasa well-known shape and a simple physical interpretation: (i)In the smallQlE, vtE limit there are three modes. These are,respectively, the two Stokes and anti-Stokes compression(sound) modes, which disperse linearly with a slope corre-sponding to the adiabatic velocity of sound, and the heatdiffusion mode, which is centered at 0-energy transfer,and has a width proportional toDTQ2, DT being the thermaldiffusion coefficient. (ii) In the highQlE limit, within theimpulse approximation, the lineshape reflects the initial

Journal of Physics and Chemistry of Solids 61 (2000) 477–483

0022-3697/00/$ - see front matterq 2000 Elsevier Science Ltd. All rights reserved.PII: S0022-3697(99)00340-6

www.elsevier.nl/locate/jpcs

* Corresponding author. Tel.:133-4768-82603; fax:133-4768-82160.

E-mail address:[email protected] (A. Cunsolo).

state momentum distribution, i.e. the Boltzmann distribu-tion. Here, theS�Q;v� reduces to a Gaussian centered atthe recoil energy"2Q2

=2M; whereM is the particle mass.The extension of the three modes description ofS�Q;v�

beyond the hydrodynamic limit whereQlE approaches unity,has been suggested by the kinetic theory [2–4] and severalmolecular dynamics studies performed with both hardspheres [5,6] and Lennard-Jones potentials [7,8]. Firmexperimental data covering in detail the whole transitionregion up toQm, the Q-value of the first maximum in thestatic structure factor, are not yet available in high densityfluids above the critical point, in spite of the interest to havea complete overview of all dynamic processes at bothmacroscopic and microscopic scales in a fluid. Up to nowonly neutron Brillouin scattering techniques have providedthe opportunity to study the dynamics in aQ region belowand aroundQm, where theS�Q;v� is deeply influenced bythe microscopic structure. This has been done in liquid neon[9,10] and in liquid and gaseous argon [11–13].

The development of the Inelastic X-ray Scattering (IXS)technique with meV energy resolution opens up new oppor-tunities to explore microscopic dynamic properties of densefluids in Q–v regions of difficult access to neutron spectro-scopy. Further, the small X-ray beam sizes allows us toinvestigate thermodynamic states which can be producedonly in small volumes, as noble gases at temperatures wellabove the critical point and at densities either comparable orhigher than the one of the triple point.

In this work, using IXS and molecular dynamics (MD)simulations, we determine theS�Q;v� of two deeply super-critical gases in the momentum transfer region bridging thehydrodynamical and single particle regimes. In particular,we show that the lineshape continuously evolves from atriplet to a complex broad feature, which cannot bedescribed as the single Gaussian expected in theQ! ∞limit. Exploiting the three modes model up to highQ-values,we show that positive dispersion of the macroscopic soundvelocity, previously observed in the liquid phase [10], doesnot exist in the gas-state. Moreover, investigating the natureof the collective excitations at highQ-values, we find that athree modes description can still be used, and it gives aminimum in the dispersion curve aroundQm, as reportedpreviously in the liquid [10].

Up to now, similar experiments on theS�Q;v� of densegases have been performed with neutrons atQ substantiallylower thanQm, and at densities lower than those of the liquid[12,13].

A previous experiment has been performed at 3 Kbarand 295 K, corresponding to a particle density of28.9 molecules/nm3 (970 kg/m3) [14] and to < 8 timesthe critical temperature. Analogous measurements havebeen done in the sameQ range for a sample of4He at.8 kbarandT � 25:5 K corresponding to a density of 30.65 mole-cules/nm3 (205 kg/m3) and < 5.4 times. The 1–25 nm21

momentum region, spanned in our experiment, allows usto follow the evolution of theS�Q;v� from QlE � 0:2 to 4

between 0.2 and 5 for helium. The exploredvtE roughlyrange from 0.05 to 0.4 for Ne and between 0.03 and 0.175for helium.

The spectra were collected at the inelastic X-ray scatter-ing beamline BL21-ID16 at the European SynchrotronRadiation Facility. The experimental assembly, based onhigh order reflections from silicon perfect crystals in back-scattering geometry, was used at the Si(11,11,11) reflectionwith incident X-ray energy of 21.748 keV (Ne) and the Si(9,9,9) reflection with incident energy of 17.747 keV (He).The resolution function, determined experimentally bymeasuring the elastic scattering from a Plexiglas sample atthe first maximum of itsS�Q� �Q� 10 nm21�; had a fullwidth at half maximum of around 1:7^ 0:1 meV for eachof the five analyzers. The energy scans were performedvarying the monochromator temperature with respect tothat of the analyzer crystals. The typical temperature scanrange wasDT � ^0:35 K with respect to the elastic line,and the step wasdT � 0:005 K: The integration time perdata point was < 300 s. The momentum transfer wasselected rotating the 7 m long spectrometer arm in the hori-zontal plane, and theQ resolution was set to 0.2 nm21.Further details on the beamline are reported elsewhere[15,16]. The sample was contained in a steel cell and thecell was connected to a cryostatic head in a vacuumenvironment. A volume of< 1 cm3 of Neon and He gaswas pressurized at two different temperature, i.e. respec-tively, to 3 kbar, 295 K, and 1.2 kbar,T � 27 K; using amembrane gas compressor. The X-ray beam passed throughtwo 1 mm thick diamond single crystal windows with a2.3 mm diameter aperture. The distance between the twowindows, i.e. the sample length along the X-ray beam was10 mm: this allowed to cover the 0–158 scattering anglerange, necessary to reachQm; the sample length matchedexactly the X-ray photoabsorption length of Neon and wasshorter of a factor 20 of that of Helium in the respectivethermodynamic points. The corresponding spannedQ/Qm

ranged between 5× 1022 and 1.25 for neon and between 4×1022 and 1.2 for helium. The pressure stability was betterthan^5 bar at 3 kbar and 5 bar at 1.2 kbar. The transversesection of the focused beam was 0:15× 0:3 mm2

: Acomparative analysis of our experimental spectra andanalogous simulatedS�Q;v� has been considered veryhelpful to have a more complete and reliable set ofresults. The MD simulations were carried out using aclassical microcanonical ensemble of 10,976 “neon”atoms in a cubic box with periodic boundary conditions,interacting via a Lennard-Jones potential�s �0:2756 nm; e=KB � 33:74 K; m� 20:18 a:u:�: The density(28.9 atoms/nm3), chosen to be the same as in theexperiment, yields a pressure in agreement with the experi-mental equation of state. The equations of motion wereintegrated using the leap frog algorithm with a time stepof 10 fs. After thermalization, the dynamics has beenfollowed for about 1300 ps and the configuration storedevery 0.05 ps, a period shorter than the inter-atomic

A. Cunsolo et al. / Journal of Physics and Chemistry of Solids 61 (2000) 477–483478

collision time. The dynamic structure factor has beencalculated as the power spectrum of the instantaneous theQ-component of the density fluctuations using the Welshmethod [17–19]. The energy resolution was always betterthan 0.1 meV. The resultingS�Q;E� was averaged over allindependent directions ofQ.

The spectra reported in Fig. 1 show, at differentQ-values,a selection of experimental data (Fig. 1, row a), the compari-son between the experimental spectra and the MD spectraafter the convolution with the experimental resolution func-tion and the proper detailed balance symmetrization (Fig. 1,row b), and the unconvoluted MD results (Fig. 1, row c),where the detailed balance has been accounted for by multi-plying the classical lineshapes byKBT="v�n�v�1 1�: Heren�v� is the Bose–Einstein population factor. TheQ depen-dence is evident from the comparison of the spectrum at thelowestQ, Q� 2 nm21

; which shows features typical of theBrillouin triplet, with those at largeQ, which are broaderand almost structureless. AtQ-values approachingQm, anarrowing of the spectrum is observed. We find at allQ-values an excellent agreement between the measured andsimulated spectra. Consequently, due to their much loweruncertainties, we study the microscopic dynamics and itsevolution at highQ from the behavior of the MDS�Q;v�:The MD spectra have been analyzed using a theoreticalexpression for theS�Q;v� given by the sum of threedynamic modes. This expression is derived within thetheory of dense gases [1], where theS�Q;v� is approximated

by:

S�Q;v� � S�Q�p

Ahzh

v2 1 z2h

"

1As�zs 1 �v 1 vs�b��v 1 vs�2 1 z2

s1

As�zs 2 �v 2 vs�b��v 2 vs�2 1 z2

s

��1�

with b� �Ahzh=As 1 zs�=vs: The first term corresponds to anonpropagating mode with a damping coefficientzh; andrepresents the central line in the spectra. The two othercontributions account for the propagating modes, and arerepresented by two side peaks with shifts respectively^v s and damping coefficientzs. A standard Levelt–Marquand x 2 minimization fitting procedure has beenapplied to MDS�Q;v� in a restricted frequency range, i.e.avoiding region where S�Q;v� is lower thanmax�S�Q;v��v=100� S�Q; 0�=100: The reason for thischoice is the failure of the used generalized hydrodynamicsmodel to describe the high frequency tails (this model has adivergent fourth moment).

The problems in reproducibility and those related to thedescription of spectral tails have been partially overcome inthe analysis of experimental spectra of4He; here due to thelower temperatureT � 27 K; the strong asymmetry inducedin the lineshape by the Bose factor allows us a more reliableevaluation of the huge inelastic peak in the Stokes side.

A. Cunsolo et al. / Journal of Physics and Chemistry of Solids 61 (2000) 477–483 479

Fig. 1. Selected IXS and MD spectra of Neon gas atT � 295 K and 3 Kbar at the indicatedQ-values. (row a): IXS data (W) at the indicatedQ-values with their error bars and their fit (solid line). The narrow dashed line, centered atE � 0; is the measured total energy resolution function.(row b): Comparison between the experimental (W) and MD spectra (solid line), after having taken into account the experimental resolutionfunction and the detailed balance. (row c): MD unconvoluted spectra.

However, an analysis performed using a different threemodes model, in place of model 1, has allowed a morereliable description of the spectral tails; The experimentalS�Q;v� of 4He, within this approach, has been approximatedby the functionF�Q;v�; consisting of ad-function for theelastic scattering, and a Damped Harmonic Oscillator(DHO) model [20] for the two side peaks:

F�Q;v� � I0�Q�v2 1 z2

h

1 I �Q� vz2sz2

s

�v2s 2 v2�2 1 z2

sv2

" #: �2�

Here I0�Q� and I �Q� are the intensities of the central peakand of the inelastic contributions, respectively, and thequantityv s, zs andzh, are respectively the shift, the inelasticand the elastic halfwidths analogous to those already defined

in Eq. (2). The detailed balance has been taken into accountin the same way discussed before for the MD spectra.

It could be shown that both the used three modesdescription are equivalent in the low�Q;v� region.

In Fig. 2 are reported some selected lineshapes obtainedas the best fit of experimental data taken at differentQ. It canbe remarked, as for MD spectra, that the unconvoluted line-shape evolves from a three peak structure towards astructureless peak, which monotonically decays aroundQm. In these spectra a triple peaks structure seems toreappear atQ higher thanQm.

The spectroscopic parametersv s, zs andzh, obtained fromthe fit of MD spectra of Neon with model 2 are reported inFig. 3 as a function ofQ. The same quantities as best fitparameters of IXS spectra of Helium are reported in Fig. 4.


Fig. 2. IXS spectra measured at some selectedQ(open circles) reported with the error bars, the corresponding experimental resolutions (dashedline) and best fit unconvoluted lineshapes (solid line). These latter have been normalized to have the same intensity as IXS spectra at 0-energytransfer.

In Fig. 2a, in the 1–8 nm21 Q range, corresponding toQlE < 0:2–1; the dispersion ofv s with Q is linear, asexpected in the hydrodynamic regime. With increasingQ,v s starts to bend down to a minimum observed atQ�22:5 nm21

; i.e. at Q < Qm: Such a minimum has alreadybeen observed in various liquid systems, and can beexplained as a manifestation of the interference betweenthe density fluctuations and the pseudo-periodicity respon-sible for the sharp feature in the static structure factor atQm.

As reported by van Well and de Graaf [10], in the liquidthere is clear evidence for a velocity of sound sensiblyhigher than that in theQ� 0 limit. This positive dispersionhas been previously predicted for liquid Argon and interpreted

in a viscoelastic framework as a manifestation of a shearrelaxation [21]. In order to compare our results with thosecharacteristic of the liquid phase more directly, in Fig. 2a wealso report the Brillouin shifts obtained in liquid Neon byVan Well and de Graaf [10] atT � 35K andP� 80 bar;corresponding to a densityr � 33:5 molecules=nm3

; i.e. adensity similar to that of our gas. The liquid values ofv s

reported in Fig. 2a have been multiplied by the factor 1.99,which corresponds to the ratio between the adiabatic soundvelocities at the two thermodynamic states. Contrary to theliquid case, noanomalousdispersion can be observed in thelow Q range. In fact the slope of the curve, 1050^ 100 m=s;agrees well with the adiabatic sound velocity in the macro-scopicQ� 0 limit, represented by the solid line.

The same trend is even more clear in Fig. 4a, where theinelastic shift of IXS spectra of Helium obtained as best fitof the model 2 have been reported, as a function ofQ,plotted together with the high-Q extension of the0-frequency inelastic shiftv0 � Qns=

��S�Q�p

:

The excellent agreement between the two quantitiessuggests that the absence of positive dispersion is adynamical feature of supercritical noble gases all over theexplored�Q;v� range and no viscoelastic transition betweena 0- and∞-frequency dynamics take place in these super-critical noble gases;

In Fig. 3b we report the halfwidth of the inelastic signalzs.A systematic increase can be observed up toQ < 10 nm21

;

the value whereQlE < 1:5 and the breakdown of the hydro-dynamic behavior is expected to occur. Here, indeed, thevalue of zs becomes comparable to that ofv s, and theexcitations acquire an increasingly overdamped nature.

In Fig. 4b we can observe an analogous behavior of IXSinelastic halfwidth in the lowQ limit; a dramatic breakdownof hydrodynamic prediction occurs around 13 nm21, corre-sponding toQlE < 1; where the curve bends down to abroad minimum or a plateau.

In Fig. 3c we report the quasielastic halfwidthzh. Theincrease at lowQ is again consistent with theQ2 hydro-dynamic behavior up toQ < 5 nm21

: The broad minimumin zh atQ < Qm is usually attributed to an increase of the selfpart of density–density correlation functions, to be expectedaround the maximum inS�Q�: This phenomenon, referred toas de Gennes narrowing, is due to a longer lifetime of thedensity fluctuations atQm. In spite of the larger statisticalfluctuation of data, the curve plotted in Fig. 4c seems toreproduce a qualitatively analogous behavior.

In conclusion, the present study on the high frequencydynamics in dense super critical noble gases shows thatthe region where hydrodynamics starts to breakdown islocated atQ-value comparable to the inverse mean freepath, i.e. between 8 and 13 nm21 for both the studiedsystems. In the Neon case the lineshape evolves from thehydrodynamicQ range into a featureless single peak, which,however, cannot be described as the single Gaussian char-acteristic of the single particle dynamics. One can success-fully model this lineshape extending to theseQ-values the


Fig. 3. Momentum dependence of the spectroscopic parametersobtained from the fitting procedure discussed in the text of theMD simulation (X) data, using the model 2: (a)v s, (b) zs, and (c)zh. Moreover, in (a), we report the dispersion relation measured inliquid Neon (open diamonds) by Van Well and de Graaf usinginelastic neutron scattering [10]. This liquid, measured atT �35 K andP� 80 bar; has a similar density to the gas consideredhere. The excitation energies in the liquid have been scaled by thefactor 1.99, corresponding to the ratio between theQ� 0 velocitiesof sound atT � 295 and 35 K, as calculated from the equation ofstate. The solid line has the slope corresponding to theQ� 0 velo-city of sound at 295 K, and is shown to emphasize the absence(presence) of positive dispersion in the gas (liquid), the dashedparabolic curves have been reported as a guide-to-eye to emphasizethe lowQ Q2 behavior.

three modes model derived in the molecular hydrodynamicstheory.

The presence of inelastic shoulders overQm has beennoted, however, in the unconvoluted spectra of Helium;this circumstance seems to point out the persistence of athree peaks lineshape all over the exploredQ range.

Finally, the dynamics of supercritical Neon and Helium athigh density shows important differences with respect to theliquid phase in almost isocore conditions. In particular, atvariance with the liquid, in the gas there is no positivedispersion of the sound velocity.

Acknowledgements

We acknowledge valuable help of L. Melesi and the ILLHigh Pressure Group during the cell set-up and test phases,and for lending us the compressor; D. Hausermann and D.

Gibson during the cell construction; and B. Gorges and J.F.Ribois for their technical assistance.

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Fig. 4. As in Fig. 3 (a)–(c) referring to best fit parameters of Helium IXS spectra, obtained using the model 3, reported with their respective errorbars. Moreover in (a) we report the 0-frequency value of inelastic shiftv0 � Qns=

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is there any evidence of a positive sound dispersion in the high frequency dynamics of noble gases?

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