observed by neutron inelastic scattering

PHYSICAL REVIEW B 1 NOVEMBER 1997-IIVOLUME 56, NUMBER 18

Collective excitations in liquid para-H2 observed by neutron inelastic scattering

F. J. Mompeań and M. Garcıá-HernandezInstituto de Ciencias de Materiales de Madrid, CSIC, Cantoblanco, E-28049 Madrid, Spain

B. FakDepartement de Recherche Fondamentale sur la Matie`re Condenseé, SPSMS/MDN, CEA Grenoble, 38054 Grenoble, France

~Received 23 May 1997!

Inelastic-neutron-scattering studies of liquid H2 enriched in para-H2 molecules show the existence of adamped mode, interpreted as a phononlike collective excitation whose wave-vector dependence is establishedfor 0.6 <Q< 2.0 Å21. Our result, in marked disagreement with previous observations in this system, is inqualitative agreement with theoretical variational calculations. A comparison with liquid normal D2 is made.The crossover from collective to single molecule response in both the solid and the liquid phases has also beendetermined.@S0163-1829~97!04242-2#

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I. INTRODUCTION

Interest in the observation of collective excitations in mlecular liquids has been focused in the liquid hydrogens bas a consequence of their significant proximity to the qutum limit and of their deviations from the correspondinstates laws followed by most liquids formed by few-atomolecules. Neutron-inelastic-scattering studies using a triaxis spectrometer carried out in the early 1970’s by Carnet al.1 are often quoted as the strongest direct experimeevidence for the existence of well-defined collective exctions far from the hydrodynamical regime in molecular liuids. Those studies, complementary to the ones conducteSchott2 using neutron time-of-flight spectroscopy, deal wthe observation of the collective excitations in liquid hydrgen samples enriched in para-H2 molecules. Hydrogen existin two modifications: para-H2 molecules have even rotationquantum number,J, and antisymmetric nuclear moleculaspin state,I 50, while ortho-H2 molecules have odd valueof J and symmetric nuclear molecular spin states,I 51. Neu-trons with energies lower than the para-ortho conversthreshold (E01514.7 meV! scatter coherently from para-H2,while for higher neutron energies the incoherent scattedominates due to the much larger incoherent cross sectioortho-H2, the incoherent scattering is always dominatinCollective excitations, seen through the coherent scattercan therefore be observed in para-H2 enriched samples usinlow-energy incident neutrons. On this basis, Carneiroet al.reported the observation of well-defined spectral featunear a neutron energy transfer ofE57.5 meV for wave vec-tors 1.1 Å21<Q<3.1 Å21, which they attributed to collective excitations of longitudinal-acoustic character. From thpublished spectra it appears as if these sharp features shvery small degree of dispersion. Additional features at lowvalues ofE were attributed to modes of transverse-acoucharacter. A more extensive account of their work is cotained in Ref. 3, including a reevaluation of the spectraobtain a revised value for the mean square displacemû2&, via the Ambegaokar, Conway, and Baym~ACB! sumrule. In particular, Ref. 3 contains details of the work on tnormal mixture~n-H2 is 75% para-H2 and 25% ortho-H2),

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which scatters mostly incoherently and thus yields informtion predominantly onSs(Q,E), the self-part of the scattering function. From these measurements information wasrived on the generalized density of collective excitations, ithe Fourier transform of the velocity autocorrelation funtion, dealt with in Ref. 4, for liquid hydrogen. More receneutron inelastic-scattering experiments in these liquids hfocused on higher incident neutron energies in order to stthe single-molecule response atQ values high enough for theimpact approximation to be valid at the molecular level.5,6 Inthese studies, for values ofQ>5 Å21, the position of therotational transitions is consistent with a single-moleculesponse already in the impact approximation in both para2and n-H2, and in the liquid and solid phases. From the broaening of the corresponding peaks the mean kinetic enerare determined for the molecules in both phases. Furwork at even higher wave vectors has concentrated receon the single-atom response yielding information eitherreal space7 or in reciprocal space.8 This experimental situa-tion contrasts vividly with the availability of a variationatheoretical estimate for the dispersion relation of the colltive excitations in liquid H2, within the single-modeapproximation.9 This prediction, prompted by the interestthe search for a superfluid phase of H2, bears a remarkablequalitative similarity to the dispersion curve for the colletive modes in liquid4He and implies noticeable dispersioof the collective modes, which should be experimentallycessible in most of theQ range with neutron-inelasticscattering measurements. Moreover, a similar set of pretions for liquid D2 in various thermodynamic states in thliquid range have been found to account qualitatively forobserved dispersion relations in n-D2.10,11

With these elements in mind, we have performed neutrscattering measurements of the inelastic response of liqH2 enriched in para-H2 molecules using a triple-axis spectrometer. The collective excitations and their dispersion wprobed by neutrons with low energies~below the para-orthoconversion threshold!. Higher-energy neutrons were usedQ54 Å21 to locate the transition from the collective to thsingle-molecule response in both the liquid and the sophases.

11 604 © 1997 The American Physical Society

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56 11 605COLLECTIVE EXCITATIONS IN LIQUID PARA-H2 . . .

II. EXPERIMENTAL DETAILS

The experiments reported in this paper were performedthe triple-axis spectrometer DN1 at the Siloe´ reactor of CEA/Grenoble. The spectrometer was operated in the ‘‘W’’ cofiguration and constant-Q scans were performed at constaneutron incident energy using the~002! reflections from avertically focused pyrolitic graphite~PG! monochromatorand a flat analyzer. A PG filter was inserted in the incidbeam to reduce the higher-order contamination. Two diffent incident energies were used:Ei5 14.2 meV andEi5 33.4 meV.

The H2 sample enriched in para-H2 was prepared as follows. High-purity commercially available~Alphagaz N99!H2 gas was condensed over ferric oxide powder at a tperature of 15 K in a sample container designed for thexperiments. The catalyst accelerates the conversion proof the initially normal mixture of H2 to a composition closeto the equilibrium concentration at the experimental tempeture ~99.98 % para-H2). This process was monitored reglarly by measuring the elastic structure factorS(Q,E50).Figure 1 shows the temporal evolution of this magnitudeone of the two samples in our study. As can be seen infigure, 26 h after condensation the shape ofS(Q,E50)clearly resembles what is expected for a molecular liqscattering mostly coherently. Repeated checks for changeS(Q,E50) over the available range ofQ were performedthroughout the duration of the experiments. These wereticularly relevant in establishing the absence of any afcondensation effects which could temporarily alter the coposition of the liquid mixture being examined and thincrease the proportion of molecules contributing to thecoherent scattering. This was detected at one occasion duthe low-energy experiment and the data for that time perwere not used. All measurements were performed at a tperature of 15 K under saturated vapor pressure, andcorresponding density was 0.0377 mol/cm3.12 In order to

FIG. 1. MeasuredS(Q,E50) at three occasions after sampcondensation. This time evolution is due to ortho-para conversEmpty circles correspond to 1 h after sample condensation, empdiamonds to 10 h after condensation and crosses linked by atinuous line to 26 h after condensation. No background subtrachas been performed as evidenced by the Al~111! Bragg reflection.

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reduce multiple scattering, the cylindrical cell~1.5 cm diam-eter, 5 cm height! was divided into five sectors of approxmately 1 cm height by means of the insertion of Cd spaclying parallel to the instrument scattering plane. Monte Casimulations using theDISCUScode13 show that combined absorption and multiple-scattering corrections in our casenegligible. The ferric oxide catalyst was masked by meana Cd strip attached to the bottom of the cell. The scatterfrom the empty cell was measured at low temperaturessubtracted from the data. As the spectrometer was operwith a fixed incident energy, the data were corrected byinstrumental factor (kf

3 cotuA)21,14 where kf is the finalwave vector anduA the analyzer Bragg angle. The instrumental energy resolution determined from the incoherscattering of vanadium was~Gaussian full width at halfmaximum! 0.68 meV atEi5 14.2 meV and 1.92 meV aEi5 33.4 meV.

A. Study below the conversion energy threshold

Our first study was conducted with an incident enerEi5 14.2 meV. The values of the horizontal in-pilemonochromator, monochromator-sample, sample-analyand analyzer-detector collimations were, respectively,30, 60, and 30 arc min. The main experimental problem wdue to the contamination from higher-order harmonics frthe monochromator, which despite the use of a PG filtethe incident beam was not sufficiently suppressed. Thhigher-order neutrons~mostly second order! have suffi-ciently high energy that they can excite a rotational levhence converting para-H2 into ortho-H2 ~i.e., J50→J51).The cross section for this process is so strong that a substial number of these neutrons can~again! be scattered insecond order~or higher! by the analyzer. This gives a spurous peak at an apparent energy transfer ofE/n253.7 meV,whereE is the real energy transfer~hereE01) andn52 is theorder of the scattering. This dispersionless peak is cleseen in all spectra shown in Fig. 2. An analysis of these dis given in Sec. IV A.

B. Study above the conversion energy threshold

A second study was performed using a higher incidenergy (Ei5 33.4 meV!, horizontal collimations of 25, 3030, and 30 arc min, and a PG filter in the incident beam. Dto the characteristics of the energy spectra of the neutrfrom the radial beam tube at the Siloe´ reactor, the amount ohigher-order contamination is considerably lower at thisergy, and no spurious peaks were observed. Data atQ54.0Å21 were taken for both the liquid phase at T5 15 K and inthe solid at T5 6 K ~fully corrected spectra forQ52.0 and4.0 Å21 are shown in Fig. 3!. The S(Q,0) measurement inthe solid phase is consistent with an hcp structure, withrameters compatible with those reported by Ishmaevet al.15

III. THEORETICAL BACKGROUND AND MODELSFOR DATA ANALYSIS

The neutron incident wavelengths employed in our sties are sufficiently close to the value of the internuclear dtance in the H2 molecule,d50.74 Å, to require a theoreticatreatment that takes into account the phase differences in

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11 606 56F. J. MOMPEAN, M. GARCIA-HERNANDEZ, AND B. FAK

scattering amplitudes corresponding to the two protons inmolecule. One treatment is due to Young and Koppel16 andis based on the assumption that collective modes arecoupled to the molecular nuclear-spin states. This assution is quite severe since the anisotropy terms in the in

FIG. 2. Representative spectra at three wave vectors withEi

514.2 meV. Filled diamonds are fully corrected experimental daThe continuous line represents the best fit to these data, consiof the convolution with the instrumental resolution of the followinmodel components: a central Lorentzian~dotted line!, a Gaussiannear 3.7 meV representing the second-order contamination~thindashed line!, a damped harmonic oscillator~thick dashed line! rep-resenting the collective mode, and an additional damped harmoscillator ~dot-dashed line! for Q>1.6 Å21.

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molecular potential in H2 depend strongly on the completmolecular wave function, which, restricting ourselves to tfundamental electronic and vibrational states, impliesstrong dependence on the rotational-nuclear quantum nbers and more specifically on the ortho-para ratio. For solthe coupling of the libron and phonon bands in ortho-H2 richmixtures has been treated by Bickermannet al.17 Since weare dealing with a mixture poor inJ51 molecules, we as-sume that the collective modes are totally decoupled frthe molecular states. Therefore, we consider that the scaing described by the formulas developed in Ref. 16, stricvalid only for a low pressure gas, will be weighted by thcollective contributions in the liquid phase in each point(Q,E) space. For ease of reference, we summarize hereresults of Young and Koppel treatment specialized tocase,

Spara~Q,E!5cpara@scohj 02~Qd/2!Scoh~Q,E!

13s incohj 12~Qd/2!Sincoh~Q,E2E01!#, ~1!

Sortho~Q,E!5corthoS scohScoh~Q,E!12

3s incohSincoh~Q,E! D

3@ j 02~Qd/2!12 j 2

2~Qd/2!#. ~2!

In the above formulas,cparaandcortho are the fractional com-position of those types of molecules,scoh5 2.05 b ands incoh5 78.7 b, andj l(x) are spherical Bessel functions oorder l . It should be remembered that, in terms of the seand distinct-scattering functions,

Scoh~Q,E!5Sd~Q,E!1Ss~Q,E!, ~3!

while

Sincoh~Q,E!5Ss~Q,E!. ~4!

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FIG. 3. Fully corrected spectra taken atEi533.4 meV. Filled diamonds are the data for the liquid atT515 K. At Q54.0 Å21, the emptysquares joined by a dotted line correspond to the solid sample studied atT56 K.

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Sd(Q,E) can be adequately modeled by a sum of damharmonic oscillator~dho! terms, each corresponding to thindividual collective modes presumed to contribute tospectra:

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is an intensity parameter and @n(E)11#5@12exp„2E/(kBT)…#21 is the Bose occupation factoSs(Q,E) requires at least two contributions that have diffeent relative weights depending on theQ value. For lowQvalues, the classical interaction time of the neutron andsystem is sufficiently long to allow the observation of tdiffusive motion of the molecules andSs(Q,E) can be ad-equately described by a Lorentzian component of intenH(Q) whose width,GQ,L , depends on the translational difusion coefficient,

Ss~Q,E!5@n~E!11#H~Q!EGQ,L

E21GQ,L2

. ~6!

For intermediate values ofQ the situation is more complesinceSs(Q,E) will contain a projection of the collective motions on the single-particle~molecule! dynamics in additionto a contribution from the broadened rotational modes.suming that there is no coupling between the two setsmodes,Sincoh(Q,E) can be related to the density of stateG(E). To obtain an estimate of the density of states froSincoh(Q,E) in the liquid phase we assume that the exprsion for a polycrystalline sample can be used without furtjustification. Thus,

G~E!}ESincoh~Q,E!

n~E!11. ~7!

For sufficiently highQ values, however, the possibility othe neutron creating multiple excitations has to be takenaccount in the corresponding evaluations. In this circustance, the simple proportionality does not hold unlessG(E)is replaced by an effective multiexcitation expansion, simto what is done in periodic media.18 Finally, we mention herethat in order to compare the model predictions with theserved intensities, it is necessary to convolve the former wthe known Gaussian instrumental resolution function.

IV. RESULTS AND DISCUSSION

A. Dispersion relation of the collective modes

A first approach to the low incident energy spectra is thevaluation in terms of a single-mode model@N51 in Eq.~5!#, suggested by results in similar systems and theorepredictions.9 The collective contribution, emerging througSd(Q,E), is described by a damped harmonic oscillator teThis contribution is supplemented by a quasielastic Lorenian centered atE50 meV to account for the diffusionlikemotions observed inSs(Q,E) and a Gaussian~centered near

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E54 meV! for the spurious peak due to higher-order sctering. This model describes satisfactorily the observatibelow Q5 1.6 Å21 but it is not adequate for spectrahigher Q values. ForQ>1.6 Å21, the fits improve signifi-cantly by using two dho terms (N52), the second beingnecessary to describe a fraction of the observed intensithigher energies. Figure 2 illustrates the good agreementtween observed and calculated spectra in bothQ ranges ob-tained after least-squares fitting of the parameters contain the corresponding model. TheQ dependence of the extracted parameters is shown in Fig. 4. Due to the broadstrumental resolution, theE50 part of the spectra cannot bused to obtain information on diffusion processes.

The interpretation of theQ dependence shown by the prameters corresponding to the single– and the two-m

FIG. 4. Q dependence of the best-fit model parameters. Filcircles refer to the first damped harmonic oscillator and emcircles to the second one. TheZi(Q) intensity parameters are showin ~a!. Renormalized energies are shown in~b! where the continu-ous line represents the extrapolation of the hydrodynamic soregime, and the dashed line the results from the variational colated density-matrix calculation. The damping coefficientsshown in~c!.

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models is by no means unique. In fact, Fig. 4 has been drunder the assumption that forQ>1.6 Å21, the dho termyielding the lower renormalized energy (i 51 with N52)represents a continuation of the single mode identified inlower Q region (i 51 with N51). This interpretation is suggested by the dispersion relation obtained from variatiocalculations and further reinforced by the obtainedQ depen-dence for the dho intensity parameter,Z1(Q), which re-sembles the static structure factor for the center-of-massclassical diatomic liquid exhibiting a first peak nearQ5 2Å21. This mode would then correspond to collective excitions of longitudinal-acoustic character. The nature ofsecond mode at higher energies andQ values is more diffi-cult to interpret. We note the smallQ dependence of therenormalized energy and the low values for the intensityrameter,Z2(Q). These characteristics are compatible wtwo possible origins for the observed scattering. First, colltive modes of optical character cannot be excluded in vof the presence of several of these branches in the solid lparticularly low towards the center of the Brillouin zone19

Molecular-dynamics simulation of other molecular liquihave identified these optical modes as responsible forobserved neutron-scattering spectra atQ values close to thefirst maximum in the static structure factor.20 Second, thescattering could also be due to the density of states or spart of S(Q,E), whose contribution will increase with increasingQ due to thej 1 Bessel function. Without recourse tpolarization analysis, there is no way to circumvent thisherent limitation of our experimental approach.

In the context of the first interpretation, the renormalizenergies for thefirst dho (i 51 with N51,2) should be com-pared both with the results from Carneiroet al. and withthose from variational calculations@see Fig. 4~b!#. Our spec-tra show a clear discrepancy with those reported by Carnet al.1,3 More specifically, our measurements do not shany trace of the strong nearly dispersionless spectral feareported by those authors nearE5 7 meV. It should be notedthat the extrapolation of the hydrodynamic limit reportedFig. 2 of Ref. 1 and in Fig. 6 of Ref. 3 does not correspoto the normally quoted values for the speed of sound inuid para-H2. At the lowerQ values covered in our experment, the renormalized energies are lower than the extralation of the hydrodynamic limit with an isothermal speedsound ofvT5 1029 m/s~derived from the adiabatic speedsound of vQ5 1235 m/s usingvT5vQ /Ag, where g5Cp /CV is the ratio of the specific heats at constant pressand constant volume and has been taken as 1.44 at thperimental temperature12!. Our results are also consistentlower than those from the variational calculations.9 However,a quantitative comparison between our experimental resand the calculations is particularly unfair to the latter sincethe present time many known ingredients of the system~suchas backflow effects and the finite lifetime of the excitation!are not included in the theoretical approach. In particulais recognized that neglecting backflow overestimates thecitation energies by a factor of 2–3 nearQ5 2 Å21. Asshown in Fig. 4~c!, the damping coefficient for thefirst dhois not greater than the renormalized energy except foQ>1.6 Å21. This implies that the observed excitations at lo

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The above comparison with measured sound velocitiethe hypersonic and ultrasonic regimes indicates negativepersion. This is at a striking difference with the observatiomade in liquid n-D2,10,11 where strong positive dispersionobserved. This difference is clearly not related to any cpling with the rotational level, since this level is much closto the collective mode in liquid D2 than in liquid H2. Themost likely origin for the effect is in terms of the intermolecular potential, which could favor anomalous dispersionliquid D 2. In fact, the measurements on liquid D2 were madeon samples of n-D2. These mixtures contain one out of evethree molecules in para-D2 states withJ51. The moleculesin this excited state contribute with anisotropic terms tointermolecular potential. This is also seen as differencesthe position of the first maximum of the single differentineutron cross sections of liquid mixtures enriched in orthand para-D2, as reported by Talhouket al.,22 and can berelated to changes in the nearest-neighbor distance and etive coordination number, probably arising from the betpacking of theJ51 molecules. The data reported in thpaper can only point out to this anomalous comparativehavior of para-H2 and n-D2 in the liquid state near theitriple points, but given the readiness with which liqusamples enriched in ortho-D2 can be prepared, an expermental approach to the dynamics of this liquid is perhapsorder.

B. Crossover to the single-molecule response regime

Instead of attempting an independent analysis of the sptra in the liquid and solid phases forQ54 Å21 with incidentenergy above the conversion threshold, we have followedalternative procedure for a comparative analysis of bphases. We start by noting that in the solid phase a narspectral feature appears centered atE514.7 meV~see Fig.3!. Given the value of energy transfer at which it appears athe fact that it is only broadened by the instrumental resotion, it can be easily assigned as theDJ51 transition corre-sponding to the para-ortho conversion and we can safelytablish that the system response is far from the domainvalidity of the impact approximation. In the impact approxmation, a Gaussian centered atEJJ81Er ~530.7 meV atQ

54 Å21) with a width of sE5A34 ErEK should have been

observed, whereEr5 \2Q2/2Mmol is the recoil energy atmomentum transferQ of a free molecule of massMmol andEK is the kinetic energy of the molecule. An alternative asignment of the spectral feature as arising from nearly-fmolecule recoil-shiftedDJ50 transitions can be dismisseon the basis of the expected line broadening~using for EKthe value of 6.560.8 meV obtained for solid para-H2 at 10 Kby Langel et al.,6 we should obtain forMmol52 a.m.u. abroadening much in excess of that observed!. Also, in thisQrange,DJ50 transitions possess considerably less intenthan DJ51 transitions, since the intramolecular structufactor for the former is small for this value ofQ, while thatfor DJ51 is close to its maximum value~see Fig. 5!. Addi-tional support for that the impact approximation is n

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reached forQ54 Å21 in solid para-H2 comes from the ob-servation of inelastic scattering at higher values ofE, remi-niscent of a fairly structured density of states. We concluthat at thisQ value we are well into the domain of validity othe incoherent approximation but far from realizing the impact approximation.

We turn now our attention to the liquid sample, where tmost striking difference found with respect to the solid is tabsence of any narrow rotational contribution. Presumathis contribution is smeared out by Doppler broadeningthe liquid and gives rise to the broad spectral feature aroE5 15 meV, which overlaps substantially with the mointense feature found at higher energies. The latter is rasimilar to the density-of-states contribution in the solid. Itclear that the observed spectrum deviates strongly fromGaussian line-shape characteristic of the impact approxition, and as the intramolecular form factor argument mtioned above for the solid is equally applicable to the liquwe conclude that the spectrum actually corresponds toobservation of rotational and collective modes of the liquphase through the incoherent scattering arising fromDJ51transitions. Hence, for both studied phases atQ54 Å21 wefind that the response is not yet in the single-particle~mol-ecule! regime. By contrast, both triple-axis and time-of-fligspectrometry results atQ55 Å21 signify the validity of theimpact approximation, at the single-molecule level, fsamples under similar thermodynamic conditions.5,6 We con-clude that in both phases, the crossover from the collectivthe single-molecule response to neutrons seems to be loized betweenQ54 andQ55 Å21. The values of the freemolecule recoil energy,Er5 \2Q2/2Mmol , corresponding totheseQ values, 16 and 25 meV, can then be employedlocate the effective binding energies of the molecules in eof the two phases. In the solid phase, it is noteworthy thatvalue for the activation energy of the thermally induced dfusion coefficient is 16.97 meV,23 reinforcing the classicalimage in which neutron scattering with wave-vector transof Q.4 Å21 leaves the molecules with an excess of tran

FIG. 5. Spherical Bessel functions entering the neutroscattering cross sections for H2 in our experimental range of energand wave-vector transfers. The internuclear distance in the mecule is taken to bed50.74 Å.

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lational energy with respect to the thermal mean value whenables them to move freely out of their equilibrium positiin the lattice. The corresponding image in the liquid phasemuch harder to establish.

A more quantitative approach to our spectra can be trunder two strong hypotheses. First, we assume that mtiphonon contributions will affect with equal weight the observed spectra in the two phases and that they can beceled by considering relative quantities. Second, we assthat the ACB sum rule can be applied to the liquid phaSince the instrumental setup is the same, and only the samthermodynamic conditions change we can express the rof the observed integrated intensities nearE01, I s,a ,a5S(solid), L(liquid) in the two phases as a function of thecorresponding Debye-Waller factors, exp@22Wa#, a5S,Land mean-square displacements as

I s,S

I s,L5exp@22~WS2WL!#

û2&S

û2&L

. ~8!

For û2&S , it is possible to get an independent estimate0.466 0.02 Å2 from the evaluations made by Nielsen19 andBickermannet al.24 In computingI s in the two phases, aftecorrecting for the two different temperature factors, we hasubtracted the rotational contribution: in the solid we hafitted and subtracted the clearly defined peak. In the liquwe have subtracted a Gaussian feature of the same integintensity and center as in the solid phase but with a wiadjusted to account for the observed slope forE valuessmaller thanE01. With these approximations we obtain aestimate, after linearizing, ofû2&L50.58 Å2. A consistencytest for this value can be performed by translating this mamplitude into the mean quantity for the corresponding mmentum distribution,25 p05A3/û2&L

1/2. From this quantity,we can estimate the mean kinetic energy for a single meculeEK5 \2p0

2/2Mmol of 60 K, which agrees well with thevalue reported in Ref. 6 of 6366 K. Consequently, theDebye-Waller factors for the two phases atQ54 Å21 areboth smaller than unity and, as in the case of solid4He,26

seem to offer a more stringent criteria than the recoil enefor the crossover to the single-molecule response.

V. CONCLUSIONS

Our experimental study on para-H2-enriched samples inthe liquid phase based on conventional triple-axis neutspectrometry and a partition of the observed scatteringmodel contributions has revealed the existence of specfeatures which are consistent with collective excitationslongitudinal-acoustic character. These excitationsdamped up to wave vectors of about 1.2 Å21, and over-damped at higher wave vectors. The dispersion seems tolow the qualitative trend of variational calculations, althouthe energies are considerably lower than the extrapolatiothe hydrodynamic sound velocity. This situation is in cotrast to the observations made in liquid n-D2, where a stronganomalous~upward! dispersion is observed at lowQ values.We suggest that this may be due to the difference indegree of anisotropy of the intermolecular potential in tstudied samples. At higher wave vectors, near the maximin S(Q,E50), additional scattered intensity is found who

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origin cannot be definitively established, but which mpoint to low-lying collective excitations of optical charactor to a density-of-states contribution. From a comparisonspectra at relatively high wave vectors,Q54 Å21, in thesolid and liquid phase with those of earlier work, we haestablished that the crossover from collective to singmolecule response takes place with recoil energies in exof 16 meV, although a better signal for this transition seeto be, for both phases, the small value of the Debye-Wafactor. Finally, we mention that our conclusions for the chacteristics of the collective excitations are based on a mointerpretation of the observed intensities. Neutron polari

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tion analysis can offer an experimental separation of the stering into coherent and incoherent nuclear-scattering cponents, from which a direct validation of our presemodels can be performed.

ACKNOWLEDGMENTS

The authors would like to thank Javier Bermejo for tinterest shown in the early stages of this work. Financsupport from Spanish DGICYT through Project No. PB90015 and from the European Union through the TMR-LProgram are greatly acknowledged.

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observed by neutron inelastic scattering

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