Role of low-frequency vibrations on sound propagation in glasses at intermediate temperature

Role o low- equency ib a ions on sound p opaga ion in glasses a in e media e empe a u e A. C iado, M. Jime ´nez-Ruiz, C. Cab illo, and F. J. Be mejo Consejo Supe io de In es igaciones Cien ı ´ icas, Se ano 123, E-28006 Mad id, Spain and Depa amen o de Fı ´sica de la Ma e ia Condensada, Uni e sidad de Se illa, P.O. Box 1065, E-41080 Se ille, Spain M. G imsdi ch A gonne Na ional Labo a o y, A gonne, Illinois 60439 H. E. Fische Ins i u Laue Lange in, Boı ˆ e Pos ale 156x, F-38042 G enoble Cedex 9, F ance S. M. Benning on and R. S. Eccles on Ru he o d Apple on Labo a o y, Chil on Didco , Oxon OX11 0QX, G ea B i ain 共Recei ed 19 Ap il 1999; e ised manusc ip ecei ed 3 Janua y 2000兲 We epo measu emen s o he empe a u e dependence o he sound a enua ion and he ac ional change in sound eloci y o he glass 共G兲and o ien a ional-glass 共OG兲phases o polymo phic e hanol. S ikingly simila beha io s a e ound o bo h phases despi e he OG’s unde lying c ys al 共bcc兲la ice. Such simila i y, which is also e ealed in dielec ic spec oscopy and inelas ic neu on sca e ing measu emen s, sugges s whole molecule small-angle lib a ions as a common mic oscopic o igin o a wide a ie y o ‘‘glassy’’ phenomena. I. INTRODUCTION The he mal,1elas ic,2and dielec ic3beha io s o glasses la gely di e om hose o bulk c ys alline solids a low and mode a ely high 共 ens o kel ins兲 empe a u e. In pa icula , some elas ic p ope ies such as sound eloci y and a enua- ion in i eous ma e ials a e known o exhibi ema kable anomalies i compa ed wi h hose exhibi ed by he pa en c ys als. I exp essed in e ms o dimensionless quan i ies such as he ac ional change in sound eloci y wi h empe a- u e ␦ / and he in e nal ic ion Q⫺1, he o me is ound o inc ease wi h Tbelow 1 K and s ongly dec ease a highe T,4whe eas Q⫺1shows an ini ial inc ease up o a ew hun- d eds o millikel ins, hen eaches a pla eau o oughly con- s an Q0 ⫺1up o a ew kel ins 共a empe a u e e e ed usually as T*), ollowed by an inc ease un il a empe a u e dubbed Tmax and inally, in some cases, a s ong d op5a e en highe Thas been epo ed. In addi ion, he measu emen s o sound a enua ion in glasses a e ound o depend qui e s ongly on he measu emen equency, which is no he usual beha io o well-o de ed c ys als. The abo e phenomena a e explained on a phenomeno- logical basis using concep s such as wo-le el sys ems6 共TLS兲, he mally ac i a ed elaxa ion,7o some in e pola ion be ween he wo such as he ‘‘so -po en ial’’ model.8The pic u e ha eme ges om mos o hese app oaches po ays he dynamics o glasses below5Kasdomina ed by cohe en mo ion o TLS’s weakly damped by elaxa ional and eso- nan in e ac ions wi h elas ic wa es. The phase cohe ence o such mo ion is los o T⬎T*mainly because o he onse o he mal mo ion. Finally, abo e T* he mal occupa ion o ex- ci ed le els o he in e ac ion po en ial leads o a b eakdown o he wo-le el app oxima ion, and he dynamics becomes go e ned by he mally ac i a ed p ocesses, which usually show an A henius dependence o he elaxa ion a e.5 I has ecen ly become clea ha a mo e mic oscopic basis is equi ed o a deepe unde s anding o he dynamics o glassy ma e . This esul s om puzzling obse a ions made on polyc ys alline me allic solids,9 ilms o e en bulk, de- o med me allic solids10 ha show ha hese ma e ials be- ha e as glasses and, e en mo e, a compa ison o such quan- i ies ca ied ou in me allic polyc ys als in hei no mal and supe conduc ing s a es9sugges s ha he TLS’s, i p esen in such ma e ials, do no in e ac wi h he conduc ion elec ons. In addi ion, o he classes o pa ially o de ed sys ems such as o a o -phase 共o ‘‘plas ic’’兲c ys als,11 mixed c ys als,12 o e en quasic ys als13 ha e shown dynamics ha esemble ha o amo phous ma e . The ideal benchma k o cla i y such appa en ly con adic o y obse a ions would hen be cons i- u ed by a ma e ial ha could be s udied unde con olled condi ions o diso de . Such an endea o has, howe e , no been ca ied ou mainly because o he ex eme di icul y in p epa ing glass samples ha ing he same composi ions as hose employed o s udies on polyc ys als o o a o phases. He e we conside a ma e ial ha can easily be p epa ed in he modynamically well-cha ac e ized phases such as he ully o de ed, o ien a ionally diso de ed, and amo phous phases o solid e hanol. A s uc u al glass 共G兲is o med upon quenching he supe cooled liquid below Tg⬃97 K. The amo phous s a e can also be achie ed using easily achie able cooling a es 共⬇6 K/min兲. Hea ing he glass abo e Tgyields a supe cooled liquid ha can be annealed o yield a o a o phase 共RP, also e e ed in he li e a u e as plas ic o glassy c ys al兲phase. Once his is o med, cooling below Tg OG ⬃97 K leads o a sluggish eezing o molecula o a ions and an o ien a ional glass 共OG, o o ien a ionally diso de ed c ys al兲. The o a ional eezing p ese es he same c ys al symme y 共an Im3 ¯ mbcc la ice兲o he RP, and om he s uc u al poin o iew only a jump in he olume expansi - i y coe icien ha amoun s ⬇4⫻10⫺4K⫺1accompanies PHYSICAL REVIEW B 1 APRIL 2000-IVOLUME 61, NUMBER 13 PRB 61 0163-1829/2000/61共13兲/8778共6兲/$15.00 8778 ©2000 The Ame ican Physical Socie y he ansi ion.15 The la e is unde s ood as a genuine glass- ansi ion phenomenon o a pu ely dynamical o igin.14 P e- ious s udies ha e compa ed he s uc u e15 and dynamics16,17 o he RP/OG and G phases, all showing e y simila glassy beha io in hei speci ic hea s, ib a ional e- quency spec a, and dielec ic spec oscopy o he s able ully o de ed 共monoclinic兲c ys al phase. Ou pu pose he e is hus o ca y a compa a i e s udy on he dynamics o a ma e ial in i s amo phous and o ien a ion- ally diso de ed o ms a meso- and mic oscopic scales. Since bo h he glass and OG solids ha e he same chemical com- posi ion as well as a he close densi ies, he di e ences in dynamic beha io , i any, will be en i ely a ibu able o he p esence in he OG o long- ange pe iodici y. The p esen s udy will hus explo e ime and leng h scales in addi ion o hose wi hin he mac oscopic ealm as a e hose al eady ex- amined using dielec ic elaxa ion17 ha ha e wi nessed he dynamic p oximi y o he ␣ and ␤ elaxa ions o he glass 共o supe cooled liquid兲and OG 共o RP兲phases. II. LIGHT SCATTERING We ha e measu ed he equency and a enua ion o hy- pe sonic wa es by B illouin ligh sca e ing.18 The echnique p obes phonons wi h wa eleng hs compa able o ha o ligh , he measu ed equency shi ␻ Bis p opo ional o he sound eloci y, and he wid h ⌫o he B illouin peak is ela ed, h ough a con olu ion wi h he ins umen unc ion, o he a enua ion. The sample was con ained in a sealed coppe cu e e e a ached o he cold inge and he empe a- u e was con olled using a con inuous low c yos a . The explo ed empe a u e in e al was 5 K–120 K, which co e s he ange o bo h glassy phases 共G and OG兲as well as ha o he RP c ys al. As in p e ious expe imen s, all he phases we e p epa ed in si u and ca e ully moni o ed. Below 5 K he low sca e ing in ensi y p ecluded he measu emen . The empe a u e ange in es iga ed also co e s he low- empe a u e anomaly 关i.e., he excess in Cp(T)/T3speci ic hea wi h espec o he o de ed c ys al兴and he egion whe e he hea conduc i i y shows i s ‘‘pla eau.’’16 The measu ed spec a show B illouin peaks a ising om longi udinal elas- ic wa es. T ans e se sound modes a e oo weak o be ob- se ed. In addi ion, da a al eady epo ed o ully hyd oge- na ed glass and supe cooled liquid samples a empe a u es nea he s uc u al glass ansi ion19 a e included o com- pa ison pu poses. Wi hin he TLS amewo k, ␻ Band ⌫a e ela ed o he dispe si e and dissipa i e pa s o he suscep ibili y ␹ ( ␻ ) by5 ␦ ␻ B ␻ B ⫽ ␦ ⫽⫺ A 2 ␹ ⬘共 ␻ 兲,Q⫺1⫽⌫ ␻ B ⫽A ␹ ⬙共 ␻ 兲, A⫽g2 ␳ 2, whe e Aincludes he ‘‘de o ma ion po en ial’’ g共i.e., a mea- su e o he coupling s eng h o TLS wi h elas ic wa es兲, mass densi y ␳ , and sound eloci y . Ou da a o bo h glass and OG phases a e shown in Fig. 1. Bo h g aphs e eal he p esence o wo a he di e en egimes o bo h solids. Be- low ⬇30 K, he a ia ion wi h To ␦ / as well as Q⫺1is a he mild, whe eas a s onge Tdependence is obse ed up o he glass ansi ion 共⬃100 K兲, and inally e en mo e ab up changes a e seen in he supe cooled-liquid o RP em- pe a u e ange. Ou da a o Q⫺1a T⬇5 K should be aken as an uppe bound since hey may be esolu ion limi ed. Taken a i s ace alue he ⬇5 K da a a e 2.7⫻10⫺3共G兲and 3.0⫻10⫺3共OG兲. I we ake hese as ep esen a i e o he ‘‘pla eau’’ alue o Q0 ⫺1, hen hey a e abo e he a e age o hose compiled by Whi e and Pohl13 bu compa e wi h da a o some me al ilms.10 I hese alues a e used o make an es ima e o he dimensionless cons an 5 C⫽2Q0 ⫺1 ␲ ⫽P ¯ g2 ␳ 2,共1兲 hen one ge s C⬇3.6⫻10⫺3and 5⫻10⫺3. These a e ce - ainly la ge han hose epo ed o oxide glasses. Howe e , ou alues come close o hose measu ed o elec oly e glasses as s udied by Reiche e al.21 Those epo ed in Table 1 o Re . 21 each alues up o (9–9.6)⫻10⫺4de- pending upon he sal con en . In pa icula , he dependence o he cons an Cupon he con en o sal 共LiCl and ZnCl2) gi en in pe cen age uni s ollows a quasilinea law wi h a ze o-sal limi o 1.34⫻10⫺3. This is he alue o Cex- pec ed o an amo phous ma e ial chemically and s uc u ally FIG. 1. 共a兲F ac ional change o sound eloci y wi h empe a u e o he glass 共open symbols兲, OG/RP phases 共 illed symbols兲, and he da a om Re . 19 共 e ical ba s兲. The inse shows da a co e ing he whole glass- ansi ion egion. The solid line shows an es ima- ion using pa ame e s gi en in he ex . 共b兲In e nal ic ion as a unc ion o empe a u e 共same symbols as abo e兲. The s aigh line depic s a p edic ion based on a he mal-ac i a ion model 共Re . 7兲. PRB 61 8779ROLE OF LOW-FREQUENCY VIBRATIONS ON SOUND . . . no e y dissimila o ou s such as amo phous wa e . In o he wo ds, ou es ima e o Ccomes close o o he s de i ed p e- iously. On he o he hand, ou alues can also be unde s ood on he basis o he linea co ela ions ound by Hunklinge 20 and Reiche e al.21 be ween he quan i ies en e ing Eq. 共1兲and he glass ansi ion empe a u e. In his espec , i is wo h men ioning ha bo h glassy e hanol and wa e show he highes densi ies o TLS s a es while he oxide glasses a e ound in he opposi e ex eme. F om ⬇30 K up o Tg,Q⫺1 o bo h solids is well ap- p oxima ed by a linea empe a u e dependence wi h coe i- cien s 1.68⫻10⫺4K⫺1共G兲and 1.53⫻10⫺4K⫺1共OG兲. The implica ion is ha da a wi hin his ange pe ain o a egime o he mally ac i a ed elaxa ion, which is go e ned by an A henius elaxa ion a e. I such alues a e aken oge he wi h he p e ious es ima e o C o make a guess o he ze o poin ene gy o he unneling pa icle E0 om Re . 7Q⫺1⫽ ␲ CkBT/E0, one ge s E0⫽18 共19.9兲K o G共OG兲, which is amazingly close o he 15 K de i ed o oxide glasses.5 Ou da a o Q⫺1do no show maxima a Tmax such as ha ound o oxide glasses by acous ic5o B illouin7spec- oscopies, p obably because he appea ance o bo h glass ansi ions a empe a u es lowe han ha expec ed o Tmax. F om he inse o Fig. 1共a兲one can, howe e , se he bounds 120 K⭐Tmax⭐140 K. III. RELATIONSHIP WITH MEASUREMENTS CARRIED OUT AT MACROSCOPIC SCALES The ␤ elaxa ion in bo h G and OG phases abo e ⬇40 K has been ecen ly s udied by dielec ic spec oscopy.17 The empe a u e dependence was desc ibable in e ms o an ac i- a ion ene gy V0, a wid h o he ene gy-ba ie dis ibu ion ␴ 0, and he in e se o he a emp equency ␶ 0, ha ing al- ues o abou 1000 K, 600 K, and 0.5⫻10⫺13 s, espec i ely. The pa ame e s o bo h phases a e he same wi hin s a is i- cal p ecision. F om da a o ␶ 0, as well as om he es ima e o Tmax gi en abo e, one ob ains an uppe bound o Vmax , which is he uppe limi o he dis ibu ion unc ion o he ba ie heigh s o s a es ha ul ill he condi ion ␻ ␶ ⫽1, as7 kBTmax⫽⫺Vmax /ln( ␻ ␶ 0) wi h ␻ being he obse a ion e- quency. The esul yields Vmax⫽720 K o a empe a u e o 130 K, in he midpoin o he in e al gi en abo e. I we now conside how ␦ / is expec ed o beha e in a he mally ac i a ed egime ul illing ␻ ␶ 0Ⰶ1, namely, ␦ / ⫽CkBTln( ␻ ␶ 0)/E0, we ind ha , using he pa ame e s e- e ed o abo e, he desc ip ion becomes alid up o T⬇60 K as Fig. 1 shows. The s onge dec ease in ␦ / abo e his empe a u e is p obably a signa u e o he inc easing impo - ance o he mal expansion e ec s no accoun ed o by he model. Ou p e ious s udy17 enabled he iden i ica ion o he - mally ac i a ed low-angle eo ien a ions 共lib a ions兲, which a e s ongly coupled o he collec i e 共phonon兲modes, as he main mic oscopic en i y gi ing ise o he dielec ic signal o bo h G and OG.17 Mo eo e , such mo ions show a peaked densi y o s a es ha accoun s o he excess in he C(T)/T3 speci ic hea plo s a abou 5–8 K as well as o he ‘‘pla eau egion’’ in he mal conduc i i y.16 F om he closeness o he he mal ac i a ion pa ame e s a bo h mac o- 共dielec ic兲and mesoscopic 共B illouin兲scales, we in e ha mechanisms simila o hose causing dielec ic elaxa ion should be e- sponsible o he obse ed beha io o ␦ / and Q⫺1. Such mo ions should hen be conside ed akin o hose ‘‘addi ional ha monic exci a ions’’ comp ising co ela ed mo ions o a ew s uc u al uni s in he well s udied case o i eous SiO2. IV. DYNAMICS AT MICROSCOPIC SCALES To explo e he na u e o he mo ions e e ed o abo e ha ha e equencies commensu a e wi h maxima in C(T)/T3, ha is, 2 meV⭐ ␻ ⭐4 meV, a numbe o measu e- men s by neu on spec oscopy we e ca ied ou using he MARI spec ome e a he ISIS pulsed neu on sou ce 共Ru- he o d Apple on Labo a o y, U.K.兲. The sample was ully deu e a ed e hanol, which allowed us o moni o con inu- ously i s s a e by inspec ion o he di ac ion pa e ns, and was con ained in a s anda d aluminum sample can. The measu emen s we e ca ied using inciden ene gies ene gy o Ei⫽100 meV and Ei⫽15 meV. This was done in o de o co e a la ge kinema ic egion so ha he depen- dence o he dynamic s uc u e ac o S(Q, ␻ ) wi h wa e ec o can be sa ely explo ed and also o de ec he possible exis ence o exci a ions occu ing a ini e equency and cons an wa e ec o . In iew o he s ong anha monic e ec s displayed in Fig. 1, he measu emen s we e ca ied a ela i ely low empe a- u es 共abou 5 K兲 ha a e close o hose whe e C(T)/T3also show hei maxima.16 As an added bonus, ca ying ou he measu emen a such empe a u es diminishes he impo - ance o mul iexci a ion con ibu ions. All h ee solids we e p epa ed in si u a e an ini ial quench o he high- empe a u e liquid in o he deep glass phase. The spec a co esponding o he amo phous solids we e he ones i s measu ed. The o ien a ionally diso de ed c ys als we e p e- pa ed by aising he empe a u e 10 K abo e Tgand a sub- sequen annealing unde such condi ions. Fo ma ion o he RP c ys al was well moni o ed by he appea ance o a s ong B agg peak in he di ac ion pa e n measu ed using he elas ic I(Q, ␻ ⫽0) channels 共in eg a ed o e esolu ion wid h兲. Finally, he s able monoclinic phase was p epa ed by annealing a empe a u es close o i s mel ing Tm⫽159 K o a ew hou s. The di ac ion pa e ns o he h ee solids a e shown as inse s o Fig. 3. Da a educ ion and analysis ol- lowed s anda d ou es. Because o he high ansmission o ou sample mul iple-sca e ing con ibu ions o he measu ed spec al pa e ns we e es ima ed o be a he small. Finally, he mul iphonon con ibu ions we e e alua ed ollowing he p ocedu e desc ibed in Re . 22 and sub ac ed om he ba e in ensi y. Figu e 2 displays a ep esen a i e se o spec a measu ed wi h Ei⫽15 meV, which enables one o compa e di ec ly he in ensi y pa e ns o he h ee solids. This allows us o see ha he e is no measu able di e ence in he spec a o he wo glassy solids, while a subs an ially di e en beha io is ollowed by he o de ed 共monoclinic兲c ys al. The la ges di e ence in in ensi ies conce n a ange o equencies ha s e ches up o equencies commensu able wi h hose whe e he monoclinic c ys al shows i s i s in ense phonon band (⬇6 meV兲. I conce ns an easily isible ‘‘excess’’ o inelas- 8780 PRB 61A. CRIADO e al. ic in ensi y o bo h glass and OG wi h espec o he o de ed c ys al, which eaches a maximum a equencies wi hin he ange 2–3 meV. Such ‘‘excess modes’’ once a e aged o e momen um ans e s will gi e ise o ib a ional equency dis ibu ions such as hose epo ed in some o ou p e ious s udies.23 Wi h espec o he wa e- ec o dependence o he inelas- ic in ensi ies shown in Fig. 2 he ollowing quali ica ions should be aken in o accoun . Below Q⬇0.5 Å⫺1 he inelas- ic in ensi ies o he glassy solids show a a he s uc u eless shape. Tha o he o de ed c ys al g ows s eeply om he ail o he elas ic line and c osses hose o he glassy solids a abou 5 meV. A small maximum s a s o de elop a abou 2.5 meV in he spec a o bo h glassy solids o Q ⫽0.7 Å⫺1. F om he e o he la ges explo ed wa e ec o s such a maximum appea s as a dispe sionless ea u e and can be iden i ied wi h he ‘‘boson peak’’ ound in s udies em- ploying incohe en -sca e ing samples.16,11 Wha seems wo h explo ing is he obse a ion ha while he cen e o such a peak does no show any clea dependence wi h wa e ec o , he amoun o excess in ensi y shown by he wo glasses wi h espec o he c ys alline g ound s a e shows some modula ion wi h Q. Fo wa e ec o s co esponding o he i s B illouin zone o he o de ed c ys al 共e.g., below 0.85 Å⫺1) he excess in ensi ies a e maximal. Such a di e - ence ge s almos supp essed when app oaching he B illouin zone bounda y and eappea s as one p oceeds u he . Ha ing measu ed S(Q, ␻ ) o ela i ely la ge Q’s enables us o conside in some de ail he quan i y S(Q, ␻ ⫽cons ), ha is, he wa e- ec o dependence o he inelas ic in ensi y o he ange o equencies whe e he excess modes com- men ed on abo e is maximal. Such quan i ies a e, in ac , inelas ic s uc u e ac o s, gi ing in o ma ion abou he ec- o displacemen s o he a oms aking place in such mo ions. A model- ee assessmen o he na u e o mo ions being sampled is p o ided by compa ison o he phases o oscilla- ions in S(Q, ␻ ⫽cons ) wi h hose shown by he s a ic S(Q) o elas ic S(Q, ␻ ⫽0) s uc u e ac o s. Pu ely ansla ional mo ions 共i.e., long-wa eleng h acous ic phonons兲lea e all dis ances unal e ed and he e o e S(Q, ␻ ⫽cons ) should show oscilla ions in phase wi h hose o S(Q)o S(Q, ␻ ⫽0). Cu s o S(Q, ␻ ⫽cons ) o 2.5 meV⭐ ␻ ⭐3.5 meV, an in e al ee om any subs an ial con amina ion by he elas- ic line, a e shown in Fig. 3. Bo h glassy solids 共G and OG兲 show a s ikingly simila in ensi y pa e n, wi h a shoulde a abou 3 Å⫺1, a b oad maximum a 7 Å⫺1, and a minimum a abou 11.2 Å⫺1. Only a small peak is seen o G and OG a Q alues co esponding o he main peak in he di ac ion pa e n Qp⬇1.7 Å⫺1. In con as , sha p peaks a 1.7 Å⫺1 and 2.5 Å⫺1iden i y he p esence o p opaga ing exci a ions in he spec um o he monoclinic c ys al, a ea u e also ep oduced om a c ys al la ice dynamics calcula ion. This shows ha e en a hese ela i ely low equencies, mos o he in ensi y in spec a o he glassy solids a ises om mo- ions no in ol ing in-phase displacemen s o he molecula cen e s o mass 共COM兲. In o he wo ds, mos o he in ensi y a equencies compa able o ha o he ‘‘boson peak’’ seem o a ise om mo ions ha should ha e a molecula eo ien- a ional componen as dominan , as explained. Now we ake ad an age o he p esence wi hin he OG phase o a c ys al bcc cell as well as he a ailabili y o he absolu e equency scale p o ided by he monoclinic c ys al. This enables he sepa a ion o pu ely ansla ional and o a- ional mo ions since he molecula cen e s o mass si a he nodes o he bcc la ice. We ha e ca ied ou a numbe o compu e simula ions o such a model o he OG c ys al. Ou poin o depa u e was o assume ha ela i e molecula o ien a ions all ha e he same s a is ical weigh , a ac g ounded on p e ious esul s.15 A andom sampling s a egy FIG. 2. Cons an -Qspec a as measu ed using an inciden ene gy Ei⫽15 meV o alues o he mo- men um ans e gi en in he in- se s. The solid line depic s he spec a o he monoclinic c ys al, he open symbols show he spec- a o he glass, and he illed symbols he spec a o he o ien a- ionally diso de ed c ys al. PRB 61 8781ROLE OF LOW-FREQUENCY VIBRATIONS ON SOUND . . . was de ised o gene a e he ini ial molecula o ien a ions as well as he small lib a ional displacemen s. Molecula COM mo ions we e simula ed by displacing he molecules along he Ca esian displacemen coo dina es o he bcc la ice. A u he a e age was hen aken o e he di e en displace- men modes and he inal a omic displacemen ec o s ui we e calcula ed. The calcula ed inelas ic s uc u e ac o I(Q, ␻ ⫽cons ) was inally e alua ed om he one-phonon pa ial s uc u e ac o s Fij(Q), I共Q, ␻ ⫽cons 兲⬀兺 i兺 j⫽iFij共Q兲G共 ␻ 兲,共2兲 Fij共Q兲/Q2⫽1 冑 MiMj 冋 1 3共ui•uj兲j0共Qdij兲 ⫹ 冉 1 3共ui•uj兲⫺1 dij 2共dij•uj兲 冊 j2共Qdij兲 册 ,共3兲 whe e Mia e a omic masses, ua e ec o displacemen s, dij a e ec o s joining wo a oms, G( ␻ ) a e spec al unc ions 共combina ion o Bose ac o s and ␦ peaks兲, and jx( ) a e sphe ical Bessel unc ions. The calcula ed cu e is compa ed wi h expe imen a e addi ion o a sel -sca e ing componen 24 and is shown in Fig. 3. The esul shows ha pu ely ansla ional mo ions con ibu e o he obse ed in ensi y a such equencies wi h only a small ea u e a Q⫽Qp. In con as , eo ien a ional excu sions wi h an a e age angle o abou 0.3° accoun o he shape o I(Q, ␻ ⫽cons ). Because o he e y close simila i y o he expe imen al inelas ic s uc u e ac o o he OG and he ully diso de ed solid, one expec s ha he esul s o he o me , he e con- side ed as a e e ence ‘‘glass,’’ should also be o ele ance o he la e . In o he wo ds, hey se e o quan i y he con- ibu ion o I(Q, ␻ ), on Å and picosecond scales, coming om pu ely acous ic mo ions ce ainly in ol ing he dis- placemen s o molecula COM’s, e sus hose a ising om addi ional 共lib a ional兲exci a ions. F om he e y small weigh o he peak a Q⫽Qpwe in e ha a high- equency cu o o he exis ence o well-de ined soundlike exci a ions should no be a beyond ␻ co⭐3 meV, a equency ha be- comes compa able o ha o he ‘‘boson peak.’’ V. CONCLUSIONS The indings epo ed he e a e o ele ance o ou unde - s anding o glassy dynamics since hey enable a cohe en desc ip ion o he mechanisms d i ing he dynamics o dis- o de ed ma e a widely di e en spa io empo al scales as explo ed by dielec ic 共mac o-兲, B illouin 共meso-兲, and neu- on 共mic oscopic兲spec oscopies. A numbe o cha ac e is- ic phenomena can be unde s ood as esul ing om he in e - ac ion o long-wa eleng h acous ic wa es wi h small- ampli ude molecula lib a ions. The la e a e known o gi e ise o he peak in C(T)/T3, cons i u e an e icien mecha- nism o sound a enua ion ia elaxa ional and esonan sca e ing mechanisms, and explain he concomi an educ- ion in sound eloci y wi h empe a u e. The p esen esul s also sugges s a se o weakly damped 共TLS兲lib a o s eso- nan ly in e ac ing wi h sound wa es as he mos easible en- i ies gi ing ise o he linea e m in he speci ic hea below 2 K ound o he bcc c ys al and glass.16 ACKNOWLEDGMENTS We acknowledge suppo om DGICYT 共Spain兲G an No. PB95-0075-C03-01 and U. S. Depa men o Ene gy, Basic Ene gy Sciences–Ma e ials Sciences unde Con ac No. W-31-109-ENG-38. FIG. 3. Neu on spec oscopy da a a cons an equency I(Q,2.5 meV⬍ ␻ ⬍3.5 meV兲 o he glass, OG, and monoclinic c ys al phases a T⫽5K关 ames 共a兲–共c兲, espec i ely兴. The solid line shown in ame 共b兲co esponds o he calcula ion made o he bcc c ys al on he basis o Eq. 共2兲. The dashed line shown in ame 共c兲depic s he esul om a ha monic la ice dynamics calcula ion o he monoclinic powde 共see Re . 16兲. Inse s display he elas ic I(Q, ␻ ⫽0) da a. 8782 PRB 61 A. CRIADO e al. 1R. C. Zelle and R. O. Pohl, Phys. Re . B 4, 2029 共1971兲. 2K. A. Topp and D. G. Cahill, Z. Phys. B: Condens. 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