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.

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