Table Of Content1
Introduction
Aglassisadisorderedmateriallikeaviscousliquidandbehavesmechanicallyasasolid
[1–6]. A glass is normally formed by supercooling the viscous liquid fast enough to
avoidcrystallization.Theliquid-glasstransitionoccursindiversemannerdependingon
materials,supercoolingprocesses,historyofmaterials,andsoon.Amongthediversity,
thereareuniversalfeaturessuchas:
1. TheKauzmannparadoxontheentropycrisis[7]:
Asthetemperaturedecreasestowardtheglasstransitiontemperature,theentropy
ofthesystemseemstodroptoanegativevalue.
2. TheVogel-Tamman-Fulcher(VTF)law[8–10]ontherelaxationandthetransport
processes:
The relaxation time and the viscosity drastically increase, while approaching the
liquid-glasstransition.Theratioofthevaluesoftherelaxationtimeandtheviscosity
attheglasstransitiontothoseathighertemperaturesbecomes1017.
3. Thejumpofspecificheatattheglasstransitiontemperature:
Thereisagapinthevaluesofthespecificheatsbetweenjustbelowandjustabove
theglasstransitiontemperature.
4. Thebosonpeak[11]:
There appears a peak of the density states at the harmonic frequency of phonons
belowtheglasstransitiontemperature.
In the crystal forming process, cooling a liquid in an equilibrium state, the specific
volumedecreaseswiththetemperaturedecreasingtillthemeltingtemperatureT ,at
m
which the specific volume decreases with the temperature keeping constant till the
completionofcrystallization,wherethefirstorderphasetransitionaccompaniedwith
theemissionofthelatentheattakesplace,andthenthespecificheatdecreaseswiththe
temperaturedecreasingasschematicallyillustratedinFigure1.1.
Intheglassformingprocesstoavoidthecrystallization,supercoolingtheliquid,the
specificvolumedecreasesbeyondT upwardconcavelywiththetemperaturedecreas-
m
ingtilltheglasstransitiontemperatureT ,wheretheliquid-glasstransitiontakesplace,
g
andthenthespecificvolumedecreasesatfirstupwardconcavely,thenlinearlywiththe
temperature decreasing as schematically illustrated in Figure 1.1. T depends on the
g
supercoolingprocesses.
In order to establish a unified theory to elucidate the universal features in such
complex systems, at the first step we must start with a simple model in a simple
liquid composed with a kind of isotropic particles and develop the theory in a most
tractableway;weconstructthetwobandmodelasasimplemodelandmakethemean
LiquidGlassTransition.DOI:http://dx.doi.org/10.1016/B978-0-12-407177-3.00001-5
©2013ElsevierInc.Allrightsreserved.
2 LiquidGlassTransition
A
Liquid
B
Supercooled
me Liquid
u E
ol Glass
V
c G
cifi D
e
p
S C
Crystal
T T
g m
Temperature
Figure1.1 Changesofspecificvolumewiththetemperaturedecreasingintheglassforming
andthecrystalformingprocesses.Inthecrystalformingprocess,coolingaliquidinanequi-
librium state, the specific volume decreases with the temperature decreasing A till B (at the
melting temperature Tm), atwhich the specific volume decreases B tillD (thecompletion of
crystallization)withthetemperatureconstant,wherethefirstordertransitionaccompaniedwith
theemissionofthelatentheattakesplaceandthenthespecificvolumedecreasingDtoCwith
thetemperaturedecreasing.Intheglassformingprocesstoavoidcrystallization,supercooling
theliquid,thespecificvolumedecreasesAtoBintheliquidstate,BtoE(attheglasstransition
temperatureTg)inthesupercooledliquidstateandthenEtoGintheglassystate.Thespecific
volumechangesupwardconcavelyaroundtheglasstransitiontemperature.
fieldapproximation(therandomphaseapproximation)asatractableway.Despitethe
simplemodelandthecrudeapproximation,oncewehaveestablishedaunifiedtheory,
it plays a guiding role in establishing more rigorous theories, rigorously simulating
thedynamicalequationsincludingnonlinearinteractions,extendingtomorecomplex
systems,arrangingexperimentalresults,andsoon.
Inordertoconstructthetwobandmodel,wemustclarifythestructureofcondensed
states.Thestructureofasystemrelateswiththelongrangeorderofthesystem,which
means that the same order holds over the whole system. The long range order spon-
taneouslybreaksthetranslationalsymmetry;thesymmetrybrokenstateisassociated
withtheNambu-Goldstone(NG)bosons(phonons)[12–15].Inthecrystallinestate,the
longrangeorderisrepresentedbytheperiodicstructure;thepairdistributionfunction
looksexactlythesametoeachobserverathisownparticle[16].Aparticlelocatesin
aperiodicallydistributedlocalizedpotentialmakingupanddowntransitionsbetween
exactly thesame twolevels foralongtime.The modes ofthepropagation oftheup
and down transitions through the interactions are a longitudinal and two transverse
phonons. Intheliquid and glassystates,although particles arerandomly distributed,
thereisakindofthelongrangeorder;thepairdistributionfunctionlooksthesameto
eachobserverathisownparticleinasenseoftheconfigurationaverage[17].Aparticle
locatesinarandomlydistributedlocalizedpotentialmakingupanddowntransitions
Introduction 3
betweenthetwolevelsandthenhopstoaneighboringvacancy.Thehoppingyieldsthe
two levels bandwidths. The modes of the propagation of the up and down transition
throughtheinteractionsarealsophonons.Intheglassystate,sincethehoppingseldom
occurs,thelongrangeorderholds;thepairdistributionfunctionholdsthesameevery
timesothatphononsareessential,whileintheliquidstate,sincethehoppingalways
occurs,thelongrangeorderdoesnotholds;thepairdistributionfunctionchangesevery
timeandthehoppingcausesthedensityfluctuationssothatthesoundmodeisessential.
ThesoundisnottheNGboson.
Thesoundplaysanessentialroleintheliquid-glasstransition.Inthetwobandmodel,
thesumofaninfiniteseriesofinteractingbubblesofintra-bandparticle-holepairscon-
stitutessoundmode,whilethesumofaninfiniteseriesofinteractingbubblesofinter-
bandparticle-holepairsconstitutesphononmodes.Thesounddisappearsatthesound
disappearingtemperature T andareciprocalparticledistanceK.Inthesupercooling
0
process, enormous bubbles of intra-band particle-hole pairs are created so that extra
bubblesofintra-bandparticle-holepairsnotconstitutingthesoundformthefluctuation
entropy,whichmixeswiththeentropyofsoundaroundthewavenumber∼K.Thismix-
ingentropyisjusttheKauzmannentropy,whichisoftheformoftheCurielawwitha
negativesignandthecriticaltemperatureT .Theglasstransitionoccursattheglasstran-
0
sitiontemperatureT ,wheretheKauzmannentropycrossesoverthefluctuationentropy
g
duetothebubblesofinter-bandparticle-holepairs.Theprobabilitycorrespondingto
theKauzmannentropyperparticlefromtheEinsteinrelationmanifeststheVTFlaw;
theprobabilityisproportionaltotheexponentoftheKauzmannentropyperparticle.
In Introduction, we briefly survey the structure of the condensed states, the two
band model, the collective modes (sound and phonons), and the mechanism of the
liquid-glasstransition.
1.1 The Structure of the Condensed States
and the Quantum Regime
A pair distribution function and an inter-particle interaction potential yield a well-
localizedpotentialofaparticle,whichcanbeapproximatedasaharmonicpotential.
Thestructureofthepairdistributionfunctiondeterminesthestructureofphonons.On
theotherhand,whenaparticlehasavacancyatthenearestneighbor,thepotentialof
theparticlehasanotherminimumpotentialinthedirectiontothevacancysothatthe
particlecanhoptothepositionofthevacancy.
Thelongrangeorderspontaneouslybreaksdownthetranslationalsymmetry,which
yieldstheNGbosons,alongitudinalandtwotransversephonons.Intheglassystate,
phononsareessential,whileintheliquidstate,soundisessentialduetothehopping
sothatthesoundisnotNGbosons.
In this section, we also verify the liquid-glass transition occurs in the quantum
regime.
4 LiquidGlassTransition
1.1.1 The Structure of Condensed States, the Pair Distribution
Function, and Hopping of Particles
In both the liquid and glassy states, particles are randomly distributed, but both the
stateshavealmostthesamepairdistributionfunctiong(R ,R )whichisdefinedas
m n
twoparticledensityatthepositions R and R undertheconfigurationaverageofthe
m n
otherparticles.Forasimpleliquid,thepairdistributionfunctiondependsonlyonthe
separation,R =|R −R |,soitissimplywrittenasg(R)calledtheradialdistribution
m n
function[17].g(R)hasapeakatameaninter-particledistanceaandwiththedistanceR
increasingitconvergesoscillatinglytotheaveragedensityρ.Randomlydistributedpar-
ticlesandthepairdistributionfunctiong(R)isschematicallyillustratedinFigure1.2.
Thepotentialbetweenparticleshasahardcoreandavalleywithaminimumneara
meaninter-particledistanceaillustratedinFigure1.3a.Aparticleatasite R feelsa
m
well-localizedpotentialmadeupbythesurroundingparticles.Inordertoseehowthe
potentialwellismadeupbythesurroundingparticles,weconsideraone-dimensional
system with the x-axis under the nearest neighbor approximation as schematically
illustratedinFigure1.3b.Thepotentialsofaparticleatthe0-site,madebytheparticles
atthe−a anda-sites,areillustratedbythetwodottedlines.Thetotalpotentialmade
upbythesurroundingparticles,(cid:3)(x),iswelllocalizedasillustratedasthickline.We
canregardthewell-localizedpotentialasaharmonicpotential.Figure1.3cshowsthe
case for the existence of a vacancy at a-site shown by a dotted circle. The potential
whichtheparticleatthe0-sitefeels,(cid:3)(x),hasdoubleminimumssothattheparticle
atthe0-sitecanhoptothevacancyatthea-site.
Thuswecanregardawell-localizedpotentialasaharmonicpotential.Theparticle
intheharmonicpotentialatarandompositionR hasarandomharmonicfrequencies,
m
ω .Inacaseoftheexistenceofavacancyat R inthex -directionofaparticleat R
m n i m
asillustratedinFigure1.4,sincethem-particlemissesthenegativepotentialatR and
m
thewallwhichshouldbemadebythen-particle,thepotentialofthem-particlehasa
Figure1.2 Randomlydistributedparticlesandthepairdistributionfunctionareschematically
illustrated.TherandomharmonicpotentialsinaCartesianaxisxi arealsoillustrated.ωn0 and
ωni aretheeigenfrequenciesofthegroundandthefirstexcitedstatesinthexi-component.
Introduction 5
(a) (b)
V (R) (x)
00
a 0 a x
a
R
0
(c)
(x)
a 0 a 2a x
Figure1.3 Thepotentialbetweenparticlesandthepotentialsmadeupbythesurroundingparti-
clesinaone-dimensionalsystemwiththex-axis.(a)ThepotentialbetweenparticlesV00(R)is
schematicallyillustrated.Ithasahardcoreandavalleywithaminimumnearameaninter-atomic
distancea.(b)Thetotalpotentialwhichaparticleatthe0-sitefeels,madeupbythesurrounding
particlesat−aandasites,(cid:3)(x),isawelllocalizedasschematicallyillustratedbyathickline.
Thepotentialmadebyeachparticleisillustratedbyitsowndottedline.(c)Forthecaseofthe
existenceofavacancyatasite,(cid:3)(x),hasdoubleminimumsasschematicallyillustratedbya
thickline.
saddlepointupwardconvexinthedirectiontothevacancyx anddownwardconvexin
i
thedirectionperpendiculartox .Them-particlecanhoptothevacancywithahopping
i
magnitude, JmμnμasillustratedinFigure1.4.
Figure1.4 RandomlydistributedparticleswithavacancyatRnandthepotentialofaparticleat
Rm inthexi-direction.Thepotentialatasaddlepointisupwardconvexinthexi-directionand
downwardconvexintheperpendiculartoxi-axis. Jm0n0and Jmini arethehoppingmagnitude
betweenthegroundstatesandthatbetweentheexcitedstates,respectively.
6 LiquidGlassTransition
Intheliquidstate,aparticlestaystemporarilyinaharmonicpotentialatR making
m
theharmonicoscillationwiththeharmonicfrequencyω andthenhopstoasurrounding
m
vacancy.Thehoppingofaparticlechangesapairdistributionfunctiontoanotherdis-
tributionfunction,buttheconfigurationallyaverageddistributiong(R)holdsthesame
overspace.Intheglassystate,sincealltheparticlesarefreezedintheirownharmonic
potentials at random positions, they scarcely hop, so the configurationally averaged
pairdistributionfunctiong(R)holdsthesameoverspaceandtime.Thus,hoppingis
essentialintheliquidstate,whiletherigidityofg(R)isessentialintheglassystate.
1.1.2 The Nambu-Goldstone Bosons, Phonons, and Sound
Inthecondensedstates,theliquid,glassy,andcrystallineones,theHamiltoniansareof
thesameforminvariantunderthespatialtranslation.However,thelongrangeorders
in the glassy and crystalline states spontaneously break down the symmetry of the
spatially translational invariance. The spontaneous breakdown of the symmetries is
associatedwiththeNambu-Goldstonebosons[12–15].TheNambu-Goldstonebosons
arealongitudinalandtwotransversephonons.TheNambu-Goldstonebosonsplaya
roleinrecoveringthesymmetries.
Inthecrystallinestate,thelongrangeorderisrepresentedbyaperiodicstructure.The
pairdistributionfunction g(R)hasdeltafunctiontypepeaksatanyintegralmultiple
of the primitive vectors; g(R) looks exactly the same to each observer at his own
particle [16]. The liquid and glassy states also have a kind of the structure, the long
rangeorderinasenseoftheconfigurationaverage.Thepairdistributionfunctiong(R)
has the structure starting with a peak at an inter-particle distance and oscillatorily
damping toward the mean particle density in distance R as illustrated in Figure 1.2;
g(R)looksthesametoeachobserverathisownparticle.Accordingtothestructureof
thepairdistributionfunction g(R)inacrystaland g(R)intheglass,thesymmetries
ofthespatiallytranslationalinvariancespontaneousbreakdown.Thisspontaneously
breakdown of the symmetry is associated with the Nambu-Goldstone bosons, which
arealongitudinalandtwotransversephonons.
Thestructureofcondensedstatesdeterminetheirstructureofphonons.Thephonon
dispersioncurvesareessentiallyobtainedbythenearestneighborparticles.Thedisper-
sioncurvesofphononsinthecrystallinestateareperiodicandclearbecauseofexactly
periodiclongrangeorder,whileintheglassystatephonondispersioncurvesarevague,
becausethelongrangeorderisonlyguaranteedbytheconfigurationallyaveragedpair
distributionfunction,g(R).Thevaguenessissubstantialinaliquidduetothehopping
ofparticles.
The hopping of particles plays an essential role in the liquid state. The hopping
ofparticlescausesdensityfluctuations.Thedensityfluctuationwavesarejustsound.
Conventionally, phonons and sound have not been discriminated clearly. Sometimes
soundhasbeenconfusedwithalongitudinalphononinthelongwavelengthregions.
We must discriminate sound and phonons clearly. Sound is not a Nambu-Goldstone
boson.Soundissubstantialinfluidsystemssuchasgaseousandliquidstates.Butthe
liquidstateisacondensedonesothatthereremainsthepairdistributionfunctiong(R)
inasenseoftheconfigurationaverage,whichyieldsvaguephonons.Soundplaysan
Introduction 7
essentialroleintheliquidstate.Thehoppingalsocausestherelaxationofsoundand
phonons.Diffusivityappearsaccordingtothedensitygradient;itisassociatedwiththe
relaxationofsound.Viscosityappearsaccordingtotheflowgradient;itisassociated
withtherelaxationofphonons.
1.1.3 The Liquid-Glass Transition Occurs in the Quantum Regime
The liquid-glass transition conventionally has been treated from the classical theory.
TheclassicalregimeisgivenbythecriterionthatthedeBlogliethermalwavelength
denotedbyλ isverymuchlessthantheinter-particledistancea:
B
λ (cid:3)a. (1.1.1)
B
λ is estimated by using the equipartition law, (cid:2)(cid:6) = ((cid:2)p)2/2M = k T/2,and the
B p B
deBlogliewavelength,λ =2π/p,as
B
h
λ = √ , (1.1.2)
B
Mk T
B
where M means the mass of a particle. This criterion (1.1.1) means that we can see
anobjectwithascaleaclassicallybyuseoflightwithwavelengths∼ λ .Thecrite-
B
rionisalsoderivedbyusingtheBoltzmanndistributionfunctionunderthecondition
e(cid:2)(μkB−T(cid:6)p) (cid:3)1forany(cid:6)p [18].Sincee(cid:2)(μkB−T(cid:6)p) <e(cid:2)(kμB−T(cid:6)0) =ek(cid:2)BμT,wehave
(cid:2)
ek(cid:2)BμT = VN (2λπ3B)23 (cid:3)1, N = (2Vπ)3 e(cid:2)(μkB−T(cid:6)p) d3p =ek(cid:2)BμTV(2λπ3B)23, (1.1.3)
where (cid:2)μ is the chemical potential, N the number of particles, and V the volume of
thesystem.Inthisbook,energyandmomentumaremeasuredbyfrequencyandwave
vector,respectively.
Near the liquid-glass transition temperature, a particle stays for a long time in a
harmonic potential making the harmonic oscillation with the averaged frequency ω
and seldom hops to a surrounding vacancy. Under the circumstances, a particle is
actually trapped inaharmonic potential. Since theglass transitiontemperature T is
g
estimatedtobethesameorderofthemeltingtemperatureT ,whichisthesameorder
m
oftheharmonicfrequencyωasshowninSection7.4[19,20].Thusweobtain
(cid:2)ω∼k T . (1.1.4)
B g
Toestimateλ neartheglasstransition,ifweput(cid:2)ω∼k T inEq.(1.1.2),weobtain
B B g
(cid:3)
h
λ ∼ nearT , (1.1.5)
B Mω g
whichisthesameorderasthewidthofthezeropointmotionfortheharmonicoscil-
latorλ ,
0
8 LiquidGlassTransition
(cid:3)
h
λ ∼ . (1.1.6)
0 Mω
FromtheharmonicpotentialillustratedinFigure1.2,wecanestimate
λ ∼a, (1.1.7)
0
sowehave
λ ∼a nearT , (1.1.8)
B g
thecriterion(1.1.1)isnotsatisfied.Thuswemusttreatthesystemquantumstatistically.
Thisisalsowhywemustdealwithphononsquantumstatistically.
1.2 The Two Band Model for the Liquid-Glass Transition
Inthissection,wedrawtheframeworkoftheunifiedtheoryoftheliquid-glasstransition.
Totreatthedynamicalprocessesinarandomsystemsuchasaliquidandaglass,we
dividethecalculationprocessesoftherandomsystemintotwostages.Atthefirststage,
weinvestigatethedynamicalmotionsintheconfigurationallyaveragedsystem.Atthe
nextstage,wetaketherandomnessofthesystemintoaccountasscatteringprocesses
byrandompotentials.
As the first stage, we construct the two band model and develop the dynamical
processesofsoundandphononsintheconfigurationallyaveragedsystem.Thesound
andphononsarerepresentedbythesumofaninfiniteseriesofbubblesofintra-band
andinter-bandparticle-holepairs,respectively.
Asthesecondstage,weobtaintherelaxationtimebytakingintoaccounttheconfig-
urationaverageofthesimultaneouslyscatteringprocessesoftwoparticlesbythesame
randompotentialsduetorandomharmonicfrequencies andrandomhoppingmagni-
tudes. We obtain the dynamical equation in the dissipative processes by introducing
therelaxationtimesandthenweobtainthediffusivityandtheviscosity.
Finally we investigate the supercooling process. The enormous bubbles of intra-
bandparticle-holepairsnotconstitutingthesoundcreatethefluctuationentropy,which
mixes with the sound entropy with wave numbers∼K. The mixing entropy leads to
theKauzmannentropy,whichyieldstheVTFlawandthegapofspecificheat.
1.2.1 The Two Band Model
Atthefirststage,weinvestigatethedynamicalmotionsintheconfigurationallyaver-
agedsystem.
Neartheglasstransitiontemperature,aparticlestaysforalongtimeinaharmonic
potentialandseldomhopstoasurroundingvacancy.Theparticleisac(cid:4)tuallyt(cid:5)rappedin
aharmonicpotential;theenergyoftheparticleisquantizedasω˜ = n+ 3 ω.Since
n 2
λ ∼ a, we consider the two levels; the ground state, ω˜ = 3ω and the first excited
0 0 2
states,ω˜ = 5ω.Thefirstexcitedstatesaredegenerated;ω˜ = ω˜ ,wherei = x,y,z
1 2 i 1
Introduction 9
and ω˜ means that ith component is excited, 3ω and otherwise, 1ω. So we denote
i 2 2
eigenfrequencies as ω˜μ, where μ = 0,i. A particle staying in a harmonic potential
makeupanddowntransitionsbetweenthetwolevels,ω˜ andω˜ .Thepropagationof
0 1
upanddowntransitionsofparticlesyieldsphonons.
Ontheotherhand,thehoppingyieldsenergybands.Weintroducethehoppingofμth
particlesfrom Rm to Rn as Jμ(Rm − Rn)≡(cid:6)Jμ(Rm,Rn)(cid:7)c,where<···>c means
the configuration average. If we denote the Fourier transformation of Jμ(Rm − Rn)
as Jμ(p), we obtain two bands, energy dispersion curves of particles illustrated in
Figure1.5:
(cid:6)μp =ω˜μ+ Jμ(p), (1.2.1)
Thus we reach the two band model. Using this two band model, we can establish a
unified theory of the liquid-glass transition to elucidate the universal features of the
liquid-glasstransition[20,21].
Thetwobandmodelisubiquitous.Itcanbeappliedtosoftmattersuchasliquids,
colloids, polymers, and a number of biological materials. A particle in a colloid can
alsobeseentolocateinashallowlocalpotentialwithalowharmonicfrequencyand
hopstoanotherplacefrequently,whichfollowsthetwobandmodelsothatthejamming
transitionincolloidscanbetreatedinasimilarmannertotheliquid-glasstransition.
1.2.2 Sound
Theelementaryexcitationsinthetwobandmodelcanbedecomposedintointra-band
andinter-bandtransitionsasillustratedinFigure1.6.Intra-bandtransitionsoccurwhen
particleshoptovacanciessothattheyareassociatedwithparticledensityfluctuation
waves.Thepropagatorofanelementaryexcitation,(cid:6)0p−(cid:6)0p−q,isabubbleofparticle-
hole pairs in the Feynman diagram. The bubble of intra-band particle-hole pairs is
Figure1.5 Energydispersioncurvesnormalizedbytheconfigurationallyaveragedfrequencyω
asafunctionofap.Hereweputg(R) = ρδ(R−a). J0(p)isisotropic,while Ji(p)depends
onthedirection,sowehaveshowntheaveragedvalueoverthedirections.Theminimumgapis
nearareciprocalinter-particledistance.Anintra-bandelementaryexcitation,(cid:6)0p−(cid:6)0p−q,and
aninter-bandelementaryexcitation,(cid:6)0p−(cid:6)ip−q,arealsoshown.
10 LiquidGlassTransition
P(q,q) V (q)
0 00
F(q q)
0,
f' 2
q
1 q02 s2q
(a)
Q (q,q) V (q)
0
D (q ,q) ( ) ( ) ( )
0
( ) 2 /
q2 2
1 ( ) 0 q
(b)
Figure1.6 Howtoconstructsoundandphononswiththerespectivebubblesareillustrated.(a)
F(q0,q):thepropagatorofsound. P(q0,q)representsthebubbleofintra-bandparticle-hole
pairs.AdottedlineindicatestheinteractionpotentialV00(q).(b) Dλ(q0,q):thepropagatorof
phononswithamodeλ.Q(q0,q)representsthepairofbubblesofinter-bandparticle-holepairs.
AdottedlineindicatesVλ(q).
definedbytheaverageofthesumofthebubblesofparticle-holepairswithrespectto
pundertheconditionβ(cid:2)J (p)(cid:3)1,andisgivenbyapropagatorofafrequencyω :
0 q
P(q ,q)= 1 (cid:6) f((cid:6)0p−q)− f((cid:6)0p) ∼=− βf(cid:8)ωq2 ,
0 N(cid:2) p q0−((cid:6)0p−(cid:6)0p−q) q02−ωq2
(cid:8)
asillustratedinFigure1.6a,whereq isafrequency,fisadistributionfunction, f isits
0
derivativeandwehaveconsideredtheinversionsymmetrywithrespecttoq underthe
summationof p;ω =ω and P(q ,q)= P(q ,q).Thesecondequationisobtained
q q 0 0
undertheconditionβ(cid:2)J (p) (cid:3) 1;thebandof(cid:6) isverynarrow,β(cid:2)(cid:6) (cid:3) 1.The
0 0p 0p
frequencyofthebubbleoftheparticle-holepairsisgivenby
ω 2 ∼= vp2q2,
q
3
√
whereω ∼=v q/ 3,wherev isanaveragevelocityofparticles.
q p p
Inordertoderivesound,weintroducetheinteractionpotentialbetweenparticle-hole
pairsV (R − R )anditsFouriertransformedone,V (q) = V (q).Soundisthe
00 m n 00 00
sumofaninfiniteseriesofinteractingbubblesofintra-bandparticle-holepairsthrough
theinteractionV (q):
00
F(q ,q)= P(q ,q)+ P(q ,q)V (q)F(q ,q)
0 0 0 00 0
1 βf(cid:8)ω2
= =− q ,
1/P(q0,q)−V00(q) q02−ωs2q
