A Noncommutative Glass Model and the Boson Peak

Master’s thesis Theoretical physics

A Noncommutative Glass Model and the Boson Peak

Juha Savolainen 2018

Advisor: Anca Tureanu Examiners: Kai Nordlund

Anca Tureanu

UNIVERSITY OF HELSINKI DEPARTMENT OF PHYSICS PL 64 (Gustaf H¨ allstr¨ omin katu 2)

00014 University of Helsinki

Matemaattis-luonnontieteellinen Fysiikan laitos Juha Savolainen

A Noncommutative Glass Model and the Boson Peak Teoreettinen fysiikka

Pro gradu -tutkielma Elokuu 2018 57 s.

Boson peak, glasses, crystals, uids, noncommutative geometry, lattices, Debye model

Työssä esitellään uusi teoreettinen malli lasien rakenteelle, ja sitä hyödynnetään lasien niin sanotun bosonipiikin tutkinnassa. Malli perustuu yksinkertaiseen kiteistä tuttuun hilaan, joka muutetaan epäjärjestyneeksi epäkommutoivista uidimalleista tutuilla menetelmillä.

Aluksi tutkielmassa käydään läpi kiteiden rakenteen kannalta tärkeitä käsitteitä, keskittyen eri- tyisesti akustisiin ja optisiin aaltoihin, hilavärähtelyjen tilatiheyteen, lämpökapasiteettiin sekä Debyen värähtelymalliin. Sen jälkeen perehdytään lyhyesti epäkommutoiviin uidimalleihin sekä epäkommutoivaan geometriaan, jotta lasimallia esitellessä nähtäisiin sen yhteys uideihin. Lopuksi esitellään itse malli lasien rakenteelle, ja siitä lasketaan lasien dispersiorelaatiot, tilatiheys sekä lämpökapasiteetti.

Tilatiheydessä havaitaan Van Hoven singulariteetti matalilla taajuuksilla, joka vastaa lasien kokeissa löydettyä bosonipiikkiä. Lasilla on sekä akustisia että optisia aaltoja, joista akustiset aallot sijaitsevat hyvin lähellä Van Hoven singulariteetin taajuutta. Löydön perusteella lasien bosonipiikki johtuu akustisista aalloista.

A new theoretical model for the structure of glasses is presented and used to study the bo- son peak found in glasses. The model is based on a simple lattice model familiar from crystals, which is disordered using techniques from noncommutative uid models.

First classical crystal models and concepts of lattice vibrations are reviewed, focusing on acoustic and optical waves, the density of vibrational states, heat capacity and the Debye model. Then noncommutative uid theory and noncommutative geometry are shortly introduced to show the connection to uids in our model. After these introductions, the glass model is formulated and used to calculate the dispersion relations, the density of vibrational states and the heat capacity.

The density of states has a Van Hove singularity at low frequencies, which generates the boson peak seen in experiments. The glass is found to have both acoustic and optical waves, and the acoustic waves are located very close to the frequency of the Van Hove singularity, which hints that the boson peak should be related to acoustic waves.

Tiedekunta/Osasto Fakultet/Sektion Faculty Laitos Institution Department

Tekijä Författare Author

Työn nimi Arbetets titel Title

Oppiaine Läroämne Subject

Työn laji Arbetets art Level Aika Datum Month and year Sivumäärä Sidoantal Number of pages

Tiivistelmä Referat Abstract

Avainsanat Nyckelord Keywords

Säilytyspaikka Förvaringsställe Where deposited

HELSINGIN YLIOPISTO HELSINGFORS UNIVERSITET UNIVERSITY OF HELSINKI

1 Glasses and the bosonpeak 1

2 Somemain features of rystals 4

2.1 Lattiestruture . . . 4

2.2 Reiproalspae . . . 5

2.2.1 Brillouinzones . . . 6

3 Vibrating latties 7 3.1 Onedimensionallattie . . . 7

3.1.1 Brillouinzonesofthe1Dmodel. . . 9

3.1.2 Optialandaoustibranhes. . . 10

3.2 Three dimensionallatties . . . 12

3.2.1 Boundaryonditions . . . 14

3.2.2 Densityofstate. . . 15

3.2.3 VanHovesingularities . . . 16

3.3 Heat apaity . . . 18

3.3.1 Debyemodel . . . 19

3.3.2 Einstein model . . . 21

4 Nonommutativeuid theory 23 4.1 Nonommutativegeometry . . . 23

4.1.1 Nonommutingoordinates . . . 24

4.2 Fluidmehanis. . . 25

5 The glass model 28 5.1 Equationsofmotion . . . 29

5.1.1 Orderedlattiesites . . . 29

5.1.2 Disorderedlattiesites . . . 34

5.2 Densityofstates . . . 37

5.2.1 Orderedlattiesite . . . 37

5.2.2 Disorderedlattiesites . . . 41

5.2.3 Wholeglass . . . 43

5.3 Dispersionrelations . . . 44

5.3.1 Orderedpartiles . . . 46

5.3.2 Disorderedpartiles . . . 47

5.4 Heat apaity . . . 51

6 Conlusions and omparisonto othermodels 55

Whenaliquid isooleddownoldenoughtomakeit solidandfastenoughto

preventitfromattainingarystallinestruture,itanreahavarietyofstates

inwhihitstilllookslikealiquidbutowstooslowlyforanexperimenttoeven

onsider it to beowing. These states arealled amorphous, and thelowest-

potential-energyamorphousstateisalledaglass. Thetransitiontemperature

anbegivendierentdenitionsdependingonhowvisousamaterialisvisous

enoughtobealled aglassfortheexperiment, andalsohowfastthematerial

wasooleddown[1℄. Severalmaterialsanbeonsideredglasses,rangingfrom

thesiliateglassesusedinwindowstofoamsandproteins. Mostliquids[2℄an

beturnedtoglassbyrapidooling.

Glassesarefamiliarfromeverydaylifeandwidelyusedinindustry,buttheir

mirosopistudy is still laking [2, 3℄. Therigidity of glassesresembles that

ofrystals, butthe disorderedmirosopi strutureis loserto liquids,and a

goodtheoretialmodelforthestrutureofglasseshasyettobedeveloped. The

theoryofrystalsandliquidshasalreadybeenwellstudied[47℄[8℄,soall the

toolsforresearhingglassesshouldbeavailable.

Severaltehniquesofstatistialphysishavebeenusedforresearhingglasses.

Fieldtheory,renormalizationgroups,far-from-equilibriumsystems,mode-oupling

theory andkineti glass models [912℄ have been triedin thelast deades, to

nameafew,butaderivationforaompleteandwell-aeptedtheoryofglasses

isstillmissing[13℄.

Inthis thesis, wewill introdueamodel forglassesstarting from arystal

modelforsolids,whihwillthenbedisorderedusingatehniquefromnonom-

mutativeuidmehanis. Thephenomenonweareinterestedinistheso-alled

bosonpeak,whihisaninreaseinthedensityofstatesofamorphousmaterials

atspei vibrationalfrequeniesinlowtemperaturesomparedtothedensity

ofstatesofrystals. Oursystemisastatiglass,soweshallnottryto answer

questionsinsomemajorglasstopislikefragilityandwhatisthetransitiontime

ortemperaturebetweenliquidsandglasses. Weareinterestedinthedensityof

statesandtheheatapaityofglasses. Themodelwasrstpublishedin [14℄.

Thebosonpeakisafeature foundinallglassesinmeasurementsofspei

heat and heat apaity and light, x-ray, and neutron sattering. The ther-

malondutivityof glassesissigniantlysmallerthanthatof rystals,andit

inreasesmonotoniallyastemperaturerises,opposedtothethermalondutiv-

ityofrystalswhihstartstodereaseafter aertainpoint. Anotherdierene

omparedtorystalsisthatthethermalondutivityofglassesdoesnotdepend

on thehemial omposition. Many glassesonsisting of ompletely dierent

moleuleshaveverysimilarthermalondutivities. [15℄

The standardmodel for the heat apaityof rystalsis theDebye model,

whih preditsa

T ³

^{dependene fortheheatapaityatlow}temperatures. In verylowtemperaturesthespei heatof glasses,ontheotherhand,riseslin-

earlywithtemperature,and

C/T ³

^asafuntionof

T

^{hasapeakatlowfrequen-}

ies,asshown ingure(1.1),takenfrom [16℄. All these anomalousbehaviours

hintthatthereisanexessofvibrationalstatesinglassesinlowtemperatures.

Figure1.1: Theheatapaity

C

^saledas

C/T ³

^{,takenfrom[16℄. Theontinu-}

ouslinesareexperimental valuesforvitreousSiO

2

^{andrystalquartz,whereas}

thelongdashedline iswhat theDebyemodelpreditsforquartz. GraphAis

I.R.Vitreosil,BisvitreoussiliaandCis

α

^-quartz.

Indeed plotting

g glass (ω) /g Debye (ω)

^{as a funtion of the frequeny}

ω

^{, where}

g glass (ω)

^{isthedensityof statesofaglass and}

g Debye (ω)

^{is thefrequenyde-}

pendene given bythe Debyemodel, gives apeak at low frequenies. This is

thebosonpeak. [17℄

There are several dierent explanations for the boson peak, but none of

them have been generally aepted and most of the experimental results t

many models. Forexample loalization of vibrations [18℄ mode-oupling [19℄

andloallyfavouredstrutures[20℄havebeentried. Onethingthatisgenerally

agreedonisthat thepeakisrelatedtothedisorderedstrutureofglasses. [17,

21℄

Ageneralmodelusedoftentoexplainlowtemperatureglassbehaviouristhe

tunnelingtwo-levelsystempublishedseparatelybyPhillips [22℄andAnderson,

Halperin andVarma [23℄. Themodelsuggeststhat there areatoms orgroups

ofatomsinglassesthat havetwonearlydegeneratestatesthattheyantunnel

glassesinlowtemperatures. MathematiallythismeansdeningaHamiltonian

withsomedistributionsforthetunnelingoeientsandthedierenesbetween

theenergy levels of the twostates. The idea is notveryrestriting, so many

variationsof themodel anbeformulated and themodelhas evolved sineit

waspublished in1972.

Thetunnelingtwo-levelsystemhasbeenthoughttobeauniqueexplanation

forthelowtemperature behaviourof glasses,but reentlythis uniqueness has

beenquestioned byLeggett andVural[24℄. Alsothemirosopioriginfor the

tunneling hasyet to be properlyexplained. Ournonommutativemodelaims

to explain glass behaviour at intermediate temperatures (1-40 K), where the

bosonpeakours,butitmightpotentiallyshedlightonlowtemperaturestoo.

Anotherapproahtoglassesandondensedmatteristheinterstitialythe-

ory. The theoryis basedoninterstitialies, whih areatoms that oupy nor-

mallyunoupied sitesin arystallattie. Aording tothetheoryliquidsare

rystalswithenoughinterstitialies, while glassesarefrozenliquids, but there

is still theoretial work to be done in developing the theory [25, 26℄. Unlike

interstitialy theory, ourmodel does notdepend onhowatoms are disloated

fromtheirusuallattiesites,butinsteadonlyonhowmuhonaveragetheyare

disloated.

In1912apaperbyLaue,FriedrihandKnippingwaspresented. InInterferene

eets with Röntgen rays it was rstshown that x-raysshould dirat when

entering matter with aperiodi struture, and then reported that rystalline

solidsauseadirationjustlikethis. Thusitwasshownthatrystalsonsist

ofaperiodilattiestrutureandsolidstatephysiswasborn. [27℄

Crystals vary from metals onsisting of a single hemial element to om-

poundmetalsorevensnowakesandanthusbeofmirosopiormarosopi

size. Thelattiestrutureofmarosopirystalsanoftenbemarosopially

visible,resultinginbeautifulobjets. Thestudyofrystalsisalledrystallog-

raphy,butitisnotthefousofthisthesis. [6℄

Laterthestudyofsolidsstartedoveringalsosolidswithoutarystalstru-

tureandafterwardsevenliquids. Solidstatephysisbeameondensedmatter

physis,thebiggesteldofphysistoday. [6℄Inthisthesiswewillrstdevelop

mathematialtehniquesused in rystalalulations,andthenuse theseteh-

niquestostudyamorphousmaterials,orglassesastheyareoftenalled,whih

liesomewherebetweensolidsandliquids.

2.1 Lattie struture

[6, hapter 1℄ Sine the dening property of a rystalis the lattie struture,

themathematialstudyofrystalsisbasially thestudyoflatties. Therefore

weshallrstreviewtheoneptofalattie.

The simplest model of arystal is a lattie struture that repeatsa nite

formationof atomsor moleules. Infor exampleopper,gold,iron and alkali

metals, the struture is formed by similar single atoms at every point of a

lattie,and in tablesalt NaClthe lattiesitesonsist of 2x2bloksofsodium

and hloride. The repeating formation of lattie points that forms the whole

rystalisalled theprimitiveellofthelattie. Theprimitiveellanalso be

verylarge,onsistingofthousandsofatomsinforexampleproteinrystals[6℄.

Howeverformarosopirystals,theprimitiveellisstillverysmallompared

tothewholerystal,sotherystallattieisusuallyassumedto beinnite.

Mathematially lattiesare desribed through vetors. Forparallelepiped

latties,thedistanebetweeneah neighbouringlattiepointisdesribedwith

three orthogonal primitive translationvetors

a 1

^,

a 2

^and

a 3

^{, whih eah tell}

thelengthof the primitiveellsin one diretion. Sinethe lattieis periodi,

theprimitivetranslationvetorshavethesamevalueforeahlattiepoint.

Usingtheprimitivetranslationvetors,theseparationbetweeneverylattie

pointan be written as

L = l a 1 + m a 2 + n a 3

^{, where}

l

^,

m

^and

n

^{are integers}

tellinghowmanylattiesitesseparatethetwopoints. Sinetheinniterystal

looks thesameat point

r

andat point

r + L

, translation by

L

isasymmetry operation. Othersymmetryoperationsforrystalsarereetionsandrotations

aroundertain points oraxes in the lattie. Compound operationsonsisting

oftranslations,rotationsandreetionsthat aresymmetryoperationsarealso

ofourse symmetries. Translations like

L

that onsistof primitivetranslation vetorsarealled justtranslationvetors.

Using symmetries it an be seenthat only ertain types of lattie shapes

arepossible. Forexampleshapessymmetriunder rotationsof

2π

5

^radians,like

pentagons in two-dimensional ases, annot form innite latties. There are

only14dierentpossibleshapesof three-dimensionallatties. Theshapesare

howevernotimportanthere,sinetheaimofthisthesisistostudysolidswith

irregularstruture.

2.2 Reiproal spae

[6,hapter2℄Crystalsanbegivendierentkindsoflatties. Thelattieson-

strutedbytheprimitivetranslationvetorsarealledBravaislattiesordiret

latties, but for eah diret lattiethere is also a lattiealled the reiproal

lattie,whihistheFouriertransformationofthediret lattie.

Let

U ( r )

^{desribeaphysialpropertyof alattie. Sinethelattieissym-}

metriunder translationsof the form

r → r + l a ₁ + m a ₂ + n a ₃

,

U ( r )

^hasto

remainunhangedunderthesetransformations,so

U ( r ) = U ( r + l a 1 + m a 2 + n a 3 ) .

^(2.1)

ThustheFourierseriesof

U ( r ) X

b

U ^b e ^{i r · b} ,

^(2.2)

where

b

isavetor,isalsoleftunhangedunder thetransformation,so

X

b

U ^b e ^{i r · b} = X

b

U ^b e ^{i( r +l a 1 +m a 2 +n a 3 )· b} .

^(2.3)

l

^,

m

^and

n

^{arearbitraryintegers,sothismeansthat foralltheomponentsof}

b

b _i · a _j = 2πδ ij .

^(2.4)

Thesolutionstothetheseequationare

b _i = 2π a _j × a _k

a _i · a _j × a _k .

^(2.5)

Thevetors

b _i

arealledreiproalprimitivetranslations,andtheyformthe reiproallattie. Sumsofreiproalprimitivetranslationsarealledreiproal

translationsorreiproalvetors. Thereiproal spaemightseemlikeavery

abstrat onstrution, but diration patterns of rystals map the reiproal

spaeoftherystal.

As isseenfrom theform of

b _i

, thereiproalvetorsareorthogonalifand onlyifthetranslationvetors

a _i

are. Thedenitionalsoshowsthattheprimitive translations are inverse to the reiproal primitive translations, and thus the

diret lattie an be alled reiproal to the reiproal lattie. Other things

worth noting are that the lengths of the reiproal primitive translations are

b i = 2π/a i

^{andtheirdimensionis}

[length] ⁻¹

^.

[6,hapter2℄Thereareseveraldierentgeometriestohoosefromwhenforming

aelloraprimitive ellforalattie. Foraprimitiveell,the simplest hoie

istohoosealattiepointandthendrawthetranslationvetorsfrom it. The

parallelepipedformedbythevetorsistheprimitiveell. Ifotherlattiepoints

areontheboundariesoftheell,theyareexludedfrom theelland inluded

intheneighbouringells.

AnotherhoieistheWigner-Seitzell. Itonsistsofahosenlattiepoint

andallthepointsbetweenlattiepointsthatarelosertothehosenpointthan

to other lattie points. The easiest way to visualize it is to drawthe lattie,

thendrawstraightlinesfromasitetothenearestneighbouringsitesandtothe

middle pointsof these lines draw straightlines normal to the rst lines. The

areainsidethenormallinesistheWigner-Seitzell. Drawingaelllikethisto

eah lattiepointllsthewhole lattiewith nogapsnoroverlapping,just like

withthe simplerell hoie. What makestheell dierent from just drawing

thetranslationvetorsisthatthisellispreservedunderrotationandreetion

symmetriesofthelattie.

Drawingdierently shaped ells is not partiularly important for applia-

tions. The importane of the Wigner-Seitz ell is unveiled by drawing it in

thereiproal spae, forming what is alled the Brillouin zones. Drawing the

Wigner-Seitz ell in the reiproal spae using nearestneighbours of a lattie

sitegivestherstBrillouinzoneofthesite,usingtheseondnearestneighbour-

ingsitesgivestheseondBrillouinzoneandsoon. DierentBrillouinzonesare

importantindiration, but asshallbe shown, onlytherstBrillouin zoneis

neededwhenstudyingvibrationsofthelattie,whihiswhatweareinterested

inhere.

Themovementofatomsinmatterisamajorpartofondensedmatterphysis.

Consideringtheatomimovementofrystals,i.e. theeets ofexternalfores

and the thermalmovement of theatoms, leadsto dierent kindsof eets in

therystal. Thermalproperties andthe transport of heat and soundinside a

rystalarebasedontheatomivibrationsofthe rystal,andvibrationsaet

also eletri and magneti properties of materials. Large movement inside a

solidobjetleadsto thebreakingoftheobjet. [6℄

We will onsider low-temperaturevibrations, whih means vibrationsthat

donotbreakthesolidand wheretheatomsanalwaysbeassumedtobenear

theirequilibriumlattiesites. Vibrationsareusuallystudied usinggeneralized

oordinates, beause there are many atoms to onsider. Here it means using

Hamiltonianmehanis.

3.1 One dimensional lattie

[5, hapter 4.2℄ A good way to start is to onsider a one-dimensional hain

of idential atoms. After this, we will onsider a three-dimensional system.

We will assume that the hain is innitely long to simplify the system. The

assumption will not have muh of an eet on the result, sine real rystals

haveanastronomialamountofatoms.

Sine weare onsidering small vibrationsaroundthe equilibrium pointsof

theatoms, eah atom in the hain is aHarmoni osillator. Let the distane

betweentheatomsinthehainbe

a

^{,sothatthepositionofthenthatomis}

x n = na + u n

^{, where}

u n

^{is the}displaementof theatomfrom itsequilibrium point.

Now the potential energy of eah atom is

ζ

2 (u n − u n+1 ) ² + ζ

2 (u n − u n−1 ) ²

^,

where

ζ

^{is alled the elasti onstant and desribes spring tension between}

eah atom. Wegetthewhole Hamiltonian

H n

^{for oneatombyadding kineti}

energyto this,so

H n = p ² _n 2m + ζ

2 (u n − u n+1 ) ² + ζ

2 (u n − u n−1 ) ² ,

^(3.1)

where

p n

^{is themomentum ofthen:th atomand}

m

^isitsmass.

TheequationsofmotionarederivedsimplybyusingHamilton'sequations

˙

q = ∂H n

∂p

^(3.2)

and

p ˙ = − ∂H n

∂q

^(3.3)

where

p

^desribesthegeneralizedmomentaand

q

^desribesthegeneralizedo- ordinates. Now

p = p n

^and

q = u n

^{. Usingtherstequationweget}

˙ u n = ∂

∂p n

p ² _n 2m + ζ

2 (u n − u n+1 ) ² + ζ

2 (u n − u n−1 ) ²

= p n

m .

^(3.4)

˙

p n = − ∂

∂u n

p ² _n 2m + ζ

2 (u n − u n+1 ) ² + ζ

2 (u n − u n−1 ) ²

= − ζ (2u n − u n+1 − u n−1 ) .

(3.5)

TheformerofHamilton'sequationsgivestheexpression

p n = m u ˙ n

^,whihwhen

insertedintothelatteroneyieldstheequationofmotion

m¨ u n = − ζ (2u n − u n+1 − u n−1 )

^(3.6)

Sine

n

^{varies from}

−∞

^to

∞

^{, wegotan inniteset of equations. Seond}

orderordinarydierentialequationswith onstant oeientsare solvedwith

exponentials,so

u n = A n e ^−iωt

^,where

A n

^and

ω

^{areonstants,shouldbeagood}

trialsolution. Pluggingitinyields

− ω ² A n e ^−iωt = ζ

m ( − 2A n + A n+1 + A n−1 ) e ^−iωt

^(3.7)

Nextwewantanansatzfortheonstants

A n

^{. Againwewantan}exponential, soletus use

A n = Ae ^ikan

^{, where}

A

^{is aonstantdesribingthe amplitudeof}

thewaveand

k

^{istheamplitudeofawavevetor. Nowwegettheequation}

− ω ² Ae ^ikan e ^−iωt = ζ m

− 2Ae ^ikan + Ae ^ika(n+1) + Ae ^ika(n−1)

e ^−iωt

^(3.8)

Dividingby

Ae ^ikan e ^−iωt

^leadsto

− ω ² = ζ

m − 2 + e ^ika + e ^−ika

= 2ζ

m ( − 1 + cos(ka)) = − 4ζ m sin ²

ka 2

(3.9)

Henethefrequeny

ω

^dependson

k

^as

ω = 2

r ζ m

sin

ka 2

(3.10)

andthesolutionsoftheequationarethewaves

u n (k) = Ae ^ikan e ^−iω(k)t .

^(3.11)

Thesolutionofourseonlydesribesasinglewave. Wavesanbein superpo-

sition,sothefull solutiontotheequationsofmotionisalinearombinationof

thewaveswegot.

Exponentsmustbedimensionless,sothedimensionof

ω

^{must be}

[time] ⁻¹

^.

Thus

ω

^{desribes the frequeny of the wave.}

a

^{is the distane between the}

equilibrium points of atoms in the lattie, so its dimension is

[length]

^{. Thus}

thedimensionof

k

^is

[length] ⁻¹

^{,soitisawavevetorin thereiproalspaeof}

thelattie. Theveryimportantequation(3.10)tellingtherelationbetweenthe

frequenyandthewavevetorisalled thedispersionrelation. Italsotellsthat

thefrequeny of atomi vibrationsis proportional to

m ^−1/2

^{, and sineatoms}

areverylight,atomsmustvibratewithhugefrequenies.

u n (k) = Ae −iω(t−kan/ω)

(3.12)

andnotingthat

an

^{tellsthepositionofeahlattiesiteshowsthattheveloity}

of the waves is

v = ω/k

^{and the wavelength is}

λ = 2π/k

^{. When}

k

^{is small,}

elastiwavesdesribesoundpropagation. Forsmall k

ω = 2 r ζ

m

sin ka

2

≈ 2

r ζ m

ka 2 = ka

r ζ

m ,

^(3.13)

sotheveloityofthewaveis

v = a r ζ

m ,

^(3.14)

whihisin fattheveloityofsound inarystal.

3.1.1 Brillouin zonesof the1D model

[5, hapter 4.2℄ Analyzingthe dispersionrelation(3.10) showsthe importane

ofBrillouinzonesinlattiedynamis. Thevibrationalfrequeny

ω

^isafuntion

of

sin ka

2

,soitisperiodiin

k

^{. Morepreiselyallthevaluesof}

ω

^arefound

inside

− π

a < k ≤ π

a .

^(3.15)

Sine the distane between sites in the reiproal lattie is

2π

a

^{and the rst}

Brillouin zone of alattie siteis dened as theset of points in the reiproal

spae that are loserto the lattie site than to its neighbouring sites, we see

thatthevaluesof

k

^{in(3.15)formtherstBrillouinzoneofthelattiesite.}

Itisimportanttonotiethat allthevaluesofthewave

u n (k) = Ae ^ikan e ^−iω(k)t

^(3.16)

areontainedwithin therstBrillouinzone. Ifforexample

k > π

a

^or

k < − π a

^,

then

k − 2mπ

a

^{lieswithin therstBrillouinzoneforsomeinteger}

m

^,and

e i(k−2mπ/a)an = e ^ikan e ^−i2mπn = e ^ikan

^(3.17)

andthusthewave

u n (k)

^{getsthesamevaluesasawavedenedwithintherst}

Brillouinzone. Therefore onlyvaluesof

k

^{within therst Brillouin zoneneed}

tobeonsidered.

[5, hapter 4.2℄ Next we shall take a look at a one dimensional lattie with

twodierentmassatoms in theunit ell, in order tointroduesome onepts

and tehniquesthat will be useful later. Let the masses of the atoms be

m 1

and

m 2

^andlet

u n

^and

v n

^bethedisplaementsoftheatomsrespetively. The Hamiltoniandesribingthenthunit ellis

H n = p ² _u;n m 1

+ p ² _v;n 2m 2

+ ζ

2 (u n − v n−1 ) ² + ζ

2 (v n − u n ) ² + ζ

2 (u n+1 − v n ) ² .

^(3.18)

UsingHamilton'sequationsresultsinthedierentialequations

˙

u n = p u;n

m 1

(3.19)

˙

v n = p v;n

m 2

(3.20)

˙

p u;n = − ζ (2u n − v n − v n−1 )

^(3.21)

˙

p v;n = − ζ (2v n − u n − u n+1 ) ,

^(3.22)

whihyield theequationsofmotion

m 1 u ¨ n = − ζ (2u n − v n − v n−1 )

^(3.23)

and

m 2 ¨ v n = − ζ (2v n − u n − u n+1 ) .

^(3.24)

Togetawavelikesolution,wewillusethetrialfuntions

u n = U e ^−iωt e ^ikan

^(3.25)

v n = V e ^−iωt e ^ikan ,

^(3.26)

where

a/2

^{isthedistanebetweenthetwoatomsoftheell. Thisresultsin the}

equations

m 1 ω ² U = ζ 2U − V − V e ^−ika

(3.27)

and

m 2 ω ² V = ζ 2V − U − U e ^ika

.

^(3.28)

Itisusefultowrite theequationin thematrixform

m 1 ω ² − ζ ζ 1 + e ^−ika ζ 1 + e ^ika

m 2 ω ² − ζ

U V

= 0.

^(3.29)

Nowwedonothavetoatuallysolvetheequationsinordertogetthedispersion

relation. Thematrixequationhasnontrivialsolutionsonlyifitsdeterminantis

equaltozero,so

m 1 ω ² − ζ

m 2 ω ² − ζ

− ζ ² 1 + e ^−ika

1 + e ^ika

= 0.

^(3.30)

Theequationisquadratiin

ω ²

^{,soitresultsintwodierentdispersionrelations,}

whihare

ω ± ² = ζ (m 1 + m 2 ) ± q

ζ ² (m 1 + m 2 ) ² − 4m 1 m 2 ζ ² (1 + e ^−ika ) (1 + e ^ika ) m 1 m 2

= ζ

m 1 m 2

m 1 + m 2 ± s

(m 1 + m 2 ) ² − 4m 1 m 2 sin ² ka

2 !

.

^(3.31)

Therelation with aplussignis alled theoptial branh, and therelation

with a minus sign is alled the aoustibranh. The reason for these names

is seen by alulatingthe amplitudes

U

^and

V

^{of the waves. It is important}

to distinguishdierent branhes, beause dierentdispersionrelations leadto

dierent properties for the system. Plugging in thedispersion relation to for

exampletherstoftheequations(3.27)yields

m 1 ω ² _± U = ζ 2U − V − V e ^−ika

,

^(3.32)

whihresultsin

U ± = ζ 1 + e ^−ika

2ζ − ω ² _± m 1 V ± .

^(3.33)

Thistellshowtheamplitudesoftheatomswithdierentmassdependoneah

other.

Usingtheverylongwavelengthlimit

k = 0

^{in(3.31)leadsto}

ω ² ₊ = 2ζ

m 1 m 2

(m 1 + m 2 )

^(3.34)

and

ω ₋ ² = 0,

^(3.35)

andthus

U + = − m 2

m 1

V +

^(3.36)

and

U − = V − .

^(3.37)

As isseen,in theaoustibranh alltheatoms vibratein the samephase.

Thisisthereasonthebranhisalledaousti,sinewhensoundpassesthrough

amedium,alltheatomsinthemediumvibrateinphase. Fortheoptialbranh

the atoms vibrate in opposite phases instead, and the enter of mass of eah

unit ellremainsstill, sinethe amplitudefortheenter ofmassof aunit ell

wheretherstatom isatloationxis

m 1 U − x + m 2 V − (x + a/2) m 1 + m 2

= a

2 (m 1 + m 2 ) V − = constant.

^(3.38)

Thus theoptialbranh desribes,forexample,thepassingofeletromagneti

waves,andthenameoptialbranhissensible.

When

k

^{is inreased from zero, the expression inside the square root in}

(3.31)startstoderease,makingtheoptialbranh

ω −

^{smallerandtheaousti}

branh

ω +

^{larger. This is an essential property separating the two kinds of}

branhes from eah other. Optial branhesstart at aonstantin the entre

oftheBrillouin zonewhere

k = 0

^{andderease towardstheedgesof thezone,}

whereas aousti branhes start at zero and inrease towards the boundaries

of the Brillouin zone. Another important dierene is that optial branhes

havealwaysahigherfrequenythanaoustibranhes,asanbeseenfromthe

dispersionrelations.

The rest of the alulations in the thesis shall only onsider latties with

partilesthathaveequalmass,buttheoneptsofoptialandaoustibranhes

willremainrelevant.

3.2 Three dimensional latties

[5,hapter4.3℄Nextwewantto introduetheoneptsofthedensityofstates

andspei heatapaity. Inthisthesisweareinterestedin three-dimensional

solids,sowewill ontinue withathree-dimensionallattiefrom nowontoget

justthetoolsthat weneed.

Inthemodelthatwilllaterbeintroduedtodesribethebosonpeak,wewill

onlyonsider nearestneighbourinterations, so that is what we will do here.

Theboson peakis alsostudied onlyinverysmall temperatures,soweneedto

onsideronlysmallosillationsofatoms.

Consideralattiethatis symmetriunderthe hangeof anytwoaxesand

hasonlyoneatomineahunit ell. Lettheprimitivetranslationvetorsofthe

lattiebe

a ₁

,

a ₂

and

a ₃

. TheHamiltoniandesribingthesystemis

H l,m,n = p ² _u;l,m,n + p ² _v;l,m,n + p ² _w;l,m,n 2m

+ ζ

2 (u l,m,n − u l+1,m,n ) ² + ζ

2 (u l,m,n − u l−1,m,n ) ² + ζ

2 (v l,m,n − v l,m+1,n ) ² + ζ

2 (v l,m,n − v l,m−1,n ) ² + ζ

2 (w l,m,n − w l,m,n+1 ) ² + ζ

2 (w l,m,n − w l,m,n−1 ) ² ,

^(3.39)

where

u l,m,n

^,

v l,m,n ,

^and

w l,m,n

^arethedisplaementsoftheatomatthelattie site

l, m, n

^and

p u;l,m,n

^,

p v;l,m,n

^and

p w;l,m,n

^{are themomenta in the three di-}

retions. Thisisasumofthreeindependentonedimensionalhainsonsidered

intheprevioussubsetion. Thustheresultshouldalsobethreeinstanesofthe

dispersionrelationin(3.10).

Letusonsideroneofthediretions. UsingHamilton'sequations

˙

q = ∂H n

∂p

^(3.40)

and

p ˙ = − ∂H n

∂q

^(3.41)

for

q = u l,m,n

^and

p = p u;l,m,n

^{yieldstheequations}

˙

u l,m,n = p u;l,m,n

m

^(3.42)

and

˙

p u;l,m,n = − ζ (2u l,m,n − u l+1,m,n − u l−1,m,n ) ,

^(3.43)

whihwhenombinedyieldthefamiliarequationofmotion

m¨ u l,m,n = − ζ (2u l,m,n − u l+1,m,n − u l−1,m,n ) .

^(3.44)

Wegotthesameequation asin theonedimensional ase,exeptthat now

ourvariableshavethreeindiesinsteadofone. Fortheothertwodiretionswe

getsimilarlytheequations

m¨ v l,m,n = − ζ (2v l,m,n − v l,m+1,n − v l,m−1,n )

^(3.45)

and

m w ¨ l,m,n = − ζ (2w l,m,n − w l,m,n+1 − w l,m,n−1 ) .

^(3.46)

Letsnowtrythesolutions

u l,m,n ( k ) = U e ^iωt e ^{i k · L l,m,n ,}

^(3.47)

v l,m,n ( k ) = V e ^iωt e ^{i k · L l,m,n}

^{and (3.48)}

w l,m,n ( k ) = W e ^iωt e ^{i k · L l,m,n} ,

^(3.49)

where

U

^,

V

^and

W

^{aretheamplitudesofthewaveinthethreediretions,}

k = (k 1 , k 2 , k 3 )

^{isthewavevetor,}

L _l,m,n = l a ₁ +m a ₂ +n a ₃

isthetranslationvetor tellingtheloationofthelattiesite and

ω

^{isthefrequenyofthewave.}

Pluggingin thefuntionsyields

− ω ² U e ^iωt e ^{i k · L l,m,n} m = − ζ 2 − e ^{ik 1 a 1} − e ^{−ik 1 a 1}

U e ^iωt e ^{i k · L l,m,n} ,

^(3.50)

− ω ² V e ^iωt e ^{i k · L l,m,n} m = − ζ 2 − e ^{ik 2 a 2} − e ^{−ik 2 a 2}

V e ^iωt e ^{i k · L l,m,n}

^{and (3.51)}

− ω ² W e ^iωt e ^{i k · L l,m,n} m = − ζ 2 − e ^{ik 3 a 3} − e ^{−ik 3 a 3}

W e ^iωt e ^{i k · L l,m,n} .

^(3.52)

Solving

ω

^{from eah of these equations gives three of the dispersionrelations}

familiarfromtheonedimensional ase:

ω = 2 r ζ

m sin

a 1 k 1

2

,

ω = 2 r ζ

m sin

a 2 k 2

2

and

ω = 2 r ζ

m sin

a 3 k 3

2

.

^(3.53)

Thethree dispersionrelations represent three dierent branhes of vibrations

goingindierentdiretions. Ifawaveispassingthroughthelattieinthedire-

tionofoneoftheprimitivetranslationvetors,thenthebranhassoiatedwith

that diretion is alled the longitudinal branh, while the twoother branhes

arealledtransverse.

[5,hapter4.2℄Nextwewanttoalulateaquantityalledthedensityofstates,

but in order to do that we need to set boundary onditions for our system.

Thereisavarietyofdierentpossibleboundaryonditions,butwearelooking

for onditions that restrit the system only a little. This is reasonable sine

systemswithanenormousamountofatomsshouldnotdependtoomuhonthe

onditionsweset forspeiatoms.

We make the popular hoie of the Born and von Karman boundary on-

ditions, whih make the system periodially symmetri. This is mathemati-

ally very onvenient. Assume that the lattie repeats a formation of

N = N 1 × N 2 × N 3

^{atoms. Thismeansthatforthewavesat thesite}

l, m, n

u l,m,n = u l+N 1 ,m,n = u l,m+N 2 ,n = u l,m,n+N 3 .

^(3.54)

The larger

N 1

^,

N 2

^and

N 3

^{are, the less the onditions restritthe system. If}

wehavethreesolutions

u l,m,n

^,

v l,m,n

^and

w l,m,n

^{forthethreediretions,weset}

thesameonditionsalsofor

v l,m,n

^and

w l,m,n

^.

Theonditionsetsarestritionon

k

. If

u l,m,n ( k ) = U e ^iωt e ^{i k · L}

^(3.55)

asitwasearlier,then

u l+N 1 ,m,n ( k ) = U e ^iωt e ^{i k · L +ia 1 k 1 N 1} = u l,m,n ( k ) = U e ^iωt e ^{i k · L}

^(3.56)

andthus

e ^{ia 1 k 1 N} = 1.

^(3.57)

Thismeansthat

k 1 = 2π

a 1 N g 1 ,

^(3.58)

where

g 1 = 1, ..., N 1 .

^Therefore

k 1

^hasonly

N 1

^{possiblevalues. Similarly}

k 2 = 2π

a 2 N g 2

^,

g = 1, ..., N 2

^(3.59)

and

k 3 = 2π

a 3 N g 3

^,

g 3 = 1, ..., N 3 ,

^(3.60)

and

k 2

^and

k 3

^haveonly

N 2

^and

N 3

^{possiblevalues.}

Thusthewholewavevetor

k

has

N 1 N 2 N 3

^{possiblevalues.}

2π a 1

,

2π a 2

and

2π a 3

arethelengthsofthereiproalprimitivetranslationvetors. Letus allthese

vetorsagain

b ₁

,

b ₂

and

b ₃

,so

k = k ₁ + k ₂ + k ₃ = g 1

N 1

b ₁ + g 2

N 2

b ₂ + g 3

N 3

b ₃ .

^(3.61)

Sine

g 1

^,

g 2

^and

g 3

^{take disrete values, we see that the expression for}

k

formsthe reiproal lattie, andthe lattiepoints mustbedistributed evenly.

Thus using the fat that there are

N 1 N 2 N 3

^{dierent valuesfor}

k

we see that eahvaluetakesthespae

∆k = V B

N 1 N 2 N 3

(3.62)

in the Brillouin zone, where

V B

^{is the volume of the zone. Let us express}

thisusingvolumesinthediretlattie. Comparingtotheonedimensionalase

showsthatthevolumeis

V B = (2π) ³ /V C

^,where

V C

^{isthevolumeofthephysial}

unitell,beausethelattieonsistsofthree onedimensionalhains. Sinewe

startedbyassumingthatthewholerystalrepeatsaformationof

N 1 × N 2 × N 3

atoms,

V C N 1 N 2 N 3 = V

^{mustgivethevolumeofthewhole rystaland}

∆k = (2π) ³

V .

^(3.63)

Thisvolumeisneededtosolvethedensityofstates.

3.2.2 Densityof state

[5, hapter 4.3.5℄After derivingthedispersionrelation,i.e. after showinghow

thefrequenyof atomiosillationsdepends onthewavevetorof theosilla-

tions,anaturalquestionistoaskwhatistheamountofosillationsinaspei

frequeny. Thisisalledthephonondensitystates.

The name phonon omes from the normal modes of the osillations. The

nameis inanalogyto photons,whihare thequantaof vibrationsoftheele-

tromagnetield,beausephonons arethequanta ofatomi vibrations. Thus

thequestionishowmanyphononsarethereofaspeifrequeny. Thederiva-

tionofthe normalmodesorphononsis aninteresting alulation,but itdoes

notbring any relevant tehniques to the main alulation of this thesis, so it

willnotbeshownhere.

If

g (ω)

^{isthedensityofstatestellingthefrequeny}distributionoftheosil- lations,then

g (ω) dω

^{tellsthenumberofphononswhosefrequeniesarebetween}

ω

^and

ω + dω

^{. Wewillderive}

g (ω)

^{byrstalulating}

g (ω) dω

^.

Considerthesurfae

S ω

^{inthereiproalspaedenedby}

ω = ω ^k = constant

foraspeibranhofthevibrations. Dierentbranhesgivedierentsurfaes,

sinetheyhaveadierentdispersionrelation.

dS ω dk ⊥

^givestheinnitesimalof thevolumeontainingthevibrationswith afrequenybetween

ω

^and

ω + dω

^,

where

dk ⊥

^isthepartof

d k

that isperpendiularto

S ω

^.

Sineeahvalueof

k

andthuseahvalueof

ω

^takesthespae

∆k = (2π) ³ /V

inthereiproalspae,there mustbe

dS ω dk ⊥

∆k = V

(2π) ³ dS ω dk ⊥

^(3.64)

phononsin thevolume

dS ω dk ⊥

^. Integratingoverthesurfaegivesthenumber ofphononsinthefrequenyrange

[ω, ω + dω]

g (ω) dω = V (2π) ³

Z Z

ω=ω k

dS ω dk ⊥

^(3.65)

This is not yet the resultwewant.

ω

^{is afuntion of}

k

and growsin the diretionperpendiularto

S ω

^,so

dω = |∇ ^k ω ^k | dk ⊥

^,and

g (ω) dω = V (2π) ³

Z Z

ω=ω k

dω

|∇ ^k ω ^k | dS ω ,

^(3.66)

sothedensityofstatesis

g (ω) = V (2π) ³

Z Z

ω=ω k

dS ω

|∇ ^k ω ^k | .

^(3.67)

Forthespeialasewhenthedispersionrelationisisotropithereisasimpler

way to alulate

g (ω)

^{. Isotropymeans that the surfaedened by}

ω = ω ^k = constant

^{isasphere. If}

V ω

^{isthevolumeofthesphere,thereare}

V V ω

(2π) ³

(3.68)

phonons with frequeny

ω ^k

^{or less. Thus when the radius of the sphere is}

inreasedby

dω

^{,theamountofphononsinside thespheregrowsby}

V

(2π) ³ dV ω

dω dω,

^(3.69)

whihmeansthatthefrequenydistribution,ordensityofstates,at

ω ^k

^is

g (ω) = V

(2π) ³ dV ω

dω .

^(3.70)

Ifthe dispersionrelationwasnot isotropi,its dependene on diretion would

haveto be onsidered when the volume inside the

ω = constant

^surfaewas

inreased,sothedensityofstateswouldneedtodependonthegradient

∇ ^k ω ^k

^.

3.2.3 VanHove singularities

[7, hapter 6℄ Sine the integral in the formula for the density of states has

adenominator, it also ould have singularities at

∇ ^k ω ^k = 0.

^The singularity points an be minima, maxima or saddle points of the frequeny. Let

k ₀

be suhapoint. Expanding

ω

^{upto theseond orderaround}

k ₀

gives

ω = ω 0 + X

i,j

1

2 C ij (k i − k 0,i ) (k j − k 0,j ) ,

^(3.71)

where

ω 0 = ω ( k ₀ )

^{. The oeients}

C ij

^{form a matrix, sowe anhoose the}

axesof the

k

spaealong theprinipal axes ofthe matrix,so that the matrix beomesdiagonaland

ω = ω 0 + 1 2

X

λ

C λ (k λ − k 0,λ ) ² .

^(3.72)

Thelast hange wewill do to this expression for

ω

^{is the hange of variables}

χ λ = p

| C λ | (k λ − k 0,λ )

^,so

ω = ω 0 + 1 2

X

λ

sgn

(C λ ) χ ² _λ ,

^(3.73)

wheresgn

(C λ )

^isthesignof

C λ .

Now theform is easier toanalyze. There are three possiblesigns sgn

(C λ )

anddierent hoiesof themgivedierenttypesofritial pointsfor

ω

^{. Ifall}

thesignsarepositive,

ω

^{hasaminimumat}

k ₀

and theisofrequenysurfaes in

χ

^{spae arespheres. Thesurfaeelementof thesphereis}

dS ω = dS ω,χ / √ C = χ ² dΩ/ √

C

^{, where}

dΩ

^{is the dierential of the solid angle and}

1/ √

C

^{is the}

Jaobianfrom thehangeofvariables. Thegradient

∇ χ ω χ = χ

^{,so thedensity}

ofstatesis

g (ω) = V (2π) ³ √

C Z Z

ω=ω k

χ ² dΩ

| χ | = g (ω 0 )+ V 2π ² √

C | χ | = g (ω 0 )+ V π ² √

2C

√ ω − ω 0 .

(3.74)

for

ω ≥ ω 0

^and

g (ω) = g (ω 0 )

^for

ω ≤ ω 0

^{. After the hange of variables to}

χ

^{-spaeweouldalsohaveusedtherule(3.70)forisotropidispersionrelations}

yieldingthesameresult.

Theasewhereallthesignssgn

(C λ )

^{arenegativeissimilar,butitdesribes}

amaximumof

ω

^{,andthedensityofstatesis}

g (ω 0 ) + V

4π ² √ C

√ ω 0 − ω

^(3.75)

for

ω ≤ ω 0

^and

g (ω) = g (ω 0 )

^for

ω ≥ ω 0

^.

Ifoneofthesignssgn

(C λ )

^{isdierentfromtheothertwo,thereisasaddle}

pointat

k ₀

. Takeasanexamplethease

ω = ω 0 + 1

2 χ ² ₁ + χ ² ₂ − χ ² ₃

.

^(3.76)

Theonstantfrequenysurfaesare nowhyperboloids. Inthease

ω < ω 0

^the

hyperboloid has two sheets and in the ase

ω > ω 0

^the hyperboloid has one sheet. Theintegralis mosteasilydone inylindrialoordinates

(χ ⊥, φ, χ 3 ) ,

where

χ ⊥ = p

χ ² ₁ + χ ² ₂

^and

φ

^{isthepolarangleinthe}

χ 1 χ 2

^{plane. When}

ω < ω 0

dS ω = 2πχ ⊥ χ

| χ 3 | dχ ⊥

^(3.77)

andagain

|∇ χ ω χ | = 2χ

^,and

g (ω) = 2

(2π) ³ √ C

Z K 0

2πχ ⊥ dχ ⊥

p χ ² _⊥ + 2 (ω 0 − ω) = 1 2π ² √

C

K − p

2 (ω 0 − ω)

= g (ω 0 ) − 1 π ² √

2C

√ ω 0 − ω,

^(3.78)

where

K

^{issomeonstant. Thease}

ω > ω 0

^yields

g (ω) = 2 (2π) ³ √

C Z K

2(ω−ω 0 )

2πχ ⊥ dχ ⊥

p χ ² _⊥ − 2 (ω 0 − ω) = K 2π ² √

C = g (ω 0 ) .

^(3.79)

Theasewheretwoofthesignsarenegativeandonepositivegivesasimilar

result,exeptthat

g (ω) = g (ω 0 ) − 1 π ² √

2C

√ ω 0 − ω

^(3.80)

for

ω > ω 0

^and

g (ω) = g (ω 0 )

^(3.81)

for

ω < ω 0

^.

3.3 Heat apaity

[5, hapter 4.5.3℄ Heat apaity isone of thepropertieswhih separate amor-

phous solids from rystals. There are several models for the heat apaity of

rystals,whihapplytodierentsituations,andwewillintroduetwoofthem,

sothatwean omparetheresultsforamorphousmaterialstothem.

Theheatapaity

C

^{ofanobjettellshowmuhtheinternalenergy}

U

^ofthe

objethanges whenthetemperate

T

^{of theobjethanges, orin otherwords}

howmuh energyisneededto hange thetemperatureofthe objetaspei

amount. Wewillusethedenition

C V = ∂U

∂T

V

(3.82)

forthe heat apaity

C V

^{, in whih wealulate the hange}

∂U/∂T

^{when the}

volume of the system

V

^{is kept onstant. Another way to dene the heat}

apaityistokeepthepressureofthesystemonstant,andthisisthedenition

usedusuallyinexperiments. [6℄

Theenergylevelsof aharmoniosillator vibratingatfrequeny

ω

^are

1

2 + n

ω, n ∈ N .

^(3.83)

Thevibrationsofarystalareasumofharmoniosillators,orphonons,indif-

ferentfrequenies,sotheinternalenergyintherystalausedbythevibrations

is

U = X

k j

1 2 + n ^k j

ω ^k j ,

^(3.84)

where

n ^k j

^{is thenumberof phonons withwavenumber}

k

in branh

j

^{. There}

isnoPauliexlusionprinipleforbosons,so

n ^k j

^{is givenbythe}Bose-Einstein distribution

n ^k j = 1

e ^ω/T − 1 .

^(3.85)

tegral,if weknowthetotaldensityofstates

g (ω)

^{ofthesystem,that inludes}

dierentbranhes:

U = Z

dωg (ω) 1

2 + 1

e ^ω/T − 1

ω

^(3.86)

Nowtakingthederivativewithrespetto

T

^weget

C V =

Z

dωg (ω)

ω/T e ^ω/T − 1

2

e ^ω/T .

^(3.87)

Thehightemperaturelimitiseasytoalulate.UsingL'Hpital'sruletwiewe

seethat

C V = Z

dωg (ω)

ω/T e ^ω/T − 1

2

e ^ω/T → Z

dωg (ω) − ω ² /T ² − 2ω/T

− e ^ω/T 2 + 2

→ Z

dωg (ω) = 3N n,

^(3.88)

whihisthenumberofpossibleosillationsin athree-dimensionalrystalof

N

primitiveellswith

n

^{atoms. Animportant remarkis thataoustivibrations}

havenothingto dowith thenumberof atomsin aunit ell,so thenumberof

aoustibranhesdependsonlyon

N

^{. Indeedthereare}

3N

^{aoustibranhesin}

athree-dimensionalrystal,sotheremaining

3N (n − 3)

^{vibrationsareoptial.}

Next,weshall onsidertwomodels that desribetheontributionsof aousti

andoptialwavestotheheatapaity.

3.3.1 Debyemodel

[5, hapter4.5.3℄Inlowtemperaturesthere areonlylowenergyexitations,so

the wavenumbers of osillations must be small. As was shown in subsetion

(3.1.2) there are only aousti waves in the

k = 0

^{limit in} one-dimensional systems,butsimplethreedimensionalsystemsarenodierentsinetheyonsist

ofonedimensionalhains. Thusweexpettondonlylongwavelengthaousti

wavesatverylowtemperatures.

Considerthe three dimensional dispersionrelations in (3.53). Taking now

thelongwavelengthlimit

| k | ≪ 1

^gives

ω j =

r ζ

m a j | k | = v j | k | ,

^(3.89)

foreahofthethreebranhes

j

^,where

v j

^{istheveloityofsoundineahofthe}

branhes,asin(3.14)intheonedimensionalase. Nowthatthethreebranhes

orrespondtothreeorthogonaldiretionsinthelattie,eah

v j

^{tellstheveloity}

ofsoundinadierentdiretion.

Theisofrequenysurfaesinthereiproalspaeofeahofthebranhesare

sphereswith radius

k j = ω/v j

^{, and the gradients}

∇ ^k ω ^k ,j = v j ˆ k

, so using the

g j (ω) = V (2π) ³

Z Z

ω=ω k

dS ω

v j

= V ω ²

2π ² v _j ³ .

^(3.90)

Addingtogetherthebranhesweget

g (ω) = V ω ² 2π ²

2 v ³ _⊥ + 1

v ³ _k

!

= 3V ω ²

2π ² v ³ _{ef f}

^(3.91)

where

2/v ⊥

^{standsforthetwotransversebranhes,}

1/v k

^{omesfromthelongitu-}

dinalbranh,and

v ef f

^{istheeetivesoundveloitythatomesfromombining}

thebranhes.

Before inserting the above expression of

g (ω)

^{to the formula for the heat}

apaity,weneedtoaddsomethingtoourmodel. Nowthedensityofstatesonly

growswiththefrequeny,resultingin aninniteamountofstatesifintegrated

to innity. We however want the maximum amount of states from aousti

osillationstobe

3N

^{,sowewill introdueautofrequeny}

ω D

^forwhih

Z ω D

0

g (ω) dω = 3N.

^(3.92)

ω D

^{is alled theDebyefrequeny. Insertingthedensityof states(3.91) to the}

aboveintegralshowsthat

V ω ³ _D

2π ² v _{ef f} ³ = 3N,

^(3.93)

sothat

ω D = v ef f

6N π ² V

1/3

= v ef f

6π ² V C

1/3

,

^(3.94)

where

V C

^{is againthevolume ofaunit ell. TheDebyefrequenyalsodenes}

atemperaturealledtheDebyetemperature,whihis

T D = ~ ω D /k B

^{, butnow}

thatwehaveset

~ = k B = 1

^{itisequivalenttothefrequeny.}

Usingtheuto theheat apaityis

C V = Z ω D

0

dω 3V ω ² 2π ² v _{ef f} ³

ω/T e ^ω/T − 1

2

e ^ω/T

^(3.95)

Openingthe squareand writingtheonstantusing theDebye frequenyleads

to

C V = 9N T ² ω _D ³

Z ω D

0

dω ω ⁴

e ^ω/T − 2 + e ^−ω/T

= 9N

4T ² ω ³ _D Z ω D

0

dω ω ⁴ sinh ² (ω/2T )

= 72N T ³ ω ³ _D

Z ω _D /2T 0

dx x ⁴

sinh ² x ,

^(3.96)

Figure3.1: TheDebyemodelheatapaitysaledas

C V

3N

^asafuntionof

T ω D

where

x = ω/2T.

^{ThisistheDebyemodel fortheheatapaityofarystal.}

Inthelowtemperaturelimit

ω D /2T → ∞

^{. Taking theupperboundof the}

integraltoinnityresultsin

C V = 72N T ³ ω ³ _D

π ⁴

30 = 12π ⁴ N T ³

5ω ³ _D .

^(3.97)

The

T ³

proportionalityisindeedingoodagreementwithexperimentsforseveral rystals[5℄.

An interesting thing is that the Debye model is also orret in the high

temperaturelimit. Inhightemperatures

ω/T ≪ 1

^,so

sinh (ω/T ) ≈ ω/T

^and

C V = 72N T ³

ω _D ³

Z ω D /2T 0

dxx ²

= 3N,

^(3.98)

whihisexatlythemaximumamountofaoustistates.

Theplotofthewhole funtion

C V

^{isin gure(3.1).}

3.3.2 Einsteinmodel

[5,hapter4.5.3℄Asimplemodeldesribingtheontributionofoptialwavesto

theheatapaityistheEinsteinmodel. Assumethatthereare

3N (n − 3)

^opti-

albranhesandtheyallhavethesamefrequeny

ω E

^,so

g (ω) = 3N (n − 3) δ (ω − ω E )

^.

Nowtheheat apaity

C V = Z

dωg (ω)

ω/T e ^ω/T − 1

2

e ^ω/T = 3N (n − 3)

ω E /T e ^{ω E /T} − 1

2

e ^{ω E /T} .

(3.99)

Inthehigh temperaturelimit

e ^{ω E /T} ≈ 1 + ω E /T

^,so

C V ≈ 3N (n − 3) ω E

T + 1

→ 3N (n − 3) as T → ∞ ,

^(3.100)

whihisindeedtheontributionoptialwavesshouldhavesothatthemaximum

amountofvibrationalmodesis

3N n

^{. Inlow}temperatures

C V ≈ 3N (n − 3) ω _E ² /T ²

e ^{ω E /T} → 0 as T → 0,

^(3.101)

so aording to this model optial waves should not ontribute to the heat

apaityin lowtemperatures. Thus when wedisuss ourmodel fortheboson

peakweshouldompareittotheDebyemodelandexpetthepeaktobereated

byaoustimodes.

Theaimofthisthesisistointrodueadisorderedlattie-likemodelforglasses,

eventhoughglassesdonotresemblealattie. Thetrikistomakepartsofthe

lattienonommutative,whihwillresultinanunertaintyforloation,similar

to the unertaintyof anonial variables in quantum mehanis, thus making

thelattielessxed. OurmodelisbasedontheworksofPolyhronakos,Jakiw,

Pi,Susskindandothers[2831℄onnonommutativeuids. Wewilloversome

ofthenonommutativeuidtheoryshortlyafterintroduingtheformalism.

Thenonommutativityin uidtheorymeansusingthenonommutativege-

ometry greatly developed by Alain Connes. In physis, nonommutative ge-

ometryhasmostly beentied tothe studyof spae-time, but thelanguagehas

found its way to ondensed matter theory too. Nonommutative spaes have

been used for quantizing spaetime sineSnyder's Quantized Spae-Time [32℄

andsuhgeometriesarisefromertainlimitsinstringtheory[33℄. Inondensed

matternonommutativityhasbeenusedforstudying,forexample,thequantum

Halleet[31℄andinsulators[34℄.

4.1 Nonommutative geometry

Fairlylittlenonommutativegeometryisneededinthisthesis,butitisreason-

able to givea short introdution to the subjet. A good introdution would

needtoberatherlengthy,sowewillnotproveresultsandtheintrodutionwill

beleftabitabstrat.

The bakground of nonommutative geometry is in C*-algebras. A C*-

algebraisaunitalBanahalgebraon

C

denedwithaninvolution

x → x ^∗

^s.t.

k x k ² = k x ^∗ x k .

^{Thesquareofthenorm}

k x k ²

^{isthespetralradiusi.e. thelargest}

absolutevalueoftheeigenvaluesof

x

^{,soitisaspetralproperty. Asusual,an}

element

x

^is self-adjoint if

x ^∗ = x

^{and unitary if}

x ⁻¹ = x ^∗

^{. C* algebrasare}

importantbothinlassialandquantum mehanis.

In physis the spae of the elements

x

^{is the phase spae. The Gelfand-}

Naimarktheoremgivesanequivalene betweenC*-algebrasand thegeometry

ofthephasespaeandthusgivesageometriapproahtothealgebra[35℄. The

theoremisanimportantstartingpointinnonommutativegeometry.

Thebasielementsinnonommutativegeometryareobservablesasinquan-

tum mehanis. The observablesobeyanonommutativeC*-algebra, and the

irreduiblerepresentations ofthe algebraform anonommutativespae, so in

thephysisontextwearetalkingofnonommutativeC*-algebrasoperatingon

Hilbertspaes.

Nonommutative geometry gives a ommon language and unies various

topisin physis. Inadditionto beingusefulin spae-timeand theondensed

mattersubjetsjust mentioned,theformalismuniesforexamplegaugeelds,

membranes,matrixmodelsandmany-bodysystems[28℄.