Table Of ContentTitelei Kossevich 17.06.2005 9:44 Uhr Seite 3
Arnold M. Kosevich
The Crystal Lattice
Phonons, Solitons, Dislocations, Superlattices
Second,Revised and Updated Edition
WILEY-VCH Verlag GmbH & Co.KGaA
Titelei Kossevich 17.06.2005 9:44 Uhr Seite 4
Author All books published by Wiley-VCH are carefully
produced. Nevertheless, authors, editors, and
Arnold M.Kosevich publisher do not warrant the information contained in
B. Verkin Institute for Low Temperature Physics and these books, including this book, to be free of errors.
Engineering Readers are advised to keep in mind that statements,
National Academy of Sciences of Ukraine data, illustrations, procedural details or other items
310164 Kharkov, Ukraine may inadvertently be inaccurate.
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© 2005 WILEY-VCH Verlag GmbH & Co. KGaA,
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ISBN-13: 978-3-527-40508-4
ISBN-10: 3-527-40508-9
Contents
Prefaces IX
Part1 Introduction 1
0 GeometryofCrystalLattice 3
0.1 TranslationalSymmetry 3
0.2 BravaisLattice 5
0.3 TheReciprocalLattice 7
0.4 UseofPenetratingRadiationtoDetermineCrystalStructure 10
0.4.1 Problems 12
Part2 ClassicalDynamicsofaCrystalLattice 15
1 MechanicsofaOne-DimensionalCrystal 17
1.1 EquationsofMotionandDispersionLaw 17
1.1.1 Problems 23
1.2 MotionofaLocalizedExcitationinaMonatomicChain 24
1.3 TransverseVibrationsofaLinearChain 29
1.4 SolitonsofBendingVibrationsofaLinearChain 33
1.5 DynamicsofBiatomic1DCrystals 36
1.6 Frenkel–KontorovaModelandsine-GordonEquation 39
1.7 SolitonasaParticlein1DCrystals 43
1.8 HarmonicVibrationsina1DCrystalContainingaCrowdion(Kink) 46
1.9 MotionoftheCrowdioninaDiscreteChain 49
1.10 PointDefectinthe1DCrystal 51
1.11 HeavyDefectsand1DSuperlattice 54
2 GeneralAnalysisofVibrationsofMonatomicLattices 59
2.1 EquationofSmallVibrationsof3DLattice 59
VI Contents
2.2 TheDispersionLawofStationaryVibrations 63
2.3 NormalModesofVibrations 66
2.4 AnalysisoftheDispersionLaw 67
2.5 SpectrumofQuasi-WaveVectorValues 70
2.6 NormalCoordinatesofCrystalVibrations 72
2.7 TheCrystalasaViolationofSpaceSymmetry 74
2.8 Long-WaveApproximationandMacroscopicEquationsforthe
DisplacementsField 75
2.9 TheTheoryofElasticity 77
2.10 VibrationsofaStronglyAnisotropicCrystal(ScalarModel) 80
2.11 “Bending”WavesinaStronglyAnisotropicCrystal 83
2.11.1 Problem 88
3 VibrationsofPolyatomicLattices 89
3.1 OpticalVibrations 89
3.2 GeneralAnalysisofVibrationsofPolyatomicLattice 94
3.3 MolecularCrystals 98
3.4 Two-DimensionalDipoleLattice 101
3.5 OpticalVibrationsofa2DLatticeofBubbles 105
3.6 Long-WaveLibrationalVibrationsofa2DDipoleLattice 109
3.7 LongitudinalVibrationsof2DElectronCrystal 112
3.8 Long-WaveVibrationsofanIonCrystal 117
3.8.1 Problems 123
4 FrequencySpectrumandItsConnectionwiththeGreenFunction 125
4.1 Constant-FrequencySurface 125
4.2 FrequencySpectrumofVibrations 129
4.3 AnalysisofVibrationalFrequencyDistribution 132
4.4 DependenceofFrequencyDistributiononCrystalDimensionality 136
4.5 GreenFunctionfortheVibrationEquation 141
4.6 RetardingandAdvancingGreenFunctions 145
4.7 RelationBetweenDensityofStatesandGreenFunction 147
4.8 TheSpectrumofEigenfrequenciesandtheGreenFunctionofaDeformed
Crystal 149
4.8.1 Problems 151
5 AcousticsofElasticSuperlattices: PhononCrystals 153
5.1 ForbiddenAreasofFrequenciesandSpecificDynamicStatesinsuch
Areas 153
5.2 AcousticsofElasticSuperlattices 155
5.3 DispersionRelationforaSimpleSuperlatticeModel 159
5.3.1 Problem 162
Contents VII
Part3 QuantumMechanicsofCrystals 163
6 QuantizationofCrystalVibrations 165
6.1 Occupation-NumberRepresentation 165
6.2 Phonons 170
6.3 Quantum-MechanicalDefinitionoftheGreenFunction 172
6.4 DisplacementCorrelatorandtheMeanSquareofAtomic
Displacement 174
6.5 AtomicLocalizationneartheCrystalLatticeSite 176
6.6 QuantizationofElasticDeformationField 178
7 InteractionofExcitationsinaCrystal 183
7.1 AnharmonicityofCrystalVibrationsandPhononInteraction 183
7.2 TheEffectiveHamiltonianforPhononInteractionandDecay
Processes 186
7.3 InelasticDiffractiononaCrystalandReproductionoftheVibration
DispersionLaw 191
7.4 EffectofThermalAtomicMotiononElasticγ-Quantum-Scattering 196
7.5 EquationofPhononMotioninaDeformedCrystal 198
8 QuantumCrystals 203
8.1 StabilityConditionofaCrystalState 203
8.2 TheGroundStateofQuantumCrystal 206
8.3 EquationsforSmallVibrationsofaQuantumCrystal 207
8.4 TheLong-WaveVibrationSpectrum 211
Part4 CrystalLatticeDefects 213
9 PointDefects 215
9.1 Point-DefectModelsintheCrystalLattice 215
9.2 DefectsinQuantumCrystals 218
9.3 MechanismsofClassicalDiffusionandQuantumDiffusionof
Defectons 222
9.4 QuantumCrowdionMotion 225
9.5 PointDefectinElasticityTheory 227
9.5.1 Problem 232
10 LinearCrystalDefects 233
10.1 Dislocations 233
10.2 DislocationsinElasticityTheory 235
10.3 GlideandClimbofaDislocation 238
10.4 Disclinations 241
10.5 DisclinationsandDislocations 244
10.5.1 Problems 246
VIII Contents
11 LocalizationofVibrations 247
11.1 LocalizationofVibrationsnearanIsolatedIsotopeDefect 247
11.2 ElasticWaveScatteringbyPointDefects 253
11.3 GreenFunctionforaCrystalwithPointDefects 259
11.4 InfluenceofDefectsontheDensityofVibrationalStatesinaCrystal 264
11.5 Quasi-LocalVibrations 267
11.6 CollectiveExcitationsinaCrystalwithHeavyImpurities 271
11.7 PossibleRearrangementoftheSpectrumofLong-WaveCrystal
Vibrations 274
11.7.1 Problems 277
12 LocalizationofVibrationsNearExtendedDefects 279
12.1 CrystalVibrationswith1DLocalInhomogeneity 279
12.2 Quasi-LocalVibrationsNearaDislocation 283
12.3 LocalizationofSmallVibrationsintheElasticFieldofaScrew
Dislocation 285
12.4 FrequencyofLocalVibrationsinthePresenceofaTwo-Dimensional
(Planar)Defect 288
13 ElasticFieldofDislocationsinaCrystal 297
13.1 EquilibriumEquationforanElasticMediumContainingDislocations 297
13.2 StressFieldActiononDislocation 299
13.3 FieldsandtheInteractionofStraightDislocations 303
13.4 ThePeierlsModel 309
13.5 DislocationFieldinaSampleofFiniteDimensions 312
13.6 Long-RangeOrderinaDislocatedCrystal 314
13.6.1 Problems 319
14 DislocationDynamics 321
14.1 ElasticFieldofMovingDislocations 321
14.2 DislocationsasPlasticityCarriers 325
14.3 EnergyandEffectiveMassofaMovingDislocation 327
14.4 EquationforDislocationMotion 331
14.5 VibrationsofaLatticeofScrewDislocations 336
Bibliography 341
Index 343
Prefaces
PrefacetotheFirstEdition
Thedesignofnewmaterialsisoneofthemostimportanttasksinpromotingprogress.
Todothisefficiently,thefundamentalpropertiesofthesimplestformsofsolids,i.e.,
singlecrystalsmustbeunderstood.
Notsolongago,materialsscienceimpliedthedevelopment,experimentalinvesti-
gation,andtheoreticaldescription,ofprimarilyconstructionmaterialswithgivenelas-
tic, plastic andresistive properties. Inthelast fewdecades, however,newmaterials,
primarilycrystalline,havebeguntobevieweddifferently:asfinished,self-contained
devices.Thisisparticularlytrueinelectronicsandoptics.
Tounderstandthepropertiesofacrystaldeviceitisnotonlynecessarytoknowits
structurebutalsothedynamicsofphysicalprocessesoccurringwithinit.Forexample,
to describethesimplest displacementof the crystalatomsalreadyrequiresa knowl-
edge of the interatomic forces, which of course, entails a knowledge of the atomic
positions.
Thedynamicsofacrystallatticeisapartofthesolid-statemechanicsthatstudies
intrinsiccrystalmotionstakingintoaccountstructure. Itinvolvesclassicalandquan-
tummechanicsofcollectiveatomicmotionsinanidealcrystal,thedynamicsofcrystal
latticedefects,atheoryoftheinteractionofarealcrystalwithpenetratingradiation,
thedescriptionofphysicalmechanismsofelasticityandstrengthofcrystalbodies.
Inthis booknew trendsin dislocationtheoryandan introductionto the nonlinear
dynamicsof1Dsystems, thatis, solitontheory,arepresented. Inparticular,thedis-
locationtheoryofmeltingof2Dcrystalsisbrieflydiscussed. Wealsoprovideanew
treatmentoftheapplicationofcrystallatticetheorytophysicalobjectsandphenomena
whoseinvestigationbeganonlyrecently,thatis,quantumcrystals,electroncrystalson
aliquid-heliumsurface,latticesofcylindricalmagneticbubblesinthin-filmferromag-
netics,andsecondsoundincrystals.
In thisbookwe treatin a simple way, notgoinginto detailsof specific cases, the
fundamentalsofthephysicsofacrystallinelattice. Tosimplifyaquantitativedescrip-
X Prefaces
tionofphysicalphenomena,asimple(scalar)modelisoftenused. Thismodeldoes
notreducethegeneralityofqualitativecalculationsandallowsustoperformalmost
allquantitativecalculations.
ThebookiswrittenonthebasisoflecturesdeliveredbytheauthorattheKharkov
University(Ukraine). Theprerequisitesforunderstandingthismaterialareageneral
undergraduate-levelknowledgeoftheoreticalphysics.
Finally,asauthor,Iwouldliketothankthemanypeoplewhohelpedmeduringthe
workonthemanuscript.
I am pleased to expressgratitudeto ProfessorPaul Zieschefor his idea to submit
the manuscriptto WILEY-VCH for publication, and for his aid in the realization of
thisproject.
IamdeeplyindebtedtoDr. SergeyFeodosievforhisinvaluablehelpinpreparinga
camera-readymanuscriptandimprovingthepresentationofsomepartsofthebook.I
amgratefultoMariaMamaluiandMariaGvozdikovafortheirassistanceinpreparing
thecomputerversionof themanuscript. I wouldlike to thankmywife Dinaforher
encouragement.
I thank Dr. Anthony Owen for his careful reading of the manuscript and useful
remarks.
KharkovJuly1999 ArnoldM.Kosevich
PrefacetotheSecondEdition
Manypartsofthisbookarenotverydifferentfromwhatwasinthefirstedition(1999).
Thisisaresultofthefactthatthebasicequationsandconclusionsofthetheoryofthe
crystallatticehavelongsincebeenestablished. Themainchanges(“reconstruction”)
ofthebookarethefollowing
1. All the questions concerning one-dimensional (1D) crystals are combined in
one chapter (Chapter 1). I consider the theory of a 1D crystal lattice as a training
and provinggroundfor studyingdynamicsof three-dimensionalstructures. The 1D
modelsallowustoformulateandsolvesimplymanycomplicatedproblemsofcrystal
mechanicsandobtainexactsolutionstoequationsnotonlyofthelineardynamicsbut
alsofordynamicsofanharmonic(nonlinear)crystals.
2.Thesecondeditionincludesanewchapterdevotedtothetheoryofelasticsuper-
lattices(Chapter5).Anewclassofmaterials,namely,phononandphotoncrystalshas
recentlybeenofthegreatinterest,andIwouldliketoproposeasimpleexplanationof
manypropertiesofsuperlatticesthatwerestudiedbeforeandknowninthetheoryof
normalcrystallattices.
3. New sections are added to the new edition concerning defects in the crystal
lattice.
Prefaces XI
Finally, I would like to thankthe peoplewho helpedme in the preparationof the
manuscript.
I am indebted to Dr. Michail Ivanov and Dr. Sergey Feodosiev for their advise
in improving the presentation of some parts of the book. I express many thanks to
AlexanderKotlyarforhisinvaluablehelpinpreparingthefiguresandelectronicver-
sion of the manuscript. The author is gratefulto Oksana Charkina for assistance in
preparingthemanuscript.IwouldliketothankmywifeDinaforherencouragement.
KharkovMarch2005 ArnoldM.Kosevich
Part 1 Introduction
TheCrystalLattice:Phonons,Solitons,Dislocations,Superlattices, SecondEdition.ArnoldM.Kosevich
Copyright(cid:1)c 2005WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
ISBN:3-527-40508-9
