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The Crystal Lattice: Phonons, Solitons, Dislocations, Superlattices, Second Edition PDF

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Titelei Kossevich 17.06.2005 9:44 Uhr Seite 1 Arnold M.Kosevich The Crystal Lattice The Crystal Lattice: Phonons, Solitons, Dislocations, Superlattices, Second Edition.Arnold M. Kosevich. Copyright © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-40508-9 Titelei 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. e-mail: [email protected] Library ofCongress Card No.:applied for. British Library Cataloging-in-Publication Data: A catalogue record for this book is available from the British Library. Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>. © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim All rights reserved (including those of translation into other languages). No part of this book may be repro- duced in any form – nor transmitted or translated into machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Printed in the Federal Republic of Germany Printed on acid-free paper Satz Uwe Krieg, Berlin Printing Strauss GmbH, Mörlenbach Bookbinding Litges & Dopf Buchbinderei GmbH, Heppenheim 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

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