Table Of ContentYi-LongChenandDe-PingYang
Mo¨ssbauerEffect
inLatticeDynamics
Mo¨ssbauerEffectinLatticeDynamics. Yi-LongChenandDe-PingYang
Copyright82007WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
ISBN:978-3-527-40712-5
Yi-Long Chen and De-Ping Yang
Mo¨ssbauer Effect in Lattice Dynamics
Experimental Techniques and Applications
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ISBN 978-3-527-40712-5
V
Contents
Preface XI
1 TheMo¨ssbauerEffect 1
1.1 ResonantScatteringofg-Rays 1
1.2 TheMo¨ssbauerEffect 4
1.2.1 CompensationforRecoilEnergy 4
1.2.2 TheDiscoveryoftheMo¨ssbauerEffect 5
1.3 TheMo¨ssbauerSpectrum 8
1.3.1 TheMeasurementofaMo¨ssbauerSpectrum 8
1.3.2 TheShapeandIntensityofaSpectralLine 9
1.4 TheClassicalTheory 14
1.5 TheQuantumTheory 16
1.5.1 CoherentStatesofaHarmonicOscillator 16
1.5.2 GammaRadiationfromaBoundNucleus 19
1.5.3 Mo¨ssbauerEffectinaSolid 21
1.5.4 AverageEnergyTransferred 25
References 26
2 HyperfineInteractions 29
2.1 ElectricMonopoleInteraction 30
2.1.1 AGeneralDescription 30
2.1.2 TheIsomerShift 31
2.1.3 CalibrationofIsomerShift 34
2.1.4 IsomerShiftandElectronicStructure 35
2.2 ElectricQuadrupoleInteraction 39
2.2.1 ElectricQuadrupoleSplitting 39
2.2.2 TheElectricFieldGradient(EFG) 43
2.2.2.1 SourcesofEFG 43
2.2.2.2 TemperatureEffectonEFG 45
2.2.3 IntensitiesoftheSpectralLines 48
2.2.4 TheSignofEFG 49
2.3 MagneticDipoleInteraction 51
2.3.1 MagneticSplitting 52
Mo¨ssbauerEffectinLatticeDynamics. Yi-LongChenandDe-PingYang
Copyright82007WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
ISBN:978-3-527-40712-5
VI Contents
2.3.2 RelativeLineIntensities 52
2.3.3 EffectiveMagneticField 53
2.4 CombinedQuadrupoleandMagneticInteractions 56
2.5 Polarizationofg-Radiation 62
2.5.1 PolarizedMo¨ssbauerSources 63
2.5.2 AbsorptionofPolarizedg-Rays 65
2.6 SaturationEffectinthePresenceofHyperfineSplittings 71
2.7 Mo¨ssbauerSpectroscopy 72
References 74
3 ExperimentalTechniques 79
3.1 TheMo¨ssbauerSpectrometer 79
3.2 RadiationSources 82
3.3 TheAbsorber 84
3.3.1 EstimationoftheOptimalThickness 84
3.3.2 SamplePreparation 90
3.4 DetectionandRecordingSystems 90
3.4.1 GasProportionalCounters 92
3.4.2 NaI(Tl)ScintillationCounters 92
3.4.3 SemiconductorDetectors 92
3.4.4 ReductionandCorrectionofBackgroundCounts 93
3.4.5 GeometricConditions 94
3.4.6 RecordingSystems 95
3.5 VelocityDriveSystem 95
3.5.1 VelocityTransducer 95
3.5.2 WaveformGenerator 96
3.5.3 DriveCircuitandFeedbackCircuit 98
3.5.4 VelocityCalibration 99
3.5.4.1 SecondaryStandardCalibration 99
3.5.4.2 AbsoluteVelocityCalibration 100
3.6 DataAnalysis 102
3.6.1 FittingIndividualLorentzianLines 102
3.6.1.1 SpectrafromCrystallineSamples 103
3.6.1.2 SpectrafromAmorphousSamples 105
3.6.2 FullHamiltonianSiteFitting 107
3.6.3 FittingThickAbsorberSpectra 108
References 109
4 TheBasicsofLatticeDynamics 113
4.1 HarmonicVibrations 113
4.1.1 AdiabaticApproximation 113
4.1.2 HarmonicApproximation 115
4.1.3 ForceConstantsandTheirProperties 117
4.1.4 NormalCoordinates 120
4.2 LatticeVibrations 123
Contents VII
4.2.1 DynamicalMatrix 123
4.2.2 ReciprocalLatticeandtheBrillouinZones 125
4.2.2.1 ReciprocalLattice 125
4.2.2.2 BrillouinZones 126
4.2.3 TheBorn–vonKarmanBoundaryCondition 128
4.2.4 AcousticandOpticalBranches 129
4.2.5 LongitudinalandTransverseWaves 133
4.2.6 ModelsofInteratomicForcesinSolids 138
4.3 QuantizationofVibrations:ThePhonons 140
4.4 FrequencyDistributionandThermodynamicProperties 141
4.4.1 TheLatticeHeatCapacity 141
4.4.2 TheDensityofStates 143
4.4.2.1 TheEinsteinModel 144
4.4.2.2 TheDebyeModel 145
4.4.3 MomentsofFrequencyDistribution 146
4.4.4 TheDebyeTemperaturey 149
D
4.4.4.1 ThePhysicalMeaningofy 149
D
4.4.4.2 ComparisonofResultsfromVariousExperimentalMethods 150
4.5 LocalizedVibrations 152
4.6 ExperimentalMethodsforStudyingLatticeDynamics 155
4.6.1 NeutronScattering 156
4.6.1.1 Theory 157
4.6.1.2 NeutronScatteringbyaCrystal 159
4.6.2 X-rayScattering 165
4.7 First-PrinciplesLatticeDynamics 165
4.7.1 LinearResponseandLatticeDynamics 166
4.7.2 TheDensity-FunctionalTheory 168
4.7.3 Exchange-CorrelationEnergyandLocal-DensityApproximation 169
4.7.4 PlaneWavesandPseudopotentials 170
4.7.5 CalculationofDOSinSolids 171
References 173
5 RecoillessFractionandSecond-OrderDopplerEffect 177
5.1 Mean-SquareDisplacementhu2iandMean-SquareVelocityhv2i 177
5.2 TemperatureDependenceoftheRecoillessFraction f 180
5.3 TheAnharmonicEffects 182
5.3.1 TheGeneralFormoftheRecoillessFraction f 183
5.3.2 CalculatingtheRecoillessFraction f UsingthePseudoharmonic
Approximation 185
5.3.3 Low-TemperatureAnharmonicEffect 188
5.4 PressureDependenceoftheRecoillessFraction f 190
5.5 TheGoldanskii–KaryaginEffect 192
5.5.1 SingleCrystals 193
5.5.2 Polycrystals 195
5.6 Second-OrderDopplerShift 196
VIII Contents
5.6.1 TransverseDopplerEffect 196
5.6.2 TheRelationbetween f andd 199
SOD
5.7 MethodsforMeasuringtheRecoillessFraction f 202
5.7.1 AbsoluteMethods 202
5.7.2 RelativeMethods 206
References 207
6 Mo¨ssbauerScatteringMethods 213
6.1 TheCharacteristicsandTypesofMo¨ssbauerg-rayScattering 213
6.1.1 TheMainCharacteristics 213
6.1.2 TypesofScatteringProcesses 215
6.2 InterferenceandDiffraction 219
6.2.1 InterferencebetweenNuclearResonanceScatteringandRayleigh
Scattering 220
6.2.2 ObservationofMo¨ssbauerDiffraction 225
6.3 CoherentElasticScatteringbyBoundNuclei 229
6.3.1 NuclearResonanceScatteringAmplitude 229
6.3.2 CoherentElasticNuclearScattering 230
6.3.2.1 ScatteringAmplitude 230
6.3.2.2 NuclearBraggScattering(NBS) 231
6.3.2.3 NuclearForwardScattering(NFS) 231
6.3.2.4 ScatteringCross-Sections 231
6.3.3 Lamb–Mo¨ssbauerFactorandDebye–WallerFactor 232
6.4 RayleighScatteringofMo¨ssbauerRadiation(RSMR) 233
6.4.1 BasicPropertiesofRSMR 233
6.4.2 SeparationofElasticandInelasticScatterings 236
6.4.3 MeasuringDynamicParametersUsingRSMR 241
6.4.3.1 TheFixedTemperatureApproach 242
6.4.3.2 TheVariableTemperatureApproach 243
6.4.4 RSMRandAnharmonicEffect 244
6.4.4.1 UsingStrongMo¨ssbauerIsotopeSources 244
6.4.4.2 UsingHigherTemperatureMeasurements 245
References 249
7 SynchrotronMo¨ssbauerSpectroscopy 253
7.1 SynchrotronRadiationandItsProperties 254
7.1.1 TheAngularDistributionofRadiation 254
7.1.2 TheTotalPowerofRadiation 256
7.1.3 TheFrequencyDistributionofRadiation 256
7.1.4 Polarization 257
7.2 SynchrotronMo¨ssbauerSources 258
7.2.1 ThemeVBandwidthSources 258
7.2.2 ThemeVBandwidthSources 261
7.3 TimeDomainMo¨ssbauerSpectroscopy 265
7.3.1 NuclearExciton 265
Contents IX
7.3.2 EnhancementofCoherentChannel 266
7.3.3 Speed-UpofInitialDecay 267
7.3.4 NuclearForwardScatteringofSR 271
7.3.5 DynamicalBeat(DB) 274
7.3.6 QuantumBeat(QB) 274
7.3.7 DistinctionsbetweenTimeDomainandEnergyDomainMethods 280
7.3.8 MeasurementoftheLamb–Mo¨ssbauerFactor 281
7.4 PhononDensityofStates 285
7.4.1 InelasticNuclearResonantScattering 286
7.4.2 MeasurementofDOSinSolids 289
7.4.3 ExtractionofLamb–Mo¨ssbauerFactor,SODShift,andForce
Constant 291
7.5 SynchrotronMethodsversusConventionalMethods 297
References 300
8 Mo¨ssbauerImpurityAtoms(I) 305
8.1 TheoryofSubstitutionalImpurityAtomVibrations 305
8.1.1 TheGeneralMethod 305
8.1.2 MassDefectApproximation 310
8.1.2.1 ResonanceModes 311
8.1.2.2 LocalizedModes 311
8.2 TheMannheimModel 313
8.3 ImpuritySiteMoments 320
8.3.1 TheEinsteinModel 323
8.3.2 TheEinstein–DebyeModel 323
8.3.3 TheMaradudin–FlinnModel 323
8.3.4 TheVisscherModel 323
8.3.5 TheMannheimModel 324
8.4 ExamplesofMo¨ssbauerStudiesof57Fe,119Sn,and197Au
Impurities 327
8.4.1 57FeImpurityAtoms 327
8.4.2 119SnImpurityAtoms 330
8.4.3 197AuImpurityAtoms 333
8.5 InterstitialImpurityAtoms 333
8.5.1 57FeImpuritiesinAu 334
8.5.2 57FeImpuritiesinDiamond 335
References 337
9 Mo¨ssbauerImpurityAtoms(II) 341
9.1 MetalsandAlloys 341
9.1.1 Metals 341
9.1.2 Alloys 345
9.1.2.1 Theb-Ti(Fe)Alloy 345
9.1.2.2 CuaZnAlloy(Brass) 347
9.2 AmorphousSolids 350
X Contents
9.2.1 TheAlloyYFe 352
2
9.2.2 TheAlloyFe B 354
80 20
9.3 MolecularCrystals 355
9.3.1 TheConceptofEffectiveVibratingMassM 355
eff
9.3.2 VibrationalDOSinMolecularCrystals 358
9.3.2.1 TheModeCompositionFactore2(l,j) 358
9.3.2.2 AnExample 359
9.4 Low-DimensionalSystems 361
9.4.1 ThinFilms 361
9.4.2 Nanocrystals 366
References 368
Appendices 373
AppendixA FractionalIntensitye(v)andAreaA(t ) 373
a
AppendixB EigenstateCalculationsinCombinedInteractions 377
B.1 ElectricQuatrupolePerturbation 377
B.2 TheCoefficientsa andb 378
i,mg j,me
AppendixC ForceConstantMatrices((cid:1)F)infccandbccLattices 380
AppendixD NearestNeighborsAroundaSubstitutionalImpurity 382
AppendixE ForceConstantsforCentralForces 383
AppendixF LatticeGreen’sFunction 385
F.1 DefinitionofGreen’sFunction 385
F.2 TheRealandImaginaryPartsofG 388
F.3 SymmetryPropertiesoftheG-Matrices 390
F.4 TheMeanSquareDisplacementhu2(0)iandtheRecoilless
Fraction f 391
0
F.5 RelationsBetweenDifferentGreen’sFunctionsG (l l,w) 391
ab
AppendixG SymmetryCoordinates 393
AppendixH MassAbsorptionCoefficients 396
CopyrightAcknowledgments 401
SubjectIndex 405
