非弹性中子散射及应用简介刘本琼课件

InelasticNeutronScatteringinCondensedMatter:Phonons,CEFExcitations,Magnons刘本琼

liuqiong414@163.com1InelasticNeutronScatteringi1.Basicprinciples2.Instrumentsintroduction3.Phonons4.Crystalfieldexcitations5.Spinwaves:MagnonsOutline21.BasicprinciplesOutline2BasicprinciplesFig.1Schematicrepresentationofthescatteringexperiment,andrelationsamongtheinitialandfinalwavevectorskiandkfandthescatteringvectorQ.2θ

isthescatteringangle,anddΩisthecollectionsolidangleofthedetector.3BasicprinciplesFig.1SchematBasicprinciplesInelasticneutronscattering

isanexperimentaltechniquecommonlyusedin

condensedmatterresearchtostudyatomicandmolecularmotionaswellasmagneticandcrystalfieldexcitations.4BasicprinciplesInelasticneutBasicprinciples5Basicprinciples5Coldtriple-axisspectrometer6monoanadetAnincidentbeamwithwavevectorkisselectedbythemonochromatorcrystal(1staxis).Themonochromaticbeamisscatteredfromthesample(2ndaxis).Theintensityofthescatteredbeamwithwavevectork’isreflectedbytheanalyzercrystal(3rdaxis)ontotheneutrondetector.Coldtriple-axisspectrometer6CTAS@CMRR:KUNPENG7SamplestageAnalyzerDetectorCTAS@CMRR:KUNPENG7SamplestagTOFspectrometers8Inneutronscatteringexperiments,neutronenergycanbedeterminedbymeasuringtheirflighttimetoveradistanceLofafewmeters.Theflighttimeoftheneutronswithwavevectorkandk’aret0=L/v,andt=L/v’,andDependingonwhetherk’andkismeasuredbyTOF,themethodiscalleddirectTOForinvertedTOF,respectively.TOFspectrometers8Inneutrons1.Basicprinciples2.Instrumentsintroduction3.Phonons4.Crystalfieldexcitations5.Spinwaves:MagnonsOutline91.BasicprinciplesOutline910Phonons/latticevibrationsTheatomsinasolidareinconstantmotionandgiverisetolatticevibrations.10Phonons/latticevibrationsTh11Phonons/latticevibrations11Phonons/latticevibrationsPhonons/latticevibrations12Fig.2Phonondispersioncurvesforaone-dimensionallineofatomswith(a)asinglemass,(b)twodifferentmassesmandM.Fig.3PhonondispersioncurvesforGealongcertainhighsymmetryaxesintheBrillouinzone.Phonons/latticevibrations12Fi13Longitudinal/transversalphononsFig.4ThemomentumtransferoftheneutronsQpointsalwaysintothedirectionoftherealdisplacementsui.(a)longitudinaland(b)transversaloscillations.ThecrosssectionofphononexcitationscontainsthescalarproductQ·uwiththepolarizationofthewaveu.Thus,anoscillationisonlyexcitedforQwithacomponentinthepolarizationdirection.13Longitudinal/transversalpho14Howtomeasurephonons?Instandardexperiments,thescansaredoneatconstantQorconstantenergytransferΔE.MostoftheBrillouinzoneisnormallymeasuredwithconstant-Q(Fig.(a)),whileforverystiffdispersionmodes,inthevicinityoftheBrillouinzonecenter,constant-Eischosen(Fig.(b)).14Howtomeasurephonons?Inst15Howtomeasurephonons?15Howtomeasurephonons?16Example:phononsinGeFig.5PhonondispersioncurvesforGealongcertainhighsymmetryaxesintheBrillouinzone.Thecharactersexpressthesymmetrytypesanddegeneraciesofthelatticemodes.Grouptheoryallowsustoidentifythehighsymmetrypointswheredegeneraciesoccur,whichmodessticktogether,whichmodescross,andwhichmodesshowanti-crossings.16Example:phononsinGeFig.517Examples:NaClFig.6(a)ThespacegroupofNaClis#225.Thefccunitcellcontains4primitiveunitcellswith4Naand4Claotms.(b)TherhombohedralprimitivecellofthefcclatticewhichcontainsoneNaatomandoneClatom.17Examples:NaClFig.6(a)The18Examples:NaClThegroupofthewavevectoratk=0fortheNaClstructureisOh.ThepointgroupoperationsforOhis18Examples:NaClThegroupoft19TheoreticalcalculationsofphononsB.Liu,etal.,ActaPhys.Sin.62,176104(2013)19Theoreticalcalculationsof20Booksandwebsites20Booksandwebsites21PhononsinCeAuAl3

B.Liu,etal.,Phys.Rev.B98,174306(2018)Fig.7(1)CrystalstructureofCeAuAl3.(b)Brillouinzoneofthebody-centeredtetragonallatticeofCeAuAl3withc>a.Fig.8PhonondispersionforCeAuAl3alonghighsymmetrylinesΓ-M-S0-Γ.21PhononsinCeAuAl3B.Liu,e22Imaginarymodeω2<022Imaginarymodeω2<023Imaginarymodeω2<0B.Liu,etal.,Phys.Chem.Chem.Phys.17,4089(2015)X.Wang,etal.,Phys.Chem.Chem.Phys.16,26974(2014)Fig.9PhonondispersionforU2MoalongsymmetrylinesinthebodycenteredtetragonalBZ.Fig.10ThestructuresofU2Mo.TheredandgreencirclesareMoandUatoms,respectively.23Imaginarymodeω2<0B.Liu,e24AcousticmodesandmacroscopicelasticityFig.11Atomicdisplacementsassociatedwithalong-wavelengthlongitudinalacousticmodepropagatingalong[100]inacubiccrystal.Consideralongitudinalacousticmodeinacubiccrystalwithwavevectoralong[100].Themagnitudeofthewavevectorissmallbutnon-zero.Each(100)planeofatomsisdisplacedinthexdirectionbyaconstantamountrelativetoitsneighbouringplanes.Thereforethedisplacementuxofeachplaneisproportionaltoitspositionx.Thiscorrespondstoauniformcompressionalstrainofthecrystal,e11=әux/әx,whichlocallymakesacubicunitcelltetragonal.24Acousticmodesandmacroscop25AcousticmodefrequenciesandtheelasticconstanttensorHowtocalculatetheslopesoftheacousticphonondispersioncurvesinthelong-wavelengthlimit,wheretheacousticmodesgiverisetostraindistortions?UsingthestandardNewtonequationofmotion,onecanobtain:ItiscommonpracticetousetheVoigtnotation,inwhichpairsofindicesarereplacedbysingleindices:25Acousticmodefrequenciesan26Example:cubicsystem26Example:cubicsystem27Example:cubicsystemWecancommentonthestabilityofthecrystal.IfC44isnegative,thecrystalisunstableagainstthesheargivenbyoneofthetransverseacousticmodeswithwavevectorsinthea*-b*plane.IfC11<C12thecrystalisunstableagainstthetransverseacousticmodewiththewavevectoralong[110]andpolarisationvectoralong[1-10].27Example:cubicsystemWecan1.Basicprinciples2.Instrumentsintroduction3.Phonons4.Crystalfieldexcitations5.Spinwaves:MagnonsOutline281.BasicprinciplesOutline2829CrystalelectricfieldexcitationsAmagneticioninacrystalexperiencestheinteractionwiththechargedsurroundingligandions,producinganelectrostaticfield(callcrystalfieldorligandfield).29Crystalelectricfieldexcit30Example:sixfoldcubiccoordination30Example:sixfoldcubiccoord31Example:sixfoldcubiccoordination31Example:sixfoldcubiccoord32Crystalelectricfieldexcitations32CrystalelectricfieldexcitForcubicpointsymmetry33CrystalelectricfieldexcitationsForhexagonalpointsymmetryFororthorhombicpointsymmetryM.T.Hutchings,Point-ChargecalculationsofenergylevelsofmagneticionsincrystallineelectricfieldsForcubicpointsymmetry33CrysCEFexcitationsinCeAuAl3Fig.132DcontourplotsofthespectralfunctionS(Q,ω)(a)ofCeAuAl3at4.5Kand(c)at50K,(b)ofLaAuAl3at4.5Kand(d)estimatedmagneticscatteringofCeAuAl3at4.5KaftersubtractingphononscatteringusingLaAuAl3data.B.Liu,etal.,Phys.Rev.B98,174306(2018)D.T.Adroja,etal.,Phys.Rev.B91,134425(2015)Fig.12CrystalstructureofCeAuAl3.34CEFexcitationsinCeAuAl3Fig.CEFexcitationsinCeAuAl335CEFparametersSteven’soperatorsTheCEFHamiltonianforatetragonalpointsymmetry(C4v)oftheCeioninCeAuAl3canbewrittenasThesixfolddegenerateCe3+(J=5/2)states,4f1,splitinto3doublets(Kramer’stheoremestablishesthatforoddnumbersoflocalizedelectrontheminimumdegeneracyshouldbeadoublet)intheparamagneticphase.Fig.14TheestimatedmagneticscatteringfromCeAuAl3atlowQ=2.43Å-1at(a)4.5K,(b)50K,and(c)250K.Thethicksolidlinesrepresentthefits(basedontheCEFmodel).(d)TheschematicCEFlevelscheme(threeCEFdoubletsat0,5.1,and24.5meV)oftheCe3+ions.CEFexcitationsinCeAuAl335CEEigenvaluesandEigenvectors36SinceJ=5/2,m=-5/2,-3/2,-1/2,1/2,3/2,5/2.Withthe|J,m>basis,theCEFHamiltonianHCEFcanbegivenasEigenvaluesandEigenvectors3637TheINSspectra37TheINSspectra38TheboundstateP.Čermák,A.Schneidewind,B.Liu,etal.,PNAS,doi/10.1073/pnas.1819664116Fig.15Keycharacteristicsoftheneutron-scatteringexcitationspectraofsingle-crystalCeAuAl3observedinreciprocalspacealongΓtoMtoStoΓatT=5K.38TheboundstateP.Čermák,A.1.Basicprinciples2.Instrumentsintroduction3.Phonons4.Crystalfieldexcitations5.Spinwaves:MagnonsOutline391.BasicprinciplesOutline39MagneticorderingMagneticordernormallyexistsbelowacriticaltemperature(TcorTN)atwhichaphasetransitiontakesplace.Forthespinorderedgroundstate,onecanimaginedifferentconfigurations,e.g.,allspinsparallel(ferromagnetism),orantiparallel(antiferromagnetism),(anti-)ferrimagneticorderingormorecomplexlikehelix-form.40MagneticorderingMagneticorde41Magneticordering41Magneticordering42Magneticordering42Magneticordering43MagneticstructureofMnWO4

MagneticstructureofMnWO4at3K(left),13K(middle),and10K(right).43MagneticstructureofMnWO444SpinWavesinFerromagnets44SpinWavesinFerromagnets45SpinWavesinFerromagnets45SpinWavesinFerromagnetsSpinWavesinAntiferromagnets46SpinWavesinAntiferromagnetsFindingtheeigenenergies47HwhereαqandβqareBoseoperators.Ifsuchatransformationexist,thediagonalelementsinthematrixωwillbetheeigenergiesofthesystemandthereforenon-negative.Findingtheeigenenergies47HwhCalculatingthespinwaveintensities

48Thedifferentialscatteringcross-sectionformagneticscatteringisgivenas:Calculatingthespinwaveinte49Calculatingthespinwaveintensities

doesnotcontributetotheinelasticscatteringcross-sectionifμorνisequaltothepreferredspindirection(thez-direction).49CalculatingthespinwaveinThespin-cantingcase:MnWO450Fig.17Themagneticsublattice(4a×2b×2c)ofAF1ofMnWO4.OnlymagneticMn2+sites(MnainpinkandMnbincyanblue)areplottedtodemonstratetwodifferentspin-cantingtexturesclearlyseenwiththerespectivecantingangles.Fig.16Thenuclearstructureunitofhuebnerite(MnWO4)containstwoindependentMnsites,denotedMnaandMnb.B.Liu,etal.,J.Phys.:Condens.Matter30,295401(2018).S.Park,B.Liu,etal.,J.Phys.:Condens.Matter30,135802(2018).Thespin-cantingcase:MnWO45051Thespin-cantingcase:MnWO4B.Liu,etal.,J.Phys.:Condens.Matter30,295401(2018).51Thespin-cantingcase:MnWO452Thespin-cantingcase:MnWO452Thespin-cantingcase:MnWO453Thespin-cantingcase:MnWO453Thespin-cantingcase:MnWO454Thespin-cantingcase:MnWO454Thespin-cantingcase:MnWO455Thespin-cantingcase:MnWO4InordertodiagonalizetheHamiltonianHq,itisneededtointroduceatransformationmatrixT,anda=Tα,sothatwhereωisadiagonalmatrixanditsdiagonalelementsareeigenvaluesofthesystem.ThetransformationmatrixTisa16×16matrixwithcolumnsthatareeigenvectorstoI1HqT=ωT,anditalsomustrespecttheBosecommutationrules.OncethecorrecttransformationmatrixTisobtained,thedifferentialscatteringcrosssectionsformagneticscatteringarecalculatedasfollows,

55Thespin-cantingcase:MnWO456Thespin-cantingcase:MnWO4Fig.18Spinwavedispersionalong[H,0.5,2H]directionthroughthemagneticpeak(0.25,0.5,0.5).Theredpointsareexperimentaldata.Fig.19Spinwavespectrumalong[H,0.5,2H]directionthroughthemagneticpeak(0.25,0.5,0.5),withGaussianfunctionconvoluted.ThecolorcodedenotestheINSintensity.56Thespin-cantingcase:MnWO457Thespin-cantingcase:MnWO457Thespin-cantingcase:MnWO4Self-developedprogram(Fortran,Matlab,…)Density-FunctionalTheorySpinW58Howtoobtaintheexchange-couplingconstants?Self-developedprogram(Fortra59DFTcalculationsFig.20Orderedspinarrangementsineach//bclayerofMn2+ionsintheFM,A1,A2,A3,A4,A5,A6,A7,A8,A9statesofMnWO4.Theup-spinanddown-spinMn2+sitesarerepresentedbyfilledandunfilledcircles,respectively.C.Tian,etal.,Phys.Rev.B80,104426(2009)59DFTcalculationsFig.20Orde60DFTcalculations60DFTcalculationsThanksforyourattention!61TheendThanksforyourattention!61ThInelasticNeutronScatteringinCondensedMatter:Phonons,CEFExcitations,Magnons刘本琼

liuqiong414@163.com62InelasticNeutronScatteringi1.Basicprinciples2.Instrumentsintroduction3.Phonons4.Crystalfieldexcitations5.Spinwaves:MagnonsOutline631.BasicprinciplesOutline2BasicprinciplesFig.1Schematicrepresentationofthescatteringexperiment,andrelationsamongtheinitialandfinalwavevectorskiandkfandthescatteringvectorQ.2θ

isthescatteringangle,anddΩisthecollectionsolidangleofthedetector.64BasicprinciplesFig.1SchematBasicprinciplesInelasticneutronscattering

isanexperimentaltechniquecommonlyusedin

condensedmatterresearchtostudyatomicandmolecularmotionaswellasmagneticandcrystalfieldexcitations.65BasicprinciplesInelasticneutBasicprinciples66Basicprinciples5Coldtriple-axisspectrometer67monoanadetAnincidentbeamwithwavevectorkisselectedbythemonochromatorcrystal(1staxis).Themonochromaticbeamisscatteredfromthesample(2ndaxis).Theintensityofthescatteredbeamwithwavevectork’isreflectedbytheanalyzercrystal(3rdaxis)ontotheneutrondetector.Coldtriple-axisspectrometer6CTAS@CMRR:KUNPENG68SamplestageAnalyzerDetectorCTAS@CMRR:KUNPENG7SamplestagTOFspectrometers69Inneutronscatteringexperiments,neutronenergycanbedeterminedbymeasuringtheirflighttimetoveradistanceLofafewmeters.Theflighttimeoftheneutronswithwavevectorkandk’aret0=L/v,andt=L/v’,andDependingonwhetherk’andkismeasuredbyTOF,themethodiscalleddirectTOForinvertedTOF,respectively.TOFspectrometers8Inneutrons1.Basicprinciples2.Instrumentsintroduction3.Phonons4.Crystalfieldexcitations5.Spinwaves:MagnonsOutline701.BasicprinciplesOutline971Phonons/latticevibrationsTheatomsinasolidareinconstantmotionandgiverisetolatticevibrations.10Phonons/latticevibrationsTh72Phonons/latticevibrations11Phonons/latticevibrationsPhonons/latticevibrations73Fig.2Phonondispersioncurvesforaone-dimensionallineofatomswith(a)asinglemass,(b)twodifferentmassesmandM.Fig.3PhonondispersioncurvesforGealongcertainhighsymmetryaxesintheBrillouinzone.Phonons/latticevibrations12Fi74Longitudinal/transversalphononsFig.4ThemomentumtransferoftheneutronsQpointsalwaysintothedirectionoftherealdisplacementsui.(a)longitudinaland(b)transversaloscillations.ThecrosssectionofphononexcitationscontainsthescalarproductQ·uwiththepolarizationofthewaveu.Thus,anoscillationisonlyexcitedforQwithacomponentinthepolarizationdirection.13Longitudinal/transversalpho75Howtomeasurephonons?Instandardexperiments,thescansaredoneatconstantQorconstantenergytransferΔE.MostoftheBrillouinzoneisnormallymeasuredwithconstant-Q(Fig.(a)),whileforverystiffdispersionmodes,inthevicinityoftheBrillouinzonecenter,constant-Eischosen(Fig.(b)).14Howtomeasurephonons?Inst76Howtomeasurephonons?15Howtomeasurephonons?77Example:phononsinGeFig.5PhonondispersioncurvesforGealongcertainhighsymmetryaxesintheBrillouinzone.Thecharactersexpressthesymmetrytypesanddegeneraciesofthelatticemodes.Grouptheoryallowsustoidentifythehighsymmetrypointswheredegeneraciesoccur,whichmodessticktogether,whichmodescross,andwhichmodesshowanti-crossings.16Example:phononsinGeFig.578Examples:NaClFig.6(a)ThespacegroupofNaClis#225.Thefccunitcellcontains4primitiveunitcellswith4Naand4Claotms.(b)TherhombohedralprimitivecellofthefcclatticewhichcontainsoneNaatomandoneClatom.17Examples:NaClFig.6(a)The79Examples:NaClThegroupofthewavevectoratk=0fortheNaClstructureisOh.ThepointgroupoperationsforOhis18Examples:NaClThegroupoft80TheoreticalcalculationsofphononsB.Liu,etal.,ActaPhys.Sin.62,176104(2013)19Theoreticalcalculationsof81Booksandwebsites20Booksandwebsites82PhononsinCeAuAl3

B.Liu,etal.,Phys.Rev.B98,174306(2018)Fig.7(1)CrystalstructureofCeAuAl3.(b)Brillouinzoneofthebody-centeredtetragonallatticeofCeAuAl3withc>a.Fig.8PhonondispersionforCeAuAl3alonghighsymmetrylinesΓ-M-S0-Γ.21PhononsinCeAuAl3B.Liu,e83Imaginarymodeω2<022Imaginarymodeω2<084Imaginarymodeω2<0B.Liu,etal.,Phys.Chem.Chem.Phys.17,4089(2015)X.Wang,etal.,Phys.Chem.Chem.Phys.16,26974(2014)Fig.9PhonondispersionforU2MoalongsymmetrylinesinthebodycenteredtetragonalBZ.Fig.10ThestructuresofU2Mo.TheredandgreencirclesareMoandUatoms,respectively.23Imaginarymodeω2<0B.Liu,e85AcousticmodesandmacroscopicelasticityFig.11Atomicdisplacementsassociatedwithalong-wavelengthlongitudinalacousticmodepropagatingalong[100]inacubiccrystal.Consideralongitudinalacousticmodeinacubiccrystalwithwavevectoralong[100].Themagnitudeofthewavevectorissmallbutnon-zero.Each(100)planeofatomsisdisplacedinthexdirectionbyaconstantamountrelativetoitsneighbouringplanes.Thereforethedisplacementuxofeachplaneisproportionaltoitspositionx.Thiscorrespondstoauniformcompressionalstrainofthecrystal,e11=әux/әx,whichlocallymakesacubicunitcelltetragonal.24Acousticmodesandmacroscop86AcousticmodefrequenciesandtheelasticconstanttensorHowtocalculatetheslopesoftheacousticphonondispersioncurvesinthelong-wavelengthlimit,wheretheacousticmodesgiverisetostraindistortions?UsingthestandardNewtonequationofmotion,onecanobtain:ItiscommonpracticetousetheVoigtnotation,inwhichpairsofindicesarereplacedbysingleindices:25Acousticmodefrequenciesan87Example:cubicsystem26Example:cubicsystem88Example:cubicsystemWecancommentonthestabilityofthecrystal.IfC44isnegative,thecrystalisunstableagainstthesheargivenbyoneofthetransverseacousticmodeswithwavevectorsinthea*-b*plane.IfC11<C12thecrystalisunstableagainstthetransverseacousticmodewiththewavevectoralong[110]andpolarisationvectoralong[1-10].27Example:cubicsystemWecan1.Basicprinciples2.Instrumentsintroduction3.Phonons4.Crystalfieldexcitations5.Spinwaves:MagnonsOutline891.BasicprinciplesOutline2890CrystalelectricfieldexcitationsAmagneticioninacrystalexperiencestheinteractionwiththechargedsurroundingligandions,producinganelectrostaticfield(callcrystalfieldorligandfield).29Crystalelectricfieldexcit91Example:sixfoldcubiccoordination30Example:sixfoldcubiccoord92Example:sixfoldcubiccoordination31Example:sixfoldcubiccoord93Crystalelectricfieldexcitations32CrystalelectricfieldexcitForcubicpointsymmetry94CrystalelectricfieldexcitationsForhexagonalpointsymmetryFororthorhombicpointsymmetryM.T.Hutchings,Point-ChargecalculationsofenergylevelsofmagneticionsincrystallineelectricfieldsForcubicpointsymmetry33CrysCEFexcitationsinCeAuAl3Fig.132DcontourplotsofthespectralfunctionS(Q,ω)(a)ofCeAuAl3at4.5Kand(c)at50K,(b)ofLaAuAl3at4.5Kand(d)estimatedmagneticscatteringofCeAuAl3at4.5KaftersubtractingphononscatteringusingLaAuAl3data.B.Liu,etal.,Phys.Rev.B98,174306(2018)D.T.Adroja,etal.,Phys.Rev.B91,134425(2015)Fig.12CrystalstructureofCeAuAl3.95CEFexcitationsinCeAuAl3Fig.CEFexcitationsinCeAuAl396CEFparametersSteven’soperatorsTheCEFHamiltonianforatetragonalpointsymmetry(C4v)oftheCeioninCeAuAl3canbewrittenasThesixfolddegenerateCe3+(J=5/2)states,4f1,splitinto3doublets(Kramer’stheoremestablishesthatforoddnumbersoflocalizedelectrontheminimumdegeneracyshouldbeadoublet)intheparamagneticphase.Fig.14TheestimatedmagneticscatteringfromCeAuAl3atlowQ=2.43Å-1at(a)4.5K,(b)50K,and(c)250K.Thethicksolidlinesrepresentthefits(basedontheCEFmodel).(d)TheschematicCEFlevelscheme(threeCEFdoubletsat0,5.1,and24.5meV)oftheCe3+ions.CEFexcitationsinCeAuAl335CEEigenvaluesandEigenvectors97SinceJ=5/2,m=-5/2,-3/2,-1/2,1/2,3/2,5/2.Withthe|J,m>basis,theCEFHamiltonianHCEFcanbegivenasEigenvaluesandEigenvectors3698TheINSspectra37TheINSspectra99TheboundstateP.Čermák,A.Schneidewind,B.Liu,etal.,PNAS,doi/10.1073/pnas.1819664116Fig.15Keycharacteristicsoftheneutron-scatteringexcitationspectraofsingle-crystalCeAuAl3observedinreciprocalspacealongΓtoMtoStoΓatT=5K.38TheboundstateP.Čermák,A.1.Basicprinciples2.Instrumentsintroduction3.Phonons4.Crystalfieldexcitations5.Spinwaves:MagnonsOutline1001.BasicprinciplesOutline39MagneticorderingMagneticordernormallyexistsbelowacriticaltemperature(TcorTN)atwhichaphasetransitiontakesplace.Forthespinorderedgroundstate,onecanimaginedifferentconfigurations,e.g.,allspinsparallel(ferromagnetism),orantiparallel(antiferromagnetism),(anti-)ferrimagneticorderingormorecomplexlikehelix-form.101MagneticorderingMagneticorde102Magneticordering41Magneticordering103Magneticordering42Magneticordering104MagneticstructureofMnWO4

MagneticstructureofMnWO4at3K(left),13K(middle),and10K(right).43MagneticstructureofMnWO4105SpinWavesinFerromagnets44SpinWavesinFerromagnets106SpinWavesinFerromagnets45SpinWavesinFerromagnetsSpinWavesinAntiferromagnets107SpinWavesinAntiferromagnetsFindingtheeigenenergies108HwhereαqandβqareBoseoperators.Ifsuchatransformationexist,thediagonalelementsinthematrixωwillbetheeigenergiesofthesystemandthereforenon-negative.Findingtheeigenenergies47HwhCalculatingthespinwaveintensities

109Thedifferentialscatteringcross-sectionformagneticscatteringisgivenas:Calculatingthespinwaveinte110Calculatingthespinwaveintensities

doesnotcontributetotheinelasticscatteringcross-sect