非弹性中子散射及应用简介

InelasticNeutronScatteringinCondensedMatter:Phonons,CEFExcitations,Magnons刘本琼

liuqiong414@163.com11.Basicprinciples2.Instrumentsintroduction3.Phonons4.Crystalfieldexcitations5.Spinwaves:MagnonsOutline2BasicprinciplesFig.1Schematicrepresentationofthescatteringexperiment,andrelationsamongtheinitialandfinalwavevectorskiandkfandthescatteringvectorQ.2θ

isthescatteringangle,anddΩisthecollectionsolidangleofthedetector.3BasicprinciplesInelasticneutronscattering

isanexperimentaltechniquecommonlyusedin

condensedmatterresearchtostudyatomicandmolecularmotionaswellasmagneticandcrystalfieldexcitations.4Basicprinciples5Coldtriple-axisspectrometer6monoanadetAnincidentbeamwithwavevectorkisselectedbythemonochromatorcrystal(1staxis).Themonochromaticbeamisscatteredfromthesample(2ndaxis).Theintensityofthescatteredbeamwithwavevectork’isreflectedbytheanalyzercrystal(3rdaxis)ontotheneutrondetector.CTAS@CMRR:KUNPENG7SamplestageAnalyzerDetectorTOFspectrometers8Inneutronscatteringexperiments,neutronenergycanbedeterminedbymeasuringtheirflighttimetoveradistanceLofafewmeters.Theflighttimeoftheneutronswithwavevectorkandk’aret0=L/v,andt=L/v’,andDependingonwhetherk’andkismeasuredbyTOF,themethodiscalleddirectTOForinvertedTOF,respectively.1.Basicprinciples2.Instrumentsintroduction3.Phonons4.Crystalfieldexcitations5.Spinwaves:MagnonsOutline910Phonons/latticevibrationsTheatomsinasolidareinconstantmotionandgiverisetolatticevibrations.11Phonons/latticevibrationsPhonons/latticevibrations12Fig.2Phonondispersioncurvesforaone-dimensionallineofatomswith(a)asinglemass,(b)twodifferentmassesmandM.Fig.3PhonondispersioncurvesforGealongcertainhighsymmetryaxesintheBrillouinzone.13Longitudinal/transversalphononsFig.4ThemomentumtransferoftheneutronsQpointsalwaysintothedirectionoftherealdisplacementsui.(a)longitudinaland(b)transversaloscillations.ThecrosssectionofphononexcitationscontainsthescalarproductQ·uwiththepolarizationofthewaveu.Thus,anoscillationisonlyexcitedforQwithacomponentinthepolarizationdirection.14Howtomeasurephonons?Instandardexperiments,thescansaredoneatconstantQorconstantenergytransferΔE.MostoftheBrillouinzoneisnormallymeasuredwithconstant-Q(Fig.(a)),whileforverystiffdispersionmodes,inthevicinityoftheBrillouinzonecenter,constant-Eischosen(Fig.(b)).15Howtomeasurephonons?16Example:phononsinGeFig.5PhonondispersioncurvesforGealongcertainhighsymmetryaxesintheBrillouinzone.Thecharactersexpressthesymmetrytypesanddegeneraciesofthelatticemodes.Grouptheoryallowsustoidentifythehighsymmetrypointswheredegeneraciesoccur,whichmodessticktogether,whichmodescross,andwhichmodesshowanti-crossings.17Examples:NaClFig.6(a)ThespacegroupofNaClis#225.Thefccunitcellcontains4primitiveunitcellswith4Naand4Claotms.(b)TherhombohedralprimitivecellofthefcclatticewhichcontainsoneNaatomandoneClatom.18Examples:NaClThegroupofthewavevectoratk=0fortheNaClstructureisOh.ThepointgroupoperationsforOhis19TheoreticalcalculationsofphononsB.Liu,etal.,ActaPhys.Sin.62,176104(2013)20Booksandwebsites21PhononsinCeAuAl3

B.Liu,etal.,Phys.Rev.B98,174306(2018)Fig.7(1)CrystalstructureofCeAuAl3.(b)Brillouinzoneofthebody-centeredtetragonallatticeofCeAuAl3withc>a.Fig.8PhonondispersionforCeAuAl3alonghighsymmetrylinesΓ-M-S0-Γ.22Imaginarymodeω2<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.24AcousticmodesandmacroscopicelasticityFig.11Atomicdisplacementsassociatedwithalong-wavelengthlongitudinalacousticmodepropagatingalong[100]inacubiccrystal.Consideralongitudinalacousticmodeinacubiccrystalwithwavevectoralong[100].Themagnitudeofthewavevectorissmallbutnon-zero.Each(100)planeofatomsisdisplacedinthexdirectionbyaconstantamountrelativetoitsneighbouringplanes.Thereforethedisplacementuxofeachplaneisproportionaltoitspositionx.Thiscorrespondstoauniformcompressionalstrainofthecrystal,e11=әux/әx,whichlocallymakesacubicunitcelltetragonal.25AcousticmodefrequenciesandtheelasticconstanttensorHowtocalculatetheslopesoftheacousticphonondispersioncurvesinthelong-wavelengthlimit,wheretheacousticmodesgiverisetostraindistortions?UsingthestandardNewtonequationofmotion,onecanobtain:ItiscommonpracticetousetheVoigtnotation,inwhichpairsofindicesarereplacedbysingleindices:26Example:cubicsystem27Example:cubicsystemWecancommentonthestabilityofthecrystal.IfC44isnegative,thecrystalisunstableagainstthesheargivenbyoneofthetransverseacousticmodeswithwavevectorsinthea*-b*plane.IfC11<C12thecrystalisunstableagainstthetransverseacousticmodewiththewavevectoralong[110]andpolarisationvectoralong[1-10].1.Basicprinciples2.Instrumentsintroduction3.Phonons4.Crystalfieldexcitations5.Spinwaves:MagnonsOutline2829CrystalelectricfieldexcitationsAmagneticioninacrystalexperiencestheinteractionwiththechargedsurroundingligandions,producinganelectrostaticfield(callcrystalfieldorligandfield).30Example:sixfoldcubiccoordination31Example:sixfoldcubiccoordination32CrystalelectricfieldexcitationsForcubicpointsymmetry33CrystalelectricfieldexcitationsForhexagonalpointsymmetryFororthorhombicpointsymmetryM.T.Hutchings,Point-ChargecalculationsofenergylevelsofmagneticionsincrystallineelectricfieldsCEFexcitationsinCeAuAl3Fig.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.12CrystalstructureofCeAuAl3.34CEFexcitationsinCeAuAl335CEFparametersSteven’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.EigenvaluesandEigenvectors36SinceJ=5/2,m=-5/2,-3/2,-1/2,1/2,3/2,5/2.Withthe|J,m>basis,theCEFHamiltonianHCEFcanbegivenas37TheINSspectra38TheboundstateP.Čermák,A.Schneidewind,B.Liu,etal.,PNAS,doi/10.1073/pnas.1819664116Fig.15Keycharacteristicsoftheneutron-scatteringexcitationspectraofsingle-crystalCeAuAl3observedinreciprocalspacealongΓtoMtoStoΓatT=5K.1.Basicprinciples2.Instrumentsintroduction3.Phonons4.Crystalfieldexcitations5.Spinwaves:MagnonsOutline39MagneticorderingMagneticordernormallyexistsbelowacriticaltemperature(TcorTN)atwhichaphasetransitiontakesplace.Forthespinorderedgroundstate,onecanimaginedifferentconfigurations,e.g.,allspinsparallel(ferromagnetism),orantiparallel(antiferromagnetism),(anti-)ferrimagneticorderingormorecomplexlikehelix-form.4041Magneticordering42Magneticordering43MagneticstructureofMnWO4

MagneticstructureofMnWO4at3K(left),13K(middle),and10K(right).44SpinWavesinFerromagnets45SpinWavesinFerromagnetsSpinWavesinAntiferromagnets46Findingtheeigenenergies47HwhereαqandβqareBoseoperators.Ifsuchatransformationexist,thediagonalelementsinthematrixωwillbetheeigenergiesofthesystemandthereforenon-negative.Calculatingthespinwaveintensities

48Thedifferentialscatteringcross-sectionformagneticscatteringisgivenas:49Calculatingthespinwaveintensities

doesnotcontributetotheinelasticscatteringcross-sectionifμorνisequaltothepreferredspindirection(thez-direction).Thespin-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).51Thespin-cantingcase:MnWO4B.Liu,etal.,J.Phys.:Condens.Matter30,295401(2018).52Thespin-cantingcase:MnWO453Thespin-cantingcase:MnWO454Thespin-cantingcase:MnWO455Thespin-cantingcase:MnWO4InordertodiagonalizetheHamiltonianHq,itisneededtointroduceatransformationmatrixT,anda=Tα,sothatwhereωisadiagonalmatrixanditsdiagonalelementsareeigenvaluesofthesystem.ThetransformationmatrixTisa16×16matrix