高等固体物理

Chapter2ReciprocalLatticeDiffractionofWavesbyCrystalsScatteredWaveAmplitudeBrillouinZonesFourierAnalysisoftheBasisQuasicrystalsChapter2ReciprocalLattice

I.DiffractionofWavesbyCrystalQuestions:

1)Wheredoesoneatomlocateinthecrystal?2)Howcanweinvestigatethestructuresofthecrystal?Distributionsoftheatomsaswellastheelectrons?3)Canwe‘see’thestructuresforonecrystaldirectly?I.DiffractionofWavesbyCrystalingbeams:(a)Photons(x-ray):

I.DiffractionofWavesbyCrystal(b)Neutrons:(c)Electrons:SeeFigure.1三种辐射的比较Xray

波长短，穿透力强，适合研究物质结构，不适合薄膜的结构研究。需高压加速电子撞击靶产生。Neutronbeam

只需0.1eV能量便可产生波长为1埃的辐射。由于具有磁矩，中子可与磁性晶体中的电子发生相互作用。常用于磁性晶体的研究。Electronbeam

散射很强，透射能力弱。主要用于薄膜结构的研究。150V电压可产生波长为1埃的电子波。I.DiffractionofWavesbyCrystalConsiderparallellatticeplanesspaceddapart.TheBragglawis2dsin=nismeasuredfromtheplane.Braggreflectioncanoccuronlyforwavelength2d.I.DiffractionofWavesbyCrystal2dsin=nAresultoftheperiodicityofthelatticethecompositionofthebasisdeterminestherelativeintensityExperimentalResultsExperimentalResultsII.ScatteredWaveAmplitude

1.FourierAnalysisTheelectronnumberdensityn(r)isaperiodicfunctionofr:(Why?Page30)n(r+T)=n(r)(3D:Withperiodsa1,a2anda3).1Dn(x)withperioda.Weexpandn(x)inaFourierseries(Why?ThemostinterestingpropertiesofcrystalsaredirectlyrelatedtotheFouriercomponentsoftheelectrondensity!)n(x)=n0

+p[Cpcos(2px/a)+Spsin(2px/a)],thep’sarepositiveintegersandCp,Sparerealconstants.1.FourierAnalysisn(x)=n0+p[Cpcos(2px/a)+Spsin(2px/a)],n(x)=n(x+a)2p/aisapointinthereciprocallatticeorFourierspaceofthecrystal.Thecompactformn(x)=pnpexp(i2px/a),wherethesumisoverallintegerspandnpiscomplexnumber.n(x)isreal,sonp=n-*p1.FourierAnalysisFor3D:n(r)=GnGexp(iG•r),nGdeterminesthex-rayscatteringamplitude.InversionofFourierseries

PleaseReadP32!nG=Vc-1celldVn(r)exp(-iG•r),hereVcisthevolumeofacellofthecrystal.For1D,np=a-10adxn(x)exp(-i2px/a),2.ReciprocalLatticeVectors2.ReciprocalLatticeVectorsEverycrystalstructurehastwolatticesassociatedwithit,thecrystallatticeandthereciprocallattice;Adiffractionpatternofacrystalisamapofthereciprocallatticeofthecrystal;Amicroscopeimageisamapofthecrystalstructureinrealspace.Dimensions[length],[1/length]ThereciprocallatticeisalatticeintheFourierspaceassociatedwiththecrystal.2.ReciprocalLatticeVectorsTheFourierrepresentationofafunctionperiodicinthecrystallatticecancontaincomponentsnGexp(iG•r)onlyatthereciprocalvectorsGasdefinedby(15).n(r+T)=GnGexp(iG•r)exp(iG•T),butexp(iG•T)=exp[i(v1b1+v2b2+v3b3)•(u1a1+u2a2+u3a3)]=exp[i2(v1u1+v2u2+v3u3)]=1Hencen(r+T)=n(r)正倒格子间的关系

正格子中一族晶面（h1h2h3）和倒格矢Gh正交倒格矢Gh的长度正比于晶面族（h1h2h3）面间距的倒数

研究倒易点阵的意义（1）利用倒易点阵的概念可以比较方便地导出晶体几何学中各种重要关系式；（2）利用倒易点阵可以方便而形象地表示晶体的衍射几何学。例如：单晶的电子衍射图相当于一个二维倒易点阵平面的投影，每一个衍射斑点与一个倒易阵点对应。因此，倒易点阵已经成为晶体衍射工作中不可缺少的分析工具。（3）倒易矢量也可以理解为波矢k，通常用波矢来描述电子在晶体中的运动状态或晶体的振动状态。由倒易点阵基矢所张的空间称为倒易空间，可理解为状态空间（k空间）。3.DiffractionConditionsTheorem:ThesetofreciprocallatticevectorsGdeterminesthepossiblex-rayreflections.3.DiffractionConditionsThescatteringamplitudeF=dVn(r)exp(-ik•r)wherek+k=k’,kmeasuresthechangeinwavevectorandiscalledthescatteringvector.F=dVnGexp[i(G-k)•r]=VnGG=k3.DiffractionConditionsFortheelasticscatteringk2=k’2G=korG+k=k’sothatthediffractionconditionis(G+k)2=(k’)2=k2,or2k•G+G2=0,or2k•G=G2,Theisthecentralresultofthetheoryofelasticscatteringofwavesinaperiodiclattice.Itisoftenusedastheconditionfordiffraction.ItisanotherstatementoftheBraggcondition.4.LaueEquationsG=ka1•k=2v1;a2•k=2v2;a3•k=2v3;Theseequationshaveasimplegeometricalinterpretation.Thefirstequationtellsusthatkliesonaconeaboutthedirectionofa1,andsoon.TheEwaldconstruction.III.BrillouinZones（布里渊区）AbrillouinzoneisdefinedasaWigner-SeitzprimitivecellinthereciprocallatticeThevalueoftheBrillouinzoneisthatitgivesavividgeometricalinterpretationofthediffractioncondition2k•G=G2k•(G/2)=(G/2)2TheBrillouinconstructionexhibitsallthewavevectorskwhichcanbeBragg-reflectedbythecrystal.III.BrillouinZonesIII.BrillouinZonesConstructionofthefirstBrillouinzone(2D)III.BrillouinZonesConstructionofthefirstBrillouinzone(1D)ReciprocalLatticetoSCLatticeTheprimitivetranslationvectorsofsc:a1=aex,a2=aey,a3=aez,thevolumeofthecell:a1•a2a3=a3.Theprimitivetranslationvectorsofthereciprocallatticeb1=(2/a)ex,b2=(2/a)ey,b3=(2/a)ez,(sc,2/a)TheboundariesofthefirstBrillouinzonesaretheplanesnormaltothesixreciprocallatticevectors±b1,±b2,±b3attheirmidpoints.Volume(2/a)3ReciprocalLatticetobccLatticeTheprimitivetranslationvectors:a1=(a/2)(-ex+ey+ez),a2=(a/2)(ex-ey+ez),a3=(a/2)(ex+ey-ez),thevolumeofthecell:a1•a2a3=a3/2.Theprimitivetranslationvectorsofthereciprocallatticeb1=(2/a)(ey+ez),b2=(2/a)(ex+ez),b3=(2/a)(ex+ey),(fcc,2/a)G=v1b1+v2b2+v3b3;the

12shortestvectors!ThefirstBrillouinzoneisaregularrhombicdodecahedron.Volumeb1•b2b3=2(2/a)3ReciprocalLatticetobccLatticeReciprocalLatticetofccLatticeTheprimitivetranslationvectors:a1=(a/2)(ey+ez),a2=(a/2)(ex+ez),a3=(a/2)(ex+ey),thevolumeofthecell:a1•a2a3=a3/4.Theprimitivetranslationvectorsofthereciprocallatticeb1=(2/a)(-ex+ey+ez),b2=(2/a)(ex-ey+ez),b3=(2/a)(ex+ey-ez),(bcc

)G,the8shortestvectors!ThefirstBrillouinzoneisatruncatedoctahedron.Volumeb1•b2b3=4(2/a)3ReciprocalLatticetofccLatticeReciprocalLatticetofccLatticeIV.FourierAnalysisoftheBasisThescatteringamplitudeforacrystalofNcellsmaybewrittenasFG=NcelldVn(r)exp(-iG•r)=NSG,whereSGiscalledthestructurefactorandisdefinedasanintegraloverasinglecell,withr=0atonecorner.Thetotalelectronconcentrationatrduetoallatomsinthecellisn(r)=j=1snj(r-rj)sumoverthesatomsofthebasis.IV.FourierAnalysisoftheBasisThestructurefactorisSG=celldVjnj(r-rj)exp(-iG•r)=jexp(-iG•rj)

dVnj()exp(-iG•),where=r-rj.Theatomicformfactor(原子形状因子)isdefinedas:fj=dVnj()exp(-iG•),integratedoverallspace.Ifnj()isanatomicproperty,fjisanatomicproperty.IV.FourierAnalysisoftheBasisThestructurefactorofthebasisisSG=jfj

exp(-iG•rj)rj=xja1+yja2+zja3,G=v1b1+v2b2+v3b3sothatSG(v1v2v3)

=jfj

exp[-2i(v1xj+v2yj+v3zj)]thescatteredintensityisS*S.AtazeroofSGthescatteredintensitywillbezero,eventhoughGisaperfectlygoodreciprocallatticevector.StructureFactorofthebcclatticeThebccbasisreferredtothecubiccellhasidenticalatomsat(0,0,0)and(1/2,1/2,1/2)ThestructurefactorisSG(v1v2v3)

=jfj

exp[-2i(v1xj+v2yj+v3zj)]=f{1+exp[-i(v1+v2+v3)]},wherefistheformfactorofanatom.S=0whenv1+v2+v3=oddinteger;S=2fwhenv1+v2+v3=eveninteger.StructureFactorofthebcclatticeForexample,metallicsodiumhasabccstructure.Thediffractionpatterndoesnotcontainlinessuchas(100),(300),(111),or(221),butlinessuchas(200),(110),and(222)willbepresent.Theindices(v1v2v3)

arereferredtoacubiccell.StructureFactorofthebcclatticeThephysicalinterpretationoftheresultthatthe(100)reflectionvanishes.StructureFactorofthefcclatticeThebasisofthefccstructurereferredtothecubiccellhasidenticalatomsat(000),(0,1/2,1/2),(1/2,0,1/2),(1/2,1/2,0).SG(v1v2v3)

=jfj

exp[-2i(v1xj+v2yj+v3zj)]=f{1+exp[-i(v2+v3)]+exp[-i(v1+v3)]+exp[-i(v1+v2)]},Ifallindicesareevenintegers,S=4f;similarlyifallindicesareoddintegers.Butifonlyoneoftheintegersiseven,Swillvanish.Ifonlyoneoftheintegersisodd,Swillalsovanish.StructureFactorofthefcclatticeInthefcclatticenoreflectionscanoccurforwhichtheindicesarepartlyevenandpartlyodd.KClandKBrhaveanfcclattice,KClsimulatesansclatticebecausetheK+andCl-ionshaveequalnumbersofelectrons.StructureFactorofthefcclatticeAtomicFormFactorfj=dVnj(r)exp(-iG•r),withtheintegralextendedovertheelectronconcentrationassociatedwithasingleatom.G•r=Grcos,iftheelectrondistributionissphericallysymmetricabouttheorigin,thenfj=4drr2nj(r)(sinGr/Gr),Ifthesametotalelectrondensitywereconcentratedatr=0.Inthislimit(sinGr/Gr)=1andfj=4drr2nj(r)=Z,thenumberofatomicelectronsIntheforwarddirectionG=0,f=Z.AtomicFormFactorQuasicrystalsIn1984quasicrystalswerefirstobserved(D.S.Schechtmanetal.Phys.Rev.Lett.53,1951(198