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1、Ⅰ.MillerIndices:Inordertodescribeaplaneorasetofplans,oradirectionorasetofdirections,andthenwecanuseMillerIndices:1.Describetheplane1)Findtheinterceptsoftheplaneonthethreeaxesdefinedbythebasisvectors(a1,a2,a3)2)Step1givesthreenumbers.Takethereciprocalofthethreenumbers.3)Dividethereciprocalsbyt
2、heirgreatestcommondivisor(whichyieldsasetofintegers).Theresultingsetofthreenumbers(h,k,l)iscalledtheMillerIndicesfortheplane.{h,k,l}meansallplanesequivalent(bysymmetry)to(h,k,l)2.DescribetheMillerindicesforthedirection1)Findanyvectorinthedesireddirection2)Expressthisvectorintermsofthebasis(a1
3、,a2,a3)3)Dividethecoefficientof(a1,a2,a3)bytheirgreatestdivisor.Theresultingsetofthreeintegers[h,k,l]definesadirection.meansallvectorsequivalentto[h,k,l].Negativesignsinanyofthenumbersareindicatedbyplacingabaroverthenumber.Note:MillerIndicesisdifferentfromthewavevector.Forthewavevector
4、,itdescribesthereciprocalspace.Forthekinthereciprocalspace,italsocontainsthreeparts(kx,ky,kz)ruuruuruurk=nk1x+nk2y+nk3zAndfortherelationbetweenkandGisasfollows:uururuuruuruururkx=bNk1/1,y=b2/Nk2,z=b3/N3AndN1N2,N3arethenumberofunitcellinthedirectionofa1,a2,a3inrealspace.SowhenwementionE-kcurve
5、in(110)direction,wemeank(110)inthereciprocalspace.Ⅱ.TheReciprocalLattice:Sincemanycharacterizationisperiodical,sowedoFourierTransformation,forexample,thechargedensityofn(r)rurrnr()=?uurnGexp(iGr·)GAccordingtotheFourierTransformation,theFouriercoefficientisdefinedasfollows:rurr-1n=VdVr()exp(-i
6、Gr·)GcòncellAndVcisthevolumeofaunitcell,andGiscalledFourierTransformationbasis,soinordertofindtheproperG,wehavethereciprocalspace.Intherealspace,wedefiner=n1a1+n2a2+n3a3,anda1,a2,a3arethebasisfortherealspace.Whencometoreciprocalspace,wedefineb1,b2,b3asthebasis,andtheycomeasfollows:uuruuruuruu
7、ruuruuruur2[2pa·a3]uur2[3pa·a1]uur2[1pa·a2]b1=uuruuruur,b2=uuruuruur,b3=uuruuruura1·a2′a3a1·a2′a3a1·a2′a3Sointhereciprocalspace,wecanwritethevectorasG=v1b1+v2b2+v3b3Forthereasonwhyweusereciprocalspace,inmyunderstanding,itisasfollows:1.Actuall
