固體物理Chapter 2Reciprocal lattice

1、固體物理固體物理Chapter 2Reciprocal Chapter 2Reciprocal latticelatticeWe study crystal structure through the diffraction of photons, neutrons, and electrons.The Bragg law:sinddndsin2 elastic scattering periodic lattice does not refer to what kind of lattice does not refer to the detail of the basis no infor

2、mation of the intensityDiffraction of waves by crystalsScattered waves amplitudeFourier analysisPeriodic structure of crystalr = T + rThe variance of the electrons density functionn(r + T)= n(r)In one dimension, consider the wave function n(x) with period a in the direction x, the Fourier transforma

3、tion of n(x) can be written as:ppapxinxn)/2exp()(whereapapxixndxan01)/2exp()( In three dimension, consider n(x+a1, y+a2, z+a3) = n(x, y, z), the Fourier transformation of n(x) can be written as:1)/2exp(),(),(11pxaxpizynzyxnGGpppxyzpppxyzppxyrGinzapyapxapinazpiaypiaxpinaypiaxpizn)exp()222(exp)/2exp()

4、/2exp()/2exp()/2exp()/2exp()(32132121,332211,332211,2211Here we define is the reciprocal lattice vector.)2,2,2(332211apapapGThe inversion of the Fourier seriescellCGrGirdVnVn)exp()(1Fourier transformationGGrGinrn)exp()(reciprocal lattice vector)2,2,2(332211apapapGReciprocal lattice (倒格子，倒易點(diǎn)陣) vector

5、sIn general case, consider n(r + T) = n(r + u1a1 + u2a2 + u3a3) = n(r).The Fourier transformation of n(r) is . )exp()(GGrGinrnThe axis vectors b1, b2, b3 of the reciprocal lattice is:.22;22;22213212131332113232321321aaVaaaaabaaVaaaaabaaVaaaaabCCCThe reciprocal lattice vector,332211bvbvbvGProperty of

6、 the reciprocal lattice: Prove:. 0 ; 0 and ,then ;221312312132132132111babaababaabaaaaaabaSame as b2 and b3.Note: bi is unnecessarily parallel to ai.)( 0)( 22jijibaijji1.mGr22. , where is a lattice vector, is a reciprocal lattice vector, and m is integer.rGProve: suppose where ui and vj are integers

7、. , and ,332211332211bvbvbvGauauaur.)()(332211332211bvbvbvauauauGrmvuvuvu2)(23322113. A reciprocal lattice vector is perpendicular to (hkl) plane in the crystal lattice. 321b lbkbhGProve: The plane (hkl) intercepts the axes at point A, B, and C. 1a2a3aGoABCha1ka2la3ha1oA = ;ka2oB = ;la3oC = .AB)(321

8、12b lbkbhhakaG 0312111232212bahlbahkbabaklbabakhABGIn the same way, we can obtain and .ACGBCGSo is perpendicular to (hkl) plane. Ghaka12AB = oB oA = .thenThe distance between two adjacent (hkl) planesGGGhaG21d(hkl) = oA1a2a3aGoABCha1ka2la3d(hkl)For a simple cubic lattice2/12222222lkhaGzalyakxahGThen

9、 2/1222)(lkhadhkl The volume of the primitive cell of the reciprocal lattice where VC is the crystal primitive cell.,)2(3*CCVVProve:The volume of the primitive cell of the reciprocal lattice)()()(2 )(2)(2)(2 )(2113323211332321*aaaaaaVaaVaaVaaVbbbVCCCCCabacbaac)()()(CVaaaaaaaaa1323212131213323)()(2aa

10、aaaaVCCCCVaVaaV31323)(2 )(2 Microscope imagee.g. STM, AFM, TEMMap of the crystal latticeDiffraction patterne.g. XRD, neutron scatteringMap of the reciprocal latticeEvery crystal structure has two lattices: crystal lattice and reciprocal latticeDiffraction conditions and Laue equationsTheorem: the se

11、t of reciprocal lattice vectors G determines the possible x-ray reflections.k kkThe difference in path length:sinrThe difference in phase:For the incident wave krkrsin2The difference in path length:sinrThe difference in phase:rkrsin2For the outgoing wave kThe total difference in phase:rkrkk) (ork kr

12、k ierk ie The scattering amplitude:)exp()( )exp()( rkirndVirndVFSubstitute in, we haveGGrGinrn)exp()(GGrkGindVF)(exp The diffraction condition is:GkThe amplitude is:GVnF For the elastic scattering, k = k.The diffraction condition is:022GGk22GGkorProject the diffraction condition to the crystal axes,

13、 we obtain the Laue equations: 3322112 ;2 ;2vkavkavkaIt can be proved that the condition for diffraction is another statement of the Bragg law.22GGkGdhkl/2)(/2k kkG2)()()/2()2/cos()/2()/2(2hklhklddsin2)(hkld(hkl) may contain a common factor n.ndhklsin2)(Brillouin zonesA Brillouin zone is defined as

14、a Wigner-Seitz primitive cell in the reciprocal lattice.1st Brillouin zone2nd Brillouin zone3rd Brillouin zone4th Brillouin zoneOCDk1k2GDGCGDk2A wave whose wavevector drawn from the origin terminates on the Brillouin zone boundary will satisfy the condition for diffraction.222GGkDGkk22Reciprocal lat

15、tice to sc latticeThe primitive translation vectors of a sc lattice are. ; ; 321zaayaaxaaThen the primitive translation vectors of the reciprocal lattice are. 2) (22; 2) (22; 2) (22323212133232113232321321zazyxayxaaaaaabyazyxaxzaaaaaabxazyxazyaaaaaabThe reciprocal lattice of a sc lattice is a sc lat

16、tice with lattice constant 2/a.The volume of the primitive cell is VC = a3.Reciprocal lattice to bcc latticeThe primitive translation vectors of a bcc lattice are).(2 );(2 );(2321zyxaazyxaazyxaaThen the primitive translation vectors of the reciprocal lattice are).(2) () (242);(2) () (242);(2) () (24

17、2232132313223321yxazyxzyxaaaaVbxzazyxzyxaaaaVbzyazyxzyxaaaaVbCCC）（）（）（）（）（）（The reciprocal lattice of a bcc lattice is a fcc lattice with the lattice constant 4/a.The volume of the primitive cell is VC = a3/2.The volume of Brillouin cell is (4/a)3/4 = 2(2/a)3.Reciprocal lattice to fcc latticeThe pri

18、mitive translation vectors of a fcc lattice are).(2 );(2 );(2321yxaaxzaazyaaThen the primitive translation vectors of the reciprocal lattice are).(2) () (282);(2) () (282);(2) () (282232132313223321zyxaxzzyaaaaVbzyxazyyxaaaaVbzyxayxxzaaaaVbCCC）（）（）（）（）（）（The reciprocal lattice of a fcc lattice is a

19、bcc lattice with the lattice constant 4/a.The volume of the primitive cell is VC = a3/4.The volume of Brillouin cell is (4/a)3/2 = 4(2/a)3.Fourier analysis of the basisStructure factor:cellGrGirndVS)exp()( Superposition of electron concentration functionscelljjjrrnrn)()(jjjrGirrndV)exp()( jjjGindVrG

20、i)exp()( )exp(jrrjjjrGif)exp()exp()( GindVfjjDefined the atomic form factor:GcellGNSrGirndVNrGirndVrkirndVF)exp()( )exp()( )exp()( Defined the structure factor:For the atom j1,0 321jjjjjjjzyxazayaxrFor the reflection labeled by v1, v2, v3332211bvbvbvGjjjjjjjjjjvvvzvyvxvifazayaxbvbvbvifS)(2exp )()(ex

21、p321321332211)(321If SG = 0, the scattered density will be zero.N1(cell)S1(basis)=N2(cell)S2(basis)Structure factor of the bcc latticeTwo identical atoms in the cubic cellf1 = f2 = f,x1 = y1 = z1 = 0 x2 = y2 = z2 = 1/2even en wh2odd when 0 )(exp1321321321)(321vvvfvvvvvvifSvvvStructure factor of the

22、fcc latticeFour identical atoms in the cubic cellf1 = f2 = f3 = f4 = f, (x1, y1 , z1) = (0, 0, 0)(x2, y2 , z2) = (0, 1/2, 1/2)(x3, y3 , z3) = (1/2, 0, 1/2)(x4, y4 , z4) = (1/2, 1/2, 0)else 0even allor odd all are ,n whe4 )(exp)(exp )(exp1321213132)(321vvvfvvivvivvifSvvvAtomic form factor:GrGrrrndriGrdrnrdriGrrnddrdrrGirndVfjjjjj)sin()( 4 )cosexp()(cos)( 2 )cosexp()( sin )exp()( 222If nj(r) = nj(r=0), or G = 0zrrndrfjj )( 42fi is a measure of the scattering power of the jt