Presentation on theme: "Reciprocal lattice How to construct reciprocal lattice"— Presentation transcript:
1Reciprocal lattice How to construct reciprocal lattice Meaning of reciprocal latticeRelation between reciprocal lattice and diffractionGeometrical relation between reciprocal lattice andoriginal lattice
2parallel d2 d1 Does it really form a lattice? Draw it to convince How to construct a reciprocal lattice from a crystal(1) Pick a set of planes in a crystalparallelPlaneset 2d2d1Plane set 1Does it really form alattice?Draw it to convinceyourself!Planeset 3d3
3Example: a monoclinic crystal Reciprocal lattice (a* and c*) on the planecontaining a and c vectors.(b is out of the plane)cba(-100)(100)(001)c(102)(001)c(002)O(002)aOOa(00-2)c(00-2)(002)Oa(001)cc*(101)c(002)(002)*Oaa*(10-1)(00-1)a(00-2)2D form a 3-D reciprocal lattice
4Lattice point in reciprocal space IntegerLattice points in real space
5Reciprocal lattice cells for cubic crystals: Simple cubic:zyx(002)?zz(010)(100)yyaaxxa(110)(001)yySimple cubicxx
7(111)(111)OO(111)FCCcornerUp and downO(222)F and BL and R
8Vector: dot and cross product v v.u = |v||u|cos You should do the same for a FCC and show it formsa BCC lattice! (Homework!)Vector: dot and cross productvv.u = |v||u|cosProjection of v onto u and timeseach other (scaler)!u|v|cosvu = |v||u|sin w|v|sin|v||u|sin is the area of theparallelogram. w v and uu
9Relationships between a, b, c and a*, b*, c*: Monoclinic: plane y-axis (b): c c*.cc*Similarly,bd001acc* = |c*|ccos, |c*| = 1/d001 ccos = d001 cc* = 1
10 The Weiss zone law or zone equation: Similarly, aa* = 1 and bb* =1.c* //ab,Define c* = k (ab), k : a constant.cc* = 1 ck(ab) = 1 k = 1/[c(ab)]=1/V.V: volume of the unit cellSimilarly, one gets The Weiss zone law or zone equation:A plane (hkl) lies in a zone [uvw] (the plane containsthe direction [uvw]). d*hkl (hkl) d*hkl ruvw = 0
11Define the unit vector in the d*hkl direction i, d*hkl nth planeruvwDefine the unit vector in the d*hkldirection i,d*hklr2uvwr1
12 Reciprocal Lattice: Fourier transform of the spatial wavefunction of the original lattice wave process (e. g. electromagnetic) in the crystalCrystal: periodicPhysical properties function of a crystalCrystal translation vectorPeriodic function Exponential Fourier Series
13If k (reciprocal) lattice ; T original lattice! u, v, w: integerTranslation vectors of the original crystal latticefor all TIfh, k, l: integerVectors of the reciprocal latticealways integerIf k (reciprocal) lattice ; T original lattice!Vice versa!
14k (in general): momentum space vector; G: reciprocal lattice points In crystallographyIn SSPk (in general): momentum space vector;G: reciprocal lattice points
15Proof: the reciprocal lattice of BCC is FCC Use primitive translation vectors onlyBCCFCCxyzcornerUp and downF and BL and R
17The vector set is the same as the FCC primitive translation vector.Unit of the reciprocal lattice is 1/length.
18Mathematics of Interference Sum of two waves:assume A1 = A2 = ARnew amplitude AR
19Geometrical analysis of Interference term rotation vectorA
20Complex Wave Representation of Interference assume A1 = A2 = A
21Diffraction conditions and reciprocal lattices: Theorem: The set of reciprocal lattice vectors Gdetermines the possible X-ray reflections.krrkk
22Complex exponential form Complex numberWhat happen to the time dependent term?X-ray wavelength ~ 0.1 nm ~ 3x1018 1/sDetectors get the average intensity!Detectors measured the intensity only!A lot of time, examining is enough!
23Fourier expansion n(r) 0 for G = k otherwise, = 0 Path differenceSimilarly,Phase angledVPhase angle= (2/)rsin= krrOFourier expansion n(r) 0 for G = kotherwise, = 0The diffraction condition is G = k. k + G = k’
25More geometric relation between direct lattice and reciprocal lattice:e1, e2, e3: contravariant basis vector of R3covariant basis vectors e1, e2, e3 (reciprocal lattice)ei and ei are not normal, but mutually orthonormal:For any vector v:
26v can be expressed in two (reciprocal) ways: Einstein’s summationconvention, omitting No proof here, but you can check whether these relationis correct or not?Use BCC or FCC lattice as examples, next page!
27Use the BCC lattice as examples AssumeYou can checkthe other wayaround.
39notWhy?(hkl) defined using unit cell!(hkl) is defined using primitive cell!(HKL)
40 Find out the relation between the (hkl) and Example Find out the relation between the (hkl) and[uvw] in the unit cell defined by and the(HKL) and [UVW] in the unit cell defined by.In terms of matrix
41Find out the relation between (hkl) and (HKL). Assume there is the first planeintersecting the a axis ata/h and the b axis at b/k.In the length of |a|, thereare h planes. In the lengthof |b|, there are k planes.How many planes canbe inserted in the length|A|? Ans. h + 2k H = 1h + 2k + 0lSimilarly, K = -1h + 1k +0l and L = 0h + 0k + 1lABb2kab/ka/hhA/(h+2k)or
45Interplanar spacing (defined based on unit cell) Cubic:Tetragonal:Orthorombic:Hexagonal:
46Get the metric tensor!Perform the inversion of the matrix!Comparing the inversion of the metric tensor indirect lattice with the metric tensor in reciprocal lattice Geometrical relation between reciprocal lattice anddirect lattice can be obtained!