Presentation on theme: "Reciprocal lattice How to construct reciprocal lattice"— Presentation transcript:
1 Reciprocal lattice How to construct reciprocal lattice Meaning of reciprocal latticeRelation between reciprocal lattice and diffractionGeometrical relation between reciprocal lattice andoriginal lattice
2 parallel 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
3 Example: 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
4 Lattice point in reciprocal space IntegerLattice points in real space
5 Reciprocal 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
8 Vector: 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
9 Relationships 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
11 Define 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
13 If 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!
14 k (in general): momentum space vector; G: reciprocal lattice points In crystallographyIn SSPk (in general): momentum space vector;G: reciprocal lattice points
15 Proof: the reciprocal lattice of BCC is FCC Use primitive translation vectors onlyBCCFCCxyzcornerUp and downF and BL and R
17 The vector set is the same as the FCC primitive translation vector.Unit of the reciprocal lattice is 1/length.
18 Mathematics of Interference Sum of two waves:assume A1 = A2 = ARnew amplitude AR
19 Geometrical analysis of Interference term rotation vectorA
20 Complex Wave Representation of Interference assume A1 = A2 = A
21 Diffraction conditions and reciprocal lattices: Theorem: The set of reciprocal lattice vectors Gdetermines the possible X-ray reflections.krrkk
22 Complex 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!
23 Fourier 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’
25 More 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:
26 v 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!
27 Use the BCC lattice as examples AssumeYou can checkthe other wayaround.
39 notWhy?(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
41 Find 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
45 Interplanar spacing (defined based on unit cell) Cubic:Tetragonal:Orthorombic:Hexagonal:
46 Get 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!
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