Presentation on theme: "VI. Reciprocal lattice 6-1. Definition of reciprocal lattice from a lattice with periodicities in real space Remind what we have learned in chapter."— Presentation transcript:
1 VI. Reciprocal lattice6-1. Definition of reciprocal lattice from a latticewith periodicities in real spaceRemind what we have learned in chapter 5Pattern Fourier transform diffractionPattern of the original pattern!
2 3-D: the Fourier transform of a function f(x,y,z) Note thatux+vy+wz: can be considered as a scalarproduct ofif the following conditions are met!What is ?Then what is ?
3 Consider the requirements for the basic translation vectors of the “reciprocal lattice”Sayi.e.In other words,
4 Similarly,＊ A translation vector in reciprocal lattice iscalled reciprocal lattice vector* orthogonality; orthornormal set
5 * In orthorhombic, tetragonal and cubic systems, * is perpendicular to the plane (h, k, l) inreal spaceCBAThe reciprocal lattice vector
6 is perpendicular to the plane (h, k, l) BASimilarly,Therefore,is perpendicular to the plane (h, k, l)
7 Moreover,interplanar spacingof the plane (h, k, l)CBAoror
9 How to construct a reciprocal lattice from a crystal Pick a set of planes in a crystal and using a directionand a magnitude to represent the planeparallelPlaneset 2d2d1Plane set 1Does it really form alattice?Draw it to convinceyourself!Planeset 3d3
10 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
11 Lattice point in reciprocal space IntegerLattice points in real space
12 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
13 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
14 6-2. Reciprocal lattices corresponding to crystal systems in real space(i) Orthorhombic ,tetragonal ,cubic(ii) Monoclinic*
15 We deal with reciprocal lattice Transformation in Miller indices. (iii) Hexagonal120o30o60o60o30oWe deal with reciprocal latticeTransformation in Miller indices.
26 Here we use primitive cell translation vector to calculate the reciprocal lattice.When we are calculating the interplanar spacing,the reciprocal lattices that we chosen is different.Contradictory?Which one is correct?
27 Using primitive translation vector to do the reciprocal lattice calculation:Case: FCC BCC
29 notWhy?(hkl) defined using unit cell!(hkl) is defined using primitive cell!(HKL)
30 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
31 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