Reciprocal lattice How to construct reciprocal lattice Meaning of reciprocal lattice Relation between reciprocal lattice and diffraction Geometrical relation between reciprocal lattice and original lattice

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 crystal parallel Plane set 2 d2 d1 Plane set 1 Does it really form a lattice? Draw it to convince yourself! Plane set 3 d3

Example: a monoclinic crystal Reciprocal lattice (a* and c*) on the plane containing a and c vectors. (b is out of the plane) c b a (-100) (100) (001) c (102) (001) c (002) O (002)  a O O a (00-2) c (00-2) (002) O a (001) c c* (101) c (002)  (002) * O  a a* (10-1) (00-1) a (00-2) 2D form a 3-D reciprocal lattice

Lattice point in reciprocal space Integer Lattice points in real space

Reciprocal lattice cells for cubic crystals: Simple cubic: z y x (002)? z z (010) (100) y y a a x x a (110) (001) y y Simple cubic x x

Base centered cubic (BCC): z y x

(111) (111) O O (111) FCC corner Up and down O (222) F and B L and R

Vector: dot and cross product v v.u = |v||u|cos  You should do the same for a FCC and show it forms a BCC lattice! (Homework!) Vector: dot and cross product v v.u = |v||u|cos  Projection of v onto u and times each other (scaler)! u |v|cos vu = |v||u|sin w |v|sin  |v||u|sin is the area of the parallelogram. w  v and u u

Relationships between a, b, c and a*, b*, c*: Monoclinic: plane  y-axis (b) : c  c*. c  c* Similarly, b d001 a cc* = |c*|ccos, |c*| = 1/d001  ccos = d001  cc* = 1

 The Weiss zone law or zone equation: Similarly, aa* = 1 and bb* =1. c* //ab, Define c* = k (ab), k : a constant. cc* = 1  ck(ab) = 1  k = 1/[c(ab)]=1/V. V: volume of the unit cell  Similarly, one gets  The Weiss zone law or zone equation: A plane (hkl) lies in a zone [uvw] (the plane contains the direction [uvw]). d*hkl  (hkl)  d*hkl ruvw = 0 

Define the unit vector in the d*hkl direction i, d*hkl nth plane ruvw Define the unit vector in the d*hkl direction i, d*hkl r2 uvw r1

 Reciprocal Lattice: Fourier transform of the spatial wavefunction of the original lattice  wave process (e. g. electromagnetic) in the crystal Crystal: periodic Physical properties function of a crystal Crystal translation vector Periodic function  Exponential Fourier Series

If k  (reciprocal) lattice ; T  original lattice! u, v, w: integer Translation vectors of the original crystal lattice for all T If h, k, l: integer Vectors of the reciprocal lattice always integer If k  (reciprocal) lattice ; T  original lattice! Vice versa!

k (in general): momentum space vector; G: reciprocal lattice points In crystallography In SSP k (in general): momentum space vector; G: reciprocal lattice points

Proof: the reciprocal lattice of BCC is FCC Use primitive translation vectors only BCC FCC x y z corner Up and down F and B L and R

BCC

The vector set is the same as the FCC primitive translation vector. Unit of the reciprocal lattice is 1/length.

Mathematics of Interference Sum of two waves: assume A1 = A2 = A R new amplitude AR

Geometrical analysis of Interference term  rotation vector A

Complex Wave Representation of Interference assume A1 = A2 = A

Diffraction conditions and reciprocal lattices: Theorem: The set of reciprocal lattice vectors G determines the possible X-ray reflections. k r r k http://en.wikipedia.org/wiki/Plane_wave k

Complex exponential form Complex number What happen to the time dependent term? X-ray wavelength ~ 0.1 nm   ~ 3x1018 1/s Detectors get the average intensity! Detectors measured the intensity only! A lot of time, examining is enough!

Fourier expansion n(r)  0 for G = k otherwise, = 0 Path difference Similarly, Phase angle dV Phase angle = (2/)rsin = kr r O  Fourier expansion n(r)  0 for G = k otherwise, = 0 The diffraction condition is G = k.  k + G = k’

G: reciprocal lattice, -G: reciprocal lattice? ____ Bragg condition? k G (hkl) plane  or k k dhkl Bragg law G = k.

More geometric relation between direct lattice and reciprocal lattice: e1, e2, e3: contravariant basis vector of R3 covariant basis vectors e1, e2, e3 (reciprocal lattice) ei and ei are not normal, but mutually orthonormal: For any vector v:

v can be expressed in two (reciprocal) ways: Einstein’s summation convention, omitting  No proof here, but you can check whether these relation is correct or not? Use BCC or FCC lattice as examples, next page!

Use the BCC lattice as examples Assume You can check the other way around.

where Similarly, Prove

gij: metric tensor in direct lattice a, b, c and , ,  : direct lattice parameters (standard definition of the Bravais lattice) det|gij| = V2.

Matrix inversion:

Inverting the matrix gij. gij : metric tensor in reciprocal lattice a*, b*, c* and *, *, *: reciprocal lattice parameters These two are the same!

One gets relation like Similarly, ……………..

d-spacing of (hkl) plane for any crystal system

Example：FCC  BCC (1) Find the primitive unit cell of the selected structure (2) Identify the unit vectors

Volume of F.C.C. is a3. There are four atoms per unit cell!  the volume for the primitive of a F.C.C. structure is ?

Similarly,  B.C.C. See page 23

Using primitive translation vector to do the reciprocal lattice calculation: Case: FCC  BCC

not Why? (hkl) defined using unit cell! (hkl) is defined using primitive cell! (HKL)

 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

Find out the relation between (hkl) and (HKL). Assume there is the first plane intersecting the a axis at a/h and the b axis at b/k. In the length of |a|, there are h planes. In the length of |b|, there are k planes. How many planes can be inserted in the length |A|? Ans. h + 2k  H = 1h + 2k + 0l Similarly, K = -1h + 1k +0l and L = 0h + 0k + 1l A B b 2k a b/k a/h h A/(h+2k) or

There are the same! Or

We proof the other way around!

Interplanar spacing (defined based on unit cell) Cubic: Tetragonal: Orthorombic: Hexagonal:

Get the metric tensor! Perform the inversion of the matrix! Comparing the inversion of the metric tensor in direct lattice with the metric tensor in reciprocal lattice  Geometrical relation between reciprocal lattice and direct lattice can be obtained!