Wave diffraction and the reciprocal lattice

Dept of Phys M.C. Chang Wave diffraction and the reciprocal lattice

Braggs’ theory of diffraction • Reciprocal lattice • von Laue’s theory of diffraction

Braggs’ view of the diffraction (1912, father and son) Treat the lattice as a stack of lattice planes 25 1915 • mirror-like reflection from crystal planes when • 2dsinθ = nλ • Difference from the usual mirror reflection: • λ > 2d, no reflection •  λ < 2d, reflection only at certain angles • Measure λ, θ→ get distance between crystal planes d

2dsinθ = nλ

Powder method For more, see www.xtal.iqfr.csic.es/Cristalografia/parte_06-en.html

k -2g -g 0 g 2g Fourier transform of the electron density of a 1-dim lattice ρ(x) Lattice in real space x a ρn Lattice in momentum space (reciprocal lattice) simplest example

important • Reciprocal lattice(倒晶格 ) • (direct) lattice reciprocal lattice • primitive vectors a1,a2,a3 primitive vectors b1,b2,b3 Def. 1 Def. 2 • The reciprocal of a reciprocal lattice is the direct lattice (obvious from Def.1)

z z 2π/a y a y x x Ex: Simple cubic lattice

BCC lattice z z 4π/a y y x x FCC lattice a

Two simple properties: 1. 2. Conversely, assume G．R=2π×integer for all R, then G must be a reciprocal lattice vector.

important • The expansion above is very general, it applies to • all types of periodic lattice (e.g. bcc, fcc, tetragonal, orthorombic...) • in all dimensions (1, 2, and 3) All you need to do is to find out the reciprocal lattice vectors G. If f(r) has lattice translation symmetry, that is, f(r)=f(r+R) for any lattice vector R, then it can be expanded as, , where G is the reciprocal lattice vector. Pf:

Summary • The reciprocal lattice is useful in • Fourier decomposition of a lattice-periodic function • von Laue’s diffraction condition k’ = k+G (later) Direct lattice Reciprocal lattice cubic (a) cubic (2π/a) fcc (a) bcc (4π/a) bcc (a) fcc (4π/a) hexagonal (a,c) hexagonal (4π/√3a,2π/c) and rotated by 30 degrees (See Prob.2)

important m/l a3 m/k a2 a1 m/h Geometrical relation between Ghkl vector and (hkl) planes Pf: v2 v1 • When the direct lattice rotates, its reciprocal lattice rotates the same amount as well.

important Ghkl R Inter-plane distance (hkl) lattice planes dhkl Given h,k,l, and n, one can always find a lattice vector RarXiv:0805.1702 [math.GM] For a cubic lattice • In general, planes with higher index have smaller inter-plane distance

Scattering from an array of atoms(Von Laue, 1912) • The same analysis applies to EM wave, electron wave, neutron wave… etc. First, scattering off an atom at the origin: 1914 • Atomic form factor: Fourier transform of atom charge distribution n() 原子結構因數

The atomic form factor (See Prob.6) 10 electrons tighter ~ Kr 36 ~ Ar 18 http://capsicum.me.utexas.edu/ChE386K/docs/29_electron_atomic_scattering.ppt

R • Scattering off an atom not at the origin A relative phase w.r.t. an atom at the origin • Two-atom scattering

N-atom scattering:one dimensional lattice http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html aΔk 2 |ψ|2

important N-atom scattering(3D lattice, neglect multiple scatterings) For asimple latticewith no basis, The lattice-sum can be separated, Laue‘s diffraction condition Number of atoms in the crystal

important a • Previous calculation is for a simple lattice, now we calculate the scattering from a crystalwith basis dj : location of the j-th atom in a unit cell Eg., atomic form factorfor the j-th atom Structure factor (of the basis)

Reciprocal of cubic lattice = 4fa when h,k,l are all odd or all even = 0 otherwise Eliminates all the points in the reciprocal cubic lattice with S=0. The result is a bcc lattice, as it should be! Example: The structure factor for fcc lattice (= cubic lattice with a 4-point basis) Cubic lattice

Atomic form factorand intensity of diffraction fK ～ fCl cubic lattice with lattice const. a/2 fK ≠ fBr fcc lattice h,k,l all even or all odd

Summary • Find out the structure factor of the honeycomb structure, then draw its reciprocal structure. Different points in the reciprocal structure may have different structure factors. Draw a larger dots if the associated |S|2 is larger.

k k’ • Laue’s diffraction condition • k’ = k+Ghkl • Given an incident k, want to find a k’ that satisfies this condition(under the constraint |k’|=|k|) • One problem: there are infinitely many Ghkl’s. • It’s convenient to solve it graphically usingthe Ewald construction • (Ewald 構圖法) More than one (or none) solutions may be found. G Reciprocal lattice

http://capsicum.me.utexas.edu/ChE386K/docs/28_The_Laue_Experiment.ppthttp://capsicum.me.utexas.edu/ChE386K/docs/28_The_Laue_Experiment.ppt

Ghkl k k’ θθ’ a(hkl)-lattice plane • It’s easy to see that θ = θ’ because |k|=|k’|. Laue’s condition = Braggs’ condition • From the Laue condition, we have Ghkl k • Given k and Ghkl, we can find the diffracted wave vector k’ Integer multiple of the smallest G is allowed Bragg’s diffraction condition

Another view of the Laue condition Ghkl k ∴ The k vector that points to the plane bi-secting a Ghkl vector will be diffracted. Reciprocal lattice

Triangle lattice direct lattice reciprocal lattice BZ Brillouin zoneDef. of the first BZ A BZ is a primitive unit cell of the reciprocal lattice

z 4π/a y x • The first BZ of bcc lattice (its reciprocal lattice is fcc lattice) 4π/a • The first BZ of fcc lattice (its reciprocal lattice is bcc lattice)

Wave diffraction and the reciprocal lattice