Wave diffraction and the reciprocal lattice Dept of

  • Slides: 31
Download presentation
Wave diffraction and the reciprocal lattice Dept of Phys M. C. Chang

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

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

• 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

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 2 dsinθ = nλ • Difference from the usual mirror reflection: λ > 2 d, no reflection λ < 2 d, reflection only at certain angles • Measure λ, θ → get distance between crystal planes d

2 dsinθ = nλ

2 dsinθ = nλ

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

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

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

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

Fourier transform of the electron density of a 1 -dim lattice ρ(x) Lattice in

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

important Reciprocal lattice (倒晶格 ) (direct) lattice primitive vectors a 1, a 2, a

important Reciprocal lattice (倒晶格 ) (direct) lattice primitive vectors a 1, a 2, a 3 reciprocal lattice primitive vectors b 1, b 2, b 3 Def. 1 Def. 2 • The reciprocal of a reciprocal lattice is the direct lattice (obvious from Def. 1)

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

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

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

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

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

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

important If f(r) has lattice translation symmetry, that is, f(r)=f(r+R) for any lattice vector

important 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: 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.

Summary The reciprocal lattice is useful in • Fourier decomposition of a lattice-periodic function

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/√ 3 a, 2π/c) and rotated by 30 degrees (See Prob. 2)

important Geometrical relation between Ghkl vector and (hkl) planes Pf: m/l a 3 v

important Geometrical relation between Ghkl vector and (hkl) planes Pf: m/l a 3 v 1 a 1 v 2 a 2 m/h • When the direct lattice rotates, its reciprocal lattice rotates the same amount as well. m/k

important Inter-plane distance (hkl) lattice planes Ghkl R dhkl Given h, k, l, and

important Inter-plane distance (hkl) lattice planes Ghkl R dhkl Given h, k, l, and n, one can always find a lattice vector R ar. Xiv: 0805. 1702 [math. GM] For a cubic lattice • In general, planes with higher index have smaller inter-plane distance

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

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

Scattering from an array of atoms (Von Laue, 1912) • The same analysis applies

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 ~

The atomic form factor (See Prob. 6) 10 electrons tighter ~ Kr 36 ~ Ar 18 http: //capsicum. me. utexas. edu/Ch. E 386 K/docs/29_electron_atomic_scattering. ppt

 • Scattering off an atom not at the origin A relative phase w.

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

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

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

important N-atom scattering (3 D lattice, neglect multiple scatterings) For a simple lattice with

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

important Previous calculation is for a simple lattice, now we calculate the scattering from

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

Example: The structure factor fcc lattice (= cubic lattice with a 4 -point basis)

Example: The structure factor fcc lattice (= cubic lattice with a 4 -point basis) Cubic lattice Reciprocal of cubic lattice = 4 fa 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!

Atomic form factor and intensity of diffraction f. K ~ f. Cl cubic lattice

Atomic form factor and intensity of diffraction f. K ~ f. Cl cubic lattice with lattice const. a/2 f. K ≠ f. Br fcc lattice h, k, l all even or all odd

Summary • Find out the structure factor of the honeycomb structure, then draw its

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.

Laue’s diffraction condition k’ = k + Ghkl • Given an incident k, want

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 using the Ewald construction (Ewald 構圖法) k k’ G Reciprocal lattice More than one (or none) solutions may be found.

http: //capsicum. me. utexas. edu/Ch. E 386 K/docs/28_The_Laue_Experiment. ppt

http: //capsicum. me. utexas. edu/Ch. E 386 K/docs/28_The_Laue_Experiment. ppt

Laue’s condition = Braggs’ condition • From the Laue condition, we have Ghkl 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’ Ghkl k θ θ’ k’ a (hkl)-lattice plane • It’s easy to see that θ = θ’ because |k|=|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

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

Brillouin zone Def. of the first BZ A BZ is a primitive unit cell

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

 • The first BZ of fcc lattice (its reciprocal lattice is bcc lattice)

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