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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.

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Wave diffraction and the reciprocal lattice

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

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

  3. 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

  4. 2dsinθ = nλ

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

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

  7. 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

  8. 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)

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

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

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

  12. 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:

  13. 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)

  14. 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.

  15. 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

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

  17. 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

  18. 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

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

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

  21. 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

  22. 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)

  23. 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

  24. 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

  25. 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.

  26. 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

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

  28. 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

  29. 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

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

  31. 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)

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