Understanding Wave Diffraction and Reciprocal Lattice in Crystal Structures

1. Dept of Phys M.C. Chang • Wave diffraction and the reciprocal lattice • reciprocal lattice (倒晶格) • definition, examples • reciprocal lattice vectors and Fourier analysis • reciprocal lattice vectors and lattice plane • diffraction of waves by crystals

2. The reciprocal lattice • (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.2)

3. z 2/a a y x Simple cubic lattice z y x Note: When the original lattice (the direct lattice) rotates, its reciprocal lattice rotates the same amount as well.

4. z z y 4/a y x x FCC lattice BCC lattice a

5. Direct lattice Reciprocal lattice cubic (a) cubic (2/a) fcc (a) bcc (4/a) bcc (a) fcc (4/a) hexagonal (a, c) hexagonal (4/3a,2/c) (See Prob.2) and rotated by 30 degrees • What’s the use of the reciprocal lattice? • Fourier decomposition of a lattice-periodic function • von Laue’s diffraction condition k’ = k+G

6. reciprocal lattice • definition, examples • reciprocal lattice vectors and Fourier analysis • reciprocal lattice vectors and lattice plane • diffraction of waves by crystals

7. Two simple properties: Conversely,assume GR=2integer for all R,

8. Fourier decomposition and reciprocal lattice vectors 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 If f(r) has lattice translation symmetry, that is, f(r)=f(r+R) for any lattice vector R [eg. f(r) can be the charge distribution in lattice], then it can be expanded as, where G is the reciprocal lattice vector.

9. A simple example: electron density of a 1-dim lattice (x) x a

10. reciprocal lattice • definition, examples • reciprocal lattice vectors and Fourier analysis • reciprocal lattice vectors and lattice plane • diffraction of waves by crystals

11. Ghkl R For a cubic lattice In general, planes with higher index have smaller inter-plane distance Geometrical relation between Ghkl and (hkl) planes (Prob. 1) (hkl) lattice planes dhkl

12. reciprocal lattice • diffraction of waves by crystals • Bragg’s condition (布拉格繞射條件) • von Laue’s condition • Brillouin zone • structure factor Applies to electron wave and neutron wave Lattice as a collection of lattice points Lattice as a collection of lattice planes

13. 1915 • mirror-like reflection from crystal planes when • 2dsin = n • Difference from the usual mirror reflection: •  > 2d, no reflection •  < 2d, reflection only at certain angles • Measure ,  get distance between crystal planes d Braggs’ view of the diffraction (1912, father and son) Treat the lattice as a stack of lattice planes

14. 2dsin = n

16. DNA, Watson and Crick, 1953 a ribosome hemoglobin Perutz Hodgkin Kendrew Braggs, 1914 Shen et al, Phys Today Mar, 2006

17. reciprocal lattice • diffraction of waves by crystals • Bragg’s condition • von Laue’s condition • Ewald construction • Equivalence with Bragg’s condition • Brillouin zone • structure factor

18. 1914 • Crystal diffraction = scattering from an array of atoms (Von Laue, 1912) You can view the same phenomena from 2 (or more) different angles, and each can get you a Nobel prize! One-atom scattering: atomic form factor (n() is the atom charge distribution) • If the wave length is much larger than the atom size, then the scattered wave would be spherical (no angle-dependence in fa). [Born approximation]

19. r1 r2 Two-atom scattering( 2-slit experiment)

20. This is actually the “scattering amplitude” in Kittel’s Eq. 2.18 • In general, for a crystalwith p-atom basis, a atomic form factor for the j-th atom where dj is the location of the j-th atom in a unit cell. structure factor N-atom scattering For a simple lattice, just sum over the waves The lattice-sum can be separated! Laue‘s diffraction condition Number of atoms in the crystal

21. One dimensional case ak 2 ||2 From http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html also see Prob. 4.

22. It’s convenient to solve it graphically using • the Ewald construction (Ewald 構圖法) More than one (or none) solution may be found. • 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.

23. Ghkl k k’ ’ a(hkl)-lattice plane It’s easy to see that =’ because |k|=|k’|. Bragg’s diffraction condition 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’

24. Another view of the diffraction Ghkl k The k vector will point to the plane bi-secting the Ghkl vector.

25. reciprocal lattice • diffraction of waves by crystals • Bragg’s condition • von Laue’s condition • Brillouin zone • structure factor

26. Triangle lattice direct lattice reciprocal lattice BZ Brillouin zone (useful later in chap 7) Def. of the first BZ A BZ is a primitive unit cell of the reciprocal lattice

27. z 4/a y x The first BZ of bcc lattice (its reciprocal lattice is fcc lattice) 4/a The first BZ of fcc lattice (its reciprocal lattice is bcc lattice)

28. reciprocal lattice • diffraction of waves by crystals • Braggs’ condition • von Laue’s condition • Brillouin zone • structure factor • atomic form factor For a crystal structure with a basis

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

30. Try: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.

31. Atomic form factor and intensity of diffraction cubic lattice with lattice const. a/2 fK  fCl fcc lattice h,k,l all even or all odd fK  fBr