Diffraction: Real Sample (From Chapter 5 of Textbook 2, Chapter 9 of reference 1,) Different sizes, strains, amorphous, ordering  Diffraction peaks

t = md hkl    +  …… m Constructive Interference    +   + . … Destructive interference: extra path difference (plane 0 and plane m/2): /2

 << 1  cos  ~ 1 and sin  ~ . = t Broadening (  ): FWHM Thickness dependent (just like slits)

2B2B  2(  B -  )2(  B +  ) Scherrer’s formula K: in general = 0.9 but very close to 1 depends on the crystal shape

22 2B2B The size of the reciprocal lattice point  1/t

Interference function:    B intensity ≠0; 2  (s – s 0 )/ ≠G. Now (s - s 0 )/ is deviation from the reciprocal lattice:

The same as the Fraunhofer diffraction for a one dimensional net

Interference function For reasonable number of unit cells, ripples beyond main peak are very weak Some define interference function as

Telling the same thing: Size of reciprocal lattice points (nodes): point  1/t

Mosaic Structure  B -  <  <  B + .

Strain b: extra broadening induced by the non uniform strain Strain  peak shift

Directly sum over all the scattered waves: Amorphous and partially crystalline samples individual atoms  electron density  (x).

individual atoms: x 1, x 2, …, x n. Density of electron cloud of the n th atom (x n )   n (x - x n ) Atomic form factor Total density of electron cloud: xnxn x x - x n } Fourier Transform

Scattering power of the object: Unit scattering power: n = n term:

n  n term: N(N-1) Identical atoms or atomic groups with structure factor F.

Expression in term of electron density Define x = u - u;

Patterson Function or Autocorrelation Function with respect to atomic density Inverse Fourier transform  N identical atoms or atomic groups: F (common scattering or structure factor)  a as the local atomic density instead of electron density

autocorrelation with itself Autocorrelation with all other atoms  pair correlation function where Pair correlation (distribution) function g(x): related to the probability of finding the center of a particle a given distance from the center of another particle

From intensity measurement  Fourier transform S(G) - 1  g(x) V: volume Example:

There are free software on the internet that can be downloaded for extracting PCF. E.g. etc.