The heart of crystalline materials

Bragg's law expresses a necessary condition for diffraction to occur, but it is not a sufficient condition. The corresponding diffracted beam must have an intensity that is not zero. This intensity is expressed as a function of integers \(h’\), \(k’\), \(l’\) such as \(h’=n.h, k’=n.k\) and \(l’=n.l\) with \(h\), \(k\) and \(l\) Miller indices for the family of diffracting planes. The diffracted intensity can be expressed as:

\(I _{h\prime k\prime l\prime} = A. {\mid F_{h\prime k\prime l\prime} \mid }^2\)

where \(F _{hkl}\) is the structure factor of the diffraction considered, \(n\) the order of diffraction and \(A\) a function of the Bragg angle and various other parameters. The structure factor is a term that takes account of the arrangement of atoms within a crystal unit cell, i.e. the pattern.

\(F _{hkl}\) is expressed as follows:

\(F_{hkl} = \sum ^s _{j=1} {f_{j} \exp{\left[ 2i\pi \left( hx_j + ky_j + lz_i \right) \right] }}\)

where \(s\) is the number of atoms per unit cell and \(f_j\) the diffusion factor, which essentially depends on the atomic number of the atoms considered.

If Bragg's law is met, there will be diffraction if \(F_{hkl}\) is not zero, which imposes conditions on \(h’\), \(k’\) and \(l’\). For example, for the \(\ce{FCC}\) structure, \(F_{hkl} \neq 0\) if \(h’\), \(k’\) and \(l’\) have the same parity; for the \(\ce{BCC}\) structure, \(F_{hkl} \neq 0\) if \(h’+ k’ + l’\) is even.

These extinction rules are summarised in the following table.