# Determining structure factors for crystals from lattice and motif

I understand that the structure factor for a crystal is given by:

$$F_{hkl} = \sum_r f_{r} e^{2{\pi}i(hx_{r}+ky_{r}+lz_{r})}$$

My textbook (''Structures of crystals'', Glazier) suggests that you can calculate structure factors if you know the lattice type and motif of the crystal, simply by finding the product of the lattice and the motif structure factors.

Can this method be used generally, and if so, do you need to define the motif relative to a centre of symmetry in order to get real structure factors for centrosymmetric crystals ?

## 1 Answer

Q1) A crystal is a convolution of a lattice and motif, so by the convolution theorem, the fourier transform of the crystal (i.e. your structure factor) is the product of the fourier transform of the lattice and the fourier transform of the motif. That is, this is general.

Q2) If the crystal is centrosymmetric* then the structure factor is real - this is a property of the fourier transform (the imaginary sine terms have opposite signs and equal other and are therefore zero). If your lattice and motif generate a centrosymmetric crystal then your fourier transform will be real. The motif doesn't need to be centrosymmetric to generate a centrosymmetric crystal.

*and the atomic form factors are real, but this isn't relevant.