Diffraction by an Ideal Crystal

Consider the elastic scattering of a parallel and monochromatic x-ray beam by an ideal periodic crystal. A rigorous treatment of the situation is given by Warren [31]. The incident x-ray is represented by the wavevector ${\bf k}_o$, the scattered x-ray by the wavevector ${\bf k}_s$. The position of a scattering centre within the crystal, relative to an arbitrary origin $O$, is given by

$${\bf R} = {\bf m} + {\bf r}_n$$ (4.9)

where m is the lattice vector of the $m$th unit cell and ${\bf r}_n$ the position of the $n$th atom in that cell, relative to the cell's origin. The resulting scattered wave is due to the superposition of many scattered waves, each with a phase shift due to the position of R. The amplitude, $A$, of the scattered wave is therefore obtained by a summation over all unit cells in the scattering volume, there being $N$ cells in total:

$$A( {\bf k}_s - {\bf k}_o) = \sum_{m=0}^N \exp(i({\bf k}_s - ... ...ggl(\sum_n f_n \exp(i({\bf k}_s - {\bf k}_o).{\bf r}_n \biggr)$$ (4.10)

Here $f_n$ is the atomic form factor of the $n$th atom, a function characteristic of the electron density for that atomic species.

The scattered amplitude is, of course, closely related to the density Fourier summation discussed in equation 2.3, i.e. the Fourier transform of the electron density. The first of the summations in equation 2.10 describes the size and geometry of the scattering volume considered. Whilst the second, the summation over $n$, is independent of such considerations and is intrinsic to a particular structure. It is called the structure factor, $S$, and is equivalent to the Fourier coefficients $n_k$ of 2.3. In an ideal periodic crystal the structure will be identical for all unit cells. The same arguments as for equation 2.3 may be applied so that $( {\bf k}_s - {\bf k}_o) = {\bf G}$ must be satisfied for a fundamental reflection, where equation 2.7 defines G as a reciprocal lattice vector.

The intensity, $I$, is the experimentally measured quantity. It must have a real value and is obtained from the complex conjugate of the scattered amplitude:

$$I = A({\bf k}_s - {\bf k}_o)A^*({\bf k}_s - {\bf k}_o)$$ (4.11)