Diffraction from a rotating crystal

It has been seen that methods of recording X-ray intensities usually involve a crystal rotating in the incident X-ray beam. We shall now look at the problem of determining the total energy in a particular diffracted beam produced during one pass of the crystal through a diffracting position. In order to do this we must make some assumptions about the geometry of the diffraction process; the configuration we shall take is that the crystal is rotating about some axis with a constant angular velocity ω and that the incident and diffracted beams are both perpendicular to the axis of rotation.

Let us first look at the situation when we have a stationary crystal in a diffracting position. Associated with the crystal, and fixed relative to it, there is a reciprocal space within which is defined the Fourier transform, Fx(s), of the electron density of the crystal. For a theoretically perfect crystal of infinite extent the value of Fx(s) would be zero everywhere except at the nodes of a δ-function reciprocal lattice, the weight associated with the point (hkl) being (l/V)Fhkl. However, if the crystal is imperfect in some way there may be non-zero Fx(s) well away from the reciprocal-lattice points and for a finite crystal there will be a small region of appreciable Fx(s) around each of the reciprocal-lattice points. The imperfect-crystal case we shall not consider here but we shall be concerned with the size of the crystal, for this is a factor which must be present in every diffraction experiment.

Consider a crystal completely bathed in an incident beam of intensity Io.