A more general expression for the average X-ray-diffraction intensity of crystals with an incommensurate one-dimensional modulation

Statistical methods are used to derive an expression for the average X-ray diffraction intensity, as a function of (sin theta)/lambda, of crystals with an incommensurate one-dimensional modulation. Displacive and density modulations are considered, as well as a combination of these two. The atomic modulation functions are given by truncated Fourier series that may contain higher-order harmonics. The resulting expression for the average X-ray diffraction intensity is valid for main reflections and low-order satellite reflections. The modulation of individual atoms is taken into account by the introduction of overall modulation amplitudes. The accuracy of this expression for the average X-ray diffraction intensity is illustrated by comparison with model structures. A definition is presented for normalized structure factors of crystals with an incommensurate one-dimensional modulation that can be used in direct-methods procedures for solving the phase problem in X-ray crystallography. A numerical fitting procedure is described that can extract a scale factor, an overall temperature parameter and overall modulation amplitudes from experimental reflection intensities.