The diffraction of X-rays by close-packed polytypic crystals containing single stacking faults. I. General theory

A general theory of X-ray diffraction by one-dimensionally disordered close-packed polytypic crystals has been developed. A random distribution of all possible single stacking faults for arbitrary n-periodical structure is taken into consideration. Exact expressions for measurable parameters of changes in the intensity distribution caused by faults are given. Initial equations of the theory have been formulated by applying Holloway's [J. Appl. Phys. (1969), 40, 4313-4321] method of analytic solution. For the mathematical description of disorder, successive layers of the perfect structure have been denoted by an additional subscript j. The probability of occurrence of the faulted layer with subscript k after the layer with subscript j has been denoted by α_jk. A set of recurrence relations is developed for average phase factors of layers with subscript j on m positions. An arbitrary sequence of layers is written in these relations by using the so-called phase-change factors (after layer with subscript j), determined by Hägg's structure symbols. Terms of the coefficients of the characteristic equation and boundary conditions which are necessary to describe the change in the intensity distribution are given for small values of α_jk. Finally, the shifts and broadenings of the reciprocal-lattice points and the changes in the intensity of peak maxima are derived in terms of α_jk.