X-Ray Diffraction in Random Layer Lattices

Random layer lattice structures are considered which consist of layers arranged parallel and equidistant, but random in translation parallel to the layer, and rotation about the normal. We call $a$ and $b$ the axes in the layer, and $c$ the axis normal to the layer. In this notation there will be crystalline reflections of type ($00l$), two-dimensional lattice reflections of type ($\mathrm{hk}$), and no general reflections ($\mathrm{hkl}$). Equations are developed for the intensity distribution in a two-dimensional powder reflection, and for the integrated intensity. Equations are also de-developed for the particle size in terms of the peak breadth, and for the displacement of the peak. The powder pattern of a heat treated carbon black is presented as an illustration of two-dimensional lattice reflections.