Simple Graphical Method of Drawing and Interpreting Parallel Planes through a Unit Crystal Cell for Any Given Set of Miller Indices

It is shown that a graphical method can provide a rapid analysis of parallel planes in various crystal structures. This method can be used to answer questions regarding relative distances between planes and whether planes with a given set of Miller indices are equivalent, as well as to determine the relative populations of each type of atom in nonequivalent planes. This in turn enables one to make some immediate inferences about x-ray scattering intensities for various orders of reflection without recourse to complicated mathematical analysis. The method involves a construction of a projectional diagram in accordance with some simple rules, which are illustrated with reference to a cubic unit cell for $CaF2$⁠, and then generalized to crystal structures of any other type.