Applications of Centroid Fractional Coordinates in Locating Interstices in Close Packings of Equal Spheres

Deep understanding of the nature and characteristics of close packings of equal spheres is fundamental for further study towards structure and property of metallic crystals. And the knowledge to number and distribution of various interstices in close packings of equal spheres is very important to help illustrate the structure and property of ionic crystals. However, the diversity and complexity of the crystal structures make it difficult in teaching and learning structural chemistry. In this paper, based on discussions on the three most common close packing models (A1, A2, A3), a method to locate centers of interstices according to calculating the centroid fractional coordinates (CFC) of particles constructing these interstices was introduced. In addition, the way using CFCs to calculated distance between vertex and interstice center and the shortest distance from interstice center to surface of the packing sphere was also illustrated in detail. Compared with the traditional solid geometry method, the CFC method is demonstrated to be much simpler, easier to learn, and most importantly, helpful for understanding number and distribution of various interstices in close packings.