Calculation of radii and atom numbers of different coordination shells in cubic crystals

Abstract

Computational problems of multilayered coordination radii and coordination numbers in cubic crystals were considered. The Diophantine equation models and the generating functions for computing the coordination radii and coordination numbers were derived for the simple cubic (SC), body-centered cubic (BCC) and face-centered cubic (FCC) crystals, respectively. Properties of the sequences of coordination radii and coordination numbers were discussed, and the results for the three types of crystals were compared. We find that generating functions can be expressed in terms of the elliptic theta functions. These results provide efficient algorithms for the multilayered coordination radii and coordination numbers by solving the Diophantine equations, or expanding the powers of series in the generating functions, or directly expanding the elliptic theta functions into power series using a computing software such as Mathematica.