A new method of calculating interatomic spacing: the equal-ratio method

Abstract

Based on a simple principle of analytical geometry, a new equal-ratio method has been developed to calculate the interatomic spacing of crystal structures. If an atom (x 2, y 2, z 2) or its equi-position atom (e + x 2, f + y 2, g + z 2) (e, f and g are integers) is located at the 1/r ≤ 1 of one interatomic spacing period d′[uvw] on the [uvw] atomic row passing through the atom (x 1, y 1, z 1), the distance between the two atoms can be calculated by the formula d [uvw] (1/r) = d′[uvw]/r, where d′[uvw] = (u 2 a 2 + v 2 b 2 + w 2 c 2 + 2uvabcosγ + 2vwbccosα + 2uwaccosβ)1/2 is the interlattice point spacing of the corresponding primary lattice of the crystal structure, 1/r is the interatomic spacing coefficient, and r is equal to the reciprocal of the common factor of (x 2 − x 1), (y 2 − y 1) and (z 2 − z 1). The reliability and advantages (no auxiliary view is required, suitable for arbitrary directions and for all crystal structures) of the equal-ratio method have been examined by calculations for the β-cristobalite SiO2 structure and Cu3Au I superstructure as well as face-centred cubic, body-centred cubic and hexagonal close-packed structures.