Distances in the face-centered cubic crystalline structure applying operational research

Abstract

The f.c.c. (face-centered cubic) grid is the structure of many crystals and minerals. It consists of four cubic lattices. It is supposed that there are two types of steps between two grid points. It is possible to step to one of the nearest neighbors of the same cubic lattice (type 1) or to step to one of the nearest neighbors of another cubic lattice (type 2). Steps belonging to the same type have the same length (weight). However, the two types have different lengths and thus may have different weights. This paper discusses the minimal path between any two points of the f.c.c. grid. The minimal paths are explicitly given, i.e. to obtain a minimal path one is required to perform only O(1) computations. The mathematical problem can be the model of different spreading phenomena in crystals having the f.c.c. structure.