Lattice Constant Systems and Graph Theory

Abstract

The two principal systems of lattice constants that have arisen in the study of cooperative phenomena and related problems on crystal lattices are the strong (low-temperature) and the weak (high-temperature) systems. The two systems are defined in terms of the concepts of graph theory, and a general theorem relevant to cluster expansions is stated. The interrelation of the two systems is studied and exploited to derive configurational data for the face-centered cubic lattice. All star graphs with up to seven points (vertices) or nine lines (edges) that are embeddable on the face-centered cubic are described. A general classification of stars with cyclomatic index 3 is given.