Systematic packing from the standpoint of the primitive cell

The systematic packing of uniform spheres is generalized by describing the primitive rhombohedral cell which characterizes the arrangement between layers. Volume and porosity are found to depend on only two angular parameters, alpha and beta: V = 8 R cubed sin alpha sin beta n = 1 - (pi/6 sin alpha sin beta). Beta is the angle between rows in a layer, and alpha is the altitude angle between members of adjacent layers. An azimuth angle gamma determines the position of the plane in which alpha is measured but does not enter into the porosity calculation. Four critical stacking arrangements are described, the porosities of which may be written as functions of the single parameter beta. The stable packings studied by Graton and Fraser (1935) are special cases of the critical positions. Typically unstable packings lie between these positions. Tables and graphs are presented which give the porosity of the primitive cell, as a function of alpha and beta, over the entire range from open to close packing for every possible layer configuration. (Author).