Close-packed structures of spheres of two different sizes II. The packing densities of likely arrangements

Abstract

The packing densities of spheres of two different sizes, regularly arranged in a few structures based on cubic or hexagonal packing, have been calculated for different ratios (γ) of the radii of the spheres. Apart from a series of structures ABn, formed by very small B spheres filling cavities in a close-packed array of A spheres, only two arrangements AB (NaCl-type) and AB2 (AlB2-type) have packing fractions greater than the 0·7405 of the separate close-packed phases. Thus it is likely that for 0·24 < γ < 0·458 a mixture of A and B spheres will contain the AB phase, and for 0·482 < γ < 0·624, AB2 should occur. An ideal cubic structure AB13, containing an icosahedral cluster of small (B) spheres cannot have a packing fraction greater than 0·738, but a small modification to the structure permits an increase in the packing to 0·76, and is suggested as the explanation for the appearance of this phase in the opal specimen described in Part I of this work.