Density and Packing in an Aggregate of Mixed Spheres

Abstract

It is shown that in close packing of spheres, two types of interstice exist, bounded by six and by four convex spherical surfaces. These are termed ``square'' and ``triangular.'' They are connected by a continuous labyrinth through which a ball not exceeding (2/√3−1)r in radius can be threaded. In both cubic and hexagonal arrangements, their shape and size are identical, but their distribution differs. Radii R for n smaller balls, which can take up patterns with cubic symmetry within each square interstice, are calculated for values of n up to 27 and plotted. The expression $$\mathrm{(R/r)=(}\sqrt{2}{\text{−1)/n}}^{\frac{1}{2}}$$ is used as a test for efficiency of packing. When R/r is plotted against the density increment attributable to this interstitial packing, a set of spires is obtained. At n=8 and n=9 (R/r being, respectively, 0.2289 and 0.2166) twin peaks of 13 percent and 12 percent appear, which are accentuated by first entry of a sphere into the triangular interstice at R/r = 0.22475; this contributes another 3 percent. The peak of 16 percent at n=21, R/r = 0.1782, is also reinforced by first entry of 4 spheres in a body‐centered tetrahedron at 0.1716, which gives an additional 6 percent density increment. Applications to bulk storage, ceramics, and interstitial compounds and solid solutions are considered. None of the special packs gives a density increment (∂Δ_square +∂Δ_triangular)/Δ even approaching the 26 percent for fine spherical filler close packed in the interstices.