Systematic Arrangements of Spheres Packing

Abstract

Several models of systematic arrangement of spheres in packing, proposed by Graton and Fraser, are often cited in hand-book and such others. But the author makes a partial amendment of these models as follow,
face angles of unit cell number of contact points porosity (%) type of packing 90°, 90°, 90° 6 47.64 Cubic 90°, 90°, 60° 8 39.54 Hexagonal 70°31′44″, 70°31′44″, 109°28′16″ 8 31.98 Body centered cubic 60°, 60°, 75°31′ 10 30.19 Tetragonal-Sphenoidal 60°, 60°, 90° 12 25.95 Face centcred cubic 60°, 60°, 60° 12 25.95 Face centered cubic
Following equation is proposed for calculating porosity “n” of pacikig by using three face angles θ₁, θ₂, θ₃ of unit cell.
n=1-π/6√1-cos²θ₁-cos²θ₂-cos²θ₃+2cosθ₁cosθ₂cosθ₃