Hard-Sphere Solid: Pair Distribution Functions at High Density

The high‐density form of the molecular pair distribution function for a system of rigid spheres near a face‐centered close‐packed configuration is calculated by a cell‐cluster technique. This technique was previously applied to the estimation of the partition function at high density and to the pair distribution functions for one‐dimensional rigid rods and two‐dimensional rigid‐disk systems. The technique is thought to be applicable as an approximation scheme for a high‐density rigid‐sphere solid or crystalline system. The primary object of this analysis is to compute the quantity Ω_N(ζ), the average number of pairs separated between centers by a distance less than or equal to ζ. For a face‐centered cubic lattice of rigid spheres the high‐density expansion (V→V₀) of Ω(ζ) for ζ→σ takes the form $$\begin{array}{c}{\text{2Ω/N=6η+bη}}^2{\mathrm{+c\eta }}^3\text{+···;\hspace{0.17em}\hspace{0.17em}η≪1}\\ \mathrm{\eta =(\zeta -\sigma )/(a-\sigma ),}\end{array}$$ where a is the distance between lattice sites and σ is the diameter of a sphere. The contributions to the constants b and c from one‐, two‐, and three‐particle cell clusters are calculated. The expansion through third order is $$\begin{array}{c}\text{b=6−2.0813+1.1811=+5.0997,}\\ \text{c=0+1.3704−4.9437=−3.5733.}\end{array}$$ The implications of a net positive value of b to considerations of the stability of long‐range crystalline order are indicated.