Moments of the Percus-Yevick hard-sphere correlation function

Abstract

A simple recursive relation is derived for the momentsM _n ,n=1, 2,..., of the Percus-Yevick correlation functionh(r) for identical hard spheres. TheM _n are rational functions of the volume fractionw occupied by the spheres; the first ten are given explicitly, and a single-term asymptotic form is obtained to suffice for the rest. Applications of theM _n(w) include testing different approximations forh by numerical integration ofh(r) r ⁿ. We compare exact moments with shell approximationsM _n [h ^s] corresponding to integration fromr=0 tos+1 fors=3−8, and with hybrid approximationsM _n [h ^s+h ^a] which supplement the shell approximations with integrals of an asymptotic tail froms+1 to ∞. For a givens, the hybrid approximation is better forw increasing than the shell approximation, andM _n [h ³+h ^a] is even better thanM _n [h ⁸]