Controlling the Short-Range Order and Packing Densities of Many-Particle Systems

TL;DR

This paper investigates the geometric constraints of particle arrangements in amorphous systems, focusing on the hard sphere model, to identify upper limits of packing densities and analyze the structure factor and pair correlation functions.

Contribution

It introduces a five-parameter model for g(r) to optimize and estimate maximum packing densities, providing new bounds and insights into amorphous particle configurations.

Findings

Maximum packing fraction around 0.58 for the model

Maximum mean contact number approximately 5.8

Derived a sharper lower bound for random sphere packings in higher dimensions

Abstract

Questions surrounding the spatial disposition of particles in various condensed-matter systems continue to pose many theoretical challenges. This paper explores the geometric availability of amorphous many-particle configurations that conform to a given pair correlation function g(r). Such a study is required to observe the basic constraints of non-negativity for g(r) as well as for its structure factor S(k). The hard sphere case receives special attention, to help identify what qualitative features play significant roles in determining upper limits to maximum amorphous packing densities. For that purpose, a five-parameter test family of g's has been considered, which incorporates the known features of core exclusion, contact pairs, and damped oscillatory short-range order beyond contact. Numerical optimization over this five-parameter set produces a maximum-packing value for the fraction of covered volume, and about 5.8 for the mean contact number, both of which are within the range of previous experimental and simulational packing results. However, the corresponding maximum-density g(r) and S(k) display some unexpected characteristics. A byproduct of our investigation is a lower bound on the maximum density for random sphere packings in $d$ dimensions, which is sharper than a well-known lower bound for regular lattice packings for d >= 3.