Hole Statistics of Equilibrium 2D and 3D Hard-Sphere Crystals

TL;DR

This paper introduces a biased-sampling method to accurately analyze large hole statistics in equilibrium 2D and 3D hard-sphere crystals, revealing oscillatory behavior linked to local order and coordination geometry.

Contribution

It presents a novel biased-sampling scheme for studying rare large holes in hard-sphere crystals, extending the understanding of $G_V(r)$ beyond previous limits.

Findings

$G_V(r)$ oscillates with increasing amplitude at higher packing fractions.

Oscillations in $G_V(r)$ correlate with local orientational order in 2D.

Transition between tetrahedral and octahedral holes in 3D states.

Abstract

The probability of finding a spherical "hole" of a given radius $r$ contains crucial structural information about many-body systems. Such hole statistics, including the void conditional nearest-neighbor probability functions $G_V(r)$, have been well studied for hard-sphere fluids in $d$-dimensional Euclidean space $\mathbb{R}^d$. However, little is known about these functions for hard-sphere crystals for values of $r$ beyond the hard-sphere diameter, as large holes are extremely rare in crystal phases. To overcome these computational challenges, we introduce a biased-sampling scheme that accurately determines hole statistics for equilibrium hard spheres on ranges of $r$ that far extend those that could be previously explored. We discover that $G_V(r)$ in crystal and hexatic states exhibits oscillations whose amplitudes increase rapidly with the packing fraction, which stands in contrast to $G_V(r)$ for fluid states that is monotonic in $r$. The oscillations in $G_V(r)$ for 2D crystals are strongly correlated with the local orientational order metric in the vicinity of the holes, and variations in $G_V(r)$ for 3D states indicate a transition between tetrahedral and octahedral holes, demonstrating the power of $G_V(r)$ as a probe of local coordination geometry. To further study the statistics of interparticle spacing in hard-sphere systems, we compute the local packing fraction distribution $f(\phi_l)$ of Delaunay cells, and find that for $d\leq 3$, the excess kurtosis of $f(\phi_l)$ switches sign at a certain transitional global packing fraction. Our investigation facilitates the study of structural and bulk properties of materials that involve the creation of rare large holes, such as the solubility of alloys.