Hole statistics of equilibrium 2D and 3D hard-sphere crystals

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 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) that monotonically increases with r for fluid states. 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(ϕ_l) of Delaunay cells and find that, for d ≤ 3, the excess kurtosis of f(ϕ_l) switches sign at a certain transitional global packing fraction. Our accurate methods to access hole statistics in hard-sphere crystals at the challenging intermediate length scales reported here can be applied to understand the important problem of solvation and hydrophobicity in water at such length scales.