# “Hole statistics of equilibrium 2D and 3D hard-sphere crystals” is Published in The Journal of Chemical Physics

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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 Rd. 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 \(f(\phi_l)\) of Delaunay cells and find that, for \(d\leq3\), the excess kurtosis of \(f(\phi_l)\) 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.