Caging of a d-dimensional sphere and its relevance for the random dense sphere packing

We analyze the caging of a hard sphere (i.e.,the complete arrest of all translational motions) by randomly distributed static contact points on the sphere surface for arbitrary dimension d greater than or equal to1, and prove that the average number of uncorrelated contacts required to cage a sphere is (N)(d)=2d+1. Computer simulations, which confirm this analytical result, are also used to model the effect of correlations between contacts that occur in real hard-sphere systems. Our analysis predicts an average coordination number of 4.79 (+/-0.02) for caged spheres, which agrees surprisingly well with the experimental coordination number for random sphere packings reported by Mason [Nature 217, 733 (1968)]. This result supports the physical picture that the coordination number in random dense sphere packings is primarily determined by caging effects. It also suggests that it should be possible to construct such packings from a local caging rule.