Enumeration of random packings for atomic substances

At fixed density, the number of distinguishable ways that N identical atoms can be packed into a fixed volume is expected to rise exponentially, exp(νN), when the number N of atoms is very large. Heretofore no satisfactory method has been available to evaluate the positive constant ν. We propose such a method in classical statistical mechanics, utilizing the formalism and some basic results from the ‘‘inherent structure’’ theory of condensed phases. It requires data concerning (a) the mean potential energy of amorphous packings, (b) the mean logarithm of the normal-mode frequencies for both crystalline and for amorphous packings, and (c) a smooth extrapolation of the liquid-phase thermodynamic energy through the supercooled regime to absolute zero. We have applied this method to the soft-sphere model with $r^{\mathrm{-}12}$ pair potentials, drawing upon published computer-simulation results. We find for this model that ν=0.07±0.06, where most of the estimated error arises from uncertainty in the data currently available for (a) above.