Enumeration of random packings for atomic substances.

Abstract

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(\ensuremath{\nu}N), when the number N of atoms is very large. Heretofore no satisfactory method has been available to evaluate the positive constant \ensuremath{\nu}. 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 ${r}^{\mathrm{\ensuremath{-}}12}$ pair potentials, drawing upon published computer-simulation results. We find for this model that \ensuremath{\nu}=0.07\ifmmode\pm\else\textpm\fi{}0.06, where most of the estimated error arises from uncertainty in the data currently available for (a) above.