The modern theory of crystallization and the Hansen-Verlet rule

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Abstract
A cluster expansion of an order parameter distribution function is introduced into the recent theory of crystallization of classical fluids. This allows us to demonstrate the purely geometric origin of the empirical Hansen-Verlet rule of crystallization (max. S(k) ⋍ 2·85 or c(k ⋍ 0·65 with S(k) = (1 - c(k)-1). Using moment expansions (whose convergence is assumed), we find that the minimum value for which bifurcation into a crystalline solution of the same thermodynamic potential as the coexisting fluid solution is possible within our perturbation scheme corresponds to c(k) = 0·69 for a f.c.c. structure and to 0·59 for a b.c.c. structure with k equal to the smallest reciprocal lattice vector of the corresponding cubic structure. To this end we have considered the limit where the fractional density change and the fluid's compressibility vanish in such a way that their ratio remains constant in order to guarantee the first order character of the transition. This limit allows us to work out the theory without any input data.

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