Statistical theory of the crystal state

A theory of the crystalline state of matter is constructed in the framework of the method of distribution functions. The system of two exact equations for the single- and two-particle distribution functions is solved by a series expansion in powers of the small parameter $\varepsilon=(n-n_0)/n_0$, where $n$ is the density of the crystal and $n_0$ is the density of the liquid at its crystallization point. In the zeroth order in $\varepsilon$, the theory leads to the Ornstein–Zernike equation, which determines all the properties of the molten state; in the first order, it leads to equations that determine the symmetry type of the crystal and the main lattice periods. Finally, in the second order in $\varepsilon$ the theory makes it possible to calculate the jump in the density on crystallization of the molten state. The proposed method of solution is valid only at temperatures above the triple point, i.e., in the region in which the crystal can be in equilibrium with the molten matter.