On a Hard-Sphere Crystal

On the basis of the effective potential method, the single-particle distribution-function in the three-dimensional hard-sphere crystal is investigated. The effective potential in a unit cell which is determined self-consistently depends not only on the displacement of a particle from the lattice point but also on the configuration of its neighboring lattice points. The effective potential is expanded in terms of the above displacement up to the quadratic terms including a linear term and its coefficients are determined by the use of the self-consistent equations. The well-localized distribution of particles around their own lattice sites is obtained below a threshold specific volume and the hard-sphere crystal cannot exist stably above that volume. The correlations between displacements of a pair of particles in the directions parallel and perpendicular to the line connecting the nearest-neighboring sites are discussed. The pressure and the isothermal compressibility as a function of volume are also studied. The effective two-body interaction potential is introduced in an intuitive way. By the use of it, the dispersion relations of the lattice vibration are investigated.