Binary distribution functions of atoms of simple crystals

Abstract

We propose a method of statistical description of simple crystals and calculation of their thermodynamic functions and equation of state. The method is based on the derivation of an exact expression for the binary distribution function of atomic displacements and a variational procedure for the determination of an effective constant of the quasielastic bond of atoms of the crystal. For rare gas crystals with Morse and Lennard-Jones potentials, we obtained the equation of state and thermodynamic parameters of the solid-state, which are in agreement with experimental data. We also found that a solid-state instability occurs near the observed melting temperature of the crystal, corresponding to a point above which there is no more an equilibrium effective parameter of the quasielastic bond.

Influence of vibrational anharmonicity and vacancies on the thermodynamic properties of rare gas crystals

“…One may notice [7] that q( ) z changes only slowly with temperature from q 0 1 87 » . at T = 0 K, to q » 2 at high temperature, where it coincides with the value for completely uncorrelated atomic states.~( ) w j K are the reduced frequencies determined by the dynamical matrix of the harmonic crystal, i.e.…”

“…The values for these parameters listed in Table were determined previously [7] in such a way that the internal energy, the lattice parameter and the bulk modulus of the RGS at zero temperature and zero pressure fitted the observed values within the framework of the statistical model [7]. The determination of the total free energy of the crystals starts from the Gibbs-Bogoliubov functional, which includes in the present case a term for the cubic anharmonicity of the atomic vibrations DF c 3 ( , ) t determined in second order perturbation theory [9]:…”

“…The integration in (3) and (4) runs over the unit cell volume of the reciprocal lattice and K represents reduced components of the wave vector, varying from 0 to 1 [7]. The average potential energy of the interatomic interaction is calculated by the use of a binary distribution functions for the atomic displacements [7] and is given within the present approximation by…”

“…These problems are not easily handled in the framework of self-consistent phonon (SCP) theory [4][5][6], widely used for numerical investigation of crystal thermodynamics. In the present study we evaluate therefore thermodynamic characteristics of both perfect RGC's and RGC's with vacancies within the framework of a statistical theory for crystals [7], whereby a crystal is represented as an ordered ensemble of particles, whose spatial distributions are described by a binary distribution function. Recently this method has been successfully applied for the investigation of the equations of state and thermodynamic properties of rare-gas solids under pressure [8].…”

“…In Sec. 2 the main results of this statistical theory [7] are outlined and the internal energy, the heat capacities, the bulk modulus, the Grüneisen parameter and the thermal expansion are calculated for perfect crystals of Ar, Kr, Xe with contributions from anharmonicity. In Sec.…”