On Calculation of Lattice Energy in Spatially Confined Domains

Abstract

Abstract- Evaluation of internal energy and the inter-atomic or ionic interactions in a crystal lattice usually requires precise calculation of lattice sums. This in the case of small nano-particles (as space-limited domains) presents several challenges, as conventional methods are usually valid only for infinite lattices, tailored for a specific potential. In this work, a new method has been developed for calculation of atomic interactions based on the radial density function with the geometric probability approach, extended to arbitrary fixed lattices and potentials in a nano-particle. The derived radial density function (RDF) combines terms for a uniform particles distribution, for non-uniform spherical symmetry and the last one for an additional, angle-dependent term. The second term originates from Walfisz-like formula for lattice sums. The RDF with these three terms is explicitly integrated for spherical lattice domains resulting in the internal energy of the system with a prescribed interaction potential. The application of the method was demonstrated for Wigner model of electrons lattice interacting with compensating positive jelly in finite lattice sphere, which interacting energy between lattice and jelly was evaluated. The excess of this energy caused by space-limitation of the lattice was explicitly expressed in the terms of absolutely convergent lattice sums.