Models of three-particle interactions and theory of nonlinear deformations of crystals

Abstract

A method of accounting for the symmetry of interaction energy of N identical atoms in terms of the theory of elastic constants has been proposed. The energy symmetry group of the cluster is GN = O(3) ⊗ PN. It has been shown that the calculation of elastic characteristics of crystals, which is based on analyzing the interaction potentials of atoms with inclusion of the symmetry, is competitive with respect to the calculations performed within the models of quantum mechanics. Nine models that depend on three parameters have been considered. In each model, the third-order elastic constants have been calculated for gold, aluminum, and copper single crystals with allowance made for the interactions of triples of atoms. The dependence of the energy of the models on the invariants forming the integral rational basis of the G3 group has the form \(\varepsilon \left( {\left. {i,k,l} \right|j} \right) = \sum\nolimits_{i,k,l} {\left[ { - A_1 r_{ik}^{ - 6} + A_2 r_{ik}^{ - 12} + Q_j I_j^{ - n} } \right]}\), where Ij is the invariant with number j (j = 1, 2, ..., 9). The parameters of the models are specified by the second-order elastic constants. The best agreement with experiment has been achieved for Cu with n = 2, j = 4; for Au with n = 1, j = 74; and for Al with n = 1, j = 9. It has been demonstrated that, for the calculation of all independent values of the second-, third-, fourth-, and fifth-order elastic constants, it is necessary and sufficient to include interactions between the clusters containing quadruples of atoms in the theory.