The applicability of discrete models to the investigation of mechanical properties of solids is limited. For continual models, this is actually not true as can be explained by a large amount of computations required if it is necessary to take into account the interaction of many particles. At the same time, discrete models better reflect the actual structure of the solid body and contemporary computers make it possible to use vast amounts of available data [ 1,2].We study two-dimensional models with square cells (Fig. 1) by using the well-known method suggested in [ 1,2]. In these models, the interatomic interactions are described by spherically symmetric potentials 9 We assume that the radius of action of the interatomic forces is bounded by a distance r = 2 r o, where r 0 is the equilibrium distance between atoms in a diatomic model 9 We define the energy of bonding of two atoms by the formula U = -A r-m + B r-n, where A and B are constants, r is a distance between the atoms, and m and n are exponents such that n > m. The minimum potential energy of a pair of atoms is attained at r = r 0 , where we have dU Am Bn --= 0 and, hence,Therefore, under the assumption that the constant A and the exponent m are equal to one, we can writeWe also assume that n = 8 and the equilibrium distance r 0 = 2.5 A. We represent the potential energy of an atom in the crystal in the form of the sum of its pairwise interactions with the first and second neighbors (Fig. 1), namely, E A = UAI + UA2 + UA3 + UA4 + UA5 + UA6 + UA7 + UA8.The potential energy E of the crystal is found as the sum of the potential energies of all its atoms.
Computation of the Parameters of the Crystal LatticeFor a crystal with N 2 atoms, the potential energy is given by the formula E = (N-2)2E3 + 4(N-2)E 2 + 4E 1, where N is the number of atoms on one side of the crystal with square lattice (this formula was established by finding the general regularity for crystals with 22, 32, 42 ..... 82 atoms), E 3 = 4U(r 1 )+ 4U(r2), E 2 = 3U(rl)+ 2 U(r 2 ), and E 1 = 2 U(r 1 ) + U (r 2 ) are, respectively, the potential energies of internal, surface, and comer atoms, r t is the distance between the nearest neighbors, r 2 is the distance between the remote atoms, and r 2 = r 1 -f2.Ivano-Frankovsk State Technical University of Oil and Gas, Ivano-Frankovsk.