On the dependence of the elastic properties of small crystals on size

Abstract

This paper aims at determining the dependence of the mechanical properties—equilibrium atomic spacing, force constant in tension and shear, and Poisson's ratios—of small crystals on their sizes. Simple crystalline atomic configurations are considered: a finite number of atoms in f.c.c. packing forming an atomic plane whose external boundaries form a parallelogram, and a finite number of such planes in closest packing forming a parallelepiped. The analysis is based on the assumption that a potential energy exists for the crystal which is a function of the atomic positions, and whose minimum determines the stable configuration. In Section 1 the relevant mechanical properties are defined and expressed in terms of the potential function and its derivatives. In Section 2 the formalism developed in Section 1 is applied to Mie potentials defined in the paper. The expressions thus obtained have been evaluated for ( m, n) equal to (6, 12) (the so-called Lennard-Jones potential) which gives good results for inert gas crystals. The results are illustrated in the paper and show that the equilibrium spacings decrease, the force constants increase and Poisson's ratios fluctuate with increasing crystal size. Similar calculations have been carried out in Section 3 using Morse potentials, where parameters have been selected to represent respectively the metals Ni, Cu, Ag, Al, Ca, Pb and Sr. These results show tendencies similar to those found in the Lennard-Jones case.