A quantum mechanical calculation of the elastic constants of monovalent metals

Abstract

The present paper is based on the results of the investigations of Wigner and Seitz and Seitz on the cohesive forces in metallic sodium and lithium and the extension of their work to copper by the present author. These calculations have shown that quantum mechanics is capable of accounting in a satisfactory way for the lattice energy and compressibility of these metals. It is the object of the present paper to give an extension of the theory from which their elastic constants can be calculated. Following Wigner and Seitz, we divide the lattice into polyhedra, one polyhedron surrounding each atom. We shall call each polyhedrons extend throughout the whole lattice and have certain periodicity properties, which lead to boundary conditions for the wave function at the surface of each unit polyhedron. For the state of lowest energy, the wave function is obtained, in each unit polyhedron, by solving the Schrödinger equation ∇ 2 ø 0 + 8π 2 m / h 2 (W-V) ø 0 =0 (1) with the boundary condition δø 0 /δ n = 0, (1A) where δ/δ n denotes the normal derivative at the boundary of the unit polyhedron. V( r ) is here the potential of the ion.