Energy changes by a variation from the crystallographic group

Abstract

The purpose of the paper may be briefly stated as follows: To investigate the possibility of energy minima by a systematic departure from the perfectly symmetrical crystalline lattice. A geometrically simple variation is considered. The surface (100-plane) of a heteropolar crystal of the face-centered type is contracted uniformly in patches, and the interior planes are likewise contracted in patches by amounts decreasing linearly with the distance from the surface. In this way a "mosaic" structure of a type previously suggested in the literature1 is obtained. A numerical calculation of the change in potential energy is carried out for rock-salt with certain simplifying assumptions. The change in energy is obtained as a power series in terms of the maximum displacement of the ions. Due to the slow convergence of this alternating series an approximate remainder term is introduced to take care of powers higher than the third. The series contains two parameters, namely the linear extension of the patches and the depth of the cracks. The function, in general, is found to have two minima with respect to the displacement of the ions, one at zero displacement, and a second at a finite displacement. The second minimum, depending on the parameters, may be negative with respect to the former. The function also has minima with respect to the two parameters, thus proving the static stability of a mosaic structure of definite dimensions. These considerations with only minor changes, are also applicable to a mosaic structure in the interior of the crystal.