Applications of the Bloch theory to the study of alloys and of the properties of bismuth

Abstract

In a recent years many binary alloys have been examined by means of X-rays and their structures have been determined at various compositions. It has been shown in this way that a pure phase consists of a single structure, whilst in the regions of mixed phases the alloy forms a simple mixture of two crystal types. Sometimes the range of composition of a pure phase is extremely narrow, as, for example, in the ε-phase of the Cu-Sn alloy. The composition within this phase corresponds almost exactly with the formula Cu 3 Sn. Examples of this kind have naturally led to the conception of intermolecular compounds, and, since the formation of a compound in chemistry is described in terms of valency bonds, it is natural that in the existing literature attempts to discuss the reason for the formation of various phases, and the properties of the alloys in these phases, have been based mainly upon the conception of the homopolar bond. It does not seem , however, that this conception is very successful in the interpretation of the formation and properties of metallic alloys. It is advantageous therefore at attempt an examination of alloys from the standpoint of he Bloch model. In Bloch’s theory a stationary state of an electron in a crystal is specified by the three components of a vector k which may be regarded as the average momentum associated with the state; k is not, however, the average velocity times the mass of the electron, but is defined in such a way that the rate of change of k is proportional to the external force acting on the electron. Since the difference in k between two successive stationary states is in general exceedingly small, the vector k may be regarded as varying continuously from state to state. It is now a wellknown result of the theory that the energy of a state is not a continuous function of position in k space, but is discontinuous across certain planes. The positions of these planes are determined by the crystal symmetry of the metal, and they divide k space into the various “Brillouin zones.”