Motion of a Particle in a Periodic Field of Zero-Range Potentials

Abstract

In all applications considered in the previous chapters of this monograph, it was assumed that the particle was in a field of a finite number of ZRPs. Now we shall consider an infinite number of potential wells. This will require the calculation of infinite sums extended over all force centers. A physically important model is an infinite number of identical wells forming a regular (periodic) structure similar to that of a crystal. The regular feature of this model facilitates the calculation of the sums. It can be considered as a generalization of the well-known Kronig-Penney model (81) . In the latter, a particle moves in the field of a one-dimensional periodic lattice in a one-dimensional space. In this and the next section, we shall consider a one-dimensional lattice in a three-dimensional space (a model of an electron moving in the field of a polymer molecule), where the result of the summation can be written in an analytical form. In Sec. 6.3 and Sec. 6.4 we shall study the case of two- and three-dimensional lattices and discuss various approximate methods of calculating the sums.