The dependence of zone axis patterns on string integrals or the number of bound states in high energy electron diffraction

Abstract

The idea of a string integral as a measure of the number of bound states, or the strength of a crystal potential when projected along a zone axis was introduced by Ozorio de Almeida (1973). We here explore its utility in distinguishing the various stages in the sequence of patterns filling the first Brillouin zone which was described in the preceding paper (Shannon and Steeds 1977) and, in particular, for locating the second critical voltage. The positions of the accidental degeneracies and of the eigenvalues having minimum attenuation coefficients (Doyle and Berry 1973) are evaluated in several cases in relation to the top of the potential wells. Critical voltages higher than the second are investigated in a few cases. One of these has a simple projected potential in the sense described by Shannon and Steeds (1977) (Mo 〈111〉) and others have more complex potentials (2H MoS₂[0001], diamond structure 〈110〉). The order of the bound states of different symmetry within the potential wells for the Mo 〈111〉 axis has been studied over a wide range of voltages.