The atomic string approximation in high energy electron diffraction

Abstract

It is often useful to study dispersion surfaces in order to understand the complicated diffraction effects which are observed in transmission electron micrographs. To obtain these dispersion surfaces accurately from the many-beam equations in cross-grating high energy electron diffraction it is necessary to diagonalise matrices at least of the order 60×60 even for the simplest structures. Such a method cannot easily be used to relate the form of the dispersion surface to the projected potential U(R) seen by the fast electrons. It is possible, however, to solve an equivalent two-dimensional band structure problem for the fast electrons in the crystal. The KKR method, which exploits the near cylindrical symmetry of the projected potential within each atomic string in order to expand the Bloch waves in terms of angular momentum basis functions eiℓθ, enables such an approach to be made and requires the solution of small secular determinants.Defining the transverse energy S to be zero at the maxima of the atomic string potentials, the atomic string approximation(ASA) reduces the KKR equations for one string per unit cell of the projected potential to the formwhere the eigenvector components C are simply related to the coefficients in the angular momentum expansion of the Bloch waves and the structure constants Mℓℓ'(K) only depend on the arrangement of the atomic strings and the orientation K.