A Simple Derivation of the Layer-Kkr Theory of Leed
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A SIMPLE DERIVATION OF THE LAYER-KKR THEORY OF LEED P.M. MARCUS IBM Research Center, Yorktown Heights, New York 10598, U.S.A.
ABSTRACT The equations of the layer-KKR method for finding the intensities of electrons diffracted by a semi-infinite layered structure are derived in a simple way by use of three identities relating spherical and plane waves and by use of a compact notation. The third identity, which transforms a sum of outgoing spherical waves on a twodimensional array into a sum of plane waves or beams, is a very useful simplification of the derivation, which does not appear in texts on LEED. Explicit linear equations are derived for amplitudes of outgoing spherical waves produced by diffraction of an incident beam first by a single Bravais net and then generalized to a composite layer made up of several Bravais nets; matrix elements of the beam reflection and transmission matrices for a single Bravais net and for a composite layer are given, and also the exact interlayer matrix equation for the reflection matrix of the semi-infinite structure.
1. INTRODUCTION The most widely-used formulation of the theory of low-energy electron diffraction (LEED) intensities is the layer-KKR method proposed by Kamb61 in 1967. The theory has been described and extended in many papers since 1967 and is presented at length in two fine monographs, by Pendry 2 in 1974 and by Van Hove, Weinberg and Chan 3 in 1986. However the mathematical development in these texts is more complicated than it needs to be. The purpose of this paper is to provide a simpler, more compact, but still reasonably complete, derivation of the basic equations of the method than in these popular texts, hence to make the theory more accessible. The simplicity is achieved by using three key mathematical identities expressed in simple compact notation. The proofs of these identities will be described, but not given in detail; however references for the proofs will be given. Simplicity is also aided by restricting the initial model in nonessential ways, then generalizing the model. The layer-KKR method separates the calculation of the scattering of a plane wave by a semi-infinite crystal into two stages. The first stage is the interlayer scattering problem assuming the scattering properties of the individual layers for plane waves are known. This stage makes use of a special plane-wave representation of the electron wavefunction called a beam representation. The wavefunction is specified by a set of beam amplitudes. The scattering properties of the separate layers and of the semiinfinite stack of layers are described by a reflection matrix R and a transmission matrix T for the beam amplitudes. The interlayer scattering problem is then described compactly and exactly by a matrix equation for the reflection matrix Rs of the stack. This matrix equation does not appear explicitly in the texts, although it is implied, since the texts describe approximate methods for its solution. The second stage of the complete layer-KKR method calculates the R and T matrices
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