Multiple scattering theory in condensed materials

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Abstract
Second-order elliptic differential equations (such as the time-independent single particle Schrödinger equation) may be solved in a finite closed disjoint region of space independently of the rest of space. The solution in all space may then be determined by solving the equations in the exterior region together with boundary conditions at the junction of the two regions. These boundary conditions are determined by the previously found interior solution. This means that such regions may be taken as ‘black boxes’ whose exact details do not matter. The simplest example of this is phase-shift scattering theory from a single scatterer where all the scattering properties are described by the phase shifts, and the exact details of the scattering potential are unimportant. In a macroscopic condensed system, however, there are many core regions and one is really concerned with the multiple scattering which takes place between these different scattering centres. Much of this article is devoted to investigating the formal properties of scattering theory when there are many non-overlapping spherical regions of radius R M, each of which is described by its own scattering matrix, or, equivalently for a spherically symmetric potential, by its phase shifts. Non-spherically symmetric and spin-dependent potentials are permitted, but for simplicity we assume initially that the interstitial region between each disjoint scattering region has zero potential. The generalization of the multiple scattering formalism for non-zero interstitial potential is also given at a later stage. It is shown that in such a system a generalized T-matrix may be defined which describes the radiation from one of the core regions when another one has been excited. It is then a many channel T-matrix in which the channels are the different disjoint scattering regions. It is shown that the formal properties of this T matrix are the same as for a normal T matrix. In § 2 we review the properties of ordinary scattering theory, and then in § 3 we show that analogous properties for the generalized T matrix hold. An exact expression for the density of particle eigenstates is derived in terms of the positions and scattering matrices of the individual scattering centres. This expression reduces to the standard KKR band structure equation for the infinite regular lattice. We also consider how to construct the density of eigenstates and the charge density for such a system. These latter quantities may be approached in two different ways: the usual way is to consider the scattering material to occupy all space, but from a multiple scattering viewpoint one must consider the total volume of condensed material to be small compared with all space, even if both limit to infinity. It is not obvious that the latter method leads to the same results as the former (formally the density of eigenvalues is identical to the free electron density of eigenvalues in the latter method) and it is shown how the differences in the two approaches are resolved. We also discuss the expansion of some of these results for a perfect lattice. While the usual expansions are pseudo-potential expansions, a manifestly ‘on-energy shell’ expansion is derived which does not contain the arbitrary parameters of the pseudo-potential expansions. Finally, in § 4, we review the most significant contributions of other authors to the theory of multiple scattering.

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