The transition firom classical to modern multiple scattering theory will be told in a very personal vein. The conjunction of a great teacher (Kronig). a classical problem (acoustical impedance of a porous plaster) the right reference (Kasterin's 1903 Moskau thesis) and my interest in the metallic state set the stage for a transfer from sound waves to Schroedinger waves. My 1947 Paper in Physica and Kohn and Rostoker's Physical Review paper of 1954 were shown to oe equivalent and constitute the basis of the KKR method, to the development of which Sam Faulkner has contributed more than anyone else. Details of these events will be presented.

The Multiple Scattering strategy for studying electrons in condensed matter is reviewed. It is emphasized that this approach is effective for the Dirac and Bogoliubov-de Gennes equations, describing relativistic and superconducting quasi-particles respectively, as well as the usual Schrödinger's equation which arises during the implementation of the Density Functional Theory.

The Quadratic Korringa-Kohn-Rostoker method is a fast band theory method in the sense that all eigenvalues for a given k are obtained from one matrix diagonalization, but it differs from other fast band theory methods in that it is derived entirely from multiple-scattering theory, without the introduction of a Rayleigh-Ritz variational step. In this theory, the atomic potentials are shifted by Δασ(r) with Δ equal to E-E0 and σ(r) equal to one when r is inside the Wigner-Seitz cell and zero otherwise, and it turns out that the matrix of coefficients is an entire function of Δ. This matrix can be terminated to give a linear KKR, quadratic KKR, cubic KKR, …, or not terminated at all to give the pivoted multiple-scattering equations. Full potentials are no harder to deal with than potentials with a shape approximation.

We review the interpretation of multiple-scattering theory in terms of muffin-tin orbitals. The use of slightly overlapping muffin-tin wells is justified rigorously. It is shown that the structure constants may be screened for a useful range of positive and negative energies, and that the screening may be chosen to yield desirable properties of the KKR matrix. Energy linearization and the linear muffin-tin-orbital method are discussed.

We describe a formalism dedicated to the realistic evaluation of correlation functions in solids and give an account of its success and its limitations by discussing various applications.

With the availability of reliable full atomic cell orbitals the possibility arises to calculate pressure or stress, restoring or relaxation driving Hellmann -Feynman forces, and total energies (especially of defects) alternatively and directly via stress tensor fields and energy densities, two local quantities. Although quantum mechanical stress field and energy density can not be defined uniquely, there is a recent interest in these quantities, because integrals with physical meaning are gauge invariant.

The mentioned fields can be defined (i) for the full many-body description with the exact one-particle density matrix and pair distribution function as well as (ii) for the Kohn-Sham one-particle description with LDA or beyond (gradient expansion approximation). If the local stress field for a special system once is constructed, then the global stress tensor and /or forces on nuclei can be calculated via the stress theorem and the force theorem by means of unit cell surface integrals. The energy density can be derived from the terms of the stress field by taking the trace and can be used to calculate defect energies without bothering about the thermodynamic limit.

The wave function in Multiple Scattering Theory (MST) is a locally exact solution to the Schrödinger equation for any ℓ truncation (ℓmax), except at the the muffin-tin boundaries (or cell boundaries in a full cell calculation) where it and its derivative generally have discontinuities. These discontinuities vanish only in the limit ℓmax → ∞. Furthermore, the MST wave function as usually calculated is not correctly normalized which means the density of states calculated from the Green function does not agree with that calculated from the Lloyd formula. Here we obtain an alternative wave function from the usual MST secular equation which is smooth and continuous everywhere and which is correctly normalized.

It appears that the most salient of the issues that a brief overview might address in this symposium concerns the relationship - the differences, the parallels, and the similarities - between research in classical wave applications of multiple scattering theory and that research whose chief focus is electronic. Each desires the solution to a linear partial differential equation, usually a wave equation such as the Schrodinger equation or Maxwell's equations, in a heterogeneous medium, and each seeks to derive bulk properties from knowledge of the properties of the constituents. As with the Schrodinger equation, Korringa-Kohn-Rostoker like equations are often used for classical waves, and expansions in spherical harmonics and outgoing and regular solutions of the wave equation in the bare medium used to represent the equations. There remain, however, substantial differences in their respective formulations and methodologies. These appear to be traceable, at least in part, to differences in the questions that are asked about the field in the heterogeneous medium, to differences in the prior information available in regard to the microstructure, and to different symmetry classes of the microphysics. It is perhaps these differences which can account for the otherwise rather surprising absence of interaction between researchers in classical applications and those in electronic structure.

The problem of calculating the mean wave in a composite with random microstructure is addressed. Exact characterizations of the problem can be given, in the form of stochastic variational principles. Substitution of simple configuration-dependent trial fields into these generates approximations which are, in a sense, ‘optimal’. It is necessary in practice to employ only trial fields which will generate, in the variational principle, no more statistical information than is actually available. Trial fields that require knowledge of two-point statistics generate equations that can also be obtained directly, through use of the QCA. The same fields can be substituted into an alternative variational principle to yield an approximation that makes use of three-point statistics – this approximation is less easy to obtain by direct reasoning. When not even two-point information is available, some more elementary approximation is needed. One such approximation, which is simple and direct in its application, is an extension to dynamics of a “self-consistent embedding” scheme which is widely used in static problems. This is also discussed, together with some illustrative results for a matrix containing inclusions and for a polycrystal.

In studying the effective medium theories, polarization is hardly given a consideration in deciding the effective properties of a composite where the host and inclusion phases follow different constitutive equations. A significant conclusion of this paper is that eventhough the composite has discrete inclusions, with the inclusion phase obeying different constitutive properties than the host, the effective medium shows a preference for the inclusion behavior rather than the host which is continuous. As an example, results on polarization study are detailed for the specific case of chiral composites. Application of similar principles is presently explored in more complex problems like the elastic wave propagation through piezoelectric composites and the acoustic wave propagation through sediments.

We applied to a one-dimensional system (1D) a recently developed diagrammatic formalism, in order to calculate the effective dielectric response of a chain of polarizable spheres embeded in an homogeneous host. The effective response is calculated within the dipolar, quasi-static approximation, through the summation of selected classes of diagrams. We compared our results with a numerical simulation, where the position of each sphere was generated at random and the induced dipole moment of each sphere was calculated by solving a set of linear equations through matrix inversion and using periodic boundary conditions.

The effective dielectric response εM of a composite with aligned spheroidal inclusions is calculated. Using the dipolar and the mean-field approximation (MFA) an analytical expression for εM as a functional of the two-particle distribution function p(2) is obtained. It is shown that previous expressions reported in the literature correspond to different choices of p(2), thus, clarifying the origin of their discrepancies. The theory is further extended beyond the MFA by including the dipolar fluctuations through a renormalization of the polarizability tensor of the inclusions. The absorption peaks are diminished and broadened by the spatial disorder, which also yields an easily identified coupling among electromagnetic modes with perpendicular polarizations.

Understanding the propagation of ultrasonic waves in polycrystals is important in nondestructive evaluation, both for the purpose of properly detecting or interpreting signals scattered from discrete flaws and for characterizing the microstructure itself. In this paper, the use of the unified theory for elastic wave propagation, developed by Stanke and Kino and based on the second order Keller approximation, is first discussed. After a brief review of the general formalism, recent extensions to the case of polycrystals with preferred grain orientation and elongation are presented. Particular attention is placed on a case commonly found in stainless steels in which the [001] crystallographic axes are aligned and the grains are elongated in the same direction. Techniques to predict the backscattered noise, based on single scattering approximation are then discussed and data motivating the need for extension to include multiple scattering is described. Finally, ways in which these theories can be used to provide improved characterizations of microstructures is indicated.

The study of ultrasonic waves in random media in a strong multiple scattering limit is becoming increasingly relevant to applied techniques of nondestructive characterization of material microstructures. This paper reviews the state of that art within the context of engineering applications and also reviews the evidence for the Anderson Localization of ultrasound.

While ultrasound has long been employed as a nondestructive probe of material microstructure, it has usually been the rapidly attenuated coherent wave, and in particular, its speed and attenuation, which has been scrutinized for that purpose. The theory of such waves is discussed here. However, recent experiments and theory are increasingly addressing the diffuse speckle wave field. This small but growing body of work is also reviewed.

While there has been much work recently on optical and microwave realizations for the study of the Anderson localization of classical waves, there has been little such investigation with sound waves. Recent work demonstrating the Anderson Localization of Ultrasound in two dimensions may be the first such for any classical waves in dimension greater than one.

A brief account of the current state or our understanding of full cell multiple scattering theory is given. The development of this understanding is also briefly outlined.

The variational principle of Kohn and Rostokcr (KR) implies that the determinant of a product of matrices vanishes at energy eigenvalues. The standard KKR method uses square matrices appropriate to a truncated spherical harmonic expansion, and sets the determinant of one matrix in this product to zero. For general nonspherical potentials in space-filling local cells, the KR variational principle requires these matrices to be rectangular, since the intermediate sum should extend over a complete set of solid harmonic functions. If extended in this way, the intermediate sum can be replaced by a Wronskian surface integral over the interfaces of adjacent space-filling atomic cells, as formulated in the variational principle of Schlosser and Marcus. E-quivalence of these two different variational principles is established here in the limit of completeness of the intermediate sum. The practical implication is that the standard KKR method is not well founded for the full-potential problem. Improved results for low-order basis function expansions can be obtained by including a Wronskian interface integral in full-potential ACO calculations, as a variational completion of the truncated intermediate K KR sum. A revision of CPA alloy theory is proposed that avoids dependence on this intermediate sum.

The multiple scattering theory (MST) method of Korringa, and of Kohn and Rostoker for determining the electronic structure of solids, originally developed in connection with potentials bounded by noa-overlapping spheres (Muffin-tin (MT) potentials), is generalized to the case of space-filling potential cells of arbitrary shape through the use of a variational formalism. This generalized version of MST retains the separability of structure and potential characteristic of the application of MST to MT potentials. However, in contrast to the MT case, different forms of MST exhibit different convergence rates for the energy and the wave function. Numerical results are presented which illustrate the differing convergence rates of the variational and nonvariatonal forms of MST for space-filling potentials.

A simple derivation (based on a derived identity of integral equation theory) of fully separable, non–muffin–tin multiple scattering theory is presented. Three multipolar representations have been proposed for use in this theory: the original representation by Williams and Morgan, a more recent one by Brown and Ciftan, and a “Bloch periodic” representation by Badralexe and Freeman (which was addressed in earlier work). We study the properties of the Williams and Morgan representation in the context of the 2D empty square lattice, where the representation of Brown and Ciftan is manifestly exact and correct (so that MST derived in terms of it is a “natural” theory). We show that the representation of Williams and Morgan contains an explict divergence predicted by its implicit use of a wrong-order Green's function expansion but that MST expressed in terms of this representation is still a fairly good asymptotic approximation to the formally exact theory.

We have developed a full-potential extension of the KKR Green's function method for impurities. The crystal potential is split into a sum of non-overlapping, cellular potentials. The solution for the individual anisotropic potentials is constructed by an iteration procedure starting from the solution for the isotropic potential. We discuss some technical details of the method and its efficiency in applications to impurities in metals. As examples we choose: i) calculations of local moments and Friedel oscillations in metals, ii) calculations of vacancy formation energies, iii) calculations of forces based on the Hellmann-Feynman theorem. For these applications we discuss the importance of the inclusion of full-potentials in comparison to atomic-sphere potentials.

We study the scattering solutions of the Dirac equation numerically for anisotropic, finite range (warped muffin-tin), potentials. In particular, we calculate the partial-wave scattering matrix, ƒAA'(ε) and S-matrix SAA′(ε), for a potential characteristic of crystalline Silicon. We illustrate the consequences of aspherical scattering with reference to Silicon.