Multiple scattering theory
Multiple scattering theory (MST) is the mathematical formalism that is used to describe the propagation of a wave through a collection of scatterers. Examples are acoustical waves traveling through porous media, light scattering from water droplets in a cloud, or x-rays scattering from a crystal. A more recent application is to the propagation of quantum matter waves like electrons or neutrons through a solid. As pointed out by Jan Korringa, the origin of this theory can be traced back to an 1892 paper by Lord Rayleigh. An important mathematical formulation of the theory was made by  Paul Peter Ewald. Korringa and Ewald acknowledged the influence on their work of the 1903 doctoral dissertation of Nikolai Kasterin, portions of which were published in German in the Proceedings of the Royal Academy of Sciences in Amsterdam under the sponsorship of Heike Kamerlingh Onnes. The MST formalism is widely used for electronic structure calculations as well as diffraction theory, and is the subject of many books. The multiple-scattering approach is the best way to derive one-electron Green's functions. These functions differ from the Green's functions used to treat the many-body problem, but they are  the best starting point for calculations of the electronic structure of condensed matter systems that cannot be treated with band theory. The terms "multiple scattering" and "multiple scattering theory" are often used in other contexts. For example, Molière's theory of the scattering of fast charged particles in matter is described in that way. Mathematical formulation. The MST equations can be derived with different wave equations, but one of the simplest and most useful ones is the Schrödinger equation for an electron moving in a solid. With the help of density functional theory, this problem can be reduced to  the solution of a one-electron equation where the effective one-electron potential, formula_2, is a functional of the density of the electrons in the system. In the Dirac notation, the wave equation can be written as an inhomogeneous equation, formula_3, where formula_4is the kinetic energy operator. The solution of the homogeneous equation is formula_5, where formula_6. A formal solution of the inhomogeneous equation is the sum of the solution of the homogeneous equation with a particular solution of the inhomogeneous equation formula_7, where formula_8. This is the Lippmann–Schwinger equation, which can also be written formula_9. The t-matrix  is defined by formula_10. Suppose that the potential formula_11 is the sum of formula_12 non-overlapping potentials, formula_13. The physical meaning of this is that it describes the interaction of the electron with a cluster of formula_12 atoms having nuclei located at positions formula_15. Define an operator formula_16 so that formula_17 can be written as a sum formula_18. Inserting the expressions for formula_11 and formula_17 into the definition of formula_17 leads to so formula_23, where formula_24 is the scattering matrix for one atom. Iterating this equation leads to The solution of the Lippmann-Schwinger equation can thus be written as  the sum of an incoming wave on any site formula_26 and the outgoing wave from that site The site formula_26 that we have chosen to focus on can be any of the sites in the cluster. The incoming wave on this site is the incoming wave on the cluster and the outgoing waves from all the other sites The outgoing wave from the site formula_26 is defined as These last two equations are the fundamental equations of multiple scattering. To apply this theory to x-ray or neutron diffraction we go back to the Lippmann–Schwinger equation, formula_34. The scattering  from a site is assumed to be very small, so formula_35 or formula_18. The Born approximation is used to calculate the t-matrix, which simply means that formula_37 is replaced with formula_38. A plane wave impinges on a site, and a spherical wave exits it. The outgoing wave from the crystal is determined by the constructive interference of the waves from the sites. Advances to this theory involve the inclusion of higher-order terms in the total scattering matrix formula_17, such asformula_40. These terms are particularly important in the scattering of charged particles treated by Molière. Multiple scattering theory of  electronic states in solids. In 1947, Korringa pointed out that the multiple scattering equations can be used to calculate stationary states in a crystal for which the number of scatterers formula_12 goes to infinity. Setting the incoming wave on the cluster and the outgoing wave from the cluster to zero, he wrote the first multiple scattering as A simple description of this process is that the electrons scatter from one atom to the other ad infinitum. Since the formula_43 are bounded in space and do not overlap, there is an interstitial region between them within which the potential is a  constant, usually taken to be zero. In this region, the Schrödinger equation becomes formula_44, where formula_45. The incoming wave on site formula_26 can thus be written in the position representation where the formula_48 are undetermined coefficients and formula_49. The Green's function may be expanded in the interstitial region and the outgoing Hankel function can be written This leads to a set of homogeneous simultaneous equations that determines the unknown coefficients formula_48 which is a solution in principle of the multiple scattering equations for stationary states. This theory is very important for studies in condensed matter physics. Periodic solids, one  atom per unit cell. The calculation of stationary states is simplified considerably for periodic solids in which all of the potentials formula_54 are the same, and the nuclear positions formula_15 form a periodic array. Bloch's theorem holds for such a system, which means that the solutions of the Schrödinger equation may be written as a Bloch wave formula_56. It is more convenient to deal with a symmetric matrix for the coefficients, and this can be done by defining These coefficients satisfy the set of linear equations formula_58, with the elements of the matrix formula_59 being and the formula_61 are  the elements of the inverse of the t-matrix. For a Bloch wave the coefficients depend on the site only through a phase factor, formula_62, and the formula_63 satisfy the homogeneous equations where formula_65 and formula_66. Walter Kohn and Norman Rostoker derived this same theory using the Kohn variational method. It is called the Korringa–Kohn–Rostoker method (KKR method) for band theory calculations. Ewald derived a mathematically sophisticated summation process that makes it possible to calculate the structure constants, formula_67. The energy eigenvalues of the periodic solid for a particular formula_68, formula_69, are the roots of the equation formula_70 . The eigenfunctions are found by solving for the formula_71 with formula_72. The dimension of these matrix equations is technically infinite, but by ignoring all contributions that correspond to an angular momentum quantum number formula_73 greater than formula_74, they have dimension formula_75. The justification for this approximation is that the matrix elements of the t-matrix formula_76 are very small when formula_73 and formula_78 are greater than formula_74, and the elements of the inverse matrix formula_80 are very large. In the original derivations of the KKR method, spherically symmetric muffin-tin potentials were used. Such potentials have the advantage that the  inverse of the scattering matrix is diagonal in formula_73 where formula_83 is the scattering phase shift that appears in the partial wave analysis in scattering theory. It is also easier to visualize the waves scattering from one atom to another, and formula_84 can be used in many applications. The muffin-tin approximation is adequate for most metals in a close-packed arrangement. It cannot be used for calculating forces between atoms, or for important systems like semiconductors. Extensions of the theory. It is now known that the KKR method can be used with space-filling non-spherical potentials. It  can be extended to treat crystals with any number of atoms in a unit cell. There are versions of the theory that can be used to calculate surface states. The arguments that lead to a multiple scattering solution for the single-particle orbital formula_85 can also be used to formulate a multiple scattering version of the single-particle Green's function formula_86 which is a solution of the equation The potential formula_88 is the same one from density functional theory that was used in the preceding discussion. With this Green's function and the Korringa–Kohn–Rostoker method, the Korringa –Kohn–Rostoker coherent potential approximation (KKR-CPA) is obtained. The KKR-CPA is used to calculate the electronic states for substitutional solid-solution alloys, for which Bloch's theorem does not hold. The electronic states for an even wider range of condensed matter structures can be found using the locally self-consistent multiple scattering (LSMS) method, which is also based on the single-particle Green's function.