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. 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. The MST equations can be derived with different wave equations, but one of the simplest and most useful ones is the Schroedinger 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, V ( r ) {displaystyle Vleft({f {r}} ight)} , 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, ( E − H 0 ) | ψ ⟩ = V | ψ ⟩ {displaystyle left({E-{H_{0}}} ight)left|psi ight angle =Vleft|psi ight angle } , where H 0 {displaystyle {H_{0}}} is the kinetic energy operator. The solution of the homogeneous equation is | φ ⟩ {displaystyle left|varphi ight angle } , where ( E − H 0 ) | φ ⟩ = 0 {displaystyle left({E-{H_{0}}} ight)left|varphi ight angle =0} . 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 | ψ ⟩ = | ϕ ⟩ + G 0 + V | ψ ⟩ {displaystyle left|psi ight angle =left|phi ight angle +{G_{0+}}Vleft|psi ight angle } , where G 0 + = lim ε → 0 ( E − H 0 + i ε ) − 1 {displaystyle {G_{0+}}={lim _{varepsilon o 0}}{left({E-{H_{0}}+ivarepsilon } ight)^{-1}}} .This is the Lippmann-Schwinger equation, which can also be written | ψ ⟩ = ( 1 + G 0 + T ) | ϕ ⟩ {displaystyle left|psi ight angle =left({1+{G_{0+}}T} ight)left|phi ight angle } . The t-matrix is defined by T = V + V G 0 + V + V G 0 + V G 0 + V + . . . {displaystyle T=V+V{G_{0+}}V+V{G_{0+}}V{G_{0+}}V+...} . Suppose that the potential V {displaystyle V} is the sum of N {displaystyle N} non-overlapping potentials, V = ∑ i = 1 N v i {displaystyle V=sum limits _{i=1}^{N}{v_{i}}} . The physical meaning of this is that it describes the interaction of the electron with a cluster of N {displaystyle N} atoms having nuclei located at positions R i {displaystyle {{f {R}}_{i}}} . Define an operator Q i {displaystyle {Q_{i}}} so that T {displaystyle T} can be written as a sum T = ∑ i = 1 N Q i {displaystyle T=sum limits _{i=1}^{N}{Q_{i}}} .Inserting the expressions for V and T into the definition of T leads to ∑ i Q i = ∑ i v i ( 1 + G 0 + ∑ j Q j ) = ∑ i [ v i G 0 + Q i + v i ( 1 + G 0 + ∑ j ≠ i Q j ) ] {displaystyle sum limits _{i}{Q_{i}}=sum limits _{i}{{v_{i}}(1+{G_{0+}}sum limits _{j}{Q_{j}}})=sum limits _{i}{left}} , so Q i = t i ( 1 + G 0 + ∑ j ≠ i Q j ) {displaystyle {Q_{i}}={t_{i}}(1+{G_{0+}}sum limits _{j eq i}{Q_{j}})} , where t i = ( 1 − v i G 0 + ) − 1 v i {displaystyle {t_{i}}={left({1-{v_{i}}{G_{0+}}} ight)^{-1}}{v_{i}}} 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 i {displaystyle i} and the outgoing wave from that site