Multiple scattering theory
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,[1] 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.
[2] 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.
[3] The MST formalism is widely used for electronic structure calculations as well as diffraction theory, and is the subject of many books.
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.
For examples, Molière's theory of the scattering of fast charged particles in matter[6], or Glauber multiple scattering theory[7] for high-energy particle multiple-scattering off nucleons in a nucleus are also denominated that way.
The physical meaning of this is that it describes the interaction of the electron with a cluster of
To apply this theory to x-ray or neutron diffraction we go back to the Lippmann–Schwinger equation,
The Born approximation is used to calculate the t-matrix, which simply means that
Advances to this theory involve the inclusion of higher-order terms in the total scattering matrix
These terms are particularly important in the scattering of charged particles treated by Molière.
[8] 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.
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.
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
which is a solution in principle of the multiple scattering equations for stationary states.
[4] [5] The calculation of stationary states is simplified considerably for periodic solids in which all of the potentials
For a Bloch wave the coefficients depend on the site only through a phase factor,
Ewald derived a mathematically sophisticated summation process that makes it possible to calculate the structure constants,
The dimension of these matrix equations is technically infinite, but by ignoring all contributions that correspond to an angular momentum quantum number
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
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.
[4][9] It can be extended to treat crystals with any number of atoms in a unit cell.
[10] The arguments that lead to a multiple scattering solution for the single-particle orbital
can also be used to formulate a multiple scattering version of the single-particle Green's function
With this Green's function and the Korringa–Kohn–Rostoker method, the Korringa–Kohn–Rostoker coherent potential approximation (KKR-CPA) is obtained.
[11] 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.
