The
muffin-tin approximation is a shape approximation of the
potential well in a
crystal lattice
In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by
: \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n ...
. It is most commonly employed in
quantum mechanical simulations of the
electronic band structure in
solids. The approximation was proposed by
John C. Slater. Augmented plane wave method (APW) is a method which uses muffin-tin approximation. It is a method to approximate the energy states of an electron in a crystal lattice. The basic approximation lies in the potential in which the potential is assumed to be spherically symmetric in the muffin-tin region and constant in the interstitial region. Wave functions (the augmented plane waves) are constructed by matching solutions of the
Schrödinger equation within each sphere with plane-wave solutions in the interstitial region, and linear combinations of these wave functions are then determined by the variational method.
[
] Many modern electronic structure methods employ the approximation.
[
][
] Among them APW method, the linear muffin-tin orbital method (LMTO) and various
Green's function methods.
[
] One application is found in the variational theory developed by
Jan Korringa
Jan Korringa (31 March 1915 – 9 October 2015) was a Dutch American theoretical physicist, specializing in theoretical condensed matter physics. He also contributed to the KKR Method.
Education and career
Korringa received his undergraduate d ...
(1947) and by
Walter Kohn and N. Rostoker (1954) referred to as the
KKR method
KKR & Co. Inc., also known as Kohlberg Kravis Roberts & Co., is an American global investment company that manages multiple alternative asset classes, including private equity, energy, infrastructure, real estate, credit, and, through its strateg ...
.
[
][
][
] This method has been adapted to treat random materials as well, where it is called the
KKR coherent potential approximation.
In its simplest form, non-overlapping spheres are centered on the atomic positions. Within these regions, the
screened potential experienced by an electron is approximated to be spherically symmetric about the given nucleus. In the remaining interstitial region, the potential is approximated as a constant. Continuity of the potential between the atom-centered spheres and interstitial region is enforced.
In the interstitial region of constant potential, the single electron wave functions can be expanded in terms of
plane waves. In the atom-centered regions, the wave functions can be expanded in terms of
spherical harmonics and the
eigenfunctions of a radial Schrödinger equation.
[
] Such use of functions other than plane waves as basis functions is termed the augmented plane-wave approach (of which there are many variations). It allows for an efficient representation of single-particle wave functions in the vicinity of the atomic cores where they can vary rapidly (and where plane waves would be a poor choice on convergence grounds in the absence of a
pseudopotential).
See also
*
Anderson's rule
*
Band gap
*
Bloch waves
*
Kohn–Sham equations
*
Kronig–Penney model
In quantum mechanics, the particle in a one-dimensional lattice is a problem that occurs in the model of a periodic crystal lattice. The potential is caused by ions in the periodic structure of the crystal creating an electromagnetic field so elec ...
*
Local-density approximation
References
{{Reflist, 2
Electronic band structures
Electronic structure methods
Computational physics
Condensed matter physics