The
muffin-tin approximation is a shape approximation of the
potential well
A potential well is the region surrounding a local minimum of potential energy. Energy captured in a potential well is unable to convert to another type of energy ( kinetic energy in the case of a gravitational potential well) because it is cap ...
in a
crystal lattice
In crystallography, crystal structure is a description of ordered arrangement of atoms, ions, or molecules in a crystal, crystalline material. Ordered structures occur from intrinsic nature of constituent particles to form symmetric patterns that ...
. It is most commonly employed in
quantum mechanical
Quantum mechanics is the fundamental physical theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is the foundation of a ...
simulations of the
electronic band structure
In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called ''band gaps'' or '' ...
in
solids
Solid is a state of matter where molecules are closely packed and can not slide past each other. Solids resist compression, expansion, or external forces that would alter its shape, with the degree to which they are resisted dependent upon the ...
. 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
The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
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
In mathematics, a Green's function (or Green function) is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if L is a linear dif ...
methods.
[
] One application is found in the variational theory developed by
Jan Korringa (1947) and by
Walter Kohn and N. Rostoker (1954) referred to as the
KKR method.
[
][
][
] 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 wave
In physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of ...
s. In the atom-centered regions, the wave functions can be expanded in terms of
spherical harmonic
In mathematics and Outline of physical science, physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. The tabl ...
s and the
eigenfunction
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
s 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
In physics, a pseudopotential or effective potential is used as an approximation for the simplified description of complex systems. Applications include atomic physics and neutron scattering. The pseudopotential approximation was first introduce ...
).
See also
*
Anderson's rule
Anderson's rule is used for the construction of energy band diagrams of the heterojunction between two semiconductor materials. Anderson's rule states that when constructing an energy band diagram, the vacuum levels of the two semiconductors on ...
*
Band gap
In solid-state physics and solid-state chemistry, a band gap, also called a bandgap or energy gap, is an energy range in a solid where no electronic states exist. In graphs of the electronic band structure of solids, the band gap refers to t ...
*
Bloch waves
In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential can be expressed as plane waves modulated by periodic functions. The theorem is named after the Swiss physicist Felix Bloch, ...
*
Kohn–Sham equations
The Kohn-Sham equations are a set of mathematical equations used in quantum mechanics to simplify the complex problem of understanding how electrons behave in atoms and molecules. They introduce fictitious non-interacting electrons and use them to ...
*
Kronig–Penney model
*
Local-density approximation
Local-density approximations (LDA) are a class of approximations to the exchange–correlation (XC) energy functional in density functional theory (DFT) that depend solely upon the value of the electronic density at each point in space (and ...
References
{{Reflist, 2
Electronic band structures
Electronic structure methods
Computational physics
Condensed matter physics