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. It is most commonly employed in

quantum mechanical Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...

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 ' ...

solids Solid is one of the four fundamental states of matter (the others being liquid, gas, and plasma). The molecules in a solid are closely packed together and contain the least amount of kinetic energy. A solid is characterized by structural ...

. The approximation was proposed by

John C. Slater John Clarke Slater (December 22, 1900 – July 25, 1976) was a noted American physicist who made major contributions to the theory of the electronic structure of atoms, molecules and solids. He also made major contributions to microwave electroni ...

. 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 linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...

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 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 \operatorname is the linear differenti ...

methods. One application is found in the variational theory developed by Jan Korringa (1947) and by

Walter Kohn Walter Kohn (; March 9, 1923 – April 19, 2016) was an Austrian-American theoretical physicist and theoretical chemist. He was awarded, with John Pople, the Nobel Prize in Chemistry in 1998. The award recognized their contributions to the unde ...

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, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space. For any position \vec x in space and any time t, ...

s. In the atom-centered regions, the wave functions can be expanded in terms of

spherical harmonic In mathematics and 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. Since the spherical harmonics form ...

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 introduced ...

References

{{Reflist, 2 Electronic band structures Electronic structure methods Computational physics Condensed matter physics

See also

References