Muffin-tin approximation
Encyclopedia
The muffin-tin
approximation is a shape approximation of the potential field in an atomistic environment. It is most commonly employed in quantum mechanical simulations of electronic band structure
in solids
. The approximation was proposed by John C. Slater
. Many modern electronic structure methods employ the approximation. Among them are the augmented plane wave (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 Korringa (1947) and by Kohn
and Rostocker (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
s. In the atom-centered regions, the wave functions can be expanded in terms of spherical harmonic
s and the eigenfunction
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
).
Muffin tin
A muffin tin is a mold in which muffins or cupcakes are baked. A single cup within a regular muffin tin is 3 and 1/2 ounces and most often has room for 12 muffins, although tins holding 6, 8, 11, 24, and 35 muffins do exist. A single cup within a mini muffin tin is 2 and 1/8 ounces, and because...
approximation is a shape approximation of the potential field in an atomistic environment. It is most commonly employed in quantum mechanical simulations of electronic band structure
Electronic band structure
In solid-state physics, the electronic band structure of a solid describes those ranges of energy an electron is "forbidden" or "allowed" to have. Band structure derives from the diffraction of the quantum mechanical electron waves in a periodic crystal lattice with a specific crystal system and...
in solids
Solid-state physics
Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the large-scale properties of solid materials result from...
. The approximation was proposed by John C. Slater
John C. Slater
John Clarke Slater was a noted American physicist who made major contributions to the theory of the electronic structure of atoms, molecules and solids. This work is of ongoing importance in chemistry, as well as in many areas of physics. He also made major contributions to microwave electronics....
. Many modern electronic structure methods employ the approximation. Among them are the augmented plane wave (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 Korringa (1947) and by Kohn
Walter Kohn
Walter Kohn is an Austrian-born American theoretical physicist.He was awarded, with John Pople, the Nobel Prize in chemistry in 1998. The award recognized their contributions to the understandings of the electronic properties of materials...
and Rostocker (1954) referred to as the KKR method
Coherent Potential Approximation
The coherent potential approximation is a method, in physics, of finding the Green's function of an effective medium. It is a useful concept in understanding how waves scatter in a material which displays spatial inhomogeneity....
. 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
Screening effect
In solids, especially in metals and semiconductors, the electrostatic screening or screening effect reduces the electrostatic field and Coulomb potential of an ion inside the solid...
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
Plane wave
In the physics of wave propagation, a plane wave is a constant-frequency wave whose wavefronts are infinite parallel planes of constant peak-to-peak amplitude normal to the phase velocity vector....
s. In the atom-centered regions, the wave functions can be expanded in terms of spherical harmonic
Spherical Harmonic
Spherical Harmonic is a science fiction novel from the Saga of the Skolian Empire by Catherine Asaro. It tells the story of Dyhianna Selei , the Ruby Pharaoh of the Skolian Imperialate, as she strives to reform her government and reunite her family in the aftermath of a devastating interstellar...
s and the eigenfunction
Eigenfunction
In mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has...
s of a radial Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
. 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
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.- Atomic physics :...
).
See also
- Anderson's ruleAnderson's ruleAnderson's rule is used for the construction of energy band diagrams of the heterojunction between two semiconductor materials. It is also referred to as the electron affinity rule. Anderson's rule was first described by R. L...
- Band gapBand gapIn solid state physics, a band gap, also called an energy gap or bandgap, is an energy range in a solid where no electron states can exist. In graphs of the electronic band structure of solids, the band gap generally refers to the energy difference between the top of the valence band and the...
- Bloch waves
- Kohn–Sham equations
- Kronig–Penney model
- Local-density approximationLocal-density approximationLocal-density approximations are a class of approximations to the exchange-correlation energy functional in density functional theory that depend solely upon the value of the electronic density at each point in space . Many approaches can yield local approximations to the XC energy...