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Muffin-tin Approximation
The muffin-tin approximation is a shape approximation of the potential well in a crystal lattice. 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 orbit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Muffin Tin
A muffin or cupcake tray is a mold in which muffins or cupcakes are baked. A single cup within a regular muffin tin is 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 , and because these are less common, there are several standard numbers of cups per tin, including 6, 12, and 24 cups per tin. A single cup within a jumbo muffin tin is , and again because these are uncommon, there are several standard numbers of cups per tin, including 4, 6, and 12 cups per tin. Muffin tins can be made out of aluminum, stainless steel, cast iron, or silicone. In addition, aluminum and stainless steel muffin tins may be coated with Teflon or other non-stick coatings. Historically, galvanized steel has been used for muffin tins but this is no longer common. See also * List of food preparation utensils A kitchen utensil is a hand-held, typically small tool that is designed for food-related functions. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coherent Potential Approximation
The coherent potential approximation (or CPA) is a method, in physics, of finding the Green's function of an effective medium. It is a useful concept in understanding how sound waves scatter in a material which displays spatial inhomogeneity. One version of the CPA is an extension to random materials of the muffin-tin approximation, used to calculate electronic band structure in solids. A variational implementation of the muffin-tin approximation to crystalline solids using Green's functions was suggested by Korringa and by Kohn and Rostoker, and is often 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 .... For random materials, the theory is applied by the introduction of an ordered lattice of effective potentials to replace the varying potentials in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electronic Structure Methods
Electronic may refer to: *Electronics, the science of how to control electric energy in semiconductor * ''Electronics'' (magazine), a defunct American trade journal *Electronic storage, the storage of data using an electronic device *Electronic commerce or e-commerce, the trading in products or services using computer networks, such as the Internet *Electronic publishing or e-publishing, the digital publication of books and magazines using computer networks, such as the Internet *Electronic engineering, an electrical engineering discipline Entertainment *Electronic (band), an English alternative dance band ** ''Electronic'' (album), the self-titled debut album by British band Electronic *Electronic music, a music genre *Electronic musical instrument *Electronic game, a game that employs electronics See also *Electronica, an electronic music genre *Consumer electronics Consumer electronics or home electronics are electronic (analog or digital) equipment intended for everyday ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electronic Band Structures
Electronic may refer to: *Electronics, the science of how to control electric energy in semiconductor * ''Electronics'' (magazine), a defunct American trade journal *Electronic storage, the storage of data using an electronic device *Electronic commerce or e-commerce, the trading in products or services using computer networks, such as the Internet *Electronic publishing or e-publishing, the digital publication of books and magazines using computer networks, such as the Internet *Electronic engineering, an electrical engineering discipline Entertainment *Electronic (band), an English alternative dance band ** ''Electronic'' (album), the self-titled debut album by British band Electronic *Electronic music, a music genre *Electronic musical instrument *Electronic game, a game that employs electronics See also *Electronica, an electronic music genre *Consumer electronics Consumer electronics or home electronics are electronic ( analog or digital) equipment intended for everyd ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 not, for example, derivatives of the density or the Kohn–Sham orbitals). Many approaches can yield local approximations to the XC energy. However, overwhelmingly successful local approximations are those that have been derived from the homogeneous electron gas (HEG) model. In this regard, LDA is generally synonymous with functionals based on the HEG approximation, which are then applied to realistic systems (molecules and solids). In general, for a spin-unpolarized system, a local-density approximation for the exchange-correlation energy is written as :E_^rho= \int \rho(\mathbf)\epsilon_(\rho(\mathbf))\ \mathrm\mathbf\ , where ''ρ'' is the electronic density and ''ε''xc is the exchange-correlation energy per particle of a homogeneous ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 electrons are subject to a regular potential inside the lattice. It is a generalization of the free electron model, which assumes zero potential inside the lattice. Problem definition When talking about solid materials, the discussion is mainly around crystals – periodic lattices. Here we will discuss a 1D lattice of positive ions. Assuming the spacing between two ions is , the potential in the lattice will look something like this: The mathematical representation of the potential is a periodic function with a period . According to Bloch's theorem, the wavefunction solution of the Schrödinger equation when the potential is periodic, can be written as: \psi (x) = e^ u(x), where is a periodic function which satisfies . It is the Bloch f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bloch Waves
In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. The theorem is named after the physicist Felix Bloch, who discovered the theorem in 1929. Mathematically, they are written where \mathbf is position, \psi is the wave function, u is a periodic function with the same periodicity as the crystal, the wave vector \mathbf is the crystal momentum vector, e is Euler's number, and i is the imaginary unit. Functions of this form are known as Bloch functions or Bloch states, and serve as a suitable basis for the wave functions or states of electrons in crystalline solids. Named after Swiss physicist Felix Bloch, the description of electrons in terms of Bloch functions, termed Bloch electrons (or less often ''Bloch Waves''), underlies the concept of electronic band structures. These eigenstates are written with subscripts as \psi_, where n is a discret ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Band Gap
In solid-state physics, a band gap, also called an energy gap, is an energy range in a solid where no electronic states can exist. In graphs of the electronic band structure of solids, the band gap generally refers to the energy difference (in electron volts) between the top of the valence band and the bottom of the conduction band in insulators and semiconductors. It is the energy required to promote a valence electron bound to an atom to become a conduction electron, which is free to move within the crystal lattice and serve as a charge carrier to conduct electric current. It is closely related to the HOMO/LUMO gap in chemistry. If the valence band is completely full and the conduction band is completely empty, then electrons cannot move within the solid because there are no available states. If the electrons are not free to move within the crystal lattice, then there is no generated current due to no net charge carrier mobility. However, if some electrons transfer from th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 either side of the heterojunction should be aligned (at the same energy). It is also referred to as the electron affinity rule, and is closely related to the Schottky–Mott rule for metal–semiconductor junctions. Anderson's rule was first described by R. L. Anderson in 1960. Constructing energy band diagrams Once the vacuum levels are aligned it is possible to use the electron affinity and band gap values for each semiconductor to calculate the conduction band and valence band offsets. The electron affinity (usually given by the symbol \chi in solid state physics) gives the energy difference between the lower edge of the conduction band and the vacuum level of the semiconductor. The band gap (usually given the symbol E_) gives the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 by Hans Hellmann in 1934. Atomic physics The pseudopotential is an attempt to replace the complicated effects of the motion of the core (i.e. non- valence) electrons of an atom and its nucleus with an effective potential, or pseudopotential, so that the Schrödinger equation contains a modified effective potential term instead of the Coulombic potential term for core electrons normally found in the Schrödinger equation. The pseudopotential is an effective potential constructed to replace the atomic all-electron potential (full-potential) such that core states are eliminated ''and'' the valence electrons are described by pseudo-wavefunctions with significantly fewer nodes. This allows the pseudo-wavefunctions to be described with far ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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, this condition can be written as Df = \lambda f for some scalar eigenvalue \lambda. The solutions to this equation may also be subject to boundary conditions that limit the allowable eigenvalues and eigenfunctions. An eigenfunction is a type of eigenvector. Eigenfunctions In general, an eigenvector of a linear operator ''D'' defined on some vector space is a nonzero vector in the domain of ''D'' that, when ''D'' acts upon it, is simply scaled by some scalar value called an eigenvalue. In the special case where ''D'' is defined on a function space, the eigenvectors are referred to as eigenfunctions. That is, a function ''f'' is an eigenfunction of ''D'' if it satisfies the equation where λ is a scalar. The solutions to Equation may also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
