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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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basis Set (chemistry)
In theoretical chemistry, theoretical and computational chemistry, a basis set is a set of Function (mathematics), functions (called basis functions) that is used to represent the Wave function, electronic wave function in the Hartree–Fock method or Density functional theory, density-functional theory in order to turn the partial differential equations of the model into algebraic equations suitable for efficient implementation on a computer. The use of basis sets is equivalent to the use of an approximate resolution of the identity: the Atomic orbital, orbitals , \psi_i\rangle are expanded within the basis set as a linear combination of the basis functions , \psi_i\rangle \approx \sum_\mu c_ , \mu\rangle, where the expansion coefficients c_ are given by c_ = \sum_\nu \langle \mu, \nu \rangle^ \langle \nu , \psi_i \rangle. The basis set can either be composed of atomic orbitals (yielding the linear combination of atomic orbitals approach), which is the usual choice within the qua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projector Augmented Wave Method
The projector augmented wave method (PAW) is a technique used in ab initio electronic structure calculations. It is a generalization of the pseudopotential and linear augmented-plane-wave methods, and allows for density functional theory calculations to be performed with greater computational efficiency. Valence wavefunctions tend to have rapid oscillations near ion cores due to the requirement that they be orthogonal to core states; this situation is problematic because it requires many Fourier components (or in the case of grid-based methods, a very fine mesh) to describe the wavefunctions accurately. The PAW approach addresses this issue by transforming these rapidly oscillating wavefunctions into smooth wavefunctions which are more computationally convenient, and provides a way to calculate all-electron properties from these smooth wavefunctions. This approach is somewhat reminiscent of a change from the Schrödinger picture to the Heisenberg picture. Transforming the wavef ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linearized Augmented-plane-wave Method
The linearized augmented-plane-wave method (LAPW) is an implementation of Kohn-Sham density functional theory (DFT) adapted to periodic materials. It typically goes along with the treatment of both valence and core electrons on the same footing in the context of DFT and the treatment of the full potential and charge density without any shape approximation. This is often referred to as the all-electron full-potential linearized augmented-plane-wave method (FLAPW). It does not rely on the pseudopotential approximation and employs a systematically extendable basis set. These features make it one of the most precise implementations of DFT, applicable to all crystalline materials, regardless of their chemical composition. It can be used as a reference for evaluating other approaches. Introduction At the core of density functional theory the Hohenberg-Kohn theorems state that every observable of an interacting many-electron system is a functional of its ground-state charge density a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atomic Form Factor
In physics, the atomic form factor, or atomic scattering factor, is a measure of the scattering amplitude of a wave by an isolated atom. The atomic form factor depends on the type of scattering, which in turn depends on the nature of the incident radiation, typically X-ray diffraction, X-ray, Electron diffraction, electron or Neutron diffraction, neutron. The common feature of all form factors is that they involve a Fourier transform of a spatial density distribution of the scattering object from space, real space to momentum space (also known as reciprocal space). For an object with spatial density distribution, \rho(\mathbf), the form factor, f(\mathbf), is defined as :f(\mathbf)=\int \rho(\mathbf) e^\mathrm^3\mathbf, where \rho(\mathbf) is the spatial density of the scatterer about its center of mass (\mathbf=0), and \mathbf is the momentum transfer. As a result of the nature of the Fourier transform, the broader the distribution of the scatterer \rho in real space \mathbf, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hans Hellmann
Hans Gustav Adolf Hellmann (14 October 1903 – 29 May 1938) was a German theoretical physicist. Biography Hellmann was born in Wilhelmshaven, Prussian Hanover. He began studying electrical engineering in Stuttgart but changed to engineering physics after a semester. Hellmann also studied at the University of Kiel. He received his diploma from the Kaiser Wilhelm Institute for Chemistry in Berlin for work on radioactive compounds under Otto Hahn and Lise Meitner. He received his Ph.D. at Stuttgart with Prof. Erich Regener for work on the decomposition of ozone. Hellmann's future spouse Victoria Bernstein was the foster daughter of Regener. In 1929 Hellmann became an assistant professor at the Leibniz University Hannover. After the Nazi rise to power, Hellmann was dismissed on 24 December 1933 as ‘undesirable’ because of his Jewish wife. He emigrated to the Soviet Union, taking up a position at the Karpov institute in Moscow working among other things on pseudopotentials ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scattering Length
The scattering length in quantum mechanics describes low-energy scattering. For potentials that decay faster than 1/r^3 as r\to \infty, it is defined as the following low-energy limit: : \lim_ k\cot\delta(k) =- \frac\;, where a is the scattering length, k is the wave number, and \delta(k) is the phase shift of the outgoing spherical wave. The elastic cross section, \sigma_e, at low energies is determined solely by the scattering length: : \lim_ \sigma_e = 4\pi a^2\;. General concept When a slow particle scatters off a short ranged scatterer (e.g. an impurity in a solid or a heavy particle) it cannot resolve the structure of the object since its de Broglie wavelength is very long. The idea is that then it should not be important what precise potential V(r) one scatters off, but only how the potential looks at long length scales. The formal way to solve this problem is to do a partial wave expansion (somewhat analogous to the multipole expansion in classical electrodyna ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neutron Scattering
Neutron scattering, the irregular dispersal of free neutrons by matter, can refer to either the naturally occurring physical process itself or to the man-made experimental techniques that use the natural process for investigating materials. The natural/physical phenomenon is of elemental importance in nuclear engineering and the nuclear sciences. Regarding the experimental technique, understanding and manipulating neutron scattering is fundamental to the applications used in crystallography, physics, physical chemistry, biophysics, and materials research. Neutron scattering is practiced at research reactors and spallation neutron sources that provide neutron radiation of varying intensities. Neutron diffraction (elastic scattering) techniques are used for analyzing structures; where inelastic neutron scattering is used in studying atomic vibrations and other excitations. Scattering of fast neutrons "Fast neutrons" (see neutron temperature) have a kinetic energy abo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Enrico Fermi
Enrico Fermi (; 29 September 1901 – 28 November 1954) was an Italian and naturalized American physicist, renowned for being the creator of the world's first artificial nuclear reactor, the Chicago Pile-1, and a member of the Manhattan Project. He has been called the "architect of the nuclear age" and the "architect of the atomic bomb". He was one of very few physicists to excel in both theoretical physics, theoretical and experimental physics. Fermi was awarded the 1938 Nobel Prize in Physics for his work on induced radioactivity by neutron bombardment and for the discovery of transuranium elements. With his colleagues, Fermi filed several patents related to the use of nuclear power, all of which were taken over by the US government. He made significant contributions to the development of statistical mechanics, Quantum mechanics, quantum theory, and nuclear physics, nuclear and particle physics. Fermi's first major contribution involved the field of statistical mechanics. Afte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Basis
In linear algebra, given a vector space V with a basis B of vectors indexed by an index set I (the cardinality of I is the dimension of V), the dual set of B is a set B^* of vectors in the dual space V^* with the same index set I such that B and B^* form a biorthogonal system. The dual set is always linearly independent but does not necessarily span V^*. If it does span V^*, then B^* is called the dual basis or reciprocal basis for the basis B. Denoting the indexed vector sets as B = \_ and B^ = \_, being biorthogonal means that the elements pair to have an inner product equal to 1 if the indexes are equal, and equal to 0 otherwise. Symbolically, evaluating a dual vector in V^* on a vector in the original space V: : v^i\cdot v_j = \delta^i_j = \begin 1 & \text i = j\\ 0 & \text i \ne j\text \end where \delta^i_j is the Kronecker delta symbol. Introduction To perform operations with a vector, we must have a straightforward method of calculating its components. In a Cartesia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sketch Pseudopotentials
Sketch or Sketches may refer to: * Sketch (drawing), a rapidly executed freehand drawing that is not usually intended as a finished work Arts, entertainment and media * Sketch comedy, a series of short scenes or vignettes called sketches Film and television * Sketch (2007 film), ''Sketch'' (2007 film), a Malayalam film * Sketch (2018 film), ''Sketch'' (2018 film), a Tamil film * Sketch (2024 film), ''Sketch'' (2024 film), an American comedy horror film * Sketch (TV series), ''Sketch'' (TV series), a 2018 South Korean series * "Sketch", a 2008 List of Skins episodes#Series 2 (2008), episode of ''Skins'' ** Sketch (Skins character), Sketch (''Skins'' character) * Sketch with Kevin McDonald, a 2006 CBC television special Literature * Sketch story, or sketch, a very short piece of writing * ''Daily Sketch'', a British newspaper 1909–1971 * ''The Sketch'', a British illustrated weekly journal 1893–1959 Music * Sketch (music), an informal document prepared by a composer to as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Wave Analysis
Partial-wave analysis, in the context of quantum mechanics, refers to a technique for solving scattering problems by decomposing each wave into its constituent angular-momentum components and solving using boundary conditions. Partial wave analysis is typically useful for low energy scattering where only a few angular momentum components dominate. At high energy were scattering is weak, an alternative called the Born approximation is used. Preliminary scattering theory A steady beam of particles scatters off a spherically symmetric potential V(r), which is short-ranged, so that for large distances r \to \infty, the particles behave like free particles. The incoming beam is assumed to be a collimated plane wave \exp(ikz) traveling along the ''z'' axis. Because the beam is switched on for times long compared to the time of interaction of the particles with the scattering potential, a steady state is assumed. This means that the stationary Schrödinger equation for the wave f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
