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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 far ... [...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 wave ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basis Set (chemistry)
In theoretical and computational chemistry, a basis set is a set of functions (called basis functions) that is used to represent the electronic wave function in the Hartree–Fock method or 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 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 quantum chemistry community; plane waves which are typically used within the solid state community, or ... [...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, electron or 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 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, the narrower the distribution of f in \mathbf; i.e., the faster the ... [...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.Über das Auftreten von Ionen beim Zerfall von Ozon und die Ionisation der Stratosphäre. he appearance of ions on the decomposition of ozone, and the ionization of the stratosphere (in German)Ann. Phys. (Leipzig) . Folge2 (1929) 707-732 (DOI: 10.1002/andp.19293940607) . A. 24:23, CAN 24:239 / Sci. Abstr. A 33 (1930) 740 / C. 1929 II, 2418/ref> Hellmann's future spouse Victoria Bernstein was the foster daughter of R ... [...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 electrodynami ... [...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 abov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Enrico Fermi
Enrico Fermi (; 29 September 1901 – 28 November 1954) was an Italian (later naturalized American) physicist and the creator of the world's first nuclear reactor, the Chicago Pile-1. 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 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 theory, and nuclear and particle physics. Fermi's first major contribution involved the field of statistical mechanics. After Wolfgang Pauli formulated his exclusion principle in 1925, Fermi followed with a paper in which he ... [...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 ... [...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), a Malayalam film * ''Sketch'' (2018 film), a Tamil film * ''Sketch'' (TV series), a 2018 South Korean series * "Sketch", a 2008 episode of ''Skins'' ** 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 assist in composition * The Sketches, a Pakistani Sufi folk rock band * ''Sketch'' (album), by Ex Norwegian, 2011 * ''Sketch'' (EP), by Hyomin, 2016 * ''Sketches'' (album), by Bert Jansch, 1990 ... [...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. Preliminary scattering theory The following description follows the canonical way of introducing elementary 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. In principle, any particle should be described by a wave packet, but we instead describe the scattering of a plane wave \exp(ikz) traveling along the ''z'' axis, since wave packets can be expanded in terms of plane waves, and this is mathematically simpler. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
