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Friedel's Law
Friedel's law, named after Georges Friedel, is a property of Fourier transforms of real functions. Given a real function f(x), its Fourier transform :F(k)=\int^_f(x)e^dx has the following properties. *F(k)=F^*(-k) \, where F^* is the complex conjugate of F. Centrosymmetric points (k,-k) are called Friedel's pairs. The squared amplitude (, F, ^2) is centrosymmetric: * , F(k), ^2=, F(-k), ^2 \, The phase \phi of F is antisymmetric: * \phi(k) = -\phi(-k) \,. Friedel's law is used in X-ray diffraction, crystallography and scattering from real potential within the Born approximation Generally in scattering theory and in particular in quantum mechanics, the Born approximation consists of taking the incident field in place of the total field as the driving field at each point in the scatterer. The Born approximation is named a .... Note that twin operation( ''Opération de maclage'') is equivalent to an inversion centre and the intensities from the individuals are equivalent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Georges Friedel
Georges Friedel (19 July 1865 – 11 December 1933) was a French Mineralogy, mineralogist and Crystallography, crystallographer. Life Georges was the son of the chemist Charles Friedel. Georges' grandfather was Georges Louis Duvernoy, Louis Georges Duvernoy who held the chair in comparative anatomy from 1850 to 1855 at the Muséum national d'histoire naturelle. Georges studied at the École Polytechnique in Paris and the École des mines de Saint-Étienne, École Nationale des Mines in St. Etienne, and was a student of François Ernest Mallard. In 1893 he obtained a professorship at the École Nationale des Mines, the director of which he would later become. After the World War I, First World War, he returned as a professor at the University of Strasbourg in Alsace. Due to ill health, he took early retirement in 1930, and died in 1933. He was married with five children. Scientific works Like his teacher Mallard, Friedel concerned himself with the theories of Auguste Bravais, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, which will output a function depending on temporal frequency or spatial frequency respectively. That process is also called ''analysis''. An example application would be decomposing the waveform of a musical chord into terms of the intensity of its constituent pitches. The term ''Fourier transform'' refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of space or time. The Fourier transform of a function is a complex-valued function representing the complex sinusoids that comprise the original function. For each frequency, the magnitude (absolute value) of the complex value represents the amplitude of a constituent complex sinusoid with that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centrosymmetric
In crystallography, a centrosymmetric point group contains an inversion center as one of its symmetry elements. In such a point group, for every point (x, y, z) in the unit cell there is an indistinguishable point (-x, -y, -z). Such point groups are also said to have ''inversion'' symmetry. Point reflection is a similar term used in geometry. Crystals with an inversion center cannot display certain properties, such as the piezoelectric effect. The following space groups have inversion symmetry: the triclinic space group 2, the monoclinic 10-15, the orthorhombic 47-74, the tetragonal 83-88 and 123-142, the trigonal 147, 148 and 162-167, the hexagonal 175, 176 and 191-194, the cubic 200-206 and 221-230. Point groups lacking an inversion center (non-centrosymmetric) can be ''polar'', ''chiral'', both, or neither. A ''polar'' point group is one whose symmetry operations leave more than one common point unmoved. A polar point group has no unique origin because each of those unmoved ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Friedel's Pairs
Friedel's law, named after Georges Friedel, is a property of Fourier transforms of real functions. Given a real function f(x), its Fourier transform :F(k)=\int^_f(x)e^dx has the following properties. *F(k)=F^*(-k) \, where F^* is the complex conjugate of F. Centrosymmetric points (k,-k) are called Friedel's pairs. The squared amplitude (, F, ^2) is centrosymmetric: * , F(k), ^2=, F(-k), ^2 \, The phase \phi of F is antisymmetric: * \phi(k) = -\phi(-k) \,. Friedel's law is used in X-ray diffraction, crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wor ... and scattering from real potential within the Born approximation. Note that twin operation( ''Opération de maclage'') is equivalent to an inversion centre and the intensities from the individuals are equivalent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Even And Odd Functions
In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power series and Fourier series. They are named for the parity of the powers of the power functions which satisfy each condition: the function f(x) = x^n is an even function if ''n'' is an even integer, and it is an odd function if ''n'' is an odd integer. Definition and examples Evenness and oddness are generally considered for real functions, that is real-valued functions of a real variable. However, the concepts may be more generally defined for functions whose domain and codomain both have a notion of additive inverse. This includes abelian groups, all rings, all fields, and all vector spaces. Thus, for example, a real function could be odd or even (or neither), as could a complex-valued function of a vector variable, and so on. The given e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
X-ray Diffraction
X-ray crystallography is the experimental science determining the atomic and molecular structure of a crystal, in which the crystalline structure causes a beam of incident X-rays to diffract into many specific directions. By measuring the angles and intensities of these diffracted beams, a crystallographer can produce a three-dimensional picture of the density of electrons within the crystal. From this electron density, the mean positions of the atoms in the crystal can be determined, as well as their chemical bonds, their crystallographic disorder, and various other information. Since many materials can form crystals—such as salts, metals, minerals, semiconductors, as well as various inorganic, organic, and biological molecules—X-ray crystallography has been fundamental in the development of many scientific fields. In its first decades of use, this method determined the size of atoms, the lengths and types of chemical bonds, and the atomic-scale differences among various mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The word "crystallography" is derived from the Greek word κρύσταλλος (''krystallos'') "clear ice, rock-crystal", with its meaning extending to all solids with some degree of transparency, and γράφειν (''graphein'') "to write". In July 2012, the United Nations recognised the importance of the science of crystallography by proclaiming that 2014 would be the International Year of Crystallography. denote a direction vector (in real space). * Coordinates in ''angle brackets'' or ''chevrons'' such as <100> denote a ''family'' of directions which are related by symmetry operations. In the cubic crystal system for example, would mean 00 10 01/nowiki> or the negative of any of those directions. * Miller indices in ''parentheses'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Born Approximation
Generally in scattering theory and in particular in quantum mechanics, the Born approximation consists of taking the incident field in place of the total field as the driving field at each point in the scatterer. The Born approximation is named after Max Born who proposed this approximation in early days of quantum theory development. It is the perturbation method applied to scattering by an extended body. It is accurate if the scattered field is small compared to the incident field on the scatterer. For example, the scattering of radio waves by a light styrofoam column can be approximated by assuming that each part of the plastic is polarized by the same electric field that would be present at that point without the column, and then calculating the scattering as a radiation integral over that polarization distribution. Born approximation to the Lippmann–Schwinger equation The Lippmann–Schwinger equation for the scattering state \vert\rangle with a momentum p and out-going ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Analysis
In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer. The subject of Fourier analysis encompasses a vast spectrum of mathematics. In the sciences and engineering, the process of decomposing a function into oscillatory components is often called Fourier analysis, while the operation of rebuilding the function from these pieces is known as Fourier synthesis. For example, determining what component frequencies are present in a musical note would involve computing the Fourier transform of a sampled musical note. One could then re-synthesize the same sound by including the frequency components as revealed in the Fourier analysis. In mathematics, the term ''Fourier ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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