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Patterson Function
The Patterson function is used to solve the phase problem in X-ray crystallography. It was introduced in 1935 by Arthur Lindo Patterson while he was a visiting researcher in the laboratory of Bertram Eugene Warren at MIT. The Patterson function is defined as :P(u,v,w) = \sum\limits_ \left, F_\^2 \;e^. It is essentially the Fourier transform of the intensities rather than the structure factors. The Patterson function is also equivalent to the electron density convolved with its inverse: :P\left(\vec\right) = \rho\left(\vec\right) * \rho\left(-\vec\right). Furthermore, a Patterson map of ''N'' points will have peaks, excluding the central (origin) peak and any overlap. The peaks' positions in the Patterson function are the interatomic distance vectors and the peak heights are proportional to the product of the number of electrons in the atoms concerned. Because for each vector between atoms ''i'' and ''j'' there is an oppositely oriented vector of the same length (between atoms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phase Problem
In physics, the phase problem is the problem of loss of information concerning the phase that can occur when making a physical measurement. The name comes from the field of X-ray crystallography, where the phase problem has to be solved for the determination of a structure from diffraction data. The phase problem is also met in the fields of imaging and signal processing. Various approaches of phase retrieval have been developed over the years. Overview Light detectors, such as photographic plates or CCDs, measure only the intensity of the light that hits them. This measurement is incomplete (even when neglecting other degrees of freedom such as polarization and angle of incidence) because a light wave has not only an amplitude (related to the intensity), but also a phase (related to the direction), and polarization which are systematically lost in a measurement. In diffraction or microscopy experiments, the phase part of the wave often contains valuable information on the stud ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
X-ray Crystallography
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] |
Arthur Lindo Patterson
Arthur Lindo Patterson (23 July 1902, Nelson, New Zealand - 6 November 1966, Philadelphia, Pennsylvania) was a pioneering British X-ray crystallographer. Patterson was born to British parents in New Zealand in 1902. Shortly afterwards the family moved to Montreal, Canada and later to London, England. In 1920 Patterson moved to Canada for college at McGill University, Montreal. Firstly he concentrated on Mathematics and but then changed his major to Physics. He received his bachelor's degree in 1923 and a master's in 1924. His master's thesis was on the production of hard X-rays by interaction of radium β rays with solids. From 1924 to 1926 he worked in London in the laboratory of W. H. Bragg, where he learnt the art of crystal structure analysis. In 1926 Patterson moved to the Kaiser Wilhelm Institute for Fibrous Materials Chemistry (later the Fritz Haber Institute) in the Dahlem neighbourhood of Berlin, where he worked on the X-ray crystallography of cellulose fibres. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bertram Eugene Warren
Bertram Eugene Warren (June 28, 1902 – June 27, 1991) was an American crystallographer. His studies of X-rays provided much knowledge and understanding of both crystalline and non-crystalline materials. He also worked on changing amorphous solid In condensed matter physics and materials science, an amorphous solid (or non-crystalline solid, glassy solid) is a solid that lacks the long-range order that is characteristic of a crystal. Etymology The term comes from the Greek ''a'' ("wit ...s to a crystalline state. References American mineralogists American crystallographers 1902 births 1991 deaths {{US-physicist-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Massachusetts Institute Of Technology
The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the most prestigious and highly ranked academic institutions in the world. Founded in response to the increasing industrialization of the United States, MIT adopted a European polytechnic university model and stressed laboratory instruction in applied science and engineering. MIT is one of three private land grant universities in the United States, the others being Cornell University and Tuskegee University. The institute has an urban campus that extends more than a mile (1.6 km) alongside the Charles River, and encompasses a number of major off-campus facilities such as the MIT Lincoln Laboratory, the Bates Center, and the Haystack Observatory, as well as affiliated laboratories such as the Broad and Whitehead Institutes. , 98 ... [...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] |
Structure Factor
In condensed matter physics and crystallography, the static structure factor (or structure factor for short) is a mathematical description of how a material scatters incident radiation. The structure factor is a critical tool in the interpretation of scattering patterns (interference patterns) obtained in X-ray, electron and neutron diffraction experiments. Confusingly, there are two different mathematical expressions in use, both called 'structure factor'. One is usually written S(\mathbf); it is more generally valid, and relates the observed diffracted intensity per atom to that produced by a single scattering unit. The other is usually written F or F_ and is only valid for systems with long-range positional order — crystals. This expression relates the amplitude and phase of the beam diffracted by the (hk\ell) planes of the crystal ((hk\ell) are the Miller indices of the planes) to that produced by a single scattering unit at the vertices of the primitive unit cell. F_ is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electron Density
In quantum chemistry, electron density or electronic density is the measure of the probability of an electron being present at an infinitesimal element of space surrounding any given point. It is a scalar quantity depending upon three spatial variables and is typically denoted as either \rho(\textbf r) or n(\textbf r). The density is determined, through definition, by the normalised N-electron wavefunction which itself depends upon 4N variables (3N spatial and N spin coordinates). Conversely, the density determines the wave function modulo up to a phase factor, providing the formal foundation of density functional theory. According to quantum mechanics, due to the uncertainty principle on an atomic scale the exact location of an electron cannot be predicted, only the probability of its being at a given position; therefore electrons in atoms and molecules act as if they are "smeared out" in space. For one-electron systems, the electron density at any point is proportional to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' refers to both the result function and to the process of computing it. It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The choice of which function is reflected and shifted before the integral does not change the integral result (see #Properties, commutativity). The integral is evaluated for all values of shift, producing the convolution function. Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, convolution (f*g) differs from cross-correlation (f \star g) only in that either or is reflected about the y-axis in convolution; thus it is a cross-c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centrosymmetry
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 un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirac Delta Function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. The current understanding of the unit impulse is as a linear functional that maps every continuous function (e.g., f(x)) to its value at zero of its domain (f(0)), or as the weak limit of a sequence of bump functions (e.g., \delta(x) = \lim_ \frace^), which are zero over most of the real line, with a tall spike at the origin. Bump functions are thus sometimes called "approximate" or "nascent" delta distributions. The delta function was introduced by physicist Paul Dirac as a tool for the normalization of state vectors. It also has uses in probability theory and signal processing. Its validity was disputed until Laurent Schwartz developed the theory of distributions where it is defined as a linear form acting on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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