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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 and the frequency doubling effect (second-harmonic generation). In addition, in such crystals, one-photon absorption (OPA) and two-photon absorption (TPA) processes are mutually exclusive, i.e., they do not occur simultaneously, and provide complementary information. 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 gro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Point Reflection
In geometry, a point reflection (also called a point inversion or central inversion) is a geometric transformation of affine space in which every point is reflected across a designated inversion center, which remains fixed. In Euclidean or pseudo-Euclidean spaces, a point reflection is an isometry (preserves distance). In the Euclidean plane, a point reflection is the same as a half-turn rotation (180° or radians), while in three-dimensional Euclidean space a point reflection is an improper rotation which preserves distances but reverses orientation. A point reflection is an involution: applying it twice is the identity transformation. An object that is invariant under a point reflection is said to possess point symmetry (also called inversion symmetry or central symmetry). A point group including a point reflection among its symmetries is called ''centrosymmetric''. Inversion symmetry is found in many crystal structures and molecules, and has a major effect upon their ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polar Point Group
In geometry, a polar point group is a point group in which there is more than one point that every symmetry operation leaves unmoved. The unmoved points will constitute a line, a plane, or all of space. While the simplest point group, C1, leaves all points invariant, most polar point groups will move some, but not all points. To describe the points which are unmoved by the symmetry operations of the point group, we draw a straight line joining two unmoved points. This line is called a polar direction. The electric polarization must be parallel to a polar direction. In polar point groups of high symmetry, the polar direction can be a unique axis of rotation, but if the symmetry operations do not allow any rotation at all, such as mirror symmetry, there can be an infinite number of such axes: in that case the only restriction on the polar direction is that it must be parallel to any mirror planes. A point group with more than one axis of rotation or with a mirror plane perpendicula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rule Of Mutual Exclusion
The rule of mutual exclusion in molecular spectroscopy relates the observation of molecular vibrations to molecular symmetry. It states that no normal modes can be both Infrared and Raman active in a molecule that possesses a center of symmetry. This is a powerful application of group theory to vibrational spectroscopy, and allows one to easily detect the presence of this symmetry element by comparison of the IR and Raman spectra generated by the same molecule. The rule arises because, in a centrosymmetric point group, a normal mode of vibration must have the same character (i.e. transform similarly, according to the same irreducible representation) under inversion as the property which generates it. IR active modes are generated by one of the components of the dipole moment vector. Vectors transform as spatial coordinates, and are thus of ''ungerade'' (u) symmetry, i.e. their character under inversion is -1. Thus, IR active modes must have character -1 under inversion. Rama ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centrosymmetric Matrix
In mathematics, especially in linear algebra and Matrix (mathematics), matrix theory, a centrosymmetric matrix is a matrix (mathematics), matrix which is symmetric about its center. Formal definition An matrix is centrosymmetric when its entries satisfy A_ = A_ \quad \texti,j \in \. Alternatively, if denotes the exchange matrix with 1 on the antidiagonal and 0 elsewhere: J_ = \begin 1, & i + j = n + 1 \\ 0, & i + j \ne n + 1\\ \end then a matrix is centrosymmetric if and only if . Examples * All 2 × 2 centrosymmetric matrices have the form \begin a & b \\ b & a \end. * All 3 × 3 centrosymmetric matrices have the form \begin a & b & c \\ d & e & d \\ c & b & a \end. * Symmetric matrix, Symmetric Toeplitz matrix, Toeplitz matrices are centrosymmetric. Algebraic structure and properties *If and are centrosymmetric matrices over a field (mathematics), field , then so are and for any in . Moreover, the matrix product is centrosymmetr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proteins
Proteins are large biomolecules and macromolecules that comprise one or more long chains of amino acid residues. Proteins perform a vast array of functions within organisms, including catalysing metabolic reactions, DNA replication, responding to stimuli, providing structure to cells and organisms, and transporting molecules from one location to another. Proteins differ from one another primarily in their sequence of amino acids, which is dictated by the nucleotide sequence of their genes, and which usually results in protein folding into a specific 3D structure that determines its activity. A linear chain of amino acid residues is called a polypeptide. A protein contains at least one long polypeptide. Short polypeptides, containing less than 20–30 residues, are rarely considered to be proteins and are commonly called peptides. The individual amino acid residues are bonded together by peptide bonds and adjacent amino acid residues. The sequence of amino acid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chirality (chemistry)
In chemistry, a molecule or ion is called chiral () if it cannot be superposed on its mirror image by any combination of rotation (geometry), rotations, translation (geometry), translations, and some Conformational isomerism, conformational changes. This geometric property is called chirality (). The terms are derived from Ancient Greek (''cheir'') 'hand'; which is the canonical example of an object with this property. A chiral molecule or ion exists in two stereoisomers that are mirror images of each other, called enantiomers; they are often distinguished as either "right-handed" or "left-handed" by their absolute configuration or some other criterion. The two enantiomers have the same chemical properties, except when reacting with other chiral compounds. They also have the same physics, physical properties, except that they often have opposite optical activity, optical activities. A homogeneous mixture of the two enantiomers in equal parts is said to be racemic mixture, racem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crystallographic Point Group
In crystallography, a crystallographic point group is a three-dimensional point group whose symmetry operations are compatible with a three-dimensional crystallographic lattice. According to the crystallographic restriction it may only contain one-, two-, three-, four- and sixfold rotations or rotoinversions. This reduces the number of crystallographic point groups to 32 (from an infinity of general point groups). These 32 groups are the same as the 32 types of morphological (external) crystalline symmetries derived in 1830 by Johann Friedrich Christian Hessel from a consideration of observed crystal forms. In 1867 Axel Gadolin, who was unaware of the previous work of Hessel, found the crystallographic point groups independently using stereographic projection to represent the symmetry elements of the 32 groups. In the classification of crystals, to each space group is associated a crystallographic point group by "forgetting" the translational components of the symmetry operatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chirality (mathematics)
In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by Rotation (mathematics), rotations and Translation (geometry), translations alone. An object that is not chiral is said to be ''achiral''. A chiral object and its mirror image are said to be enantiomorphs. The word ''chirality'' is derived from the Greek (cheir), the hand, the most familiar chiral object; the word ''enantiomorph'' stems from the Greek (enantios) 'opposite' + (morphe) 'form'. Examples Some chiral three-dimensional objects, such as the helix, can be assigned a right or left handedness, according to the right-hand rule. Many other familiar objects exhibit the same chiral symmetry of the human body, such as gloves and shoes. Right shoes differ from left shoes only by being mirror images of each other. In contrast thin gloves may not be considered chiral if you can wear them wiktionary:inside ou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Space Group
In mathematics, physics and chemistry, a space group is the symmetry group of a repeating pattern in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of the pattern that leave it unchanged. In three dimensions, space groups are classified into 219 distinct types, or 230 types if chiral copies are considered distinct. Space groups are discrete cocompact groups of isometries of an oriented Euclidean space in any number of dimensions. In dimensions other than 3, they are sometimes called Bieberbach groups. In crystallography, space groups are also called the crystallographic or Fedorov groups, and represent a description of the symmetry of the crystal. A definitive source regarding 3-dimensional space groups is the ''International Tables for Crystallography'' . History Space groups in 2 dimensions are the 17 wallpaper groups which have been known for several centuries, though the proof that the list ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crystallography
Crystallography is the branch of science devoted to the study of molecular and crystalline structure and properties. The word ''crystallography'' is derived from the Ancient Greek word (; "clear ice, rock-crystal"), and (; "to write"). In July 2012, the United Nations recognised the importance of the science of crystallography by proclaiming 2014 the International Year of Crystallography.UN announcement "International Year of Crystallography" iycr2014.org. 12 July 2012 Crystallography is a broad topic, and many of its subareas, such as X-ray crystallography, are themselves important scientific topics. Crystallography ranges from the fundamentals of crystal structure to the mathematics of Crystal system, crystal geometry, including those that are Aperiodic crystal, not periodic or quasi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Two-photon Absorption
In atomic physics, two-photon absorption (TPA or 2PA), also called two-photon excitation or non-linear absorption, is the (almost) simultaneous Absorption (electromagnetic radiation), absorption of two photons of identical or different frequencies in order to Electron excitation, excite an atom or a molecule from one state (usually the ground state), via a virtual energy level, to a higher energy, most commonly an excited electronic state. Absorption of two photons with the same frequency is called degenerate two-photon absorption, while absorption of two photons with different frequencies is called non-degenerate two-photon absorption. The energy difference between the involved lower and upper states is equal or smaller than the sum of the photon energy, photon energies of the two photons absorbed. Since TPA depends on the simultaneous absorption of two photons, the probability of two-photon absorption is proportional to the photon dose (), which is proportional to the square of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
