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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 perpendicular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Point Group
In geometry, a point group is a mathematical group of symmetry operations ( isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every point group in dimension ''d'' is then a subgroup of the orthogonal group O(''d''). Point groups are used to describe the symmetries of geometric figures and physical objects such as molecules. Each point group can be represented as sets of orthogonal matrices ''M'' that transform point ''x'' into point ''y'' according to Each element of a point group is either a rotation (determinant of ''M'' = 1), or it is a reflection or improper rotation (determinant of ''M'' = −1). The geometric symmetries of crystals are described by space groups, which allow translations and contain point groups as subgroups. Discrete point groups in more than one dimension come in infinite families, but from the crystallographic restriction theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonal
In crystallography, the hexagonal crystal family is one of the six crystal families, which includes two crystal systems (hexagonal and trigonal) and two lattice systems (hexagonal and rhombohedral). While commonly confused, the trigonal crystal system and the rhombohedral lattice system are not equivalent (see section crystal systems below). In particular, there are crystals that have trigonal symmetry but belong to the hexagonal lattice (such as α-quartz). The hexagonal crystal family consists of the 12 point groups such that at least one of their space groups has the hexagonal lattice as underlying lattice, and is the union of the hexagonal crystal system and the trigonal crystal system. There are 52 space groups associated with it, which are exactly those whose Bravais lattice is either hexagonal or rhombohedral. __TOC__ Lattice systems The hexagonal crystal family consists of two lattice systems: hexagonal and rhombohedral. Each lattice system consists of one Bravais ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definition, and is usually used to refer to an object that is invariant under some transformations; including translation, reflection, rotation or scaling. Although these two meanings of "symmetry" can sometimes be told apart, they are intricately related, and hence are discussed together in this article. Mathematical symmetry may be observed with respect to the passage of time; as a spatial relationship; through geometric transformations; through other kinds of functional transformations; and as an aspect of abstract objects, including theoretic models, language, and music. This article describes symmetry from three perspectives: in mathematics, including geometry, the most familiar type of symmetry for many people; in science and natu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triboluminescence
Triboluminescence is a phenomenon in which light is generated when a material is mechanically pulled apart, ripped, scratched, crushed, or rubbed (see tribology). The phenomenon is not fully understood, but appears to be caused by the separation and reunification of static electrical charges. The term comes from the Greek τρίβειν ("to rub"; see tribology) and the Latin ''lumen'' (light). Triboluminescence can be observed when breaking sugar crystals and peeling adhesive tapes. ''Triboluminescence'' is often used as a synonym for ''fractoluminescence'' (a term sometimes used when referring only to light emitted from fractured crystals). Triboluminescence differs from piezoluminescence in that a piezoluminescent material emits light when it is deformed, as opposed to broken. These are examples of mechanoluminescence, which is luminescence resulting from any mechanical action on a solid. History Uncompahgre Ute indigenous people The Uncompahgre Ute indigenous people ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sucrose
Sucrose, a disaccharide, is a sugar composed of glucose and fructose subunits. It is produced naturally in plants and is the main constituent of white sugar. It has the molecular formula . For human consumption, sucrose is extracted and refined from either sugarcane or sugar beet. Sugar mills – typically located in tropical regions near where sugarcane is grown – crush the cane and produce raw sugar which is shipped to other factories for refining into pure sucrose. Sugar beet factories are located in temperate climates where the beet is grown, and process the beets directly into refined sugar. The sugar-refining process involves washing the raw sugar crystals before dissolving them into a sugar syrup which is filtered and then passed over carbon to remove any residual colour. The sugar syrup is then concentrated by boiling under a vacuum and crystallized as the final purification process to produce crystals of pure sucrose that are clear, odorless, and sweet. Su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pyroelectricity
Pyroelectricity (from the two Greek words ''pyr'' meaning fire, and electricity) is a property of certain crystals which are naturally electrically polarized and as a result contain large electric fields. Pyroelectricity can be described as the ability of certain materials to generate a temporary voltage when they are heated or cooled. The change in temperature modifies the positions of the atoms slightly within the crystal structure, such that the polarization of the material changes. This polarization change gives rise to a voltage across the crystal. If the temperature stays constant at its new value, the pyroelectric voltage gradually disappears due to leakage current. The leakage can be due to electrons moving through the crystal, ions moving through the air, or current leaking through a voltmeter attached across the crystal. Explanation Pyroelectric charge in minerals develops on the opposite faces of asymmetric crystals. The direction in which the propagation of the cha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Space Group
In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of an object 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 was complete was ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic Crystal System
In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals. There are three main varieties of these crystals: *Primitive cubic (abbreviated ''cP'' and alternatively called simple cubic) *Body-centered cubic (abbreviated ''cI'' or bcc) *Face-centered cubic (abbreviated ''cF'' or fcc, and alternatively called ''cubic close-packed'' or ccp) Each is subdivided into other variants listed below. Although the ''unit cells'' in these crystals are conventionally taken to be cubes, the primitive unit cells often are not. Bravais lattices The three Bravais lattices in the cubic crystal system are: The primitive cubic lattice (cP) consists of one lattice point on each corner of the cube; this means each simple cubic unit cell has in total one lattice point. Each atom at a lattice point is then shared equally between eight adjacent cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hexagonal
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has Schläfli symbol and can also be constructed as a truncated equilateral triangle, t, which alternates two types of edges. A regular hexagon is defined as a hexagon that is both equilateral and equiangular. It is bicentric, meaning that it is both cyclic (has a circumscribed circle) and tangential (has an inscribed circle). The common length of the sides equals the radius of the circumscribed circle or circumcircle, which equals \tfrac times the apothem (radius of the inscribed circle). All internal angles are 120 degrees. A regular hexagon has six rotational symmetries (''rotational symmetry of order six'') and six reflection symmetries (''six lines of symmetry''), making up the dihedral group D6. The longest diagonals of a r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetragonal
In crystallography, the tetragonal crystal system is one of the 7 crystal systems. Tetragonal crystal lattices result from stretching a cubic lattice along one of its lattice vectors, so that the cube becomes a rectangular prism with a square base (''a'' by ''a'') and height (''c'', which is different from ''a''). Bravais lattices There are two tetragonal Bravais lattices: the primitive tetragonal and the body-centered tetragonal. The base-centered tetragonal lattice is equivalent to the primitive tetragonal lattice with a smaller unit cell, while the face-centered tetragonal lattice is equivalent to the body-centered tetragonal lattice with a smaller unit cell. Crystal classes The point groups that fall under this crystal system are listed below, followed by their representations in international notation, Schoenflies notation, orbifold notation, Coxeter notation and mineral examples.Hurlbut, Cornelius S.; Klein, Cornelis, 1985, ''Manual of Mineralogy'', 20th ed., ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthorhombic
In crystallography, the orthorhombic crystal system is one of the 7 crystal systems. Orthorhombic lattices result from stretching a cubic lattice along two of its orthogonal pairs by two different factors, resulting in a rectangular prism with a rectangular base (''a'' by ''b'') and height (''c''), such that ''a'', ''b'', and ''c'' are distinct. All three bases intersect at 90° angles, so the three lattice vectors remain mutually orthogonal. Bravais lattices There are four orthorhombic Bravais lattices: primitive orthorhombic, base-centered orthorhombic, body-centered orthorhombic, and face-centered orthorhombic. For the base-centered orthorhombic lattice, the primitive cell has the shape of a right rhombic prism;See , row oC, column Primitive, where the cell parameters are given as a1 = a2, α = β = 90° it can be constructed because the two-dimensional centered rectangular base layer can also be described with primitive rhombic axes. Note that the length a of the prim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetry Operation
In group theory, geometry, representation theory and molecular symmetry, a symmetry operation is a transformation of an object that leaves an object looking the same after it has been carried out. For example, as transformations of an object in space, rotations, reflections and inversions are all symmetry operations. Such symmetry operations are performed with respect to symmetry elements (for example, a point, line or plane). In the context of molecular symmetry, a symmetry operation is a permutation of atoms such that the molecule or crystal is transformed into a state indistinguishable from the starting state. Two basic facts follow from this definition, which emphasizes its usefulness. # Physical properties must be invariant with respect to symmetry operations. # Symmetry operations can be collected together in groups which are isomorphic to permutation groups. In the context of molecular symmetry, quantum wavefunctions need not be invariant, because the operation can multiply t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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