1.
Crystallography
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Crystallography is the experimental science of determining the arrangement of atoms in the crystalline solids. The word crystallography derives from the Greek words crystallon cold drop, frozen drop, with its meaning extending to all solids with some degree of transparency, and 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, X-ray crystallography is used to determine the structure of large biomolecules such as proteins. Before the development of X-ray diffraction crystallography, the study of crystals was based on measurements of their geometry. This involved measuring the angles of crystal faces relative to other and to theoretical reference axes. This physical measurement is carried out using a goniometer, the position in 3D space of each crystal face is plotted on a stereographic net such as a Wulff net or Lambert net. The pole to face is plotted on the net. Each point is labelled with its Miller index, the final plot allows the symmetry of the crystal to be established. Crystallographic methods now depend on analysis of the patterns of a sample targeted by a beam of some type. X-rays are most commonly used, other beams used include electrons or neutrons and this is facilitated by the wave properties of the particles. Crystallographers often explicitly state the type of beam used, as in the terms X-ray crystallography and these three types of radiation interact with the specimen in different ways. X-rays interact with the distribution of electrons in the sample. Electrons are charged particles and therefore interact with the charge distribution of both the atomic nuclei and the electrons of the sample. Neutrons are scattered by the atomic nuclei through the nuclear forces, but in addition. They are therefore also scattered by magnetic fields, when neutrons are scattered from hydrogen-containing materials, they produce diffraction patterns with high noise levels. However, the material can sometimes be treated to substitute deuterium for hydrogen, because of these different forms of interaction, the three types of radiation are suitable for different crystallographic studies. An image of an object is made using a lens to focus the beam. However, the wavelength of light is three orders of magnitude longer than the length of typical atomic bonds and atoms themselves
2.
Crystal system
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In crystallography, the terms crystal system, crystal family and lattice system each refer to one of several classes of space groups, lattices, point groups or crystals. Informally, two crystals are in the crystal system if they have similar symmetries, though there are many exceptions to this. Space groups and crystals are divided into seven crystal systems according to their point groups, five of the crystal systems are essentially the same as five of the lattice systems, but the hexagonal and trigonal crystal systems differ from the hexagonal and rhombohedral lattice systems. The six crystal families are formed by combining the hexagonal and trigonal crystal systems into one hexagonal family, a lattice system is a class of lattices with the same set of lattice point groups, which are subgroups of the arithmetic crystal classes. The 14 Bravais lattices are grouped into seven lattice systems, triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, in a crystal system, a set of point groups and their corresponding space groups are assigned to a lattice system. Of the 32 point groups that exist in three dimensions, most are assigned to only one system, in which case both the crystal and lattice systems have the same name. However, five point groups are assigned to two systems, rhombohedral and hexagonal, because both exhibit threefold rotational symmetry. These point groups are assigned to the crystal system. In total there are seven crystal systems, triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, a crystal family is determined by lattices and point groups. It is formed by combining crystal systems which have space groups assigned to a lattice system. In three dimensions, the families and systems are identical, except the hexagonal and trigonal crystal systems. In total there are six families, triclinic, monoclinic, orthorhombic, tetragonal, hexagonal. Spaces with less than three dimensions have the number of crystal systems, crystal families and lattice systems. In one-dimensional space, there is one crystal system, in 2D space, there are four crystal systems, oblique, rectangular, square and hexagonal. The relation between three-dimensional crystal families, crystal systems and lattice systems is shown in the table, Note. To avoid confusion of terminology, the term trigonal lattice is not used, if the original structure and inverted structure are identical, then the structure is centrosymmetric. Still, even for non-centrosymmetric case, inverted structure in some cases can be rotated to align with the original structure and this is the case of non-centrosymmetric achiral structure. If the inverted structure cannot be rotated to align with the structure, then the structure is chiral
3.
Lattice (group)
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In geometry and group theory, a lattice in R n is a subgroup of R n which is isomorphic to Z n, and which spans the real vector space R n. In other words, for any basis of R n, the subgroup of all linear combinations with integer coefficients of the basis vectors forms a lattice, a lattice may be viewed as a regular tiling of a space by a primitive cell. Lattices have many significant applications in mathematics, particularly in connection to Lie algebras, number theory. More generally, lattice models are studied in physics, often by the techniques of computational physics, a lattice is the symmetry group of discrete translational symmetry in n directions. A pattern with this lattice of translational symmetry cannot have more, as a group a lattice is a finitely-generated free abelian group, and thus isomorphic to Z n. A lattice in the sense of a 3-dimensional array of regularly spaced points coinciding with e. g, a simple example of a lattice in R n is the subgroup Z n. More complicated examples include the E8 lattice, which is a lattice in R8, the period lattice in R2 is central to the study of elliptic functions, developed in nineteenth century mathematics, it generalises to higher dimensions in the theory of abelian functions. Lattices called root lattices are important in the theory of simple Lie algebras, for example, a typical lattice Λ in R n thus has the form Λ = where is a basis for R n. Different bases can generate the lattice, but the absolute value of the determinant of the vectors vi is uniquely determined by Λ. If one thinks of a lattice as dividing the whole of R n into equal polyhedra and this is why d is sometimes called the covolume of the lattice. If this equals 1, the lattice is called unimodular, minkowskis theorem relates the number d and the volume of a symmetric convex set S to the number of lattice points contained in S. The number of lattice points contained in an all of whose vertices are elements of the lattice is described by the polytopes Ehrhart polynomial. Formulas for some of the coefficients of this polynomial involve d as well, Lattice basis reduction is the problem of finding a short and nearly orthogonal lattice basis. The Lenstra-Lenstra-Lovász lattice basis reduction algorithm approximates such a basis in polynomial time, it has found numerous applications. There are five 2D lattice types as given by the crystallographic restriction theorem, below, the wallpaper group of the lattice is given in IUC notation, Orbifold notation, and Coxeter notation, along with a wallpaper diagram showing the symmetry domains. Note that a pattern with this lattice of translational symmetry cannot have more, a full list of subgroups is available. For example below the hexagonal/triangular lattice is given twice, with full 6-fold, if the symmetry group of a pattern contains an n-fold rotation then the lattice has n-fold symmetry for even n and 2n-fold for odd n. For the classification of a lattice, start with one point
4.
Cubic crystal system
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In crystallography, the cubic 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 Body-centered cubic, Face-centered cubic Each is subdivided into other variants listed below. Note that although the cell in these crystals is conventionally taken to be a cube. This is related to the fact that in most cubic crystal systems, a classic isometric crystal has square or pentagonal faces. The three Bravais lattices in the crystal system are, The primitive cubic system consists of one lattice point on each corner of the cube. Each atom at a point is then shared equally between eight adjacent cubes, and the unit cell therefore contains in total one atom. The body-centered cubic system has one point in the center of the unit cell in addition to the eight corner points. It has a net total of 2 lattice points per unit cell, Each sphere in a cF lattice has coordination number 12. The face-centered cubic system is related to the hexagonal close packed system. The plane of a cubic system is a hexagonal grid. Attempting to create a C-centered cubic crystal system would result in a simple tetragonal Bravais lattice, there are a total 36 cubic space groups. Other terms for hexoctahedral are, normal class, holohedral, ditesseral central class, a simple cubic unit cell has a single cubic void in the center. Additionally, there are 24 tetrahedral voids located in a square spacing around each octahedral void and these tetrahedral voids are not local maxima and are not technically voids, but they do occasionally appear in multi-atom unit cells. A face-centered cubic unit cell has eight tetrahedral voids located midway between each corner and the center of the cell, for a total of eight net tetrahedral voids. One important characteristic of a structure is its atomic packing factor. This is calculated by assuming all the atoms are identical spheres. The atomic packing factor is the proportion of space filled by these spheres, assuming one atom per lattice point, in a primitive cubic lattice with cube side length a, the sphere radius would be a⁄2 and the atomic packing factor turns out to be about 0.524. Similarly, in a bcc lattice, the atomic packing factor is 0.680, as a rule, since atoms in a solid attract each other, the more tightly packed arrangements of atoms tend to be more common
5.
Prism (geometry)
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In geometry, a prism is a polyhedron comprising an n-sided polygonal base, a second base which is a translated copy of the first, and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the bases are translations of the bases, prisms are named for their bases, so a prism with a pentagonal base is called a pentagonal prism. The prisms are a subclass of the prismatoids, a right prism is a prism in which the joining edges and faces are perpendicular to the base faces. This applies if the faces are rectangular. If the joining edges and faces are not perpendicular to the base faces, for example a parallelepiped is an oblique prism of which the base is a parallelogram, or equivalently a polyhedron with six faces which are all parallelograms. A truncated prism is a prism with nonparallel top and bottom faces, some texts may apply the term rectangular prism or square prism to both a right rectangular-sided prism and a right square-sided prism. A right p-gonal prism with rectangular sides has a Schläfli symbol ×, a right rectangular prism is also called a cuboid, or informally a rectangular box. A right square prism is simply a box, and may also be called a square cuboid. A right rectangular prism has Schläfli symbol ××, an n-prism, having regular polygon ends and rectangular sides, approaches a cylindrical solid as n approaches infinity. The term uniform prism or semiregular prism can be used for a prism with square sides. A uniform p-gonal prism has a Schläfli symbol t, right prisms with regular bases and equal edge lengths form one of the two infinite series of semiregular polyhedra, the other series being the antiprisms. The dual of a prism is a bipyramid. The volume of a prism is the product of the area of the base, the volume is therefore, V = B ⋅ h where B is the base area and h is the height. The volume of a prism whose base is a regular n-sided polygon with side s is therefore. The surface area of a prism is 2 · B + P · h, where B is the area of the base, h the height. The surface area of a prism whose base is a regular n-sided polygon with side length s and height h is therefore. The rotation group is Dn of order 2n, except in the case of a cube, which has the symmetry group O of order 24. The symmetry group Dnh contains inversion iff n is even, a prismatic polytope is a higher-dimensional generalization of a prism
6.
Radix
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In mathematical numeral systems, the radix or base is the number of unique digits, including zero, used to represent numbers in a positional numeral system. For example, for the system the radix is ten. For example,10 represents the one hundred, while 2 represents the number four. Radix is a Latin word for root, root can be considered a synonym for base in the arithmetical sense. In the system with radix 13, for example, a string of such as 398 denotes the number 3 ×132 +9 ×131 +8 ×130. More generally, in a system with radix b, a string of digits d1 … dn denotes the number d1bn−1 + d2bn−2 + … + dnb0, commonly used numeral systems include, For a larger list, see List of numeral systems. The octal and hexadecimal systems are used in computing because of their ease as shorthand for binary. Every hexadecimal digit corresponds to a sequence of four binary digits, a similar relationship holds between every octal digit and every possible sequence of three binary digits, since eight is the cube of two. However, other systems are possible, e. g. golden ratio base. Base Radix economy Non-standard positional numeral systems Base Convert, a floating-point base calculator MathWorld entry on base
7.
Orthogonality
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The concept of orthogonality has been broadly generalized in mathematics, as well as in areas such as chemistry, and engineering. The word comes from the Greek ὀρθός, meaning upright, and γωνία, the ancient Greek ὀρθογώνιον orthogōnion and classical Latin orthogonium originally denoted a rectangle. Later, they came to mean a right triangle, in the 12th century, the post-classical Latin word orthogonalis came to mean a right angle or something related to a right angle. In geometry, two Euclidean vectors are orthogonal if they are perpendicular, i. e. they form a right angle, two vectors, x and y, in an inner product space, V, are orthogonal if their inner product ⟨ x, y ⟩ is zero. This relationship is denoted x ⊥ y, two vector subspaces, A and B, of an inner product space, V, are called orthogonal subspaces if each vector in A is orthogonal to each vector in B. The largest subspace of V that is orthogonal to a subspace is its orthogonal complement. Given a module M and its dual M∗, an element m′ of M∗, two sets S′ ⊆ M∗ and S ⊆ M are orthogonal if each element of S′ is orthogonal to each element of S. A term rewriting system is said to be if it is left-linear and is non-ambiguous. Orthogonal term rewriting systems are confluent, a set of vectors in an inner product space is called pairwise orthogonal if each pairing of them is orthogonal. Such a set is called an orthogonal set, nonzero pairwise orthogonal vectors are always linearly independent. In certain cases, the normal is used to mean orthogonal. For example, the y-axis is normal to the curve y = x2 at the origin, however, normal may also refer to the magnitude of a vector. In particular, a set is called if it is an orthogonal set of unit vectors. As a result, use of the normal to mean orthogonal is often avoided. The word normal also has a different meaning in probability and statistics, a vector space with a bilinear form generalizes the case of an inner product. When the bilinear form applied to two results in zero, then they are orthogonal. The case of a pseudo-Euclidean plane uses the term hyperbolic orthogonality, in the diagram, axes x′ and t′ are hyperbolic-orthogonal for any given ϕ. In 2-D or higher-dimensional Euclidean space, two vectors are orthogonal if and only if their dot product is zero, i. e. they make an angle of 90°, hence orthogonality of vectors is an extension of the concept of perpendicular vectors into higher-dimensional spaces
8.
Bravais lattice
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This discrete set of vectors must be closed under vector addition and subtraction. For any choice of position vector R, the lattice looks exactly the same, when the discrete points are atoms, ions, or polymer strings of solid matter, the Bravais lattice concept is used to formally define a crystalline arrangement and its frontiers. A crystal is made up of an arrangement of one or more atoms repeated at each lattice point. Consequently, the crystal looks the same when viewed from any equivalent lattice point, two Bravais lattices are often considered equivalent if they have isomorphic symmetry groups. In this sense, there are 14 possible Bravais lattices in three-dimensional space, the 14 possible symmetry groups of Bravais lattices are 14 of the 230 space groups. In two-dimensional space, there are 5 Bravais lattices, grouped into four crystal families, the unit cells are specified according to the relative lengths of the cell edges and the angle between them. The area of the cell can be calculated by evaluating the norm || a × b ||. The properties of the families are given below, In three-dimensional space. These are obtained by combining one of the six families with one of the centering types. For example, the monoclinic I lattice can be described by a monoclinic C lattice by different choice of crystal axes, similarly, all A- or B-centred lattices can be described either by a C- or P-centering. This reduces the number of combinations to 14 conventional Bravais lattices, the unit cells are specified according to the relative lengths of the cell edges and the angles between them. The volume of the cell can be calculated by evaluating the triple product a ·, where a, b. The properties of the families are given below, In four dimensions. Of these,23 are primitive and 41 are centered, ten Bravais lattices split into enantiomorphic pairs. Bravais, A. Mémoire sur les systèmes formés par les points distribués régulièrement sur un plan ou dans lespace, hahn, Theo, ed. International Tables for Crystallography, Volume A, Space Group Symmetry. Catalogue of Lattices Smith, Walter Fox
9.
Crystal structure
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In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystalline material. Ordered structures occur from the nature of the constituent particles to form symmetric patterns that repeat along the principal directions of three-dimensional space in matter. The smallest group of particles in the material that constitutes the pattern is the unit cell of the structure. The unit cell completely defines the symmetry and structure of the crystal lattice. The repeating patterns are said to be located at the points of the Bravais lattice, the lengths of the principal axes, or edges, of the unit cell and the angles between them are the lattice constants, also called lattice parameters. The symmetry properties of the crystal are described by the concept of space groups, all possible symmetric arrangements of particles in three-dimensional space may be described by the 230 space groups. The crystal structure and symmetry play a role in determining many physical properties, such as cleavage, electronic band structure. The crystal structure of a material can be described in terms of its unit cell, the unit cell is a box containing one or more atoms arranged in three dimensions. The unit cells stacked in three-dimensional space describe the arrangement of atoms of the crystal. Commonly, atomic positions are represented in terms of fractional coordinates, the atom positions within the unit cell can be calculated through application of symmetry operations to the asymmetric unit. The asymmetric unit refers to the smallest possible occupation of space within the unit cell and this does not, however imply that the entirety of the asymmetric unit must lie within the boundaries of the unit cell. Symmetric transformations of atom positions are calculated from the group of the crystal structure. Vectors and planes in a lattice are described by the three-value Miller index notation. It uses the indices ℓ, m, and n as directional parameters, which are separated by 90°, by definition, the syntax denotes a plane that intercepts the three points a1/ℓ, a2/m, and a3/n, or some multiple thereof. That is, the Miller indices are proportional to the inverses of the intercepts of the plane with the unit cell, if one or more of the indices is zero, it means that the planes do not intersect that axis. A plane containing a coordinate axis is translated so that it no longer contains that axis before its Miller indices are determined, the Miller indices for a plane are integers with no common factors. Negative indices are indicated with horizontal bars, as in, in an orthogonal coordinate system for a cubic cell, the Miller indices of a plane are the Cartesian components of a vector normal to the plane. Likewise, the planes are geometric planes linking nodes
10.
Rhombus
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In Euclidean geometry, a rhombus is a simple quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length, every rhombus is a parallelogram and a kite. A rhombus with right angles is a square, the word rhombus comes from Greek ῥόμβος, meaning something that spins, which derives from the verb ῥέμβω, meaning to turn round and round. The word was used both by Euclid and Archimedes, who used the term solid rhombus for two right circular cones sharing a common base, the surface we refer to as rhombus today is a cross section of this solid rhombus through the apex of each of the two cones. This is a case of the superellipse, with exponent 1. Every rhombus has two diagonals connecting pairs of vertices, and two pairs of parallel sides. Using congruent triangles, one can prove that the rhombus is symmetric across each of these diagonals and it follows that any rhombus has the following properties, Opposite angles of a rhombus have equal measure. The two diagonals of a rhombus are perpendicular, that is, a rhombus is an orthodiagonal quadrilateral, the first property implies that every rhombus is a parallelogram. Thus denoting the common side as a and the diagonals as p and q, not every parallelogram is a rhombus, though any parallelogram with perpendicular diagonals is a rhombus. In general, any quadrilateral with perpendicular diagonals, one of which is a line of symmetry, is a kite, every rhombus is a kite, and any quadrilateral that is both a kite and parallelogram is a rhombus. A rhombus is a tangential quadrilateral and that is, it has an inscribed circle that is tangent to all four sides. As for all parallelograms, the area K of a rhombus is the product of its base, the base is simply any side length a, K = a ⋅ h. The inradius, denoted by r, can be expressed in terms of the p and q as. The dual polygon of a rhombus is a rectangle, A rhombus has all sides equal, a rhombus has opposite angles equal, while a rectangle has opposite sides equal. A rhombus has a circle, while a rectangle has a circumcircle. A rhombus has an axis of symmetry through each pair of opposite vertex angles, the diagonals of a rhombus intersect at equal angles, while the diagonals of a rectangle are equal in length. The figure formed by joining the midpoints of the sides of a rhombus is a rectangle, a rhombohedron is a three-dimensional figure like a cube, except that its six faces are rhombi instead of squares. The rhombic dodecahedron is a polyhedron with 12 congruent rhombi as its faces
11.
Crystallographic point group
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For a periodic crystal, the group must also be consistent with maintenance of the three-dimensional translational symmetry that defines crystallinity. The macroscopic properties of a crystal would look exactly the same before, in the classification of crystals, each point group is also known as a crystal class. There are infinitely many three-dimensional point groups, however, the crystallographic restriction of the infinite families of general point groups results in there being only 32 crystallographic point groups. These 32 point groups are one-and-the same as the 32 types of morphological crystalline symmetries derived in 1830 by Johann Friedrich Christian Hessel from a consideration of observed crystal forms, the point groups are denoted by their component symmetries. There are a few standard notations used by crystallographers, mineralogists, for the correspondence of the two systems below, see crystal system. In Schoenflies notation, point groups are denoted by a symbol with a subscript. The symbols used in crystallography mean the following, Cn indicates that the group has a rotation axis. Cnh is Cn with the addition of a plane perpendicular to the axis of rotation. Cnv is Cn with the addition of n mirror planes parallel to the axis of rotation, s2n denotes a group that contains only a 2n-fold rotation-reflection axis. Dn indicates that the group has a rotation axis plus n twofold axes perpendicular to that axis. Dnh has, in addition, a plane perpendicular to the n-fold axis. Dnd has, in addition to the elements of Dn, mirror planes parallel to the n-fold axis, the letter T indicates that the group has the symmetry of a tetrahedron. Td includes improper rotation operations, T excludes improper rotation operations, the letter O indicates that the group has the symmetry of an octahedron, with or without improper operations. Due to the crystallographic restriction theorem, n =1,2,3,4, d4d and D6d are actually forbidden because they contain improper rotations with n=8 and 12 respectively. The 27 point groups in the table plus T, Td, Th, O, an abbreviated form of the Hermann–Mauguin notation commonly used for space groups also serves to describe crystallographic point groups. Group names are Molecular symmetry Point group Space group Point groups in three dimensions Crystal system Point-group symbols in International Tables for Crystallography,12.1, pp. 818-820 Names and symbols of the 32 crystal classes in International Tables for Crystallography. 10.1, p.794 Pictorial overview of the 32 groups Point Groups - Flow Chart Inorganic Chemistry Group Theory Practice Problems
12.
Orbifold notation
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Groups representable in this notation include the point groups on the sphere, the frieze groups and wallpaper groups of the Euclidean plane, and their analogues on the hyperbolic plane. e. All translations which occur are assumed to form a subgroup of the group symmetries being described. The symbol ×, which is called a miracle and represents a topological crosscap where a pattern repeats as an image without crossing a mirror line. A string written in boldface represents a group of symmetries of Euclidean 3-space, a string not written in boldface represents a group of symmetries of the Euclidean plane, which is assumed to contain two independent translations. By abuse of language, we say that such a group is a subgroup of symmetries of the Euclidean plane with only one independent translation. The frieze groups occur in this way, the exceptional symbol o indicates that there are precisely two linearly independent translations. An orbifold symbol is called if it is not one of the following, p, pq, *p, *pq, for p, q>=2. An object is chiral if its symmetry group contains no reflections, the corresponding orbifold is orientable in the chiral case and non-orientable otherwise. The Euler characteristic of an orbifold can be read from its Conway symbol, as follows. Each feature has a value, n without or before an asterisk counts as n −1 n n after an asterisk counts as n −12 n asterisk, subtracting the sum of these values from 2 gives the Euler characteristic. If the sum of the values is 2, the order is infinite. Indeed, Conways Magic Theorem indicates that the 17 wallpaper groups are exactly those with the sum of the feature values equal to 2, otherwise, the order is 2 divided by the Euler characteristic. The following groups are isomorphic, 1* and *1122 and 221 *22 and *221 2* and this is because 1-fold rotation is the empty rotation. The symmetry of a 2D object without translational symmetry can be described by the 3D symmetry type by adding a dimension to the object which does not add or spoil symmetry. The bullet is added on one- and two-dimensional groups to imply the existence of a fixed point, thus the discrete symmetry groups in one dimension are *•, *1•, ∞• and *∞•. Another way of constructing a 3D object from a 1D or 2D object for describing the symmetry is taking the Cartesian product of the object, on Three-dimensional Orbifolds and Space Groups. Contributions to Algebra and Geometry,42, 475-507,2001, J. H. Conway, D. H. Huson. The Orbifold Notation for Two-Dimensional Groups, structural Chemistry,13, 247-257, August 2002
13.
Space groups
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In mathematics, physics and chemistry, a space group is the symmetry group of a configuration in space, usually in three dimensions. In three dimensions, there are 219 distinct types, or 230 if chiral copies are considered distinct, Space groups are also studied in dimensions other than 3 where they are sometimes called Bieberbach groups, and are discrete cocompact groups of isometries of an oriented Euclidean space. In crystallography, space groups are called the crystallographic or Fedorov groups. A definitive source regarding 3-dimensional space groups is the International Tables for Crystallography, in 1879 Leonhard Sohncke listed the 65 space groups whose elements preserve the orientation. More accurately, he listed 66 groups, but Fedorov and Schönflies both noticed that two of them were really the same, the space groups in 3 dimensions were first enumerated by Fedorov, and shortly afterwards were independently enumerated by Schönflies. The correct list of 230 space groups was found by 1892 during correspondence between Fedorov and Schönflies, burckhardt describes the history of the discovery of the space groups in detail. The space groups in three dimensions are made from combinations of the 32 crystallographic point groups with the 14 Bravais lattices, the combination of all these symmetry operations results in a total of 230 different space groups describing all possible crystal symmetries. The elements of the space group fixing a point of space are rotations, reflections, the identity element, the translations form a normal abelian subgroup of rank 3, called the Bravais lattice. There are 14 possible types of Bravais lattice, the quotient of the space group by the Bravais lattice is a finite group which is one of the 32 possible point groups. Translation is defined as the moves from one point to another point. A glide plane is a reflection in a plane, followed by a parallel with that plane. This is noted by a, b or c, depending on which axis the glide is along. There is also the n glide, which is a glide along the half of a diagonal of a face, and the d glide, the latter is called the diamond glide plane as it features in the diamond structure. In 17 space groups, due to the centering of the cell, the glides occur in two directions simultaneously, i. e. the same glide plane can be called b or c, a or b. For example, group Abm2 could be also called Acm2, group Ccca could be called Cccb, in 1992, it was suggested to use symbol e for such planes. The symbols for five groups have been modified, A screw axis is a rotation about an axis. These are noted by a number, n, to describe the degree of rotation, the degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. So,21 is a rotation followed by a translation of 1/2 of the lattice vector
14.
Space group
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In mathematics, physics and chemistry, a space group is the symmetry group of a configuration in space, usually in three dimensions. In three dimensions, there are 219 distinct types, or 230 if chiral copies are considered distinct, Space groups are also studied in dimensions other than 3 where they are sometimes called Bieberbach groups, and are discrete cocompact groups of isometries of an oriented Euclidean space. In crystallography, space groups are called the crystallographic or Fedorov groups. A definitive source regarding 3-dimensional space groups is the International Tables for Crystallography, in 1879 Leonhard Sohncke listed the 65 space groups whose elements preserve the orientation. More accurately, he listed 66 groups, but Fedorov and Schönflies both noticed that two of them were really the same, the space groups in 3 dimensions were first enumerated by Fedorov, and shortly afterwards were independently enumerated by Schönflies. The correct list of 230 space groups was found by 1892 during correspondence between Fedorov and Schönflies, burckhardt describes the history of the discovery of the space groups in detail. The space groups in three dimensions are made from combinations of the 32 crystallographic point groups with the 14 Bravais lattices, the combination of all these symmetry operations results in a total of 230 different space groups describing all possible crystal symmetries. The elements of the space group fixing a point of space are rotations, reflections, the identity element, the translations form a normal abelian subgroup of rank 3, called the Bravais lattice. There are 14 possible types of Bravais lattice, the quotient of the space group by the Bravais lattice is a finite group which is one of the 32 possible point groups. Translation is defined as the moves from one point to another point. A glide plane is a reflection in a plane, followed by a parallel with that plane. This is noted by a, b or c, depending on which axis the glide is along. There is also the n glide, which is a glide along the half of a diagonal of a face, and the d glide, the latter is called the diamond glide plane as it features in the diamond structure. In 17 space groups, due to the centering of the cell, the glides occur in two directions simultaneously, i. e. the same glide plane can be called b or c, a or b. For example, group Abm2 could be also called Acm2, group Ccca could be called Cccb, in 1992, it was suggested to use symbol e for such planes. The symbols for five groups have been modified, A screw axis is a rotation about an axis. These are noted by a number, n, to describe the degree of rotation, the degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. So,21 is a rotation followed by a translation of 1/2 of the lattice vector
15.
Coxeter notation
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The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson. For Coxeter groups defined by pure reflections, there is a correspondence between the bracket notation and Coxeter-Dynkin diagram. The numbers in the notation represent the mirror reflection orders in the branches of the Coxeter diagram. It uses the same simplification, suppressing 2s between orthogonal mirrors, the Coxeter notation is simplified with exponents to represent the number of branches in a row for linear diagram. So the An group is represented by, to imply n nodes connected by n-1 order-3 branches, example A2 = = or represents diagrams or. Coxeter initially represented bifurcating diagrams with vertical positioning of numbers, but later abbreviated with an exponent notation, like, Coxeter allowed for zeros as special cases to fit the An family, like A3 = = = =, like = =. Coxeter groups formed by cyclic diagrams are represented by parenthesese inside of brackets, if the branch orders are equal, they can be grouped as an exponent as the length the cycle in brackets, like =, representing Coxeter diagram or. More complicated looping diagrams can also be expressed with care, the paracompact complete graph diagram or, is represented as with the superscript as the symmetry of its regular tetrahedron coxeter diagram. The Coxeter diagram usually leaves order-2 branches undrawn, but the bracket notation includes an explicit 2 to connect the subgraphs, so the Coxeter diagram = A2×A2 = 2A2 can be represented by × =2 =. For the affine and hyperbolic groups, the subscript is one less than the number of nodes in each case, Coxeters notation represents rotational/translational symmetry by adding a + superscript operator outside the brackets which cuts the order of the group in half. This is called a direct subgroup because what remains are only direct isometries without reflective symmetry, + operators can also be applied inside of the brackets, and creates semidirect subgroups that include both reflective and nonreflective generators. Semidirect subgroups can only apply to Coxeter group subgroups that have even order branches next to it, the subgroup index is 2n for n + operators. So the snub cube, has symmetry +, and the tetrahedron, has symmetry. Johnson extends the + operator to work with a placeholder 1 nodes, in general this operation only applies to mirrors bounded by all even-order branches. The 1 represents a mirror so can be seen as, or, like diagram or, the effect of a mirror removal is to duplicate connecting nodes, which can be seen in the Coxeter diagrams, =, or in bracket notation, = =. Each of these mirrors can be removed so h = = = and this can be shown in a Coxeter diagram by adding a + symbol above the node, = =. If both mirrors are removed, a subgroup is generated, with the branch order becoming a gyration point of half the order, q = = +. For example, = = = ×, order 4. = +, the opposite to halving is doubling which adds a mirror, bisecting a fundamental domain, and doubling the group order
16.
Chirality (chemistry)
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Chirality /kaɪˈrælɪti/ is a geometric property of some molecules and ions. A chiral molecule/ion is non-superposable on its mirror image, the presence of an asymmetric carbon center is one of several structural features that induce chirality in organic and inorganic molecules. The term chirality is derived from the Greek word for hand, the mirror images of a chiral molecule/ion are called enantiomers or optical isomers. Individual enantiomers are often designated as either right- or left-handed, Chirality is an essential consideration when discussing the stereochemistry in organic and inorganic chemistry. The concept is of practical importance because most biomolecules and pharmaceuticals are chiral. Chirality is based on molecular symmetry elements, specifically, a chiral compound can contain no improper axis of rotation, which includes planes of symmetry and inversion center. Chiral molecules are always dissymmetric but not always asymmetric, in general, chiral molecules have point chirality at a single stereogenic atom, which has four different substituents. The two enantiomers of such compounds are said to have different absolute configurations at this center, the stereogenic atom is usually carbon, as in many biological molecules. However chirality can exist in any atom, including metals, phosphorus, Chiral nitrogen is equally possible, although the effects of nitrogen inversion can make many of these compounds impossible to isolate. While the presence of a stereogenic atom describes the great majority of cases, for instance it is not necessary for the chiral substance to have a stereogenic atom. Examples include 1-bromo-1-chloro-1-fluoroadamantane, methylethylphenyltetrahedrane, certain calixarenes and fullerenes, which have inherent chirality, the C2-symmetric species 1, 1-bi-2-naphthol,1, 3-dichloro-allene have axial chirality. -cyclooctene and many ferrocenes have planar chirality, when the optical rotation for an enantiomer is too low for practical measurement, the species is said to exhibit cryptochirality. Even isotopic differences must be considered when examining chirality, illustrative is the derivative of benzyl alcohol PhCHDOH is chiral. The S enantiomer has D = +0. 715°, many biologically active molecules are chiral, including the naturally occurring amino acids and sugars. In biological systems, most of these compounds are of the chirality, most amino acids are levorotatory. Typical naturally occurring proteins, made of L amino acids, are known as left-handed proteins, d-amino acids are very rare in nature and have only been found in small peptides attached to bacteria cell walls. The origin of this homochirality in biology is the subject of much debate, however, there is some suggestion that early amino acids could have formed in comet dust. Enzymes, which are chiral, often distinguish between the two enantiomers of a chiral substrate, one could imagine an enzyme as having a glove-like cavity that binds a substrate
17.
Epsomite
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Epsomite is a hydrous magnesium sulfate mineral with formula MgSO4·7H2O. Epsomite crystallizes in the system as rarely found acicular or fibrous crystals. It is colorless to white with tints of yellow, green, the Mohs hardness is 2 to 2.5 and it has a low specific gravity of 1.67. Epsomite is the same as the chemical, Epsom salt. It absorbs water from the air and converts to hexahydrate with the loss of one water molecule, epsomite forms as encrustations or efflorescences on limestone cavern walls and mine timbers and walls, rarely as volcanic fumarole deposits, and as rare beds in evaporite layers. It was first systematically described in 1806 for an occurrence near Epsom, Surrey, England and it occurs in association with melanterite, gypsum, halotrichite, pickeringite, alunogen, rozenite and mirabilite. The epsomite group includes solid solution series with morenosite and goslarite Kieserite is a hydrated magnesium sulfate
18.
Hemimorphite
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Hemimorphite, is Zn42·H2O, a component of calamine. It is a mineral which has been historically mined from the upper parts of zinc and lead ores, chiefly associated with smithsonite. They were assumed to be the mineral and both were classed under the same name of calamine. In the second half of the 18th century it was discovered that two different minerals were both present in calamine. The silicate was the rarer of the two, and was named hemimorphite, because of the development of its crystals. This unusual form, which is typical of only a few minerals, some specimens show strong green fluorescence in shortwave ultraviolet light and weak light pink fluorescence in longwave UV. Hemimorphite is an important ore of zinc and contains up to 54. 2% of the metal, together with silicon, the crystals are blunt at one end and sharp at the other. The regions on the Belgian-German border are known for their deposits of hemimorphite of metasomatic origin, especially Vieille Montagne in Belgium. Hurlbut, Cornelius S. Klein, Cornelis,1985, Manual of Mineralogy, 20th ed. ISBN 0-471-80580-7 Boni, M. Gilg, H. A. Aversa, G. and Balassone, G.2003, The Calamine of southwest Sardinia, Geology, mineralogy, and stable isotope geochemistry of supergene Zn mineralization, Economic Geology, v.98, p. 731-748. Reynolds, N. A. Chisnall, T. W. Kaewsang, K. Keesaneyabutr, C. and Taksavasu, T.2003, The Padaeng supergene nonsulfide zinc deposit, Mae Sod, Thailand, Economic Geology, v.98, p. 773-785
19.
Bertrandite
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Bertrandite is a beryllium sorosilicate hydroxide mineral with composition, Be4Si2O72. Bertrandite is a colorless to pale yellow orthorhombic mineral with a hardness of 6-7 and it is commonly found in beryllium rich pegmatites and is in part an alteration of beryl. Bertrandite often occurs as a replacement of beryl. Associated minerals include beryl, phenakite, herderite, tourmaline, muscovite, fluorite and it, with beryl, are ores of beryllium. It was discovered near Nantes, France in 1883 and named after French mineralogist, List of minerals List of minerals named after people
20.
Centrosymmetry
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In crystallography, a point group which contains an inversion center as one of its symmetry elements is centrosymmetric. In such a point group, for point in the unit cell there is an indistinguishable point. Such point groups are said to have inversion symmetry. Point reflection is a term used in geometry. Crystals with an inversion center cannot display certain properties, such as the piezoelectric effect, point groups lacking an inversion center are further divided into polar and chiral types. A chiral point group is one without any rotoinversion symmetry elements, rotoinversion is rotation followed by inversion, for example, a mirror reflection corresponds to a twofold rotoinversion. Chiral point groups must therefore only contain rotational symmetry and these arise from the crystal point groups 1,2,3,4,6,222,422,622,32,23, and 432. Chiral molecules such as proteins crystallize in chiral point groups, the term polar is often used for those point groups which are neither centrosymmetric nor chiral. However, the term is more used for any point group containing a unique anisotropic axis. These occur in crystal point groups 1,2,3,4,6, m, mm2, 3m, 4mm, thus some chiral space groups are also polar
21.
Olivine
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The mineral olivine is a magnesium iron silicate with the formula 2SiO4. Thus it is a type of nesosilicate or orthosilicate and it is a common mineral in the Earths subsurface but weathers quickly on the surface. The ratio of magnesium and iron varies between the two endmembers of the solid solution series, forsterite and fayalite, compositions of olivine are commonly expressed as molar percentages of forsterite and fayalite. Forsterite has a high melting temperature at atmospheric pressure, almost 1,900 °C. The melting temperature varies smoothly between the two endmembers, as do other properties, olivine incorporates only minor amounts of elements other than oxygen, silicon, magnesium and iron. Manganese and nickel commonly are the elements present in highest concentrations. Olivine gives its name to the group of minerals with a structure which includes tephroite, monticellite and kirschsteinite. It has a structure similar to magnetite but uses one quadravalent. Olivine gemstones are called peridot and chrysolite, olivine is named for its typically olive-green color, though it may alter to a reddish color from the oxidation of iron. Translucent olivine is sometimes used as a gemstone called peridot, some of the finest gem-quality olivine has been obtained from a body of mantle rocks on Zabargad island in the Red Sea. Olivine occurs in mafic and ultramafic igneous rocks and as a primary mineral in certain metamorphic rocks. Mg-rich olivine crystallizes from magma that is rich in magnesium and low in silica and that magma crystallizes to mafic rocks such as gabbro and basalt. Ultramafic rocks such as peridotite and dunite can be left after extraction of magmas. Olivine and high pressure structural variants constitute over 50% of the Earths upper mantle, the metamorphism of impure dolomite or other sedimentary rocks with high magnesium and low silica content also produces Mg-rich olivine, or forsterite. In contrast, Mg-rich olivine does not occur stably with silica minerals, Mg-rich olivine is stable to pressures equivalent to a depth of about 410 km within Earth. Mg-rich olivine has also discovered in meteorites, on the Moon and Mars, falling into infant stars. Such meteorites include chondrites, collections of debris from the early Solar System, the spectral signature of olivine has been seen in the dust disks around young stars. The tails of comets often have the signature of olivine
22.
Aragonite
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Aragonite is a carbonate mineral, one of the two most common, naturally occurring, crystal forms of calcium carbonate, CaCO3. It is formed by biological and physical processes, including precipitation from marine, aragonites crystal lattice differs from that of calcite, resulting in a different crystal shape, an orthorhombic crystal system with acicular crystal. Repeated twinning results in pseudo-hexagonal forms, Aragonite may be columnar or fibrous, occasionally in branching stalactitic forms called flos-ferri from their association with the ores at the Carinthian iron mines. The type location for aragonite is Molina de Aragón,25 km from Aragon for which it was named in 1797, an aragonite cave, the Ochtinská Aragonite Cave, is situated in Slovakia. In the US, aragonite in the form of stalactites and cave flowers is known from Carlsbad Caverns, massive deposits of oolitic aragonite sand are found on the seabed in the Bahamas. Aragonite is the high pressure polymorph of calcium carbonate, as such, it occurs in high pressure metamorphic rocks such as those formed at subduction zones. Aragonite forms naturally in almost all mollusk shells, and as the calcareous endoskeleton of warm-, because the mineral deposition in mollusk shells is strongly biologically controlled, some crystal forms are distinctively different from those of inorganic aragonite. In some mollusks, the shell is aragonite, in others. The nacreous layer of the fossil shells of some extinct ammonites forms an iridescent material called ammolite. Aragonite also forms in the ocean and in caves as inorganic precipitates called marine cements and speleothems, Aragonite is not uncommon in serpentinites where high Mg in pore solutions apparently inhibits calcite growth and promotes aragonite precipitation. Aragonite is metastable at the low pressures near the Earths surface and is commonly replaced by calcite in fossils. Aragonite older than the Carboniferous is essentially unknown and it can also be synthesized by adding a calcium chloride solution to a sodium carbonate solution at temperatures above 60 °C or in water-ethanol mixtures at ambient temperatures. Aragonite is thermodynamically unstable at temperature and pressure, and tends to alter to calcite on scales of 107 to 108 years. The mineral vaterite, also known as μ-CaCO3, is another phase of calcium carbonate that is metastable at ambient conditions typical of Earths surface, in aquaria, aragonite is considered essential for the replication of reef conditions. Aragonite provides the necessary for much sea life and also keeps the pH of the water close to its natural level. Aragonite has been tested for the removal of pollutants like zinc, cobalt. Aragonite sea Ikaite, CaCO3·6H2O Monohydrocalcite, CaCO3·H2O Nacre, otherwise known as Mother-of-Pearl Oolitic aragonite sand The Ochtinska aragonite cave Kosovo Caves Aragonite Formations
23.
Marcasite
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The mineral marcasite, sometimes called white iron pyrite, is iron sulfide with orthorhombic crystal structure. It is physically and crystallographically distinct from pyrite, which is iron sulfide with cubic crystal structure, both structures do have in common that they contain the disulfide S22− ion having a short bonding distance between the sulfur atoms. The structures differ in how these di-anions are arranged around the Fe2+ cations, marcasite is lighter and more brittle than pyrite. Specimens of marcasite often crumble and break up due to the crystal structure. On fresh surfaces it is yellow to almost white and has a bright metallic luster. It tarnishes to a yellowish or brownish color and gives a black streak and it is a brittle material that cannot be scratched with a knife. The thin, flat, tabular crystals, when joined in groups, are called cockscombs, in marcasite jewellery, pyrite used as a gemstone is termed marcasite – that is, marcasite jewellery is made from pyrite, not from the mineral marcasite. In the late medieval and early modern eras the word marcasite meant both pyrite and the mineral marcasite, the narrower, modern scientific definition for marcasite as orthorhombic iron sulfide dates from 1845. The jewellery sense for the word pre-dates this 1845 scientific redefinition, marcasite in the scientific sense is not used as a gem due to its brittleness. Marcasite can be formed as both a primary or a secondary mineral and it typically forms under low-temperature highly acidic conditions. It occurs in rocks as well as in low temperature hydrothermal veins. Commonly associated minerals include pyrite, pyrrhotite, galena, sphalerite, fluorite, dolomite and calcite, as a secondary mineral it forms by chemical alteration of a primary mineral such as pyrrhotite or chalcopyrite. Blueite, Nickel variety of marcasite, found in Denison Drury and Townships, lonchidite, Arsenic variety of marcasite, found at Churprinz Friedrich August Erbstolln Mine, Großschirma Freiberg, Erzgebirge, Saxony, Germany, ideal formula Fe2. Synonyms for this variety, kausimkies, kyrosite, lonchandite, metalonchidite described at Bernhard Mine near Hausach, sperkise, designates a marcasite having twin spearhead crystal on. Sperkise derives from the German Speerkies and this twin is very common in the marcasite of a chalky origin, particularly those from the Cap Blanc Nez. Marcasite reacts more readily than pyrite under conditions of high humidity, the product of this disintegration is iron sulfate and sulfuric acid. The hydrous iron sulfate forms a white powder consisting of the mineral melanterite and this disintegration of marcasite in mineral collections is known as pyrite decay. Low humidity storage conditions prevents or slows the reaction, how Minerals Form and Change Pyrite oxidation under room conditions
24.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
25.
Springer Science+Business Media
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Springer also hosts a number of scientific databases, including SpringerLink, Springer Protocols, and SpringerImages. Book publications include major works, textbooks, monographs and book series. Springer has major offices in Berlin, Heidelberg, Dordrecht, on 15 January 2015, Holtzbrinck Publishing Group / Nature Publishing Group and Springer Science+Business Media announced a merger. In 1964, Springer expanded its business internationally, opening an office in New York City, offices in Tokyo, Paris, Milan, Hong Kong, and Delhi soon followed. The academic publishing company BertelsmannSpringer was formed after Bertelsmann bought a majority stake in Springer-Verlag in 1999, the British investment groups Cinven and Candover bought BertelsmannSpringer from Bertelsmann in 2003. They merged the company in 2004 with the Dutch publisher Kluwer Academic Publishers which they bought from Wolters Kluwer in 2002, Springer acquired the open-access publisher BioMed Central in October 2008 for an undisclosed amount. In 2009, Cinven and Candover sold Springer to two private equity firms, EQT Partners and Government of Singapore Investment Corporation, the closing of the sale was confirmed in February 2010 after the competition authorities in the USA and in Europe approved the transfer. In 2011, Springer acquired Pharma Marketing and Publishing Services from Wolters Kluwer, in 2013, the London-based private equity firm BC Partners acquired a majority stake in Springer from EQT and GIC for $4.4 billion. In 2014, it was revealed that Springer had published 16 fake papers in its journals that had been computer-generated using SCIgen, Springer subsequently removed all the papers from these journals. IEEE had also done the thing by removing more than 100 fake papers from its conference proceedings. In 2015, Springer retracted 64 of the papers it had published after it was found that they had gone through a fraudulent peer review process, Springer provides its electronic book and journal content on its SpringerLink site, which launched in 1996. SpringerProtocols is home to a collection of protocols, recipes which provide step-by-step instructions for conducting experiments in research labs, SpringerImages was launched in 2008 and offers a collection of currently 1.8 million images spanning science, technology, and medicine. SpringerMaterials was launched in 2009 and is a platform for accessing the Landolt-Börnstein database of research and information on materials, authorMapper is a free online tool for visualizing scientific research that enables document discovery based on author locations and geographic maps. The tool helps users explore patterns in scientific research, identify trends, discover collaborative relationships. While open-access publishing typically requires the author to pay a fee for copyright retention, for example, a national institution in Poland allows authors to publish in open-access journals without incurring any personal cost - but using public funds. Springer is a member of the Open Access Scholarly Publishers Association, the Academic Publishing Industry, A Story of Merger and Acquisition – via Northern Illinois University
26.
Triclinic crystal system
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In crystallography, the triclinic crystal system is one of the 7 crystal systems. A crystal system is described by three basis vectors, in the triclinic system, the crystal is described by vectors of unequal length, as in the orthorhombic system. In addition, no vector is at right angles orthogonal to another, the triclinic lattice is the least symmetric of the 14 three-dimensional Bravais lattices. It is the lattice type that itself has no mirror planes. There are a total 2 space groups, with each only one space group is associated. Pinacoidal is also known as triclinic normal, pedial is also triclinic hemihedral Mineral examples include plagioclase, microcline, rhodonite, turquoise, wollastonite and amblygonite, all in triclinic normal. Crystal structure Hurlbut, Cornelius S. Klein, Cornelis,1985, Manual of Mineralogy, 20th ed. pp.64 –65, ISBN 0-471-80580-7
27.
Monoclinic crystal system
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In crystallography, the monoclinic crystal system is one of the 7 crystal systems. A crystal system is described by three vectors, in the monoclinic system, the crystal is described by vectors of unequal lengths, as in the orthorhombic system. They form a rectangular prism with a parallelogram as its base, hence two vectors are perpendicular, while the third vector meets the other two at an angle other than 90°. There is only one monoclinic Bravais lattice in two dimensions, the oblique lattice, two monoclinic Bravais lattices exist, the primitive monoclinic and the centered monoclinic lattices. In this axis setting, the primitive and base-centered lattices interchange in centering type, sphenoidal is also monoclinic hemimorphic, Domatic is also monoclinic hemihedral, Prismatic is also monoclinic normal. Crystal structure Hurlbut, Cornelius S. Klein, Cornelis, hahn, Theo, ed. International Tables for Crystallography, Volume A, Space Group Symmetry
28.
Tetragonal crystal system
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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 vectors, so that the cube becomes a rectangular prism with a square base. There is only one tetragonal Bravais lattice in two dimensions, the square lattice, there are two tetragonal Bravais lattices, the simple tetragonal and the centered tetragonal. One might suppose stretching face-centered cubic would result in face-centered tetragonal, BCT is considered more fundamental, so that is the standard terminology. Crystal structure point groups Bravais lattices
29.
Hexagonal crystal family
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In crystallography, the hexagonal crystal family is one of the 6 crystal families. In the hexagonal family, the crystal is described by a right rhombic prism unit cell with two equal axes, an included angle of 120° and a height perpendicular to the two base axes. There are 52 space groups associated with it, which are exactly those whose Bravais lattice is either hexagonal or rhombohedral, the hexagonal crystal family consists of two lattice systems, hexagonal and rhombohedral. Each lattice system consists of one Bravais lattice, hence, there are 3 lattice points per unit cell in total and the lattice is non-primitive. The Bravais lattices in the hexagonal crystal family can also be described by rhombohedral axes, the unit cell is a rhombohedron. This is a cell with parameters a = b = c, α = β = γ ≠ 90°. In practice, the description is more commonly used because it is easier to deal with a coordinate system with two 90° angles. However, the axes are often shown in textbooks because this cell reveals 3m symmetry of crystal lattice. However, such a description is rarely used, the hexagonal crystal family consists of two crystal systems, trigonal and hexagonal. A crystal system is a set of point groups in which the point groups themselves, the trigonal crystal system consists of the 5 point groups that have a single three-fold rotation axis. The crystal structures of alpha-quartz in the example are described by two of those 18 space groups associated with the hexagonal lattice system. The hexagonal crystal system consists of the seven point groups such that all their groups have the hexagonal lattice as underlying lattice. Graphite is an example of a crystal that crystallizes in the crystal system. Note that the atom in the center of the HCP unit cell in the hexagonal lattice system does not appear in the unit cell of the hexagonal lattice. It is part of the two atom motif associated with each point in the underlying lattice. The trigonal crystal system is the crystal system whose point groups have more than one lattice system associated with their space groups. The 5 point groups in this system are listed below, with their international number and notation, their space groups in name. The point groups in this system are listed below, followed by their representations in Hermann–Mauguin or international notation and Schoenflies notation