1.
Crystal
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A crystal or crystalline solid is a solid material whose constituents are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macroscopic single crystals are usually identifiable by their geometrical shape, the scientific study of crystals and crystal formation is known as crystallography. The process of crystal formation via mechanisms of crystal growth is called crystallization or solidification, the word crystal derives from the Ancient Greek word κρύσταλλος, meaning both ice and rock crystal, from κρύος, icy cold, frost. Examples of large crystals include snowflakes, diamonds, and table salt, most inorganic solids are not crystals but polycrystals, i. e. many microscopic crystals fused together into a single solid. Examples of polycrystals include most metals, rocks, ceramics, a third category of solids is amorphous solids, where the atoms have no periodic structure whatsoever. Examples of amorphous solids include glass, wax, and many plastics, Crystals are often used in pseudoscientific practices such as crystal therapy, and, along with gemstones, are sometimes associated with spellwork in Wiccan beliefs and related religious movements. The scientific definition of a crystal is based on the arrangement of atoms inside it. A crystal is a solid where the form a periodic arrangement. For example, when liquid water starts freezing, the change begins with small ice crystals that grow until they fuse. Most macroscopic inorganic solids are polycrystalline, including almost all metals, ceramics, ice, rocks, solids that are neither crystalline nor polycrystalline, such as glass, are called amorphous solids, also called glassy, vitreous, or noncrystalline. These have no periodic order, even microscopically, there are distinct differences between crystalline solids and amorphous solids, most notably, the process of forming a glass does not release the latent heat of fusion, but forming a crystal does. A crystal structure is characterized by its cell, a small imaginary box containing one or more atoms in a specific spatial arrangement. The unit cells are stacked in three-dimensional space to form the crystal, the symmetry of a crystal is constrained by the requirement that the unit cells stack perfectly with no gaps. There are 219 possible crystal symmetries, called space groups. These are grouped into 7 crystal systems, such as cubic crystal system or hexagonal crystal system, Crystals are commonly recognized by their shape, consisting of flat faces with sharp angles. Euhedral crystals are those with obvious, well-formed flat faces, anhedral crystals do not, usually because the crystal is one grain in a polycrystalline solid. The flat faces of a crystal are oriented in a specific way relative to the underlying atomic arrangement of the crystal. This occurs because some surface orientations are more stable than others, as a crystal grows, new atoms attach easily to the rougher and less stable parts of the surface, but less easily to the flat, stable surfaces
2.
Gypsum
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Gypsum is a soft sulfate mineral composed of calcium sulfate dihydrate, with the chemical formula CaSO4·2H2O. It is widely mined and is used as a fertilizer, and as the constituent in many forms of plaster, blackboard chalk. Mohs scale of hardness, based on scratch Hardness comparison. It forms as a mineral and as a hydration product of anhydrite. The word gypsum is derived from the Greek word γύψος, plaster, because the quarries of the Montmartre district of Paris have long furnished burnt gypsum used for various purposes, this dehydrated gypsum became known as plaster of Paris. Upon addition of water, after a few tens of minutes plaster of Paris becomes regular gypsum again, causing the material to harden or set in ways that are useful for casting, Gypsum was known in Old English as spærstān, spear stone, referring to its crystalline projections. Gypsum may act as a source of sulfur for plant growth, which was discovered by J. M. Mayer, american farmers were so anxious to acquire it that a lively smuggling trade with Nova Scotia evolved, resulting in the so-called Plaster War of 1820. In the 19th century, it was known as lime sulfate or sulfate of lime. Gypsum is moderately water-soluble and, in contrast to most other salts, it exhibits retrograde solubility, when gypsum is heated in air it loses water and converts first to calcium sulfate hemihydrate, and, if heated further, to anhydrous calcium sulfate. As for anhydrite, its solubility in saline solutions and in brines is also dependent on NaCl concentration. Gypsum crystals are found to contain water and hydrogen bonding. Gypsum occurs in nature as flattened and often twinned crystals, and transparent, selenite contains no significant selenium, rather, both substances were named for the ancient Greek word for the Moon. Selenite may also occur in a silky, fibrous form, in case it is commonly called satin spar. Finally, it may also be granular or quite compact, in hand-sized samples, it can be anywhere from transparent to opaque. A very fine-grained white or lightly tinted variety of gypsum, called alabaster, is prized for ornamental work of various sorts, in arid areas, gypsum can occur in a flower-like form, typically opaque, with embedded sand grains called desert rose. It also forms some of the largest crystals found in nature, up to 12 m long, Gypsum is a common mineral, with thick and extensive evaporite beds in association with sedimentary rocks. Deposits are known to occur in strata from as far back as the Archaean eon, Gypsum is deposited from lake and sea water, as well as in hot springs, from volcanic vapors, and sulfate solutions in veins. Hydrothermal anhydrite in veins is commonly hydrated to gypsum by groundwater in near-surface exposures and it is often associated with the minerals halite and sulfur
3.
Wallpaper group
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A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art, There are 17 possible distinct groups. Wallpaper groups are two-dimensional symmetry groups, intermediate in complexity between the simpler frieze groups and the space groups. Wallpaper groups categorize patterns by their symmetries, subtle differences may place similar patterns in different groups, while patterns that are very different in style, color, scale or orientation may belong to the same group. Consider the following examples, Examples A and B have the same group, it is called p4m in the IUC notation. Example C has a different wallpaper group, called p4g or 4*2, a complete list of all seventeen possible wallpaper groups can be found below. A symmetry of a pattern is, loosely speaking, a way of transforming the pattern so that it exactly the same after the transformation. For example, translational symmetry is present when the pattern can be translated some finite distance, think of shifting a set of vertical stripes horizontally by one stripe. Strictly speaking, a true symmetry only exists in patterns that repeat exactly, a set of only, say, five stripes does not have translational symmetry—when shifted, the stripe on one end disappears and a new stripe is added at the other end. In practice, however, classification is applied to finite patterns, sometimes two categorizations are meaningful, one based on shapes alone and one also including colors. When colors are ignored there may be more symmetry, the types of transformations that are relevant here are called Euclidean plane isometries. This type of symmetry is called a translation, Examples A and C are similar, except that the smallest possible shifts are in diagonal directions. If we turn example B clockwise by 90°, around the centre of one of the squares, Examples A and C also have 90° rotations, although it requires a little more ingenuity to find the correct centre of rotation for C. We can also flip example B across a horizontal axis that runs across the middle of the image, example B also has reflections across a vertical axis, and across two diagonal axes. The same can be said for A. However, example C is different and it only has reflections in horizontal and vertical directions, not across diagonal axes. If we flip across a line, we do not get the same pattern back. This is part of the reason that the group of A and B is different from the wallpaper group of C. A proof that there were only 17 possible patterns was first carried out by Evgraf Fedorov in 1891, the proof that the list of wallpaper groups was complete only came after the much harder case of space groups had been done
4.
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
5.
Halotrichite
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Halotrichite, also known as feather alum, is a highly hydrated sulfate of aluminium and iron. It is formed by the weathering and decomposition of pyrite commonly near or in volcanic vents, the locations of natural occurrences include, the Atacama Desert, Chile, Dresden in Saxony, Germany, San Juan County, Utah, Iceland and Mont Saint-Hilaire, Canada. The name is from Latin, halotrichum for salt hair which accurately describes the precipitate/evaporite mineral
6.
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
7.
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
8.
Orthoclase
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Orthoclase, or orthoclase feldspar, is an important tectosilicate mineral which forms igneous rock. The name is from the Ancient Greek for straight fracture, because its two planes are at right angles to each other. It is a type of potassium feldspar, also known as K-feldspar, the gem known as moonstone is largely composed of orthoclase. Orthoclase is a constituent of most granites and other felsic igneous rocks and often forms huge crystals. Typically, the pure potassium endmember of orthoclase forms a solution with albite. While slowly cooling within the earth, sodium-rich albite lamellae form by exsolution, the resulting intergrowth of the two feldspars is called perthite. The higher-temperature polymorph of KAlSi3O8 is sanidine, sanidine is common in rapidly cooled volcanic rocks such as obsidian and felsic pyroclastic rocks, and is notably found in trachytes of the Drachenfels, Germany. The lower-temperature polymorph of KAlSi3O8 is microcline, adularia is a low temperature form of either microcline or orthoclase originally reported from the low temperature hydrothermal deposits in the Adula Alps of Switzerland. It was first described by Ermenegildo Pini in 1781, the optical effect of adularescence in moonstone is typically due to adularia. The largest documented crystal of orthoclase was found in Ural mountains. It measured ~10×10×0.4 m and weighed ~100 tons, together with the other potassium feldspars, orthoclase is a common raw material for the manufacture of some glasses and some ceramics such as porcelain, and as a constituent of scouring powder. Some intergrowths of orthoclase and albite have an attractive pale luster and are called moonstone when used in jewellery, most moonstones are translucent and white, although grey and peach-colored varieties also occur. In gemology, their luster is called adularescence and is described as creamy or silvery white with a billowy quality. It is the gem of Florida. Orthoclase is one of the ten defining minerals of the Mohs scale of mineral hardness, nASAs Curiosity Rover discovery of high levels of orthoclase in Martian sandstones suggested that some Martian rocks may have experienced complex geological processing, such as repeated melting
9.
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
10.
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
11.
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
12.
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