1.
Calcite
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Calcite is a carbonate mineral and the most stable polymorph of calcium carbonate. The Mohs scale of hardness, based on scratch hardness comparison. Other polymorphs of calcium carbonate are the minerals aragonite and vaterite, aragonite will change to calcite at 380–470 °C, and vaterite is even less stable. Calcite is derived from the German Calcit, a term coined in the 19th century from the Latin word for lime and it is thus etymologically related to chalk. Calcite crystals are trigonal-rhombohedral, though actual calcite rhombohedra are rare as natural crystals, however, they show a remarkable variety of habits including acute to obtuse rhombohedra, tabular forms, prisms, or various scalenohedra. Calcite exhibits several twinning types adding to the variety of observed forms and it may occur as fibrous, granular, lamellar, or compact. Cleavage is usually in three directions parallel to the rhombohedron form and its fracture is conchoidal, but difficult to obtain. It has a defining Mohs hardness of 3, a gravity of 2.71. Color is white or none, though shades of gray, red, orange, yellow, green, blue, violet, brown, calcite is transparent to opaque and may occasionally show phosphorescence or fluorescence. A transparent variety called Iceland spar is used for optical purposes, acute scalenohedral crystals are sometimes referred to as dogtooth spar while the rhombohedral form is sometimes referred to as nailhead spar. Single calcite crystals display an optical property called birefringence and this strong birefringence causes objects viewed through a clear piece of calcite to appear doubled. The birefringent effect was first described by the Danish scientist Rasmus Bartholin in 1669, at a wavelength of ~590 nm calcite has ordinary and extraordinary refractive indices of 1.658 and 1.486, respectively. Between 190 and 1700 nm, the refractive index varies roughly between 1.9 and 1.5, while the extraordinary refractive index varies between 1.6 and 1.4. Calcite, like most carbonates, will dissolve with most forms of acid, calcite can be either dissolved by groundwater or precipitated by groundwater, depending on several factors including the water temperature, pH, and dissolved ion concentrations. Although calcite is fairly insoluble in water, acidity can cause dissolution of calcite. Ambient carbon dioxide, due to its acidity, has a slight solubilizing effect on calcite, calcite exhibits an unusual characteristic called retrograde solubility in which it becomes less soluble in water as the temperature increases. When conditions are right for precipitation, calcite forms mineral coatings that cement the existing rock grains together or it can fill fractures. On a landscape scale, continued dissolution of calcium carbonate-rich rocks can lead to the expansion and eventual collapse of cave systems, high-grade optical calcite was used in World War II for gun sights, specifically in bomb sights and anti-aircraft weaponry

2.
Silicate minerals
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Silicate minerals are rock-forming minerals made up of silicate groups. They are the largest and most important class of rock-forming minerals and they are classified based on the structure of their silicate groups, which contain different ratios of silicon and oxygen. Nesosilicates, or orthosilicates, have the orthosilicate ion, which constitute isolated 4− tetrahedra that are connected only by interstitial cations and these exist as 3-member 6− and 6-member 12− rings, where T stands for a tetrahedrally coordinated cation. Inosilicates, or chain silicates, have interlocking chains of silicate tetrahedra with either SiO3,1,3 ratio, for single chains or Si4O11,4,11 ratio, for double chains. Nickel–Strunz classification,09. D Pyroxene group Enstatite – orthoferrosilite series Enstatite – MgSiO3 Ferrosilite – FeSiO3 Pigeonite – Ca0.251, all phyllosilicate minerals are hydrated, with either water or hydroxyl groups attached. Serpentine subgroup Antigorite – Mg3Si2O54 Chrysotile – Mg3Si2O54 Lizardite – Mg3Si2O54 Clay minerals group Halloysite – Al2Si2O54 Kaolinite – Al2Si2O54 Illite – 24O10 Montmorillonite –0 and this group comprises nearly 75% of the crust of the Earth. Tectosilicates, with the exception of the group, are aluminosilicates. Nickel–Strunz classification,09. F and 09. G,04. A, an introduction to the rock-forming minerals. Wise, W. S. Zussman, J. Rock-forming minerals, P.982 pp. Hurlbut, Cornelius S. Danas Manual of Mineralogy. Mindat. org, Dana classification Webmineral, Danas New Silicate Classification Media related to Silicates at Wikimedia Commons

3.
Chemical formula
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These are limited to a single typographic line of symbols, which may include subscripts and superscripts. A chemical formula is not a name, and it contains no words. Although a chemical formula may imply certain simple chemical structures, it is not the same as a full chemical structural formula. Chemical formulas can fully specify the structure of only the simplest of molecules and chemical substances, the simplest types of chemical formulas are called empirical formulas, which use letters and numbers indicating the numerical proportions of atoms of each type. Molecular formulas indicate the numbers of each type of atom in a molecule. For example, the formula for glucose is CH2O, while its molecular formula is C6H12O6. This is possible if the relevant bonding is easy to show in one dimension, an example is the condensed molecular/chemical formula for ethanol, which is CH3-CH2-OH or CH3CH2OH. For reasons of structural complexity, there is no condensed chemical formula that specifies glucose, chemical formulas may be used in chemical equations to describe chemical reactions and other chemical transformations, such as the dissolving of ionic compounds into solution. A chemical formula identifies each constituent element by its chemical symbol, in empirical formulas, these proportions begin with a key element and then assign numbers of atoms of the other elements in the compound, as ratios to the key element. For molecular compounds, these numbers can all be expressed as whole numbers. For example, the formula of ethanol may be written C2H6O because the molecules of ethanol all contain two carbon atoms, six hydrogen atoms, and one oxygen atom. Some types of compounds, however, cannot be written with entirely whole-number empirical formulas. An example is boron carbide, whose formula of CBn is a variable non-whole number ratio with n ranging from over 4 to more than 6.5. When the chemical compound of the consists of simple molecules. These types of formulas are known as molecular formulas and condensed formulas. A molecular formula enumerates the number of atoms to reflect those in the molecule, so that the formula for glucose is C6H12O6 rather than the glucose empirical formula. However, except for very simple substances, molecular chemical formulas lack needed structural information, for simple molecules, a condensed formula is a type of chemical formula that may fully imply a correct structural formula. For example, ethanol may be represented by the chemical formula CH3CH2OH

4.
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

5.
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

6.
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

7.
H-M symbol
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In geometry, Hermann–Mauguin notation is used to represent the symmetry elements in point groups, plane groups and space groups. It is named after the German crystallographer Carl Hermann and the French mineralogist Charles-Victor Mauguin and this notation is sometimes called international notation, because it was adopted as standard by the International Tables For Crystallography since their first edition in 1935. Rotation axes are denoted by a number n —1,2,3,4,5,6,7,8, for improper rotations, Hermann–Mauguin symbols show rotoinversion axes, unlike Schoenflies and Shubnikov notations, where the preference is given to rotation-reflection axes. The rotoinversion axes are represented by the number with a macron. The symbol for a plane is m. The direction of the plane is defined as the direction of perpendicular to the face. Hermann–Mauguin symbols show symmetrically non-equivalent axes and planes, the direction of a symmetry element is represented by its position in the Hermann–Mauguin symbol. If a rotation axis n and a mirror plane m have the same direction, if two or more axes have the same direction, the axis with higher symmetry is shown. Higher symmetry means that the axis generates a pattern with more points, for example, rotation axes 3,4,5,6,7,8 generate 3-, 4-, 5-, 6-, 7-, 8-point patterns, respectively. Improper rotation axes 3,4,5,6,7,8 generate 6-, 4-, 10-, 6-, 14-, 8-point patterns, if both, the rotation and rotoinversion axes satisfy the previous rule, the rotation axis should be chosen. For example, 3/m combination is equivalent to 6, since 6 generates 6 points, and 3 generates only 3,6 should be written instead of 3/m. Analogously, in the case when both 3 and 3 axes are present,3 should be written, however we write 4/m, not 4/m, because both 4 and 4 generate four points. Finally, the Hermann–Mauguin symbol depends on the type of the group and these groups may contain only two-fold axes, mirror planes, and inversion center. These are the point groups 1 and 1,2, m, and 2/m, and 222, 2/m2/m2/m. If the symbol contains three positions, then they denote symmetry elements in the x, y, z direction, First position — primary direction — z direction, assigned to the higher-order axis. Second position — symmetrically equivalent secondary directions, which are perpendicular to the z-axis and these can be 2, m, or 2/m. Third position — symmetrically equivalent tertiary directions, passing between secondary directions and these can be 2, m, or 2/m. These are the crystallographic groups 3,32, 3m,3, and 32/m,4,422, 4mm,4, 42m, 4/m, and 4/m2/m2/m, and 6,622, 6mm,6, 6m2, 6/m, and 6/m2/m2/m

8.
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

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