1.
Dolomite
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Dolomite is an anhydrous carbonate mineral composed of calcium magnesium carbonate, ideally CaMg2. The term is used for a sedimentary carbonate rock composed mostly of the mineral dolomite. An alternative name used for the dolomitic rock type is dolostone. Most probably the mineral dolomite was first described by Carl Linnaeus in 1768, nicolas-Théodore de Saussure first named the mineral in March 1792. The mineral dolomite crystallizes in the trigonal-rhombohedral system and it forms white, tan, gray, or pink crystals. Dolomite is a carbonate, having an alternating structural arrangement of calcium and magnesium ions. It does not rapidly dissolve or effervesce in dilute hydrochloric acid as calcite does, solid solution exists between dolomite, the iron-dominant ankerite and the manganese-dominant kutnohorite. Small amounts of iron in the give the crystals a yellow to brown tint. Manganese substitutes in the structure also up to three percent MnO. A high manganese content gives the crystals a rosy pink color, lead, zinc, and cobalt also substitute in the structure for magnesium. The mineral dolomite is closely related to huntite Mg3Ca4, because dolomite can be dissolved by slightly acidic water, areas of dolomite are important as aquifers and contribute to karst terrain formation. Modern dolomite formation has been found to occur under conditions in supersaturated saline lagoons along the Rio de Janeiro coast of Brazil, namely, Lagoa Vermelha. It is often thought that dolomite will develop only with the help of sulfate-reducing bacteria, however, low-temperature dolomite may occur in natural environments rich in organic matter and microbial cell surfaces. This occurs as a result of magnesium complexation by carboxyl groups associated with organic matter, vast deposits of dolomite are present in the geological record, but the mineral is relatively rare in modern environments. Reproducible, inorganic low-temperature syntheses of dolomite and magnesite were published for the first time in 1999, the general principle governing the course of this irreversible geochemical reaction has been coined breaking Ostwalds step rule. There is some evidence for an occurrence of dolomite. One example is that of the formation of dolomite in the bladder of a Dalmatian dog. In 2015, it was discovered that the direct crystallization of dolomite can occur from solution at temperatures between 60 and 220 °C
2.
Cinnabar
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Cinnabar generally occurs as a vein-filling mineral associated with recent volcanic activity and alkaline hot springs. The mineral resembles quartz in symmetry and in its exhibiting birefringence, cinnabar has a refractive index of ~3.2. The color and properties derive from a structure that is a crystalline lattice belonging to the hexagonal crystal system. Associated modern precautions for use and handling of cinnabar arise from the toxicity of the mercury component, the name comes from Ancient Greek, κιννάβαρι, a Greek word most likely applied by Theophrastus to several distinct substances. Other sources say the word comes from the Persian, شنگرف shangarf, in Latin it was sometimes known as minium, meaning also red cinnamon, though both of these terms now refer specifically to lead tetroxide. Cinnabar is generally found in a massive, granular or earthy form and is bright scarlet to brick-red in color and it resembles quartz in its symmetry. It exhibits birefringence, and it has the highest refractive index of any mineral and its mean refractive index is 3.08, versus the indices for diamond and the non-mineral gallium arsenide, which are 2.42 and 3.93, respectively. The hardness of cinnabar is 2. 0–2.5 on the Mohs scale, structurally, cinnabar belongs to the trigonal crystal system. It occurs as thick tabular or slender prismatic crystals or as granular to massive incrustations, crystal twinning occurs as simple contact twins. Note, mercury sulfide, HgS, adopts the structure described. Cinnabar is the stable form, and is a structure akin to that of HgO. In addition, HgS is found in a black, non-cinnabar polymorph that has the zincblende structure, Cinnabar generally occurs as a vein-filling mineral associated with recent volcanic activity and alkaline hot springs. Cinnabar is deposited by epithermal ascending aqueous solutions far removed from their igneous source and it is associated with native mercury, stibnite, realgar, pyrite, marcasite, opal, quartz, chalcedony, dolomite, calcite and barite. Cinnabar is essentially found in all mineral extraction localities that yield mercury, notably Puerto Princesa, Almadén, New Almaden, Hastings Mine and it was also mined near Red Devil, Alaska on the middle Kuskokwim River. Red Devil was named after the Red Devil cinnabar mine, a source of mercury. It has been found in Dominica near its sulfur springs at the end of the island along the west coast. Cinnabar is still being deposited, e. g. at the present day from the hot waters of Sulphur Bank Mine in California and Steamboat Springs, Nevada. As the most common source of mercury in nature, cinnabar has been mined for thousands of years, during the Roman Empire it was mined both as a pigment, and for its mercury content
3.
Beryl
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Beryl is a mineral composed of beryllium aluminium cyclosilicate with the chemical formula Be3Al26. Well-known varieties of beryl include emerald and aquamarine, naturally occurring, hexagonal crystals of beryl can be up to several meters in size, but terminated crystals are relatively rare. Pure beryl is colorless, but it is tinted by impurities, possible colors are green, blue, yellow, red. The name beryl is derived from Greek βήρυλλος beryllos which referred to a precious blue-green color-of-sea-water stone, akin to Prakrit verulia, the term was later adopted for the mineral beryl more exclusively. When the first eyeglasses were constructed in 13th century Italy, the lenses were made of beryl as glass could not be clear enough. Consequently glasses were named Brillen in German, beryl of various colors is found most commonly in granitic pegmatites, but also occurs in mica schists in the Ural Mountains, and limestone in Colombia. Beryl is often associated with tin and tungsten ore bodies, beryl is found in Europe in Norway, Austria, Germany, Sweden, Ireland and Russia, as well as Brazil, Colombia, Madagascar, Mozambique, Pakistan, South Africa, the United States, and Zambia. US beryl locations are in California, Colorado, Connecticut, Georgia, Idaho, Maine, New Hampshire, North Carolina, South Dakota and Utah. As of 1999, the worlds largest known naturally occurring crystal of any mineral is a crystal of beryl from Malakialina, Madagascar,18 m long and 3.5 m in diameter, aquamarine is a blue or cyan variety of beryl. It occurs at most localities which yield ordinary beryl, the gem-gravel placer deposits of Sri Lanka contain aquamarine. Clear yellow beryl, such as occurring in Brazil, is sometimes called aquamarine chrysolite. The deep blue version of aquamarine is called maxixe, maxixe is commonly found in the country of Madagascar. Its color fades to white when exposed to sunlight or is subjected to heat treatment, the pale blue color of aquamarine is attributed to Fe2+. Fe3+ ions produce golden-yellow color, and when both Fe2+ and Fe3+ are present, the color is a darker blue as in maxixe, decoloration of maxixe by light or heat thus may be due to the charge transfer between Fe3+ and Fe2+. Dark-blue maxixe color can be produced in green, pink or yellow beryl by irradiating it with high-energy particles, in the United States, aquamarines can be found at the summit of Mt. Antero in the Sawatch Range in central Colorado. In Wyoming, aquamarine has been discovered in the Big Horn Mountains, another location within the United States is the Sawtooth Range near Stanley, Idaho, although the minerals are within a wilderness area which prevents collecting. In Brazil, there are mines in the states of Minas Gerais, Espírito Santo, and Bahia, the mines of Colombia, Zambia, Madagascar, Malawi, Tanzania and Kenya also produce aquamarine. The largest aquamarine of gemstone quality ever mined was found in Marambaia, Minas Gerais, Brazil and it weighed over 110 kg, and its dimensions were 48.5 cm long and 42 cm in diameter
4.
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
5.
Crystal family
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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
6.
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
7.
Lattice 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
8.
Hexagon
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In geometry, a hexagon is a six sided polygon or 6-gon. The total of the angles of any hexagon is 720°. A regular hexagon has Schläfli symbol and can also be constructed as an equilateral triangle, t. A regular hexagon is defined as a hexagon that is both equilateral and equiangular and it is bicentric, meaning that it is both cyclic and tangential. The common length of the sides equals the radius of the circumscribed circle, all internal angles are 120 degrees. A regular hexagon has 6 rotational symmetries and 6 reflection symmetries, the longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane, the cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a triangular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral, the maximal diameter, D is twice the maximal radius or circumradius, R, which equals the side length, t. The minimal diameter or the diameter of the circle, d, is twice the minimal radius or inradius. If a regular hexagon has successive vertices A, B, C, D, E, F, the regular hexagon has Dih6 symmetry, order 12. There are 3 dihedral subgroups, Dih3, Dih2, and Dih1, and 4 cyclic subgroups, Z6, Z3, Z2 and these symmetries express 9 distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order, r12 is full symmetry, and a1 is no symmetry. These two forms are duals of each other and have half the order of the regular hexagon. The i4 forms are regular hexagons flattened or stretched along one symmetry direction and it can be seen as an elongated rhombus, while d2 and p2 can be seen as horizontally and vertically elongated kites. G2 hexagons, with sides parallel are also called hexagonal parallelogons. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g6 subgroup has no degrees of freedom but can seen as directed edges. Hexagons of symmetry g2, i4, and r12, as parallelogons can tessellate the Euclidean plane by translation, other hexagon shapes can tile the plane with different orientations
9.
Rhombohedron
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In geometry, a rhombohedron is a three-dimensional figure like a cube, except that its faces are not squares but rhombi. It is a case of a parallelepiped where all edges are the same length. It can be used to define the rhombohedral system, a honeycomb with rhombohedral cells. In general the rhombohedron can have three types of faces in congruent opposite pairs, Ci symmetry, order 2. Four points forming non-adjacent vertices of a rhombohedron necessarily form the four vertices of an orthocentric tetrahedron, the rhombohedral lattice system has rhombohedral cells, with 3 pairs of unique rhombic faces, Cube, with Oh symmetry, order 48. Trigonal trapezohedron, with D3d symmetry, order 12, if all of the non-obtuse internal angles of the faces are equal. This can be see by stretching a cube on its body-digonal axis, for example, a regular octahedron with two tetrahedra attached on opposite faces constructs a 60 degree trigonal trapezohedron. Right rhombic prism, with D2h symmetry, order 8 and it constructed by two rhombi and 4 squares. This can be see by stretching a cube on its face-digonal axis, for example, two triangular prisms attached together makes a 60 degree rhombic prism. A general rhombic prism, With C2h symmetry, order 4 and it has only one plane of symmetry through four vertices, and 6 rhombic faces. The height, h, is given by h = a sin θ The internal diagonals of the shown in Fig. SG1 are interesting. Three of the diagonals are all the same length. They are easily calculated from coordinate geometry once the coordinates of each vertex is known. Distance in a 3-d space is given by, d =2 +2 +2 For example, for the unit rhombohedron with rhombic acute angle of 72. The volume of this rhombohedron is 0.8789, and the height is 0.9242