Cubic crystal system
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 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 unit cell in these crystals is conventionally taken to be a cube, the primitive unit cell is not; the three Bravais lattices in the cubic crystal system are: The primitive cubic system consists of one lattice point on each corner of the cube. Each atom at a lattice point is shared between eight adjacent cubes, the unit cell therefore contains in total one atom; the body-centered cubic system has one lattice 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; the face-centered cubic system has lattice points on the faces of the cube, that each gives one half contribution, in addition to the corner lattice points, giving a total of 4 lattice points per unit cell.
Each sphere in a cF lattice has coordination number 12. Coordination number is the number of nearest neighbours of a central atom in the structure; the face-centered cubic system is related to the hexagonal close packed system, where two systems differ only in the relative placements of their hexagonal layers. The plane of a face-centered cubic system is a hexagonal grid. Attempting to create a C-centered cubic crystal system would result in a simple tetragonal Bravais lattice; the isometric crystal system class names, point groups, examples, International Tables for Crystallography space group number, space groups are listed in the table below. There are a total 36 cubic space groups. Other terms for hexoctahedral are: normal class, ditesseral central class, galena type. A simple cubic unit cell has a single cubic void in the center. A body-centered cubic unit cell has six octahedral voids located at the center of each face of the unit cell, twelve further ones located at the midpoint of each edge of the same cell, for a total of six net octahedral voids.
Additionally, there are 24 tetrahedral voids located in a square spacing around each octahedral void, for a total of twelve net tetrahedral voids. These tetrahedral voids are not local maxima and are not technically voids, but they do 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 unit cell, for a total of eight net tetrahedral voids. Additionally, there are twelve octahedral voids located at the midpoints of the edges of the unit cell as well as one octahedral hole in the center of the cell, for a total of four net octahedral voids. One important characteristic of a crystalline structure is its atomic packing factor; this is calculated by assuming that all the atoms are identical spheres, with a radius large enough that each sphere abuts on the next. 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.
In a bcc lattice, the atomic packing factor is 0.680, in fcc it is 0.740. The fcc value is the highest theoretically possible value for any lattice, although there are other lattices which achieve the same value, such as hexagonal close packed and one version of tetrahedral bcc; as a rule, since atoms in a solid attract each other, the more packed arrangements of atoms tend to be more common. Accordingly, the primitive cubic structure, with low atomic packing factor, is rare in nature, but is found in polonium; the bcc and fcc, with their higher densities, are both quite common in nature. Examples of bcc include iron, chromium and niobium. Examples of fcc include aluminium, copper and silver. Compounds that consist of more than one element have crystal structures based on a cubic crystal system; some of the more common ones are listed here. The space group of the caesium chloride structure is called Pm3m, or "221"; the Strukturbericht designation is "B2". One structure is the "interpenetrating primitive cubic" structure called the "caesium chloride" structure.
Each of the two atom types forms a separate primitive cubic lattice, with an atom of one type at the center of each cube of the other type. Altogether, the arrangement of atoms is the same as body-centered cubic, but with alternating types of atoms at the different lattice sites. Alternately, one could view this lattice as a simple cubic structure with a secondary atom in its cubic void. In addition to caesium chloride itself, the structure appears in certain other alkali halides when prepared at low temperatures or high pressures; this structure is more to be formed from two elements whose ions are of the same size. The coordination
The mineral marcasite, sometimes called white iron pyrite, is iron sulfide with orthorhombic crystal structure. It is physically and crystallographically distinct from pyrite, 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. Marcasite is more brittle than pyrite. Specimens of marcasite crumble and break up due to the unstable crystal structure. On fresh surfaces it is pale yellow to white and has a bright metallic luster, it gives a black streak. It is a brittle material; the thin, 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 a secondary mineral, it forms under low-temperature acidic conditions. It occurs in sedimentary rocks as well as in low temperature hydrothermal veins. Associated minerals include pyrite, galena, fluorite and calcite; as a primary mineral it forms nodules and crystals in a variety of sedimentary rock, such as in the chalk layers found on both sides of the English Channel at Dover, England and at Cap Blanc Nez, Pas De Calais, where it forms as sharp individual crystals and crystal groups, nodules. 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, Sudbury Dist. Ontario, Canada. Lonchidite: Arsenic variety of marcasite, found at Churprinz Friedrich August Erbstolln Mine, Großschirma Freiberg, Saxony, Germany.
Synonyms for this variety: kausimkies, lonchandite, metalonchidite described at Bernhard Mine near Hausach, Germany. Sperkise: designates a marcasite having twin spearhead crystal on. Sperkise derives from the German Speerkies; this twin is common in the marcasite of a chalky origin those from the Cap Blanc Nez. Marcasite reacts more than pyrite under conditions of high humidity; the product of this disintegration is sulfuric acid. The hydrous iron sulfate forms a white powder consisting of the mineral melanterite, FeSO4·7H2O; this disintegration of marcasite in mineral collections is known as "pyrite decay". When a specimen goes through pyrite decay, the marcasite reacts with moisture and oxygen in the air, the sulfur oxidizing and combining with water to produce sulfuric acid that attacks other sulfide minerals and mineral labels. Low humidity storage conditions slows the reaction. How Minerals Form and Change "Pyrite oxidation under room conditions"
In geometry, orbifold notation is a system, invented by William Thurston and popularized by the mathematician John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advantage of the notation is that it describes these groups in a way which indicates many of the groups' properties: in particular, it describes the orbifold obtained by taking the quotient of Euclidean space by the group under consideration. Groups representable in this notation include the point groups on the sphere, the frieze groups and wallpaper groups of the Euclidean plane, their analogues on the hyperbolic plane; the following types of Euclidean transformation can occur in a group described by orbifold notation: reflection through a line translation by a vector rotation of finite order around a point infinite rotation around a line in 3-space glide-reflection, i.e. reflection followed by translation. All translations which occur are assumed to form a discrete subgroup of the group symmetries being described.
Each group is denoted in orbifold notation by a finite string made up from the following symbols: positive integers 1, 2, 3, … the infinity symbol, ∞ the asterisk, * the symbol o, called a wonder and a handle because it topologically represents a torus closed surface. Patterns repeat by two translation; the symbol ×, called a miracle and represents a topological crosscap where a pattern repeats as a mirror 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, assumed to contain two independent translations; each symbol corresponds to a distinct transformation: an integer n to the left of an asterisk indicates a rotation of order n around a gyration point an integer n to the right of an asterisk indicates a transformation of order 2n which rotates around a kaleidoscopic point and reflects through a line an × indicates a glide reflection the symbol ∞ indicates infinite rotational symmetry around a line.
By abuse of language, we might 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 two linearly independent translations. An orbifold symbol is called good if it is not one of the following: p, pq, *p, *pq, for p,q>=2, p≠q. An object is chiral; the corresponding orbifold is non-orientable otherwise. The Euler characteristic of an orbifold can be read from its Conway symbol; each feature has a value: n without or before an asterisk counts as n − 1 n n after an asterisk counts as n − 1 2 n asterisk and × count as 1 o counts as 2. Subtracting the sum of these values from 2 gives the Euler characteristic. If the sum of the feature values is 2, the order is infinite, i.e. the notation represents a wallpaper group or a frieze group. Indeed, Conway's "Magic Theorem" indicates that the 17 wallpaper groups are 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 *11 22 and 221 *22 and *221 2* and 2*1; this is. The symmetry of a 2D object without translational symmetry can be described by the 3D symmetry type by adding a third dimension to the object which does not add or spoil symmetry. For example, for a 2D image we can consider a piece of carton with that image displayed on one side, thus we have n• and *n•. The bullet is added on one- and two-dimensional groups to imply the existence of a fixed point. A 1D image can be drawn horizontally on a piece of carton, with a provision to avoid additional symmetry with respect to the line of the image, e.g. by drawing a horizontal bar under the image. 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 and an asymmetric 2D or 1D object, respectively.
*Schönflies's point group notation is extended here as infinite cases of the equivalent dihedral points symmetries §The diagram shows one fundamental domain in yellow, with reflection lines in blue, glide reflection lines in dashed green, translation normals in red, 2-fold gyration points as small green squares. A first few hyperbolic groups, ordered by their Euler characteristic are: Mutation of orbifolds Fibrifold notation - an extension of orbifold notation for 3d space groups John H. Conway, Olaf Delgado Friedrichs, Daniel H. Huson, W
In crystallography, the terms crystal system, crystal family, lattice system each refer to one of several classes of space groups, point groups, or crystals. Informally, two crystals are in the same crystal system if they have similar symmetries, although there are many exceptions to this. Crystal systems, crystal families and lattice systems are similar but different, there is widespread confusion between them: in particular the trigonal crystal system is confused with the rhombohedral lattice system, the term "crystal system" is sometimes used to mean "lattice system" or "crystal family". Space groups and crystals are divided into seven crystal systems according to their point groups, into seven lattice systems according to their Bravais lattices. Five of the crystal systems are 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, in order to eliminate this confusion.
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, orthorhombic, rhombohedral and cubic. 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 lattice system, in which case both the crystal and lattice systems have the same name. However, five point groups are assigned to two lattice systems and hexagonal, because both exhibit threefold rotational symmetry; these point groups are assigned to the trigonal crystal system. In total there are seven crystal systems: triclinic, orthorhombic, trigonal and cubic. A crystal family is determined by lattices and point groups, it is formed by combining crystal systems which have space groups assigned to a common lattice system. In three dimensions, the crystal families and systems are identical, except the hexagonal and trigonal crystal systems, which are combined into one hexagonal crystal family.
In total there are six crystal families: triclinic, orthorhombic, tetragonal and cubic. Spaces with less than three dimensions have the same 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 and hexagonal; the relation between three-dimensional crystal families, crystal systems and lattice systems is shown in the following table: Note: there is no "trigonal" lattice system. To avoid confusion of terminology, the term "trigonal lattice" is not used; the 7 crystal systems consist of 32 crystal classes as shown in the following table: The point symmetry of a structure can be further described as follows. Consider the points that make up the structure, reflect them all through a single point, so that becomes; this is the'inverted structure'. If the original structure and inverted structure are identical the structure is centrosymmetric. Otherwise it is non-centrosymmetric.
Still in the non-centrosymmetric case, the inverted structure can in some cases be rotated to align with the original structure. This is a non-centrosymmetric achiral structure. If the inverted structure cannot be rotated to align with the original structure the structure is chiral or enantiomorphic and its symmetry group is enantiomorphic. A direction is called polar if its two directional senses are physically different. A symmetry direction of a crystal, polar is called a polar axis. Groups containing a polar axis are called polar. A polar crystal possesses a unique polar axis; some geometrical or physical property is different at the two ends of this axis: for example, there might develop a dielectric polarization as in pyroelectric crystals. A polar axis can occur only in non-centrosymmetric structures. There cannot be a mirror plane or twofold axis perpendicular to the polar axis, because they would make the two directions of the axis equivalent; the crystal structures of chiral biological molecules can only occur in the 65 enantiomorphic space groups.
The distribution of the 14 Bravais lattices into lattice systems and crystal families is given in the following table. In geometry and crystallography, a Bravais lattice is a category of translative symmetry groups in three directions; such symmetry groups consist of translations by vectors of the form R = n1a1 + n2a2 + n3a3,where n1, n2, n3 are integers and a1, a2, a3 are three non-coplanar vectors, called primitive vectors. These lattices are classified by the space group of the lattice itself, viewed as a collection of points, they represent the maximum symmetry. All crystalline materials must, by definition, fit into one of these arrangements. For convenience a Bravais lattice is depicted by a unit cell, a factor 1, 2, 3 or 4 larger than the primitive cell. Depending on the symmetry of a crystal or other pattern, the fundamental domain is again smaller, up to a factor 48; the Bravais lattices were studied by Moritz Ludwig Frankenheim in 1842, who found that there we
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, it is found in beryllium rich pegmatites and is in part an alteration of beryl. Bertrandite occurs as a pseudomorphic replacement of beryl. Associated minerals include beryl, herderite, muscovite and quartz. It, with beryl, are ores of beryllium, it was named after French mineralogist, Emile Bertrand. One of the world's largest deposits of bertrandite is Spor Mountain, Thomas Range, Utah, the source of most of the world's beryllium production. List of minerals List of minerals named after people
Hexagonal crystal family
In crystallography, the hexagonal crystal family is one of the 6 crystal families, which includes 2 crystal systems and 2 lattice systems. The hexagonal crystal family consists of the 12 point groups such that at least one of their space groups has the hexagonal lattice as underlying lattice, is the union of the hexagonal crystal system and the trigonal crystal system. There are 52 space groups associated with it, which are those whose Bravais lattice is either hexagonal or rhombohedral; the hexagonal crystal family consists of two lattice systems: rhombohedral. Each lattice system consists of one Bravais lattice. In the hexagonal family, the crystal is conventionally 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; the hexagonal unit cell for the rhombohedral Bravais lattice is the R-centered cell, consisting of two additional lattice points which occupy one body diagonal of the unit cell with coordinates and.
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 be described by rhombohedral axes; the unit cell is a rhombohedron. This is a unit cell with parameters a = b = c. In practice, the hexagonal description is more used because it is easier to deal with a coordinate system with two 90° angles. However, the rhombohedral axes are shown in textbooks because this cell reveals 3m symmetry of crystal lattice; the rhombohedral unit cell for the hexagonal Bravais lattice is the D-centered cell, consisting of two additional lattice points which occupy one body diagonal of the unit cell with coordinates and. However, such a description is used; the hexagonal crystal family consists of two crystal systems: hexagonal. A crystal system is a set of point groups in which the point groups themselves and their corresponding space groups are assigned to a lattice system; the trigonal crystal system consists of the 5 point groups that have a single three-fold rotation axis.
These 5 point groups have 7 corresponding space groups assigned to the rhombohedral lattice system and 18 corresponding space groups assigned to the hexagonal lattice system. The hexagonal crystal system consists of the 7 point groups that have a single six-fold rotation axis; these 7 point groups have 27 space groups, all of which are assigned to the hexagonal lattice system. Graphite is an example of a crystal; the trigonal crystal system is the only crystal system whose point groups have more than one lattice system associated with their space groups: the hexagonal and rhombohedral lattices both appear. The 5 point groups in this crystal system are listed below, with their international number and notation, their space groups in name and example crystals; the point groups in this crystal system are listed below, followed by their representations in Hermann–Mauguin or international notation and Schoenflies notation, mineral examples, if they exist. Hexagonal close packed is one of the two simple types of atomic packing with the highest density, the other being the face centered cubic.
However, unlike the fcc, it is not a Bravais lattice as there are two nonequivalent sets of lattice points. Instead, it can be constructed from the hexagonal Bravais lattice by using a two atom motif associated with each lattice point. Quartz is a crystal that belongs to the hexagonal lattice system but exists in two polymorphs that are in two different crystal systems; the crystal structures of α-quartz are described by two of the 18 space groups associated with the trigonal crystal system, while the crystal structures of β-quartz are described by two of the 27 space groups associated with the hexagonal crystal system. The lattice angles and the lengths of the lattice vectors are all the same for both the cubic and rhombohedral lattice systems; the lattice angles for simple cubic, face-centered cubic, body-centered cubic lattices are π/2 radians, π/3 radians, arccos radians, respectively. A rhombohedral lattice will result from lattice angles other than these. Crystal structure Close-packing Wurtzite Hahn, Theo, ed..
International Tables for Crystallography, Volume A: Space Group Symmetry. A. Berlin, New York: Springer-Verlag. Doi:10.1107/97809553602060000100. ISBN 978-0-7923-6590-7. Media related to Trigonal lattices at Wikimedia Commons Mineralogy database