Coordination number
In chemistry and materials science the'coordination number' called ligancy, of a central atom in a molecule or crystal is the number of atoms, molecules or ions bonded to it. The ion/molecule/atom surrounding the central ion/molecule/atom is called a ligand; this number is determined somewhat differently for molecules than for crystals. For molecules and polyatomic ions the coordination number of an atom is determined by counting the other atoms to which it is bonded. For example, − has Cr3+ as its central cation, which has a coordination number of 6 and is described as hexacoordinate; however the solid-state structures of crystals have less defined bonds, in these cases a count of neighboring atoms is employed. The simplest method is one used in materials science; the usual value of the coordination number for a given structure refers to an atom in the interior of a crystal lattice with neighbors in all directions. In contexts where crystal surfaces are important, such as materials science and heterogeneous catalysis, the number of neighbors of an interior atom is the bulk coordination number, while the number of surface neighbors of an atom at the surface of the crystal is the surface coordination number.
In chemistry, coordination number, defined in 1893 by Alfred Werner, is the total number of neighbors of a central atom in a molecule or ion. Although a carbon atom has four chemical bonds in most stable molecules, the coordination number of each carbon is four in methane, three in ethylene, two in acetylene. In effect we count the first bond to each neighboring atom, but not the other bonds. In coordination complexes, only the first or sigma bond between each ligand and the central atom counts, but not any pi bonds to the same ligands. In tungsten hexacarbonyl, W6, the coordination number of tungsten is counted as six although pi as well as sigma bonding is important in such metal carbonyls; the most common coordination number for d-block transition metal complexes is 6, with an octahedral geometry. The observed range is 2 to 9. Metals in the f-block can accommodate higher coordination number due to their greater ionic radii and availability of more orbitals for bonding. Coordination numbers of 8 to 12 are observed for f-block elements.
For example, with bidentate nitrate ions as ligands, CeIV and ThIV form the 12-coordinate ions 2− and 2−. When the surrounding ligands are much smaller than the central atom higher coordination numbers may be possible. One computational chemistry study predicted a stable PbHe2+15 ion composed of a central lead ion coordinated with no fewer than 15 helium atoms. At the opposite extreme, steric shielding can give rise to unusually low coordination numbers. An rare instance of a metal adopting a coordination number of 1 occurs in the terphenyl-based arylthallium complex 2,6-Tipp2C6H3Tl, where Tipp is the 2,4,6-triisopropylphenyl group. For π-electron ligands such as the cyclopentadienide ion −, alkenes and the cyclooctatetraenide ion 2−, the number of atoms in the π-electron system that bind to the central atom is termed the hapticity. In ferrocene the hapticity, η, of each cyclopentadienide anion is five, Fe2. There are various ways of assigning the contribution made to the coordination number of the central iron atom by each cyclopentadienide ligand.
The contribution could be assigned as one since there is one ligand, or as five since there are five neighbouring atoms, or as three since there are three electron pairs involved. The count of electron pairs is taken. In order to determine the coordination number of an atom in a crystal, the crystal structure has first to be determined; this is achieved using neutron or electron diffraction. Once the positions of the atoms within the unit cell of the crystal are known the coordination number of an atom can be determined. For molecular solids or coordination complexes the units of the polyatomic species can be detected and a count of the bonds can be performed. Solids with lattice structures which includes metals and many inorganic solids can have regular structures where coordinating atoms are all at the same distance and they form the vertices of a coordination polyhedron. However, there are many such solids where the structures are irregular. In materials science, the bulk coordination number of a given atom in the interior of a crystal lattice is the number of nearest neighbours to a given atom.
For an atom at a surface of a crystal, the surface coordination number is always less than the bulk coordination number. The surface coordination number is dependent on the Miller indices of the surface. In a body-centered cubic crystal, the bulk coordination number is 8, for the surface, the surface coordination number is 4.α-Aluminium has a regular cubic close packed structure, where each aluminium atom has 12 nearest neighbors, 6 in the same plane and 3 above and below and the coordination polyhedron is a cuboctahedron. Α-Iron has a body centered cubic structure where each iron atom has 8 nearest neighbors situated at the corners of a cube. The two most common allotropes of carbon have different coordination numbers. In diamond, each carbon atom is at the centre of a regular tetrahedron formed by four other carbon atoms, the coordination number is four, as for methane. Graphite is made of two-dimensional layers in which each carbon is covalently bonded to three other carbons.
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
Cuboctahedron
In geometry, a cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, 24 identical edges, each separating a triangle from a square; as such, it is a quasiregular polyhedron, i.e. an Archimedean solid, not only vertex-transitive but edge-transitive. It is the only radially equilateral convex polyhedron, its dual polyhedron is the rhombic dodecahedron. The cuboctahedron was known to Plato: Heron's Definitiones quotes Archimedes as saying that Plato knew of a solid made of 8 triangles and 6 squares. Heptaparallelohedron Fuller applied the name "Dymaxion" to this shape, used in an early version of the Dymaxion map, he called it the "Vector Equilibrium" because of its radial equilateral symmetry. He called a cuboctahedron consisting of rigid struts connected by flexible vertices a "jitterbug". With Oh symmetry, order 48, it is a rectified cube or rectified octahedron With Td symmetry, order 24, it is a cantellated tetrahedron or rhombitetratetrahedron.
With D3d symmetry, order 12, it is a triangular gyrobicupola. The area A and the volume V of the cuboctahedron of edge length a are: A = a 2 ≈ 9.464 1016 a 2 V = 5 3 2 a 3 ≈ 2.357 0226 a 3. The cuboctahedron has four special orthogonal projections, centered on a vertex, an edge, the two types of faces and square; the last two correspond to the B2 and A2 Coxeter planes. The skew projections show a hexagon passing through the center of the cuboctahedron; the cuboctahedron can be represented as a spherical tiling, projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not lengths. Straight lines on the sphere are projected as circular arcs on the plane; the Cartesian coordinates for the vertices of a cuboctahedron centered at the origin are: An alternate set of coordinates can be made in 4-space, as 12 permutations of: This construction exists as one of 16 orthant facets of the cantellated 16-cell. The cuboctahedron's 12 vertices can represent the root vectors of the simple Lie group A3.
With the addition of 6 vertices of the octahedron, these vertices represent the 18 root vectors of the simple Lie group B3. The cuboctahedron can be dissected into two triangular cupolas by a common hexagon passing through the center of the cuboctahedron. If these two triangular cupolas are twisted so triangles and squares line up, Johnson solid J27, the triangular orthobicupola, is created; the cuboctahedron can be dissected into 6 square pyramids and 8 tetrahedra meeting at a central point. This dissection is expressed in the alternated cubic honeycomb where pairs of square pyramids are combined into octahedra; the cuboctahedron is the unique convex polyhedron in which the long radius is the same as the edge length. This radial equilateral symmetry is a property of only a few polytopes, including the two-dimensional hexagon, the three-dimensional cuboctahedron, the four-dimensional 24-cell and 8-cell. Radially equilateral polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.
Therefore, all the interior elements which meet at the center of these polytopes have equilateral triangle inward faces, as in the dissection of the cuboctahedron into 6 square pyramids and 8 tetrahedra. Each of these radially equilateral polytopes occurs as cells of a characteristic space-filling tessellation: the tiling of regular hexagons, the rectified cubic honeycomb, the 24-cell honeycomb and the tesseractic honeycomb, respectively; each tessellation has a dual tessellation. The densest known regular sphere-packing in two and four dimensions uses the cell centers of one of these tessellations as sphere centers. A cuboctahedron has octahedral symmetry, its first stellation is the compound of a cube and its dual octahedron, with the vertices of the cuboctahedron located at the midpoints of the edges of either. A cuboctahedron can be obtained by taking an equatorial cross section of a four-dimensional 24-cell or 16-cell. A hexagon can be obtained by taking an equatorial cross section of a cuboctahedron.
The cuboctahedron is a rectified cube and a rectified octahedron. It is a cantellated tetrahedron. With this construction it is given the Wythoff symbol: 3 3 | 2. A skew cantellation of the tetrahedron produces a solid with faces parallel to those of the cuboctahedron, namely eight triangles of two sizes, six rectangles. While its edges are unequal, this solid remains vertex-uniform: the solid has the full tetrahedral symmet
Symmetry
Symmetry in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definition, that an object is invariant to any of various transformations. Although these two meanings of "symmetry" can sometimes be told apart, they are related, so in this article they are discussed together. Mathematical symmetry may be observed with respect to the passage of time; this article describes symmetry from three perspectives: in mathematics, including geometry, the most familiar type of symmetry for many people. The opposite of symmetry is asymmetry. A geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion; this means that an object is symmetric if there is a transformation that moves individual pieces of the object but doesn't change the overall shape. The type of symmetry is determined by the way the pieces are organized, or by the type of transformation: An object has reflectional symmetry if there is a line going through it which divides it into two pieces which are mirror images of each other.
An object has rotational symmetry if the object can be rotated about a fixed point without changing the overall shape. An object has translational symmetry. An object has helical symmetry if it can be translated and rotated in three-dimensional space along a line known as a screw axis. An object contracted. Fractals exhibit a form of scale symmetry, where small portions of the fractal are similar in shape to large portions. Other symmetries include glide reflection rotoreflection symmetry. A dyadic relation R is only if, whenever it's true that Rab, it's true that Rba. Thus, "is the same age as" is symmetrical, for if Paul is the same age as Mary Mary is the same age as Paul. Symmetric binary logical connectives are and, or, nand and nor. Generalizing from geometrical symmetry in the previous section, we say that a mathematical object is symmetric with respect to a given mathematical operation, if, when applied to the object, this operation preserves some property of the object; the set of operations that preserve a given property of the object form a group.
In general, every kind of structure in mathematics will have its own kind of symmetry. Examples include and odd functions in calculus. In statistics, it appears as symmetric probability distributions, as skewness, asymmetry of distributions. Symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations; this concept has become one of the most powerful tools of theoretical physics, as it has become evident that all laws of nature originate in symmetries. In fact, this role inspired the Nobel laureate PW Anderson to write in his read 1972 article More is Different that "it is only overstating the case to say that physics is the study of symmetry." See Noether's theorem. Important symmetries in physics include discrete symmetries of spacetime. In biology, the notion of symmetry is used explicitly to describe body shapes. Bilateral animals, including humans, are more or less symmetric with respect to the sagittal plane which divides the body into left and right halves.
Animals that move in one direction have upper and lower sides and tail ends, therefore a left and a right. The head becomes specialized with a mouth and sense organs, the body becomes bilaterally symmetric for the purpose of movement, with symmetrical pairs of muscles and skeletal elements, though internal organs remain asymmetric. Plants and sessile animals such as sea anemones have radial or rotational symmetry, which suits them because food or threats may arrive from any direction. Fivefold symmetry is found in the echinoderms, the group that includes starfish, sea urchins, sea lilies. In biology, the notion of symmetry is used as in physics, to say to describe the properties of the objects studied, including their interactions. A remarkable property of biological evolution is the changes of symmetry corresponding to the appearance of new parts and dynamics. Symmetry is important to chemistry because it undergirds all specific interactions between molecules in nature; the control of the symmetry of molecules produced in modern chemical synthesis contributes to the ability of scientists to offer the
Walter Raleigh
Sir Walter Raleigh spelled Ralegh, was an English landed gentleman, poet, politician, courtier and explorer. He was cousin to younger half-brother of Sir Humphrey Gilbert, he is well known for popularising tobacco in England. Raleigh was one of the most notable figures of the Elizabethan era. Raleigh was born to a Protestant family in Devon, the son of Walter Raleigh and Catherine Champernowne. Little is known of his early life, though in his late teens he spent some time in France taking part in the religious civil wars. In his 20s he took part in the suppression of rebellion in Ireland participating in the Siege of Smerwick, he became a landlord of property confiscated from the native Irish. He rose in the favour of Queen Elizabeth I and was knighted in 1585. Raleigh was instrumental in the English colonisation of North America and was granted a royal patent to explore Virginia, paving the way for future English settlements. In 1591, he secretly married Elizabeth Throckmorton, one of the Queen's ladies-in-waiting, without the Queen's permission, for which he and his wife were sent to the Tower of London.
After his release, they retired to his estate at Dorset. In 1594, Raleigh heard of a "City of Gold" in South America and sailed to find it, publishing an exaggerated account of his experiences in a book that contributed to the legend of "El Dorado". After Queen Elizabeth died in 1603, Raleigh was again imprisoned in the Tower, this time for being involved in the Main Plot against King James I, not favourably disposed towards him. In 1616, he was released to lead a second expedition in search of El Dorado. During the expedition, men led by his top commander ransacked a Spanish outpost, in violation of both the terms of his pardon and the 1604 peace treaty with Spain. Raleigh returned to England and, to appease the Spanish, he was arrested and executed in 1618. Little is known about Raleigh's birth but he is believed to have been born on 22 January 1552, he grew up in the parish of East Budleigh in South Devon. He was the youngest of the five sons of Walter Raleigh of Fardel Manor in the parish of Cornwood, in South Devon.
His family is assumed to have been a junior branch of the de Raleigh family, 11th century lords of the manor of Raleigh, Pilton in North Devon, although the two branches are known to have borne dissimilar coats of arms, adopted at the start of the age of heraldry. His mother was Katherine Champernowne, his father's 3rd wife, the 4th daughter of Sir Philip Champernowne, lord of the manor of Modbury, Devon, by his wife Catherine Carew, a daughter of Sir Edmund Carew of Mohuns Ottery in the parish of Luppitt and widow of Otes Gilbert of Greenway in the parish of Brixham and of Compton Castle in the parish of Marldon, both in Devon. Katherine Champernowne's paternal aunt was Kat Ashley, governess of Queen Elizabeth I, who introduced the young men at court; the coat of arms of Otes Gilbert and Katherine Champernowne survives in a stained glass window in Churston Ferrers Church, near Greenway. Sir Walter's half-brothers John Gilbert, Humphrey Gilbert, Adrian Gilbert, his full brother Carew Raleigh were prominent during the reigns of Queen Elizabeth I and King James I.
Raleigh's family was Protestant in religious orientation and had a number of near escapes during the reign of Roman Catholic Queen Mary I of England. In the most notable of these, his father had to hide in a tower to avoid execution; as a result, Raleigh developed a hatred of Roman Catholicism during his childhood, proved himself quick to express it after Protestant Queen Elizabeth I came to the throne in 1558. In matters of religion, Elizabeth was more moderate than her half sister Mary. In 1569, Raleigh left for France to serve with the Huguenots in the French religious civil wars. In 1572, Raleigh was registered as an undergraduate at Oriel College, but he left a year without a degree. Raleigh proceeded to finish his education in the Inns of Court. In 1575, he was registered at the Middle Temple. At his trial in 1603, he stated, his life is uncertain between 1569 and 1575, but in his History of the World he claimed to have been an eyewitness at the Battle of Moncontour in France. In 1575 or 1576, Raleigh returned to England.
Between 1579 and 1583, Raleigh took part in the suppression of the Desmond Rebellions. He was present at the Siege of Smerwick, where he led the party that beheaded some 600 Spanish and Italian soldiers. Raleigh received 40,000 acres upon the seizure and distribution of land following the attainders arising from the rebellion, including the coastal walled town of Youghal and, further up the Blackwater River, the village of Lismore; this made him one of the principal landowners in Munster, but he had limited success inducing English tenants to settle on his estates. Raleigh made the town of Youghal his occasional home during his 17 years as an Irish landlord being domiciled at Killua Castle, County Westmeath, he was mayor there from 1588 to 1589. His town mansion of Myrtle Grove is assumed to be the setting for the story that his servant doused him with a bucket of water after seeing clouds of smoke coming from Raleigh's pipe, in the belief that he had been set alight, but this story is told of other places associated with Raleigh: the Virginia Ash Inn in Henstridge near Sherborne, Sherborne Castle, South Wraxall Manor in Wiltshire, home of Raleigh's friend Sir Walter Long.
Amongst Raleigh's acquai
Rhombic dodecahedral honeycomb
The rhombic dodecahedral honeycomb is a space-filling tessellation in Euclidean 3-space. It is the Voronoi diagram of the face-centered cubic sphere-packing, which has the densest possible packing of equal spheres in ordinary space, it consists of copies of the rhombic dodecahedron. All faces are rhombi, with diagonals in the ratio 1:√2. Three cells meet at each edge; the honeycomb is thus face-transitive and edge-transitive. The vertices with the obtuse rhombic face angles have 4 cells; the vertices with the acute rhombic face angles have 6 cells. The rhombic dodecahedron can be twisted on one of its hexagonal cross-sections to form a trapezo-rhombic dodecahedron, the cell of a somewhat similar tessellation, the Voronoi diagram of hexagonal close-packing. Cells can be given 4 colors in square layers of 2-colors where neighboring faces have different colors, 6 colors in hexagonal layers of 3 colors where same-colored cells have no contact at all; the rhombic dodecahedral honeycomb can be dissected into a trigonal trapezohedral honeycomb with each rhombic dodecahedron dissected into 4 trigonal trapezohedrons.
Each rhombic dodecahedra can be dissected with a center point into 12 rhombic pyramids of the rhombic pyramidal honeycomb. The trapezo-rhombic dodecahedral honeycomb is a space-filling tessellation in Euclidean 3-space, it consists of copies of the trapezo-rhombic dodecahedron. It is similar to the higher symmetric rhombic dodecahedral honeycomb which has all 12 faces as rhombi, it is a dual to the vertex-transitive gyrated tetrahedral-octahedral honeycomb. The rhombic pyramidal honeycomb or half oblate octahedrille is a uniform space-filling tessellation in Euclidean 3-space; this honeycomb can be seen as a rhombic dodecahedral honeycomb, with the rhombic dodecahedra dissected with its center into 12 rhombic pyramids. It is dual to the cantic cubic honeycomb: catoptric tessellation Williams, Robert; the Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 168. ISBN 0-486-23729-X. Weisstein, Eric W. "Space-filling polyhedron". MathWorld. Examples of Housing Construction using this geometry
Sphere
A sphere is a round geometrical object in three-dimensional space, the surface of a round ball. Like a circle in a two-dimensional space, a sphere is defined mathematically as the set of points that are all at the same distance r from a given point, but in a three-dimensional space; this distance r is the radius of the ball, made up from all points with a distance less than r from the given point, the center of the mathematical ball. These are referred to as the radius and center of the sphere, respectively; the longest straight line segment through the ball, connecting two points of the sphere, passes through the center and its length is thus twice the radius. While outside mathematics the terms "sphere" and "ball" are sometimes used interchangeably, in mathematics the above distinction is made between a sphere, a two-dimensional closed surface, embedded in a three-dimensional Euclidean space, a ball, a three-dimensional shape that includes the sphere and everything inside the sphere, or, more just the points inside, but not on the sphere.
The distinction between ball and sphere has not always been maintained and older mathematical references talk about a sphere as a solid. This is analogous to the situation in the plane, where the terms "circle" and "disk" can be confounded. In analytic geometry, a sphere with center and radius r is the locus of all points such that 2 + 2 + 2 = r 2. Let a, b, c, d, e be real numbers with a ≠ 0 and put x 0 = − b a, y 0 = − c a, z 0 = − d a, ρ = b 2 + c 2 + d 2 − a e a 2; the equation f = a + 2 + e = 0 has no real points as solutions if ρ < 0 and is called the equation of an imaginary sphere. If ρ = 0 the only solution of f = 0 is the point P 0 = and the equation is said to be the equation of a point sphere. In the case ρ > 0, f = 0 is an equation of a sphere whose center is P 0 and whose radius is ρ. If a in the above equation is zero f = 0 is the equation of a plane. Thus, a plane may be thought of as a sphere of infinite radius; the points on the sphere with radius r > 0 and center can be parameterized via x = x 0 + r sin θ cos φ y = y 0 + r sin θ sin φ z = z 0 + r cos θ The parameter θ {
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