Crystallography is the experimental science of determining the arrangement of atoms in 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, 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 physical measurements of their geometry; this involved measuring the angles of crystal faces relative to each other and to theoretical reference axes, establishing the symmetry of the crystal in question. 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 each 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 diffraction patterns of a sample targeted by a beam of some type. X-rays are most used; this is facilitated by the wave properties of the particles. Crystallographers explicitly state the type of beam used, as in the terms X-ray crystallography, neutron diffraction and electron diffraction; these three types of radiation interact with the specimen in different ways. X-rays interact with the spatial distribution of electrons in the sample. Electrons are charged particles and therefore interact with the total charge distribution of both the atomic nuclei and the electrons of the sample. Neutrons are scattered by the atomic nuclei through the strong nuclear forces, but in addition, the magnetic moment of neutrons is non-zero, they are therefore 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 a small object is made using a lens to focus the beam, similar to a lens in a microscope. However, the wavelength of visible light is three orders of magnitude longer than the length of typical atomic bonds and atoms themselves. Therefore, obtaining information about the spatial arrangement of atoms requires the use of radiation with shorter wavelengths, such as X-ray or neutron beams. Employing shorter wavelengths implied abandoning microscopy and true imaging, because there exists no material from which a lens capable of focusing this type of radiation can be created. Scientists have had some success focusing X-rays with microscopic Fresnel zone plates made from gold, by critical-angle reflection inside long tapered capillaries. Diffracted X-ray or neutron beams cannot be focused to produce images, so the sample structure must be reconstructed from the diffraction pattern.
Sharp features in the diffraction pattern arise from periodic, repeating structure in the sample, which are very strong due to coherent reflection of many photons from many spaced instances of similar structure, while non-periodic components of the structure result in diffuse diffraction features - areas with a higher density and repetition of atom order tend to reflect more light toward one point in space when compared to those areas with fewer atoms and less repetition. Because of their ordered and repetitive structure, crystals give diffraction patterns of sharp Bragg reflection spots, are ideal for analyzing the structure of solids. Coordinates in square brackets such as denote a direction vector. Coordinates in angle brackets or chevrons such as <100> denote a family of directions which are related by symmetry operations. In the cubic crystal system for example, <100> would mean, or the negative of any of those directions. Miller indices in parentheses such as denote a plane of the crystal structure, regular repetitions of that plane with a particular spacing.
In the cubic system, the normal to the plane is the direction, but in lower-symmetry cases, the normal to is not parallel to. Indices in curly brackets or braces such as denote a family of planes and their normals which are equivalent in cubic materials due to symmetry operations, much the way angle brackets denote a family of directions. In non-cubic materials, <hkl> is not perpendicular to. Some materials that have been analyzed crystallographically, such as proteins, do not occur as crystals; such molecules are placed in solution and allowed to crystallize through vapor diffusion. A drop of solution containing the molecule and precipitants is sealed in a container with a reservoir containing a hygroscopic solution. Water in the drop diffuses to the reservoir increasing the concentration and allowing a crystal to form. If the concentration were to rise more the molecule would precipitate out of solution, resulting in disorderly granules rather than an orderly and hence usable crystal. Once a crystal is obtained, data can be collected using a beam of radiation.
Although many universities that engage in crystallographic research have their own X-ray producing equipment, synchrotrons are used as X-ray sources, bec
In geometry, crystallography and solid state physics, a primitive cell is a minimum-volume cell corresponding to a single lattice point of a structure with discrete translational symmetry. The concept is used in describing crystal structure in two and three dimensions, though it makes sense in all dimensions. A lattice can be characterized by the geometry of its primitive cell; the primitive cell is a primitive unit. A primitive unit is a section of the tiling that generates the whole tiling using only translations, is as small as possible; the primitive cell is a fundamental domain with respect to translational symmetry only. In the case of additional symmetries a fundamental domain is smaller. A crystal can be categorized by the atoms that lie in a primitive cell. A cell will fill all the lattice space without leaving gaps by repetition of crystal translation operations. By definition, a primitive cell must contain one and only one lattice point. For unit cells lattice points that are shared by n cells are counted as 1/n of the lattice points contained in each of those cells.
A 2-dimensional primitive cell is a parallelogram, which in special cases may have orthogonal angles, or equal lengths, or both. The primitive translation vectors a→1, a→2, a→3 span a lattice cell of smallest volume for a particular three-dimensional lattice, are used to define a crystal translation vector T → = u 1 a → 1 + u 2 a → 2 + u 3 a → 3, where u1, u2, u3 are integers, translation by which leaves the lattice invariant; that is, for a point in the lattice r, the arrangement of points appears the same from r′ = r + T→ as from r. Since the primitive cell is defined by the primitive axes a→1, a→2, a→3, the volume Vp of the primitive cell is given by the parallelepiped from the above axes as V p = | a → 1 ⋅ |. For any 3-dimensional lattice, you can find primitive cells which are parallelepipeds, which in special cases may have orthogonal angles, or equal lengths, or both. While not mathematically required, by convention, one defines the parallelepiped primitive cell so that there is a lattice point on each corner.
When the lattice points are on the corner, each lattice point is shared by eight different primitive cells, so each lattice point will contribute only 1/8 of a lattice point to each of those cells. However, there are eight corners, so there is still a total of one lattice point per cell, as required by definition; some of the fourteen three-dimensional Bravais lattices are represented using such parallelepiped primitive cells, as shown below. The other Bravais lattices have a primitive unit cell in the shape of a parallelepiped, but in order to allow easy discrimination on the basis of symmetry, they are conventionally represented by a non-primitive unit cell which contains more than one lattice point. An alternative to the unit cell, for every Bravais lattice there is another kind of primitive cell called the Wigner–Seitz cell. In the Wigner–Seitz cell, the lattice point is at the center of the cell, for most Bravais lattices, the shape is not a parallelogram or parallelepiped; this is a type of Voronoi cell.
The Wigner–Seitz cell of the reciprocal lattice in momentum space is called the Brillouin zone. Wigner–Seitz cell Bravais lattice Wallpaper group Space group
In mathematics and chemistry, a space group is the symmetry group of a configuration in space in three dimensions. In three dimensions, there are 230 if chiral copies are considered distinct. Space groups are studied in dimensions other than 3 where they are sometimes called Bieberbach groups, are discrete cocompact groups of isometries of an oriented Euclidean space. In crystallography, space groups are called the crystallographic or Fedorov groups, represent a description of the symmetry of the crystal. A definitive source regarding 3-dimensional space groups is the International Tables for Crystallography. Space groups in 2 dimensions are the 17 wallpaper groups which have been known for several centuries, though the proof that the list was complete was only given in 1891, after the much more difficult classification of space groups had been completed. In 1879 Leonhard Sohncke listed the 65 space groups. More he listed 66 groups, but Fedorov and Schönflies both noticed that two of them were the same.
The space groups in three dimensions were first enumerated by Fedorov, shortly afterwards were independently enumerated by Schönflies. The correct list of 230 space groups was found by 1892 during correspondence between Fedorov and Schönflies. Barlow enumerated the groups with a different method, but omitted four groups though he had the correct list of 230 groups from 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, each of the latter belonging to one of 7 lattice systems. This results in a space group being some combination of the translational symmetry of a unit cell including lattice centering, the point group symmetry operations of reflection and improper rotation, the screw axis and glide plane symmetry operations; 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 the identity element, reflections and improper rotations. 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, one of the 32 possible point groups. Translation is defined as the face moves from one point to another point. A glide plane is a reflection in a plane, followed by a translation parallel with that plane; this is noted depending on which axis the glide is along. There is the n glide, a glide along the half of a diagonal of a face, the d glide, a fourth of the way along either a face or space diagonal of the unit cell; the latter is called the diamond glide plane. In 17 space groups, due to the centering of the cell, the glides occur in two perpendicular directions i.e. the same glide plane can be called b or c, a or b, a or c. For example, group Abm2 could be called Acm2, group Ccca could be called Cccb.
In 1992, it was suggested to use symbol e for such planes. The symbols for five space groups have been modified: A screw axis is a rotation about an axis, followed by a translation along the direction of the axis; these are noted by a number, n, to describe the degree of rotation, where the number is how many operations must be applied to complete a full rotation. The degree of translation is 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 twofold rotation followed by a translation of 1/2 of the lattice vector; the general formula for the action of an element of a space group is y = M.x + D where M is its matrix, D is its vector, where the element transforms point x into point y. In general, D = D + D, where D is a unique function of M, zero for M being the identity; the matrices M form a point group, a basis of the space group. The lattice dimension can be less than the overall dimension, resulting in a "subperiodic" space group.
For:: One-dimensional line groups: Two-dimensional line groups: frieze groups: Wallpaper groups: Three-dimensional line groups. Some of these methods can assign several different names to the same space group, so altogether there are many thousands of different names. Number; the International Union of Crystallography publishes tables of all space group types, assigns each a unique number from 1 to 230. The numbering is arbitrary, except that groups with the same crystal system or point group are given consecutive numbers. International symbol or Hermann–Mauguin notation; the Hermann–Mauguin notation describes the lattice and some generators for the group. It has a shortened form called the international short symbol, the one most used in crystallography
Galena called lead glance, is the natural mineral form of lead sulfide. It is an important source of silver. Galena is one of the most abundant and distributed sulfide minerals, it crystallizes in the cubic crystal system showing octahedral forms. It is associated with the minerals sphalerite and fluorite. Galena is the main ore of lead, used since ancient times; because of its somewhat low melting point, it was easy to liberate by smelting. It forms in low-temperature sedimentary deposits. In some deposits the galena contains about 1–2% silver, a byproduct that far outweighs the main lead ore in revenue. In these deposits significant amounts of silver occur as included silver sulfide mineral phases or as limited silver in solid solution within the galena structure; these argentiferous galenas have long been an important ore of silver. Galena deposits are found worldwide in various environments. Noted deposits include those at Freiberg in Saxony. In the United States, it occurs most notably in the Mississippi Valley type deposits of the Lead Belt in southeastern Missouri, in the Driftless Area of Illinois and Wisconsin.
Galena was a major mineral of the zinc-lead mines of the tri-state district around Joplin in southwestern Missouri and the adjoining areas of Kansas and Oklahoma. Galena is an important ore mineral in the silver mining regions of Colorado, Idaho and Montana. Of the latter, the Coeur d'Alene district of northern Idaho was most prominent. Galena is the official state mineral of the U. S. states of Wisconsin. The largest documented crystal of galena is composite cubo-octahedra from the Great Laxey Mine, Isle of Man, measuring 25 cm × 25 cm × 25 cm. Galena belongs to the octahedral sulfide group of minerals that have metal ions in octahedral positions, such as the iron sulfide pyrrhotite and the nickel arsenide niccolite; the galena group is named after its most common member, with other isometric members that include manganese bearing alabandite and niningerite. Divalent lead cations and sulfur anions form a close-packed cubic unit cell much like the mineral halite of the halide mineral group. Zinc, iron, antimony, arsenic and selenium occur in variable amounts in galena.
Selenium substitutes for sulfur in the structure constituting a solid solution series. The lead telluride mineral altaite has the same crystal structure as galena. Within the weathering or oxidation zone galena alters to cerussite. Galena exposed to acid mine drainage can be oxidized to anglesite by occurring bacteria and archaea, in a process similar to bioleaching. One of the oldest uses of galena was in the eye cosmetic kohl. In Ancient Egypt, this was applied around the eyes to reduce the glare of the desert sun and to repel flies, which were a potential source of disease. Galena is the primary ore of lead, used in making lead–acid batteries. Galena is mined for its silver content, such as at the Galena Mine in northern Idaho. Known as "potter's ore", galena is used in a green glaze applied to pottery. Galena is a semiconductor with a small band gap of about 0.4 eV, which found use in early wireless communication systems. It was used as the crystal in crystal radio receivers, in which it was used as a point-contact diode capable of rectifying alternating current to detect the radio signals.
The galena crystal was used with a sharp wire, known as a "cat's whisker" in contact with it. The operation of the radio required that the point of contact on the galena be shifted about to find a part of the crystal that acted as a rectifying diode. Making such wireless receivers was a popular home hobby in Britain and other European countries during the 1930s. Scientists associated with the investigation of the diode effect are Karl Ferdinand Braun and Jagadish Bose. In modern wireless communication systems, galena detectors have been replaced by more reliable semiconductor devices. List of minerals Lead smelter Klein, Cornelis. Manual of Mineralogy. Wiley. Pp. 274–276. ISBN 0-471-80580-7. Case Studies in Environmental Medicine: Lead Toxicity. ToxFAQs: Lead. Mineral Information Institute entry for lead
A crystal or crystalline solid is a solid material whose constituents are arranged in a ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macroscopic single crystals are identifiable by their geometrical shape, consisting of flat faces with specific, characteristic orientations; the scientific study of crystals and crystal formation is known as crystallography. The process of crystal formation via mechanisms of crystal growth is called crystallization or solidification; the word crystal derives from the Ancient Greek word κρύσταλλος, meaning both "ice" and "rock crystal", from κρύος, "icy cold, frost". Examples of large crystals include snowflakes and table salt. Most inorganic solids are not crystals but polycrystals, i.e. many microscopic crystals fused together into a single solid. Examples of polycrystals include most metals, rocks and ice. A third category of solids is amorphous solids, where the atoms have no periodic structure whatsoever.
Examples of amorphous solids include glass and many plastics. Despite the name, lead crystal, crystal glass, related products are not crystals, but rather types of glass, i.e. amorphous solids. Crystals are used in pseudoscientific practices such as crystal therapy, along with gemstones, are sometimes associated with spellwork in Wiccan beliefs and related religious movements; the scientific definition of a "crystal" is based on the microscopic arrangement of atoms inside it, called the crystal structure. A crystal is a solid where the atoms form a periodic arrangement.. Not all solids are crystals. For example, when liquid water starts freezing, the phase change begins with small ice crystals that grow until they fuse, forming a polycrystalline structure. In the final block of ice, each of the small crystals is a true crystal with a periodic arrangement of atoms, but the whole polycrystal does not have a periodic arrangement of atoms, because the periodic pattern is broken at the grain boundaries.
Most macroscopic inorganic solids are polycrystalline, including all metals, ice, etc. Solids that are neither crystalline nor polycrystalline, such as glass, are called amorphous solids called glassy, vitreous, or noncrystalline; these have no periodic order microscopically. There are distinct differences between crystalline solids and amorphous solids: most notably, the process of forming a glass does not release the latent heat of fusion, but forming a crystal does. A crystal structure is characterized by its unit cell, a small imaginary box containing one or more atoms in a specific spatial arrangement; the unit cells are stacked in three-dimensional space to form the crystal. The symmetry of a crystal is constrained by the requirement that the unit cells stack with no gaps. There are 219 possible crystal symmetries, called crystallographic space groups; these are grouped into 7 crystal systems, such as hexagonal crystal system. Crystals are recognized by their shape, consisting of flat faces with sharp angles.
These shape characteristics are not necessary for a crystal—a crystal is scientifically defined by its microscopic atomic arrangement, not its macroscopic shape—but the characteristic macroscopic shape is present and easy to see. Euhedral crystals are those with well-formed flat faces. Anhedral crystals do not because the crystal is one grain in a polycrystalline solid; the flat faces of a euhedral crystal are oriented in a specific way relative to the underlying atomic arrangement of the crystal: they are planes of low Miller index. This occurs; as a crystal grows, new atoms attach to the rougher and less stable parts of the surface, but less to the flat, stable surfaces. Therefore, the flat surfaces tend to grow larger and smoother, until the whole crystal surface consists of these plane surfaces. One of the oldest techniques in the science of crystallography consists of measuring the three-dimensional orientations of the faces of a crystal, using them to infer the underlying crystal symmetry.
A crystal's habit is its visible external shape. This is determined by the crystal structure, the specific crystal chemistry and bonding, the conditions under which the crystal formed. By volume and weight, the largest concentrations of crystals in the Earth are part of its solid bedrock. Crystals found in rocks range in size from a fraction of a millimetre to several centimetres across, although exceptionally large crystals are found; as of 1999, the world's largest known occurring crystal is a crystal of beryl from Malakialina, Madagascar, 18 m long and 3.5 m in diameter, weighing 380,000 kg. Some crystals have formed by magmatic and metamorphic processes, giving origin to large masses of crystalline rock; the vast majority of igneous rocks are formed from molten magma and the degree of crystallization depends on the conditions under which they solidified. Such rocks as granite, which have cooled slowly and under great pressures, have crystallized.
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.