1.
Albite
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Albite is a plagioclase feldspar mineral. It is the sodium endmember of the solid solution series. As such it represents a plagioclase with less than 10% anorthite content, the pure albite endmember has the formula NaAlSi3O8. Its color is pure white, hence its name from Latin albus. It is a constituent in felsic rocks. Albite crystallizes with triclinic pinacoidal forms and its specific gravity is about 2.62 and it has a Mohs hardness of 6 -6.5. Albite almost always exhibits crystal twinning often as minute parallel striations on the crystal face, Albite often occurs as fine parallel segregations alternating with pink microcline in perthite as a result of exolution on cooling. There are two variants of albite, which are referred to as low albite and high albite, the latter is known as analbite. Although both variants are triclinic, they differ in the volume of their cell, which is slightly larger for the high form. The high form can be produced from the low form by heating above c.750 °C, upon further heating to more than c.1050 °C the crystal symmetry changes from triclinic to monoclinic, this variant is also known as monalbite. It occurs in granitic and pegmatite masses, in hydrothermal vein deposits. It was first reported in 1815 for an occurrence in Finnbo, Falun, Dalarna and it is used as a gemstone. Mineral galleries This article incorporates text from a now in the public domain, Chisholm, Hugh
2.
Basal plane
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In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystalline material. Ordered structures occur from the nature of the constituent particles to form symmetric patterns that repeat along the principal directions of three-dimensional space in matter. The smallest group of particles in the material that constitutes the pattern is the unit cell of the structure. The unit cell completely defines the symmetry and structure of the crystal lattice. The repeating patterns are said to be located at the points of the Bravais lattice, the lengths of the principal axes, or edges, of the unit cell and the angles between them are the lattice constants, also called lattice parameters. The symmetry properties of the crystal are described by the concept of space groups, all possible symmetric arrangements of particles in three-dimensional space may be described by the 230 space groups. The crystal structure and symmetry play a role in determining many physical properties, such as cleavage, electronic band structure. The crystal structure of a material can be described in terms of its unit cell, the unit cell is a box containing one or more atoms arranged in three dimensions. The unit cells stacked in three-dimensional space describe the arrangement of atoms of the crystal. Commonly, atomic positions are represented in terms of fractional coordinates, the atom positions within the unit cell can be calculated through application of symmetry operations to the asymmetric unit. The asymmetric unit refers to the smallest possible occupation of space within the unit cell and this does not, however imply that the entirety of the asymmetric unit must lie within the boundaries of the unit cell. Symmetric transformations of atom positions are calculated from the group of the crystal structure. Vectors and planes in a lattice are described by the three-value Miller index notation. It uses the indices ℓ, m, and n as directional parameters, which are separated by 90°, by definition, the syntax denotes a plane that intercepts the three points a1/ℓ, a2/m, and a3/n, or some multiple thereof. That is, the Miller indices are proportional to the inverses of the intercepts of the plane with the unit cell, if one or more of the indices is zero, it means that the planes do not intersect that axis. A plane containing a coordinate axis is translated so that it no longer contains that axis before its Miller indices are determined, the Miller indices for a plane are integers with no common factors. Negative indices are indicated with horizontal bars, as in, in an orthogonal coordinate system for a cubic cell, the Miller indices of a plane are the Cartesian components of a vector normal to the plane. Likewise, the planes are geometric planes linking nodes
3.
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
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 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
6.
X-ray crystallography
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By measuring the angles and intensities of these diffracted beams, a crystallographer can produce a three-dimensional picture of the density of electrons within the crystal. From this electron density, the positions of the atoms in the crystal can be determined, as well as their chemical bonds, their disorder. The method also revealed the structure and function of biological molecules, including vitamins, drugs, proteins. X-ray crystallography is still the method for characterizing the atomic structure of new materials. In a single-crystal X-ray diffraction measurement, a crystal is mounted on a goniometer, the goniometer is used to position the crystal at selected orientations. The crystal is illuminated with a finely focused monochromatic beam of X-rays, poor resolution or even errors may result if the crystals are too small, or not uniform enough in their internal makeup. X-ray crystallography is related to other methods for determining atomic structures. Similar diffraction patterns can be produced by scattering electrons or neutrons, for all above mentioned X-ray diffraction methods, the scattering is elastic, the scattered X-rays have the same wavelength as the incoming X-ray. By contrast, inelastic X-ray scattering methods are useful in studying excitations of the sample, Crystals, though long admired for their regularity and symmetry, were not investigated scientifically until the 17th century. Johannes Kepler hypothesized in his work Strena seu de Nive Sexangula that the symmetry of snowflake crystals was due to a regular packing of spherical water particles. The Danish scientist Nicolas Steno pioneered experimental investigations of crystal symmetry, hence, William Hallowes Miller in 1839 was able to give each face a unique label of three small integers, the Miller indices which remain in use today for identifying crystal faces. In the 19th century, a catalog of the possible symmetries of a crystal was worked out by Johan Hessel, Auguste Bravais, Evgraf Fedorov, Arthur Schönflies. Wilhelm Röntgen discovered X-rays in 1895, just as the studies of crystal symmetry were being concluded, physicists were initially uncertain of the nature of X-rays, but soon suspected that they were waves of electromagnetic radiation, in other words, another form of light. Single-slit experiments in the laboratory of Arnold Sommerfeld suggested that X-rays had a wavelength of about 1 angstrom, however, X-rays are composed of photons, and thus are not only waves of electromagnetic radiation but also exhibit particle-like properties. Albert Einstein introduced the concept in 1905, but it was not broadly accepted until 1922. Therefore, these properties of X-rays, such as their ionization of gases. Nevertheless, Braggs view was not broadly accepted and the observation of X-ray diffraction by Max von Laue in 1912 confirmed for most scientists that X-rays were a form of electromagnetic radiation, Crystals are regular arrays of atoms, and X-rays can be considered waves of electromagnetic radiation. Atoms scatter X-ray waves, primarily through the atoms electrons and this phenomenon is known as elastic scattering, and the electron is known as the scatterer
7.
X-ray scattering techniques
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These techniques are based on observing the scattered intensity of an X-ray beam hitting a sample as a function of incident and scattered angle, polarization, and wavelength or energy. However, both scattering and diffraction are related general phenomena and the distinction has not always existed, thus Guiniers classic text from 1963 is titled X-ray diffraction in Crystals, Imperfect Crystals and Amorphous Bodies so diffraction was clearly not restricted to crystals at that time. X-ray reflectivity is a technique for determining thickness, roughness. Wide-angle X-ray scattering, a technique concentrating on scattering angles 2θ larger than 5°, in IXS the energy and angle of inelastically scattered X-rays are monitored, giving the dynamic structure factor S. From this many properties of materials can be obtained, the specific property depending on the scale of the energy transfer, the table below, listing techniques, is adapted from. Inelastically scattered X-rays have intermediate phases and so in principle are not useful for X-ray crystallography
8.
Octahedron
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In geometry, an octahedron is a polyhedron with eight faces, twelve edges, and six vertices. A regular octahedron is a Platonic solid composed of eight equilateral triangles, a regular octahedron is the dual polyhedron of a cube. It is a square bipyramid in any of three orthogonal orientations and it is also a triangular antiprism in any of four orientations. An octahedron is the case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan metric, the second and third correspond to the B2 and A2 Coxeter planes. The octahedron can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. An octahedron with edge length √2 can be placed with its center at the origin and its vertices on the coordinate axes, the Cartesian coordinates of the vertices are then. In an x–y–z Cartesian coordinate system, the octahedron with center coordinates, additionally the inertia tensor of the stretched octahedron is I =. These reduce to the equations for the regular octahedron when x m = y m = z m = a 22, the interior of the compound of two dual tetrahedra is an octahedron, and this compound, called the stella octangula, is its first and only stellation. Correspondingly, an octahedron is the result of cutting off from a regular tetrahedron. One can also divide the edges of an octahedron in the ratio of the mean to define the vertices of an icosahedron. There are five octahedra that define any given icosahedron in this fashion, octahedra and tetrahedra can be alternated to form a vertex, edge, and face-uniform tessellation of space, called the octet truss by Buckminster Fuller. This is the only such tiling save the regular tessellation of cubes, another is a tessellation of octahedra and cuboctahedra. The octahedron is unique among the Platonic solids in having a number of faces meeting at each vertex. Consequently, it is the member of that group to possess mirror planes that do not pass through any of the faces. Using the standard nomenclature for Johnson solids, an octahedron would be called a square bipyramid, truncation of two opposite vertices results in a square bifrustum. The octahedron is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices and it is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size
9.
Dodecahedron
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In geometry, a dodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the dodecahedron, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form, all of these have icosahedral symmetry, order 120. The pyritohedron is a pentagonal dodecahedron, having the same topology as the regular one. The rhombic dodecahedron, seen as a case of the pyritohedron has octahedral symmetry. The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra are space-filling, there are a large number of other dodecahedra. The convex regular dodecahedron is one of the five regular Platonic solids, the dual polyhedron is the regular icosahedron, having five equilateral triangles around each vertex. Like the regular dodecahedron, it has twelve pentagonal faces. However, the pentagons are not constrained to be regular, and its 30 edges are divided into two sets – containing 24 and 6 edges of the same length. The only axes of symmetry are three mutually perpendicular twofold axes and four threefold axes. Note that the regular dodecahedron can occur as a shape for quasicrystals with icosahedral symmetry. Its name comes from one of the two common crystal habits shown by pyrite, the one being the cube. The coordinates of the eight vertices of the cube are, The coordinates of the 12 vertices of the cross-edges are. When h =1, the six cross-edges degenerate to points, when h =0, the cross-edges are absorbed in the facets of the cube, and the pyritohedron reduces to a cube. When h = √5 − 1/2, the inverse of the golden ratio, a reflected pyritohedron is made by swapping the nonzero coordinates above. The two pyritohedra can be superimposed to give the compound of two dodecahedra as seen in the image here, the regular dodecahedron represents a special intermediate case where all edges and angles are equal. A tetartoid is a dodecahedron with chiral tetrahedral symmetry, like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular and the figure has no fivefold symmetry axes, although regular dodecahedra do not exist in crystals, the tetartoid form does
10.
Calcite
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Calcite is a carbonate mineral and the most stable polymorph of calcium carbonate. The Mohs scale of hardness, based on scratch hardness comparison. Other polymorphs of calcium carbonate are the minerals aragonite and vaterite, aragonite will change to calcite at 380–470 °C, and vaterite is even less stable. Calcite is derived from the German Calcit, a term coined in the 19th century from the Latin word for lime and it is thus etymologically related to chalk. Calcite crystals are trigonal-rhombohedral, though actual calcite rhombohedra are rare as natural crystals, however, they show a remarkable variety of habits including acute to obtuse rhombohedra, tabular forms, prisms, or various scalenohedra. Calcite exhibits several twinning types adding to the variety of observed forms and it may occur as fibrous, granular, lamellar, or compact. Cleavage is usually in three directions parallel to the rhombohedron form and its fracture is conchoidal, but difficult to obtain. It has a defining Mohs hardness of 3, a gravity of 2.71. Color is white or none, though shades of gray, red, orange, yellow, green, blue, violet, brown, calcite is transparent to opaque and may occasionally show phosphorescence or fluorescence. A transparent variety called Iceland spar is used for optical purposes, acute scalenohedral crystals are sometimes referred to as dogtooth spar while the rhombohedral form is sometimes referred to as nailhead spar. Single calcite crystals display an optical property called birefringence and this strong birefringence causes objects viewed through a clear piece of calcite to appear doubled. The birefringent effect was first described by the Danish scientist Rasmus Bartholin in 1669, at a wavelength of ~590 nm calcite has ordinary and extraordinary refractive indices of 1.658 and 1.486, respectively. Between 190 and 1700 nm, the refractive index varies roughly between 1.9 and 1.5, while the extraordinary refractive index varies between 1.6 and 1.4. Calcite, like most carbonates, will dissolve with most forms of acid, calcite can be either dissolved by groundwater or precipitated by groundwater, depending on several factors including the water temperature, pH, and dissolved ion concentrations. Although calcite is fairly insoluble in water, acidity can cause dissolution of calcite. Ambient carbon dioxide, due to its acidity, has a slight solubilizing effect on calcite, calcite exhibits an unusual characteristic called retrograde solubility in which it becomes less soluble in water as the temperature increases. When conditions are right for precipitation, calcite forms mineral coatings that cement the existing rock grains together or it can fill fractures. On a landscape scale, continued dissolution of calcium carbonate-rich rocks can lead to the expansion and eventual collapse of cave systems, high-grade optical calcite was used in World War II for gun sights, specifically in bomb sights and anti-aircraft weaponry
Spencer, Leonard James (1911). "Crystallography". Encyclopædia Britannica. 7 (11th ed.). pp. 569–591. 
This article incorporates text from this source, which is in the public domain.
Spencer, Leonard James (1911). "
This article incorporates text from this source, which is in the