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Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics. The word "crystallography" is derived from the Greek words κρύσταλλος (krystallos) "clear ice, rock-crystal", with its meaning extending to all solids with some degree of transparency, and γράφειν (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.
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 intrinsic nature of the constituent particles to form symmetric patterns that repeat along the principal directions of three-dimensional space in matter.
In geometry, biology, mineralogy, and solid state physics, a unit cell is a repeating unit formed by the vectors spanning the points of a lattice. Despite its suggestive name, the unit cell does not necessarily have unit size, or even a particular size at all. Rather, the primitive cell is the closest analogy to a unit vector, since it has a determined size for a given lattice and is the basic building block from which larger cells are constructed.
In mathematics, a frieze or frieze pattern is a two-dimensional design that repeats in one direction. Such patterns occur frequently in architecture and decorative art. Frieze patterns can be classified into seven types according to their symmetries. The set of symmetries of a frieze pattern is called a frieze group.
A wallpaper is a mathematical object we imagine covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. To a given wallpaper there corresponds a group of such congruent transformations, with function composition as the group operation. Thus, a wallpaper group is in a mathematical classification of a two‑dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art, especially in textiles, tessellations and tiles as well as wallpaper.
In crystallography, a crystal system is a set of point groups. A lattice system is a set of Bravais lattices. Space groups are classified into crystal systems according to their point groups, and into lattice systems according to their Bravais lattices. Crystal systems that have space groups assigned to a common lattice system are combined into a crystal family.
In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group are the rigid transformations of an object that leave it unchanged. In three dimensions, space groups are classified into 219 distinct types, or 230 types if chiral copies are considered distinct. Space groups are discrete cocompact groups of isometries of an oriented Euclidean space in any number of dimensions. In dimensions other than 3, they are sometimes called Bieberbach groups.
In crystallography, the monoclinic crystal system is one of the seven crystal systems. A crystal system is described by three vectors. In the monoclinic system, the crystal is described by vectors of unequal lengths, as in the orthorhombic system. They form a rectangular prism with a parallelogram as its base. Hence two pairs of vectors are perpendicular, while the third pair makes an angle other than 90°.
In crystallography, the orthorhombic crystal system is one of the 7 crystal systems. Orthorhombic lattices result from stretching a cubic lattice along two of its orthogonal pairs by two different factors, resulting in a rectangular prism with a rectangular base and height (c), such that a, b, and c are distinct. All three bases intersect at 90° angles, so the three lattice vectors remain mutually orthogonal.
In geometry and crystallography, a Bravais lattice, named after Auguste Bravais (1850), is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by
In crystallography, a crystallographic point group is a set of symmetry operations, corresponding to one of the point groups in three dimensions, such that each operation would leave the structure of a crystal unchanged i.e. the same kinds of atoms would be placed in similar positions as before the transformation. For example, in many crystals in the cubic crystal system, a rotation of the unit cell by 90 degree around an axis that is perpendicular to one of the faces of the cube is a symmetry operation that moves each atom to the location of another atom of the same kind, leaving the overall structure of the crystal unaffected.
In geometry, a point group is a mathematical group of symmetry operations that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every point group in dimension d is then a subgroup of the orthogonal group O(d). Point groups are used to describe the symmetries of geometric figures and physical objects such as molecules.
In geometry, Hermann–Mauguin notation is used to represent the symmetry elements in point groups, plane groups and space groups. It is named after the German crystallographer Carl Hermann and the French mineralogist Charles-Victor Mauguin. This notation is sometimes called international notation, because it was adopted as standard by the International Tables For Crystallography since their first edition in 1935.
A crystallographic database is a database specifically designed to store information about the structure of molecules and crystals. Crystals are solids having, in all three dimensions of space, a regularly repeating arrangement of atoms, ions, or molecules. They are characterized by symmetry, morphology, and directionally dependent physical properties. A crystal structure describes the arrangement of atoms, ions, or molecules in a crystal.
Crystallographic image processing (CIP) is traditionally understood as being a set of key steps in the determination of the atomic structure of crystalline matter from high-resolution electron microscopy (HREM) images obtained in a transmission electron microscope (TEM) that is run in the parallel illumination mode. The term was created in the research group of Sven Hovmöller at Stockholm University during the early 1980s and became rapidly a label for the "3D crystal structure from 2D transmission/projection images" approach. Since the late 1990s, analogous and complementary image processing techniques that are directed towards the achieving of goals with are either complementary or entirely beyond the scope of the original inception of CIP have been developed independently by members of the computational symmetry/geometry, scanning transmission electron microscopy, scanning probe microscopy communities, and applied crystallography communities.
A line group is a mathematical way of describing symmetries associated with moving along a line. These symmetries include repeating along that line, making that line a one-dimensional lattice. However, line groups may have more than one dimension, and they may involve those dimensions in its isometries or symmetry transformations.
In mathematics, a layer group is a three-dimensional extension of a wallpaper group, with reflections in the third dimension. It is a space group with a two-dimensional lattice, meaning that it is symmetric over repeats in the two lattice directions. The symmetry group at each lattice point is an axial crystallographic point group with the main axis being perpendicular to the lattice plane.
Bilbao Crystallographic Server is an open access website offering online crystallographic database and programs aimed at analyzing, calculating and visualizing problems of structural and mathematical crystallography, solid state physics and structural chemistry. Initiated in 1997 by the Materials Laboratory of the Department of Condensed Matter Physics at the University of the Basque Country, Bilbao, Spain, the Bilbao Crystallographic Server is developed and maintained by academics.
In solid state physics, the magnetic space groups, or Shubnikov groups, are the symmetry groups which classify the symmetries of a crystal both in space, and in a two-valued property such as electron spin. To represent such a property, each lattice point is colored black or white, and in addition to the usual three-dimensional symmetry operations, there is a so-called "antisymmetry" operation which turns all black lattice points white and all white lattice points black. Thus, the magnetic space groups serve as an extension to the crystallographic space groups which describe spatial symmetry alone.
References
- Hitzer, E.S.M.; Ichikawa, D. (2008), "Representation of crystallographic subperiodic groups by geometric algebra" (PDF), Electronic Proc. Of AGACSE, Leipzig, Germany (3, 17–19 Aug. 2008), archived from the original (PDF) on 2012-03-14
- Kopsky, V.; Litvin, D.B., eds. (2002), International Tables for Crystallography, Volume E: Subperiodic groups, International Tables for Crystallography, vol. E (5th ed.), Berlin, New York: Springer-Verlag, doi:10.1107/97809553602060000105, ISBN 978-1-4020-0715-6