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Topic: Crystallographic point group

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Symmetry group

Cyclic group

Space group

Crystallographic group

Group action

Symmetry

Crystal structure

Monoclinic

Bravais lattice

Orthorhombic

Crystal optics

Crystallography

Crystal

Tetrahedron

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	Crystallographic point group - Wikipedia, the free encyclopedia
		In crystallography, a crystallographic point group is a set of symmetry operations, like rotations or reflections, that leave a point fixed while moving each atom of the crystal to the position of an atom of the same kind.
		This crystallographic restriction of the infinite families of general point groups results in 32 crystallographic point groups.
		The point group of a crystal, among other things, determines some of the crystal's optical properties, such as whether it is birefringent, or whether it shows the Pockels effect.
	en.wikipedia.org /wiki/Crystallographic_point_group (440 words)

	Point groups in three dimensions - Wikipedia, the free encyclopedia
		In geometry a point group in 3D is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere.
		It is a subgroup of the orthogonal group O(3), the group of all isometries which leave the origin fixed, or correspondingly, the group of orthogonal matrices.
		, and is the symmetry group of the cube and octahedron.
	en.wikipedia.org /wiki/Schoenflies_notation (3180 words)

	Point group - Wikipedia, the free encyclopedia
		A point group is a group of geometric symmetries (isometries) leaving a point fixed.
		However, in crystallography, 3D point groups are restricted to be compatible with the discrete translation symmetries of a crystal lattice.
		Point groups in 2D fall into two distinct families, according to whether they consist of rotations only, or include reflections.
	www.wikipedia.org /wiki/Point_group (663 words)

	CONK! Encyclopedia: Crystal_structure (Site not responding. Last check: 2007-11-06)
		The crystal system is the point group of the lattice (the set of rotation and reflection symmetries which leave a lattice point fixed), not including the positions of the atoms in the unit cell.
		The crystallographic point group or crystal class is the set of non-translational symmetries that leave a point in the crystal fixed.
		The space group of the crystal structure is composed of the translational symmetries in addition to the symmetries of the point group.
	www.conk.com /search/encyclopedia.cgi?q=Crystal_structure (553 words)

	Crystal system - Wikipedia, the free encyclopedia
		A crystal system is a category of space groups, which characterize symmetry of structures in three dimensions with translational symmetry in three directions, having a discrete symmetry group.
		The 73 symmorphic space groups (see space group) are largely combinations, within each crystal system, of each applicable point group with each applicable Bravais lattice: there are 2, 6, 12, 14, 5, 7, and 15 combinations, respectively, together 61.
		A symmetry group consists of isometric affine transformations; each is given by an orthogonal matrix and a translation vector (which may be the zero vector).
	en.wikipedia.org /wiki/Crystal_system (582 words)

	Crystal Structures (Site not responding. Last check: 2007-11-06)
		For a complex structure the identification of the symmetry group may be a rather nontrivial task.
		Considering the examples in Figure 1.1, the rotations around P and the mirror line m are point group symmetries, but the combination of mirroring around m' and the subsequent shift is not a point group operation.
		Finally, when the point group symmetries of the Bravais lattices are considered, the choices are further limited, and in three dimensions only seven distinct groups are left.
	solidstate.physics.sunysb.edu /book/prob/node3.html (2041 words)

	Science Fair Projects - Space group
		The space group of a crystal is a mathematical description of the symmetry inherent in the structure.
		The word 'group' in the name comes from the mathematical notion of a group, which is used to build the set of space groups.
		The set of all 230 possible space groups in 3D is made from the combination of the 32 crystallographic point groups with the 14 Bravais lattices which belong to one of 7 crystal systems.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Space_group (732 words)

	Crystallographic Topology 101 - Orbifold 1
		The relations of primary interest among the crystallographic groups are that space groups projected along their primary axes of symmetry become plane groups and that space groups "projected" along the space of all translations (i.e., all translations deleted) become point groups.
		Point groups are simply discrete symmetries about a point, limited crystallographically to the 2-, 3-, 4-, and 6-fold symmetries of cyclic, dihedral, tetrahedral, and octahedral groups.
		The bottom symbol under each orbifold drawing is the international short crystallographic notation for the point group from which the orbifold is derived, with overbars and m's denoting inversion centers and mirrors, respectively, and with 2, 3, 4, and 6 describing the order of rotation axes.
	www.ornl.gov /sci/ortep/topology/orbfld1.html (2624 words)

	Crystallographic point group (Site not responding. Last check: 2007-11-06)
		Crystallographic Studies Information about the research carried out by Luca Jovine on crystallographic studies of RNA and DNA and protein structures, with contact details, request details for a copy of the PhD thesis and links to scientific publications.
		Symmetry The main point group symmetries of interest to defect physics by operation (reflection, rotations etc) and classification (trigonal, and cubic).
		Point to Point Protocol Extensions Working Group (PPPEXT) An IETF group working on PPP's extensions.
	www.serebella.com /encyclopedia/article-Crystallographic_point_group.html (673 words)

	Crystal Systems and Bravais Lattices
		Both are composite symmetry operations that involve first a point symmetry operation (reflection for a glide plane, and rotation for a screw axis) followed by a translation.
		The point group symmetry describes the non-translational symmetry of the crystal.
		For the purposes of determining the crystallographic point group glide planes are treated as mirror planes, and screw axes as rotation axes.
	www.chemistry.ohio-state.edu /~woodward/ch754/sym_3d.htm (1216 words)

	Space group classification: Massimo Nespolo's Research Themes - LCM3B UHP Nancy 1 (Site not responding. Last check: 2007-11-06)
		A space group is characterized not only by the translational and point symmetry, but also by the metric of the lattice (in other words, the cell parameters enter in the definition of space group).
		An arithmetic crystal class is indicated by the symbol of the corresponding crystallographic point group, followed by the symbol of the lattice.
		A geometric crystal class is indicated by the symbol of the corresponding crystallographic point group (but they are NOT the same thing: a geometric crystal class is a set of space groups, and this whole set is in 1:1 correspondence with a crystallographic point group).
	www.lcm3b.uhp-nancy.fr /lcm3b/Pages_Perso/Nespolo/spacegroups.htm (1110 words)

	Crystallographic Topology - Cubic Groups
		The 36 cubic crystallographic space groups are different from the remaining 194 space groups in that they each have body diagonal 3-fold axes arising from their tetrahedral and octahedral point groups.
		The seven rhombohedral trigonal subgroups of the cubic groups are shown in the bottom row of the figure with their space group symbols and simplest lattice complex in the top row of each box.
		Crystallographic color groups and the concept of antisymmetry are discussed in Appendix A. In the bicolor space groups, there are 1191 out of the total of 1651 group members that have mixed symmetry and antisymmetry group elements.
	www.ornl.gov /sci/ortep/topology/cubicgsg.html (1827 words)

	Crystallographic Topology - Appendix A
		There are 1651 groups in the complete bicolor space group family, 230 of which are the regular space groups with fl symmetry operators, another 230 have simultaneously fl and white (gray) symmetry operators, and the remaining 1191 nontrivial bicolor space groups have mixed fl and white operators.
		Bicolor groups are called magnetic groups when used to describe simultaneously the arrangement of atoms (regular symmetry) and the up/down magnetic spin vector orientations (antisymmetry) for magnetic atoms in a crystal (Opechowski and Guccione, 1965).
		Since the group generators for space groups are given explicitly in the crystallographic space group nomenclature, the bicolor space group nomenclature is given by simply placing flags, such as , on the antisymmetry operators in the group* name (see Table A.1).
	www.ornl.gov /sci/ortep/topology/app_a.html (1795 words)

	Cubic (crystal system) - Wikipedia, the free encyclopedia
		For bcc the primitive cells have a volume of 1/2 of the cube, e.g.
		Both scales are "special", allowing a cubic symmetry: for bcc the middle layer has a height of 1/2 of the grid size of the square grid of each layer, while for fcc the middle layer has a height of 1/2 √2 of that grid size.
		The point groups that fall under this crystal system are listed below, followed by their representations in international notation and Schoenflies notation, and mineral examples.
	www.wikipedia.org /wiki/Face-centered_cubic (472 words)

	W0111 (Site not responding. Last check: 2007-11-06)
		Crystallographic Education: The Role of the Point Groups.
		An exposition of the crystallographic point groups is an ideal vehicle for explaining the nature of the crystal systems, the Laue groups and other groupings that are useful to the diffractionist.
		Beyond its pedagogical utility, however, mastery of the point groups and the layout of the point group table, together with an understanding of point-group/space-group isogonality and its consequences, provide the diffractionist with a readily accessible tool of everyday utility in structural crystallography.
	www.hwi.buffalo.edu /ACA/ACA03/abstracts/text/W0111.html (99 words)

	MaThCryst Twinned crystal- International Union of Crystallography, Commission on Mathematical and Theoretical ...
		holohedral and its lattice does not have a specialized metric, then the point groups of the lattice and of the motif coincide: twinning by merohedry is not possible.
		sublattice does not coincide with the point group of the original lattice, and its translation group is a subgroup of the translation group of the original lattice.
		D(L): the point group of the individual lattice (in the case of merohedry, the lattice of the individual and that of the twin coincide by definition)
	www.lcm3b.uhp-nancy.fr /mathcryst/twins.htm (3368 words)

	Crystallography Collection Instructor's Notes
		Point Group I, II, and III have been used by the author in a student-centered instructional setting, where students use the programs and the instructor is available to help as required; and for independent study, with the instructor meeting with the students between units to clarify material as needed.
		Crystallographic CourseWare has been used in a student-centered instructional setting, with students using the programs and the instructor available to help students as required; and for independent study, with the instructor meeting with the students between units to clarify material as needed.
		Although designed for individual or group student use, some units could be projected for use in lecture courses.
	jchemed.chem.wisc.edu /jcesoft/programs/cc/CC_inst.html (354 words)

	Journal of Research of the National Institute of Standards and Technology: Quasicrystals
		Quasiperiodic objects can have any of the infinite set of point group symmetries listed as non-crystallographic in the International Tables for Crystallography (4); because they have a single rotation axis of order 5, or one greater than or equal to 7, or have icosahedral symmetry with its six intersecting 5-fold axes.
		Cubes, octahedra, and tetrahedra, for instance, are examples of special forms belonging to the cubic point groups, octahedra to point groups 432, m3, and m3m, tetrahedra to 23 and 43m, and cubes to all five.
		Magnetic structures and their 1609 Shubnikov space groups are an example of such an extension in which spins, up or down (or two colors), are treated as if in a fourth dimension (8).
	www.findarticles.com /p/articles/mi_m0IKZ/is_6_106/ai_86041893 (1458 words)

	Crystallographic Point-Group Symmetry (Site not responding. Last check: 2007-11-06)
		The crystallographic point-group symmetry associated with each space group is given on the top line of the space-group diagrams.
		The 11 centrosymmetric point groups are shown in
		The symbols for point groups -42m, 32, 3m, -3m, and -62m may be written in alternative forms so as to indicate symmetry with respect to the unit-cell axes, e.g.
	img.cryst.bbk.ac.uk /sgp/MISC/POINTGRP.HTM (111 words)

	CRYSTAL SYSTEM FACTS AND INFORMATION (Site not responding. Last check: 2007-11-06)
		The 73 symmorphic space groups (see space_group) are largely combinations, within each crystal system, of each applicable point group with each applicable Bravais lattice: there are 2, 6, 12, 14, 5, 7, and 15 combinations, respectively, together 61.
		A symmetry_group consists of isometric affine_transformations; each is given by an orthogonal_matrix and a translation vector (which may be the zero vector).
		In geometry and crystallography, a Bravais lattice is a category of symmetry_groups for translational_symmetry in three directions, or correspondingly, a category of translation lattices.
	www.flowergods.com /crystal_system (527 words)

	Crystallographic point group (Site not responding. Last check: 2007-11-06)
		In crystallography, a crystallographic pointgroup or crystal class is a set of symmetry operationsthat leave a point fixed, like rotations or reflections, which leave the crystal unchanged.
		For instance, one knows whether it is birefringent, or whether it shows the Pockelseffect, simply by knowing its point group.
		The letter T (for tetrahedron) indicates that the group has thesymmetry of a tetrahedron.
	www.therfcc.org /crystallographic-point-group-127028.html (271 words)

	Crystal.m - a Mathematica package for drawing crystals shapes (Site not responding. Last check: 2007-11-06)
		It is an amazing fact that all these minerals fit into only 32 crystallographic point groups constituting seven symmetry classes.
		These crystallographic point groups, the groups of rotations and inversions compatible with translational symmetries, mainly determine the macroscopic morphology of mineral crystals.
		Due to the symmetries of the point group of a crystal, one needs to know the indices of only some of the crystal's faces, since the remaining faces are determined by the symmetry operations of the point group.
	www.joerg-enderlein.de /crystal/crystal.html (235 words)

	Crystal structure - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-06)
		A unit cell is a spatial arrangement of atoms which is tiled in three-dimensional space to describe the crystal.
		The positions of the atoms inside the unit cell are described by the asymmetric unit or basis, the set of atomic positions $(x_i, y_i, z_i) measured from a lattice$ point.
		For each crystal structure there is a conventional unit cell, usually chosen to make the resulting lattice as symmetric as possible.
	www.americancanyon.us /project/wikipedia/index.php/Crystal_structure (636 words)

	The MOE Crystal Builder (Site not responding. Last check: 2007-11-06)
		All points in the lattice have identical "environments" --- the view from every point in the lattice is identical to that from any other point in the lattice.
		The absolute positions of the points of a lattice, and hence the unit cell, are arbitrary with respect to a pattern.
		We have extended the expression of planar (2D) groups to 3D molecules by interpreting two-sidedness as the conformation of the asymmetric unit above and below the slice of the plane.
	www.chemcomp.com /feature/crysbld.htm (1784 words)

	Bravais Lattice (Site not responding. Last check: 2007-11-06)
		In geometry and crystallography, a Bravais lattice is an infinite set of points generated by a set of discrete translation operations.
		When classified by space group, there are 14 unique Bravais lattices in three dimensions.
		These can be grouped according to their crystal system, or crystallographic point group.
	www.wikiverse.org /bravais-lattice (130 words)

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