Tetragonal crystal system

Encyclopedia

In crystallography

, the

is one of the 7 lattice point group

s. Tetragonal crystal lattices result from stretching a cubic lattice along one of its lattice vectors, so that the cube becomes a rectangular prism

with a square base (

There are two tetragonal Bravais lattices: the simple tetragonal (from stretching the simple-cubic lattice) and the centered tetragonal (from stretching either the face-centered or the body-centered cubic lattice).

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.

Crystallography

Crystallography is the experimental science of the arrangement of atoms in 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 grapho = write.Before the development of...

, the

**tetragonal**crystal systemCrystal system

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...

is one of the 7 lattice point group

Point group

In geometry, a point group is a group of geometric symmetries that keep at least one point fixed. Point groups can exist in a Euclidean space with any dimension, and every point group in dimension d is a subgroup of the orthogonal group O...

s. Tetragonal crystal lattices result from stretching a cubic lattice along one of its lattice vectors, so that the cube becomes a rectangular prism

Prism (geometry)

In geometry, a prism is a polyhedron with an n-sided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a...

with a square base (

*a*by*a*) and height (*c*, which is different from*a*).There are two tetragonal Bravais lattices: the simple tetragonal (from stretching the simple-cubic lattice) and the centered tetragonal (from stretching either the face-centered or the body-centered cubic lattice).

Primitive | Body-centered |
---|---|

The point groups

Crystallographic point group

In crystallography, a crystallographic point group is a set of symmetry operations, like rotations or reflections, that leave a central point fixed while moving other directions and faces of the crystal to the positions of features of the same kind...

that fall under this crystal system are listed below, followed by their representations in international notation and Schoenflies notation

Schoenflies notation

The Schoenflies notation or Schönflies notation, named after the German mathematician Arthur Moritz Schoenflies, is one of two conventions commonly used to describe Point groups. This notation is used in spectroscopy. The other convention is the Hermann–Mauguin notation, also known as the...

, and mineral

Mineral

A mineral is a naturally occurring solid chemical substance formed through biogeochemical processes, having characteristic chemical composition, highly ordered atomic structure, and specific physical properties. By comparison, a rock is an aggregate of minerals and/or mineraloids and does not...

examples.

# | Point group | Example | Space group Space group In mathematics and geometry, a space group is a symmetry group, usually for three dimensions, that divides space into discrete repeatable domains.In three dimensions, there are 219 unique types, or counted as 230 if chiral copies are considered distinct... s |
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Class | Intl Hermann-Mauguin notation 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... |
Schoenflies Schoenflies notation The Schoenflies notation or Schönflies notation, named after the German mathematician Arthur Moritz Schoenflies, is one of two conventions commonly used to describe Point groups. This notation is used in spectroscopy. The other convention is the Hermann–Mauguin notation, also known as the... |
Orbifold Orbifold In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold is a generalization of a manifold... |
Coxeter Coxeter notation In geometry, Coxeter notation is a system of classifying symmetry groups, describing the angles between with fundamental reflections of a Coxeter group. It uses a bracketed notation, with modifiers to indicate certain subgroups. The notation is named after H. S. M... |
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75-80 | Tetragonal pyramidal | 4 | C_{4} |
44 | [4]^{+} |
pinnoite, richellite | P4, P4_{1}, P4_{2}, P4_{3}, I4, I4_{1} |

81-82 | Tetragonal disphenoidal | S_{4} |
2x | [2^{+},4^{+}] |
cahnite Cahnite Cahnite is a brittle white or colorless mineral that has perfect cleavage and is usually transparent. It usually forms tetragonal-shaped crystals and it has a hardness of 3 mohs. Cahnite was discovered in the year 1921. It was named Cahnite to honor Lazard Cahn, who was a mineral collector and... , tugtupite Tugtupite Tugtupite is a rare beryllium aluminium tectosilicate. It also contains sodium and chlorine and has the formula Na4AlBeSi4O12Cl. Tugtupite is a member of the silica deficient feldspathoid mineral group... |
P, I | |

83-88 | Tetragonal dipyramidal | 4/m | C_{4h} |
4* | [2,4^{+}] |
scheelite Scheelite Scheelite is a calcium tungstate mineral with the chemical formula CaWO4. It is an important ore of tungsten. Well-formed crystals are sought by collectors and are occasionally fashioned into gemstones when suitably free of flaws... , wulfenite Wulfenite Wulfenite is a lead molybdate mineral with the formula PbMoO4. It can be most often found as thin tabular crystals with a bright orange-red to yellow-orange color, sometimes brown, although the color can be highly variable. In its yellow form it is sometimes called "yellow lead ore".It crystallizes... , leucite Leucite Leucite is a rock-forming mineral composed of potassium and aluminium tectosilicate K[AlSi2O6]. Crystals have the form of cubic icositetrahedra but, as first observed by Sir David Brewster in 1821, they are not optically isotropic, and are therefore pseudo-cubic. Goniometric measurements made by... |
P4/m, P4_{2}/m, P4/n, P4_{2}/n, I4/m, I4_{1}/a |

89-98 | Tetragonal trapezohedral | 422 | D_{4} |
224 | [2,4]^{+} |
cristobalite Cristobalite The mineral cristobalite is a high-temperature polymorph of silica, meaning that it has the same chemical formula, SiO2, but a distinct crystal structure. Both quartz and cristobalite are polymorphs with all the members of the quartz group which also include coesite, tridymite and stishovite... , wardite Wardite Wardite is a hydrous sodium aluminium phosphate hydroxide mineral with formula: NaAl324·2. Wardite is of interest for its rare crystallography. It crystallizes in the tetragonal trapezohedral class and is one of only a few minerals in that class. Wardite forms vitreous green to bluish green to... |
P422, P42_{1}2, P4_{1}22, P4_{1}2_{1}2, P4_{2}22, P4_{2}2_{1}2, P4_{3}22, P4_{3}2_{1}2, I422, I4_{1}22 |

99-110 | Ditetragonal pyramidal | 4mm | C_{4v} |
*44 | [4] | diaboleite | P4mm, P4bm, P4_{2}cm, P4_{2}nm, P4cc, P4nc, P4_{2}mc, P4_{2}bc, I4mm, I4cm, I4_{1}md, I4_{1}cd |

111-122 | Tetragonal scalenohedral | 2m | D_{2d} |
2*2 | [2^{+},4] |
chalcopyrite Chalcopyrite Chalcopyrite is a copper iron sulfide mineral that crystallizes in the tetragonal system. It has the chemical composition CuFeS2. It has a brassy to golden yellow color and a hardness of 3.5 to 4 on the Mohs scale. Its streak is diagnostic as green tinged black.On exposure to air, chalcopyrite... , stannite Stannite Stannite is a mineral, a sulfide of copper, iron, and tin. The chemical formula Cu2FeSnS4. Zinc commonly occurs with the iron and trace germanium may be present. Stannite is used as an ore of tin, consisting of approximately 28% tin, 13% iron, 30% copper, 30% sulfur by mass... |
P2m, P2c, P2_{1}m, P2_{1}c, Pm2, Pc2, Pb2, Pn2, Im2, Ic2, I2m, I2d |

123-142 | Ditetragonal dipyramidal | 4/mmm | D_{4h} |
*224 | [2,4] | rutile Rutile Rutile is a mineral composed primarily of titanium dioxide, TiO2.Rutile is the most common natural form of TiO2. Two rarer polymorphs of TiO2 are known:... , pyrolusite Pyrolusite Pyrolusite is a mineral consisting essentially of manganese dioxide and is important as an ore of manganese. It is a black, amorphous appearing mineral, often with a granular, fibrous or columnar structure, sometimes forming reniform crusts. It has a metallic luster, a black or bluish-black... , zircon Zircon Zircon is a mineral belonging to the group of nesosilicates. Its chemical name is zirconium silicate and its corresponding chemical formula is ZrSiO4. A common empirical formula showing some of the range of substitution in zircon is 1–x4x–y... |
P4/mmm, P4/mcc, P4/nbm, P4/nnc, P4/mbm, P4/mnc, P4/nmm, P4/ncc, P4_{2}/mmc, P4_{2}/mcm, P4_{2}/nbc, P4_{2}/nnm, P4_{2}/mbc, P4_{2}/mnm, P4_{2}/nmc, P4_{2}/ncm, I4/mmm, I4/mcm, I4_{1}/amd, I4_{1}/acd |