Space group in the context of "Crystallographic point group"

In crystallography, a crystal system is a set of point groups (a group of geometric symmetries with at least one fixed point). A lattice system is a set of Bravais lattices (an infinite array of discrete points). Space groups (symmetry groups of a configuration in space) 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.

The seven crystal systems are triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. Informally, two crystals are in the same crystal system if they have similar symmetries (though there are many exceptions).

A sesquioxide is an oxide of an element (or radical), where the ratio between the number of atoms of that element and the number of atoms of oxygen is 2:3. For example, aluminium oxide Al₂O₃ and phosphorus(III) oxide P₄O₆ are sesquioxides.Many sesquioxides contain a metal in the +3 oxidation state and the oxide ion O, e.g., aluminium oxide Al₂O₃, lanthanum(III) oxide La₂O₃ and iron(III) oxide Fe₂O₃. Sesquioxides of iron and aluminium are found in soil. The alkali metal sesquioxides are exceptions because they contain both peroxide O2−2 and superoxide O−2 ions, e.g., rubidium sesquioxide Rb₄O₆ is formulated (Rb)₄(O2−2)(O−2)₂. Sesquioxides of metalloids and nonmetals are better formulated as covalent, e.g. boron trioxide B₂O₃, dinitrogen trioxide N₂O₃ and phosphorus(III) oxide P₄O₆; chlorine trioxide Cl₂O₃ and bromine trioxide Br₂O₃ do not have oxidation state +3 on the halogen.

Many transition metal oxides crystallize in the corundum structure type, with space group R3c. Sesquioxides of rare earth elements crystalize into one or more of three crystal structures: hexagonal (type A, space group P3m1), monoclinic (type B, space group C2/m), or body-centered cubic (type C, space group Ia3).

In crystallography, a centrosymmetric point group contains an inversion center as one of its symmetry elements. In such a point group, for every point (x, y, z) in the unit cell there is an indistinguishable point (-x, -y, -z). Such point groups are also said to have inversion symmetry. Point reflection is a similar term used in geometry.Crystals with an inversion center cannot display certain properties, such as the piezoelectric effect and the frequency doubling effect (second-harmonic generation). In addition, in such crystals, one-photon absorption (OPA) and two-photon absorption (TPA) processes are mutually exclusive, i.e., they do not occur simultaneously, and provide complementary information.

The following space groups have inversion symmetry: the triclinic space group 2, the monoclinic 10-15, the orthorhombic 47-74, the tetragonal 83-88 and 123-142, the trigonal 147, 148 and 162-167, the hexagonal 175, 176 and 191-194, the cubic 200-206 and 221-230.

Forsterite (Mg₂SiO₄; commonly abbreviated as Fo; also known as white olivine) is the magnesium-rich end-member of the olivine solid solution series. It is isomorphous with the iron-rich end-member, fayalite. Forsterite crystallizes in the orthorhombic system (space group Pbnm) with cell parameters a 4.75 Å (0.475 nm), b 10.20 Å (1.020 nm) and c 5.98 Å (0.598 nm).

Forsterite is associated with igneous and metamorphic rocks and has also been found in meteorites. In 2005 it was also found in cometary dust returned by the Stardust probe. In 2011 it was observed as tiny crystals in the dusty clouds of gas around a forming star.

Fayalite (Fe
₂SiO
₄, commonly abbreviated to Fa) is the iron-rich end-member of the olivine solid-solution series. In common with all minerals in the olivine group, fayalite crystallizes in the orthorhombic system (space group Pbnm) with cell parameters a = 4.82 Å, b = 10.48 Å and c = 6.09 Å.

Fayalite forms solid solution series with the magnesium olivine endmember forsterite (Mg₂SiO₄) and also with the manganese rich olivine endmember tephroite (Mn₂SiO₄).