Crystallography, mathematical

The totality of methods for describing the external forms of crystals and their internal spatial structure. Mathematical crystallography is based on the conception that the particles forming the crystal lattice are arranged in an ordered, periodic three-dimensional configuration. Crystals grown under equilibrium conditions have the form of regular convex polyhedra with some sort of symmetry. The symmetry groups are classified according to the dimension $n$ of the space in which they are defined, according to the dimension $m$ of the space in which the object is periodic (the groups are accordingly denoted by $G_m^n$), and according to various other criteria. In order to describe crystals one uses various symmetry groups, of which the most important are the spatial groups $G_3^3$ describing the atomic structure of crystals, and the point symmetry groups $G_0^3$ describing their external form.

The operations of a point symmetry group are the following: rotations through $360^\circ/N$ about an axis of symmetry of order $N$; reflection in a plane of symmetry (a mirror reflection); inversion $T$ (a symmetry about a point); and inversive rotations $\overline N$ (a combination of a rotation through $360^\circ/N$ followed by an inversion). Inversive rotations are sometimes replaced by reflectional rotations $\widetilde N$. The number of groups $G_0^3$ is infinite. In crystals, however, because of the existence of a crystal lattice, the only possible operations are the following. There can only be rotations (and corresponding axes of symmetry) of orders up to 6, excluding 5. These are denoted by 1, 2, 3, 4, 6. There can be also inversion axes I (this is a centre of symmetry) and there can be inversive rotations $\overline2=m$ (this is a plane of symmetry), $\overline3,\overline4,\overline6$. The number of point crystallographic groups describing the external forms of crystals is bounded: 32 groups.

Groups containing only rotations describe crystals that consist exclusively of compatible equal particles. These are known as groups of the first kind. Groups containing reflections, or inversive rotations, describe crystals in which there are mirror-image particles (though there may also be compatible equal particles). These are known as groups of the second kind. Crystals described by groups of the first kind may crystallize in two enantiomorphic forms, arbitrarily named "right" and "left" , each of which does not contain second-kind symmetry elements, though they are mirror images of one another.

Many properties of crystals belonging to certain classes are described by limit point groups, which contain axes of symmetry of an infinite order. The existence of such an axis means that the object coincides with itself under rotation through any angle — including an infinitely small one. There are seven such groups.

The spatial symmetry group (or space group) of a crystal lattice is described by the groups $G_3^3$. The operations characteristic for the lattice are three non-coplanar translations $\mathbf a,\mathbf b,\mathbf c$, which correspond to the three-dimensional periodicity of the atomic structure of crystals.

Thanks to the possibility of combining translations and point symmetry operations in the lattice, the groups $G_3^3$ also contain operations and corresponding symmetry elements with a translation component — screw axes of various orders and sliding planes.

All in all, there are 230 known spatial symmetry groups $G_3^3$ (Fedorov groups), and any crystal belongs to one of these groups. The translation components of the microsymmetry elements do not appear macroscopically; therefore, each of the 230 groups $G_3^3$ is macroscopically similar to one of the 32 point groups. The set of translations in a given space group is a translation subgroup of it, or a Bravais lattice; there exist 14 such lattices. See also Crystallographic group.

#### References

[1] |  A.V. Shubnikov, E.E. Flint, G.B. Bokii, "Fundamentals of crystallography" , Moscow-Leningrad (1940) (In Russian)

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[2] |  E.S. Fedorov, "The symmetry and structure of crystals. Fundamental works" , Moscow (1949) (In Russian)
[3] |  M. Shaskol'skaya, "Crystals" , Moscow (1959) (In Russian)

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From The Encyclopedia of Math: Crystallography, mathematical.

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