Group of space

The group of space of a Cristal is a mathematical description of the Symétrie of a crystalline Structure. It is about a group with the mathematical direction of the term.

Any group of space results from the combination of a specific Groupe of symmetry and of a network of Faced.

Various notations are used to represent a group of space: the principal ones are the notations of Hermann - Mauguin and of Schoenflies .

The International union of crystallography publishes international Tables of crystallography where each group of space and its operations of symmetry are represented graphically and mathematically.

Principle of determination of the groups of space

The whole of the groups of space results from the combination from a basic unit (or reason) with specific operations of symmetry (reflection, rotation and inversion), to which operations of translation are added, translation in the plan or combined with a reflection or a rotation.

However the number of distinct groups is lower than that of the combinations, some being isomorphous, i.e. leading to the same group of space. This result can be shown mathematically by the Théorie of the groups.

The operations of translation include/understand:

* translation according to the basic vectors of the network, which makes pass from a mesh to the close mesh;

* translations combined with the reflections and rotations:

** axe hélicoïdal: a rotation along an axis, combined with a translation according to the direction of the axis, and whose amplitude is a fraction of the basic vectors. They are noted by a number N describing the degree of rotation, where N is the number of times where rotation must be applied to obtain the identity (3 thus represent for example a rotation of a third of turn, that is to say 2π/3). The degree of translation is then noted by an index which indicates to which fraction of the vector of the network corresponds the translation. For example, 2₁ represents a rotation of a half-turn followed by a translation of a half-vector of the network.

** to miroir translatoire: a reflection followed by a translation parallel with the plan, like defined in the following table:

*Le plane of type E exists only in groups having a centered network (nonprimitive) and the two slips are connected by the vector of translation to fractional components

In a group of space, various elements of the symmetry of the same dimensionality can coexist in parallel orientation. For example, of the axes 2₁ can be parallel to axes 2; mirrors of the type m can be parallel to mirrors of the type has ; etc In the symbol of the group of space, the choice of the representative element follows a set of priorities, which is the following:

the axes without slip take precedence over the helicoid axes;
the priority in the choice of the representative mirror is: m > E > has > B > C > N > D.

However, some exceptions exist (see the International Tables for Crystallography , Volume has, 2002, 4.1.2.3 section). For example, the groups I 222 and I 2₁2₁2₁ contain axes 2₁ parallel with axes 2, but in the first groups three axes 2 have common intersection and the three axes 2₁ also, while in the second group it is not the case. The rule of priority does not apply here, otherwise the two groups would have the same symbol.

230 types of groups of space

All 230 type of groups of space in three dimensions results from the combination of the 32 specific groups of symmetry with the 14 types of Réseaux from Faced.

By Isomorphism, the combinations of a type of network of Faced and of a specific Groupe of symmetry (32*14 = 448) are reduced finally to 230 types of distinct groups of space.

The list of these types of groups of space, classified by class of symmetry, is presented in the article crystalline Système.

See too

specific Group of symmetry

Group of space (4D)

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