Space group

Mirror symmetry in the crystal structure of ice

A crystallographic space group or short space group mathematically describes the symmetry of the arrangement of atoms, ions and molecules in a crystal structure . The term "group" comes from group theory .

For example, a component (such as a sulfate ion) of the structure can be obtained by mirroring or rotating another component (in this case another sulfate ion). To describe the complete crystal structure, only the description of the first ion is then necessary, the second ion is obtained by the symmetry operation of mirroring or rotation. The illustration shows this using the example of the crystal structure of ice. The right six-ring is the mirror image of the left six-ring; the space group reproduces (among others) this property of symmetry. The symbols that are used for this are described in detail under Hermann Mauguin symbols .

The space group is a discrete subgroup of the Euclidean motion group of an Euclidean (affine) space with a limited fundamental domain . The space groups belong to the symmetry groups and are usually described using the Hermann Mauguin symbolism or sometimes also in the Schoenflies symbolism .

While the crystallographic point groups are made up of non-translative symmetry operations (e.g. rotations or reflections), when determining the different spatial groups, this requirement is relaxed in favor of translative symmetry operations (this results in e.g. sliding mirror planes and screw axes ) and the lattice translations . This results in a large number of new symmetry groups, the space groups.

Mathematical definition

The isometric group of -dimensional Euclidean space is the group ${\ displaystyle \ operatorname {isom} (\ mathbb {R} ^ {n})}$ ${\ displaystyle n}$ ${\ displaystyle \ mathbb {R} ^ {n}}$

{\ displaystyle \ operatorname {Isom} (\ mathbb {R} ^ {n}) = O (n) \ ltimes \ mathbb {R} ^ {n}}

,

where the orthogonal group consists of reflections and rotations around the zero point and is understood as the group of the displacements of the . ${\ displaystyle O (n)}$ ${\ displaystyle \ mathbb {R} ^ {n}}$ ${\ displaystyle \ mathbb {R} ^ {n}}$

A crystallographic group of rank ${\ displaystyle n}$ is a discrete and co-compact subgroup of . (A subgroup is called discrete if there is no sequence with and . It is called cocompact if the quotient space is compact .) ${\ displaystyle \ operatorname {isom} (\ mathbb {R} ^ {n})}$ ${\ displaystyle \ Gamma \ subset \ operatorname {isom} (\ mathbb {R} ^ {n})}$ ${\ displaystyle \ gamma \ in \ Gamma}$ ${\ displaystyle (\ gamma _ {n}) _ {n} \ subset \ Gamma}$ ${\ displaystyle \ gamma _ {n} \ not = \ gamma}$ ${\ displaystyle \ lim _ {n \ rightarrow \ infty} \ gamma _ {n} = \ gamma}$ ${\ displaystyle \ Gamma \ backslash \ mathbb {R} ^ {n}}$

A Bieberbach group is a torsion-free crystallographic group. (A group with a neutral element is called torsion-free if it always follows from and .) ${\ displaystyle \ Gamma}$ ${\ displaystyle e}$ ${\ displaystyle \ gamma \ not = e}$ ${\ displaystyle n \ not = 0}$ ${\ displaystyle \ gamma ^ {n} \ not = e}$

Number of possible room groups

Number of room groups (without taking the room orientation into account)
dimension
1	2	3	4th	5	6th
2	17th	219	4,783	222.018	28,927,915

The number of possible room groups depends on the dimension and orientation of the room under consideration. In three-dimensional space , crystallographic space groups describe the symmetries of an infinitely extended crystal . Symmetry operations in a crystal are (apart from the identity operation, which maps each point onto itself) point reflection, reflection on a plane, rotation around an axis, displacement (the so-called translation ) and combinations of these operations. If one understands the execution of symmetry operations one after the other as a multiplicative operation, one recognizes that a set of symmetry operations is a group (usually not commutative ).

The determination of the 230 possible room groups (or room group types ) in three dimensions was carried out in 1891 independently of one another in laborious sorting work by Arthur Moritz Schoenflies and Evgraf Stepanowitsch Fjodorow . William Barlow also managed this independently, although he did not publish it until 1894. The 230 space groups (and the crystals, which have the symmetry elements of one of these space groups) can u. a. can be divided into the seven crystal systems , the 14 Bravais lattices and the 32 crystal classes.

	Bravais lattice - basic objects with spherical symmetry	Crystal structure - basic objects with any symmetry
Number of point groups	7 crystal systems	32 crystallographic point groups
Number of room groups	14 Bravais grid	230 room groups

If the orientation of the room is not taken into account , the number is reduced to 219 different room groups. This results in the existence of eleven pairs of enantiomorphic space groups. In these pairs, the arrangements of the symmetry elements such as image and mirror image differ, which cannot be converted into one another by rotation.

An algebraic method for classifying the space groups (also in higher dimensions) comes from Johann Jakob Burckhardt in the 1930s, who also dealt with the history of the problem.

designation

The designation of the room groups is usually done in the Hermann Mauguin symbolism , in some departments the Schoenflies symbolism is still used today as an alternative. In the Hermann Mauguin symbolism, the space group symbol consists of a capital letter, which indicates the type of Bravais, as well as a sequence of symbols (numbers and small letters which indicate the presence of further symmetry elements), which are based closely on the symbolism for point groups , but also takes into account that combined symmetry operations from translation and rotation or mirroring can also exist.

A complete list of the 230 three-dimensional room groups can be found in the list of room groups .

literature

Johann Jakob Burckhardt : The movement groups of crystallography. 2nd edition, Springer, 1966, ISBN 978-3-0348-6931-7 .
John Horton Conway , Olaf Delgado Friedrichs, Daniel Huson, William Thurston : On three dimensional space groups. In: Contributions to Algebra and Geometry. 42, 2001, pp. 475-507. ( Online ).
Hans Zassenhaus : About an algorithm for determining the room groups. In: Comm. Math. Helveticae. 21, 1948, pp. 117-141. ( Online ).
Harold Brown, J. Neubüser, Hans Wondratschek , R. Bülow, Hans Zassenhaus : Crystallographic groups of four-dimensional space. Wiley 1978, ISBN 978-0-471-03095-9 .
Joachim Neubüser , Hans Wondratschek, Rolf Bülow: On crystallography in higher dimensions. (Part 1-3) In: Acta Crystallographica A. Volume 27, 1971, pp. 517-535 (especially 4 dimensions).
- J. Neubüser, H. Wondratschek, R. Bülow: On crystallography in higher dimensions. I. General definitions . In: Acta Crystallographica Section A . tape 27 , no. 6 , November 1, 1971, p. 517-520 , doi : 10.1107 / S0567739471001165 .
- R. Bülow, J. Neubüser, H. Wondratschek: On crystallography in higher dimensions. II. Procedure of computation in R 4 . In: Acta Crystallographica Section A . tape 27 , no. 6 , November 1, 1971, p. 520-523 , doi : 10.1107 / S0567739471001177 .
- H. Wondratschek, R. Bülow, J. Neubüser: On crystallography in higher dimensions. III. Results in R 4 . In: Acta Crystallographica Section A . tape 27 , no. 6 , November 1, 1971, p. 523-535 , doi : 10.1107 / S0567739471001189 .
Harold Brown: An algorithm for the determination of space groups. In: Mathematics of Computation. Volume 23, 1969, pp. 499-514. ( PDF; 1.25 MB ).

Web links

Interactive illustration of the 17 room groups on the level:
- Drawing ornaments , Java applet and application. Retains drawn lines when changing groups.
- Escher Web Sketch , Java Applet. In addition to freehand drawing, it also allows the use of individual other objects.

Individual evidence

^ ^A ^b ^c ^d Will Kleber , Hans-Joachim Bautsch , Joachim Bohm : Introduction to Crystallography . Oldenbourg Wissenschaftsverlag, Munich 2010, ISBN 978-3-486-59075-3 , p. 101 ff . ( limited preview in Google Book search).

[Kleber-1] A ^b ^c ^d Will Kleber , Hans-Joachim Bautsch , Joachim Bohm : Introduction to Crystallography . Oldenbourg Wissenschaftsverlag, Munich 2010, ISBN 978-3-486-59075-3 , p. 101 ff . ( limited preview in Google Book search).