Crystallographic group

Crystallographic group (Fedorov group) - a discrete group of movements $n$ ${\ displaystyle n}$ $n$ - dimensional Euclidean space having a limited fundamental region .

Content

1 Bieberbach's Theorem
- 1.1 Number of groups
2 Possible symmetries
- 2.1 Point Elements
- 2.2 Broadcasts
- 2.3 Complex symmetry operations
3 Designations
- 3.1 Numbering
- 3.2 Symbols of Herman - Mogen
- 3.3 Schönflis Symbol
4 History
5 See also
6 notes
7 Literature
8 References

Bieberbach's Theorem

Two crystallographic groups are considered equivalent if they are conjugated in the group of affine transformations of Euclidean space.

Bieberbach's theorems

Every $n$ ${\ displaystyle n}$ -dimensional crystallographic group $\Gamma$ ${\ displaystyle \ Gamma}$ contains $n$ ${\ displaystyle n}$ linearly independent parallel hyphenation ; Group $G$ ${\ displaystyle G}$ linear parts of transformations (i.e. image $\Gamma$ ${\ displaystyle \ Gamma}$ at $GL_{n}$ ${\ displaystyle GL_ {n}}$ ) is finite.
Two crystallographic groups are equivalent if and only if they are isomorphic as abstract groups.
For any $n$ ${\ displaystyle n}$ there is only a finite number $n$ ${\ displaystyle n}$ -dimensional crystallographic groups considered up to equivalence (which is a solution to the 18th Hilbert problem ).

The theorem allows us to give the following description of the structure of crystallographic groups as abstract groups: Let $L$ ${\ displaystyle L}$ - the set of all parallel transfers belonging to the crystallographic group $\Gamma$ ${\ displaystyle \ Gamma}$ . Then $L$ ${\ displaystyle L}$ Is a normal subgroup of finite index isomorphic $\mathbb {Z} ^{n}$ ${\ displaystyle \ mathbb {Z} ^ {n}}$ and coinciding with its centralizer in $\Gamma$ ${\ displaystyle \ Gamma}$ . The presence of such a normal subgroup in an abstract group $\Gamma$ ${\ displaystyle \ Gamma}$ is also a sufficient condition for the group $\Gamma$ ${\ displaystyle \ Gamma}$ was isomorphic to the crystallographic group.

Group $G$ ${\ displaystyle G}$ linear parts of the crystallographic group $\Gamma$ ${\ displaystyle \ Gamma}$ saves the grill $L$ ${\ displaystyle L}$ ; in other words, in the basis of the lattice $L$ ${\ displaystyle L}$ conversions from $G$ ${\ displaystyle G}$ written by integer matrices.

Number of groups

The number of crystallographic groups $n$ ${\ displaystyle n}$ -dimensional space with or without orientation is given by sequences A004029 and A006227 . Up to equivalence, there is

17 flat crystallographic groups ^[1]
219 spatial crystallographic groups;
- if spatial groups are considered up to conjugation using affine transformations that preserve orientation , then there will be 230 of them.
In dimension 4, there are 4894 crystallographic groups with orientation preservation, or 4783 without orientation preservation ^[2] ^[3] .

Possible Symmetries

Point Elements

Symmetry elements of finite figures that leave at least one point stationary.

Rotary axis of symmetry, mirror plane of symmetry, center of inversion (center of symmetry) and improper rotations - inversion axes and mirror-rotary axes. Own rotations are defined as the sequential execution of rotation and inversion (or reflection in a perpendicular plane). Any mirror rotary axis can be replaced by an inversion axis and vice versa. When describing spatial groups, preference is usually given to inversion axes (while Schönflis symbols use mirror-rotary axes). In 2-dimensional and 3-dimensional crystallographic groups, only rotations around the axis of symmetry can be present at angles of 180 ° (axis of symmetry of the second order), 120 ° (third order), 90 ° (fourth order) and 60 ° (6th order). The axis of symmetry in the Brava symbols is denoted by the letter L with a lower numerical index n corresponding to the order of the axis ( $L_{n}$ ${\ displaystyle L_ {n}}$ ), in international symbolism (Herman-Mogen symbolism), in Arabic numerals indicating the order of the axis (for example, $L_{2}$ ${\ displaystyle L_ {2}}$ = 2, $L_{3}$ ${\ displaystyle L_ {3}}$ = 3 and $L_{4}$ ${\ displaystyle L_ {4}}$ = 4). Inversion axes in the Brava symbolism are denoted by the letter Ł with a lower numerical index n corresponding to the order of the rotary axis ( Ł _n ), in international symbols, by a digital index with a dash above n (for example, Ł ₃ = 3 , Ł ₄ = 4 , Ł ₆ = 6 ) More information about improper rotations and their notation is written here . The symmetry axes L ₃ , L ₄ , L ₆ are called higher order symmetry axes ^[4] . The mirror plane of symmetry is denoted by P according to the Brava and m in international symbolism. The center of inversion is denoted by C according to Brava and 1 in international symbolism.

All possible combinations of point symmetry elements lead to 10 turned symmetry groups in 2-dimensional space and 32 point groups in 3-dimensional space.

A new type of symmetry elements appears in 4-dimensional space - double rotations in two absolutely perpendicular planes . Due to this, the number of symmetry elements compatible with translational symmetry increases. For spaces of dimensions 4 and 5 in the crystal, point symmetry elements with orders of 1, 2, 3, 4, 5, 6, 8, 10, and 12 are possible. Moreover, since rotations in each of the absolutely perpendicular planes can occur in different directions, appear enantiomorphic pairs of point elements of symmetry (for example, fourth-order double rotation, where 90 ° rotations in the first plane and 90 ° in the second plane are combined enantiomorphic fourth-order double rotation, where 90 ° rotations in the first plane and −90 ° rotate are combined second ). All possible combinations of point symmetry elements in 4-dimensional space lead to 227 4-dimensional point groups, of which 44 are enantiomorphic (that is, a total of 271 point symmetry groups is obtained).

In 6-dimensional and 7-dimensional spaces in a crystal, point symmetry elements with orders of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24, and 30 are possible ^[5] . See also en: Crystallographic restriction theorem .

Broadcasts

In crystallographic groups there are always translations - parallel transfers , with a shift by which the crystalline structure will combine with itself. The translational symmetry of the crystal is characterized by the Bravais lattice . In the 3-dimensional case, a total of 14 types of Bravais lattices are possible. In dimensions 4, 5, and 6, the number of types of Bravais lattices is 64, 189, and 841, respectively ^[6] . From the point of view of group theory, a translation group is a normal Abelian subgroup of a space group, and a space group is an extension of its translation subgroup. The factor group of the space group in the translation subgroup is one of the point groups.

Complex symmetry operations

Rotations around the axes with simultaneous transfer to a vector in the direction of this axis (helical axis) and reflection relative to the plane with a simultaneous shift by some vector parallel to this plane (plane of sliding reflection). In international symbolism, helical axes are indicated by the number of the corresponding rotary axis with an index characterizing the amount of transport along the axis while rotating. Possible screw axes in the 3-dimensional case: 2 ₁ (180 ° rotation and 1/2 translation shift), 3 ₁ (120 ° rotation and 1/3 translation shift), 3 ₂ (120 ° rotation and shift 2/3 broadcasts), 4 ₁ (90 ° rotation and 1/4 broadcast shift), 4 ₂ (90 ° rotation and 1/2 broadcast shift), 4 ₃ (90 ° rotation and 3 shift / 4 broadcasts), 6 ₁ , 6 ₂ , 6 ₃ , 6 ₄ , 6 ₅ (rotation by 60 ° and a shift by 1/6, 2/6, 3/6, 4/6, and 5/6 broadcasts, respectively ) The axes 3 ₂ , 4 ₃ , 6 ₄ , and 6 _{5 are} enantiomorphic to the axes 3 ₁ , 4 ₁ , 6 ₂ , and 6 ₁ , respectively. It is due to these axes that there are 11 enantiomorphic pairs of space groups - in each pair, one group is a mirror image of the other.

Sliding reflection planes are indicated depending on the direction of sliding with respect to the axes of the crystal cell. If sliding occurs along one of the axes, then the plane is denoted by the corresponding Latin letter a , b or c . In this case, the slip value is always equal to half the translation. If the slip is directed along the diagonal of the face or the spatial diagonal of the cell, then the plane is indicated by the letter n in the case of slip equal to half the diagonal, or d in the case of slip equal to a quarter of the diagonal (this is possible only if the diagonal is centered). The n and d planes are also called wedge planes. d planes are sometimes called diamond planes, because they are present in the structure of diamond (eng. diamond - diamond).

In some spatial groups, there are planes where gliding occurs both along one axis and along the second axis of the cell (i.e., the plane is simultaneously a and b or a and c or b and c ). This is due to the centering of the face parallel to the slip plane. In 1992, the symbol e was introduced for such planes. ^[7] Nikolai Vasilievich Belov also proposed introducing the notation r for planes with sliding along the spatial diagonal in a rhombohedral cell. However, r planes always coincide with ordinary mirror planes, and the term has not taken root.

Conventions

Numbering

Crystallographic (spatial) groups with all their symmetry elements are summarized in the international directory International Crystallographic Tables , published by the International Union of Crystallography. The use of the numbering given in this manual is accepted. Groups are numbered from 1 to 230 in order of increasing symmetry.

Symbols of Herman - Mogen

The space group symbol contains the Bravais lattice symbol (capital letter P, A, B, C, I, R or F) and the international dot group symbol. The Bravais lattice symbol indicates the presence of additional translation nodes within the unit cell: P (primitive) - primitive cell; A, B, C (A-centered, B-centered, C-centered) - an additional node in the center of the face A, B or C, respectively; I (I-centered) - body-centered (additional node in the center of the cell), R (R-centered) - twice body-centered (two additional nodes on the large diagonal of the unit cell), F (F-centered) - face-centered (additional nodes in the centers of all faces).

The international symbol of a point group in the general case is formed of three symbols denoting symmetry elements that correspond to three main directions in a crystal cell. By a symmetry element corresponding to a direction, we mean either the axis of symmetry passing in this direction, or the plane of symmetry perpendicular to it, or both (in this case, they are written through a fraction, for example, 2 / c is the second-order symmetry axis and a plane of sliding reflection perpendicular to it with a shift in the direction c ). Under the main areas understand:

directions of the cell base vectors in the case of triclinic, monoclinic, and rhombic syngony;
the direction of the 4th order axis, the direction of one of the basis vectors at the base of the unit cell and the diagonal direction of the base of the cell in the case of tetragonal syngony;
the direction of the axis of the 3rd order or 6th order, the direction of one of the basis vectors at the base of the unit cell and the direction of the vector along the diagonal of the unit cell at an angle of 60 ° to the previous one in the case of hexagonal syngony (this also includes trigonal syngony, which in this case is given to the hexagonal orientation of the unit cell);
the direction of one of the basis vectors, the direction along the spatial diagonal of the unit cell and the direction along the bisector of the angle between the basis vectors.

The Herman - Mogen symbols are usually abbreviated by removing the designation of missing symmetry elements in separate directions when this does not create ambiguity, for example, write P4 instead of P411. Also, in the absence of ambiguity, the designations of the second-order axes are omitted, which are perpendicular to the plane of symmetry, for example, replace C ${\tfrac {2}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$ ${\ displaystyle {\ tfrac {2} {m}} {\ tfrac {2} {m}} {\ tfrac {2} {m}}}$ on $C m m m$ ${\ displaystyle Cmmm}$ .

Schönflis Symbol

The Schönflis symbol defines a symmetry class (main symbol and subscript) and a conditional group number within this class (superscript).

C _n — cyclic groups — groups with a single special direction represented by a rotary axis of symmetry — are denoted by the letter C , with a lower numerical index n corresponding to the order of this axis.

With _ni , groups with a single inverse axis of symmetry are followed by a subscript i .
C _nv (from German vertical - vertical) - also has a plane of symmetry located along a single or main axis of symmetry, which is always thought to be vertical.
C _nh (from German horizontal - horizontal) - also has a plane of symmetry perpendicular to the main axis of symmetry.

S ₂ , S ₄ , S ₆ (from German spiegel - mirror) - groups with a single mirror axis of symmetry.

C _s - for a plane of uncertain orientation, that is, not fixed due to the absence of other symmetry elements in the group.

D _n - is a group C _n with additional n second-order symmetry axes perpendicular to the original axis.

D _nh - also has a horizontal plane of symmetry.
D _nd (from German diagonal - diagonal) - also has vertical diagonal planes of symmetry that go between the symmetry axes of the second order.

O, T - symmetry groups with several higher order axes - groups of cubic syngony. They are indicated by the letter O if they contain the complete set of symmetry axes of the octahedron, or by the letter T if they contain the complete set of symmetry axes of the octahedron.

O _h and T _h - also contain a horizontal plane of symmetry
T _d - also contains the diagonal plane of symmetry

n may be 1, 2, 3, 4, 6.

History

The origin of the theory of crystallographic groups is associated with the study of the symmetry of ornaments ( $n=2$ ${\ displaystyle n = 2}$ ) and crystal structures ( $n=3$ ${\ displaystyle n = 3}$ ) The classification of all planar (two-dimensional) and spatial (three-dimensional) crystallographic groups was independently obtained by Fedorov (1885), Schönflis (1891) and Barlow (1894). The main results for multidimensional crystallographic groups were obtained by Bieberbach ^[8] .

Notes

↑ Wallpaper Groups - from Wolfram MathWorld
↑ H. Brown, R. Bülow, J. Neubüser, H. Wondratschek and H. Zassenhaus, Crystallographic Groups of Four-Dimensional Space. Wiley, NY, 1978, p. 52.
↑ J. Neubüser, B. Souvignier and H. Wondratschek, Corrections to Crystallographic Groups of Four-Dimensional Space by Brown et al. (1978) [New York: Wiley and Sons], Acta Cryst (2002) A58, 301. http://journals.iucr.org/a/issues/2002/03/00/au0290/index.html
↑ Yu. K. Egorov-Tismenko, G.P. Litvinskaya, Yu. G. Zagalskaya, Crystallography, ed. Moscow State University, 1992, p. 22.
↑ T. Janssen, JL Birman, VA Koptsik, M. Senechal, D. Weigel, A. Yamamoto, SC Abrahams and T. Hahn, Acta Cryst. (1999). A55, 761-782
↑ Opgenorth, J; Plesken, W; Schulz, T (1998), "Crystallographic Algorithms and Tables", Acta Cryst. A 54 (5): 517-531
↑ PM de Wolff, Y. Billiet, JDH Donnay, W. Fischer, RB Galiulin, AM Glazer, Th. Hahn, M. Senechal, DP Shoemaker, H. Wondratschek, AJC Wilson, & SC Abrahams, 1992, Acta Cryst., A48, 727-732.
↑ Bieberbach L. Über die Bewegungsgruppen der Euklidischen Raume I. — Math. Ann., 1911, 70, S. 297-336; 1912, 72, S. 400-412.

Literature

J. Wolf, Spaces of constant curvature. Translation from English. Moscow: “Science”, Main Edition of Physics and Mathematics, 1982.
Yu.K. Egorov-Tismenko, G.P. Litvinskaya, Theory of crystal symmetry, M. GEOS, 2000 (available on-line http://geo.web.ru/db/msg.html?mid=1163834 )
Crystallographic group // Mathematical Encyclopedia / I. M. Vinogradov (Ch. Ed.). - M .: Soviet Encyclopedia, 1982. - T. 3. - S. 106-108. - 592 p. - 150,000 copies.

Links

International Tables for Crystallography
Spatial group - an article from the Great Soviet Encyclopedia .

[1] Wallpaper Groups - from Wolfram MathWorld

[2] H. Brown, R. Bülow, J. Neubüser, H. Wondratschek and H. Zassenhaus, Crystallographic Groups of Four-Dimensional Space. Wiley, NY, 1978, p. 52.

[3] J. Neubüser, B. Souvignier and H. Wondratschek, Corrections to Crystallographic Groups of Four-Dimensional Space by Brown et al. (1978) [New York: Wiley and Sons], Acta Cryst (2002) A58, 301. http://journals.iucr.org/a/issues/2002/03/00/au0290/index.html

[4] Yu. K. Egorov-Tismenko, G.P. Litvinskaya, Yu. G. Zagalskaya, Crystallography, ed. Moscow State University, 1992, p. 22.

[5] T. Janssen, JL Birman, VA Koptsik, M. Senechal, D. Weigel, A. Yamamoto, SC Abrahams and T. Hahn, Acta Cryst. (1999). A55, 761-782

[6] Opgenorth, J; Plesken, W; Schulz, T (1998), "Crystallographic Algorithms and Tables", Acta Cryst. A 54 (5): 517-531

[7] PM de Wolff, Y. Billiet, JDH Donnay, W. Fischer, RB Galiulin, AM Glazer, Th. Hahn, M. Senechal, DP Shoemaker, H. Wondratschek, AJC Wilson, & SC Abrahams, 1992, Acta Cryst., A48, 727-732.

[8] Bieberbach L. Über die Bewegungsgruppen der Euklidischen Raume I. — Math. Ann., 1911, 70, S. 297-336; 1912, 72, S. 400-412.