Symbols of Hermann - Mogen

Hermann - Mogen symbols are used to denote the symmetry of point groups (along with Schönflies symbols ), flat groups and space groups. They were proposed by the German crystallographer Carl Hermann ( born Carl Hermann ) in 1928 and modified by the French mineralogist Charles-Victor Mogen ( fr. Charles Victor Mauguin ) in 1931. Also called international symbols because they are used in International Tables for Crystallography ^[1] since their first edition in 1935. Prior to this, to denote point and space groups, Schenflies symbols were used, as a rule.

In the Hermann - Mogen symbol, symmetrically nonequivalent symmetry elements are designated. Rotary axes of symmetry are denoted by Arabic numerals - 1, 2, 3, 4 and 6. Inversion axes are denoted by Arabic numerals with a dash above - 1 , 3 , 4 and 6 . In this case, axis 2 , which is simply the plane of symmetry, is denoted by the symbol m (English mirror). The direction of the plane is the direction of the perpendicular to it (i.e. axis 2 ). Mirror axes are not used in international symbolism.

The orientation of the element relative to the coordinate axes is determined by the position of the element in the group symbol. If the direction of the axis of symmetry is perpendicular to the direction of the plane, then they are written in one position as a fraction. If the inversion axis has a greater symmetry value (multiplying ability) than a rotary one coinciding with it, then it is precisely this symbol that is indicated in the symbol (i.e. $\color {Black}{\tfrac {3}{m}}$ ${\ displaystyle \ color {Black} {\ tfrac {3} {m}}}$ $\color {Black}{\tfrac {3}{m}}$ , and 6 ; if there is an inversion center in the group, not 3, but 3 ).

The lowest category is point groups in which the maximum order of any axis (rotary or improper rotation) is two. It includes groups 1, 1 , 2, m, $\color {Black}{\tfrac {2}{m}}$ ${\ displaystyle \ color {Black} {\ tfrac {2} {m}}}$ $\color {Black}{\tfrac {2}{m}}$ , 222, mm2 and $\color {Black}{\tfrac {2}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$ ${\ displaystyle \ color {Black} {\ tfrac {2} {m}} {\ tfrac {2} {m}} {\ tfrac {2} {m}}$ $\color {Black}{\tfrac {2}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$ . If there are three positions in the group symbol, then

1st position - direction along the X axis

at the 2nd position - the direction along the Y axis

on the 3rd position - the direction along the Z axis

In a non-standard installation, the mm2 group can be written as m2m or 2mm. Similarly, groups 2, m and $\color {Black}{\tfrac {2}{m}}$ ${\ displaystyle \ color {Black} {\ tfrac {2} {m}}}$ can be written in more detail - with an indication along which coordinate axis the direction of the second order axis and / or the plane goes. For example, 11m, 1m1 or m11. This feature of symbolism is used to unambiguously describe spatial groups with a different choice of coordinate system, since the symbols of spatial groups are derived from the symbols of the corresponding point groups.

The middle category is point groups in which there is one axis of order higher than two (the axis of higher order). It should be noted here that crystallographic coordinate system is used in crystallography, which is associated with crystal symmetry. In this system, special axes in the crystal are selected by the axes (the directions along which the symmetry or translation axes go). Therefore, if there is one axis of order 3 or 6, the angle between the X and Y directions is 120 °, not 90 ° as in the usual Cartesian coordinate system .

on the 1st position - the direction of the main axis, that is, the Z axis

on the 2nd position - the secondary direction. That is, the direction along the X axis and its equivalent Y axis

on the 3rd position - the diagonal direction between symmetrically equivalent side directions

This category includes groups of 3, 4, 6, 3 , 4 , 6 , 32, 422, 622, 3m, 4mm, 6mm, 3 $\color {Black}{\tfrac {2}{m}}$ ${\ displaystyle \ color {Black} {\ tfrac {2} {m}}}$ , 4 2m, 6 m2, $\color {Black}{\tfrac {4}{m}}$ ${\ displaystyle \ color {Black} {\ tfrac {4} {m}}}$ , $\color {Black}{\tfrac {6}{m}}$ ${\ displaystyle \ color {Black} {\ tfrac {6} {m}}}$ , $\color {Black}{\tfrac {4}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$ ${\ displaystyle \ color {Black} {\ tfrac {4} {m}} {\ tfrac {2} {m}} {\ tfrac {2} {m}}$ and $\color {Black}{\tfrac {6}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$ ${\ displaystyle \ color {Black} {\ tfrac {6} {m}} {\ tfrac {2} {m}} {\ tfrac {2} {m}}$ .

Since the axis 3 and the plane perpendicular to it are equivalent to axis 6 , then $\color {Black}{\tfrac {3}{m}}$ ${\ displaystyle \ color {Black} {\ tfrac {3} {m}}}$ = 6 and $\color {Black}{\tfrac {3}{m}}$ ${\ displaystyle \ color {Black} {\ tfrac {3} {m}}}$ m2 = 6 m2, but it is recommended to use the notation with inversion axis 6 , since its symmetry is higher than that of axis 3. Groups 4 2m and 6 m2 can be written as 4 m2 and 6 2m. Above were given the notation adopted in the Russian-language literature. The sequence of symbols 2 and m in these groups becomes important when describing space groups derived from them, since the element in the second position is directed along the axis of the Bravais cell, and the element in the third position is directed along the diagonal of the face. For example, the symbols P 4 2m and P 4 m2 denote two different space groups. Group 32 can also be written in more detail as 321 or 312 for different orientations of axis 2. Similarly, different orientations lead to two different space groups P321 and P312. The same applies to the 3m groups (alternative 3m1 and 31m entries) and 3 $\color {Black}{\tfrac {2}{m}}$ ${\ displaystyle \ color {Black} {\ tfrac {2} {m}}}$ (alternative entries 3 $\color {Black}{\tfrac {2}{m}}$ ${\ displaystyle \ color {Black} {\ tfrac {2} {m}}}$ 1 and 3 1 $\color {Black}{\tfrac {2}{m}}$ ${\ displaystyle \ color {Black} {\ tfrac {2} {m}}}$ ).

The highest category is point groups in which several higher order axes are present.

on the 1st position - equivalent directions X, Y, Z

on the 2nd position - four axes 3 or 3 always present there

on the 3rd position - the diagonal direction between the coordinate axes

This category includes five groups - 23, 432, $\color {Black}{\tfrac {2}{m}}$ ${\ displaystyle \ color {Black} {\ tfrac {2} {m}}}$ 3 , 4 3m and $\color {Black}{\tfrac {4}{m}}$ ${\ displaystyle \ color {Black} {\ tfrac {4} {m}}}$ 3 $\color {Black}{\tfrac {2}{m}}$ ${\ displaystyle \ color {Black} {\ tfrac {2} {m}}}$

International characters usually simplify by replacing $\color {Black}{\tfrac {n}{m}}$ ${\ displaystyle \ color {Black} {\ tfrac {n} {m}}}$ by m , if the n axis is generated by other symmetry elements indicated in the symbol. It is impossible to remove only the designation of the main axis in the middle category. For example, $\color {Black}{\tfrac {2}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$ ${\ displaystyle \ color {Black} {\ tfrac {2} {m}} {\ tfrac {2} {m}} {\ tfrac {2} {m}}$ recorded as mmm $\color {Black}{\tfrac {4}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$ ${\ displaystyle \ color {Black} {\ tfrac {4} {m}} {\ tfrac {2} {m}} {\ tfrac {2} {m}}$ as $\color {Black}{\tfrac {4}{m}}$ ${\ displaystyle \ color {Black} {\ tfrac {4} {m}}}$ mm, and $\color {Black}{\tfrac {4}{m}}$ ${\ displaystyle \ color {Black} {\ tfrac {4} {m}}}$ 3 $\color {Black}{\tfrac {2}{m}}$ ${\ displaystyle \ color {Black} {\ tfrac {2} {m}}}$ as m 3 m.

Designation of point groups

Groups with a single axis of higher order are recorded according to the same principles as the crystallographic groups of the middle category. They can be arranged in the following table.

Schonflies	HM symbol	3	four	five	6	7	eight	9	ten	eleven	12	13	14	...	$\infty$ ${\ displaystyle \ infty}$
$C_{n}$ ${\ displaystyle C_ {n}}$	$n$ ${\ displaystyle n}$	3	four	five	6	7	eight	9	ten	eleven	12	13	14	...	$\infty$ ${\ displaystyle \ infty}$
$C_{nv}$ ${\ displaystyle C_ {nv}}$	$n$ ${\ displaystyle n}$ m	3m		5m		7m		9m		11m		13m			$\infty$ ${\ displaystyle \ infty}$ m
$C_{nv}$ ${\ displaystyle C_ {nv}}$	$n$ ${\ displaystyle n}$ mm		4mm		6mm		8mm		10mm		12mm		14mm		$\infty$ ${\ displaystyle \ infty}$ m
$S_{2n}$ ${\ displaystyle S_ {2n}}$	${\bar {n}}$ ${\ displaystyle {\ bar {n}}}$	3		five		7		9		eleven		13			${\tfrac {\infty }{m}}$ ${\ displaystyle {\ tfrac {\ infty} {m}}}$
$S_{n}$ ${\ displaystyle S_ {n}}$			four				eight				12
$C_{{\tfrac {n}{2}}h}$ ${\ displaystyle C _ {{\ tfrac {n} {2}} h}}$					6				ten				14
$C_{nh}$ ${\ displaystyle C_ {nh}}$	${\tfrac {n}{m}}$ ${\ displaystyle {\ tfrac {n} {m}}}$		${\tfrac {4}{m}}$ ${\ displaystyle {\ tfrac {4} {m}}}$		${\tfrac {6}{m}}$ ${\ displaystyle {\ tfrac {6} {m}}}$		${\tfrac {8}{m}}$ ${\ displaystyle {\ tfrac {8} {m}}}$		${\tfrac {10}{m}}$ ${\ displaystyle {\ tfrac {10} {m}}}$		${\tfrac {12}{m}}$ ${\ displaystyle {\ tfrac {12} {m}}}$		${\tfrac {14}{m}}$ ${\ displaystyle {\ tfrac {14} {m}}}$
$D_{n}$ ${\ displaystyle D_ {n}}$	$n$ ${\ displaystyle n}$ 2	3 2		5 2		7 2		9 2		11 2		13 2			$\infty$ ${\ displaystyle \ infty}$ 2
$D_{n}$ ${\ displaystyle D_ {n}}$	$n$ ${\ displaystyle n}$ 22		4 22		6 22		8 22		10 22		12 22		14 22		$\infty$ ${\ displaystyle \ infty}$ 2
$D_{nd}$ ${\ displaystyle D_ {nd}}$	${\bar {n}}{\tfrac {2}{m}}$ ${\ displaystyle {\ bar {n}} {\ tfrac {2} {m}}}$	3 ${\tfrac {2}{m}}$ ${\ displaystyle {\ tfrac {2} {m}}}$		five ${\tfrac {2}{m}}$ ${\ displaystyle {\ tfrac {2} {m}}}$		7 ${\tfrac {2}{m}}$ ${\ displaystyle {\ tfrac {2} {m}}}$		9 ${\tfrac {2}{m}}$ ${\ displaystyle {\ tfrac {2} {m}}}$		eleven ${\tfrac {2}{m}}$ ${\ displaystyle {\ tfrac {2} {m}}}$		13 ${\tfrac {2}{m}}$ ${\ displaystyle {\ tfrac {2} {m}}}$			${\tfrac {\infty }{m}}m$ ${\ displaystyle {\ tfrac {\ infty} {m}} m}$
$D_{{\tfrac {n}{2}}d}$ ${\ displaystyle D _ {{\ tfrac {n} {2}} d}}$	${\bar {n}}2m=$ ${\ displaystyle {\ bar {n}} 2m =}$ $={\bar {n}}m2$ ${\ displaystyle = {\ bar {n}} m2}$		4 2m				8 2m				12 2m
$D_{{\tfrac {n}{2}}h}$ ${\ displaystyle D _ {{\ tfrac {n} {2}} h}}$					6 m2				10 m2				14 m2
$D_{nh}$ ${\ displaystyle D_ {nh}}$	${\tfrac {n}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$ ${\ displaystyle {\ tfrac {n} {m}} {\ tfrac {2} {m}} {\ tfrac {2} {m}}}$		${\tfrac {4}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$ ${\ displaystyle {\ tfrac {4} {m}} {\ tfrac {2} {m}} {\ tfrac {2} {m}}}$		${\tfrac {6}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$ ${\ displaystyle {\ tfrac {6} {m}} {\ tfrac {2} {m}} {\ tfrac {2} {m}}}$		${\tfrac {8}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$ ${\ displaystyle {\ tfrac {8} {m}} {\ tfrac {2} {m}} {\ tfrac {2} {m}}}$		${\tfrac {10}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$ ${\ displaystyle {\ tfrac {10} {m}} {\ tfrac {2} {m}} {\ tfrac {2} {m}}}$		${\tfrac {12}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$ ${\ displaystyle {\ tfrac {12} {m}} {\ tfrac {2} {m}} {\ tfrac {2} {m}}}$		${\tfrac {14}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$ ${\ displaystyle {\ tfrac {14} {m}} {\ tfrac {2} {m}} {\ tfrac {2} {m}}}$

Of the final non-crystallographic ones, only two groups remain, containing several higher order axes. This is the icosahedron symmetry group and its subgroup — the icosahedral axial symmetry group (a combination of six axes of the 5th order, ten axes of the 3rd order and 15 axes of the 2nd order). Since the symbolism of Hermann - Mogen was originally intended only for crystallographic groups, the symbols of these groups are rather arbitrary and are built like symbols of crystallographic groups of the highest category. Also for these groups there is no standard installation of the coordinate system (and the international symbol depends on it). Below are a few character choices.

^[2] $\color {Black}{\tfrac {2}{m}}$ ${\ displaystyle \ color {Black} {\ tfrac {2} {m}}}$ 3 5 (abbreviated m 3 5 ) and 235.

^[3] ^[4] $\color {Black}{\tfrac {2}{m}}$ ${\ displaystyle \ color {Black} {\ tfrac {2} {m}}}$ 5 (abbreviated m 5 ) and 25 - by analogy with the symbols $\color {Black}{\tfrac {2}{m}}$ ${\ displaystyle \ color {Black} {\ tfrac {2} {m}}}$ 3 (abbreviated m 3 ) and 23.

^[5] m 5 m and 532 - by analogy with the symbols m 3 m and 432.

^[6] 5 3 $\color {Black}{\tfrac {2}{m}}$ ${\ displaystyle \ color {Black} {\ tfrac {2} {m}}}$ (abbreviated 5 3 m) and 532 - by analogy with the symbols $\color {Black}{\tfrac {4}{m}}$ ${\ displaystyle \ color {Black} {\ tfrac {4} {m}}}$ 3 $\color {Black}{\tfrac {2}{m}}$ ${\ displaystyle \ color {Black} {\ tfrac {2} {m}}}$ and 432.

^[7] 5 3m and 53.

In practice, as a rule, to denote these groups, Schönflies symbols I _h and I are used .

Five groups from the table with $n=\infty$ ${\ displaystyle n = \ infty}$ are called limit groups ^[8] or Curie groups. These include two more groups that are not presented in the table. This is a group of all possible rotations around all axes passing through a point, $\infty \infty$ ${\ displaystyle \ infty \ infty}$ - a group of rotations, as well as a group $\color {Black}{\tfrac {\infty }{m}}\infty$ ${\ displaystyle \ color {Black} {\ tfrac {\ infty} {m}} \ infty}$ , which describes the symmetry of the ball - the maximum possible point symmetry in three-dimensional space; All point groups are subgroups of a group. $\color {Black}{\tfrac {\infty }{m}}\infty$ ${\ displaystyle \ color {Black} {\ tfrac {\ infty} {m}} \ infty}$ . Again, as with icosahedral symmetry groups, there are several designations for these groups ( $2\infty$ ${\ displaystyle 2 \ infty}$ and $m{\bar {\infty }}$ ${\ displaystyle m {\ bar {\ infty}}}$ , $\infty \infty$ ${\ displaystyle \ infty \ infty}$ and $\infty \infty m$ ${\ displaystyle \ infty \ infty m}$ ). In mathematics and theoretical physics, they are usually referred to as SO (3) and O (3) (a special orthogonal group in three-dimensional space and an orthogonal group in three-dimensional space).

Designation of space groups

The symbol of Hermann - Moguen for a space group is constructed according to the same principles as the symbol of a crystallographic point group, plus the type of centering of the cell is added to the beginning of the symbol. The following types of centering are possible.

P - primitive
I - body-centered (additional node in the center of the cell).
F - face-centered (additional nodes in the centers of all faces).
C, A, or B is base-centered (additional node in the center of the face C, A, or B) A and B cells are also called bocentric.
R - twice body-centered (two additional nodes on the large diagonal of the cell).

Mirror planes are designated the same as in point groups - the symbol m . Glide reflection planes are designated depending on the direction of sliding with respect to the axes of the crystal cell. If the slide occurs along one of the axes, then the plane is indicated by the corresponding Latin letter a , b or c . In this case, the amount of slip is always equal to half the broadcast. If the slip is directed along the diagonal of the face or spatial diagonal of the cell, then the plane is denoted by the letter n in the case of a slip equal to half the diagonal, or d in the case of a slip equal to a quarter of the diagonal (this is possible only if the diagonal is centered). Planes n and d are also called clinoplanes. D planes are sometimes called diamond planes, since they are present in the structure of a diamond.

Nikolai Vasilievich Belov also proposed to introduce the notation r for planes with sliding along the spatial diagonal in the rhombohedral cell. However, the r planes always coincide with ordinary mirror planes, and the term has not taken root. In the five space groups, there are planes where the slip occurs both along one axis and along the second axis of the cell (that is, the plane is both 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. ^[9]

Group number	39	41	64	67	68
Old character	Abm2	Aba2	Cmca	Cmma	Ccca
New character	Aem2	Aea2	Cmce	Cmme	Ccce

Ordinary rotary axes of the nth order are designated the same as in point groups - by the Arabic numeral n . Screw axes are designated by the number of the corresponding rotary axis with an index characterizing the magnitude of the transfer along the axis while rotating. Possible screw axes in the 3-dimensional case: 2 ₁ (rotate by 180 ° and shift by 1/2 broadcasts), 3 ₁ (rotate by 120 ° and shift by 1/3 broadcast), 3 ₂ (rotate by 120 ° and shift 2/3 broadcast), 4 ₁ (rotate by 90 ° and shift by 1/4 broadcast), 4 ₂ (rotate by 90 ° and shift by 1/2 broadcast), 4 ₃ (rotate by 90 ° and shift by 3 / 4 broadcast), 6 ₁ , 6 ₂ , 6 ₃ , 6 ₄ , 6 ₅ (rotate 60 ° and shift to 1/6, 2/6, 3/6, 4/6, and 5/6 broadcast, 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. Due to these axes, there are 11 enantiomorphic pairs of space groups - in each pair one group is a mirror image of the other.

P4 ₁	P4 ₁ 22	P4 ₁ 2 ₁ 2	P3 ₁	P3 ₁ 12	P3 ₁ 21	P6 ₁	P6 ₂	P6 ₁ 22	P6 ₂ 22	P4 ₁ 32
P4 ₃	P4 ₃ 22	P4 ₃ 2 ₁ 2	P3 ₂	P3 ₂ 12	P3 ₂ 21	P6 ₅	P6 ₄	P6 ₅ 22	P6 ₄ 22	P4 ₃ 32

Setting the space group and selecting the Bravais cell

The Hermann - Mogen symbol depends on the installation of the space group, that is, on how the symmetry elements (axes, planes, translation) are directed relative to the selected coordinate system. This is especially important in the case of space groups, when the coordinate system, that is, the choice of the Bravais cell, affects the designation of the plane of the sliding reflection ( a, b, c, n, d ) and the type of centering of the cell. In groups in which one direction differs from the other two (for example, point groups 3, 4, 6, mm2, 3m 4mm, 6mm, 32, 422, 622 and space groups derived from them), this particular direction is chosen for the Z axis ( Brave cell c vector). An important exception is the monoclinic syngony groups (point groups 2, m, 2 / m and space groups derived from them) in which this particular direction is chosen for the Y axis (vector b of the Bravais cell). The reason for this is purely historical and comes from mineralogy. As Belov writes, “the classical crystallographer and, above all, the mineralogist well knows that the elongation of the crystal, with which he, without thinking, links the vertical axis Z , in most cases does not coincide with the special direction of the monoclinic crystal, to which the morphologist provides the second axis Y. ” ^[10] Thus, the expanded international symbols for these groups will be as follows.

Group number	3	four	five	6	7	eight	9	ten	eleven	12	13	14	15
Symbol	P2	P2 ₁	C2	Pm	Pc	Cm	Cc	P2 / m	P2 ₁ / m	C2 / m	P2 / c	P2 ₁ / c	C2 / c
Expanded character	P121	P12 ₁ 1	C121	P1m1	P1c1	C1m1	C1c1	P1 $\color {Black}{\tfrac {2}{m}}$ ${\ displaystyle \ color {Black} {\ tfrac {2} {m}}}$ one	P1 $\color {Black}{\tfrac {2_{1}}{m}}$ ${\ displaystyle \ color {Black} {\ tfrac {2_ {1}} {m}}}$ one	C1 $\color {Black}{\tfrac {2}{m}}$ ${\ displaystyle \ color {Black} {\ tfrac {2} {m}}}$ one	P1 $\color {Black}{\tfrac {2}{c}}$ ${\ displaystyle \ color {Black} {\ tfrac {2} {c}}}$ one	P1 $\color {Black}{\tfrac {2_{1}}{c}}$ ${\ displaystyle \ color {Black} {\ tfrac {2_ {1}} {c}}}$ one	C1 $\color {Black}{\tfrac {2}{c}}$ ${\ displaystyle \ color {Black} {\ tfrac {2} {c}}}$ one

In a standard installation, the slip plane in the monoclinic system cannot be b , since the slip direction cannot be perpendicular to the plane itself. Also, the centering of the cell cannot be B, since in this case it would be possible to go to a primitive cell of half the volume and the same symmetry.

Notes

↑ (International Tables) Home page
↑ Wiley Online Library: IUCR ITL Access Denied (inaccessible link)
↑ P. M. Zorky. Symmetry of molecules and crystal structures, Moscow State University, 1986, p. 42.
↑ Families of point groups
↑ B. K. Weinstein, V. M. Fridkin, V. L. Indenbom. Modern crystallography. Volume 1. M .: Science, 1979, p. 97.
↑ Point groups in three dimensions
↑ A.V. Shubnikov. Symmetry and antisymmetry of finite figures, Publishing House of the Academy of Sciences of the USSR, 1951
↑ Limit point groups
↑ 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.
↑ N.V. Belov, G.P. Litvinskaya, On the installation of crystals of lower systems. - In the book: Problems of Crystalology. M .: Publishing House of Moscow State University, 1976. p. 13-14

Literature

Point groups

P. M. Zorky. Symmetry of molecules and crystal structures. M .: Publishing House of Moscow State University, 1986 (available on-line http://www.chem.msu.su/rus/teaching/zorkii2/welcome.html )
Yu. K. Egorov-Tismenko, G. P. Litvinskaya. The theory of symmetry of crystals. M .: Geos, 2000 (available on-line http://geo.web.ru/db/msg.html?mid=1163834 )
Yu. K. Egorov-Tismenko, G. P. Litvinskaya, Yu. G. Zagalskaya. Crystallography. M .: Publishing House of Moscow State University, 1992
Yu. G. Zagalskaya, G.P. Litvinskaya. Geometric crystallography. M .: Publishing House of Moscow State University, 1973

Spatial Groups

Yu. K. Egorov-Tismenko, G. P. Litvinskaya. The theory of symmetry of crystals. M .: Geos, 2000 (available on-line http://geo.web.ru/db/msg.html?mid=1163834 )
Yu. G. Zagalskaya, G.P. Litvinskaya. Geometric microcrystallography. M .: Moscow State University Publishing House, 1976
N.V. Belov. A cool method for deriving spatial symmetry groups. Works Ying-that Crystallography of the Academy of Sciences of the USSR. 1951. № 6. S. 25-62.
N.V. Belov. Essays on structural crystallography and Fedorov symmetry groups. M .: Science. 1986

[1] (International Tables) Home page

[2] Wiley Online Library: IUCR ITL Access Denied (inaccessible link)

[3] P. M. Zorky. Symmetry of molecules and crystal structures, Moscow State University, 1986, p. 42.

[4] Families of point groups

[5] B. K. Weinstein, V. M. Fridkin, V. L. Indenbom. Modern crystallography. Volume 1. M .: Science, 1979, p. 97.

[6] Point groups in three dimensions

[7] A.V. Shubnikov. Symmetry and antisymmetry of finite figures, Publishing House of the Academy of Sciences of the USSR, 1951

[8] Limit point groups

[9] 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.

[10] N.V. Belov, G.P. Litvinskaya, On the installation of crystals of lower systems. - In the book: Problems of Crystalology. M .: Publishing House of Moscow State University, 1976. p. 13-14

Symbols of Hermann - Mogen

Contents

Designation of crystallographic point groups