Copeland–Erdős constant

In geometry, Hermann–Mauguin notation is used to represent the symmetry elements in point groups, plane groups and space groups. It is named after the German crystallographer Carl Hermann (who introduced it in 1928) and the French mineralogist Charles-Victor Mauguin (who modified it in 1931). This notation is sometimes called international notation, because it was adopted as standard by the International Tables For Crystallography since their first edition in 1935.

The Hermann–Mauguin notation, compared with the Schoenflies notation, is preferred in crystallography because it can easily be used to include translational symmetry elements, and it specifies the directions of the symmetry axes.

Point groups

Rotation axes are denoted by a number n — 1, 2, 3, 4, 5, 6, 7, 8 ... (angle of rotation φ = 360° / n) For improper rotations Herman-Mauguin symbols show rotoinversion axes, unlike Schoenflies and Shubnikov notations, where preference is given to rotation-reflection axes. The rotoinversion axes are represented by the corresponding number with a macron, Template:Overline — Template:Overline, Template:Overline, Template:Overline, Template:Overline, Template:Overline, Template:Overline, Template:Overline ... The symbol for a mirror plane (rotoinversion axis Template:Overline) is m. The direction of a mirror plane is defined as direction perpendicular to it (the direction of Template:Overline axis).

Herman-Mauguin symbols show symmetrically non-equivalent axes and planes. The direction of a symmetry element is represented by its position in the Herman-Mauguin symbol. If a rotation axis n and a mirror plane m have the same direction (i.e. the plane is perpendicular to axis n), then they are denoted as fraction $\color {Black}{\tfrac {n}{m}}$ or n/m.

If two or more axes have the same direction, the axis with higher symmetry is shown. Higher symmetry here means that the axis generates a pattern with more points. For example, rotation axes 3, 4, 5, 6, 7, 8 generate 3-, 4-, 5-, 6-, 7-, 8-point patterns, respectively. Improper rotation axes Template:Overline, Template:Overline, Template:Overline, Template:Overline, Template:Overline, Template:Overline generate 6-, 4-, 10-, 6-, 14-, 8-point patterns, respectively. If both the rotation and rotoinversion axes satisfy the previous rule, the rotation axis should be chosen. For example, $\color {Black}{\tfrac {3}{m}}$ combination is equivalent to Template:Overline. Because Template:Overline generates 6 points, and 3 generates only 3, Template:Overline should be written instead of $\color {Black}{\tfrac {3}{m}}$ (not $\color {Black}{\tfrac {\bar {6}}{m}}$, because Template:Overline already contains mirror plane m). The same situations is in the case when both 3 and Template:Overline axes are present — Template:Overline should be written. However we write $\color {Black}{\tfrac {4}{m}}$, not $\color {Black}{\tfrac {\bar {4}}{m}}$, because both 4 and Template:Overline generate four points. The analogous case is the $\color {Black}{\tfrac {6}{m}}$ combination, where 2, 3, 6, Template:Overline, and Template:Overline axes are present; axes Template:Overline, Template:Overline, and 6 all generate 6-point patterns, but the latter should be used because it is a rotation axis — the symbol will be $\color {Black}{\tfrac {6}{m}}$.

Finally, the Herman-Mauguin symbol depends on the type of the group.

Groups without higher-order axes (axes of order three or more)

These groups may contain only two-fold axes, mirror planes, and inversion center. These are the crystallographic point groups 1 and Template:Overline (triclinic crystal system), 2, m, and $\color {Black}{\tfrac {2}{m}}$ (monoclinic), and 222, $\color {Black}{\tfrac {2}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$, and mm2 (orthorhombic). If the symbol contains three positions, then they denote symmetry elements in the X, Y, Z directions, respectively.

Groups with one higher-order axis

• First position — primary direction — Z direction, assigned to the higher-order axis.
• Second position — symmetrically equivalent secondary directions, which are perpendicular to Z-axis. These can be 2, m, or $\color {Black}{\tfrac {2}{m}}$.
• Third position — symmetrically equivalent tertiary directions, passing between secondary directions. These can be 2, m, or $\color {Black}{\tfrac {2}{m}}$.

These are the crystallographic groups 3, 32, 3m, Template:Overline, and Template:Overline$\color {Black}{\tfrac {2}{m}}$ (trigonal crystal system), 4, 422, 4mm, Template:Overline, Template:Overline2m, $\color {Black}{\tfrac {4}{m}}$, and $\color {Black}{\tfrac {4}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$ (tetragonal), and 6, 622, 6mm, Template:Overline, Template:Overlinem2, $\color {Black}{\tfrac {6}{m}}$, and $\color {Black}{\tfrac {6}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$ (hexagonal). Analogously, symbols of non-crystallographic groups (with axis of order 5, 7, 8, 9 ...) can be constructed. These groups can be arranged in the following table

n	3	4	5	6	7	8	...	$\infty$
n	3	4	5	6	7	8	...	$\infty$
n2 or n22	32	422	52	622	72	822	...	$\infty 2$
nm or nmm	3m	4mm	5m	6mm	7m	8mm	...	$\infty m$
$\mathbf {\tfrac {n}{m}}$	${\color {Red}{\tfrac {3}{m}}}$ = Template:Overline	$\color {Black}{\tfrac {4}{m}}$	${\color {Red}{\tfrac {5}{m}}}$ = Template:Overline	$\color {Black}{\tfrac {6}{m}}$	${\color {Red}{\tfrac {7}{m}}}$ = Template:Overline	$\color {Black}{\tfrac {8}{m}}$	...	$\color {Black}{\tfrac {\infty }{m}}$
$\mathbf {{\tfrac {n}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}}$	${\color {Red}{\tfrac {3}{m}}}$m2 = Template:Overlinem2	$\color {Black}{\tfrac {4}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$	${\color {Red}{\tfrac {5}{m}}}$m2 = Template:Overlinem2	$\color {Black}{\tfrac {6}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$	${\color {Red}{\tfrac {7}{m}}}$m2 = Template:Overlinem2	$\color {Black}{\tfrac {8}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$	...	$\color {Black}{\tfrac {\infty }{m}}m$
$\mathbf {\bar {n}}$	${\bar {3}}$	${\bar {4}}$	${\bar {5}}$	${\bar {6}}$	${\bar {7}}$	${\bar {8}}$	...	${\color {Red}{\bar {\infty }}}={\tfrac {\infty }{m}}$
$\mathbf {{\bar {n}}{\tfrac {2}{m}}}$ or $\mathbf {{\bar {n}}2m}$	Template:Overline$\color {Black}{\tfrac {2}{m}}$	Template:Overline2m	Template:Overline$\color {Black}{\tfrac {2}{m}}$	Template:Overlinem2	Template:Overline$\color {Black}{\tfrac {2}{m}}$	Template:Overline2m	...	${\color {Red}{\bar {\infty }}m}={\tfrac {\infty }{m}}m$

The symbols that shouldn't be used are shown in red text.

It can be noticed that in groups with odd-order axes n and Template:Overline the third position in symbol is always absent, because all n directions, perpendicular to higher-order axis, are symmetrically equivalent. For example, in the picture of a triangle all three mirror planes (S₀, S₁, S₂) are equivalent — all of them pass through one vertex and the center of the opposite side. For even-order axes n and Template:Overline there are $\color {Black}{\tfrac {n}{2}}$ secondary directions and $\color {Black}{\tfrac {n}{2}}$ tertiary directions. For example, in the picture of a regular hexagon one can distinguish two sets of mirror planes — three planes go through two opposite vertexes, and three other planes go though the centers of opposite sides. In this case any of two sets can be chosen as secondary directions, the rest set will be tertiary directions. Hence groups Template:Overline2m, Template:Overline2m, Template:Overline2m, ... can be written as Template:Overlinem2, Template:Overlinem2, Template:Overlinem2, ... For symbols of point groups this order usually doesn't matter; however, it will be important for Herman-Mauguin symbols of corresponding space groups, where secondary directions are directions of symmetry elements along unit cell translations b and c, while the tertiary directions correspond to the direction between unit cell translations b and c. For example, symbols PTemplate:Overlinem2 and PTemplate:Overline2m denote two different space groups. This also applies to symbols of space groups with odd-order axes 3 and Template:Overline. The perpendicular symmetry elements can go along unit cell translations b and c or between them. Space groups P321 and P312 are examples of the former and the latter cases, respectively.

The symbol of point group Template:Overline$\color {Black}{\tfrac {2}{m}}$ may be confusing; the corresponding Schoenflies symbol is D_3d, which means that the group consists of 3-fold axis, three perpendicular 2-fold axes, and 3 vertical diagonal planes passing between these 2-fold axes, so it seems that the group can be denoted as 32m or 3m2. However, one should remember that, unlike Schoenflies notation, the direction of a plane in a Herman-Mauguin symbol is defined as the direction perpendicular to the plane, and in the D_3d group all mirror planes are perpendicular to 2-fold axes, so they should be written in the same position as $\color {Black}{\tfrac {2}{m}}$. Second, these $\color {Black}{\tfrac {2}{m}}$ complexes generate an inversion center, which combining with the 3-fold rotation axis generates a Template:Overline rotoinversion axis.

Groups with $n=\infty$ are called limit groups or Curie groups.

Groups with several higher-order axes

These are the crystallographic groups of a cubic crystal system: 23, 432, $\color {Black}{\tfrac {2}{m}}$Template:Overline, Template:Overline3m, and $\color {Black}{\tfrac {4}{m}}$Template:Overline$\color {Black}{\tfrac {2}{m}}$. All of them contain four diagonal 3-fold axes. These axes are arranged as 3-fold axes in a cube, directed along its four space diagonals (the cube has $\color {Black}{\tfrac {4}{m}}$Template:Overline$\color {Black}{\tfrac {2}{m}}$ symmetry). These symbols are constructed the following way:

• First position — symmetrically equivalent directions of coordinate axes X, Y, Z. They are equivalent due to the presence of diagonal 3-fold axes.
• Second position — diagonal 3 or Template:Overline axes.
• Third position — diagonal directions between any two of three coordinate axes X, Y, and Z. These can be 2, m, or $\color {Black}{\tfrac {2}{m}}$.

All Herman-Mauguin symbols presented above are called full symbols. For many groups they can be simplified by omitting n-fold rotation axes in $\color {Black}{\tfrac {n}{m}}$ positions. This can be done if the rotation axis can be unambiguously obtained from the combination of symmetry elements presented in the symbol. For example, the short symbol for $\color {Black}{\tfrac {2}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$ is mmm, for $\color {Black}{\tfrac {4}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$ is $\color {Black}{\tfrac {4}{m}}$mm, and for $\color {Black}{\tfrac {4}{m}}$Template:Overline$\color {Black}{\tfrac {2}{m}}$ is mTemplate:Overlinem. In groups containing one higher-order axis, this higher-order axis cannot be omitted. For example, symbols $\color {Black}{\tfrac {4}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$ and $\color {Black}{\tfrac {6}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$ can be simplified to 4/mmm (or $\color {Black}{\tfrac {4}{m}}$mm) and 6/mmm (or $\color {Black}{\tfrac {6}{m}}$mm), but not to mmm; the short symbol for Template:Overline$\color {Black}{\tfrac {2}{m}}$ is Template:Overlinem. The full and short symbols for all 32 crystallographic point groups are given in crystallographic point groups page.

Besides five cubic groups, there are two more non-crystallographic icosahedral groups (I and I_h in Schoenflies notation) and two limit groups (K and K_h in Schoenflies notation). The Herman-Mauguin symbols were not designed for non-crystallographic groups, so their symbols are rather nominal and based on similarity to symbols of the crystallographic groups of a cubic crystal system. Group I can be denoted as 235, 25, 532, 53. The possible short symbols for I_h are mTemplate:Overline, mTemplate:Overline, mTemplate:Overlinem, Template:Overlinem. The possible symbols for limit groups are $\infty \infty$ or $2\infty$ for K and $\color {Black}{\tfrac {\infty }{m}}\infty$ or $m{\bar {\infty }}$ or $\infty \infty m$ for K_h.

Plane groups

Plane groups can be depicted using the Hermann-Mauguin system. The first letter is either lowercase p or c to represent primitive or centered unit cells. The next number is the rotational symmetry, as given above. The presence of mirror planes are denoted m, while glide reflections are denoted g.

Space groups

Space groups can be defined by combining the point group identifier with the uppercase letters describing the lattice. Translations within the lattice in the form of screw axes and glide planes are also noted, giving a complete crystallographic space group. An example of a space group would be IaTemplate:Overlined describing symmetry of Garnet crystal structure.

Lattice types

Template:Image Template:Image Template:Image These are the Bravais lattices in three dimensions:

• P primitive
• I body centered (from the German "Innenzentriert")
• F face centered (from the German "Flächenzentriert")
• A nodes on A faces only
• B nodes on B faces only
• C centered on the a,b or A,B face
• R rhombohedral

Screw axis

The screw axis is noted by a number, n, where the angle of rotation is $\color {Black}{\tfrac {360^{\circ }}{n}}$. The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. For example, 2₁ is a 180° (twofold) rotation followed by a translation of ½ of the lattice vector. 3₁ is a 120° (threefold) rotation followed by a translation of ⅓ of the lattice vector.

The possible screw axis are: 2₁, 3₁, 3₂, 4₁, 4₂, 4₃, 6₁, 6₂, 6₃, 6₄, and 6₅. There are 4 enantiomorphic pairs of axes: (3₁ - 3₂), (4₁ - 4₃), (6₁ - 6₅), and (6₂ - 6₄). This enantiomorphism results in 11 pairs of enantiomorphic space groups, namely

Crystal system	Tetragonal			Trigonal			Hexagonal				Cubic
First	P41 (76)	P4122 (91)	P41212 (92)	P31 (144)	P3112 (152)	P3121 (151)	P61 (169)	P62 (171)	P6122 (168)	P6222 (181)	P4132 (213)
Second	P43 (78)	P4322 (95)	P43212 (96)	P32 (145)	P3212 (154)	P3221 (153)	P65 (170)	P64 (172)	P6522 (169)	P6422 (182)	P4332 (212)

Glide planes

Glide planes are noted by a, b, or c depending on which axis the glide is along. There is also the n glide, which is a glide along the half of a diagonal of a face, and the d glide, which is along a quarter of either a face or space diagonal of the unit cell. The d glide is often called the diamond glide plane as it features in the diamond structure.

• $a$, $b$, or $c$ glide translation along half the lattice vector of this face
• $n$ glide translation along with half a face diagonal
• $d$ glide planes with translation along a quarter of a face diagonal.
• $e$ two glides with the same glide plane and translation along two (different) half-lattice vectors.

References

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