This appendix is my attempt to demystify the crazy symbolism used by the Hermann-Maguin and Schoenflies conventions. This is by no means an adequate explanation of the rich and beautiful field of crystallography. For that, I recommend one of the books in the bibliography.
An important part of the demystification process is to define some of the important terms used to describe crystal symmetries. The words system, Bravais lattice, crystal class, and space group have well-defined meanings. The symbols used in each of the notation conventions specifically relate the various symmetries of crystals. In crystallography, a symmetry operation is defined as a sequence of reflections, translations, and/or rotations that map the crystal back onto itself in such a way that the crystal after the mapping is indistinguishable from the crystal before the mapping.
To start, here are some definitions. These will be elaborated below.
There are seven systems of crystals. The system refers to the shape of the undecorated unit cell. They are:
There are fourteen Bravais lattices. The Bravais lattices are
constructed from the simplest translational symmetries applied to the
seven crystal systems. A P
lattice has decoration only at the
corners of the unit cell. An I
lattice has decoration at the
body center of the cell as well as at the corners. An F
lattice
has decoration at the face centers as well as at the corners. A
C
lattice has decoration at the center of the (001) face as well
as at the corners. Likewise A
and B
lattices have
decoration at the centers of the (100) and (010) faces respectively.
R
lattices are a special type in the trigonal system which
possess rhombohedral symmetry.
All seven crystal systems have P
lattices, but not all the
classes have the other type of Bravais lattices. This is because
there is degeneracy when all the Bravais lattice types are applied to
all the crystal systems. For example, a face centered tetragonal cell
can be expressed as a body centered tetragonal cell by rotating the
two equivalent axes by 45 deg and shortening them by a factor of
square root of 2. Considering such degeneracies reduces the possible
decorations of the seven systems to these 14 unique three dimensional
lattices:
Triclinic P
Monoclinic P, C
Orthorhombic P, C, I, F
Tetragonal P, I
Hexagonal P
Trigonal P, R
Cubic P, I, F
For historic reasons, hexagonal cells are sometimes called C
lattices. atoms will recognize hexagonal P
cells denoted in
atoms.inp
by the letter C
. Modern literature usually uses
the P
designation.
The decorations placed on the Bravais lattices come in 32 flavors called classes or point groups which represent the possible symmetries within the decorations. Each type of symmetry is defined either by a reflection plane, a rotation axis, or a rotary inversion axis. A reflection plane can either be a simple mirror plane or a glide plane, which defines the symmetry operation of reflecting through a mirror followed by translating along a direction in the plane. A rotation axis can either define a simple rotation or a screw rotation, which is the symmetry operation of rotating about the axis followed by translating along that axis. A rotary inversion axis defines the symmetry operation of reflecting through a plane followed by rotating about an axis in that plane.
These three symmetry types, reflection plane, rotation axis, and rotary inversion axis, can be combined in 32 non-degenerate ways. (An example degeneracy: the symmetry operation of combining a 180 deg rotary inversion with a mirror reflection is identical to the operation of a simple 180 deg rotation.) It would seem that the 32 classes could decorate the 14 Bravais lattices in 458 ways. In fact, the number might be larger as there are numerous types of screw axes and glide planes. Again, considering degeneracies reduces the total number of combinations, leaving 230 unique decorations of the Bravais lattices. These are called space groups. The 230 space groups are a rigorously complete set of descriptions of crystal symmetries in three dimensional space. That is, there may be new crystals but there are no new space groups. Here I am only considering space-filling crystals with translational periodicity. 3-D Penrose structures and quasi-crystals are outside the realm of this appendix and of the code.
The Hermann-Maguin notation uses a set of two to four symbols to completely specify the symmetries of a space group. The first symbol is always a single letter specifying the Bravais lattice. The next three symbols specify the class of the space group. These three symbols are some combination of the following characters:
1 2 3 4 5 6 A B C D M N / -
These are sufficient to completely specify the various planar and axial symmetries of the classes and sub-classes. The following is a discussion of the most important rules of this convention. Some details are neglected but sufficient information is provided to appreciate the information contained in the notation.
The second symbol in the Hermann-Maguin notation, i.e. the one after the Bravais lattice symbol, tells about symmetries involving the primary axis of the cell and/or of the plane normal to the primary axis. The primary axis is defined as follows:
In cubic or rhombohedral lattices the axes are equivalent, thus the primary axis is arbitrary. For orthorhombic lattices the third and fourth symbols specify the symmetries of the a and b axes respectively. In other lattices, the last two symbols encode the remaining symmetries as described below.
A space filling crystal will always show a symmetry when rotated
through (360/n) degrees, where n is one of {1,2,3,4, or 6}.
The second symbol often tells the rotational symmetry properties of
the primary axis. Notice that all trigonal, tetragonal, and hexagonal
groups have a 3
, 4
, or 6
respectively in their
designations. Many orthorhombic and monoclinic groups have a 2
,
which is the highest degree of rotational symmetry available to those
lattices. Cubic groups may possess 2- or 4-fold rotational symmetry
about the cell axes, thus have 2
or 4
in the second symbol.
Many second symbols contain a second number. This is the subscripted
number when the Hermann-Maguin notation is typeset. This refers to
the type of screw symmetry associated with the axis. A screw
symmetric lattice is mapped onto itself by an anti-clockwise rotation
through m * (360/n) degrees and a translation of 1/n
up the primary axis. Here n is the degree of rotational symmetry, m
is the type of screw, and the definition of rotation and direction is
right-handed. Two types of screw symmetry that are different only in
handedness of rotation are called enantiomorphous. The
enantiomorphous pairs are 31
and 32
, 41
and 43
,
61
and 65
, and 62
and 64
.
Several of the second symbols are one or two numbers followed by a
slash and a letter, e.g. P 63/M M C
. The letter specifies
the type of reflection plane that is normal to the rotation axis.
There are several types of reflection planes. The simplest is a
mirror plane, denoted by the letter M
. This says the crystal is
mapped onto itself by reflecting all atoms through a mirror placed in
an appropriate plane in the crystal. The letters A
, B
, or
C
denote glide planes. These map the crystal onto itself by
reflecting through the plane then translating elements of the crystal
by half the length of the cell axis normal to the reflection plane. A
D
glide plane is similar but involves translations of a quarter of the
cell axis length. Finally, the letter N
denotes a diagonal glide
plane, which is a reflection through a plane followed by a translation
in the same plane of half the length of both cell axes in that plane.
The symbol -
before a number indicates a rotary inversion axis.
This maps the crystal back onto itself by rotating through
(360/n) deg then reflecting through a plane parallel to the
rotation axis.
A final word about the Hermann-Maguin notation. All cubic space groups have four three-fold rotational axes through the body diagonals. Thus all cubic groups have the number 3 as the third symbol.
The Schoenflies notation uses a set of three symbols to classify sets
of space groups by their dominant symmetry features. The letters
C
, D
, S
, T
, and O
denote the character of the
center of symmetry. The symbol after the underscore (the subscript
when typeset) indicates the presence of symmetry planes and additional
symmetry axes. The number after the caret (the superscript when
typeset) is simply an indexing of all the distinct space groups that
share major symmetry properties. In the older literature, D
symmetry centers are occasionally referred to as V
. atoms
will probably understand a space group referred to by the letter
V
, but using the D
notation is recommended.
The letter C
indicates an rotation axis where the crystal is mapped
onto itself when rotated by (360/n) deg, where n is the number after
the underscore. An H after the underscore indicates the presence of a
plane of symmetry normal to the rotation axis. A V
after the
underscore indicates one or two planes of symmetry parallel to the
rotation axis. The letter S
after the underscore indicates a normal
plane of symmetry in a crystal where the degree of rotational symmetry
is 1. The letter I
after the underscore indicates the presence of a
point center of symmetry.
The letter S
indicates a rotary inversion axis. The degree of rotation
is the number after the underscore.
The letter D
denotes a primary rotation axis with another
rotation axis normal to it. The degree of rotation of both axes is
the number after the underscore. The letters H
and V
have
the same meanings as they did in groups beginning with the letter
C
. The letter D
indicates the presence of a diagonal
symmetry plane.
Cubic groups are all specified by the letters T
and O
.
T
indicates tetrahedral symmetry, that is, the presence of the
four three-fold axes and three two-fold axes. O
indicates
octahedral symmetry, i.e. four three-fold axes with three four-fold
axes. H
and D
after the underscore carry the same meaning
as before.
2 Triclinic and 13 Monoclinic Space Groups
P 1 P -1 P 2 P 21 C 2 P M
P C C M C C P 2/M P 21/M C 2/M
P 2/C P 21/C C 2/C
59 Orthorhombic Space Groups
P 2 2 2 P 2 2 21 P 21 21 2 P 21 21 21 C 2 2 21 C 2 2 2
F 2 2 2 I 2 2 2 I 21 21 21 P M M 2 P M C 21 P C C 2
P M A 2 P C A 21 P N C 2 P M N 21 P B A 2 P N A 21
P N N 2 C M M 2 C M C 21 C C C 2 A M M 2 A B M 2
A M A 2 A B A 2 F M M 2 F D D 2 I M M 2 I B A 2
I M A 2 P M M M P N N N P C C M P B A N P M M A
P N N A P M N A P C C A P B A M P C C N P B C M
P N N M P M M N P B C N P B C A P N M A C M C M
C M C A C M M M C C C M C M M A C C C A F M M M
F D D D I M M M I B A M I B C A I M M A
68 Tetragonal Space Groups
P 4 P 41 P 42 P 43 I 4 I 41
P -4 I -4 P 4/M P 42/M P 4/N P 42/N
I 4/M I 41/A P 4 2 2 P 4 21 2 P 41 2 2 P 41 21 2
P 42 2 2 P 42 21 2 P 43 2 2 P 43 21 2 I 4 2 2 I 41 2 2
P 4 M M P 4 B M P 42 C M P 42 N M P 4 C C P 4 N C
P 42 M C P 42 B C I 4 M M I 4 C M I 41 M D I 41 C D
P -4 2 M P -4 2 C P -4 21 M P -4 21 C P -4 M 2 P -4 C 2
P -4 B 2 P -4N2 I -4 M 2 I -4 C 2 I -42 M I -42 D
P 4/M M M P 4/M C C P 4/N B M P 4/N N C P 4/M B M P 4/M N C
P 4/N M M P 4/N C C P 42/M M C P 42/M C M P 42/N B C P 42/N N M
P 42/M B C P 42/M N M P 42/N M C P 42/N C M I 4/M M M I 4/M C M
I 41/A M D I 41/A C D
25 Trigonal Space Groups
P 3 P 3 1 P 32 R3 P -3 R -3
P 3 1 2 P 3 2 1 P 31 1 2 P 31 2 1 P 32 1 2 P 32 2 1
R 32 P 3 M 1 P 3 1 M P 3 C 1 P 3 1 C R 3 M
R 3C P -3 1 M P -3 1 C P -3 M 1 P -3 C 1 R -3 M
R -3 C
27 Hexagonal Space Groups
P 6 P 61 P 65 P 62 P 64 P 63
P -6 P 6/M P 63/M P 62 2 P 61 2 2 P 65 2 2
P 62 2 2 P 64 2 2 P 63 2 2 P 6 M M P 6 C C P 63 C M
P 63 M C P -6 M 2 P -6 C 2 P -6 2 M P -62 C P 6/M M M
P 6/M C C P 63/M C M P 63/M M C
36 Cubic Space Groups
P 2 3 F 2 3 I 2 3 P 21 3 I 21 3 P M 3
P N 3 F M 3 F D 3 I M 3 P A 3 I A 3
P 4 3 2 P 42 3 2 F 4 3 2 F 41 3 2 I 4 3 2 P 43 3 2
P 41 3 2 I 41 3 2 P -4 3 M F -4 3 M I -4 3 M P -4 3 N
F -4 3 C I -4 3 D P M 3 M P N 3 N P M 3 N P N 3 M
F M 3 M F M 3 C F D 3 M F D 3 C I M 3 M I A 3 D
Here are the notations for the alternate settings of the monoclinic and orthorhombic space groups. Also presented are the notations for tetragonal space groups that have been rotated by 45 degrees resulting in a unit cell of doubled volume and of a different Bravais type.
In an monoclinic or orthorhombic space group, the Hermann-Maguin symbols are identical for the various settings if none of the three axes possess special symmetry properties. In this case the three axes are distinguished only by length and the symbol is the same for all settings.
The column headings below indicate the orientations of the alternative
settings relative to the standard setting. For instance, cab
is
a setting with axes and coordinates cyclically permuted from the
standard setting. This is equivalent to a rotation of 120 degrees
about an axis in a <111> direction relative to the Cartesian
axes. The setting a-cb
is rotated by 90 degrees about the A
axis. Thus the B
and C
axes are swapped and the y
and
z
coordinates in the standard setting map onto the z
and
-y
coordinates of the alternate setting. In atoms, when an
alternative setting is specified in atoms.inp
, the axes and
coordinates are multiplied by the appropriate permutation matrix onto
the standard setting. The positions in the unit cell are expanded
according to the Hermann-Maguin symbol for the standard setting. The
contents of the unit cell are then permuted back to the specified
setting.
Symbols for Monoclinic Groups of Various Settings
standard | standard
abc bca | abc bca
-----------------------------------------------------------------
P 2 both settings | P 21 both settings
B 2 or C 2 | P M both settings
P B or P C | B M or C M
B B or C C | P 2/M both settings
P 21/M both settings | B 2/M or C 2/M
P 2/B or P 2/C | P 21/B or P 2/C
B 2/B or C 2/C |
Symbols for Orthorhombic Groups of Various Settings
standard
abc cab bca a-cb ba-c -cab
--------------------------------------------------------------------
P 2 2 2 each setting
P 2 2 21 P 21 2 2 P 2 21 2 P 2 21 2 P 2 2 21 P 21 2 2
P 21 21 2 P 2 21 21 P 21 2 21 P 21 2 21 P 21 21 2 P 2 21 21
P 21 21 21 each setting
C 2 2 21 A 21 2 2 B 2 21 2 B 2 21 2 C 2 2 21 A 21 2 2
C 2 2 2 A 2 2 2 B 2 2 2 B 2 2 2 C 2 2 2 A 2 2 2
F 2 2 2 each setting
I 2 2 2 each setting
I 21 21 21 each setting
P M M 2 P 2 M M P M 2 M P M 2 M P M M 2 P 2 M M
P M C 21 P 21 M A P B 21 M P M 21 B P C M 21 P 21 A M
P C C 2 P 2 A A P B 2 B P B 2 B P C C 2 P 2 A A
P M A 2 P 2 M B P C 2 M P M 2 A P B M 2 P 2 C M
P C A 21 P 21 A B P C 21 B P B 21 A P B C 21 P 21 C A
P N C 2 P 2 N A P B 2 N P N 2 B P C N 2 P 2 A N
P M N 21 P 21 M N P N 21 M P M 21 N P N M 21 P 2 N M
P B A 2 P 2 C B P C 2 A P C 2 A P B A 2 P 2 C B
P N A 21 P 21 N B P C 21 N P N 21 A P B N 21 P 2 C N
P N N 2 P 2 N N P N 2 N P N 2 N P N N 2 P 2 N N
C M M 2 A 2 M M B M 2 M B M 2 M C M M 2 A 2 M M
C M C 21 A 21 M A B B 21 M B M 21 B C C M 21 A 21 A M
C C C 2 A 2 C A B B 2 C B B 2 B C C C 2 A 2 A A
A M M 2 B 2 M M C M 2 M A M 2 M B M M 2 C 2 M M
A B M 2 B 2 C M C M 2 A A C 2 M B M A 2 C 2 M B
A M A 2 B 2 M B C C 2 M A M 2 A B B M 2 C 2 C M
A B A 2 B 2 C B C C 2 A A C 2 A B B A 2 C 2 C B
F M M 2 F 2 M M F M 2 M F M 2 M F M M 2 F 2 M M
F D D 2 F 2 D D F D 2 D F D 2 D F D D 2 F 2 D D
I M M 2 I 2 M M I M 2 M I M 2 M I M M 2 I 2 M M
I B A 2 I 2 C B I C 2 A I C 2 A I B A 2 I 2 C B
I M A 2 I 2 M B I C 2 M I M 2 A I B M 2 I 2 C M
P M M M each setting
P N N N each setting
P C C M P M A A P B M B P B M B P C C M P M A A
P B A N P N C B P C N A P C N A P B A N P N C B
P M M A P B M M P M C M P M A M P M M B P C M M
P N N A P B N N P N C N P N A N P N N B P C N N
P M N A P B M N P N C M P M A N P N M B P C N M
P C C A P B A A P B C B P B A B P C C B P C A A
P B A M P M C B P C M A P C M A P B A M P M C B
P C C N P N A A P B N B P B N B P C C N P N A A
P B C M P M C A P B M A P C M B P C A M P M A B
P N N M P M N N P N M N P N M N P N N M P M N N
P M M N P N M M P M N M P M N M P M M N P N M M
P B C N P N C A P B N A P C N B P C A N P N A B
P B C A P B C A P B C A P C A B P C A B P C A B
P N M A P B N M P M C N P N A M P M N B P C M N
C M C M A M M A B B M M B M M B C C M M A M A M
C M C A A B M A B B C M B M A B C C M B A C A M
C M M M A M M M B M M M B M M M C M M M A M M M
C C C M A M A A B B M B B B M B C C C M A M A A
C M M A A B M M B M C M B M A M C M M B A C M M
C C C A A B A A B B C B B B A B C C C B A C A A
F M M M each setting
F D D D each setting
I M M M each setting
I B A M I M C B I C M A I C M A I B A M I M C B
I B C A I B C A I B C A I C A B I C A B I C A B
I M M A I B M M I M C M I M A M I M M B I C M M
Symbols for Tetragonal Groups of Various Orientations
standard | standard
abc (a+b)(b-a)c | abc (a+b)(b-a)c
---------------------------------------------------------------------------
P 4 or C 4 | P 41 or C 41
P 42 or C 42 | P 43 or C 43
I 4 or F 4 | I 41 or F 41
P -4 or C -4 | I -4 or F -4
P 4/M or C 4/M | P 42/M or C 42/M
P 4/N or C 4/A | P 42/M or C 42/A
I 4/M or F 4/M | I 41/A or F 41/D
P 4 2 2 or C 4 2 2 | P 4 2 21 or C 4 2 21
P 41 2 2 or C 41 2 2 | P 41 2 21 or C 41 2 21
P 42 2 2 or C 42 2 2 | P 42 2 21 or C 42 2 21
P 43 2 2 or C 43 2 2 | P 43 2 21 or C 43 2 21
I 4 2 2 or F 4 2 2 | I 41 2 2 or F 41 2 2
P 4 M M or C 4 M M | P 4 B M or C 4 M B
P 42 C M or C 42 M C | P 42 N M or C 42 M N
P 4 C C or C 4 C C | P 4 N C or C 4 C N
P 42 M C or C 42 C M | P 42 B C or C 42 C B
I 4 M M or F 4 M M | I 4 C M or F 4 M C
I 41 M D or F 41 D M | I 41 C D or F 41 D C
P -4 2 M or C -4 M 2 | P -4 2 C or C -4 C 2
P -4 21 M or C -4 M 21 | P -4 21 C or C -4 C 21
P -4 M 2 or C -4 2 M | P -4 C 2 or C -4 2 C
P -4 B 2 or C -4 2 B | P -4 N 2 or C -4 2 N
I -4 M 2 or F -4 2 M | I -4 C 2 or F -4 2 C
I -4 2 M or F -4 M 2 | I -4 2 D or F -4 D 2
P 4/M M M or C 4/M M M | P 4/M C C or C 4/M C C
P 4/N B M or C 4/A M B | P 4/N N C or C 4/A C N
P 4/M B M or C 4/M M B | P 4/M N C or C 4/M C N
P 4/N M M or C 4/A M M | P 4/N C C or C 4/A C C
P 42/M M C or C 42/M C M | P 42/M C M or C 42/M M C
P 42/N B C or C 42/A C B | P 42/N N M or C 42/A M N
P 42/M B C or C 42/M C B | P 42/M N M or C 42/M M N
P 42/N M C or C 42/A C M | P 42/N C M or C 42/A M C
I 4/M M M or F 4/M M M | I 4/M C M or F 4/M M C
I 41/A M D or F 41/D D M | I 41/A C D or F 41/D D C
2 Triclinic and 13 Monoclinic Space Groups
C_1^1 C_I^1 C_2^1 C_2^2 C_2^3 C_S^1
C_S^2 C_S^3 C_S^4 C_2H^1 C_2H^2 C_2H^3
C_2H^4 C_2H^5 C_2H^6
59 orthorhombic space groups
D_2^1 D_2^2 D_2^3 D_2^4 D_2^5 D_2^6
D_2^7 D_2^8 D_2^9 C_2V^1 C_2V^2 C_2V^3
C_2V^4 C_2V^5 C_2V^6 C_2V^7 C_2V^8 C_2V^9
C_2V^10 C_2V^11 C_2V^12 C_2V^13 C_2V^14 C_2V^15
C_2V^16 C_2V^17 C_2V^18 C_2V^19 C_2V^20 C_2V^21
C_2V^22 D_2H^1 D_2H^2 D_2H^3 D_2H^4 D_2H^5
D_2H^6 D_2H^7 D_2H^8 D_2H^9 D_2H^10 D_2H^11
D_2H^12 D_2H^13 D_2H^14 D_2H^15 D_2H^16 D_2H^17
D_2H^18 D_2H^19 D_2H^20 D_2H^21 D_2H^22 D_2H^23
D_2H^24 D_2H^25 D_2H^26 D_2H^27 D_2H^28
68 Tetragonal space groups
C_4^1 C_4^2 C_4^3 C_4^4 C_4^5 C_4^6
S_4^1 S_4^2 C_4H^1 C_4H^2 C_4H^3 C_4H^4
C_4H^5 C_4H^6 D_4^1 D_4^2 D_4^3 D_4^4
D_4^5 D_4^6 D_4^7 D_4^8 D_4^9 D_4^10
C_4V^1 C_4V^2 C_4V^3 C_4V^4 C_4V^5 C_4V^6
C_4V^7 C_4V^8 C_4V^9 C_4V^10 C_4V^11 C_4V^12
D_2D^1 D_2D^2 D_2D^3 D_2D^4 D_2D^5 D_2D^6
D_2D^7 D_2D^8 D_2D^9 D_2D^10 D_2D^11 D_2D^12
D_4H^1 D_4H^2 D_4H^3 D_4H^4 D_4H^5 D_4H^6
D_4H^7 D_4H^8 D_4H^9 D_4H^10 D_4H^11 D_4H^12
D_4H^13 D_4H^14 D_4H^15 D_4H^16 D_4H^17 D_4H^18
D_4H^19 D_4H^20
25 Trigonal space groups
C_3^1 C_3^2 C_3^3 C_3^4 C_3I^1 C_3I^2
D_3^1 D_3^2 D_3^3 D_3^4 D_3^5 D_3^6
D_3^7 C_3V^1 C_3V^2 C_3V^3 C_3V^4 C_3V^5
C_3V^6 D_3D^1 D_3D^2 D_3D^3 D_3D^4 D_3D^5
D_3D^6
27 Hexagonal space groups
C_6^1 C_6^2 C_6^3 C_6^4 C_6^5 C_6^6
C_3H^1 C_6H^1 C_6H^2 D_6^1 D_6^2 D_6^3
D_6^4 D_6^5 D_6^6 C_6V^1 C_6V^2 C_6V^3
C_6V^4 D_3H^1 D_3H^2 D_3H^3 D_3H^4 D_6H^1
D_6H^2 D_6H^3 D_6H^4
36 Cubic space groups
T^1 T^2 T^3 T^4 T^5 T_H^1
T_H^2 T_H^3 T_H^4 T_H^5 T_H^6 T_H^7
O^1 O^2 O^3 O^4 O^5 O^6
O^7 O^8 T_D^1 T_D^2 T_D^3 T_D^4
T_D^5 T_D^6 O_H^1 O_H^2 O_H^3 O_H^4
O_H^5 O_H^6 O_H^7 O_H^8 O_H^9 O_H^10
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