The two most commonly used standards for the designation of three dimensional space groups are the Hermann-Maguin and Schoenflies conventions. ATOMS recognizes both conventions. Each of the 230 space groups as designated in each convention is listed later in this chapter.
The Hermann-Maguin system uses four symbols to uniquely specify the group properties of each of the 230 space groups. The first symbol is a single letter P, I, R, F, A, B, or C which refers to the Bravais lattice type. The remaining three letters refer to the point group of the crystal.
Some modifications to the notation convention are made for use with a keyboard. Spaces must separate each of the four symbols. Subscripted numbers are printed next to the number being modified (e.g. 63 is printed as 63). A bar above a number is entered with a minus sign.
Occasionally there are variations in how space groups are referenced. For example, the hausmannite structure of Mn3 O4 is placed in space group I 41/A M D by the conventions laid out in . In v. 3, Wyckoff denotes this space group as I 4/A M D. This sort of incongruity is unfortunate. The list of Hermann-Maguin space group designations as recognized by ATOMS is shown below. If you cannot resolve the incongruity using this list, try using the Schoenflies notation.
The Schoenflies conventions are also recognized by ATOMS. In the literature there is less variation in the application of these conventions. The Schoenflies convention is, in fact, less precise than the Hermann-Maguin in that the complete symmetry characteristics of the crystal are not encoded in the space group designation. Adaptations to the keyboard have been made here as well. Subscripts are denoted with an underscore (_) and superscripts are denoted with a caret (^. Spaces are not allowed in the keyboard designation. A couple of examples: d_4^8, and O_5. The underscore does not need to precede to superscript. C_2V^9 can also be written C_9^2V. Each of the 230 space groups as designated by the Schoenflies notation is listed below in the same order as the listing of the Hermann-Maguin notation. The two conventions are equally supported in the code.
The atoms list in atoms.inp is a list of the unique crystallographic sites in the unit cell. A unique site is one (and only one) of a group of equivalent positions. The equivalent positions are related to one another by the symmetry properties of the crystal. ATOMS determines the symmetry properties of the crystal from the name of the space group and applies those symmetry operations to each unique site to generate all of the equivalent positions.
If you include more than one of a group of equivalent positions in the atom list, then a few odd things will happen. A series of run-time messages will issued telling you that atom positions were found that were coincident in space. This is because each of the equivalent positions generated the same set of points in the unit cell. ATOMS removes these redundancies from the atom list. The atom list and the potentials list written to feff.inp will be correct and FEFF can be run correctly using this output. However, the site tags and the indexing of the atoms will certainly make no sense. Also the density of the crystal will be calculated incorrectly, thus the absorption calculation and the self-absorption correction will be calculated incorrectly as well. The McMaster correction is unaffected.
For some common crystal types it is convenient to have a shorthand way of designating the space group. For instance, one might remember that copper is an fcc crystal, but not that it is in space group F M 3 M (or O_H^5). In this spirit, ATOMS will recognize the following words for common crystal types. These words may be used as the value of the keyword space and ATOMS will supply the correct space group. Note that several of the common crystal types are in the same space groups. For copper it will still be necessary to specify that an atom lies at (0,0,0), but it isn't necessary to remember that the space group is F M 3 M.
_______________________________________________________ cubic | cubic | P M 3 M body-centered cubic | bcc | I M 3 M face-centered cubic | fcc | F M 3 M halite | salt or nacl | F M 3 M zincblende | zincblende or zns | F -4 3 M cesium chloride | cscl | P M 3 M perovskite | perovskite | P M 3 M diamond | diamond | F D 3 M hexagonal close pack | hex or hcp | P 63/M M C graphite | graphite | P 63 M C -------------------------------------------------------
When space is set to hex or graphite, γ is automatically set to 120.
ATOMS assumes certain conventions for each of the Bravais lattice types. Listed here are the labeling conventions for the axes and angles in each Bravais lattice.
Triclinic: All axes and angles must be specified.
Monoclinic: B is the perpendicular axis, thus β is the angle not equal to 90.
Orthorhombic: A, B, and C must all be specified.
Tetragonal: The C axis is the unique axis in a tetragonal cell. The A and B axes are equivalent. Specify A and C in atoms.inp.
Trigonal: If the cell is rhombohedral then the three axes are equivalent as are the three angles. Specify A and α. If the cell has hexagonal axes, specify A and C. γ will be set to 120 by the program.
Hexagonal: The equivalent axes are A and B. Specify A and C in atoms.inp. γ will be set to 120 by the program.
Cubic: Specify A in atoms.inp. The other axes will be set equal to A and the angles will all be set to 90.
In three dimensional space there is an ambiguity in choice of right handed coordinate systems. Given a set of mutually orthogonal axes, there are six choices for how to label the positive x, y, and z directions. For some specific physical problem, the crystallographer might choose a non-standard setting for a crystal. The choice of standard setting is described in detail in The International Tables. The Hermann-Maguin symbol describes the symmetries of the space group relative to this choice of coordinate system.
The symbols for triclinic crystals and for crystals of high symmetry are insensitive to choice of axes. Monoclinic and orthorhombic notations reflect the choice of axes for those groups that possess a unique axis. Tetragonal crystals may be rotated by 45 degrees about the z axis to produce a unit cell of doubled volume and of a different Bravais type. Alternative symbols for those space groups that have them are listed in Appendix A.
ATOMS recognizes those non-standard notations for these crystal classes that are tabulated in The International Tables. atoms.inp may use any of these alternate notations so long as the specified cell dimensions and atomic positions are consistent with the choice of notation. Any notation not tabulated in chapter 6 of the 1969 edition of The International Tables will not be recognized by ATOMS.
This resolution of ambiguity in choice of coordinate system is one of the main advantages of the Hermann-Maguin notation system over that of Shoenflies. In a situation where a non-standard setting has been chosen in the literature, use of the Schoenflies notation will, for many space groups, result in unsatisfactory output from ATOMS. In these situations, ATOMS requires the use of the Hermann-Maguinn notation to resolve the choice of axes.
Here is an example. In the literature, La2CuO4 was given in the non-standard b m a b setting rather than the standard c m c a. As you can see from the axes and coordinates, these settings differ by a 90 degree rotation about the A axis. The coordination geometry of the output atom list will be the same with either of these input files, but the actual coordinates will reflect this 90 degree rotation.
title La2CuO4 structure at 10K from Radaelli et al. title standard setting space c m c a a= 5.3269 b= 13.1640 c= 5.3819 rmax= 8.0 core= la atom la 0 0.3611 0.0074 Cu 0 0 0 O 0.25 -0.0068 -0.25 o1 O 0 0.1835 -0.0332 o2 --------------------------------------
title La2CuO4 structure at 10K from Radaelli et al. title non standard setting, rotated by 90 degrees about A axis space b m a b a= 5.3269 b= 5.3819 c= 13.1640 rmax= 8.0 core= la atom la 0 -0.0074 0.3611 Cu 0 0 0 O 0.25 0.25 -0.0068 o1 O 0 0.0332 0.1835 o2 --------------------------------------
There are seven rhombohedral space groups. Crystals in any of these space groups that may be represented as either monomolecular rhombohedral cells or as trimolecular hexagonal cells. These two representations are entirely equivalent. The rhombohedral space groups are the ones beginning with the letter R in the Hermann-Maguin notation. ATOMS does not care which representation you use, but a simple convention must be maintained. If the rhombohedral representation is used then the keyword α must be specified in atoms.inp to designate the angle between the rhombohedral axes and the keyword a must be specified to designate the length of the rhombohedral axes. If the hexagonal representation is used, then a and c must be specified in atoms.inp. γ will be set to 120 by the code. Atomic coordinates consistent with the choice of axes must be used.
Some space groups in The International Tables are listed with two possible origins. The difference is only in which symmetry point is placed at (0,0,0). ATOMS always wants the orientation labeled “origin-at-centre”. This orientation places (0,0,0) at a point of highest crystallographic symmetry. Wyckoff and other authors have the unfortunate habit of not choosing the “origin-at-centre” orientation when there is a choice. Again Mn3 O4 is an example. Wyckoff uses the “origin at -4m2” option, which places one Mn atom at (0,0,0) and another at (0,1/4,5/8). ATOMS wants the “origin-at-centre” orientation which places these atoms at (0,3/4,1/8) and (0,0,1/2). Admittedly, this is an arcane and frustrating limitation of the code, but it is not possible to conclusively check if the “origin-at-centre” orientation has been chosen.
Twenty one of the space groups are listed with two origins in The International Tables. ATOMS knows which groups these are and by how much the two origins are offset, but cannot know if you chose the correct one for your crystal. If you use one of these groups, ATOMS will print a run-time message warning you of the potential problem and telling you by how much to shift the atomic coordinates in atoms.inp if the incorrect orientation was used. This warning will also be printed at the top of the feff.inp file. If you use the “origin-at-center” orientation, you may ignore this message.
If you use one of these space groups, it usually isn't hard to know if you have used the incorrect orientation. Some common problems include atoms in the atom list that are very close together (less than 1 Å), unphysically large densities (see section 4.1), and interatomic distances that do not agree with values published in the crystallography literature. Because it is tedious to edit the atomic coordinates in the input file every time this problem is encountered and because forcing the user to do arithmetic invites trouble, there is a useful keyword called shift. For the Mn3 O4 example discussed above, simply insert this line in atoms.inp if you have supplied coordinates referenced to the incorrect origin:
shift = 0.0 0.25 -0.125This vector will be added to all of the coordinates in the atom list after the input file is read.
Here is the input file for Mn3O4 using the shift keyword:
title Mn3O4, hausmannite structure, using the shift keyword a 5.75 c 9.42 core Mn2 rmax 7.0 Space i 41/a m d shift 0.0 0.25 -0.125 atom * At x y z tag Mn 0.0 0.0 0.0 Mn1 Mn 0.0 0.25 0.625 Mn2 O 0.0 0.25 0.375
The above input file gives the same output as the following. Here the shift keyword has been removed and the shift vector has been added to all of the fractional coordinates. These two input files give equivalent output.
title Mn3O4, hausmannite structure, no shift keyword a 5.75 c 9.42 core Mn2 rmax 7.0 Space i 41/a m d atom * At x y z tag Mn 0.0 0.25 -0.125 Mn1 Mn 0.0 0.50 0.50 Mn2 O 0.0 0.50 0.25
The following 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 a real crystallography text.
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.
System: The undecorated shape of the unit cell.
Bravais Lattice: An undecorated lattice of equivalent points.
Crystal Class: The description of the symmetries about a point.
Space Group: The complete description of three dimensional crystal symmetries.
There are seven systems of crystals. The system refers to the shape of the undecorated unit cell. They are:
Triclinic: a ≠ b ≠ c, α ≠ β ≠ γ ≠ 90°
Monoclinic: a ≠ b ≠ c, α=γ=90°, β ≠ 90°
Orthorhombic: a ≠ b ≠ c, α=β=γ=90°
Tetragonal: a = b ≠ c, α=β=γ=90°
Hexagonal: a = b ≠ c, α=β=90°, γ=120°
a = b = c, α=β=γ < 120° ≠ 90°
(hexagonal axes): a = b ≠ c, α=β=90°, γ=120°
Cubic: a = b = c, α=β=γ=90°
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° 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° rotary inversion with a mirror reflection is identical to the operation of a simple 180° 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:
Monoclinic: the B axis
Orthorhombic: the C axis
Tetragonal: the C axis
Hexagonal: the C axis
Trigonal: the A axis
Cubic: the A axis
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) degrees 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, e.g. F M 3 M.
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
standard abc cab bca a-cb ba-c -cab ----------------------------------------------------------------- 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