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 in appendix A.
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. 6_3 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 The International Tables.
In Crystal Structures, 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
included in appendix A. 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 (ˆ). Spaces are not allowed in the keyboard designation. A couple
of examples: d_4ˆ8, and O_5. The underscore does not
need to precede to superscript. C_2ˆV9 can also be written
CˆV9_2. Each of the 230 space groups as designated by the
Schoenflies notation is listed in Appendix A in the same order as the listing of
the Hermann-Maguin notation. The two conventions are equally supported in the
code.
The atom or basis 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
or basis list, then a few odd things will happen. A series of run-time messages
will be printed to the screen 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 (section 4.1) and the
self-absorption correction (section 4.3) will be calculated incorrectly as well.
The McMaster correction (section 4.2) 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_Hˆ5). 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, gamma 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.
B is the perpendicular axis, thus
beta is the angle not equal to 90.
A, B, and C
must all be specified.
C axis is the unique axis in a
tetragonal cell. The A and B axes are equivalent.
Specify A and C in atoms.inp.
A and
alpha. If the cell has hexagonal axes, specify A and
C. gamma will be set to 120 by the program.
A and
B. Specify A and C in
atoms.inp. Gamma will be set to 120 by the program.
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 reference, La2 Cu O4 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 alpha 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. Gamma 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 for X-Ray Crystallography. 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 that 1 angstrom), 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 (any good scientist's
bugaboo!) 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 -0.125
Here is the input file for Mn3 O4 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.type 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.type 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
-------------------------------------------------
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