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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