Overview of symmetry information (CCP4: General)
General references
Definitive guides are the
International Tables,
particularly volumes A and F.
Other CCP4 documentation
- alternate_origins - lists of possible
alternative origins for spacegroups.
- cheshirecell - a description of the use
of Cheshire cells in Molecular Replacement, and a table of Cheshire cells.
- twinning - a description of twinning together
with look-up tables for twinning operators
- CSYM library - documentation for
the C API to the symmetry library
- symlib library - documentation for
the Fortran API to the symmetry library
syminfo.lib
From CCP4 5.0, information for all spacegroups in a variety of settings
is tabulated in syminfo.lib
(which is distributed in the directory $CCP4/lib/data and replaces the
previous symop.lib). A description of the contents of this file is
given in the documentation of the Fortran API.
hexagonal vs. rhombohedral settings e.g. H3 vs. R3
From CCP4 4.2, trigonal spacegroups in a centered hexagonal setting are
denoted by "H". The symbol "R" is reserved for these spacegroups in
a rhombohedral setting.
Explanation: All trigonal spacegroups (groups 143 to 167) can
be described by hexagonal axes with a = b, alpha = beta = 90, gamma = 120
(they have hexagonal metric symmetry but are not in the hexagonal crystal
system because they lack
a 6-fold symmetry operator). A subset of trigonal groups possess two
centering operators: x+2/3,y+1/3,z+1/3 and x+1/3,y+2/3,z+2/3 These groups
are denoted by "H" e.g. "H 3", as opposed to the primitive spacegroups
such as "P 3". In the extended Hermann Mauguin symbol, it is denoted
by ":H" e.g. "R 3 :H".
These trigonal spacegroups which are centered in the hexagonal representation
can also be represented in terms of rhombohedral axes. The rhombohedral
unit cell has a = b = c, alpha = beta = gamma. The spacegroup lacks the
centering operators, and the unit cell is a third of the volume of that
in the hexagonal representation. This representation of the spacegroup
is denoted by "R" e.g. "R 3". In the extended Hermann Mauguin symbol, it is
denoted by ":R" e.g. "R 3 :R".
Symmetry of reflection data
The quoted spacegroup describes the symmetry of the protein crystal in
real space, i.e. the spatial distribution of molecules in the crystal.
The diffraction pattern produced by the crystal has, in general, point
group symmetry about (0,0,0). The point group is derived from the
spacegroup by removing the centering letter (P, A, H, etc.), replacing
screw axes by rotation axes, and replacing glide planes by mirror planes.
For example:
- "P 21" -> "2"
- "P 21 21 21" -> "222"
- "P 43 21 2" -> "422"
- "P 3 1 2" -> "312"
There are 32 point groups in total.
If Friedel's Law is obeyed (i.e. if anomalous scattering is neglected) then
the diffraction pattern has inversion symmetry about (0,0,0) since
I(h, k, l) = I(-h, -k, -l). Adding the inversion operator gives the Laue
class:
- "P 21" -> "2/m"
- "P 21 21 21" -> "2/m 2/m 2/m" or "mmm"
- "P 43 21 2" -> "4/mmm"
- "P 3 1 2" -> "-31m"
There are 11 possible Laue classes, one each for triclinic ("-1"), monoclinic ("2/m")
and orthorhombic ("mmm"), and two each for tetragonal ("4/m" and "4/mmm"),
trigonal ("-3" and "-3m"), hexagonal ("6/m" and "6/mmm") and cubic
("m-3" and "m-3m").
However, to take account of different spacegroup settings (e.g. choice of unique
axis), further Laue classes may be defined. CCP4 uses the 11 standard classes
but divides "-3m" into "-31m" and "-3m1"
The standard MTZ file holds a list of all reflections in the asymmetric unit
of the Laue class (see e.g. the program unique).
In the general case, different point groups can be distinguished when
anomalous data is available, since there is no longer necessarily inversion
symmetry. In practice, for protein crystals, there is only one possible
point group for each Laue class, so the choice is automatic.
Martyn Winn, Daresbury