Centred Space Groups |

**Centred Space Groups**

So far, the only space considered in detail have been those with a primitive
lattice, *P*. Centred space groups involve additional lattice translations
in addition to those of the whole unit-cell lattice translations as exhibited
by the simplest space group *P*1. A frequently observed centred space
group is the monoclinic space group *C*2/*c*. The properties of
this space group, which is shown below, will be used to demonstrate some of
the properties that result from a *C*-centred lattice. By extension,
similar properties arise from the other lattice centrings.
You should note that there is no symmetry symbol in the diagram to indicate
a centred lattice: however, the presence of it can be inferred from the
combination of symmetry elements present as will be explained.

The space group *C*2/*c* can be considered as a combination of
a *C*-centred lattice with space group *P*2/*c*
(or alternatively space group *P*2_{1}/*n*).

Space group *P*2/*c* has an inversion centre at the origin plus 7
others per unit cell (as for space group
*P*-1 as discussed earlier). Considering
the inversion centre at the origin, then the combination of
1/2+*x*,1/2+*y*,*z*
with -*x*,-*y*,-*z* yields the symmetry operator
1/2-*x*,1/2-*y*,-*z*, which corresponds to an
inversion centre at the point (1/4,1/4,0) as shown in the figure.
This approach can be extended to the other 7 inversion centres of the
primitive space group, yielding a total of 16 per unit cell in the
*C*-centred space group. These 16 should be considered as two groups
of 8 since each set of eight is related as in *P*-1.

Furthermore, space group *P*2/*c* has a twofold rotation axis
at (0,*y*,1/4) plus 3 others per unit cell (each separated by
either half a unit cell in *a* or *c* similar to the
two-one screw axes as discussed earlier). Considering
the twofold rotation at (0,*y*,1/4), then the combination of
1/2+*x*,1/2+*y*,*z* with -*x*,*y*,1/2-*z*
yields the symmetry operator
1/2-*x*,1/2+*y*,1/2-*z*, which corresponds to a
two-one screw axis along the line (1/4,*y*,1/4) as shown in the figure.
This approach can be extended to the other 3 rotation axes of the
primitive space group, yielding a total of 4 screw axes per unit cell in the
*C*-centred space group. Thus one consequence of the *C*-centred
lattice is an equal number of twofold rotation axes and two-one
screw axes.

Finally, considering the *c*-glide plane with symmetry operator
*x*,-*y*,1/2+*z*, then the combination of this
with the *C*-centring results in an *n*-glide plane located
at *y* = 1/4 with symmetry operator
1/2+*x*,1/2-*y*,1/2+*z*. Thus, another consequence of the
*C*-centred lattice is a set of interleaving planes, separated
by 1/4. These can be seen more clearly in the final interactive exercise
that follows in the projections as viewed down either the *a* or
*c* axes.

Finally, an alternative view of this space group is a combination of
the two primitive monoclinic space groups *P*2/*c*
and *P*2_{1}/*n*, which generate a
*C*-centred lattice translations when combined.

© Copyright 1995-2006. Birkbeck College, University of London. |
Author(s):
Jeremy Karl Cockcroft |