# Symmetries in silicon

by Kreck
Tags: silicon, symmetries
 P: 3 As I'm interested in the simplifications of property tensors due to crystal symmetry, I have been trying to find the symmetries of silicon (i.e. the diamond structure). As silicon belongs to the m3m point group I would e.g. expect to find a mirror plane perpendicular to the [100], [010] and [001] directions. I have, however, not been able to locate the coordinate system in which these (and all the other m3m symmetries) exists. The standard cubic crystal axes do as far as I can see not include these symmetries. Do anybody know which coordinate system one should use in order to obtain all symmetries of the m3m point group?
 Emeritus Sci Advisor PF Gold P: 11,154 As far as I can see, the principal axes are normals to the {110} family of planes (i.e., lines joining edge-center and opposite edge-center) as they have a 4-fold rotation symmetry. That automatically makes the 3 planes in the {110} family the $\sigma _h$ planes. I don't see any reason that the {100} family needs to make up the $\sigma _h$ planes. The point group is defined by the symmetry elements, which are independent of the co-ordinate system.
 P: 3 Thanks. The {110} normals work out fine, my only problem with these vectors is that they are not orthogonal. I'm not sure whether that is in fact a requirement for the coordinate system in this case, but I would have expected them to be orthogonal as we are considering a cubic crystal.
Its not....the vectors will not be orthogonal in the {110} set of planes. Remember that silicon $\Gamma$ -point is 48 fold symmetric.