Symmetries of Silicon: M3M Point Group

[Kreck]

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?

 

[Gokul43201]

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.

 

[Kreck]

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.

 

[Dr Transport]

Its not...the vectors will not be orthogonal in the {110} set of planes. Remember that silicon $\Gamma$ -point is 48 fold symmetric.

 

[Kreck]

Okay, so I will use the {110} planes. Now I just have to locate the m3m symmetries, ie. two mirror planes and a threefold rotation around [111]. The rotation is easy and [111] is the same direction both in the crystal and the {110} coordinate system, but the two mirror planes either seem to be equivalent, i.e. I'm not getting any information from the second one, or non-existent. Some books list the two mirror planes as perpendicular to one of the axis and to [110], respectively. In the {110} coordinate system the one perpendicular to the axis is simply the [110] in the crystal coordinate system, and the one perpendicular to the [110] direction in the {110} coordinate system, which is equivalent to [112] in the crystal coordinate system, does not exist (as far as I can see). I have been wondering if you simply neglect depth information when looking at point groups. I mean, the stereograms are all 2D. If so, why?