
#1
Dec307, 04:04 PM

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?




#2
Dec307, 06:15 PM

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 edgecenter and opposite edgecenter) as they have a 4fold rotation symmetry. That automatically makes the 3 planes in the {110} family the [itex]\sigma _h[/itex] planes. I don't see any reason that the {100} family needs to make up the [itex]\sigma _h[/itex] planes.
The point group is defined by the symmetry elements, which are independent of the coordinate system. 



#3
Dec407, 02:12 AM

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.




#4
Dec407, 09:03 PM

Sci Advisor
PF Gold
P: 1,456

Symmetries in silicon
Its not....the vectors will not be orthogonal in the {110} set of planes. Remember that silicon [itex] \Gamma [/itex] point is 48 fold symmetric.




#5
Dec507, 12:59 PM

P: 3

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 nonexistent. 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?



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