Finding the star of a wave vector using group theory

[jocke_x1]

I'm working on a problem where I have to find the little co-group and star of two wave vectors for a diamond structure (space group 227). I know I have to act on the vector by the symmetry operations in the group (perhaps only the ones in the isogonal point group, O_h?) and see if it remains the same or at a point in the reciprocal lattice. My problem is that i don't know how to do it explicitly. How do I find the matrix representations of the operations?

 

[M Quack]

Since k-vectors are in reciprocal space, you use the point group operations, in this case Oh.

Doing this by matrix multiplication is tedious. It is much easier to do graphically.

You know the operations are

* 4-fold (90 deg) rotations about the face normals of the cube

* 3-fold (120 deg) rotations about the body diagonals of the cube

* 2-fold (180 deg) rotations about the face diagonal (translated so it goes through the center of the cube).

* Inversion symmetry k--> -k.

* all of the above followed by inversion symmetry.

Find the symmetry elements that leave k in place. These form the little co-group.
Once you have found all of them, you can check that they form a group (sub-group of Oh).

Be careful with vectors on the border of the Brillouin zone boundary. They may change place to a new positions that is related to the old one by a reciprocal space vector. Such positions are equivalent and the symmetry element belongs to the little co-group.

Whatever is left throws k elsewhere. All the positions you get in this way form the star of k.

You can check your results on the Bilbao crystallography server

http://www.cryst.ehu.es/cgi-bin/cryst/programs/nph-kv-list

(you will understand the output once you understand how to do this by hand :-) )