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A Brillouin zone is defined as a Wigner~Secitz primitive cell in the reciprocal lattice [1,2].

The construction of the first Brillouin zone for two different 2D lattices are shown below:

First Brillouin zone for a 2D lattice

The construction procedure for 3D lattices is basically the same as for 2D lattices. In the image below, the first Brillouin zone of FCC lattice, a truncated octahedron, showing symmetry labels for high symmetry lines and points are shown:

First Brillouin zone of FCC lattice

My question is: how the labels of the high symmetry points are defined and associated with the vertices, edges, faces and lines of the surface?

References:

[1] Ashcroft, Neil W. & Mermin, N. David, Solid State Physics

[2] Kittel, Charles, Introduction to Solid State Physics

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I think the naming conventions are standardized and people computing band structures employ them directly. I will break down the high-symmetry points for a 2-D brillouin zone and post very good references for you to explore 3-D BZs.

Take a rectangular Brillouin zone as the one depicted below: K-points path in rectangular Brillouin zone

The center of the brillouin zone is unequivocally marked as $\Gamma$. For the other high-symmetry points, it's easy to reconcile in this case - 'X' and 'Y' just represent points on the rectangle along $ k_x $ an $ k_y $. 'S' is the other high-symmetry point on the corner of the rectangle, an intermediate point in the $ \Gamma $ - X - S - Y - $ \Gamma $ path.

For a very comprehensive treatment of 3-D brillouin zones, refer to this paper. In 3-D brillouin zones, you will encounter more high-symmetry points - $ \Gamma $ will still refer to the center of the BZ and points like 'M' refer to mid-points i.e. $ k/2 $ . If you just want to find the high-symmetry points, I've found this table very helpful. It directly gives you the high-symmetry points and their coordinates.

References

  1. https://i.stack.imgur.com/Iatur.png
  2. https://arxiv.org/pdf/1004.2974.pdf
  3. https://msu.edu/~dodat/files/Brillouin_zone.pdf
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  • $\begingroup$ Thanks for your answer. Looking figs. 1, 2, 4 and 5, for example, the point X is not over b1 (kx). In figs. 12 and 13, the points M and S are out of middle/vertices points. There must be something I am missing. By the way, the paper by Stefano Curtarolo is a must in band structure calculations. $\endgroup$
    – Camps ♦
    May 26, 2020 at 18:37
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    $\begingroup$ @I.Camps The labeling might slightly vary between different reciprocal lattices but it is standard. The example I used is only for 2-D BZ. For 3-D BZ, 'X' need not point directly along either of the reciprocal lattice vectors. It will point along one of those only for the orthorhombic case - just imagine extending the above example to 3-D with a lattice vector along 'z'. For other types of reciprocal cells (like the more complicated ones in the paper) , the lattice vectors defining the cell are not mutually orthogonal, so 'X' for example, cannot point exactly along one lattice vector. $\endgroup$
    – Xivi76
    May 26, 2020 at 18:43
  • $\begingroup$ In figs. 1 and 4, the lattices are cubic and thetragonal and the label X is over b2, that's why I am confused. $\endgroup$
    – Camps ♦
    May 26, 2020 at 18:53
  • $\begingroup$ Good evening, I am glad to find this discussion.I was wondering how for example they would get 0.5,0,0 for X or 0,0.5,0 for Y ? The document was not clear enough. I am interested in the monoclinic system at least. Hope to hearing from you soon. $\endgroup$
    – abed haidar
    Sep 11, 2020 at 14:56

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