# High symmetry points and lines in Brillioun Zone

by Log
Tags: brillioun, brillioun zone, high symmetry points, lines, points, symmetry, zone
 P: 23 Hi, I've seen pictures like this one: Is there any good explanation somewhere on this subject? I'm using Kittel's book but there's nothing in there on this.
Mentor
P: 28,453
 Quote by Log Is there any good explanation somewhere on this subject? I'm using Kittel's book but there's nothing in there on this.
What explanation exactly do you need? An explanation on what a Brillouin zone is?

Zz.
 P: 23 I know that a Brillioun Zone is a Wigner Seitz cell in k-space, but what are the symmetry points and lines? How are these used and what physical significance do they have? How are they chosen? I've read the first 6 chapters in Kittel. I don't think we're required to know this in the course I'm taking, just asking out of curiosity. :)
P: 12

## High symmetry points and lines in Brillioun Zone

You need to study a little group theory as it applies to crystallographic symmetries. It is surprisingly easy to understand. I leaned it from the book by Micheal Tinkham "Group Theory in Quantum Mechanics". Basically, the geometric structure of the Wigner Seitz cell is subsumed to an irreducible representation of the geometry by the symmetry group operators of rotation, reflection, and inversion.

This technique is fundamental to the interpretation of almost all solid state spectroscopic experiments (i.e. x-ray diffraction, EPR etc).
 P: 23 I have been thinking about buying that book actually, it seems quite interesting. How much mathematics and QM is required to understand it? I have Ashcroft as well. There's a section on point groups and such but the notation is different, is this the same thing as high symmetry points? I didn't bother reading it yet as the subject seemed to be different. I bought the book as a supplement but haven't been using it that much. My understanding is only basic so far. I know some basic QM and I'm studying Kittel.
 P: 12 The first four chapters or so of Tinkham are related to crystallographic symmetry groups. The math is not hard at all. If you can do the problems in Kittel, you can work through Tinkham. I suggest you read the first few chapters and try to work the problems. The time spent studying group theory will be enormously beneficial to your understanding of solid state physics.
 P: 627 Look at the point group of the crystal. Then pick a point within the BZ, e.g. one of the special points (Gamma, X, L, K, U, W) or along one of the special lines (Sigma, Lambda, Delta), or any other point. Then figure out which of the symmetry operations of the point group project that point onto itself (or itself+reciprocal lattice vector). What do you get?
 Sci Advisor P: 3,260 I found this script excellent: http://stuff.mit.edu/afs/athena/cour...oup-full02.pdf

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