Hey all, I have a question with first Brillouin zone. It's: It says that it's enough to consider values of K within the first brillouin zone, K= [-pi/a,pi/a). why is this easy to see, and why don't we choose the interval K= (0 , 2pi/a) instead? Why is the point K=pi/a not included (open interval)? I know that according to periodicity we get that if we solve for the first Billiouin zone then we know the solution in the whole lattice. But why just in this interval?
It can be any interval with length 2pi/a that you want. But what you call the first Brillouin zone is the set of unique points closest to k = 0, so that would be [-pi/a, pi/a). You have an interval of length 2pi/a (the periodicity) centered around k = 0. You don't include the end point pi/a because it would be included twice, since -pi/a is the same point due to the translational symmetry. You could just as well do (-pi/a, pi/a]. If you're writing code to model a periodic solid, it will not matter what interval you use as long as it's the right size.