How to think about a Brillouin Zone?

[Jesse Keeper]

Isn't a Brillouin zone a Wigner Seitz cell in reciprocal space? Is it just a collection of wave vectors?

Will you have Brillouin zone boundaries in many different places in your real space crystal, and hence standing waves there? What causes the standing waves at the zone boundaries? Isn't a standing wave a superposition of two identical waves traveling in opposite directions? What are the waves that are superposed at the boundary? And is the boundary a physical point?

What is the connection between a Bloch wave and Brillouin zone?

Why does Bragg scattering happen at the Brillouin zone boundary, and what is it?

Is it possible to view Brillouin zone in another physical way?

Thanks in advance!

Jess

 

[jedishrfu]

Welcome to PF!

That's a lot of questions to answer. Perhaps this article will answer some of them:

https://en.m.wikipedia.org/wiki/Brillouin_zone

 

[DeathbyGreen]

A Brillouin zone is just a plot of the possible momentum values electrons can take in a system. Bloch waves are directly connected to this concept; a Bloch wave follows a periodic behavior, $\Psi \propto u(r)e^{ikr}$, with $u(r)$ some periodic function. A periodic function will be proportional to some sine or cosine type thing. So since a Bloch wave repeats itself according to this function, the repetitive behavior is reflected in the Brillouin zone. The (1st) BZ ranges from $\pi \rightarrow -\pi$, and since it is proportional to a periodic function it repeats itself. Regarding your question about physical boundaries, the BZ reflects the possible momentum values in a unit cell, so the symmetry is reflected in real space and momentum space.