Proof that the n brillouin zones are of equal areas?

[roya]

proof that the "n" brillouin zones are of equal areas?

i'm trying to find a way to prove that the brillouin zones are indeed of equal areas.
if i draw, for examle, the first 3 or 4 brillouin zones of a cubic 2-dimensional lattice, then it is relatively easy to show geometrically how the parts of each of the higher order zones can be combined and put together in order to form the 1st zone. but how can i prove that this is correct for all additional zones as well, up to the nth zone?

 

[roya]

maybe instead of proof, perhaps some suggestions on how to approach the matter in general. basically any direction or anything that could help build some intuition will be very much appreciated...

 

[marcusl]

I think it comes from the periodicity of the crystal lattice. Think about a one-dimensional lattice of identical atoms with spacing d, for simplicity. A propagating wave is thus essentially sampled at intervals d. The spatial frequency domain spectrum is the convolution of a) the wave's Fourier spectrum with b) the Fourier transform of a series of delta functions with spacing d. But the latter is a series of delta functions spaced with spatial frequency intervals 2*pi/d. Each interval is a Brillouin zone; by definition they are identical, and all have the same "area" (linear spacing in the case of this 1D problem).

This might help in visualizing the multiple dimension case.