Volume of Brillouin zone is the same as Fourier primitive parallelepip

[ZetaOfThree]

In Kittel's solid state text, problem 2.3, he says that the volume of the Brillouin zone is the same as a primitive parallelepiped in Fourier space. Somehow I can't see why this is true. Can someone help me see why this is true? Also, is the same relationship true between Wigner-Seitz cells and primitive parallelepiped in real space?

Thanks

 

[M Quack]

The same is true for the cells in real space, and it is perhaps easier to see in real space.

By construction, the Wigner-Seitz cell contains the basis of the crystal structure exactly once. This is also true for any primitive unit cell. It follows that it must be possible to convert the WS-cell into any primitive cell by cutting, translating by a lattice vector, and pasting parts.

Hope this helps.