[Physics] Volume of Brillouin zone is the same as Fourier primitive cell

crystalsfourier transformsolid-state-physicsvolume

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?

Best Answer

This has to be true by construction, either in the real space or in the reciprocal space. There is one primitive parallelepiped and one Wigner-Seitz cell per lattice point, and both of them tile the whole space.

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