Brillouin Zone Volume

exercise 3_4942

Show that the volume of the first Brillouin zone (1 st B.Z.) is (2 p ) 3 / V c , where V c is the volume of a crystal primitive cell in the real or direct lattice. Hint: The volume of a B.Z. is equal to the volume of the Wigner-Seitz cell and to the volume of a primitive cell in the reciprocal lattice. Also, the vector identity is useful: ( c x a )x( a x b ) = ( c a x b ) a .