 Solution

The first Brillouin zone is the origin-centered Wigner-Seitz cell of the reciprocal lattice. Its nearest reciprocal points are $\pm {\mathbf{b}}_1$ and $\pm {\mathbf{b}}_2$. The perpendicular bisectors follow fromConsequently the zone isIt is the diamond with verticesThe reciprocal-space sketch above shows these boundaries. Its area is $2{\pi }^2/a^2=(2\pi {)}^2/A_c$, where $A_c=2a^2$. This area check rules out mistakenly using the nonprimitive conventional cell to define a smaller first zone.