Past exam of the condensed matter physics course of the University of Oxford / 2017 / 1 / b / Solution

The cubic Miller indices give an interplanar spacing $d_{hkl}=a/\sqrt{h^2+k^2+l^2}$. Equivalently, the reciprocal lattice vector normal to these lattice planes has length $2\pi /d_{hkl}$. HenceThe eight $\{111\}$ reciprocal lattice vectors are the nearest nonzero reciprocal points. Their perpendicular bisectors form the eight hexagonal faces of the first Brillouin zone, a truncated octahedron. For the positive $(111)$ direction, that Bragg plane is $\mathbf{k}\cdot {\mathbf{G}}_{111}=|{\mathbf{G}}_{111}{|}^2/2$, or $k_x+k_y+k_z=3\pi /a$. It is parallel to the direct $(111)$ lattice planes. The six remaining square zone faces correspond to the $\{200\}$ reciprocal points. The zone boundary is halfway to a reciprocal point, not the point itself.