Past exam of the condensed matter physics major option course of the University of Oxford / 2019 / 1 / c / ii

Let the reciprocal lattice vector be $\mathbf{G}=h_r{\mathbf{a}}_r^{\ast }+k_r{\mathbf{b}}_r^{\ast }=h_p{\mathbf{a}}_p^{\ast }+k_p{\mathbf{b}}_p^{\ast }$, with each dual pair obeying $\mathbf{a}\cdot {\mathbf{a}}^{\ast }=\mathbf{b}\cdot {\mathbf{b}}^{\ast }=2\pi$ and zero cross products. Its coefficients are obtained by dotting with the corresponding direct-space primitive lattice translation vectors:HenceFor actual integer-indexed reciprocal lattice vectors, this also explains the centred-rectangular condition that $h_r$ and $k_r$ have the same parity. General phonon wave vectors need not have integer coefficients.