= Solution
{c}
Let the <reciprocal lattice vector> be $\mathbf G=h_r\mathbf a_r^*+k_r\mathbf b_r^*=h_p\mathbf a_p^*+k_p\mathbf b_p^*$, with each dual pair obeying $\mathbf a\cdot\mathbf a^*=\mathbf b\cdot\mathbf b^*=2\pi$ and zero cross products. Its coefficients are obtained by dotting with the corresponding direct-space <primitive lattice translation vectors>:
$$
\begin{aligned}
h_p&=\frac{\mathbf G\cdot\mathbf a_p}{2\pi}=\frac{h_r-k_r}{2},\\
k_p&=\frac{\mathbf G\cdot\mathbf b_p}{2\pi}=\frac{h_r+k_r}{2}.
\end{aligned}
$$
Hence
$$
\boxed{h_p=\tfrac12h_r-\tfrac12k_r,\qquad k_p=\tfrac12h_r+\tfrac12k_r.}
$$
For 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.
Back to article page
