 Solution

ID: past-exam-of-the-condensed-matter-physics-major-option-course-of-the-university-of-oxford/2019/1/c/ii/solution

G = h_{r} a_{r}^{*} + k_{r} b_{r}^{*} = h_{p} a_{p}^{*} + k_{p} b_{p}^{*}

, with each dual pair obeying

a \cdot a^{*} = b \cdot b^{*} = 2 π

and zero cross products. Its coefficients are obtained by dotting with the corresponding direct-space primitive lattice translation vectors:

h_{p} k_{p} = \frac{G \cdot a _{p}}{2 π} = \frac{h _{r} - k _{r}}{2}, = \frac{G \cdot b _{p}}{2 π} = \frac{h _{r} + k _{r}}{2} .

Hence

h_{p} = \frac{1}{2} h_{r} - \frac{1}{2} k_{r}, k_{p} = \frac{1}{2} h_{r} + \frac{1}{2} k_{r} .

For actual integer-indexed reciprocal lattice vectors, this also explains the centred-rectangular condition that

h_{r}

and

k_{r}

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