Show that the reciprocal lattice of a simple hexagonal lattice (see Table 1.1) is also a simple hexagonal, with lattice constants 2π/c and 4π/√3a, rotated through 30° about the c-axis with respect to the real-space lattice.

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Table 1.1 Common crystal structures Structure Simple cubic (sc) Body-centered cubic (bcc) Face-centered cubic (fcc) Standard primitive vectors and basis ax, ay, az ax, ay, a(x + y + 2) or sc lattice with the basis (0, a( +ŷ + 2)) {a(ŷ + 2), ¼a(2+x), ¼a(x +ŷ) or sc lattice with the basis (0, ½a(î + ŷ), ⁄a(ŷ + 2), ½a(2 + x)) Diamond fcc lattice with the basis (0, a(+ŷ + 2)) ax, (¹ax +ay), c² Simple hexagonal (sh) Hexagonal close-packed (hcp) sh lattice with the basis (0, -ax+ 2√3aŷ + ½cz), where c = 8 V za