= Solution
For a <Bravais lattice> with primitive translation vectors $\mathbf a_1,\mathbf a_2,\mathbf a_3$, define the <reciprocal lattice> by vectors $\mathbf G$ satisfying $e^{i\mathbf G\cdot\mathbf R}=1$ for every direct-lattice translation $\mathbf R$. A convenient basis is
$$
\boxed{\mathbf b_1=2\pi\frac{\mathbf a_2\times\mathbf a_3}{\mathbf a_1\cdot(\mathbf a_2\times\mathbf a_3)}}
$$
with cyclic formulas for $\mathbf b_2,\mathbf b_3$. Then $\mathbf a_i\cdot\mathbf b_j=2\pi\delta_{ij}$ and $\mathbf G=h\mathbf b_1+k\mathbf b_2+l\mathbf b_3$ for integers $h,k,l$. The reciprocal primitive volume is $(2\pi)^3/V_{\rm cell}$.
These vectors are the spatial <Fourier series> frequencies of a periodic lattice potential or density. Elastic coherent crystal <diffraction> occurs when the scattering <wavevector> transfer equals a <reciprocal lattice> vector, with intensities additionally controlled by the basis <crystal lattice structure factor>. This is the reciprocal-space version of <Bragg's law>. A lattice of points and its basis should not be confused: a basis can make allowed reciprocal-lattice <diffraction> intensities vanish.
The first <Brillouin zone> is the <Wigner-Seitz cell> around the reciprocal origin: it consists of points closer to $\mathbf G=0$ than to any other reciprocal point. Its faces lie on the perpendicular bisectors
$$
\boxed{\mathbf k\cdot\mathbf G=|\mathbf G|^2/2}.
$$
To see the condition, equate $|\mathbf k|^2$ and $|\mathbf k-\mathbf G|^2$. These are also free-electron Bragg-degeneracy planes, where a periodic potential can open a <band gap>. Higher zones classify regions beyond successive Bragg planes; the reduced-zone description folds them into the first zone while retaining a band label.
In one dimension with spacing $a$, reciprocal points occur at $2\pi j/a$, and the first zone is $[-\pi/a,\pi/a]$. A square lattice of spacing $a$ has reciprocal basis $2\pi\hat{\mathbf x}/a,2\pi\hat{\mathbf y}/a$, and a square first zone with edges at $k_x,k_y=\pm\pi/a$.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2017/paper-2-d13-reciprocal-zone.png]
{title=Reciprocal points and the first Brillouin zone of a two-dimensional square lattice}
The reciprocal of a <face-centered cubic lattice> is a <body-centered cubic lattice> and vice versa, with the scale fixed by the primitive-vector definition. In a <Bloch state>, <wavevectors> differing by $\mathbf G$ describe equivalent crystal momenta modulo the lattice periodicity. A crystal with $N$ primitive cells has $N$ allowed <wavevectors> in one Brillouin zone, hence $N$ states in each band per spin. This state count links reciprocal geometry to <band filling>, metallic conduction and band insulating behaviour. Reciprocal space also labels <phonons> and makes conservation of <crystal momentum> modulo $\hbar\mathbf G$ transparent.
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