Define the reciprocal lattice for a lattice $L$ with lattice vectors $\ell$.

A beam of electrons, with wave vector $\mathbf{k}$, is incident on a Bravais lattice $L$ with a large number of atoms, $N$. If the scattering amplitude for scattering on an individual atom in the direction $\hat{\mathbf{k}}^{\prime}$ is $f\left(\hat{\mathbf{k}}^{\prime}\right)$, show that the scattering amplitude for the whole lattice

$$\sum_{\ell \in L} e^{i \mathbf{q} \cdot \ell} f\left(\hat{\mathbf{k}}^{\prime}\right), \quad \mathbf{q}=\mathbf{k}-|\mathbf{k}| \hat{\mathbf{k}}^{\prime}$$

Derive the formula for the differential cross section

$$\frac{\mathrm{d} \sigma}{\mathrm{d} \Omega}=N\left|f\left(\hat{\mathbf{k}}^{\prime}\right)\right|^{2} \Delta(\mathbf{q}),$$

obtaining an explicit form for $\Delta(\mathbf{q})$. Show that $\Delta(\mathbf{q})$ is strongly peaked when $\mathbf{q}=\mathbf{g}$, a reciprocal lattice vector. Show that this leads to the Bragg formula $2 d \sin \frac{\theta}{2}=\lambda$, where $\theta$ is the scattering angle, $\lambda$ the electron wavelength and $d$ the separation between planes of atoms in the lattice.