Define a Bravais lattice $\Lambda$ in three dimensions. Define the reciprocal lattice $\Lambda^{\star}$. Define the Brillouin zone.

An FCC lattice has a basis of primitive vectors given by

$$\mathbf{a}_{1}=\frac{a}{2}\left(\mathbf{e}_{2}+\mathbf{e}_{3}\right), \quad \mathbf{a}_{2}=\frac{a}{2}\left(\mathbf{e}_{1}+\mathbf{e}_{3}\right), \quad \mathbf{a}_{3}=\frac{a}{2}\left(\mathbf{e}_{1}+\mathbf{e}_{2}\right),$$

where $\mathbf{e}_{i}$ is an orthonormal basis of $\mathbb{R}^{3}$. Find a basis of reciprocal lattice vectors. What is the volume of the Brillouin zone?

The asymptotic wavefunction for a particle, of wavevector $\mathbf{k}$, scattering off a potential $V(\mathbf{r})$ is

$$\psi(\mathbf{r}) \sim e^{i \mathbf{k} \cdot \mathbf{r}}+f_{\mathrm{V}}\left(\mathbf{k} ; \mathbf{k}^{\prime}\right) \frac{e^{i k r}}{r}$$

where $\mathbf{k}^{\prime}=k \hat{\mathbf{r}}$ and $f_{\mathrm{V}}\left(\mathbf{k} ; \mathbf{k}^{\prime}\right)$ is the scattering amplitude. Give a formula for the Born approximation to the scattering amplitude.

Scattering of a particle off a single atom is modelled by a potential $V(\mathbf{r})=V_{0} \delta(r-d)$ with $\delta$-function support on a spherical shell, $r=|\mathbf{r}|=d$ centred at the origin. Calculate the Born approximation to the scattering amplitude, denoting the resulting expression as $\tilde{f}_{V}\left(\mathbf{k} ; \mathbf{k}^{\prime}\right)$.

Scattering of a particle off a crystal consisting of atoms located at the vertices of a lattice $\Lambda$ is modelled by a potential

$$V_{\Lambda}=\sum_{\mathbf{R} \in \Lambda} V(\mathbf{r}-\mathbf{R})$$

where $V(\mathbf{r})=V_{0} \delta(r-d)$ as above. Calculate the Born approximation to the scattering amplitude giving your answer in terms of your approximate expression $\tilde{f}_{\mathrm{V}}$ for scattering off a single atom. Show that the resulting amplitude vanishes unless the momentum transfer $\mathbf{q}=\mathbf{k}-\mathbf{k}^{\prime}$ lies in the reciprocal lattice $\Lambda^{\star}$.

For the particular FCC lattice considered above, show that, when $k=|\mathbf{k}|>2 \pi / a$, scattering occurs for two values of the scattering angle, $\theta_{1}$ and $\theta_{2}$, related by

$$\frac{\sin \left(\frac{\theta_{1}}{2}\right)}{\sin \left(\frac{\theta_{2}}{2}\right)}=\frac{2}{\sqrt{3}}$$