Let $\Lambda$ be a Bravais lattice in three dimensions. Define the reciprocal lattice $\Lambda^{*}$.

State and prove Bloch's theorem for a particle moving in a potential $V(\mathbf{x})$ obeying

$$V(\mathbf{x}+\ell)=V(\mathbf{x}) \quad \forall \ell \in \Lambda, \mathbf{x} \in \mathbb{R}^{3}$$

Explain what is meant by a Brillouin zone for this potential and how it is related to the reciprocal lattice.

A simple cubic lattice $\Lambda_{1}$ is given by the set of points

$$\Lambda_{1}=\left\{\ell \in \mathbb{R}^{3}: \ell=n_{1} \hat{\mathbf{i}}+n_{2} \hat{\mathbf{j}}+n_{3} \hat{\mathbf{k}}, n_{1}, n_{2}, n_{3} \in \mathbb{Z}\right\}$$

where $\hat{\mathbf{i}}, \hat{\mathbf{j}}$ and $\hat{\mathbf{k}}$ are unit vectors parallel to the Cartesian coordinate axes in $\mathbb{R}^{3}$. A bodycentred cubic $\left(\mathrm{BCC}\right.$ ) lattice $\Lambda_{B C C}$ is obtained by adding to $\Lambda_{1}$ the points at the centre of each cube, i.e. all points of the form

$$\ell+\frac{1}{2}(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}), \quad \ell \in \Lambda_{1}$$

Show that $\Lambda_{B C C}$ is Bravais with primitive vectors

$$\begin{aligned} \mathbf{a}_{1} &=\frac{1}{2}(\hat{\mathbf{j}}+\hat{\mathbf{k}}-\hat{\mathbf{i}}) \\ \mathbf{a}_{2} &=\frac{1}{2}(\hat{\mathbf{k}}+\hat{\mathbf{i}}-\hat{\mathbf{j}}) \\ \mathbf{a}_{3} &=\frac{1}{2}(\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}) \end{aligned}$$

Find the reciprocal lattice $\Lambda_{B C C}^{*}$. Hence find a consistent choice for the first Brillouin zone of a potential $V(\mathbf{x})$ obeying

$$\begin{aligned} & V(\mathbf{x}+\ell)=V(\mathbf{x}) \quad \forall \ell \in \Lambda_{B C C}, \mathbf{x} \in \mathbb{R}^{3} \\ & \text { [Hint: The matrix } \left.M=\frac{1}{2}\left(\begin{array}{rrr}-1 & 1 & 1 \\1 & -1 & 1 \\1 & 1 & -1\end{array}\right) \text { has inverse } M^{-1}=\left(\begin{array}{lll}0 & 1 & 1 \\1 & 0 & 1 \\1 & 1 & 0\end{array}\right) .\right] \end{aligned}$$