(i) Each particle in a system of $N$ identical fermions has a set of energy levels, $E_{i}$, with degeneracy $g_{i}$, where $1 \leq i<\infty$. Explain why, in thermal equilibrium, the average number of particles with energy $E_{i}$ is

$$N_{i}=\frac{g_{i}}{e^{\beta\left(E_{i}-\mu\right)}+1} .$$

The physical significance of the parameters $\beta$ and $\mu$ should be made clear.

(ii) A simple model of a crystal consists of a linear array of sites with separation $a$. At the $n$th site an electron may occupy either of two states with probability amplitudes $b_{n}$ and $c_{n}$, respectively. The time-dependent Schrödinger equation governing the amplitudes gives

$$\begin{aligned} &i \hbar \dot{b}_{n}=E_{0} b_{n}-A\left(b_{n+1}+b_{n-1}+c_{n+1}+c_{n-1}\right), \\ &i \hbar \dot{c}_{n}=E_{1} c_{n}-A\left(b_{n+1}+b_{n-1}+c_{n+1}+c_{n-1}\right) \end{aligned}$$

where $A>0$.

By examining solutions of the form

$$\left(\begin{array}{l} b_{n} \\ c_{n} \end{array}\right)=\left(\begin{array}{l} B \\ C \end{array}\right) e^{i(k n a-E t / \hbar)},$$

show that the energies of the electron fall into two bands given by

$$E=\frac{1}{2}\left(E_{0}+E_{1}-4 A \cos k a\right) \pm \frac{1}{2} \sqrt{\left(E_{0}-E_{1}\right)^{2}+16 A^{2} \cos ^{2} k a}$$

Describe briefly how the energy band structure for electrons in real crystalline materials can be used to explain why they are insulators, conductors or semiconductors.