Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 3 / 10E / i / Solution

As $T\downarrow 0$, the Fermi-Dirac distribution tends to a step function: $\bar{n}(p)=1$ for $ϵ(p)<{\mu }_0$, and $0$ for $ϵ(p)>{\mu }_0$. The value at equality is immaterial to the integrals. The Fermi momentum is defined by $ϵ(p_F)={\mu }_0$: the Pauli exclusion principle fills the lowest available states and forbids multiple occupation of a state with a fixed spin. Thus the sketch is a horizontal line at one up to $p_F$, followed by a jump to zero.

In the ultrarelativistic, $ϵ(p)\simeq cp$, givingEliminate $p_F$ to obtainThe supplied expression for $ϵ$ includes rest energy; in this limit its leading term is also the leading kinetic energy.