5.3 Ideal Fermi gas

Figure 5.2 shows the fermion population density (see equ. 5.4 with the positive sign in the denominator.) It is evidently quite different from its classical counterpart, the Boltzmann factor. In particular, when the temperature is low then it approaches a step function: in the limit $kT \rightarrow 0$ all states having an energy below a certain value are inhabited, while the states with higher energies remain empty. The threshold energy $E_{F}=\mu$ is called ``Fermi energy''.

Figure 5.2: Mean population numbers of the states in a Fermi-Dirac system with $\mu =0$. For comparison we have also drawn the classical Boltzmann factor $\exp \left[ -E / kT\right]$ for a temperature of $kT=5$.

It is only for high temperatures and small values of $\mu$ that the Fermi-Dirac density approaches the classical Boltzmann density; and as it does so it strictly obeys the inequality $f_{FD} < f_{Bm}$. For the purpose of comparison we have included a graph of $f_{Bm} = \exp \left[ - E_{\vec{p}} /kT\right]$ in Figure 5.2.

ELECTRONS IN METALS
Conduction electrons in metallic solids may be considered as an ideal fermion gas - with an additional feature: since electrons have two possible spin states the maximum number of particles in a state $\vec{p}$ is $2$ instead of $1$.

At low temperatures all states $\vec{p}$ with $E_{\vec{p}} \leq E_{F} = \mu$ are populated. The number of such states is, as we can see from eq. 1.21,

$$N = \frac{8 \pi}{3} V \left( \frac{2m \mu}{h^{2}}\right)^{3/2}$$ (5.11)

Assuming a value of $\mu \approx 5 \cdot 10^{-19} J$ typical for conduction electrons we find $N/V \approx 3 \cdot 10^{27} m^{-3}$.

Applying the general formulae 4.51 and 4.53 we may easily derive the pressure and internal energy of the electron gas. We may also recapture the relation $PV = 2 U / 3$, consistent with an earlier result for the classical ideal gas.