In the previous piece we saw that a fermion admits at most one particle per level. That rule starts to bite where the particles are packed tight. The electrons in a metal are such a case, and we call that collection a Fermi gas.
Put the electrons in one at a time. As the figure shows, the levels fill from the bottom up. No second particle joins a level that is taken, so the next electron has nowhere to go but higher. Put in N of them and the lowest N levels are full.
The height at which the filling stops is called the Fermi energy \varepsilon_{\mathrm{F}}. Where bosons would all drop into the lowest level, fermions cannot. Even at absolute zero the electrons still carry kinetic energy all the way up to \varepsilon_{\mathrm{F}}.
How large \varepsilon_{\mathrm{F}} turns out to be follows from counting. Counting in momentum space, the filling starts near the origin, and the boundary of what is filled is a sphere. Its radius is the Fermi momentum p_{\mathrm{F}}. The figure is a section through it.
The number of states goes as the volume of the sphere, so N goes as p_{\mathrm{F}} to the 3rd power. Hence p_{\mathrm{F}} goes as \dfrac{N}{V} to the \dfrac{1}{3}, and since the energy goes as its 2nd power, \varepsilon_{\mathrm{F}} goes as \dfrac{N}{V} to the \dfrac{2}{3}.
All that enters is how crowded it is. Temperature appears nowhere. A metal packs some 10^{22} electrons into a cubic centimetre, which puts \varepsilon_{\mathrm{F}} at a few electronvolts. In temperature that is of order 10^{4} kelvin. Melting the metal does not come close.
Now let temperature in. The occupation is the Fermi distribution of the previous piece, \dfrac{1}{e^{\beta(\varepsilon - \mu)} + 1}, which at absolute zero was a step with its edge at \varepsilon_{\mathrm{F}}.
The figure warms it up. The only thing that moves is the corner of the step, blurred over a width of about kT. The electrons deep down cannot move at all: every level just above them is taken. The only free seats are near \varepsilon_{\mathrm{F}}.
This is the defining feature of a Fermi gas. There may be 10^{23} electrons, and yet only a fraction of about \dfrac{kT}{\varepsilon_{\mathrm{F}}} of them can respond to temperature at all. In a metal at room temperature that is less than one part in 100.
Now the heat capacity. Only the electrons at the edge can take up heat, and their fraction is \dfrac{T}{T_{\mathrm{F}}}, where T_{\mathrm{F}} is the Fermi temperature, fixed by \varepsilon_{\mathrm{F}} = kT_{\mathrm{F}}.
Each of those takes up something like k, so the heat capacity is of order k\dfrac{T}{T_{\mathrm{F}}} per electron. Doing the integral properly attaches a coefficient: \dfrac{\pi^2}{2}k\dfrac{T}{T_{\mathrm{F}}}. What matters is not the coefficient but the shape, that it goes as the temperature.
The horizontal line in the figure is the equipartition answer, \dfrac{3}{2}k. Classically each electron would carry kinetic energy in 3 directions and give a flat line with no temperature in it. The truth falls nowhere near it, and comes as a straight line through the origin.
Dulong and Petit put the heat capacity of a solid at 3R, an answer that counts nothing but the vibrations of the atoms. A metal holds nearly as many freely moving electrons as it holds atoms, and yet their share is not added in. Counted classically, the electrons alone should add \dfrac{3}{2}R.
The answer of the previous scene is the reason. The right bar in the figure is the electronic heat capacity at room temperature, nowhere near the equipartition answer on the left. With \dfrac{T}{T_{\mathrm{F}}} of order 10^{-2}, the electrons take up only about 1\% of what classical counting promises. It hardly shows up in a measurement, so the law that counts only the atoms comes out right.
It does not disappear altogether, though. Cool the solid and the lattice vibrations fall away as T^{3}, as we saw in the piece on Debye, while the electrons fall away only as T. Down at a few kelvin the electrons are what is left, and a measurement of the heat capacity shows this straight line. Next we go back to the boson side and count the light shut inside a cavity.