Editing Free electron model (section)

=== Compressibility of metals and degeneracy pressure ===
The total energy per unit volume (at <math display="inline">T = 0</math>) can also be calculated by integrating over the [[phase space]] of the system, we obtain<ref name=":3" group="Ashcroft & Mermin">{{Harvnb|Ashcroft|Mermin|1976|pp=38-39|p=}}</ref>
:<math>u(0) = \frac{3}{5}nE_{\rm F},</math>

which does not depend on temperature. Compare with the energy per electron of an ideal gas: <math display="inline">\frac{3}{2}k_{\rm B}T</math>, which is null at zero temperature. For an ideal gas to have the same energy as the electron gas, the temperatures would need to be of the order of the Fermi temperature. Thermodynamically, this energy of the electron gas corresponds to a zero-temperature pressure given by<ref name=":3" group="Ashcroft & Mermin" />
: <math>P = -\left(\frac{\partial U}{\partial V}\right)_{T,\mu} = \frac{2}{3}u(0),</math>

where <math display="inline">V</math> is the volume and <math display="inline">U(T) = u(T) V</math> is the total energy, the derivative performed at temperature and chemical potential constant. This pressure is called the [[electron degeneracy pressure]] and does not come from repulsion or motion of the electrons but from the restriction that no more than two electrons (due to the two values of spin) can occupy the same energy level. This pressure defines the compressibility or [[bulk modulus]] of the metal<ref name=":3" group="Ashcroft & Mermin" />
:<math>B = -V\left(\frac{\partial P}{\partial V}\right)_{T,\mu} = \frac{5}{3}P = \frac{2}{3}nE_{\rm F}.</math>

This expression gives the right order of magnitude for the bulk modulus for alkali metals and noble metals, which show that this pressure is as important as other effects inside the metal. For other metals the crystalline structure has to be taken into account.