Physics Friday 90

Let us consider a metal. When metal atoms are brought together, the few electrons in the outermost shell break away from the atoms, and can move (mostly) freely through the solid. On average, the electrical repulsive forces between these electrons tend to be neutralized by the electrical interactions with the background positive charges of the fixed ions. Thus, these “conduction band” electrons can be approximated by an ideal spin-½ fermi fluid. For an alkali metal, such as sodium or potassium (where the approximation works best), we have a concentration of (approximately) one conduction electron per ion. With an interionic distance of ~5 Å, we have an electron density on the order of 1028 to 1029 electrons per cubic meter.

Now, let us find the Fermi energy μ0, which you may remember from here is the value of the Fermi level at T=0. Here, the occupation probability is 1 for  and zero for . Using our integral formula for the number of particles in an ideal Fermi fluid (see here), we see:
,
and thus:

For our metal, this is

The temperature TF for which μ0=kTF (the Fermi temperature), is then on the order of 104 to 105 K. For room temperature, we are at T much lower than this, placing us clearly in the quantum regime
(For 300 K,  is approximately 43 Å). A Fermi fluid in such a state is referred to as a “degenerate” Fermi gas.

Now, the energy of the Fermi fluid at absolute zero is
,
so the energy per particle is . For our electron gas in our metal, this is on the order of 104 K in equivalent temperature units. For nucleons in the nucleus of an atom of a heavy element, the fermi energy even higher, on the order of 30 MeV, so the average kinetic energy is on the order of 18 MeV, which is still in the non-relativistic domain.

Note that the density of states increases as energy increases, so, using our result from here that the Fermi level is “repelled” by energies with higher densities of states, we should expect the Fermi level to decrease as the temperature rises. Similarly, an increasing number of electrons at energies just below the Fermi energy will be promoted to states with energies above the Fermi energy.

For an arbitrary function , we consider the integral
.
Using the fact that  resembles a step function at low temperatures, one can, with some math, expand this integral in a power series in temperature, getting:
,
where the upper limit in the right-hand side integral and the points of evaluation in φ‘ and φ”’ is the temperature dependent Fermi level μ, not the Fermi energy μ0.

To find the dependence of the Fermi level on temperature, we first use . This latter integral is the above integral with , meaning
.
To find μ to second order in T, let us replace μ by μ0 in the second-order term of the above expansion, and using , we solve for μ to get:
,
confirming the Fermi level decreases with energy.

Similarly, we can develop a series expansion for internal energy as a function of temperature:
,
This tells us that the heat capacity at constant volume is
.
, where the term  is the classical value; the factor in parentheses is thus a quantum correction factor. For our metal at room temperature, this factor is on the order of 10-1. This drastic reduction in the heat capacity of the conduction band electrons from what classical theory predicts was one of the motivations in the development of the Fermi-Dirac distribuion discribing the occupancy of states in a Fermi gas (see here).

Now, recall that for our ideal Fermi fluid, we have equation of state . For small T, the internal energy will be roughly the value at absolute zero, ; thus,
, where n is the particle density. Note that this is substantially larger than the P→0 as T→0 of the classical ideal gas; for our metal, this pressure is on the order of 1010 Pa! (In our metal, this enormous pressure is countered by the Coulomb attraction between the electrons and the positively charged fixed ions).
This additional, non-classical pressure, which ultimately derives from the Pauli exclusion principle, is known as degeneracy pressure. For massive astronomical objects without internal fusion, most notably white dwarfs, the object is supported against gravitational collapse by the degeneracy pressure of a degenerate electron gas. Similarly, neutron stars are supported by the degeneracy pressure of a degenerate neutron gas.

Combining  and , we see ; this gives the dependency of the degeneracy pressure on density (for non-relativistic particle velocities). Further, for , the degeneracy pressure is effectively independent of the temperature.