The particles of the standard model of particle physics carry an intrinsic angular momentum known as spin, represented by the spin quantum number \(S\). The magnitude of the angular momentum vector is

$$ |\mathbf{L}|^2 = S(S+1)\hbar^2 $$

where \(\hbar = h/(2\pi)\) and \(S\) can be either zero or a half-integer. Particles with half-integer spin are called fermions, and those with integer spin are called bosons.

1. Fermions: electrons, protons, neutrons, neutrinos, and any nucleus with an odd number of nucleons
2. Bosons: photons, gluons, and nuclei such as \(^{4}\)He and \(^{12}\)C

The Pauli exclusion principle states that no two identical fermions can occupy the same quantum state simultaneously. In the one-dimensional phase space shown above, an area element \(\Delta A = \Delta x\,\Delta p_x = h\) corresponds to one quantum state. Because electrons can have two spin orientations (up and down), at most two electrons can occupy a single phase-space cell.

In full six-dimensional phase space,

$$ \Delta x\,\Delta y\,\Delta z\,\Delta p_x\,\Delta p_y\,\Delta p_z = h^3, $$

and each state can hold only two fermions of opposite spins.

At high temperature or low density, the number of available states greatly exceeds the number of particles, and the exclusion principle has negligible effect. The distribution of particles then follows Maxwell–Boltzmann statistics (MBS).

At very low temperatures or high densities, however, nearly all the low-energy states become filled. This regime is described by Fermi–Dirac statistics (FDS), and the gas is said to be degenerate.

1. In MBS: most particles occupy low-energy states, with a long tail at high energies.
2. In FDS: all states below a certain energy are filled, and very few particles occupy higher levels.

This creates an effective pressure even at zero temperature, called degeneracy pressure. It arises not from thermal motion, but from the Pauli principle itself: the particles cannot all occupy the lowest state.

The Fermi–Dirac occupation probability for a particle of energy \(E\) is

$$ F(E) = \frac{1}{e^{(E - E_F)/(kT)} + 1}, $$

where \(E_F\) is the Fermi energy — the highest occupied energy level at \(T=0\). At absolute zero, all states with \(E < E_F\) are filled and those with \(E > E_F\) are empty, giving a nearly rectangular distribution.

The general form of the phase-space distribution function is

$$ f_{FD} = \frac{2}{h^3}\,F(E) $$

where the prefactor \(2/h^3\) accounts for spin degeneracy (two electrons per quantum state). In the non-degenerate limit (\(E_F \ll kT\)), the exponential term dominates and FDS reduces to MBS.

In three dimensions, the occupied region in momentum space at \(T=0\) is a sphere of radius \(p_F\), called the Fermi sphere. The total number of electrons within this sphere is

$$ N_e = \frac{2}{h^3}\,\frac{4}{3}\pi p_F^3 V_x, $$

where \(V_x\) is the physical volume. The number density of electrons is therefore

$$ n_e = \frac{N_e}{V_x} = \frac{8\pi p_F^3}{3h^3}. $$

Hence the Fermi momentum depends only on the electron density:

$$ p_F = h\left(\frac{3n_e}{8\pi}\right)^{1/3}. $$

At zero temperature, all states up to \(p_F\) are filled. The Fermi energy follows as

$$ E_F = \frac{p_F^2}{2m_e} \quad \text{(nonrelativistic)} $$

or, for relativistic electrons,

$$ E_F = \sqrt{p_F^2 c^2 + m_e^2 c^4} - m_e c^2. $$

In a degenerate gas:

1. Pressure arises from the quantum mechanical restriction on fermions.
2. The pressure remains nonzero even at \(T=0\).
3. It depends only on the density, not on the temperature.
4. Electrons dominate the pressure, while protons and ions dominate the mass.

Degeneracy pressure supports compact objects such as white dwarfs, and partially degenerate matter occurs in brown dwarfs and planetary cores.

1. Fermi–Dirac statistics governs particles that obey the Pauli exclusion principle.
2. The Fermi energy defines the boundary between filled and empty states at \(T=0\).
3. Degeneracy pressure is independent of temperature.
4. FDS reduces to MBS in the high-temperature or low-density limit.

1. Derive the expression for \(p_F\) from the number density \(n_e\).
2. What happens to the Fermi distribution as \(T \to 0\)?
3. Why does degeneracy pressure persist even when temperature is zero?
4. How does Fermi–Dirac statistics differ from Maxwell–Boltzmann statistics?