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 \(^{4}\mathrm{He}\) and \(^{12}\mathrm{C}\)

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 the full six-dimensional phase space,

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

and each quantum 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 filling of quantum states creates an effective pressure even at zero temperature, called degeneracy pressure. It arises not from thermal motion, but from the Pauli exclusion itself: the particles cannot all occupy the same low-energy state.

The Fermi–Dirac occupation probability for a particle of energy \(E\) at temperature \(T\) is given by

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

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

The figure above shows how \(F(E)\) changes with temperature. At absolute zero, the probability drops sharply from 1 to 0 at \(E = E_F\). As the temperature increases, the sharp step softens — a few particles gain enough energy to occupy states with \(E > E_F\), while some lower-energy states become partially empty. However, even for moderate temperatures (\(kT \ll E_F\)), the curve remains steep around \(E_F\), meaning that most fermions stay below the Fermi energy.

The general phase-space distribution function is

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

where the factor \(2/h^3\) accounts for spin degeneracy — two allowed spin orientations per quantum state. In the non-degenerate limit (\(E_F \ll kT\)), the exponential term dominates and Fermi–Dirac statistics reduce to Maxwell–Boltzmann statistics.

In three dimensions, all occupied states at \(T=0\) form a Fermi sphere in momentum space of radius \(p_F\), called the Fermi momentum. The total number of electrons inside 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. Hence the number density of electrons is

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

Rearranging, the Fermi momentum is expressed in terms of the electron density as

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

At zero temperature, all states up to \(p_F\) are occupied. The corresponding Fermi energy is

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

and for relativistic electrons,

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

The diagram above visualizes the Fermi sphere in momentum space. Each point inside the sphere represents an occupied quantum state of momentum \(\mathbf{p}\). At \(T=0\), all states with \(|\mathbf{p}| < p_F\) are filled, while those outside remain empty. The radius of this sphere, \(p_F\), depends only on the density of electrons and determines the Fermi energy, which separates occupied and unoccupied states even in the absence of thermal motion.

In a degenerate fermion gas:

1. Pressure arises from the quantum mechanical restriction on identical particles.
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 ions provide most of the mass.

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

1. Fermi–Dirac statistics describe systems where the Pauli exclusion principle is active.
2. The Fermi energy defines the boundary between filled and empty states at \(T=0\).
3. The occupation probability \(F(E)\) becomes smoother with temperature but remains steep near \(E_F\).
4. Degeneracy pressure originates from the exclusion principle, not from thermal motion.
5. FDS converges to MBS in the high-temperature or low-density limit.

1. Derive the expression for \(p_F\) from the number density \(n_e\).
2. Sketch \(F(E)\) at \(T=0\) and at a finite temperature, labeling \(E_F\).
3. Why does degeneracy pressure persist even when \(T = 0\)?
4. How does the Fermi–Dirac distribution differ from the Maxwell–Boltzmann distribution?
5. Explain the physical significance of the Fermi sphere and its radius \(p_F\).