The Cluster's Fermi-Dirac Statistics
A page about Fermi-Dirac statistics — the quantum statistics of indistinguishable fermions.
The Fermi-Dirac distribution
The Fermi-Dirac distribution gives the probability that a single-particle state with energy E is occupied: f(E) = 1 / (exp((E - mu) / (k_B T)) + 1), where mu is the chemical potential. At T = 0, f(E) = 1 for E < mu (= E_F, the Fermi energy) and f(E) = 0 for E > E_F. The distribution is step-like at low T. In the cluster, the edit Fermi-Dirac distribution gives the edit probability that an edit single-particle state with edit energy E is occupied.
The Fermi sea
At T = 0, the ground state of a non-interacting fermion system is the Fermi sea — all states with E < E_F filled, all states with E > E_F empty. The Fermi momentum is p_F = (3 pi^2 n)^{1/3} for a gas of density n. The Fermi energy is E_F = p_F^2 / (2m) = hbar^2 / (2m) (3 pi^2 n)^{2/3}. The total energy is U = (3/5) N E_F at T = 0. In the cluster, the edit Fermi sea is the ground state of an edit non-interacting edit fermion system.
The Sommerfeld expansion
At low T, thermodynamic quantities can be expanded in powers of T/T_F: integral f(E) phi(E) dE = integral_0^{mu} phi(E) dE + (pi^2/6) (k_B T)^2 phi'(mu) + O(T^4). For a Fermi gas: C_V = (pi^2/2) N k_B (T / T_F), N mu = U + (pi^2/12) (k_B T)^2 g(E_F). In the cluster, the edit Sommerfeld expansion gives edit thermodynamic quantities.
The applications
Fermi-Dirac statistics apply to:
- Electrons in metals (free electron model)
- Neutron stars (degenerate neutron gas)
- White dwarfs (degenerate electron gas, electron degeneracy pressure)
- Semiconductors (Fermi level, carrier statistics)
- Nuclei (Fermi gas model of nucleons)
In the cluster, edit Fermi-Dirac statistics apply to:
- edit Electrons in edit metals
- edit Neutron stars (edit degenerate edit neutron gas)
- edit White dwarfs (edit degenerate edit electron gas)
- edit Semiconductors (edit Fermi level)
- edit Nuclei (edit Fermi gas model)
This statistics
This page is about Fermi-Dirac statistics. f(E) = 1 / (exp((E-mu)/(k_B T)) + 1). At T=0: step function. Fermi momentum: p_F = (3 pi^2 n)^{1/3}. C_V = (pi^2/2) N k_B (T/T_F). The statistics is real.