Fermi–Dirac Statistics

Fermi–Dirac statistics is the branch of statistical mechanics that describes identical fermions, including electrons, through their allowed quantum-state occupations. Its defining restriction is the Pauli exclusion principle: each complete single-particle state can contain at most one identical fermion. For independent fermions in thermodynamic equilibrium, it yields the Fermi–Dirac distribution, which gives the mean occupation of a state as a function of its energy, temperature, and chemical potential. (damtp.cam.ac.uk)

Quantum basis

In quantum mechanics, identical particles cannot be distinguished by permanent individual labels. For fermions, exchanging the complete coordinates of two particles, including their spin variables, changes the sign of the many-particle wave function:

\Psi(\ldots,x_i,\ldots,x_j,\ldots) = -\Psi(\ldots,x_j,\ldots,x_i,\ldots).

An antisymmetric wave function vanishes if two identical particles are assigned the same single-particle state. This is the quantum-mechanical origin of the occupation restriction. The spin–statistics theorem connects fermionic exchange behavior with half-integer spin under the assumptions of relativistic quantum field theory. (damtp.cam.ac.uk)

A complete state includes all relevant quantum labels. Consequently, two electrons can occupy the same spatial orbital if their spin states differ: they then occupy different complete states. Exclusion does not mean that electrons must have different energies, since several distinct states can share one energy. (ocw.mit.edu)

The occupation distribution

For a single-particle state of energy $\varepsilon$ , the mean occupation is

f(\varepsilon) = \frac{1}{\exp[(\varepsilon-\mu)/(k_{\mathrm B}T)]+1},

where $T$ is absolute temperature, $k_{\mathrm B}$ is the Boltzmann constant, and $\mu$ is the chemical potential. The actual occupation number is either zero or one; $f$ is its expected value and, equivalently, the probability that the state is occupied. (fab.cba.mit.edu)

For $T>0$ , direct consequences of the formula are:

$0<f(\varepsilon)<1$ .
$f(\mu)=1/2$ .
States well below $\mu$ are almost fully occupied.
States well above $\mu$ are almost empty.

The function describes occupation per state, not a normalized probability distribution over energy. To obtain particle numbers, it must be combined with the number of available states. (fab.cba.mit.edu)

Derivation from the grand canonical ensemble

The distribution follows particularly simply in the grand canonical ensemble, where the system exchanges energy and particles with a reservoir. Define $\beta=1/(k_{\mathrm B}T)$ . For an independent state $i$ , exclusion permits only $n_i=0$ and $n_i=1$ . Its grand partition function is therefore

\Xi_i = \sum_{n_i=0}^{1} e^{-\beta n_i(\varepsilon_i-\mu)} = 1+e^{-\beta(\varepsilon_i-\mu)}.

The probability of the occupied configuration is

\langle n_i\rangle = \frac{e^{-\beta(\varepsilon_i-\mu)}} {1+e^{-\beta(\varepsilon_i-\mu)}} = f(\varepsilon_i).

For independent states, the total partition function factorizes:

\Xi=\prod_i\Xi_i.

Thus the $+1$ in the distribution’s denominator follows directly from having precisely two possible occupations for each state. (fab.cba.mit.edu)

Density of states and particle numbers

The density of states $D(\varepsilon)$ specifies how many single-particle states are available per energy interval. If it counts states for the whole system, then

N=\int D(\varepsilon)f(\varepsilon)\,d\varepsilon, \qquad U=\int \varepsilon D(\varepsilon)f(\varepsilon)\,d\varepsilon

give the mean particle number and internal energy of independent particles. Spin and other degeneracies must either be included in $D$ or supplied as separate factors, but not counted twice. At fixed $N$ , the number equation determines $\mu(T)$ . (damtp.cam.ac.uk)

For a uniform, three-dimensional gas of nonrelativistic free particles,

\varepsilon=\frac{p^2}{2m}, \qquad D(\varepsilon) = \frac{g_sV}{4\pi^2} \left(\frac{2m}{\hbar^2}\right)^{3/2} \sqrt{\varepsilon}.

Here $p$ is momentum, $m$ is particle mass, $V$ is volume, $g_s$ counts internal states of equal energy, and $\hbar=h/(2\pi)$ , with $h$ the Planck constant. Different dimensions, confinement, and energy–momentum relations produce different densities of states without changing the independent-fermion occupation formula. (ocw.mit.edu)

Zero temperature and degeneracy pressure

As $T$ approaches absolute zero, the distribution becomes a step:

f(\varepsilon)\longrightarrow \begin{cases} 1,&\varepsilon<\mu,\\ 0,&\varepsilon>\mu. \end{cases}

For a macroscopic ideal Fermi gas at fixed density, the zero-temperature chemical potential is the Fermi energy $E_{\mathrm F}$ . Particles fill the lowest available states up to this energy. In a uniform free gas, occupied momentum states form a sphere; its boundary is the Fermi surface. (damtp.cam.ac.uk)

Writing the number density as $n=N/V$ ,

k_{\mathrm F} = \left(\frac{6\pi^2n}{g_s}\right)^{1/3}, \qquad E_{\mathrm F} = \frac{\hbar^2k_{\mathrm F}^2}{2m}, \qquad T_{\mathrm F}=\frac{E_{\mathrm F}}{k_{\mathrm B}}.

The gas is strongly quantum-degenerate when $T\ll T_{\mathrm F}$ . For the three-dimensional nonrelativistic ideal gas at zero temperature,

U=\frac35NE_{\mathrm F}, \qquad P=\frac25nE_{\mathrm F}.

This nonzero degeneracy pressure arises because exclusion forces particles to occupy increasingly energetic momentum states. It does not require thermal excitation or a repulsive interparticle force. (damtp.cam.ac.uk)

Low-temperature behavior

At small nonzero temperature, the sharp occupation edge is rounded over an energy interval of order $k_{\mathrm B}T$ . States far below the chemical potential remain almost occupied, while those far above remain almost empty. Thermal changes therefore primarily involve states near the Fermi surface. (damtp.cam.ac.uk)

The Sommerfeld expansion evaluates low-temperature integrals involving $f$ . For a sufficiently smooth function $\phi$ , with $\mu$ well above the lower energy boundary,

\int_0^\infty \phi(\varepsilon)f(\varepsilon)\,d\varepsilon = \int_0^\mu\phi(\varepsilon)\,d\varepsilon + \frac{\pi^2}{6}(k_{\mathrm B}T)^2\phi'(\mu) +\cdots.

For the three-dimensional nonrelativistic ideal gas at fixed density, this gives

\mu(T) = E_{\mathrm F} \left[ 1-\frac{\pi^2}{12} \left(\frac{T}{T_{\mathrm F}}\right)^2 +\cdots \right],

and the leading constant-volume heat capacity is

C_V = \frac{\pi^2}{2}Nk_{\mathrm B}\frac{T}{T_{\mathrm F}} +\cdots.

The linear dependence on temperature contrasts with the temperature-independent heat capacity of a classical monatomic ideal gas. (damtp.cam.ac.uk)

Comparison with other statistics

Bose–Einstein statistics describes identical bosons, whose state occupations can be any nonnegative integer. For independent bosons,

f_{\mathrm{BE}}(\varepsilon) = \frac{1}{e^{\beta(\varepsilon-\mu)}-1}.

The denominator’s minus sign, rather than the fermionic plus sign, permits large occupation of a single state. (fab.cba.mit.edu)

Both quantum distributions approach Maxwell–Boltzmann statistics when occupation per state is small. For fermions, if $e^{\beta(\varepsilon-\mu)}\gg1$ ,

f(\varepsilon)\simeq e^{-\beta(\varepsilon-\mu)}.

For a uniform nonrelativistic gas, the classical regime can be expressed as

\frac{n\lambda_T^3}{g_s}\ll1, \qquad \lambda_T=\frac{h}{\sqrt{2\pi mk_{\mathrm B}T}}.

Thus “high temperature” is relative to density and particle mass, rather than a universal temperature threshold. A sufficiently dilute fermion gas can be classical even at low absolute temperature. (damtp.cam.ac.uk)

Applications

In metals, the occupation of electronic states explains why only a small fraction of electrons contributes substantially to low-temperature thermal excitations. It provides the basis for the electronic heat capacity and other properties of a degenerate electron gas. (damtp.cam.ac.uk)

In semiconductors, conduction-electron and hole concentrations are calculated by integrating the appropriate band densities of states against $f$ and $1-f$ , respectively. The chemical potential controls the occupation of the available states; a value $f=1/2$ does not imply that a state actually exists at that energy. This distinction matters when the chemical potential lies within a band gap. (ocw.mit.edu)

In white dwarfs, electron degeneracy pressure supplies support against gravitational compression. At sufficiently high density, relativistic electron energies must replace the nonrelativistic dispersion relation. The occupation rule remains Fermi–Dirac, while the resulting pressure–density relation changes. (damtp.cam.ac.uk)

Historical development

Enrico Fermi introduced the statistical law in 1926 by applying exclusion to a monatomic ideal gas. His original treatment used harmonic confinement and the quantization methods available at the time. An English translation of that paper preserves this early formulation. (arxiv.org)

Paul Dirac developed the quantum-mechanical treatment of identical particles in “On the Theory of Quantum Mechanics,” published on October 1, 1926. His discussion distinguished symmetric from antisymmetric many-particle states and connected antisymmetry with exclusion. The combined name reflects these contributions to quantum statistics. (ethw.org)

Scope and limitations

Fermionic exchange symmetry and exclusion remain valid for interacting systems. However, the simple Fermi–Dirac formula for occupation of specified single-particle energy levels is exact for independent particles; it is not a general solution of an interacting many-body problem. Interactions can change energies, correlations, and excitation spectra. (damtp.cam.ac.uk)

In Fermi-liquid theory, some interacting systems can be described at low energy using fermionic quasiparticles. This explains why ideal-gas reasoning remains useful beyond strictly noninteracting systems, but it requires an effective excitation description rather than identifying every physical particle with a free-particle state. (damtp.cam.ac.uk)

Likewise, an arbitrary nonequilibrium fermion population need not have the equilibrium occupation function. Exclusion constrains possible occupations, but does not by itself establish a common temperature and chemical potential. (fab.cba.mit.edu)

References

Semiconductor Materials and Devicesfab.cba.mit.edu
Lecture 14 — Electronic, Optical and Magnetic Properties of Materialsocw.mit.edu
Toward a 1D Device Model, Part I: Device Fundamentalsocw.mit.edu
On the Quantization of the Monoatomic Ideal Gasarxiv.org
Fermi-liquid ground state of interacting Dirac fermions in two dimensionsarxiv.org