In quantum statistics, a branch of physics, Fermi–Dirac statistics describes distribution of particles in certain systems comprising many identical particles that obey the Pauli exclusion principle. It is named after Enrico Fermi and Paul Dirac, who each discovered it independently, although Enrico Fermi defined the statistics earlier than Paul Dirac.^{[1]}^{[2]}
Fermi–Dirac (F–D) statistics applies to identical particles with half-odd-integer spin in a system in thermodynamic equilibrium. Additionally, the particles in this system are assumed to have negligible mutual interaction. This allows the many-particle system to be described in terms of single-particle energy states. The result is the F–D distribution of particles over these states and includes the condition that no two particles can occupy the same state, which has a considerable effect on the properties of the system. Since F–D statistics applies to particles with half-integer spin, these particles have come to be called fermions. It is most commonly applied to electrons, which are fermions with spin 1/2. Fermi–Dirac statistics is a part of the more general field of statistical mechanics and uses the principles of quantum mechanics.
History
Before the introduction of Fermi–Dirac statistics in 1926, understanding some aspects of electron behavior was difficult due to seemingly contradictory phenomena. For example, the electronic heat capacity of a metal at room temperature seemed to come from 100 times fewer electrons than were in the electric current.^{[3]} It was also difficult to understand why the emission currents, generated by applying high electric fields to metals at room temperature, were almost independent of temperature.
The difficulty encountered by the electronic theory of metals at that time was due to considering that electrons were (according to classical statistics theory) all equivalent. In other words it was believed that each electron contributed to the specific heat an amount on the order of the Boltzmann constant k.
This statistical problem remained unsolved until the discovery of F–D statistics.
F–D statistics was first published in 1926 by Enrico Fermi^{[1]} and Paul Dirac.^{[2]} According to an account, Pascual Jordan developed in 1925 the same statistics which he called Pauli statistics, but it was not published in a timely manner.^{[4]} According to Dirac, it was first studied by Fermi, and Dirac called it Fermi statistics and the corresponding particles fermions.^{[5]}
F–D statistics was applied in 1926 by Fowler to describe the collapse of a star to a white dwarf.^{[6]} In 1927 Sommerfeld applied it to electrons in metals^{[7]} and in 1928 Fowler and Nordheim applied it to field electron emission from metals.^{[8]} Fermi–Dirac statistics continues to be an important part of physics.
Fermi–Dirac distribution
For a system of identical fermions, the average number of fermions in a single-particle state $i$, is given by the Fermi–Dirac (F–D) distribution,^{[9]}
- $\backslash bar\{n\}\_i\; =\; \backslash frac\{1\}\{e^\{(\backslash epsilon\_i-\backslash mu)\; /\; k\; T\}\; +\; 1\}$
where k is Boltzmann's constant, T is the absolute temperature, $\backslash epsilon\_i\; \backslash $ is the energy of the single-particle state $i$, and μ is the total chemical potential. At zero temperature, μ is equal to the Fermi energy plus the potential energy per electron. For the case of electrons in a semiconductor, $\backslash mu\backslash $ is typically called the Fermi level or electrochemical potential.^{[10]}^{[11]}
The F–D distribution is only valid if the number of fermions in the system is large enough so that adding one more fermion to the system has negligible effect on $\backslash mu\backslash $.^{[12]} Since the F–D distribution was derived using the Pauli exclusion principle, which allows at most one electron to occupy each possible state, a result is that $0\; <\; \backslash bar\{n\}\_i\; <\; 1$ .^{[13]}
- Fermi–Dirac distribution
Energy dependence. More gradual at higher T. $\backslash bar\{n\}$ = 0.5 when $\backslash epsilon\; \backslash ;$ = $\backslash mu\; \backslash ;$. Not shown is that $\backslash mu\; \backslash $ decreases for higher T.^{[14]}
Temperature dependence for $\backslash epsilon\; >\; \backslash mu\; \backslash $ .
(Click on a figure to enlarge.)
Distribution of particles over energy
The above Fermi–Dirac distribution gives the distribution of identical fermions over single-particle energy states, where no more than one fermion can occupy a state. Using the F–D distribution, one can find the distribution of identical fermions over energy, where more than one fermion can have the same energy.^{[15]}
The average number of fermions with energy $\backslash epsilon\_i\; \backslash $ can be found by multiplying the F–D distribution $\backslash bar\{n\}\_i\; \backslash $ by the degeneracy $g\_i\; \backslash $ (i.e. the number of states with energy $\backslash epsilon\_i\; \backslash $ ),^{[16]}
- $\backslash begin\{alignat\}\{2\}$
\bar{n}(\epsilon_i) & = g_i \ \bar{n}_i \\
& = \frac{g_i}{e^{(\epsilon_i-\mu) / k T} + 1} \\
\end{alignat}
When $g\_i\; \backslash ge\; 2\; \backslash $, it is possible that $\backslash \; \backslash bar\{n\}(\backslash epsilon\_i)\; >\; 1$ since there is more than one state that can be occupied by fermions with the same energy $\backslash epsilon\_i\; \backslash $.
When a quasi-continuum of energies $\backslash epsilon\; \backslash $ has an associated density of states $g(\; \backslash epsilon\; )\; \backslash $ (i.e. the number of states per unit energy range per unit volume ^{[17]}) the average number of fermions per unit energy range per unit volume is,
- $\backslash bar\; \{\; \backslash mathcal\{N\}\; \}(\backslash epsilon)\; =\; g(\backslash epsilon)\; \backslash \; F(\backslash epsilon)$
where $F(\backslash epsilon)\; \backslash $ is called the Fermi function and is the same function that is used for the F–D distribution $\backslash bar\{n\}\_i$,^{[18]}
- $F(\backslash epsilon)\; =\; \backslash frac\{1\}\{e^\{(\backslash epsilon-\backslash mu)\; /\; k\; T\}\; +\; 1\}$
so that,
- $\backslash bar\; \{\; \backslash mathcal\{N\}\; \}(\backslash epsilon)\; =\; \backslash frac\{g(\backslash epsilon)\}\{e^\{(\backslash epsilon-\backslash mu)\; /\; k\; T\}\; +\; 1\}$ .
Quantum and classical regimes
The classical regime, where Maxwell–Boltzmann statistics can be used as an approximation to Fermi–Dirac statistics, is found by considering the situation that is far from the limit imposed by the Heisenberg uncertainty principle for a particle's position and momentum. Using this approach, it can be shown that the classical situation occurs if the concentration of particles corresponds to an average interparticle separation $\backslash bar\{R\}$ that is much greater than the average de Broglie wavelength $\backslash bar\{\backslash lambda\}$ of the particles,^{[19]}
- $\backslash bar\{R\}\; \backslash \; \backslash gg\; \backslash \; \backslash bar\{\backslash lambda\}\; \backslash \; \backslash approx\; \backslash \; \backslash frac\{h\}\{\backslash sqrt\{3mkT\}\}$
where $h$ is Planck's constant, and $m$ is the mass of a particle.
For the case of conduction electrons in a typical metal at T = 300K (i.e. approximately room temperature), the system is far from the classical regime because $\backslash bar\{R\}\; \backslash approx\; \backslash bar\{\backslash lambda\}/25$ . This is due to the small mass of the electron and the high concentration (i.e. small $\backslash bar\{R\}$) of conduction electrons in the metal. Thus Fermi–Dirac statistics is needed for conduction electrons in a typical metal.^{[19]}
Another example of a system that is not in the classical regime is the system that consists of the electrons of a star that has collapsed to a white dwarf. Although the white dwarf's temperature is high (typically T = 10,000K on its surface^{[20]}), its high electron concentration and the small mass of each electron precludes using a classical approximation, and again Fermi–Dirac statistics is required.^{[6]}
Three derivations of the Fermi–Dirac distribution
Derivation starting with grand canonical ensemble
The Fermi-Dirac distribution, which applies only to a quantum system of non-interacting fermions, is easily derived from the grand canonical ensemble.^{[21]} In this ensemble, the system is able to exchange energy and exchange particles with a reservoir (temperature T and chemical potential µ fixed by the reservoir).
Due to the non-interacting quality, each available single-particle level (with energy level ϵ) forms a separate thermodynamic system in contact with the reservoir.
In other words, each single-particle level is a separate, tiny grand canonical ensemble.
By the Pauli exclusion principle there are only two possible microstates for the single-particle level: no particle (energy E=0), or one particle (energy E=ϵ). The resulting partition function for that single-particle level therefore has just two terms:
- $\backslash begin\{align\}\backslash mathcal\; Z\; \&\; =\; \backslash exp(0(\backslash mu\; -\; 0)/k\_B\; T)\; +\; \backslash exp(1(\backslash mu\; -\; \backslash epsilon)/k\_B\; T)\; \backslash \backslash \; \&\; =\; 1\; +\; \backslash exp((\backslash mu\; -\; \backslash epsilon)/k\_B\; T)\backslash end\{align\}$
and the average particle number for that single-particle substate is given by
- $\backslash langle\; N\backslash rangle\; =\; k\_B\; T\; \backslash frac\{1\}\{\backslash mathcal\; Z\}\; \backslash left(\backslash frac\{\backslash partial\; \backslash mathcal\; Z\}\{\backslash partial\; \backslash mu\}\backslash right)\_\{V,T\}\; =\; \backslash frac\{1\}\{\backslash exp((\backslash epsilon-\backslash mu)/k\_B\; T)+1\}$
This result applies for each single-particle level, and thus gives the Fermi-Dirac distribution for the entire state of the system.^{[21]}
The variance in particle number (due to thermal fluctuations) may also be derived (the particle number has a simple Bernoulli distribution):
- $\backslash langle\; (\backslash Delta\; N)^2\; \backslash rangle\; =\; k\_B\; T\; \backslash left(\backslash frac\{d\backslash langle\; N\backslash rangle\}\{d\backslash mu\}\backslash right)\_\{V,T\}\; =\; \backslash langle\; N\backslash rangle\; (1\; -\; \backslash langle\; N\backslash rangle)$
This quantity is important in transport phenomena such as the Mott relations for electrical conductivity and thermoelectric coefficient for an electron gas,^{[22]} where the ability of an energy level to contribute to transport phenomena is proportional to $\backslash langle\; (\backslash Delta\; N)^2\; \backslash rangle$.
Derivations starting with canonical distribution
It is also possible to derive Fermi–Dirac statistics in the canonical ensemble.
Standard derivation
Consider a many-particle system composed of N identical fermions that have negligible mutual interaction and are in thermal equilibrium.^{[12]} Since there is negligible interaction between the fermions, the energy $E\_R$ of a state $R$ of the many-particle system can be expressed as a sum of single-particle energies,
- $E\_R\; =\; \backslash sum\_\{r\}\; n\_r\; \backslash epsilon\_r\; \backslash ;$
where $n\_r$ is called the occupancy number and is the number of particles in the single-particle state $r$ with energy $\backslash epsilon\_r\; \backslash ;$. The summation is over all possible single-particle states $r$.
The probability that the many-particle system is in the state $R$, is given by the normalized canonical distribution,^{[23]}
- $P\_R\; =\; \backslash frac\; \{\; e^\{-\backslash beta\; E\_R\}\; \}$
{ \displaystyle \sum_{R'} e^{-\beta E_{R'}} }
where $\backslash beta\backslash ;$$=\; 1/kT$, $k$ is Boltzmann's constant, $T$ is the absolute temperature, e^{$\backslash scriptstyle\; -\backslash beta\; E\_R$} is called the Boltzmann factor, and the summation is over all possible states $R\text{'}$ of the many-particle system. The average value for an occupancy number $n\_i\; \backslash ;$ is^{[23]}
- $\backslash bar\{n\}\_i\; \backslash \; =\; \backslash \; \backslash sum\_\{R\}\; n\_i\; \backslash \; P\_R$
Note that the state $R$ of the many-particle system can be specified by the particle occupancy of the single-particle states, i.e. by specifying $n\_1,\backslash ,\; n\_2,\backslash ,\; ...\; \backslash ;,$ so that
- $P\_R\; =\; P\_\{n\_1,n\_2,...\}\; =\; \backslash frac\{\; e^\{-\backslash beta\; (n\_1\; \backslash epsilon\_1+n\_2\; \backslash epsilon\_2+...)\}\; \}$
{\displaystyle \sum_\sum_{n_1,n_2,\dots} e^{-\beta (n_1\epsilon_1+n_2\epsilon_2+\cdots)} }
{\displaystyle \sum_{n_i=0} ^1 e^{-\beta (n_i\epsilon_i)} \qquad \sideset{ }{^{(i)}}\sum_{n_1,n_2,\dots} e^{-\beta (n_1\epsilon_1+n_2\epsilon_2+\cdots)} }
where the $^\{(i)\}$ on the summation sign indicates that the sum is not over $n\_i$ and is subject to the constraint that the total number of particles associated with the summation is $N\_i\; =\; N-n\_i$ . Note that $\backslash Sigma^\{(i)\}$ still depends on $n\_i$ through the $N\_i$ constraint, since in one case $n\_i=0$ and $\backslash Sigma^\{(i)\}$ is evaluated with $N\_i=N\; ,$ while in the other case $n\_i=1$ and $\backslash Sigma^\{(i)\}$ is evaluated with $N\_i=N-1\; .$ To simplify the notation and to clearly indicate that $\backslash Sigma^\{(i)\}$ still depends on $n\_i$ through $N-n\_i$ , define
- $Z\_i(N-n\_i)\; \backslash equiv\; \backslash \; \backslash sideset\{\; \}\{^\{(i)\}\}\backslash sum\_\{n\_1,n\_2,...\}\; e^\{-\backslash beta\; (n\_1\backslash epsilon\_1+n\_2\backslash epsilon\_2+\backslash cdots)\}\; \backslash ;$
so that the previous expression for $\backslash bar\{n\}\_i$ can be rewritten and evaluated in terms of the $Z\_i$,
- $\backslash begin\{alignat\}\; \{3\}$
\bar{n}_i \ & = \frac{ \displaystyle \sum_{n_i=0} ^1 n_i \ e^{-\beta (n_i\epsilon_i)} \ \ Z_i(N-n_i)}
{ \displaystyle \sum_{n_i=0} ^1 e^{-\beta (n_i\epsilon_i)} \qquad Z_i(N-n_i)} \\
\\
& = \ \frac { \quad 0 \quad \; + e^{-\beta\epsilon_i}\; Z_i(N-1)} {Z_i(N) + e^{-\beta\epsilon_i}\; Z_i(N-1)} \\
& = \ \frac {1} ; width:15em; margin:.3em">
$\backslash bar\{n\}\_i\; =\; \backslash \; \backslash frac\; \{1\}\; \{e^\{(\backslash epsilon\_i\; -\; \backslash mu)/kT\; \}+1\}$
A result can be achieved by directly analyzing the multiplicities of the system and using Lagrange multipliers.^{[27]}
For example, distributing two particles in three sublevels will give population numbers of 110, 101, or 011 for a total of three ways which equals 3!/(2!1!). The number of ways that a set of occupation numbers n_{i} can be realized is the product of the ways that each individual energy level can be populated: