# Fermi–Dirac statistics

In quantum statistics, a branch of physics, Fermi–Dirac statistics describes distribution of particles in a system 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 thermal 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.

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## 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]

$\bar{n}_i = \frac{1}{e^{(\epsilon_i-\mu) / k T} + 1}$

where k is Boltzmann's constant, T is the absolute temperature, $\epsilon_i \$ 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, $\mu\$ is typically called the Fermi level or electrochemical potential.[10][11]

The F–D distribution is only valid when the fermions do not significantly interact with each other, so that the addition of a fermion does not disrupt the values of $\epsilon_i \$. 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 $0 < \bar{n}_i < 1$ .[12]

(Click on a figure to enlarge.)

### Distribution of particles over energy

Fermi function F($\epsilon \$) vs. energy $\epsilon \$, with μ = 0.55 eV and for various temperatures in the range 50K ≤ T ≤ 375K.

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.[14]

The average number of fermions with energy $\epsilon_i \$ can be found by multiplying the F–D distribution $\bar{n}_i \$ by the degeneracy $g_i \$ (i.e. the number of states with energy $\epsilon_i \$ ),[15]

\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 \ge 2 \$, it is possible that $\ \bar{n}(\epsilon_i) > 1$ since there is more than one state that can be occupied by fermions with the same energy $\epsilon_i \$.

When a quasi-continuum of energies $\epsilon \$ has an associated density of states $g( \epsilon ) \$ (i.e. the number of states per unit energy range per unit volume [16]) the average number of fermions per unit energy range per unit volume is,

$\bar { \mathcal{N} }(\epsilon) = g(\epsilon) \ F(\epsilon)$

where $F(\epsilon) \$ is called the Fermi function and is the same function that is used for the F–D distribution $\bar{n}_i$,[17]

$F(\epsilon) = \frac{1}{e^{(\epsilon-\mu) / k T} + 1}$

so that,

$\bar { \mathcal{N} }(\epsilon) = \frac{g(\epsilon)}{e^{(\epsilon-\mu) / k T} + 1}$ .
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## Quantum and classical regimes

The classical regime, where Maxwell–Boltzmann (M–B) statistics can be used as an approximation to F–D 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 $\bar{R}$ that is much greater than the average de Broglie wavelength $\bar{\lambda}$ of the particles,[18]

$\bar{R} \ \gg \ \bar{\lambda} \ \approx \ \frac{h}{\sqrt{3mkT}}$

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 since $\bar{R} \approx \bar{\lambda}/25$ . This is due to the small mass of the electron and the high concentration (i.e. small $\bar{R}$) of conduction electrons in the metal. Thus F–D statistics is needed for conduction electrons in a typical metal.[18]

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[19]), its high electron concentration and the small mass of each electron precludes using a classical approximation, and again F–D statistics is required.[6]

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## Three derivations of the Fermi–Dirac distribution

### Derivation starting with grand canonical distribution

The Fermi-Dirac distribution, which applies only to a quantum system of non-interacting fermions, is easily derived from the grand canonical ensemble. 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:

\begin{align}\mathcal Z & = \exp(0(\mu - 0)/k_B T) + \exp(1(\mu - \epsilon)/k_B T) \\ & = 1 + \exp((\mu - \epsilon)/k_B T)\end{align}

and the average particle number for that single-particle substate is given by

$\langle N\rangle = k_B T \frac{1}{\mathcal Z} \left(\frac{\partial \mathcal Z}{\partial \mu}\right)_{V,T} = \frac{1}{\exp((\epsilon-\mu)/k_B T)+1}$

This result applies for each single-particle level, and thus gives the exact Fermi-Dirac distribution for the entire state of the system.

The variance in particle number (due to thermal fluctuations) may also be derived:

$\langle (\Delta N)^2 \rangle = k_B T \left(\frac{d\langle N\rangle}{d\mu}\right)_{V,T} = \langle N\rangle (1 - \langle N\rangle)$

This quantity is important in transport phenomena such as the Mott relations for electrical conductivity and thermoelectric coefficient for an electron gas[20]. The ability of an energy level to contribute to transport phenomena is proportional to $\langle (\Delta N)^2 \rangle$.

### Derivations starting with canonical distribution

It is also possible to derive approximate Fermi-Dirac statistics in the canonical ensemble. These derivations are lengthy and only yield the above results in the asymptotic limit of a large number of particles. The reason for the inaccuracy is that the total number of fermions is conserved in the canonical ensemble, which contradicts the implication in Fermi-Dirac statistics that each energy level is filled independently from the others (which would require the number of particles to be flexible).

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## References

1. Reif, F. (1965). Fundamentals of Statistical and Thermal Physics. McGraw–Hill. ISBN 978-0-07-051800-1.
2. Blakemore, J. S. (2002). Semiconductor Statistics. Dover. ISBN 978-0-486-49502-6.
3. Kittel, Charles (1971). Introduction to Solid State Physics (4th ed.). New York: John Wiley & Sons. ISBN 0-471-14286-7. OCLC 300039591.
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## Footnotes

1. ^ a b Fermi, Enrico (1926). "Sulla quantizzazione del gas perfetto monoatomico". Rendiconti Lincei (in Italian) 3: 145–9., translated as Zannoni, Alberto (transl.) (1999-12-14). "On the Quantization of the Monoatomic Ideal Gas". arXiv:cond-mat/9912229 [cond-mat.stat-mech].
2. ^ a b Dirac, Paul A. M. (1926). "On the Theory of Quantum Mechanics". Proceedings of the Royal Society, Series A 112 (762): 661–77. Bibcode:1926RSPSA.112..661D. doi:10.1098/rspa.1926.0133. JSTOR 94692.
3. ^ (Kittel 1971, pp. 249–50)
4. ^ "History of Science: The Puzzle of the Bohr–Heisenberg Copenhagen Meeting". Science-Week (Chicago) 4 (20). 2000-05-19. OCLC 43626035. Retrieved 2009-01-20.
5. ^ Dirac, Paul A. M. (1967). Principles of Quantum Mechanics (revised 4th ed.). London: Oxford University Press. pp. 210–1. ISBN 978-0-19-852011-5.
6. ^ a b Fowler, Ralph H. (December 1926). "On dense matter". Monthly Notices of the Royal Astronomical Society 87: 114–22. Bibcode:1926MNRAS..87..114F.
7. ^ Sommerfeld, Arnold (1927-10-14). "Zur Elektronentheorie der Metalle". Naturwissenschaften 15 (41): 824–32. Bibcode:1927NW.....15..825S. doi:10.1007/BF01505083.
8. ^ Fowler, Ralph H.; Nordheim, Lothar W. (1928-05-01). "Electron Emission in Intense Electric Fields" (PDF). Proceedings of the Royal Society A 119 (781): 173–81. Bibcode:1928RSPSA.119..173F. doi:10.1098/rspa.1928.0091. JSTOR 95023.
9. ^ (Reif 1965, p. 341)
10. ^ (Blakemore 2002, p. 11)
11. ^ Kittel, Charles; Kroemer, Herbert (1980). Thermal Physics (2nd ed.). San Francisco: W. H. Freeman. p. 357. ISBN 978-0-7167-1088-2. More than one of |authorlink=, |authorlink=, and |author-link= specified (help)
12. ^ Note that $\bar{n}_i$ is also the probability that the state $i$ is occupied, since no more than one fermion can occupy the same state at the same time and $0 < \bar{n}_i < 1$.
13. ^ (Kittel 1971, p. 245, Figs. 4 and 5)
14. ^ These distributions over energies, rather than states, are sometimes called the Fermi–Dirac distribution too, but that terminology will not be used in this article.
15. ^ Leighton, Robert B. (1959). Principles of Modern Physics. McGraw-Hill. p. 340. ISBN 978-0-07-037130-9.
Note that in Eq. (1), $n(\epsilon) \,$ and $n_s \,$ correspond respectively to $\bar{n}_i$ and $\bar{n}(\epsilon_i)$ in this article. See also Eq. (32) on p. 339.
16. ^ (Blakemore 2002, p. 8)
17. ^ (Reif 1965, p. 389)
18. ^ a b (Reif 1965, pp. 246–8)
19. ^ Mukai, Koji; Jim Lochner (1997). "Ask an Astrophysicist". NASA's Imagine the Universe. NASA Goddard Space Flight Center. Archived from the original on 2009-01-20.
20. ^ doi:10.1103/PhysRev.181.1336
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21. ^ (Reif 1965, pp. 340–2)
22. ^ a b (Reif 1965, pp. 203–6)
23. ^ See for example, Derivative - Definition via difference quotients, which gives the approximation f(a+h) ≈ f(a) + f '(a) h .
24. ^ (Reif 1965, pp. 341–2) See Eq. 9.3.17 and Remark concerning the validity of the approximation.
25. ^ By definition, the base e antilog of A is eA.
26. ^ (Blakemore 2002, pp. 343–5)
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