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72-482: (Redirected from Fermi-Dirac ) Fermi–Dirac may refer to: Fermi–Dirac statistics or Fermi–Dirac distribution Fermi–Dirac integral (disambiguation) Complete Fermi–Dirac integral Incomplete Fermi–Dirac integral See also [ edit ] Fermi (disambiguation) Dirac (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with

144-427: A heuristic tool for solving problems. There was no agreement whether chemical molecules, as measured by atomic weights , were the same as physical molecules, as measured by kinetic theory . Planck's 1920 lecture continued: Nothing can better illustrate the positive and hectic pace of progress which the art of experimenters has made over the past twenty years, than the fact that since that time, not only one, but

216-458: A thermodynamic system at an absolute temperature T , the average thermal energy carried by each microscopic degree of freedom in the system is ⁠ 1 / 2 ⁠   k T (i.e., about 2.07 × 10  J , or 0.013  eV , at room temperature). This is generally true only for classical systems with a large number of particles , and in which quantum effects are negligible. In classical statistical mechanics , this average

288-535: A form of the gas constant R , and macroscopic energies for macroscopic quantities of the substance. The iconic terse form of the equation S = k ln W on Boltzmann's tombstone is in fact due to Planck, not Boltzmann. Planck actually introduced it in the same work as his eponymous h . In 1920, Planck wrote in his Nobel Prize lecture: This constant is often referred to as Boltzmann's constant, although, to my knowledge, Boltzmann himself never introduced it—a peculiar state of affairs, which can be explained by

360-567: A great number of methods have been discovered for measuring the mass of a molecule with practically the same accuracy as that attained for a planet. In versions of SI prior to the 2019 revision of the SI , the Boltzmann constant was a measured quantity rather than a fixed value. Its exact definition also varied over the years due to redefinitions of the kelvin (see Kelvin § History ) and other SI base units (see Joule § History ). In 2017,

432-464: A positive or negative value. At zero absolute temperature, μ is equal to the Fermi energy plus the potential energy per fermion, provided it is in a neighbourhood of positive spectral density. In the case of a spectral gap, such as for electrons in a semiconductor, the point of symmetry μ is typically called the Fermi level or—for electrons—the electrochemical potential , and will be located in

504-477: A rescaled dimensionless entropy in microscopic terms such that S ′ = ln ⁡ W , Δ S ′ = ∫ d Q k T . {\displaystyle {S'=\ln W},\quad \Delta S'=\int {\frac {\mathrm {d} Q}{kT}}.} This is a more natural form and this rescaled entropy exactly corresponds to Shannon's subsequent information entropy . The characteristic energy kT

576-451: A single-particle state i is given by the Fermi–Dirac (F–D) distribution : n ¯ i = 1 e ( ε i − μ ) / k B T + 1 , {\displaystyle {\bar {n}}_{i}={\frac {1}{e^{(\varepsilon _{i}-\mu )/k_{\text{B}}T}+1}},} where k B

648-478: A state i with energy E weighted by the corresponding Boltzmann factor : P i ∝ exp ⁡ ( − E k T ) Z , {\displaystyle P_{i}\propto {\frac {\exp \left(-{\frac {E}{kT}}\right)}{Z}},} where Z is the partition function . Again, it is the energy-like quantity k T that takes central importance. Consequences of this include (in addition to

720-407: A total of n i particles. Suppose each level contains g i distinct sublevels, all of which have the same energy, and which are distinguishable. For example, two particles may have different momenta (i.e. their momenta may be along different directions), in which case they are distinguishable from each other, yet they can still have the same energy. The value of g i associated with level i

792-472: A total of six degrees of simple freedom per molecule that are related to atomic motion (three translational, two rotational, and one vibrational). At lower temperatures, not all these degrees of freedom may fully participate in the gas heat capacity, due to quantum mechanical limits on the availability of excited states at the relevant thermal energy per molecule. More generally, systems in equilibrium at temperature T have probability P i of occupying

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864-402: 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: Boltzmann constant The Boltzmann constant ( k B or k ) is the proportionality factor that relates the average relative thermal energy of particles in a gas with

936-585: Is N i = N − n i {\displaystyle N_{i}=N-n_{i}} . Note that ∑ ( i ) {\displaystyle \textstyle \sum ^{(i)}} still depends on n i {\displaystyle n_{i}} through the N i {\displaystyle N_{i}} constraint, since in one case n i = 0 {\displaystyle n_{i}=0} and ∑ ( i ) {\displaystyle \textstyle \sum ^{(i)}}

1008-421: Is Bose–Einstein statistics , which applies to identical and indistinguishable particles with integer spin (0, 1, 2, etc.) called bosons . In classical physics, Maxwell–Boltzmann statistics is used to describe particles that are identical and treated as distinguishable. For both Bose–Einstein and Maxwell–Boltzmann statistics, more than one particle can occupy the same state, unlike Fermi–Dirac statistics. Before

1080-460: Is a proportionality factor between temperature and energy, its numerical value depends on the choice of units for energy and temperature. The small numerical value of the Boltzmann constant in SI units means a change in temperature by 1 K only changes a particle's energy by a small amount. A change of 1  °C is defined to be the same as a change of 1 K . The characteristic energy kT

1152-520: Is a term encountered in many physical relationships. The Boltzmann constant sets up a relationship between wavelength and temperature (dividing hc / k by a wavelength gives a temperature) with one micrometer being related to 14 387 .777 K , and also a relationship between voltage and temperature ( kT in units of eV corresponds to a voltage) with one volt being related to 11 604 .518 K . The ratio of these two temperatures, 14 387 .777 K  /  11 604 .518 K  ≈ 1.239842,

1224-662: Is approximately 25.85 mV which can be derived by plugging in the values as follows: V T = k T q = 1.38 × 10 − 23   J ⋅ K − 1 × 300   K 1.6 × 10 − 19   C ≃ 25.85   m V {\displaystyle V_{\mathrm {T} }={kT \over q}={\frac {1.38\times 10^{-23}\ \mathrm {J{\cdot }K^{-1}} \times 300\ \mathrm {K} }{1.6\times 10^{-19}\ \mathrm {C} }}\simeq 25.85\ \mathrm {mV} } At

1296-478: Is called the Boltzmann factor , and the summation is over all possible states R ′ {\displaystyle R'} of the many-particle system. The average value for an occupancy number n i {\displaystyle n_{i}} is Note that the state R {\displaystyle R} of the many-particle system can be specified by the particle occupancy of

1368-503: Is called the "degeneracy" of that energy level. The Pauli exclusion principle states that only one fermion can occupy any such sublevel. The number of ways of distributing n i indistinguishable particles among the g i sublevels of an energy level, with a maximum of one particle per sublevel, is given by the binomial coefficient , using its combinatorial interpretation : For example, distributing two particles in three sublevels will give population numbers of 110, 101, or 011 for

1440-436: Is evaluated with N i = N , {\displaystyle N_{i}=N,} while in the other case n i = 1 , {\displaystyle n_{i}=1,} and ∑ ( i ) {\displaystyle \textstyle \sum ^{(i)}} is evaluated with N i = N − 1. {\displaystyle N_{i}=N-1.} To simplify

1512-578: Is negligible interaction between the fermions, the energy E R {\displaystyle E_{R}} of a state R {\displaystyle R} of the many-particle system can be expressed as a sum of single-particle energies: where n r {\displaystyle n_{r}} is called the occupancy number and is the number of particles in the single-particle state r {\displaystyle r} with energy ε r {\displaystyle \varepsilon _{r}} . The summation

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1584-485: Is over all possible single-particle states r {\displaystyle r} . The probability that the many-particle system is in the state R {\displaystyle R} is given by the normalized canonical distribution : where β = 1 / k B T {\displaystyle \beta =1/k_{\text{B}}T} , e − β E R {\displaystyle e^{-\beta E_{R}}}

1656-570: Is possible that n ¯ ( ε i ) > 1 {\displaystyle {\bar {n}}(\varepsilon _{i})>1} , since there is more than one state that can be occupied by fermions with the same energy ε i {\displaystyle \varepsilon _{i}} . When a quasi-continuum of energies ε {\displaystyle \varepsilon } has an associated density of states g ( ε ) {\displaystyle g(\varepsilon )} (i.e.

1728-432: Is predicted to hold exactly for homogeneous ideal gases . Monatomic ideal gases (the six noble gases) possess three degrees of freedom per atom, corresponding to the three spatial directions. According to the equipartition of energy this means that there is a thermal energy of ⁠ 3 / 2 ⁠   k T per atom. This corresponds very well with experimental data. The thermal energy can be used to calculate

1800-420: Is proportional to ⟨ ( Δ N ) 2 ⟩ {\displaystyle {\big \langle }(\Delta N)^{2}{\big \rangle }} . It is also possible to derive Fermi–Dirac statistics in the canonical ensemble . Consider a many-particle system composed of N identical fermions that have negligible mutual interaction and are in thermal equilibrium. Since there

1872-436: Is the Boltzmann constant , T is the absolute temperature , ε i is the energy of the single-particle state i , and μ is the total chemical potential . The distribution is normalized by the condition that can be used to express μ = μ ( T , N ) {\displaystyle \mu =\mu (T,N)} in that μ {\displaystyle \mu } can assume either

1944-408: Is the mass of a particle . For the case of conduction electrons in a typical metal at T = 300  K (i.e. approximately room temperature), the system is far from the classical regime because R ¯ ≈ λ ¯ / 25 {\displaystyle {\bar {R}}\approx {\bar {\lambda }}/25} . This is due to the small mass of

2016-460: Is the molar gas constant ( 8.314 462 618 153 24  J⋅K ⋅ mol ). Introducing the Boltzmann constant as the gas constant per molecule k = R / N A ( N A being the Avogadro constant ) transforms the ideal gas law into an alternative form: p V = N k T , {\displaystyle pV=NkT,} where N is the number of molecules of gas. Given

2088-488: Is the magnitude of the electrical charge on the electron with a value 1.602 176 634 × 10  C . Equivalently, V T T = k q ≈ 8.617333262 × 10 − 5   V / K . {\displaystyle {V_{\mathrm {T} } \over T}={k \over q}\approx 8.617333262\times 10^{-5}\ \mathrm {V/K} .} At room temperature 300 K (27 °C; 80 °F), V T

2160-399: Is the numerical value of hc in units of eV⋅μm. The Boltzmann constant provides a mapping from the characteristic microscopic energy E to the macroscopic temperature scale T = ⁠ E / k ⁠ . In fundamental physics, this mapping is often simplified by using the natural units of setting k to unity. This convention means that temperature and energy quantities have

2232-631: Is thus the energy required to increase the rescaled entropy by one nat . In semiconductors , the Shockley diode equation —the relationship between the flow of electric current and the electrostatic potential across a p–n junction —depends on a characteristic voltage called the thermal voltage , denoted by V T . The thermal voltage depends on absolute temperature T as V T = k T q = R T F , {\displaystyle V_{\mathrm {T} }={kT \over q}={RT \over F},} where q

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2304-562: The Z i {\displaystyle Z_{i}} : The following approximation will be used to find an expression to substitute for Z i ( N ) / Z i ( N − 1 ) {\displaystyle Z_{i}(N)/Z_{i}(N-1)} : where α i ≡ ∂ ln ⁡ Z i ( N ) ∂ N . {\displaystyle \alpha _{i}\equiv {\frac {\partial \ln Z_{i}(N)}{\partial N}}.} If

2376-499: The 2019 revision of the SI , the Boltzmann constant is one of the seven " defining constants " that have been given exact definitions. They are used in various combinations to define the seven SI base units. The Boltzmann constant is defined to be exactly 1.380 649 × 10 joules per kelvin. Boltzmann constant : The Boltzmann constant, k , is one of seven fixed constants defining the International System of Units,

2448-620: The CODATA recommended 1.380 649 × 10  J/K to be the final fixed value of the Boltzmann constant to be used for the International System of Units . As a precondition for redefining the Boltzmann constant, there must be one experimental value with a relative uncertainty below 1 ppm , and at least one measurement from a second technique with a relative uncertainty below 3 ppm. The acoustic gas thermometry reached 0.2 ppm, and Johnson noise thermometry reached 2.8 ppm. Since k

2520-472: The Drude model , 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 B . This problem remained unsolved until the development of Fermi–Dirac statistics. Fermi–Dirac statistics

2592-581: The Maxwell–Boltzmann distribution in the limit of high temperature and low particle density, without the need for any ad hoc assumptions: 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 . For example, in physics of semiconductor, when

2664-560: The root-mean-square speed of the atoms, which turns out to be inversely proportional to the square root of the atomic mass . The root mean square speeds found at room temperature accurately reflect this, ranging from 1370 m/s for helium , down to 240 m/s for xenon . Kinetic theory gives the average pressure p for an ideal gas as p = 1 3 N V m v 2 ¯ . {\displaystyle p={\frac {1}{3}}{\frac {N}{V}}m{\overline {v^{2}}}.} Combination with

2736-585: The standard state temperature of 298.15 K (25.00 °C; 77.00 °F), it is approximately 25.69 mV . The thermal voltage is also important in plasmas and electrolyte solutions (e.g. the Nernst equation ); in both cases it provides a measure of how much the spatial distribution of electrons or ions is affected by a boundary held at a fixed voltage. The Boltzmann constant is named after its 19th century Austrian discoverer, Ludwig Boltzmann . Although Boltzmann first linked entropy and probability in 1877,

2808-558: The thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the gas constant , in Planck's law of black-body radiation and Boltzmann's entropy formula , and is used in calculating thermal noise in resistors . The Boltzmann constant has dimensions of energy divided by temperature , the same as entropy and heat capacity . It is named after the Austrian scientist Ludwig Boltzmann . As part of

2880-484: The Fermi–Dirac distribution n ¯ i {\displaystyle {\bar {n}}_{i}} by the degeneracy g i {\displaystyle g_{i}} (i.e. the number of states with energy ε i {\displaystyle \varepsilon _{i}} ), When g i ≥ 2 {\displaystyle g_{i}\geq 2} , it

2952-502: The Fermi–Dirac distribution for the entire state of the system. The variance in particle number (due to thermal fluctuations ) may also be derived (the particle number has a simple Bernoulli distribution ): This quantity is important in transport phenomena such as the Mott relations for electrical conductivity and thermoelectric coefficient for an electron gas , where the ability of an energy level to contribute to transport phenomena

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3024-550: The Fermi–Dirac distribution: Like the Maxwell–Boltzmann distribution and the Bose–Einstein distribution , the Fermi–Dirac distribution can also be derived by the Darwin–Fowler method of mean values. A result can be achieved by directly analyzing the multiplicities of the system and using Lagrange multipliers . Suppose we have a number of energy levels, labeled by index i , each level having energy ε i and containing

3096-461: The Pauli exclusion principle, and n r = 0 {\displaystyle n_{r}=0} = 0 or 1 {\displaystyle 1} for each r {\displaystyle r} . Furthermore, each combination of values of n 1 , n 2 , … {\displaystyle n_{1},n_{2},\ldots } satisfies the constraint that

3168-420: 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: and the average particle number for that single-particle level substate is given by This result applies for each single-particle level, and thus gives

3240-477: The SI, with k = 1.380 649 x 10 J K . The Boltzmann constant is a proportionality constant between the quantities temperature (with unit kelvin) and energy (with unit joule). Macroscopically, the ideal gas law states that, for an ideal gas , the product of pressure p and volume V is proportional to the product of amount of substance n and absolute temperature T : p V = n R T , {\displaystyle pV=nRT,} where R

3312-432: The average energy per degree of freedom equal to one third of that, i.e. ⁠ 1 / 2 ⁠   k T . The ideal gas equation is also obeyed closely by molecular gases; but the form for the heat capacity is more complicated, because the molecules possess additional internal degrees of freedom, as well as the three degrees of freedom for movement of the molecule as a whole. Diatomic gases, for example, possess

3384-459: The case of negligible interaction between particles, the system can be described in terms of single-particle energy states . A result is the Fermi–Dirac distribution of particles over these states where no two particles can occupy the same state, which has a considerable effect on the properties of the system. Fermi–Dirac statistics is most commonly applied to electrons , a type of fermion with spin 1/2 . A counterpart to Fermi–Dirac statistics

3456-424: The classical situation prevails when the concentration of particles corresponds to an average interparticle separation R ¯ {\displaystyle {\bar {R}}} that is much greater than the average de Broglie wavelength λ ¯ {\displaystyle {\bar {\lambda }}} of the particles: where h is the Planck constant , and m

3528-413: The collapse of a star to a white dwarf . In 1927 Arnold Sommerfeld applied it to electrons in metals and developed the free electron model , and in 1928 Fowler and Lothar Nordheim applied it to field electron emission from metals. Fermi–Dirac statistics continue to be an important part of physics. For a system of identical fermions in thermodynamic equilibrium, the average number of fermions in

3600-408: The density of states of conduction band is much higher than the doping concentration, the energy gap between conduction band and fermi level could be calculated using Maxwell-Boltzmann statistics. Otherwise, if the doping concentration is not negligible compared to density of states of conduction band, the Fermi–Dirac distribution should be used instead for accurate calculation. It can then be shown that

3672-421: The electron and the high concentration (i.e. small R ¯ {\displaystyle {\bar {R}}} ) of conduction electrons in the metal. Thus Fermi–Dirac statistics is needed for conduction electrons in a typical metal. 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

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3744-607: The energy associated with each classical degree of freedom ( 1 2 k T {\displaystyle {\tfrac {1}{2}}kT} above) becomes E d o f = 1 2 T {\displaystyle E_{\mathrm {dof} }={\tfrac {1}{2}}T} As another example, the definition of thermodynamic entropy coincides with the form of information entropy : S = − ∑ i P i ln ⁡ P i . {\displaystyle S=-\sum _{i}P_{i}\ln P_{i}.} where P i

3816-569: The exponential function to both sides, substituting for α i {\displaystyle \alpha _{i}} and rearranging, Substituting the above into the equation for n ¯ i {\displaystyle {\bar {n}}_{i}} and using a previous definition of β {\displaystyle \beta } to substitute 1 / k B T {\displaystyle 1/k_{\text{B}}T} for β {\displaystyle \beta } , results in

3888-426: The fact that Boltzmann, as appears from his occasional utterances, never gave thought to the possibility of carrying out an exact measurement of the constant. This "peculiar state of affairs" is illustrated by reference to one of the great scientific debates of the time. There was considerable disagreement in the second half of the nineteenth century as to whether atoms and molecules were real or whether they were simply

3960-474: The ideal gas law p V = N k T {\displaystyle pV=NkT} shows that the average translational kinetic energy is 1 2 m v 2 ¯ = 3 2 k T . {\displaystyle {\tfrac {1}{2}}m{\overline {v^{2}}}={\tfrac {3}{2}}kT.} Considering that the translational motion velocity vector v has three degrees of freedom (one for each dimension) gives

4032-527: 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 . 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

4104-556: The microscopic details, or microstates, of the system (via W ) to its macroscopic state (via the entropy S ), is the central idea of statistical mechanics. Such is its importance that it is inscribed on Boltzmann's tombstone. The constant of proportionality k serves to make the statistical mechanical entropy equal to the classical thermodynamic entropy of Clausius : Δ S = ∫ d Q T . {\displaystyle \Delta S=\int {\frac {{\rm {d}}Q}{T}}.} One could choose instead

4176-575: The middle of the gap. The Fermi–Dirac 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 μ . Since the Fermi–Dirac distribution was derived using the Pauli exclusion principle , which allows at most one fermion to occupy each possible state, a result is that 0 < n ¯ i < 1 {\displaystyle 0<{\bar {n}}_{i}<1} . The variance of

4248-408: The most accurate measures of the Boltzmann constant were obtained by acoustic gas thermometry, which determines the speed of sound of a monatomic gas in a triaxial ellipsoid chamber using microwave and acoustic resonances. This decade-long effort was undertaken with different techniques by several laboratories; it is one of the cornerstones of the 2019 revision of the SI. Based on these measurements,

4320-505: The notation and to clearly indicate that ∑ ( i ) {\displaystyle \textstyle \sum ^{(i)}} still depends on n i {\displaystyle n_{i}} through N − n i , {\displaystyle N-n_{i},} define so that the previous expression for n ¯ i {\displaystyle {\bar {n}}_{i}} can be rewritten and evaluated in terms of

4392-436: The number of particles N {\displaystyle N} is large enough so that the change in the chemical potential μ {\displaystyle \mu } is very small when a particle is added to the system, then α i ≃ − μ / k B T . {\displaystyle \alpha _{i}\simeq -\mu /k_{\text{B}}T.} Applying

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4464-456: The number of particles in state i can be calculated from the above expression for n ¯ i {\displaystyle {\bar {n}}_{i}} : From the Fermi–Dirac distribution of particles over states, one can find the distribution of particles over energy. The average number of fermions with energy ε i {\displaystyle \varepsilon _{i}} can be found by multiplying

4536-479: The number of states per unit energy range per unit volume ), the average number of fermions per unit energy range per unit volume is where F ( ε ) {\displaystyle F(\varepsilon )} is called the Fermi function and is the same function that is used for the Fermi–Dirac distribution n ¯ i {\displaystyle {\bar {n}}_{i}} : so that The Fermi–Dirac distribution approaches

4608-414: The relation was never expressed with a specific constant until Max Planck first introduced k , and gave a more precise value for it ( 1.346 × 10  J/K , about 2.5% lower than today's figure), in his derivation of the law of black-body radiation in 1900–1901. Before 1900, equations involving Boltzmann factors were not written using the energies per molecule and the Boltzmann constant, but rather using

4680-549: The results for ideal gases above) the Arrhenius equation in chemical kinetics . In statistical mechanics, the entropy S of an isolated system at thermodynamic equilibrium is defined as the natural logarithm of W , the number of distinct microscopic states available to the system given the macroscopic constraints (such as a fixed total energy E ): S = k ln ⁡ W . {\displaystyle S=k\,\ln W.} This equation, which relates

4752-401: The same dimensions . In particular, the SI unit kelvin becomes superfluous, being defined in terms of joules as 1 K = 1.380 649 × 10  J . With this convention, temperature is always given in units of energy, and the Boltzmann constant is not explicitly needed in formulas. This convention simplifies many physical relationships and formulas. For example, the equipartition formula for

4824-495: The single-particle states, i.e. by specifying n 1 , n 2 , … , {\displaystyle n_{1},n_{2},\ldots ,} so that and the equation for n ¯ i {\displaystyle {\bar {n}}_{i}} becomes where the summation is over all combinations of values of n 1 , n 2 , … {\displaystyle n_{1},n_{2},\ldots } which obey

4896-406: 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

4968-422: The temperature of white dwarf is high (typically T = 10 000  K on its surface ), its high electron concentration and the small mass of each electron precludes using a classical approximation, and again Fermi–Dirac statistics is required. 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,

5040-646: The title Fermi–Dirac . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Fermi–Dirac&oldid=688953654 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Fermi%E2%80%93Dirac statistics Fermi–Dirac statistics applies to identical and indistinguishable particles with half-integer spin (1/2, 3/2, etc.), called fermions , in thermodynamic equilibrium . For

5112-409: The total number of particles is N {\displaystyle N} : Rearranging the summations, where the upper index ( i ) {\displaystyle (i)} on the summation sign indicates that the sum is not over n i {\displaystyle n_{i}} and is subject to the constraint that the total number of particles associated with the summation

5184-442: Was first published in 1926 by Enrico Fermi and Paul Dirac . According to Max Born , Pascual Jordan developed in 1925 the same statistics, which he called Pauli statistics , but it was not published in a timely manner. According to Dirac, it was first studied by Fermi, and Dirac called it "Fermi statistics" and the corresponding particles "fermions". Fermi–Dirac statistics was applied in 1926 by Ralph Fowler to describe

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