Fermi–Dirac statistics is a part of the science of
physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
that describes the energies of single particles in a
systemSystem is a set of interacting or interdependent components forming an integrated whole....
comprising many
identical particlesIdentical particles, or indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. Species of identical particles include elementary particles such as electrons, and, with some clauses, composite particles such as atoms and molecules.There are two...
that obey the
Pauli Exclusion PrincipleThe Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...
. It is named after
Enrico FermiEnrico Fermi was an Italian-born, naturalized American physicist particularly known for his work on the development of the first nuclear reactor, Chicago Pile-1, and for his contributions to the development of quantum theory, nuclear and particle physics, and statistical mechanics...
and
Paul DiracPaul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...
, who each discovered it independently.
Fermi–Dirac (F–D) statistics applies to identical particles with
half-odd-integer spinIn quantum mechanics, spin is an intrinsic property of all elementary particles. Fermions, the particles that constitute ordinary matter, have half-integer spin. Spin-½ particles constitute an important subset of such fermions. All known elementary fermions have a spin of ½.- Overview :Particles...
in a system in
thermal equilibriumThermal equilibrium is a theoretical physical concept, used especially in theoretical texts, that means that all temperatures of interest are unchanging in time and uniform in space...
. Additionally, the particles in this system are assumed to have negligible mutual
interactionIn particle physics, fundamental interactions are the ways that elementary particles interact with one another...
. 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, they 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 mechanicsStatistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
and uses the principles of
quantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
.
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 capacityHeat capacity , or thermal capacity, is the measurable physical quantity that characterizes the amount of heat required to change a substance's temperature by a given amount...
of a
metalA metal , is an element, compound, or alloy that is a good conductor of both electricity and heat. Metals are usually malleable and shiny, that is they reflect most of incident light...
at
room temperature-Comfort levels:The American Society of Heating, Refrigerating and Air-Conditioning Engineers has listings for suggested temperatures and air flow rates in different types of buildings and different environmental circumstances. For example, a single office in a building has an occupancy ratio per...
seemed to come from 100 times fewer
electronThe electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s than were in the
electric currentElectric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...
. 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 were first published in 1926 by
Enrico FermiEnrico Fermi was an Italian-born, naturalized American physicist particularly known for his work on the development of the first nuclear reactor, Chicago Pile-1, and for his contributions to the development of quantum theory, nuclear and particle physics, and statistical mechanics...
and
Paul DiracPaul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...
. According to an account,
Pascual Jordan
developed in 1925 the same statistics which he called
PauliWolfgang Ernst Pauli was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after being nominated by Albert Einstein, he received the Nobel Prize in Physics for his "decisive contribution through his discovery of a new law of Nature, the exclusion principle or...
statistics, but it was not published in a timely manner. Whereas according to Dirac, it was first studied by Fermi, and Dirac called it Fermi statistics and the corresponding particles fermions.
F–D statistics was applied in 1926 by Fowler to describe the collapse of a
starA star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...
to a
white dwarfA white dwarf, also called a degenerate dwarf, is a small star composed mostly of electron-degenerate matter. They are very dense; a white dwarf's mass is comparable to that of the Sun and its volume is comparable to that of the Earth. Its faint luminosity comes from the emission of stored...
. In 1927
SommerfeldArnold Johannes Wilhelm Sommerfeld was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and groomed a large number of students for the new era of theoretical physics...
applied it to electrons in metals and in 1928 Fowler and
NordheimLothar Wolfgang Nordheim was a German-born Jewish theoretical physicist...
applied it to field electron emission from metals. 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
, is given by the Fermi–Dirac (F–D) distribution,
where
k is Boltzmann's constant,
T is the absolute
temperatureTemperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
,
is the energy of the single-particle state
, and
is the
chemical potentialChemical potential, symbolized by μ, is a measure first described by the American engineer, chemist and mathematical physicist Josiah Willard Gibbs. It is the potential that a substance has to produce in order to alter a system...
. the chemical potential is equal to the
Fermi energyThe Fermi energy is a concept in quantum mechanics usually referring to the energy of the highest occupied quantum state in a system of fermions at absolute zero temperature....
. For the case of electrons in a semiconductor,
is also called the
Fermi levelThe Fermi level is a hypothetical level of potential energy for an electron inside a crystalline solid. Occupying such a level would give an electron a potential energy \epsilon equal to its chemical potential \mu as they both appear in the Fermi-Dirac distribution function,which...
.
The F–D distribution is valid only if the number of fermions in the system is large enough so that adding one more fermion to the system has negligible effect on
. Since the F–D distribution was derived using the
Pauli exclusion principleThe Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...
, which allows at most one electron to occupy each possible state, a result is that
.
(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.
The average number of fermions with energy
can be found by multiplying the F–D distribution
by the
degeneracyIn physics, two or more different quantum states are said to be degenerate if they are all at the same energy level. Statistically this means that they are all equally probable of being filled, and in Quantum Mechanics it is represented mathematically by the Hamiltonian for the system having more...
(i.e. the number of states with energy
),
When
, it is possible that
since there is more than one state that can be occupied by fermions with the same energy
.
When a quasi-continuum of energies
has an associated
density of statesIn solid-state and condensed matter physics, the density of states of a system describes the number of states per interval of energy at each energy level that are available to be occupied by electrons. Unlike isolated systems, like atoms or molecules in gas phase, the density distributions are not...
(i.e. 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
is called the Fermi function and is the same
functionIn mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
that is used for the F–D distribution
,
so that,
.
Quantum and classical regimes
The classical regime, where
Maxwell–Boltzmann (M–B) statisticsIn statistical mechanics, Maxwell–Boltzmann statistics describes the statistical distribution of material particles over various energy states in thermal equilibrium, when the temperature is high enough and density is low enough to render quantum effects negligible.The expected number of particles...
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
momentumIn classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
. Using this approach, it can be shown that the classical situation occurs if the concentration of particles corresponds to an average interparticle separation
that is much greater than the average de Broglie wavelength
of the particles,
where
is Planck's constant, and
is the
massMass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
of a particle.
For the case of conduction electrons in a typical metal at
T=300
KThe kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...
(i.e. approximately room temperature), the system is far from the classical regime since
. This is due to the small mass of the electron and the high concentration (i.e. small
) of conduction electrons in the metal. Thus F–D 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 the white dwarf's temperature is high (typically
T=10,000K on its surface), its high electron concentration and the small mass of each electron, precludes using a classical approximation, and again F–D statistics is required.
Derivation starting with canonical distribution
Consider a many-particle system composed of
N identical fermions that have negligible mutual interaction and are in thermal equilibrium. Since there is negligible interaction between the fermions, the energy
of a state
of the many-particle system can be expressed as a sum of single-particle energies,
where
is called the occupancy number and is the number of particles in the single-particle state
with energy
. The summation is over all possible single-particle states
.
The probability that the many-particle system is in the state
, is given by the normalized canonical distribution,
where
,
is Boltzmann's constant,
is the absolute
temperatureTemperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
,
e
is called the
Boltzmann factorIn physics, the Boltzmann factor is a weighting factor that determines the relative probability of a particle to be in a state i in a multi-state system in thermodynamic equilibrium at temperature T...
, and the summation is over all possible states
of the many-particle system. The average value for an occupancy number
is
Note that the state
of the many-particle system can be specified by the particle occupancy of the single-particle states, i.e. by specifying
so that
and the equation for
becomes
where the summation is over all combinations of values of
which obey the Pauli exclusion principle, and for each
. Furthermore, each combination of values of
satisfies the constraint that the total number of particles is
,
.
Rearranging the summations,
where the
on the summation sign indicates that the sum is not over
and is subject to the constraint that the total number of particles associated with the summation is
. Note that
still depends on
through the
constraint, since in one case
and
is evaluated with
while in the other case
and
is evaluated with
To simplify the notation and to clearly indicate that
still depends on
through
, define
so that the previous expression for
can be rewritten and evaluated in terms of the
,
The following approximation will be used to find an expression to substitute for
.
where
If the number of particles
is large enough so that the change in the chemical potential
is very small when a particle is added to the system, then
Taking the base
e antilog of both sides, substituting for
, and rearranging,
.
Substituting the above into the equation for
, and using a previous definition of
to substitute
for
, results in the Fermi–Dirac distribution.
Derivation using Lagrange multipliers
A result can be achieved by directly analyzing the multiplicities of the system and using
Lagrange multipliersIn mathematical optimization, the method of Lagrange multipliers provides a strategy for finding the maxima and minima of a function subject to constraints.For instance , consider the optimization problem...
.
Suppose we have a number of energy levels, labeled by index
i, each level
having energy ε
i and containing a total of
ni particles. Suppose each level contains
gi 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
gi associated with level
i is called the "degeneracy" of that energy level. The
Pauli exclusion principleThe Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...
states that only one fermion can occupy any such sublevel.
The number of ways of distributing
ni indistinguishable particles among the
gi sublevels of an energy level, with a maximum of one particle per sublevel, is given by the
binomial coefficientIn mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written \tbinom nk , and it is the coefficient of the x k term in...
, using its combinatorial interpretation
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
ni can be realized is the product of the ways that each individual energy level can be populated:
Following the same procedure used in deriving the
Maxwell–Boltzmann statisticsIn statistical mechanics, Maxwell–Boltzmann statistics describes the statistical distribution of material particles over various energy states in thermal equilibrium, when the temperature is high enough and density is low enough to render quantum effects negligible.The expected number of particles...
,
we wish to find the set of
ni for which
W is maximized, subject to the constraint that there be a fixed number of particles, and a fixed energy. We constrain our solution using
Lagrange multipliersIn mathematical optimization, the method of Lagrange multipliers provides a strategy for finding the maxima and minima of a function subject to constraints.For instance , consider the optimization problem...
forming the function:
Using
Stirling's approximationIn mathematics, Stirling's approximation is an approximation for large factorials. It is named after James Stirling.The formula as typically used in applications is\ln n! = n\ln n - n +O\...
for the factorials, taking the derivative with respect to
ni, setting the result to zero, and solving for
ni yields the Fermi–Dirac population numbers:
By a process similar to that outlined in the Maxwell-Boltzmann statistics article, it can be shown thermodynamically that
and
where
is the
chemical potentialChemical potential, symbolized by μ, is a measure first described by the American engineer, chemist and mathematical physicist Josiah Willard Gibbs. It is the potential that a substance has to produce in order to alter a system...
,
k is Boltzmann's constant and
T is the
temperatureTemperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
, so that finally, the probability that a state will be occupied is:
See also
- Fermi energy
The Fermi energy is a concept in quantum mechanics usually referring to the energy of the highest occupied quantum state in a system of fermions at absolute zero temperature....
- Maxwell–Boltzmann statistics
In statistical mechanics, Maxwell–Boltzmann statistics describes the statistical distribution of material particles over various energy states in thermal equilibrium, when the temperature is high enough and density is low enough to render quantum effects negligible.The expected number of particles...
- Bose–Einstein statistics
In statistical mechanics, Bose–Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium.-Concept:...
- Parastatistics
In quantum mechanics and statistical mechanics, parastatistics is one of several alternatives to the better known particle statistics models...
Footnotes