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Fermi-Dirac statistics
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Fermi-Dirac statistics (F-D statistics) is a part of the science of physics, that applies to a system comprised of many particles (for example electrons) that obey the Pauli Exclusion Principle. It is named after Enrico Fermi and Paul Dirac, who each discovered it independently.
F-D statistics describes the energies of identical particles with half-integer spin which comprise a many-particle system in thermal equilibrium. Additionally, the particles in this system are assumed to have negligible mutual interaction.
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Fermi-Dirac statistics (F-D statistics) is a part of the science of physics, that applies to a system comprised of many particles (for example electrons) that obey the Pauli Exclusion Principle. It is named after Enrico Fermi and Paul Dirac, who each discovered it independently.
F-D statistics describes the energies of identical particles with half-integer spin which comprise a many-particle 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. F-D statistics gives the 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 Fermi-Dirac statistics applies to particles with half-integer spin, they have come to be called fermions. F-D statistics 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, understanding some aspects of electron behavior was difficult due to seemingly contradictory phenomena. It was difficult to understand, for example, why electrons in a metal can move freely to conduct electric current, while their contribution in the same metal to the specific heat was negligible, as if there were considerably fewer electrons.
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 of the order of the Boltzmann constant k.
This statistical problem remained unsolved until the discovery of F-D statistics.
The discovery of F-D statistics was first published in 1926 by Enrico Fermi and Paul Dirac, independently of each other. According to an account, Pascual Jordan developed in 1925 the same statistics which he called Pauli statistics, but it wasn't published in a timely manner because the journal editor Max Born forgot the paper for six months until after the publications by Fermi and Dirac. Whereas according to Dirac, Fermi-Dirac statistics was first studied by Fermi, and Dirac called it Fermi statistics and the corresponding particles fermions.
F-D statistics was applied in 1926 by Ralph Fowler to describe the collapse of a star to a white dwarf. In 1927 Arnold Sommerfeld applied it to electrons in metals. Fermi-Dirac statistics continues to be an important part of physics.
Fermi-Dirac distribution
The average number of fermions in a single-particle state , is given by the Fermi-Dirac (F-D) distribution,
where is the energy of state , is the chemical potential, k is Boltzmann's constant, and T is the absolute temperature. (See figures below.) Since the F-D distribution was derived using the Pauli exclusion principle, a result is that 0 < < 1 . Note that is also the probability that the state is occupied since no more than one fermion can occupy the same state at the same time. 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 the chemical potential .
Image:FD e mu.jpg|Energy dependence for different temperatures.
Energy dependence is more gradual at higher temperatures.
(Click on figure to enlarge.)
Image:FD kT e.jpg|Temperature dependence for .
(Click on figure to enlarge.)
Derivation
Derivation of the Fermi-Dirac Distribution
Consider a many-particle system comprised of N identical fermions that have negligible mutual interaction and are in thermal equilibrium. For this case of negligible interaction between the fermions, the energy of a state of the many-particle system can be expressed as a sum of the energies of particles in single-particle states ,
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 temperature, is called the Boltzmann factor, and the summation is over all possible states of the many-particle system. Using , 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, = 0 or 1 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 as,
-
Evaluating the summations over ,
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 anti-log of both sides and substituting for ,
and rearranging,
.
Substituting the above into the equation for , and substituting for ,
-
which is the Fermi-Dirac distribution.
Another Derivation of the Fermi-Dirac Distribution
A result can be achieved by directly analyzing the multiplicities of the system.
Suppose we have a number of energy levels, labeled by index i, each level
having energy ei 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, 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 principle states that only one fermion can occupy any such sublevel.
Let w(n, g) be the number of ways of distributing n particles among the g sublevels of an energy level. It's clear that there are g ways of putting one particle into a level with g sublevels, so that w(1, g) = g which
we will write as:
We can distribute 2 particles in g sublevels by putting one in the first sublevel and then distributing the remaining (n − 1) particles in the remaining (g − 1) sublevels, or we could put one in the second sublevel and then distribute the remaining (n − 1) particles in the remaining (g − 2) sublevels, etc. so that w'(2, g) = w(1, g − 1) + w(1,g − 2) + ... + w(1, 1) or
where we have used the following theorem involving binomial coefficients:
Continuing this process, we can see that w(n, g) is just a binomial coefficient
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 statistics,
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 multipliers forming the function:
Again, using Stirling's approximation for the factorials and taking the derivative with respect to ni, and setting the result to zero and solving for ni yields the Fermi-Dirac population numbers:
It can be shown thermodynamically that ß = 1/kT where k is Boltzmann's constant and T is the temperature, and that a = -µ/kT where µ is the chemical potential, so that finally:
Note that the above formula is sometimes written:
where is the fugacity.
Distribution of particles over energy
The above Fermi-Dirac distribution gives the distribution of particles over single-particle energy states, where no more than one particle can occupy a state. Using the F-D distribution, one can find the distribution of particles over energy, where more than one particle can have the same energy.
The average number of fermions with energy can be found by multiplying the F-D distribution by the degeneracy (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 states (i.e. the number of states per unit energy range) the average number of fermions per unit energy range is,
-
where is called the Fermi function and is the same function that is used for the F-D distribution ,
-
so that,
- .
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 that is much greater than the average de Broglie wavelength of the particles,
-
where is Planck's constant, and 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 . 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 metal.
Another example of a system that is not in the classical regime is the system comprised 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.
See also
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