Fermi-Dirac statistics
Encyclopedia : F : FE : FER : Fermi-Dirac statistics
| Particle statistics |
|---|
| Maxwell-Boltzmann statistics |
| Bose-Einstein statistics |
| Fermi-Dirac statistics |
| Parastatistics |
| Anyonic statistics |
| Braid statistics |
| [edit] |
In statistical mechanics, Fermi-Dirac statistics is a particular case of particle statistics developed by Enrico Fermi and Paul Dirac that determines the statistical distribution of fermions over the energy states for a system in thermal equilibrium. In other words, it is a probability of a given energy level to be occupied by a fermion. Fermions are particles which are indistinguishable and obey the Pauli exclusion principle, i.e., no more than one particle may occupy the same quantum state at the same time. Statistical thermodynamics is used to describe the behaviour of large numbers of particles. A collection of non-interacting fermions is called a Fermi gas.
F-D statistics was introduced in 1926 by Enrico Fermi and Paul Dirac and applied in 1927 by Arnold Sommerfeld to electrons in metals.
For F-D statistics, the expected number of particles in states with energy [\epsilon _i] is
- [ n_i = \frac + 1} ]
- [n_i \ ] is the number of particles in state i,
- [\epsilon_i \ ] is the energy of state i,
- [g_i \ ] is the degeneracy of state i (the number of states with energy [\epsilon_i \ ]),
- [\mu ] is the chemical potential (Sometimes the Fermi energy [E_F \ ] is used instead, as a low-temperature approximation),
- k is Boltzmann's constant, and
- T is absolute temperature.
- [ F(E) = \frac + 1} ]
Which distribution to use
Fermi-Dirac and Bose-Einstein statistics apply when quantum effects have to be taken into account and the particles are considered "indistinguishable". The quantum effects appear if the concentration of particles (N/V) ≥ nq (where nq is the quantum concentration). The quantum concentration is when the interparticle distance is equal to the thermal de Broglie wavelength i.e. when the wavefunctions of the particles are touching but not overlapping. As the quantum concentration depends on temperature; high temperatures will put most systems in the classical limit unless they have a very high density e.g. a White dwarf. Fermi-Dirac statistics apply to fermions (particles that obey the Pauli exclusion principle), Bose-Einstein statistics apply to bosons. Both Fermi-Dirac and Bose-Einstein become Maxwell-Boltzmann statistics at high temperatures or low concentrations.Maxwell-Boltzmann statistics are often described as the statistics of "distinguishable" classical particles. In other words the configuration of particle A in state 1 and particle B in state 2 is different from the case where particle B is in state 1 and particle A is in state 2. When this idea is carried out fully, it yields the proper (Boltzmann) distribution of particles in the energy states, but yields non-physical results for the entropy, as embodied in Gibbs paradox. These problems disappear when it is realized that all particles are in fact indistinguishable. Both of these distributions approach the Maxwell-Boltzmann distribution in the limit of high temperature and low density, without the need for any ad hoc assumptions. Maxwell-Boltzmann statistics are particularly useful for studying gases F-D statistics are most often used for the study of electrons in solids. As such, they form the basis of semiconductor device theory and electronics.
A derivation
Consider a single-particle state of a multiparticle system, whose energy is [\mathbf]. For example, if our system is some quantum gas in a box, then a state might be a particular single-particle wave function. Recall that, for a grand canonical ensemble in general, the grand partition function is
- :[Z \;= \sum_s e^ ]
- [E(s) \ ] is the energy of a state s,
- [N(s) \ ] is the number of particles possessed by the system when in the state s,
- [\mu] denotes the chemical potential, and
- s is an index that runs through all possible microstates of the system.
For fermions, a state can only be either occupied by a single particle or unoccupied. Therefore our system has multiplicity two: occupied by one particle, or unoccupied, called [s_1] and [s_2] respectively. We see that [E(s_1) = \; \epsilon], [N(s_1) = \; 1], and [E(s_2) = \; 0], [N(s_2) = \; 0]. The partition function is therefore
- [Z = \sum_ ^2 e^ = e^ + 1 ].
- [P( s_ ) = \frac) - \mu N(s_)} }].
- [\bar = P( s_1 ) = \frac } = \frac} + 1}= \frac + 1}].
[\bar] is called the Fermi-Dirac distribution. For a fixed temperature T, [\bar(\epsilon)] is the probability that a state with energy ε will be occupied by a fermion. Notice [\bar] is a decreasing function in ε. This is consistent with our expectation that higher energy states are less likely to be occupied.
Note that if the energy level ε has degeneracy [\; g_], then we would make the simple modification:
- [\bar = g_ \cdot \frac + 1}].
For all temperature T, [\bar(\mu) = \frac] , that is, the states whose energy is μ will always have equal probability of being occupied or unoccupied.
In the limit [T \rightarrow 0], [\bar] becomes a step function (see graph above). All states whose energy is below the chemical potential will be occupied with probability 1 and those states with energy above μ will be unoccupied. The chemical potential at zero temperature is called Fermi energy, denoted by [E _F], i.e.
[ E _F = \; \mu(T = 0)].
It may be of interest here to note that, in general the chemical potential is temperature-dependent. However, for systems well below the Fermi temperature [T_F = \frac], it is often sufficient to use the approximation [\mathbf] ≈ [\; E_F] .
Another derivation
In the previous derivation, we have made use of the grand partition function (or Gibbs sum over states). Equivalently, the same result can be achieved by directly analysing the multiplicities of the system.
Suppose there are two fermions placed in a system with four energy levels. There are six possible arrangements of such a system, which are shown in the diagram below.
ε1 ε2 ε3 ε4 A * * B * * C * * D * * E * * F * *Each of these arrangements is called a microstate of the system. Assume that, at thermal equilibrium, each of these microstates will be equally likely, subject to the constraints that there be a fixed total energy and a fixed number of particles.
Depending on the values of the energy for each state, it may be that total energy for some of these six combinations is the same as others. Indeed, if we assume that the energies are multiples of some fixed value ε, the energies of each of the microstates become:
- A: 3ε
- B: 4ε
- C: 5ε
- D: 5ε
- E: 6ε
- F: 7ε
Now suppose we have a number of energy levels, labelled by index i , each level having energy εi 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. Its 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:
- [w(1,g)=\frac]
- [w(2,g)=\sum_^w(1,g-k) =\sum_^\frac=\frac]
- [\sum_^g \frac=\frac]
- [w(n,g)=\frac]
- [W = \prod_i w(n_i,g_i) = \prod_i \frac]
- [f(n_i)=\ln(W)+\alpha(N-\sum n_i)+\beta(E-\sum n_i \epsilon_i)]
- [n_i = \frac+1} ]
- [n_i = \frac+1} ]
- [n_i = \frac/z+1} ]
See also
From Wikipedia, the Free Encyclopedia. Original article here. Support Wikipedia by contributing or donating.
All text is available under the terms of the GNU Free Documentation License See Wikipedia Copyrights for details.