Fermi Dirac distribution

One of the simplest procedures to get the expression for the Fermi-Dirac (F-D) and the Bose-Einstein (B-E) distributions, is to apply the grand canonical ensemble methodology for a system of noninteracting indistinguishable particles, that is, fermions for the Fermi-Dirac distribution and bosons for the Bose-Einstein distribution. For these systems, the grand canonical partition function can be expressed as follows [12]  

X = e kT, in which p is the chemical potential of the system of N indistinguishable noninteracting particles 

The summation over At means that we are summing the particle distributions in the energy states accessible to the system where 

The Physical Chemistry of Materials Energy and Environmental Applications 

We know from the Pauli principle that for fermions Nk = 0 and Nk = 1. Consequently, 

---

At thermal equiUbrium characterized by temperature, T, the distribution of electrons over the allowed band of energies is given by a Fermi-Dirac distribution function ... 

Fig. 2. (a) Energy, E, versus wave vector, k, for free particle-like conduction band and valence band electrons (b) the corresponding density of available electron states, DOS, where Ep is Fermi energy (c) the Fermi-Dirac distribution, ie, the probabiUty P(E) that a state is occupied, where Kis the Boltzmann constant and Tis absolute temperature ia Kelvin. The tails of this distribution are exponential. The product of P(E) and DOS yields the energy distribution... 

Intrinsic Semiconductors. For semiconductors in thermal equiHbrium, (Ai( )), the average number of electrons occupying a state with energy E is governed by the Fermi-Dirac distribution. Because, by the Pauli exclusion principle, at most one electron (fermion) can occupy a state, this average number is also the probabiHty, P E), that this state is occupied (see Fig. 2c). In equation 2, K... 

Noise. So fat, as indicated at the beginning of this section on semiconductor statistics, equihbtium statistics have been considered. Actually, there ate fluctuations about equihbtium values, AN = N— < N >. For electrons, the mean-square fluctuation is given by < ANf >=< N > 1- ) where (Ai(D)) is the Fermi-Dirac distribution. This mean-square fluctuation has a maximum of one-fourth when E = E-. These statistical fluctuations act as electrical noise and limit minimum signal levels. 

One can actually prove a stronger result all nondeterministic LG models that satisfy semi-detailed balance and possess no spurious conservation laws have universal equilibrium solutions whose mean populations are given by the Fermi-Dirac distribution (equation 9.93) [frishc87]. 

The electronic contribution to the energy is obtained by integrating over all occupied states. To a good approximation, the Fermi-Dirac distribution can be replaced by a step function, and the integral can be performed up to the Fermi level ... 

Examination of the Fermi-Dirac distribution function Eq. (2.41) shows that the condition for applicability of the ideal-gas distribution to electron velocities is... 

The Fermi-Dirac distribution law5 gives the probability that a single-... 

Their values depend on the overpotential. Show that for r) = 0 a+/3 / 1. This (small) error arises because the Fermi-Dirac distribution has been replaced by a step function. 

The last approximation is valid if Ec — Ep kT (i.e., if the band edge is at least a few kT above the Fermi level), and the Fermi-Dirac distribution /(e) can be replaced by the Boltzmann distribution. Similarly, the concentration of holes in the valence band is ... 

The principle of the computation is to use the expressions of the forward and backward rate constant as being those of individual rate constants and sum these individual rate constants over all electronic states weighting the contribution of each state according to the Fermi-Dirac distribution.44 Assuming that H, and the density of states and therefore Kei, are independent of the energy of the electronic states,45 the results are expressed by the following equations (see Section 6.1.8) ... 

The following procedure may be used if more precision is desired to take into account integration over the Fermi-Dirac distribution in the electrode, which may be necessary for low reorganization energies. ks in equation (1.37) is converted into... 

Overhauser s original derivation of the effect employed the Fermi-Dirac distribution functions for electrons and was an involved calculation. Kittel ISl), Slichter 1S2), and others supplied simple derivations for this effect and Abragam 133) extended it to nonmetallic systems. 

Extension to finite temperature T can be made by using the Fermi-Dirac distribution fimction for fk in Eq. (82)... 

The significance of the electrochemical potential is apparent when related to the concepts of the usual stati.stical model of free electrons in a body where there are a large number of quantum states e populated by noninteracting electrons. If the electronic energy is measured from zero for electrons at rest at infinity, the Fermi-Dirac distribution determines the probability P(e) that an electron occupies a state of energy e given by... 

The full Fermi-Dirac distribution law (as distinct from the Fermi-Dirac probability of occupancy expression) is therefore... 

Here u fl" and E " are the periodic part of the Bloch function, energy and Fermi-Dirac distribution functions for the n-th carrier spin subband. In the case of cubic symmetry, the susceptibility tensor is isotropic, Xcj) = Xc ij- It has been checked within the 4 x 4 Luttinger model that the values of 7c, determined from eqs (13) and (12), which do not involve explicitly u and from eqs (14) and (15) in the limit q - 0, are identical (Ferrand et al. 2001). Such a comparison demonstrates that almost 30% of the contribution to 7c originates from interband polarization, i.e. from virtual transitions between heavy and light hole subbands. 

Neutrino Temperature Assuming the neutrino spectrum to be Fermi-Dirac distribution with the vanishing chemical potential, we got the Fe temperature, T [3,4]. For the Kamiokande data, T is 2.6 3.lMeV, for IMB 3.9 5.3 MeV. These values are a little close to the values by those who insist the late time neutrino heating mechanism of the explosion. 

We assumed that tau neutrinos are emitted from their neutrino-sphere with the speotrum of Fermi-Dirac distribution with zero ohemical potential determined by the temperature at the neutrino-sphere. This temperature is estimated to be 5 MeV from the spectrum calculated by Wilson and his collaborators. 

---