Fermi-Dirac distribution function, for

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. 

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. 

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

Modification of this model to get the potential function is obtained considering the Fermi-Dirac distribution function for the electron density and the Boltzmann distribution for the ionic density. This was done by Stewart and Pyatt [58] to get the energy levels and the spectroscopic properties of several atoms under various plasma conditions. Here the electron density was given by... 

In (124)/(e) is the Fermi-Dirac distribution function for the probability that a state k in the electrode with an energy e(k) is occupied, and l//(e)l is an electron wave number k-weighted interaction coupling element [186, 187],... 

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... 

Now the Fermi-Dirac distribution function of Eq. (8.2.2) will be proved. For a system of N particles in equilibrium at a finite temperature T (where N is very large, of the order of Avogadro s number), statistical mechanics suggests that the statistical weight PN(E) for the energy state E is given by... 

For metal electrodes, the Fermi level is embedded within a broad distribution of closely spaced electronic levels. The Fermi level describes the occupancy of energy levels of a system at equilibrium and can simply be thought of as the chemical potential of electrons in the solid [10]. When employed in the Fermi-Dirac distribution function, Eq. 3 results ... 

In the case of electrochemical ET, the relevant overlap was between the gaussian density-of-states function of the reacting species in solution and the Fermi-Dirac distribution function of the charge carriers in the electrode (Fig. 4.21). Figure 4B. 1 shows the analogous density-of-states functions for a homogeneous ET reaction. The rate... 

The intrinsic carrier (for instance, hole) concentration Ni t in valance band can be given using the Fermi-Dirac distribution function... 

For typical semiconductors the bandgap is between 1 and 3 eV. The electronic population of the energy bands is determined by the Fermi-Dirac distribution function, i.e. 

In the above equations, rip can be identified as the Fermi-Dirac distribution function denoting the average thermal excitation number of the spin-bath. This distribution does not contain any contribution due to the chemical potential, which implies that our stating Hamiltonian Equation 9.1 does not conserve the spin -1/2 particle number. Ideally, the typical system/spin-bath model for dissipation may be considered as an ion in an environment of two-level quantum dots (Bras et al. 2002 Favero et al. 2007 Santori and Yamamoto 2009 Xu and Teichert 2005) with characteristic frequencies governed by the size distributions of the dots. To preclude the possibility of any recurrence and to ensure irreversibility associated with the notion of dissipation, a large number is an essential requirement. 

The probability that a given energy level is occupied by an electron is provided by the Fermi-Dirac distribution function/= exp[(e eiTj/feBT] + 1 derived in Section 10.1. Near T=0, f is essentially a rectangular distribution function, with /= 1 for e < ep and/= 0 for e > ep. 

Fowler proposed a theory in 1931 which showed that the photoelectric current variation with light frequency could be accounted for by the effect of temperature on the number of electrons available for emission, in accordance with the distribution law of Sommerfeld s theory of metals. Sommerfeld s theory (1928) had resolved some of the problems surrounding the original models for electrons in metals. In classical Drude theory, a metal had been envisaged as a three-dimensional potential well (or box) containing a gas of freely mobile electrons. This adequately explained their high electrical and thermal conductivities. However, because experimentally it is found that metallic electrons do not show a gaslike heat capacity, the Boltzman distribution law is inappropriate. A Fermi-Dirac distribution function is required, consistent with the need that the electrons obey the Pauli exclusion principle, and this distribution function has the form... 

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. 

Figure 5.4.5-1 Probability functions as a function of energy-photon wavelength. Lower curves show the probability profiles of two individual energy bands. Normally, the n-band is full and the 7i -band is empty. The probability of an electron transferring between these bands when excited by a photon is a function of the energy of the photon. The upper curve illustrates this probability as a function of photon energy and wavelength for the long wavelength photoreceptor of vision. This form is frequently described as the Fermi-Dirac distribution or function.

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] ... 

For the chromatographic process, we discuss Fermi-Dirac s distribution function for characterizing the energetics of a solid surface, because it is known that this distribution is very powerful, especially, for the conditions such as weakly van der Waals attraction and relatively low temperature systems, as compared with B.E and/or M.B distributions [215]. 

Fig. V-2.—Distribution functions for Fermi-Dirac statistics (a) Maxwell-Boltzmann statistics (b) and Finstoin-Bose statistics (c).

From Eq. (4.7) we can find the distribution functions for the Einstein-Bose and Fermi-Dirac statistics. We rewrite the equation in the form... 

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