Fermi-Dirac distribution law

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

First, let us see that the Fermi-Dirac distribution law reduces to the Maxwell-Boltzinann law in the limit when the Nt s are small. In that case, it must be that the denominator in Eq. (3.5) is large compared to... 

The Fermi-Dirac distribution law in the forms analogous to Equations 49-1, 49-3, and 49-4 is... 

Problem 49-2. Show that at very low temperatures the Fermi-Dirac distribution law places one system part in each of the N lowest states. 

The Fermi-Dirac distribution law for the kinetic energy of the particles of a gas would be obtained by replacing p W) by the expression of Equation 49-5 for point particles (without spin) or molecules all of which are in the same non-degenerate state (aside from translation), or by this expression multiplied by the appropriate degeneracy factor, which is 2 for electrons or protons (with spin quantum number ), or in general 21 + 1 for spin quantum number I. This law can be used, for example, in discussing the behavior of a gas of electrons. The principal application which has been made of it is in the theory of metals,1 a metal being considered as a first approximation as a gas of electrons in a volume equal to the volume of the metal. 

In a non-degenerate semicondnctor - where there are few electrons or holes compared to the available locations in the condnction or valence bands - the Fermi level is located on the forbidden band and there is enough spacing between it and the conduction and valence bands. In such a case, the energy levels E of the bands are such that e - Ep kT, and the Fermi-Dirac distribution laws, which apply to electrons and holes, are reduced to Boltzmarm statistics. 

The Bose-Einstein distribution law was derived by S.N. Bose in 1924 to describe a photon gas. Einstein extended it to material gasses. Fermi developed the Fermi-Dirac distribution law in 1926 by exploring the Pauli exclusion principle, and Dirac obtained it independently in the same year by considering antisymmetric wavefimctions. 

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]. 

For this work, we [109] proposed an analytical method for the correlation between infinite dilution and finite dilution in adsorbate-adsorbent interaction system. And we reported the results of the experimental adsorption isotherm calculated by a distribution function of adsorption site energies on the basis of Fermi-Dirac s law. 

FIG. 18 Distribution functions of adsorption site energies based on the Fermi-Dirac s law, measured at 30 °C. 

We can show, as we did with the Fermi-Dirac statistics, that the distribution (6.5) approaches the Maxwell-Boltzmann distribution law at high temperatures. It is no easier to make detailed calculations with the Einstein-Bose law than with the Fermi-Dirac distribution, and on account of its smaller practical importance we shall not carry through a detailed... 

S. J. Park and J. B. Donnet, Evaluation of the distribution function of adsorption site energies based on the Fermi-Dirac s law in a monolayer, J. Colloid Interface Sci. 200 (1998) pp. 46-51. 

Writing combined wavefunctions of the form of Eq. (B2.6.1) becomes problematic if the individual components of the system are indistinguishable. As we have discussed, wavefunctions that are symmetric for interchange of two identical particles are not available to fermions, while wavefunctions that are antisymmetric for such interchanges are not available to bosons. One consequence of this is that systems of fermiOTis or bosons follow different distribution laws. Fermions obey the Fermi-Dirac distribution. 

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... 

In conclusion we must mention that a necessary condition for the validity of Eq, (3), and consequently of other formulas derived from Eq. (3) is that Ni < 1 for the state (or states) of lowest energy and a fortiori for all other states, When this inequality does not hold. Boltzmann s distribution law must be replaced by a more general and more precise distribution law, either that of Fermi and Dirac or that of Bose and Einstein according to the nature of the molecules. See also Statistical Mechanics. 

Many interesting and useful concepts follow from classical statistical con side rations (eg, the Boltzmann distribution law) and their later modifications to take into account quantum mechanical effects (Bose-Einstein and Fermi-Dirac statistics). These concepts are quite beyond the scope of the present article, and the reader should consult Refs 14 16. A brief excursion into this area is appropriate, however. A very useful concept is the so-called partition function, Z, which is defined as ... 

The previously described theory in its original form assumes that the classical kinetic theory of gases is applicable to the electron gas, that is, electrons are expected to have velocities that are temperature dependent according to the Maxwell-Boltzmann distribution law. But, the Maxwell-Boltzmann energy distribution has no restrictions to the number of species allowed to have exactly the same energy. However, in the case of electrons, there are restrictions to the number of electrons with identical energy, that is, the Pauli exclusion principle consequently, we have to apply a different form of statistics, the Fermi-Dirac statistics. 

In most physical applications of statistical mechanics, we deal with a system composed of a great number of identical atoms or molecules, and are interested in the distribution of energy between these molecules. The simplest case, which we shall take up in this chapter, is that of the perfect gas, in which the molecules exert no forces on each other. We shall be led to the Maxwell-Boltzmann distribution law, and later to the two forms of quantum statistics of perfect gases, the Fermi-Dirac and Einstein-Bose statistics. 

Equation (6.4) expresses the Einstein-Bose distribution law. As with the Fermi-Dirac law, the constant o is to be determined by the condition... 

The subject of statistical mechanics is a branch of mechanics which has been found very useful in the discussion of the properties of complicated systems, such as a gas. In the following sections we shall give a brief discussion of the fundamental theorem of statistical quantum mechanics (Sec. 49a), its application to a simple system (Sec. 496), the Boltzmann distribution law (Sec. 49c), Fermi-Dirac and Bose-Einstein statistics (Sec. 49d), the rotational and vibrational energy of molecules (Sec. 49e), and the dielectric constant of a diatomic dipole gas (Sec. 49/). The discussion in these sections is mainly descriptive and elementary we have made no effort to carry through the difficult derivations or to enter into the refined arguments needed in a... 

The Maxwell-Boltzmann Statistics, 282. The Fermi-Dirac Statistics, 285. The Bose-Einstein Statistics, 287. Relation of Statistical Mechanics to Thermodynamics, 289. Approximate Molecular Partition Functions, 292. An Alternative Formulation of the Distribution Law, 296. 

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