Fermi-Dirac statistics derivation

Maxwell-Boltzmann particles are distinguishable, and a partition function, or distribution, of these particles can be derived from classical considerations. Real systems exist in which individual particles are indistinguishable. For example, individual electrons in a solid metal do not maintain positional proximity to specific atoms. These electrons obey Fermi-Dirac statistics (133). In contrast, the quantum effects observed for most normal gases can be correlated with Bose-Einstein statistics (117). The approach to statistical thermodynamics described thus far is referred to as wave mechanics. An equivalent quantum theory is referred to as matrix mechanics (134—136). 

The Fermi energy (level) is derived from the Fermi-Dirac statistics which describe the distribution of indistinguishable, noninteracting particles in n available... 

In this derivation, it was assumed that the charge-transfer complex is formed between the p- or n- primary dopant and the gas acts as a secondary dopant. In fact, the interaction of the secondary dopant with any energy state in the matrix is possible that would lead to the same result, as long as the exchanged electron density becomes part of the electron population governed by the Fermi-Dirac statistics. The analytical utility of this relationship has been shown for several inorganic gases (Janata and Josowicz, 2003). 

In the Einstein-Bose statistics, there is no such clear physical way to find the revised law of collisions as in the Fermi-Dirac statistics. The law can be derived from the quantum theory but not in a simple enough way to describe here. In contrast to the Fermi-Dirac statistics, in which the presence of one molecule in a cell prevents another from entering the same cell, the situation with the Einstein-Bose statistics is that the presence of a molecule in a cell increases the probability that another one should enter the same cell. In fact, the number of molecules going into the fcth cell per second turns out to have a factor (1 + Nk), increasing linearly with the mean number Nk of molecules in that cell. Thus, the law of collisions for the Einstein-Bose statistics is just like Eq. (4.1), only with + signs replacing the — signs. In fact, we may write the law of collisions for both forms of statistics in the form... 

Enrico Fermi (1901-1954), Italian physicist, professor at universities in Florence, Rome, New York, and in 1941-1946 at the University of Chicago. Fermi introduced the notion of statistics for the particles with a half-integer spin number (called fermions) during the Florence period. Dirac made the same discovery independently, hence this property is called the Fermi-Dirac statistics. Young Fermi was notorious for being able to derive a formula from any domain of physics faster than someone sent to find it in textbooks. His main topic was nuclear physics. He played an important role in the A bomb construction in Los Alamos, and in 1942 he built the world s first nuclear reactor on a tennis court at the University of Chicago. 

At absolute zero, the highest occupied energy level is referred to as the Fermi level (in 3-D Fermi surface), derived from Fermi-Dirac statistics.The Fermi-Dirac distribution function, f(E), describes the probability that a given available energy state will be occupied at a given temperature ... 

Fowler s derivation for a single photon does not explicitly involve the quantum mechanical form of current instead, a semiclassical flux of electrons arriving at tbe metal surface is used. The electron gas in a metal will obey Fermi-Dirac statistics, and the number of electrons per unit volume having velocity components in the ranges , w + du, v,v + dv, w, and w + dw is given by the formula... 

The occupation of the electron energy levels under thermal equilibrium at T > 0 K can be derived from statistical thermodynamics [11]. The reasoning results in the well-known Fermi-Dirac occupation function. The probability that an electron energy level at E is occupied by an electron is given by... 

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... 

Here, i and stand for initial states, k and I for final states and a i(gfi,4>) dQ, defined for a rearrangement collision in Eq. (161) has been substituted for the classical 2nb db. Equation (230) may be derived from the quantum mechanical Boltzmann equation of Uehling and Uhlenbeck with the assumption of Boltzmann statistics, i.e., that the low-temperature symmetry effects of Fermi-Dirac or Bose-Einstein statistics are negligible, but that the quantum mechanical collision cross-section should be retained. We should note that although one may heuristically introduce the quantum mechanical cross-section as in Eq. (230), the Uehling-Uhlenbeck or a similar equation is strictly derived for special interactions only. In this connection it is interesting that the method given later in Section V-C yields the same result [Eq. (330)] as that of this section [Eq. (251)] only when an approximation equivalent to the Bom approximation is made. 

A chapter introducing the Bose-Einstein, Maxwell-Boltzman, Planck, and Fermi-Dirac distribution fimctions follows before discussing the thermal, electronic, magnetic, and optical properties for the benefit of students who have not been exposed to quantum statistical mechanics. This chapter is a logical beginning for the second half of this book since these concepts are essential to an imderstanding of these properties. Similarly, the Maxwell equations are used to derive the equations for absorption and normal reflection of electromagnetic waves in the chapter on optical properties. The band structure of metals... 

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. 

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