Fermi-Dirac

As pointed out in the previous paragraph, the total wave function of a molecule consists of an electronic and a nuclear parts. The electrons have a different intrinsic nature from nuclei, and hence can be treated separately when one considers the issue of permutational symmetry. First, let us consider the case of electrons. These are fermions with spin and hence the subsystem of electrons obeys the Fermi-Dirac statistics the total electronic wave function... 

Thomas-Fermi total energy Eg.j.p [p] gives the so-called Thomas-Fermi-Dirac (TFD) energy functional. 

Fermi-Dirac statistics Fermi-Dirac systems Fermi level Fermi levels Fermion Fermions... 

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. 

In the above-mentioned 1980 symposium (p. 8), the historians Hoddeson and Baym outline the development of the quantum-mechanical electron theory of metals from 1900 to 1928, most of it in the last two years of that period. The topic took off when Pauli, in 1926, examined the theory of paramagnetism in metals and proved, in a famous paper (Pauli 1926) that the observations of weak paramagnetism in various metals implied that metals obeyed Fermi-Dirac statistics - i.e., that the electrons in... 

Density functional theory-based methods ultimately derive from quantum mechanics research from the 1920 s, especially the Thomas-Fermi-Dirac model, and from Slater s fundamental work in quantum chemistry in the 1950 s. The DFT approach is based upon a strategy of modeling electron correlation via general functionals of the electron density. 

TF) theory, including the Ko[p exchange part (first derived by Block but commonly associated with the name of Dirac" (constitutes the Thomas-Fermi-Dirac (TFD) model. 

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]. 

In formulating the second-quantized description of a system of noninteracting fermions, we shall, therefore, have to introduce distinct creation and annihilation operators for particle and antiparticle. Furthermore, since all the fermions that have been discovered thus far obey the Pauli Exclusion principle we shall have to make sure that the formalism describes a many particle system in terms of properly antisymmetrized amplitudes so that the particles obey Fermi-Dirac statistics. For definiteness, we shall in the present section consider only the negaton-positon system, and call the negaton the particle and the positon the antiparticle. 

The commutations (9-416)-( 9-419) guarantee that the state vectors are antisymmetric and that the occupation number operators N (p,s) and N+(q,t) can have only eigenvalues 0 and 1 (which is, of course, what is meant by the statement that particles and antiparticles separately obey Fermi-Dirac etatistios). In fact one readily verifies that... 

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 ... 

The behavior of a multi-particle system with a symmetric wave function differs markedly from the behavior of a system with an antisymmetric wave function. Particles with integral spin and therefore symmetric wave functions satisfy Bose-Einstein statistics and are called bosons, while particles with antisymmetric wave functions satisfy Fermi-Dirac statistics and are called fermions. Systems of " He atoms (helium-4) and of He atoms (helium-3) provide an excellent illustration. The " He atom is a boson with spin 0 because the spins of the two protons and the two neutrons in the nucleus and of the two electrons are paired. The He atom is a fermion with spin because the single neutron in the nucleus is unpaired. Because these two atoms obey different statistics, the thermodynamic and other macroscopic properties of liquid helium-4 and liquid helium-3 are dramatically different. 

Of these, only J[p] is known, while the explicit forms of the other two contributions remain a mystery. The Thomas-Fermi and Thomas-Fermi-Dirac approximations that we briefly touched upon in Chapter 3 are actually realizations of this very concept. All terms present in these models, i. e., the kinetic energy, the potential due to the nuclei, the classical... 

As a simple illustration of Eq. (77), consider a system composed of three particles of the same energy that can occupy four quantum states. The case of Fermi-Dirac statistics is shown in Fig. 2a. 

For liquid metals, one has to set up density functionals for the electrons and for the particles making up the positive background (ion cores). Since the electrons are to be treated quantum mechanically, their density functional will not be the same as that used for the ions. The simplest quantum statistical theories of electrons, such as the Thomas-Fermi and Thomas-Fermi-Dirac theories, write the electronic energy as the integral of an energy density e(n), a function of the local density n. Then, the actual density is found by minimizing e(n) + vn, where v is the potential energy. Such... 

Bloch (1933a,b) first pointed out that in the Thomas-Fermi-Dirac statistical model the spectral distribution of atomic oscillator strength has the same shape for all atoms if the transition energy is scaled by Z. Therefore, in this model, I< Z Bloch estimated the constant of proportionality approximately as 10-15 eV. Another calculation using the Thomas-Fermi-Dirac model gives I tZ = a + bZ-2/3 with a = 9.2 and b = 4.5 as best adjusted values (Turner, 1964). This expression agrees rather well with experiments. Figure 2.3 shows the variation of IIZ vs. Z. 

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