Statistics Fermi-Dirac

Under these conditions, the thermodynamic probabihty is said of a Fermi-Dirac (FD) type, or in a few words, of fermionic type (particle with half-integer spin) and is genuinely written as 

TABLE 1.4 The Illustration of a Mode (from the possible ones) for the Quantum Distribution for Fermionic Type Particles with Half-Integer Spin on an Energetic Level with Sub-Levels g. (Putz, 2010) 

It is nevertheless equivalently written through the application of Stirling transformation of permutation statistics (see Appendix A.2) 

Furthermore, the thermodynamic function of the macrostate is formed, using the last form of Fermi-Dirac probability to look like 

Quantum Nanochemistry-Volume I Quantum Theory and Observability 

The density of states is foimd as before, except that we have made the cells in phase space half as small to assure no more than single occupancy. Thus Equation 15.18 becomes 

However, if we put the numbers corresponding to electrons in a metal, say Al, the electron density n = 1.8 x 10 electrons/cm and m = 9.11 x 10 g into Equation 15.20, we see that the value for A computed in the classical manner is 1.6 x 10, which of course violates the assumption A C 1 that was made in taking the classical limit. Therefore, F-D statistics must be used in treating electrons in a metal. 

---

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... 

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

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... 

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 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. 

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. 

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. 

For a metal, the negative of the work function gives the position of the Fermi level with respect to the vacuum outside the metal. Similarly, the negative of the work function of an electrochemical reaction is referred to as the Fermi level Ep (redox) of this reaction, measured with respect to the vacuum in this context Fermi level is used as a synonym for electrochemical potential. If the same reference point is used for the metal s,nd the redox couple, the equilibrium condition for the redox reaction is simply Ep (metal)= Ep(redox). So the notion of a Fermi level for a redox couple is a convenient concept however, this terminology does not imply that there are free electrons in the solution which obey Fermi-Dirac statistics, a misconception sometimes found in the literature. 

Fermentor(s), 10 266 11 1 Fermentor agitators, 11 34 Fermentor vent, 11 40 Fermi-1 fast-breeder reactor, 17 586 Fermi-Dirac statistics, silicon-based semiconductors and, 22 235-236, 237... 

To calculate the total energy of the cell with AN electrons, one needs fis), the probability for the state with energy s to be occupied by an electron. This is given by Fermi-Dirac statistics as... 

Electrons in thermal equilibrium in the solid state obey Fermi-Dirac statistics as given by... 

As mentioned above, we assume that the molecular energy does not depend on the nuclear spin state For the initial rovibronic state nuclear spin functions available, for which the product function 4 i) in equation (2) is an allowed complete internal state for the molecule in question, because it obeys Fermi-Dirac statistics by permutations of identical fermion nuclei, and Bose-Einstein statistics by permutations of identical boson nuclei (see Chapter 8 in Ref. [3]). By necessity [3], the same nuclear spin functions can be combined with the final rovibronic state form allowed complete... 

Another subtlety is that the assumption nuclei behave as Dirac particles, amounts to assuming that all nuclei have spin 1/2. However, it is not uncommon to have nuclei with spin as high as 9/2 worse nuclei with integer spins are bosons and do not obey Fermi-Dirac statistics. The only justification to use equation (75) for such a case is that the resulting theory agrees with experiment. Under the assumption, we are in a position to extend our many-fermion Hamiltonian to molecules assuming that the nuclei are Dirac particles with anomalous spin. The molecular Hamiltonian may then be written as... 

In a scheme of available energy states, a population of electrons distributes according the Fermi-Dirac statistics The probability f(E) of having an electron in a state of energy E, is, at temperature T... 

The two populations however, must have the same chemical potential pp, in a Fermi-Dirac statistics (Eq. (13)). 

III.3 GENERALIZATION TO TREAT BOSE-EINSTEEV AND FERMI-DIRAC STATISTICS... 

It is not absolutely necessary to have accurate interatomic potentials to perform reasonably good calculations because the many collisions involved tend to obscure the details of the interaction. This, together with the fact that accurate potentials are only known for a few systems makes the Thomas-Fermi approach quite attractive. The Thomas-Fermi statistical model assumes that the atomic potential V(r) varies slowly enough within an electron wavelength so that many electrons can be localized within a volume over which the potential changes by a fraction of itself. The electrons can then be treated by statistical mechanics and obey Fermi-Dirac statistics. In this approximation, the potential in the atom is given by ... 

---