Fermi particle

We have assumed so far, implicitly, that the interactions are strictly local between neighboring atoms and that long-ranged forces are unimportant. Of course the atom-atom interaction is based on quantum mechanics and is mediated by the electron as a Fermi particle. Therefore the assumption of short-range interaction is in principle a simplification. For many relevant questions on crystal growth it turns out to be a good and reasonable approximation but nevertheless it is not always permissible. For example, the surface of a crystal shows a superstructure which cannot be explained with our simple lattice models. 

From the well-known statistical requirement for an assembly of Fermi particles, W 2--------N) is subject to a limitation in its form of anti-... 

Electrons with their half-integral spins are known as Fermi particles or fermions and no more than two electrons can occupy a quantum state. At absolute zero the electrons occupy energy levels from zero to a maximum value of f F, defined by... 

Particles that obey Fermi statistics are called Fermi particles or fermions. The probability density of Fermi particles in their energy levels is thus represented by the Fermi function, fiz), that gives the probability of fermion occupation in an energy level, e, as shown in Eqn. 1-1 ... 

The particles we will deal with in this textbook are mainly electrons and ions in condensed solid and liquid phases. In condensed phases ions are the classical Boltzmann particles and electrons are the degenerated Fermi particles. 

Electrode reactions can be classified into two groups one in which an electron transfer takes place across the electrode interface, such as ferric-ferrous redox reaction (Fet, + e = Fe ) and the other in which an ion transfer takes place across the electrode interface, such as iron dissolution-deposition reaction (Fe M = FeVq). Since electrons are Fermi particles in contrast to ions that obey the Boltzmann statistics as described in Chap. 1, the reaction kinetics of the two groups differ in their electrode reactions. 

An electrode is called an electronic electrode when the transfer of electrons occurs, while it is called an ionic electrode when the transfer of ions occurs at the electrode interface. Although electrons and ions are in the same category of charged particles, they are different in electrochemical behavior due to a difference in the type of statistics that governs them. Electrons are Fermi particles which obey the Fermi statistics, whereas ions are Boltzmann particles which obey the Boltzmann statistics. 

For a symmetrical (D ) diatomic or linear polyatomic molecule with two, or any even number, of identical nuclei having the nuclear spin quantum number (see Equation 1.47) I = n + where n is zero or an integer, exchange of any two which are equidistant from the centre of the molecule results in a change of sign of i/c which is then said to be antisymmetric to nuclear exchange. In addition the nuclei are said to be Fermi particles (or fermions) and obey Fermi Dirac statistics. However, if / = , p is symmetric to nuclear exchange and the nuclei are said to be Bose particles (or bosons) and obey Bose-Einstein statistics. 

It is interesting to draw a distinction between the two aspects of correlation which we have considered so far in terms of the second-quantization method for systems of N identical Fermi particles. Those methods (which are but a more effective and general way of formulating Cl) rest upon the occupation-number representation given the set of all possible single-particle states (spinorbitals), one builds a complete set of N-particle states. .. > by constructing Slater determinants or... 

The sum over rij can now be performed, but this depends on the statistics that the particles in the ideal gas obey. Fermi particles obey the Pauli exclusion principle, which allows only two possible values rij = 0, 1. For Bose particles, rij can be any integer between zero and infinity. Thus the grand partition function is... 

Eermions (particles with a spin of n ft/2 and n = odd integer) play an important role in many fields of physics, because aU stable elementary particles, such as electron, proton and neutron are Fermi particles with a spin of ft/2. Also atoms with a total angular momentum of n ft/2 (n = odd) are Fermions, such as the lithium isotope Li with a nuclear spin / = 1 ft and an electron spin s = ft/2, resulting in a total angular momentum of F = 3/2 ft or F = 1/2 ft. 

It is also possible to trap Fermi-particles in an optical lattice. Such a system has many similarities with a correlated electron gas in a solid. 

Since electrons, as a kind of Fermi particles, obey Pauli exclusion principle, the overlap of the orbitals of outer electrons when two atoms approach each other induces a strong short range repulsion. For this reason, we can consider an atom or monoatomic ion as a spherical globe having definite radius in some kind of environments. This is of course only an approximation, since the inter-atomic charge transfer surely changes the size of atoms significantly. But we cannot consider this value as a variable in atomic parameter method otherwise we cannot... 

The expression for the pair (n = 2) distribution function in three (d = 3) dimension is well known [1,2]. However, the general one for any n and d is much less known. Interestingly, the distribution of Fermi particles in one d = 1) dimension has a mathematical structure similar to those found for the eigenvalues of the random matrices [3-5] and for the zeros of the Riemann zeta function [6,7], as shown below. In the following Sects. 14.2 and 14.3, explicit expressions for the pair and ternary distribution functions of the ideal Fermi gas system in any dimension are derived. We then find an expression for the n-particle distribution function as a determinant form in Sect. 14.4. Another representation for the multiparticle distribution for finite IV is given in terms of density matrix in Sect. 14.5. The explicit formula for correlation kernel which plays an essential role for the description of the multiparticle correlations in the Fermi system is derived in Sect. 14.6. The relationship with the theories for the random matrices and the Riemann zeta function is addressed in Sect. 14.7. 

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