Free electron model, Pauli exclusion

The resolution of this issue is based on the application of the Pauli exclusion principle and Femii-Dirac statistics. From the free electron model, the total electronic energy, U, can be written as... 

In the free electron model, the electrons are presumed to be loosely bound to the atoms, making them free to move throughout the metal. The development of this model requires the use of quantum statistics that apply to particles (such as electrons) that have half integral spin. These particles, known as fermions, obey the Pauli exclusion principle. In a metal, the electrons are treated as if they were particles in a three-dimensional box represented by the surfaces of the metal. For such a system when considering a cubic box, the energy of a particle is given by... 

Optical properties of metal nanoparticles embedded in dielectric media can be derived from the electrodynamic calculations within solid state theory. A simple model of electrons in metals, based on the gas kinetic theory, was presented by Drude in 1900 [9]. It assumes independent and free electrons with a common relaxation time. The theory was further corrected by Sommerfeld [10], who incorporated corrections originating from the Pauli exclusion principle (Fermi-Dirac velocity distribution). This so-called free-electron model was later modified to include minor corrections from the band structure of matter (effective mass) and termed quasi-free-electron model. Within this simple model electrons in metals are described as... 

The DOS at 0 K is shown in Figure 11.18 using the free electron model. However, the Drude-Lorentz model employs the classical equipartition of energy and does not take into account the fact that quantum mechanics places restrictions on the placement of the electrons (as a result of the Pauli exclusion principle). A revised theory, known as the Sommerfield model, allows for this modification. At temperatures above 0 K, the fraction, f(E), of allowed energy levels with energy E follows the... 

The qualitative interpretation of these results in terms of conduction —> covalent electronic transformation model is based on the following principles (1) covalent electrons are localized and therefore are identifiable with a group of ions, whereas conduction ( free ) electrons are delocalized and are simultaneously shared by all ions. (2) thus, covalent electrons having no Fermi surface whereas conduction electrons (because of the Pauli exclusion principle) having well defined Fermi surface, and (3) electrons are needed in forming covalent bonds, (i.e., under no circumstances can holes be substituted for electrons in forming bonds) in sharp contrast, holes behave in much the same way as electrons in band structure. 

The characteristic properties of the metallic state of matter follow from two overwhelmingly important physical effects the overlap of valence electron wave functions on neighboring atoms and the Pauli exclusion principle. These effects are embodied in the nearly-free-electron (NFE) model of metals. 

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