Crystal Pauli principle

Identify the ligands and the geometiy of the coordination complex, construct the crystal field energy level diagram, count d electrons from the metal and place them according to the Pauli principle and Hund s rule. 

Sommerfeld suggested that the potential in a metal crystal could be assumed constant. This assumption implies that the forces acting on an electron cancel to zero and that the electrons in a metal can be described like a non-interacting gas of electrons, confined to a box that represents the metal. The only restriction on electronic motion would be the Pauli principle. The electronic energy in a three-dimensional rectangular box is known as... 

For two and three dimensions, it provides a crude but useful picture for electronic states on surfaces or in crystals, respectively. Free motion within a spherical volume gives rise to eigenfunctions that are used in nuclear physics to describe the motions of neutrons and protons in nuclei. In the so-called shell model of nuclei, the neutrons and protons fill separate s, p, d, etc orbitals with each type of nucleon forced to obey the Pauli principle. These orbitals are not the same in their radial shapes as the s, p, d, etc orbitals of atoms because, in atoms, there is an additional radial potential V(r) = -Ze2/r present. However, their angular shapes are the same as in atomic structure because, in both cases, the potential is independent of 0 and (f>. This same spherical box model has been used to describe the orbitals of valence electrons in clusters of mono-valent metal atoms such as Csn, Cu , Na and their positive and negative ions. Because of the metallic nature of these species, their valence electrons are sufficiently delocalized to render this simple model rather effective (see T. P. Martin, T. Bergmann, H. Gohlich, and T. Lange, J. Phys. Chem. 95, 6421 (1991)). 

An important achievement of quantum mechanics is the concept of band structure for the energies that electrons can assume, see fig. 3.68. Just as in atoms, where electrons are confined to certain orbitals, between which they can "Jump" across a forbidden zone, in crystals there are bands of a given width between which there are forbidden zones that under some conditions can be crossed. As is the case with atomic orbitals the number of electrons that can be accommodated in a band is limited by the Pauli principle. As displacement of electrons under the influence of an applied electric field is not possible for a completely filled band, such bands do not contribute to the conductivity. In the... 

The Pauli principle accounts for crystal and molecular structures and for the fact that covalent bonding is directed in space. Its quantum-mechanical basis is explained with a simple orbital approach the wavefunction (l, 2) for two electrons that have the same spin (and together form a triplet state) is described by the Slater determinant ab in which a and b indicate the atomic orbitals on atoms a and b. This notation is shorthand for... 

However, electrons in solids obey Fermi statistics. Now, Boltzmann statistics makes no assertions regarding the number of particles which are permitted in the cells in phase space, but rather, it examines the probability of finding a given number of particles in the individual cells. Fermi statistics, on the other hand, asserts from the start that, because of the Pauli principle, only two particles are permitted in any cell of phase space [7]. In order to show how Fermi statistics affects the formulation of the chemical potentials of electronic defects in crystals, and how the mass action laws are thereby affected, let us take the compound silver sulfide Aga+ S as an example [15]. 

Therefore, in order to satisfy the Pauli principle of the orbital occupancies, for the multi-electrons orbitals in the crystal field there appears two effects of electronic repelling ... 

Short-range repulsive forces are a direct result of the Pauli exclusion principle and are thus quantum mechanical in nature. Kitaigorodskii (1961) has emphasized that such short-range repulsive forces play a major role in determining the packing in molecular crystals. The size and shape of molecules is determined by the repulsive forces, and the molecules pack as closely as is permitted by these forces. 

ELECTRON GAS. The term electron gas is used to denote a system of mobile electrons, as. for example, the electrons in a metal that are free to move. In the free electron theory of metals, these electrons move through the metal in the region of nearly uniform positive potential created by the ions of the crystal lattice. This theory when modified by the Pauli exclusion principle, serves to explain many properties of metals, especially the alkali metals. For metals with more complex electronic structure, and semiconductors, the band theory of solids gives a better picture. 

We must add Nq electrons to our crystal, in accordance with the Pauli Exclusion Principle. Each level of Equation (5.16) can hold two electrons of opposite spin. The volume element (27r/a) defines a primitive unit cell in k space, each cell contains one energy level. The ground state will fill all levels from A = 0 to a limiting value, k. The Nq electrons will need Nq/2 unit cells, or the number lying in a sphere of radius k-p. 

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