Energy states of solids

Energy Band Models Used to Depict Energy States of Solids Insulator Semi-Conductor... 

EXPERIMENTAL TECHNIQUES USED TO DETERMINE THE COMPOSITION, STRUCTURE, AND ENERGY STATES OF SOLIDS AND LIQUIDS... 

Many phenomena in solid-state physics can be understood by resort to energy band calculations. Conductivity trends, photoemission spectra, and optical properties can all be understood by examining the quantum states or energy bands of solids. In addition, electronic structure methods can be used to extract a wide variety of properties such as structural energies, mechanical properties and thennodynamic properties. 

Consider Figure la, which shows the electronic energy states of a solid having broadened valence and conduction bands as well as sharp core-level states X, Y, and Z. An incoming electron with energy Eq may excite an electron ftom any occupied state to any unoccupied state, where the Fermi energy Ap separates the two... 

The inherent problems associated with the computation of the properties of solids have been reduced by a computational technique called Density Functional Theory. This approach to the calculation of the properties of solids again stems from solid-state physics. In Hartree-Fock equations the N electrons need to be specified by 3/V variables, indicating the position of each electron in space. The density functional theory replaces these with just the electron density at a point, specified by just three variables. In the commonest formalism of the theory, due to Kohn and Sham, called the local density approximation (LDA), noninteracting electrons move in an effective potential that is described in terms of a uniform electron gas. Density functional theory is now widely used for many chemical calculations, including the stabilities and bulk properties of solids, as well as defect formation energies and configurations in materials such as silicon, GaN, and Agl. At present, the excited states of solids are not well treated in this way. 

Figure 3.6 shows the various relationships between the energy levels of solids and liquids. In electrolytes three energy levels exist, Ep, redox, Eox and Ered- The energy levels of a redox couple in an electrolyte is controlled by the ionization energy of the reduced species Ered, and the electron affinity of the oxidized species Eox in solution in their most probable state of solvation due to varying interaction with the surrounding electrolyte, a considerable... 

The wetting experiments to be described were performed on Pb single-crystals. Interpretation of the results requires a knowledge of the surface energy anisotropy of solid Pb, as well as the atomic scale physical state of Pb surfaces of different orientations. These features of Pb surfaces have been studied by several authors in great detail and are described in the following seetion. 

The energy states of gaseous atoms split because of the overlap between electron clouds. Obviously, therefore, atoms must come much closer before the clouds of the core electrons begin to overlap compared with the distance at which the clouds of outer (or valence) electrons overlap (Fig. 6.119). Hence, at the equilibrium interatomic distances, the energy levels of the core electrons (in contrast to the valence electrons) do not show any band structure and therefore will be neglected in the following discussion. This simplified picture of the band theory of solids will now be used to explain the differences in conductivity of metals, semiconductors, and insulators. 

Compensation behavior found for the decomposition of hydrogen peroxide on preparations of chromium (III) oxide, which had previously been annealed to various temperatures, was attributed to variations in the energy states of the active centers (here e 0.165). Compensation behavior has also been observed (284) in the decomposition of hydrogen peroxide on cobalt-iron spinels the kinetic characteristics of reactions on these catalysts were ascribed to the electronic structures of the solids concerned. 

The electronic structure of a solid metal or semiconductor is described by the band theory that considers the possible energy states of delocalized electrons in the crystal lattice. An apparent difficulty for the application of band theory to solid state catalysis is that the theory describes the situation in an infinitely extended lattice whereas the catalytic process is located on an external crystal surface where the lattice ends. In attempting to develop a correlation between catalytic surface processes and the bulk electronic properties of catalysts as described by the band theory, the approach taken in the following pages will be to assume a correlation between bulk and surface electronic properties. For example, it is assumed that lack of electrons in the bulk results in empty orbitals in the surface conversely, excess electrons in the bulk should result in occupied orbitals in the surface (7). This principle gains strong support from the consistency of the description thus achieved. In the following, the principle will be applied to supported catalysts. 

Mo (g). Jones, Langmuir, and Mackay1 reported values for the vapor pressure of solid molybdenum. The energy states of gaseous monatomic molybdenum are evaluated from the data of Catalan1 and Meggers and Kiess.1 See also Bacher and Goudsmit.1... 

Ca (g). Vapor pressure data on solid and liquid calcium were reported by Pilling,1 Ruff and Hartman,1 and Hartman and Schneider.1 See also Randall and Tamale.1 The values for the energy states of gaseous monatomic calcium are from Fowler1 and Bowen.5... 

The absorbed photons transmit their energy hv to electrons in the lattice of the solid near the surface. If the photon energy is sufficiently high, photoelectric emission is observed. This method of direct observation of the excited electrons enables information to be gained about the nature of the excited electrons and the initial states of the electron transitions. Thus, a complete analysis of the energy states of a semiconductor surface becomes possible. 

A most important aspect of the tunneling mechanism as applied to low-temperature irradiation-induced chain conversions in a solid is the assumption of energy equilibrium in a reacting solid-state system. However, the universality of kinetic models of the above processes based on the Arrhenius equilibrium law has neven been considered an axiom—especially in studies of chemical reactions stimulated externally by ionizing radiation, that is, under conditions where temperature was no longer the only parameter characterizing the energy state of the system. 

As mentioned previously, the probability of emission (escape) of an electron from a solid surface is inversely proportional to the strength of interaction between the electron and the solid. In other terms, the higher the energy state of an electron in a solid, the lower the energy necessary for it to be removed from the surface. The population of these electronic energy states is proportional to temperature, and can be approximated by a Boltzmann distribution. 

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