Electron gas model

Simple metals like alkalis, or ones with only s and p valence electrons, can often be described by a free electron gas model, whereas transition metals and rare earth metals which have d and f valence electrons camiot. Transition metal and rare earth metals do not have energy band structures which resemble free electron models. The fonned bonds from d and f states often have some strong covalent character. This character strongly modulates the free-electron-like bands. 

K). T is the measurement temperature and Tq is the "degeneracy temperature," equal to kEo, where k is the Boltzmann constant. According to a two-dimensional electron gas model for graphitic carbons (see ref. 2a), is the energy "shift" from the Fermi level (Ep), to the top of the valence band. Small values of To ( <344 K) and consequently of Eq signify a more perfect graphite... 

Figure 7. The occupation number densities as functions of wave vector for Na. The thick curves labeled (100), (110) and (111) represent the three principal directions within the first Brillouin zone, obtained by the FLAPW-GWA. The thin solid curve is obtained from an interacting electron-gas model [27]. The dash-dotted line represents the Fermi momentum. 

A theoretically well-founded theory of two-photon absorption using the free electron-gas model of dye molecules 50> is to be found in 49>. 

The sp-valent metals such as sodium, magnesium and aluminium constitute the simplest form of condensed matter. They are archetypal of the textbook metallic bond in which the outer shell of electrons form a gas of free particles that are only very weakly perturbed by the underlying ionic lattice. The classical free-electron gas model of Drude accounted very well for the electrical and thermal conductivities of metals, linking their ratio in the very simple form of the Wiedemann-Franz law. However, we shall now see that a proper quantum mechanical treatment is required in order to explain not only the binding properties of a free-electron gas at zero temperature but also the observed linear temperature dependence of its heat capacity. According to classical mechanics the heat capacity should be temperature-independent, taking the constant value of kB per free particle. 

The free-electron gas model is a good starting point for the sp-valent metals where the loosely bound valence electrons are stripped off from their ion cores as the atoms are brought together to form the solid. However, bonding in the majority of elements and compounds takes place through saturated... 

Here J, JQ and Ja are the statistical sums of activated complex and gas-phase molecules and of adsorbed atom (adatom), respectively, sA and eD the adsorption and desorption activation energies, a the area of adatom localization, h Planck s constant, and f. the parameters of the activated complex-adatom and adatom-adatom interactions (e < 0 for repulsion and e > 0 for attraction), A the contribution to the complete drop of adsorption heat AQ from the electron subsystem (for a two-dimensional free-electron gas model), x = exp (ej — e) — 1, jc, = = 0), / = 1/kT (k is the Boltzmann con-... 

The electron gas model adequately describes the conduction of electrons in metals however, it has a problem, that is, the electrons with energy near the Fermi level have wavelength values comparable to the lattice parameters of the crystal. Consequently, strong diffraction effects must be present (see below the diffraction condition (Equation 1.47). A more realistic description of the state of the electrons inside solids is necessary. This more accurate description is carried out with the help of the Bloch and Wilson band model [18],... 

An alternative to the perturbation theory approach is the approximate method of Gordon and Kim.60 In this method the electron density is first calculated as the sum of the densities of the separate atoms and the energy is then obtained as the sum of a Coulomb term calculated exactly, and kinetic energy, exchange, and correlation terms calculated from the free electron gas model. Though it worked well for larger... 

In the free-electron gas model of metals, it can be shown that the contribution to the bulk modulus owing to the kinetic energy of the free electrons can be approximated by... 

It is possible to exactly identify and characterize the radical species and chain structures of the reaction intermediates, which are determined by their different reactive or unreactive chain ends. The reactive intermediates are best described by diradical (DR), asymmetric carbene (AC) and dicarbene (DC) oligomer molecules of different lengths. The respective singlet (S = 0), triplet (S = I) or quintet (S = 1) states and their roles in the polymerization process are investigated in detail by solid state spectroscopy. A one-dimensional electron gas model is successfully applied to the optical absorption series of the DR and AC intermediates as well as on the different stable oligomer SO molecules obtained after final chain termination reactions. 

Table 3. Data used in the one-dimensional electron gas model calculation following Eq. (2)...

Following the example of a Fermi liquid, however, a selected emphasis is put on the electronic states close to the Fermi level where the argument E JT of the logarithm goes to infinity at T— 0. In the framework of the electron gas model, this amounts to write the one-particle kinetic term of the Hamiltonian in the form... 

Figure 21 g-ology decomposition of the electron-electron interaction close to the Fermi points in the electron gas model. 

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