Completely free electron model

Figure 2.14 Free electron in a one-dimensional box of length / as function of wave number k (a) Completely free electron model and (b) nearly free electron model, reflection of the electrons on the ions.

Once more, free-electron models correctly predict many qualitative trends and demonstrate the appropriateness of the general concept of electron delocalization in molecules. Free electron models are strictly one-electron simulations. The energy levels that are used to predict the distribution of several delocalized electrons are likewise one-electron levels. Interelectronic effects are therefore completely ignored and modelling the behaviour of many-electron systems in the same crude potential field is ndt feasible. Whatever level of sophistication may be aimed for when performing more realistic calculations, the basic fact of delocalized electronic waves in molecular systems remains of central importance... 

The electronic heat capacity for the free electron model is a linear function of temperature only for T Tp = p / kp. Nevertheless, the Fermi temperature Tp is of the order of 105 K and eq. (8.46) holds for most practical purposes. The population of the electronic states at different temperatures as well as the variation of the electronic heat capacity with temperature for a free electron gas is shown in Figure 8.20. Complete excitation is only expected at very high temperatures, T>Tp. Here the limiting value for a gas of structureless mass points 3/2/ is approached. 

Band theory started with a gross approximation the idea that a metal contained a gas of completely free electrons, uninfluenced by the other electrons or anything else. The model, an adaptation of the kinetic theory of gases, was very successful. The electrons were supposed to be flying about in the... 

We have seen in previous chapters that the good examples of metallic crystals are the alkali metals, which can be correctly described by the near-free electron model. The valence electrons in these metals are completely separated from their ion cores and form a nearly uniform gas. 

In the previous chapters, we discussed various models of bonding for covalent and polar covalent molecules (the VSEPR and LCP models, valence bond theory, and molecular orbital theory). We shall now turn our focus to a discussion of models describing metallic bonding. We begin with the free electron model, which assumes that the ionized electrons in a metallic solid have been completely removed from the influence of the atoms in the crystal and exist essentially as an electron gas. Freshman chemistry books typically describe this simplified version of metallic bonding as a sea of electrons that is delocalized over all the metal atoms in the crystalline solid. We shall then progress to the band theory of solids, which results from introducing the periodic potential of the crystalline lattice. 

Chapter 2 introduces the band theory of solids. The main approach is via the tight binding model, seen as an extension of the molecular orbital theory familiar to chemists. Physicists more often develop the band model via the free electron theory, which is included here for completeness. This chapter also discusses electronic condnctivity in solids and in particular properties and applications of semiconductors. 

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-... 

In this model molecules are represented by BE-matrices (see Section 2.1.), reactions are taken as transformations of BE-matrices of isomeric ensembles of molecules by R-matrices (see Section 2.2.). These A-matrices indicate the changes in the distribution of bonds and free electrons occurring in a reaction. The / -matrices can be generated quite formally thereby obtaining all conceivable reactions irrespective of whether a reaction is already known or is completely novel, i.e., without any precedence. 

However, the model in which the (valence) electrons are completely free and are neither feeling the attraction nor the repulsion is certain not properly describing the nature of the chemical bond. In fact, this limitation was also the main objection brought to Thomas-Fermi model and to the atomic or molecular approximation of the homogeneous electronic gas or helium model in solids. Nevertheless, the lesson is well served because Thomas-Fermi description may be regarded as the inferior extreme in quantum known structures while further exchange-correlation effects may be added in a perturbation manner. 

In this chapter we consider a gas that consists of free electrons. The homogeneous gas of interacting electrons, in which the positive charges are assumed to be uniformly distributed in space, is the simplest model representing condensed matter. This model is completely specified by the density n of the electron gas or by the average distance Vs between electrons. 

In 1933, Arnold Sommerfeld and Hans Bethe revised the Drude model. Their more complete, quantum mechanical theory goes under the name of the free electron... 

Alternatively, one could assume that the atoms are completely dissociated at all concentrations and the electrons behave like a free-electron gas. This would seem attractive from a consideration of Muster s data on saturated sodium-ammonia solutions. Mow-ever the observation by Muster in dilute sodium-ammonia solutions and by Freed and Sugarman in dilute potassium-ammonia solutions that the atomic susceptibility tends to (ji jkT (and not 2/3 (i jkT as expected from free-electron gas model) is evidence against the free electron gas model. The strongest point against the free-electron gas model is the finite photoelectric threshold observed for these solutions by Masing and Teal, which indicates that the electron is not free but is bound to some center. The next step is the cavity model to be discussed in the next section. 

The partial equilibrium assumptions by themselves in conjunction with Eqs. (9.1), (9.2), and (9.3), and a reaction mechanism as outlined above, do not permit construction of a complete model from which an eigenvalue burning velocity and full profiles may be computed ab initio. On the other hand the assumptions are extremely useful when dealing with H-N-C-O ffame systems, since their application to reactions (i), (ii), (iii), and (xviii) above allows us to calculate many of the species profiles, and particularly the free radical profiles, on close approach to full equilibrium. The time-dependent computation does not economically do this directly. The computations require an input mass flux or burning velocity which must be either a measured or a separately calculated value. For composite flux calculation purposes the overall radical pool is chosen so as to represent a total flux of free electron spins, that is, spins belonging to H, O, OH, and O2 (Dixon-Lewis et al, 1975). 

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