Free 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. 

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-... 

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... 

The common feature of most of the theoretical approaches presented here is the use of the free electron gas model. In this model the only parameter used is the mean electronic density of the system hq. In this respect, it is customary to use the one-electron radius defined as l/ o = 4/3TTrs-Finally, in different parts of this work atomic units (a.u.) are used, in which h = = I where nig and e are the electron mass and charge,... 

Nevertheless, except in these extreme cases (very slow ions and grazing incidence geometry) in which the ion does not get close enough to the surface, it can be considered a weak perturbation, no effect related to the insulator character of the target could be observed. In other words, the same free electron gas model that is used to obtain the stopping power of slow ions in metals showed to be applicable in the case of insulators [65,66]. 

In Figs 8 and 9 we compare experimental results obtained in LiF to the predictions of the free electron gas model. Figure 8 shows the measured energy loss as a function of atomic number of the projectile, for specularly reflected ions at the same velocity v = 0.5 a.u. under = 1° angle of incidence. oscillations similar to those in metals are observed, reflecting the shell structure of the projectile levels a minimum appears at Z = 12... 

Local-scaling transformations made their appearance in density functional theory (although in a disguised manner) in the works of Macke [58, 59]. Because the Thomas-Fermi theory corresponded to a free-electron gas model, and as such it was cast in terms of plane waves, any improvement on this theory required the introduction of deformed plane waves. Thus, initially local-scaling transformations were implicitly used when plane waves (defined in the volume V in ft3 and having uniform density p0 = N/V) ... 

Although there has been much theoretical study of the spectra of porphin and its metal derivatives (137, 138, 230, 283), comparatively little has been said about the metal phthalocyanines. It has been generally assumed that the theory developed for porphin will apply equally to the phthalocyanines. Two general approaches have been used to interpret the spectra of the phthalocyanines. In the Free Electron Gas model, developed by Kuhn (201-206, 309) in a manner similar to that of Simpson (327), the phthalocyanine unit is regarded as a polyene in which the ir electrons are constrained to move in a closed ring-shaped path, in a field of constant... 

According to the elementary free-electron gas model of a metal, the metal electrons move in a flat, position-independent potenti2d. The average charge density due to the positive nuclear charges is equal to that of the negative charge of the electrons. At the surface the attractive potential is zero. The potential model is illustrated in Fig.(2.14). 

In order to derive the results in this section we have often used arguments that apply to the free-electron gas model, because electrostatic expressions then can readily be derived. In the final section we will return to more rigorous theoretical studies based on free-electron theory. 

This can be seen schematically in Figure 2.3. With Eq. (2.5), this simple description of a bulk material can be concluded. The possible states in which an electron can be found are quasi-continuous, with the density of states scaling with the square-root of the energy. Further details regarding the free electron gas model, and more refined descriptions of electrons in solids may be found in any solid-state physics textbook [15]. 

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. 

In addition for the noble metals further extension of the free electron gas model is necessary for higher frequencies co > cop. Though the response is essentially determined by free s-electrons, the filled d-band close to the Fermi surface represent a highly polarizable background which can be described by a dielectric constant s c (typically 1 < oo < 10), and we can write ... 

In addition to the failure of the free electron gas model to properly account for the observed temperature behavior of resistivity of metals, there are serious problems when other properties are considered. First, let us make a few simple calculations to illustrate one of its major difficulties. Multiplying the collision time of 2.5 x 10 s by the thermal velocity 1 x 10 m/s, one gets a mean free path of 2.5 nm, which is on the order of a few lattice spacing. This seems incredibly small, especially considering that Cu is one of the... 

The simplest treatment of a many-electron system is obtained using the free-electron gas model. For a free-electron gas we calculate the electron density from the number of states in a system with potential equal to zero, i.e the electron has only kinetic energy. Thus the electron can be described by the simple-particle-in-a-box model. 

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