Nearly free electron

An ideal metal crystal consists of a regular array of ion cores with the valence electrons nearly free to move throughout the whole mass, as the conduction electrons. In addition to the interaction of conduction electrons, there is mutual interaction of ion cores for each other Repulsive, ion cores have a net positive charge attractive, dispersion force of electrons in the ion cores. 

Wang Y A, Govind N and Carter E A 1998 Orbital-free kinetic energy functionals for the nearly-free electron gas Phys. Rev. B 58 13 465... 

Conversely, there are compounds in which nearly free rotation is possible around what are formally C=C double bonds. These compounds, called push-pull or captodative ethylenes, have two electron-withdrawing groups on one carbon and two electron-donating groups on the other (66). The contribution of di-ionic... 

The valence band structure of very small metal crystallites is expected to differ from that of an infinite crystal for a number of reasons (a) with a ratio of surface to bulk atoms approaching unity (ca. 2 nm diameter), the potential seen by the nearly free valence electrons will be very different from the periodic potential of an infinite crystal (b) surface states, if they exist, would be expected to dominate the electronic density of states (DOS) (c) the electronic DOS of very small metal crystallites on a support surface will be affected by the metal-support interactions. It is essential to determine at what crystallite size (or number of atoms per crystallite) the electronic density of sates begins to depart from that of the infinite crystal, as the material state of the catalyst particle can affect changes in the surface thermodynamics which may control the catalysis and electro-catalysis of heterogeneous reactions as well as the physical properties of the catalyst particle [26]. 

Only the most simple form of metallic bonding will be considered here. In its simple form a metal is a dense plasma of nearly free electrons and positive ions. The ions are condensed into close-packed 3-D face-centered arrays. Metallic bonding results from a balance between attractive potential energy and repulsive kinetic energy. 

Quantitative data on local structure can be obtained via an analysis of the decaying slope next to the absorption edge. The absorption of an X-ray photon boosts a core electron up into an unoccupied band of the material which, in a metal, is the conduction band above the Fermi level. Electrons in such a band behave as if nearly free and no fine structure would be expected on the absorption tail . However, fine structure is observed up to 500 to 1000eV above the edge (see Figure 2.73(b)). The ripples are known as the Kronig fine structure or extended X-ray absorption fine structure (EX AFS). 

The situation described here is based on a simple one-electron model which can hardly be expected to predict the behaviour of complex many-electron systems in quantitative detail. There can be no doubt however, that the qualitative picture is convincing and probably that the broad principles of electronic behaviour in solids have been identified. The most significant feature of the model is the band structure that makes no sense except in terms of the electron as a wave. Important, but largely unexplored aspects of solid-state reactions and heterogeneous catalysis must also relate to the nearly-free models of electrons in solids. 

This notion of occasional ion hops, apparently at random, forms the basis of random walk theory which is widely used to provide a semi-quantitative analysis or description of ionic conductivity (Goodenough, 1983 see Chapter 3 for a more detailed treatment of conduction). There is very little evidence in most solid electrolytes that the ions are instead able to move around without thermal activation in a true liquid-like motion. Nor is there much evidence of a free-ion state in which a particular ion can be activated to a state in which it is completely free to move, i.e. there appears to be no ionic equivalent of free or nearly free electron motion. 

Various astoichiometric components (hydrogen, carbon, and others, for example, silicium and aluminum) present may interact with localized and nearly free electrons to differing extents. According to the localized free electron interplay model of metal catalysts developed by Knor 163, 164) the ratio of the two types of electrons may influence the catalytic properties considerably. For example, a subsurface proton attracts nearly free electrons and thus uncovers some localized orbitals. Carbon may interact first with localized electrons 164). This may be one of the reasons why their effects are of opposite character. The collective efforts of catalytic and surface chemists are necessary to bring some clarity to the multitude of problems arising here. 

In the simplest form, the Thomas-Fermi-Dirac model, the functionals are those which are valid for an electronic gas with slow spatial variations (the nearly free electron gas ). In this approximation, the kinetic energy T is given by... 

The outer electrons in metals such as Li and Na have a very low ionization energy, and are largely delocalized. Such electrons are described as constituting a nearly free electron gas. It may be noted, though, that this description is somewhat misleading as the behavior of the electrons is dominated by the exclusion principle, while the molecules in normal gases can be described by classical statistical mechanics. 

The first successful first-principle theoretical studies of the electronic structure of solid surfaces were conducted by Appelbaum and Hamann on Na (1972) and A1 (1973). Within a few years, first-principles calculations for a number of important materials, from nearly free-electron metals to f-band metals and semiconductors, were published, as summarized in the first review article by Appelbaum and Hamann (1976). Extensive reviews of the first-principles calculations for metal surfaces (Inglesfeld, 1982) and semiconductors (Lieske, 1984) are published. A current interest is the reconstruction of surfaces. Because of the refinement of the calculation of total energy of surfaces, tiny differences of the energies of different reconstructions can be assessed accurately. As examples, there are the study of bonding and reconstruction of the W(OOl) surface by Singh and Krakauer (1988), and the study of the surface reconstruction of Ag(llO) by Fu and Ho (1989). 

The modification of an x-wave sample state due to the existence of the tip is similar to the case of the hydrogen molecule ion. For nearly free-electron metals, the surface electron density can be considered as the superposition of the x-wave electron densities of individual atoms. In the presence of an exotic atom, the tip, the electron density of each atom is multiplied by a numerical constant, 4/e 1.472. Therefore, the total density of the valence electron of the metal surface in the gap is multiplied by the same constant, 1.472. Consequently, the corrugation amplitude remains unchanged. 

Under electron impact ionization, when the energy transfer greatly exceeds the orbital ionization energy, the process resembles a Rutherford-type collision between two nearly free electrons. On this Mott [17] imposed the condition of indistinguishability of the outgoing electrons and obtained in the nonrelativistic limit... 

In the context of the spherical shell model for nearly-free electrons, assuming that Au, Sc, and Ti contribute with 1,3,5 delocalized valence electrons, respectively, it appears that AueSc+ and Au5Ti+ are magic clusters with 8 valence electrons. However, for the other TM impurities, delocalization of valence charge is restricted to the 4s electrons if this model is forced to explain the observed drops of intensity, at n=5 and 7. We obtain the experimental magic numbers ° of Au TM+ clusters without resorting to the empirical shell-model of delocalized electrons. 

In addition to these examples, an electron-transfer free radical photoinitiator H-Nu 470 (5,7-diiodo-3-butoxy-6-fluorone) has been also successfully used for 3D microfabrication by near-IR two-photon induced polymerization... 

The curves of Fig. 12.17 nicely illustrate the varied optical effects exhibited by small metallic particles in the surface mode region, both those explained by Mie theory with bulk optical constants and those requiring modification of the electron mean free path (see Section 12.1). Absorption by particles with radii between about 26 and 100 A peaks near the Frohlich frequency (XF — 5200 A), which is independent of size. Absorption decreases markedly at longer... 

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