Solids metallic electrons

In this chapter, the foundations of equilibrium statistical mechanics are introduced and applied to ideal and weakly interacting systems. The coimection between statistical mechanics and thennodynamics is made by introducing ensemble methods. The role of mechanics, both quantum and classical, is described. In particular, the concept and use of the density of states is utilized. Applications are made to ideal quantum and classical gases, ideal gas of diatomic molecules, photons and the black body radiation, phonons in a hannonic solid, conduction electrons in metals and the Bose—Einstein condensation. Introductory aspects of the density... 

Slater s Xa method is now regarded as so much history, but it gave an important stepping stone towards modem density functional theory. In Chapter 12, I discussed the free-electron model of the conduction electrons in a solid. The electrons were assumed to occupy a volume of space that we identified with the dimensions of the metal under smdy, and the electrons were taken to be non-interacting. 

A value close to 4.8 V has been obtained in four different laboratories using quite different approaches (solid metal/solution Ay, 44 emersed electrodes,40,47 work function changes48), and is apparently supported by indirect estimates of electronic energy levels. The consistency of results around 4.8 V suggests that the value of 4.44 V is probably due to the value of 0 not reflecting the actual state of an Hg jet or pool. According to some authors,44 the actual value of 0 for Hg in the stream should be 4.8 V in that the metal surface would be oxidized. 

As described in Section 10-, the bonding in solid metals comes from electrons in highly delocalized valence orbitals. There are so many such orbitals that they form energy bands, giving the valence electrons high mobility. Consequently, each metal atom can be viewed as a cation embedded in a sea of mobile valence electrons. The properties of metals can be explained on the basis of this picture. Section 10- describes the most obvious of these properties, electrical conductivity. 

As we can see from the last entry in this table, we have deduced only a rule. In InBi there are Bi-Bi contacts and it has metallic properties. Further examples that do not fulfill the rule are LiPb (Pb atoms surrounded only by Li) and K8Ge46. In the latter, all Ge atoms have four covalent bonds they form a wide-meshed framework that encloses the K+ ions (Fig. 16.26, p. 188) the electrons donated by the potassium atoms are not taken over by the germanium, and instead they form a band. In a way, this is a kind of a solid solution, with germanium as solvent for K+ and solvated electrons. K8Ge46 has metallic properties. In the sense of the 8-A rule the metallic electrons can be captured in K8Ga8Ge38, which has the same structure, all the electrons of the potassium are required for the framework, and it is a semiconductor. In spite of the exceptions, the concept has turned out to be very fruitful, especially in the context of understanding the Zintl phases. 

We begin with a presentation of the ideas of the electronic structure of metals. A liquid or solid metal of course consists of positively charged nuclei and electrons. However, since most of the electrons are tightly bound to individual nuclei, one can treat a system of positive ions or ion cores (nuclei plus core electrons) and free electrons, bound to the metal as a whole. In a simple metal, the electrons of the latter type, which are treated explicitly, are the conduction electrons, whose parentage is the valence electrons of the metal atoms all others are considered as part of the cores. In some metals, such as the transition elements, the distinction between core and conduction electrons is not as sharp. 

In addition to the effect of the nonideality of the metal on the electrolyte phase, one must consider the influence of the electrolyte phase on the metal. This requires a model for the interaction between conduction electrons and electrolyte species. Indeed, this interaction is what determines the position of electrolyte species relative to the metal in the interface. Some of the work described below is concerned with investigating models for the electrolyte-electron interaction. Although we shall not discuss it, the penetration of water molecules between the atoms of the metal surface may be related3 to the different values of the free-charge or ionic contribution to the inner-layer capacitance found for different crystal faces of solid metals. Rough calculations have been done to... 

Modern theories of electronic structure at a metal surface, which have proved their accuracy for bare metal surfaces, have now been applied to the calculation of electron density profiles in the presence of adsorbed species or other external sources of potential. The spillover of the negative (electronic) charge density from the positive (ionic) background and the overlap of the former with the electrolyte are the crucial effects. Self-consistent calculations, in which the electronic kinetic energy is correctly taken into account, may have to replace the simpler density-functional treatments which have been used most often. The situation for liquid metals, for which the density profile for the positive (ionic) charge density is required, is not as satisfactory as for solid metals, for which the crystal structure is known. 

With the surface ionization source it is generally assumed that the reactant ion internal state distribution is characterized by the source temperature and that the majority of the reactant ions are in their ground electronic state. This contrasts with the uncertainty in reactant state distributions when transition metal ions are generated by electron impact fragmentation of volatile organometallic precursors (10) or by laser evaporation and ionization of solid metal targets (11). Many examples... 

The apparatus and techniques of ion cyclotron resonance spectroscopy have been described in detail elsewhere. Ions are formed, either by electron impact from a volatile precursor, or by laser evaporation and ionization of a solid metal target (14), and allowed to interact with neutral reactants. Freiser and co-workers have refined this experimental methodology with the use of elegant collision induced dissociation experiments for reactant preparation and the selective introduction of neutral reactants using pulsed gas valves (15). Irradiation of the ions with either lasers or conventional light sources during selected portions of the trapped ion cycle makes it possible to study ion photochemical processes... 

The electronic states in the cluster (or in the solid metal) are expressed as a linear combination of atomic states (LCAO)... 

A platinum electrode is needed in some half-cells to provide a surface at which electrons can be exchanged. Such a surface is lacking where there is no solid metal as part of the half-reaction, such as with the ferrous-ferric half-reaction. 

In metallic solids, metal atoms occupying the crystal lattice are held together by metallic bonding. In metallic bonding, the electrons of the atoms are delocalized and are free to move throughout the entire solid. This explains electrical and thermal conductivity, as well as many other properties of metals. 

For high density electron ensembles such as free valence electrons in solid metals where electrons are in the state of degeneracy, the distribution of electron energy follows the Fermi function of Eqn. 1-1. According to quantum statistical dynamics [Davidson, 1962], the electrochemical potential, P., of electrons is represented by the Fermi level, ep, as shown in Eqn. 1-10 ... 

The metal ion sublimation of Eqn. S—4 is also equivalent to the process that consists of the sublimation of a siuface metal atom M. followed by both the ionization of a gaseous metal atom Motd) and the injection of a gaseous electron e(STD) into the metal phase to produce a standard gaseous metal ion leaving an electron eu in the solid metal as shown in Eqn. 3-5 and in Fig. 3-3 (h) ... 

Next, we consider the interface M/S of a nonpolarizable electrode where electron or ion transfer is in equilibrium between a solid metal M and an aqueous solution S. Here, the interfadal potential is determined by the charge transfer equilibrium. As shown in Fig. 4-9, the electron transfer equilibrium equates the Fermi level, Enn) (= P (M)), of electrons in the metal with the Fermi level, erredox) (= P s)), of redox electrons in hydrated redox particles in the solution this gives rise to the inner and the outer potential differences, and respectively, as shown in Eqn. 4-10 ... 

In physics the electron level in an isolated solid metal is conventionally represented by the negative work function, - 4, which corresponds to the real potential, o-,auv), of electrons in the isolated metal. Similarly, in electrochemistry. 

Fig. 4-10. Electron energy levels in (a) an isolated solid metal and in (b) a metal electrode immersed in an electrolyte solution M = metal S = electrolyte solution e(STD) = gaseous electrons in the standard state e

Fig. 4-16. Energy levels of metal ion and electron in an ionic electrode of metal ion transfer 4Cjn i = sublimation energy of solid metal /m" = ionization energy of gaseous metal atoms > >s = outer potential of electrolyte solution E s electrode potential (absolute electrode potential).

It, thus, follows that the electrode potential in equilibrium of metal ion transfer is given by the free enthalpy for the formation of a solid metal from both hydrated metal ions and standard gaseous electrons as shown in Eqn. 4—25 ... 

Fig. 5-3. Electron energy levels in an isolated adsorbate particle and an adsorbent solid metal M = metal R = isolated particle LUMO = lowest unoccupied molecular orbital (lowest vacant electron level) HOMO = highest occupied molecular orbital (highest occupied electron level).

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