Metals and the Electron Gas

So far, our quantum mechanics examples have been rather far afield from questions of immediate interest to our study of solids. In particular, we aim to see what light quantum mechanics sheds on the origin of bonding and cohesion in solids. To do so, in this introductory chapter we consider two opposite extremes. Presently, we consider the limit in which each atom is imagined to donate one or more electrons to the solid which are then distributed throughout the solid. The resulting model is the so-called electron gas model and will be returned to repeatedly in coming 

Within the model of a metal proposed above, the Schrodinger equation becomes 

The set of allowed points in -space is depicted schematically in fig. 3.9. The isotropy of the energy spectrum implies a wide range of degeneracies which are of interest as we fill these states. 

For the sake of concreteness, we assume that the confining box has cube edges of dimension L. Further, we assume that N atoms occupy this box and that each has 

Upon explicitly writing the integral given above in spherical coordinates, we find 

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If we can assume that the electrode material is a good metal, and the electronic gas is fully degenerate, the chemical potential of the electrons is given by the Fermi level, EP, which can be written as... 

For the hydrogen electrode, the interfadal potential between the electrode metal and the hydrogen gas film is determined by the electron transfer equilibrium and the interfacial potential between the hydrogen gas film and the aqueous... 

In Chap. XX, Sec. 3, we spoke about the detachment of electrons from atoms, and in Sec. 4 of that chapter we took up the resulting chemical equilibrium, similar to chemical equilibrium in gases. But electrons can be detached not only from atoms but from matter in bulk, and particularly from metals. If the detachment is produced by heat, we have thermionic emission, a process very similar to the vaporization of a solid to form a gas. The equilibrium concerned is very similar to the equilibrium in problems of vapor pressure, and the equilibrium relations can be used, along with a direct calculation of the rate of condensation, to find the rate of thermionic emission. In connection with the equilibrium of a metal and its electron gas, we can find relations between the electrical potentials near two metals in an electron gas and derive information about the so-called Volta difference of potential, or contact potential difference, between the metals. We begin by a kinetic discussion of the collisions of electrons with metallic surfaces. 

One of the earliest treatments of a metal surface was based upon a jellium model (Bardeen, 19.36). If the electron gas terminated abruptly at the surface of the jellium there would be no net potential to contain the electrons in the metal. Therefore the electron gas extends beyond the metal, giving a dipole layer, as illustrated in Fig. 17-5. Bardeen attempted the self-consistent calculation of the resulting potential. It should be mentioned that the Fermi-Thomas approximation is not adequate for this task and was not used by Bardeen it is not difficult to see that it would predict the Fermi energy to be at the vacuum level, corresponding to a vanishing work function. 

The first successful theory of the metallic state may be said to have arisen from the work of Drude and Lorentz in the early years of the present century. On this theory a metal is to be regarded as an assemblage of positive ions immersed in a gas of free electrons. A potential gradient exists at the surface of the metal to imprison the electrons, but within the metal the potential is uniform.. Attraction between the positive ions and the electron ga gives, the structure its coherence, and the free mobility of this electron gas under the influence... 

Metallic bonding is the primary bond in metals and can be thought of as an electrostatic interaction between the delocalized valence electrons and the positively charged ion cores. It is the delocalized electron gas that gives rise to many of the characteristic properties of metals such as high electrical and high thermal conductivities. Metallic bonds do not require a balance of the electric charge between the elements the electrostatic equilibrium is between the metal ions and the electron gas. For this reason different elements can mix in metallic alloys in arbitrary ratios. 

The cohesion above is for ionic solids covalent and metallic bonds are different. In covalent bonds, electrons between atoms are shared, whereas in metallic solids, atoms of the same (or different) elements donate their valence electrons to form an electron gas throughout the space occupied by the atoms. Giving up then-electrons to a common pool, known as an electron cloud or electron gas , these atoms actually become positive (similar to positive ions). They are held together by forces similar to those of ionic bonds, but acting between ions and electrons. The electrostatic interaction between the positive ions and the electron gas holds metals together. Unlike other crystals, metals may be deformed without fracture, because the electron gas permits atoms to slide past one another, acting as a lubricant. In non-ductile materials, such as most ceramics, this is not possible and renders them brittle. 

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... 

This rule conforms with the principle of equipartition of energy, first enunciated by Maxwell, that the heat capacity of an elemental solid, which reflected the vibrational energy of a tliree-dimensional solid, should be equal to 3f JK moH The anomaly that the free electron dreory of metals described a metal as having a tliree-dimensional sUmcture of ion-cores with a three-dimensional gas of free electrons required that the electron gas should add anodier (3/2)7 to the heat capacity if the electrons behaved like a normal gas as described in Maxwell s kinetic theory, whereas die quanmtii theory of free electrons shows that diese quantum particles do not contribute to the heat capacity to the classical extent, and only add a very small component to the heat capacity. 

The logaritlrmic law is also observed when the oxide him is an electrical insulator such as AI2O3. The transport of elecuons tlrrough the oxide is mainly due to a space charge which develops between tire metal-oxide interface and the oxide-gas interface. The incorporation of oxygen in the surface of tire oxide requhes the addition of electrons, and if this occurs by a charging process... 

Metallic materials consist of one or more metallic phases, depending on their composition, and very small amounts of nonmetallic inclusions. In the metallic state, atoms donate some of their outer electrons to the electron gas that permeates the entire volume of the metal and is responsible for good electrical conductivity (10 S cm ). Pure elements do not react electrochemically as a single component. A mesomeric state can be approximately assumed... 

Another progress in our understanding of the ideally polarizable electrode came from theoretical works showing that the metal side of the interface cannot be considered just as an ideal charged plane. A simple quantum-mechanical approach shows that the distribution of the electron gas depends both on the charge of the electrode and on the metal-solution coupling [12,13]. 

The transition metals lie in the d block, at the center of the periodic table, between the s-block metals and the elements in the p block, as Figure 20-1 shows. As we describe in Chapter 8, most transition metal atoms in the gas phase have valence electron configurations of, where x is the group number of the metal. Titanium, for... 

Differences in the parameters of the electron gas between fine crystallites and the compact metal or large crystals. 

Metals are immune to radiation damage by ionization. This is also a consequence of the free electron structure. Fast charged particles and ionizing rays can knock off electrons from the atoms they encounter. In metals, the positive vacancies so formed are immediately filled up by the electron gas, leaving no sign of damage apart from a small amount of heat. 

If V is localized, say, near the origin, then for locations far from the origin, this behaves like j 2kFr)/r2, which means as cos(2kFr)/ r3. These damped oscillations of frequency 2kF are the Friedel oscillations, which always arise when an electron gas is perturbed the frequency of oscillation comes from the kink in the dielectric function at 2kF. We see the Friedel oscillations (in planar rather than in spherical geometry) for the electron gas at a hard wall [Eq. (12) et seq.] and for the electron density at the surface of a metal. 

In the past the theoretical model of the metal was constructed according to the above-mentioned rules, taking into account mainly the experimental results of the study of bulk properties (in the very beginning only electrical and heat conductivity were considered as typical properties of the metallic state). This model (one-, two-, or three-dimensional), represented by the electron gas in a constant or periodic potential, where additionally the influence of exchange and correlation has been taken into account, is still used even in the surface studies. This model was particularly successful in explaining the bulk properties of metals. However, the question still persists whether this model is applicable also for the case where the chemical reactivity of the transition metal surface has to be considered. 

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