Electrons in a metal

Each level will be occupied by two electrons with opposite spins. If there are n atoms in the metal strip with n conduction electrons altogether (as in the alkali metals, copper group), 

When we remove the separating wall and place the four electrons in the twice enlarged box, a new level has become available below the lowest level k = 1 in the original boxes, which now has become the level k = 2 in the large box with V = 2/. The total energy of the four electrons, previously 

In the example of the box the lowering of the energy is exclusively due to a decrease of kinetic energy. In the formation of molecules like H2+ and H2, however, it is the decrease of the potential energy that is the important factor the kinetic energy increases. 

The same model of a metal has been applied by O. Schmidt, H. Kuhn and others for the calculation of the light absorption of coloured substances (p. 253). 

In place of the above intuitive solution the result can also be deduced from Schrodinger s wave equation. 

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Fig. VIII-5. Schematic potential energy diagram for electrons in a metal with and without an applied field , work function Ep, depth of the Fermi level. (From Ref. 62.)...

This type of argument leads us to picture a metal as an array of positive ions located at the crystal lattice sites, immersed in a sea of mobile electrons. The idea of a more or less uniform electron sea emphasizes an important difference between metallic bonding and ordinary covalent bonding. In molecular covalent bonds the electrons are localized in a way that fixes the positions of the atoms quite rigidly. We say that the bonds have directional character— the electrons tend to remain concentrated in certain regions of space. In contrast, the valence electrons in a metal are spread almost uniformly throughout the crystal, so the metallic bond does not exert the directional influence of the ordinary covalent bond. 

An entirely different approach to the correlation problem is taken in the plasma model (Bohm and Pines 1953, Pines 1954, 1955), in which the electrons in a metal are approximated by a free-electron gas moving in a uniform positive background. According to classical discharge theory, such a plasma is characterized by an oscillatory behavior having a frequency... 

As already noted the electrochemical potential of electrons in a metal, jl, is related to the Galvani potential q> via ... 

The mobility of the valence electrons in a metal accounts for its electrical... 

Some years later a more thorough discussion of the motion of pairs of electrons in a metal was given by Cooper,7 as well as by Abrikosov8 and Gor kov,9 who emphasized that the effective charge in superconductivity is 2e, rather than e. The quantization of flux in units hc/2e in superconducting metals has been verified by direct experimental measurement of the magnetic moments induced in thin films.10 Cooper s discussion of the motion of electron pairs in interaction with phonons led to the development of the Bardeen-Cooper-Schrieffer (BCS) theory, which has introduced great clarification in the field of superconductivity.2... 

Iron and other metals have tremendous mechanical strength, which suggests that the bonds between their atoms must be strong. At the same time, most metals are malleable, which means they can be shaped into thin sheets to make objects such as aluminum cans. Metals are also ductile, which means they can be drawn into wires. The properties of malleability and ductility suggest that atoms in metals can be moved about without weakening the bonding. Finally, metals conduct electricity, which shows that some of the electrons in a metal are free to move throughout the solid. 

The concept of electrons not belonging to any particular atom in a molecule brings us back to resonance structures. The electrons in a metal are also delocalized. An electron in a bar of sodium is not associated with any particular atom, just as the electrons in the double bonds of benzene are not associated with any particular atom. 

For electrons in a metal, these velocities are on the order of 10 cm s . The Fermi temperature Tp is defined by the relation... 

One can expect that the electron density corresponding to the electronic state of lowest energy is roughly constant in the interior of the metal and decreases to zero outside the metal. This means that the potential seen by an electron, due to the ion cores and the other electrons, is roughly constant inside the metal, with a value significantly lower than the potential outside. The simplest model for electrons in a metal, the Sommerfeld38 model, takes this potential as -V0 inside and 0 outside. One is then led to consider the one-dimensional Schrodinger equation... 

Treating the free electrons in a metal as a collection of zero-frequency oscillators gives rise51 to a complex frequency-dependent dielectric constant of 1 - a>2/(co2 - ia>/r), with (op = (47me2/m)l/2 the plasma frequency and r a collision time. For metals like Ag and Au, and with frequencies (o corresponding to visible or ultraviolet light, this simplifies to give a real part... 

Quantum mechanical calculations are appropriate for the electrons in a metal, and, for the electrolyte, modern statistical mechanical theories may be used instead of the traditional Gouy-Chapman plus orienting dipoles description. The potential and electric field at any point in the interface can then be calculated, and all measurable electrical properties can be evaluated for comparison with experiment. 

One source of EM enhancement may be attributed to the excitation of surface plasmons (SP) in the metal. A plasmon is a collective excitation in which all of the conduction electrons in a metal oscillate in phase. In the bulk, there is essentially only one allowed fundamental plasmon frequency. 

Sommerfeld suggested that the potential in a metal crystal could be assumed constant. This assumption implies that the forces acting on an electron cancel to zero and that the electrons in a metal can be described like a non-interacting gas of electrons, confined to a box that represents the metal. The only restriction on electronic motion would be the Pauli principle. The electronic energy in a three-dimensional rectangular box is known as... 

According to the Sommerfeld model electrons in a metal electrode are free to move through the bulk of the metal at a constant potential, but not to escape at the edge. Within the metal electrons have to penetrate the potential barriers that exist between atoms, as shown schematically below. 

For electrons in a metal the work function is defined as the minimum work required to take an electron from inside the metal to a place just outside (c.f. the preceding definition of the outer potential). In taking the electron across the metal surface, work is done against the surface dipole potential x So the work function contains a surface term, and it may hence be different for different surfaces of a single crystal. The work function is the negative of the Fermi level, provided the reference point for the latter is chosen just outside the metal surface. If the reference point for the Fermi level is taken to be the vacuum level instead, then Ep = —, since an extra work —eoV> is required to take the electron from the vacuum level to the surface of the metal. The relations of the electrochemical potential to the work function and the Fermi level are important because one may want to relate electrochemical and solid-state properties. 

However, few or none of those 1012 electrons per second were colliding with the nuclei of the molecule, therefore all the heat was dissipated in the contact. Note that the mean tree path of an electron in a metal is hundreds of angstroms. Hence, it is not surprising that collisions, within a small molecule, did not take place Most importantly, since most computing instruments operate on microamps of current, the prospects for molecular scale electronics are quite intriguing. 

In the case of nonpolaiizable interfaces, the inner and the outer potential differences, 4>a/b and v a/b, are determined by the equilibrium of chai transfer that occurs across the interface. Figure 4—8 shows the electron energy levels in two sohd metals A and B before and after they are brought into contact with each other. As a result of contact, electrons in a metal B of the hi er electron level (the lower work function ) move into a metal A of the lower electron level (the higher work fiuiction), and the Fermi levels of the two metals finally become equal to each other in the state of electron transfer equilibrium. The electrochemical... 

Since the electrons in a metallic lattice are in a gas, we must use the core electrons and nuclei to determine the structure in metals. This will be true of most solids we will describe, regardless of the type of bonding, since the electrons occupy such a small volume compared to the nucleus. For ease of visualization, we consider the atomic cores to be hard spheres. Because the electrons are delocalized, there is little in the way of electronic hindrance to restrict the number of neighbors a metallic atom may have. As a result, the atoms tend to pack in a close-packed arrangement, or one in which the maximum number of nearest neighbors (atoms directly in contact) is satisfied. 

Figure 6.3 The Fermi distribution function (a) at absolute zero and (b) at a finite temperature, (c) The population density of electrons in a metal as a function of energy. From Z. Jastrzebski, The Nature and Properties of Engineering Materials, 2nd ed. Copyright 1976 by John Wiley Sons, Inc. This material is used by permission of John Wiley Sons, Inc.

FIGURE 4.4 Electrons in a metal in the absence of an electric field. They move in all directions, but, overall, no net motion occurs in any direction. 

Since electronics is all about moving electrical currents around, it may seem strange that a semiconductor rather than a metal is used to make the electrical components on silicon chips. But silicon s paucity of conduction electrons is the whole point here. It means that the conductivity can be delicately fine-tuned by sprinkling the crystal lattice with atoms of other elements, which increase or decrease the number of mobile electrons. In a metal, awash with... 

The energy of a free electron in a metal is essentially the kinetic energy due to its motion, given by... 

Fig. la. Schematic potential energy diagrams for electrons in a metal with and without applied field. Clean metal, no image potential assumed, x = work function, fi = depth of Fermi sea. 

Conduction electrons in a metal are nearly free to move within the metal in response to an applied electric field. A surface plasma wave, also called a surface plasmon, is an electromagnetic wave that propagates along the boundary between a metal and a dielectric (an electrical insulator). The electromagnetic field decreases exponentially into both layers but is concentrated in the dielectric layer. 

The mobility of electrons in a metal accounts for the metals significant ability to conduct electricity and heat. Also, metals are opaque and shiny because the free electrons easily vibrate to the oscillations of any light felling on them, reflecting most of it. Furthermore, the metal ions are not rigidly held to fixed positions, as ions are in an ionic crystal. Rather, because the metal ions are held together by a fluid of electrons, these ions can move into various orientations relative to one another, which is what happens when a metal is pounded, pulled, or molded into a different shape. 

This current is conducted by electrons in a metal electrode, electrons and other charge carriers in a semiconductor, and by ions in the electrolyte. The conduction process provides an additional impediment, represented by the ohmic resistance Rn. Its effect is added to the interfacial potential difference, E, so that the total voltage will be... 

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