Theories of the Metallic State

We now need to know how the probability of finding an electron of specified energy varies across the permitted band. In the first and simplest version of the Electron Band Theory, electrons were assumed to move in a field of uniform positive potential (i.e. ion cores were neglected), and mutual electrostatic repulsion was ignored. Application of the Schrodinger equation and Fermi-Dirac statistics leads to the conclusion that a collection of N electrons at the absolute zero occupies the N/2 lowest levels, those at the maximum being said to be at the Fermi surface Ef. 

Extension of the Band Theory to the metals of the Transition Series required the introduction of ion cores into the argument. While for sodium the ion core is about 10% of the atomic volume, for Transition metals it is a much larger fraction, and cannot be ignored. Although in the case of an alkali metal the nature of the ion core is unambiguously defined, with Transition metals the core will not always have an inert gas configuration, and its structure has to be assumed before the potential field of the crystal can be defined. Moreover the location of the nuclei has to be precisely defined. It is a major weakness of the Band Theory that it does not address the directional nature of bonding between metal atoms 

There have been many attempts to correlate the outstanding chemisorptive and catalytic properties of the Groups 8-10 metals with the presence of an incomplete d-band or unfilled d-orbitals. According to the Band Theory, electrical conduction requires excitation to energy levels above the Fermi surface, so that substances that have only completely filled bands will be insulators. A metal such as magnesium for example is a good conductor because it possesses a partly filled hybrid sp band. By the same token, it is easier to carry a full bottle of mercury than a half-full one, because it doesn t slop about so much. 

It needs to be stressed that models of the metallic state on which the Band Theory is based suppose an infinite three-dimensional array of ion cores, so that the band structure cannot be expected to persist unchanged to the surface. Moreover the ion cores must be precisely located before theoretical analysis starts, and we shall shortly see that interatomic distances and vibrational amplitudes in the surface differ somewhat from those in the interior. These factors certainly complicate the useful application of Band Theory to the properties of surfaces. 

Of the five r/-orbitals, it is assumed that only 2.56 are capable of bonding, the remaining 2.44 being localised atomic d-orbitals, which are non-bonding, and capable of receiving electrons with parallel spins as long as is permitted by Hund s Rule. With chromium the sixth electron is divided as shown in Table 1.1. Now the r/ip-hybrid orbital should in theory accommodate 6.56 electrons 

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Any prospective quantitative theory of the metallic state must include a proper treatment of the two (at least) types of bound electron levels present in the nonmetallic state. In addition, a proper theory of electron transport must be developed for situations wherein electron-electron interactions are important. 

Any theory of the metallic state worth its salt must account for the phenomenon of superconductivity, in which the resistance of a substance is so small as not to be measurable. Indeed, anyone who has ever seen the manifestation of superconductivity will never forget his or her experience [104],... 

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... 

It is not surprising that the application of such a powerful method of investigation should have led, on the experimental side, to a vast extension of our knowledge of the properties of alloy systems. Even more important, however, is the fact that it has also laid the foundations of the modem theory of the metallic state, for, as we have seen in chapter 5, the basic concept on which this theory is based is that of the periodic field in a crystal structure. The development of metallurgy in the past has been hampered by attempts to make metal systems conform to the laws of chemical combination established by observations on bodies in which forces of an entirely different character are operative. Alloys differ profoundly in many of their properties from... 

Because present-day theory of the metallic state does not treat the situation, the electron configuration of alloys made up of two or more transition metals with relation to their passive behavior is not as well understood as for the copper-nickel system. Nevertheless, useful simphfying assumptions can be made. For example, the most passive component of an alloy is assumed to be the acceptor element, which tends to share electrons donated by the less passive components. 

For a meaningful discussion of electronic factors in catalysis it is necessary to briefly review the nature of chemisorption bonds. Two theories of the metallic state have been accepted, the electron band theory and the valence bond theory. Both theories recognize the existence of two separate functions for valence electrons in metals one function is to bind the atoms together and the other is to account for magnetic and conductive properties. In the electron band theory, as particularly applied to the transition metals, the s-electron energy band is broad with a low maximum... 

The actinide metals pose some of the most interesting problems in actinide research. Many actinide compounds behave in a perfectly conventional way (except for radioactivity), and have properties that can be safely inferred from lanthanide chemistry or the chemistry of similar compounds of well-studied elements. For no category of materials is this less true than for the actinide metals. The actinide elements in their elemental state are unique. They have metallurgical properties that are unprecedented in conventional metals, and their properties cannot be accounted for by conventional theories of the metallic state. The theoretical framework of the metallic state has had to be broadened to accommodate this group of unusual metals, and this has led to a better understanding of the metallic state in general [25]. 

A theory for the metallic state proposed by Drude at the turn of this century explained many characteristic features of metals. In this model, called the free-electron theory, all the atoms in a metallic crystal are assumed to take part collectively in bonding, each atom providing a certain number of (valence) electrons to the bond. These free electrons belong to the crystal as a whole. The crystal is considered to be... 

The metallic nature of concentrated metal-ammonia solutions is usually called "well known." However, few detailed studies of this system have been aimed at correlating the properties of the solution with theories of the liquid metallic state. The role of the solvated electron in the metallic conduction processes is not yet established. Recent measurements of optical reflectivity and Hall coefficient provide direct determinations of electron density and mobility. Electronic properties of the solution, including electrical and thermal conductivities, Hall effect, thermoelectric power, and magnetic susceptibility, can be compared with recent models of the metallic state. 

Structural models for molten salts have been proposed by several authors Bockris [1-3], Stillinger [4], Zarzycki [5], Janz [6], Kleppa [7], Blander [8], and others. These structural models are based on the older theories of the liquid state, which were applied for molecular liquids, liquified gases, molten metals, etc. Some of these models will be treated in the following. 

Much has still to be clarified in the resonating-valence-bond theory of the metals, especially is it not yet clear how the various valences and properties of atoms in different states could ever be derived in an independent way. 

It follows that the evaluation of the extent to which one-dimensional physics is relevant has always played an important part in the debate surrounding the theoretical description of the normal state of these materials. One point of view expressed is that the amplitude of in the b direction is large enough for a FL component to develop in the ab plane, thereby governing most properties of the normal phase attainable below say room temperature. In this scenario, the anisotropic Fermi liquid then constitutes the basic electronic state from which various instabilities of the metallic state, like spin-density-wave, superconductivity, etc., arise [29]. Following the example of the BCS theory of superconductivity in conventional superconductors, it is the critical domain of the transition that ultimately limits the validity of the Fermi liquid picture in the low temperature domain. 

The major observations which a theory of the solid state must explain are (1) the freedom of motion of electrons through a metal and their inability to move in an insulating crystal and (2) the breadth of the allowed energy states (as deduced from spectroscopic observations on solids) relative to those of the free atoms or ions of which the crystal is composed. 

A simple explanation for the many characteristic features of the metallic state is given by free-electron theory. In metallic crystals the atoms are assumed to take part collectively in bonding, where each atom provides electrons from outer electron energy levels to the bond. The crystal... 

Electron transfers and the formation of stable ions depend upon the Coulomb forces, and upon the regulating character of the Pauli principle. The nature of atomic stability has already been discussed, and it can be said that an ion is simply a more stable form of atom. The problem of how it disposes itself with other ions to achieve an overall electrical neutrality is, from the point of view of the theory of atomic structure, a secondary matter but from another point of view it presents us with a quite fundamental question. Prom what has been said so far it cannot be concluded that the nature of the metallic state is at all obvious, and yet this is one of the commonest conditions which matter assumes. The constitution of the metallic state therefore calls for special consideration. 

In the earliest theories of the solid state, electrons were perceived as being free and non-interacting - an electron gas. In this model, the atomic orbitals of the component atoms are spread out into energy bands, the detailed form of which depends upon the crystal structure of the phase. An upper energy band which is only partly filled with electrons characterises a metal with itinerant (freely moving) non-interacting electrons. 

On the one hand the chemical theorj was strengthened by Faraday s discovery of the equivalence of the current produced to the amount of chemical action in the cell and also by the discovery of the relation between the electrical energy produced and the energy change in the chemical reaction stated incompletely by Kelvin in 1851 and correctly by Helmholtz in 1882. Nernst s theory of the metal electrode process (1889) also added weight to the chemical theory. 

Color from Transition-Metal Compounds and Impurities. The energy levels of the excited states of the unpaked electrons of transition-metal ions in crystals are controlled by the field of the surrounding cations or cationic groups. Erom a purely ionic point of view, this is explained by the electrostatic interactions of crystal field theory ligand field theory is a more advanced approach also incorporating molecular orbital concepts. 

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