Theories of the Solid State

The molecular orbital treatment of a crystalline solid considers the outer electrons as belonging to the crystal as a whole (10,11). Sommer-feld s early free electron theory of metals neglected the field resulting 

In the consideration of the momentum of a large number of particles restricted to a volume V, it is often convenient to describe the system by an assembly of points in a momentum diagram (Fig. 1). The length OA represents the magnitude of momentum of the particle A, and its direction is OA. The application of Heisenberg s uncertainty principle leads to the 

A typical energy distribution curve is represented by Fig. 3. An elec- 

The effect of an external electric field is to produce an acceleration of the electrons in the direction of the field, and this causes a shift of the Fermi surface. It is a necessary condition for the movement of electrons in the fc-space that there are allowed empty states at the Fermi surface hence electrical conductivity is dependent on partially filled bands. An insulating crystal is one in which the electron bands are either completely full or completely empty. If the energy gap between a completely filled band and an empty band is small, it is possible that thermal excitation of electrons from the filled to the empty band will result in a conducting crystal. Such substances are usually referred to as intrinsic semiconductors. A much larger class of semiconductors arises from impurities 

Hume-Rothery (12,13) has pointed out that in some alloys the structure of the intermetallic phases are determined by the electron concentration (E.C.). The work of Hume-Rothery and others has shown that the series of changes (i.e. a phase — (3 phase — 7 phase — phase), which occurs as the composition of an alloy is varied continuously, takes place at electron-atom ratios of 3/2, 21/13, and 7/4, respectively. The interpretation of these changes in terms of the Brillouin zone theory has been made by H. Jones (14) and can be understood from the A (E)-curves for typical face centered cubic (a) and body centered cubic (6) structures as 

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Callaway J 1974 Quantum Theory of the Solid State (Boston Aoademio)... 

The elucidation of the factors determining the relative stability of alternative crystalline structures of a substance would be of the greatest significance in the development of the theory of the solid state. Why, for example, do some of the alkali halides crystallize with the sodium chloride structure and some with the cesium chloride structure Why does titanium dioxide under different conditions assume the different structures of rutile, brookite and anatase Why does aluminum fluosilicate, AljSiCV F2, crystallize with the structure of topaz and not with some other structure These questions are answered formally by the statement that in each case the structure with the minimum free energy is stable. This answer, however, is not satisfying what is desired in our atomistic and quantum theoretical era is the explanation of this minimum free energy in terms of atoms or ions and their properties. 

The electron theory of catalysis cannot as yet be regarded as a complete theory. It resembles a building from which the scaffolding has not yet been removed. It is being erected on the foundation of the modern theory of the solid state and thus introduces new concepts and ideas into the theory of catalysis. This does not mean, of course, that it excludes other concepts and ideas prevalent today in other theories of catalysis. On the contrary, it makes use of these while attempting to disclose their physical content. 

In the general case, as has also been shown, the same chemisorbed particle on the same adsorbent may simultaneously be both acceptor and donor, possessing a definite affinity both for a free electron and a free hole. We observe that structural defects which are simultaneously acceptors and donors are well known in the theory of the solid state. Take, for example. 

The free valencies of a crystal can form pairs, each such pair wandering through the crystal as an entity until it breaks up. Such formations are well known in the theory of the solid state. A pair of opposite valencies in an ionic crystal (electron - - hole bound by Coulomb interaction) forms what is called a Mott exciton. A pair of like valencies (election + electron or hole + hole bound by exchange interactions) forms a so-called doublon. Such formations have recently been investigated, 12, IS). 

Theoretical investigations of the modification phenomenon have been carried out by Roginskil and Vol kenshteln (342,418,419,420,421,423). Their work has been based upon the electronic theory of catalysis which utilizes greatly present-day knowledge of quantum chemistry and those theories of the solid state which deal with processes occurring within crystalline materials. On the basis of these concepts it is possible to treat the problems involved in the conversion of molecules adsorbed on a solid surface. The adsorbed molecule and the solid are treated as a unified system, the electrons of the lattice participating in bonding and subsequently chemical reaction. The velocity of the chemical reaction is dependent upon the electronic properties of the solid and reactants involved. 

Cerium, praseodymium, and terbium oxides display homologous series of ordered phases of narrow composition range, disordered phases of wide composition range, and the phenomenon of chemical hysteresis among phases which are structurally related to the fluorite-type dioxides. Hence they must play an essential role in the satisfactory development of a comprehensive theory of the solid state. All the actinide elements form fluorite-related oxides, and the trend from ThOx to CmOx is toward behavior similar to that of the lanthanides already mentioned. The relationships among all these fluorite-related oxides must be recognized and clarified to provide the broad base on which a satisfactory theory can be built. 

Real solid bodies, therefore, differ considerably from the perfect solid at higher temperatures, but appear to approach asymptotically to the perfect condition as the temperature is lowered. The conception of a perfect solid body hke that of a perfect gas is only true in the limiting case. It will, perhaps, be possible to build up a complete theory of the solid state on the basis of Einstein s hypothesis, as van der Waals theory was evolved from the conceptions of the classical theory of... 

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. 

We shall see in the next chapter that the results obtained have given a decisive bent to our conceptions of the energy content of solids, and hence also of the theory of the solid state in general. 

These disadvantages may all be avoided, and are in fact considerably reduced by the use of lead instead of platinum wire, but a simpler procedure is still desirable. The problem of establishing the variation of specific heat for a large number of substances as completely as possible is extremely important, both in the light of any future theory of the solid state, which will apparently be developed from a consideration of low-temperature properties, and also in view of the applications of our Heat Theorem and the study of chemical affinity. 

Critical.—Upon the measurements which have been carried out by the methods described in this chapter depend, in part, the recent theories of the solid state (cf. Chapter IV), and also to a large extent the exact tests of our Heat Theorem. It will therefore be of use to devote a few words to the reliability of the results obtained. 

I have given a review in four lectures on The Theory of the Solid State," delivered at the University of London (University of London Press, 1914). 

Callaway, J. (1976), Quantum Theory of the Solid State, Academic Press, New York. 508 Callaway, J and Hughes, A. J. (1967) Phys. 

Fortunately, during the last twenty years or so, two developments have taken place which enable one to define complicated catalyst systems in a fairly sophisticated manner. First, the theory of the solid state has advanced to the point where, despite its well-known limitations and ambiguities, it can serve as a reasonable foundation for the construction of a functional model of a catalyst system. Second, there has been the development of several research techniques, principally of a spectroscopic nature, which permit a more detailed study of catalysts than has hitherto been possible. The present chapter consists essentially of an illustration of the application of these two developments to the elucidation of the physical-chemical structure of a fairly complicated catalyst of practical importance, namely chromia-alumina. While it cannot be claimed that a complete description of the chromia-alumina catalyst system has been realized, it is fair to say that, by the application of a variety of modern experimental techniques to this single catalyst, a molecular picture has emerged which is more detailed than had been previously available. This alone encourages cautious optimism concerning the future. 

Plane waves used historically in the theory of the solid state, these functions are being used increasingly in molecular theories in conjunction with the density functional method discussed in Chapter 32. These functions are not dependent on the positions of the nuclei and offer considerable simplifications in gradient calculations. 

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. 

Experimental diffusivities, solubilities, and permeabilities. Accurate prediction of diffusivities in solids is generally not possible because of the lack of knowledge of the theory of the solid state. Hence, experimental values are needed. Some experimental data for diffusivities, solubilities, and permeabilities are given in Table 6.5-1 for gases diffusing in solids and solids diffusing in solids. 

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