The perfect solid

The perfect solid does not exist in Nature and its reactive properties are determined, to a great extent, by the defects present in its structure". 

Non-stoichiometry is a very important property of actinide dioxides. Small departures from stoichiometric compositions, are due to point-defects in anion sublattice (vacancies for AnOa-x and interstitials for An02+x )- A lattice defect is a point perturbation of the periodicity of the perfect solid and, in an ionic picture, it constitutes a point charge with respect to the lattice, since it is a point of accumulation of electrons or electron holes. This point charge must be compensated, in order to preserve electroneutrality of the total lattice. Actinide ions having usually two or more oxidation states within a narrow range of stability, the neutralization of the point charges is achieved through a Redox process, i.e. oxidation or reduction of the cation. This is in fact the main reason for the existence of non-stoichiometry. In this respect, actinide compounds are similar to transition metals oxides and to some lanthanide dioxides. 

The atoms that comprise a solid can be considered for many purposes to be hard balls which rest against each other in a regular repetitive pattern called the crystal structure. Most elements have relatively simple crystal structures of high symmetry, but many compounds have complex crystal structures of low symmetry. The determination of crystal structures, of atom location in the crystal, and of the dependence of many physical properties upon the inherent charactensdcs of the perfect solid is an absorbing study, one that has occupied the lives of numerous geologists, mineralogists, physicists, and other scientists for many years. 

To our knowledge there have been no reported measurements of equilibrium defect concentrations in soft-sphere models. Similarly, relatively few measurements have been reported of defect free energies in models for real systems. Those that exist rely on integration methods to connect the defective solid to the perfect solid. In ab initio studies the computational cost of this procedure can be high, although results have recently started to appear, most notably for vacancies and interstitial defects in silicon. For a review see Ref. 109. 

A less orthodox line of attack, as yet not explored to its full potential, applies from the beginning the recursion method to the solid-plus-impurity system. The direct use of memory function methods to the perturbed solid is no more difficult than for the perfect solid, with the advantage of overcoming the traditional separation of the actual Hamiltonian into a perfect part and a perturbed part. In fact, such a separation, to make any practical sense, requires that the perturbed part be localized in real space, a restriction hardly met when treating impurities with a coulombic tail. 

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... 

II Gruneisen has calculated some of the consequences of the theory, assuming that the frequency / is a function of the volume v, hut independent of the temperature (Zeit. f. Elektrochem. 17, 732, 1911). See also Haber, Verh. der deutach. phyaik. Geadlach. 13, 1117, 1911. An important advance in the theory of the perfect solid is due to Debye (see Chap. II. p. 37). 

Much has been written about the failure of classical theory in interpretating the specific heat of metals and the subsequent development of a quantum theory of the perfect solid state. It would be quite impossible to give an adequate account of this development here, and the... 

On the theoretical side cur problem is just as great. Quite apart from the purely mathematical difficulties involved there are principles which are lacking. There are many unjustifiable assumptions made m the thermodynamic arguments, on which any estimation of defect concentration is inevitably based. Over and above this there is only a very approximate treatment of the quantum mechanics of the perfect solid state available the zone theory and the method of atomic orbitals give widely different solutions to the same problem. The most significant theoretical advance here has come from James, and more particularly Slater, who has provided a useful theorem on which to base study of the defect state. 

By applying his knowledge of the statistical mechanics of the perfect solid and of the perfect gas, Eyring was able to write a partition function for the liquid state on the assumption that a liquid consists of an intermixing of crystalline and gaseous fractions. This culminated in the theory of significant liquid structures. ... 

Note that we are using the Gibbs free energy rather than the Helmholtz free energy at this point. For the defect solid, we must define free-energy in terms of the free-energy of the perfect solid. Go, as related to the free-energy of the defect solid, vis ... 

Note that due to increased entropy, the energy of the defect state is higher than that of the perfect solid. For this reason, we must distinguish between the sources which contribute to the total entropy, and multiply by the number of defects present. The applicable equation is ... 

This value can now be compared with the k value calculated from the data from Ex. 12.2-3, to gauge the appropriateness of the perfect solid mixing approximation for a commercial fluidiz bed. Retaining Eq. (d) for k(0) (i.e., complete mixing for the gas phase), the following integral has to be evaluated from the data ... 

The integrals over y extend over the entire system, which has length in the y direction that we denote as L. The difference in energy between this system and the contact energy of the perfect solid is... 

The three-dimensional order of the crystallized solids is expensive in entropy its reason is thus in the maximization of interactions (van der Waals, hydrogen bond, electrostatic, delocalization of the electrons in a metal) between structure elements, considering the geometrical constraints related to the shape of the molecules or the size of the ions. The whole of the sohd thus creates a periodic potential, in particular in space surrounding a stracture element, which at 0 K occupies the position of minimum energy in the perfect solid. 

The ideal solid would be perfectly ordered and geometrically regular, chemically pure and stoichiometric. No real solid is found in this state, and some of the most interesting and important properties of solids depend upon departure from the ideal. The study and control of these deviations from the perfect solid are a principal concern of solid-state chemists. Later volumes of the Treatise will be concerned with defects in solids in relation to physical and chemical properties. Here we lay the foundation by examining the nature of defects and the equilibria controlling their concentrations. In addition, the characterization of solids is included it would seem almost self-evident that the measurement of departures of a solid from the ideal chemical and geometrical state is a sine qua non in the study of the imperfect solid, yet it has all too often been overlooked or treated casually. 

The first four chapters dealing with the perfect state are more introductory. According to Fig. 1.2 we formally construct our real solids by superposition of the perfect solid ( chemical groundstate ) and the defects ( chemi excitations ). Both ensembles are not independent of each other but strictly coupled in equilibrium. For this reason we start with a concise treatment of the chemically perfect solid. Firstly, there is a discussion (Chapter 2) of the chemical bonding and then of the formation of the solid state, followed by a discussion of lattice vibrations (Chapter 3). 

The purpose of the chapter dealing with equilibrium thermodynamics of the perfect solid (Chapter 4) is to elaborate, on the one hand, simple expressions for the thermodynamic functions of the chemical ground state and, on the other hand, to make the reader familiar with questions of internal and external equilibria, not least with the intention to provide the equipment to deal with the thermodynamics of defect formation. (The major portion of the free enthalpy at absolute zero consists of bonding energy, while the temperature dependence is largely determined by the vibration properties.)... 

According to Fig. 1.2 we decompose thermodynamic functions into contributions that arise from (chemically) perfect solids and contributions that are brought in by defects. At this point we are now interested in the equilibrium thermodynamics of the (chemically) perfect state. Our aim is to sketch the fi ee enthalpy of the perfect solid with the aid of the previous chapters on chemical bonding and phonons, as well as to consider relevant aspects of the thermodynamic formalism and its apphcation to solids, in particular in view of interactions with the chemical environment. 

The addendum chemical is intended to emphasize that phonons are elements of the perfect solid as it is defined here. On the other hand, chemical at this point also includes effects that can, with some justification, be regarded as crystallographic. 

Equation (4.4) neglects the work contributions resulting from changes in the surface area (A), which strictly speaking always play a role even in the perfect solid in equilibrium (7dA, 7 surface tension). Amongst other contributions left out in Eq. (4.4) are electrical work terms ( Q electrical potential, Q electrical charge) which will become important when we deal with charge carriers in boundary zones. (In open systems we also have to take account of external material input (/rk eUk ... 

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