Perfect solid

It is found in this way that crystalline hydrogen at temperatures somewhat below the melting point is a nearly perfect solid solution of symmetric and antisymmetric molecules, the latter retaining the quantum weight 3 for the state with j = 1 as well as the spin quantum weight 3. This leads to the expression... 

In crystals for which n0 is large, such as iodine, the lowest symmetric and the lowest antisymmetric state have practically the same energy and properties, and each corresponds to one eigenfunction only. As a result a mixture of symmetric and antisymmetric molecules at low temperatures will behave as a perfect solid solution, each molecule having just its spin quantum weight, and the entropy of the solid will be the translational entropy plus the same entropy of mixing and spin entropy as that of the gas. This has been verified for I2 by Giauque.17 Only at extremely low temperatures will these entropy quantities be lost. 

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". 

We have shown that by stacking atoms or propagation units together, a solid with specific symmetry results. If we have done this properly, a perfect solid should result with no holes or defects in it. Yet, the 2nd law of thermod5mamics demands that a certain number of point defects (vacancies) appear in the lattice. It is impossible to obtain a solid without some sort of defects. A perfect solid would violate this law. The 2nd law states that zero entropy is only possible at absolute zero temperature. Since most solids exist at temperatures far from absolute zero, those that we encounter are defect-solids. It is natural to ask what the nature of these defects might be. 

Of course, you don t have to use either of the above standards at all. In the case of samples run in deutero chloroform/methanol and dimethyl sulfoxide, it is perfectly acceptable, and arguably preferable, to reference your spectra to the residual solvent signal (e.g., CD2HOH) which is unavoidable and always present in your spectrum (see Table 2.2). These signals are perfectly solid in terms of their shifts (in pure solvent systems) though the same cannot be said for the residual HOD signal in D2O and for this reason, we would advise adhering to TSP for all samples run in D20. 

Figure 4.5 shows a conventional unit cell of an fee crystal. It consists of atoms at the eight edges of a cube and at the centers of the six sides. The length a of the side of the cube is the lattice constant-, for our present purpose we may assume that it is unity. The lattice of an infinite, perfect solid is obtained by repeating this cubic cell periodically in all three directions of space. 

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. 

A recirculation design (Fig. 10) returns the gas to the classifier through the fan after the fine particles are removed from the gas stream. Such an arrangement requires an excellent solid/gas separator otherwise the classification becomes less efficient. A perfect solid/gas separator would be a device having a = 1. If the recirculated gas is entered through a secondary coarse stream classification section, then the classification is not less efficient unless the secondary classification is very inefficient. 

Three pounds of olive oil gave a block of soap weighing six pounds ten ounces, which, an exposure to the air for two months, became four pounds fifteen ounces. It was then dry and perfectly solid, of excellent color, resembling that of Marseilles soap. By further exposure in a dry place it became still lighter, owing to loss of water. 

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. 

The tetrahedron is another of Plato s perfect solids. The hydrocarbon having this shape is known as tetrahedrane. Because of its three-membered rings, it has considerably more strain than cubane and has, so far, resisted many attempts to prepare it. However, tetrahedrane substituted with fert-butyl groups at its vertices was prepared in 1981. It is a stable solid at room temperature. 

The entropy in a solid arises first from corrfignrational terms that for a perfect solid are zero. However, for a solid showing orientational or translational disorder, corrfignrational expressions based on the Boltzmann expression S = khi(W)may be used. In this section, we shall pay more attention to the second term, which is arises from the population of the vibrational degrees of freedom of the solid. Thus the entropy of a solid may be written as ... 

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. 

As in the case of electrical failure in random conductor-insulator networks in the earlier chapter, we first discuss here the concept of stress concentration in an otherwise perfect solid, which is stressed and contains a single crack inside. Here, the stresses concentrate at the sharp edges of the crack, where it can become much larger compared to the external force. As one increases the external force, the crack starts propagating from such... 

Fracture strength of a perfect solid containing a single crack Griffith s law... 

The above results are all for a perfect solid under stress, with a single microcrack inside. For randomly disordered solids, the appropriate modification of the above Mott formula has not been developed yet. However, some quantitative features of the fracture propagation process in extremely disordered solids, like the percolating solid near its percolation threshold, are quite obvious and interesting. Although the (equilibrium) strength erf of the solid vanishes near the percolation threshold Pc erf (Ap) ), the... 

This equation leads us to some important and strikingly simple results, much as we were led to draw conclusions as to the physical behaviour of perfect gases from their entropy equation. In analogy with the term perfect gas, the author proposes the term perfect solid for substances which conform to Einstein s assumptions. 

The entropy of a perfect solid is therefore independent of its volume. From this it follows that if solid solutions can be formed at all, no diffusion can take place in them. 

The significance of these equations is as follows The co-efiSicient of expansion of a perfect solid body is zero, and its compressibility is independent of the temperature. In the compression of a perfect solid, no rise in temperature is produced. The work done in the compression does not contribute to the kinetic energy of the atoms and merely increases their potential energy. 

If we compare the actual behaviour of solid bodies with those of the hypothetical perfect solid, we find that the coefiicient of expansion is very small for all sofids, and appears to approach zero as we diminish the temperature (according to Thiesen, Gruneisen,t and Lindeman J). The variation of compressibifity with temperature is also very small, and appears hkewise to approach zero as the temperature is diminished. ... 

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). 

There are relatively few examples of perfect solutions. A perfect solid... 

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. 

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