Defect interactions in solids

STATISTICAL MECHANICS OF POINT-DEFECT INTERACTIONS IN SOLIDS... 

Statistical Mechanics of Point-Defect Interactions in Solids... 

A. R. Allnatt, Equilibrium Statistical Mechanics. Statistical Mechanics of Point-defect Interaction in Solids , Advances in Chemical Physics, vol. XI, ed. I. Prigogine, J. Wiley and Sons, New York, 1967. 

RAU2] Rau, H., Range of homogeneity and defect interaction in high temperature nickel sulfide Nii. S, J. Phys. Chem. Solids, 36, (1975), 1199-1204. Cited on pages 164, 170, 171,375. 

Charged defects in solids can interact with one another in an analogous way to the interactions between ions (or between ions and electrons) in a solution. In the solid-state situation, the crystal may be viewed as a neutral medium into which the charged defects are dissolved. This similarity between solution chemical interactions and defect interactions in the solid state has resulted in the field of defect chemistry, which provides basic methods for studying the effects of point defects in solids. The methods are normally applicable to fairly low defect concentrations. Generally, a broad distinction is made between intrinsic defects that are thermally gena-ated in pure compounds and extrinsic defects produced by external influences such as impurities and gaseous atmospheres. References 2 and 3 provide a detailed discussion of point defects and defect chemistry in metal oxides. 

The SCF method for molecules has been extended into the Crystal Orbital (CO) method for systems with ID- or 3D- translational periodicityiMi). The CO method is in fact the band theory method of solid state theory applied in the spirit of molecular orbital methods. It is used to obtain the band structure as a means to explain the conductivity in these materials, and we have done so in our study of polyacetylene. There are however some difficulties associated with the use of the CO method to describe impurities or defects in polymers. The periodicity assumed in the CO formalism implies that impurities have the same periodicity. Thus the unit cell on which the translational periodicity is applied must be chosen carefully in such a way that the repeating impurities do not interact. In general this requirement implies that the unit cell be very large, a feature which results in extremely demanding computations and thus hinders the use of the CO method for the study of impurities. 

These effects can all be enhanced if the point defects interact to form defect clusters or similar structures, as in Fej xO above or U02, (Section 4.4). Such clusters can suppress phase changes at low temperatures. Under circumstances in which the clusters dissociate, such as those found in solid oxide fuel cells, the volume change can be considerable, leading to failure of the component. 

Besides the 29Si and 27 A1 NMR studies of zeolites mentioned above, other nuclei such as H, 13C, nO, 23Na, 31P, and 51V have been used to study physical chemistry properties such as solid acidity and defect sites in specific catalysts [123,124], 129Xe NMR has also been applied for the characterization of pore sizes, pore shapes, and cation distributions in zeolites [125,126], Finally, less common but also possible is the study of adsorbates with NMR. For instance, the interactions between solid acid surfaces and probe molecules such as pyridine, ammonia, and P(CH3)3 have been investigated by 13C, 15N, and 31P NMR [124], In situ 13C MAS NMR has also been adopted to follow the chemistry of reactants, intermediates, and products on solid catalysts [127,128],... 

This article is concerned with the statistical mechanics of interactions between point defects in solids at thermodynamic equilibrium. The review is made entirely from the point of view of the... 

The preceding paragraphs illustrate that analogies between point defects in a crystal and solute molecules in a solution have been used previously but in a fairly elementary way. However, the implications of the existence of such analogies in the formulation of the statistical mechanics of interacting defects has not been considered in detail apart from an early paper by Mayer,69 who was interested primarily in the relation of defect interactions, to the solid-liquid phase transition in crystals with short-range forces. The... 

Another defect problem to which the ion-pair theory of electrolyte solutions has been applied is that of interactions to acceptor and donor impurities in solid solution in germanium and silicon. Reiss73>74 pointed out certain difficulties in the Fuoss formulation. His kinetic approach to the problem gave results numerically very similar to that of the Fuoss theory. A novel aspect of this method was that the negative ions were treated as randomly distributed but immobile while the positive ions could move freely. 

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