Higher-dimensional defects

Further characteristic solid state properties following from the high bond strength compared to kT, are the anisotropies of structure and properties (transport coefficients e.g. are tensors) as well as the occurrence of higher-dimensional defects such as dislocations and internal boundaries. 

Since temperature and partial pressure are often fixed, as far as application is concerned, the most powerful means in modifying a given structure and compound is the homogeneous doping. (In the following we will also see that frozen-in higher dimensional defects as well as frozen-in native defects can act similarly). 

From a more conceptual point of view, it is the introduction of higher-dimensional defects that allows the transition to a soft materials science , characterized by an enhanced information content even in systems in which the atomic bonds are not covalent. The future will be witness to increased research and applications in the field of metastable materials characterized by increased local complexity, with the possibility of further systematic collaboration with semiconductor physics and biology. 

In a crystal lattice there is translation symmetry but in a polycrystalline solid it exists only approximately within one grain. Similar to the outer surfaces of the crystallites, the grain boundaries are two-dimensional defects the crystal lattice stops there. Dislocations are one-dimensional defects and pores are defects in solids having a dimension that is usually three but can be lower. Such higher-dimensional defects (Table 10.1) determine many properties the dislocations in metals affect plasticity and the porosity, if open, determines gas permeability. 

A suitable classification of crystalline defects can be achieved by first considering the so-called point defects and then proceeding to higher-dimensional defects. Point defects are atomic defects whose effect is limited only to their immediate surroundings. Examples are vacancies in the regular lattice, or interstitial atoms. Dislocations are classified as linear or one-dimensional defects. Grain boundaries, phase boundaries, stacking faults, and surfaces are two-dimensional defects. Finally, inclusions or precipitates in the crystal matrix can be classified as three-dimensional defects. 

If we leave the regime of elastic deformations we introduce irreversible changes, e.g. in the form of dislocations. This is considered in more detail in Section 5.4.3 when we consider higher-dimensional defects. 

We distingiiish between point defects (zero dimensional defects) — these are atomic and electronic imperfections line (one-dimensional) defects — these are essentially dislocations plane (two-dimensional) defects — i.e. surfaces and basically internal interfaces and pores or inclusions as three-dimensional defects. We will not discuss other variants of higher-dimensional disorder, which can be very compleot, particularly in multiphase systems. Since we concentrate on the equilibrium state in this chapter, we are primarily interested in point defects and surfaces. Point defects exist at equilibrium on accoimt of entropy surfaces are a necessary consequence of the requirement that the amoimt of substance is finite. Defects of other types are necessarily nonequilibrium phenomena", which will be demonstrated in Section 5.4. Nonetheless, the higher-dimensional defects will, as metastable structure elements, be important for us later (see Sections 5.4, 5.8). 

Let us now consider the same situation from another point of view and regard the grain boundary as an aggregation of v point defects d to a higher-dimensional defect (d). ... 

On account of this dependence on the sample s pre-history we meet a whole range of different cases with respect to higher-dimensional defects. Even though the absolute G minimum is not normally achievable, more or less marked local minima in the free enthalpy can still be realized. We will content ourselves with a few remarks concerning structure and energetics at this point. 

Now, after this excursion, let us focus on point defects, which are the heart of the text. We will return in Section 5.8 to higher-dimensional defects in the form of boundary conditions for the distribution of point defects. 

Higher dimensional defects such as dislocations and grain bonndaries are thermodynamically unstable, and their behavior mnst be left for a class that covers kinetics. 

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