Solid Frenkel defects

Dislocation theory as a portion of the subject of solid-state physics is somewhat beyond the scope of this book, but it is desirable to examine the subject briefly in terms of its implications in surface chemistry. Perhaps the most elementary type of defect is that of an extra or interstitial atom—Frenkel defect [110]—or a missing atom or vacancy—Schottky defect [111]. Such point defects play an important role in the treatment of diffusion and electrical conductivities in solids and the solubility of a salt in the host lattice of another or different valence type [112]. Point defects have a thermodynamic basis for their existence in terms of the energy and entropy of their formation, the situation is similar to the formation of isolated holes and erratic atoms on a surface. Dislocations, on the other hand, may be viewed as an organized concentration of point defects they are lattice defects and play an important role in the mechanism of the plastic deformation of solids. Lattice defects or dislocations are not thermodynamic in the sense of the point defects their formation is intimately connected with the mechanism of nucleation and crystal growth (see Section IX-4), and they constitute an important source of surface imperfection. 

Thus, if Frenkel Defects predominate in a given solid, other defects are usually not present. Likewise, for the Schottky Defect. Note that this applies for associated defects. If these are not present, there will still be 2 types of defects present, each having an opposite effect upon stoichiometry. 

Diffusion and migration in solid crystalline electrolytes depend on the presence of defects in the crystal lattice (Fig. 2.16). Frenkel defects originate from some ions leaving the regular lattice positions and coming to interstitial positions. In this way empty sites (holes or vacancies) are formed, somewhat analogous to the holes appearing in the band theory of electronic conductors (see Section 2.4.1). 

A somewhat different situation is found in the type of point defect known as a Frenkel defect. In this case, an atom or ion is found in an interstitial position rather than in a normal lattice site as is shown in Figure 7.17. In order to position an atom or ion in an interstitial position, it must be possible for it to be close to other lattice members. This is facilitated when there is some degree of covalence in the bonding as is the case for silver halides and metals. Accordingly, Frenkel defects are the dominant type of defect in these types of solids. 

A point defect is a localized defect that consists of a mistake at a single atom site in a solid. The simplest point defects that can occur in pure crystals are missing atoms, called vacancies, or atoms displaced from the correct site into positions not normally occupied in the crystal, called self-interstitials. Additionally atoms of an impurity can occupy a normal atom site to form substitutional defects or can occupy a normally vacant position in the crystal structure to form an interstitial. Other point defects can be characterized in pure compounds that contain more than one atom. The best known of these are Frenkel defects, Schottky defects, and antisite defects. 

It is important that the copper is in the monovalent state and incorporated into the silver hahde crystals as an impurity. Because the Cu+ has the same valence as the Ag+, some Cu+ will replace Ag+ in the AgX crystal, to form a dilute solid solution Cu Agi- X (Fig. 2.6d). The defects in this material are substitutional CuAg point defects and cation Frenkel defects. These crystallites are precipitated in the complete absence of light, after which a finished glass blank will look clear because the silver hahde grains are so small that they do not scatter light. 

In densely packed solids without obvious open channels, the transport number depends upon the defects present, a feature well illustrated by the mostly ionic halides. Lithium halides are characterized by small mobile Li+ ions that usually migrate via vacancies due to Schottky defects and have tc for Li+ close to 1. Similarly, silver halides with Frenkel defects on the cation sublattice have lc for Ag+ close to 1. Barium and lead halides, with very large cations and that contain... 

Figure 5.1 Point defects in ionic solids Schottky defect, vacancy pair, Frenkel defect and aliovalent impurity (for definitions see Section 5.2).

The existence of Schottky or Frenkel defects, or both, within an ionic solid provides a mechanism for significant electrical conductance through ion migration from site to empty site (leaving, of course, a fresh empty site behind).4 Solid /3-AgI provides a classic example of a nonmetallic solid with substantial electrical conductivity at elevated temperatures at 147 °C, it undergoes a transition to a-Agl in which the silver ion sublattice is disordered and consequently allows for relatively free movement of Ag+ and... 

In the examples considered above traps B were assumed to serve as sinks of the infinite capacity. Its simple modification for the case of saturable traps (the policeman who caught some toper is no longer on his post but is bringing him to the police station) leads us immediately to the new class of reactions known as A + B —> C (C is a neutral reaction product - the policeman bound to a toper is out of his duty) or A+B —> 0 (reaction product leaves the system). This latter reaction is typical for the so-called Frenkel defects in solids when complementary defects A and B (interstitial atoms and vacancies) annihilate each other, thus giving no products and restoring the perfect crystalline lattice. 

Consider now the fluctuations of the order parameter in the system possessing the chemical reaction this problem could be perfectly illustrated by computer simulations on lattices. We start with the bimolecular A + B -y 0 reaction discussed above, and first of all froze particle diffusion. Let the recombination event happen instantly when a pair AB of dissimilar particles occupies the nearest lattice sites (assume lattice to be squared). Immobile particles enter into reaction as a result of their creation with the equal probabilities in empty lattice sites from time to time a newly created particle A(B) finds itself nearby pre-created B(A) and they recombine. (Since this recombination event is instant, the creation rate is of no importance.) This model describes, in particular, Frenkel defect accumulation in solids under... 

Equation (2.2.24) means homogeneous generation of particles A and B with the rate p (per unit time and volume), whereas (2.2.25) comes from the statistical independence of sources of a different-kind particles. Physical analog of this model is accumulation of the complementary Frenkel radiation defects in solids. Note that depending on the irradiation type and chemical nature of solids (metal or insulator), dissimilar Frenkel defects could be either spatially correlated in the so-called geminate pairs (see Chapter 3) or distributed at random. We will focus our attention on the latter case. 

Irradiation of all kinds of solids (metals, semiconductors, insulators) is known to produce pairs of the point Frenkel defects - vacancies, v, and interstitial atoms, i, which are most often spatially well-correlated [1-9]. In many ionic crystals these Frenkel defects form the so-called F and H centres (anion vacancy with trapped electron and interstitial halide atom X° forming the chemical bonding in a form of quasimolecule X2 with some of the nearest regular anions, X-) - Fig. 3.1. In metals the analog of the latter is called the dumbbell interstitial. 

However, in the last two decades it has been shown experimentally [1,7, 8,12-14] and theoretically [15-18] that in many wide-gap insulators including alkali halides the primary mechanism of the Frenkel defect formation is subthreshold, i.e., lattice defects arise from the non-radiative decay of excitons whose formation energy is less than the forbidden gap of solids, typically 10 eV. These excitons are created easily by X-rays and UV light. Under ionic or electron beam irradiations the main portion of the incident particle... 

As it was said in Section 3.1, the survival probability of the geminate pairs oj 1 [ oo) directly defines the efficiency of the Frenkel defect accumulation in solids. Let us assume that the concentration of radiation-created defects is n. Then after a transient period during which mobile interstitials recombine with their geminate partners or leave them thus surviving, we have nacc of stable defects obeying the relation nacc = rjn, where r = cu(oo) is accumulation efficiency, which could be identified with the survival probability of a single mobile interstitial (provided that different geminate pairs do not mix). 

Up to now we have been discussing in this Chapter many-particle effects in bimolecular reactions between non-interacting particles. However, it is well known that point defects in solids interact with each other even if they are not charged with respect to the crystalline lattice, as it was discussed in Section 3.1. It should be reminded here that this elastic interaction arises due to overlap of displacement fields of the two close defects and falls off with a distance r between them as U(r) = — Ar 6 for two symmetric (isotropic) defects in an isotropic crystal or as U(r) = -Afaqjr-3, if the crystal is weakly anisotropic [50, 51] ([0 4] is an angular dependent cubic harmonic with l = 4). In the latter case, due to the presence of the cubic harmonic 0 4 an interaction is attractive in some directions but turns out to be repulsive in other directions. Finally, if one or both defects are anisotropic, the angular dependence of U(f) cannot be presented in an analytic form [52]. The role of the elastic interaction within pairs of the complementary radiation the Frenkel defects in metals (vacancy-interstitial atom) was studied in [53-55] it was shown to have considerable impact on the kinetics of their recombination, A + B -> 0. 

As it has been said above, accumulation of radiation-induced (Frenkel) defects takes place in all kinds of solids irrespective which of the two basic recombination mechanisms - annihilation or tunnelling recombination - occurs [9, 11, 13, 17-20, 39, 42, 99-107]. In a good approximation this process could be considered as the A + B —> 0 reaction with the particle input. In many cases strong arguments exist for the clustering of radiation-induced... 

It is important to note that it was K. Dettmann [34] who first created the solid basis of a rigourous theory of the kinetics of accumulation of the Frenkel defects in crystals. He showed that in the absence of diffusion the accumulation curve, n = n(t), is determined by the infinite sum of the correlation functions pm o> m — 2,... oo that describe the spatial correlations of all orders of similar defects only. 

A simulation has been carried out [105, 106, 119] of the process of accumulation of the immobile Frenkel defects restricted by tunnelling recombination of dissimilar defects, as it is observed in many solid insulators. As follows from Chapter 3, in contrast to the ionic process of instant annihilation of close pairs of the vacancy-atom type, it is characterized by a broad spectrum of recombination times. Thus, the probability for a pair of chosen defects that lie at the relative distance r to survive for r seconds is... 

The analysis conducted in this Chapter dealing with different theoretical approaches to the kinetics of accumulation of the Frenkel defects in irradiated solids (the bimolecular A + B —> 0 reaction with a permanent particle source) with account taken of many-particle effects has shown that all the theories confirm the effect of low-temperature radiation-stimulated aggregation of similar neutral defects and its substantial influence on the spatial distribution of defects and their concentration at saturation in the region of large radiation doses. The aggregation effect must be taken into account in a quantitative analysis of the experimental curves of the low-temperature kinetics of accumulation of the Frenkel defects in crystals of the most varied nature - from metals to wide-gap insulators it is universal, and does not depend on the micro-mechanism of recombination of dissimilar defects - whether by annihilation of atom-vacancy pairs (in metals) or tunnelling recombination (charge transfer) in insulators. 

If the semiconductor is an ionic solid, then electrical conduction can be electronic and ionic, the latter being due to the existence of defects within the crystal that can undergo movement, especially Frenkel defects (an ion vacancy balanced by an interstitial ion of the same type) and Schottky defects (cation and anion vacancies with ion migration to the surface). This will be discussed further in Chapter 13, as ionic crystals are the sensing components of an important class of ion selective electrodes. 

A variety of defect formation mechanisms (lattice disorder) are known. Classical cases include the - Schottky and -> Frenkel mechanisms. For the Schottky defects, an anion vacancy and a cation vacancy are formed in an ionic crystal due to replacing two atoms at the surface. The Frenkel defect involves one atom displaced from its lattice site into an interstitial position, which is normally empty. The Schottky and Frenkel defects are both stoichiometric, i.e., can be formed without a change in the crystal composition. The structural disorder, characteristic of -> superionics (fast -> ion conductors), relates to crystals where the stoichiometric number of mobile ions is significantly lower than the number of positions available for these ions. Examples of structurally disordered solids are -> f-alumina, -> NASICON, and d-phase of - bismuth oxide. The antistructural disorder, typical for - intermetallic and essentially covalent phases, appears due to mixing of atoms between their regular sites. In many cases important for practice, the defects are formed to compensate charge of dopant ions due to the crystal electroneutrality rule (doping-induced disorder) (see also -> electroneutrality condition). 

Exciton — A bound and thus localized pair of an - electron in the conduction band and a hole in the valence band. An exciton is an elementary excitation and quasiparticle in solids. See also Frenkel, defects in solids. Ref [i] Eliott SR (1998) The physics and chemistry of solids. Wiley, Chichester... 

Figure 28. a) Ion redistribution process at the contact MX/MX concentration effects.117 b) Variation of the potentials, charge densities, and dielectric displacements at the contact of two Frenkel defect ionic conductors.94 (Reprinted from J. Maier, Ionic Conduction in Space Charge Regions. Prog. Solid St. Chem. 171-263, 23, Copyright 1995 with permission from Elsevier.)... 

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