Temperature Schottky defects

Crystals with Frenkel or Schottky defects are reasonably ion-conducting only at rather high temperatures. On the other hand, there exist several crystals (sometimes called soft framework crystals ), which show surprisingly high ionic conductivities even at the room or slightly elevated temperatures. This effect was revealed by G. Bruni in 1913 two well known examples are Agl and Cul. For instance, the ar-modification of Agl (stable above 146°C, sometimes denoted also as y-modification ) exhibits at this temperature an Ag+ conductivity (t+ = 1) comparable to that of a 0.1m aqueous solution. (The solid-state Ag+ conductivity of a-Agl at the melting point is actually higher than that of the melt.) This unusual behaviour can hardly be explained by the above-discussed defect mechanism. It has been anticipated that the conductivity of ar-Agl and similar crystals is described... 

It is possible to create a population of Schottky defects that is much higher than the equilibrium population that is based on Eq. (7.32). If a crystal is heated to high temperature, lattice vibrations become more pronounced, and eventually ions begin to migrate from their lattice sites. If the crystal is quickly cooled, the extent of the motion of ions decreases rapidly so that ions that have moved from their lattice sites cannot return. As a result, the crystal will contain a population of Schottky defects that is much higher than the equilibrium population at the lower temperature. If a crystal of KC1 is prepared so that it contains some CaCl2 as an impurity, incorporating a Ca2+ ion in the crystal at a K+ site... 

At all temperatures above 0°K Schottky, Frenkel, and antisite point defects are present in thermodynamic equilibrium, and it will not be possible to remove them by annealing or other thermal treatments. Unfortunately, it is not possible to predict, from knowledge of crystal structure alone, which defect type will be present in any crystal. However, it is possible to say that rather close-packed compounds, such as those with the NaCl structure, tend to contain Schottky defects. The important exceptions are the silver halides. More open structures, on the other hand, will be more receptive to the presence of Frenkel defects. Semiconductor crystals are more amenable to antisite defects. 

The formation energy of Schottky defects in NiO has been estimated at 198 kJ mol-1. The lattice parameter of the sodium chloride structure unit cell is 0.417 nm. (a) Calculate the number of Schottky defects per cubic meter in NiO at 1000°C. (b) How many vacancies are there at this temperature (c) Estimate the density of NiO and hence the number of Schottky defects per gram of NiO. 

The following table gives the values of the fraction of Schottky defects, S/N, in a crystal of NaBr, with the sodium chloride structure, as a function of temperature. Estimate the formation enthalpy of the defects. 

The fraction of vacancies in a crystal of NaCl, riy/N due to a population of Schottky defects, is 5 x 10-5 at 1000 K. In a diffusion experiment at this temperature, the activation energy for self-diffusion of Na was found to be 173.2 kJ mol-1. Determine the potential barrier that the diffusing ions have to surmount. 

Frenkel and Schottky defect equilibria are temperature sensitive and at higher temperatures defect concentrations rise, so that values of Ks and Kv, increase with temperature. The same is true of the intrinsic electrons and holes present, and Kc also increases with temperature. This implies that the defect concentrations in the central part of a Brouwer diagram will move upward at higher temperatures with respect to that at lower temperatures, and the whole diagram will be shifted vertically. 

In a crystal of overall composition MX, suppose that ns is the number of Schottky defects in the crystal at a temperature T (K), that is, there are ns vacant cation sites and ns vacant anion sites present, distributed over N possible cation sites and N possible anion sites. The configurational entropy change, ASs, due to the distribution of the... 

The equilibrium concentration of intrinsic defects in a structure depends on temperature. For the Schottky defect, the equilibrium constant K for the defect-generation reaction is... 

Table 5.9 gives the variation of defect concentration with temperature for Csl. Determine the enthalpy of formation for one Schottky defect in this crystal. 

TABLE 5.10 Schottky defect concentration in MX compound at various temperatures... 

This expression is not strictly correct, only at absolute zero is this state realized. With an increase of temperature, however, the free energy of the system decreases with increasing entropy. Therefore at higher temperatures, the crystal develops lattice defects in both the metal and oxygen sites, known as Schottky defects (see Section 1.3.2). 

One of the causes of point defects is a temperature increase which results in an increased thermal movement of the atoms which can subsequently lead to the atoms escaping from their place in the lattice. Other causes are the effects of radiation and inbuilt, foreign atoms. In an atomic lattice a vacancy can occur due to the movement of an atom, an absence of an atom or molecule from a point which it would normally occupy in a crystal. In addition to this vacancy an atomic will form elsewhere. This combination of an atomic pair and a vacancy is called the Frenkel defect. In ionic crystals an anion and a cation have to leave the lattice simultaneously due to the charge balance. As a result a vacancy pair remains and this is called the Schottky defect. Both defects can be seen in figure 4.8. 

There are about 106 schottky pairs per cc at room temperature in NaCl. In one cc of NaCl there are about 1022 ions, so there will be one schottky defect per 1016 ions. 

Gibbs energy of formation of the Schottky defects, R is the gas constant, and T the absolute temperature. Because Ucv is equal to n y and to the number of Schottky defects present, ns ... 

A population of vacancies on one subset of atoms created by displacing some atoms into normally unoccupied interstitial sites constitute a second arrangement of paired point defects, termed Frenkel defects (Figure 2(b), (c)). Because one species of atom or ion is simply being redistributed in the crystal, charge balance is not an issue. A Frenkel defect in a crystal of formula MX consists of one interstitial cation plus one cation vacancy, or one interstitial anion plus one anion vacancy. Equally, a Frenkel defect in a crystal of formula MX2 can consist of one interstitial cation plus one cation vacancy, or one interstitial anion plus one anion vacancy. As with the other point defects, it is found that the free energy of a crystal is lowered by the presence of Frenkel defects and so a popnlation of these intrinsic defects is to be expected at temperatures above 0 K. The calculation of the number of Frenkel defects in a crystal can proceed along lines parallel to those for Schottky defects. The appropriate chemical equilibrium for cation defects is ... 

Schottky defects occur when sites that are normally occupied by atoms or ions are left vacant. In order that the crystal structure maintain its electrical neutrality, for every cation-site vacancy there must be an anion-site vacancy. At room temperatures, one in 10 sites is typically vacant, but this adds up to 10 Schottky defects in a 1 mg crystal. A less commonly observed defect is a Frenkel defect, in which an atom or ion is displaced from its site to an interstitial site that is normally unoccupied. In so doing the number of nearest neighbors of one component of the crystal is changed. This type of defect is seen in... 

Point defects in a pure crystalline substance include vacancies, in which atoms are missing from lattice sites, and interstitials, in which atoms are inserted in sites different from their normal sites. In real crystals, a small fraction of the normal atom sites remain unoccupied. Such vacancies are called Schottky defects, and their concentration depends on temperature ... 

Point defects where atoms are missing from lattice sites are called vacancies, or Schottky defects. Their nnmber density depends on the temperature and on the Gibbs free energy of formation of defects. 

Point (microscopic) defects in contrast from the macroscopic are compatible with the atomic distances between the neighboring atoms. The initial cause of appearance of the point defects in the first place is the local energy fluctuations, owing to the temperature fluctuations. Point defects can be divided into Frenkel defects and Schottky defects, and these often occur in ionic crystals. The former are due to misplacement of ions and vacancies. Charges are balanced in the whole crystal despite the presence of interstitial or extra ions and vacancies. If an atom leaves its site in the lattice (thereby creating a vacancy) and then moves to the surface of the crystal, it becomes a Schottky defect. On the other hand, an atom that vacates its position in the lattice and transfers to an interstitial position in the crystal is known as a Frenkel defect. The formation of a Frenkel defect therefore produces two defects within the lattice—a vacancy and the interstitial defect—while the formation of a Schottky defect leaves only one defect within the lattice, that is, a vacancy. Aside from the formation of Schottky and Frenkel defects, there is a third mechanism by which an intrinsic point defect may be formed, that is, the movement of a surface atom into an interstitial site. Considering the electroneutrality condition for the stoichiometric solid solution, the ratio of mole parts of the anion and cation vacancies is simply defined by the valence of atoms (ions). Therefore, for solid solution M X, the ratio of the anion vacancies is equal to mJn. 

Usually the concentration of Frenkel and Schottky defects is relatively small. The maximum mole part of these defects does not exceed several tenths of a percent. Thus, the electroconductivity of such solid solutions is minimal even at the temperatures close to their melting point. 

The energies of Schottky defects in KF, KC1, KBr and KI have been calculated as 244, 241, 219 and 210 kJ mol-1, respectively. Which halide will have the most defects at room temperature ... 

So far in this chapter, we have assumed implicitly that all the pure substances considered have ideal lattices in which every site is occupied by the correct type of atom or ion. This state appertains only at OK, and above this temperature, lattice defects are always present the energy required to create a defect is more than compensated for by the resulting increase in entropy of the structure. There are various types of lattice defects, but we shall introduce only the Schottky and Erenkel defects. Solid state defects are discussed further in Chapter... 

---