Defect equations

Observe that this is a geometric property, not to be confused with the modulus of the material, which is a material property. I, c, Z, and the cross-sectional areas of some common cross-sections are given in Fig. 3-1, and the mechanical engineering handbooks provide many more. The maximum stress and defection equations for some common beamloading and support geometries are given in Fig. 3-2. Note that for the T- and U-shaped sections in Fig. 3-1 the distance from the neutral surface is not the same for the top and bottom of the beam. It may occasionally be desirable to determine the maximum stress on the other nonneutral surface, particularly if it is in tension. For this reason, Z is provided for these two sections. 

Additionally, we can have at least four (4) types of charged vacancies and interstitial sites as given in 3.2.2. (Charged surface sites are not common but are included here for the sake of completeness). This gives rise to 12 more charge defect equations or a total of 42 defect equations that we can write ... 

In the heterogeneous solid, a different mechanism concerning charge dominates. If there are associated vacancies, a different type of electronic defect, called the "M-center", prevails. In this case, a mechanism similcu to that already given for F-centers operates, except that two (2) electrons occupy neighboring sites in the crystal. The defect equation for formation of the M-center is ... 

We have given defect-equations for edl nine types of defects, and the Equilibrium Constants thereby associated. However, calculation of these equilibria would require values in terms of energy at each site, values which are difficult to determine. A better method is to convert these EC equations to those involving numbers of each Qrpe of intrinsic defect, as a ratio to an intrinsic cation or einion. This would allow us to calculate the actual number of intrinsic defects present in the crystal, at a specified temperature. 

We have shown that defect equations and equilibria can be written for the MXs compound, both for stoichiometric and non-... 

The principles described above apply equally well to oxides with more complex formulas. In these materials, however, there are generally a number of different cations or anions present. Generally, only one of the ionic species will be affected by the defect forming reaction while (ideally) others will remain unaltered. The reactant, on the other hand, can be introduced into any of the suitable ion sites. This leads to a certain amount of complexity in writing the defect equations that apply. The simplest way to bypass this difficulty is to decompose the complex oxide into its major components and treat these separately. Two examples, using the perovskite structure, can illustrate this. 

The oxides with general formulas A20, AO, A203, A02, and A2Os are doped into MgO so that the cation substitutes for Mg. Write general defect equations for the reactions, assuming cation vacancies rather than electronic compensation occurs. 

Write the defect equations and the ionic formula of the phases produced by doping of BaTiO PCT material with (a) Y3+ as a Ba2+ substituent and (b) Nb5+ as a Ti4+ substituent. 

The average valance of the Ti component can be pushed toward 4.0 by valence induction, simply by increasing the amount of Li present relative to that of titanium, to give a formula Li1+xTi2 x04. This is equivalent to acceptor doping of the phase. The additional Li+ ions enter the octahedral sites, as the tetrahedral sites are already occupied, to a maximum of Li [Li (nTi nJCL and the acceptor doping is balanced by holes. In order to write formal defect equations for this situation, it is convenient... 

Write defect equations for (a) TiC>2 doped with A1203 (b) Ti02 doped with Nb2Os (c) Mn304 doped with NiO (d) MgO doped with Cr203 and (e) M Os doped with Fe203. 

LaCo03 can be donor doped with Ce02 to give a mixed conductor, (a) Write defect equations for the reaction. A sample of nominal composition Lao.99Ce0.oiCo03 has a Seebeck coefficient of approximately — 300 pVK-1. 

Now that we know how to write defect equations, let s look at Frenkel and Schottky defects in more detail. 

Kroger-Vink notation is used for defect equations). 

The defect equation for formation of the M-center is also given as follows ... 

Defect Type Defect Pair Exam pie Defect Equation Equilib. Constant... 

We have given defect-equations for all nine types of defects, and the Equilibrium Constant (EC) thereby associated. However, calculation of these equilibria would require values in terms of energy at each site, values which are sometimes difficult to determine. 

Note the various mechanisms which give rise to the specific combinations of defects. These mechanisms have been thoroughly studied as a function of specific compounds. It is sufficient for us, at this point, to observe which defect equations govern both the equilibria and the nonstoichiometry of the general compound, MXs 6. 

Here, the defect equations for our MX compound (containing Schottky defects) are plotted for the case Kion Ksh. In obtaining this plot, we have derived the following equations from equations 2.7.1. and 2.7.2. ... 

Byrewriting the equilibrium constants of Table 2-1 and 2.7.20. (ionization of vacancies) as logarithms, we obtained linear relations among the set of defect equations. 

We have already discussed the high-pressure measurements of Chan et al. (1965) which show that the static conductivity dielectric relaxation time t is controlled by orientational defects. Equations (9.32) and (9.35) therefore lead us to the conclusion that, for pure ice, (T 4. ctjjl. This is borne out by the work on ice crystals containing impurities which we shall discuss in the next section, and indeed that work led to this conclusion several years before the high-pressure results were available. 

The presence of a small number of Frenkel defects reduces the Gibbs energy of a crystal and so Frenkel defects are intrinsic defects. The formula for the equilibrium concentration of Frenkel defects in a crystal is similar to that for Schottky defects. There is one small difference compared with the Schottky defect equations the number of interstitial positions that are available to a displaced ion, N, need not be the same as the number of normally occupied positions, N, from which the ion moves. The number of Frenkel defects, np. present in a crystal of formula MX at equilibrium is given by ... 

Calculations based on the above four defect equations prove the oxygen pressure dependence of the electrical conductivity to be Pq, Pq and respectively. 

The point defect chemistry of ZnO is different partly because there are two charge states of Zn, namely 2+ (as in ZnO) and 1+. When we heat ZnO in Zn vapor, we form a Zn-rich oxide, Zni+jf). Experimentally it is found that the excess Zn sits on interstitial sites (as you would expect from the crystal structure). We can write the basic defect equation for the singly charged interstitial as... 

Doping Zr02 with Ca is a special example of a Zr02 solid solution. We can incorporate -15% CaO in the structure to form Ca-stabilized cubic zirconia (CSZ). (CZ is the general abbreviation for cubic zirconia.) The special feature here is that the cubic (fluorite structure) phase is not stable at room temperature unless the Zr02 is heavily doped. However, we write the point defect equations as if it were always stable. The Ca " cation substitutes for the Zr" cation as shown in Figure 11.5. Since the charges are different, we must compensate with other point defects. 

The exponent P < 1 depends for example on the swelling degree and/or on the number of network defects. Equation (21) resolves the contradictions observed for swollen or imperfect networks (see Sect. 5.3). The mechanisms leading to Eq. (21) are not yet eompletely dear. Some hints come from a paper of Gordon and Scho-walter who obtained relation (21) by treating elements sliding relative to deformed surroundings. 

The hydration process in all proton conducting perovskites can be formally written as a defect equation ... 

It is very difficult to separate the broadening of a reflection into components due to small crystallite size and to structural defects. Equation (65) implies that PcosO should be constant for all glancing angles, if the broadening results entirely from small crystallite size. Because L is the dimension peipendicular to the diffracting net planes, this statement cannot hold rigorously, except for the diffraction orders of one and the same net plane. If line broadening is the consequence of crystal structure defects alone, the width increases with 0. 

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