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Chapter 12: Problem 35

What point defects are possible for \(\mathrm{Al}_{2} \mathrm{O}_{3}\) as an impurity in \(\mathrm{MgO}\) ? How many \(\mathrm{Al}^{3+}\) ions must be added to form each of these defects?

Short Answer

Expert verified
Short Answer: When Al₂O₃ is introduced as an impurity in MgO, the possible point defects that can occur are Schottky defects and Frenkel defects. In Schottky defects, 2 Al³⁺ ions replace 4 Mg²⁺ ions while maintaining charge neutrality, whereas in Frenkel defects, 1 Al³⁺ ion replaces 1 Mg²⁺ ion while maintaining charge neutrality.

Step by step solution

01

Identify the ions in the lattice

First, let's identify the ions present in the crystal lattice of the undoped \(\mathrm{MgO}\) compound. In \(\mathrm{MgO}\), we have \(\text{Mg}^{2+}\) and \(\text{O}^{2-}\) ions. When we introduce \(\mathrm{Al}_{2}\mathrm{O}_{3}\) as an impurity, we have additional \(\mathrm{Al}^{3+}\) and \(\mathrm{O}^{2-}\) ions.
02

Analyze the charge balance in the lattice

Since \(\mathrm{Al}^{3+}\) ions have a higher positive charge than the \(\text{Mg}^{2+}\) ions, charge balance must be maintained in the lattice when these ions are introduced. This can be achieved by the formation of point defects, which include vacancies and interstitials. These defects maintain charge neutrality when impurities are introduced.
03

Determine possible defect formations

There are two possible defect formations when \(\mathrm{Al}^{3+}\) ions are added to \(\mathrm{MgO}\): 1. Schottky defect: This defect formation involves the replacement of two \(\text{Mg}^{2+}\) ions with one \(\mathrm{Al}^{3+}\) ion while maintaining charge neutrality. In this case, we need 2 \(\mathrm{Al}^{3+}\) ions to replace 4 \(\text{Mg}^{2+}\) ions, along with 2 vacancies. 2. Frenkel defect: This defect involves an \(\mathrm{Al}^{3+}\) ion taking an interstitial site while an \(\text{Mg}^{2+}\) ion leaves its lattice site and forms a vacancy. In this case, we need 1 \(\mathrm{Al}^{3+}\) ion to replace 1 \(\text{Mg}^{2+}\) ion while maintaining charge neutrality.
04

Calculate the number of \(\mathrm{Al}^{3+}\) ions needed for each defect

For each of the possible defects: 1. Schottky defect: We need 2 \(\mathrm{Al}^{3+}\) ions to maintain charge balance while replacing 4 \(\text{Mg}^{2+}\) ions. 2. Frenkel defect: We need 1 \(\mathrm{Al}^{3+}\) ion to maintain charge balance while replacing 1 \(\text{Mg}^{2+}\) ion. In conclusion, when \(\mathrm{Al}_{2}\mathrm{O}_{3}\) is introduced as an impurity in \(\mathrm{MgO}\), there are two possible point defects: Schottky defects and Frenkel defects. For Schottky defects, we need 2 \(\mathrm{Al}^{3+}\) ions for every 4 \(\text{Mg}^{2+}\) ions that are replaced, and for Frenkel defects, we need 1 \(\mathrm{Al}^{3+}\) ion for every 1 \(\text{Mg}^{2+}\) ion that is replaced.

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Most popular questions from this chapter

A circular specimen of \(\mathrm{MgO}\) is loaded using a three-point bending mode. Compute the minimum possible radius of the specimen without fracture, given that the applied load is \(425 \mathrm{~N}\left(95.5 \mathrm{lb}_{\mathrm{f}}\right)\), the flexural strength is \(105 \mathrm{MPa}(15,000 \mathrm{psi})\), and the separation between load points is \(50 \mathrm{~mm}\) (2.0 in.).

Would you expect Frenkel defects for anions to exist in ionic ceramics in relatively large concentrations? Why or why not?

For each of the following crystal structures, represent the indicated plane in the manner. of Figures \(3.11\) and \(3.12\), showing both anions and cations: (a) (100) plane for the rock salt crystal structure (b) \((110)\) plane for the cesium chloride crystal structure (c) (111) plane for the zinc blende crystal structure (d) (110) plane for the perovskite crystal structure

The modulus of elasticity for beryllium oxide (BeO) having 5 vol\% porosity is 310 GPa \(\left(45 \times 10^{6} \mathrm{psi}\right)\) (a) Compute the modulus of elasticity for the nonporous material. (b) Compute the modulus of elasticity for 10 vol \% porosity.

The flexural strength and associated volume fraction porosity for two specimens of the same ceramic material are as follows: \begin{tabular}{cc} \hline\(\sigma_{f_{s}}(M P a)\) & \(P\) \\ \hline 100 & \(0.05\) \\ 50 & \(0.20\) \\ \hline \end{tabular} (a) Compute the flexural strength for a completely nonporous specimen of this material. (b) Compute the flexural strength for a \(0.10\) volume fraction porosity.
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