Using the data given below that relate to the formation of Schottky defects in some oxide ceramic (having the chemical formula \(\mathrm{MO}\) ), determine the following: (a) The energy for defect formation (in eV), (b) the equilibrium number of Schottky defects per cubic meter at \(1000^{\circ} \mathrm{C},\) and (c) the identity of the oxide (i.e., what is the metal M?) $$\begin{array}{rcc} \hline \boldsymbol{T}\left(^{\circ} \boldsymbol{C}\right) & \boldsymbol{\rho}\left(\boldsymbol{g} / \mathrm{cm}^{3}\right) & \boldsymbol{N}_{\boldsymbol{s}}\left(\boldsymbol{m}^{-3}\right) \\ \hline 750 & 3.50 & 5.7 \times 10^{9} \\ 1000 & 3.45 & ? \\ 1500 & 3.40 & 5.8 \times 10^{17} \\ \hline \end{array}$$

The defect formation energy is a critical concept in the study of crystallography and materials science. It refers to the amount of energy required to create a defect in a crystalline structure. In the context of Schottky defects in oxide ceramics, it specifically relates to the energy needed to remove a pair of oppositely charged ions from their positions in the lattice, creating vacancies.

According to statistical thermodynamics, the probability of defect formation is not random but depends on the temperature and the intrinsic defect formation energy of the material. The relationship between the number of defects and the formation energy is expressed by the Arrhenius equation, which shows that as the temperature increases, the number of defects tends to increase as well due to the higher thermal energy available to overcome the formation energy barrier.

The exercise involves calculating this energy using the proportionality of the equilibrium number of Schottky defects at two different temperatures, which are given in the exercise. It's an application of the Arrhenius equation, where a plot of the natural logarithm of the number of defects versus the reciprocal of the temperature would yield a straight line. The slope of this line is related to the defect formation energy. Understanding how to extract and use this value is vital in predicting how a material’s properties may change with temperature.

The equilibrium number of defects in a crystalline material is the number of defects that naturally exist within a crystalline lattice at a given temperature. At equilibrium, the rate at which defects are formed is equal to the rate at which they are eliminated. This concept stems from the dynamic equilibrium set in a crystalline lattice, where defects are continuously generated and recombined.

The equilibrium number of defects increases with the temperature, given the increased thermal energy that allows more atoms or ions to leave their lattice sites. In the exercise, the calculation of the equilibrium number of Schottky defects at a specific temperature, such as 1000°C, requires knowledge of the defect formation energy, the Boltzmann constant, and the absolute temperatures involved. Simplified, the equation showing this relationship can be used to predict the number of defects at any temperature once the defect formation energy is known.

Understanding the factors that impact the equilibrium number of defects is crucial for designing materials with specific properties. For example, in oxide ceramics, the number of defects can significantly affect electrical conductivity, mechanical strength, and diffusion characteristics.

Oxide ceramics are a class of materials known for their high temperature stability, chemical inertness, and electrical properties, which make them suitable for a wide range of applications, including electronics, refractories, and as structural materials. The term ‘oxide ceramics’ usually refers to compounds composed of metal ions and oxygen, often forming part of a crystalline lattice.

Schottky defects are a common feature in oxide ceramics, where they can influence material properties. These defects occur when pairs of cations and anions are missing from the structure, leading to a stoichiometrically balanced but less dense material. The occurrence of Schottky defects is intrinsic to these materials and depends on factors such as temperature, as shown in the exercise.

Understanding the formation and behavior of Schottky defects helps scientists and engineers tailor the properties of oxide ceramics for specific applications. For example, controlling defect concentrations can modify the electrical conductivity of semiconducting ceramics or improve the ionic conductivity in solid oxide fuel cells.