16. Point Defects

So it has the same units as energy; but while the energy of formation is the heat evolved when a reaction is run at constant volume, enthalpy takes into account that work must be performed against an outer pressure. Enthalpy is the total energy of a system that can be converted into heat, and the enthalpy of formation is the heat evolved during a chemical reaction at constant pressure. Accordingly, the difference between ΔE and ΔH is small for reactions involving only liquids and solids because there is little if any change in volume during such reactions; however, the difference can be significant for reactions that involve gases and/or high pressures, especially if they involve a change in the amount of gas. The sign of ΔH determines whether a reaction is exothermic (ΔH < 0) or endothermic (ΔH > 0). The Gibbs free energy, G, is the total energy available to perform thermodynamic work, and the sign of ΔG determines the spontaneity of a reaction (a reaction is spontaneous when ΔG < 0).

Theoretically, this means that at absolute zero there would be no defects, as ΔH and therefore ΔG would always be positive; however, thermodynamics and kinetics are two different things! Even though a defect may not be thermodynamically stable at very low temperatures, the kinetics of diffusion which would be required to remove it might be so slow at those temperatures that the defect is effectively “frozen in.”

ln x! = x ln x – x (valid for large x).

It is left to the reader to show that \( {\updelta}^2G/\updelta {n}^2=\frac{knT}{n\left(n+N\right)}>0 \).

Equation 16.13 is often incorrectly rewritten as \( \ln \left[\mathrm{V}\right]=\left(\frac{-\Delta {g}_f}{k}\right)\frac{1}{T} \), where Δg_f is the energy of vacancy formation (Δg_f ~ 10 eV for ionic oxides), in which case the intercept goes to 0 and the slope of the ln[V] vs 1/T line would be related to the energy of formation rather than the enthalpy of formation. The source of the error is the implicit assumption that Δs_vib = Δs, which neglects the considerable contribution of Δs_config.

“Dope” comes from the Latin dotare (to endow) and “doping” refers to the practice of deliberately adding impurities (dopants), either additively or substitutionally, to affect the structure and thereby properties of a material. It has been used in this sense since at least 1898. In the mid-1950s, a narrower definition was introduced to refer exclusively to the practice of introducing very low concentrations of impurities to semiconductors in order to produce a desired electrical characteristic.

The Wigner effect was a cause of the Windscale fire of 10 October 1957, which remains the UK’s worst nuclear accident. It burned for 3 days at the plant near Seascale on the Cumbrian coast and released radioactive fallout across Europe. For the next month, milk from the nearby countryside had to be diluted and destroyed. The disaster made Windscale a target of hate for opponents of nuclear energy and prompted British Nuclear Fuels (BNFL) to change the plant’s name to Sellafield in 1981. Electricity production finally ceased at Sellafield in 2003, but fuel reprocessing, nuclear-waste storage, and nuclear decommissioning activities are still carried out there. The Nuclear Decommissioning Authority has been overseeing the gradual closure of the site since 2005, but the UK’s National Nuclear Laboratory still maintains its Central Laboratory and headquarters there. It should be noted that current operational reactors operate well above 250°C and so are no longer susceptible to disasters like the one at Windscale.

In Al³⁺, there are no partially filled energy levels or orbitals (the electron configuration of Ar³⁺ = [Ne]), so no light absorption is possible. For this reason, pure α-Al₂O₃ (sapphire) is transparent. On the other hand, in Cr³⁺, there are partially filled orbitals (the electron configuration is [Ar]3d³) and the 3d electrons can be excited and so cause light absorption. The frequency (and so energy) of the photons absorbed is increased with compression, resulting in observed colours with decreased frequencies, i.e., red.

For the chemically inquisitive, naphthopyran is any organic heterotricyclic compound with a skeleton consisting of a pyran ring ortho-fused to a naphthalene.

A symmetric tensor is one which is equal to its own transpose, A = A^T.