Summary and Terms

Summary

That vacancies exist wasn't always a widely accepted fact, but with cutting edge techniques we were able to verify their existence in the mid 20th century. The visualization of atomic terraces we showed early in the chapter, coupled with some other very sensitive measurements confirmed that vacancies were indeed real. They are also very important to understanding the behavior of crystalline materials, from joint implants to nuclear reactors.

The presence of vacancies alters the energetics of a crystal by increasing the enthalpy and increasing the configurational entropy. These competing influences result in a temperature at which the change in Gibbs free energy is zero, in other words, equilibrium. Thus, we expect all crystals above 0K to have some number of vacancies present, and we also expect this number of vacancies to be related to the temperature of the crystal in some way. We can describe the concentration of vacancies using an Arrhenius behavior, provided we know the activation energy required to form a vacancy and the number of sites where a vacancy could form (effectively the concentration of lattice atoms in the crystal). This is incredibly powerful - with a little thermodynamics we can model the behavior of vacancies in an incredible variety of crystalline systems, something that will come in very handy when we model phenomena like vacancy-mediated diffusion.

Terms

Gibb's Free Energy: The change in Gibbs Free energy ($\Delta G$) for a reaction or process describes whether, under those conditions, the process will occur spontaneously. $\Delta G$ has two contributions and is given by:

$\Delta G = \Delta H - T \Delta S$

The first contribution is enthalpic (relating to the bonding of the system in our simplified systems) while the second is entropic (relating to the amount of disorder in the system). Spontaneous reactions have a negative $\Delta G$ while $\Delta G$ is zero for systems at equilibrium. Section 8.6.2

Enthalpy $\Delta H$: For Crystals, a measure of the internal energy due to bond-formation. If bonds are broken, enthalpy should go up, while forming bonds will lower it. We thus expect the formation of a vacancy, which by definition breaks bonds in a crystal, to increase the enthalpy of the system. Section 8.6.3

Configurational Entropy $\Delta S$: For the purposes of this class, we can think of $\Delta S$ as related to the number of available microstates - that is, the number of unique ways the system can be arranged. Adding vacancies allows for more possible arrangements, so vacancy formation increases the configurational entropy of the system (at least until we get to half of our sites being vacancies, by which point the crystal will certainly have fallen apart already). Section 8.6.3

Equilibrium: a system state which has $\Delta G = 0$. In the case of vacancies, this describes a state in which vacancies are created exactly as fast as they annihilate, leading to no net change in their concentration. Section 8.6.5

Arrhenius Behavior: A useful exponential equation which gives us information about the speed and direction of a reaction. For the purposes of describing vacancy concentration, the relevant Arrhenius behavior is given by:

$N_{\mathrm{v}}= N\exp{\frac{-Q_{\mathrm{v}}}{k_{\mathrm{B}}T}}$

This equation is powerful in that it can predict the concentration of vacancies in a crystal at any temperature provided we know the activation energy for vacancy creation, $Q_{\mathrm{v}}$, and the number of possible sites for vacancy formation $N$. Section 8.7.5