To formulate quantitative or often even qualitative models for defect processes in materials, it is essential that lattice relaxation be effected. Without lattice relaxation, the total energies calculated for defect reactions would be so great that we would have to conclude that no point defects would ever form in the material.9

Each defect has an associated defect volume. That is, each defect, when introduced into the lattice, causes a distortion in its surroundings, which is
manifested as a volume change arising as a result of the way in which the lattice responds, that is, how the lattice ions relax around the defect. For example, vacancies in ionic materials usually result in positive defect volumes. Consider the example of a vacancy in MgO. The nearest neighbor cations are displaced outward, away from the vacant site, causing an increase in volume, whereas the second neighbor anions move inward, albeit to a lesser extent (see Figure 8).

What drives these ion relaxations is the change in the Coulombic interactions due to defect formation. We say that the oxygen vacancy carries an effective positive charge because an O2~ has been removed and thus, an electrostatic attraction between O2~ and Mg2+ is removed. As ionic forces are balanced in a crystal, the outer O2~ ions now attract the Mg2+ away from the VO* defect site. In covalent materials, vacant sites result in atomic relaxations that are due to the formation of an incomplete complement of bonds, often termed ‘dangling bonds.’ In this case, the net result can be different from those in ionic solids and by way of an example, in silicon, a vacancy results in a volume decrease. On the other hand, an arsenic substitutional atom causes an increase in vol — ume.10 Finally, in a material such as ZrN or TiN, which exhibits both covalent and metallic bonding, the volume of a nitrogen vacancy is practically zero.1

Clearly, the overall response of the lattice can be rather complicated. However, the defect volume can be determined fairly easily by applying the rela­tionship

Figure 8 Schematic of the lattice relaxations around an oxygen vacancy in MgO.

where uP is the defect volume (in A3); KT the iso­thermal compressibility (in eV/A3); V the volume of the unit cell (in A3); andfy the Helmholz free energy of formation of the defect (in eV).

Finally, defect associations can also (but not nec­essarily) have a significant effect on defect volumes for a given solution reaction. For example, for the Al2O3 solution in MgO if we assume isolated AlMg and VMMg has the least effect on lattice parameter as a function of AlMg whereas the formation of neutral AlMg • VMMgX has the greatest effect (in fact ten times the reduction in lattice parameter).12