Microscopic Theories of Point Defects

An alternative perspective on the subject of point defects to the continuum analysis advanced above is offered by atomic-level analysis. Perhaps the simplest microscopic model of point defect formation is that of the formation energy for vacancies within a pair potential description of the total energy. This calculation is revealing in two respects first, it illustrates the conceptual basis for evaluating the vacancy formation energy, even within schemes that are energetically more accurate. Secondly, it reveals additional conceptual shortcomings associated with 

If a vacancy is created in an otherwise perfect lattice, the implication is that an additional atom must now occupy the surface of the crystal. The basic idea behind the vacancy formation energy is that it is a measure of the difference in energy between two states one being the perfect crystal and the other being that in which an atom has been plucked from the bulk of the crystal and attached to the surface. From a computational perspective, the vacancy formation energy may be defined as 

If we now form the difference between the energy of the perfect lattice and that of the defected lattice, we are left with 

What is remarkable about this result is that we have found that the vacancy formation energy is with opposite sign equal to the cohesive energy per atom. An assessment of the accuracy of this conclusion is given in table 7.1. We note 

We may again adopt a simple near-neighbor model to elucidate the fundamental ideas associated with a pair functional treatment of the vacancy. As noted earlier, within a pair functional description of the total energy, the energy of the perfect fee crystal is given by 

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