A1 An Effective Medium Approximation

The crystal defects that we consider here are mainly of microscopic size ranging from single atoms to

clusters of several hundreds of atomic defects, and the volume fraction they occupy ranges from a part per million to perhaps a tenth of a percent. So if p is the typical radius of a defect under consideration and 2R the average distance between them, then the defect volume fraction

S = (p/R)3 [A1]

is a small number. As a result, each defect is sur­rounded by a cell that contains no other defect, and the solid can be viewed as composed of these cells. If the solid as a whole is free of external stresses on its macroscopic surface, then the normal stress component averaged over the surface of each cell is also zero, since on the cell boundary, the stress fields of two neighboring defects overlap and cancel on average. In the final step in this cellular approach, we approximate the typical cell by a spherical region with radius R with a defect of radius p at its center. It is this effective medium approximation that elevates the analysis of a sphere with a defect at its center to more than just an academic exercise, and its results are more generally valid for defects in solids of finite extent.

Atomistic computer simulations using either semi-empirical potentials or first-principle methods are usually carried out in a periodically repeated cell that contains a finite number of atoms. However, in this case the medium is in fact infinite, and the results are different from those obtained for a finite crystal on whose external surface the stress field created by the defect satisfies the boundary condition of zero tractions, that is, the vanishing of the stress compo­nent normal to the external surface. Satisfying this boundary condition is different from having a defect stress field that approaches zero at infinity. This difference is often attributed to the so-called image stresses that exist in a finite crystal, but not in an infinite one. The model of a defect at the center of a finite sphere ofradius R captures this difference, and as we shall see, this difference becomes independent of the radius R as it approaches infinity but such that S, the defect concentration, remains small but constant.