Volume changes and dipole tensors for point defects in crystals

Results for the volume change and dipole tensor for point defects are derived using a generalisation of the Betti reciprocity theorem. These results incorporate the correction terms derived by Flynn, by Lidiard and by Gillan in a simpler way, and allow some more general results to be obtained straightforwardly. Calculations of volume changes caused by defects in solids often use an expression involving the virial of the defect forces (Hardy 1968, Temkin 1970): axes sites with s the elastic compliance, and FI the defect force on the neighbouring ion at site Ri . The expression has, however, been consistently unsuccessful in its predictions. Whilst some of the weaknesses can be shown to result from inadequacies in working approximations commonly made in evaluating the virial (Schober and Ingle 1980, Gillan 1983), the main problem is, in fact, an inconsistency in the expression itself. Whilst this point is comprehensively discussed by Gillan, it proves both useful and revealing to give a re-analysis parallel to the elegant discussion of Temkin. Temkin exploited the so-called Betti reciprocity theorem (for fuller references see Stoneham 1975). We cannot use the same theorem, for it relies on an assumption of a harmonic host, and so it specifically excludes some of the critical corrections noted by Flynn (1971), Schober and Ingle (1980), Lidiard (1981) and Gillan (1983). Fortunately, we can exploit the central idea used in the proof of the Betti theorem, and hence we can produce a generalised theorem as a by-product. The central thesis is this: the final state reached by applying two sets of forces F1 and F2 to a solid is independent of their order of application. In our case, the forces F 1 will be the defect forces, and the forces F2 will correspond to an external pressure p . The two sequences of application of these forces are shown schematically in figure 1, together with the displacements (represented in the diagram as a scalar, essentially a projection of the displacements corresponding to the volume change) to which they give rise. We are free to choose the pressurep (i.e. F 2 ) such that XII %= X I , XIII. Quantitatively, this requires a pressure p producing a volume change for the whole crystal which greatly exceeds the volume change due to a single defect, i.e. a fractional volume change in excess of lo-** typically. Our final result will be independent @ 1983 The Institute of Physics L925 L926 Letter to the Editor