Spatial oscillations in strain fields due to paraelastic defects

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Abstract
Analytic expressions based on a phonon rather than an elastic continuum treatment are given for lattice relaxation displacements and strains in the vicinity of paraelastic defects in cubic crystals. Relaxation energies are also given in closed form. The approximations involved in deriving these results are the use of the long-wave strain limiting form of the defect-phonon interaction and the use of a Debye model for the phonons as well as a modified Debye model with appropriate Van Hove singularity at the Debye cutoff. The elastic continuum limit in which the Debye frequency is allowed to go to infinity is investigated. Only the modified Debye model gives sensible results in this limit. The analytic expressions can be evaluated in terms of host-crystal parameters and measured stress-coupling parameters of the defect. Displacements and energies are calculated for nine different defect systems. It is found that the modified Debye model predicts defect interactions which are of longer range than does the elastic continuum model and that these interactions show a spatially oscillatory character at large defect separations.