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98), the frequency dispersion and the defept lattice Green's function are required for evaluation. Similar relations can be written for the other correlation functions. In the equal-time case, it is easily shown that <«a(kH(/v)> = -β-1Σ υχβ^ικ, - ηή (99) 42 WILBUR M. FRANKLIN where Ω„ = 2nn/ßh. Thus, evaluation of the correlation functions is reduced to evaluation of the elements of the Green's function matrix for the defect crystal, which may be done without deriving the eigenvectors. An additional application of lattice vibration theory to the theory of diffusion is that of the scattering of phonons by defects.

Consequently, the aspects of the independent defects only can be discussed. Since the isolated vacancy problem is applicable to self-diffusion, the results of the solution of the determinantal equation for the frequencies of three-dimensional lattices are useful to diffusion. Ludwig (1966) found that a vacancy in a SC lattice does not show localized vibrations. Land and Goodman (1967) got a similar result for Fe and Na but did find a weak localized mode (frequency slightly above the band modes) in Cu.

The detailed calculations for realistic three-dimensional models are highly mathematical and are often too complex to solve in a reasonable amount of time with any degree of precision using contemporary techniques. However, calculations have been done on defect systems in which results have been obtained which are directly applicable to diffusion. These include calculations of the defect 1. CLASSICAL AND QUANTUM THEORY OF DIFFUSION 33 modes at vacancies and interstitials in three-dimensional systems.

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