Accurate and approximate results for the time correlation function and memory function associated with tracer diffusion in a pure crystal

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Abstract
The time correlation function associated with the tracer diffusion coefficient for the vacaacy mechanism in a pure simple cubic substance has been calculated accurately from an integral equation in time derived from the kinetic equation for tracer migration. Results for different tracer to host jump frequency ratios are compared with approximate solutions valid at short and long times. The memory function of the time correlation function has been calculated. Simple approximations to the memory function and the time correlation function generated from the Mori continued fraction representation of the memory function are tested. The Fourier transform of the time correlation function, which is proportional to the frequency dependent part of the frequency dependent diffusion coefficient, is also described.