Abstract
A sum rule is derived for the hopping conductivity ( sigma ( omega )) in the rate equation formalism. It is shown that the pair approximation to ( sigma ( omega )) satisfies the sum rule exactly in the limit of low site densities. Detailed comparison of the exact high-frequency expansion of ( sigma ( omega )) with that derived from the pair approximation shows that the pair approximation becomes exact for all omega in the limit of low site densities. The real and imaginary parts of ( sigma ( omega )) are calculated numerically in this limit on the basis of model assumptions about the site statistics and the intersite transition rates which are appropriate to doped crystalline semiconductors. The magnitude of the ratio of the imaginary to the real part of ( sigma ( omega )) is shown to be 0.293 lg ( omega tau 0)-1 in this case where tau 0 is the minimum pair relaxation time.