Your question is not exactly precisely defined, but it is clear that you refer to the problem of impurity scattering in semiconductors. So the problem is the problem of electric conductivity through a semiconductor. In semiconductor physics the following relation is used:

\sigma = n e \mu

where \sigma is the conductivity and \mu is the mobility. For the mobility is used:

\mu = e / m_Eff

where n is the charge density, e the electronic charge and m_Eff the effective mass. These expressions are all for one type of charge carrier, here electrons, but they can be expanded. For your question the , the relaxation time is the key parameter. It is part of the Boltzmann transport equation. You have to study that first, but it will turn out, that you need the average relaxation time and for that you have to integrate over k. So that should answer you question about the wave number k. Usually for the relative dielectric constant, the measured value is used and that is typically about 10. For further study use any text about semiconductor physics, like e.g. A.I. Anselm, "Introduction to semiconductor theory" or McKelvey, "Solid State and Semiconductor Physics"

Thanks a lot for your answer C.M.J. Wijers. As you said, I have to integrate over k to get the average relaxation time . What can I do to integrate over k? And in the literature, ‘the latter can be calculated for arbitrary degeneracy’, and I think 'the latter' refers to the wave number k.

You ask two things. First you want to know how to do the integration. This is part of the standard impurity scattering theory, known as Herring-Brooks theory. Key is the assumption of isotropic scattering, so integration over the angles yields 4 \pi and the integration over the modulus k becomes k^2 dk. To do the whole derivation here, will be too much. Secondly you talk about degeneracy. This question is ambiguous. Each state in a bandstructure calculation is defined by a wave vector k and an energy E(k). Let us call this a state. This state itself can be degenerate for reasons of symmetry and in practice the degeneracy can be 1 (general), 2 or 3. Correspondingly there can be 2, 4 or 6 charge carriers, electrons or holes in the state. This is the occupancy. Whether these states will really be occupied is determined by the statistics, the influence of temperature. For semiconductors this is described by Fermi-Dirac. If the difference between FD and Boltzmann-statistics is negligible, the statistics is called degenerate, again.. However, please study the literature. To explain this in detail will take pages.