Approximate mean free path relaxation time

The Fermi surface is assumed to be spherical. In the above equations, is the Fermi wave vector, / is the electron mean free path, m is the electron mass and x is the relaxation time, x = ml/Pikp. As the disorder increases, more and more states get localized and Ec and Ec move toward the centers of the respective bands. The mean free path (/) also decreases and in the limit, I = a which is the lattice distance (loffe-Regel limit). The conductivity also reaches the limit and is e 3nha), since kfl becomes approximately equal to I. Introduction of any further disorder only broadens the band and does not affect /, it alters N ( ). The minimum metallic conductivity, csm (all a values like am, , a, afO) etc. refer only to d.c. conductivities in this chapter the subscript d.c. is dropped to make the notation less cumbersome. A.c. conductivities will be referred to as cr(eo)), before the disorder localizes all the states and the conductivity drops to zero for the three dimensional problem may be approximated as 2... 

For an approximate estimate of the mobilities, either a constant, isotropic relaxation time r or a constant, isotropic mean free path X can be assumed. In these two approximations, the components pty of the mobility tensor are found from the Einstein relation (Eq. (8.22)) to be... 

The above-defined approximate relaxation times can be used to evaluate the mean free path of the charge carriers, averaging over the states of a band using Boltzmann statistics. 

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