In the RTA model for thermal conductivity calculation, N-processes are treated the same as other resistive processes. However, it is well known that N-proceses conserve momentum and do not pose resistance to the heat transport directly; they only contribute to the resistance indirectly by scattering the low-frequency phonons into the high frequency region, where the phonon group velocity is much smaller and Umklapp scattering rate is much higher. Correction to the RTA thermal conductivity is needed, especally at low temperatures.

   Callaway has proposed the expression for the correction of thermal conductivity due to the non-resistive nature of N-processes half a century ago. However, his original model was based on many simplifications including no dispersion, no optical phonons and no differentiation between LA and TA branches.

   Here I rederive his model in full dispersion and propose the expressions of thermal conductivity as:

with the constant u as: