Collision cross sections for the thermalization of cold gases

The collision cross section that controls thermalization of gas mixtures is the transport cross section $\sigma_\eta^{(1)}$ and not the elastic cross section $\sigma_{\rm el}$. The two are the same for pure s-wave scattering but not when higher partial waves contribute. We investigate the differences between them for prototype atomic mixtures and show that the distinction is important at energies above 100 $\mu$K for LiYb and 3 $\mu$K for RbYb and RbCs. For simple systems both $\sigma_\eta^{(1)}$ and $\sigma_{\rm el}$ follow universal energy dependence that depends only on the s-wave scattering length when expressed in reduced length and energy units.