  We investigate the effects of collective modes on the temperature relaxation
rates in fully coupled electron-ion systems. Firstly, the well-understood limit
of weakly coupled plasmas is considered and the coupled mode formula within the
random phase approximation is derived starting from the Lenard-Balescu kinetic
equation. We show how the frequency integration can be performed by standard
methods without applying additional approximations. Due to the well-defined
approximation scheme, the results can serve as a benchmark for more approximate
theories and numerical simulations in this limit. The coupled mode electron-ion
transfer rates show a considerable reduction compared to the Fermi-Golden Rule
approach for certain parameters and very small changes for other systems. We
demonstrate how these coupled mode effects are connected to the occurrence of
ion acoustic modes and under which conditions they occur. Interestingly,
coupled mode effects can also occur for plasmas with very high electron
temperatures; a regime, where the Landau-Spitzer approach is believed to give
accurate results. Finally, we extend the approach to systems with strongly
coupled ions by applying static local field corrections. This extension can
substantially increase the coupled mode effects.
