  The accurate evaluation of the second-order Doppler frequency shift (SODFS)
of trapped ions in a linear Paul trap has been studied with experiments and
molecular dynamics (MD) simulations. The motion of trapped ions in the trap has
three contributions, and we focus on the ion excess micromotion, which is
rarely discussed when evaluating the SODFS. Based on the hypothesis that the
ion density is uniformly distributed in the radial direction, we propose a new
model to accurately evaluate the total SODFS for ion microwave clocks. The
effectiveness of the model has been verified both in simulation and experiment,
especially for ion ensemble with temperature less than 100 mK. We believe that
our new model offers advantages in accurately evaluating the SODFS for the ion
trap, especially those of laser-cooled ion microwave clocks based on large ion
clouds.
