  The epoch of reionization (EoR) marks the end of the Cosmic Dawn and the
beginning of large-scale structure formation in the universe. The impulsive
ionization fronts (I-fronts) heat and ionize the gas within the reionization
bubbles in the intergalactic medium (IGM). The temperature during this process
is a key yet uncertain ingredient in current models. Typically, reionization
simulations assume that all baryonic species are in instantaneous thermal
equilibrium with each other during the passage of an I-front. Here we present a
new model of the temperature evolution for the ionization front by studying
non-equilibrium effects. In particular, we include the energy transfer between
major baryon species ($e^{-}$, \HI, \HII, \HeI, and \HeII) and investigate
their impacts on the post-ionization front temperature $T_{\mathrm{re}}$. For a
better step-size control when solving the stiff equations, we implement an
implicit method and construct an energy transfer rate matrix. We find that the
assumption of equilibration is valid for a low-speed ionization front
($\lessapprox\ 10^9~\mathrm{cm}/\mathrm{s}$), but deviations from equilibrium
occur for faster fronts. The post-front temperature $T_{\mathrm{re}}$ is lower
by up to 19.7\% (at $3\times 10^9$ cm/s) or 30.8\% (at $10^{10}$ cm/s) relative
to the equilibrium case.
