  We examine the temperature structure of the IGM due to the passage of
individual ionisation fronts using a radiative transfer (RT) code coupled to a
particle-mesh (PM) N-body code. Multiple simulations were performed with
different spectra of ionising radiation: a power law (goes as nu^{-0.5}),
miniquasar, starburst, and a time-varying spectrum that evolves from a
starburst spectrum to a power law. The RT is sufficiently resolved in time and
space to correctly model both the ionisation state and the temperature across
the ionisation front. We find the post-ionisation temperature of the reionised
intergalactic medium (IGM) is sensitive to the spectrum of the source of
ionising radiation, which may be used to place strong constraints on the nature
of the sources of reionisation. Radiative transfer effects also produce large
fluctuations in the HeII to HI number density ratio eta. The spread in values
is smaller than measured, except for the time-varying spectrum. For this case,
the spread evolves as the spectral nature of the ionising background changes.
Large values for eta are found in partially ionised HeII as the power-law
spectrum begins to dominate the starburst, suggesting that the large eta values
measured may be indicating the onset of the HeII reionisation epoch.
