NGC 6752 AGB Stars Revisited: I. Improved AGB temperatures remove
  apparent overionisation of Fe I
  A recent study reported a strong apparent depression of Fe I, relative to Fe
II, in the AGB stars of NGC 6752. This depression is much greater than that
expected from the neglect of non-local thermodynamic equilibrium effects, in
particular the dominant effect of overionisation. Here we attempt to reproduce
the apparent Fe discrepancy, and investigate differences in reported sodium
abundances. We compare in detail the methods and results of the recent study
with those of an earlier study of NGC 6752 AGB stars. Iron and sodium
abundances are derived using Fe I, Fe II, and Na I lines. Various uncertainties
are explored. We reproduce the large Fe I depression found by the recent study,
using different observational data and computational tools. Further
investigation shows that the degree of the apparent Fe I depression is strongly
dependent on the adopted stellar effective temperature. To minimise
uncertainties in Fe I we derive temperatures for each star individually using
the infrared flux method (IRFM). We find that the $T_{\rm{eff}}$ scales used by
both the previous studies are cooler, by up to 100 K; such underestimated
temperatures amplify the apparent Fe I depression. Our IRFM temperatures result
in negligible apparent depression, consistent with theory. We also re-derived
sodium abundances and, remarkably, found them to be unaffected by the new
temperature scale. [Na/H] in the AGB stars is consistent between all studies.
Since Fe is constant, it follows that [Na/Fe] is also consistent between
studies, apart from any systematic offsets in Fe. We recommend the use of
$(V-K)$ relations for AGB stars. We plan to investigate the effect of the
improved temperature scale on other elements, and re-evaluate the subpopulation
distributions on the AGB, in the next paper of this series. [abridged]
