  The EUV Imaging Spectrometer (EIS) instrument on board the Hinode satellite
has access to some of the best coronal density diagnostics and the high
sensitivity of the instrument now allows electron number density, N_e,
measurements to an unprecedented precision of up to +/-5 % in active regions.
This paper gives a thorough overview of data analysis issues for the best
diagnostics of Fe XII and Fe XIII and assesses the accuracy of the
measurements. Two density diagnostics each from Fe XII (186.88/195.12 and
196.64/195.12) and Fe XIII (196.54/202.04 and 203.82/202.04) are analysed in
two active region data-sets from 2007 May 3 and 6 that yield densities in the
range 8.5 < log N_e < 11.0. The densities are derived using v5.2 of the CHIANTI
atomic database. The Fe XII and Fe XIII diagnostics show broadly the same trend
in density across the active region, consistent with their similar temperatures
of formation. However the high precision of the EIS measurements demonstrates
significant discrepancies of up to 0.5 dex in derived log N_e values, with Fe
XII always giving higher densities than Fe XIII. The discrepancies may partly
be due to real physical differences between the emitting regions of the two
plasmas, but the dominant factor lies in the atomic models of the two ions. Two
specific problems are identified for Fe XII 196.64 and Fe XIII 203.82: the
former is found to be under-estimated in strength by the CHIANTI atomic model,
while the high density limit of the 203.82/202.04 is suggested to be inaccurate
in the CHIANTI atomic model. The small grating tilt of the EIS instrument is
found to be very significant when deriving densities from emission lines
separated by more than a few angstroms.
