High precision density measurements in the solar corona: I. Analysis   methods and results for Fe XII and Fe XIII
  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. 
P.R. Young, T. Watanabe, H. Hara, J.T. Mariska