Recent fully relativistic calculations of radiative rates and electron impact excitation cross sections for Fe {sc xiii} are used to generate emission-line ratios involving 3s$^{2}$3p$^{2}$--3s3p$^{3}$ and 3s$^{2}$3p$^{2}$--3s$^{2}$3p3d transitions in the 170--225 AA and 235--450 AA wavelength ranges covered by the Solar Extreme-Ultraviolet Research Telescope and Spectrograph (SERTS). A comparison of these line ratios with SERTS active region observations from rocket flights in 1989 and 1995 reveals generally very good agreement between theory and experiment. Several new Fe {sc xiii} emission features are identified, at wavelengths of 203.79, 259.94, 288.56 and 290.81 AA. However, major discrepancies between theory and observation remain for several Fe {sc xiii} transitions, as previously found by Landi (2002) and others, which cannot be explained by blending. Errors in the adopted atomic data appear to be the most likely explanation, in particular for transitions which have 3s$^{2}$3p3d $^{1}$D$_{2}$ as their upper level. The most useful Fe {sc xiii} electron density diagnostics in the SERTS spectral regions are assessed, in terms of the line pairs involved being (i) apparently free of atomic physics problems and blends, (ii) close in wavelength to reduce the effects of possible errors in the instrumental intensity calibration, and (iii) very sensitive to changes in N$_{e}$ over the range 10$^{8}$--10$^{11}$ cm$^{-3}$. It is concluded that the ratios which best satisfy these conditions are 200.03/202.04 and 203.17/202.04 for the 170--225 AA wavelength region, and 348.18/320.80, 348.18/368.16, 359.64/348.18 and 359.83/368.16 for 235--450 AA.
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