We have obtained deep optical, long-slit spectrophotometry of the Galactic HII regions M 17, NGC 3576 and of the Magellanic Cloud HII regions 30 Doradus, LMC N11B and SMC N66, recording the optical recombination lines (ORLs) of CII, NII and OII. Temperature-insensitive ORL C2+/O2+ and N2+/O2 ratios are obtained for all nebulae except SMC N66. The ORL C2+/O2+ ratios show remarkable agreement within each galactic system, while also being in agreement with the corresponding CEL ratios. For all five nebulae, the O2+/H+ abundance derived from multiple OII ORLs is found to be higher than the corresponding value derived from the strong [OIII] 4959, 5007A CELs, by factors of 1.8--2.7 for four of the nebulae. The LMC N11B nebula exhibits a more extreme discrepancy factor for the O2+ ion, ~5. Thus these HII regions exhibit ORL/CEL abundance discrepancy factors that are similar to those previously encountered amongst planetary nebulae. Our optical CEL O2+/H+ abundances agree to within 20-30 per cent with published O2+/H+ abundances that were obtained from observations of infrared fine-structure lines. Since the low excitation energies of the latter make them insensitive to variations about typical nebular temperatures, fluctuations in temperature are ruled out as the cause of the observed ORL/CEL O2+ abundance discrepancies. We present evidence that the observed OII ORLs from these HII regions originate from gas of very similar density (<3500 cm-3) to that emitting the observed heavy-element optical and infrared CELs, ruling out models that employ high-density ionized inclusions in order to explain the abundance discrepancy. We consider a scenario whereby much of the heavy-element ORL emission originates from cold (<=500 K) metal-rich ionized regions.
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