There is considerable confusion about the difference between these two ways of referring to a spectrum or ion, for example C III or C+2.  These have very definite different physical meanings and cannot be used interchangeably.

C+2 is a baryon and C III is a set of photons. 

C+2 refers to carbon with two electrons removed, so that it is doubly ionized, with a net charge of +2.   

C III is the spectrum produced by carbon with two electrons removed.  The C III spectrum will be produced by impact excitation of C+2 (C+2 + e- -> C+2* + e- -> C+2 +hn) or by recombination of C+3 (C+3 + e- -> C+2 + hn).  So, depending on how the spectrum is formed, C III may be emitted by C+2 or C+3

There is no ambiguity in absorption line studies - only C+2 can produce a C III absorption line.  This had caused many people to think that C III refers to the matter rather than the spectrum.  But this notation is ambiguous in the case of emission lines.

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