A comparative study of optical spectra of Type Ia supernovae (SNe Ia) obtained near 1 week, 3 weeks, and 3 months after maximum light is presented. Most members of the four groups that were defined on the basis of maximum light spectra in Paper II (core normal, broad line, cool, and shallow silicon) develop highly homogeneous postmaximum spectra, although there are interesting exceptions. Comparisons with SYNOW synthetic spectra show that most of the spectral features can be accounted for in a plausible way. The fits show that 3 months after maximum light, when SN Ia spectra are often said to be in the nebular phase and to consist of forbidden emission lines, the spectra actually remain dominated by resonance scattering features of permitted lines, primarily those of Fe II. Even in SN 1991bg, which is said to have made a very early transition to the nebular phase, there is no need to appeal to forbidden lines at 3 weeks postmaximum, and at 3 months postmaximum the only clear identification of a forbidden line is [Ca II] 7291, 7324. Recent studies of SN Ia rates indicate that most of the SNe Ia that have ever occurred have been prompt SNe Ia, produced by young (100,000,000 yr) stellar populations, while most of the SNe Ia that occur at low redshift today are tardy, produced by an older (several Gyrs) population. We suggest that the shallow silicon SNe Ia tend to be the prompt ones.
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