(abridged) We measure spectral indices for 1823 galaxies in the CNOC1 sample of fifteen X-ray luminous clusters at 0.18<z<0.55, to investigate the mechanisms responsible for differential evolution between the galaxy cluster and field environments. The radial trends of D4000, Hdelta and [OII] are all consistent with an age sequence, in the sense that the last episode of star formation occurred more recently in galaxies farthest from the cluster center. Throughout the cluster environment, galaxies show evidence for older mean stellar populations than field galaxies. From the subsample of galaxies more luminous than M_r=-18.8 + 5log h, we identify a sample of K+A galaxies, which may result from recently terminated star formation. Corrected for a systematic effect which results from the large uncertainties on individual spectral index measurements, we estimate that K+A galaxies make up only 1.5 +/- 0.8 % of the cluster sample, and 1.2 +/- 0.8 % of the field. We compare our data with spectrophotometric models and conclude that up to 1.9 +/- 0.8 % of the cluster population may have had its star formation recently truncated without a starburst. However, this is still not significantly greater than the fraction of such galaxies in the field, 3.1 +/- 1.0 %. Furthermore, we do not detect an excess of cluster galaxies that have unambiguously undergone a starburst within the last 1 Gyr. Our results imply that these cluster environments are not responsible for inducing starbursts; thus, the increase in cluster blue galaxy fraction with redshift may not be a strictly cluster--specific phenomenon. We suggest that the truncation of star formation in clusters may largely be a gradual process, perhaps due to the exhaustion of gas in the galactic disk over fairly long timescales.
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