Combining Hubble Space Telescope WFPC2 mosaics with extensive ground-based spectroscopy, we study the nature of E+A galaxies in three intermediate-redshift clusters (z = 0.33, 0.58, and 0.83). From a sample of ~500 confirmed cluster members, we isolate 46 E+A candidates to determine the E+A fraction and study their physical properties. Spectral types are assigned using Balmer (Hδ, Hγ, Hβ) and [O II] λ3727 equivalent widths. For all members, we have galaxy colors, luminosities, Hubble types, and quantitative structural parameters. We also include measured internal velocity dispersions for 120 cluster members and estimate velocity dispersions for the rest of the cluster sample using the fundamental plane. We find that E+A galaxies comprise a nonnegligible component (~7%-13%) of the cluster population at these redshifts, and their diverse nature indicates a heterogeneous parent population. While cluster E+A's are predominantly disk-dominated systems, they span the range in Hubble type and bulge-to-total fraction to include even early-type members. Cluster E+A's also cover a wide range in luminosity [LB ~ (0.2-2.5)L], internal velocity dispersion (σ ~ 30-220 km s-1), and half-light radius [r1/2 ~ (0.4-4.3)h-1 kpc]. From their velocity dispersions and half-light radii, we infer that the descendants of E+A's in our highest redshift cluster are massive early-type galaxies. In contrast to the wide range of luminosity and internal velocity dispersion spanned by E+A's at higher redshift, only low-mass E+A's are found in nearby clusters, e.g., Coma. The observed decrease in the characteristic E+A mass is similar to the decrease in the luminosity of rapidly star-forming field galaxies since z ~ 1, i.e., galaxy "downsizing." In addition, we argue that our statistics imply that 30% of the E-S0 members have undergone an E+A phase; the true fraction could be 100% if the effects of E+A downsizing, an increasing E+A fraction with redshift, and the conversion of spiral galaxies into early type galaxies are also considered. Thus, the E+A phase may indeed be an important stage in the transformation of star-forming galaxies into early-type members.
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