We analyze high-resolution, high signal-to-noise ratio spectra of six red giant branch (RGB) stars in the globular cluster M3 (NGC 5272) and three in M13 (NGC 6205) that were obtained with the 4 m Mayall Telescope and echelle spectrometer on Kitt Peak. The spectra include lines of O, Na, Mg, Al, Si, Ca, Ti, V, Mn, Fe, and Ni. We also analyze the [Al/Fe] values of 96 RGB stars in M13 covering the brightest 3.5 mag, which include 66 measurements that were derived from moderate-resolution, low signal-to-noise ratio spectra obtained with the WIYN 3.5 m telescope and Hydra multiobject spectro graph, also on Kitt Peak. In addition, we compile from the literature and inspect the [Na/Fe] values of 119 RGB stars in M13. We test for bimodality in the [Al/Fe] and [Na/Fe] distributions using the KMM algorithm and find that the [Al/Fe] values in M13 are distributed bimodally at all points along the RGB that were observed, while the [Na/Fe] values are bimodal only over the brightest 2 mag. The ratios of Al-enhanced to Al-normal and Na-enhanced to Na-normal giants increase toward the tip of the RGB in M13, which is suggestive of deep mixing in this cluster. The limited M3 data exhibit a bimodal distribution of [Al/Fe] values and are suggestive of no deep mixing; however, they are too few to be conclusive. We further test for a relationship between deep mixing on the RGB and a second parameter that can create the extended blue tail seen along the horizontal branches of some clusters by using an "instantaneous" mixing algorithm, which we develop here. We conclude that the data for both clusters are consistent with deep mixing as a "blue-tail second parameter," and we suggest future observations to further constrain the results. Finally, we offer a solution to the problem of overproducing sodium during deep mixing that is based on the depletion of 22Ne in asymptotic giant branch stars and suggest that pollution might best be traced by s-process elements in the Sr-Y-Zr peak.
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