  Among stars in Galactic globular clusters the carbon abundance tends to
decrease with increasing luminosity on the upper red giant branch, particularly
within the lowest metallicity clusters. While such a phenomena is not predicted
by canonical models of stellar interiors and evolution, it is widely held to be
the result of some extra mixing operating during red giant branch ascent which
transports material exposed to the CN(O)-cycle across the radiative zone in the
stellar interior and into the base of the convective envelope, whereupon it is
brought rapidly to the stellar surface. Here we present measurements of [C/Fe]
abundances among 67 red giants in 19 globular clusters within the Milky Way.
Building on the work of Martell et al., we have concentrated on giants with
absolute magnitudes of $M_\mathrm{V} \sim -1.5$ within clusters encompassing a
range of metallicity (-2.4 $<$ [Fe/H] $<$ -0.3). The Kitt Peak National
Observatory (KPNO) 4 m and Southern Astrophysical Research (SOAR) 4.1 m
telescopes were used to obtain spectra covering the $\lambda$4300 CH and
$\lambda$3883 CN bands. The CH absorption features in these spectra have been
analyzed via synthetic spectra in order to obtain [C/Fe] abundances. These
abundances and the luminosities of the observed stars were used to infer the
rate at which C abundances change with time during upper red giant branch
evolution (i.e., the mixing efficiency). By establishing rates over a range of
metallicity, the dependence of deep mixing on metallicity is explored. We find
that the inferred carbon depletion rate decreases as a function of metallicity,
although our results are dependent on the initial [C/Fe] composition assumed
for each star.
