The bright [CI] 1-0 and [CI] 2-1 lines of atomic carbon are becoming more and more widely employed tracers of the cold neutral gas in high-redshift galaxies. Here we present observations of these lines in the 11 galaxies of the set of Plancks Dusty GEMS, the brightest gravitationally lensed galaxies on the extragalactic submillimeter sky probed by the Planck satellite. We have [CI] 1-0 measurements for seven, and [CI] 2-1 measurements for eight galaxies, including four galaxies where both lines are measured. We use our observations to constrain the gas excitation mechanism, excitation temperatures, optical depths, atomic carbon and molecular gas masses, and carbon abundances. Ratios of L_CI/L_ FIR are similar to those found in the local Universe, and suggest that the total cooling budget through atomic carbon has not strongly changed in the last 12 Gyr. Both lines are optically thin and trace 1 - 6 x 10^7 M_sun of atomic carbon. Carbon abundance ratios with H_2, X_CI, are between 2.5 and 4 x 10^-5, for a ULIRG CO-to-H_2 conversion factor of alpha_CO=0.8 M_sun/ [K km s^-1 pc^2]. Ratios of molecular gas masses derived from [CI] 1-0 and CO agree within the measurement uncertainties for five galaxies, and to better than a factor of 2 for another two with [CI] 1-0 measurements, after taking CO excitation carefully into account. This does not support the idea that intense, high-redshift starburst galaxies host large quantities of CO-dark gas. These results also support the common assumptions underlying most molecular gas mass estimates made for massive, dusty, high-redshift starburst galaxies, although the good agreement between the masses obtained with both tracers cannot be taken as an independent confirmation of either alpha_CO or X_CI.
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