We systematically analyze total reaction cross sections of carbon isotopes with N=6--16 on a $^{12}$C target for wide range of incident energy. The intrinsic structure of the carbon isotope is described by a Slater determinant generated from a phenomenological mean-field potential, which reasonably well reproduces the ground state properties for most of the even $N$ isotopes. We need separate studies not only for odd nuclei but also for $^{16}$C and $^{22}$C. The density of the carbon isotope is constructed by eliminating the effect of the center of mass motion. For the calculations of the cross sections, we take two schemes: one is the Glauber approximation, and the other is the eikonal model using a global optical potential. We find that both of the schemes successfully reproduce low and high incident energy data on the cross sections of $^{12}$C, $^{13}$C and $^{16}$C on $^{12}$C. The calculated reaction cross sections of $^{15}$C are found to be considerably smaller than the empirical values observed at low energy. We find a consistent parameterization of the nucleon-nucleon scattering amplitude, differently from previous ones. Finally, we predict the total reaction cross section of $^{22}$C on $^{12}$C.
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