Cluster radioactivity has been successfully described as a superasymmetric fission mode within the microscopic self-consistent Gogny Hartree-Fock-Bogoliubov approximation [M. Warda and L. M. Robledo, Phys. Rev. C 84, 044608 (2011)]. For nuclei preserving the neutron-to-proton $N/Z$ ratio of the doubly magic ${}^{208}\mathrm{Pb}$, a cluster radioactivity fission valley has been identified. Such a valley can also be found both in actinides and superheavy nuclei. In this paper, chains of isotopes and isotones are examined to determine the limits of existence of the cluster radioactivity fission mode. It is shown that the superasymmetric valley can be found in a wide range of the nuclear chart. Nevertheless, the valley flattens more and more when diverging from the isospin asymmetry of ${}^{208}\mathrm{Pb}$. For neutron-deficient nuclei with $N/Z<$ 1.41, it is found that the valley diminishes before reaching the scission point, and cluster radioactivity cannot be observed.

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• Fission
• Nuclear fragmentation
• Scission point
• Nuclear density functional theory