Ion clock and search for the variation of the fine-structure constant using optical transitions in Nd${}^{13+}$ and Sm${}^{15+}$

We study ultranarrow $5s_{1/2}$-$4f_{5/2}$ transitions in Nd${}^{13+}$ and Sm${}^{15+}$ and demonstrate that they lie in the optical region. The transitions are insensitive to external perturbations. At the same time they are sensitive to the variation of the fine structure constant $\alpha$. The fractional accuracy of the frequency of the transitions can be smaller than ${10}^{-19}$, which may provide a basis for atomic clocks of superb accuracy. Sensitivity to the variation of $\alpha$ approaches ${10}^{-20}$ per year.