Decay of the low-energy nuclear $^{229m}$Th isomer via atomic Rydberg states.
In the paper, a unique process of the decay of the $^{229m}$Th($3/2^+,8.28\pm 0.17$ eV) low energy nuclear isomer via the internal conversion (IC) channel on Rydberg states is considered for the first time. The Rydberg atom $^{229m}$Th$^{+}+e^-_{Ry}$ is a unique object where IC is possible exclusively on the Rydberg electron $e^-_{Ry}$. It is shown that in the $^{229m}$Th$^{4+}+e^-_{Ry}$ system a) IC on the electron states with relatively small values of the principal quantum number $n$ and the orbital moment $l$ is practically completely suppressed, b) IC probability, $W_{IC}$, on the $\psi_{ns_{1/2}}({\bf{r}})$ states is proportional to $|\psi_{ns_{1/2}}(0)|^2$ and can be related with the energy of the hyperfine interaction of the Rydberg electron with the nucleus, c) $W_{IC}$ decreases rapidly with the increase of $n$ in the range from 10 to 50, and in the range $n\gtrsim 150$ $W_{IC}$ changes as ${\text{Const}}/n^3$ typical for hydrogen-like ions, d) at $n\thickapprox 10$--30, $W_{IC}$ as a function of $l$ has a characteristic "knee" between $l=3$ and $l=4$, i.e. a three order of magnitude decrease of $W_{IC}$ due to the qualitative change in the ratio between the centrifugal and shielding potentials.