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Comparative study of high-spin isomers in semi-magic $Z$=50 isotopic and $N$=82 isotonic chains

We present a comparative study of the high-spin ${11/2}^-$, ${10}^+$ and ${27/2}^-$ nuclear isomers, observed commonly in the full semi-magic $Z=50$ isotopic and $N=82$ isotonic chains, which are found to display nearly identical systematics in excitation energy and half-lives. A constant and particle number independent energy gap of $\sim$ 4 MeV between the $0^+$ ground states and ${10}^+$ isomers and, the ${11/2}^-$ and ${27/2}^-$ isomers exists before the mid-shell, which declines near the mid-shell, and then becomes constant at $\sim$ 3 MeV after the mid-shell. Large scale shell model calculations are carried out for the full chains of Sn-isotopes and $N=82$ isotones, which reproduce the observed energy systematics reasonably well. The role of seniority, generalized seniority and alignment in the high-spin isomeric states as well as in the states populating the high spin isomers has been discussed. The empirical features suggest that the generalized seniority, known to be valid in the low lying states of even-even nuclei before the mid-shell, remains good up to the ${10}^+$ isomeric states; we further extend this interpretation to the odd-A nuclei before the mid-shell, particularly the ${27/2}^-$ isomers. The seniority becomes good after the mid-shell. The change in energy gap around the mid-shell, therefore, marks a transition from the generalized seniority regime to the seniority regime. From the alignment considerations, the high-spin isomers after the mid-shell emerge as maximally aligned decoupled states. We also note that the similar features of the isomeric half-lives in both the chains may be understood in terms of their similar seniorities and configurations. An overview of the $0^+$ to ${10}^+$ yrast states in the full chain of even-even Sn-isotopes has also been presented. We also make some predictions of possible new isomers based on the empirical systematics.