Recent studies of exotic nuclei near the self-conjugate doubly-magic $^{\mathrm{100}}$Sn nucleus

The exotic proton-rich self-conjugate doubly-magic nucleus $^{\mathrm{100}}$Sn is one of the corner stones of nuclear structure. The $^{\mathrm{100}}$Sn region provides a stringent test for the shell model far away from the line of stability. The $^{\mathrm{100}}$Sn nucleus is the fastest known Gamow-Teller $\beta $ emitter. Its large binding energy is signaled by the existence of an island of proton and $\alpha $ emitters decaying towards the N$=$Z$=$50 closed shells. Also, the astrophysical rp-process was proposed to terminate with $\alpha $ decays of light Te isotopes. Despite prohibitively small production cross sections, several exotic nuclei near $^{\mathrm{100}}$Sn have been studied recently using various probes at the ATLAS facility at the Argonne National Laboratory. 1) First evidence for the $\alpha $-decay chain $^{\mathrm{108}}$Xe-$^{\mathrm{104}}$Te into $^{\mathrm{100}}$Sn was observed. This is only the second case of $\alpha $ decay into a doubly-magic nucleus besides $^{\mathrm{212}}$Po, which has been a benchmark of microscopic models of $\alpha $ decay. The reduced $\alpha $-decay widths deduced for $^{\mathrm{108}}$Xe and $^{\mathrm{104}}$Te are larger than that for $^{\mathrm{212}}$Po supporting the expectation that the enhanced interaction between protons and neutrons, which occupy the same orbitals, leads to a larger $\alpha $-particle preformation, which results in the so-called superallowed $\alpha $ decay. 2) A small proton-decay branch was found in $^{\mathrm{108}}$I. The proton separation energy in $^{\mathrm{104}}$Sb, deduced using the measured $^{\mathrm{108}}$I proton energies, indicates that the rp-process does not form a Sn-Sb-Te cycle at $^{\mathrm{103}}$Sn which is delayed until heavier Sn isotopes. 3) Excited states in the fast $^{\mathrm{105}}$Te $\alpha $ emitter were studied for the first time using in-beam $\gamma $-ray spectroscopy to shed light on the long standing issue of the ordering of the d$_{\mathrm{5/2}}$ and g$_{\mathrm{7/2}}$ single-neutron orbitals in $^{\mathrm{101}}$Sn.