Quantifying neutron-proton interactions in $N=51$ isotones: from NEEC candidate $^{93}$Mo to $^{99}$Cd

Abstract: We present a shell-model analysis of $N=51$ isotones, $<sup>{93}$Mo, $<sup>{95}$Ru, $<sup>{97}$Pd, and $<sup>{99}$Cd, to quantify the role of neutron-proton interactions in shaping the location and half-life of isomeric states. The study is motivated by the anomalous behavior of the ${21/2}<sup>+$ isomeric state in $<sup>{93}$Mo, a prominent candidate for nuclear excitation by electron capture (NEEC), which misses an $E2$ decay branch due to a higher-lying ${17/2}<sup>+$ state and instead proceeds via a long-lived $E4$ isomeric transition. Employing a consistent configuration space and empirically derived effective interaction, we extract and compare the proton-proton and neutron-proton matrix elements for the four $N=51$ isotones. Our results show a distinct dominance of the neutron-proton interaction in $<sup>{93}$Mo, in contrast to its neighbors--$<sup>{95}$Ru, $<sup>{97}$Pd, and $<sup>{99}$Cd--where no analogous isomeric behavior emerges due to structural evolution. These findings reveal that the favorable structure for NEEC in $<sup>{93}$Mo stems from subtle interaction systematics that do not persist across the chain. We find that the $E2$ strength of the key NEEC transition is reduced by 40\% compared to the previously estimated value. The analysis provides microscopic insights into the origin of long-lived isomerism in medium-mass nuclei and outlines a framework for identifying future candidates in other mass regions for exploiting the potential energy storage capacities of isomeric states.