Background: Even-even tellurium isotopes, located just above the $Z=50$ shell closure, are expected to exhibit near-spherical vibrational character. However, previous measurements have indicated nonzero quadrupole moments and possible shape coexistence, motivating precise studies of their electromagnetic properties. In particular, ${}^{130}\mathrm{Te}$ is crucial for understanding collectivity near $N=82$ and for constraining nuclear matrix elements relevant for the ${}^{130}\mathrm{Te} \to {}^{130}\mathrm{Xe} 0\nu \beta \beta$ decay.

Purpose: To determine the electromagnetic matrix elements governing the low-lying structure of ${}^{130}\mathrm{Te}$ with special emphasis on the spectroscopic quadrupole moment of the $2_1^{+}$ state and the evolution of quadrupole collectivity in this transitional nucleus.

Methods: A multistep Coulomb-excitation experiment was performed at INFN-LNL using a 231-MeV ${}^{82}\mathrm{Se}$ beam impinging on an enriched ${}^{130}\mathrm{Te}$ target. De-exciting $\gamma$ rays were detected with the AGATA array in coincidence with scattered ejectiles measured by the SPIDER detector. Electromagnetic matrix elements were extracted via a least-squares analysis of $\gamma$-ray yields using the $\mathrm{GOSIA}$ code, incorporating available spectroscopic constraints. The results were interpreted within the framework of quadrupole sum rules and large-scale shell-model calculations.

Results: A comprehensive set of $E2$ and $M1$ matrix elements connecting low-lying states in ${}^{130}\mathrm{Te}$ was determined, from which level lifetimes were deduced. The diagonal matrix element of the $2_1^{+}$ state was extracted with both magnitude and sign, yielding $Q_s\left(2_1^{+}\right)=+0.35\left(2\right)e\mathrm{b}$, consistent with a weak oblate deformation. The deduced quadrupole invariant corresponds to a small deformation parameter $\langle {\beta }_2\rangle \approx 0.12$, consistent with near-spherical structure. Shell-model calculations reproduce the overall collectivity and the sign of the quadrupole moment, although discrepancies persist for transitions involving excited $2^{+}$ states. A systematic study of even-even Te isotopes shows that the evolution of excitation energies, transition strengths, and quadrupole moments is reasonably well described, with particularly good agreement for ${}^{130}\mathrm{Te}$, while larger deviations toward the lighter isotopes (${}^{124,122}\mathrm{Te}$) may indicate missing correlations in the present theoretical framework.

Conclusions: The results provide stringent constraints on the electromagnetic structure of ${}^{130}\mathrm{Te}$, confirming its weakly collective, near-spherical character. In addition, they provide an important experimental benchmark for microscopic descriptions of ${}^{130}\mathrm{Te}$, a nucleus of particular relevance to searches for $0\nu \beta \beta$ decay.

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• Coulomb excitation
• Electromagnetic transitions
• Nuclear spin & parity
• Nuclear structure & decays
• 90 ≤ A ≤ 149
• Shell model
• Spectrometers & spectroscopic techniques