Testing the predictive power of realistic shell model calculations via lifetime measurement of the $11/2^{+}$ state in ${}^{131}\mathrm{Sb}$

The lifetime of the $11/2_1^{+}$ state in the ${}^{131}\mathrm{Sb}$ nucleus was measured at the LOHENGRIN spectrometer of the Institut Laue-Langevin via neutron-induced fission of ${}^{235}\mathrm{U}$ using $\gamma$-ray fast-timing techniques. The obtained value of $T_{1/2}=3$(2) ps, at the edge of the sensitivity of the experimental method, is the first result for the $11/2_1^{+}$ state half-life in neutron-rich Sb isotopes. The corresponding quadrupole reduced transition probability to the ground state is $B\left(E2\right)=1.4_{-0.6}^{+1.5}$W.u., indicating a noncollective nature of this state. Realistic shell-model calculations performed in a large valence space reproduce well the experimental value and point to a dominant $2^{+}\left({}^{130}\mathrm{Sn}\right)\otimes \pi {\mathrm{g}}_{7/2}$ configuration for the $11/2_1^{+}$ state, as expected in a weak-coupling scenario. At the same time, the sum of the quadrupole strength of the multiplet states is predicted to exceed the one of the ${}^{130}\mathrm{Sn}$ core as a consequence of the equal contribution of the proton and the proton-neutron quadrupole matrix elements, pointing to possible development of collectivity already in the close neighborhood of ${}^{132}\mathrm{Sn}$.

Figure 1

Total projections of the $\gamma \text{−}\gamma$ matrices measured by HPGe detectors (black) and ${\mathrm{LaBr}}_3$(Ce) scintillators (red). The 450- and 1226-keV transitions in ${}^{131}\mathrm{Sb}$, populating and depopulating the $11/2^{+}$ state, respectively, are highlighted. The $\gamma$ lines coming from ${\beta }^{-}$ decays are also marked (see text for details). (Inset) Projection of the $\gamma \text{−}\gamma$ matrices gated on the 1226-keV transition in ${}^{131}\mathrm{Sb}$, showing a clean coincidence with the 450-keV line deexciting the 64-$\mu \mathrm{s}$ isomer.

Figure 2

(left) Partial experimental level scheme of ${}^{131}\mathrm{Sb}$, as observed in this work, with the $11/2_1^{+}$ state and its $E2$ decay (of interest here) shown in red. The $T_{1/2}=3$(2) ps half-life of the $11/2^{+}$ state and the $T_{1/2}=1.35$(17) ns half-life of the $19/2^{+}$ measured in this experiment are reported in bold, together with literature values for the microsecond isomers, taken from Ref. [18]. (right) Realistic shell-model predictions obtained using the ${\mathrm{V}}_{\text{low-}k}$ approach (see text for details). The experimental and theoretical $B(E2;11/2_1^{+}\to 7/2_1^{+})$ values are also displayed.

Figure 3

(a) Start-stop $\gamma \text{−}\gamma$ matrix measured by ${\mathrm{LaBr}}_3$(Ce) scintillators. The energy gate around the 450–1226-keV, p|p coincidence is marked in red and corresponds to the delayed sector. The p|b, b|p, and b|b background gates are shown in green, purple, and gold, respectively (see text for details). (b) Delayed time distributions for the 450–1226-keV cascade. Vertical bars represent the centroid positions of the p|p (red), p|b (green), b|p (purple), and b|b (gold) components, with their height being proportional to the number of counts in each time distribution.

Figure 4

(top) Systematic of the energy of the $2_1^{+}$ states in Sn isotopes (blue diamonds) compared with the energy of the $11/2_1^{+}$ states in Sb nuclei (red squares). Realistic shell-model calculations are shown as open diamonds and squares for Sn and Sb nuclei, respectively. (bottom) Systematic of the $B(E2;2_1^{+}\to 0_1^{+})$ values in Sn isotopes (blue diamonds) compared with the $B(E2;11/2_1^{+}\to 7/2_1^{+})$ value in ${}^{131}\mathrm{Sb}$ obtained in this work (red square). Result for ${}^{129}\mathrm{Sb}$ are also shown [8]. Realistic shell-model calculations are presented as open squares for Sb nuclei. Calculations for $B(E2;2_1^{+}\to 0_1^{+})$ values in Sn isotopes are not reported since the effective charges were fit to reproduce experimental data (see text for details).

Figure 5

(left) Shell-model calculations of the $2^{+}$ state and of the $B(E2;0_1^{+}\to 2_1^{+})$ in ${}^{130}\mathrm{Sn}$. (right) Shell-model calculations of the $2^{+}\left({}^{130}\mathrm{Sn}\right)\otimes \pi {\mathrm{g}}_{7/2}$ multiplet in ${}^{131}\mathrm{Sb}$ and of the $B(E2,7/2_1^{+}\to {\mathrm{j}}^{\pi })$ in ${}^{131}\mathrm{Sb}$. The $5/2_1^{+}$ state of $0^{+}\left({}^{130}\mathrm{Sn}\right)\otimes \pi {\mathrm{d}}_{5/2}$ character, located at 942 keV, is also reported. The percentages of the main wave-function components are indicated in red. Reduced transition probabilities are in Weisskopf units (see text for details).