Cross-shell excited configurations in the structure of ${}^{34}\mathrm{Si}$

The cross-shell excited states of ${}^{34}\mathrm{Si}$ have been investigated via $\beta$ decays of the $4^{-}$ ground state and the $1^{+}$ isomeric state of ${}^{34}\mathrm{Al}$. Since the valence protons and valence neutrons occupy different major shells in the ground state as well as the intruder $1^{+}$ isomeric state of ${}^{34}\mathrm{Al}$, intruder levels of ${}^{34}\mathrm{Si}$ are populated via allowed $\beta$ decays. Spin assignments to such intruder levels of ${}^{34}\mathrm{Si}$ were established through $\gamma \text{−}\gamma$ angular correlation analysis for the negative-parity states with dominant configurations ${\left(\nu d_{3/2}\right)}^{-1}\otimes {\left(\nu f_{7/2}\right)}^1$ as well as the positive-parity states with dominant configurations ${\left(\nu sd\right)}^{-2}\otimes {\left(\nu f_{7/2}{p}_{3/2}\right)}^2$. The configurations of such intruder states play crucial roles in our understanding of the $N=20$ shell gap evolution. A configuration interaction model derived from the FSU Hamiltonian was utilized in order to interpret the intruder states in ${}^{34}\mathrm{Si}$. Shell model interaction derived from a more fundamental theory with the valence space in medium similarity renormalization group method was also employed to interpret the structure of ${}^{34}\mathrm{Si}$.

Figure 1

Level scheme of ${}^{34}\mathrm{Si}$ built from the current analysis of the $\beta$ decay (both the ground state $4^{-}$ and the isomeric state $1^{+}$) of ${}^{34}\mathrm{Al}$. The transitions and levels shown in red were observed for the first time in this work. The $\gamma$-ray branching ratios from the individual levels are listed in Table 1.

Figure 2

$\beta$-gated $\gamma$-ray energy spectrum in coincidence with the SCEPTAR and ZDS detectors. Peaks belonging to ${}^{34}\mathrm{Si}$ are labeled in black, to ${}^{33}\mathrm{Si}$ are labeled in magenta, to ${}^{34}\mathrm{P}$ are labeled in blue, to ${}^{33}\mathrm{P}$ are labeled in green, and to ${}^{34}\mathrm{S}$ are labeled in orange. Peaks labeled with s.e. are the single-escape peaks and those labeled with d.e. are the double-escape peaks.

Figure 3

(a) 1053-keV peak in the singles $\beta$-gated $\gamma$-ray spectrum (b) projection of the $\gamma \text{−}\gamma$ ${180}^{\circ }$ angle matrix gated on the 124-keV transition showing the 929-keV peak, which is in immediate coincidence with the gating transition. See Ref. [17] for details about the summing correction method.

Figure 4

Two newly observed ground-state $\gamma$-ray transitions in ${}^{34}\mathrm{Si}$ and the corresponding coincidence. The red text indicates that the $\gamma$-ray peaks were newly identified.

Figure 5

(a) Measured $\gamma \text{−}\gamma$ angular correlation for the 929- to 3326-keV transitions associated with the 4255-3326-0 keV cascade. The linear combination of the simulated correlation performed with geant4 is shown in red. (b) ${\chi }^2$ fit to the measured $\gamma \text{−}\gamma$ angular correlation for different spin hypotheses. The mixing ratio of the 3326-keV transition was kept fixed at 0 and $\delta$; along the $x$ axis in this plot is the mixing of the 929-keV $\gamma$-ray transition. The value of $\delta$ is extracted as 0.02(1). The ${\chi }^2$ analysis confirms the well-established spin 3 to the 4255-keV level. All the 51 angular bins have been grouped into 7 as mentioned in the text.

Figure 6

(a) Measured $\gamma \text{−}\gamma$ angular correlation for the 124- to 929-keV transitions associated with the 4379-4255-3326 keV cascade. The linear combination of the simulated correlation performed with geant4 is shown in red. (b) ${\chi }^2$ fit to the measured $\gamma \text{−}\gamma$ angular correlation for different spin hypotheses. In this plot the mixing ratio of the 929-keV transition is kept fixed at 0.02 and $\delta$ is the mixing of the 124-keV $\gamma$-ray transition. The ${\delta }_{\gamma 1}$ value for the 124-keV transition is extracted as 0.012(17) for the spin hypothesis of 4. The uncertainty in ${\delta }_{\gamma 1}$ includes the uncertainty in ${\delta }_{\gamma 2}$ determined for the 929-keV transition. The spin assignment for the corresponding level at 4379 keV is discussed in the main text.

Figure 7

(a) Measured $\gamma \text{−}\gamma$ angular correlation for the 591- to 124-keV transitions associated with the 4970-4379-4255 keV cascade. The linear combination of the simulated correlation performed with geant4 is shown in red. (b) ${\chi }^2$ fit to the measured $\gamma \text{−}\gamma$ angular correlation for different spin hypotheses. In this plot the mixing ratio of the 124-keV transition is kept fixed at $-0.012$ and $\delta$ is the mixing of the 591-keV $\gamma$-ray transition. The uncertainty in ${\delta }_{\gamma 1}$ includes the uncertainty in ${\delta }_{\gamma 2}$ determined for the 124-keV transition. The value of $\delta$ is extracted as 0.01(4) for the spin hypothesis 5, which is consistent with a pure $M1$ type, and $-0.09\left(35\right)$ for the spin hypothesis of 3 to the 4970-keV level.

Figure 8

(a) Measured $\gamma \text{−}\gamma$ angular correlation for the 1194- to 3326-keV transitions associated with the 4519-3326-0 keV cascade. The linear combination of the simulated correlation performed with geant4 is shown in red. (b) ${\chi }^2$ fit to the measured $\gamma \text{−}\gamma$ angular correlation for different spin hypotheses. The mixing ratio of the 3326-keV transition is kept fixed at 0 and $\delta$ in this plot is the mixing of the 1194-keV $\gamma$-ray transition. The value of $\delta$ is extracted as 0.43(2) for the spin hypothesis of 2 to the 4519-keV level. Discussions are in the main text.

Figure 9

(a) Measured $\gamma \text{−}\gamma$ angular correlation for the 4151- to 3326-keV transitions associated with the 7477-3326-0 keV cascade. The linear combination of the simulated correlation performed with geant4 is shown in red. (b) ${\chi }^2$ fit to the measured $\gamma \text{−}\gamma$ angular correlation for different spin hypotheses of the 7477-keV level. From this plot, it is clear that spin 0 is the only possible option for this state, which is in very good agreement with the theoretical predictions made in the current work.

Figure 10

Some observed excited energy states of ${}^{34}\mathrm{Si}$ are compared to those calculated with the FSU interaction and the VS-IMSRG method. The middle panel, which consists of the experimental states, has the same energy scaling as that of the theory panels. In the calculated levels, the red portion represents the occupancy of the $0f_{7/2}$ orbital and the blue portion represents that of the $1p_{3/2}$ orbital.

Figure 11

Systematics of the negative-parity states in the even-mass silicon isotopes. The lowest-lying (a) $3^{-}$, (b) $4^{-}$, and (c) $5^{-}$ states, both experimental and predicted with the FSU and the IMSRG-derived interactions, are shown here. The experimental levels, except for those of ${}^{34}\mathrm{Si}$, are compiled from Refs. [40, 41].