The ${}^{30}\mathrm{Mg}(t,p){}^{32}\mathrm{Mg}$ “puzzle” reexamined

Background: Competing interpretations of the results of a ${}^{30}\mathrm{Mg}$($t,p$)${}^{32}\mathrm{Mg}$ measurement populating the ground state and $0_2^{+}$ state in ${}^{32}\mathrm{Mg}$, both limited to a two-state mixing description, have left an open question regarding the nature of the ${}^{32}\mathrm{Mg}$ ground state.

Purpose: Inspired by recent shell-model calculations, we explore the possibility of a consistent interpretation of the available data for the low-lying $0^{+}$ states in ${}^{32}\mathrm{Mg}$ by expanding the description from two-level to three-level mixing.

Methods: A phenomenological three-level mixing model of unperturbed 0p0h, 2p2h, and 4p4h states is applied to describe both the excitation energies in ${}^{32}\mathrm{Mg}$ and the transfer reaction cross sections.

Results: Within this approach, self-consistent solutions exist that provide good agreement with the available experimental information obtained from the ${}^{30}\mathrm{Mg}$($t,p$)${}^{32}\mathrm{Mg}$ reaction.

Conclusion: The inclusion of the third state, namely the 4p4h configuration, resolves the “puzzle” that results from a two-levelmodel interpretation of the same data. In our analysis, the ${}^{32}\mathrm{Mg}$ ground state emerges naturally as dominated by intruder (2p2h and 4p4h) configurations, at the 95% level.

Figure 1

Graphical illustration of the solutions for diagonalization of the matrix of Eq. (1) as a function of $V$. The top panel (a) shows the evolution of the energy levels for the mixed $0^{+}$ states. The dot-dashed line at $V=0.58$ MeV represents the interaction strength at which the excitation energy for $0_2^{+}$ equals the experimental value. The bottom panel (b) shows the evolution of the wave-function amplitudes for the three $0^{+}$ states.

Figure 2

Schematic representation of the two-neutron transfers under consideration. The components of the ${}^{30}\mathrm{Mg}$ ground-state wave function connect to components of the ${}^{32}\mathrm{Mg}{0}^{+}$ wave functions as shown. The solid connecting lines correspond to transfer of $sd$ shell neutrons, while the dotted connecting lines correspond to transfer of a pair of $fp$ shell neutrons. Open circles represent holes in the $sd$ shell, while the crosses represent particles in the $fp$ shell.

Figure 3

Cross-section ratio ${\sigma }_{0_2^{+}}/{\sigma }_{0_1^{+}}$ calculated according to Eq. (5) based on the ${}^{32}\mathrm{Mg}$ wave-function amplitudes as plotted in Fig. 1 as a function of $V$ and the $|0\text{p}0\mathrm{h}\rangle$ squared amplitude in the ${}^{30}\mathrm{Mg}$ ground state $[{\epsilon }^2$ in Eq. (3)]. The hashed area represents the results of Ref. [11], while the dashed line and shaded error band indicate the experimental range of the cross-section ratio. The black data point indicates the mixing strength, $V$, which reproduces the experimental $E\left(0_2^{+}\right)$.

Figure 4

Percent squared amplitude of the $0\text{p}0\text{h}$ component in the ${}^{30}\mathrm{Mg}$ ground state $[{\epsilon }^2$ in Eq. (3)] required to explain the experimental data as a function of the energies of the unperturbed states. Point A shows the solution [Eq. (6)] with the energies fixed at the unperturbed values discussed in Sec. II. The dashed lines indicate possible solutions with a $0\text{p}0\text{h}$ amplitude for the ${}^{30}\mathrm{Mg}$ ground state at 80% and 90%. As discussed in the text, points B and C are used as a reference with respect to the ${}^{32}\mathrm{Mg}$ wave functions.