Spectroscopic factors in the $N=20$ island of inversion: The Nilsson strong-coupling limit

Spectroscopic factors, extracted from one-neutron knockout and Coulomb dissociation reactions, for transitions from the ground state of ${}^{33}\mathrm{Mg}$ to the ground-state rotational band in ${}^{32}\mathrm{Mg}$, and from ${}^{32}\mathrm{Mg}$ to low-lying negative-parity states in ${}^{31}\mathrm{Mg}$, are interpreted within the rotational model. Associating the ground state of ${}^{33}\mathrm{Mg}$ and the negative-parity states in ${}^{31}\mathrm{Mg}$ with the $\frac{3}{2}\left[321\right]$ Nilsson level, the strong coupling limit gives simple expressions that relate the amplitudes ($C_{j\ell }$) of this wave function with the measured cross sections and derived spectroscopic factors ($S_{j\ell }$). To obtain a consistent agreement with the data within this framework, we find that one requires a modified $\frac{3}{2}\left[321\right]$ wave function with an increased contribution from the spherical $2p_{3/2}$ orbit as compared to a standard Nilsson calculation. This is consistent with the findings of large-scale shell model calculations and can be traced to weak binding effects that lower the energy of low-$\ell$ orbitals.

Figure 1

Schematic representation of the possible transitions for the reactions analyzed in this work. Both $\ell =1$ and $\ell =3$ transitions are allowed in both cases considered.

Figure 2

Two-dimensional plot showing the possible solutions of Eqs. (3) that reproduce the experimental magnetic moment (red line). The intersection with the normalization condition (black line and blue shaded area reflecting the uncertainty in $C_{3/2,1}$) together with the requirement that $|C_{7/2,3}|>|C_{5/2,3}|$ (shaded gray area) determines our adopted solution (solid circle).