Structure of 33 Mg sheds new light on the N = 20 island of inversion

Structure of 33Mg sheds new light on the N = 20 island of inversion R. Kanungo a,∗, C. Nociforo b, A. Prochazka b,c, Y. Utsuno d, T. Aumann b, D. Boutin c, D. Cortina-Gil e, B. Davids f, M. Diakaki g, F. Farinon b,c, H. Geissel b, R. Gernhäuser h, J. Gerl b, R. Janik i, B. Jonson j, B. Kindler b, R. Knöbel b,c, R. Krücken h, M. Lantz j, H. Lenske c, Y. Litvinov b,k, K. Mahata b, P. Maierbeck h, A. Musumarra l,m, T. Nilsson j, T. Otsuka n, C. Perro a, C. Scheidenberger b, B. Sitar i, P. Strmen i, B. Sun b, I. Szarka i, I. Tanihata o, H. Weick b, M. Winkler b

For a complete understanding on the evolution of the island of inversion it is crucial to understand how the nuclear orbitals across the broken N = 20 gap are arranged. We report here the first investigation of the structure of 33 Mg through the longitudinal momentum distribution from one-neutron removal from 33 Mg. The shape of the distribution is much narrower than predictions from the shell model with the SDPF-M interaction, suggesting a larger occupancy of neutrons in the 2p 3/2 orbital. This signals a lowering of the 2p 3/2 orbital. The increased p-wave occupancy has not been discussed in the recent works that debate the ground state spin of 33 Mg [15][16][17]. The present finding therefore opens a new dimension in our attempt to understand the island of inversion. A recent calculation using a Wood-Saxon potential for neutron-rich Mg isotopes discusses the near-degeneracy of the 1 f 7/2 and 2p 3/2 orbitals in a spherical potential to be a basic element in producing deformation [18].
In-beam and beta-delayed gamma spectroscopy experiments have tried to assess the boundary of the island of inversion. Among the excited states observed for [25][26][27][28][29] Ne, those for 28 Ne have been interpreted to show intruder signatures, while [25][26][27] Ne were in agreement with normal sd-shell predictions [19]. On the other hand proton inelastic scattering has identified low-lying intruder states in 27 Ne [20] that are in agreement with a study of the 26 Ne(d,p) reaction [21]. The level scheme of 29,30 Na constructed on the basis of the beta decay of 29,30 Ne reflected the dominance of intruder configurations in the low-lying states [22,23]. These experiments therefore, provide signatures of intruder configurations in the excited states around N = 20.
It is worthwhile to point out here that the island of inversion refers to intruder orbitals in the ground states of nuclei [1,11,13]. Intruder configurations were reported for the ground state of 28 Ne [26]. The 3/2 − intruder 'core' state was found to have a spectroscopic factor (SF) of 0.32(4), in fair agreement with SDPF-M shell model predictions. The momentum distribution for this intruder component was explained by a combination of l = 0 and 1 contributions. A measurement of the neutron knockout from 32 Mg found that the ground state consists dominantly of 2p-2h N = 20 cross-shell configurations [27]. The SF's for the intruder 2p 3/2 and the 1 f 7/2 orbitals were found to be 0.59 (11) and 1.19(36) respectively. The abnormal spin of 1/2 + observed for 31 Mg showed a deformed ground state with an intruder configuration [28].
On the neutron-rich side of the N = 20 gap, the beta decay of 33 Mg suggested an abnormal spin of 3/2 + due to a 1p-1h configuration resulting from an inversion of the d 3/2 and f 7/2 orbitals [15]. The intermediate energy Coulomb excitation of 33 Mg populated a state at 485 keV that was attributed to a rotational excitation and suggested a 5/2 + spin for this state [29]. Such an excitation would support 33 Mg having similar deformation as observed for other nuclei in the island of inversion. The measurement of a negative magnetic moment for the 33 Mg ground state has however suggested a spin of 3/2 − with a 2p-2h configuration [16]. In this configuration, it was noted that there are an odd number of neutrons in the f p shell. Very recently the large branching ratio observed for the beta decay of 33 Mg to the 5/2 + ground state of 33 Al suggests the 33 Mg ground state to be 3/2 + [17]. It was proposed to have an admixture of 1p-1h with two neutrons in the f 7/2 orbital and 3p-3h with four neutrons in the f 7/2 orbital configurations for 33 Mg. In the proposed configurations ( Fig. 4 [17]) the odd neutron occupies the d 3/2 orbital and no occupancy in the 2p 3/2 orbital is discussed. In this work the ground state configuration of 33 Mg is investigated by measuring the momentum distribution after removal of one neutron.
The experiment was performed at the fragment separator, FRS, at GSI, Darmstadt, Germany. The schematic view of the experi- ment is shown in Fig. 1 where the first three foci F1-F3 of the FRS are dispersive while the last F4 is achromatic. The 33 Mg secondary beam was produced via the fragmentation of a 1 A GeV 48 Ca primary beam on a 6.347 g/cm 2 thick Be target. The nuclei produced were then separated and identified by transporting them to the second dispersive focus of the FRS. Detectors placed at F1 and before the target at F2 served to identify the secondary beam on target. The plastic scintillators at F1 and F2 measure the time-offlight (TOF) between these locations. The position on the reaction target was determined by tracking using the time projection chambers (TPC) at F2. The momentum of the incident particles can be determined from the position and the central magnetic rigidity of the first half of the FRS. In addition, it can be derived from the TOF measurement. The energy-loss ( E) measured using the multisampling ionization chamber (MUSIC) provided the charge ( Z ) of the nuclei. The combined information from Bρ-E-TOF allowed a complete event-by-event identification of the 33 Mg secondary beam with less than 1% contamination. A 4.05 g/cm 2 carbon reaction target was placed at the F2 dispersive mid-plane. The 32 Mg fragments from the one-neutron removal reaction were transported to F4 using the second half of the FRS. Here they were identified using the same principle as above with the TPC, MUSIC and scintillator detectors placed at F4.
To determine the momentum distribution due to one-neutron removal at the target, the FRS was operated in dispersion-matched mode. In this mode, the position distribution at F4 is a measure of the momentum distribution due to the reaction, independent of the initial momentum spread of the 33 Mg secondary beam. The energy loss in and the shape of the reaction target at F2 leads to some deviation from this ideal achromatic image condition as observed through a correlation of the horizontal position at the reaction target at F2 and the horizontal position at the F4 focus. This correlation could then be accounted for to improve the achromatic condition by making the position at F4 independent of that at F2.
Reactions occurring in the residual matter (detectors, vacuum windows) were separately measured in a setting where the carbon reaction target was removed. The magnetic rigidity of the second half of FRS was set to transport 32 Mg as the central fragment at F4. To subtract these background contributions, the number of 32 Mg fragments detected without the carbon target were normalized by the ratio of the incident beam ( 33 Mg) particles measured with and without the target. Fig. 2 shows the normalized background distribution (open circles). This distribution was then subtracted from the distribution with the carbon target (Fig. 2, open squares). The background subtracted momentum distribution is shown in Figs. 3-5. The momentum acceptance of the FRS is shown by the vertical dashed lines in Fig. 2. The momentum resolution was determined in a separate setting, where the second half of the FRS was set to transport the unreacted 33 Mg beam after the reaction target to F4. The observed momentum distribution fitted by a Gaussian function gives a measure of the resolution as 29.0 ± 0.2 MeV/c (FWHM). The width of the background subtracted distribution for 33 Mg → 32 Mg, shown by the black data points in Fig. 3, was 150 ± 3 MeV/c (FWHM). Accounting for the momentum resolution, the intrinsic width (assuming gaussian shape) is 147 ± 2 MeV/c (FWHM). The one-neutron removal cross-section is 74 ± 4 mb. The error bars shown in Fig. 3 include both statistical and systematic errors. The systematic errors arise from the 1.2% uncertainty in the measured target thickness and the estimated 5% uncertainty in transmission.
The data are interpreted in the framework of eikonal model calculations [30] based on a 32 Mg + n description of 33 Mg. The distributions calculated in this model have been folded with the experimental momentum resolution. The wavefunctions for the neutrons were obtained from a Wood-Saxon binding potential, the depth of which was adjusted to reproduce the effective neutron separation energy, including the excitation energy of the 32 Mg core for each configuration.
Following the conventional shell model with a shell closure at N = 20, the last neutron in 33 Mg should occupy the 1 f 7/2 orbital. The dotted line in Fig. 3(a) shows the distribution for the 32 Mg gs + n(1 f 7/2 ) configuration with the spectroscopic factor S = 1. It can be seen that this configuration does not describe the data either in shape or in magnitude thereby confirming that neutron(s) are distributed differently between the sd-shell and the f p-shell. This rules out a 7/2 − spin assignment in a 0p-0h description of the ground state of 33 Mg. The beta decay studies suggest the ground state of 33 Mg to be 3/2 + with an odd number of neutron(s) in the 1d 3/2 orbital either in a 1p-1h or a 3p-3h configuration. The solid line in Fig. 3(a) shows the calculated momentum distribution for 32 Mg gs +n(1d 3/2 ) with a spectroscopic factor of 1.0. This distribution fails as well in reproducing the shape and magnitude of the data.
The relatively narrow width of the momentum distribution suggests neutron removal from orbitals with low angular momentum. We consider neutron removal from the 2p 3/2 orbital, that may be lowered both due to deformation as well as the decreasing neutron separation energy for neutron-rich nuclei. The momentum distribution for the configuration with the 32 Mg core in its ground state and one neutron in the 2p 3/2 orbital resulting in a ground state spin of 3/2 − , is shown by the dashed line in Fig. 3(a) with a  Table 1). The MCSMmod has the 2p 3/2 orbital lowered by 1 MeV. spectroscopic factor of unity. Its width of 108 MeV/c (FWHM) is narrower than the data, suggesting that the core 32 Mg should also be in an excited state. This would increase the effective separation energy, thereby making the momentum distribution wider.
The Monte Carlo shell model (MCSM) calculation with the SDPF-M interaction, that has been considered to be very successful in describing the island of inversion, predicts the ground state of 33 Mg to be 3/2 − . A highly mixed configuration is predicted with the configurations and spectroscopic factors listed in Table 1. Fig. 3(b). shows that the calculated momentum distribution (dashed line) with the MCSM configurations (Table 1) does not explain the data. The experiment clearly shows the necessity for an enhanced low angular momentum component in the ground state of 33 Mg. The modified MCSM calculations with the 2p 3/2 orbital lowered by 1 MeV are shown in Table 1 (MCSMmod) and in Fig. 3(b) by the solid line. This leads to a better reproduction of the momentum distribution. The magnetic moment values predicted by the MCSM is −0.750 while that with the modified MCSM is −0.763 showing that it is minimally affected by the lowering of the 2p 3/2 orbital. Both these values are consistent with the experiment [16]. The present data thus provides a sensitive way to understand the gap between the 1 f 7/2 and the 2p 3/2 orbitals which is otherwise not known experimentally yet. The influence of the 2p 3/2 orbital is also discussed to be important towards understanding the magnetic moment of 34 Al [31].
We performed a search for a best fit to the data from χ 2 minimization with the spectroscopic factors as free parameters for the following configurations in the 3/2 − ground state of 33 Mg.  The ground state is described by α 1 · 32 Mg 2 + i is the spectroscopic factor. The maximum allowed values of the spectroscopic factor are restricted to the respective sum rule values for each case.
The 2s 1/2 orbital in the 32 Mg core is occupied in the ground state as well as in the 2 + excited states. The average occupation number predicted by MCSM is 1.899 and 1.891 for the first two 2 + states. This would mean that in the configuration for 33 Mg there is not much possibility of placing the odd-neutron in the 2s 1/2 orbital outside the 32 Mg core. In the configuration mixing considered in the fitting we have therefore restricted the allowed occupancy for the valence neutron of 33 Mg to be in the 2s 1/2 orbital to be 0.1.
−0.5 . Inclusion of higher core excited states in the fitting did not lead to any strength in them.
Therefore, the important observation in this work is that the narrow observed momentum distribution is a clear signature of significant occupation of the 2p 3/2 orbital. It is much larger than the prediction from the MCSM, which is unable to describe the data. The p-wave spectroscopic strength is larger than that observed for 32 Mg from one-neutron removal [27]. In 32 Mg the occupancy of the neutron in the 1 f 7/2 orbital (1.19(36) [27]) is larger than the 2p 3/2 orbital (0.59(11) [27] Although these nuclei are deformed, in this case we do not employ Nilsson orbitals to interpret the data due to several reasons. It is already known from the study of 11,12 Be [24,25] where deformation plays a key role in the inversion of the 2s 1/2 and 1p 1/2 orbitals that the momentum distributions are well explained using spherical single-particle wavefunctions. It was also applied successfully to the study of 28 Ne [26] and 32 Mg [27]. This reflects the fact that these distributions clearly exhibit the orbital angular momentum of the single-particle orbital. Moreover, the deformation in these neutron-rich nuclei has been understood in the context of particle-hole excitations within the spherical shell model. A Nilsson model that includes orbital re-arrangement in spherical configurations for neutron-rich nuclei is yet to be developed. Furthermore, its extension to reaction theory also needs to be developed. Moreover the excited states of 32 Mg do not exhibit the spectrum of a rigid rotor.
In summary, the longitudinal momentum distribution of oneneutron removal from 33 Mg was measured at 898 A MeV at the FRS of GSI. The one-neutron removal cross-section was measured to be 74 ± 4 mb. The narrow structure of the momentum distribution clearly supports significant occupancy in low angular momentum orbitals. Within the eikonal model, the data could be explained with a large spectroscopic strength for the 2p 3/2 orbital. The data could not be described by the spectroscopic strengths predicted by the MCSM with the SDPF-M interaction. The modified MCSM calculation with the 2p 3/2 orbital lowered by 1 MeV shows an improved agreement with the data. The magnetic moment is found to be less sensitive to this lowering and is consistent with experimental observation both for MCSM and modified MCSM. This shows the lowering of the 2p 3/2 orbital due to deformation and weaker binding in 33 Mg. Thus the island of inversion becomes more pronounced across the N = 20 gap. The present data has some sensitivity to the relative placing of the 1 f 7/2 and 2p 3/2 orbitals which is otherwise not known experimentally yet.