Intruder features in the island of inversion: The case of 33 Mg

The 33 Na b decay was studied online using mass separation techniques and a ﬁrst description of the level structure of the neutron-rich isotope 33 Mg, with N 5 21, has been obtained. The experiment involved the measurement of b - g , b - g - g , and b - n - g coincidences as well as neutron spectra by time-of-ﬂight technique. The ﬁrst low energy level scheme for the daughter nucleus 33 Mg is given with ﬁve bound states. Spin and parity assignments are proposed according to b feedings and g -ray multipolarities. b -strength distribution is evaluated, taking into account 1 n - and 2 n -emission channels and it is compared with the calculated GT strength distribution. In particular, the 1 p -1 h and 2 p -2 h excitations are shown to result in substantial contribution to the low energy conﬁgurations. Allowed b branch to the 33 Mg g.s. @ log ft 5 5.27(26) # gives direct evidence for the inversion of n ( f 7/2 ) and n ( d 3/2 ) states.


I. INTRODUCTION
The first signs, indicating the inversion in shell ordering at Nϭ20, were observed as irregularities in the binding energies of neutron-rich AХ32 nuclei ͓1͔.Theoretical calculations, both Hartree-Fock ͓2͔ and shell model ͓3-6͔, were able to reproduce this unexpected increase of binding energy by allowing neutron excitations from the sd into the fp shell, which lead to deformed ground-state configurations.This particular region of deformation, the so-called island of inversion, has now become accessible through many experimental methods.Out of these, Coulomb excitation has allowed a systematic study of B(E2;0 1 ϩ →2 1 ϩ ) values for even Si nuclei ͓7͔, as well as for few other close-by nuclei at the borders of the island of inversion ͓8,9͔.Also, recent mass measurements for neutron-rich 29ϽAϽ47 nuclei have revealed a similar effect around Nϭ28, where prolate deformations of the ground state are predicted ͓10͔.Another powerful method to study the detailed structure of neutron-rich nuclei is the ␤ decay.In our earlier work ͓11͔ we used this method to determine the level schemes of 34 Si(Nϭ20) and 35 Si(Nϭ21) of which the latter provided new information on the evolution of neutron orbits outside the Nϭ20 closed shell.These two nuclei have, as expected, a ground-state configuration with a main 0p-0h component.
In the present study we have advanced one step further and aim to determine the low-energy level scheme of 33 Mg(Nϭ21) via the ␤ decay of 33 Na.Now we enter the island of inversion ͓5,6͔ where predictions using a simple shell model are no longer valid since 2p-2h, 1p-1h, and 0p-0h configurations are in close competition at low energy.
The 33 Na→ 33 Mg decay has been observed already in 1984 ͓12,13͔, however, no information on the decay scheme, or on the level scheme of 33 Mg, has been available up to now.
Apart from interest in the shell structure there is also an interest related to the vicinity of the neutron drip line.New efforts have been made to redefine the limit of particle stability in the Nϭ21 region.Recent mass evaluations ͓14͔ predict 31 Ne unbound, while the finite-range droplet model ͑FRDM͒ ͓15͔ predicts the stability of 31 Ne in agreement with an experimental result ͓16͔. 31 Ne is expected to be the lightest bound Nϭ21 isotone since clear evidence for the instability of 30 F has been reported ͓17͔, while 31 F with 22 neutrons was recently observed ͓18͔.
In this paper we present ␤-␥ measurements on 33 Na sources and we compare the results with those of shell model calculations.The analysis of the ␤-delayed neutron emission, registered in the same experiment and the results of which are used in our discussion, are presented in a separate paper ͓19͔.

II. EXPERIMENTAL PROCEDURES
The 33 Na activity was produced in fragmentation reactions by 1.4 GeV protons, provided by the PS/Booster at CERN, impinging on a uranium carbide target.The target ͑46 g/cm 2 for U͒ and its temperature ͑above 2000 °C͒ were selected in order to optimize the release of activities with short half-lives.The intensity of the primary beam was 3 ϫ10 13 protons/pulse and the interval between pulses was a multiple of 1.2 s.A typical average yield for 33 Na was 2 atoms/s.The reaction products were ionized by a surfaceionization source, accelerated and mass separated by the ISOLDE facility.During the first 20 ms, starting from each proton pulse, the separator beam gate was open and ions were transported to the experimental setup on to a collection tape.For the remaining time until the next arriving proton pulse the separator beam was deflected off in order to optimize the collection of only the short-living components and to obtain a pure decay period for half-life determination.In order to reduce the amount of contaminants and longer-lived daughter activities, the tape was moved periodically.For detection, we adopted a similar setup as in our previous study of the 34,35 Al decays ͓11͔.The collection point was surrounded by a thin plastic scintillator detector, covering about 70% of the total solid angle.This provided a trigger signal for ␤-␥ coincidences and a start signal for the neutron timeof-flight measurements.␥ rays were detected with two largevolume Ge detectors allowing to record both ␤-␥ and ␤-␥-␥ coincidences.Furthermore, due to the large energy window for ␤-delayed neutrons in the 33 Na decay, neutron detection played an important role.In fact, more than 20 MeV is available for the ␤ decay of 33 Na, out of which only 2.07 MeV corresponds to particle-bound levels in 33 Mg.Therefore, eight low-threshold neutron detectors ͓20͔ of 15% intrinsic efficiency were located around the collection point ͓19͔, with a flight path of 50.8 cm.Neutrons were detected as ␤-n coincidences to obtain neutron time-of-flight spectra.In some cases, ␤-n-␥ coincidences could be detected as well.

III. RESULTS
Primary identification of ␥ rays belonging to the 33 Na decay was possible as the 33 Na lifetime ͓T 1/2 ϭ8.2(4) ms, ͓12͔͔ is short compared to the daughter isotopes ( 31-33 Mg) and contaminants, such as multiply-charged ions ͑e.g.,

198
Bi,qϭ6 ϩ ͒.Since the decay of 33 Na consists of several decay channels, including pure ␤ and ␤-delayed one-, two-, and even three-neutron decays, an important parameter in the analysis was the time of detection of a ␤-␥ or ␤-n event.A half-life measurement, performed from ␤, ␥ and delayed neutron activities, resulted in an average value, T 1/2 ϭ8.0(3) ms, in good agreement with the reported one ͓12͔.In Fig. 1 two ␤-gated ␥ spectra are presented for Aϭ33, corresponding to two time windows relative to the proton pulse.The first spectrum corresponds to a time interval up to 50 ms after the arriving protons pulse ͓͑1a͔͒ and the second corresponds to the following 50-500 ms ͓͑1b͔͒.Clearly only the first spectrum contains ␥ rays from the 33 Na decay.
Altogether, 19 ␥ transitions could be assigned to the ␤ decay of 33 Na based on the decay rate of their intensity.Six have been reported in Ref. ͓12͔, out of which four are deexcitations in 33 Mg and two in 32 Mg.All transitions, observed in the first time window of this measurement, are listed in Table I together with the information on coincident lines.In this table there are five ␥ transitions following the ␤-delayed 1n emission, thus belonging to the level scheme of 32 Mg ͓21-23͔.Among them, the 885 keV line is the most intense one and corresponds to the known 2 1 ϩ →0 1 ϩ transition.This transition was clearly observed in coincidence with neutrons.Three ␥ transitions belong to the 31 Mg level scheme ͓23͔ and correspond to ␤-delayed 2n emission.All three were seen in coincidence with neutrons, although two of them weakly.Special attention should be paid to the 221 keV transition which can be partly explained by a transition in 31 Mg but is also related to the 33 Mg and 32 Al level schemes as it results from coincidences observed with 484 and 735 keV, respectively.The latter has been observed previously ͓24,25͔.
Out of the remaining 11 ␥ rays 6 are used for building the level scheme of 33 Mg.All these transitions, except the 546 keV one, are placed in the scheme according to ␥-␥ coincidence data or according to the sum of ␥ ray energies.The 546 keV ␥ transition is the second most intense transition belonging to the 33 Na decay scheme, but it cannot be associated to any other transition on the basis of coincidences or energy balance.Its origin could be explained by three possible choices.First, it could de-excitate a level at 546 keV.Second, it could correspond to a de-excitation of one of the levels established through ␥-␥ coincidences, but with a cascade ␥ transition outside our observation limits ͑time or en-ergy͒.Third, it could belong to the 32 Mg level scheme, related to ␤-delayed 1n emission.The third hypothesis can be ruled out as this strong line has not been observed in coincidence with neutrons in our experiment and it was not seen in previous experiments on 32 Mg ͓7,23͔.Therefore, only the first two assumptions for the 546 keV placement in the level scheme will be discussed later.
In order to evaluate the ␤ branchings of the 33 Na decay to 33 Mg, absolute ␥ intensities together with the P n values had to be determined.For the evaluation of absolute ␥-transition intensities, a use was made of the opportunity to measure, with the same setup and in the same experiment, the activity of a well-known ␤ emitter 26 Na.This procedure has already been used previously and is described in Ref. ͓26͔.Then, with a 33 Na beam collected at the same position, ␤-multiscaling and ␥-ray energy spectra are simultaneously recorded and the 33 Na contribution clearly separated from competing activities.Using the ratio of ␤ and ␥ efficiencies obtained with 26 Na, we obtain, after introducing corrections for the differences in ␥-ray energy, the absolute intensity of the transitions associated with 33 Na.
The ␤-delayed neutron emission probability of 33 Na could not be directly determined from ␥ intensity analysis only because the decay scheme of 33 Mg and 33 Al are still unsettled and also because in our experiment the direct production of 33 Al from the target is not negligible.Intensities of the ␥ transitions in 33 Mg were compared with those of ␥ transitions following the ␤ decay of the daughter activities 32 Al(1941 keV) and 31 Mg(947 keV).An upper limit for the P n value could be evaluated by comparing the ␥ intensities in the ␤-delayed neutron branch and in the direct ␤ decay channel.It resulted in the total P n less than 86% with P n ϭ P 1n ϩ2 P 2n .For more specific P n analysis, the P 1n / P 2n ratio was determined from the relative 32 Al and 31 Mg activities, resulting in 3.6͑9͒.The determination of the P n value could be done with independent ␤ and neutron measurements by comparing the absolute ␤ and ␤-coincident activities.The details of this analysis are described in Ref. ͓19͔.The total neutron emission probability resulted in P n ϭ73(6)%.Finally, the absolute P 1n and P 2n values were obtained from the P 1n / P 2n ratio, resulting in 47͑6͒ and 13͑3͒ %, respectively.The previously reported values in Ref.
The decay scheme of 33 Na is given in Fig. 2. For all transitions placed in 33 Mg, except for the 546 keV one, supporting evidence is provided by the ␥-␥ coincidences.The difficulty to assign the strong 546 keV line in the level scheme required additional steps.Its decay rate was carefully measured ͓T 1/2 ϭ7.7(0.9)ms͔ and found in excellent agreement with the 33 Na half-life.As mentioned previously, the absence of neutron-␥ coincidences rules out attribution to Aϭ32 or 31.Consequently, there are two possible interpretations for the 546 keV transition.First of all, if this transi-  tion would originate directly from a postulated level at 546 keV, it should be preceded by an allowed GT transition that, according to the intensity analysis, is also the case for the levels at 705 and 1243 keV.However, there are no connecting transitions observed between this hypothetical 546 keV level and the other two mentioned here which have the same parity and for which the angular momentum difference ⌬J should be low.As there are no reasonable explanations why these transitions should be inhibited, or beyond our observation, this interpretation is very unlikely.
In the remaining alternative, the 546 keV would be associated with an unobserved transition in the decay of one of the other levels validated by ␥-␥ coincidences, 1243 or 705 keV.Let us first discuss the case of the 1243 keV level.In order to include the 546 keV transition in its decay, an unobserved transition of 697 keV is needed.Yet, at this energy, any transition with ⌬Jр2, even with a change of parity, would satisfy the timing and energy requirements of our experiment.The last possibility is then to associate the 546 keV transition with the decay of the level at 705 keV.This fixes the unobserved transition to 159 keV.At this energy, a transition with multipolarity Ͼ1 and involving a parity change would be unobserved as our ␤-␥ coincidence window was set at 500 ns.Therefore we are led to retain this last assumption and place the 546 keV transition decaying from the 705 keV level and followed by an unobservable 159 keV transition.Only a dedicated experiment can validate these conclusions.
Once we have the level scheme, the absolute ␥ intensities allow to determine the ␤ branches and the corresponding log ft values ͑Table II͒.The intensity of the ground-state branch is obtained from the difference between P 0n ϭ(1 Ϫ P 1n Ϫ P 2n ) and the sum of the feedings to ␥-emitting levels.The quoted errors result mainly from the uncertainty of the estimated Q ␤ value ͓14͔ and the experimental intensities.From Table II, evidence is found for three allowed ␤ transitions to 1243, 705 keV and g.s. of 33 Mg, the branch to the 1243 keV level being at the limit of the recommended range of allowed transitions ͓27͔.
In the decay scheme of Fig. 2, proposed J values result from ␤and ␥-decay selection rules and simple shell model configuration arguments.In this respect, the positive parity deduced from the allowed ␤ branch to the ground state can be most easily explained with a two-particle-one-hole neutron state in 33    2.9 ͑1.2͒ 5.97 ͑25͒ a The feeding of this level is assumed to be neglible, see text.III.Total ␤ Ϫ strength in each spin channel from the first two levels of 33 Na.The half-life obtained in these two cases has to be compared with the experimental one: 8.0͑3͒ ms.According to the proposed level scheme, the 159 keV level decays by a long-lived, and therefore unobserved, M 2 transition.For the 484 keV level, the E1 g.s.transition dominates completely the competing E2 branch to the 159 keV level.For the strongly ␤-populated 705 keV level, the three possible ␥ transitions are observed with comparable intensities, thus of the same multipole order.In our case, this order is restricted to dipoles as their decay time is short.In addition, dipoles to 3 2 and 7 2 levels fix J ϭ 5 2 ϩ to the 705 keV level.Finally, for the 1243 keV level, the lack of transitions to levels at 705 and 159 keV suggests J ϭ 1 2 ϩ , the observed decay modes being then M 1 and E1, respectively.
In the decay scheme ͑Fig.2͒, we have also reported the main result of the analysis of the ␤-delayed neutrons, emitted in the 33 Na decay, and discussed in a separate paper ͓19͔.Two strong peaks have been observed in the time-of-flight spectrum at E n ϭ800(60) and 1020͑80͒ keV and attributed to 33 Na decay with exclusion of the other delayed neutron emitters contributing to the spectrum ͑ 33 Mg, 32 Mg, and 33 Al͒.From our analysis ͓19͔, it appears that these neutrons populate the 2 ϩ state at 885 keV in 32 Mg locating around 4 MeV excitation energy in 33 Mg, the main part of the Gamow-Teller strength measured in this experiment.

IV. DISCUSSION
The ␤ decay of 33 Na has been compared to shell model calculations.For n-rich Aϭ33 nuclei, the situation is complicated since they are situated in the so-called island of inversion where many kinds of excitations (0p0h,1p1h,2p2h) are coexisting.The discussion proceeds in three steps: first we discuss the nature of the parent nucleus 33 Na, then we analyze its beta decay and embed the final states produced by the shell model into the low-lying level spectrum observed for 33 Mg.

A. Shell model structure of 33 Na
Following our previous study of 34,35 Si ͓11͔, we work in the sd-p f valence space.Three different kind of excitations are considered, 0ប, 1ប, and 2ប and no mixing is assumed between 0ប and 2ប states.The interaction used is the one from Ref. ͓28͔, which was adjusted in our recent publication ͓11͔ to take into account the reduction of the gap between the 1 f 7/2 and 2p 3/2 neutron orbitals in 35 Si.The calculated low-lying level spectrum of 33 Na is shown in Fig. 3.No ambiguity on the 0ប nature of the ground state is seen.The first 1ប and 2ប states are lying around 1 MeV.Nevertheless, the first two states, 3 2 ϩ and 5 2 ϩ appear to be very close in energy.This degeneracy originates from the three protons in the the 1d 5/2 orbital where the seniority 0 and 2 states coexist as can be already seen in the spectrum of 19 O.We will then consider both cases ͑ 3 2 ϩ and 5 2 ϩ ͒ for the ground state in the discussion of the ␤ decay.

B. Gamow-Teller ␤ decay of 33 Na
The schematic decay of 33 Na is shown in Fig. 4. Two paths can be considered for the decay: one is to transform an sd neutron into an sd proton, the other is to transform an fp neutron into an fp proton.In both cases, the final state obtained in 33 Mg is a 1ប excitation.But the important difference is that in the second case, the final state will be located at much higher excitation energy than in the first case.As we consider no sd-f p neutron excitations for the 33 Na groundstate, the ␤ Ϫ decay exhauts the full 3(NϪZ) Ikeda sum rule, which amounts here to 33 GT units.Table III gives the B͑GT͒ values in the different spin channels.
In Figs. 5 and 6 we show the detailed Gamow-Teller strength functions obtained with J ϭ 3 5 ϩ or 5 2 ϩ as the ground states of 33 Na.The standard quenching factor of 0.77 ͓29͔ was used to renormalize the operator.In these figures no real distinction can be observed between the two spin-parity states.This is not surprising since both parent states are similar in nature.Both calculations show ␤ decay branches to the 33 Mg ground-state, to the first 5 2 ϩ state at 0.811 MeV and to a multiplet of states located around 4 MeV.This is in a full agreement with the experimental results exhibiting the strong ␤ branches deduced from the delayed neutron measurement ͓19͔.The computed half-lives are also very similar, 2.70 and 2.76 ms.The agreement with the experimental half-life can be considered fair since this observable is determined solely by the small fraction of the strength function leading to lowlying states.In particular, few states determining the half-life carry only a few percent of the whole strength and such an accuracy is evidently very difficult to achieve in the theoretical calculation.The key feature is indeed the nature of the ground state of this nucleus which is predicted to be a J ϭ 3 2 ϩ state resulting from a 1p1h excitation, i.e., a 2p1h configuration in this Nϭ21 nucleus.This spin-parity value appears to be confirmed by the experiment.For the excited levels, the predicted coexistence is supported by the presence of many states at low energy.This relatively high number of states, compared to 35 Si, makes a one to one assignment difficult between the observed and the calculated states.Nevertheless, the main features of the 33 Na decay and the level structure of 33 Mg is understood.The drawback of the present calculation was the absence of mixing which was not taken into account for 0p0h and 2p2h configurations.In particular, the close proximity of ( 7 2 Ϫ ) 1,2 states does not support such a strong approximation.Therefore, following the procedure previously applied to 34 Si ͓11͔, we performed a two by two mixing by calculating the nondiagonal matrix element between these fixed configurations.The resulting spectrum is shown is Fig. 8.The repulsion between the two 7 2 Ϫ states has split the degeneracy and lowered strongly the first 7 2 Ϫ level.This gives a theoretical candidate for the long-lived M 2 isomer suggested by the present experiment ͑see Sec.III͒.In conclusion, the case of 33 Mg is of particular interest as it shows the first observation of inversion of states in this region.It provides a nice example of the selectivity of ␤ decay measurements that can provide a direct modelindependent evidence for a positive parity ground state of 33 Mg.

FIG. 1 .
FIG. 1. Gamma spectra measured during the first 50 ms ͑a͒ and the next 450 ms ͑b͒ after a proton pulse.
FIG. 2. Experimental decay scheme for 33 Na.Reported values for the energy of the levels and relative intensity in the different channels are from the present work and from the analysis of the neutron activity reported in Ref. ͓19͔.J values indicated for 33 Mg levels, correspond to most plausible values according to beta feedings and gamma intensities ͑see text͒.Q ␤ value is from Ref. ͓14͔.

ϩ
FIG.3.Calculated low-lying states of 33 Na.The indicated npnh correspond to a neutron excitation of the core, to be coupled to the odd nucleon.

FIG. 5 .
FIG. 5. ␤ decay of 33 Na.Calculated values of Gamow-Teller strength distribution versus excitation energy in the final nucleus, assuming 3 2 ϩ for 33 Na ground state.

ϪϪ
are expected at low energy.The value proposed for E x ϭ159 keV) are supported by the ␥-ray results.

C.
The structure of 33 Mg within the shell model Shell model calculations for the states of 33 Mg are shown in Fig. 7.The complexity of the spectrum is evident and derives from the coexistence of different kind of excitations.

TABLE I .
Energy and intensity of the ␥ rays attributed to the 33 Na decay.Energy ͑keV͒ Intensity a ͑relative͒ a Intensities are relative to the 885 keV ␥ ray.The intensity per 100

TABLE II .
␤ intensities and logft values in the 33 Na ␤ decay to bound levels in 33 Mg.