Competing particle–hole excitations in 30 Na: Constraining state-of-the-art effective interactions

The odd–odd nucleus 30 Na is studied via a one-proton, one-proton–one-neutron and one-neutron removal reaction using an intermediate-energy 31 Mg, 32 Mg and 31 Na radioactive ion beam, respectively. Combining high-resolution γ -ray spectroscopy with the selectivity of the three reaction mechanisms, we are able to distinguish multiple particle–hole conﬁgurations. Negative-parity states in 30 Na are observed for the ﬁrst time, providing an important measure of the excitation of the 1p1h/3p3h conﬁguration and hence the sd – pf shell gap. The extracted band structures and level energies serve as invaluable input for the theoretical reﬁnement of the effective interactions used in this region. © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP 3 .


Introduction
The nuclear shell structure and the corresponding magic numbers change dramatically as one moves away from the valley of beta stability, see Ref. [1] for a recent review. The breakdown of the N = 20 magic number was the first indication of this shell evolution when an excess in the binding energy of 31 Na was observed by Thibault et al. [2] in a mass measurement experiment. This was attributed to a deformed ground-state configuration [3][4][5]. Around 31 Na and N = 20 a number of other nuclei with the same properties (deformed ground state) have been observed, forming a region of deformed nuclei commonly referred to as the "Island of Inver-sion"; a collection of examples and references is given in Ref. [1]. In this region, the expected filling of the neutron orbitals (i.e. the N = 20 nuclei fully occupying the neutron sd shell) is not fulfilled and the ground-state configuration of these nuclei is dominated by the pf neutron orbitals. The driving mechanism behind this inversion is the gain in correlation energy as neutrons are promoted from the sd to the pf shell coupled with a reduced N = 20 shell gap, as recently discussed in Refs. [6][7][8][9].
For neutron number N = 20 and proton numbers Z = 10-12 (e.g. 30 Ne, 31 Na, 32 Mg), the gain in correlation energy by promoting two neutrons in the pf shell is large and therefore such intruder configurations become the ground state even in the presence of a relatively large sd-pf shell gap [10]. However, for nuclei with N < 20 the dominance of intruder configurations becomes less favored, because of the large correlation energy that an open-shell nucleus already gains from the interaction between protons and neutrons at normal states [10]. In the neutron-rich Na isotopic chain, 30 Na is of particular interest. The probability of intruder 2p2h 4 configurations in the lowest positive-parity state of 30 Na, as calculated by Utsuno et al. [10], varies dramatically as a function of the sd-pf shell gap and there is a sharp transition between a "normal" and "intruder" dominated configuration within a small change in the shell gap. This is not the case for 29,31 Na, where the transition between normal (0p0h) and intruder (2p2h) configurations is far more gradual. The sensitivity of the structure of this nucleus to the sd-pf shell gap turns 30 Na into a key nucleus and motivated this experimental investigation with particular emphasis on negative-parity states, arising from 1p1h and 3p3h excitations, which have not been experimentally observed so far.
The deviation of the low-lying properties of 30 Na from the USD predictions was already evident in 1978, when Huber et al. [11] measured a smaller magnetic moment in 30 Na than expected. Recent studies of excited states in 30 Na include intermediateenergy Coulomb excitation [12,13], proton inelastic scattering [14], a β-decay study [15] and a low-energy Coulomb excitation experiment [16]. All of these studies (except from [12]) agree that the ground state of 30 Na contains 2p2h excitations. In these experiments, only a few excited states in 30 Na have been observed, most of them in the β-decay experiment; all of them are considered to be of positive parity.
In this paper, we report on the low-lying structure of 30 Na, populated by three different nucleon-removal reactions. The oneneutron and one-proton removal reactions proceed as direct reactions and will lead to different final state populations. We have used this feature to associate the observed band structures to specific dominant npnh configurations. Specifically, we identify the key negative parity states for the first time.

Experimental details
The experiment was performed at the National Superconducting Cyclotron Laboratory at Michigan State University. 30 Na was populated via three nucleon-removal reactions, a one-proton (1p) knockout reaction from a secondary 31 Mg beam, a one-protonone-neutron (1p1n) removal reaction from a secondary 32 Mg beam and a one-neutron (1n) knockout reaction from a secondary 31 Na beam. 5 These secondary beams were produced by the fragmentation of a 140 MeV/u primary 48 Ca beam (delivered from the Coupled Cyclotron Facility) on a thick 9 Be production target (888 mg/cm 2 ). The secondary beams of interest were selected in the A1900 separator [17] and identified on an event-by-event basis through time-of-flight measurements. They were delivered to the experimental hall (S3 Vault) with an energy of 93 MeV/u for 31 Mg, 87 MeV/u for 32 Mg, and 94 MeV/u for 31 Na, where they reacted with the secondary 9 Be target (376 mg/cm 2 ). The rate of the secondary beams, for a momentum acceptance of 2%, was ∼2 × 10 4 pps for 31 Mg, ∼1 × 10 4 pps for 32 Mg and ∼ 700 pps for 31 Na. Gamma rays emitted from the reactions 9 Be( 31 Mg, 30 Na + γ )X, 9 Be( 32 Mg, 30 Na + γ )X and 9 Be( 31 Na, 30 Na + γ )X were detected with SeGA [18], an array of 16 32-fold segmented high-purity germanium detectors. For this measurement SeGA consisted of seven detectors at 37 • and nine detectors at 90 • surrounding the S800 target position. Outgoing 30 Na fragments were identified by energy-loss and time-of-flight measurements in the S800 spectrometer [19]. 4 The term npnh (n = 0, 1, 2, 3, 4) denotes the number of particles (np) promoted to the next oscillator shell and the number of holes (nh) that consequently emerge. 5 A distinction has been made to the terminology used for the different nucleonremoval reactions due to the direct nature of the 1p and 1n reactions. The latter two will be called knockout reactions hereafter.

Results
Doppler-corrected γ -ray spectra in coincidence with 30 Na fragments from the three reactions are shown in Fig. 1. These are assigned as γ rays de-exciting excited states in 30 Na. The 172 (2) [15]. The energy shift is due to the long lifetime of this transition (of the order of 500 ps), evident from its low-energy tail in the γ -ray spectrum. From a γ − γ coincidence analysis the observed level scheme of 30 Na is deduced in each reaction and shown in Fig. 2. We have identified four distinct structures (bands) based on the different intensities with which these are populated in each reaction. Band I: This forms the ground-state band of 30 Na. The state at 424 keV has already been established [13,12,14] and assigned to be rows are proportional to the intensities of the γ rays as measured in the singles spectra. Dotted arrows indicate γ rays observed in the singles spectrum but not in coincidence with other γ rays. In (a) this is because these γ rays are not in coincidence with any other γ ray. In (b) and (c) this is due to the limited statistics of our γ -γ matrix; however, these transitions have been firmly placed in the level scheme from the 1p knockout reaction. Tentative spin assignments (see Section 3) are shown in parentheses. In (d) the calculated levels have been adapted from Ref. [10]. The γ rays are shown for easy comparison between theory and experiment. the J = 3 member of the K = 2 rotational band built upon the 2 + ground state. A clear transition at 501 keV in coincidence with the 424 keV is observed, verifying and supporting the suggestion by Seidlitz et al. [16] that this level at 925 keV forms the J = 4 mem-ber of the ground state rotational band. In the singles spectrum we observe a 925 keV γ ray, which is not in coincidence with any other transition. We assign this γ ray as the one de-exciting the 925 keV level. The experimentally observed branching ratio of the 4 + 1 → 2 + 1 and 4 + 1 → 3 + 1 transitions is 60(8)% and 40(5)%, respectively, yielding smaller uncertainties compared to the one deduced in Ref. [16] (of 67(10)% and 33(10)%).
Band II: The first two transitions (151 and 365 keV) of this band are consistent with the results from the β-decay experiment of Ref. [15], therefore we adopt their spin assignments of 1 + and 2 + , respectively. The 151 keV level has a long lifetime (of the order of 500 ps) resulting in a de-exciting γ ray with a characteristic tail towards lower energies and a shift in its energy (148 keV). The states at 1032 and 1263 keV are observed for the first time together with their de-exciting γ rays.
Band III: This band has been observed for the first time. It is strongly populated both in the 1p1n removal and in the 1n knockout reactions. These reactions involve removing a neutron from the sd and pf shells, with the latter leading to the population of negative-parity states. We therefore argue that Band III is of negative parity, see discussion in Section 4.
Band IV: This band has been identified as a separate structure, despite the fact that its level lies very close to the 925 keV level of Band I, since it has been populated only in the 1p1n removal reaction. According to the experiment of Ref. [15], the 410 keV transition belongs to the 0p0h 1 + state. We believe the reason we populate this state only in the 1p1n reaction is due to the fact that this reaction is not a purely direct two-nucleon knockout reaction. It rather includes evaporation channels, i.e. a neutron evaporates after one-proton removal. Therefore, the decay will also have a statistical character and levels not observed in the direct 1p ( 31 Mg → 30 Na) and 1n ( 31 Na → 30 Na) reactions will be populated in this reaction.

Discussion
State-of-the art shell model calculations using the SDPF-M interaction, which allow neutron excitations across the N = 20 shell gap and reproduce the spectroscopic observables of nuclei in this region, suggest that bands built on 1p1h, 2p2h, 3p3h and 0p0h neutron configurations should coexist at low energies [10]. Here we argue that exploiting the different reaction mechanisms, such configurations can be preferentially and selectively populated and identified. We will only use the 1p and 1n knockout reactions, since our arguments are based on the direct character of the reaction. In a simple picture, the reaction mechanism and the states that are populated in 30 Na are shown in Fig. 3.

Structure assignment
1p knockout ( 31 Mg → 30 Na): In the ground state of 31 Mg, the 12 protons are filling the sd shell and two of the 19 neutrons lie in the pf shell [20]. The 1p knockout reaction thus will result in populating 2p2h states in 30 Na, as is illustrated in Fig. 3(a).
1n knockout ( 31 Na → 30 Na): The ground state of 31 Na may be considered a proton hole coupled to a 32 Mg core, where 2p2h and 4p4h configurations dominate its ground state [6]. The 1n knockout thus will result in probing 1p1h, 2p2h, and 3p3h states in 30 Na, depending on whether the neutron was knocked out from the pf shell in a 2p2h configuration, the sd shell in a 2p2h configuration or the pf shell in a 4p4h configuration, respectively, as shown in Fig. 3(b). So in this reaction, all 2p2h, 1p1h and 3p3h configurations may be populated.
By analyzing the number of γ rays and the number of 30 Na fragments, we were able to establish the strength with which each Knocking out 1n from 31 Na can either populate 3p3h states in 30 Na when the neutron is knocked out from a 4p4h ground state in 31 Na, or 2p2h and 1p1h configurations when the neutron is knocked out from the 31 Na ground state dominated by 2p2h excitations.

Table 1
Intensity of each structure as was populated in the 1p and 1n knockout reactions. The npnh configuration is assigned according to the argumentation made in the text. The K assignments are taken from Ref. [10]. band (I, II, III) is populated in either the 1p or 1n knockout reaction. In the population of Band I, the ground state contribution is also included, i.e. its population comes from the intensity of the 424 and 925 keV γ rays plus the direct population of the ground state. 6 These numbers are shown in Table 1.
In the 1p knockout reaction, Bands I and II take most of the intensity with 90% population. We assign these bands as 2p2h excitations. The lower population of Band III (10%) in the 1p knockout suggests this structure does not have a large 2p2h component. We interpret the small population of this band as being due to side feeding and configuration mixing.
In the 1n knockout reaction the picture is different. Here we strongly populate all three bands, leading us to conclude that Band III corresponds to 1p1h and 3p3h excitations, since we have already assigned Bands I and II as of 2p2h nature. Hence Band III is of negative parity. 6 The direct ground state population is calculated by subtracting the absolute intensity of the γ rays that feed the ground state from the total number of 30 Na fragments.

Comparison to shell model predictions
The theoretical predictions in this region depend strongly on the effective interaction. The current experimental results with the observation of a rather complete set of coexisting configurations provide an important test for the theoretical predictions.
Here, we compare our experimental findings with the calculations of Ref. [10], as this is the last published theoretical calculations regarding this odd-odd nucleus. Nevertheless, the present experimental work will serve as an invaluable reference for future theoretical calculations and their refinement.
Utsuno et al. [10] predict the co-existence of different particlehole configurations and we confirm this experimentally; Bands I, II, III and IV have been assigned as being of 2p2h, 2p2h, 1p1h/3p3h and 0p0h nature, respectively. Our assignment of a dominant 1p1h/3p3h configuration to Band III provides the first evidence for the location of the negative parity states in this key ("transitional") nucleus.
As shown in Fig. 6(d) of Ref. [10], the relative excitations between the various particle-hole configurations and specifically that separating the 2p2h and 1p1h configurations can constrain the sd-pf shell gap. The measured excitation energy of 338 keV for the 1p1h/3p3h band (Band III) above the 2p2h ground state (Band I) is close to the calculated value in Ref. [10] of ∼140 keV.
Our data provides important confirmation of the calculation, supporting the notion of a reduced sd-pf shell gap (3.3 MeV) at While likely within expected uncertainties for shell model calculations, when comparing the measured level energies (Fig. 2(a)-(c)) with the calculated ones ( Fig. 2(d)), we find systematically an overestimation of the level energy for the positive-parity states by a few 100 keV, while the negative-parity states are lower in energy than what is observed in the present experiment. It is interesting to speculate whether these systematic differences reflect a further reduction in the sd-pf shell gap and can then help refine the effective interactions used in these shell model calculations.

Summary
We have studied the odd-odd nucleus 30 Na via three different nucleon-removal reactions, which allowed us to identify multiple npnh configurations, co-existing at low energies. The relative excitation of each configuration constrains the sd-pf shell gap at N = 19 and helps refine the phenomenological interactions used in this region of the nuclear chart.