Shell evolution approaching the N=20 island of inversion: structure of 26Na

The levels in 26Na with single particle character have been observed for the first time using the d(25Na,p gamma) reaction at 5 MeV/nucleon. The measured excitation energies and the deduced spectroscopic factors are in good overall agreement with (0+1) hbar-omega shell model calculations performed in a complete spsdfp basis and incorporating a reduction in the N=20 gap. Notably, the 1p3/2 neutron configuration was found to play an enhanced role in the structure of the low-lying negative parity states in 26Na, compared to the isotone 28Al. Thus, the lowering of the 1p3/2 orbital relative to the 0f7/2 occurring in the neighbouring Z=10 and 12 nuclei -- 25,27Ne and 27,29Mg -- is seen also to occur at Z=11 and further strengthens the constraints on the modelling of the transition into the island of inversion.

excitation energies and the deduced spectroscopic factors are in good overall agreement with (0+1) ω shell model calculations performed in a complete spsdf p basis and incorporating a reduction in the N = 20 gap. Notably, the 1p 3/2 neutron configuration was found to play an enhanced role in the structure of the low-lying negative parity states in 26 Na, compared to the isotone 28 Al. Thus, the lowering of the 1p 3/2 orbital relative to the 0f 7/2 occuring in the neighbouring Z=10 and 12 nuclei - 25,27 Ne and 27,29 Mg -is seen also to occur at Z=11 and further strengthens the constraints on the modelling of the transition into the island of inversion.
Keywords: 81V35, nuclear structure, 26 Na, γ-Ray transitions, transfer reactions, shell migration The breakdown of the shell model magic number N = 20 for neutrons, in the case of neutron-rich isotopes near 31 Na and 32 Mg [1,2], has led to the concept of an "island of inversion", where neutrons preferentially occupy orbitals above the normal N = 20 gap, leaving vacancies below it [3,4]. The mechanism is now understood in terms of the shell model to involve specific valence nucleon interactions [3,5] as well as a monopole shift in the effective single particle energies [6]. In a wider picture, the monopole migration of levels is understood as arising from the tensor and three-body components of the nucleon-nucleon force between valence nucleons in partially filled orbitals [7,8].
The level schemes of nuclei with odd N , such as N = 15 and N = 17 ( Figure 1), provide important insight into the evolution of the single particle structure approaching the island of inversion. Between Z = 8 and 14, the proton d 5/2 orbital is filled, and this affects the energies of the neutron orbitals. The reference energy is the 7/2 − level, as it is expected that the energy of the f 7/2 neutron orbital (a j > orbital, i.e. j = + 1/2) will depend relatively weakly on the filling of the proton d 5/2 orbital (also j > [7]). In the case of 21,23 O the J π = 7/2 − assignments are tentative [14,15] but the small width observed in 23 O [15] supports the 7/2 − assignment. The systematics and also the shell model [15] suggest a near degeneracy between the 3/2 + and 3/2 − states in oxygen. As may be seen, 26 Na falls in a key region where the shell closures at N = 20 and 16 are weakening and strengthening respectively. As such, compared to the isotone 28 Al, the levels in 26 Na corresponding to a neutron in the 0d 3/2 orbital would be expected to lie higher in energy, whereas states with a neutron occupying the 1p 3/2 orbital may be de-generate with the 0f 7/2 levels or even lie below them 3 . In order to explore this behaviour the present study has been undertaken to locate the corresponding levels using neutron transfer onto 25 Na, produced as a secondary beam. As will be shown, the detection of coincident gamma-rays has dramatically enhanced the excitation energy resolution and provided vital complementary information on the spins and parities of the populated levels.
Previous studies have identified the ground state of 26 Na as having J π = 3 + [16,17] and there are three 1 + states fed by the beta-decay of 26 Ne [18,19]. Apart from the low-lying quartet of states (3 + , 1 + , 2 + , 2 + ) below 407 keV [20], which have been very recently probed via Coulomb excitation [21], the many observed states [20,22,23,19] have a structure that is almost completely unknown. The (d,p) reaction employed here will selectively populate the levels with a predominantly single-particle structure, which are those most useful for testing shell model predictions.
The experiment was undertaken at the ISAC-II facility at TRIUMF. A pure beam of 5.0 MeV/nucleon 25 Na ions of intensity 3×10 7 pps was employed to bombard a self-supporting (CD 2 ) n deuterated polythene foil of thickness 0.5 mg/cm 2 . The target was mounted at the centre of the SHARC silicon strip detector array [24] which was surrounded by 8 segmented germanium clover gamma-ray detectors of the TIGRESS array [25]. Four clovers were centred at 90 • in the laboratory and four at 135 • , giving a total absolute photopeak efficiency of some 3% at 1.33 MeV. In order to identify events in SHARC arising from fusion-evaporation reactions initiated on the carbon in the target (a source of significant background), an Al foil followed by a thin plastic scintillator detector (the TRIFOIL [26]) were installed 400 mm downstream of the target. The Al thickness of 30 µm was chosen to stop fusion-evaporation residues whilst transmitting the beam and the products of direct reactions [27]. Further, the scintillator foil was thin enough (10 µm) to transmit the radioactive beam.
Events were recorded whenever a particle was registered in SHARC, including a logic signal to indicate whether the TRIFOIL had fired during the same beam bunch. The TRIFOIL allowed the events corresponding to the production of 26 Na via the (d,p) reaction to be highlighted in the analysis in order to optimise the software gating. In the final determination of the absolute differential cross sections of the (d,p) reaction, the TRIFOIL requirement was not imposed, however, owing to its efficiency being dependent on the position on the foil, and hence on the angle of the recoil proton [28]. The coincident gamma-rays were recorded in TIGRESS and their energies were corrected for Doppler shift (v/c ≈ 0.1) [28].
The excitation energy spectrum for states populated in 26 Na, reconstructed from the observed energy and angle of the protons, is shown in Figure 2 where the resolution is 350 keV (FWHM). This figure illustrates how closely spaced states could be distinguished using the gamma-ray data. The FWHM resolution in gamma-ray energy at 1.8 MeV, after Doppler correction, was 18 keV at 135 • and 23 keV at 90 • . This resolution, rather than that of the proton-derived excitation energy spectrum, defined the precision with which individual states could be selected. On the other hand, the protonderived excitation energy was critical in determining the energy at which the 26 Na was populated in the (d,p) reaction (i.e. prior to gamma-decay of the states). In addition, gating on this excitation energy was employed in the determination of the gamma-ray branching ratios [28].
Elastically scattered deuterons from the target were prominent in the spectrum of particle energy versus angle and the corresponding differential cross section was extracted [28]. Optical model parameters for d+ 26 Mg at an almost identical centre of mass energy [29] were employed to construct a theoretical angular distribution and this was found to reproduced well the form of the measured distribution (including the observed minimum). The best-fit normalisation thus gave the product of the deuteron target thickness and the total integrated beam flux. The uncertainty associated with this fitting was estimated to be 3%. An estimate of systematic uncertainties in this analysis was obtained by repeating the procedure with parameters taken from d+ 28 Si scattering at the same energy [30]. The variation was less than 2%. Figure 3 shows the Doppler corrected gamma-ray energy (E γ ) plotted against the excitation energy in 26 Na (E x ) as derived from the proton energy and angle. The superior resolution of the gamma-ray energy is clearly apparent, and the states that overlap in E x can be distinguished using the gamma-ray energy. A number of levels with ground-state gamma-ray branches (E γ = E x ) can be seen, including many of the more strongly populated states. The neutron separation energy is S n = 5.57 MeV [31]. The ground state branch generally has a lower and better defined underlying background than any other peak in Figure 3 and hence for the most part only these peaks were chosen for initial analysis 4 (Table 1). With suitable background subtraction [28], the yield of protons could be deduced for individual states in 26 Na as a function of laboratory angle. These distributions were converted to absolute differential cross sections by taking into account the geometry of SHARC, the elastic scattering normalisation, the measured gamma-ray branching ratios and the gamma-ray detection efficiency (corrected for Doppler and relativistic angular abberation effects). The overall systematic error associated with these effects was typically 5-6%.
Differential cross sections for states populated in 26 Na are shown in Figure  4, plotted in terms of the laboratory scattering angle. These are compared with reaction calculations performed using the code TWOFNR [33] employing the Adiabatic Distorted Wave Approximation (ADWA) of Johnson and Soper [34] with standard input parameters [35] including the Chapel-Hill (CH89) nucleon-nucleus optical potential [36]. This formalism benefits inter alia from having no requirement for a deuteron-nucleus optical potential. The magnetic substate populations from TWOFNR were used to check that the gamma-ray coincidence requirement did not alter the shape of the differential cross sections by more than a few percent across the full range of proton angles [28]. Spectroscopic factors, S, were extracted by normalising the calculated differential cross sections to the data using the full range of angles shown in Figure 4. The transferred angular momentum, L, that best describes the shape of the measured angular distribution for each state is listed in Table 1.
States for which more than one value is possible for the transferred angular momentum have been analysed by fitting a linear combination of the calculated cross sections using the two possible values L 1 and L 2 (where L 1 =L). It is important to note that the contribution from the larger of L 1 and L 2 is suppressed (for a similar spectroscopic factor) due to poorer kinematic matching. As a result, the spectroscopic factor deduced for the higher L is seen generally to exhibit a larger statistical uncertainty. As may be seen in Table 1, the addition of L 2 to the fit does not significantly change the spectroscopic factor deduced using just L 1 , i.e. S 1 ≈ S. The overall uncertainties assigned to the spectroscopic factors are some 20% (dominated overwhelmingly by the reaction calculations) [35]. The inferred spin and parity assignments are discussed below.
The 25 Na projectile has a structure, in the simplest shell model picture, of three 0d 5/2 protons coupled to 5/2 + and a closed 0d 5/2 neutron orbital. The positive parity orbitals 1s 1/2 and 0d 3/2 , as well as the negative parity orbitals 0f 7/2 , 1p 3/2 , 1p 1/2 . . . are thus available for neutron transfer. As shown in Figure 2, the known low-lying positive parity states below 450 keV [16,20] are populated. As discussed below, the data also show clearly that there is population of negative parity states, as would be expected from the previous (d,p) studies of 25,27 Ne [12,13], 27 Mg [37] and 28 Al [38,37,39]. In order to interpret the results, shell model calculations have been performed including all 0 ω sd-shell configurations for positive parity states and all 1 ω excitations in an spsdpf basis for negative parity states. The program OXBASH [40] was used with the USD-A [32] and WBP-M [13] interactions respectively. The WBP-M interaction, which describes the neighbouring nuclei 25,27 Ne and 29 Mg in a consistent manner [13], is a modification of the WBP interaction [41] that shifts the energies of the f p-shell orbitals down by 0.7 MeV. It was previously noted [13] that for the neon and magnesium isotopes the strongly populated 3/2 − and 7/2 − states (S ≈ 0.4-0.6) appear to track fairly closely the shift in the energy of the f p-shell orbitals. In fact, earlier work by Bender and coworkers also applied a similar interaction called WBP-a that improved the agreement with the shell model for states in nearby (and only slightly higher Z) isotopes of Al [42] and P [43,44]. Excitation energies for the WBP-M and USD-A calculations were computed relative to the (positive parity) ground state calculated with the same interaction. The association of experimentally observed states with shell model states, indicated in Table  1 and Figure 5, was based on the observed L-transfer, the excitation energy and the gamma-decay selectivity ( Figure 6) as discussed below.
Above ∼2.5 MeV excitation, no states of positive parity are expected to be strongly populated, according to the shell model calculations. The states seen here at higher energies are therefore likely to have negative parity. Of these, the most easily identified is the strongly populated state at 3.511 MeV which is observed to have a yield dominated by L= 1 neutron transfer at large laboratory angles (Figure 4). The only observed gamma-decay branch for this state is to the 3 + ground state, in line with the favoured decay pattern for the lowest 4 − state in the isotone 28 Al [38]. This state is thus assigned to be the 4 − 1 state in 26 Na. There are two strongly populated higher lying levels that gamma-decay via the 3.511 MeV level ( Figure 6). In terms of predicted levels in this  shown for each fitted orbital (nlj) and, for completeness, its spin-orbit partner also. Numbering of states as in Figure 4.
single L analysis two L analysis (where applicable) No. E inferred in present work (see text) except g.s. [16] and first four excited states [20,18,19] d) excitation energy of this isomeric state deduced from gamma-decays feeding both this level and the g.s. (see Figure 6) energy range, the obvious candidates are the lowest 5 − and 6 − states (Table  1). These could reasonably be expected to gamma-decay via the 4 − state, according to the pattern observed [39] in 28 Al. The decays of both the proposed 5 − and 6 − states proceed in part (20% and 31% respectively [28]) via the state at 2.116 MeV. Given that this level is populated via an L= 2 transfer (Figure 4) and that it is fed by these higher lying negative parity states, it has a likely J π assignment of greater than 4, and a comparison of possible spins with shell model calculations indicates an assignment of 5 + (supported by both the energy and the weak spectroscopic factor). Indeed, there is an analogous state at 2.581 MeV in 28 Al, perhaps weakly populated in the decay of the 6 − state [39], that has a 5 + assignment [45,46].
The two reasonably strong states seen just below the 4 − are most naturally associated with the 2 − and 3 − states predicted in the shell model (see Table 1). In particular, the identification of the 3 − at 3.135 MeV is confirmed by its gamma-decay branch to the lowest lying 1 + state at 1.507 MeV as occurs in 28 Al (Table 28.6 of ref. [46]). This is further supported by the mixed L 1 = 1 plus L 2 = 3 differential cross section (Figure 4).
The spectroscopic factors extracted from the proton differential cross sections are compared to the shell model predictions in Table 1 and Figure 5. The 1s 1/2 strength that leads to 2 + states is concentrated in the two known states below 0.5 MeV and sums to approximately 0.45 in both theory and experiment. The tentative identifications of 3 + 2 and 4 + 2 states are based on a comparison with the shell model calculations as follows. The observed 0d 3/2 strength leading to J π = 3 + is concentrated in one state near 1.8 MeV with a spectroscopic strength in very good agreement with the predictions. The 0d 3/2 strength leading to the 4 + states has a comparable magnitude in theory and experiment and is located close to 2 MeV, so that the experiment is compatible with the strength being concentrated in the second 4 + state. We note that the USD-A interaction [32] predicts the excitation energies for positive parity states to within ∼150 keV, in line with typical shell model accuracy. When we instead used the WBP-M interaction to calculate the energies of states with a strong 0d 3/2 neutron character (namely the 3 + and 2 + states), then the predictions lay about 0.4 MeV lower than experiment. This is because the WBP-M incorporates the USD interaction [47] to compute the 0 ω positive parity levels and this is known to underestimate 0d 3/2 neutron energies [12].
The spectroscopic factors deduced for the 4 − state indicate a comparable Table 2: Mixing between 1p 3/2 and 0f 7/2 configurations in 26 Na, compared with the isotone 28 Al. Values of (2J+1)S from [48] are converted to S using the confirmed spins [46]. 26 Na, present work 28 Al, ref. [48] J π E x a) strength to that predicted for the 1p 3/2 transfer, albeit somewhat weaker. However, the shell model fails completely to reproduce the substantial 0f 7/2 strength. The excitation energy is given accurately by the WBP-M calculation, as are those of the other negative parity states, within 100 keV (J π = 2 − , 3 − , 5 − , 6 − ). These latter states are all predicted to have a structure that overlaps substantially with that of a neutron in the 0f 7/2 orbital coupled to the 0d 5/2 proton of the 25 Na ground state. It is worthwhile noting that the 3 − state is predicted also to have a similar spectroscopic strength for the coupling with a 1p 3/2 neutron. Because of a better kinematic matching, the L= 1 transfer dominates the yield; a similar situation is observed for the lowest 3 − state in 28 Al as populated in (d,p) [48,49,45]. In 28 Al, the 0f 7/2 measured spectroscopic factor actually exceeds that of the 1p 3/2 orbital by a factor of three [48] (Table 2). Another 3 − state in 26 Na is predicted by the WBP-M calculations to have approximately equal mixing and to lie at 4.462 MeV, whilst a predominantly 1p 3/2 neutron state occurs at 4.774 MeV. These states may be populated but have not been identified in the present analysis.
In the case of the 4 − state in 26 Na, the deduced 1p 3/2 spectroscopic factor is twice as large as that reported for 28 Al whereas the 0f 7/2 spectroscopic factor is effectively unchanged ( Table 2). In contrast, for the 3 − state, the spectroscopic factors are three times and six times lower, respectively, than in 28 Al [48,46]. A clearer picture emerges from Table 2 if the relative magnitudes of the 1p 3/2 and 0f 7/2 spectroscopic factors for each of the 26 Na states are compared with the behaviour in 28 Al. For both the 3 − and 4 − states in 26 Na, the 1p 3/2 spectroscopic factor is half the magnitude of that for 0f 7/2 . In contrast, the analogous states in 28 Al exhibit 1p 3/2 spectroscopic factors that are three to five times smaller. This demonstrates an enhanced role emerging for the 1p 3/2 orbital in the structure of the low-lying negative parity states in 26 Na as compared to 28 Al. Indeed, in the case of 28 Al the spectroscopic factors ( Table 2) indicate that the states with predominantly 1p 3/2 structure lie in the region of 4.8 MeV, significantly above the 0f 7/2 -dominated states which are closer to 4.0 MeV ( Table 2 of ref. [48]). It would be very interesting if the spectroscopic factors for the higher lying 2 − and 3 − states in 26 Na could be measured and compared. As is evident in Figure 5, a theorybased comparison between the 1p 3/2 and 0f 7/2 strengths in 26 Na indicates that the levels with a 1p 3/2 structure are on average around 1 MeV below those with 0f 7/2 structure (in fact, 0.83 MeV when weighted by the spectroscopic factors), an inversion that is in accord with the systematics of Figure  1.
In conclusion, as noted in the introduction and illustrated in Figure 1, the ordering of levels in nuclei with A 25-30 evolves dramatically as they become more neutron-rich, driven largely by the interaction between protons in the 0d 5/2 orbital and the valence neutrons. Importantly in this context, the results presented here confirm that the evolution is also manifest in 26 Na. In particular, the low-lying 3 − and 4 − states in 26 Na are found to exhibit an enhanced influence of the 1p 3/2 neutron orbital, compared to the isotone 28 Al. In addition the WBP-M shell model calculations, in which the f pshell orbitals are lowered to reduce the N = 20 gap by 0.7 MeV, succeed in reproducing the energies of the negative parity states in 26 Na as they do in the neighbouring neon and magnesium isotopes [12,13]. From a theoretical perspective, a less ad hoc description of the transition into the island of inversion represents an interesting and important challenge.
The assistance of our late colleague and friend R.M. Churchman in preparing the experiment is gratefully acknowledged. The efforts of the ISAC operations team in supplying the 25 Na beam are appreciated. Illuminating discussions with B.A. Brown are also acknowledged. We acknowledge support from the Science and Technologies Facility Council (UK) (grants ST/J000051/1, STFC-EP/D060575/1) and the Natural Sciences and Engineering Research Council of Canada; TRIUMF is funded via a contribution agreement with the National Research Council of Canada.  [9,10,11,12,13] and compilations, whilst those for oxygen are from refs. [14,15]. The energy reference is chosen to be the 7/2 − level (see text) and the Z position of 26 Na is indicated.     Table 1 (dashed line = fit with single angular momentum, labelled as L or as L 1 where another fit is also shown, full line = sum of two contributions L 1 and L 2 ).