A New Measurement of the Intruder Configuration in 12Be

A new $^{11}$Be($d,p$)$^{12}$Be transfer reaction experiment was carried out in inverse kinematics at 26.9$A$ MeV, with special efforts devoted to the determination of the deuteron target thickness and of the required optical potentials from the present elastic scattering data. In addition, a direct measurement of the cross sections for the 0$_2^+$ state was realized by applying an isomer-tagging technique. The s-wave spectroscopic factors of 0.20(0.04) and 0.41(0.11) were extracted for the 0$_1^+$ and 0$_2^+$ states, respectively, in $^{12}$Be. Using the ratio of these spectroscopic factors, together with the previously reported results for the p-wave components, the single-particle component intensities in the bound 0$^+$ states of $^{12}$Be were deduced, allowing a direct comparison with the theoretical predictions. It is evidenced that the ground-state configuration of $^{12}$Be is dominated by the d-wave intruder, exhibiting a dramatic evolution of the intruding mechanism from $^{11}$Be to $^{12}$Be, with a persistence of the $N = 8$ magic number broken.


Introduction
According to the well-established mean field framework for nuclear structure, nucleons (protons or neutrons) are filling in the single-particle orbitals grouped into shells characterized by the conventional magic numbers [1]. However, for nuclei far from the β-stability line, especially those in the region of light nuclei where the concept of a mean field is less robust, the exotic rearrangement of the single-particle configuration often appears and may result in vanishing or changing of the magic numbers. One widely-noted example is the ground state (g.s.) of the one-neutron-halo nucleus 11 Be, which possesses an unusual spin-parity of 1/2 + , being dominated (∼ 71%) by an intruding 1s 1/2 neutron coupled to a 10 Be(0 + ) core [2,3]. Obviously the prominent appearance of the s-wave in the g.s. of 11 Be is responsible for the formation of its novel halo structure [4].
The immediate question goes into the single-particle configuration of 12 Be, having one more valence neutron outside the 10 Be core. This neutron-rich nucleus has four particle-bound states, namely the g.s.(0 + ), and the excited states at 2.107 (2 + ), 2.251 (0 + ) and 2.710 MeV (1 − ) [8]. The relatively low energies of the latter three states imply the breakdown of the N = 8 magic number and the strong intruder from the upper sd-shell [5][6][7][8][9], leading to the growth of other non-shell-like structure in this nucleus [10,11]. Since Barker's early work in describing the isospin T = 2 states of the mass A = 12 nuclei with a mixed configuration [12], substantial theoretical studies have been devoted to the spectroscopic studies of the low-lying states in 12 Be. To date most studies agree on the large probability ( 60%) of intruder from the sd-shell, but the relative importance of the s-and d-components remains a subject of active investigation [1]. A standard way to describe the intruding effects around N = 8 is to use the configuration mixing α(s 2 ) + β(d 2 ) + γ(p 2 ), with α, β and γ the normalized intensities (percentages) of the respective components for valence neutrons in 0 + -states outside the 10 Be core [13,14]. In principal there should be three 0 + states in this p − sd model space, but only the lowest two have been found in the bound region. The third 0 + state was predicted to appear in a wide energy range of 3∼9 MeV [12,13,15,16], but to date it has not been identified experimentally. Therefore in the present work we focus on the lowest two 0 + states only. Table 1 (upper panel) summarizes the individual intensities from the shell model calculations by Barker [12] and Fortune et al. [15], the three-body model predictions by Nunes et al. [17] [23][24][25], respectively, which are normalized to their sum to give the intensities [13]. c p-wave intensities extracted from a charge-exchange experiment [28].
and Redondo et al. [18], the nuclear field theory approach by Gori et al. [19], and the random-phase approximation by Blanchon et al. [20]. The results are quite disparate in terms of the dominant component of each state. For instance the s-wave intensity in the 0 + 1 g.s. ranges from 23% up to 76%, resulting in active disputing [13,14]. In fact the model calculation of the configuration admixture depends on various basic physics ingredients, such as the particle-separation energy, the deformation of the nucleus, the core-nucleon potential and wave functions, the effective pair interaction, the interplay between the collective motion and the valence nucleons, and so on [1,19]. Particularly the ratio of s 2 to d 2 is sensitively regulated by the core-nucleon Hamiltonian and the nucleon-nucleon residual interaction [21].
As discussed in detail in Refs. [1,13,14,22], various experiments have been carried out to quantify the intruder strengths. Here in Table 1 (lower panel) are listed only those sensitive to individual structure component. Oneneutron knockout reactions were performed for 12 Be to extract spectroscopic factor (SF) of each single-particle orbit [23,24]. The comparison to the theoretical intensities can be made by normalizing to the sum of the three SFs, similar to the way used in row N of Table I in Ref. [13].
The obtained values show almost equivalent intensities for the s-, d-and p-orbital in the g.s. of 12 Be. It was noticed that the 12 Be beam used in the knockout reaction may be in both the g.s. and the long-lived isomeric 0 + 2 state, leading to a reduced strength difference between the two 0 + states [25]. One-neutron transfer reaction, namely 11 Be(d, p) 12 Be at 5A MeV, was carried out to populate the s-component in the first two 0 + states of 12 Be. The obtained SFs are 0.28 +0.03 −0.07 and 0.73 +0.27 −0.40 , respectively. This experiment was later on questioned for the possible contamination of the (CD 2 ) n target and the large uncertainties in extracting SFs from the undistinguishable 0 + 2 and 2 + states [22]. Another one-neutron transfer experiment at 2.8A MeV was then performed with a clear separation of all low-lying excited states by incorporating the γ-ray detection [26]. The extracted SFs (set III) are 0.15 +0.03 −0.05 and 0.40 +0. 13 −0.09 , respectively, for two low-lying 0 + states. This experiment suffered from a very low beam energy, leading to an effective detection outside the most sensitive angular range, especially for the 0 + 2 state. Due to the lack of proper normalization procedures for these transfer reactions, it would be difficult to compare their SF results with other measurements or to each other [27]. Recently the p-wave intensities for the two low-lying 0 + states were determined from a charge-exchange experiment [28], which are listed also in Table 1. It is evident that more measurements are urgently needed to clarify the theoretical deviations and the experimental ambiguities [1,22]. In this letter, we report on a new measurement of the 11 Be(d, p) 12 Be transfer reaction, with special measures taken to deal with the questioned experimental uncertainties.

Experimental setup
The experiment was carried out at the EN-course beam line, RCNP (Research Center for Nuclear Physics), Osaka University [29]. A 11 Be secondary beam at 26.9A MeV with an intensity of 10 4 particles per second (pps) and a purity of about 95% was produced from a 13 C primary beam impinging on a Be production target with a thickness of 456 mg/cm 2 . The energy of the secondary beam was chosen considering the effective detection of the recoil protons at backward angles, the availability of the primary beam, and the validation of the transfer reaction mechanism. A schematic view of the detection system is shown in Fig.1 (with more details in Ref. [30]). Elastic scattering of 11 Be from protons or deuterons was measured by using a (CH 2 ) n (4.00 mg/cm 2 ) or a (CD 2 ) n (4.00 mg/cm 2 ) target, respectively, with the background subtraction provided by C-target runs [30,31]. The inevitable hydrogen contamination in the (CD 2 ) n target was found to be 9.5 ± 0.6% out of the total deuterium contents, determined by the number of recoil protons relative to those from the known (CH 2 ) n target [31]. The incident angle and the hit position on the target were determined by two parallel-plate avalanche counters (PPAC) placed upstream of the target (not shown in the figure), with resolutions (FWHM) less than 0.3 • and 2.0 mm, respectively. The backward emitted protons were detected using a set of the annular doublesided silicon-strip detector (ADSSD in Fig.1) composed of six sectors, each divided into sixteen 6.4-mm-wide rings on one side and 8 wedge-shaped regions on the other side. This annular detector has an inner and an outer radii of 32.5 mm and 135 mm, respectively, covering laboratory angles of 165 • ∼ 135 • relative to the beam direction. The energy detection threshold was set at 1.0 MeV, allowing to cut off the noise while retaining a high sensitivity for protons related to interested excited states in 12 Be. The ADSSD provided also good timing signals with a resolution (∼ 2 ns) good enough to reject protons not coming from the target. The forward moving projectile-like fragments were detected and identified by a set of charged-particle telescope (TELE0 in Fig.1) composed of a double sided silicon-strip detector (DSSD) of 1000 µm thick and two layers of large size silicon detector (SSD), each having a thickness of 1500 µm. This telescope has an active area of 62.5 × 62.5 mm 2 (32 × 32 strips) and was centered at the beam direction (0 0 ) at a distance of 200 mm down stream from the target. A particle identification (PID) spectrum, taken by the TELE0 and in coincidence with protons in the ADSSD, is shown in Fig. 2(a). 12 Be in the figure must come from the (d, p) transfer reaction, whereas 11 Be and 10 Be, with much broader energy spread, are most likely related to the neutron decay following the population of unbound states in 12 Be. The coincidence with the backward-emission protons is essential here to avoid the large background arising from the direct beam [24]. Gated on 12 Be in TELE0, protons are the only charged particles being detected in the ADSSD and therefore their kinematics can be mapped out based on the detected energies and angles. The excitation energy in 12 Be can then be deduced from the recoil protons, as shown in Fig.2 g.s. peak (centered at 0.0 MeV) in Fig.2(b). Although the number of counts in this peak is relatively small, its significance is clear due to the very low background. A large and broad peak stands between 1 and 4 MeV, contributed from the unsolved three states at 2.107, 2.251 and 2.710 MeV in 12 Be. It is worth noting that protons belonging to the g.s. peak in Fig.2(b) have higher energies in the ADSSD detector and thus are almost free from the detection loss. Also background counts were checked by employing a carbon target.
A special isomer-tagging method was used to discriminate the 0 + 2 state from the broad excitation-energy peak ( Fig.2(b)). The method relies on its well-known isomeric property: a life-time of 331 ± 12 ns [8] and an E0-decay (via e + e − pair emission) branching ratio of 83 ± 2 % [7]. 12 Be(0 + 2 ) isomers were stopped in the TELE0 and the subsequently emitting γ-rays, particularly the 0.511 MeV ones from the e + -annihilations, were measured by an array of six large-size NaI(Tl) scintillation detectors surrounding or at the back of the TELE0 (Fig.1). This kind of decay-tagging method has been successfully applied in many particle-emission experiments [32][33][34]. The 12 Be + p + γ triple-coincidence was realized based on the good timing signals generated from the strips in the TELE0 and the ADSSD, and from the scintillation detectors, respectively. A time window of 3 µs for the triple-coincidence was applied, which covers about 9 times of the decay half-live (331 ns) of the 0 + 2 state. The γ-energy spectrum of these triple-coincidence events is presented in Fig.3(b), with the 0.511 MeV γ-ray peak (between 0.4 and 0.6 MeV) standing well above the background. The time distribution of these 0.511 MeV γ-rays follows approximately the exponentialdecay curve with an extracted half-life of 270 ± 120 ns, being consistent with the reported value [8] within the error bar. The source of these coincidentally observed 0.511 MeV γ-rays were checked against all possible contamina- tions, such as the random or accidental coincidences, target impurities, event mixing and so on. These can be realized by selecting various event samples, other than the targeted one, to build the similar coincidences. Furthermore, the possible 0.511 MeV γ-rays cascaded, or indirectly produced, from other decay-chains in 12 Be were analyzed by realistic Monte Carlo simulations. It turned out that all these backgrounds are negligible, mostly attributed to the strict triple-coincidence 12 Be + p + γ and the detectorsetup scheme. Of course this strict coincidence would lead to a reduction of the detection efficiency and hence the event statistics. However the observation of the 0 + 2 isomerdecay is still at very high significance (at least 3.4σ, or > 99.9% confidence level), owing to the very low background. The triple-coincidence events are presented in the insert of Fig.2(b)(dotted curve), which do concentrate around the excitation energy of the 0 + 2 state (2.251 MeV). The detection efficiency in the present work for the 0.511 MeV γ-rays, produced from the positron annihilation, is determined to be 23 ± 1 %, based on realistic Monte Carlo simulations using the GEANT4 code [35].

Experimental result
Differential cross sections for the 11 Be(d, p) 12 Be transfer reaction at 26.9A MeV are presented in Fig. 3, deduced from the recoil protons and gated on the excited state in 12 Be. The g.s. events are selected by a cut from -1.0 to 0.6 MeV on the excitation energy spectrum (Fig.2(b)). A gate between 0.4 and 0.6 MeV on the γ-ray energy spectrum ( Fig. 3(b)) is applied to select the isomeric 0 + 2 state. 2 + and 1 − states are still indistinguishable from the excitation energy spectrum (Fig. 2(b)) and the summed cross sections are plotted in Fig. 3(d) with those for 0 + 2 state subtracted. The error bars in the figure are statistical only. The systematic error is less than 10%, taking into consideration the uncertainties in the detection efficiency determination(∼ 5%), the (CD 2 ) n target thickness (∼ 2%), and the cuts on the PID spectrum (∼ 4%) and on the excitation energy spectrum (∼ 5%).
To extract the SFs, theoretical calculations were performed by using the code FRESCO [36], which incorporates approaches such as the distorted wave Born approximation (DWBA) or the finite-range adiabatic distorted wave approximation (FR-ADWA). Due to the uncertainties in DWBA calculation associated with the applied optical potentials (OPs) [2,26,37], we adopt the FR-ADWA method, which uses nucleonic potentials, includes explicitly the deuteron breakup process and can provide consistent results for (d, p) transfer reactions [2]. In the present work the p + n potential is given by the Reid soft-core interaction [38]. A Woods-Saxon form was used for the 11 Be + n binding potential, with a fixed radius and diffuseness of 1.25 fm and 0.65 fm, respectively. These geometrical parameters were widely adopted for loosely-bound states in light nuclei [2,[39][40][41]. The well depth of this binding potential was adjusted to reproduce the correct excitation energies [25], and the obtained values are 65.18 MeV and 56.49 MeV, respectively, for the 0 + 1 and 0 + 2 states. The entrance channel OP is obtained by folding the 11 Be + p and 11 Be + n potentials, with the former extracted from the present elastic-scattering data [30] and the latter from global potentials [42,43]. As a matter of fact the currently extracted potential is just the global one (CH89) but with two normalization factors, namely 0.78 and 1.02, applied to the depths of the real and imaginary parts of the potential, respectively. These normalization factors are necessary for weakly-bound nuclei and the currently adopted factors are close to the averaged ones in the literature [30]. The exit channel OP is extracted from the data reported in Ref. [45] by using the same method as for the 11 Be + p elastic-scattering data.
The results of FR-ADWA calculations, multiplied by the SFs for the selected single-particle component, are fitted to the experimental data by the standard χ 2 minimization method [25], and the results are shown in Fig. 3. Data in Fig. 3(d) for the mixed 2 + and 1 − states are fitted by the weighted sum of S1 · ( 11 Be ⊗n(1d 5/2 )) + S2 · ( 11 Be ⊗n(1p 1/2 )), where S1 and S2 are SFs for the d-wave and p-wave neutrons in the low-lying 2 + and 1 − states in 12 Be, respectively. The best fit (red solid curve in Fig.3d) is obtained by S1 = 0.26 ± 0.05 and S2 = 0.76 ± 0.17, with the error bars corresponding to a 68.3% confidence level [25]. If only one component was used, the result is represented by the dotted or dashed curve for a pure d-wave with SF = 0.5 or a pure p-wave with SF = 1.4, respectively. We notice that the 2 + state was resolved in an previous measurement [26], but the unfavorable angular coverage of the data did not allow a unique extraction of the SF. Our SF result for the d-wave component in the 2 + state, 0.26 ± 0.05, is consistent with two out of four sets of results reported in Ref. [26] for various selections of optical potentials, namely 0.30 ± 0.10 (set II), and 0.40 ± 0.10 (set III).
The extracted s-wave SFs for the 0 + 1 and 0 + 2 states are 0.20 +0.03 −0.04 and 0.41 +0.11 −0.11 , respectively, with the error bars corresponding to a 68.3% confidence level [25]. These results are compatible with those obtained from the previous transfer experiments within the error bars [25,26], although the normalization of the SFs for each measurement was not obtained. Since we have resolved the 0 + 2 state by using the implantation-decay technique and applied the more suitable FR-ADWA analysis [2], the currently extracted SF should be more reliable. It should be worth noting that, although the cross section for the 0 + 2 state looks smaller than that for the 0 + 1 state, its SF is two times as big as that of the latter one. This is essentially attributed to the large reduction of the calculated cross sections for the halo-like states. This behavior was also clearly exhibited in the similar reaction 15 C(d, p), in which the s-wave SFs of 0.60 ± 0.13 and 1.40 ± 0.31 were extracted for the first and second 0 + states in 16 C [46]. This difference in cross sections for various final states may depend also on the incident energy [47] due naturally to the match of the internal and external waves. However, since this energy dependence happens for both the measurement and the proper calculation, the SFs, at least for its relative or normalized values, should be stable within a relevant energy range [47].
In order to compare our SF results with those from theoretical calculations and from other measurements, the conversion into relative intensities (percentages) is required [27]. Since the necessary quantities related to the sum rule were not measured, we rely on the ratio of SFs for the 0 + 1 and 0 + 2 states, which is independent of the normalization factors. Using the standard method proposed by Barker [12], the wave functions of the two low-lying 0 + states can be written as From the present measurement we have α 1 /α 2 = 0.20/0.41 = 0.49 +0.15 −0.16 . The errors are statistic only. The systematic uncertainty of this ratio is estimated to be less than ± 7%, due basically to the possible choices of the optical potentials. Previously the 0p 1/2 -wave strengths in the two low-lying 0 + states of 12 Be were investigated via a charge-exchange reaction experiment 12 B( 7 Li, 7 Be) 12 Be [28]. The extracted values are γ 1 = 0.24 and γ 2 = 0.59 within the p-sd model space. Combining all these conditions, the intensities in the above normalization equations can be deduced: α 1 = 0.19±0.07, β 1 = 0.57 ± 0.07, γ 1 = 0.24 ± 0.05, α 2 = 0.39 ± 0.02 , β 2 = 0.02 ± 0.02, γ 2 = 0.59 ± 0.05. These results are also listed in Table 1. The error bars are deduced from the statistic uncertainties of the SFs extracted in the present work. According to the experimental as well as the theoretical definition of the intensity (I) [13], which is the SF divided by the adopted sum rule and hence sums up to 100%, and by using the expression of Ref. [27], we have I = SF exp /[F q * (2j + 1)]. Based on the presently determined SFs (0.20 or 0.41) and intensities (0.19 or 0.39, respectively), the quenching factor F q can easily be deduced to be 0.53 for the s-wave (j = 1/2) components in the low-lying 0 + states of 12 Be, fairly within the range of the nominal values [27].
We have applied the shell model calculations, with the latest YSOX interaction [48,49], to reproduce the experimentally observed spectroscopic strengths. This approach works in a full p-sd model space, including (0-3) ω excitations, and may give good descriptions of the energy, electric quadrupole and spin properties of low-lying states in B, C, N, and O isotopes. The calculated individual s-, d-, and p-wave strengths for the first two 0 + states in 12 Be, denoted by Case1 in Fig.4(c), are compared to the experimental results shown in Fig.4(b). The calculated s-wave intensities for these two 0 + states are in good agreement with the experimental ones, whereas the calculated p-wave intensity for the 0 + 1 (0 + 2 ) state is slightly larger (smaller) than the experimental value [28]. This deviation in p-wave is opposite to the d-wave components. Despite a generally good description of intensities by the Case1 calculation, it does not give the correct level order of the low-lying excited states as demonstrated in Fig.4(a), neither does with the WBP interaction [25]. A decrease of 0.5 MeV for the d-orbit in the calculation would lead to the restoration of the level order (a relative decrease of the 2 + state), and also a better reproduction of the p-wave intensities, as dis-played by Case2 in Fig.4(d). Case2 parametrization allows also a good description of the ground and low-lying excited states in 11 Be. The meaning of this shift for d-orbit needs to be understood by further theoretical investigations.

Summary
In summary, a new measurement of the 11 Be(d,p) 12 Be transfer reaction was performed with a 11 Be beam at 26.9A MeV. Special measures were taken in determining the deuteron target thickness and in separating the 0 + 2 isomeric state from the mixed excitation-energy peak. Elastic scattering of 11 Be + p was simultaneously measured to estimate the hydrogen contamination in the (CD 2 ) n target and to obtain the reliable OP to be used in the analysis of the transfer reaction. FR-ADWA calculations were employed to extract the SFs for the low-lying states in 12 Be. The ratio between the SFs of the two low-lying 0 + states, together with the previously reported results for the pwave components, was used to deduce the single-particle component intensities in the two bound 0 + states of 12 Be, which are to be compared directly to the theoretical predictions. The results show a clear d-wave predominance in the g.s. of 12 Be, which is dramatically different from the g.s. of 11 Be dominated by a intruding s-wave. This exotic intruding phenomenon was also observed in a latest 12 Be(p, pn) knockout reaction experiment [50]. The present results are also compatible with those obtained from the previous transfer reaction measurements, considering the reported uncertainties. This work demonstrates the importance of measuring the individual SFs in the lowlying states in order to fix the configuration-mixing mechanism.