I. INTRODUCTION

During the past decades, studies on exotic nuclei have been stimulated considerably owing to the enormous development of radioactive ion beam (RIB) technique. Many peculiarities of exotic nuclei have been revealed, such as the halo/skin-like structure [1], the cluster structure [2] and the breakdown of shell closure [3, 4]. Studies on these peculiarities greatly improved understanding of the exotic nuclei, e.g. the beryllium isotopes. Theoretical researches on properties of Be isotopes are based on the three-body model [5, 6] and the density-dependent relativistic mean-field model [7, 8]. The neutron halo of ¹⁴Be has been well explained using these models [5, 6, 7, 8]. Located between the halo nuclei ¹¹Be and ¹⁴Be, ¹²Be is an interesting combination of the peculiarities and plays a key role in the beryllium chain. In shell model, ¹²Be is supposed to be a "magic nucleus" with simple structure. But recent experiments [4, 9, 10] provided direct evidence for breakdown of the N = 8 shell closure in ¹²Be and the (s, d) intruder states. In principle this intruder-state configuration can cause halo-like structure, yet neither the wide momentum distribution [11] nor the relative large two-neutron separation energy (S_2n = 3.67 MeV [12]) indicates the signature for a halo. There exists controversy between the recent theoretical and experimental results on halo-like structure of ¹²Be. The giant deformation in the ground state of ¹²Be [13] is also predicted by Antisymmetrized Molecular Dynamics (AMD), the ground state structure of ¹²Be is still ambiguous. Also, explorations in the level structure, parity, spin and deformation of the excited states of ¹²Be with molecular cluster model draws quite a lot of attentions [14-18]. The ground state structure properties of ¹²Be are indispensable in the studies because it provides fundamental information for the studies of the excited states. Therefore, it is of significance to study the ground state structure of ¹²Be.

As an essential property of the ground state structure, the nuclear-matter density distribution of ¹²Be not only provides basic structure information such as nuclear-matter radius, but also helps to determine whether the ¹²Be ground state has a halo structure. Recently, particle-particle random-phase approximation (pp-RPA) [19] and microscopic no-core shell-model (NCSM) calculations [18] have shown that the ground state wave functions of ¹²Be are dominated by the p shell configuration, which is in conflict with the previous calculations [20, 21, 22, 23, 24] and the knockout measurements [4, 9]. The nuclear-matter density distribution may be used to determine the configuration mixing between the 1p_1/2, 1d_5/2 and 2s_1/2 orbitals, thereby help to resolve the inconsistencies between current data and various theoretical models. Therefore, we were motivated to determine the nuclear-matter density distribution of ¹²Be ground state.

In the following text, the experimental method of the σR measurement is introduced in Sec.II, the feasibility analysis and optimization of the reaction cross section (σR) measurement are elaborated in Sec.III, the way to determine the s orbital component of ¹²Be ground state through σR is illustrated in Sec.IV, and a summary is given in Sec.V.

II. GENERAL SCHEME

Typically the nuclear-matter density distribution is investigated by measuring the σR (or the total interaction cross sections σI). This involves theoretical models, and several methods were developed to study the total reaction cross section, such as the multi-step scattering theory of Glauber [25], the transport model method of Ma et al. [26, 27] and the semi-empirical formula of Kox et al. [28] and Shen et al. [29] A series of investigations to ¹²Be were carried out in the past decades. By measuring the σI on Be, C and Al at 790 MeV/nucleon, Tanihata et al. successfully determined the effective root-mean-square (RMS) radius of ¹²Be through a Glauber model in 1988 [30]. Liatard et al. measured the σR of radioactive Be isotopes on Cu at around 25 to 65 MeV/nucleon and deduced the radii by using a simple microscopic model [31]. Later, Warner et al. measured the σR of ¹²Be on Pb and Si at about 30 to 60 MeV/nucleon and obtained the radius of ¹²Be [32]. However, the studies have several drawbacks. First, their energy region did not cover both low and high energy regions. Data at low energy region give constraint to the outer structure, and data at high energy region provide more information on the core part, hence low and high energy region data are both needed to obtain accurate nuclear-matter density distribution. Second, generally, their targets were too heavy. This makes σR insensitive to the surface structure of ¹²Be. Third, their models were too simple to interpret the data especially at low energies. The results did not give detailed nuclear-matter density distribution of ¹²Be especially for the surface area.

In obtaining detailed outer structure of the nuclei, two improvements were made recently for extracting accurate nuclear-matter density distribution of nuclei. First, the method of proton elastic scattering at intermediate energies was developed. Ilieva et al. applied proton-scattering method in 2012 to ¹²Be [33] and determined the nuclear-matter density distribution of ¹²Be ground state. But the result has considerable uncertainty because of various parametrizations. So the method is not well established yet for studing the surface structure of especially unstable nuclei. Second, the applicability of Glauber model in the whole energy region was studied and a Modified Optical Limit Glauber model (MOL[FM]) was developed by incorporating the Fermi motion of nucleons in the finite-range MOL [34]. The MOL[FM] has reduced the discrepancy between the data of calculation and measurement to just 1%–2% for the whole energy region. By measuring the σR on Be, C and Al targets at intermediate energies, the nuclear-matter density distributions of ²²C [35] and ¹⁷Ne [36] were determined accurately with MOL[FM]. Through this method we obtained the nuclear-matter density distribution of ⁸Li [37] precisely and the relevant paper about our further study is in preparation. Therefore, MOL[FM] provides a powerful tool for interpreting the σI or σR data, and measuring the σI or σR determining the nuclear-matter density distribution with MOL[FM] is still a good method at present.

Thus, we were motivated to precisely measure σR of ¹²Be at low energy region and extract the nuclear-matter density distribution of ¹²Be ground state using MOL[FM].

Refer to the measurements of σR, transmission method is usually used. Typical experimental procedures are given in Ref. [38]. The σR is obtained by Eq. (1),

where E is the energy point; t is the target thickness expressed by the target particles numbers per unit area; and R=N_out/N_in is the ratio of outgoing projectile particles number to incident projectile particles. Since energy of the projectile particles decreases as passing through the target, we determine the energy point of σR by mean energy E_mean, which is given by

where t is the target thickness, and E(x) is the residual energy of incident beam travelling along the path by distance x. E(x) can be calculated by the improved Bethe-Bloch formula [39].

The main relative error of σR can be written as

where Δt/t stands for uncertainty of the target thickness, the subscripts "sys" and "stat" denote the systematic and statistical error of R, respectively, and the statistical error of R is given by

because it follows the binomial distribution. In practice, R_in/R_out is taken as R in order to remove the events interact outside the target, R_in and R_out are the ratios of N_out/N_in corresponding to the target-in and target-out measurements, respectively. Accordingly, the relative error of σR becomes

Besides the contribution of statistical error, the systemic uncertainty of σR is mainly from the correction of the number of inelastic-scattering events merged into the non-reaction events. By using the method in Ref. [40] and Monte Carlo simulation, the systematic error of σR can be limited within 1%–2% [34]. So 1%–2% total uncertainty can be achieved if sufficient reaction events are recorded. In fact, this was achieved recently in most experiments of the kind.

Based on available nuclear-matter density distributions of ¹²Be from experiments and theories (Fig. 1), we calculated the σR (Fig. 2) at different energies by MOL[FM]. The momentum width and the finite-range parameter β were taken from Ref. [34]. From Fig. 2 one sees that below ∼50 MeV/nucleon, σR data with uncertainty around 2% is sufficient to distinguish several previous results and determine whether ¹²Be has a clear halo-like structure.

Fig. 1.Full-screen View

(Color online) Current results of the nuclear-matter density distributions of ¹²Be (XB97: [41], SI12: [33], GG05: [42], IT88: [30]). The solid line is given by the sum of the nuclear-matter density distribution of ¹⁰Be core in ref. [33] and that of two valence neutrons calculated in section 4 with the configuration in ref. [4].

Fig. 2.Full-screen View

(Color online) The calculations of σR(E) corresponds to current nuclear-matter density distributions of ¹²Be (XB97: [41], SI12: [33], GG05: [42], IT88: [30]). The upper group of lines corresponds to the σR of ¹²Be + Al, and the lower group of lines corresponds to the σR of ¹²Be + C.

However, the specific requirement on experimental conditions for a 2% precision of σR is still unknown, and investigation is needed for further discussion of the method’s feasibility.

III. FEASIBILITY STUDY AND OPTIMIZATION OF σR MEASUREMENT

To make sure that the experiment is practical, we have studied the feasibility and optimized the experimental method. The experiment feasibility mainly relies on detecting system, reaction targets and ¹²Be beam. In the energy region below 50 MeV/nucleon, ΔE - E method is often used for particle identification. The required energy deposition of ¹²Be in the ΔE detector is about 10 MeV/nucleon, while the required energy in the E detector shall be above 10 MeV/nucleon, just for ensuring validity of the particle identification. So the energy of ¹²Be right before the ΔE detector is supposed to be over 20 MeV/nucleon. Usually Si detector and scintillator detector are used as the ΔE and E detector, respectively. Their thicknesses depend on the specific energy of ¹²Be right before the ΔE detector. As for the particle identification before the reaction target, Bρ-ΔE-TOF technique is often used. Typical layout of the detecting system is shown in Fig. 3.

Fig. 3.Full-screen View

Schematic diagram of typical experimental setup.

Besides the detection system, the target material and thickness t and incident energy of the ¹²Be beam E_in shall be determined before the experiment. Regarding the target material, ¹²C is a good candidate, because its nuclear-matter density distribution is well determined and its mass number is comparable with ¹²Be, so its σR is more sensitive to the ¹²Be surface structure. Another target, ²⁷Al, is also needed for reducing the target-dependence of the result.

For the t and E_in, there are direct impact factors, such as the transmission rate R, ¹²Be outgoing energy E_out after the reaction target, and energy point E_mean with certain restriction for each of these factors, hence it is difficult to consider t and E_in separately. We studied the relations between these parameters through calculations, which includes:

(1) E_out for certain E_in and t, E_out was calculated by LISE++ [43];

(2) E_mean could be determined through (E_in + E_out)/2 approximately;

(3) σR(E_mean) was calculated using MOL[FM] based on the nuclear-matter density distribution of the expectation in Fig. 1;

(4) R is calculated by Eq. (1) with certain t and corresponding σR(E_mean).

Then, the trends of R varying with E_in under different t and Eout were obtained. Fig. 4 shows the calculation results of ¹²Be+C. Each dash line corresponds to the same t, each dash dot line corresponds to the same E_out. Subsequently, we took into account the restrictions of R, E_out and E_mean to give proper t and E_in. As mentioned before, E_mean should be less than 50 MeV/n to ensure the physical goal, and E_out (viz. the energy of ¹²Be right before the ΔE detector) should be above 20 MeV/n to ensure the availability of ΔE - E method.

Fig. 4.Full-screen View

The change trends of R with E_in under different t and E_out.

R is restricted by systematic error if the statistical error is small enough. It is of certain difficulty to achieve 0.1% systematic error for a directly measured quantity, like R. To ensure a <2.5% precision of σR, the factor 1//ln R before the systematic error of R in Eq. (5) shall be less than 25. Accordingly R should be less than 0.96. Therefore, proper ranges of t and E_in are indicated as the hatched area in Fig. 4.

In the energy region of 30–50 MeV/nucleon we intend to obtain three data points by using subtraction method. As shown by Conditions (1) and (2) in Fig. 5, at incident beam energy of E₁ and E₂, with target thickness of t₁ and t₂, reaction cross sections σ₁ and σ₂ are measured by the transmission method, respectively, and by adjusting E₁ and E₂, both the outgoing beam energies E_out can be of the same energy. Then, the reaction rate in the target in Condition (2) shall be equal to that of the corresponding thickness of t₂ in the target in Condition (1). By subtracting the two data, one obtains another σR as,

Fig. 5.Full-screen View

(Color online) Schematic diagram of the subtraction method.

where R_in-1 and R_in-2 denote the transmission rates of the target-in experiment in Conditions (1) and (2), respectively. The transmission rates of the target-out experiments do not appear in Eq. (6) because the reaction rate outside the target is canceled between the two measurements. The error of this deduced σR was determined in Ref. [40]. Through this method, three data points can be obtained with only two measurements. This greatly improves the efficiency of the experiment. The key point is that the outgoing energies E_out in the two measurements should be the same. And the interval between the two adjacent incident beam energies E_in should be over 15 MeV/nucleon in order to make sure the energy points are evenly distributed between 30–50 MeV/nucleon.

Thus we have decided the range at E_in, and C and Al target thicknesses, and the detecting system. Next, we are to determine the number of events we need according to the requirement of the statistical error, and estimate the requisite beam intensity.

According to Eq. (3) the statistical error should be less than 0.5%, so that it will not be a main error source, which gives

Since R is at most 0.96, N_in of 10⁶ is already sufficient. So, it will take only 10³ seconds of beam time under typical condition of 10³ s^-1 beam intensity, which is very practical.

To sum up, for the experiment a ¹²Be beam of about 10³ s^-1 intensity at 20–70 MeV/nucleon is needed to provide the projectile particles, and the outgoing energy of ¹²Be after the reaction targets in the two measurements should be the same. In addition, a Si and a scintillator detector are needed for particle identification. Finally, we select two energy points, find the corresponding target thickness, evaluate the R and decide corresponding requirement of N_in. The detailed experimental scheme is given in Table 1.

We find that the Radioactive Ion Beam Line in Lanzhou (RIBLL) [44] is a suitable candidate for providing ¹²Be beam. The schematic diagram of the layout of RIBLL is shown in Fig. 6.

Fig. 6.Full-screen View

Schematic diagram of the layout of RIBLL.

The primary beam ¹⁸O⁸⁺ is accelerated by the Heavy Ion Research Facility of Lanzhou (HIRFL) and introduced to RIBLL. It bombards the production target of Be at T0 and generates the secondary beam of ¹²Be. An Al degrader at C1 is used for an energy-loss analysis of the secondary beam separation. Another Al degrader at T1 is used for energy degradation. Two slits at C1 and C2 is used for momentum acceptance controlling. Two plastic scintillation counters at focal points T1 and T2 can provide the TOF information. Si solid state detector (SSD) at T2 can be used to provide the ΔE signals for particle identification before the reaction target. Energy E_in of the incident beam right before the reaction target is determined by the $B\rho$ value of the fourth dipole magnet D4. Based on this configuration, the ¹²Be beam condition is simulated by LISE++ using a typical 100 enA primary beam of ¹⁸O⁸⁺ at 80 MeV/nucleon. Major parameters of the simulation result given in Table 2. It’s worth mentioning that similar experiments have been performed on RIBLL since 2000 [45, 46, 47, 48, 49]. From their results we infer that typically the error of σR is up to about 5% including 3%–4% systematic error. So we need further consideration based on the real performance of RIBLL to reduce the error especially the systematic error.

For the detecting system, typically a Si detectors of 1500 μm thickness is competent for the ΔE measurement, and a CsI(Tl) scintillator detector of 30 mm thickness is adequate for the E measurement. It should be noted that the Si detector is made of single crystal, hence the concern of channeling effect to the particles being detected. By tilting the Si detector against the beam axis at a certain angle, the fraction of channeling events can be reduced to $<1\%$ [40]. The tilting angle can be determined by studying the angle dependence of the channeling events in advance.

Based on the above feasibility study, the σR measurement is reasonable and valid, and the requisite conditions can be satisfied. Through the measurement the nuclear-matter density distribution of ¹²Be ground state can be determined and then used to extract the component of ¹²Be ground state. In Sec.IV, we elaborate that how to determine the s orbital component through σR.

IV. EXTRACTION OF THE S ORBITAL COMPONENT OF ¹²BE GROUND STATE

As shown by the fermionic molecular dynamics (FMD) calculations (see the inset (b) in Fig. 10 of Ref. [33]), different configuration mixings of the two valence neutrons in ¹²Be lead to different nuclear-matter density distributions of ¹²Be, and the difference is obviously indicated in the surface structure of ¹²Be. Inspired by this result, we bring forward a method to extract the ground state component of ¹²Be.

In the method, we treat ¹²Be as a system of ¹⁰Be core plus two valence neutrons as usual. The ¹⁰Be determines the nuclear-matter density distribution of the core part of ¹²Be, while the two valence neutrons determine the outer part density distribution. According to the intruder configuration of ¹²Be [4], the two valence neutrons are populated in 1p_1/2, 1d_5/2 and 2s_1/2 states at certain occupation probabilities. So we can construct the outer structure of ¹²Be according to a certain configuration as long as a model can give preferable density distribution of the two valence neutrons corresponding to different states. By using an appropriate function to describe the core structure of ¹²Be, we can construct a nuclear-matter density distribution of the ground state of ¹²Be. By adjusting the proportion of the components, we can find a nuclear-matter density distribution which is consistent with the experimental result. Then the corresponding proportion can provide configuration information of ¹²Be ground state.

We have tried to extract the ground state component of ¹²Be in order to verify the feasibility of the method. As mentioned above, a precondition of the method is to obtain the valence neutron density distributions corresponding to 1p_1/2, 1d_5/2 and 2s_1/2 states. Usually single particle model (SPM), three-body-model, cluster model, shell model, etc. are used to calculate the density distribution of the valence neutron. In this article, the SPM [35] is used. In this SPM, the Woods-Saxon potential, the Coulomb barrier and the centrifugal barrier are taken into account. The nuclear part of the potential assumed is written as

where V₀ is the depth of the Woods-Saxon potential, V₁= 17 MeV is the l·s strength taken from Ref. [50], rl·s = 1.1 fm is the radius of spin-orbit potential, Rc=r₀A^1/3 (r₀= 1.2 fm) is the radius of the Woods-Saxon potential, and a= 0.6 fm is the diffuseness parameter. V₀ is adjusted to reproduce the separation energy of the valence neutron. Here we treat the two valence neutrons as equal and set the separation energy of single valence neutron to be a half of the two-neutron separation energy of ¹²Be. The corresponding nuclear-matter density distributions of the two valence neutrons are shown in Fig. 7. We can see that the 2s_1/2 state has larger density in the surface. Although different models will not give exactly the same density distributions of the two valence neutrons, we infer that our result is less model-dependant based on the fact that s intruder valence neutron configuration is the chief cause of the halo-like structure in light nuclei just as the cases of ¹¹Li and ¹¹Be.

Fig. 7.Full-screen View

Density distributions of the two valence neutrons corresponding to 1p_1/2, 1d_5/2 and 2s_1/2 states.

Then we consider a configuration mixing of 1p_1/2, 1d_5/2 and 2s_1/2 for the two valence neutrons as follows:

where α and β denote the occupation probability of (1p_1/2)² and (1d_5/2)² configuration, respectively. The integrals of the single configuration density and mixed density are all normalized to be two nucleons. Combining the ¹⁰Be core distribution of Gaussian-Gaussian (GG) parametrzation given in Ref. [33], we calculated the corresponding σR and obtained a range of σR accordingly, as indicated in Fig. 8. We can see clearly that the σR is sensitive to the s orbital occupancy, and it is difficult to derive the p and d orbital occupancies from the σR data. This is because the 2s_1/2 component contributes much more to the surface structure of ¹²Be than the other components. By using the σI of ¹²Be on C in Ref. [30], we extract the upper limit of the s orbital occupation probability to be about 56%. The upper limits are indicated by the vertical line in Fig. 8(c).

Fig. 8.Full-screen View

The reaction cross section (σR) of ¹²Be calculated at 33.2 MeV/nucleon and 790 MeV/nucleon, plotted against the occupation probability of the valence neutrons configuration in p, d and s orbit.

The result indicates that the s orbital is not a dominant component in ¹²Be ground state. It is supposed to be the reason why the halo-like structure in ¹²Be is not evident, because the s orbital component is the major contribution to the halo-like structure. Compared with the calculations given by Ref. [4-20, 22-24], our result is relatively small (Table 3). Although our result is consistent with the β decay result given by Ref. [42], large uncertainty exists in both results. Therefore, higher precision extraction is needed.

By comparing the increasing trend of σR in insets (c) and (f) of Fig. 8, one can sees that at low energy region, σR is even more sensitive to the s orbital occupancy. This implies that the extraction uncertainty can be reduced if σR at low energy region is used. So, we calculate σR based on an s orbital occupation of 28% (half the value of the upper limit) and a p orbital occupation probability of 25%, as the percentage of 1p_1/2 component was determined as 25 ± 5% [10]. The result is shown in Fig. 8(f), in which the hatched area indicates 2% error band. Thereby we can extracted that uncertainty of the s orbital occupation probability is about 23%. And take into account that six data points will be available from the experiment introduced in previous section, the precision of the extracted s orbital occupation probability is expected to be around 9%. This makes the experimental measurement of σR at low energy region more desirable.

Since the percentage of the 1p_1/2 component has been determined by Gamow-Teller transition strengths [10], if either 1d_5/2 or 2s_1/2 component is extracted precisely, the ground state configuration of ¹²Be shall be reliably determined. Our method provides a promising approach to extract the 2s_1/2 component of ¹²Be ground state and to determine the ground state configuration of ¹²Be. Through the proposed σR measurements, we expect to extract the s orbital occupation probability with 9% uncertainty.

V. SUMMARY

In summary, the ground state structure of ¹²Be is of great significance. The ambiguity of the configuration of ¹²Be ground state has attracted our attention. We bring forward to determine the s orbital component of ¹²Be ground state by measuring the σR of ¹²Be on C and Al at several tens of MeV/nucleon. By using existed interaction cross section data of ¹²Be on C at 790 MeV/nucleon, we roughly determine the upper limit of the s orbital occupation probability of ¹²Be ground state to be 56% with SPM calculations. The precision of the s orbital occupation probability is expected to be 9% by using the proposed 2% σR data. The feasibility of the σR measurement is carefully studied and concrete procedures of the experiment are given.