Measurement of Elastic Scattering Angular Distributions for Proton-Rich Nuclei ^21,22Na on Double-Magic Nucleus ⁴⁰Ca

Abstract

Present and future rare isotope accelerator facilities provide new opportunities to explore the structure of unstable nuclei. We report the measurements of the elastic scattering angular distributions of ²¹Na and ²²Na on the doubly magic ⁴⁰Ca above the Coulomb barrier energies, using high-purity post-accelerated ISOL beams from Beijing Radioactive Ion Beam Facility (BRIF). Angular distributions were measured with a silicon detector telescope array, and relative cross sections were determined with a CaF₂ target on Au backing. The data were well reproduced by optical model calculations with Woods–Saxon and USNP potentials, the latter giving better agreement. These results confirm the stable operation and performance of the BRIF ISOL production and post-acceleration system, demonstrate its capability to provide radioactive beams of useful intensity and purity for future investigations of reaction dynamics and astrophysically relevant processes involving proton-rich nuclei, and simultaneously extend proton-rich elastic scattering studies to heavier systems.

1. Introduction

When nuclear studies extend toward the proton-rich or neutron-rich sides of the chart of nuclides—particularly near the drip lines—the resulting isotopes exhibit strong instabilities that impose substantial experimental challenges [1,2]. Studies of exotic nuclei are essential for understanding nucleosynthesis pathways and stellar evolution. In addition, weakly bound nuclei often present structural features markedly different from those of stable isotopes, such as spatially extended matter distributions or halo-like configurations arising from their extremely low separation energies [3]. These unique characteristics make reaction dynamics involving drip-line nuclei an important topic in low-energy nuclear physics [4,5,6,7].

The advent of rare-isotope beam (RIB) technology has enabled systematic studies of such exotic systems. As early as 1985, Tanihata et al. employed RIBs at LBNL’s BEVALAC to investigate the structure of neutron-rich nuclei such as ¹¹Li, leading to the landmark discovery of neutron halos [8]. Since then, RIBs have become powerful tools for probing novel nuclear structures and reaction mechanisms far from the stable line, stimulating broad scientific interest worldwide [9].

At present, two principal methods are used to produce RIBs. Projectile fragmentation (PF), or the in-flight method, can generate extremely short-lived nuclei (lifetimes down to several hundred nanoseconds) but typically yields beams with large energy spreads, sizable emittances, and high energies. In contrast, the isotope-separation online (ISOL) method provides beams with adjustable energies and excellent purity, albeit with limitations in producing very short-lived nuclides. Major PF facilities include FRS [10] at GSI, RIBF [11] at RIKEN, RIBLL [12] at IMP CAS, and FRIB [13] at MSU, while ISOL facilities such as ISOLDE [14] at CERN, SPIRAL [15] at GANIL, ISAC [16] at TRIUMF, and BRIF [17,18] at CIAE represent the complementary approach. New large-scale facilities, including FAIR [19], HIAF [20], and RAON [21], are expected to expand global capabilities for studying rare isotopes further.

Although extensive experimental efforts have been devoted to the study of weakly bound nuclei, the availability of post-accelerated ISOL beams for medium-mass proton-rich isotopes remains comparatively limited worldwide. In this context, it is crucial not only to pursue new physics cases but also to validate the performance and reliability of emerging radioactive ion beam facilities capable of delivering such beams. The commissioning of post-acceleration systems, in particular, requires benchmark reaction measurements that can simultaneously assess beam purity, intensity, energy definition, and stability under realistic experimental conditions. Therefore, demonstration experiments using well-understood reaction channels provide an effective approach to verify facility performance while extending the accessible region of exotic nuclei. The Beijing Radioactive Ion Beam Facility (BRIF) [17,18], located at the China Institute of Atomic Energy, is an ISOL-based complex designed for the production and post-acceleration of high-purity radioactive ion beams. The facility integrates several major components: a compact 100-MeV proton cyclotron capable of delivering 200 µA beam currents [22]; a high-efficiency production target and ion-source assembly optimized for rapid release and ionization of reaction products [23]; an ISOL mass-separation system with a resolving power approaching 2 × 10⁴ [24]; the 13 MV HI-13 tandem accelerator used for charge breeding and post-acceleration [25]; and a superconducting linear accelerator capable of accelerating ions up to 13 MeV/q. A schematic overview of the facility is presented in Figure 1.

Figure 1. The layout of BRIF and the experimental setup in the target chamber at the R60 terminal. The two MCPs are used to measure TOF signals, and the QSD was used to monitor the intensity of the beam. The silicon detector array was used to detect the scattered particles. The two MCPs were dismantled to perform the elastic scattering angular distribution measurements.

Since entering operation in 2015, BRIF has delivered a variety of radioactive beams—including ^20,21,22Na and ^37,38K—and has supported several milestone experiments. These include the first direct observation of the exotic $\beta$-$\gamma$-$\alpha$ decay sequence in ²⁰Na [26], initial reaction studies demonstrating the feasibility of precision measurements with post-accelerated beams [27], and the first online collinear laser spectroscopy of radioactive nuclei in China [28]. At present, the facility can provide 58 different species of RIBs with half-lives down to 58 ms. The combination of high beam purity, broad species coverage, and a flexible acceleration scheme makes BRIF a powerful platform for detailed investigations of reaction dynamics and structure phenomena in unstable nuclei.

Elastic scattering, as a process occurring at the periphery of the nucleus, cannot directly provide information about the internal structure of the nucleus. However, it can probe the tail of the wave function to study the properties of the nuclear surface. Moreover, constraining the radii and diffuseness parameters of rare isotopes through elastic scattering is of significant importance for studying r-process nucleosynthesis. In weakly bound nuclei, due to the tunneling effect, weakly bound nucleons can penetrate the classically forbidden region, forming a halo-like structure surrounding a tightly bound core. The angular distribution data and optical potential parameters derived from elastic scattering between weakly bound and tightly bound nuclei allow for in-depth investigation of these properties [29]. There are many experiments performed for the measurement of elastic-scattering angular distributions of weakly bound nuclei, including neutron-rich nuclei [30,31,32,33,34,35,36,37,38,39] and neutron-deficient nuclei [29,33,40,41,42,43,44,45,46,47,48,49,50], as shown in Figure 2. Based on existing experiments, scientifically rigorous experimental and analytical methodologies have been established, enabling multi-faceted investigations into the properties of weakly bound nuclei. However, in the aforementioned studies, the heaviest nuclide examined is only ¹⁷F, which somewhat limits the generalizability of findings on weakly bound nuclei. Therefore, research on heavier weakly bound nuclei is highly desirable.

Figure 2. Radioactive nuclei whose reaction mechanisms have been studied so far with measurement of elastic scattering cross sections [5,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50]. Other works can be found from the references in Refs. [5,32]. The blue and orange squares represent the nuclei studied in other works and present work, respectively.

This experiment, utilizing the ²¹Na and ²²Na beams provided by the BRIF facility, successfully performed the elastic scattering measurements of proton-rich nuclei ²¹Na and ²²Na with the double-magic nucleus ⁴⁰Ca at energies above the Coulomb barrier. The results not only demonstrate the excellent and reliable performance of the BRIF facility equipped with its post-acceleration system but also expand the scope of exotic nuclear. The main results of this study were previously reported in a short communication in Ref. [51]. In this article, we present a comprehensive description of the experiment and a detailed analysis of the data.

2. Experimental Setup

The measurements were performed at BRIF of CIAE. Radioactive Na isotopes were generated by bombarding an MgO production target with a 100 MeV proton beam delivered by the on-site compact cyclotron operating at a current of 10 eµA. After diffusing out of the heated target matrix, the reaction products were ionized and extracted into the ISOL system. A two-stage mass-separation scheme provided the required beam purity, isolating the Na isotopes from other reaction fragments prior to post-acceleration.

Following mass selection, the radioactive ions were injected into the HI-13 tandem accelerator, which boosted the ²¹Na and ²²Na beams to kinetic energies suitable for scattering measurements. These beams were then transported to the R60 reaction station via the standard BRIF beamline optics. The R60 station houses a large cylindrical chamber with a diameter of 1.2 m, allowing the installation of extended detector arrays and accommodating a wide range of reaction geometries. In this experiment, the ²¹Na and ²²Na beam energies were 91 MeV and 87 MeV, respectively, with a beam energy spread of 1% and an emittance of 40 mm·mrad·MeV^1/2 [52]. With the assistance of an analyzing magnet, the beam purity could reach nearly 100%. Their typical intensities were 1.2 × 10⁴ pps for ²¹Na and 2.5 × 10⁵ pps for ²²Na. The measured ²²Na beam spot at the scattering target exhibited FWHM values of 2.49 mm (horizontal) and 2.36 mm (vertical), demonstrating good beam focusing at the target position.

Beam identification at the reaction chamber entrance relied on a pair of micro-channel plates (MCPs) timing detectors positioned upstream and downstream of the target. The fast timing signals from the MCPs provided time-of-flight (TOF) information, which, combined with the total deposited energy, enabled isotope separation. Representative TOF–energy correlations are shown in Figure 3, including both the reference ²³Na spectrum from Ref. [27] and the new ²²Na data. A 1000 µm quadrant silicon detector (QSD) located downstream monitored the beam rate and independently confirmed the beam energy. After the identity and stability of the Na beams were verified, the MCP detectors were withdrawn from the beam path so as not to obstruct the scattering measurements.

Figure 3. The left panel illustrates the identification of the beam particles. The horizontal axis represents the total energy determined with the QSD, and the vertical axis corresponds to the TOF signal measured with the MCPs for the ²³Na beam in (a) from Ref. [27] and ²²Na from this work in (b). (c) Presents the measurement of the residual energy (E_r) versus the energy lost ($\Delta E$) in the first two planes of the silicon detector array (300 and 20 µm, respectively). The upper panel of (c) corresponds to the elastic scattering of ²²Na(⁴⁰Ca, el), while the lower panel shows the data for the elastic scattering of ²¹Na(⁴⁰Ca, el). This figure also shows the elastic scattering data of ^21,22Na beams on Au (target substrate).

The scattering target consisted of a natural CaF₂ layer with an areal density of 1120 ± 55 µg/cm², deposited onto a 400 µg/cm² Au foil that served as a mechanical backing and provided a secondary scattering reference. Target thickness was determined using two methods. During CaF₂ vacuum evaporation, the thickness was monitored in real time with a quartz crystal film thickness monitor. For verification, a 10 µm aluminum foil comparator was placed next to the Au substrate so that CaF₂ was deposited simultaneously on both. After deposition, the foil was removed and weighed with an ultra-precision electronic balance (with an accuracy of 10⁻⁷ g), and the measured CaF₂ thickness was used to determine the thickness of the target on the Au substrate. A silicon-detector telescope array placed 10 cm downstream of the target detected the reaction products. The array consisted of four identical telescopes, each composed of three detection layers. The first two layers were single-sided silicon-strip detectors (SSSDs) of 20 µm and 300 µm thickness, respectively. Each SSSD comprised 16 parallel strips 3 mm wide with 0.1 mm inter-strip spacing; the two SSSDs in each telescope were oriented orthogonally to provide two-dimensional position determination with an energy resolution of roughly 0.3%. The final layer was a 1000 µm QSD with an active area of 50 × 50 mm², primarily used to identify light charged particles such as protons and $\alpha$ particles. Altogether, the strip detectors provided 1024 effective pixels for angular reconstruction. The array covered laboratory scattering angles from 10.2^∘ to 39.6^∘.

Energy-loss calculations show that by the time the beams reached the center of the CaF₂ target, their energies were reduced to 86.46 MeV for ²¹Na and 82.27 MeV for ²²Na. For the elastic scattering analysis, only the first two layers of each telescope were required: the 20 µm SSSD recorded the energy loss $\Delta E$, and the 300 µm SSSD measured the residual energy $E_{\mathrm{r}}$. Typical $\Delta E$-$E_{\mathrm{r}}$ distributions and total-energy spectra are presented in the right panel of Figure 3. The locations corresponding to scattering from ⁴⁰Ca are clearly distinguished from those associated with reactions on the Au backing, allowing the events of interest to be selected unambiguously. No reliable particles originating from breakup events were observed in the spectra of this experiment.

3. Data Analysis and Discussion

The differential cross section of a reaction can be expressed as

$$d\sigma /d\Omega ={\displaystyle \frac{N}{I{N}_s\Omega }},$$

(1)

where N and I represent the numbers of the emitted ions and the incident ions, respectively; $N_s$ denotes the number of the target atom per unit area, and $\Omega$ represents the solid angle.

To suppress uncertainties associated with the solid angle and the absolute beam normalization, a relative-normalization technique was employed in this work. At the beam energies of interest, the elastic scattering of Na ions from ¹⁹⁷Au is purely Coulombic; therefore, its differential cross section is given exactly by the Rutherford formula. By simultaneously measuring elastic events from both ⁴⁰Ca and ¹⁹⁷Au in the composite CaF₂/Au target, the ratio of the Na+⁴⁰Ca differential cross section to the Rutherford prediction can be obtained from

$${\displaystyle \frac{{(d\sigma /d\Omega )}_{\mathrm{Ca}}}{{(d\sigma /d\Omega )}_{\mathrm{Ca}}^{\mathrm{Ruth}}}}={\displaystyle \frac{Y_{\mathrm{Ca}}\times n^{\mathrm{Au}}\times {(d\sigma /d\Omega )}_{\mathrm{Au}}^{\mathrm{Ruth}}}{Y_{\mathrm{Au}}\times n^{\mathrm{Ca}}\times {(d\sigma /d\Omega )}_{\mathrm{Ca}}^{\mathrm{Ruth}}}}.$$

(2)

Here, $Y_{\mathrm{Ca}}$ and $Y_{\mathrm{Au}}$ represent the numbers of scattered ^21,22Na ions arising from interactions with ⁴⁰Ca and ¹⁹⁷Au, respectively; $n^{\mathrm{Ca}}$ and $n^{\mathrm{Au}}$ are the corresponding areal densities of Ca and Au atoms in the target. ${(d\sigma /d\Omega )}_{\mathrm{X}}^{\mathrm{Ruth}}$ denotes the Rutherford differential cross section for Na+X scattering at the appropriate energy in the laboratory reference system. As is evident from the structure of the expression, both the solid-angle factor and the absolute beam-intensity normalization cancel out, leaving the ratio dependent only on measurable yields and well-defined target parameters. For the angular distribution analysis, data recorded by detectors within an interval of 2^∘ in laboratory angle were merged to form a single experimental data point.

In Figure 4, the experimental elastic-scattering angular distributions for the ^21,22Na+⁴⁰Ca systems measured in the present work are displayed. For comparison, the corresponding data for the stable nucleus ²³Na+⁴⁰Ca from Ref. [27] are also included. The error bars shown in the figure represent statistical uncertainties only. To characterize the reaction dynamics, the experimental angular distributions were compared with optical-model calculations using two different global interaction parameterizations: a Woods–Saxon (WS) systematic potential [53], the updated systematic single-folding model potential (USNP) [54], and the São Paulo potential (SPP) [55]. The ²³Na+⁴⁰Ca data from Ref. [27] were analyzed using the same approach to provide a consistent reference. The root mean square (rms) matter radii of ^20,21,22Na are listed in Table 1, along with comparisons to calculations from the Hartree–Fock calculations with the SkX interactions (HFSkX) and other experimental results [56]. The proton and neutron density distributions of the projectile are obtained from Hartree–Fock calculations based on the SkX parameterization for ²¹Na and ²³Na, considering the good agreement between data and results of optical model calculations with USNP. However, the rms radii of the matter distributions of ²²Na is 2.819 fm, which is larger than the experimental value of 2.72 ± 0.04 fm. Therefore, the original density distribution of ²²Na is rescaled so that the rms radii agree with its experimental value. The predicted angular distribution with this changed density distribution provides a better description of the elastic scattering data for the ²²Na+⁴⁰Ca system. This suggests that it is possible to extract the rms radii of heavy ions from the optical model analysis of the elastic scattering data with the USNP systematics. As shown in Figure 4, both potentials reproduce the general features of the measured angular distributions reasonably well; moreover, the calculations employing the USNP potential show systematically better agreement with the data over the full angular range.

Figure 4. (a–c) Elastic scattering angular distributions of ^21,22,23Na+⁴⁰Ca and the optical model calculations. The points represent the results obtained in ^21,22Na+⁴⁰Ca experiments and the experimental results of ²³Na+⁴⁰Ca from Ref. [27]. The black solid, red dashed, and blue dotted curves represent the results of optical model calculations for ^21,22,23Na+⁴⁰Ca with USNP [54], WS [53], and SPP [55], respectively.

Table 1. HFSkX, EXP, and USNP denote the rms matter radii of ^20,21,22Na obtained from HFSkX calculations, previous experimental data [56], and the recommended values in this work, respectively.

	HFSkX	EXP	USNP
21Na	2.795	2.75(2)	2.795
22Na	2.819	2.72(4)	2.720
23Na	2.845	2.83(3)	2.845

For the proton-rich nuclei of interest, although ²¹Na and ²²Na have relatively different proton separation energies (2.432 MeV for ²¹Na and 6.739 MeV for ²²Na), the consistency of their angular distributions indicates that this level of separation energy difference is insufficient to manifest an enhancement in coupling effects within this energy region. Meanwhile, it is noteworthy that the USNP potential describes the present ^21,22Na data without any adjustment of its parameters, despite being developed primarily from optical model analysis of elastic scattering data of stable nuclei. The energy dependence of USNP is a predominant factor, though the extraction of rms radii from the optical model analysis of elastic scattering is affected by many factors. This result demonstrates the robustness and predictive power of the folding-type potential when extrapolated to proton-rich systems in the medium-mass region. Combining the present results with previous experimental data on ⁸B [29], ⁹C [50], ¹²N [32], and ¹³O [33], it is observed that at incident energies of two to three times the Coulomb barrier, the breakup effect has a negligible influence on elastic scattering cross section (although the breakup cross section is known to be affected by structural effects, as documented in Ref. [50]). This phenomenon may be attributed to two primary factors: most importantly, this is the centrifugal and Coulomb barriers experienced by their valence protons that suppressed the breakup coupling effects to the elastic scattering channel [57]. Secondly, the influence of the Coulomb interaction is also diminished due to the relatively high incident energy. The excellent agreement between the present experimental data and theoretical calculations provides strong evidence for the reliability of the BRIF facility and its post-acceleration system. In the future, we plan to conduct a series of experiments at the BRIF facility to further advance the understanding of exotic nuclei from multiple perspectives.

4. Conclusions

In summary, we have carried out elastic-scattering measurements of the proton-rich nuclei ^21,22Na on the doubly magic nucleus ⁴⁰Ca using post-accelerated radioactive ion beams delivered by BRIF. Beyond the specific reaction study, this experiment serves as a commissioning and validation benchmark for the BRIF ISOL production and post-acceleration system.

The successful production, mass separation, and post-acceleration of medium-mass proton-rich beams with typical intensities of 10⁴∼10⁵ pps, high purity, and an energy spread of approximately 1% demonstrate the stable and reliable operation of the facility. The well-defined angular distributions, clear Fresnel structures, and consistent agreement with optical-model calculations confirm the accuracy of beam energy determination, angular resolution, and normalization procedures. These results collectively establish BRIF as a capable platform for precision reaction measurements with unstable nuclei in the A ≈ 20 mass region.

From the physics perspective, no pronounced long-range suppression or strong breakup effects are observed at the present incident energies, indicating that continuum couplings play only a minor role under these conditions. Furthermore, optical-model analyses based on WS potential, USNP, and SPP showed low parameter sensitivity and yielded tightly constrained interaction strengths, suggesting that the surface properties of these Na isotopes remain relatively compact despite their modest separation energies. The present results extended the systematics of elastic scattering of weakly bound nuclei beyond the light-mass region (A ≤ 17), providing new benchmarks for reaction dynamics involving medium-mass proton-rich isotopes. These findings also highlighted the suitability of ⁴⁰Ca as a clean probe nucleus for isolating weak-binding effects in exotic systems.

With the demonstrated beam quality and post-acceleration performance, BRIF is well positioned to support future experimental programs involving direct reactions, transfer reactions, and measurements of astrophysical relevance, including studies of proton-rich nuclei participating in explosive nucleosynthesis processes. The present work thus represents an important step toward expanding the experimental reach of BRIF in both nuclear structure and nuclear astrophysics research.