Introduction

Nuclei with extreme neutron-to-proton ratios, located far from the line of β stability, play a key role in modern nuclear physics. With major advances in developing next-generation radioactive isotope (RI) beam facilities, an increasing number of exotic nuclei have become accessible for experimental studies. As a consequence, new phenomena—particularly among neutron-rich nuclei—have been observed, including shape coexistence [1], shape deformation [2] and shell evolution [3], thereby reshaping our understanding of nuclear structure. In addition, studies on structural properties and decay properties in heavy neutron-rich nuclei provide essential insights into nuclear astrophysics, particularly the rapid neutron-capture process (r-process), which is thought to synthesize approximately half of the elements heavier than iron in the universe. For instance, efforts have been made to produce neutron-rich nuclei that lie within the r-process abundance regions around [4] and [5].

In recent years, considerable attention has been devoted to the neutron-rich nuclei with Z=61 - 71 and in the rare-earth region, owing to their significance in both nuclear structure and nucleosynthesis. These neutron-rich nuclei are expected to exhibit strong deformation [6]. Moreover, their decay properties affect the small, broad peak at in the r-process abundance distribution [7]. Despite their importance, experimental studies of these neutron-rich nuclei remain limited due to their low production yields. The current experimental frontier in this region [8-12] is shown in Fig. 1.

Fig. 1Full-screen View

(Color online) Part of nuclear chart showing the forefront of the rare-earth region. The black squares represent the stable nuclei. Colored squares show the isotopes can be produced by the fragmentation of ¹⁹⁸Pt at 800 MeV/nucleon on Be (4 g/cm²) with an intensity of 1.0×10¹¹ ppp, corresponding to pps in the slow extraction mode. The estimated production yields are given by different colors. The neutron-rich isotopes identified experimentally so far are highlighted by the thick black lines. Highlighted by the green triangles and red circles are the new neutron-rich isotopes expected to be observed by the fragmentation of ¹⁹⁸Pt at HIAF. The numerical data for the production yields of these new isotopes are listed in Table 1 in the appendix. See text for details

To study these neutron-rich rare-earth nuclei in the laboratory, it is critical to identify the most effective routes to produce them with radioactive isotope beam facilities. Over the past decades, the rare-earth region has been accessed mainly through fission-based methods, such as spontaneous fission of ²⁵²Cf [13-16], thermal neutron-induced fission of ²³⁵U [17-19], as well as multinucleon-transfer reactions [20-22]. These methods, however, are largely restricted to nuclei near the β stability line. Thanks to advances in in-flight separators, in-flight fission of a ²³⁸U beam has been used successfully to produce neutron-rich rare-earth nuclei. For instance, new neutron-rich isotopes ranging from Nd (Z=60) to Pt (Z=78) [23], and from Ba (Z=56) to Er (Z=68) [9, 12, 24] were discovered at GSI and RIBF by means of in-flight fission of ²³⁸U at 1 GeV/nucleon and 345 MeV/nucleon, respectively. Toward still more neutron-rich rare-earth isotopes, however, in-flight fission is hindered by the rapid decrease in cross section, which limits further access to this region.

To overcome this limitation, it is essential to explore new pathways for the production of neutron-rich nuclei. Projectile fragmentation of heavy ions has proven to be a powerful tool for producing such nuclei [25, 26]. By selecting a suitable projectile, large production cross sections can be achieved through fragmentation reactions. Indeed, fragmentation of a recently introduced projectile, ¹⁹⁸Pt, was performed at 85 [10] and 186 MeV/nucleon [11], leading to the identification of new isotopes ^182,183Tm, ^186,187Yb,^189,190Lu, and ^191,192Hf. These discoveries demonstrate the strong potential of ¹⁹⁸Pt for accessing even more neutron-rich isotopes in the rare-earth region.

In this work, the production potential of several projectiles is evaluated for producing new neutron-rich rare-earth nuclei at the High-rigidity Radioactive Ion Beam Line (HIRIBL) of the High-Intensity Heavy-ion Accelerator Facility (HIAF).

Experimental setup at HIRIBL

As a next-generation RI facility, HIAF is expected to provide a variety of beams ranging from hydrogen to uranium with high intensities [27], thereby providing new opportunities for the production and investigation of heavy neutron-rich nuclei. Primary beam intensities up to particles per pulse (ppp) can be expected in both fast- and slow-extraction modes after acceleration in the main synchrotron [27]. The HIRIBL is HIAF’s primary in-flight separator for rare-isotope studies. HIRIBL has a two-stage structure, namely a Pre-Separator (from PF0 to PF4) and a Main-Separator (from MF0 to MF6), presenting a concept similar to that of other advanced in-flight separators worldwide, such as BigRIPS [28] at RIBF, SuperFRS at FAIR [29] and ARIS [30] at FRIB. HIRIBL offers a high magnetic rigidity of up to 25 Tm and a large acceptance, with angular acceptances of ±30 mrad horizontally and ± 15 mrad vertically, and a momentum acceptance of ±2% [27, 31, 32]. These capabilities enable HIRIBL to produce, separate, and identify nuclei far away from the β-stability line. The schematic and detailed characteristics of the HIRIBL separator are presented in Refs. [31-33].

To search for new isotopes produced by fragmentation of primary beams, HIRIBL will be operated in the achromatic mode [31]. The primary beams will be delivered in the slow-extraction mode with an extraction time of 3 s and a repetition period of 13 s [33]. In this mode, the Pre-Separator is used for separating and purifying the fragments, and the Main-Separator serves as a spectrometer to identify the isotopes event by event by measuring the time of flight (TOF), energy loss (ΔE) and magnetic rigidity (Bρ). The time of flight will be measured by plastic scintillators installed at the MF0 and MF6 focal planes. The magnetic rigidity will be determined by the trajectory reconstruction using the position and angle information measured at MF0, MF4, and MF6. The energy loss ΔE will be measured with an ionization chamber at MF6. The atomic number Z and the mass-to-charge ratio A/Q of nuclei produced by fragmentation will be derived from the TOF-ΔE and TOF-Bρ correlations, respectively. To identify particles unambiguously, the isomer-tagging technique [34] will be used through detection of the delayed γ-rays emitted from known short-lived isomers with Ge detectors at MF6. The experimental setup is included in a LISE++ [35] simulation for the transmission efficiency of HIRIBL.

At in-flight separators, one challenge for the production of heavy nuclei is the charge-state distribution [36]. Indeed, at intermediate energies, a wide charge-state distribution [9, 11] causes difficulties in the identification and purification of heavy reaction products. To distinguish different charge states clearly for heavy nuclei at HIRIBL, a high A/Q resolution is required. This is because a fully-stripped ion with A and the hydrogen-like one with are very close in the A/Q spectrum, for the same Z. Taking ¹⁸¹Er as an example, the A/Q resolution should be better than 8.5×10^-4 [given here as root mean square (RMS)] to separate ¹⁸¹Er⁶⁸⁺(A∕Q = 2.6618) and ¹⁷⁸Er⁶⁷⁺(A∕Q = 2.6567) at the 6σ level.

With such a high requirement on A/Q separation, the resolution specifications for the beam-line detectors at HIRIBL were estimated. Based on the TOF method, A/Q can be obtained as(1)(2)(3)where x and x₀ represent the measured positions, and represents the magnetic rigidity of the central trajectory. The coefficients and are the ion-optical transfer-matrix elements, which are given in Ref. [31].

According to Eqs. 1 and 2, the resolution of A/Q can be expressed as(4)where σBρ and σ_TOF represent the RMS resolutions of Bρ and TOF, respectively. From Eq. 3, the resolution of Bρ can be expressed as(5)σx and are the position and angle resolutions, respectively. Position and angle information will be measured using PPAC [37], MWDC [38] or GEM-TPC [39] detectors. For these detectors, a position resolution of σx=300 μm can be achieved. Thus, is estimated to be 0.015%. Assuming an energy of approximately 600 MeV/nucleon for ¹⁸¹Er (), the time resolution should be better than 1.05×10^-4 to separate the charge states ¹⁸¹Er⁶⁸⁺ and ¹⁷⁸Er⁶⁷⁺ at the 6σ level. The flight length from MF0 to MF6 is 118 m, corresponding to a flight time of 500 ns, and thus requiring a TOF resolution of 53 ps (RMS).

Evaluation of production methods

To produce neutron-rich nuclei in the rare-earth region using projectile fragmentation, it is necessary to select a suitable primary-beam projectile. Here, ²⁰⁸Pb(Z=82, N=126), ¹⁹⁸Pt(Z=78, N=120), and ¹⁹⁷Au(Z=79, N=118) can be plausible candidates, and their fragmentation cross sections are critical for evaluating their production potential. The experimental results on the production cross sections via the fragmentation of ²⁰⁸Pb at 1 GeV/nucleon [25], ¹⁹⁸Pt at 85 MeV/nucleon [10] and ¹⁹⁷Au [26] at 950 MeV/nucleon on ⁹Be are compared in Fig. 3. In general, fragmentation of ¹⁹⁸Pt yields larger cross sections than fragmentation of both ¹⁹⁷Au and ²⁰⁸Pb. Cross sections for the fragmentation of ¹⁹⁷Au are higher than those for ²⁰⁸Pb at Z=76 and 77 as plotted in Fig. 3(g) and (h), but become smaller for as shown in Fig. 3(a)-(f). For ¹⁹⁸Pt and ²⁰⁸Pb, the cross-section difference becomes smaller towards N=120 along an isotopic chain, because of the neutron-number limit of ¹⁹⁸Pt. For instance, similar cross sections can be found around N=120 between these two projectiles for Lu, Hf and Ta, leading to a crossover of the two isotopic distributions, as shown in Fig. 3(b)-(d). Beyond N>120, fragmentation of ²⁰⁸Pb presents larger cross sections than those for ¹⁹⁸Pt and ¹⁹⁷Au, because ²⁰⁸Pb has more neutrons.

Fig. 2Full-screen View

(Color online) Layout of HIRIBL separator

Fig. 3Full-screen View

(Color online) Measured cross sections of isotopes from ₇₀Yb (a) to ₇₇Ir (h) produced by fragmentation of ²⁰⁸Pb at 1 GeV/nucleon [25], ¹⁹⁷Au at 950 MeV/nucleon [26] and ¹⁹⁸Pt at 85 MeV/nucleon [10] on beryllium. The COFRA (dotted lines), EPAX2 (dashed lines), and EPAX3 (solid lines) calculations were plotted for comparison

As shown in Fig. 3, cross-section data are available for isotopes with . To estimate production of more neutron-rich isotopes with smaller Z, the available data are first used to assess the predictive power of the calculations, and the validated models are then applied to nuclei with Z<70. So far, many models have been developed to predict cross sections of fragmentation reactions [40-44]. To examine their predictive power, three sets of calculations were chosen for comparison with the data, as shown in Fig. 3. They are EPAX2 [45], EPAX3 [46] and COFRA [26], which are widely used to estimate the production yields at in-flight separators.

EPAX is an empirical parameterization that provides a universal description of fragmentation cross sections in high-energy heavy-ion reactions [45, 46]. The cross sections are assumed to be energy independent. EPAX2 was developed by fitting data from stable projectiles, such as ⁴⁰Ar, ⁵⁸Ni, ⁸⁶Kr, ¹²⁹Xe, and ²⁰⁸Pb [47-51]. In EPAX2, the description of fragmentation yields close to the projectile and the memory effect of neutron-deficient projectiles were improved significantly. However, large discrepancies with measured data still remain for the production of extremely neutron-rich nuclei in EPAX2 predictions. EPAX3 was developed using a more comprehensive data set, such as ^40,48Ca,^58,64Ni,¹³²Sn [52, 53], among others. EPAX3 introduces several improvements: a revised treatment of the memory effect, an additional correction for the residue corridor, an empirical downscale factor to avoid the overestimation for the yields of extremely neutron-rich fragments, and improved parameterization of the charge dispersion width. As a result, EPAX3 provides a much better description of the most neutron-rich fragments while retaining the good performance of EPAX2 on the neutron-deficient side. EPAX3 has been applied successfully to predict cross sections in many cases, such as the fragmentation of ¹³⁶Xe, ⁸¹Ga, ⁸²Ge, ¹³²Sn [4, 5, 54] and related systems. The detailed formulas and parameters of EPAX2 and EPAX3 are described in Ref. [45] and Ref. [46], respectively.

The COFRA code is a simplified version of the abrasion-ablation reaction model. In the first stage, the abrasion process determines the size of the pre-fragment based on the geometric overlap between projectile and target nuclei. The excitation energy gained during this stage is estimated from the energy of particle-hole excitations produced by the abraded nucleons. In the second stage, the ablation process is described within the framework of the statistical model using the Weisskopf formalism. It is treated under the assumption that only neutron evaporation occurs, which is a good approximation for nuclei with a large neutron excess. This assumption allows the statistical evaporation stage to be simplified using an analytical formulation. The detailed formalism of COFRA is given in Ref. [26].

The comparison results are shown in Fig. 3. EPAX2 cannot reproduce all three data sets. For ¹⁹⁷Au, EPAX2 reproduces the measured cross sections reasonably well overall. For ¹⁹⁸Pt and ²⁰⁸Pb, however, it underestimates the cross sections for products close in Z to the projectile, such as Re(Z=75), Os(Z=76), and Ir(Z=77) isotopes produced by ¹⁹⁸Pt(Z=78) and ¹⁹⁷Au(Z=79), as presented in Fig. 3(f)-(h), whereas it overestimates the cross sections for products with Z far from the projectile, such as Lu(Z=71) and Hf(Z=72) isotopes produced by ¹⁹⁸Pt, and especially Yb(Z=70), Lu(Z=71), Hf(Z=72), and Ta(Z=73) isotopes produced by ²⁰⁸Pb, as displayed in Fig. 3(a)-(d). Such a large overestimation of the cross sections for low-Z neutron-rich nuclei was also reported in the ¹³⁶Xe+⁹Be at 1 GeV/nucleon [54]. The limitation of EPAX2 on the neutron-rich side could be attributed to the limited availability of experimental cross-section data in that region during its development [46]. Both EPAX3 and COFRA show good agreement with the measured cross sections for all three projectiles. For products close to the projectile, such as Os and Ir isotopes produced by ²⁰⁸Pb in Fig. 3(g)-(h), and W, Re, Os and Ir isotopes produced by ¹⁹⁸Pt and ¹⁹⁷Au in Fig. 3(e)-(h), COFRA gives a better description than EPAX3. For the products far from the projectile, COFRA gives larger cross sections than EPAX3 but predicts a rapid decrease toward the very neutron-rich side along the isotopic chains, leading to crossover points between the two calculations, as shown in Fig. 3(a)-(c). This behavior may be related to the treatment of the excitation energy and the neutron separation energy of the pre-fragments in the COFRA model [26, 53]. Because of their good agreement with data, both COFRA and EPAX3 were used to calculate production cross sections for neutron-rich nuclei with Z<70 in the rare-earth region and to identify a suitable projectile.

To evaluate the production potential for the neutron-rich isotopes in the rare-earth region, the production cross sections for the isotopes with Z=61-71, produced via fragmentation of ¹⁹⁸Pt, ¹⁹⁷Au, and ²⁰⁸Pb on a Be target, are calculated with COFRA and EPAX3, as plotted in Fig. 4. For fragmentation reactions, the cross sections from ¹⁹⁸Pt are much larger than those from ²⁰⁸Pb and ¹⁹⁷Au, especially for isotopes with as presented in Fig. 4(e)-(k), demonstrating the advantage of using ¹⁹⁸Pt to produce neutron-rich rare-earth isotopes.

Fig. 4Full-screen View

(Color online) Comparison of production cross sections for isotopes from ₆₁Pm(a) to ₇₁Lu(k) by fragmentation of ¹⁹⁸Pt (red solid lines), ¹⁹⁷Au (blue dotted lines), and ²⁰⁸Pb (green dashed lines), as well as by fission of ²³⁸U. The fragmentation cross sections were calculated by EPAX 3 and COFRA. It is noted that the cross sections for Pm isotopes produced by fragmentation of ²⁰⁸Pb can not be reached by COFRA. The fission cross sections were adopted from the measured data by RIBF [12, 24] (black circles) and GSI [23] (purple triangles), respectively

The measured cross sections for the ²³⁸U fission [9, 23, 24] are also shown in Fig. 4 for comparison. For isotopes with , the fragmentation cross sections of ¹⁹⁸Pt are smaller on the less neutron-rich side, as shown in Fig. 4(a)-(e). However, the cross-section difference between the two production mechanisms decreases with increasing neutron number, because the fission cross sections decrease rapidly [9]. This tendency leads to a crossover point, beyond which the ¹⁹⁸Pt fragmentation cross sections exceed those from the ²³⁸U fission. Indeed, such a crossover is observed for N>108 in the Dy, Ho and Er isotopic chains, as shown in Fig. 4(f)-(h). For the Tm, Yb and Lu isotopes, the fragmentation cross sections of ¹⁹⁸Pt are 1–2 orders higher than those from the ²³⁸U fission, as displayed in Fig. 4(i)-(k). These results suggest that ¹⁹⁸Pt fragmentation may offer a promising alternative to ²³⁸U fission in accessing very neutron-rich rare-earth isotopes, particularly for those with Z>65.

Optimization of target thickness

To maximize the yield of neutron-rich rare-earth isotopes via the fragmentation of ¹⁹⁸Pt, it is essential to optimize the target thickness. An energy of 800 MeV/nucleon was adopted for ¹⁹⁸Pt, as it is the typical energy suggested for heavy-ion beams delivered to the HIRIBL beamline at HIAF [27, 55]. While increasing the target thickness enhances the initial production yield, it also introduces two counteracting effects: the fraction of fully stripped ions decreases with increasing thickness, and reaction losses increase due to multi-step reactions within the thick target. Therefore, it is necessary to balance these competing factors to identify an optimal target thickness.

Taking the production yield of ¹⁸¹Er by an 800-MeV/nucleon ¹⁹⁸Pt beam on Be as an example, the dependence of the survival probability (P_surv) and the fully-stripped fraction (R_FS) of ¹⁸¹Er on the target thickness (T_tar) are shown in Fig. 5(a) and (b), respectively. The charge -state distributions were calculated using the GLOBAL [36] code. GLOBAL is a numerical program developed to simulate the charge-state evolution of relativistic heavy ions as they traverse matter. The model takes into account a broad distribution of charge states and equilibrium conditions for a wide range of projectiles and target materials at energies above 100 MeV/nucleon. A detailed description of the GLOBAL code can be found in Ref. [36]. Both the fully-stripped fraction and the survival probability decrease as the target thickness increases. By combining these effects, Fig. 5(c) shows the product of the target thickness, the fully stripped fraction, and the survival probability as a function of the target thickness, which provides an effective measure of yield estimation. As demonstrated in Fig. 5(c), the maximum value of the yield measure appears near 4 g/cm², corresponding to an energy of 580 MeV/nucleon for ¹⁸¹Er. In this case, the survival probability and fully-stripped fraction for ¹⁸¹Er are approximately 40% and 80%, respectively, as displayed in Fig. 5(a) and (b). It is noted that the thickness of 4 g/cm² is also applicable to the production of other nuclei in the rare-earth region. Thus, an optimal thickness of 4 g/cm² for Be was adopted to estimate the production of neutron-rich isotopes in the rare-earth region from fragmentation of ¹⁹⁸Pt.

Fig. 5Full-screen View

(Color online) (a) Survival probability of ¹⁸¹Er as a function of Be target thickness, calculated by LISE++ [35]. (b) Charge state fractions of fully-stripped (blue solid line) and H-like (magenta dashed line) ions of ¹⁸¹Er with the thickness of Be target, calculated by GLOBAL [36] code. (c) The product of the fully-stripped fraction, the survival probability and the target thickness as a function of target thickness

Production yield

Based on the discussion above, the calculated production yields of neutron-rich nuclei in the rare-earth region from fragmentation of ¹⁹⁸Pt at HIRIBL are plotted in Fig. 1, using a beam intensity of ppp, corresponding to pps in the slow extraction mode. The production cross sections were calculated using EPAX3 and COFRA. A 4 g/cm² Be target was selected as the reaction target. The transmission efficiencies and production yields were simulated with LISE++ [35]. It predicts beam intensity, purity, and transmission for experiment planning by incorporating physical processes such as projectile fragmentation, fission, and related reactions. LISE++ provides flexibility to design and optimize separator configurations for experiment planning and beamline tuning at rare-isotope facilities. In the present work, the configuration of the HIRIBL separator at HIAF, which is presented in Fig. 2, was implemented to calculate the transmission efficiency. According to the LISE++ simulation, the transmission efficiency is approximately 90%. As shown in Fig. 1, about 36 and 27 new isotopes with can be detected with a production rate higher than 1.5×10^-5 pps, corresponding to 1 event per day, according to the calculations by EPAX3 and COFRA, respectively. The yields for these new isotopes expected to be discovered at HIAF are listed in Table 1. This result demonstrates strong potential for the production of new neutron-rich rare-earth isotopes at the HIRIBL in-flight separator at HIAF.

Summary

In summary, the production of new neutron-rich rare-earth isotopes through fragmentation reactions of primary beams at 800 MeV/nucleon at HIRIBL has been evaluated. By comparing the fragmentation of ¹⁹⁸Pt with that of ²⁰⁸Pb and ¹⁹⁷Au as well as with the fission of ²³⁸U, it was found that ¹⁹⁸Pt may offer advantages for producing new isotopes in this region. The measured production cross sections for the heavy projectiles ²⁰⁸Pb, ¹⁹⁷Au, and ¹⁹⁸Pt are well reproduced by EPAX3 and COFRA. Using EPAX3 and COFRA, the production yields of new isotopes from the fragmentation of ¹⁹⁸Pt were estimated. About 30 new isotopes could be discovered through the fragmentation of ¹⁹⁸Pt at HIRIBL at HIAF, providing a valuable opportunity to extend experimental access to the rare-earth region for studies in both nuclear physics and nuclear astrophysics.