npsm 새물리 New Physics : Sae Mulli

pISSN 0374-4914 eISSN 2289-0041


Research Paper

New Phys.: Sae Mulli 2024; 74: 35-39

Published online January 31, 2024

Copyright © New Physics: Sae Mulli.

The p+238U Reaction for the RI Beam based on the Langevin Method

Chang-hoon Song1*, Chang-Hwan Lee1†, Ik Jae Shin2, Youngman Kim3, Yoshihiro Aritomo4

1Department of Physics, Pusan National University, Busan 46241, Korea
2Institute for Rare Isotope Science, Institute for Basic Science, Daejeon 34000, Korea
3Center for Exotic Nuclear Studies, Institute for Basic Science, Daejeon 34126, Korea
4Graduate School of Science and Engineering, Kindai University, Higashi-Osaka 577-8502, Japan

Correspondence to:*

Received: October 19, 2023; Revised: October 31, 2023; Accepted: October 31, 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License( which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

The Rare isotope Accelerator complex for ON-line experiments (RAON) utilizes the ISOL method to produce rare isotopes. In this process, protons can be accelerated and directed to collide with a uranium-238 target, leading to the extraction of rare isotopes from the resulting fragments. This research involves the prediction of proton-induced uranium fission nuclear reactions through calculations based on the Langevin method. To evaluate the predictive accuracy, we employ a well-established multimodal empirical formula for comparison. A discrepancy in the MDFF become evident in high-proton beam fission, emphasizing the need for model improvements. By visualizing the potential energy surface of the compound nucleus, the significance of shell effects is recognized. This study serves as motivation to identify and implement improvements in our approach, with the goal of enhancing the accuracy of fission predictions in the future.

Keywords: RAON, Proton-induced fission, Langevin method, Multimodal formula, MDFF

Rare isotopes, with lifetimes lasting only a few seconds or less, briefly appear during the evolution of the universe and then vanish, leaving minimal traces on Earth. Nevertheless, nature reveals its characteristics through infrequent and small-scale events, such as supernova explosions. The field of rare isotope research has substantial suggestion, not only in astrophysics but also in nuclear physics. This is because it has the potential to account for the processes involved in the formation of heavy elements. Unlike in the past, recent scientific advancements have equipped us with the means to artificially create rare isotopes. This enables us to explore their essential properties and investigate the natural world in greater detail[1-3].

While many accelerators worldwide operate either the Isotope Separator On-Line (ISOL) or In-flight fragmentation methods for rare isotope production, RAON, a Korean heavy ion accelerator, stands out as the first to integrate both approaches. Specifically, the ISOL line at RAON employs a method that accelerates protons to collide with uranium targets, a well-recognized and promising approach for producing neutron-rich rare isotopes. However, as it is essentially challenging to directly observe the complex processes involved in nuclear reactions, this type of experiment requires a theoretical model for explanation. Consequently, dynamic calculations have been used with the primary goal of demonstrating the evolutionary mechanisms that nuclear materials undergo during deformation[4,5].

In this research, we adopted a fluctuation-dissipation model based on the Langevin equation. This model has successfully elucidated the comprehensive process in the reaction of two nuclei, such as fusion or fission, as demonstrated in previous studies[6,7]. To assess the suitability of the model for application to proton-induced fission, we compared it with a well-established empirical formula commonly used to estimate fission reactions within the actinides.

1. Langevin method

The Langevin equation is a probabilistic differential equation that characterizes the evolution of a system in classical mechanics, influenced by both deterministic and fluctuating forces. In the context of nuclear systems, it explains the temporal changes in the shape of the compound nucleus (CN) following the collision of two nuclei, offering a macroscopic depiction of this process as follows[8]:

dpidt=V(q)qi12 jkm jk1qipipj jkηij(q)m jk1(q)pk+ jgijRj(t),

where qi={z,δ,α} incorporates three strain parameters, and pi is the conjugate momentum to qi. Here, z represents the separation distance between the two potential centers, δ quantifies the deformation of the nuclear component, and α is defined as the mass asymmetry between the two constituent pieces. In this work, the nuclear shape is parameterized by the Two-Center Shell Model (TCSM)[9,10]. Additionally, mij represents the collective inertia parameters, which depend on the nuclear shape adopted following the Werner–Wheeler approximation. V denotes the system's potential energy, defined as the sum of the liquid-drop model (LDM) and microscopic shell correction (SH) for a given system temperature, T. The dissipation tensors denoted as ηij, derived using the wall-and-window one-body formula. In last term of Eq. (2), Rj(t) is the normalized random noise, and its strength gij is given by ηijT=kgikgjk, where T is the temperature of the CN calculated from the intrinsic energy of the composite system expressed as Eint=aT2 with a denoting the level density parameter[11-14].

In the case of low-energy nuclear reactions, especially those occurring near the Coulomb barrier, the emphasis is on interactions between nuclei rather than search for the complex details of the internal structure of the nucleus. This is why we can use the Langevin equation to describe the nuclear shape during the reaction.

2. Multimodal method

In many cases, low-energy actinide fission experiments show that mass distribution of fission fragments (MDFF) is well-described as a multimodal fission. This can be represented by a linear combination of symmetric and asymmetric modes. Each mode of fission is influenced by the fission barrier which is determined by the energy of the induced proton. Each mode also may be expressed in Gaussian shape[15,16]:


where Gaussian parameters CSM, CASYM, σSM, and σASYM are employed to describe the amplitudes and widths of the symmetric (SM) and asymmetric (ASYM) fission modes. ASM represents the most probable mass value for the symmetric fission mode, while ASM±DASYM correspond to the most probable masses of the lighter and complementary heavier fragments, respectively, in the asymmetric fission mode[17,18]. The Gaussian parameters were fitted to experimental data available in the experimental nuclear reaction data[19,20]. The center of the symmetric peak was set to ASM=119, while the centers of both asymmetric peaks, ASM±DASYM, were set to 100 and 138, respectively.

In this study, we conduct a thorough comparison of two methodologies: the Langevin approach and an empirical formula. Our primary goal is to enhance our understanding of nuclear fission processes and the distribution of mass among fission fragments, a critical factor known as MDFF.

In the Langevin approach, we focus on the CN as a representative nucleus for calculations, specifically 239Np in an excited state. In this calculation, we have excluded the rotational potential due to the significantly smaller proton mass compared to uranium. Figure 1(a) presents the MDFF results obtained from our Langevin calculations, with a specific focus on a proton beam energy range of 20–70 MeV. This range corresponds with RAON's initial proton beam energy extraction. Simultaneously, we employ an empirical formula influenced by data from uranium nucleus fission to compute another set of MDFF values. This empirical approach involves interpolation and extrapolation, allowing us to make predictions and analyze mass distribution based on available experimental data. Figure 1(b) provides a visual representation of the MDFF derived from our multimodal calculations, also within a proton beam energy range of 20–70 MeV.

Figure 1. (Color online) MDFF variation depending on proton beam energy. (a) The Langevin model, (b) The multimodal fission model.

When comparing the outcomes of these two distinct approaches, it appears that they exhibit similar distributions at low proton beam energies. However differences become noticeable, particularly at high proton beam energies. Given that the multimodal results serve as the standard for comparison based on experimental data, these disparities suggest that the Langevin approach may not fully account for specific variables or inherent effects within the nuclear fission process at high beam energy.

To further investigate these findings, we have plotted the potential energy surfaces of the CN using the LDM+SH method which is integral in Langevin calculation. Figure 2 presents these potential energy surfaces, with each coordinate axis representing the nucleus deformation along the TCSM. They exhibit variations depending on the excitation energy of the CN. This analysis shows the significant impact of shell effects within the Langevin approach. Consequently, it becomes evident that we must consider shell effects during the de-excitation process and also consider the shell damping energy. These enhancements are crucial for improving the precision of our model and bridging the gap between theoretical predictions and experimental data.

In conclusion, we have employed the Langevin method to predict proton-induced nuclear fission reactions. For our calculations, we consider the compound nucleus as 239 Np, without incorporating a rotational potential. Using the LDM and SH methods, we construct the potential energy surface of the compound nucleus and perform dynamic calculations, resulting in the MDFF. To assess the accuracy of our calculations, we utilize a well-established multimodal empirical formula as a comparative benchmark for describing the nuclear fission distribution of actinide. The disparities observed between the Langevin approach and the empirical formula implies the ongoing need for refinements and enhancements in fission modeling, particularly at high excitation energies. To address this discrepancy and enhance the accuracy of the MDFF in the Langevin method, we have confirmed that the consideration of shell effects is essential by plotting the potential energy surface of the CN. In the future, these improvements will bridge the gap between theoretical predictions and experimental observations, and will further enable more accurate prediction of the fission process.

Figure 2. (Color online) The variation in shell correction effects of the CN, 239Np, with fixed shell damping energy at 20 MeV (a) at an excitation energy of 0 MeV, (b) at an excitation energy of 50 MeV.

This work was supported by a 2-Year Research Grant of Pusan National University.

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