Abstract
We present an original algorithm in the MAPLE system for solving the scattering problem in single-channel approximation of the coupled-channel method of the optical model (OM) described by a second-order ordinary differential equation (ODE) with a complex-valued potential and regular boundary conditions. The complex-valued potential consists of the known real part, which is a sum of the nuclear potential, the Coulomb potential, and the centrifugal potential, and the imaginary part, which is a product of the unknown coupling constant g(E), depending on the collision energy E of a pair of ions, and the derivative of the real part of the known nuclear potential with respect to the ODE independent variable.
The presented algorithm implements the solution of the inverse problem, i.e., calculates the unknown coupling constant g(E) and scattering matrix S(g(E), E) from condition \(|S(g(E),E)|^{2}=1-|T(E)|^{2}\) by means of the secant method. The required amplitudes of transmission T(E) and reflection R(E) subject also to the condition \(|R(E)|^{2}=1 -|T(E)|^{2}\) of the model with incoming wave boundary conditions (IWBCs) are previously calculated by the standard MAPLE implemented KANTBP 4M program.
The algorithm provides a one-to-one correspondence between the OM with a complex-valued potential and the model of IWBCs with a real-valued potential.
The efficiency of the proposed approach is shown by solving numerically the scattering problem and calculating the reference fusion cross section for a pair of heavy ions \(^{16}\)O+\(^{144}\)Sm in the single-channel approximation of the close-coupling method.
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Acknowledgments
The present research benefited from computational resources of the HybriLIT heterogeneous platform of the JINR. This publication has been supported by the Russian Foundation for Basic Research and Ministry of Education, Culture, Science and Sports of Mongolia (the grant 20-51-44001) and the Peoples’ Friendship University of Russia (RUDN) Strategic Academic Leadership Program, project No.021934-0-000. This research is funded by Ho Chi Minh City University of Education Foundation for Science and Technology (grant No. CS.2021.19.47).
OCH acknowledges financial support from the Ministry of Education and Science of Mongolia (grant No. ShuG 2021/137). The work of PWW, CJL, and HMJ is supported by the National Key R &D Program of China (Contract No. 2022YFA1602302), the National Natural Science Foundation of China (Grants Nos. 12235020, 12275360, 12175314, 12175313, and U2167204), the Leading Innovation Project (Grant No. LC192209000701), and the project supported by the Directors Foundation of Department of Nuclear Physics, China Institute of Atomic Energy (12SZJJ-202305).
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Gusev, A.A. et al. (2023). Symbolic-Numerical Algorithm for Solving the Problem of Heavy Ion Collisions in an Optical Model with a Complex Potential. In: Boulier, F., England, M., Kotsireas, I., Sadykov, T.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2023. Lecture Notes in Computer Science, vol 14139. Springer, Cham. https://doi.org/10.1007/978-3-031-41724-5_7
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