KANTBP 3.1: A program for computing energy levels, reflection and transmission matrices, and corresponding wave functions in the coupled-channel and adiabatic approaches☆,☆☆,☆☆☆
Introduction
The KANTBP (KANTorovich Boundary Problem) versions 1.0 [1] and 2.0 [2] were intended only to calculate the energy levels, reaction matrix and radial wave functions of the bound state problem and the elastic scattering problem in the coupled-channel hyperspherical adiabatic approach, in which the original problems were reduced to a system of coupled-channel second order differential equations (CCSODEs) with respect to a radial variable on a semi-axis. The KANTBP version 3.0 [3] was intended for calculate of the energy levels, reflection and transmission amplitude matrices and corresponding wave functions of the bound state problem and scattering problem for the system of CCSODEs on a whole axis. Moreover, the scattering problem is solved under the condition that potential matrix elements in left and right asymptotic regions have only a “almost” diagonal form, and the left and right thresholds are the same. However a wider range of physical scattering problems are reduced to a system of CCSODEs with non-diagonal potential matrix elements in the left and/or right asymptotic regions and different left and right thresholds. The purpose of this version is to provide a computational program for calculating the reflection and transmission amplitude matrices and corresponding wave functions of the scattering spectrum problem thus covering a wider range of physical scattering problems.
The KANTBP 3.1 extends the framework of the previous versions of KANTBP for the case of the scattering problem. As in [3], it calculates the reflection and transmission amplitude matrices and corresponding wave functions of the continuous spectrum of the boundary-value problem for the system of CCSODEs on finite intervals of the variable using a general homogeneous boundary condition of the third-type at and . The third-type boundary conditions are formulated for the continuous problems under consideration by using known asymptotes for a set of linearly independent asymptotic regular and irregular solutions in the open channels and a set of linearly independent regular asymptotic solutions in the closed channels, respectively. We have considered more general cases, namely, in left and right asymptotic regions and the potential matrix elements are non-diagonal and constant or weakly dependent on the variable z; the left and right thresholds are different, and the left and right threshold values may not be known in advance.
We have applied the new approach to the computation of sub-barrier and above-barrier fusion cross sections as well as the astrophysical S factor of some reactions, to study the deep sub-barrier fusion hindrance phenomenon in [5], [6], and study of fast fission and quasifission in the 40Ca+208Pb reaction leading to the formation of the transfermium nucleus 248No [7]. The results obtained using KANTBP 3.1 and the modified Numerov method in the CCFULL program [8], the Gauss reduction method in the NRV project [9], [10], [11] are compared.
Benchmark calculations for the fusion cross sections of 36S+48Ca, 64Ni+100Mo reactions, which are studied in [5], are presented. In our previous study, the fusion cross sections are calculated at the experimental incident energy in order to compare with the data for 36S+48Ca, 64Ni+100Mo reactions, while in this work they are calculated at MeV for a strict test. The calculated fusion cross sections are also compared with those obtained by the CCFULL program [8], and the advantage of KANTBP 3.1 is more prominent. As a test desk, the program is applied to the calculation of the reflection and transmission matrices and corresponding wave functions of the exact solvable wave-guide model considered in [12], and the fusion cross sections and mean angular momenta of the 16O+144Sm reaction.
The paper is organized as follows. In Section 2 we give a brief overview of the problem. A description of the KANTBP 3.1 program is given in Section 3. Benchmark calculations and test desk are given in Section 4.
Section snippets
Physical scattering asymptotic forms of solutions in longitudinal coordinates and the scattering matrix
In the Kantorovich approach [1], [4], a multidimensional Schrödinger equation is reduced to a finite set of N ordinary second-order differential equations on the finite interval for the partial solution Here I, and are the unit, real valued symmetric and antisymmetric matrices, respectively. Below we consider only the scattering problem with .
The matrix-solution
Description of the KANTBP 3.1 program
The KANTBP program is called from the main routine (supplied by the user) which sets the dimensions of the arrays and is responsible for the input data. The KANTBP program does not require installation. A description of all subroutines can be found in comments in the program source code. Users can also find instructions on how to compile KANTBP in the README file.
The calling sequence for the KANTBP subroutine is:
CALL KANTBP(TITLE,IPTYPE,ISC,ISCAT,NROOT,MDIM,IDIM,NPOL,
1
Exact solvable wave-guide model
For example we consider the multichannel scattering problem for the Schrödinger equation of the exact solvable wave-guide problem [12] in the 2D domain , with the potential We seek for the solution in the form of the expansion in the set of basis functions , leading to the system of Eq. (1) with and the effective potentials :
CRediT authorship contribution statement
O. Chuluunbaatar: Conceptualization, Software, Supervision, Visualization, Writing – original draft, Writing – review & editing. A.A. Gusev: Conceptualization, Software, Visualization. S.I. Vinitsky: Conceptualization, Software, Writing – original draft. A.G. Abrashkevich: Conceptualization, Software, Visualization, Writing – original draft, Writing – review & editing. P.W. Wen: Conceptualization, Software, Visualization, Writing – original draft. C.J. Lin: Conceptualization, Software,
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The work was partially supported by the grant of the Ministry of Science and Higher Education of the Russian Federation 075-10-2020-117, the grant of the Foundation of Science and Technology of Mongolia SST_18/2018, the National Key R&D Program of China (Contract No. 2018YFA0404404), the National Natural Science Foundation of China (Grants Nos. 11635015, 11805280), the Continuous Basic Scientific Research Project (No. WDJC-2019-13), the Young Talent Development Foundation (Grant No.
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The review of this paper was arranged by Prof. N.S. Scott.
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This paper and its associated computer program are available via the Computer Physics Communications homepage on ScienceDirect (http://www.sciencedirect.com/science/journal/00104655).
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We dedicate this article to the 75th anniversary of the birth of professor M.S. Kashchiev. He was one of the creators of the first versions of the KANTBP program.