KANTBP 3.1: A program for computing energy levels, reflection and transmission matrices, and corresponding wave functions in the coupled-channel and adiabatic approaches,☆☆,☆☆☆

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Abstract

A FORTRAN program for calculating energy values, reflection and transmission matrices, and corresponding wave functions in a coupled-channel approximation of the adiabatic approach is presented. In this approach, a multidimensional Schrödinger equation is reduced to a system of the coupled second-order ordinary differential equations on a finite interval with the homogeneous boundary conditions of the third type at left- and right-boundary points for the discrete spectrum and scattering problems. The resulting system of such equations, containing potential matrix elements and first-derivative coupling terms is solved using high-order accuracy approximations of the finite element method. The scattering problem is solved with non-diagonal potential matrix elements in the left and/or right asymptotic regions and different left and right threshold values. Benchmark calculations for the fusion cross sections of 36S+48Ca, 64Ni+100Mo reactions are presented. 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, and also the fusion cross sections and mean angular momenta of the 16O+144Sm reaction.

Program summary

Program Title: KANTBP

CPC Library link to program files: https://doi.org/10.17632/4vm9fhyvh3.1

Licensing provisions: CC BY NC 3.0

Programming language: FORTRAN

Nature of problem: In the adiabatic approach [1], a multidimensional Schrödinger equation for quantum reflection [2], the photoionization and recombination of a hydrogen atom in a homogeneous magnetic field [3–6], the three-dimensional tunneling of a diatomic molecule incident upon a potential barrier [7], wave-guide models [8], the fusion model of the collision of heavy ions [9–11], and low-energy fusion reactions of light- and medium mass nuclei [12] is reduced by separating the longitudinal coordinate, labeled as z, from transversal variables to a system of second-order ordinary differential equations containing the potential matrix elements and first-derivative coupling terms. The purpose of this paper is to present a program based on the use of high-order accuracy approximations of the finite element method (FEM) for calculating energy levels, reflection and transmission matrices and wave functions for such systems of coupled-channel second order differential equations (CCSODEs) on finite intervals of the variable z[zmin,zmax] with homogeneous boundary conditions of the third-type at the left- and right-boundary points, which follow from the discrete spectrum and scattering problems.

Solution method: The boundary-value problems for the system of CCSODEs are solved by the FEM using high-order accuracy approximations [13,14]. The generalized algebraic eigenvalue problem AF=EBF with respect to pair unknowns (E,F), arising after the replacement of the differential eigenvalue problem by the finite-element approximation, is solved by the subspace iteration method [14]. The generalized algebraic eigenvalue problem of a special form (AEB)F=DF with respect to pair unknowns (D,F) arising after the corresponding replacement of the scattering boundary problem in open channels at fixed energy value, E, is solved by the L DLT factorization of the symmetric matrix and back-substitution methods [14].

Additional comments including restrictions and unusual features: The user must supply subroutine POTCAL for evaluating potential matrix elements. The user should also supply subroutines ASYMEV (when solving the eigenvalue problem) or ASYMSL and ASYMSR (when solving the scattering problem) which evaluate asymptotics of the wave functions at boundary points in the case of a boundary conditions of the third-type for the above problems.

References

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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 z[zmin,zmax] using a general homogeneous boundary condition of the third-type at z=zmin and z=zmax. 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 zzmin and zzmax 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 ΔE=0.1 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 z(zmin,zmax) for the partial solution χ(j)(z)=(χ1(j)(z),,χN(j)(z))T(I1zd1ddzzd1ddz+V(z)+Q(z)ddz+1zd1dzd1Q(z)dz2EI)χ(j)(z)=0. Here I, V(z) and Q(z) are the unit, real valued symmetric and antisymmetric N×N matrices, respectively. Below we consider only the scattering problem with d=1.

The matrix-solution Φv(z)=Φ(z)

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](2y22z2+U(y,z)2E)Ψ(y,z)=0 in the 2D domain Ωyz={y(0,π),z(,+)}, with the potentialU(y,z)={0,z<2;2y,|z|2;2y,z>2}. We seek for the solution in the form of the expansionΨ(y,z)=i=1NBi(y)Φi(z) in the set of basis functions Bi(y)=2/πsin(iy), leading to the system of Eq. (1) with Qij(z)=0 and the effective potentials Vij(z):Vij(z)=0πdyBi(y)U(y,z)Bj(y)=i2

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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Cited by (2)

The review of this paper was arranged by Prof. N.S. Scott.

☆☆

This paper and its associated computer program are available via the Computer Physics Communications homepage on ScienceDirect (http://www.sciencedirect.com/science/journal/00104655).

☆☆☆

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.

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