Proceedings of the 15th International Symposium on Origin of Matter and Evolution of Galaxies (OMEG15)
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Proc. 15th Int. Symp. Origin of Matter and Evolution of Galaxies (OMEG15)JPS Conf. Proc. 31, 011018 (2020)
https://doi.org/10.7566/JPSCP.31.011018

Application of absorbing boundary condition to few-bodycluster dynamics
M. Ito, M. Iwasaki, R. Otani, M. Nakao, M. Tomita
Deparment of Pure and Applied Physics, Kansai University, 3-3-35 Yamatecho, Suita, Osaka
564-8680, Japan
E-mail: itomk@kansai-u.ac.jp(Received October 31, 2019)
We show the formulation of the absorbing boundary condition (ABC) and demonstrate its applicability to resonant phenomena induced by reactions of few-body cluster systems. The resonanceparameter and the strength function, which are important quantities in astrophysics, are calculatedby the ABC method, and the results are compared with the complex scaling method, which is another
tool to handle the resonant phenomena. We will show the realistic applications of ABC to the 3/2+
resonance in 19 Ne and discuss the prescription of ABC in the three-body problem.
KEYWORDS: non-Hermite systems, few-body cluster systems, resonance parameters

1.

Introduction

Dynamics induced by reactions of few-body nuclear systems in the unbound energy region gives
an important impact on the research field about nuclear astrophysics. For example, a resonance formation induced by the binary collision and dissociation of a nucleus into two- or three-body fragments in continua are typical examples of the microscopic dynamical process, which are essential inmacroscopic astrophysical phenomena. From the viewpoint of the nuclear physics, it is important toevaluate the resonance parameters, such as the resonant energy and the decay width, and the so-calledstrength function in the few-body dynamical reaction.
Since the resonant phenomena occur at the particle unbound and continuum energy region, in anaive treatment, a scattering boundary condition should be imposed among the interacting particles.
However, imposing the scattering boundary condition is complicate if three particles participate in theconsidered reaction process. In order to avoid the complexity of the three-body scattering boundary
condition, the scattering problem is often reduced to the bound-state-like problem by transforming
the Hamiltonian of a total system into a non-Hermite form. There are two representative methodsof the non-Hermitian transformations: the complex scaling method (CSM) [1] and the method of theabsorbing boundary condition (ABC), or the method of the complex absorbing potential (CAP) [2–6].
In CSM, the radial coordinates and the respective momenta in the Hamiltonian of the total system
are simply rotated with the angle of θ in the complex plane, such as r → reiθ and k → keiθ , while, in
ABC, the negative imaginary potential is placed at the outer region of the system. These two procedures commonly give rise to the damping of the wave function in the asymptotic region. Therefore,
the scattering problem can be transformed into the bound-state-like problem, in which the variational
calculation using the basis expansion technique is available.
There are many sophisticated applications of CSM to the unbound continuum states [1]. However,
the application of CSM is diﬃcult in some special problems, for example, the microscopic clustermodel called the generator coordinate method (GCM) [4]. Therefore, it is instructive to develop the
ABC method as another tool to handle the unbound resonant phenomena. We have advanced theapplication of the ABC method to various few-body cluster systems. In the present report, we will1■■■
011018-1©2020 The Author(s)This article is published by the Physical Society of Japan under the terms of the Creative Commons Attribution 4.0 License. Any further distribution of this work
must maintain attribution to the author(s) and the title of the article, journal citation, and DOI.

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explain the basic aspects of the ABC method and show the several applications to the dynamical
reaction in the two- or three-body cluster systems: a binary cluster system and a three-body problem
composed of identical three-bosons. All of the results in this article are compressed from Refs. [4–8].

2.

Framework of absorbing boundary condition

2.1

Calculation of energy eigenvalues
In the ABC calculation, we solve the eigenvalue problem of the total Hamiltonian H with thenegative imaginary potential −iηW, such as( H − iηW) Ψη = E(η) · Ψη .

(1)Here η means the strength of the imaginary potential and hence, the wave function Ψη and the energy
eigenvalues E(η) depend on the strength of η. Eq. (1) is possible to be solved by the basis expansion
method because the solution with the absorber, Ψη , has a damping behavior in the asymptotic region.
In Eq. (1), no transformations are imposed on H itself and hence, the computational technique ofthe Hamiltonian matrix is unchanged in introducing absorber. This simple implementation of ABC
especially matches the microscopic cluster model, such as the generator coordinate method (GCM)[4, 12]. This is a great advantage in ABC and marked contrast to the complex scaling method (CSM),
where H must be transformed by the complex rotation [4].
The functional form of W is not arbitrary and various conditions are required as discussed inRef. [2]. In a simple two-body problem, for instance, we introduce the shifted polynomial function in
the radial coordinate of r such as
W → W(r − ra ) = θ(r − ra ) · (r − ra ) p .

(2)

The polynomial absorber is standard and has been usually employed in previous studies [2, 3]. A
starting point of the absorber ra should be taken to the outside of the physical interaction region.
The power of the polynomial is set to p = 4 in the present calculation although the power should be
carefully chosen by considering the computational situation [4].
Since the energy eigenvalues and the wave function depend on the absorber strength, η, we should
optimize η to determine the resonance parameters such as resonance energy and decay width. If thebasis set is complete, the correct resonance parameters can be obtained as a limit of η → 0. In thefinite basis set calculations, however, a finite η gives approximate resonance parameters.
In the optimization of η, we impose the following condition on the energy eigenvalue of E(η):ηdE(η)= min .
dη η=ηo(3)This condition means that the energy eigenvalue becomes stationary at the optimal strength of η =ηo . In the pragmatic calculation, the derivatives η · dE(η)/dη are obtained simply by employing thegeneralized Hellmann-Feynman theorem [9] like
η

dE(η)
η
η
= η < χ̃L |W|χL > ≡ < W(η) > ,dη(4)

An argument to justify the condition of Eq. (3) was presented in Refs. [2, 4, 5].
2.2

Strength function
We also calculate the strength function of the breakup into the continuum. A general definitionof the strength function is given by
∑⟨
⟩2
(5)
Ψν |Oˆλ |Ψi δ(E − Eν ),S(E) =
ν2■■■

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where Ψi and Ψν are the initial bound state and the ν-th final state having the energy eigenvalue Eν inthe continuum, respectively. In Eq. (5), Ôλ denotes the operator of an external field with the rank λ.
The calculation of the strength function in Eq. (5) can be achieved by assuming the extended
completeness relation (ECR) for the discrete solutions under the absorbing boundary,
1̂ =

N
∑

η

η

|Ψn >< Ψ̃n | =

n

∑

η

η

|Ψn >< Ψ̃n | +

n∈B

∑

η

η

|Ψn >< Ψ̃n | +

n∈R

∑

η

η

|Ψn >< Ψ̃n | .

(6)

n∈C

η

Here Ψn denotes the n-th solution of Eq. (1), which is obtained by the basis expansion method. In
the last equation, the summation is decomposed into three parts: the bound states (B), the resonant
states (R), and the non-resonant continuum states (C). The final expression of the strength function,
on which the ECR is implemented, is given byηSλ (E)ηRλ (E)]
1 [− ℑ Rη λ (E)
π

=

(7)

η
η
N
∑
< Ψi |Ô† |Ψn >< Ψ̃n |Ôλ |Ψi >λ=n

E − En (η)

η

η

η

= RλB (E) + RλR (E) + RλC (E) .

(8)

η

In Eqs. (7) and (8), Rλ (E) is called the response function, which is decomposed into three parts inaccordance with Eq. (6). This decomposition is very useful in the analysis of the reaction mechanism[1,6]. For example, we can understand the dominant breakup process, such as the resonance decay or
the direct breakup, in which the ground state directly goes into the continuum states. In this response
function, the denominator contains a complex number of En (η), which is the n-th eigenvalue in the
ABC solution for Eq. (1). Thus, this function is converged to a finite value even if the excitationenergy E is close to the real part of En (η).

3.

Results

3.1

Simple two-body system
We demonstrate the ABC calculation by showing the simple application to the two-body problem.
A model Hamiltonian with a schematic potential [1] of

H

=

V(r)

=

ℏ2 2
∇ + V(r),2m2
2−V1 e−µ1 r + V2 e−µ2 r .
−

(9)
(10)

is used in the present demonstration. Here the parameters of the potential V(r) is chosen to V1,2 =
2
8.0 (MeV), 4.0 (MeV) and µ1,2 = 0.16 (fm−2 ), 0.04 (fm−2 ). In the kinetic energy part, we set ℏm(MeV·fm2 ) =1 for simplicity. Thus, the dimension of the energy equals the (length)−2 . The complex scaling method (CSM) was applied to the resonance problem in this potential, and the resonant
energies are deeply analyzed in Ref. [1].
The energy distribution of these states plotted in the complex energy plane is shown in the left
panel of Fig. 1 [6]. In this energy calculation, the strength of the absorber is set to be η = 10−5 . In theenergy distribution of the left panel, the bound state (open square), first resonance (solid circle) andsecond resonance (solid diamond) are clearly separated from the non-resonant continuum (asterisks).
The negative energy state around ℜ[E] = ER ∼ −0.7 fm−2 is a bound state, while the positive energy
state around ER ∼ 1.17 fm−2 is the first resonant state. The width of this resonance is quite small (ΓR /ER ∼ 10−2 ) and hence, it appears quite close to the real axis. The second resonance appears at ER ∼3■■■

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2 fm−2 and ΓR /2 ∼ 0.5 fm−2 with a broad width (ΓR /ER ∼ 0.5). These two resonances are clearly
separated from the continuum states (solid circles) because the imaginary part of the continuum states
is systematically larger than that of the separated resonant states. Here all the resonance parameters
are calculated at the optimal strength of η = ηo , and resultant parameters are consistent to the results
obtained by CSM [1].
0

−4[X10 ]++Im E [MeV]2

strength function [fm ]

2.5

−1

2
1.5
1
0.5

−2
0

−1

0

1

2

3
4
5
Re E [MeV]

6

7

8

−0.5
0

1

2

3
4
5
Energy [MeV]

6

7

8

Fig. 1. (Left panel) Energy distribution calculated by ABC. The parameters of the absorber are p = 4 andη = 10−5 . The open square shows the bound state, while the solid circle and the solid diamond represent thefirst and second resonances, respectively. All the non-resonant continuum states are plotted by the asterisks.
(Right panel) Decomposed strength function into the individual contributions of the resonance and continuum.
The symbols shown in the right panel (the circle, the diamond, and the asterisk) are the same as the symbols
plotted in the left panel. These figures are taken from Ref. [6].

In the right panel of Fig. 1, the strength function is shown by the solid curve. According to theenergy distribution in the left panel, the strength function is decomposed, and the results are also
shown in the same figure. The individual contributions in the decomposed strength function (rightpanel) are plotted by the respective symbols shown in the energy eigenvalues (left panel). In the rightpanel, the contributions from the first resonance (dashed curve with the solid circle) and the second
resonance (dotted curve with the diamond) are predominant, but the continuum states (two-dotteddashed curve with the asterisk) provide a considerable contribution in the total strength function (thesolid curve).
J π = 3/2+ resonances in 19 Ne
We have applied the ABC method to the coupled-channel problem for the binary system of He+ O in 19 Ne [8]. There is an important reason why we focus on the 19 Ne nucleus. The 15 O(α,γ)19 Ne
reaction is known to play a crucial role in the advanced stages of astrophysical hydrogen burning,
and the most crucial resonance is known to arise through the J π = 3/2+ resonant level at 504 keVwith respect to the α + 15 O threshold (E x = 4.03 MeV). Unfortunately, no theoretical calculation was
successful in reproducing this resonant state. The experimental analyses in Ref. [13] have pointed outthat the intrinsic structure of the resonance at 504 keV in 19 Ne is not the α + 15 O cluster structure,
but the five particle–two hole (5p–2h) shell-model configuration with the 14 Og.s. core. The 5p–2hconfiguration has a large overlap with the shell model limit of the 5 He + 14 O configuration. Therefore,
the coupling of 5 He + 14 O on the J π = 3/2+ state is important in the cluster model approach.
3.2

4
■■■

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In fact, it is well known that 3 He + 16 O and α +15 O cluster configurations are insuﬃcient toreproduce the 3/2+ resonance [14]. Therefore, in the present analysis, we have performed the microscopic calculation of (3 He+16 O) + (α+15 O) + (5 He+16 O) in J π = 3/2+ . In this calculation, theinternal wave function of the individual clusters are described by the microscopic shell model configuration, and the anti-symmetrization eﬀect among the nucleons is completely taken into account. The
total Hamiltonian is composed of the kinetic energy and the two-body nucleon-nucleon interaction.
The calculation is performed by applying the generator coordinate method (GCM) [11] plus ABC[8]. The details of the computational procedure is explained in Refs. [4, 12].
In the strict treatment, the threshold energies, which means the dissociation energy of 19 Ne into
individual He + O, partitions, should be simultaneously reproduced. However, this reproduction isdiﬃcult in the GCM treatment. This problem always arises in the full microscopic treatment. In order
to avoid this threshold problem, we replace the theoretical thresholds in the 3 He + 16 O and 5 He +14 O to the experimental ones (8.4 and 14 MeV, respectively). In this modified GCM calculation, we
have found that the 3/2+ resonance appears around E ∼ 2 MeV with respect to the α threshold [8].
Therefore, we can conclude that the 5 He + 14 O partition plays an important role for the formation of
3/2+ resonance just above the α threshold.
3.3

Application to three-body system
In this section, we present the formulation of ABC in the three-body system. In the variational
treatment of the three-body systems, the coordinate rearrangement among the three particles is important to obtain the fast conversion in binding energy [16]. The variational calculation with the coordinate rearrangement is called the coupled-rearrangement-channel variational method (CRCVM). The
formulation of the ABC method in CRCVM was unclear although the CSM + CRCVM calculation
has already been established in the three-body problem [1].
In order to establish the formulation of ABC in the three-body system, we have considered theidentical three-bosons, which interact by the Gaussian potential in Eq. (10) with the condition ofℏ2 /m = 1 [15]. In this boson system, we consider the S wave state for simplicity, and the symmetrization among three bosons is explicitly considered by performing the coordinate rearrangement. As for
the ABC treatment, we consider two ways of introducing the absorbing potential. In a naive treatment,
the absorber should be included in accordance with the Jacobi coordinate in each of the rearrangement
channel, such as∑(W(rc − ra ) + W(Rc − Ra )) .
W →(11)
c=1,2,3Here rc and Rc represent the absolute values of the Jacobi vectors in the rearrangement channel ofc, while ra and Ra show the respective starting points of the absorber. In Eq. (11), the absorbing
potentials are introduced along the Jacobi coordinate in the c-th rearrangement channels, and they
are symmetrized on the channel suﬃx c. Another possible way of introducing the absorber is theimaginary potentials among the pair of two particles,∑
W →W(si j − ra ),
(12)
i> j

where si j denotes the magnitude of the relative coordinate of the i-th and j-th particles. In Eqs. (11)and (12), the function form of W is given by Eq. (2) with a setting of p = 4 and ra = Ra =14 fm.
In the ABC + CRCVM calculation, we have confirmed one bound state and two resonant states.
The resonance parameters for the first resonance are summarized in Table I. In the second row fromthe top, the resonance parameters calculated from the Jacobi absorber ( Eq. (11) ) are shown, while
the parameters obtained from the pair absorber ( Eq. (12) ) are shown in the third row. The result of
the Jacobi absorber is the case as those of the pair absorber. These two results of ABC are consistent
with the CSM calculation, shown in the bottom row. As for the second resonance, the results are also
consistent to CSM. Thus, our formulation is considered to be successful in the three-body system.
5■■■

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Table I. Resonance parameters for the first resonance calculated at the optimal strength ηo . The resonanceenergy (ER ) and the resonance width (ΓR ) are shown in the unit of MeV. < W(ηo ) > represent the expectation
value of the absorber at ηo . In the second row, the result of the ABC method is shown, while the results of CSM
is shown in the lowest row. This table is taken from Ref. [7].
MethodERΓR /2< W(ηo ) >ηoABC (Jacobi)

0.424

5.239×10−3

6.942×10−7

1.78×10−6

ABC (pair)

0.424

5.239×10−3

7.118×10−7

8.0×10−6

CSM

0.424

5.260×10−3

XX

4. Summary
In summary, we have shown the basic formulation and application of the absorbing boundary
condition (ABC) to the resonant phenomena in the few-body cluster systems. Here we have handledthree problems: simple two-body system, microscopic two-body system, and identical three-boson
system. As for the simple two-body problem, we have obtained the numerical results the same as thecomplex scaling method (CSM), which is an alternative method to handle the resonant phenomena.
This result means that ABC is basically a powerful tool in handling resonant phenomena as well as
the CSM treatment. Since the application of CSM becomes diﬃcult in the microscopic cluster model,such as the generator coordinate method (GCM), the successful application of ABC to GCM is quite
meaningful as we have shown in the 19 Ne problem.
Furthermore, we have tried to formulate the ABC method in the three-body problem with thecoordinate rearrangement. Two possible ways in the absorbing potential have been checked, such as
the Jacobi absorber and the relative absorber, and both of the calculations give the same results as thatobtained by the CSM calculation. Thus, we believe that our prescription to introduce the absorber issuccessful in handling general three-body problems. The important subject of ABC is the application
to full microscopic three-body problem, such as the GCM calculation of 12 C = 3α. The applicationof ABC to the 3α GCM is now being prepared.
References[1] S. Aoyama, T. Myo, K. Katõ, and K. Ikeda, Prog. Theor. Phys. 116, 1-35 (2006), and references therein.
[2] U. V. Riss and H-D. Meyer, Jour. Phys. B26,4503 (1993).
[3] U. V. Riss and H-D. Meyer, Jour. Phys. B28,1475 (1995); ibid, Jour. Phys. B31, 2279 (1998).
[4] M. Ito and K. Yabana, Prog. Theor. Phys. 113, 1047 (2005).
[5] Y. Takenaka, R. Otani, M. Iwasaki, K. Mimura, and M. Ito, Prog. Theor. Exp. Phys. 2014, 113D04 (2014).
[6] M. Iwasaki, R. Otani, Y. Takenaka, and M. Ito, Prog. Theor. Exp. Phys. 2015, 023D01 (2015).
[7] M. Iwasaki, R. Otani, M. Ito and M. Kamimura, EPJ Web of Conferences 122, 06005 (2016).
[8] R. Otani, M. Iwasaki, and M. Ito, PoS (INPC2016) 036 (2016).
[9] G. Jolicard, C. Leforestier and E. J. Austin, Jour. Chem. Phys. 88, 1026, (1988).
[10] B. Gyaramati, K. T. Kuruppa, K. F. Pal., Phys. Rev. A41, 3469 (1990).
[11] H. Horiuchi et al., Prog. Theor. Phys. 62, 1,(1962).
[12] M. Ito and K. Ikeda, Rep. Prog. Phys. 77, 096301 (2014).
[13] Z. Q. Mao et al., Phys. Rev. Lett. 74, 3760 (1995).
[14] T. Sakuda and F. Nemoto, Prog. Theor. Phys. 62, 1274 (1979).
[15] H. Masui and Y. K. Ho, Phys. Rev. C 65, 054305 (2002).
[16] E. Hiyama et al., Prog. Part. Nucl. Phys. 51, 223 (2003).

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