Evaluation Activities at JCPRGZHOU Bo, EBATA Shuichiro, DAGVADORJ Ichinkhorloo
Meme Media Laboratory, Hokkaido University
ODSUREN Myagmarjav
Nuclear Research Center, National University of Mongolia
KATŌ Kiyoshi, AIKAWA Masayuki
Faculty of Science, Hokkaido University

Abstract
Nuclear structure and reaction theory always plays a very important role to study
nuclear physics. In this report, we introduce briefly some evaluation activities inJCPRG in 2014. These theoretical calculations can help us to evaluate some nuclear
data from experiments and also provide us with a good prediction for some required
nuclear data.

1

Introduction

The JCPRG’s central task is to compile the data obtained in Japan on charged-particle and photoinduced nuclear reactions [1]. As the member of the International Network of Nuclear Reaction
Data Centres (NRDCs), JCPRG has contributed about 10 percent of the data on charged-particle
nuclear reactions in the database. On the other hand, some members of JCPRG are also doingsome theoretical calculations for studying nuclear reaction and structure. By using diﬀerent nuclear
models, we can get useful information on the nuclear reaction and structure. Now, to develop further
the theory-based approaches for the evaluation of nuclear reaction data has became one important
objective in JCPRG.
In this report, our activities on the evaluation of the nuclear data are presented. First, wewill show a systematic study on electric dipole (E1) modes of even-even nuclei obtained from thecanonical-basis time-dependent Hartree-Fock-Bogoliubov theory (Cb-TDHFB). Then, we will show
some results on the continuum discretized coupled channel (CDCC) analysis to the integrated elasticand inelastic scattering cross sections for 6,7 Li. Next, nuclear scattering problems in the complex
scaling method are discussed. Finally, 2-alpha correlation of the ground state of 12 C is studied ina container picture.

2

Systematic investigation of low-energy electric dipole modes ofeven-even nuclei

We systematically investigate the electric dipole (E1) modes of even-even nuclei using the canonicalbasis time-dependent Hartree-Fock-Bogoliubov theory (Cb-TDHFB). The Cb-TDHFB can describe- 16 -the nuclear dynamics in the three-dimensional Cartesian coordinate space dealing with nuclear superfluidity [2]. Therefore we can carry out systematic investigation without any restriction formass number, open shell structure and deformation. The linear response calculation with theCb-TDHFB for E1 and quadrupole excitation reproduces the strength function obtained by thequasi-particle random phase approximation which is a small amplitude limit of full TDHFB. Actually the photoabsorption reaction cross section of 172 Yb is well reproduced by the Cb-TDHFBcalculation [3].

628Ni30Zn32Ge

534SefPDR [%]36Kr38Sr

440Zr
42Mo
44Ru

346Pd
48Cd
50Sn2102836445260

6876

8492

Neutron Number
Figure 1: Neutron number dependence of the low-lying E1 modes in total E1 strength which istaken from Refs. [4, 5].
To understand the structure of unstable nuclei and the contribution of them to the nucleosynthesis related with neutron-rich region, we investigate especially the systematics of low-lying E1
mode which is often called Pygmy dipole resonance (PDR). Figure 1 shows the neutron number
dependence of the low-lying E1 strength, which is evaluated as the ratio in the energy weighted
total E1 strength. There are kinks of the ration which are reflected in shell structure at N=50 and82, and also the kinks at N=60 and 70 corresponding to the shape transition region [4, 5]. Thiswork shows that the shell eﬀects for PDR of heavy nucleus remain also around N=82 corresponding to f -orbits occupation, and the change of chemical potential due to the deformation induces
the PDR suppression (N=60) and enhancement (N=70). In our future work, we will constructthe theoretical database for medical, astrophysics and nuclear engineering from these systematic
calculation and simulations.

3Analysis of 6,7 Li + n reactions using CDCC method

We have studied applicability of the continuum discretized coupled channel (CDCC) method forneutron scattering on lithium isotope [6–8]. In this fiscal year, we extended the CDCC [9] analysis
to the integrated elastic and inelastic scattering cross sections for 6,7 Li at incident neutron energies
below 10 MeV by using optical model potential (OMP) [10, 11] and above 10 MeV by usingJLM [12]. Energy dependence of the normalization factors, for the OMP cluster-folding potential isintroduced and determined from measured integrated elastic cross sections, respectively, at incident- 17 -neutron energies below 10 MeV. We adjust the normalization constants for the OMP, because theagreement of the calculated cross sections data at very low incident energies of the neutron is
insuﬃcient without any adjustments. The energy dependent normalization constants, real part λv
and imaginary part λw , of the OMP and JLM are determined explicitly from integrated elastic
cross section data, respectively. Comparing the results of calculations and experimental data, wediscuss that the present CDCC calculations, which reproduce the experimental data observed inincident energies higher than 10 MeV with the single folding potential of the JLM and in lower
energies with introducing the normalization factors for the cluster folding potential of the OMP.
1

Scattering Cross Section [barn]

10R.Batchelor (1963)
F.Merchez (1966)
J.A.Cookson (1967)
J.C.Hopkins (1968)
H.H.Hogue (1979)P.W.Lisowski (1980)
S.Chiba (1985)
M.Adel−Fawzy (1987)
G.Chen (2009)

Result with OMP
Result with JLM
JENDL−3.30103 45678 9 10 11 12 13 14 15 16 17 18 19 20

Incident neutron energy [MeV]

Figure 2: The integrated elastic scattering cross sections of 6 Li, in comparison with the evaluateddata and experimental data.
010R.Batchelor (1963)
F.Merchez (1966)
J.A.Cookson (1967)
J.C.Hopkins (1968)
A.B.Smith (1982)
H.H.Hogue (1979)
P.W.Lisowski (1980)
S.Chiba (1985)
M.Adel−Fawzy (1987)
S.Chiba(1988)
M.Ibaraki(1998)
G.Chen (2009)

Scattering Cross Section [barn]

Result with OMP
Result with JLM
JENDL−3.3

10−110−24681012141618202224

Incident neutron energy [MeV]

Figure 3: The integrated inelastic scattering cross sections for the 2.186-MeV state of 6 Li, incomparison with the evaluated data and experimental data.

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For example, the integrated elastic cross sections for 6 Li agree with the evaluated data (JENDL3.3) and other measurements within the experimental uncertainties, as shown in Fig. 2. In Fig. 3,the integrated inelastic cross-section values for the 2.186-MeV state of 6 Li are almost in good
agreement with the evaluation data of JENDL-3.3 and the experimental data.

4

Nuclear scattering problems in the complex scaling method

Our long standing problem is to understand the scattering phenomena together with nuclear structure problems. For this purpose, the complex scaling method is very promising, because it has thefollowing advantages: 1) We can obtain resonant states by solving a eigen-value problem with anL2 basis functions in the same way as a bound-state problem. 2) The discretized continuum states
are also obtained as complex energy solutions on the rotated branch cuts separately from isolatedresonances. 3) Not only two-body but also many-body resonances can easily be obtained. In 2013,
we had developed results of the following two problems:i) Decomposition of scattering phase shifts and reaction cross sections using the complex scaling
method [13]. In this work, we have shown a new method to calculate the scattering phase shiftusing complex scaled eigen-value solutions. This method provides us with a useful decompositionof the phase shift into resonance and background terms. We presented the applications for several
two-body systems: (a) a schematic model with the Gyarmati potential, which produces manyresonances, (b) the α+α system, which has a Coulomb barrier potential in addition to an attractive
nuclear interaction, and (c) the α+n system, which has no barrier potential. Using diﬀerent kindsof potentials, we discuss the reliability of this method to investigate the resonance structure in thephase shifts and cross sections.
ii) Analysis of Three Body Resonances in the Complex Scaled Orthogonal Condition Model [14].
Although the resonance structures of α+α+n have been studied experimentally and theoretically,
it is still necessary to have more accurate and comprehensive understandings of the structure anddecay of the low-lying excited states in 9 Be. To perform calculations of an α+α+n system, weinvestigate five resonant states of α+α subsystem by utilizing diﬀerent potential parameters andbasis functions. In addition, two resonance states of α+n subsystem are computed.
These studies have shown that the CSM provides us with a useful way for evaluations of nuclear
data through analyses of the observed scattering data using resonance solutions of the CSM. On
the basis of these results, it is planned to carry out an analysis of the photodisintegration crosssections of 9 Be in which some discrepancies are observed in the experimental data.

5

Study of the ground state of 12 C in the container picture

Recently, we proposed a container picture for the description of the cluster structures in light nuclei,
which has been very successful for the description of the inversion-doublet band states of 20 Ne [15]and the hoyle state of 12 C. To develop the container picture on the firmer ground, we construct the2-alpha+alpha THSR (Tohsaki-Horiuchi-Schuck-Röpke) wave function [16], in which the 2-alphacorrelation can be included in a natural way. Thus, we can get the single optimum THSR wave
function by variation calculations and then explore whether the compact ground state of 12 C canbe described well by this extended THSR wave function.
In the container picture, the 2-alpha+alpha THSR wave function can be written as follows,22∑riyr2r2
Φ(β 1 , β 2 ) ∝ ϕG A{exp[−( ix2 + 2 + iz2 )]ϕ(α1 )ϕ(α2 )ϕ(α3 )},Bix Biy
Biz
i=1- 19 -(1)2 = b2 + β 2 , B 2 = 3 b2 + β 2 , and β ≡ (β , β , β ). b is the size parameter of theWhere B1kix iy iz
i2k1k2k
4
harmonic-oscillator wave function. ϕ(αi ) represents the ith-α-cluster intrinsic wave function andX i is its corresponding center-of-mass coordinate. r 1 = X 2 − X 1 , r 2 = X 3 − (X 1 + X 2 )/2. ϕG is12 C, which can be expressed as, exp(−6X 2 /b2 ). If we make thethe center-of-mass waveG√ function of √replacement, β 1 → 2β 0 and β 2 → 3/2β 0 in Eq. (1), this 2-alpha+alpha THSR wave function
becomes the 3-alpha THSR wave function with single β 0 parameter used by Funaki et al. in Ref[17].
First, we find that the single extended two-β THSR wave function for the ground state of 12 C
is very close to the full solution results of the 3-alpha cluster models [18, 19], e.g., the calculatedsquared overlaps between Φ̂GCM (β 1 , β 2 ) and the single normalized 2-alpha+alpha THSR wave
functions corresponding to their minimum energies are as high as 98%. Thus, the container picture
is proved to be also very successful for describing this compact three-body cluster structure of theground state with normal density. Furthermore, the squared overlap between the single 3-alpha
THSR wave function and the THSR-GCM wave function for the ground state is about 93%, byintroducing the 2-alpha correlation, the corresponding squared overlap increases to 98%. This
provides a strong support for the existence of the 2-alpha correlation in the ground state of 12 C.
See details in Ref. [20].
Compared with the traditional microscopic RGM (resonating group method)/GCM(generatorcoordinate method) wave function for 12 C, the THSR wave function is very simple and flexible.
Next, this extended THSR wave function can be used for the evaluation of nuclear structure data+
related with 12 C, e.g., the recently observed 0+
3 and 04 states from experiments [21].

6Summary

In summary, firstly, we show the systematic study for electric dipole (E1) modes of even-even nuclei
obtained with Cb-TDHFB. Based on the Cb-TDHFB theory, the evaluation database for medical,
astrophysics and nuclear engineering can be built from the systematic calculation and simulationsin the future. Secondly, it is shown that the integrated inelastic and elastic cross-section values forthe 2.186-MeV state of 6 Li by CDCC are in good agreement with the experimental data. Thirdly,
we discuss the complex scaled orthogonal condition model, which is a useful way for evaluations ofnuclear data through analyses of the observed scattering data. Last, it is shown that the groundstate of 12 C has been described very well in a container picture, thus, the related excited states
information of 12 C are promising to be evaluated in this theoretical framework in the future.

Acknowledgment
We would like to acknowledge the support by ’R&D’ Platform Formation of Nuclear Reaction Datain Asian Countries (2010-2013)”, Asia-Africa Science Platform, Japan Society for the Promotionof Science and the support by the collaboration project between Faculty of Science, Hokkaido
University and RIKEN.

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