AAPPS Bulletin
> home > Research and Review
Gaussian Expansion Method and its Application to Nuclear Physic with Strangeness
Emiko Hiyama
File 1 : Vol30_No3_Review And Research-34~36.pdf (0 byte)

DOI: 10.22661/AAPPSBL.2020.30.3.34

Gaussian Expansion Method and its Application to Nuclear Physic with Strangeness


Communicated by Tohru Motobayashi


We have proposed one of the most powerful methods to calculate few-body problems in physics, called Gaussian Expansion Method. The method has been successfully applied to various hypernuclei such as Λ, double Λ and Ξ hypernuclei. In this paper, we describe the structure of He, which is a neutron-rich hypernucleus, as a successful example.


Many important problems in physics can be treated by solving accurately the Schrödinger equation for three- and four-body interacting particles. By solving this equation we can predict various observable, before performing its measurement, and obtain new understandings by comparing the observed data and our theoretical prediction.

For this purpose, it is necessary to develop a method to calculate precisely the three- and four-body problems, and to apply it to various fields such as nuclear physics as well as atomic physics. Along this line, we have been developing our few-body calculation method, Gaussian Expansion method (GEM), which is a kind of variational approach. This method fist was proposed by Kamimura and collaborators [1] and has been successfully applied to various three-body systems such as muonic molecules, [1, 2] three-nucleon bound states (3H, 3He), [3] neutron-rich nuclei [4] and antiprotonic helium atoms. [5] Afterwards, the present author and collaborators proposed to modify the GEM with a new-type of basis functions, called infinitesimally-shift-Gaussian Lobe basis function, [6] and it became in this way much easier to apply the GEM to the solution of the four- and five-body problems. To show the accuracy of our method, we performed a four-body benchmark test calculation of the 4He ground state with realistic NN interaction among seven few-body groups using different approaches. The calculated binding energy, radii and density distribution were found in good agreement among each other. [7]

GEM has been most extensively applied to the study of hypernuclear physics by the present author and collaborators. One of the important goals in the hypernuclear physics is to produce neutron-rich Λ hypernuclei and to study the dynamical change of the structure by comparing the core nuclei. Since the Λ hyperon is free from the nuclear Pauli principle, it is considered to be responsible for a sizeable dynamical contraction of hypernuclear systems due to the addition of a Λ. This is currently known as the "glue-like" role of the Λ particle.

Using this phenomena, the present author has studied with some detail several neutronrich Λ hypernuclei. In the light nuclei sector, near the neutron drip line, some new interesting phenomena concerning the neutron halo have been observed. If a Λ particle is added to such a halo nucleus, which is a very weakly bound system, the resulting hypernucleus will become stable against the neutron decay. To investigate the structure of neutron-rich Λ hypernuclei, we have been studied several systems, like nnΛ, [12] He, [11] A = 7 hypernuclei, [8-10] etc.


Fig. 1: Three sets of Jacobi coordinates of three-body system.


We shortly explain in this section the formalism of GEM for the three-body system. We consider the case of central forces only for simplicity of the expressions (see Ref. [6] for more complicated cases including the solutions of four-body systems). The Schrödinger equation is written as


with obvious notation. The three-body total wave function ΨJM is described as a sum of amplitudes of three rearrangement channels c = 1 - 3 (Fig. 1).


Each amplitude is expanded in terms of the Gaussian basis functions written in Jacobi coordinates rc and Rc:




Nnl (NNL) denotes the normalization constant. The Gaussian ranges νn and λN are given by a geometric progression.

The eigenenergies E and the coefficients A are determined by means of the Rayleigh-Ritz variational principle.

Since the GEM obtained results are reliable, our predictions have been in good agreement with the data. In the next section, a successful example is shown.


The observed data of the core nucleus 6He reported a 0 ground state and the second 2 state. In 2012, see Ref. [13], a second 2+ resonant state was observed in the charge-exchange-reaction 6Li(t, 3He)6He with parameters Ex = 2.6 ± 0.3 and Γ = 1.6 ± 0.4 MeV.

From the theoretical point of view, many authors have studied this system. [14-16] Among these studies, Myo et al. [16] calculated the 6He spectrum using the complex scaling method (CSM) to obtain the energy together with the decay width of the low-lying states. They reproduced the observed ground state and the first 2+ state. They also obtained a second 2+ state. Then, it is interesting to add a Λ particle in 6He. Due to the "glue-like" role of Λ, it is likely to have narrower 3/2 and 5/2 resonant states in He. It was also observed that other states, the 1/2+ ground and the 3/2 and 5/2 excited states, become more stable than the corresponding states of the core nucleus 6He.

To investigate the structure of He, we have performed a four-body cluster model calculation of this system described as α+Λ+n+n and using GEM. In particular, we expected to found some resonant states in this hypernucleus. In order to obtain these resonant states, we employed the complex scaling method, in a similar way of how was applied by Myo et al. to determine the resonant states of the core nucleus 6He.

In 6He, we found a bound ground state with 0+ which is in good agreement with the observed energy. We found also two 2+ states at Er = 0.96 MeV with Γ = 0.14MeV and Er = 2.81 MeV with Γ = 4.63 MeV, respectively. The calculated energy of the first 2+ is also in good agreement with the data. The SPIRAL data reported an energy for the second 2+ state Er = 1.63 MeV with Γ = 1.6 MeV, which is much lower than the calculated Er energy and had narrower decay width than the calculated one.

In the He system, and due to the "glue-like" role of Λ particle, all the states become more stable than the corresponding states of 6He. The calculated Λ-separation energy of BΛ = E(6He)-E(He) = 5.36 MeV, which is consistent with the recent observed data of JLab E01011 experiment: B = 5.68±0.03(stat.)±0.25(sysm.) MeV. The recent analysis of the JLab E05-E115 experiment [17] also reports an excited-state B = 3.65±0.20(stat.)±0.11(sysm.) MeV, which is also in good agreement with our calculations (see Fig. 2). In addition, we predict additional 5/2+ and 3/2+ states having respectively Er = 0.03 MeV with Γ = 1.13 MeV and Er = 0.07 MeV with Γ = 1.01 MeV.

To encourage future experimental activity at JLab, we propose to perform the experiments 7Li(γ, K+)He producing these states. The calculated cross sections are 3.4 and 4.3 nb+sr for 3/2 and 5/2, respectively, which seem to be an accessible value for a future experiment. This experiment is of crucial importance in order to confirm the existence of the second 2+ state in 6He.


Fig. 2: The calculated energy spectra of 6He and He. The states of 2 , 2 and 1+ of 6He and 3/2 and 5/2 are obtained using CSM. The values in parentheses are decay widths Γ in MeV. The figure is taken from Ref. [10]


[1] M. Kamimura, Phys. Rev., A38, 621 (1988)
[2] M. Kamimura, Muon Catal. Fusion 3, 335 (1988).
[3] H. Kameyama, M. Kamimura, and Y. Fukushima, Phy. Rev. C40, 974 (1989).
[4] S. Funada H. Kameyama, and Y. Sakuragi, Nucl. Phys. A 575, 93 (1994).
[5] Y. Kino, M. Kamimura, and H. Kudo, Nucl. Phys. A 631, 649 (1998).
[6] E. Hiyama, Y. Kino and M. Kamimura, Prog. Part. Nucl. Phys. 51, 223 (2003).
[7] H. Kamada et al., Phys. Rev. C 64, 044001 (2001).
[8] E. Hiyama, M. Kamimura, T. Motoba, T. Yamada, and Y. Yamamoto, Phys. Rev. C53, 2075 (1996).
[9] E. Hiyama, M. Kamimura, K. Miyazaki, and T. Motoba, Phys. Rev. C 59, 2351 (1999).
[10] E. Hiyama, M. Isaka, M. Kamimura, T. Myo and T. Motoba, Phys. Rev. C 91, 054316 (2016).
[11] E. Hiyama, S. Ohnishi, M. Kamimura, and Y. Yamamoto, Nucl. Phys. 908, 29 (2013).
[12] E. Hiyama, S. Ohnishi, B.F. Gibson, and Th. A. Rijken, Phys. Rev. C 89, 061302(R) (2014).
[13] X. Mougeot et al., Phys. Lett. B718, 441 (2012).
[14] S. C. Pieper, R. B. Wiringa, and J. Carlson, Phys. Rev. C 70, 054325 (2004).
[15] A. Volya and V. Zelevinsky, Phys. Rev. Lett. 94, 0542501 (2005).
[16] T. Myo, K. Kato and K. Ikeda Phys. Rev. C 76, 054309 (2007).
[17] T. Gogami et al., Phys. Rev. C94, 021302(R) (2016).


AAPPS Bulletin        ISSN: 2309-4710
Copyright © 2018 Association of Asia Pacific Physical Societies. All Rights Reserved.
Hogil Kim Memorial Building #501 POSTECH, 67 Cheongam-ro, Nam-gu, Pohang-si, Gyeongsangbuk-do, 37673, Korea
Tel: +82-54-279-8663 Fax: +82-54-279-8679 e-mail: aapps@apctp.org