Three-Body Structure of \(^9_{\varLambda }\)Be with \(\alpha \alpha \varLambda \) Cluster Model


In the framework of \(\alpha +\alpha +\varLambda \) three-body cluster model, we calculate energy spectra from bound energy region to resonant energy region up to around 20 MeV with respect to \(\alpha \alpha \varLambda \) threshold. To calculate resonant states, we employ Complex Scaling method which is one of the powerful method. We obtain the states of \(^8\)Be analogue and genuine hypernuclear analogue, which are consistent with those by Yamada et al. (Prog Theor Phys Suppl 81:104, 1985). However, the calculated ordering of \(^9\)Be analogue states is quite different with their calculation. We also obtain \(2^+\) and \(4^+\) resonant states of which have been never pointed out.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6


  1. 1.

    T. Motoba, H. Bandō, K. Ikeda, Prog. Theor. Phys. Suppl. 70, 189 (1983)

    Article  Google Scholar 

  2. 2.

    T. Yamada, T. Motoba, K. Ikeda, H. Bandō, Structure study of typical light hypernuclei. Prog. Theor. Phys. Suppl. 81, 104 (1985)

    ADS  Article  Google Scholar 

  3. 3.

    E. Hiyama, M. Kamimura, T. Motoba, T. Yamada, Y. Yamamoto, Three-body model study of \(A=6 {-} 7\) hypernuclei: halo and skin structures. Phys. Rev. C 53, 2075 (1996)

    ADS  Article  Google Scholar 

  4. 4.

    E. Hiyama, M. Kamimura, K. Miyazaki, T. Motoba, \(\gamma \) transitions in \(A=7\) hypernuclei and a possible derivation of hypernuclear size. Phys. Rev. C 59, 2351 (1999)

    ADS  Article  Google Scholar 

  5. 5.

    B.-N. Lu, E. Hiyama, H. Sagawa, S.-G. Zhou, Superdeformed \(\varLambda \) hypernuclei within relativstic mean field models. Phys. Rev. C 89, 044307 (2014)

    ADS  Article  Google Scholar 

  6. 6.

    M. Isaka, K. Fukukawa, M. Kimura, E. Hiyama, H. Sagawa, Y. Yamamoto, Superdeformed \(\varLambda \) hypernuclei with antisymmetrized molecular dynamics. Phys. Rev. C 89, 024310 (2014)

    ADS  Article  Google Scholar 

  7. 7.

    Y. Funaki, M. Isaka, E. Hiyama, T. Yamada, K. Ikeda, Multi-cluster dynamics in \(^{13}_{\varLambda }\)C and analogy to clustering in \(^{12}\)C. Phys. Lett. B 773, 336 (2017)

    ADS  Article  Google Scholar 

  8. 8.

    K. Tanida et al., Measurement of the \(B(E2)\) of \(^7_{\varLambda }\)Li and shirinkage of the hypernucelar size. Phys. Rev. Lett. 86, 1982 (2001)

    ADS  Article  Google Scholar 

  9. 9.

    R.H. Dalitz, A. Gal, Supersymmetric and strangeness analog states in p-shell \(\varLambda \) hypernuclei. Phys. Rev. Lett. 36, 362 (1976)

    ADS  Article  Google Scholar 

  10. 10.

    R.H. Dalitz, A. Gal, Strangeness analogue states in \(_\varLambda ^9\)Be. Ann. Phys. 131, 314 (1981)

    ADS  Article  Google Scholar 

  11. 11.

    H. Akikawa et al., Hypernuclear fine structure in \(^9_{\varLambda }\)Be. Phys. Rev. Lett. 88, 082501 (2002)

    ADS  Article  Google Scholar 

  12. 12.

    H. Tamura et al., \(\gamma \)-ray spectroscopy in \(\varLambda \) hypernuclei. Nucl. Phys. A 754, 58c (2005)

    ADS  Article  Google Scholar 

  13. 13.

    S. Ajimura et al., Observation of spin–orbit splitting in \(\varLambda \) single-particle states. Phys. Rev. Lett. 86, 4255 (2001)

    ADS  Article  Google Scholar 

  14. 14.

    H. Tamura et al., Observation of a spin-flip M1 transition in \(_\varLambda ^7\)Li. Phys. Rev. Lett. 84, 5963 (2000)

    ADS  Article  Google Scholar 

  15. 15.

    M. Ukai et al., Hypernuclear fine structure in \(^{16}_\varLambda \)O and the \({\varLambda }N\) tensor interaction. Phys. Rev. Lett. 93, 232501 (2004)

    ADS  Article  Google Scholar 

  16. 16.

    M. Ukai et al., Cascade \(\gamma \) decay in the \(_\varLambda ^7\)Li hypernucleus. Phys. Rev. C 73, 012501(R) (2006)

    ADS  Article  Google Scholar 

  17. 17.

    O. Hashimoto, H. Tamura, Spectroscopy of \(\varLambda \) hypernuclei. Prog. Part. Nucl. Phys. 57, 564 (2006)

    ADS  Article  Google Scholar 

  18. 18.

    E. Hiyama, Y. Kino, M. Kamimura, Gaussian expansion method for few-body systems. Prog. Part. Nucl. Phys. 51, 223 (2003)

    ADS  Article  Google Scholar 

  19. 19.

    E. Hiyama, T. Yamada, Structure of light hypernuclei. Prog. Part. Nucl. Phys. 63, 339 (2009)

    ADS  Article  Google Scholar 

  20. 20.

    D.J. Millener, Shell-model interpretation of \(\gamma \)-ray transitions in p-shell hypernuclei. Nucl. Phys. A 804, 84 (2008)

    ADS  Article  Google Scholar 

  21. 21.

    D.J. Millener, Lect. Notes Phys. 724, 31 (2007)

    ADS  Article  Google Scholar 

  22. 22.

    J. Aguilar, J.M. Combes, Commun. Math. Phys. 22, 269 (1971)

    ADS  Article  Google Scholar 

  23. 23.

    E. Balslev, J.M. Combes, Commun. Math. Phys. 22, 280 (1971)

    ADS  Article  Google Scholar 

  24. 24.

    B. Simon, Commun. Math. Phys. 27, 1 (1972)

    ADS  Article  Google Scholar 

  25. 25.

    Y.K. Ho, Phys. Rep. 99, 1 (1983)

    ADS  Article  Google Scholar 

  26. 26.

    N. Moiseyev, Phys. Rep. 302, 211 (1998)

    ADS  Article  Google Scholar 

  27. 27.

    S. Aoyama, T. Myo, K. Kato, K. Ikeda, Prog. Theor. Phys. 116, 1 (2006)

    ADS  Article  Google Scholar 

  28. 28.

    T. Myo, Y. Kikuchi, H. Masui, K. Kato, Prog. Part. Nucl. Phys. 79, 1 (2014)

    ADS  Article  Google Scholar 

  29. 29.

    S. Saito, Effect of Pauli principle in scattering of two clusters. Prog. Theor. Phys. 40, 893 (1968)

    ADS  Article  Google Scholar 

  30. 30.

    S. Saito, Effect of Pauli principle in scattering of two clusters. Prog. Theor. Phys. 41, 705 (1969)

    ADS  Article  Google Scholar 

  31. 31.

    E. Hiyama, M. Kamimura, T. Motoba, T. Yamada, Y. Yamamoto, Three- and four-body cluster models of hypernuclei using the G-matrix \({\varLambda }N\) interaction—\(_\varLambda ^9\)Be, \(_\varLambda ^{13}\)C, \(_{\varLambda \varLambda }^6\)He and \(_{\varLambda \varLambda }^{10}\)Be -. Prog. Theor. Phys. 97, 881 (1997)

    ADS  Article  Google Scholar 

  32. 32.

    E. Hiyama, M. Isaka, M. Kamimura, T. Myo, T. Motoba, Resonant states of the neutron-rich \(\varLambda \) hypernucleus \(^7_{\varLambda }\)He. Phys. Rev. C 91, 054316 (2015)

    ADS  Article  Google Scholar 

  33. 33.

    E. Hiyama, R. Lazauskas, J. Carbonell, M. Kamimura, Possibility of generating a 4-neutron resonance with a \(T=3/2\) isospin 3-neutron force. Phys. Rev. 93, 044004 (2016)

    ADS  MathSciNet  Google Scholar 

  34. 34.

    M. May et al., Observation of hypernuclear gamma-ray transitions in \(_\varLambda ^{7}\)Li and \(_\varLambda ^{9}\)Be. Phys. Rev. Lett. 51, 2085 (1983)

    ADS  Article  Google Scholar 

Download references


The authors would like to thank to Prof. M. Kamimura for useful discussion. This work is supported by RIKEN-MOST project. One of the authors (E. H.) is supported by a Grants-in-Aid for Scientific Research from Monbukagakusho of Japan (18H05407, 16H03995 and 16H02180). One of the authors (Y. F.) is supported by JSPS KAKENHI Grant Number JP18K03658. The numerical calculations were performed at the Yukawa Institute Computer Facility. One of authors (Q.W.) is supported by National Natural Science Foundation of China (11475085, 11535005, 11690030) and National Major state Basic Research and Development of China (2016YFE0129300).

Author information



Corresponding author

Correspondence to Emiko Hiyama.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article belongs to the Topical Collection “Ludwig Faddeev Memorial Issue”.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lee, J., Wu, Q., Funaki, Y. et al. Three-Body Structure of \(^9_{\varLambda }\)Be with \(\alpha \alpha \varLambda \) Cluster Model. Few-Body Syst 60, 30 (2019).

Download citation