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Shape of \(^{12}\textrm{C}\)

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Abstract

We have examined the hypothesis by Bijker and Iachello who asserted that \(^{12}\textrm{C}\) has an internal structure with three \(\alpha \) particles arranged in a triangular shape, leading to the formation of the ground rotational band consisting of \(0^+\), \(2^+\), \(3^-\), \(4^\pm \) and \(5^-\) states. Following this idea, we reconstructed the intrinsic shape of \(^{12}\textrm{C}\) using experimental electron scattering data with minimal theoretical assumption. Our sole assumption was that the observed \(0^+_1\), \(2^+_1\), \(3^-_1\), and \(4^+_2\) states share a common internal structure, forming a rotational spectrum. The reconstructed intrinsic density showed a beautiful triangular shape with three peaks implying \(\alpha \) cluster formation in the ground band.

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Data availability

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: All the data used in this paper are available in the Refs. [29, 31,32,33,34,35,36,37,38].]

Notes

  1. Note that to convert the plots in Fig. 4 to densities, you need to multiply by spherical harmonics according to Eq. (1). For example, the central density of \(L=0\) component is approximately \(0.25/\sqrt{4\pi }\simeq 0.07\ \mathrm fm^{-3}\). Thus, combined with the neutron density, the central matter density is about \(0.14\ \mathrm fm^{-3}\), slightly lower than the normal density.

  2. It is evident from the theoretical point of view [1, 18], but it is not easy to prove it by experiments.

References

  1. A. Tohsaki, H. Horiuchi, P. Schuck, G. Röpke, Alpha cluster condensation in C12 and O16. Phys. Rev. Lett. 87, 192501 (2001)

    Article  ADS  Google Scholar 

  2. Y. Funaki, H. Horiuchi, W. von Oertzen, G. Röpke, P. Schuck, A. Tohsaki, T. Yamada, Concepts of nuclear \(\alpha \) -particle condensation. Phys. Rev. C 80(6), 064326 (2009). https://doi.org/10.1103/PhysRevC.80.064326

    Article  ADS  Google Scholar 

  3. Y. Funaki, Hoyle band and \(\alpha \) condensation in C12. Phys. Rev. C 92, 021302 (2015)

    Article  ADS  Google Scholar 

  4. P. Schuck, Y. Funaki, H. Horiuchi, G. Röpke, A. Tohsaki, T. Yamada, Alpha particle clusters and their condensation in nuclear systems. Physica Scripta 91, 123001 (2016)

    Article  ADS  Google Scholar 

  5. M. Itoh, H. Akimune, M. Fujiwara, U. Garg, N. Hashimoto, T. Kawabata, K. Kawase, S. Kishi, T. Murakami, K. Nakanishi, Y. Nakatsugawa, B.K. Nayak, S. Okumura, H. Sakaguchi, H. Takeda, S. Terashima, M. Uchida, Y. Yasuda, M. Yosoi, J. Zenihiro, Candidate for the 2+ excited Hoyle state at Ex=10 MeV in 12C. Phys. Rev. C 84, 054308 (2011)

  6. M. Freer, M. Itoh, T. Kawabata, H. Fujita, H. Akimune, Z. Buthelezi, J. Carter, R.W. Fearick, S.V. Förtsch, M. Fujiwara, U. Garg, N. Hashimoto, K. Kawase, S. Kishi, T. Murakami, K. Nakanishi, Y. Nakatsugawa, B.K. Nayak, R. Neveling, S. Okumura, S.M. Perez, P. Papka, H. Sakaguchi, Y. Sasamoto, F.D. Smit, J.A. Swartz, H. Takeda, S. Terashima, M. Uchida, I. Usman, Y. Yasuda, M. Yosoi, J. Zenihiro, Consistent analysis of the 2 + excitation of the 12 C Hoyle state populated in proton and \(\alpha \)-particle inelastic scattering. Phys. Rev. C 86(3), 034320 (2012). https://doi.org/10.1103/PhysRevC.86.034320

  7. M. Itoh, H. Akimune, M. Fujiwara, U. Garg, T. Kawabata, K. Kawase, T. Murakami, K. Nakanishi, Y. Nakatsugawa, H. Sakaguchi, S. Terashima, M. Uchida, Y. Yasuda, M. Yosoi, J. Zenihiro, Nature of 10 MeV state in 12 C. J. Phys.: Conf. Ser. 436(1), 012006 (2013). https://doi.org/10.1088/1742-6596/436/1/012006. http://stacks.iop.org/1742-6596/436/i=1/a=012006?key=crossref.66f3eb5d27716d2a8697b3dd2ef6d8d0

  8. T. Wakasa, E. Ihara, K. Fujita, Y. Funaki, K. Hatanaka, H. Horiuchi, M. Itoh, J. Kamiya, G. Röpke, H. Sakaguchi, N. Sakamoto, Y. Sakemi, P. Schuck, Y. Shimizu, M. Takashina, S. Terashima, A. Tohsaki, M. Uchida, H. Yoshida, M. Yosoi, New candidate for an alpha cluster condensed state in at 400 MeV. Phys. Lett. B 653, 173–177 (2007). https://doi.org/10.1016/j.physletb.2007.08.016.http://linkinghub.elsevier.com/retrieve/pii/S0370269307009690

  9. Y. Funaki, T. Yamada, H. Horiuchi, G. Röpke, P. Schuck, A. Tohsaki, \(\alpha \) -particle condensation in O16 studied with a full four-body orthogonality condition model calculation. Phys. Rev. Lett. 101, 082502 (2008). https://doi.org/10.1103/PhysRevLett.101.082502

  10. Y. Funaki, Container evolution for cluster structures in O 16. Phys. Rev. C 97(2), 021304 (2018). https://doi.org/10.1103/PhysRevC.97.021304

  11. S. Adachi, T. Kawabata, K. Minomo, T. Kadoya, N. Yokota, H. Akimune, T. Baba, H. Fujimura, M. Fujiwara, Y. Funaki, T. Furuno, T. Hashimoto, K. Hatanaka, K. Inaba, Y. Ishii, M. Itoh, C. Iwamoto, K. Kawase, Y. Maeda, H. Matsubara, Y. Matsuda, H. Matsuno, T. Morimoto, H. Morita, M. Murata, T. Nanamura, I. Ou, S. Sakaguchi, Y. Sasamoto, R. Sawada, Y. Shimizu, K. Suda, A. Tamii, Y. Tameshige, M. Tsumura, M. Uchida, T. Uesaka, H.P. Yoshida, S. Yoshida, Systematic analysis of inelastic \(\alpha \) scattering off self-conjugate A = 4 n nuclei. Phys. Rev. C 97(1), 014601 (2018). https://doi.org/10.1103/PhysRevC.97.014601

  12. S. Adachi, Y. Fujikawa, T. Kawabata, H. Akimune, T. Doi, T. Furuno, T. Harada, K. Inaba, S. Ishida, M. Itoh, C. Iwamoto, N. Kobayashi, Y. Maeda, Y. Matsuda, M. Murata, S. Okamoto, A. Sakaue, R. Sekiya, A. Tamii, M. Tsumura, Candidates for the 5\(\alpha \) condensed state in 20Ne. Phys. Lett. B 819, 136411 (2021). https://doi.org/10.1016/j.physletb.2021.136411. https://linkinghub.elsevier.com/retrieve/pii/S0370269321003518

  13. B. Zhou, Y. Funaki, H. Horiuchi, Y.G. Ma, G. Röpke, P. Schuck, A. Tohsaki, T. Yamada, The 5alpha condensate state in 20Ne. Nat. Commun. 14(1), 8206 (2023). https://doi.org/10.1038/s41467-023-43816-9

  14. Y. Fujikawa et al., Phys. Lett. B 848, 138384 (2024)

  15. E. Epelbaum, H. Krebs, D. Lee, U.G. Meißner, Ab-initio calculation of the Hoyle state. Phys. Rev. Lett. 106(19), 192501 (2011). https://doi.org/10.1103/PhysRevLett.106.192501

    Article  ADS  Google Scholar 

  16. E. Epelbaum, H. Krebs, T.A. Lähde, D. Lee, U.G. Meißner, U.G. Meiner, Structure and rotations of the Hoyle state. Phys. Rev. Lett. 109(25), 252501 (2012). https://doi.org/10.1103/PhysRevLett.109.252501

    Article  ADS  Google Scholar 

  17. A. Lovato, S. Gandolfi, J. Carlson, S.C. Pieper, R. Schiavilla, Electromagnetic response of 12C: a first-principles calculation. Phys. Rev. Lett. 117(8), 082501 (2016). https://doi.org/10.1103/PhysRevLett.117.082501

    Article  ADS  Google Scholar 

  18. T. Otsuka, T. Abe, T. Yoshida, Y. Tsunoda, N. Shimizu, N. Itagaki, Y. Utsuno, J. Vary, P. Maris, H. Ueno, \(\alpha \)-Clustering in atomic nuclei from first principles with statistical learning and the Hoyle state character. Nat. Commun. 13(1), 2234 (2022). https://doi.org/10.1038/s41467-022-29582-0

    Article  ADS  Google Scholar 

  19. S. Shen, S. Elhatisari, T. A. Lähde, D. Lee, B. N. Lu, and U.-G. Meißner, Nat. Commun. 14, 2777 (2023)

  20. G. Stellin, L. Fortunato, A. Vitturi, Electromagnetic selection rules in the triangular \(\alpha \) -cluster model of 12 C. J. Phys. G: Nuclear Part. Phys. 43(8), 085104 (2016). https://doi.org/10.1088/0954-3899/43/8/085104

    Article  ADS  Google Scholar 

  21. P.O. Hess, 12C within the Semimicroscopic algebraic cluster model. Eur. Phys. J. A 54(3), 32 (2018). https://doi.org/10.1140/epja/i2018-12468-7

    Article  ADS  Google Scholar 

  22. A. Vitturi, J. Casal, L. Fortunato, E.G. Lanza, Transition densities and form factors in the triangular \(\alpha \)-cluster model of C12. Phys. Rev. C 101(1), 014315 (2020). https://doi.org/10.1103/PhysRevC.101.014315

    Article  ADS  Google Scholar 

  23. J. Casal, L. Fortunato, E.G. Lanza, A. Vitturi, Alpha-induced inelastic scattering and alpha-transfer reactions in12C and 16O within the Algebraic Cluster Model. Eur. Phys. J. A 57(1), 33 (2021). https://doi.org/10.1140/epja/s10050-021-00347-5

    Article  ADS  Google Scholar 

  24. R. Bijker, F. Iachello, Cluster states in nuclei as representations of a U(nu+1) group. Phys. Rev. C 61(6), 067305 (2000). https://doi.org/10.1103/PhysRevC.61.067305

    Article  ADS  Google Scholar 

  25. R. Bijker, F. Iachello, Cluster structure of light nuclei. Progress Part. Nuclear Phys. 110, 103735 (2020). https://doi.org/10.1016/j.ppnp.2019.103735

    Article  Google Scholar 

  26. D.J. Marín-Lámbarri, R. Bijker, M. Freer, M. Gai, T. Kokalova, D.J. Parker, C. Wheldon, Evidence for triangular D3h symmetry in C12. Phys. Rev. Lett. 113(1), 012502 (2014). https://doi.org/10.1103/PhysRevLett.113.012502

    Article  ADS  Google Scholar 

  27. A. Nakada, Y. Torizuka, Y. Horikawa, Determination of the deformation in 12C from electron scattering. Phys. Rev. Lett. 27(11), 745–748 (1971). https://doi.org/10.1103/PhysRevLett.27.745

    Article  ADS  Google Scholar 

  28. M. Kamimura, Transition densities between states in 12C derived from the three-alpha resonating-group wave functions. Nucl. Phys. A 351, 456–480 (1981)

    Article  ADS  Google Scholar 

  29. J.H. Fregeau, Elastic and inelastic scattering of 187-Mev electrons from Carbon-12. Phys. Rev. 104(1), 225–236 (1956). https://doi.org/10.1103/PhysRev.104.225

    Article  ADS  Google Scholar 

  30. H.L. Crannell, T.A. Griffy, Determination of radiative transition C12. Phys. Rev. 136(6B), B1580–B1584 (1964). https://doi.org/10.1103/PhysRev.136.B1580

    Article  ADS  Google Scholar 

  31. H. Crannell, Elastic and inelastic electron scattering from C12 and O16. Phys. Rev. 148(3), 1107–1118 (1966). https://doi.org/10.1103/PhysRev.148.1107

    Article  ADS  Google Scholar 

  32. P. Strehl, T. Schucan, Study of monopole transitions in 12C, 24Mg, 28Si, 32S and 40Ca by inelastic electron scattering. Phys. Lett. B 27(10), 641–643 (1968). https://doi.org/10.1016/0370-2693(68)90303-1

    Article  ADS  Google Scholar 

  33. I. Sick, J. McCarthy, Elastic electron scattering from 12C and 16O. Nucl. Phys. A 150(3), 631–654 (1970). https://doi.org/10.1016/0375-9474(70)90423-9

    Article  ADS  Google Scholar 

  34. F. Kline, H. Crannell, J.T. O’Brien, J. McCarthy, R. Whitney, Elastic electron scatterring from 14C. Nucl. Phys. A 209(2), 381–395 (1973). https://doi.org/10.1016/0375-9474(73)90585-X

    Article  ADS  Google Scholar 

  35. W. Reuter, G. Fricke, K. Merle, H. Miska, Nuclear charge distribution and RMS radius of C12 from absolute elastic electron scattering measurements. Phys. Rev. C 26(3), 806–818 (1982). https://doi.org/10.1103/PhysRevC.26.806

  36. H. De Vries, C. De Jager, C. De Vries, Nuclear charge-density-distribution parameters from elastic electron scattering. At. Data Nucl. Data Tables 36(3), 495–536 (1987). https://doi.org/10.1016/0092-640X(87)90013-1

  37. A. Bohr, B. Mottelson, Nuclear Structure, vol. 2 (Benjamin Inc., New York, 1975)

    Google Scholar 

  38. R. Imai, T. Tada, M. Kimura, Real-time evolution method and its application to the 3\(\alpha \) cluster system. Phys. Rev. C 99, 064327 (2019)

    Article  ADS  Google Scholar 

  39. B. Pritychenko, M. Birch, B. Singh, M. Horoi, Tables of E2 transition probabilities from the first 2+ states in even-even nuclei. At. Data Nucl. Data Tables 107, 1–139 (2016). https://doi.org/10.1016/j.adt.2015.10.001.https://linkinghub.elsevier.com/retrieve/pii/S0092640X15000406

    Article  ADS  Google Scholar 

  40. H. Crannell, T. Griffy, L. Suelzle, M. Yearian, A determination of the transition widths of some excited states in 12C. Nucl. Phys. A 90(1), 152–158 (1967). https://doi.org/10.1016/0375-9474(67)90745-2

    Article  ADS  Google Scholar 

  41. W. Vermeer, M. Esat, J. Kuehner, R. Spear, A. Baxter, S. Hinds, Electric quadrupole moment of the first excited state of 12C. Phys. Lett. B 122(1), 23–26 (1983). https://doi.org/10.1016/0370-2693(83)91160-7

    Article  ADS  Google Scholar 

  42. G. Ripka, Advances in Nuclear Physics (Springer, Boston, 1968), pp.183–259. https://doi.org/10.1007/978-1-4757-0103-6_3

    Book  Google Scholar 

  43. Y. Kanada-En’yo, Variation after angular momentum projection for the study of excited states based on antisymmetrized molecular dynamics. Phys. Rev. Lett. 81, 5291–5293 (1998)

    Article  ADS  Google Scholar 

  44. M. Chernykh, H. Feldmeier, T. Neff, P. von Neumann-Cosel, A. Richter, Structure of the Hoyle State in C 12. Phys. Rev. Lett. 98, 032501 (2007)

    Article  ADS  Google Scholar 

  45. M. Chernykh, H. Feldmeier, T. Neff, P. von Neumann-Cosel, A. Richter, Pair decay width of the Hoyle state and its role for stellar carbon production. Phys. Rev. Lett. 105(2), 022501 (2010). https://doi.org/10.1103/PhysRevLett.105.022501

    Article  ADS  Google Scholar 

  46. Y. Kanada-En’yo, The structure of ground and excited states of 12C. Progress Theoret. Phys. 117(4), 655–680 (2007). https://doi.org/10.1143/PTP.117.655

    Article  ADS  Google Scholar 

  47. B. Foris, C.N. Papanicolas, Electron scattering and nuclear structure. Annu. Rev. Nucl. Part. Sci. 37(1), 133–176 (1987). https://doi.org/10.1146/annurev.ns.37.120187.001025

    Article  ADS  Google Scholar 

  48. L.S. Cardman, D.H. Dowell, R.L. Gulbranson, D.G. Ravenhall, R.L. Mercer, Coupled-channel analysis of high-energy electron scattering by 152Sm. Phys. Rev. C 18(3), 1388–1417 (1978). https://doi.org/10.1103/PhysRevC.18.1388

    Article  ADS  Google Scholar 

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Acknowledgements

The authors acknowledge the fruitful discussions with Dr. Q. Zhao, Dr. B. Zhou and Dr. Y. Funaki. They also thank to Dr. T. Neff and Dr. P. von Neumann-Cossel for the discussion and providing us the latest form factor data. We dedicate this work to Prof. Peter Schuck, whose pioneering study of \(\alpha \) particle condensation ignited our interest in nuclear cluster physics. One of the authors fondly remembers the discussion with him on the results presented here. It was just before the onset of COVID-19, marking our last conversation in person. Although he might not have fully endorsed this idea, it felt like a fitting topic to pay tribute to his lasting impact on our community.

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Authors and Affiliations

  1. RIKEN Nishina Center, Wako, Saitama, 351-0198, Japan

    Masaaki Kimura & Yasutaka Taniguchi

  2. Department of Computer Science, Fukuyama University, Fukuyama, Hiroshima, 729-0292, Japan

    Yasutaka Taniguchi

  3. Department of Information Engineering, National Institute of Technology (KOSEN), Kagawa College, Mitoyo, 769-1192, Kagawa, Japan

    Yasutaka Taniguchi

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  1. Masaaki Kimura
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Correspondence to Masaaki Kimura.

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Kimura, M., Taniguchi, Y. Shape of \(^{12}\textrm{C}\). Eur. Phys. J. A 60, 77 (2024). https://doi.org/10.1140/epja/s10050-024-01302-w

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