Geometric symmetries in light nuclei

Geometric symmetries in light nuclei

Abstract

The algebraic cluster model is is applied to study cluster states in the nuclei C and O. The observed level sequences can be understood in terms of the underlying discrete symmetry that characterizes the geometrical configuration of the -particles, i.e. an equilateral triangle for C, and a regular tetrahedron for O. The structure of rotational bands provides a fingerprint of the underlying geometrical configuration of -particles.

1 Introduction

Ever since the early days of nuclear physics the structure of C has been extensively investigated both experimentally and theoretically [1, 2, 3, 4]. In recent years, the measurement of new rotational excitations of both the ground state [5, 6, 7] and the Hoyle state [8, 9, 10, 11] has generated a lot of renewed interest to understand the structure of C and that of cluster nuclei in general. Especially the (collective) nature of the Hoyle state at 7.65 MeV which is of crucial importance in stellar nucleosynthesis to explain the observed abundance of C, has presented a challenge to nuclear structure calculations, such as -cluster models [12], antisymmetrized molecular dynamics [13], fermionic molecular dynamics [14], BEC-like cluster model [15], (no-core) shell models [16, 17], ab initio calculations based on lattice effective field theory [18, 19], and the algebraic cluster model [7, 20, 21].

In this contribution, I discuss some properties of the -cluster nuclei C and O in the framework of the algebraic cluster model.

2 Algebraic Cluster Model

The Algebraic Cluster Model (ACM) describes the relative motion of the -body clusters in terms of a spectrum generating algebra of where represents the number of relative spatial degrees of freedom. For the two-body problem the ACM reduces to the vibron model [22], for three-body clusters to the model [20, 23] and for four-body clusters to the model [21, 24]. In the application to -cluster nuclei the Hamiltonian has to be invariant under the permuation group for the identical particles. Since one does not consider the excitations of the particles themselves, the allowed cluster states have to be symmetric under the permutation group.

ACM
Point group
Geometry Linear Triangle Tetrahedron
G.s. band
Table 1: Algebraic Cluster Model for two-, three- and four-body clusters

The potential energy surface corresponding to the invariant ACM Hamiltonian gives rise to several possible equilibrium shapes. In addition to the harmonic oscillator (or limit) and the deformed oscillator (or limit), there are other solutions which are of special interest for the applications to -cluster nuclei. These cases correspond to a geometrical configuration of particles located at the vertices of an equilateral triangle for C and of a regular tetrahedron for O. Even though they do not correspond to dynamical symmetries of the ACM Hamiltonian, one can still obtain approximate solutions for the rotation-vibration spectrum

The rotational structure of the ground-state band depends on the point group symmetry of the geometrical configuration of the particles and is summarized in Table 1.

The triangular configuration with three particles has point group symmetry [20]. Since , the transformation properties under are labeled by parity and the representations of which is isomorphic to the permutation group . The corresponding rotation-vibration spectrum is that of an oblate top: represents the vibrational quantum number for a symmetric stretching vibration, denotes a doubly degenerate vibration. The rotational band structure of C is shown in the left panel of Fig. 1.

Figure 1: (Color online) Rotational band structure of the ground-state band, the Hoyle band (or vibration) and the bending vibration (or vibration) in C (left) [7], and the ground-state band (closed circles), the vibration (closed squares), the vibration (open circles) and the vibration (open triangles) in O (right) [21].

The tetrahedral group is isomorphic to the permutation group . In this case, there are three fundamental vibrations: represents the vibrational quantum number for a symmetric stretching vibration, denotes a doubly degenerate vibration, and a three-fold degenerate vibration. The right panel of Fig. 1 shows the rotational band structure of O.

3 Electromagnetic transitions

C Th Exp Ref
8.4 [25, 26, 27]
44 [25, 26, 27]
73 [25, 26, 27]
0.4 [25, 26, 27]
O Th Exp Ref
215 [28]
425 [28]
9626 [28]
0.54 [28]
Table 2: values in C (top) and O (bottom).

For transitions along the ground state band the transition form factors are given in terms of a product of a spherical Bessel function and an exponential factor arising from a Gaussian distribution of the electric charges, [20]. The charge radius can be obtained from the slope of the elastic form factor in the origin . The transition form factors depend on the parameters and which can be determined from the first minimum in the elastic form factor and the charge radius.

The transition probabilities along the ground state band can be extracted from the form factors in the long wavelength limit

with

The good agreement for the values for the ground band in Table 2 shows that both in C and in O the positive and negative parity states merge into a single rotational band. Moreover, the large values of indicate a collectivity which is not predicted for simple shell model states. The large deviation for the between the first excited (Hoyle) state and the ground state indicates that the state cannot be interpreted as a simple vibrational excitation of a rigid triangular (C) or tetrahedral (O) configuration, but rather corresponds to a more floppy configuration with large rotation-vibration couplings. A more detailed study of the electromagnetic properties of -cluster nuclei in the ACM for non-rigid configurations is in progress.

4 Summary and conclusions

In this contribution, the cluster states in C and O were interpreted in the framework of the ACM as arising from the rotations and vibrations of a triangular and tetrahedral configuration of particles, respectively. In both cases, the ground state band consist of positive and negative parity states which coalesce to form a single rotational band. This interpretation is validated by the observance of strong values. The rotational sequences can be considered as the fingerprints of the underlying geometric configuration (or point-group symmetry) of particles.

For the Hoyle band in C there are several interpretations for the geometrical configuration of the three particles. In order to determine whether the geometrical configuration of the -particles for the Hoyle band is linear, bent or triangular, the measurement of a possible Hoyle state is crucial, since its presence would indicate a triangular configuration, just as for the ground state band.

Finally, the results presented here for C and O emphasize the occurrence of -cluster states in light nuclei with and point group symmetries, respectively.

\ack

This work was supported in part by research grant IN107314 from PAPIIT-DGAPA.

References

References

  1. Wheeler J A 1937 Phys. Rev. 52 1083
  2. Hafstad L R and Teller E 1938 Phys. Rev. 54 681
  3. Von Oertzen W, Freer M and Kanada-En’yo Y 2006 Phys. Rep. 432 43
  4. Freer M and Fynbo H O U 2014 Prog. Part. Nucl. Phys. 78 1
  5. Freer M et al. 2007 Phys. Rev. C 76 034320
  6. Kirsebom O S et al. 2010 Phys. Rev. C 81 064313
  7. Marín-Lámbarri D J, Bijker R, Freer M, Gai M, Kokalova T, Parker D J and Wheldon C 2014 Phys. Rev. Lett. 113 012502
  8. Itoh M et al. 2011 Phys. Rev. C 84 054308
  9. Freer M et al. 2012 Phys. Rev. C 83 034314
  10. Zimmerman W R et al. 2013 Phys. Rev. Lett. 110 152502
  11. Freer M et al. 2007 Phys. Rev. C 76 034320
  12. Robson D 1978 Nucl. Phys. A 308 381
  13. Kanada-En’yo Y 2007 Prog. Theor. Phys. 117 655
  14. Chernykh M, Feldmeier H, Neff H, Von Neumann-Cosel P and Richter A 2007Phys. Rev. Lett. 98 032501
  15. Funaki Y, Horiuchi H, Von Oertzen W, Ropke G, Schuck P, Tohsaki A and Yamada T 2009 Phys. Rev. C 80 64326
  16. Roth R, Langhammer J, Calci A, Binder S and Navrátil P Phys. Rev. Lett. 107 072501
  17. Dreyfuss A C, Launey K D, Dytrych T, Draayer J P and Bahri C 2013 Phys. Lett. B 727 511
  18. Epelbaum E, Krebs H, Lee D and Meissner U G 2011 Phys. Rev. Lett. 106 192501
  19. Epelbaum E, Krebs H, Lähde T, Lee D and Meissner U G 2012 Phys. Rev. Lett. 109 252501
  20. Bijker R and Iachello F 2000 Phys. Rev. C 61 067305
    Bijker R and Iachello F 2002 Ann. Phys. (N.Y.) 298 334
  21. Bijker R and Iachello F 2014 Phys. Rev. Lett. 112 152501
  22. Iachello F 1981 Chem. Phys. Lett. 78 581
  23. Bijker R, Iachello F and Leviatan A 1994 Ann. Phys. (N.Y.) 236 69
    Bijker R, Iachello F and Leviatan A 2000 Ann. Phys. (N.Y.) 284 89
  24. Bijker R 2010 AIP Conference Proceedings 1323 28
    Bijker R 2012 J. Phys.: Conf. Ser. 380 012003
  25. Ajzenberg-Selove F 1990 Nucl. Phys. A 506 1
  26. Reuter W, Fricke G, Merle K and Miska H 1982 Phys. Rev. C 26 806
  27. Strehl P and Schucan Th H 1968 Phys. Lett. B 27 641
  28. Tilley D R, Weller H R and Cheves C M 1993 Nucl. Phys. A 564 1
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
""
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
   
Add comment
Cancel
Loading ...
332419
This is a comment super asjknd jkasnjk adsnkj
Upvote
Downvote
""
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters
Submit
Cancel

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test
Test description