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], BEClike cluster model [15], (nocore) 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 twobody problem the ACM reduces to the vibron model [22], for threebody clusters to the model [20, 23] and for fourbody 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  
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 rotationvibration spectrum
The rotational structure of the groundstate 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 rotationvibration 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.
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 threefold 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] 
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 rotationvibration couplings. A more detailed study of the electromagnetic properties of cluster nuclei in the ACM for nonrigid 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 pointgroup 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.
This work was supported in part by research grant IN107314 from PAPIITDGAPA.
References
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