Discrete symmetries in the cluster shell model

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part of Springer Nature, 2020 https://doi.org/10.1140/epjst/e2020-000006-0

THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS

Regular Article

Discrete symmetries in the cluster shell model A.H. Santana Vald´es and R. Bijkera Instituto de Ciencias Nucleares, Universidad Nacional Aut´ onoma de M´exico, A.P. 70-543, 04510 M´exico D.F., Mexico Received 18 January 2020 / Accepted 28 August 2020 Published online 23 October 2020 Abstract. The role of discrete (or point-group) symmetries is discussed in the framework of the cluster shell model which describes the splitting of single-particle levels in the deformed field of cluster potentials. We discuss the classification of the eigenstates for the cases of a triangular and tetrahedral configuration of α-particles in terms of the irreducible 0 representations of the double point groups D3h and Td0 , respectively, and show how the discrete symmetry of a given eigenstate can be determined. Finally, we derive the Coriolis coupling for each one of these geometrical configurations.

1 Introduction Discrete symmetries have been used in nuclear physics in the context of collective models to characterize the intrinsic shape of the nucleus, such as axial symmetry for quadrupole deformations [1], and tetrahedral [2,3] and octahedral [3,4] symmetries for deformations of higher multipoles. A different application of the concept of discrete symmetries is found in the context of α-particle clustering in light nuclei to describe the geometric configuration of the α-particles. Early work on α-cluster models goes back to the 1930s with studies by Wheeler [5], and Hafstad and Teller [6], followed by later work by Brink [7,8] and Robson [9]. The measurements in recent years of new rotational excitations of the ground state of 12 C with LP = 4− and 5− [10–12] and of the Hoyle state with LP = 2+ and 4+ [13–16] have generated a large renewed interest in the structure of 12 C, and of α-cluster nuclei in general [17–19]. The experimental verification of the existence of discrete symmetries in nuclei consists in the study of the structure of rotational bands as fingerprints of the underlying discrete symmetry involving both the angular momentum and parity content of rotational bands and electromagnetic transitions and moments [20–24]. An analysis of the available experimental data of kα nuclei has provided evidence for the existence of triangular D3h symmetry in 12 C and tetrahedral Td symmetry in 16 O [12,20–23,25]. An interesting question is to what extent these geometric configurations are manifested in the neighboring odd-mass nuclei, and what are their characteristic signatures. Hereto the Cluster Shell Model (CSM1 ) has been developed which describes the splitting of a

e-mail: [email protected] 1 Not

to be confused with the Cranked Shell Model.

2354

The European Physical Journal Special Topics

single-particle levels in deformed cluster potentials [26] with applications to 9 Be and 9 B [27], and 13 C [28]. The aim of this contribution is to discuss the classification of the eigenstate

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