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Symmetries and order in cluster nuclei

It is shown that the rotational band structure of the cluster states in 12C and 16O can be understood in terms of the underlying discrete symmetry that characterizes the geometrical configuration of the alpha-particles, i.e. an equilateral triangle for 12C, and a regular tetrahedron for 16O. The structure of rotational bands provides a fingerprint of the underlying geometrical configuration of alpha-particles. Finally, some first results are presented for odd-cluster nuclei.


INTRODUCTION
The concept of symmetries has played an important role in nuclear structure physics, both continuous and discrete symmetries. Examples of continuous symmetries are isospin symmetry [1], Wigners combined spin-isospin symmetry [2], the (generalized) seniority scheme [3,4], the Elliott model [5] and the interacting boson model [6]. Discrete symmetries have been used in the context of collective models to characterize the intrinsic shape of the nucleus, such as axial symmetry for quadrupole deformations [7], octupole [8], tetrahedral [9,10] and octahedral [10,11] symmetries for deformations of higher multipoles.
A different application is found in 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 1930's with studies by Wheeler [12], and Hafstad and Teller [13], followed by later work by Brink [14,15] and Robson [16]. Recently, there has been a lot of renewed interest in the structure of α-cluster nuclei, especially for the nucleus 12 C [17]. The measurement of new rotational excitations of the ground state [18,19,20] and the Hoyle state [21,22,23,24] has stimulated a large theoretical effort to understand the structure of 12 C ranging from studies based on the semi-microscopic algebraic cluster model [25], antisymmetrized molecular dynamics [26], fermionic molecular dynamics [27], BEC-like cluster model [28], (no-core) shell models [29,30], ab initio calculations based on lattice effective field theory [31,32], and the algebraic cluster model [20,33,34].
In this contribution, I discuss some properties of the α-cluster nuclei 12 C and 16 O in the algebraic cluster model, and present some first results for odd-cluster nuclei in the framework of the cluster shell model.

ALGEBRAIC CLUSTER MODEL
The Algebraic Cluster Model (ACM) is an interacting boson model (IBM) to describe the relative motion of n-body clusters in terms of a spectrum generating algebra (SGA) of U(ν + 1) where ν = 3(n − 1) represents the number of relative spatial degrees of freedom. In the IBM, ν denotes the five quadrupole degrees of freedom, leading to a SGA of U (6). For the two-body problem the ACM reduces to the U(4) vibron model [35], for three-body clusters to the U(7) model [33,36] and for four-body clusters to the U(10) model [34,37]. The ACM has a very rich symmetry structure. In addition to continuous symmetries like the angular momentum, in case of α-cluster nuclei the Hamiltonian has to be invariant under the permuation of the n identical α particles. Since one does not consider the excitations of the α particles themselves, the allowed cluster states have to be symmetric under the permutation group S n .
The potential energy surface corresponding to the S n invariant ACM Hamiltonian gives rise to several possible equilibrium shapes. In addition to the harmonic oscillator (or U(3n−3) limit) and the deformed oscillator (or S O(3n−2) limit), there are other solutions which are of special interest for the applications to α-cluster nuclei. These cases correspond to a geometrical configuration of α particles located at the vertices of an equilateral triangle for 12 C and of a regular tetrahedron for 16 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 α particles and is summarized in Table 1.
The triangular configuration with three α particles has point group symmetry D 3h [33]. Since D 3h ∼ D 3 × P, the transformation properties under D 3h are labeled by parity P and the representations of D 3 which is isomorphic to the permutation group S 3 . The corresponding rotation-vibration spectrum is that of an oblate top: v 1 represents the vibrational quantum number for a symmetric stretching A vibration, v 2 denotes a doubly degenerate E vibration. The rotational band structure of 12   The tetrahedral group T d is isomorphic to the permutation group S 4 . In this case, there are three fundamental vibrations: v 1 represents the vibrational quantum number for a symmetric stretching A vibration, v 2 denotes a doubly degenerate E vibration, and v 3 a three-fold degenerate F vibration. The right panel of Fig. 1 shows the rotational band structure of 16

ELECTROMAGNETIC TRANSITIONS
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 [33] The transition form factors depend on the parameters α and β. The value of β is determined from the first minimum in the elastic form factor giving β = 1.74 (fm) [33] for 12 C and β = 2.07 (fm) [34] for 16 O. The value of α is determined by the size of the α-particle to be α = 0.56 (fm) −2 [38]. The transition probabilities B(EL) along the ground state band can be extracted from the form factors in the long wavelength limit This means that the transition probabilities B(EL) for different multipolarities L are related by the product of a numerical factor and a power of β, for 12 C, and for 16 O. The good agreement for the B(EL) values for the ground band in Table 2 shows that both in 12 C and in 16 O the positive and negative parity states indeed merge into a single rotational band. The large values of B(EL; L P 1 → 0 + 1 ) indicate a collectivity which is not predicted for simple shell model states. The relation between octupole and quadrupole transitions in 12 C, and that between octupole and hexadecupole transitions in 16 O are beautifully explained by Eqs. (2) and (3). The charge radii can be obtained from the slope of the elastic form factors in the origin, and are found to be in good agreement with the experimental value.
The quadrupole moment for 12 C can be calculated as i.e. a positive value, as is to be expected for a oblate deformation, and is found to be in good agreement with a recent measurement [42].

CLUSTER SHELL MODEL
The next interesting question is how is the cluster configuration of α-particles reflected in the neighboring odd-mass nuclei? What are the signatures of the underlying geometric symmetry in odd-cluster nuclei? Hereto the Cluster Shell Model (CSM) was developed [44]. In this model nuclei with Z = N = 2n are treated as a cluster of n α-particles whose matter and charge density are given by a gaussian form Here r i = (β, θ i , φ i ) where β denotes the relative distance of the α particles to the center of mass, and θ i and φ i are the angles. An important factor is the cluster factor, n i=1 Y * λµ (θ i , φ i ), which contains the information about the geometrical configuration of the alpha particles. The potential is obtained by convoluting the density with a Volkov potential [45] to obtain The spin-orbit interaction is taken as Finally, for protons one has to include the Coulomb potential which is obtained by convoluting the carge density with the Green's function as We then obtain the single-particle energy levels and intrinsic states as a function of β for each configuration (n = 2, 3, 4) by solving the single-particle Schrödinger equation In the case of neutrons V C r = 0. FIGURE 2. Single-particle energies in a cluster potential with D ′ 3h triangular symmetry (left) and with T ′ d tetrahedral symmetry (right). In the left panel, the single-particle levels are labeled by E 1/2 (black), E 5/2 (red) and E 3/2 (blue), and in the right panel by E 1/2 (black), E 5/2 (red) and G 3/2 (blue). For β = 0 the ordering of the single-particle orbits is 1s 1/2 , 1p 3/2 , 1p 1/2 and (almost degenerate) 1d 5/2 , 2s 1/2 . Fig. 2 shows how the single-particle orbits are split as a function of the deformation parameter β, the relative distance of the α particles with respect to the center of mass. The single-particle levels can be classified according to the irreducible representations (irreps) of the corresponding double point group, E 1/2 , E 5/2 and E 3/2 for the case of triangular symmetry (point group D ′ 3h ) and E 1/2 , E 5/2 and G 3/2 for tetrahedral symmetry (point group T ′ d ), in the notation of Herzberg [46]. The E states are double degenerate and the G states fourfold degenerate. In neither case, angular momentum and parity are good quantum numbers (with the exception for β = 0).
The construction of representations of the double groups D ′ 3h and T ′ d was carried out for applications to molecular physics by Herzberg [46] and to crystal physics by Koster et al. [47]. Here I discuss an application in nuclear physics [48]. Table 3 shows the decomposition of single-particle orbits into irreps double point groups D ′ 3h and T ′ d . For example, both the p 3/2 and d 3/2 orbits transform as G 3/2 under T ′ d , whereas under D ′ 3h the p 3/2 orbit splits into E 5/2 and E 3/2 , and the d 3/2 into E 1/2 and E 3/2 . Fig. 3 shows the rotational bands for the case of a single-particle coupled to a triangular configuration of α particles. The left panel shows the result for the ground-state band of even-cluster nuclei, as has been observed in 12 C. The ground-state band consists of a series of K-bands with K = 3k (k = 0, 1, 2, . . . ,) and angular momenta L = 0, 2, 4, . . . , for K = 0 and L = K, K + 1, K + 2, . . . , for K 0. The parity is given by P = (−1) K = (−1) k .

SUMMARY AND CONCLUSIONS
In this contribution, I reviewed a study of the cluster states in 12 C and 16 O in the framework of the ACM in which they were interpreted as arising from the rotations and vibrations of a triangular and tetrahedral configuration of α particles, respectively. An analysis of both the rotation-vibration spectra and electromagnetic transition rates shows strong evidence for the occurrence of D 3h and T d symmetry in in 12 C and 16 O, respectively. In both cases, the ground state band consist of positive and negative parity states which merge to form a single rotational band. This interpretation is validated by the observance of strong B(EL) values. The rotational sequences can be considered as the fingerprints of the underlying geometric configuration (or point-group symmetry) of α particles.
In the second part, I discussed the Cluster Shell Model which is very similar in spirit to the Nilsson model, in which the single-particle levels are split in the deformed field of the core nucleus. In the CSM, the levels are split in the deformed field generated by the cluster potential with triangular D 3h or tetrahedral T d symmetry. As a first result, I presented the quantum numbers of the deformed single-particle levels and, for the case of 12 C plus a nucleon, the angular momentum content of the rotational bands. The study of odd-cluster nuclei is currently in progress. The first results for 9 Be and 9 B have been published [49], and the systems with three-and four-α particles plus a nucleon (particle or hole) are underway.

ACKNOWLEDGMENTS
This contribution was presented at a the Symposium 'Symmetries and Order', in honor of Franco Iachello on the occasion of his retirement from Yale University. The title of the Symposium is most fitting to describe Franco's legacy. His work is characterized by an uncanny ability to observe patterns, regularity, order and symmetries where most other people would simply see data and numbers. His work has inspired many generations of physicists all around the world and continues to do so. In appreciation of his support, guidance and inspiration during my entire career I wish Franco a happy and productive retirement! This work is the result of a collaboration between UNAM and Yale University, and has benefitted greatly from discussions with Franco, as well as with Valeria Della Rocca, Omar Alejandro Díaz Caballero and Adrian Horacio Santana Valdés. This work was supported in part by research grant IN109017 from PAPIIT-DGAPA.