On the symmetries of the {}^{12}C nucleus

On the symmetries of the C nucleus

J. Cseh and R. Trencsényi
Institute for Nuclear Research, Hungarian Academy of Sciences, Debrecen, Pf. 51, Hungary-4001
July 20, 2019

The consequences of some symmetries of the three-alpha system are discussed. In particular, the recent description of the low-energy spectrum of the C nucleus in terms of the Algebraic Cluster Model (ACM) is compared to that of the Semimicroscopic Algebraic Cluster Model (SACM). The previous one applies interactions of a geometric symmetry evid (), while the latter one has a U(3) multichannel dynamical symmetry, that connects the shell and cluster pictures. The available data is in line with both descriptions.

alpha-clustering, geometrical, dynamical, and permutational symmetries
21.60.Fw, 21.60.Cs, 27.20.+h

The C nucleus plays a crucial role in the evolution of stars, it is an important test ground of different models of atomic nuclei, furthermore, it is a rich laboratory of symmetries. Therefore, it is in the focus of both the experimental and the theoretical research. New states or transitions detected in this nucleus can choose between competitive models.

The recent observation of the 5 state by the Birmingham group evid () was interpreted as an evidence for the symmetry in nuclear structure. The algebraic cluster model bipr (); biap () describes the detailed rotational-vibrational spectrum of the three-alpha system, and the application of its limit results in an energy spectrum that is in good agreement with the experimental observation evid ().

Alpha-cluster models have been applied to the C very extensively exte (), from the beginning of nuclear structure studies begi () until very recently rece (). It seems to be a general agreement of these cluster studies that the ground state of the C nucleus has a triangular shape at the vertices with the three alpha-particles. The novel feature of the algebraic cluster model interpretation evid (); bipr (); biap () is twofold. First: in this model not only the ground state, or the ground band is considered to have the symmetry, but the excitation spectrum, too, including the Hoyle-band, and several others. Second: the algebraic cluster model describes in detail the spectrum of the three-alpha system with the triangular symmetry, as opposed to e.g. the Bloch–Brink alpha-cluster model, which gives this shape for the ground state, but it does not provide us with detailed spectrum blbr ().

In Table 1 we show the symmetries of some alpha-cluster models applied to the C nucleus, including the present work. The interactions applied in the calculations are also listed. The microscopic cluster model (MCM) in the last line actually indicates a family of models (GCM, RGM, OCM…). From the viewpoint of the applied symmetries they are similar to each other. For their detailed discussion and comparison we refer to hoik ().

A new chapter of the structure studies of C is the application of those microscopic calculations, which describe each nucleonic degrees of freedom without supposing any cluster structure amd (); fumi (); luis (), and in some cases realistic nucleon-nucleon forces are applied, i.e. real ab initio calculations are carried out abin (). Obviously, for the understanding of the structure, these approaches are the most promising.

In this paper we investigate the three-alpha cluster system from a different angle. In particular, we compare the consequences of different symmetries of the system. The point symmetry, mentioned beforehand, is a geometrical one. In addition to this, the permutational symmetry of the three identical alpha-particles is also essential. It is involved both in the algebraic and in the other cluster models. A further basic symmetry is the antisymmetry of the 12 nucleons building up the C, resulting in the Pauli-exclusion principle. This is taken into account in the microscopic (and semimicroscopic) cluster models, but it is not involved in the phenomenologic ones, like the algebraic cluster model. Here we make a comparison between the performance of the algebraic cluster model with symmetry and that of a U(3) multichannel dynamical symmetry mus1 (); mus2 (). This latter one is the connecting symmetry of the cluster and shell (quartet) models, which is formulated in the semimicroscopic algebraic approach sacm (); saqm ().

Model Interaction Pp Spectrum
SACM U(3) MUSY + +
Bl-Br +
MCM + +
Table 1: The symmetries of some 3-alpha-models of the C nucleus. ACM: algebraic cluster model, SACM: semimicroscopic algebraic cluster model, Bl-Br: Bloch–Brink alpha-cluster model, MCM: microscopic cluster model. stands for the permutational symmetry of the three identical particles, is the geometrical symmetry of the equilateral triangle, while Pp means Pauli-principle. The last column indicates if detailed spectrum is provided by the model.

In what follows, first we introduce the semimicroscopic algebraic quartet (SAQM) and cluster (SACM) models and their connecting multichannel dynamical symmetry (MUSY). Then we present the U(3) MUSY spectrum in comparison with the experimental one and with the spectrum of the model. Finally, some conclusions are drawn.

The semimicroscopic algebraic quartet model (SAQM) saqm () is a symmetry-governed truncation of the no-core shell model nocore (), that describes the quartet excitations in a nucleus. A quartet is formed by two protons and two neutrons, which interact with each other very strongly, as a consequence of the short-range attractive forces between the nucleons inside a nucleus arima (). The interaction between the different quartets is weaker. In this approach the L-S coupling is applied, the model space has a spin-isospin sector characterized by Wigner’s U(4) group wigner (), and a space part described by Elliott’s U(3) group elliott (). Four nucleons form a quartet harvey () when their spin-isospin symmetry is {1,1,1,1}, and their permutational symmetry is {4}. This definition allows two protons and two neutrons to form a quartet even if they sit in different shells. As a consequence, the quartet model space incorporates 0, 1, 2, 3, 4, … major shell excitations (in the language of the shell model), contrary to the original interpretation of arima (), when the four nucleons had to occupy the same single-particle orbital, therefore, only 0, 4, 8, … major shell excitations could be described.

The model is fully algebraic, therefore, group theoretical methods can be applied in calculating the matrix elements. The operators contain parameters to fit to the experimental data, that is why the model is called semimicroscopic: phenomenologic operators are combined with microscopic model spaces. Due to the quartet symmetry, only a single {1,1,1,1} U(4) sector plays a role in the calculation of the physical quantities, thus the U(3) space-group and its subgroups are sufficient for characterizing the situation:


In Eq. (1) we have indicated also the representation labels of the groups, which serve as quantum numbers of the basis states. Here is the number of the oscillator quanta, and . The angular momentum content of a representation is as follows elliott (): , with the exception of , for which . In the limiting case of the dynamical symmetry, when the Hamiltonian is expressed in terms of the invariant operators of this group-chain, an analytical solution is available for the energy-eigenvalue problem (an example is shown below).

The SAQM can be considered as an effective model in the sense of effective (): the bands of different quadrupole shapes are described by their lowest-grade U(3) irreducible representations (irreps) without taking into account the giant-resonance excitations, built upon them, and the model parameters are renormalized for the subspace of the lowest U(3) irreps.

The semimicroscopic algebraic cluster model (SACM) sacm (), just like the other cluster models, classifies the relevant degrees of freedom of the nucleus into two categories: they belong either to the internal structure of the clusters, or to their relative motion. In other words, the description is based on a molecule-like picture. The internal structure of the clusters is handled in terms of Elliott’s shell model elliott () with U(4)U(3) group structure (as discussed beforehand). The relative motion is taken care of by algebraic models with a U(3) basis. In particular, for a binary configuration it is the (modified) vibron model of U(4) dynamical algebra vibron (), which is a group-theoretical model of the dipole motion. (The modification means a truncation of the basis due to the Pauli-principle sacm ().) For a ternary configuration the two independent Jacobi-coordinates are described by the U(7) model bipr (); biap (); u7ha (); u7di (). For a three-cluster configuration this model has a group-structure of U(4)U(3) U(4)U(3) U(4)U(3) U(7). The exclusion of the Pauli-forbidden states requires the truncation of the basis of the U(7) model, that determines the lowest allowed major shell, and in addition one needs to distinguish between the Pauli-allowed and forbidden states within a major shell, too. Different methods can be applied to this purpose; e.g. one can make an intersection with the U(3) shell model basis of the nucleus, which is constructed to be free from the forbidden states.

The SACM is fully algebraic, and semimicroscopic in the sense discussed above.

When we are interested only in spin-isospin zero states of the nucleus (a typical problem in cluster studies, and being our case here, too), then only the space symmetries are relevant (apart from the construction of the model space). For a ternary cluster configuration the corresponding group-chain is


The basis defined by this chain is especially useful for treating the exclusion principle, since the U(3) generators commute with those of the permutation group, therefore, all the basis states of an irrep are either Pauli-allowed, or forbidden horisup (). By applying basis (2) we can pick up the allowed cluster states from the U(3) shell model basis (1).

A Hamiltonian corresponding to the dynamical symmetry of group-chain (2) reads as


We note here that the first part


is an operator that corresponds to the pure cluster picture, while the second part


is a shell model Hamiltonian (of the united nucleus).

The multichannel dynamical symmetry (MUSY) mus1 (); mus2 () connects different cluster configurations (including the shell model limit) in a nucleus. Here the word channel refers to the reaction channel that defines the cluster configuration.

The simplest case is a two-channel symmetry connecting two different clusterizations. It holds when both cluster configurations can be described by a U(3) dynamical symmetry, and in addition a further symmetry connects them to each other. This latter symmetry acts in the pseudo space of the particle indices hori (); mus2 (). The Hamiltonian of Eq. (5) is symmetric with respect to these transformations, therefore, it is invariant under the changes from one clusterization to the other. The cluster part of the Hamiltonian, is affected by the transformation from one configuration to the other, of course. Nevertheless, it may remain invariant, which is the case for simple operators, like the harmonic oscillator Hamiltonian, or the quadrupole operator mus2 (). Due to this symmetry of the quadrupole operator, the transitions of different clusterizations also coincide, when the MUSY holds, just like the energy eigenvalues of the symmetric Hamiltonians mus2 ().

The MUSY is a composite symmetry of a composite system. Its logical structure is somewhat similar to that of the dynamical supersymmetry (SUSY) of nuclear spectroscopy susy (). In the SUSY case the system has two components, a bosonic and a fermionic one, each of them showing a dynamical symmetry, and a further symmetry connects them to each other. The connecting symmetry is that of the supertransformations which change bosons into fermions, or vice versa. In the MUSY case the system has two (or more) different clusterizations, each of them having dynamical symmetries which are connected to each other by the symmetry of the pseudo space of the particle indices that change from one configuration to the other.

When the multichannel dynamical symmetry holds then the spectra of different clusterizations are related to each other by very strong constraints. The MUSY provides us with a unified multiplet structure of different cluster configurations, furthermore the corresponding energies and transitions coincide exactly. Of course, it can not be decided a priori whether the MUSY holds or not, rather one can suppose the symmetry and compare its consequences with the experimental data.

Figure 1: Comparison between the experimental and model spectra of the C nucleus. The energy scale is given in MeV, and the bands of the experimental part are labelled by the values. The (as well as the experimental spectrum) is taken from evid (), and the model bands are labelled by the rotational-vibrational quantum numbers of the model. In the U(3) MUSY spectrum the bands have the quantum numbers.

The energy spectrum of Figure 1 was calculated from the formula


In the first term is the number of oscillator quanta. The second term is the expectation value of the second-order invariant operator of the SU(3) algebra, which represents quadrupole-quadrupole interaction. The third one is the eigenvalue of the third-order invariant distinguishing between the prolate and oblate shapes. The -dependent term splits the bands belonging to the same SU(3) representation. (The corresponding operator is determined by the operators of the integrity basis of the SU(3) algebra, and is very nearly diagonal in the SU(3) basis states kdep ().) In the last part is the moment of inertia calculated classically for the rigid shape determined by the U(3) quantum numbers (for a rotor with axial symmetry) saqm (). The parameters were fitted to the experimental data: MeV, MeV, MeV, MeV, MeV.

As it is discussed above, the U(3) MUSY connects the cluster and quartet (i.e. shell) descriptions. Therefore, in determining its parameters some shell-model constraints (e.g. systematics) can be, and in some cases has been applied csri (). In the present study, however, we determined the parameters from the experimental spectrum, like in the work evid (), in order to treat the two descriptions (based on the and U(3) MUSY) on an equal footing. For comparison we show in the lower part of Figure 1 the result of the ACM, too from evid (). The number of parameters is comparable in the two cases: 5 in our Eq. (6), and 6 in Eq. (2) of evid ().

In comparing our semimicroscopic description to that of the no-core symplectic shell model luis (), it is worth noting that the lowest-grade SU(3) symmetries we associate with the ground-state band and to the Hoyle-band are the dominating ones in the fully microscopic description, too. In particular, the (0,4) basis has the largest contribution to the ground state, and (12,0) symmetry is the head of the symplectic band, dominating the Hoyle-states.

To sum up: In Figure 1 we compared the model spectra of two algebraic descriptions to the experimental spectrum of the C nucleus. The result illustrates both the usefulness of the U(7) dynamical algebra in the treatment of the three-cluster problem, and the fact that it incorporates different models. In particular, the algebraic cluster model with interaction and the semimicroscopic algebraic cluster model with the U(3) multichannel dynamical symmetry give very similar descriptions. Therefore, further experimental details (on the “missing states” and on the transitions) seem to be essential in order to decide which symmetry is realized to a better approximation. (At the same time they can deepen our understanding in terms of the fully microscopic theories.) From the viewpoint of the symmetry studies the combination of the two algebraic methods can also be very informative: the operators with the symmetry of the ACM could be applied on the model space of the SACM, which incorporates the Pauli-principle. This algebraic treatment would include all the symmetries applied only partly so far by different models (Table 1).

This work was supported by the Hungarian Scientific Research Fund - OTKA (Grant No. K106035). Helpful discussions on this topic with M. Freer, F. Iachello, K. Katō, G. Lévai, Tz. Kokalova, and A. Merchant, as well as the technical help by G. Riczu are kindly acknowledged. The encouragement of the Birmingham group for preparing this publication is also highly appreciated.


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