Clusters in light stable and exotic nuclei

TL;DR

This paper reviews the progress and challenges in understanding alpha-clustering in light nuclei, exploring exotic shapes, Bose-Einstein Condensates, and astrophysical implications of nuclear molecules.

Contribution

It provides a comprehensive overview of recent experimental and theoretical developments in nuclear clustering, including exotic shapes and astrophysical reaction rates.

Findings

Evidence of exotic shapes and Bose-Einstein Condensates in light nuclei

Analysis of superdeformed and hyperdeformed structures linked to quasimolecular resonances

Extraction of astrophysical reaction rates for 12C+12C at deep subbarrier energies

Abstract

Since the discovery of molecular resonances in $^{12}$C+$^{12}$C in the early sixties a great deal of research work has been undertaken to study alpha-clustering. Our knowledge on physics of nuclear molecules has increased considerably and nuclear clustering remains one of the most fruitful domains of nuclear physics, facing some of the greatest challenges and opportunities in the years ahead. Occurrence of "exotic" shapes and Bose-Einstein Condensates in light alpha-cluster nuclei are investigated. Various approaches of superdeformed/hyperdeformed shapes associated with quasimolecular resonant structures are discussed. The astrophysical reaction rate of 12C+12C is extracted from recent fusion measurements at deep subbarrier energies near the Gamov window. Evolution of clustering from stability to the drip-lines is examined.

Full text

1 Introduction

In the last decades, one of the greatest challenges in nuclear science is the understanding of the clustered structure of nuclei from both experimental and theoretical perspectives [1, 2, 3, 4, 5, 6, 7, 8]. Our knowledge on physics of nuclear molecules has increased considerably and nuclear clustering remains one of the most fruitful domains of nuclear physics. Fig. 1 summarizes the different types of clustering [4]: most of these structures were investigated in an experimental context by using either some new approaches [5] or developments of older methods. The search for resonant structures in the excitation functions for various combinations of light $\alpha$-cluster ($N$=$Z$) nuclei in the energy regime from the Coulomb barrier up to regions with excitation energies of $E_{x}$=20$-$50 MeV remains a subject of contemporary debate.

The question of how nuclear molecules may reflect continuous transitions from scattering states in the ion-ion potential to true cluster states in the compound systems is still unresolved. Clustering in light $\alpha$-like nuclei is observed as a general phenomenon at high excitation energy close to the $\alpha$-decay thresholds [4]. This exotic behavior has been perfectly illustrated 50 years ago by the famous ”Ikeda-diagram” for $N$=$Z$ nuclei [9], which has been modified and recently extended by von Oertzen [6] for neutron-rich nuclei, as shown in the left panel of Fig. 2. Despite the early inception of cluster studies, it is only recently that radioactive ion beams experiments, with great helps from advanced theoretical works, enabled new generation of studies, in which data with variable excess neutron numbers or decay thresholds are compared to predictions with least or no assumptions of cluster cores. Some of the predicted but elusive phenomena, such as molecular orbitals or linear chain structures, are now gradually coming to light.

2 12C nucleus ”Hoyle” state and BEC in light nuclei

The ground state of 8Be is the most simple and convincing example of $\alpha$-clustering in light nuclei as suggested by several theoretical models and appears naturally in ab initio calculations [4, 8]. The picture of the 8Be nucleus prediced by the No Core Shell model [8] as being a dumbbell-shaped configuration of two $\alpha$ particles closely resembles the superdeformed (SD) shapes known to arise in heavier nuclei in the actinide mass region. This dumbbell-like structure gives rise to a rotational band, from which the moment of inertia is found to be commensurate with an axial deformation of 2:1. According to the schematic picture of the ”Ikeda-Diagram” [9] the nuclear cluster structure of 12C may also induce axial deformation close to 3:1 of a hyperdeformed (HD) shape. The large deformations of light $\alpha$-conjugate nuclei with SD, HD and linear-chain configurations are under discussion.

The renewed interest in 12C was mainly focused to a better understanding of the nature of the so called ”Hoyle” state [10, 11], the excited 0+ state at 7.654 MeV that can be described in terms of a bosonic condensate, a cluster state and/or a $\alpha$-particle gas [12]. The resonant ”Hoyle” state [10] is regarded as the prototypical $\alpha$-cluster state whose existence is of great importance for the nucleosynthesis of 12C within stars. Further knowledge of the ”Hoyle” state [10, 11] and its rotational excitations would help not only to understand the debated structure of the 12C nucleus in the “Hoyle state”, but also to determine the high-temperature (T $\approx$ 1 GK) reaction rate of the triple $\alpha$ process more precisely. The structure of this state has been thoroughly investigated with theoretically modelled with both ab initio and cluster models [4, 8]. Much experimental progress has been achieved recently as far as the spectroscopy of 12C near and above the $\alpha$-decay threshold is concerned [13]. More particularly, the the second 2${}^{+}_{2}$ ”Hoyle” rotational excitation in 12C has been observed [14]. Another experiment [15] populates a new state compatible with an equilateral triangle configuration of three $\alpha$ particles. Still, the structure of the ”Hoyle” state remained controversial as experimental results of its direct decay into three $\alpha$ particles are found to be in disagreement until two experiments provided the most precise picture of how a 12C excited state decays into three He nuclei [13, 16, 17].

In the study of Bose-Einstein Condensation (BEC), the $\alpha$-particle states in light $N$=$Z$ nuclei [12], are of great interest. the search for an experimental signature of BEC in 16O is of highest priority. Furthermore, ab initio calculations [4, 8] predict that nucleons are arranged in a tetrahedral configuration of $\alpha$ clusters. A state with the structure of the ”Hoyle” state [10] in 12C coupled to one $\alpha$ particle is predicted in 16O at about 15.1 MeV (the 0${}^{+}_{6}$ state), the energy of which is $\approx$ 700 keV above the 4$\alpha$-particle breakup threshold. However, any state in 16O equivalent to the ”Hoyle” state [10] in 12C is most certainly going to decay exclusively by particle emission with very small $\gamma$-decay branches, thus, very efficient particle-$\gamma$ coincidence techniques [5] will have to be used in the near future to search for them.

3 Nuclear molecules, 12C+12C reaction rate and carbon burning

in massive stars

The real link between superdeformation/hyperdeformation (SD/HD), nuclear molecules and $\alpha$-clustering [4] is of particular interest, since nuclear shapes with major-to-minor axis ratios of 2:1–3:1 have the typical ellipsoidal elongation for light nuclei. A further area where electromagnetic transitions would be of great interest in support of cluster models is in the case of the quasi-molecular resonances observed in the 12C+12C reaction. The widths of these resonances were $\approx$ 100 keV, indicating the formation of a 24Mg intermediate system with a lifetime significantly longer than the nuclear crossing time. These resonances were subsequently interpreted as 12C+12C cluster states. There has been only one valient attempt to directly observe transitions in this reaction [4] focussing on transitions between 10*+* and 8*+* resonant states at a bombarding energy E(12C) = 32 MeV chosen to populate a known and isolated 10+ resonance. However, the measurement reported only an upper limit (for the radiative partial width of 1.2 $\pm$ 10*-5*) given the extreme challenges of eliminating all background.

The role of cluster configurations in stellar helium burning is well established and, discussion about the nature and the role of resonance structures that characterize the low-energy cross section of the 12C+12C fusion process is underway in recent experimental investigations [18, 19, 20, 21, 22, 23, 24, 25]. The resonant structures at very low energies have still been identified as molecular 12C+12C configurations in the 24Mg compound nucleus [4, 18]. However, the reaction rate is calculated on the basis of an average cross section integrating over the molecular resonance components. Indications of possible existence of a pronounced low-energy resonance at Ecm = 2.14 MeV [18, 25] that can only be explained by strong 12C cluster configurations of the corresponding state in 24Mg.

There have been also predictions based on phenomenological considerations of explosive stellar events, such as X-ray superbursts, type Ia supernovae, stellar evolution etc…, that suggest a strong 12C+12C cluster resonance around Ecm = 1.5 MeV in 24Mg that would drastically enhance the energy production and may provide a direct nuclear driver for the superburst phenomenon [28, 29]. However, no indication for such a state was reported. Much of the data collected to date [18, 31, 32] are shown in Fig. 3 taken from Ref. [25]. First direct 12C+12C measurement [18] seemed to indicate such a resonance. Recent measurements were performed at deep subbarrier energies using the newly developed Stella apparatus [24] associated with the UK FATIMA detectors [26] for the exploration of fusion cross sections of astrophysical interest [25]. Gamma-rays have been detected in an array of LaBr3 scintillators whereas proton and $\alpha$-particles were identified in double-sided silicon-strip detectors. A novel rotating target system has been developed in order to be capable to sustain high-intensity carbon beams delivered by the Andomède facility of the University Paris-Saclay and IPN Orsay, France [27]. The particle-$\gamma$ coincidence technique as well as nanosecond timing conditions have been used in the data analysis in order to minimize the background as much as possible. Our preliminary results [25] obtained with Stella [24] confirm the possible occurence of such a resonant structure in the $\alpha$ channel but not in the proton channel.

At higher energies the 12C+12C cross sections expressed for Stella [24] in terms of the modified astrophysical S-factor are typically in fair agreement with those measured at Argonne [21] with similar coincidence techniques [30]. The comparisons with previous data obtained by Becker et al. [31] (open triangles), E. Aguilera et al. [32] (open stars), and Spillane et al. [18] (open circles), respectively, show a perfect agreement each other. All sets of chosen data lie in between the Fowler model [33] (black dotted line) and the hindrance model [34] (red dashed line).

Our first conclusions might be summarized such as we confirm the possible fusion hindrance plus persisting resonances near the Gamow energy window. Furthermore, the preliminary Stella S-factors appear to be in qualitative agreement with either classical coupled-channel calculations of Esbensen [35, 36] or more recent theoretical investigations [37, 38, 39]. It is not clear that recent studies [22] using the Trojan Horse Method technique (THM) confirmed the cluster level at Ecm = 2.1 MeV but rather suggested the existence of the predicted state at Ecm = 1.5 MeV for the fusion reaction. Such a discovery of a low energy 12C+12C cluster state would indeed have significant impact on the reaction rate; but some doubts [40, 41] have been raised to our attention as far as the validity of the indirect THM is concerned [22, 42]. Obviously an experimental confirmation through direct fusion studies would be of utmost importance and several experiments are underway [43]. On the other hand, it is expected that if a strong resonance indeed exists around the Gamow energy, then the theoretical structure studies should be able to predict a 0+ excited state of 24Mg or an L = 0 12C+12C resonance at this particular energy, which plays about the same igniting role as that of the ”Hoyle” state [10] in the triple-$\alpha$ process of 12C formation [11]. A multichannel folding model [44] demonstrates the importance of inelastic channels involving the “Hoyle state” especially in the low-energy range relevant in astrophysics in the vicinity of the Gamow region.

4 Clustering in light neutron-rich nuclei

Clustering is a general phenomenon observed also in nuclei with extra neutrons as it is presented in the ”Extended Ikeda-diagram” [9] proposed by von Oertzen [6] (see the left panel of Fig. 2). With additional neutrons, specific molecular structures appear with binding effects based on covalent molecular neutron orbitals. In these diagrams $\alpha$-clusters and 16O-clusters (as shown by the middle panel of the diagram of Fig. 2) are the main ingredients. Actually, the 14C nucleus may play similar role in clusterization as the 16O one since it has similar properties as a cluster: i) it has closed neutron p-shells, ii) first excited states are well above E*∗* = 6 MeV, and iii) it has high binding energies for $\alpha$ particles.

The possibility of extending molecular structures from dimers (Be isotopes) to trimers [6] has been investigated in detail for C and O isotopes [45]. For C isotopes the neutrons would be exchanged between the three centers ($\alpha$ particles). It is possible that the three $\alpha$-particle configuration can align themselves in a linaer fashion, or alternative collapse into a triangle arrangment - in either case the neutrons being localised across the three centers. Possibly the best case for the linear arrangement is 16C.

A general picture of clustering and molecular configurations in light nuclei can also be drawn from the detailed investigation of the light O isotopes [45]. The bands of 20O [45] compared with the ones of 18O clearly establishes parity inversion doublets predicted by both the Generator-Coordinate-Method (GCM) and the Antisymmetrized Molecular Dynamics (AMD) [7] calculations for the 14C–6He cluster and 14C–2n–$\alpha$ molecular structures. The corresponding moments of inertia are suggesting large deformations for the cluster structures.

We may conclude that the reduction of the moments of inertia of the lowest bands of 20O is consistent with the assumption that the strongly bound 14C nucleus having equivalent properties to 16O, has a similar role as 16O in relevant, less neutron rich nuclei. Therefore, the ”Ikeda-Diagram [9] and the ”extended Ikeda-Diagram” consisting of 16O cluster cores with covalently bound neutrons must be further extended to include also the 14C cluster cores as illustrated in Fig. 2.

5 Summary and outlook

The link of $\alpha$-clustering, quasimolecular resonances and extreme deformations (SD, HD etc…) has been discussed. Several examples emphasize the general connection between molecular structure and deformation effects within ab initio models and/or cluster models [8]. We have also presented the BEC picture of light (and medium-light) $\alpha$-like nuclei that appears to be an alternate way of understanding most of properties of nuclear clusters [4]. New results regarding cluster and molecular states in neutron-rich oxygen isotopes in agreement with AMD predictions are summarized [45]. Consequently, the ”Extended Ikeda-diagram” has been further modified for light neutron-rich nuclei by inclusion of the 14C cluster, similarly to the 16O one. Marked progress has been made in many traditional and novels subjects of nuclear cluster physics and astrophysics (stellar He burning [24, 25, 28, 29]).

The developments in these subjects show the importance of clustering among the basic modes of motion of nuclear many-body systems. All these open questions will require precise coincidence measurements [5] coupled with state-of-the-art theory [4, 7, 8].

6 Dedication and acknowledgements

This written contribution is dedicated to the memory of my friends Alex Szanto de Toledo, Valery Zagrebaev, Walter Greiner and Paulo Gomes who unexpectelly passed away since early 2015. I am very pleased to first acknowledge Walter Greiner for his continuous support of the cluster physics [46, 47, 48, 49, 50]. I would like to thank Christian Caron (Springer) for initiating in 2008 the series of the three volumes of Lecture Notes in Physics entitled ”Clusters in Nuclei” and edited between 2010 and 2014 [1, 2, 3]. All the 37 authors of the 19 chapters of these volumes are warmly thanked for their fruitfull collaboration during the course of the project which is still in progress [5, 7, 45, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64]. Thanks also to Udo Schroeder for the edition of the volume ”Nuclear Particle Correlations and Cluster Physics” that inspired [4] so much several aspects presented at the EXON2018 Symposium in September 2018 as well as at the ISPUN2017 Symposium held in September 2017 in Halong Bay, Vietnam. Special thanks to all the members of the Stella collaboration [24], in particular Sandrine Courtin, Guillaume Fruet, Marcel Heine, Mohamad Moukaddam, Dominique Curien, et al. from the IPHC Strasbourg, Serge Della Negra (Andromède accelerator [27]) et al. from IPN Orsay, David Jenkins, Paddy Regan (UK FATIMA collaboration [26]) et al. from the UK and Christelle Stodel from GANIL. The Stella collaboration is supported by the french “Investissements d’avenir” program, the University of Strasbourg “IdEx Attractivity” program and the USIAS, Strasbourg, France. Finally, Dao Khoa, Le Hoang Chien, Alexis Diaz-Torres, Cheng-Lie Jiang, Akram Mukhamedzhanov and Xiadong Tang are acknowledged for their carefull reading of the manuscript.