Elsevier

Physics Reports

Volume 432, Issue 2, September 2006, Pages 43-113
Physics Reports

Nuclear clusters and nuclear molecules

https://doi.org/10.1016/j.physrep.2006.07.001Get rights and content

Abstract

Clustering has long been known to be influential in the structure of ground and excited states of N=Z nuclei. States close to the decay thresholds are of particular interest, as clustering becomes dominant. Recent studies of loosely bound light neutron-rich nuclei have focused attention on structures based on clusters and additional valence neutrons, which give rise to covalent molecular binding effects. These nuclear molecules appear only at the extremes of deformation, in the deformed shell model they are referred to as super- and hyper-deformed. The beryllium isotopes provide the first examples of such states in nuclear physics. Further nuclear molecules consisting of unequal cores and also with three centres can be considered. These arise in the isotopes of neon and carbon, respectively. Molecular states in intrinsically asymmetric configurations give rise to parity (inversion) doublets. Examples of recent experiments demonstrating the molecular structure of the rotational bands in beryllium isotopes are presented. Further experimental evidence for bands as parity doublets in nuclei with valence neutrons in molecular orbits is also analysed. Work on chain states (nuclear polymers) in the carbon isotopes is discussed. These are the first examples of hyper-deformed structures in nuclei with an axis ratio of 3:1. Future perspectives are outlined based on a threshold diagram for covalent nuclear molecules with clusters bound via neutrons in covalent molecular configurations.

Introduction

The subject of clustering transcends many areas of science, from clusters of galaxies to clusters of micro-organisms, and in each instance there is some evolutionary advantage. In nuclear physics, clusterisation enhances, in certain circumstances, the binding energy of the system. The concept has a history of more than 40 years when detailed studies started, but is actually one of the oldest models of the nucleus, since the α-cluster model was developed even before the discovery of the neutron [292], [293], [294], [295]. However, after this discovery, the single-particle description of nuclei based on the concept of a mean field for all nucleons became the prime focus.

The last 30 years have seen the discovery of complex cluster structures composed of alpha particles in what are called α-conjugate nuclei, that is nuclei with N=Z, which have an even, and equal, number of protons and neutrons [105]. In general, this precipitation of the nuclear liquid drop into more weakly interacting strongly bound droplets reveals highly symmetric structures, and the preponderance of α-clusters is due to their high stability. The binding energy per nucleon of the α-particle is significantly larger than in all other neighbouring light nuclei, and the first excited state of the 4He resides at 20.21 MeV. Together with a strong, and repulsive αα interaction, arising due to the Pauli exclusion principle [90], [251], α-cluster states are rather robust against the collapse into more compact shell model-like configurations. This realization led to a strong revival of the α-particle model in the 1960s [132], [262], [297], [298] with the use of the resonating group method [295], [298]. States in nuclei based on α-particles and other strongly bound sub-structures with N=Z (e.g. 16O) are typically not found in ground states, but are observed as excited states close to the decay thresholds into clusters, as was suggested in 1968 by Ikeda. The Ikeda diagram [144], [136] is shown in Fig. 1, this links the energy required to liberate the cluster constituents to the excitation energy at which the cluster structures prevail in the host nucleus. The clear prediction, which is borne out experimentally, is that cluster structures are mainly found close to cluster decay thresholds.

The formation of clusters is a fundamental aspect of nuclear many-body dynamics, which must exist simultaneously with the formation of a mean-field. Under the assumption of spherical symmetry this gives rise to the nuclear shell structure. Importantly, clustering gives rise to states in light nuclei which are not reproduced by the shell model. The nuclear shell model does, however, play an important role in the emergence of nuclear clusters, and also in the description of special deformed nuclear shapes, which are stabilised by the quantal effects of the many-body system, namely the deformed shell gaps (as opposed to the spherical shell gaps).

This connection is illustrated by the behaviour of the energy levels in the deformed harmonic oscillator [46], shown in Fig. 2. The numbers in the circles correspond to the number of nucleons, which can be placed into the crossing points of orbits. At zero deformation there is the familiar sequence of magic numbers which would be associated with spherical shell closures, and the associated degeneracies. At a deformation of the potential, where the ratios of the axes are 2:1, these same magic degeneracies reappear, but are repeated twice. This establishes an explicit link between deformed shell closures and clustering. As an example, at a deformation of 2:1 the occupancy by both protons and neutrons of the levels labelled by the degeneracies 2+2+6=10 would correspond to the deformed 20Ne nucleus. The degeneracies 2+6=8 are associated with the formation of 16O and thus an 16O+α cluster (8+2) structure would be predicted for 20Ne. This concept, fundamental for the understanding of the appearance of clustering within the nucleus, has been discussed before in detail [3], [54], [214], [215], [238] (see also Section 2.2). In particular the work of Rae was seminal in crystallising the discussion. To illustrate this point, we show in Table 1 the compilation made by Rae [238], following an examination of the properties of the deformed harmonic oscillator as shown in Fig. 2. We see that the deformed magic structures with special stability (and corresponding magic numbers) are expected for particular combinations of spherical (shell-model) clusters. For example, for super-deformed structures (2:1) the magic numbers have a decomposition into two magic numbers, of two spherical clusters, e.g. 20Ne(16O+α). Thus, one would expect clusterisation not only to appear at a particular excitation energy (the Ikeda picture), but also at a specific deformation. These structures give rise to not only rotational bands, but also to exotic vibrational modes, e.g. the butterfly mode described for the 24Mg+24Mg scattering resonances [302], [273] and also those in the 28Si+28Si system [219].

In the present review, it will become evident that additional valence neutrons do not destroy these structures, instead interesting nuclear structures described by molecular concepts will emerge.

An extension of the above discussion will show, that the intrinsically reflection symmetric states with hyper-deformation (3:1) are related to cluster structures consisting of three clusters. In going to larger deformations and placing particles in the orbits in which the oscillator quanta are only along the deformation direction, longer α-chain states are produced. For example, the linear 3α configuration corresponds to the filling of the three lowest levels at 3:1 labelled with degeneracy numbers 2 in Fig. 2 (see also Fig. 8).

Cluster structures with intrinsically reflection asymmetric shapes will consist of clusters of different size and magic numbers, for example in the 20Ne nucleus. They are then related to octupole shapes [3], [54], [134]. The octupole deformations give rise to the observation of rotational bands with parity inversion doublets [46], [54].

The interest in nuclear clustering has been pushed strongly due to the study of neutron-rich and of exotic weakly-bound nuclei. This is a field which has attracted worldwide attention, because the weakly-bound nuclear systems exhibit unique features related to the quantal properties of the many body systems, like halos [165] and clustering.

More explicitly, the strong clustering in weakly-bound systems can give rise to two-centre and multi-centre nuclear configurations, whose structures can be described by the concepts of molecular physics. Valence neutrons can exist in molecular orbitals, their role becomes analogous to that of electrons in covalent bonds in atomic molecules. In the nuclear case, these covalent neutrons stabilize the unstable multi-cluster states. The form of the covalent orbits for p-states is illustrated in Fig. 3. The figure shows the result of combining two orientations of the p-orbits, which in the atomic case would be found in the covalent binding of oxygen or carbon atoms. One linear combination, with the single particle orbits aligned perpendicular to the separation axis (a), gives rise to a π-type bonding orbit (b). The other alignment, illustrated in (d), gives rise to the σ-bonding orbit (e). The other arrangements, (c) and (f), give anti-bonding configurations.

These concepts can also be used to describe the exchange of valence neutrons between cluster cores on the nuclear scale. For example, 9Be may be considered to be composed of two α-particles and a valence neutron, forming, at larger α+α separations 5He nuclei, where the neutron resides in a p3/2-orbit(see Section 3.3). The linear combinations of two such orbits give rise to nuclear molecular π- and σ-bonds in 9Be [283]. It should be noted, however, that unlike in atomic systems, in nuclei no “ionic” molecular binding effect can occur (with valence neutrons of different binding energies at the asymptotic centres).

The structures based on such nuclear multi-centre configurations are difficult to obtain in the shell model approach (even with an extremely expanded basis). On the other hand, they are very well reproduced by a “model independent” approach, Antisymmetrized Molecular Dynamics (AMD, see in particular [172], [176], and Section 5.4). These calculations actually illustrate the origin of the molecular cluster structure: the nuclear forces are saturated in spin–isospin space in the α-clusters and in other N=Z nuclei, the remaining interactions are weak and give rise to unique quantal structures for the weakly bound nucleons.

A qualitative argument can be used to illustrate the relative strength of the molecular interaction between the cluster constituents in multi-cluster structures and of the strength of the mean field: only in a strongly deformed weakly bound system can the quantal (molecular) binding effects compete with the mean field aspect of nuclear forces, because the latter are saturated within the N=Z clusters.

A new threshold diagram is required, in order to describe the structure of non-alpha conjugate nuclei, i.e. those with valence neutrons, which reside in covalent orbits. Following the arguments summarised in Table 1, covalent molecular structures should be built mainly from α and 16O components. The states close to the thresholds for the decomposition into clusters and valence neutrons are expected to be bound by the covalent neutrons. This extended Ikeda-diagram appears in Fig. 4, it shows some of the combinations for which covalently bound shape-isomeric structures are expected. The relevant threshold energies for the decomposition into the constituents [282], [285] are given. Many of the structures that appear in this figure will be considered in the present review.

This review concentrates on recent work on the structure of excited states in light nuclei related to molecular structures consisting of clusters and valence nucleons. It is organised as follows. In Section 2 we give a historical introduction. In Section 3 the important question of the nucleus–nucleus potentials is addressed, the potentials have special properties for strongly bound clusters and the presence of valence neutrons gives important effects. We review the status and recent developments for N=Z cluster nuclei in Section 4. Section 5 starts with an examination of the different theoretical approaches for the N=Z nuclei. Then a brief description of the Bloch–Brink α-cluster model is given and of the models used for the study of the structures including valence particles, such as the generator coordinate method (GCM), the AMD-approach, and the molecular orbital (MO) theories. We review the experimental evidence for molecular structures in nuclei, first for beryllium isotopes in Section 6. This is extended to three centre systems in carbon isotopes in Section 7 and in Section 8. Section 9 presents perspectives for future research in weakly bound exotic nuclei.

Section snippets

Clusters and valence nucleons in light nuclei

In this Section some basic principles of molecular concepts in nuclear physics are shown.

Nucleus–nucleus potentials, relation to threshold diagrams

There is a strong relationship between clustering, molecular resonances and the properties of the nucleus–nucleus potentials. In the Hauser–Feshbach (HF) picture, with a binary channel consisting of two clusters, the formation or decay of resonances is governed by the real and imaginary parts of the optical potential. The imaginary part is responsible for the width or the life time of the cluster resonances. In part this was realized in the early stages of heavy-ion scattering by Feshbach [94],

Cluster states for N=Z, recent results

There has been an enormous effort devoted to exploring and understanding the appearance of clustering in N=Z systems, much of the experimental work probing quasi-molecular resonances via the collision of two clusters [49], [91], [120]. These studies suggest that two-body cluster structures exist in a broad variety of systems extending to 28Si+28Si [219]. These resonances were most prominently observed in the 12C+12C system [11], [49], [52], [53], [91] where their energies coincided with the

Overview of microscopic cluster models

The microscopic cluster model [132], [262], [297], [298] developed remarkably in the early 1960s with the realization of the resonating group method (RGM) [295]. The evolution of cluster physics in the two subsequent decades was strongly influenced by the RGM, but also new models such as the GCM [51], [121], [130] and the orthogonality condition method (OCM) [243], [244], [245], which were applied to light p-shell nuclei (see Refs. in [138]). These three methods are now the traditional

The structure of beryllium isotopes—complete spectroscopy

The first deformed structure in nuclear physics with an axis ratio of 2:1 is the unbound (by 92 keV) 8Be-nucleus, consisting of two α-particles in an L=0 resonant state. Furthermore, it has been known for more than two decades, that the ground state of the 9Be nucleus can be explained by covalent molecular binding, where two α-particles are bound by the p3/2 valence neutron (Fig. 28).

This was determined by molecular orbital models based on the Born–Oppenheimer approximation [98], [223], [248]

Chain states in nuclei: nuclear polymers

In this section the basic concepts behind the formation of chain states and other three-centre systems in nuclei will be outlined. A more detailed discussion and a comparison with recent data will be given in Section 8.

The first structures which can be built from many α-particles are the linear chain states, since such configurations minimize the Coulomb repulsion between the constituents. These are unique shapes in nuclear physics. We immediately realize when inspecting Fig. 8, that they are

Oblate and prolate states in carbon isotopes

In order to identify the strongly deformed states in carbon isotopes all spectroscopic information accumulated in the last decades can be used. The carbon isotopes provide excellent examples for testing the concept of complete spectroscopy, since for these nuclei a large variety of reactions have been studied. The first step is to identify the states with single-particle properties. By “removing” these states from the spectrum, multi-particle–multi-hole configurations remain as candidates for

Future perspectives

The evidence for real (not “quasi”) covalently bound nuclear cluster states is well established for dimers in the beryllium isotopes and partially for the carbon isotopes 13C–14C. We feel that the study of nuclear states built from clusters bound by valence neutrons in their molecular configurations is a field with a much larger scope. The most important cluster structures to consider are based on α-particles and 16O, as was summarised in Fig. 4. Experimentally, this research will rely on a

Acknowledgements

It is a pleasure to thank many colleagues, who were involved and helped to revive the work on molecular structure in nuclei. For contributions to this review and for checking many details, we thank, in particular, M. Milin and H.G. Bohlen, and referees for their detailed criticism and suggestions. M. Freer thanks the A.v. Humboldt foundation for their support.

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