# Nucleon Clustering in Light Nuclei

# Nucleon Clustering in Light Nuclei

- Martin FreerMartin FreerSchool of Physics and Astronomy, University of Birmingham

### Summary

The ability to model the nature of the strong interaction at the nuclear scale using ab initio approaches and the development of high-performance computing is allowing a greater understanding of the details of the structure of light nuclei. The nature of the nucleon–nucleon interaction is such that it promotes the creation of clusters, mainly α-particles, inside the nuclear medium. The emergence of these clusters and understanding the resultant structures they create has been a long-standing area of study. At low excitation energies, close to the ground state, there is a strong connection between symmetries associated with mean-field, single-particle behavior and the geometric arrangement of the clusters, while at higher excitation energies, when the cluster decay threshold is reached, there is a transition to a more gas-like cluster behavior. State-of-the-art calculations now guide the thinking in these two regimes, but there are some key underpinning principles that they reflect. Building from the simple ideas to the state of the art, a thread is created by which the more complex calculations have a foundation, developing a description of the evolution of clustering from α-particle to ^{16}O clusters.

### Keywords

### Subjects

- Nuclear Physics

### 1. Introduction

The nature of the four fundamental forces is a basic ingredient to understanding nature across length scales from the subatomic to the galactic. All interactions drive the formation of structure in the universe either seeded by an element of randomness in the distribution of objects on which the force acts or through the details of the force itself. In the latter instance, a detailed understanding of the nature of the interaction is required. The pervasive gravitational interaction, which even at the human scale plays a key role in balancing out the electromagnetic force in determining the three-dimensional forms of life on Earth (Rees, 2000), is not well understood in terms of the force carrier or the form at short-length scales where gravity meets quantum. In this regime, the strong interaction is represented at the sub-nucleonic scale through the exchange of gluons between constituent quarks as captured in the highly successful standard model. However, the detailed nature of the strong force is complex in that the force carrier, the gluon, can interact with other gluons, and indeed, 99% of the mass of a nucleon is ascribed to the strong interaction rather than the Higgs mechanism—the bare mass of the up and down quarks is a few mega electron volts (MeV) and the mass of a nucleon is a giga electron volts (GeV).

At the nuclear scale, the relevant degrees of freedom cease to be quarks and gluons but emerge as nucleons and mesons, with the longer range components associated with pions and shorter length scales with heavier mesons. The variety of exchange particles and the fact that the exchange can be one, two, three, or more mesons result in a field-theory approach and many-body components to the nuclear force. Chiral effective field methods have had a key role to play in the evolution of nuclear theory as it has progressed from mean-field to effective forces and, in the early 21st century, to ab initio–motivated approaches (Freer et al., 2018). This complexity implies that both the calculations themselves are computationally challenging and the nuclear force is hard to precisely characterize. Nevertheless, the significant advances that are taking place in this field shed new light on the emergence of nuclear structure and nuclear correlations and in particular nuclear clustering. The nucleus is a complex many-body problem that has strong analogies to condensed-matter physics, which is rich in phenomenology.

The success of the mean-field approach, which has dominated nuclear physics for nearly a century since its first inception in 1932, indicates that even in systems of small numbers of nucleons, the intricacy of the nuclear force average away. Indeed, the nuclear shell model can successfully describe a whole series of nuclear properties, from energies to spins and from electric and magnetic moments to transition rates between quantum states (Brown & Wildenthal, 1998). However, there are some well-documented examples, for example, the 7.65 MeV, ${0}_{2}^{+}$, Hoyle state in ^{12}C, where the mean-field method does not readily reproduce the experimental spectrum. This signals that there are instances where the details of the nuclear force and the bound or unbound nature of the nuclear environment matter. It is clear that the correlations that arise from the details of the strong interaction are important, as illustrated by the very high binding energies of light nuclei that have equal, and even, numbers of protons and neutrons (Figure 1). This enhancement illustrates that the force is maximal when two protons, or two neutrons, reside in the same orbit with their spins anti-aligned. This is similar to the well-known nuclear pairing interaction, but the effect is enhanced if in addition the neutrons and protons are in identical orbits giving rise to enhanced proton–neutron interactions.

These correlations are both spatial and momentum and result in the α-particle having 2.5 times the degree of binding per nucleon than the proton–neutron pair experience in the deuteron, where the neutron and proton spins are aligned. This enhancement in the binding of the four-nucleon system with spins coupled to zero and total angular-momentum zero is also evident in the excited states of the ^{4}He nucleus, in which the configuration of one, or two, nucleons are disturbed with respect to the ground state. The first two excited states are 20.21 MeV 0^{+} and 21.01 MeV 0^{−} and only 7–8 MeV from the decay threshold for the system to disaggregate into its constituent parts, signaling that the four-nucleon correlation energy is of the order of 20 MeV. In heavier systems and in the semi-empirical mass formula (von Weizsäcker, 1935), this correlation energy is captured in the symmetry-energy term, but the question then arises as to the extent to which the α-particles themselves are also present.

*Source*: Freer (2007).

This observation was the foundation for the remarkable work of Hafstad and Teller (1938), which established that the α-particle had a dominant role to play in light nuclei and that ground and excited states could be represented by α-particle structures and rotations associated with the dynamical symmetries. In their description, nuclei were not exclusively constructed from α-particles but from the excess particles (or equivalently holes) being exchanged between the α-particle *clusters*. These states were the nuclear equivalent of atomic molecules, where instead of electrons protons, neutrons, or even their corresponding holes are the covalent objects. In the early 21st-century, these ideas and principles provide some of the foundations for understanding nuclear structure.

Within the modern description of light nuclei, a duality exists between the mean field and the appearance of clustering, and state-of-the-art models are allowing new insights into the emergence of cluster correlations. Questions that are wrestled with in the early 21st-century discussion are how is it the nuclear shell model can be so successful when averaging away much of the detail of the nucleon–nucleon interaction and assuming sphericity for the nuclear potential? Yet, why does it fail for certain specific nuclei, where do the α-particles enter into the description, and with the best calculations available, what is the best characterization of the nuclear force?

The discussion starts with constructing some simple building blocks, mainly using a harmonic oscillator (HO) and a deformed harmonic oscillator (DHO) to develop some basic principles. These simple ideas translate into more complex calculations and attempts to set out a basis for understanding the emergence of clustering in nuclei both below and above the cluster decay threshold.

### 2. Building Blocks

#### 2.1 Cluster Symmetries in the Mean Field

The starting point for assembling an understanding of the essential ingredients underpinning the structure of light nuclei will, perhaps surprisingly, be the mean field. However, given the success of this approach in capturing the vast majority of nuclear properties, it must somehow contain the essential physics.

In order to simplify the discussion, an approximation to the nuclear shell model will be used; the HO. Moreover, to recognize that in reality, nuclei are not spherical but deformed, this will be extended to the DHO. Before progressing, it is important to reflect on the differences between the energy-level schemes of the nuclear shell model and the DHO. Broadly, the schemes are similar, but the spin–orbit interaction present in the nuclear shell model lifts the degeneracy. For example, the *N = 1* HO level is split into the 1*p*_{3/2} and 1*p*_{1/2} levels. Here, the spin of the individual nucleon (*s* = 1/2), is either aligned or anti-aligned with the orbital angular momentum (*l* = 1). Two protons, or two neutrons, for example, in the 1*p*_{1/2} level will both have *j* = 1/2, with spins anti-aligned and a total orbital-angular momentum of zero. Thus, two neutrons and two protons in this orbit will have features that resemble an α-particle. This will be a key feature in the later discussion.

*Source*: Adapted from Leander and Larsson (1975).

The same is true for a deformed nucleus, which in terms of the mean-field approach is well represented by the Nilsson-level scheme (Figure 2). The Nilsson scheme shows zero deformation ($\u03f5=0$) at the shell-model levels, which then diverge as the potential is deformed. Aside from the spin–orbit splitting, this is replicated in the DHO (Figure 3). Here, the energy levels are given by the usual formula

with ${\omega}_{x,y,z}$ being the characteristic angular frequencies in the three Cartesian coordinate directions and ${n}_{x,y,z}$ being the associated oscillator quanta, and

and imposing axial symmetry ${\omega}_{x}={\omega}_{y}={\omega}_{\perp}$such that the deformation parameter is given by

Here,$\phantom{\rule{0.25em}{0ex}}{\delta}_{\mathit{osc}}=\u03f5$; that is, the deformation parameters have been used interchangeably, historically.

In Figure 2, the circled numbers represent magic numbers of protons or neutrons associated with shell closures. Shell closures are linked to nuclear configurations with higher stability. This stability not only arises from the energy cost to disturb the nucleon configuration through the promotion of nucleons to the next available energy level, but the degeneracy also allows more configurations to contribute to the observed quantum state, enhancing binding.

*Source*: Freer et al. (2022).

Broadly, the same shell structure is observed in Figure 3, where instead the circled numbers represent the numbers of protons, or neutrons, that can be placed in each level consistent with the Pauli exclusion principle. Thus, for example, at ${\delta}_{\mathit{osc}}=0$, the shell closures and magic numbers will occur at 2, 8, 20, and so on. For positive and negative values of ${\delta}_{\mathit{osc}}$, corresponding to prolate and oblate nuclei, there is a similar agreement between the DHO and Nilsson-level schemes. This is not surprising, since the Nilsson calculations are built from the DHO. The main point is that the influence of other components in the calculations does not destroy the pattern.

The point of plotting degeneracies is that certain patterns, or symmetries, emerge that lead to a geometric interpretation. Note at ${\delta}_{\mathit{osc}}=$ 0.6 and 0.86, the spherical degeneracies are repeated either two and three times, which coincides with an axial deformation of 2:1 and 3:1. Hence, the spherical representation is preserved at these prolate deformations (Nazarewicz & Dobaczewski, 1992) and invites a discussion of two- or threefold clustering.

The nature of the oblate, ${\delta}_{\mathit{osc}}<0$, degeneracies in Figure 3 is less obvious but is revealed in the diagram shown in Figure 4 (Freer et al., 2022), where the pattern of 2, 6, 12, 20, 30, . . . is shown to be universal.

*Source*: Freer et al. (2022).

The interpretation of these patterns is relatively simple. The preceding sequence corresponds to the sequential filling of spherical levels to produce the closed-shell nuclei ^{4}He, ^{16}O, and ^{40}Ca, among others, and the prolate-repeated patterns correspond to the assembly of these spherical nuclei along the axis of deformation (*z*-axis), while the oblate deformations are the union of prolate-deformed nuclei orientated along the *x*-direction (or *y*-direction) but assembled along the *z*-axis.

Taking the simplest case as an example, at a deformation of 2:1, 2 + 2 corresponds to the prolate-deformed ^{8}Be nucleus, which would then possess a 2α structure. The 1:2 deformation 2 + 2 + 2 structure would correspond to this ^{8}Be nucleus being fused with a further α-particle in a triangular arrangement. The next most complex example would be for the 2:1 deformation 2 + 2 + 6 + 6, which is two ^{16}O clusters, and at 1:2 2 + 2 + 2 + 6 + 6 + 6, three ^{16}O nuclei in a triangular arrangement. Of course, just because symmetries exist does not mean that the nuclei exist, a point we return to later. Similarly, the more realistic Nilsson scheme washes out these patterns for heavier nuclei and more extreme deformations.

These patterns are, however, found in the densities that would be calculated from the HO wave functions associated with the cluster configurations. These densities also have strong similarities to those from more detailed nuclear calculations. This is illustrated by the rather complex case of ^{24}Mg in Figure 5. The triangular contour plot at the center of each part of the figure represents the results of Nilsson-Strutinsky (NS)-type calculations (Leander & Larsson, 1975). In these calculations, the macroscopic deformation energy is calculated by deforming a liquid drop, as motivated by a droplet of nuclear fluid. As the droplet is deformed, the surface area increases, as does the potential energy. This potential-energy surface is then modulated by a correction term that traces out the magnitude of the separation of the energy levels (the size of the shell gap) for a fixed nucleon number as a function of deformation. The bigger the gap to the next energy level, the higher the assumed stability. At the points in the Nilsson diagram where the shell gap is largest, this gives rise to minima in the potential energy and associated quasi-stable configurations. The minima are hatched in the NS calculations.

The plots explore a variety of deformations from prolate ($\gamma ={0}^{\circ},{\u03f5}_{2}$ increasing) to oblate ($\gamma ={60}^{\circ},{\u03f5}_{2}$ increasing), through intermediate triaxial deformations. The bottom panel on these calculations also explores the octupole, the pear shape, and the degrees of freedom ($\gamma ={0}^{\circ},{\u03f5}_{3}$ increasing). These calculations reveal five minima, one global and four local, which are identified with stable or quasi-stable configurations.

These results were compared with those of alpha-cluster model (ACM) calculations by Fulton and Rae (1990). The ACM, often called the Bloch–Brink ACM (Brink, 1966), is constructed from a starting point of the existence of the α-particles and that the overall wave function, ${\mathrm{\Phi}}_{\mathrm{ACM}}$, of the system is then constructed as an anti-symmetrization of the product of the α-particle, Gaussian, wave functions, ${\varphi}_{{\alpha}_{i}}\left({r}_{i}\right)$. This process respects the fact that although the α-particles are bosons, their internal constituents are fermions and should therefore obey the Pauli exclusion principle. The overall wave function is thus given by a Slater determinant:

where ${K}_{N}$ is a normalization constant and $\mathcal{A}$ is the anti-symmetrization operator.

In this multi-center approach, the positions of the α-particles are varied so as to minimize the total energy of the system under the influence of an effective α–α interaction. The ^{24}Mg nucleus can be described in terms of six α-particles, the arrangements of which can be seen on the left-hand side of Figure 5 of the NS calculations. Remarkably, there is a one-to-one correlation between the stable configurations found in the NS and mean-field calculations and those in the ACM, although these are completely different approaches. Not only do the number of minima match, but so, too, do the shapes predicted.

The connection is further reinforced when the DHO densities are computed, where the HO configuration is taken from the energy levels populated in the NS calculations associated with each minimum in the potential-energy surface. These calculations are shown on the right-hand side of Figure 5 (Freer & Merchant, 1997). These mean-field densities reveal the shapes, which are almost precisely those found in the ACM calculations. It is worth noting that these conclusions are not unique to ^{24}Mg but extend to the vast majority of light nuclei, where they can be decomposed into α-particle subunits, so-called alpha-conjugate nuclei.

It is worth pausing at this point to reflect on what this comparison reveals:

the mean-field calculations predict that at certain deformations quasi-stable structures exist;

these quasi-stable deformations can be associated with symmetries in the DHO;

these symmetries can be traced to geometric arrangements of α-particles (two protons and two neutrons in the same energy level); and

the same geometric arrangements are found in ACM calculations.

In other words, the density patterns of the mean field enhance the probabilities for the nucleons to be in regions where they then behave as α-particles. Given the very high binding of the α-particle, these spatial enhancements are further promoted and clusterization develops.

#### 2.2 Clusters of Different Sizes

The main focus of the discussion to this point has revolved around the formation of α-particle clusters. However, there are other closed-shell configurations, which could be argued would have enhanced stability and hence would be good cluster candidates. Indeed, examination of the octupole stabilized structure in ^{24}Mg (bottom of Figure 5) shows a density that has the form α-^{16}O-α. Here, the central ^{16}O nucleus in principle could be decomposed into a tetrahedral structure of four α-particles or, alternatively, be thought of as a single entity.

There are two considerations to reflect on: (a) Is there an energy advantage in the nucleons and α-particles coalescing into the cluster, and (b) Do the symmetries exist to enhance the clusterization?

Figure 6 illustrates the binding energies per nucleon, BE/A, and excitation energies of the first excited states (how inert the nucleus is) for light nuclei. It is clear that the ^{4}He nucleus stands out in terms of its properties. However, the relatively high BE/A of ^{16}O and highish first excited state, 6.05 MeV, mean that the nucleus has the potential to form a cluster. Indeed, the decay energy into four α-particles is 14.4 MeV, not so dissimilar to the four-nucleon decay energy of the ^{4}He nucleus.

*Source*: Freer (2007).

Figure 7 illustrates the HO densities at the deformations of 2:1 and 3:1, which would correspond to shell closures associated with ^{4}He + ^{4}He and ^{16}O + ^{16}O clusters at 2:1 and ^{4}He + ^{4}He + ^{4}He and ^{16}O + ^{16}O + ^{16}O clusters at 3:1. It is evident that the densities reveal the two-centered nature at 2:1 and three-centered structure at 3:1 for both cluster types. In other words, the ingredients for these larger clusters exist, and the symmetries are robust.

*Source*: Author.

The robustness of this approach can be further tested through comparison with more accurate calculations. The anti-symmetrized molecular dynamics (AMD) method is one that is similar to that described with ACM, but rather than there being an implicit assumption of the existence of α-clusters, the full complexity of the *A*-body wave function is considered while utilizing a realistic nucleon–nucleon interaction, albeit an effective interaction. In the example that follows, the Gogny D1S (Berger et al., 1991) force has been utilized. The full wave function is anti-symmetrized, and a variational approach is used to project out configurations and densities.

The AMD calculations reported by Kimura and Horiuchi (2004) for the nucleus ^{32}S are presented in Figure 8. There are a series of solutions for the binding energy of the nucleus as a function of deformation and of angular momentum. The variation in the binding energy is sensitive to both the increased surface area and the stabilizing effects associated with energetically preferred configurations, just as in the case of the NS method. A stable structure is observed at a deformation value of 0.7 (here, $\beta $ is equivalent to ${\delta}_{\mathit{osc}}$ described earlier), and the associated density is shown as the inset. The density clearly demonstrates the ^{16}O + ^{16}O cluster structure and should be compared with the bottom-left-hand image in Figure 7. The ab initio AMD approach produces almost exactly the same result as found in the HO calculations. The symmetries are robust, as is the cluster structure.

In summary, the symmetries exist that allow the formation of heavier clusters; however, their binding and stability are less than that of the α-particles, and some duality will exist between the heavier cluster and its α-particle substructure.

*Source*: Kimura and Horiuchi (2004).

#### 2.3 Multi-Center Approaches

The question remains as to if this coincidence between the cluster description and the mean field is an accident or if there is a deeper connection between these two descriptions. The vehicle for exploring this is the multi-center shell model, or equivalently and simpler, the multi-center HO. This is encapsulated in the Harvey prescription (Harvey, 1975), and only a summary is presented here (see Freer et al., 2022, and references therein for further details).

The starting point is the sequence of energy levels in the spherical harmonic oscillator. These are given by quanta configurations $N:\left({n}_{x},{n}_{y},{n}_{z}\right):$

$N=0:\left(0,0,0\right)$

$N=1:\left(1,0,0\right),\left(0,1,0\right),\left(0,0,1\right)$

$N=2:\left(2,0,0\right),\left(0,2,0\right),\left(0,0,2\right),\left(1,1,0\right),\left(1,0,1\right),\left(0,1,1\right)$

$N=3:\left(3,0,0\right),\left(0,3,0\right),\left(0,0,3\right),\left(2,1,0\right),\left(2,0,1\right),\left(1,2,0\right),\left(1,0,2\right),$

$\left(1,1,1\right),\left(0,2,1\right),\left(0,1,2\right)$, and so on,

where $N\phantom{\rule{0.25em}{0ex}}$is the total number of oscillator quanta and ${n}_{x,y,z}$ is the projection of these quanta onto the three Cartesian coordinate axes. In the spherical case, levels with the same $N$ are degenerate. The Harvey prescription considers what happens when two spherical harmonic oscillators merge into a single oscillator. The rules follow from the details of solving the Schrödinger equation for the two oscillators as a function of their separation from infinity to zero. They can be summarized simply.

When two initial levels in the separate potentials are merged along the $z$-axis, then the resultant two levels in the single-center potential have quantum numbers $\left({n}_{x},{n}_{y},2{n}_{z}\right)$ and $\left({n}_{x},{n}_{y},2{n}_{z}+1\right)$. If three potentials are combined by merging along the $z$-axis, the resultant levels have quantum numbers $\left({n}_{x},{n}_{y},3{n}_{z}\right)$, $\left({n}_{x},{n}_{y},3{n}_{z}+1\right)$, and $\left({n}_{x},{n}_{y},3{n}_{z}+2\right)$, among others. These rules follow from the linear combinations of the individual wave functions, for example,

and

The symmetric combination, ${\mathrm{\Psi}}_{s}$, preserves the number of nodes in the *z*-direction, and the wave functions are continuous in the other two dimensions, $\left({n}_{x},{n}_{y},2{n}_{z}\right)$, and the asymmetric combination, ${\mathrm{\Psi}}_{a}$, creates an additional node in the *z*-direction, $\left({n}_{x},{n}_{y},2{n}_{z}+1\right)$. As a simple example, the fusion of two α-particles starts with two protons and two neutrons in the separate HO potentials in the level $\left(0,0,0\right)$, and then in the merged case, these move to the levels $\left(0,0,0\right)$ and $\left(0,0,1\right)$. This respects the Pauli exclusion principle as in the fused system, the $\left(0,0,0\right)$ level is fully occupied with a spin-up and spin-down proton and neutron pair.

As a more complex example of the application of the rules, the case of the super-deformed ^{16}O + ^{16}O cluster state in ^{32}S is considered (as shown in Figure 8), although the principles are general. The first step is to assemble the tetrahedral arrangement of α-particles in ^{16}O. This can be formed from first a triangle and then the addition of another particle above the plane of the triangle. The process is as follows:

Assemble two α-particles into ^{8}Be along the *z*-axis:

$\left(0,0,0\right)+\left(0,0,0\right)\underset{z}{\underset{\u23df}{\to}}\text{}[\left(0,0,0\right)+\left(0,0,1\right)]$

Join ^{8}Be with an α-particle along the *x*-axis to make ^{12}C triangle:

$[\left(0,0,0\right)+\left(0,0,1\right)]+\left(0,0,0\right)\underset{x}{\underset{\u23df}{\to}}[\left(0,0,0\right)\left(0,0,1\right)+\left(1,0,0\right)]$

Join ^{12}C with an α-particle along the *y*-axis to make ^{16}O tetrahedron:

$[\left(0,0,0\right)+\left(0,0,1\right)+\left(1,0,0\right)]+\left(0,0,0\right)\underset{y}{\underset{\u23df}{\to}}[\left(0,0,0\right)+\left(0,0,1\right)+\left(0,1,0\right)+\left(1,0,0\right)]$

The oscillator levels in bold are the ones are transformed through the Harvey rules. This tetrahedral arrangement corresponds to a spherical ^{16}O nucleus. The next step is then to fuse two ^{16}O clusters. This is illustrated in Figure 9.

The resulting ^{32}S HO configuration is what would have been concluded by populating the lowest eight energy levels at a deformation of 2:1 in Figure 3 and is indeed the configuration used to calculate the density in the ^{16}O + ^{16}O (bottom left) plot in Figure 7.

In summary, the relationship between the symmetries and densities of the DHO, and indeed more complex models, and those associated with geometric arrangements of α-particles are encoded in the quantum mechanics of the construction of complex objects from their constituent parts.

*Source*: Author.

There are some underlying assumptions, not least of which is that the two protons and two neutrons in each orbit remain as an α-particle in this process, but as illustrated in the AMD calculations in Figure 8, in the case of the complex nucleus ^{32}S, and as will be seen later, these approximations are valid.

#### 2.4 Collective Rotations

The rotations of clusters, under the assumption that the clusters remain intact, generate the excited states of the system. These rotations respect the symmetry, and the excitation energy of the collective states are given by the quantum versions of the classical equations for rotating systems. In the case of ^{8}Be, there are two identical axes around which the rotations may occur, and the equation for the rotational energy is (Hafstad & Teller, 1938)

where ${I}_{\mathit{Be}}$ is the moment of inertia of two touching α-particles. This produces a set of states ${J}^{\pi}$ = 0^{+}, 2^{+}, 4^{+}, . . . up to a maximum angular momentum the system can sustain. Since in the ^{8}Be nucleus there are four particles in *p*-orbitals (*l* = 1), then the maximum total angular momentum that can be generated is ${J}^{\pi}$ = 4^{+}.

*Source*: Author.

For ^{12}C, the situation is more complex as there are two different symmetry axes. The first has a threefold rotational symmetry (perpendicular to the plane of the triangle), and the second has a twofold symmetry (in the plane of the triangle). The second of these corresponds to a rotation of the two α-particles in the base of the triangle; that is, the moment of inertia is given by ${I}_{\mathit{Be}}$. This symmetry is designated ${D}_{3h}$. The rotations around the threefold symmetry axis are labeled by the quantum number $K$, and ${K}^{\pi}$ can take values of 0^{+}, 3^{−}, 6^{+}, . . . Collective rotations are labeled by the ${K}^{\pi}$ and *J* values, and the rotational energy is given by (Hafstad & Teller, 1938)

For ${K}^{\pi}\phantom{\rule{0.25em}{0ex}}$= 0^{+}, the rotations will be around an axis that lies in the plane of the three α-particles (in fact passing through the center of one α-particle and bisecting the other two), generating a series of states 0^{+}, 2^{+}, 4^{+}. The next set of rotations are associated with the rotation around an axis perpendicular to the plane of the triangle, with each α-particle having one unit of angular momentum, giving *L* = 3 *×* 1$\hslash $; ${K}^{\pi}=$3^{−}. Rotations around this axis and that are parallel to the plane combine to give a series of states 3^{−}, 4^{−}, 5^{−}, . . . The next set of collective states then correspond to each α-particle having *L* = 2$\hslash $; ${K}^{\pi}=$6^{+}, corresponding to *L* = 3 *×* 2$\hslash $.

For the tetrahedral arrangement of clusters in ^{16}O, there is one common symmetry axis, and the rotational energies are given by (note the denominator is the same as the second term in Equation (8))

The symmetry then dictates that all values of *J* are permitted except *J* = 1, 2, and 5; states with *J* = 0, 4, and 8 have even parity; and *J* = 3, 7, and 11 have negative parity. The relevant symmetry is here ${T}_{d}$.

These principles are the basis for the ACM approach which has been used for ^{12}C (Marın-Lambarri et al., 2014) and ^{16}O (Bijker & Iachello, 2014), where the rotations are coupled with vibrational modes, and the α-particles oscillate with respect to the center of mass. For example, a breathing mode is associated with a monopole type, *L* = 0, oscillation. This generates a richer spectrum of states. However, this spectrum is driven by the symmetries alone and does not always properly recognize that the individual nucleons must respect the Pauli exclusion principle, and some of the levels will be forbidden (Hess, 2018; Hess et al., 2019).

#### 2.5 Molecules

Until this point, the discussion has focused on the limited description of nuclei that have an even and equal numbers of protons and neutrons. As described by Hafstad and Teller (1938), it is possible to construct other types of structures while retaining the underlying principles developed to this point. In this early work, nuclei such as ^{7}Be and ^{9}Be were described in terms of the underlying ^{8}Be, 2α, structure but with the exchange of a neutron hole, or neutron particle, between the two clusters. This is completely analogous to the exchange of electrons in covalently bound atomic molecules, except that care must be taken over the Pauli exclusion principle, the neutrons being exchanged, and the existence of neutrons within the cluster cores.

The nature of such molecular states has been described extensively in earlier works (Freer, 2007; von Oertzen et al., 2006). There is extensive evidence for molecular states in nuclei, such as ^{9}Be and ^{10}Be (Freer, 2007), whereby neutrons are exchanged between the cluster cores in $\sigma $- or $\pi $-type molecular orbitals formed from the 1*p* orbitals that the valence neutrons occupy around the α-particles. These molecular structures also exist in systems with asymmetric cluster cores, for example, ^{4}He + ^{16}O (Kimura, 2007).

The question of the nature of these molecular states is explored, that is, is it an accident that molecular states exist, or are they somehow related to the fundamental symmetries that have been described earlier (Canavan & Freer, 2020). As set out in Canavan & Freer (2020), the existence of the molecular states is precisely related to ideas developed here and is a universal property related to the existence of clustering.

In line with the discussion of the ^{16}O + ^{16}O cluster states in the 2:1, super-deformed configuration in ^{32}S, nuclear molecules would be formed from the linear combinations of the valence orbits of the neutrons in the nucleus ^{17}O. In the HO representation, these are the

$N=2:\left(2,0,0\right),\left(0,2,0\right),\left(0,0,2\right),\left(1,1,0\right),\left(1,0,1\right),\left(0,1,1\right)$

levels. The linear combinations of identical orbits at the two centers are then given by

The symmetric, ${\mathrm{\Psi}}_{s}$, version is preferred as the asymmetric one results in an additional node in the two-center wave function and hence higher energy. So here, only the lower energy solutions are considered. In terms of the Harvey rules, this is the $\left({n}_{x},{n}_{y},2{n}_{z}\right)$ solution.

*Source*: Canavan and Freer (2020).

Thus, the $N=2$ levels in potentials in each of the two separated centers map to the set of levels in the single center:

$\left(2,0,0\right),\left(0,2,0\right),\left(0,0,4\right),\left(1,1,0\right),\left(1,0,2\right),\left(0,1,2\right)$.

The ^{16}O + ^{16}O super-deformed state in ^{32}S corresponds to the levels at 2:1 being filled up to those labeled and circled 2 and 6 in Figure 3. The next available states would be those labeled and circled 12. If a ^{33}S molecular is created then the neutron would occupy these levels. Tracing their origin they are found to be

$N=2:\left(2,0,0\right),\left(0,2,0\right),\left(1,1,0\right)$;

$N=3:\left(1,0,2\right),0,1,2)$;

$N=4:\left(0,0,4\right)$.

These reproduce identically those from the two-centered approach. Figure 12 (left) shows the wave functions produced either in the multi-center and the single center are essentially the same. In other words, the molecular orbitals are the next available orbitals for the neutron to occupy once the ^{16}O + ^{16}O cluster structure is formed.

As described earlier, it is possible to have systems in which there are combinations of asymmetric clusters; for example, ^{4}He + ^{16}O is famously linked to the octupole-deformed state in ^{20}Ne (Nemoto et al., 1975). This nucleus would lie at a 2:1 deformation with the levels with degeneracy 2 + 2 + 6 occupied, in Figure 3. The two center orbitals around the ^{16}O and ^{4}He clusters are

^{16}O*:* $N=2:\left(2,0,0\right),\left(0,2,0\right),\left(0,0,2\right),\left(1,1,0\right),\left(1,0,1\right),\left(0,1,1\right)$

^{4}He*:* $N=1:\left(1,0,0\right),\left(0,1,0\right),\left(0,0,1\right)$

In order to construct molecular orbitals, the quantum numbers in the *x*- and *y*-directions should match; that is, the matching combinations are $\left(1,0,0\right)+\left(1,0,1\right)$, $\left(0,1,0\right)+\left(0,1,1\right)$, and $\left(0,0,1\right)+\left(0,0,2\right)$. Adding the ${n}_{z}$ values as this reflects the nodes in the contributing wave functions produces the three molecular orbitals $\left(1,0,1\right)$, $\left(0,1,1\right)$, and $\left(0,0,3\right)$. Again, examining the preceding set of HO levels, the 2 + 2 + 6 occupied levels give

$N=2:\left(1,0,1\right),\left(0,1,1\right)$*;*

$N=3:\left(0,0,3\right)$.

The similarity between the single-center and multi-center molecular wave functions is again illustrated in Figure 12.

Thus, the universality of the concept of molecular structures in these prolate-deformed nuclei can be seen. The details of such structures on the oblate side of deformation are still being worked through.

### 3. State of the Art

Up to this point, the overarching principles that underpin the appearance of nucleon clustering in nuclei have been explored using the harmonic oscillator. Examples of other types of calculations have been presented that show that the ideas developed through the deformed harmonic oscillator reproduce features of these more complex calculations. State-of-the-art theory has moved closer to being able to describe the nucleon–nucleon interaction from first principles. These now give an even greater insight into the nature of the correlations that give rise to not only clustering but also a sense of confidence as to the robustness of the symmetries driving the formation of clusters.

#### 3.1 AMD

The starting point for this discussion is the AMD calculations, compared with experimental data, for the nucleus ^{12}C. As described earlier, ^{12}C is expected to have a triangular arrangement of α-particles in the ground state. This is the configuration in Figure 3 at a deformation of 1:2, with the three levels labeled 2 + 2 + 2.

The lowest energy rotational structure would have states of ${J}^{\pi}$= 0^{+}, 2^{+} and 4^{+}, ${K}^{\pi}$=0^{+}, associated with rotations about the line of symmetry passing through the plane of the triangular structure. Figure 13 shows a series of densities obtained from AMD calculations with a Volkov No. 1 (MV1) force (Kanada-En’yo, 2007, 2009). The densities reveal the 3α structure that is preserved, and enhanced, through the rotational states. The percentages shown reflect the weight of the density displayed in the overall wavefunction for the state; a high percentage shows a single dominant component. In all cases, there is not a single configuration that contributes to the states, and there will be some mixing of the HO structures.

As observed, the theory follows the experiment to the point of the 4^{+} state. The structure reveals the dominance of the triangular nature of ^{12}C, although the ^{8}Be + α structure is more evident than the 3α. The rotational behavior is close to that shown in Figure 11 and described by the ${D}_{3h}$ dynamical symmetries.

The top three densities in Figure 13 show the nature of the excited states that are created through what would be described in terms of monopole breathing in the ACM approach (Figure 11). The density shown for the ${0}_{2}^{+}$ state is indeed an expanded version of the ${0}_{1}^{+}$ configuration, but it is also observed that there are many similar configurations contributing to this state, given the 49% weighting. A similar conclusion is reached for this state in what are called fermionic molecular dynamics (FMD) calculations, which are very similar in their spirit to AMD (Chernykh et al., 2007). The need for a variety of configurations to describe this state has been interpreted as the structure having a loose assembly, that is more weakly bound, of α-particles, which may be considered a more gas-like state rather than a crystalline arrangement of constituents.

The ^{12}C nucleus, and in particular the ${0}_{2}^{+}$ state, has become a touchstone for understanding of the nuclear structure, the nature of the strong nuclear force, the emergence of clusterization, and the competition between single-particle, mean-field characteristics, and clustering. The Hoyle state has a determining role in the formation of the element carbon in stellar nucleosynthesis. In a process that mirrors the ^{8}Be + α structure of the Hoyle state, ^{12}C is synthesized through a two-stage helium burning process in which, first, ^{8}Be is created and then ^{8}Be captures an α-particle to make ^{12}C in the Hoyle state, which then decays through electromagnetic processes to the ground state (Freer & Fynbo, 2014). As such, understanding the nature of the Hoyle state in ^{12}C is important for understanding both the origin of carbon and, therefore, carbon-based life and nuclear physics.

*Source*: Kanada-En’yo (2007).

#### 3.2 Continuum Shell Model and Clustering

As the 7.65 MeV, ${0}_{2}^{+}$ state lies above the decay threshold for both the 3α and ^{8}Be + α decay channels, that is, the state sits above the top of the nuclear potential, the state is embedded in, and couples to, the continuum. In this approach, it is found that nuclear clustering is an emergent, near-threshold phenomenon, which arises from a coupling of shell-model-like states to the continuum, where the continuum is strongly representative of the available scattering states (Okołowicz & Nazarewicz, 2013), that is, ^{8}Be + α. This amplifies any cluster-like components in the state. This threshold description is similar to the Ikeda picture (Ikeda et al., 1968), where clustering is postulated to appear near to decay thresholds, although the premise is different. In the Ikeda description, it is argued that in order for the clusters to be liberated, an internal energy is required which is equivalent to the mass difference between the parent and the cluster components, that is, the decay Q-value or threshold. At this point, the clusters can be liberated.

It should be observed that nuclear-structure calculations, such as AMD and FMD, predict the most evolved cluster states to be close to decay thresholds as the nuclear interactions used also respect nuclear binding energies. Thus, it is likely that the evolution of clustering away from the ground state of nuclei is influenced by the underlying nuclear symmetries, the associated dynamics, and indeed the role of the continuum coupling.

#### 3.3 No-Core Shell Model

The slightly complex nature of the Hoyle state has proved a challenge for the shell model, which fails to produce a low-enough energy anywhere. As illustrated in Figure 14, the no-core shell model (NCSM) calculations produce a Hoyle state at about 18 MeV, 10 MeV above the experimental energy (Navratil et al., 2000). This no-core approach embraces the computational power of the early 21st century to perform a no-holds-barred shell model calculation, where the basis is expanded to include higher and higher energy shell-model (HO) states. In order to get close to reproducing the energy of the Hoyle state, a shell-model space including 20$\phantom{\rule{0.25em}{0ex}}\hslash \mathrm{\Omega}$ is required. This is taken as a strong vindication of the cluster nature of the Hoyle state. A constrained version of the NCSM, called the no-core symplectic shell model, has been used that employs symmetry arguments to select the states featured in the calculations to reproduce the ^{12}C spectrum (Dreyfuss et al., 2013). This points to the key role of symmetries in these cluster states and has been extended to heavier nuclei with similar conclusions (Tobin et al., 2014).

*Source*: Navratil et al. (2000).

The conclusion from the NCSM approach is that the cluster symmetries observed in the low-lying states are well reproduced and have a dual existence between a cluster and a mean-field description. They may be thought of in terms of a more crystalline geometric arrangement of clusters, whereas the cluster states above the decay threshold have an existence that lies beyond, but is still influenced by, the cluster symmetries. As such these clusters have a more gas-like structure and may be considered to be a phase transition as the decay threshold is crossed.

As an example, Figure 7 illustrates the HO configuration (0,0,0), (0,0,1), and (0,0,2), which would place the three α-particles in a linear configuration. If the Hoyle state had this structure, it would readily be captured by the NCSM and other approaches. However, as shown in the AMD calculations, Figure 13, an extended triangular arrangement is found. In other words, above the decay thresholds, in this gas-like regime, the symmetries of the mean field are no longer sufficient to shape the structure, but the clusters gain the ability to reconfigure themselves into more stable structures, and moreover, the linear arrangement is inherently unstable.

A final insight into the structure of ^{12}C, which is consistent with the preceding arguments, comes from the Monte Carlo shell model approach (Figure 15), performed on a supercomputer (Otsuka et al., 2022) with a statistical learning approach. The extensive sampling of the shell-model states of this approach allows a characterization of the ground $\left({0}_{1}^{+}\right)$ and Hoyle $\left({0}_{2}^{+}\right)$ states in ^{12}C. The two dominant configurations are a closed triangular structure and a more open triangular structure, as found in the AMD calculations. The weighting of these components in the ground and Hoyle states show that the ground state is dominated by the 3α triangle. The purity of the Hoyle state is less but with a major component from the open triangle but, again, a structure that is consistent with a gas-like nature.

*Source*: Otsuka et al. (2022).

#### 3.4 Chiral Effective Field Theory on a Lattice

The study of nuclear forces and clustering has a long history, in particular the role of the tensor component in enhancing clusterization (Matsuno et al., 2018; Yamamoto, 1974). However, although these interactions, including the Gogny and Volkov forces, do well in terms of reproducing the experimental characteristics, they are effective interactions inasmuch that they include parameterizations of potentials representing exchange processes.

An alternative approach that recognizes that ultimately the nuclear strong force emerges from the quantum chromo dynamics (QCD) degrees of freedom and that the meson exchange is a manifestation of this is to use chiral perturbation theory (Bernard & Meißner, 2007). The QCD description is simplest where the strong coupling constant, α_{S}, is weakest. At low energies that are appropriate for nuclear matter, the regime is such that the coupling is so strong that it cannot be treated as a perturbation. One approach has been to calculate properties based on constituents being fixed on a lattice (lattice QCD). A second has been to use an effective field theory, which draws on ideas developed for solid-state theory: chiral effective field theory (Bernard & Meißner, 2007). This draws on the fact that the up and down quarks display a chiral-like symmetry, which is broken only by their non-zero mass. In this description, the properties of both the hadrons (including protons and neutrons) and the mass of the pion emerge. In analogy with condensed-matter chiral-symmetry breaking, the pion should be a spinless Goldstone boson, which couples only weakly. This last aspect is essential as it then permits a tractable, perturbative approach (Bernard & Meißner, 2007).

This effective field theory then results in a series of exchange diagrams that relate to the perturbative expansion in orders of (Q/Λ)^{n}, where Q is of the order of the pion mass (∼140 MeV) and Λ is the chiral-symmetry-breaking scale (∼1 GeV). In leading order (n = 0), there is a one-pion exchange component, plus what is called a contact term. This term essentially accounts for the heavier exchange mesons (ρ, σ, and ω), whose range is significantly less than that of the pion. The next term (n = 1) vanishes due to parity and time invariance, leaving the n = 2 term as the next-to-leading order (NLO). This includes processes such as two-pion exchange. The complexity increases as more orders are added, meaning that although the contribution from each term to the interaction is less by the factor (Q/Λ), there are more processes to consider. The aim is to calculate the interaction to all orders, but this is clearly a challenge, and NNNLO (next-to, next-to, next-to leading order) is the current state of the art (N3LO).

The great advantage of this approach is that three-body forces emerge naturally, first appearing in the NNLO term, and that higher order terms, such as NNNLO, introduce four-body interactions. Here the two-body and n-body interactions emerge in more or less a consistent fashion. Such interactions have been deployed in a number of calculations, including the NCSM.

The application of these ab initio–type interactions to light nuclei has also been through the calculation of nuclear properties on the lattice. An example of this is shown in Figure 16 for the lowest energy configuration in ^{16}O. What is observed is that under this most advanced interaction, the nucleons cluster into α-particles at four lattice locations, with the pairs of protons and neutrons with their spins anti-aligned. The four α-particles are then found to occupy locations, within the constraints of the lattice, which reproduce the tetrahedral structure of ^{16}O, as found in the HO and ACM analyses. Again, the symmetries of the mean field are verified by this most advanced nuclear calculation.

This nuclear lattice effective field theory methodology has also been applied (Epelbaum et al., 2012) to the nature of ^{12}C, where again the triangular structure for the ground state is reproduced and a more open triangular arrangement for the Hoyle state.

*Source*: Freer et al. (2018).

This method has also been used to probe the nature of the quantum phase transition that occurs in nuclear matter between a nucleon liquid and an α-particle gas (Elhatisari et al., 2016), which was observed to be first order in its nature and consistent with the earlier description.

### 4. Summary and Outlook

The topic of clustering in nuclei has a history that is as long as that of nuclear physics itself, with the speculation that α-particles were preformed inside the nucleus following observations of spontaneous α-decay. The early contributions of Hafstad and Teller (1938) were really remarkable in that they established a series of principles and ideas that have set the direction for the field subsequently. Almost a century has passed, and it is clear that clustering and cluster geometries do indeed have a determining influence on the structure of light nuclei and their collective properties.

*Source*: Author.

The emergence of clustering from the mean field can be traced to the symmetries that are present and most clearly observed through the deformed harmonic oscillator. These symmetries have a direct link to the assembly of cluster structures into a composite nucleus and are responsible for the coexistence of cluster-model- and shell-model-like solutions at low energies and below-cluster-decay thresholds. Here, the idea of geometric assemblies of clusters may have some validity. However, the clusters overlap, and the Pauli exclusion principle plays a significant role, and the cluster components should not be confused with their free counterparts.

As the cluster decay thresholds are reached, as set out by Ikeda et al. (1968), the clusters become closer to these asymptotically free counterparts, some of the rigidity of the cluster arrangements is lost and the cluster systems become superpositions of a variety of geometries, more akin to a gas-like state. At this point, the symmetries of the mean field impose themselves more weakly, with clusters optimizing their geometric configurations to maximize the binding energy of the gas-like state, with a remaining tension between the symmetry and the binding.

Figure 17 illustrates this conclusion with the Ikeda diagram, which illustrates the decay thresholds that have been the traditional demarcation for the appearance of clustering. This is the point of transition from the nuclear states that have cluster-like behavior and are constrained by the harmonic oscillator symmetries to the gas-like behavior of the clusters above the water level. A first-order transition takes place in nuclear lattice effective field theory calculations (Elhatisari et al., 2016).

Not only is the decay threshold a frontier for the nature of clustering, but it really also represents a challenge for a theoretical description of nuclei beyond the threshold. The states are a complex interplay between nuclear configurations and hence highly fragmented in terms of a shell-model-like description; they are embedded and influenced by the continuum and have become open quantum systems. There are some signatures of this complexity experimentally in ^{12}C; as the experimental spectrum above the Hoyle state is explored, there is a collection of broad resonances whose properties, judging from theory, are strongly mixed. Certainly, these states are highly clustered, but the traditional spectroscopic tools for understanding their properties will not be sufficient. This is a new frontier for nuclear science, and it is likely that ^{12}C will continue to be a touchstone for some time to come. As a key next step, the full mapping of the 0^{+} and 2^{+} strength above the Hoyle state, a correlation with the integrated strength found in calculations and a detailed understanding of the role of the continuum will be important.

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