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 1p_3/2 and 1p_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 1p_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.

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Figure 2. Nilsson-level scheme. The parameter $ϵ$ is the deformation of the nuclear potential, and a value of $ϵ=+0.6$ represents a deformation in which the axis of deformation is extended by twice those in the perpendicular direction (which are assumed to be the same). See reference for the notation.

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 ($ϵ=0$) 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,${\delta }_{\mathit{osc}}=ϵ$; 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.

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Figure 3. The energy levels of the deformed harmonic oscillator as a function of deformation, ${\delta }_{\mathit{osc}}$, in units of $\hslash {\omega }_0$. The red and blue lines correspond to even and odd total oscillator quanta, and the numbers in the circles represent the numbers of protons, or neutrons, that can be placed in the levels at the points of high degeneracy.

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.

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Figure 4. The symmetries of the deformed harmonic oscillator. The spherical pattern of degeneracies (how many protons or neutron may be placed in orbits) 2, 6, 12, 20, 42, . . . are seen to repeat at integer deformations of the harmonic oscillator potential.

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 ⁴He, ¹⁶O, and ⁴⁰Ca, 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 ⁸Be nucleus, which would then possess a 2α‎ structure. The 1:2 deformation 2 + 2 + 2 structure would correspond to this ⁸Be 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 ¹⁶O clusters, and at 1:2 2 + 2 + 2 + 6 + 6 + 6, three ¹⁶O 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 ²⁴Mg 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 ($\gamma =0^{\circ },{ϵ}_2$ increasing) to oblate ($\gamma ={60}^{\circ },{ϵ}_2$ increasing), through intermediate triaxial deformations. The bottom panel on these calculations also explores the octupole, the pear shape, and the degrees of freedom ($\gamma =0^{\circ },{ϵ}_3$ increasing). These calculations reveal five minima, one global and four local, which are identified with stable or quasi-stable configurations.

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Figure 5. (Left) Nilsson-Strutinsky calculations (center) with alpha-cluster model (ACM) density calculations for ²⁴Mg. The deformations of the nucleus in the Nilsson-Strutinsky calculations are labeled as $ϵ$ (deformation) and $\gamma$ (triaxiality). $\gamma$ = 0 is prolate deformation and $\gamma$ = 60 is oblate. ${ϵ}_3$ gives the degree of octupole deformation. The ACM calculations reproduce the shapes predicted by the mean-field model. (Right) Same as the left-hand figure but with the ACM calculations replaced by the deformed harmonic oscillator calculations. Different shapes are labeled.

Source: Freer and Merchant (1997), adapted from Fulton and Rae (1990).

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 ²⁴Mg 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 ²⁴Mg 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:

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 ²⁴Mg (bottom of Figure 5) shows a density that has the form α‎-¹⁶O-α‎. Here, the central ¹⁶O 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 ⁴He nucleus stands out in terms of its properties. However, the relatively high BE/A of ¹⁶O 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 ⁴He nucleus.

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Figure 6. The excitation energy of first excited states of isotopes plotted against binding energy per nucleon for light nuclei. The α‎-particle stands out as the ideal cluster. The box drawn illustrates a group of nuclei ¹²C, ¹⁴O, ¹⁴C, ¹⁵N, and ¹⁶O, which have potential to form clusters inside the nuclear medium.

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 ⁴He + ⁴He and ¹⁶O + ¹⁶O clusters at 2:1 and ⁴He + ⁴He + ⁴He and ¹⁶O + ¹⁶O + ¹⁶O 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.

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Figure 7. Harmonic oscillator densities for the configurations at 2:1 and 3:1 deformations corresponding to the 2 + 2 and 2 + 2 + 6 + 6 shell closures at 2:1 and ⁴He + ⁴He and ¹⁶O + ¹⁶O clusters and 2 + 2 + 2 and 2 + 2 + 2 + 6 + 6 + 6 shell closures at 3:1 and ⁴He + ⁴He + ⁴He and ¹⁶O + ¹⁶O + ¹⁶O clusters.

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 ³²S 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 ¹⁶O + ¹⁶O 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.

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Figure 8. Anti-symmetrized molecular dynamics calculations for the ³²S nucleus. The lines show the binding energy evolution as a function of the prolate deformation of the system for different values of angular momentum, J. The inset plots the density corresponding to the super-deformed, SD, minimum at $\beta =0.7$, J = 0. The black dots correspond to the location of the nucleons inside the ¹⁶O clusters.

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 $N: \left(n_x,n_y,n_z\right):$

where $N$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 $\left(n_x,n_y,2n_z\right)$ and $\left(n_x,n_y,2n_z+1\right)$. If three potentials are combined by merging along the $z$-axis, the resultant levels have quantum numbers $\left(n_x,n_y,3n_z\right)$, $\left(n_x,n_y,3n_z+1\right)$, and $\left(n_x,n_y,3n_z+2\right)$, 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, $\left(n_x,n_y,2n_z\right)$, and the asymmetric combination, ${\mathrm{\Psi }}_a$, creates an additional node in the z-direction, $\left(n_x,n_y,2n_z+1\right)$. 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 ¹⁶O + ¹⁶O cluster state in ³²S 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 ¹⁶O. 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:

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

The resulting ³²S 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 ¹⁶O + ¹⁶O (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.

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Figure 9. The application of the Harvey prescription (two-center oscillator) to the fusion of two ¹⁶O clusters to form a super deformed excited configuration in ³²S. Levels are labeled by their HO quanta: $n_x{n}_y{n}_z$, and fusion is along the z-direction.

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 ³²S, 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 ⁸Be, 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 ⁸Be nucleus there are four particles in p-orbitals (l = 1), then the maximum total angular momentum that can be generated is $J^{\pi }$ = 4⁺.

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Figure 10. Collective rotations of the 2α‎ (dumbbell), 3α‎ (triangle), and 4α‎ (tetrahedron) systems.

Source: Author.

For ¹²C, 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 $K^{\pi }$= 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 ¹⁶O, 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 ¹²C (Marın-Lambarri et al., 2014) and ¹⁶O (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).

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Figure 11. Algebraic cluster model calculations for ¹²C, 3α‎ (triangle), and ¹⁶O, 4α‎ (tetrahedron) where the rotational and vibrational modes are illustrated.

Source: Left: Marın-Lambarri et al. (2014). Right: Bijker and Iachello (2014).

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 ⁷Be and ⁹Be were described in terms of the underlying ⁸Be, 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 ⁹Be and ¹⁰Be (Freer, 2007), whereby neutrons are exchanged between the cluster cores in $\sigma$- or $\pi$-type molecular orbitals formed from the 1p orbitals that the valence neutrons occupy around the α‎-particles. These molecular structures also exist in systems with asymmetric cluster cores, for example, ⁴He + ¹⁶O (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 ¹⁶O + ¹⁶O cluster states in the 2:1, super-deformed configuration in ³²S, nuclear molecules would be formed from the linear combinations of the valence orbits of the neutrons in the nucleus ¹⁷O. In the HO representation, these are the

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 $\left(n_x,n_y,2n_z\right)$ solution.

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Figure 12. (Left) Five of the six possible molecular orbitals for the nucleus ³³S. The right column is the harmonic oscillator wave function, and the left column is the result of the linear combination of the orbits shown around the two ¹⁶O clusters. (Right) Molecular wave functions for ²¹Ne. Left column are the single-center wave functions; right are linear combinations of the two single-center orbitals.

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:

The ¹⁶O + ¹⁶O super-deformed state in ³²S 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 ³³S molecular is created then the neutron would occupy these levels. Tracing their origin they are found to be

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 ¹⁶O + ¹⁶O cluster structure is formed.

As described earlier, it is possible to have systems in which there are combinations of asymmetric clusters; for example, ⁴He + ¹⁶O is famously linked to the octupole-deformed state in ²⁰Ne (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 ¹⁶O and ⁴He clusters are

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

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.

3.1 AMD

The starting point for this discussion is the AMD calculations, compared with experimental data, for the nucleus ¹²C. As described earlier, ¹²C 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 ¹²C, although the ⁸Be + 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 ¹²C 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 ⁸Be + α‎ structure of the Hoyle state, ¹²C is synthesized through a two-stage helium burning process in which, first, ⁸Be is created and then ⁸Be captures an α‎-particle to make ¹²C 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 ¹²C is important for understanding both the origin of carbon and, therefore, carbon-based life and nuclear physics.

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Figure 13. Energy levels of the 0⁺, 2⁺, and 4⁺ states in ¹²C and density distributions calculated using the anti-symmetrized molecular dynamics method (asterisk symbols). The calculations are compared with the experimental energy levels (other symbols). The intrinsic density distributions are shown together with percentages of the dominant component in the final wave functions.

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 ⁸Be + 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, ⁸Be + 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$\hslash \mathrm{\Omega }$ 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 ¹²C 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).

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Figure 14. No-core shell model calculations for the ¹²C nucleus. The left-hand column shows the experimental results. The calculations using the CD Bonn N–N interaction with different numbers of oscillator orbits are shown. States are labeled by $J^{\pi }\text{and}T$, spin and isospin, respectively.

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 ¹²C, 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 ¹²C. 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.

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Figure 15. Monte Carlo shell model calculations of the ground $\left(0_1^{+}\right)$ and Hoyle $\left(0_2^{+}\right)$ states in ¹²C. The result for the $0_3^{+}$is also shown.

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 ¹⁶O. 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 ¹⁶O, 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 ¹²C, where again the triangular structure for the ground state is reproduced and a more open triangular arrangement for the Hoyle state.

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Figure 16. Nuclear lattice effective field theory calculations for the ground state of ¹⁶O.

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.