# Symmetries in Collisions as Explored through the Harmonic Oscillator

^{1}

^{2}

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## Abstract

**:**

## 1. Nuclear Clustering and Symmetries

^{4}He nucleus. In fact, the binding of

^{4}He is so large that the

^{8}Be nucleus finds itself unbound to decay into two

^{4}He nuclei. This is a feature that is important in limiting the rate of helium burning in the red giant phase of stars and subsequently the rate of formation of

^{12}C [7]. As nuclei such as

^{8}Be,

^{12}C,

^{16}O,… can be decomposed into α-particles,

^{4}He nuclei subunits invite the idea that these nuclei might be described in terms of α-particle clusters.

^{8}Be, there are two identical axes around which the rotations may occur, and the equation for the rotational energy is [8]:

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

^{12}C, there are two different symmetry axes. The first has a three-fold rotational symmetry (perpendicular to the plane of the triangle) and the second has a two-fold 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, i.e., the moment of inertia is given by ${I}_{Be}$. This symmetry is designated ${D}_{3h}$. The rotations around the three-fold symmetry axis are labelled by the quantum number K, and ${K}^{\pi}$ can take values of ${0}^{+},{3}^{-},{6}^{+}$ … Collective rotations are labelled by the ${K}^{\pi}$ and J values and the rotational energy is given by [8]:

^{16}O, there is one common symmetry axis and the rotational energies are given by:

_{d}.

^{12}C and

^{16}O systems [10,11]. These cluster symmetries assume that the α-particles are boson-like and that the internal structure can be neglected. However, one should recognize that the full wavefunction of the system needs to be fully antisymmetrized, given the fermionic nucleon components. This process of antisymmetrization precludes certain states that would appear otherwise, as demonstrated in Refs. [12,13].

^{4}He,

^{16}O, and

^{40}Ca have high binding energies and comparatively high excitation energies for the first excited state. These are associated with magic proton and neutron numbers 2, 8, and 20. Figure 3 illustrates that at high levels of degeneracy, shell gaps appear in the deformed energy level scheme at both prolate $({\delta}_{osc}>0)$ and oblate $({\delta}_{osc}<0)$ deformations associated with integer ratios of axial deformation parameters/frequencies. This produces new sets of magic numbers particular to that deformation. As demonstrated in Ref. [15], these new magic numbers can be represented by a single sequence of numbers, 2, 6, 12, 20, 30,…, as illustrated in Figure 4.

^{8}Be. For the 1:2 oblate deformation, it is seen that the sequence of degeneracies is created by combining two 2:1 prolate sequences, but with one offset by 1$\hslash \omega $. As discussed in Ref. [15], this sequence can reproduce the structure of oblate, deformed, clustered nuclei such as

^{12}C and

^{28}Si and the associated D

_{3h}structure of

^{12}C. The T

_{d}structure of

^{16}O corresponds to the spherical 1:1 deformation with an α-particle stacked on top of the 3α, triangular, D

_{3h}structure of

^{12}C.

^{12}C [10].

## 2. Clustering and Collisions

^{12}C +

^{12}C,

^{12}C +

^{16}O, and

^{16}O +

^{16}O reveal a series of resonances associated with the intermediate nuclei

^{24}Mg,

^{28}Si, and

^{32}S, respectively. The widths of these resonances are $\Gamma \simeq 100$ keV, indicating lifetimes that are much greater than the collision time and point to the formation of a series of special states, which preserve the original structure of the colliding partners, exciting di-nuclear structures in the intermediate system with a series of rotational bands identified. The

^{12}C +

^{12}C resonances have been interpreted in terms of an array of cluster-like structures, which can be found in both the mean-field [21,22] and the alpha cluster models [23] with resonances observed in different reaction channels being ascribed to different types of cluster structures [20,24].

#### The Double-Center Oscillator

^{12}C +

^{12}C, the location of the two potentials along the collision, $z$, axis is given by ${z}_{1}={z}_{2}={z}_{0}$ (the centers are the same distance from the origin), and for asymmetric cases, ${z}_{1}\ne {z}_{2}$. This can be solved analytically, as shown by Holzer [25], and the derivation here closely follows the original derivation. The Hamiltonian used is a direct generalization of the Nilsson Hamiltonian for two centers:

- (i)
- The two functions must be single valued (unique for any given value). This ensures a single probability for a given state. This is naturally satisfied as ${{}_{1}F}_{1}$ does not contain any branch points (in contrast to ${{}_{2}F}_{1}$, which does).
- (ii)
- The two functions must be continuous at z = 0.
- (iii)
- The first derivatives of the two functions must be continuous at z = 0.
- (iv)
- The functions must be square integrable and therefore have the correct asymptotic values such that they vanish as $z\to \pm \infty .$ This is ensured through the asymptotic definition of the confluent hypergeometric function.

- Positive parity: ${C}_{1}^{+}=-{C}_{1}^{-}$ and ${C}_{2}^{+}={C}_{2}^{-}$
- Negative parity: ${C}_{1}^{+}={C}_{1}^{-}$ and ${C}_{2}^{+}=-{C}_{2}^{-}$

## 3. Results

^{12}C nuclei.

^{12}C nucleus in its ground state can be represented by the HO configuration $\left({n}_{x},{n}_{y},{n}_{z}\right)=$ (0,0,0)

^{4}, (0,0,1)

^{4}, and (0,1,0)

^{4}, which is associated with a triangular structure orientated in the y–z plane. Different orientations of the 3α structure can be created by populating the (1,0,0), (0,1,0), and (0,0,1) levels with different pairs. As is shown in Figure 8, when these nuclei merge, following solutions of the DCHO and as depicted by the Harvey scheme, different final

^{24}Mg structures are produced. The interesting observation is that the final structure that is produced, as represented by the DHO densities, retains the symmetries of the original arrangements of the

^{12}C clusters. For example, the left-hand side of Figure 8 shows the merger of two

^{12}C nuclei with all α-particles in the y–z plane and this results in a

^{24}Mg cluster structure in which 6α-particles are arranged, where two 3α triangles are clear. Different orientations give different cluster structures with the central example corresponding to an α +

^{16}O + α structure and the right-hand image would be a compact

^{24}Mg structure associated with an arrangement similar to the

^{24}Mg ground state.

^{16}O + α structure is intermediate). This can be thought of as a Pauli repulsion effect, which would add to the Coulomb repulsion associated with the like proton charges in the two

^{12}C nuclei. Thus, the formation of the compact structure would proceed at lower energies and the planar structure at higher energies, and the difference in barriers would drive re-orientation of the

^{12}C nuclei in the collision process. This has been demonstrated, for example, in coupled-reaction-channel calculations by Boztosun and Rae [26].

^{12}C nuclei. However, using the DCHO and the two-center wavefunctions, it is possible to calculate the evolution of the densities from the point where the two potentials begin to merge, through the point of closest contact and the formation of the intermediate structure to the point where the two nuclei move apart. In these calculations there is full consideration of the incident kinetic energy and the repulsive effect of the Pauli repulsion as particles trace out orbitals that climb in energy and the repulsive effect of the Coulomb repulsion. This is shown in Figure 9. Here, it is assumed that in the DCHO calculations, the line joining the two

^{12}C nuclei remains as the z-axis.

^{12}C and the 6α structure of

^{24}Mg is preserved, and the symmetries described earlier in this paper not only affect the static properties of nuclei, but also the dynamical ones as well.

## 4. Discussion and Conclusions

^{12}C nuclei in the collision is particularly important in understanding

^{12}C +

^{12}C burning in massive stars; changes to the structure that create low-energy resonances could lower the temperature and density required to ignite carbon burning, and change the burning rate. The rate of carbon burning (along with the stellar mass) can dictate whether the star will stay as a white dwarf or evolve into a type 1A supernova: a standard candle. In

^{12}C+

^{12}C burning, the two primary reaction channels are

^{12}C(

^{12}C, α)

^{20}Ne and

^{12}C(

^{12}C, p)

^{23}Na, with the former calculated using antisymmetrized molecular dynamics (AMD) highlighting the α substructure of

^{12}C [27]. Similar conclusions have been reached in time-dependent Hartree Fock (TDHF) calculations [28]. The emergence of similar conclusions from these more complex models and the more simplified DCHO method demonstrates the robustness of the symmetries in the calculations and their deeper impact on the synthesis of elements in stars.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Binding energy per nucleon (BE/A) plotted as a function of nucleon number. Different isotopes have different color lines, e.g., the helium isotopes are red, and lithium and beryllium isotopes are green and yellow, respectively.

**Figure 3.**The energy levels of the deformed harmonic oscillator as a function of deformation, ${\delta}_{osc}$, 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 [15].

**Figure 4.**The symmetries of the deformed harmonic oscillator. The spherical pattern of degeneracies (how many protons or neutrons may be placed in orbits) 2, 6, 12, 20, 30, 42… are seen to repeat at integer deformations of the HO potential [15].

**Figure 5.**The DCO potential shape (arbitrary axes): the colors indicate different values for ${n}_{z}$ (${z}_{0}$ = 0). ${n}_{z}\left(0\right)=0$ is given by red; ${n}_{z}\left(0\right)=1$ is given by blue; ${n}_{z}\left(0\right)=2$ is given by green. During deformation of the potential, these energy level positions move (with respect to one another). (

**a**) shows when the potential acts as two independent (D)Hos, (

**b**) shows partial merging of these potentials in the true DCO regime, and (

**c**) shows how the (D)HO is recovered in the limit $\Delta {z}_{0}\to 0$.

**Figure 6.**The evolution of the ${n}_{z}$ values as a function of the separation of the two DHO potentials. In this representation the two potentials merge at ${z}_{0}=0$ with potential 1 approaching the origin from the negative z-direction and potential 2 from the positive z-direction. The blue and red lines show the evolution of the ${n}_{z}$ values such that when the two potentials are merged then the asymptotic ${n}_{z}$ values become ${2n}_{z}$ and ${2n}_{z}+1.$ The green line illustrates the point when the two separate potentials begin to merge.

**Figure 7.**The energy levels of the DCHO given by Equations (19) and (24). The blue lines provide the symmetric solutions and the red lines provide asymmetric solutions as a function of the separation of the two potentials (z0).

**Figure 8.**Harvey characterization of the fusion of two

^{12}C nuclei with three different relative orientations. See the text for details.

**Figure 9.**Collision of two

^{12}C nuclei calculated using DCHO wavefunctions for a small impact parameter. The bottom panels show the evolving densities, and for different separations of the two potentials (labelled by $\left|Z\right|$), the middle panel shows the classical trajectories in the center-of-mass frame for the two nuclei and the top panel shows how the particles follow the energy solutions of the DCHO via the green dots.

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**MDPI and ACS Style**

Freer, M.; Davies, M.
Symmetries in Collisions as Explored through the Harmonic Oscillator. *Symmetry* **2024**, *16*, 231.
https://doi.org/10.3390/sym16020231

**AMA Style**

Freer M, Davies M.
Symmetries in Collisions as Explored through the Harmonic Oscillator. *Symmetry*. 2024; 16(2):231.
https://doi.org/10.3390/sym16020231

**Chicago/Turabian Style**

Freer, Martin, and Miriam Davies.
2024. "Symmetries in Collisions as Explored through the Harmonic Oscillator" *Symmetry* 16, no. 2: 231.
https://doi.org/10.3390/sym16020231