Alpha clustering in nuclear astrophysics and topology

Abstract

When we think of clustering in nuclear physics, the astrophysical importance within light nuclei and structural manifestations with classical analogs immediately come to mind. ⁴He, also known as the alpha particle, is the most abundant nucleus in the Universe, being quite tightly bound for its mass, with a first excited state of over 20 MeV. The nature of the alpha particle places it in a unique position within nuclear astrophysics and structure (including geometry). The plurality of energy release from stellar hydrogen fusion—whether quiescent or explosive—comes from the conversion of hydrogen to helium. Within more complex nuclei, the alpha particles are continuously arranged, leading to fascinating phenomena such as excited rotational bands, Borromean ring ground states, and linear structures. Nuclei with an equal and even number of protons and neutrons are colloquially referred to as “alpha conjugate nuclei,” where such special properties are the most pronounced and easiest to spot. However, when a single nucleon or a pair of nucleons is added to the system, alpha clustering not only remains evident but it may also be enhanced. Excited states with large alpha partial widths are a signature of clustering behavior, and these states can have a profound effect on the reaction rates in astrophysical systems when the excitation energy aligns with the so-called Gamow energy—the preferential thermal energy to statistically overcome the Coulomb barrier. In this article, we will consider in detail the specific ramifications of alpha clustering in selected scenarios for both nuclear astrophysics and topology. In particular, we discussed the astrophysical reactions of ⁷Li (α, γ), ⁷Be+α, ¹¹C (α, p), and ³⁰S (α, p), where α-clusters may increase the reaction rates from 10% to an order of magnitude; large α resonances make the astrophysical rate of ¹⁸F (p, α) quite uncertain. We also focused on the α rotational bands of both positive and negative parities of ¹¹B and ¹¹C, and finally on the strongest evidence for the linear-chain cluster state observed in ¹⁴C.

1 Introduction

The clustering of α particles in nuclei plays a critical role in both nuclear astrophysics and nuclear structure. Before delving into the topic, we will review the synthesis of several elements in the light and medium mass regions as they pertain to α clusters. We will then discuss by what empirical measurements and which types of experiments are most suitable to discuss α-clusterization. Once the groundwork is laid, we will review works published over the last dozen years on α-clusterization in the following nuclei: ¹¹B, ¹¹C, ¹⁴C, ¹⁵O, ¹⁹Ne, and ³⁴Ar.

Most of the aforementioned nuclei are radioactive, which often requires rare isotope beams (RIBs). However, clusterization has a long history even in stable isotopes. In particular, during the quiescent phases of stellar evolution of a massive star leading to an iron core in pre-collapse Type II supernovae, clusterization may not be limited to α-clusters, as shown in the Cluster Nucleosynthesis Diagram Figure 1, inspired by the Ikeda diagram [1]. In a massive star (10 M_⊙) according to one model, the core burning temperature, density, and lifetime are as follows: hydrogen, 37 × 10⁶ K, 3.8 g/cm³, and 7.3 × 10⁶ years; helium, 1.80 × 10⁸ K, 620 g/cm³, and 0.72 × 10⁶ years; carbon, 7.20 × 10⁸ K, 6.4 × 10⁵ g/cm³, and 320 years; neon, 1.9 × 10⁹ K,  g/cm³, and  years; oxygen, 1.8 × 10⁹ K, 1.3 × 10⁷ g/cm³, and 0.5 years; silicon, 3.4 × 10⁹ K, 1.1 × 10⁸ g/cm³, and  day; and collapse, 8.3 × 10⁹ K,  g/cm³, and 0.45 s [2]. From this example, it can be seen that α-clusters shape the evolution of stars more than any other clusterization burning, e.g., the Hoyle state is responsible for the nucleosynthesis of all heavier elements and helps set the timescale [3]. Although we will mainly discuss the ν-process, νp-process, hot carbon–nitrogen–oxygen (HCNO) cycles, and the αp process, the aforementioned stellar nucleosynthesis shows us that α clusters generally play an important role in such scenarios.

FIGURE 1

The Wigner limit, W, is frequently encountered in the literature, although it is strongly dependent on the radius, R, without any specific structural information, and there are several definitions. In this article, the maximum width of a resonance can be estimated as follows:where μ is the reduced mass of the channel, R is the channel radius, and P_ℓ is the channel penetrability for channel i [4,5]. The penetrability is calculated as where includes the phase space factor k, and F_ℓ and G_ℓ are the regular and irregular Coulomb functions, respectively. The dimensionless partial width is then , which can be used like a spectroscopic factor for a single-particle state. A typical radius can be calculated as , where R₀ is typically chosen from 1.2 to 1.6 fm and A is the atomic mass number. A more refined concept of α-clusterization can instead be calculated with the asymptotic normalization coefficient (ANC), but for basic discussions, the Wigner limit is sufficient. Although the Wigner limit is model-dependent, it is not widely agreed upon which θ² defines an α cluster (e.g., it might be 10%, 25%, or 50%). The basic idea is to obtain Γ_α from an experiment and simply compare it to Γ_W.

There are several experiments for determining Γ_α. Elastic α-scattering has a large cross section, although owing to the Coulomb barrier, elastic scattering of a charged particle rarely shows a resonant structure near the α separation energy, S_α, because of the Rutherford peak. The Trojan horse method is able to measure two-body reactions with extremely low cross sections via a two-to-three-body reaction in a quasi-free kinematics condition [6–8]. α-transfer reactions, such as (⁶Li, d) or (⁷Li, t), are also quite good near S_α and even below it for sub-threshold energies to obtain the ANCs of peripheral reactions [9]. Photodisintegration is useful to study the radiative capture on an unstable species when the recoil is stable (Utsunomiya et al. [10]), and moreover, the type and origin of the background differ from those of the radiative capture (Smith et al. [11]). Coulomb dissociation (Baur et al. [12]) and time reversal studies have sizable cross sections and can be used at higher beam energies. In the case when Γ_α is smaller than another competing partial width, say (α, p) or (α, n) direct measurements, application of the isolated, narrow-resonance condition for the reaction A (i, j)B aswhere γ is the reduced width [5] and the smaller partial width controls the contribution to the reaction rate.

Most of the experiments discussed in this paper use an RIB on gaseous helium targets using the thick target in the inverse kinematics (TTIK) technique [13], which has several advantages. First, the intensity of short-lived RIBs is several orders of magnitude less than the primary beam, so the statistics would be very poor using a thin target of implanted helium (Lennard et al. [14]). One way to increase the total yield is obviously using a thicker target. Second, the impinging beam loses significant energy within the target (often times being stopped entirely), and using the kinematic equations, the measured energy of the reaction point and, hence, the beam energy at that point can be deduced if the energy loss functions and the specific reaction are known. Because the reaction of interest is measured over a large range of energy, it, moreover, prevents the time-consuming retuning of the RIB. In this manuscript, all reactions are written in the astrophysical, normal kinematics nomenclature for convenience despite the fact that most of the experiments were performed in inverse kinematics.

2 Rotational bands and astrophysics of ¹¹B and ¹¹C

Lithium, beryllium, and boron are relatively less abundant than other light and medium mass nuclei, and the astrophysical sites of their production and destruction continue to pose challenges today, for example, the cosmological lithium problem (Hou et al. [15]; Hayakawa et al. [16] and references therein). The nuclei ¹¹B and ¹¹C are mirror nuclei, and some excited states show a dilute form of alpha-clusterization as either 2α + t or 2α +³He, respectively [17–19]. Furthermore, both have been observed to have positive- and negative-spin rotational bands, represented as α-clusterization (Yamaguchi et al. [20], Yamaguchi et al. [21] and references therein). These α-clusters can play an essential role in many explosive environments: ν-process, pp-V, rap (II, III, and IV), and the νp-process. The aforementioned processes may be found in a number of stellar environments: supermassive stars with low metallicity, novae, big-bang nucleosynthesis, and core-collapse supernovae. Considering the number of theoretical and experimental studies, the diverse manifestation of α-clusters, and a large range of astrophysical impact, the present review will mainly focus on ⁷Li (α, α) and ⁷Be (α, α).

The ν-process occurs in core-collapse supernovae where neutrinos are emitted from the proto–neutron star [22]. In the ν-process, ¹¹B is mainly produced via ⁴He (ν, ν′p)³H (α, γ)⁷Li (α, γ)¹¹B, although it can be enhanced by (anti-)neutrino inelastic and breakup reactions on ¹²C as well. Therefore, a precise measurement of ⁷Li (α, γ)¹¹B at GK can provide valuable information to reactions and oscillations of neutrinos. The ⁷Be (α, γ) reaction can either take place in the pp-V chain as ⁷Be (α, γ)¹¹C (β⁺ν)¹¹B (p, 2α)⁴He or in a number of the rap-processes and also the νp-process [23–25] (Section 4). Except in the case where ¹¹C decays ( min), this reaction can bridge the gap from the pp region to the CNO region, similar to the triple-α process. Although the (α, γ) reactions at high temperature are important on ⁷Li and ⁷Be, looking at Eq. 2, it is mainly Γ_γ (typically on order eV), spin-parity, and energy which dominate the analytic reaction rate from these alpha-clusters, not Γ_α. Experiments on α-scattering can determine or constrain the energy and spin-parity but not Γ_γ. The dominance of Γ_γ was recently confirmed by a direct measurement of ⁷Be (α, γ) [26, 27] when compared with the evaluation by Kelley et al. [28]. However, in converse to α scattering, no information was obtained on Γ_α and hence the cluster structure of ¹¹C.

An experiment on ⁷Li+α in inverse kinematics showed only ground-state transitions [20], and the total cross section of ⁷Li (α, p) was comparable to a normal kinematics experiment [29]. The best-fit R-Matrix parameters are shown in Table 1, although a major enhancement in the reaction rate compared to the Nuclear Astrophysics Compilation of REaction rates (NACRE) was not found [30]. Conversely, in the ⁷Be+α inverse kinematic experiment, several channels were seen, (α, α₀), (α, α₁), (α, p₀), and (α, p₁), measuring the charged particles and gamma rays in coincidence [21]. Freer et al. [31] also carried out a similar experiment, except not separating out the inelastic component. Although the two spectra show a basic similarity of the peak shape and relative location, Freer et al. [31] showed a factor of two to three lower cross sections, and the centroids of the peak structure were shifted down by approximately 500 keV. Yamaguchi et al. [21] were able to explain nearly all the resonances with known peaks rather than introducing new resonances. A factor of approximately 10% enhancement in the reaction rate over NACRE could be expected between 1.5 and 3 GK [30]. Although the experiments do not agree on the exact quantum values, it is clear alpha-clusterization is prominent in both sets of experiments, with the best-fit R-Matrix parameters shown in Table 2. Most recently, Psaltis et al. confirmed the astrophysical reaction rate for ⁷Be (α, γ) suggested in NACRE II [26, 27, 32] with a significantly improved precision.

Although the astrophysical reaction rate was not significantly changed by any of these measurements, the rotational bands in ¹¹B and ¹¹C are interesting from a structure perspective. Previously known rotational bands, as well as those found by Yamaguchi et al. are shown in Figures 2, 3. Both the K = 5/2⁺ and K = 3/2⁺ bands show a linear dependence of E_ex on J (J + 1) and may be related to their oblate, multi-centered α-cluster structure (Soić et al. [33] and references therein). Inspection of the figures indicates mirror levels offset by approximately 500 keV, which diminishes at a higher excitation energy, possibly from a phase transition from the shell model to cluster states (Freer et al. [31] and references therein). It might be strange that the ground state of ¹¹B, which is 3/2⁻, appears above the 1/2⁻ state, but Ragnarsson et al. [34] found a similar energy of the two states, suggesting the 3/2⁻ state is lower than expected. As the figures show, the negative parity bands in both ¹¹B and ¹¹C do not show a perfect linear correlation, and there is a commonality that the energies of two of the bandheads are lower than expected. A calculation using antisymmetrized molecular dynamics (AMD) qualitatively reproduced these findings, attributing this difference to a weaker interaction between the three-center cluster bandhead compared to a more rigid structure of higher-energy states [35]. Finally, the 9/2⁺, 12.4-MeV state in ¹¹C observed may be a missing member of the K = 3/2⁺ band.

FIGURE 2

FIGURE 3

3 Linear state and rotation in ¹⁴C

Cluster states where all the clusters fall in a line—the linear-chain cluster state (LCCS)—have been of theoretical interest since the 1950s [36]. Linear clusters are seen in other quantum mechanical systems, such as chemistry. Carbon dioxide (CO₂) shows a linear geometry with an atom of carbon covalently double-bonded between two oxygen atoms at 180°; the molecule can rotate and/or vibrate, which are also observed in the nuclear structure. The shapes of nuclei and their excited states have been frequently studied (and even used to explain certain shape isomers), but until the last several years, no experiment had found strong evidence to verify the LCCS in nuclei. From an intuitive perspective, we may think of modeling the nucleus as a pair of atoms, where most of them are joined into one of the clusters and only a small number of nucleons are used to bond them together. Efforts in the cluster shell model of adding a single nucleon to α-conjugate nuclei may, in the future, add multiple nucleons to gain better understanding of this phenomenon [37]. Initial theories on the LCCS looked at α-nuclei like ¹²C and ¹⁶O with no bonding nucleons, but other theoretical topologies better fit the experimental data. For example, the Hoyle state in ¹²C was found to be either a molecular-like state of ⁸Be+α or a weak coupling of three α-particles [38], and the 0⁺ state just above the four-α threshold in ¹⁶O showed quenching of the moment of inertia consistent with a superfluid state [39]. Until now, ¹⁴C is the only species where the experimental work shows a strong indication of a theoretical LCCS.

Suhara and Kanada-En’Yo [40] proposed a prolate band starting a few MeV above the α threshold of ¹⁴C as an LCCS using the AMD method. The wavefunction dominating the LCCS and an intuitive cartoon are shown in Figure 4. The LCCS was demonstrated to be stabilized by orthogonality to underlying states, and it has a nature of J^π = 0⁺, 2⁺, and 4⁺. Independently, several groups around the world began looking for this LCCS using a ¹⁰Be beam on a helium target for elastic scattering [41–43]. Similar to the ⁷Be case mentioned previously, Freer et al. [41] and Yamaguchi et al. [43] had a reasonable agreement in the spectral shape but the absolute cross sections did not agree. Unlike the case of ⁷Be, Freer et al. [41] found a cross section about a factor of four higher than Yamaguchi et al. [43]. As Freer et al. [41] used a normalization factor for the absolute cross section and Yamaguchi et al. [43] only relied on the experimental parameters, the latter might be considered more reliable. Similar to ⁷Be, the peak structure showed less consistency toward the lower energy side, where the energy loss function plays an important role. Although Freer et al. [41] did not find evidence for a prolate rotational band, Yamaguchi et al. [43] found striking similarity in the energy and spin for the LCCS of Suhara and Kanada-En’Yo [40], as shown in Figure 5. Fritsch et al. [42] used an active target but showed deviation from these results, such as much smaller absolute cross sections (even than Rutherford scattering), no goodness of fit between their theoretical and measured J^π, and a sharp E_{c. m.} locus for inelastic scattering.

FIGURE 4

FIGURE 5

Table 3 shows a strong indication of an LCCS, only plotting the relevant states. Although the widths show some differences, their average downward trend is the same. Moreover, it is not straightforward to ascertain from the AMD calculation, and so, the theoretical width was extracted by an overlap of the AMD and Brink wavefunctions [44]. The cause of this disagreement is an open question, which could be an experimental issue, a theoretical ambiguity, the physical states themselves, or some combination of the aforementioned issues. As the 4⁺ state is in a doublet with a 5⁻ state, it is possible that the 4⁺ component is actually stronger due to limited orthogonality between these two resonances. As Γ_α(5⁺) = 9.4% was measured experimentally, the maximum limit of this effect was evaluated as . Another possibility is that the 4⁺ resonance has a large Γ_n ≥ 300 keV, resulting in —although there are no predictions available, nearby resonances having Γ_n of a similar order. The AMD calculation also did not include several factors such as the rotational motion of ¹⁰Be, the radial motion of the α particle, and the possible fragmentation of the state; the first point is most relevant for the 0⁺ state, while the second aspect may explain the 2⁺ and 4⁺ widths. More work should be carried out to determine the values both experimentally and theoretically to determine if it is a true LCCS.

4 Astrophysics of the compound nucleus ¹⁵O

The νp-process occurs in the neutrino-driven wind of core-collapse supernovae [24, 25, 45], where owing to the neutrino interactions, the material is actually proton rich similar to the rp-process (Section 6). Light p-nuclei such as ^92,94Mo and ^96,98Ru, which cannot be explained by other astrophysical sites, might originate from this process. Because of the environment, a sequence like ⁷Be (α, γ)¹¹C (α, p) can compete with the triple-alpha process to bridge the gap between the pp-chains and the CNO region. In this case, it begins the αp-process, which can increase the energy generation by two or three orders of magnitude in less than one second. However, unlike the rp-process, some longer-lived species such as ⁵⁶Ni and ⁶⁴Ge can be bypassed by (n, p) reactions as a small amount of protons are turned into neutrons by neutrino interactions. The amount of the material to be processed depends on bridging the A = 5, 8 mass gaps, requiring knowledge of the ¹¹C (α, p) reaction. Most recently, the ¹¹C (α, p) reaction was found to be one of the two most important processes in the ν-process in core-collapse supernovae Yao et al. [46], as described in Section 2.

Prior to the work on a direct measurement, the compilations relied on the time-reversed reaction ¹⁴N (p, α)¹¹C via the activation method [30, 47]. The method is quite successful and determines the ¹¹C (α, p₀) rate at all angles, but it does not give any information on excited-state transitions. With a fairly simple setup (Figure 6), Hayakawa et al. [48] measured the (α, p₀), (α, p₁), and (α, p₂) reactions on ¹¹C without gamma-detectors (such as used to separate states in Section 2). The experiment used beam tracking monitors to get a system time-of-flight (TOF) for each incoming beam that coincided with the proton detected using a silicon strip detector. Using the TTIK method, the depth of the beam’s interaction in the target gas determines E_cm, which can be measured by the system TOF. The events are shown in Figure 7 for both high and low energies, where the difference in beam energy is from the energy loss in the second beamline monitor used (multi-channel plate or parallel plate avalanche counter). Protons show loci and the correct gradient along the calculated lines for p₀₋₂, except for broadening toward lower energy, owing to the resolution of the silicon detector.

FIGURE 6

FIGURE 7

The results show that the (α, p₁₋₂) cross sections were about an order of magnitude lower (maximum 20%) than the (α, p₀) rate. The summed reaction rate was higher than compilations as much as 40% [30, 47], specifically due to resonances at 0.9 and 1.35 MeV which were not considered. Conversely, the Hauser–Feshbach statistical rate over-estimates the total experimental rate at low energies but begins to match at higher temperatures [49]. As the Hauser–Feshbach rate suggests only a small difference between (α, p₀₋₂) and the total rate (α_all, p_all), the experiment has likely measured the main transitions.

No Γ_α partial widths were previously known. However, the statistics, resolution, and overlapping widths suggested that a new R-Matrix fit to the data was unlikely to bring new information. Some α-elastic scattering was observed at rather high energies but not reported since it was well over the three GK region of known astrophysical importance. Approximately 50 years ago, the α-cluster structure of ¹⁵O was studied up to 8 MeV over the α-threshold via the ¹²C (³He, α)¹¹C and ¹⁴N (p, α)¹¹C reactions (Weller [50], Weller [51]; Huttlin and Rollefson [52], and references therein). Weller assumed an α-particle of ℓ = 1 orbital angular momentum couples to one of the states in the ¹¹C core. The lowest-lying cluster states in ¹⁵O would have an excitation associated with the α-decay threshold, while higher-lying states are based on excitations of the ¹¹C core and different couplings of the α-particle with the core’s intrinsic angular momenta. To test this theory, Huttlin and Rollefson made higher-energy measurements of ¹⁴N (p, α)¹¹C and found some similarities with ¹²C (³He, α); the study was ultimately inconclusive as definitive spin-parity assignments were unavailable for many states, which began to overlap. Unfortunately, further investigation seems to have waned in both theoretical and experimental work on the α-cluster structure of ¹⁵O in the intervening years. Considering the continued astrophysical attention to the ¹¹C (α, p) reaction since the mid-2000s, additional studies of the α-cluster structure of ¹⁵O—including reanalysis of old data with modern methods—may be warranted.

5 Subthreshold alpha states in ¹⁹Ne

Typically, when a given channel has a significantly smaller width than its competing astrophysical channel, the smaller width dominates the analytic reaction rate, as shown in Eq. 2. However, when a nearby resonance has the same J^π, these resonances can interfere depending on the widths and sometimes drastically change the reaction probability. In general, it requires a direct measurement as the interference sign cannot otherwise be deduced, but measurements of the widths, spin-parity, and energy can reduce the uncertainty. Such resonances are often outside the Gamow window and can also be the subthreshold. A key example of this case is the α-cluster structure of ¹⁹Ne which affects the ¹⁸F (p, α) reaction, where the proton threshold is above the α-threshold, but the ℓ = 0 resonances interfere with each other (J^π = 1/2⁺, 3/2⁺) for temperatures below 0.3 GK [53, 54].

Classical novae occur in explosive binary star systems, typically with recurrence times greater than a thousand years. One partner is a white dwarf which accretes matter from a low-mass, less-evolved companion until the accreted material builds up enough pressure to induce thermonuclear runaway in the compact object’s envelope. In addition to supernovae and neutron star mergers, novae are the most powerful thermonuclear explosions in the universe. The ¹⁸F (p, α) reaction is one of the few remaining nuclear structure cases where the uncertainty is large enough to influence the outcome of classical novae simulations; the other two cases are ²⁵Al (p, γ) and ³⁰P (p, γ). The 2-h half-life of ¹⁸F is comparable to the timescale for the nova ejecta to become transparent to γ radiation, making it an ideal radiotracer of the explosion. Its decay to ¹⁸O emits a neutrino and positron, the latter of which annihilates with an electron to form a pair of γ-rays, and is thought to be the main source of 511-keV γ-rays from a nova explosion. However, thus far, no such signal from nearby novae has been detected by a satellite telescope.

From a nuclear physics perspective, it is difficult to calculate the maximum distance to a nova to determine the expected 511-keV flux because the ¹⁸F (p, α) rate remains quite uncertain despite nearly three decades of research since ¹⁸F was produced as a radioactive beam. Interference between ℓ = 0 states (J^π = 1/2⁺, 3/2⁺) near the proton threshold with higher-lying states is a major source of this uncertainty. There is a well-known 3/2⁺ resonance at E_r = 664.7 keV (E_ex = 7.0747 MeV) with Γ_p = 15.2 keV and Γ_α = 23.8 keV [55]. A broad 1/2⁺ state was predicted [56] and also observed by several works near E_r ≈ 1.5 MeV but not resolved [53, 57, 58]. A state with ℓ = 0 was resolved at E_r = 1.38 (3) with a smaller total width Γ = 130 (10) keV, as shown in Figure 8 [59]. The partial widths of the 1/2⁺ state remain uncertain, but it was shown that as long as the width is broad, the partial widths of Γ_p and Γ_α do not affect the interference pattern [60]. The uncertainty of 30 keV of the state also does influence the magnitude or location of the interference [60]. Enough information on the higher-lying states in ¹⁹Ne to constrain the experiment is extant, and so the focus of the community is looking at ℓ = 0 states nearby the proton threshold in addition to pushing the direct measurement to lower energies. According to mirror symmetry, there are two 3/2⁺ states and one 1/2⁺ state near the proton separation energy of ¹⁹Ne in ¹⁹F.

FIGURE 8

Several experiments were published in the last several years [59, 61, 62], which were analyzed by Kahl et al. [60]. Since then, new experiments [63, 64] and further analysis of the existing data were also presented [65, 66]. The subthreshold state at 6.132 (5) MeV was unambiguously identified as an ℓ = 0 transition by a model-independent method, as shown in Figure 8 [59], and its spin-parity in combination with that of Kahl et al. was determined as 3/2⁺ [64]. Another subthreshold state is approximately 6.29 MeV and was measured to be 1/2⁺ [67], which is now confirmed as a doublet [59, 61, 64, 68, 69]. It is noted that for low-spin states in this region, the calculated mirror energy displacement (MED) is 70 ± 50 keV [64], and pairing the 6.13 MeV state with a known 3/2⁺ state in ¹⁹F requires an MED of over 350 keV; this may suggest an unknown 3/2⁺ state in ¹⁹F. The final 3/2⁺ state(s) in this region should be above the proton separation energy, and two such states with tentative assignments were provided by Hall et al. [61] at 6.423 (3) and 6.441 (3) MeV.

The clusters in ¹⁹F and ¹⁹Ne have been described in terms of oscillator shells including s, p, and sd harmonics [56, 66]. The ℓ = 1 α cluster states (J^π = 1/2⁺, 3/2⁺) are interesting in a comparison with ²⁰Ne, where n = 3 nodes are Pauli-blocked in ¹⁶O+α but not in the A = 19 mirror case [66]. A broad 1/2⁺ state was paired with the 5.94-MeV state in ¹⁹F by the generator coordinate method (GCM) [56], and experiments looking for a broad state near 6 MeV in ¹⁹Ne have been reported [63]. Twenty years ago, Descouvemont and Baye [70] used the GCM and paired the same state with a resonance at 5.34 MeV in ¹⁹F rather than the 5.94-MeV state as they found the energies were up to 1 MeV smaller in experiment vs. theory. If one takes the experiments from Di Leva et al. [71] on ¹⁹F and Torresi et al. [72] for ¹⁹Ne, it can be seen there may be no missing state from GCM studies, as shown in Table 4. These and theoretical states are strongly coupled to the n = 4 node α channel, which were also assigned to the experimental energies shown here [66]. Assuming the aforementioned analysis is correct, the “missing,” broad, 1/2⁺ state in ¹⁹Ne has no significant interference effect on ¹⁸F (p, α), being shielded by another resonance, having significantly smaller Γ_α, and being far from the proton threshold.

The ¹⁸F (p, α) reaction requires not only knowledge of ¹⁹Ne but also further investigation of its mirror ¹⁹F.

6 Astrophysics of the compound nucleus ³⁴Ar

Type I X-ray bursts (XRBs) are similar to classical novae, except the compact object is a neutron star rather than a white dwarf. As such, XRBs are a class of astronomical phenomena where in a binary pair, a low mass star accretes matter onto a neutron star and regular explosions take place. Indeed, there are over 100 known XRBs within our galaxy. Characteristic bursting behavior shows a very sharp rise time (one to 10 s) and then a thermal decline. The energy output is around 10³⁹–10⁴⁰ ergs for 10–100 s, and the bursts repeat much more frequently than in classical novae, in the order of hours to days, owing to the compactness of the neutron star. The amount of the accreted material per time, , is inversely proportional to the luminosity and time between bursts because the HCNO cycle can operate longer between bursts and increase the amount of helium. The sharp rise time and maximum luminosity are modeled as coming from a series of (α, p) (p, γ) reactions [73], for example, starting from ¹⁴O and going to around ³⁸Ca. Schematically, this is pure helium burning without waiting for β-decays as the hydrogen content is not changed.

³⁴Ar is, like ¹⁴O, an nucleus and important for X-ray bursts in the αp-process as the compound reaction for ³⁰S+α. The species is an α nucleus (²⁸Si) with two additional protons. The ³⁰S (α, p) reaction is identified as a potentially important reaction, contributing more than 5% to the total energy generation (Parikh et al. [74]), influencing the elemental abundances in the burst ashes (Parikh et al. [74]) relevant to compositional inertia (Taam [75] for a description of this phenomenon), moving the material away from the ³⁰S waiting point (Iliadis et al. [76]), and possibly accounting for double peaked XRBs (Fisker et al. [77]). Another study found the ³⁰S (α, p) reaction sensitivity in XRBs among the top four in a single-zone model (Cyburt et al. [78]), as well as having a prominent (but unquantified) impact on the burst light curve in a multizone model.

The only experimental paper to measure Γ_α in ³⁴Ar is by Kahl et al. [79] using an active target. The center-of-mass energies of that study are near 3 GK, being rather higher than the peak XRB temperature of 1.3–2.0 GK. There have been several theoretical works to determine ³⁰S (α, p), but here, we focus mainly on the experimentally determined values for Γ_α. Three states with the proposed α-cluster structure were observed between approximately 4.3 and 5.5 MeV, which had ranging from 8% to 100%, as shown in Table 5. The dip-like structures are shown in Figure 9 along with χ². It may be unusual if all the strong resonances have J^π = 2⁺, and if the first resonance observed is 4⁺, the order is the opposite for a rotational band. However, a measurement of ³⁶Ar (⁶Li, d)⁴⁰Ca showed several resonances of the same spin-parity emerged at similar energies with large spectroscopic factors (Yamaya et al. [80]), including two multiplets with 2⁺ each, and ⁴⁰Ca is nearby in mass to ³⁴Ar.

FIGURE 9

The ³³Cl (p, α) time-reversal reaction was measured previously to constrain the ³⁰S (α, p) reaction around 3 GK [81]. They found a comparable cross section from the NON-SMOKER code, whereas Kahl et al. [79] found a factor of 10 times increase in the reaction rate compared to the statistical model [49]. The difference could be attributed to Deibel et al [81] measuring the ³⁰S (α₀, p₀) rate, similar to the case of ¹¹C (α, p). These rates are, as mentioned previously, outside the astrophysical region of interest for XRBs. However, Long et al. [82] performed an ³⁶Ar (p, t) spectroscopy experiment over the XRB Gamow window with 30-keV resolution. No ³⁴Ar states were observed with a width larger than the resolution. Therefore, they estimated Γ_α for the levels they observed from a shell-model calculation and found a cross section in the astrophysical region of interest significantly lower than that of the NON-SMOKER statistical model. Unfortunately, as the experiment did not overlap with the other two studies, a comparison between their results at different energies is not possible. More experiments to measure Γ_α in ³⁴Ar and also ³⁰S (α, p) at astrophysical energies are certainly warranted. According to the data in Table 5, Γ_α either dominates or contributes strongly to the ³⁰S (α, p) resonant reaction rate (at least at higher temperatures). A confirmation of the total widths of these states would be helpful to rule out contributions from Γ_p.

7 Conclusion

We presented studies on several nuclei both from a nuclear astrophysics and/or structural perspective. Alpha-clusters are found in a variety of lighter-mass species on both sides of the valley of stability. Although most of the works call for future studies, we emphasize that it is not necessarily finding alpha-clusters in the atomic nuclei but why. It is hard to imagine that alpha-clusters do not play an essential role in the nuclear structure, particularly in light nuclei. Yet, simply identifying them without an overarching framework would likely lead to a preponderance of data, looking for a good model or scenario to justify. We showed that the application of alpha-clusters in the selected nuclei spread over a large, somewhat mutually exclusive, domain.

For high-temperature astrophysical reactions like (α, γ), α, clusters may be involved, but typically, Γ_γ has the smallest width and thus, dominates the rate. On the other hand, for (α, p) reactions, despite clusterization, which leads to large Γ_α values, these reduced widths may be quite important even if Γ_p is just a few percentage of the Wigner limit. We infer a similar situation is true with (α, n) reactions but did not review them. Sometimes, from the interference between states of the same J^π, often from the α-width, it can have a huge impact on the lower energy rate like ¹⁸F (p, α), and the interference sign cannot be predicted by theory. We found that positive parity, α-cluster rotational bands show an excellent linearity, whereas negative parity bands require deeper understanding for the non-linear structure. Finally, we reviewed the strongest experimental evidence for an experimental linear-chain cluster state in ¹⁴C.

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