Cluster structure of 3$\alpha$+p states in ${}^{13}\mathrm{N}$

The most well-known instance of $\alpha$ clustering in light nuclei is perhaps that of the Hoyle state in ¹²C Freer and Fynbo (2014). When additional protons or neutrons are added to this system, the propensity for clustering is of interest to study to understand the phenomenon of clustering. While clustering in ¹³C and ¹⁴C has been examined through resonant scattering - the unbound nature of ⁹B means that these data for ¹³N are not accessible. Instead, one must use different mechanisms to populate these highly-excited exotic states. Combined with the experimental difficulties of observing the 3$\alpha$+p decay of these states (as the characteristic clustered decay mode), a high-sensitivity experimental approach is required.

To probe these $\alpha$-clustered states in ¹³N, $\beta$-delayed charged-particle spectroscopy was used to populate states in ¹³N via ¹³O and decays to a final state of 3$\alpha$+p were then measured. To achieve this, the Texas Active Target Time Projection Chamber (TexAT TPC) was used to perform one-at-a-time implantation and decay which has been demonstrated previously to have a very high sensitivity to rare decays due to the absence of background Bishop et al. (2020a, b, 2021). This paper provides more details on this approach and deeper insight into the states observed in a previous paper detailing the first observation of the $\beta 3\alpha p$ decay channel Bishop et al. (2023).

The experimental setup utilized for this experiment follows that successfully applied to studying the rare decay modes of near-threshold $\alpha$-clustered states in ¹²C Bishop et al. (2020a, b, 2021) via the $\beta$-delayed 3$\alpha$-particle decay of ${}^{12}\mathrm{N}$ via ¹²C^⋆. Instead, the $\beta$-delayed 3$\alpha p$-particle decay of ${}^{13}\mathrm{O}$ via ¹³N^⋆ is studied here.

The K500 Cyclotron at Texas A&M University was used to produce a beam of ¹⁴N which was incident on a ³He gas cell to produce a secondary beam of ¹³O via the ${}^{14}\mathrm{N}(^{3}\mathrm{He},xn)^{13}$O reaction. The ¹³O of interest was then selected using the MARS (Momentum Achromat Recoil Separator) Tribble et al. (1989) with a typical intensity of 5 pps and an energy of 15.1 MeV/u. The beam was then degraded to 2 MeV/u using an aluminum foil in order to stop the beam inside of the sensitive area of the TexAT TPC Koshchiy et al. (2020) in 50 Torr of CO₂ gas. The one-at-a-time $\beta$-delayed charged-particle spectroscopy technique requires the implantation of the $\beta$-decaying nucleus ¹³O into the active area of the TexAT TPC (with $t_{\frac{1}{2}}$ = 8.58 ms). The implanted ion then subsequently decays with the TPC volume being insensitive to the $\beta^{+}$ in the gas (and the subsequent $\beta^{+}$ from the decay of ¹³N for some events). When states in ¹³N are populated above the particle decay threshold (1.944 MeV for the p-threshold and 9.496 MeV for the $\alpha$-threshold), the daughter nucleus can undergo particle decay and the recoil products are measured inside the TPC.

As the ion is incident into the TPC active volume, the implantation beam track triggers the GET electronics setup Pollacco et al. (2018) which sends a 30-ms-long busy signal to the K500 phase shifter which blocks the transmission of the primary ¹⁴N through the cyclotron and therefore prevents any subsequent implantations. During the 30 ms busy signal, the data acquisition then awaits an additional event corresponding to the decay of ¹³N^⋆ through the proton or $\alpha$ decay channels. For the majority of events, this second trigger is not generated as the ¹³O decays to states in ¹³N that are particle bound. After either 30 ms has elapsed or a decay event has registered, the data acquisition is ready for a new implantation event and the K500 phase shifter signal is disabled. This setup is known as the ‘2p-mode’ in the GET system Mutant module and allows for the beam implant and decay events to be correlated. The implantation from the beam is referred to as the L1A trigger and requires 10 channels in TexAT to be above threshold. The decay event is known as the L1B trigger and requires only a multiplicity of 1 to allow for very low-energy recoils to trigger the data acquisition. The time between the L1A and L1B event (known as the d2p time) is also recorded in the GET system and corresponds to the decay time of the $\beta$-delaying particle. Partial events where an L1A trigger was not followed by an L1B trigger were also recorded for normalization and beam characterization. An overview of this experimental setup is shown in Fig. 1.

To select events of interest, the full L1A (implant) + L1B (decay) events were selected where the time between the two was between 1 and 30 ms (with small times omitted to remove double trigger events due to sudden beam-induced noise). In addition to the secondary ¹³O of interest, there were also some other beam contaminants. Therefore, to ensure the implanted ion corresponded to ${}^{13}\mathrm{O}$, the energy deposited by the beam implant event in the Micromegas “Jr” (MM Jr) beam tracker Holmes et al. (2020) at the entrance to the TexAT chamber was recorded. The beam contaminants were ${}^{7}\mathrm{Be}$ and ${}^{10}\mathrm{C}$, dominated by ${}^{7}\mathrm{Be}$ at $\approx$ 28% of the beam intensity. The energy spectrum from the Micromegas Jr is shown in Fig. 2.

Following a selection of ${}^{13}\mathrm{O}$ implants, the stopping position of the beam was evaluated event-by-event and events where the beam stopped inside the active area of the Micromegas were taken, events where a signal was observed in the zero-degree silicon detector were automatically vetoed as signifying the beam did not stop in the gas. Because of the proximity of some events to the edge of the active region, only events which were more than 31.5 mm from the edge of the Micromegas were taken for further analysis. Due to the large energy degradation required to stop inside TexAT, the spread in the ${}^{13}\mathrm{O}$ stopping position was 67.5 mm.

To ensure the implant and decay events are correlated, the 2D distance (in the plane of the Micromegas) between the stopping position of the implanted beam and the decay vertex (identified by the highest energy deposition inside the active volume) is then evaluated. Following previous results Bishop et al. (2020a), the separation for correlated events is given by:

with $\sigma$ = 4.9 mm for these data which agrees well with a predicted diffusion value of $\approx 4$ mm from the ideal gas law. A selection of events with a displacement of $<5$ mm were taken as correlated. To determine the purity of this selection, the time between the implantation and decay for these events was accumulated (the d2p time). A background-free exponential fit yielded a half-life value of 8.55$\pm$0.09 (stat.) ms compared to the adopted value of 8.58$\pm$0.05 ms ($\tilde{\chi}^{2}$ = 1.26) which is shown in Fig. 3. The excellent agreement without the need for any background demonstrates the purity of the selection of ${}^{13}\mathrm{O}$ in the system.

Tracks from these events were fit with a single track segment using a randomly-sampled $\chi$-squared minimization algorithm (more detail on this fitting technique is covered in Sec. V). For those tracks for which the reduced $\chi$-squared was good, these events were identified as single proton events and are discussed in Sec. IV. For those which gave a poor reduced $\chi$-squared, these events were fit with four track segments as candidate 3$\alpha$+p events using randomly-sampled $\chi$-squared minimization and were visually inspected to evaluate the quality of fit and also identify any events which were unable to be fit with the algorithm (given the complexity of the fit required). These 3$\alpha$+p events are discussed in Sec. V.

Due to the low gas pressure used, the majority of proton event tracks escape the TPC active volume. In order to evaluate the state populated in ${}^{13}\mathrm{N}^{\star}$, the recoiling ${}^{12}\mathrm{C}$ was therefore used instead which has 1/12^th of the proton energy and a much higher dE/dx therefore stops safely inside the active volume.

In order to calculate the ${}^{12}\mathrm{C}$ recoil energy, the decay event is fit with a single line segment defined by two points: the end point of the proton track and the end point of the ${}^{12}\mathrm{C}$ track. The decay vertex then dissects this line segment and the energy in the point cloud that is on the recoil side of the fit was also added to the ${}^{12}\mathrm{C}$ recoil energy. Fig. 4a shows the relationship between the recoil energy and range from the track overlaid with the expected TRIM result which allowed for an additional cut on the data. The events that lie outside of this cut constitute events where the separation between the proton and ${}^{12}\mathrm{C}$ recoil could not be accurately attained. The number of events lost through this cut is $<4\%$ which has little effect on the branching ratios but removes all background at higher excitation energies. Given the typical energy of the recoil is from 0.12 to 1.1 MeV (corresponding to track lengths of 6.7 to 30.5 mm), the resolution of the energy spectrum using this technique is poor and the proton energies and relative intensities from previous studies Knudsen et al. (2005) are used to demonstrate the compatibility of our result with those previously observed. The excitation function obtained is shown in Fig. 4b with the expected yield overlaid from previous studies Knudsen et al. (2005) convoluted with a Gaussian response. Any deviation can be attributed to the dynamics of the gas avalanche that is better characterized as a Breit-Wigner convoluted with a Gamma distribution (representing the gas gain dynamics), convoluted with a Gaussian (representing the resolution associated with noise and difficulties associated with fitting the ¹²C recoil track). This also confirms the implanted number of ${}^{13}\mathrm{O}$ that make it through our cuts is 1.90$\times 10^{5}$ from 1.86$\times 10^{4}$ proton decay events. An independent branching ratio measurement from the number of implants was not reliable during this experiment due to a sizeable noise contribution which adversely affected the L1A/L1B ratio (corresponding to the branching ratio to particle-unbound states) but was necessary to ensure 100% trigger efficiency on genuine proton events.

A total of 149 3$\alpha$+p events were identified. Due to the size of the TPC and limitations on reconstruction in parts of the TexAT TPC, only 102 out of 149 of these events could be fully-reconstructed. These events that are lost are almost exclusively from $\alpha$-decay as this produces a high-energy $\alpha$-particle that may escape. The efficiency for the $\alpha_{0}$ decay starts to deviate from 100% at $E_{x}$ = 10 MeV, slowly drops to around 60% at $E_{x}$ = 14 MeV. The efficiency for $\alpha_{1}$ and $\alpha_{3}$ are less affected and only decrease to 70% at $E_{x}$ = 14 MeV. For events that proton decay to the Hoyle state, the majority of the energy is taken by the proton which is not required for reconstruction. Corrections to the yields obtained during this work are made to account for this effect in Table 1.

In order to accurately fit the four-track events, a highly-robust fitting technique is required. In order to achieve this, the ansatz for the reaction vertex is identified by the stopping position of the beam and then the point cloud for the decay events is fit with 15 parameters: the decay vertex and the endpoint of each of the four tracks (with each being a 3D vector). Due to the highly complex nature of the fit with many free parameters and a ‘noisy’ fitting space, a modified version of RANSAC Fischler and Bolles (1987) was used that has been successfully employed in several other TexAT experiments whereby the four parameters are randomly selected from the set of points in the pointcloud (as per RANSAC) and then a goodness of fit is evaluated by the sum of the distance-squared for all points to the nearest line defined by the vertex to each of the three endpoints. It is better characterized as RANdomly-Sampled Chi-Squared Minimization, referred to here as RANSChiSM. To reduce the influence of outliers, where the distance to all three lines exceeded 10 mm, the distance was saturated to be 10 mm. This functional form is given by:

where $\vec{P_{i}}$ is the i^th point in the pointcloud, $\vec{V_{j}}$ is the vector for the endpoint of the $j^{th}$ track, and $\vec{L_{j}}$ is the vector from the decay vertex to the endpoint of one of the three tracks. The vector product here merely calculates the shortest 3D distance of the point $\vec{p_{i}}$ to the line $\vec{L_{j}}$ To ensure the point that is chosen is at the end of the track, if the distance between the point and the vertex exceeds the distance between the vertex and the end of the arm, then the distance between the point and the end of the arm is taken instead, weighted by a factor of 10. This was seen to be very effective at ensuring the proper selection of the end of the tracks which is necessary for the conversion between range in the gas and the energy. To reduce the number of random samples of endpoints and the decay vertex, one endpoint was defined as that which had the largest distance from the decay vertex and a weighting scheme was used where two of the remaining three endpoints were randomly sampled with a weight given by the distance of the point in the pointcloud to the randomly-chosen decay vertex squared. Similarly, only points within 5 mm of the known beam stopping point were allowed to be selected for the decay vertex point. A schematic of this fitting is shown in Fig. 5. For the $\beta 3\alpha p$ events, 20,000 iterations were sufficient to converge on the best fit. Each decay was examined manually and those that were not perfectly fitted were reanalyzed and, if needed, the parameters for certain tracks were constrained manually and the fit reevaluated. For some extremely short tracks, this was necessary due to the presence of many ‘fake’ minima in the chi-squared phase space.

In order to identify the parent state in ${}^{13}\mathrm{N}^{\star}$, the lowest energy deposition arm was identified as the proton track and the momentum of the 3 $\alpha$-particles was determined by the length and direction of the track in the gas. As the proton almost always escapes the TPC sensitive volume, the proton momentum is reconstructed from momentum conservation. The decay energy is then the sum of the three $\alpha$-particles and proton energy. From here, the ${}^{8}\mathrm{Be}$ (Fig. 6), ${}^{9}\mathrm{B}$ (Fig. 7) relative energies and ${}^{12}\mathrm{C}$ (Fig. 8) excitation energies were determined from the invariant mass. This allowed for a selection of events which proceeded to decay via p+${}^{12}\mathrm{C}(0_{2}^{+})$ [$p_{2}$], $\alpha$+${}^{9}\mathrm{B}$(g.s) [$\alpha_{0}$], $\alpha$+${}^{9}\mathrm{B}(\frac{1}{2}^{+})$ [$\alpha_{1}$] and $\alpha$+${}^{9}\mathrm{B}(\frac{5}{2}^{+})$ [$\alpha_{3}$]. An example $p_{2}$ event is shown in Fig. 9 and an example $\alpha_{0}$ event is shown in Fig. 10. It is remarkable to see in Fig. 7 evidence of strength in ${}^{9}\mathrm{B}$ between 1 and 2.4 MeV that cannot be explained without the long-sought after $\frac{1}{2}^{+}$ state in ${}^{9}\mathrm{B}$ that is the mirror of the well-studied ${}^{9}\mathrm{Be}$ $\frac{1}{2}^{+}$. The ground state contribution (from Monte Carlo simulations) and the higher-lying states from a single-channel Breit-Wigner convoluted with a Gaussian ($\sigma$=0.23 MeV) were fitted to the spectrum. Attempting to fit the spectrum with and without the $\frac{1}{2}^{+}$ contribution shows the probability such a peak can occur by chance is 0.01% demonstrating this contribution can only occur due to a contribution from the $\frac{1}{2}^{+}$ due to the absence of any background. The $\frac{1}{2}^{+}$ state in ${}^{9}\mathrm{B}$ was selected by taking an excitation energy of between 1.4 and 2.4 MeV in ${}^{9}\mathrm{B}$ (following the centroid and width as observed via ${}^{9}\mathrm{Be}(^{3}\mathrm{He},t)$ Wheldon et al. (2015) which is consistent with our current results) and the $\frac{5}{2}^{+}$ was taken as having an excitation energy of above 2.4 MeV. A more recent experimental study has suggested the possibility that discrepancies in experimental results are the result of a doublet of $\frac{1}{2}^{+}$ states Odsuren et al. (2023). Any contribution from the relatively-narrow 2.345 MeV $\frac{5}{2}^{-}$ is not present in the presented plots as this state decays almost exclusively via ${}^{5}\mathrm{Li}$ and therefore would not correspond to a peak in the ⁸Be spectrum. There were only 3 events associated with this decay to ${}^{5}\mathrm{Li}$ hence the statistics were insufficient to incorporate into the analysis.

Following the channel selection, the excitation energy in ${}^{13}\mathrm{N}$ was calculated and is shown in Fig. 11. Despite low statistics, a number of states can be seen and will be discussed individually. A summary of the properties of these states observed is then shown in Table 1. A GEANT4 simulation was performed to test the variation in experimental resolution as a function of excitation energy for the $\alpha_{0}$ channel which, as shown in Fig. 12, is typically around $\sigma$ = 200 keV. The $p_{2}$ channel resolution is almost entirely dominated by discrepancies between the expected and real stopping powers for the $\alpha$-particles and therefore cannot be accurately determined and is extremely sensitive to small changes. For all excitation energies, it is realistically greater than $\sigma$ = 160 keV however.

Three new states and a previously-tentative state have been observed with a strong $3\alpha+p$ nature with their excitation energies relative to the thresholds shown in Fig. 13. The first is a narrow potential $\frac{3}{2}^{-}$ state at $E_{x}$ = 11.3(1) MeV with mixed ${}^{9}\mathrm{B}(\mathrm{g.s})\bigotimes\alpha$ and $p+^{12}\mathrm{C}(0_{2}^{+})$ nature.

Another previously-observed $\frac{3}{2}^{-}$ was seen to have mixed $p+^{12}\mathrm{C}(\mathrm{g.s.})$ and $p+^{12}\mathrm{C}(0_{2}^{+})$ nature at 11.8 MeV with around half of the total strength as $p+^{12}\mathrm{C}(\mathrm{g.s.})$.

At higher excitation, another strong $\alpha$-decaying state was seen at $E_{x}$ = 12.4(1) MeV although this state has a much stronger ${}^{9}\mathrm{B}({\frac{1}{2}^{+}})\bigotimes\alpha$ nature.

A revised excitation energy of 13.1(1) MeV is suggested for a previously-seen state at 13.26 MeV. The ${}^{9}\mathrm{B}({\frac{5}{2}^{+}})\bigotimes\alpha$ structure dominates in this state and a spin assignment of $J^{\pi}$ = ${\frac{1}{2}^{-}}$ or ${\frac{5}{2}^{-}}$ are therefore preferred.

Finally, another $\frac{3}{2}^{-}$ is seen at 13.7 MeV which is also dominated by ${}^{9}\mathrm{B}({\frac{5}{2}^{+}})\bigotimes\alpha$.

The inability to extract the width of these narrow states means that the magnitude of clustering cannot be fully evaluated however the type (channel) of clustering can be determined without this information. Higher resolution data focusing on the proton channel may provide further information on the magnitude of this clustering phenomenon. From our current data, one may conclude however when comparing the reduced widths that the clustered channels are very competitive against the single-particle $p_{0}$ channel. Configuration mixing may however be very strong in these states and therefore quench the single-cluster nature of these resonances.

One can compare these states with $3\alpha$+p events observed via other reactions such as those populated by single proton decay from highly-excited states in ¹⁴O^⋆ in complete kinematics Charity et al. (2009). Due to the spin-parity selectivity of $\beta$-decay and the unusual population method of states in ¹³N^⋆ in the previous work, it is perhaps not surprising that the same states are not observed in the current work.

Work using ¹³C(³He,t) by Fujimura et al. has also been performed which, at high energy, is a better analogue to inverse $\beta$-decay and should preferentially populate similar states although the ground state spin-parity of ¹³C is $J^{\pi}=\frac{1}{2}^{-}$ rather than $J^{\pi}=\frac{3}{2}^{-}$ as for ¹³O Fujimura et al. (2004). This work relies on the excitation energy being extracted by missing mass and the decay channels tagged solely by protons to separate $p_{0}$, $p_{1}$, $p_{2}$ etc. A broad peak at $E_{x}$ = 13.5 MeV can be seen in the $p_{2}$ channel. This lies close in energy to our observed peak at 13.8 MeV which, with a preferred $J^{\pi}=\frac{3}{2}^{-}$ agrees with the state being observed with (³He,t). The 11.8 MeV states, also observed previously by Knudsen was also seen to be very strongly populated in the Fujimura data. Their data show a smaller $p_{0}$ and $p_{2}$ yield and a dominant $p_{1}$ yield which is at odds with the Knudsen result which ascribed the yield in this region to the $J^{\pi}=\frac{5}{2}^{-}$ which would not be populated strongly by Fujimara. Is it therefore possible that in the current work and that of Knudsen, the $J^{\pi}=\frac{3}{2}^{-}$ and $J^{\pi}=\frac{5}{2}^{-}$ are both contributing strength here in different channels. Alternatively, in the work by Knudsen the $p_{1}$ yield could have been partially obscured by the strong $p_{0}$ yield for the 7.376 MeV state. Further studies with higher spin-sensitivity are necessary to disentangle these two contributions however.

We hope this experimental work will motivate further theoretical studies for ¹³N including using the Algebraic Cluster Model (D${}^{{}^{\prime}}_{3h}$) as performed for ¹³C Bijker and Iachello (2019) and AMD calculations.

This work was supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Science under Award No. DE-FG02-93ER40773 and by the National Nuclear Security Administration through the Center for Excellence in Nuclear Training and University Based Research (CENTAUR) under Grant No. DE-NA0003841. G.V.R. also acknowledges the support of the Nuclear Solutions Institute. S.A., S.M.C., C.K., D.K., S.K. and C.P. also acknowledge travel support from the IBS grant, funded by the Korean Government under grant number IBS-R031-D1. C.H.K acknowledges travel support from the National Research Foundation of Korea(NRF) grant, funded by the Korea government(MSIT) (No. 2020R1A2C1005981 and 2013M7A1A1075764).