A New Reaction Rate of the ²⁷Al(p/α)²⁴Mg Reaction Based on Indirect Measurements at Astrophysical Energies and Implications for ²⁷Al Yields of Intermediate-mass Stars

The indirect label is used to indicate those techniques making it possible to deduce the astrophysical S-factor (Iliadis 2015) of a reaction by measuring the cross section of a closely related process, and by applying nuclear reaction theory to link the two. In the present work, we have investigated the ²⁷Al(p,α)²⁴Mg reaction using the THM, which is an experimental indirect technique successfully applied to study several astrophysically relevant reactions by appropriate three-body quasi-free (QF) processes (Tribble et al. 2014; Spitaleri et al. 2019). One can briefly describe the technique as follows (see Figure 1), for the case under investigation. A ²⁷Al projectile is accelerated onto a deuteron target nucleus, having a very simple p + n structure, the p − n motion taking place mostly in an s-wave (Lamia et al. 2012). Under selected kinematic conditions, ²⁷Al interacts only with the proton (or neutron), which is the participant in the Trojan Horse (TH) reaction, then the neutron (or proton) is emitted without taking part in the ²⁷Al + p(n) interaction (therefore it is labeled as spectator). This is enforced by selecting small p − n relative momenta, corresponding to large p − n relative distances. Following the QF interaction, ²⁸Si excited states are populated (lower vertex of Figure 1), later decaying into the ²⁴Mg + α channel to be selected in the offline analysis.

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If the bombarding energy is chosen to overcome the Coulomb barrier in the entrance channel of the d(²⁷Al, α²⁴Mg)n reaction, the participant proton is brought into the nuclear field of ²⁷Al to induce the ²⁷Al(p,α)²⁴Mg reaction. Moreover, since the beam energy can be compensated for by the deuteron binding energy, the two-body reaction can take place in the energy region of astrophysical relevance of the ²⁷Al + p interaction. Since the scrutiny of the resonance list in Iliadis et al. (2010) shows that the low-energy cross section of the ²⁷Al(p,α)²⁴Mg reaction is dominated by narrow resonances, in comparison with the typical THM energy resolution of a few tens of keV (FWHM), the THM approach for narrow resonances thoroughly discussed in La Cognata et al. (2010) and the corresponding equations have been adopted in the present work. Here, we review the basic ones.

The TH double differential cross section for the d(²⁷Al, α²⁴Mg)n reaction, integrated over $d{{\rm{\Omega }}}_{{\hat{{\bf{k}}}}_{\alpha {}^{24}{Mg}}}$, is given by (see Equation (6) of La Cognata et al. 2010):

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{d}^{2}{\sigma }^{\mathrm{TH}}}{{{dE}}_{\alpha {}^{24}{Mg}}\,d{{\rm{\Omega }}}_{{\hat{{\bf{k}}}}_{n{}^{28}\mathrm{Si}}}}\\ =\,\displaystyle \frac{1}{2\pi }\displaystyle \frac{{{\rm{\Gamma }}}_{\alpha {}^{24}{Mg}}\left({E}_{\alpha {}^{24}{Mg}}\right)}{{\left({E}_{\alpha {}^{24}{Mg}}-{E}_{R}\right)}^{2}+\tfrac{1}{4}\,{{\rm{\Gamma }}}^{2}({E}_{\alpha {}^{24}{Mg}})}\\ \times \,\displaystyle \frac{d{\sigma }_{d{\left({}^{27}\mathrm{Al},n\right)}^{28}\mathrm{Si}}}{d{{\rm{\Omega }}}_{{\hat{{\bf{k}}}}_{n{}^{28}\mathrm{Si}}}}\end{array}\end{eqnarray} \tag{ 1 }$$

under the simplifying assumption that a single resonant state at energy E_R is populated; in the equation above, $\displaystyle \frac{d{\sigma }_{d{{(}^{27}\mathrm{Al},n)}^{28}\mathrm{Si}}}{d{{\rm{\Omega }}}_{{\hat{{\bf{k}}}}_{n{}^{28}\mathrm{Si}}}}$ is the differential cross section for the stripping $d{{(}^{27}\mathrm{Al},n)}^{28}\mathrm{Si}$ to the resonant state of ²⁸Si, ${E}_{\alpha {}^{24}{Mg}}$ is the α−²⁴Mg relative energy, and Γ and ${{\rm{\Gamma }}}_{\alpha {}^{24}{Mg}}$ are the total and partial width for the resonance under examination. From the comparison with the resonant cross section in the Breit–Wigner approximation for a direct measurement, it is clear that a straightforward connection with the resonance strength (Iliadis 2015) can be established. This is a very important point since, in the case of reactions controlled by narrow resonances, the reaction rate is essentially determined by the resonance energies and strengths. See La Cognata et al. (2010b) for the description of the broad-resonance case.

Following the discussion and using the same notation as in La Cognata et al. (2010), if N_i is the area of the ith peak in the THM cross section described by Equation (1) and ω_i is the statistical factor (Iliadis 2015), the strength (ω γ)_i of the ith resonance is given in arbitrary units by:

$$\begin{eqnarray}&&{\left(\omega \gamma \right)}_{i}=\displaystyle \frac{1}{2\pi }{\omega }_{i}{N}_{i}\displaystyle \frac{{{\rm{\Gamma }}}_{{p}^{27}\mathrm{Al}}^{s.p.}}{\tfrac{d{\sigma }_{d{\left({}^{27}\mathrm{Al},n\right)}^{28}\mathrm{Si}}^{s.p}}{d{{\rm{\Omega }}}_{{\hat{{\bf{k}}}}_{n{}^{28}\mathrm{Si}}}}},\end{eqnarray} \tag{ 2 }$$

where the superscript s. p. is used to indicate that single-particle wave functions should be used in the calculation of the quantities therein, therefore devoid of explicit spectroscopic factors. As discussed in La Cognata et al. (2010), the rightmost factor is then an easily calculable and unambiguous quantity.

While, in principle, the strengths could be deduced in absolute units (see, e.g., the discussion in La Cognata et al. 2009), in the THM application to resonant reactions high accuracy is achieved by scaling the indirectly deduced strengths to measured ones at high energies. This makes it possible to use the simplifying plane wave approximation, while the double ratio (see Equation 22 of La Cognata et al. 2010) compensates for possible systematic errors coming from the model assumptions. Furthermore, additional sources of systematic errors due to the determination of collected charge, of the target thickness, of solid angles, and dead time are removed or strongly reduced. Finally, it has been shown that extending the normalization procedure to more than one resonance greatly reduces possible systematic errors affecting directly measured strengths (see La Cognata et al. 2015 for details).

The ²H(²⁷Al, α²⁴Mg)n reaction (Q = −0.6 MeV) sketched in Figure 1, used to study the ²⁷Al(p,α)²⁴Mg reaction (Q = 1.6 MeV), was measured at the INFN-LNS Tandem (Catania, Italy). A 80 MeV ²⁷Al beam, ∼1 pnA intensity was delivered onto a CD₂ target (isotopically enriched to 98%) about 100 μ g cm⁻² thick, placed at 90° with respect to the beam axis. The beam energy was chosen to span the ²⁷Al−p energy range between the threshold and ∼1.5 MeV for normalization to measured resonance strengths. In this work, a higher beam energy than in the previous run (see Palmerini et al. 2021 for details) was considered to explore a phase-space region where the contribution of the ²⁵Mg compound-nucleus decay to the reaction yield is negligible.

The experimental setup was made of four 1000 μm double-sided silicon strip detectors (DSSSD); two (named B and C) were placed at 700 mm from the target, centered at 55 and covering each about 4° on opposite sides with respect to the beam direction. The other two DSSSDs (A and D, A on the same side as B) were placed at about 135 mm from the target at about 25°, each covering about 105 on opposite sides of the beam axis. The use of DSSSDs in the place of PSDs as in Palmerini et al. (2021) made it possible to increase statistics, each DSSSD covering a solid angle 5 times larger than a PSD. Also, the symmetric setup made it possible to double statistics. Thanks to a reanalysis of the data in Palmerini et al. (2021), it turned out that no ΔE–E technique was necessary to perform particle selection, as will be clear later in this paper. The nonuse of an ionization chamber as a ΔE detector, as in Palmerini et al. (2021), made it possible to improve the energy resolution in the particle detection and to improve the accuracy of measurement by reducing the systematic errors due, for instance, to the measurement of the pressure in the ionization chambers. Given the reaction kinematics, the forward detectors were used to detect the heavy fragment emitted in the reaction, while the other two detectors, placed closer to the target, were used to measure the light fragment. The trigger of the data acquisition was then the logical AND of the signals from the more forward and the more backward detectors.

Since we aim to measure a reaction with three particles in the exit channel, only two of them have to be detected. Indeed, from their measured energies and emission angles, it is possible to deduce all the kinematic variables with no need to use neutron detectors to observe the candidate spectator particle.

Detector calibration was carried out using reactions and scattering of ²⁷Al, at energies of between 40 and 80 MeV, off Au and CD₂ targets, as well as standard calibration alpha sources. Elastic scattering at the lowest energy of 40 MeV was used to check detectors' positions and solid angles by comparison of the measured yields with the result of a Monte Carlo simulation, the cross section at the angles of the measurement closely following the Rutherford trend.

As discussed above, the results from Palmerini et al. (2021) showed us that no particle identification was necessary using the ΔE–E technique. Instead, we carried out channel identification using the kinematic techniques discussed in, e.g., La Cognata et al. (2010). Indeed, we need to verify that trigger logic is enough to remove competing reaction channels, as foreseen in the previous experiment. The first step was the application of the procedure discussed in Costanzo et al. (1990) and Badala et al. (2022). In brief, for a generic a + b → c + d + e reaction the following X and Y variables are introduced:

$$\begin{eqnarray}&&X=\displaystyle \frac{{p}_{e}^{2}}{2u}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&Y={E}_{a}-{E}_{c}-{E}_{d}\end{eqnarray} \tag{ 4 }$$

assuming that b is at rest in the laboratory frame (target), p_e is the particle e momentum, and u the unit mass in a.m.u. It was shown (Costanzo et al. 1990) that the value of p_e is calculated with no need to introduce any hypothesis on its mass. E_a is the beam energy, and E_c and E_d are the energies measured by the detectors. Then, the two quantities above are related through the simple relation:

$$\begin{eqnarray}&&Y=\displaystyle \frac{1}{{A}_{e}}X-Q.\end{eqnarray} \tag{ 5 }$$

Therefore, events from the a + b → c + d + e reaction should fall along a straight line, defined by Equation (5), if data are drawn in the X–Y plane. This line intercepts the Y-axis at Y = −Q and has a slope given by 1/A_e (A_e being the mass of the undetected particle in a.m.u.), making it possible to identify the reaction channels feeding the reaction yield.

The X–Y plot for the present experiment is shown in Figure 2. The picture clearly shows two loci only, with apparently no background, making us confident about the absence of parasitic processes since they are both linked to the ²H(²⁷Al,α²⁴Mg)n reaction. A closer examination and the comparison with the theoretical lines demonstrate that the events gathering along the short-dashed line correspond to the ²H(²⁷Al,α²⁴Mg)n reaction populating the ²⁴Mg ground state (Q₀ = −0.624 MeV, 0 for ground state, A_e = 1). The other locus, marked by the dotted–short-dashed line, is instead due to the ²H(²⁷Al,α²⁴Mg)n reaction populating the ²⁴Mg first excited state at 1.369 MeV (Q₁ = −1.993 MeV, 1 for the first excited state, A_e = 1). The slopes of the lines clearly define the mass of the undetected particle to be about 1 a.m.u., while the intercepts prove the accuracy of the detector calibration we have carried out. This is also confirmed by the Q-value spectrum (Figure 3), obtained by consequently assuming that the undetected particle is a neutron. It exhibits two distinctive peaks, the arrows indicating the theoretical values (−0.624 MeV and −1.993 MeV for ²⁴Mg ground and first excited state, respectively). While no other peak is present, suggesting that no other reaction is populating the spectrum, a skewness is definitely present toward smaller Q. Simulations show that this can be attributed to a less efficient charge collection in the interstrip region of the Si detectors (Torresi et al. 2013). Therefore, in the following analysis a gate on the Q-value spectrum was set, rejecting events having Q ≤ −1 MeV or Q ≥ −0.2 MeV to improve our energy resolution and to discard events corresponding to the population of the ²⁴ Mg first excited state.

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Finally, Figure 4 shows the experimental kinematic locus, obtained by charting the energies detected in detectors A and C in coincidence events, for the angular couple θ_A = 255 ± 05, θ_C = 55 ± 05 (similar results are obtained also for other angular couples and for the B–D detectors couple). In agreement with what is expected from simulations (see, e.g., Palmerini et al. 2021), events from the ²H(²⁷ Al,α²⁴Mg)n three-body reaction gather along a characteristic locus in the E_A−E_C plane, while no additional spots are visible. A striking feature of the present experiment is the much-improved statistics with respect to that discussed in Palmerini et al. (2021; compare to Figure 4 therein).

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Given a unique identification of the events of interest and the absence of any observable background, we carried out the investigation of the reaction mechanisms leading to the population of the exit channel. Indeed, the equations briefly sketched in Section 3.1 can be applied only under the assumption that the ²H(²⁷Al,α²⁴Mg)n reaction can be described as a QF process in some phase-space regions. Following, e.g., the procedure outlined in La Cognata et al. (2010), we first tested the relative energy spectra to exclude the presence of sequential decay processes. In the cases in which the ²H(²⁷Al,α²⁴Mg)n three-body reaction takes place as a two-step process, where ⁵He or ²⁵Mg are formed, subsequently emitting a neutron to populate the α + ²⁴ Mg + n final channel, the neutron cannot be regarded as a spectator to the ²⁷Al(p,α)²⁴Mg lower vertex of Figure 1. Therefore, deuteron breakup cannot be direct, and the ²H(²⁷Al,α²⁴Mg)n does not proceed through a QF process. If ⁵He or ²⁵Mg are formed, the relative energy spectra would clearly bear a signature, through the formation of resonances in the coincidence yield in correspondence of ⁵He or ²⁵Mg excited states (the excitation is easily calculated from the ⁴He−n and ²⁴Mg−n relative energies by summing the corresponding neutron emission threshold energies).

Figure 5 presents the ⁴He−n versus ²⁴Mg−n relative energy 2D spectrum. It does not exhibit horizontal or vertical loci that would signal the formation of ⁵He or ²⁵Mg intermediate nuclei, respectively, at least above ${E}_{{}^{4}\mathrm{He}-n}\sim 1.5\,{MeV}$. Instead, oblique loci are clearly seen and an energy conservation consideration suggests to attribute them to the formation of ²⁸Si excited states. The correlation plot in Figure 5 was obtained for the A–C couple of detectors; similar results are obtained for B–D as well. At ${E}_{{}^{4}\mathrm{He}-n}\sim 1.5\,{MeV}$ and ${E}_{{}^{24}\mathrm{Mg}-n}\sim 3.5\,{MeV}$, a cluster of events is recognizable, which is not clearly associated with sequential processes due to the formation of ⁵He or ²⁵Mg compound systems. We will show that such a cluster does not affect the extraction of the resonance strengths.

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Using a similar approach, it is possible to find out whether ²⁸Si states are populated in the ²H(²⁷Alα²⁴Mg)n reaction. We know from Palmerini et al. (2021) that hints of the presence of ²⁸Si states were reported, yet further work was necessary to extract quantitative information. In Figure 6, we illustrate the ${E}_{{}^{24}\mathrm{Mg}-n}$ versus ${E}_{{}^{24}\mathrm{Mg}-\alpha }$ 2D spectrum, where the feeding of ²⁸Si states right above the α separation energy is apparent. Clear vertical loci are present below ∼3.5 MeV (corresponding to the sloping ones in Figure 5), corresponding to ²⁸Si states. Such a picture demonstrates that reaction mechanisms leading to the population of ⁵He or ²⁵Mg excited states, if any, can only affect the ${E}_{{}^{24}\mathrm{Mg}-\alpha }$ energies well above the α threshold (≥3 MeV), far from the interval of astrophysical interest, so its influence will be neglected in the forthcoming analysis. Comparison with the previous experiment results (Palmerini et al. 2021) proves that the drawbacks therein pointed out have been solved in the present one.

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The analysis above cannot establish whether ²⁸Si states are populated in a two-step process (sequential mechanism) or following QF deuteron breakup. Therefore, additional tests are carried out, in particular the study of the experimental neutron momentum distribution, ${\left|{\rm{\Phi }}({p}_{3})\right|}^{2}$. Indeed, if the process is direct, neutrons should keep the same momentum as inside deuterons, so deviation from the theoretical trend (Lamia et al. 2012) would signal the prevalence of a non-QF reaction mechanism. The experimental ${\left|{\rm{\Phi }}({p}_{3})\right|}^{2}$ was obtained following the prescriptions in Lamia et al. (2012), selecting a narrow ${E}_{{}^{24}\mathrm{Mg}-\alpha }$ energy range (smaller than the widths of the bands in Figure 6) to keep the cross section almost constant within the interval, and then correcting the three-body reaction cross section for the phase-space factor. The resulting experimental momentum distribution is displayed as solid circles in Figure 7 for an ${E}_{{}^{24}\mathrm{Mg}-\alpha }$ energy window 50 keV wide, centered at 2.2 MeV. Similar results are obtained for other energy values. In the construction of Figure 7, we attributed a positive or negative sign to p₃ depending on the sign of the cosine of the azimuthal angles $\cos {\phi }_{3}$. In this figure, the theoretical p−n relative-momentum distribution is rendered with a red solid line; the theoretical line was scaled to match the arbitrary normalization of the experimental data. Good agreement is present up to about $\left|{p}_{3}\right|\leqslant 40\,{\rm{MeV}}\,{c}^{-1}$, as should be expected (see Spitaleri et al. 2019 for an extensive discussion). This makes us confident about the occurrence of the QF mechanism in the ²H(²⁷Al,α²⁴Mg)n reaction at the 80 MeV beam energy, in agreement with what was found by Palmerini et al. (2021), and suggests the most suitable cut to introduce in the next steps to single out the QF reaction process.

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From our analysis, we concluded that no background affects our reaction and that there is a well-defined phase-space region where the reaction mechanism is QF, so the THM standard equations can be applied (La Cognata et al. 2010). Since the ²⁷Al + p relative energy E_cm is calculated as ${E}_{{}^{24}\mathrm{Mg}-\alpha }-{\text{}}Q$ (Q = 1.6 MeV), we obtained the raw E_cm spectra for the ²H(²⁷Al,α²⁴Mg)n reaction, shown as solid circles in Figure 8. The lower and upper panels refer to the A–C and B–D detector couples, respectively. In agreement with our scrutiny of Figure 6, for both couples we see pronounced resonances, which were only suggested in Palmerini et al. (2021). Superimposed onto the experimental spectra (solid circles), we present the result of a Monte Carlo simulation (red histogram), obtained including all experimental effects such as beam spot size and beam divergence, energy and angular straggling in the targets, strip size in detectors, and measured energy resolutions. Also, the number of events is fixed to the experimental one in the simulations, the only free parameters being the resonances' relative heights. Only the strongest resonances as listed in Iliadis et al. 2010b (see Table 1 of Palmerini et al. 2021 for more details) were considered in the simulations, and a single resonance at 84 keV was adequate to reproduce the lowest energy peak. No states leading to resonances above about 1.6 MeV were considered in the analysis, since they lie well beyond the energy region of astrophysical interest (this is why the histograms in Figure 8 do not follow the trend of experimental data above about E_cm = 1.6 MeV).

Table 1. List of the Levels Showing the Highest Strength in the ²⁷Al(p,α)²⁴Mg Reaction Below About 2 MeV

	${{\rm{E}}}_{i}^{R}$ (keV)	Jπ	ω γ (eV)	δ ω γ (eV)
(1)	71.5	2+	2.47 × 10−14	-
(2)	84.3	1−	2.60 × 10−13	-
(3)	193.5	2+	3.74 × 10−7	-
(4)	214.7	3−	1.13 × 10−7	-
(5)	486.74	2+	0.11	0.05
(6)	609.49	3−	0.275	0.069
(7)	705.08	2+	0.52	0.13
(8)	855.85	2+	0.83	0.21
(9)	903.54	3−	4.3	0.4
(10)	1140.88	2+	79	27
(11)	1316.7	2+	137	47
(12)	1388.8	1−	54	15
(13)	1661.4	3−	12.5	2.5
(14)	1838.8	1−	30.0	6.0

Note. States for which only an upper limit for the strengths is available are reported in italics. Resonance parameters (resonance energies, strengths and their uncertainties) are taken from Iliadis et al. (2010b). J^π 's are taken from NuDat2 database 2.8 (2021).

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The simulation shows that experimental effects are well understood while indicating that most peaks in the spectra are the superposition of two or more ²⁸Si resonances. Therefore, the analysis of the angular distributions will help to determine the dominant orbital angular momentum of the populated resonances in the different peaks of Figure 8, which will make it possible to determine which resonances are dominantly populated in this work. To carry out the angular distribution integration, we adopted the phenomenological approach as in La Cognata et al. (2008, 2010). In brief, we have expressed the coincidence yield obtained by gating on each peak as a function of the α emission angle in the p−²⁷Al center-of-mass system:

$$\begin{eqnarray}&&{\theta }_{{\rm{c}}.{\rm{m}}.}=\arccos \left({\hat{{\bf{k}}}}_{p{}^{27}\mathrm{Al}}{\hat{{\bf{k}}}}_{\alpha {}^{24}{Mg}}\right)\end{eqnarray} \tag{ 6 }$$

where ${\hat{{\bf{k}}}}_{{ij}}={{\bf{k}}}_{{ij}}/{k}_{{ij}}$ are the (Galilean invariant) relative momenta. The resulting THM angular distributions are displayed in Figure 9. Each panel, from (a) to (e), is relative to a peak in Figure 8, from the one at the lowest energy (about 90 keV) to the one at the highest ones (about 1.6 MeV). The experimental distributions in the pictures are the sum of the results for the A–C and B–D couples for each peak, since they show an identical trend and covered θ_c.m. range. Following the prescription of Evans (1955), we fitted the angular distributions (solid/open symbols) with the polynomial:

$$\begin{eqnarray}&&\displaystyle \frac{d\sigma }{d{\rm{\Omega }}}=N\left(1+a{\cos }^{2}{\theta }_{{\rm{c}}.{\rm{m}}.}+b{\cos }^{4}{\theta }_{{\rm{c}}.{\rm{m}}.}\right)\end{eqnarray} \tag{ 7 }$$

(we restricted the order to 4 since it is always enough to achieve a good reproduction of the experimental data), leading to the red curves in Figure 9. According to the considerations in Evans (1955), since b ≠ 0 is necessary to fit the spectra in Figures 9(d) and (e), we can conclude that only for the two highest energy resonances the dominant wave is ℓ = 2, the other peaks being mostly controlled by p-waves (in both p and α channels). Such results are in very good agreement with what is listed in the compilations of ²⁸Si resonances (Endt 1998; Iliadis et al. 2010b; NuDat2 database 2.8, 2021). The most intense resonances below about 2 MeV were already identified by Palmerini et al. (2021) and are listed in Table 1. Above about 1.6 MeV the contribution of strong tails from higher energy resonances cannot be neglected, as seen in Figure 8, contributing the ℓ = 2 wave in Figure 9(e).

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Regarding Figure 9(a), the dominant contribution is clearly linked to excitation of the ${E}_{2}^{R}=84.3$ keV resonance (see Table 1), which has been measured for the first time in this ²⁷Al(p,α)²⁴Mg study. However, the E_c.m. energy resolution deduced from the Monte Carlo simulation (σ = 95 keV) does not make it possible to separate the possible contribution of the ${E}_{1}^{R}=71.5$ keV resonance. The shape of measured angular distribution still makes it possible to set an upper limit to it, conservatively established to be equal to 36% of the total, taking into account the error bars affecting the experimental angular distribution under inspection. The next two angular distributions (Figures 9(b) and (c)) are also dominated by p-waves, matching the presence of the two strong 3⁻ states at ${E}_{6}^{R}=609.49\,\mathrm{keV}$ and ${E}_{9}^{R}=903.54\,\mathrm{keV}$. At higher energies, at about 1.4 MeV (Figure 9(d)), where the d-wave is dominating angular distributions, the shape of the angular distribution may be likely controlled by the ${E}_{11}^{R}=1316.7\,\mathrm{keV}$ resonance. The situation is less clear at about 1.7 MeV (Figure 9(e)), where tails of higher energy resonances could give a substantial contribution to the reaction yield (as is apparent from Figure 8). Since these energies are already well beyond the ones of astrophysical interest, we will not dwell on their discussion in the present work.

An evident feature of the measured angular distributions is that we could not cover the whole θ_c.m. range with the present choice of detector configuration. Therefore, for angular integration, angular distributions were extrapolated beyond the measured regions using the fitting curves. Uncertainties on the fitting parameters were used to determine the contribution to the total error budget from the integration of the angular distributions. However, in the experiment preparation we took care to span the θ_c.m. region around π/2, which is the most influential for angular distribution integration, strongly reducing the systematic error due to the extrapolation of the trend into the θ_c.m. region where data are not available. As a result, such an uncertainty contributes to the overall error budget from 2% up to 12% for the highest energy peak.

The integrated cross sections are shown in Figure 10, still for the two separated couples of detectors, even if the integration formulas are the same. Indeed, we preferred to carry out independent analyses as far as possible to check compatibility and reduce systematic errors. The figures clearly show that uncertainties are larger for higher energies, due to the larger uncertainty on the angular distribution fitting.

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Since the A–C and the B–D cross sections are in perfect agreement with one another, we took the sum and the resulting ²H(²⁷Al,α²⁴Mg)n cross section is shown in Figure 11 (solid circles). According to La Cognata et al. (2010), since the widths in the literature are much narrower than our estimated energy resolution, we carried out a fit with Equation (A5) therein:

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{d}^{2}\sigma }{{{dE}}_{\mathrm{cm}}d{{\rm{\Omega }}}_{n}}=\displaystyle \sum _{i=1}^{N}{N}_{i}\\ \times \,\exp \,\left[-\displaystyle \frac{1}{2}{\left(\displaystyle \frac{{E}_{\mathrm{cm}}-{E}_{{R}_{i}}}{\sigma }\right)}^{2}\right]+{a}_{0}+{a}_{1}{E}_{\mathrm{cm}}\end{array}\end{eqnarray} \tag{ 8 }$$

to deduce the THM resonance strengths N_i (see also Equation (2) and the discussion therein). In the fitting, the only free parameters are the N_i coefficients, the a_i parameters used to reproduce the combinatorial background being unnecessary to fit the experimental data. Also, the resonance energies were taken from Table 1 and the Gaussian σs were taken all equal to 95 keV (the experimental energy resolution) as deduced from the Monte Carlo simulation (which is devoid of free parameters). The resulting fitting curve is shown as a red solid line in Figure 11, while the individual contribution from each resonance is shown as a red dashed line. The quality of the fitting is worse above about 1.5 MeV owing to tails of higher energy resonances and larger error bars, yet this region is not going to be discussed in the present work, since it is quite remote from the energy range of astrophysical interest, around about 100 keV.

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Focusing first on the energy region of astrophysical interest, we could fit very well the lowest energy peak with a single resonance centered at about 84 keV. This state is very important given that the energy region of greatest astrophysical relevance is around 100 keV. A small tail of a subthreshold state is also recognizable, as it is visible in Figure 6. Even if its contribution is negligible (a few percent), it is accounted for in the fitting procedure. Besides the state at 71.5 keV, for which we could set an upper limit, the two additional resonances at 193.5 and 214.7 keV (states (3) and (4) in Table 1) may influence the reaction rate at T₉ ∼ 0.055. If the corresponding N_i are left as free parameters, values compatible with zero are inferred from the data fitting. Therefore, to set an upper limit we shifted the 84.3 keV peak by 1 energy bin (the energy uncertainty) and added a Gaussian as large as possible, within the total error shown in the Figure 11. In this way, we could set an upper limit to the THM strength summed over the contributions of the resonances around 200 keV (states (3) and (4) in Table 1) equal to 8.6% of the strength of the 84.3 keV resonance.

Then, since the THM strengths are deduced in arbitrary units, we need to single out one or more measured resonance strengths for normalization, following the procedure outlined in La Cognata et al. (2010). These resonances have to sit at energies larger than those for which electron screening effects are expected (see Sergi et al. 2015 for a discussion about electron screening in resonant reactions) and, obviously, a smaller uncertainty would imply a lower normalization error. Moreover, if normalization is extended to more than one resonance, La Cognata et al. (2015) shows that a significant reduction in unpredictable systematic errors affecting direct measurements can be achieved. That said, model dependence in this application of the THM is strongly reduced thanks to the use of double ratios as shown in La Cognata et al. (2010). Given their resonance energies and the uncertainties affecting their strengths (Table 1), and since they are prominent among other peaks in Figure 11, two resonances at 903.54 keV (9) and 1388.8 keV (12) are particularly suited for normalization of the THM strengths. The normalized strengths are shown in Table 2, together with the resulting weighted average.

Table 2. Strengths of the Levels in the ²⁷Al(p,α)²⁴Mg Reaction

	${E}_{i}^{R}$ (keV)	Jπ	ω γ (eV) a	δ ω γ (eV) a	ω γ (eV) b	δ ω γ (eV) b	ω γ (eV) c	δ ω γ (eV) c
(1)	71.5	2+	9.28 × 10−15	⋯	7.17 × 10−15	⋯	8.23 × 10−15	⋯
(2)	84.3	1−	1.90 × 10−14	4.7 × 10−15	1.47 × 10−14	4.3 × 10−15	1.67 × 10−14	3.2 × 10−15
(3)	193.5	2+	2.82 × 10−7	⋯	2.18 × 10−7	⋯	2.50 × 10−7	⋯
(4)	214.7	3−	4.92 × 10−8	⋯	3.80 × 10−8	⋯	4.36 × 10−8	⋯
(5)	486.74	2+	0.122	0.031	0.094	0.028	0.107	0.021
(6)	609.49	3−	0.282	0.082	0.218	0.071	0.245	0.054
(7)	705.08	1− d	0.30	0.10	0.232	0.086	0.261	0.065
(8)	855.85	3− e	0.71	0.56	0.55	0.44	0.61	0.35
(9)	903.54	3−	4.3	0.4	3.3	1.2	4.20	0.38
(10)	1140.88	2+	83	21	64	19	73	14
(11)	1316.7	2+	142	43	110	37	124	28
(12)	1388.8	1−	70	18	54	15	61	12

Notes. Best values are given in the two rightmost columns. Resonance energies and spin-parities are the same as in Table 1 if not otherwise noted. Strengths in italic are upper limits.

^a Strengths normalized to the resonance at 903.54 keV and corresponding error. ^b Strengths normalized to the resonance at 1388.8 keV and corresponding error. ^c Weighted average of the two results and corresponding error. ^d Our analysis suggests this state is more compatible with a 1⁻ instead of a 2⁺ assignment, in agreement with Nann (1982). ^e Our analysis suggests this state is more compatible with a 3⁻ instead of a 2⁺ assignment. In the NuDat2 database 2.8 (2021), the attribution is uncertain, while Endt (1998) lists a nearby possible 3⁻ state that we would not be able to resolve.

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In the determination of the strengths of the unresolved states at 193.5 and 214.7 keV ((3) and (4)), the upper limits recorded in Table 2 are calculated by attributing the whole THM strength to each of the states. Therefore, we definitively overestimate the upper limit we could set from our data. However, a simplified hypothesis such as attributing half of the THM strength to each state might lead to an underestimate of the constraint. Indeed, from La Cognata et al. (2010) we know that the ruling factor entering the determination of the strength from the THM one is the penetration factor P_l (kr) for the relevant wave l in the entrance channel. For the mentioned states we have s- and p-waves, respectively, so even weights in the THM strengths would result in very asymmetrical on-energy-shell strengths. We will address the calculation of the reaction rate in the corresponding temperature region in the next section.

From the comparison with Table 1, our calculations point to a change in the attribution of the spin-parity of the resonances at 705.08 keV and 855.85 keV. While the most recent tabulations in the NuDat2 database 2.8 (2021) prefer (yet the attribution is tentative) J^π = 2⁺ for both, the sensitivity to orbital momentum of formulas used to deduce strengths from THM data leads us to a different choice for the spin-parities, even if uncertainties are quite large, due to the relative weakness of these states. Regarding the state at 705.08 keV (7), the 1⁻ preference is supported by the (³He, d) transfer data in Nann (1982). In the case of the 855.85 keV state, Endt (1998) lists a nearby 3⁻ state we would not be able to separate owing to the present-experiment energy resolution.

The error budget enumerated in Table 2 incorporates all the sources of uncertainty. They include the statistical error, ranging from about 3% to 10%, the angular distribution integration uncertainty discussed in Section 4.3, the error arising from the disentanglement of the peaks in the THM cross section of Figure 11 (up to 80% for the resonance at 855.85 keV (8)) and the normalization error, corresponding to the uncertainties affecting the resonances at 903.54 keV and 1388.8 keV ((9) and (12)) used for normalization. This latest source of uncertainty is generally the largest one (except in the case of the 855.85 keV resonance (8), as mentioned earlier); since the errors on the normalization resonances are the dominant ones and they are independent, we use standard formulas for error propagation on the weighted averaged strengths in the rightmost column of Table 2. Resonances between the ones at 903.54 keV (9) and 1388.8 keV (12) were not used in the normalization procedure since their strengths have a 34% error, larger than the one affecting the peaks used in our normalization (see Table 2), so in the weighted mean such resonances have a minor effect. Further, resonances (10) and (11) are observed as shoulders for which the quality of the fit might be not as good as for the others. The comparison between Tables 1 and 2 shows a very good agreement between the THM resonant strengths and the ones in the literature, except in the case of the resonance at 705.08 keV (7), but for this resonance a different spin-parity assignment might justify the origin of the difference (that is, however, less than 2 standard deviations).

To summarize the present section, the most important achievement of the present work is the observation and accurate assessment of the strength of the 84.3 keV resonance in the ²⁷Al(p,α)²⁴Mg reaction, sitting right at astrophysical energies. Such a resonance has an important impact on the reaction rate that is discussed in the next section. Furthermore, thanks to the application of the THM, the deduced resonance strength is devoid of electron screening effects, as requested for astrophysical applications.