The astrophysical $S-$factor and reaction rate for ¹⁵N($p,\gamma$)¹⁶O within the modified potential cluster model

Stellar burning depend on the star’s initial mass and can proceed either through the $p-p$ chain or through the Carbon-Nitrogen-Oxygen (CNO) cycle, fusing hydrogen to helium through a chain fusion processes, $-$ sequence of thermonuclear reactions that provides most of the energy radiated by the hot stars [1, 2, 3]. Unlike the $p-p$ chain, the CNO cycle is a catalytic one, that converts 4 protons into one helium nucleus but does so via reactions on the pre-existent seed isotopes of carbon, nitrogen, and oxygen nuclei. The carbon, nitrogen, and oxygen isotopes act just as catalysts in the CNO cycle. The CNO bi-cycle involves the following chains of nuclear reactions:

Therefore, the CNO bi-cycle produces three electron neutrinos from $\beta^{+}$ decay of ${}^{\text{13}}$N, ${}^{\text{15}}$O, and ${}^{\text{17}}$F and is also referred to as the “cold” CNO cycle [4]. The CN cycle contains no stable ${}^{\text{13}}$N and ${}^{\text{15}}$O isotopes, that decay to the stable isotopes ${}^{\text{13}}$C and ${}^{\text{15}}$N, respectively.  The catalytic nuclei are lost from the process via the leak reaction ${}^{\text{15}}$N($p,\gamma$)${}^{\text{16}}$O and the subsequent reactions in (1) restore the catalytic material, generating ${}^{\text{16}}$O and heavier isotopes leading to the accumulation of the ${}^{\text{4}}$He and ${}^{\text{14}}$N nuclei. This second branch produces ${}^{\text{17}}$F, which $\beta^{+}$ decay with the emission of the 1.74 MeV electron neutrinos. Thus, the ${}^{\text{15}}$N($p,\gamma$)${}^{\text{16}}$O process represents a branching reaction linking the alternative NO channel of the CNO cycle that produces the stable oxygen isotopes [5]. Therefore, in the CNO cycle, the proton capture reaction on ¹⁵N allows two possible channels: the branch of the cycle ¹⁵N($p,^{4}$He)¹²C and the branch of the cycle ¹⁵N($p,\gamma$)¹⁶O, reactions and they intersect at the ${}^{\text{15}}$N nucleus.

The rate of the CN with respect to the NO cycle depends on the branching ratio of the ${}^{\text{15}}$N($p,\gamma$)${}^{\text{16}}$O and ${}^{\text{15}}$N($p,\alpha$)${}^{\text{12}}$C reaction cross sections. The probability for the ${}^{\text{15}}$N($p,\gamma$)${}^{\text{16}}$O process to occur is about one for every thousand of the second [6], thus the contribution to the overall nuclear energy production is negligible, while the consequences on the nucleosynthesis are critical [7]. Therefore, in the case of an active NO cycle, the correct evaluation of the ${}^{\text{15}}$N($p,\gamma$)${}^{\text{16}}$O reaction is crucial to properly predict the abundances of all the stable ${}^{\text{16}}$O, ${}^{\text{17}}$O, and ${}^{\text{18}}$O isotopes and their relative ratios [8, 9, 10]. The reaction rates ratio determines on how much nucleosynthesis of ${}^{\text{16}}$O, ${}^{\text{17}}$O, and ${}^{\text{18}}$O takes place during CNO burning [8].

Since the first experimental study of ¹⁵N($p,\gamma$)¹⁶O reaction in 1952 [11] experimental data [12, 14, 13, 15, 16, 17, 6] for total cross sections of the radiative $p^{15}$N capture in the energy region from 80 keV to 2.5 MeV were collected [18, 19]. Analysis of existing experimental measurements of the low-energy ${}^{\text{15}}$N($p,\gamma$)${}^{\text{16}}$O reaction shows that cross section data differ substantially at lower energies.

In the past, a variety of theoretical approaches from potential cluster models to multilevel $R-$matrix formalisms [20, 21, 16, 23, 5, 22] were used to describe the ${}^{\text{15}}$N($p,\gamma$)${}^{\text{16}}$O reaction cross section at the stellar energies and astrophysical $S-$factor that is the main characteristic of this process at low energies. In the framework of the selective resonant tunneling model [24] ${}^{\text{15}}$N($p,\gamma$)${}^{\text{16}}$O cross section and $S-$factor have been studied [25]. Most recently, the astrophysical $S-$factor for the radiative proton capture process on the ¹⁵N nucleus at stellar energies are studied within the framework of the cluster effective field theory [26, 27]. The authors perform the single channel calculations where only the contribution of the first resonance into a cross section was considered [26] and then reported the results by including two low-energy resonances [27].

In this paper we are continuing the study of the reactions of radiative capture of protons on light atomic nuclei [28, 29] and consider the radiative proton capture on ¹⁵N at astrophysical energies in the framework of a modified potential cluster model (MPCM). Within the MPCM over thirty reactions of radiative capture of protons, neutrons and other charged particles on light atomic nuclei were considered and results are summarized in [30, 31]. References [28, 29] provide the basic theoretical framework of the MPCM approach for the description of a charged-particle-induced radiative capture reactions. Calculation methods based on the MPCM of light nuclei with forbidden states (FS) are used [32]. The presence of allowed states (AS) and FS are determined based on the classification of the orbital states of clusters according to Young diagrams [33]. Our analysis is data driven: the potentials of intercluster interactions for scattering processes are constructed based on the reproduction of elastic scattering phase shifts, taking into account their resonance behavior or the spectra of the levels of the final nucleus. For the bound states (BS) or ground state (GS) of nuclei in cluster channels, intercluster potentials are built based on a description of the binding energy, asymptotic normalization coefficient (ANC) and mean square radius [31, 28, 29].

The modified model includes a classification of orbital states according to Young’s diagrams. This classification leads to the concept of “forbidden states” in interaction potentials. In particular, for the GS of the $p^{15}$N system, the potential has a deeply bound state, which is the FS, and the second bound state is the GS of the ¹⁶O nucleus in this channel. The concept of forbidden states makes it possible to effectively take into account the Pauli principle for multinucleon systems when solving problems in cluster models for a single-channel approximation. The current study is based on the detailed classification of orbital states in $p\ +^{15}$N channel by Young’s diagrams we performed early in Ref. [22].

In this work we study a radiative $p^{15}$N capture on the ground state of ¹⁶O within the framework the MPCM. For the first time the contribution of ${}^{3}P_{1}$ scattering wave in $p$ + ¹⁵N channel due to ${}^{3}P_{1}\longrightarrow$ ${}^{3}P_{0}$ $M1$ transition is considered. Our approach allows to analyze the explicit contribution of each transition into the $S-$factor and show the origin of the interference for $E1$ transitions.

This paper is organized as follows. Section II presents a structure of resonance states and construction of interaction potentials based on the scattering phase shifts, mean square radius, asymptotic constant and bound states or ground state of ¹⁶O nucleus. Results of calculations of the astrophysical $S-$factor and reaction rate for the proton radiative capture on ¹⁵N are presented in Secs. III and IV, respectively. In the same sections we discuss a comparison of our calculation for $S-$factor and reaction rate with existing experimental and theoretical data. Conclusions follow in Sec. V.

The $E1$ transitions from resonant ${}^{3}S_{1}-$scattering states are the main contributions to the total cross section of the radiative proton capture on ¹⁵N to the ground state of ¹⁶O [5]. In the channel of $p$ + ¹⁵N in continuum there are two ${}^{3}S_{1}$ resonances:

1.  The first resonance appears at an energy of 335(4) keV with a width of 110(4) keV in the laboratory frame and has a quantum numbers $J^{\pi}$, $T=1^{-}$, $0$ (see Table 16.22 in [34]). This resonance is due to the triplet ${}^{3}S_{1}$ scattering state and leads to $E1$ transition to the GS. In the center-of-mass (c.m.) frame this resonance is at an energy of 312(2) keV with a width of 91(6) keV and corresponds to the resonant state of the ¹⁶O at an excitation energy of $E_{\text{x}}=12.440(2)$ MeV (see Table 16.13 in [34]). However, in the new database [35], for this resonance, the excitation energy of $E_{\text{x}}=12.445(2)$ MeV and the width of $\Gamma=101(10)$ keV in the c.m. are reported.

2.  The second resonance is at an energy of 1028(10) keV with a width of 140(10) keV in laboratory frame and has a quantum numbers $J^{\pi}=1^{-}$ and $T=1$ [34]. This resonance is also due to the triplet ${}^{3}S_{1}$ scattering and leads to $E1$ transition to the GS of ¹⁶O. In the c.m. the resonance emerges at an energy of 962(2) keV with a width of $\Gamma=139(2)$ keV and corresponds to the excitation energy of $E_{\text{x}}=13.090(2)$ MeV of ¹⁶O in a new database [35]. In the database [34] for this resonance, the excitation energy of $E_{\text{x}}=13.090(8)$ MeV and width of $\Gamma=130(5)$ keV in the c.m. are reported.

The compilation of experimental data on the ${}^{3}S_{1}$ resonances is presented in Table 1.

In databases [34, 35] are reported the other resonances as well. The third resonance has an energy of 1640(3) keV with a width of 68(3) keV in the laboratory frame and quantum numbers $J^{\pi}$,$T=1^{+},0$. This resonance can be due to the triplet ${}^{3}P_{1}$ scattering and leads to $M1$ transition to the GS. The resonance is at an energy of 1536(3) keV with a width of 64(3) keV in the c.m. that corresponds to the excitation energy 13.664(3) MeV of the ¹⁶O [34] and in Ref. [35] the excitation energy of 13.665(3) MeV and the width of 72(6) keV in the c.m. are reported. However, this resonance was observed only in measurements [13], and in the later measurements [16, 17] the resonance is absent. Therefore, we will not consider it in our calculations. The next resonance is excited at the energy of 16.20(90) MeV ($J^{\pi}$,$T=1^{-},0$) has a larger width of 580(60) keV in the c.m. and its contribution to the reaction rate will be small. In addition, in the spectra of ¹⁶O [34], another resonance is observed at an excitation energy of 16.209(2) MeV ($J^{\pi}$,$T=1^{+},1$) with a width of 19(3) keV in the c.m. However, the resonance energy is too large and its width too small to make a noticeable contribution to the reaction rates.

The cascade transitions via two very narrow 2^- resonances as well as the 0^- and 3^- resonances in the 0.40 $\lesssim E_{R}\lesssim$ 1.14 MeV range [36, 17, 37] are considered for the reaction rate calculations. These cascading transitions are included in the NACRE II [19] reaction rate calculations and appear at high $T_{9}$. There are two 2^- resonances [17], at the excitation energies of 12.53 Mev and 12.9686 MeV with the widths of 97(10) eV and 1.34(4) keV [35], respectively. When one considers transitions to the ground state with 0⁺, only $M2$ transitions are possible here, which have a very small cross section, and we do not consider them. In addition, due to such small widths, their contribution to the reaction rate will be very small. The measurement of the excitation functions of the three dominant cascade transitions allows one to estimate of the contributions from these transitions. In Ref. [5] capture processes to the GS and three excited states $E_{\text{x}}$ $=6.049$ MeV, 6.130 and 7.117 MeV are considered, that gives $S(0)=$ 41(3) keV${\cdot}$b in total while 40(3) keV$\cdot$b for the GS. It is shown that the 1^- and two 2⁺ resonances do not affect the value of the $S-$factor, two 3^- resonances at $E_{\text{x}}$ = 13.142 MeV and 13.265 MeV decay into the GS due to the $E3$ transition and their contribution is negligible [5].

On the background of strong $E1$ transitions the next one in the long-wave expansion of the electromagnetic Hamiltonian is magnetic dipole $M1$ transition [38]. In the case of ¹⁵N($p,\gamma_{0}$)¹⁶O, $M1$ transition is allowed by selection rules. It occurs as a direct capture from the non-resonance scattering wave ${}^{3}P_{1}$ to the ¹⁶O ${}^{3}P_{0}$ state. The intensity of the $M1$ partial transition depends on the distorted ${}^{3}P_{1}$ wave only and is related to the corresponding interaction potential. Our estimations of the $M1$ partial cross section in plane-wave approximation shows near order of magnitude suppression. The inclusion of the $p-$wave interaction in $p+^{15}$N channel enhances the $M1$ transition. The interaction potential could be constructed based on an elastic ¹⁵N($p,p$)¹⁵N scattering to describe the ${}^{3}P_{1}$ phase shift. The phase shifts should satisfy the following conditions: i. at $E=0$ the $\delta_{{}^{3}P_{1}}=180^{0}$ according to the generalized Levinson theorem [32]; ii. fit the exsisting $p-$wave ¹⁵N($p,p$)¹⁵N scattereng data. iii. have a non-resonace behaviour. Below we found potential parameters that provides a reasonable phase shifts. Therefore, in calculations we are considering only the above two ${}^{3}S_{1}$ resonance transitions and non-resonance ${}^{3}P_{1}$ scattering for the $M1$ transition to the ¹⁶O GS.

We should distinguish $M1$ capture from the non-resonance ${}^{3}P_{1}$ scattering wave to the ¹⁶O ${}^{3}P_{0}$ state and $M1$ de-excitation of 1⁺ level at 13.665 MeV to the ground state. As pointed by deBoer et al. [5]: “The 1⁺ level at $E_{x}$ = 13.66 MeV could decay by $M1$ de-excitation to the ground state but no evidence for this is observed (contrary to Ref. [13]).” As mentioned above we are not considering the contribution of $M1$ de-excitation (1⁺ level at 13.665 MeV) to the $S(E)$.

For calculations of the total radiative capture cross sections, the nuclear part of the $p^{15}$N interaction potential is contracted using Gaussian form [31, 28, 29]:

The parameters $\alpha$ and $V_{0}$ in Eq. (2) are the interaction range and the strength of the potential, respectively.

The strength and the interaction range of the potential (2) depend on the total and angular momenta, the spin, $JLS$, and Young diagrams $\{f\}$ [30, 31]. For description of the ${}^{3}S_{1}$ scattering states we use the corresponding experimental energies and widths from Table 1. Coulomb potential is chosen as point-like interaction potential.

Construction of the potentials that give the energies and widths of ${}^{3}S_{1}$(312) and ${}^{3}S_{1}$(962) resonances reported in the literature is a challenging task. One has to find the optimal parameters of the potentials for the description of $E1$ transitions that lead to the fitting of the experimental resonance energies and the widths of both interfering resonances. For the ${}^{3}S_{1}$(962) resonance the found optimal parameters of the interaction potential that allow to reproduce the resonance energy $E_{\text{res}}=962(1)$ keV and width $\Gamma_{\text{res}}=131$ keV are reported in Table 2. The situation is more complicated with the ${}^{3}S_{1}$(312) resonance. While it is possible to reproduce rather accurately the position and the width of the ${}^{3}S_{1}$(312) resonance, the consideration of the interference of ${}^{3}S_{1}$ resonances gives different sets of optimal parameters for the potential. We found three sets I $-$ III of the optimal values for $V_{0}$ and $\alpha$ parameters reproducing exactly the energy of the first resonance $E_{\text{theory}}=312(1)$ keV, but the width $\Gamma_{\text{theory}}$ varies in the range of $125-141$ keV.

The dependence of the elastic $p^{15}$N scattering phase shifts for the $E1$ transitions on the energy is shown in Fig. 1$a$. The result of the calculation of the ${}^{3}S_{1}$ phase shifts with the parameters for the $S$ scattering potential without FS from Table 2 leads to the value of 90^∘(1) at the energies 312(1) and 962(1) keV, respectively. Our calculations of the phase shifts for the ${}^{3}S_{1}$(312) resonance with the parameter sets I - III show very close energy dependence in the entire energy range up to 5 MeV at a fixed resonance position.

In elastic $p^{15}$N scattering spectra at energies up to 5 MeV, there are no resonance levels with $J^{\pi}=0^{+},1^{+},2$ except for the mentioned above, and widths greater than 10 keV [34]. Therefore, for potentials of non-resonance ${}^{3}P-$waves with one bound FS parameters can be determined based on the assumption that in the energy region under consideration their phase shifts are practically zero or have a gradually declining character [29]. For such potential the optimal parameters are: $V_{p}=14.4$ MeV, $\alpha_{p}=0.025$ fm${}^{\text{-2}}$. The result of the calculation of $P-$phase shifts with such potential at an energy of up to 5 MeV is shown in Fig. 1$a$. To determine the values of phase shifts at zero energy, we use the generalized Levinson theorem [32], so the phase shifts of the potential with one bound FS should begin from 180^∘. In the energy region $E_{\text{c.m.}}<5$ MeV the ${}^{3}P_{1}$ phase shift has gradual energy dependence and it is almost constant up to $E_{\text{c.m.}}\lesssim 2.2$ MeV.

It is interesting to compare experimentally determined phase shifts with our calculations. While in Ref. [39] the elastic scattering of protons from ¹⁵N was studied and authors measured the excitation functions of ¹⁵N($p,p$)¹⁵N over the proton energy range from 0.6 to 1.8 MeV at some laboratory angles, there been no phase shifts reported. There has been no systematic experimentally determined phase shifts at the energies of the astrophysical interest. Absolute differential cross sections were measured for the reactions ¹⁵N($p,p$)¹⁵N [40] and spins and parities are discussed where resonances suggest the existence of excited states in ¹⁶O. In Ref. [41] authors carried out a phase-shift analysis of cross section for the energy interval 8$-$15 MeV. Angular distributions of cross section and analyzing power for elastic scattering of protons from ¹⁵N have been measured for the energy interval 2.7$-$7 MeV [42], and the authors gave a phase-shift analysis of data. Results of our calculations for the ${}^{3}P_{1}$ phase shift along with the experimental data [42] are presented in Fig. 1$a$.

The main assumption in all previous studies of the reaction ¹⁵N($p,\gamma$)¹⁶O was that a direct and resonant radiative capture cross sections and their interferences contribute to the total cross section. The direct radiative capture process is considered assuming a potential peripheral radiative capture for a hard sphere scattering [43, 13, 44, 45, 46]. However, in general, phase shifts are extracted from experimental data analysis without the separation on the potential and resonant terms [47, 48]. We follow this Ansatz. Consequently, the $E1$ transition amplitudes are constructed based on a single radial scattering wave function resembling continuous, both smooth and resonance energy dependence. Both ${}^{3}S_{1}$ phase shifts obtained with these scattering wave functions depend on the energy as shown in Fig. 1$a$. Therefore, we have only the interference between two partial $E1$ matrix elements in contrast to all previous considerations.

The resonant phase shift is given by the usual expression [13, 48]

where $\Gamma$ is the width of a resonance. In Fig. 1$b$ the energy dependence of the resonant $\delta_{R1}$ and $\delta_{R2}$ phase shifts for the resonance width 312 keV and 962 keV is presented, respectively. The comparison of the phase shifts for the $E1$ transitions via the ${}^{3}S_{1}$(312) and ${}^{3}S_{1}$(962) resonances with the resonant phase shifts $\delta_{R1}$ and $\delta_{R2}$ shows their different energy dependence.

We construct the potential for ¹⁶O in GS with J^π,T = 0⁺,0 in $p^{15}$N-channel based on the following characteristics: the binding energy of 12.1276 MeV, the experimental values of 2.710(15) fm and 2.612(9) fm [34] for the root mean square radii of ¹⁶O and ¹⁵N of [49], respectively, and a charge and matter radius of a proton 0.8414 fm [50]. The potential also should reproduce the AC. The corresponding potential includes the FS and refers to the ${}^{3}P_{0}$ state.

Usually for a proton radiative capture reaction of astrophysical interest one assumes that it is peripheral, occurring at the surface of the nucleus. If the nuclear process is purely peripheral, then the final bound-state wave function can be replaced by its asymptotic form, so the capture occurs through the tail of the nuclear overlap function in the corresponding two-body channel. The shape of this tail is determined by the Coulomb interaction and is proportional to the asymptotic normalization coefficient. The role of the ANC in nuclear astrophysics was first discussed by Mukhamedzhanov and Timofeyuk [51] and in Ref. [52]. These works paved the way for using the ANC approach as an indirect technique in nuclear astrophysics. See Refs. [53, 54, 55, 56, 57, 58, 59, 60, 45] and citation herein and the most recent review [46].

We construct a potential with the FS ${}^{3}P_{0}$ state using the experimental ANC given in Ref. [21] that relates to the asymptotics of radial wave function as $\chi_{L}(R)=CW_{-\eta L+1/2}(2k_{0}R)$. The dimensional constant $C$ is linked with the ANC via the spectroscopic factor $S_{F}$. In our calculations we exploited the dimensionless constant [61] $C_{W}$, which is defined in [29] as $C_{W}=C/\sqrt{2k_{0}}$, where $k_{0}$ is wave number related to the binding energy. In Ref. [21] the values$\ 192(26)$ fm^-1 and $2.1$ were reported for the ANC and spectroscopic factor, respectively. In Ref. [23] the dimensional ANC includes the antisymmetrization factor $N$ into the radial overlap function as it was clarified by authors in [59]. The factor $N$ is defined as $N=\left(\begin{matrix}A\\ x\end{matrix}\right)^{1/2}=\sqrt{\dfrac{A!}{(A-x)!x!}},$ where ${x}$ and ${A}$ are the atomic mass numbers of the constituent nucleus from ${x}$ and ${A-x}$ nucleons, respectively [62]. If $x=1$, then $N=\sqrt{A}$ and for the reaction ¹⁵N($p,\gamma_{0}$)¹⁶O $N=4$. Thus, using the experimental square of the ANC $192\pm 26$ fm^-1 [21, 23], we obtained the interval for the dimensionless AC used in our calculations: $C_{W}=1.82-2.09$ that corresponds to the ANC of $12.88-14.76$ fm^-1/2. In the present calculations, we use for the proton mass $m_{p}=1.00727646677$ amu [50], ¹⁵N mass 15.000108 amu [63], and the constant $\hbar^{2}/m_{0}=41.4686$ MeV·fm², where $m_{0}=931.494$ MeV is the atomic mass unit (amu).

The ¹⁵N($p,\gamma$)¹⁶O is the astrophysical radiative capture process, in which the role of the ANC is elucidated [46]. In Table 2 three sets of parameters for the ${}^{3}P_{0}$ GS potential and AC $C_{W}$ are listed. The asymptotic constant $C_{W}$ is calculated over averaging at the interval $5-10$ fm. Each set leads to the binding energy of 12.12760 MeV, the root mean square charge radius of 2.54 fm and the matter radius of 2.58 fm, but the sets of $C_{W}$ lead to the different widths of the ${}^{3}S_{1}$(312) resonance.

Note, that there is one important benchmark for the choice of optimal sets for the parameters of interaction potentials for the first $E1$(312) resonance. There are the experimental values of the total cross section $\sigma_{\text{exp}}(312)=6.0\pm 0.6$ $\mu$b [6] and $6.5\pm 0.6$ $\mu$b [16], which are in excellent agreement with earlier data $6.3$ $\mu$b [14] and $6.5\pm 0.7$ $\mu$b [12]. Simultaneous variation of $C_{W}$ for the GS and parameters $V_{0}$ and $\alpha$ for the ${}^{3}S_{1}$(312) was implemented to keep the value of the cross section $\sigma_{\text{theory}}(312)=5.8-5.9$ $\mu$b matching the experimental data. The result of this optimization is presented in Table 2 as sets I $-$ III.

Table 2 summarizes the potential parameters used in the case where the MPCM works reasonably well for a radiative proton capture in the ${}^{\text{15}}$N($p,\gamma_{0}$)${}^{\text{16}}$O reaction.

We sum up the procedure and choice of potential parameters as:

1. we construct the nucleon-nuclei potentials that give the channel binding energy with the requested accuracy 10^-5 MeV. There are a few such potentials;

2. the experimental ANC is used as a criterion for the choice of the potential that provides the required asymptotic behavior of the radial wave function at the fixed binding energy. Thus, the variety of wave function is constrained within the upper and lower limits for the ANC: $12.88-14.76$ fm^-1/2 for ¹⁵N($p,\gamma$)¹⁶O;

3. an additional test of the wave functions is a reproduction of the matter and charge radii with a precision of $\sim$ 5% and the $\sigma_{\text{exp}}(312)$ cross section within experimental uncertainties;

4. for the continuous spectrum the parameters of the potential are fixed by the resonance energy and width above threshold. An additional source of the $S-$factor uncertainty relates to uncertainties of the resonance energy and width;

5. this procedure gives the model’s uncertainty bands for the $S-$factor.

The astrophysical $S-$factor is the main characteristic of any thermonuclear reaction at low energies. The present analysis focuses primarily on extrapolating the low-energy $S-$factor of the reaction ¹⁵N($p,\gamma$)¹⁶O into the stellar energy range. Since the first experimental study of ¹⁵N($p,\gamma_{0}$)¹⁶O reaction in 1960 [12], experimental data [13, 16, 17, 6, 19] for total cross sections of the radiative $p^{15}$N capture in the energy region from 80 keV to 2.5 MeV have been collected. These experimental studies verified and confirmed that the radiative $p^{15}$N capture is dominated by the first two interfering resonances at 312 keV and 962 keV with the quantum numbers $J^{\pi}$, $T=1^{-}$,$0$ and $J^{\pi}$, $T=1^{-}$,$1$, respectively.

The reaction rates for nuclear fusion are the critical component for stellar-burning processes and the study of stellar evolution [4]. In stellar interiors, where the interacting particles follow a Maxwell-Boltzmann distribution, the reaction rate describes the probability of nuclear interaction between two particles with an energy-dependent reaction cross section $\sigma(E)$. The reaction rate of proton capture processes can be written as [74, 48].

In Eq. (6) $N_{A}$ is the Avogadro number, $\mu$ is the reduced mass of two interacting particles, $k_{B}$ is the Boltzmann constant, $T$ is the temperature of the stellar environment, and the factor $e^{-2\pi\eta}$ approximates the permeability of the Coulomb barrier between two point-like charged particles.