In 1982, Mahoney et al.^[1] discovered the $ {\rm{\gamma }} $ rays with an energy of 1.809 MeV in the interstellar space due to the decay of ²⁶Al. ²⁶Al is an unstable nucleus (with a half-life of about $ 7.2\times 10^{5} $a), which decays into the excited states of ²⁶Mg by ${\rm{\beta }}^{+}$ decay or electron capture, and a 1 809 MeV $ {\rm{\gamma }} $ ray is emitted during the deexcitation process of the first 2⁺ state of ²⁶Mg. So, by detecting the flux of this $ {\rm{\gamma }} $ ray, the abundance of ²⁶Al in the interstellar can be reckoned^[2]. Based on astronomical observations, Diehl et al.^[3] estimated that the abundance of ²⁶Al in the interstellar is $(2.8\pm 0.8)M_{\odot}$. The origin of such a large amount of aluminum has attracted widespread attention^[4] and become an open problem in nuclear astrophysics.

According to the theoretical studies of nuclear astrophysics, the abundance of interstellar aluminum is mainly contributed by the nucleosynthesis in the following sites, type II supernovae^[5-6], AGB stars^[7-8], nova^[9-12], Wolf–Rayet stars^[13], and low energy heavy cosmic rays^[14-15]. It is worth noting that Peng et al.^[16-17] proposed an alternative way of nucleosynthesis for ²⁶Al from both SNII and SNI$ _{a} $. The Gamma-Ray Imaging Spectrometer(GRIS) data indicate that AGB stars are a non-negligible site. Two evidences support that AGB stars play an important role in the origin of interstellar ²⁶Al. The first one, the high ²⁶Al/²⁷Al ratios observed in the silicon carbide(SiC) grains around the majority of AGB stars^[18]. The second one, circumstellar spectroscopy observed the ²⁶Al around the nearest carbon star IRC+10216^[19].

The nucleosynthesis study of AGB stars shows that ²⁶Al is synthesized at the bottom of the H combustion shell of the AGB stars(Hot Bottom Burning, referred to as HBB) through the NeNa cycle and the MgAl cycle^[4]. Therefore, Mowlavi and Meynet^[7] proposed that AGB stars might also be the main contributor to the interstellar ²⁶Al. They gave a detailed research on the production and destruction process of ²⁶Al in AGB stars and found that at the HBB of massive ($M \!> \!4M_{\odot}$) AGB stars a large amount of ²⁶Al could be synthesized, so HBB is likely to be the main site for the synthesis of ²⁶Al. In addition, according to Wasserburg et al.^[20], AGB stars with initial mass $ M $=$ 1.5 \sim 3M_{\odot} $ can also effectively synthesize ²⁶Al. Due to the existence of divergence and the update of nuclear reaction rates, it is necessary to re-analyze and calculate the nucleosynthesis of ²⁶Al in AGB stars, which is also one of the main purposes of this article. Another main purpose of this article is to analyze the sensitivity of the nuclear reaction rates associated with ²⁶Al nucleosynthesis, and to identify which nuclear reactions having more impact on the yield of ²⁶Al. It should be pointed out that the sensitivity of the nuclear reactions relevant to the ²⁶Al nucleosynthesis in the stage of explosive neon–carbon burning in the massive stars has been systematically studied by Iliadis et al.^[21]. In this work, we focus on the steady combustion stage of the 3$ M_{\odot} $ AGB stars.

This paper studies the nucleosynthesis of ²⁶Al and the sensitivity of related nuclear reactions in 3$ M_{\odot} $ AGB stars. In order to calculate the abundance of ²⁶Al, first of all, an entire reaction network from carbon to silicon isotopes is constructed. This reaction network contains 25 nuclides and 102 nuclear reactions ranging from ¹²C to ⁴⁰Si. Furthermore, a related nuclear reaction network equation system is established, which is a highly stiff differential equation system^[22], and the coefficient of the network equations is the reaction rates of nuclear reactions. Using the semi-implicit Runge-Kutan method^[23], we establish the relevant calculation program and give the numerical solution of the network equations. Our calculation results show that ²⁶Al can be effectively synthesized and subsequently consumed by a series of nuclear reactions in AGB stars, and reaction flow calculations show that the MgAl cycle has a positive effect on the synthesis of ²⁶Al. Then, the main nuclear reactions involved in the nucleosynthesis of ²⁶Al are divided into three types, namely (n, $ {\rm{\gamma }} $), (p, $ {\rm{\gamma }} $) and ($ \alpha ,\, {\rm{\gamma }} $), and the sensitivity analysis of the reaction rates of these nuclear reactions is carried out in detail by using our calculation program. With the variable-controlling approach, the reaction rate of one nuclear reaction is changed from $-20$% to 20% each time, with each step increasing by 5%. By changing the step once, the corresponding differential equations can be solved to obtain the ²⁶Al abundance variation curve. We define sensitivity as the sum of the product of the function value of the ²⁶Al abundance curve node and the step length of the differential equation system. In fact, this definition of sensitivity is equivalent to the Euclidean norm of the function. Therefore, the sensitivity of the corresponding curve after changing the step length can be obtained, and the sensitivity change curve can be obtained. In this way, we have identified the most influential reactions and they are: ²⁵Mg(n, $ {\rm{\gamma }} $)²⁶Mg, ²⁵Mg(p, $ {\rm{\gamma }} $)²⁶Al, ²⁶Mg(p, $ {\rm{\gamma }} $)²⁷Al, ²¹Ne(p, $ {\rm{\gamma }} $)²²Na, ¹⁸O($ \alpha,\, {\rm{\gamma }} $)²²Ne and ²²Ne($ \alpha,\, {\rm{\gamma }}$)²⁶Mg. Among all reactions, ²⁵Mg(p, $ {\rm{\gamma }} $)²⁶Al is the reaction that has the greatest impact on the yield of ²⁶Al.

This paper is organized as follows. In Section 2, models and a nuclear reaction network are set up. In Section 3, the evolution chart of various nuclides and the MgAl cycle flow graph are presented. Section 4 concentrates on reaction rate sensitivity for the three types of nuclear reactions (n, $ {\rm{\gamma }} $), (p, $ {\rm{\gamma }} $) and ($ \alpha ,\,{\rm{\gamma }}$). The nuclear reactions having the large influence upon ²⁶Al abundance are distinguished. A summary and some conclusions are made at the end.

The nuclear reaction(Fig. 1 for details) network contains 25 nuclides from ¹²C to ⁴⁰Si and 102 nuclear reactions. The latest nuclear reactions and nuclear structure data come from the JINA Reaclib Database and CINA Database. All the data in the two databases are ongoing updates, publicly available and can be downloaded via web. The former is maintained by the Joint Institute for Nuclear Astrophysics(http://groups.nscl.msu.edu/jina/reaclib/db/), the latter is managed by the Computational Infrastructure for Nuclear Astrophysics(http://www.nucastrodata.org).

Figure 1.  Nuclear reaction network.

The various nuclear reactions in the stellar interior arouse chemical composition variation in the reaction zone, which is followed by the structural changes of pressure, temperature, density, and so on. Therefore, the fundamental cause of the stellar evolution is thermonuclear reactions. Unquestionably, it is crucial to investigate the variation of chemical composition, resulting from nuclear reactions, for the research of the evolution of the structure inside the stars. Assuming in the volume of 1 cm³ per second, the number of the element $ i $ particle varies as follows:

where $ r_{ij}(m) $ is the total number of nuclear reactions in which $ i $ target nuclides interact with $ j $ incident particles to generate $ m $ particles. In other words, the total number of $ i $ particles consumes in 1 cm³ volume per second. $ r_{kl}(i) $ is similar to $ r_{ij}(m) $, $ r_{kl}(i) $ represents generation. In Eq. $ (1) $, $ a_{i} $ denotes the number of particles involving and consuming in a nuclear reaction, and similar to $ a_{i} $, $ b_{i} $ denotes production. By introducting a new variable, $ Y_{i} = X_{i}/A_{i} $, where Y_i is the element abundance, the particles density can be written as:

where, $ \rho $, $ X_{i} $, $ N_{A} $ and $ A_{i} $ are density, elemental mass fraction, Avogadro constant and the atomic weight of the element, respectively. In Eq. $ (1) $, $ r_{ij}(m) $ and $ r_{kl}(i) $ can be expressed as follows:

where, $ \langle\sigma\nu\rangle_{ij} $ denotes the nuclear reaction rate of $ i $ particles and $ j $ particles. Combining Eq. $ (2) $ and Eq. $ (3) $, Eq. $ (1) $ can be expressed as:

Here, $ a_{i} = b_{i} = 1 $ and $ \delta_{ij} = \delta_{kl} = 0 $. The abundance of $ _{2}^{4} $He particles is almost unchanged in 3$ M_{\odot} $ AGB stars. Therefore, this Eq. $ (4) $ can be expressed as follows:

where, $ N_{\rm n} $ and $ N_{\rm p} $ are the density of neutrons and protons, respectively. In Eq. $ (5) $, each term represents the effect on nuclear reactions with $ \alpha $ particles, $ \beta^{\pm} $, the neutron and the proton.

All equations are established on the basis of the nuclear reaction network, in which the coefficient is the nuclear reaction rates. The attribute of equations, high stiffness, is determined by greatly different reaction rates’ magnitude. The numerical solution of the semi-implicit Runge-Kuta method is as follows:

For a problem^[24]:

the numercial solution format can be determined by the following forms

where, $ h $ represents the step length. $ a $, $ b $, $ c $, $ w $ are all coefficients, which have a lot of groups of values. The parameters are chosen here

Rosenbrock’s semi-implicit Runge-Kutta method, which avoids iteration and retains the stability of the calculation format, is chosen to solve the equations. It greatly improves the efficiency of code execution.

In HBB of 3$ M_{\odot} $ AGB stars, the key physical parameters are the density $ \rho $=$ 1\;500 $ g/cm³, the burning temperature $ T $=3×10⁸ K and the initial mole abundances of $ Y_{0} $(⁴He)=0.175 and $ Y_{0} $(¹²C)=0.014, and other relevant nuclide abundance values are given according to the Ref. [25]. The injection way of ¹³C is gradual, which keeps neutron density stable^[26]. In low-mass AGB stars, the neutron value is taken as $ n $=10⁷ cm^–3 that is mainly from nuclear reactions ¹³C($ \alpha $, n)¹⁶O and ¹⁸O($ \alpha $, n)²¹Ne^[27]. The proton source is included in the reaction network, especially the nuclear reactions ¹⁴N(n, p)¹⁴C and ²⁶Al(n, p)²⁶Mg.

Fig. 2 shows, the abundance of ¹⁹F firstly increases over time due to the role of reaction ¹⁴N($ \alpha ,\,{\rm{\gamma }}$)¹⁹F. The curve of ¹⁹F quickly begins to decline because it primarily produces ²²Ne. It can be seen that the abundance curve of ²⁵Mg is stable at first, and then it increases mainly because of the ²²Ne($ \alpha $, n)²⁵Mg reaction. The abundance curve of ²⁶Al gradually decreases via ²⁶Al(n, p)²⁶Mg and ²⁶Al(${\rm{\beta }}^{+}$)²⁶Mg, while the content of ²⁶Mg subsequently increases. ²³Na is the seed for the synthesis of ²⁷Al, so ²⁷Al nuclide abundance remains stable. ²⁴Mg abundance can also rise and form the MgAl cycle by the reaction ²⁷Al(p, $ \alpha $)²⁴Mg. Finally, the reduction of raw materials leads the ²⁶Al consumption more than production, and its abundance decreases rapidly.

Figure 2.  (color online) The abundances of various nuclides evolving over the time.

The nuclear reaction network equations describe the dynamic changes of each nuclide in the network. By calculating the reaction flow between two nuclides, we can see the dynamic changes between the various nuclides more clearly.

For two nuclides, the reaction flow is defined as follows:

where, $ F_{ij} $ denotes reaction flow, and $ Y_{i}(i\rightarrow j) $ is the abundance of nuclide $ i $ converted to nuclide $ j $. The entire time span is integrated to find the main path for the synthesis of ²⁶Al. Within $ 10^{8}s $, the flow chart of the MgAl cycle is shown in Fig. 3.

Figure 3.  (color online) The calculated the MgAl cycle reaction flow.

As shown in Fig. 3, the MgAl cycle is presented as ²⁴Mg(n, $ {\rm{\gamma }} $)²⁵Mg(p, $ {\rm{\gamma }} $)²⁶Al($ \beta^{+} $)²⁶Mg(p, $ {\rm{\gamma }} $)²⁷Al(p, $ \alpha $)²⁴Mg.

In the nuclear reaction network equations, the decisive factor is the reaction rates. The reaction rates determine the yield of each nuclide and the direction of reaction flow. The reaction rates can be obtained by experiments or by theoretical models if the experimental data are scarce. Whether it is experimental measurements or theoretical estimations, there are certain errors in the rates. Therefore, sensitivity analysis of the reaction rates in the network is very important, especially through sensitivity analysis, we can find out nuclear reactions that have a relatively large impact on the abundance of ²⁶Al. This can help the experimentalists to choose the important reactions which need more precise measurements. As for sensitivity calculation, under the variable-controlling approach, only the rate of one nuclear reaction is altered from –20% to 20%, increasing by 5% per time for eight times to measure the sensitivity of ²⁶Al. By drawing, the impact of each nuclear reaction on ²⁶Al abundance can be compared intuitively. Expression of the sensitivity metric is as follows:

where, $ D $ represents the sensitivity metric; $ F^{0} $ is the ²⁶Al abundance from the original reaction rate; $ F $ is the ²⁶Al abundance with varying the reaction rate; $ h $ is the step size. To compare the sensitivity of each nuclear reaction more clearly, the y-coordinate sensitivity measure D is logarithm base $ 10 $ and the x-coordinate is –20%~20%, increasing by 5% each time. The sensitivity analysis for three types of nuclear reactions (n, $ {\rm{\gamma }} $), (p, $ {\rm{\gamma }} $) and ($ \alpha,\, {\rm{\gamma }}$) is studied. The results are presented Fig. 4~7 as showing below.

Figure 4.  (color online) (n, $ {{\gamma }} $) sensitivity analysis. Only eight reactions are shown, and the remaining sensitivities are too small to show in the Fig. 4 (Similar to Fig. 5 and Fig. 6)

Figure 6.  (color online) ($ \alpha,\, {{\gamma }} $) sensitivity analysis.

Fig. 4 shows that ²⁵Mg(n, $ {\rm{\gamma }} $)²⁶Mg has the greatest impact on the synthesis of ²⁶Al and ²⁶Al(n, $ {\rm{\gamma }}) $²⁷Al takes the second place. Compared with other nuclear reactions, the impact of ²⁵Mg(n, $ {\rm{\gamma }} $)²⁶Mg on ²⁶Al is larger at least one order of magnitude than that of the other (n, $ {\rm{\gamma }} $) reactions. Among all the (n, $ {\rm{\gamma }} $) reactions, the sensitivity of ²⁶Al(n, $ {\rm{\gamma }} $)²⁷Al is in second place. Thus, it can be considered that ²⁶Al(n, $ {\rm{\gamma }} $)²⁷Al is one of the main nuclear reactions for consuming ²⁶Al.

Fig. 5 shows that the sensitivity of ²⁵Mg(p, $ {\rm{\gamma }} $)²⁶Al is the largest. It is found that the production of ²⁵Mg(p, $ {\rm{\gamma }} $)²⁶Al to ²⁶Al is at least five orders of magnitude higher than ²²Na($ \alpha,\, {\rm{\gamma }}$)²⁶Al. The secondary important reactions are ²⁶Mg(p, $ {\rm{\gamma }} $)²⁷Al, ²¹Ne(p, $ {\rm{\gamma }} $)²²Na, ²²Ne(p, $ {\rm{\gamma }} $)²³Na, ¹²C(p, $ {\rm{\gamma }} $)¹³N and ¹⁶O(p, $ {\rm{\gamma }} $)¹⁷F($ {\rm{\beta }}^{+} $)¹⁷O. Thereinto, ²⁵Mg(p, $ {\rm{\gamma }} $)²⁶Al should be the prime target for future measurement.

Figure 5.  (color online) (p, $ {{\gamma }} $) sensitivity analysis

Fig. 6 shows that, in the type of ($ \alpha,\, {\rm{\gamma }}$), the nuclear reactions that have a relatively large impact on ²⁶Al are ¹⁸O($ \alpha ,\,{\rm{\gamma }}$)²²Ne, ²²Ne($ \alpha,\, {\rm{\gamma }} $)²⁶Mg, ¹⁵N($ \alpha ,\, {\rm{\gamma }}$)¹⁹F and ¹⁹F($ \alpha ,\, {\rm{\gamma }}$)²³Na.

Finally, in order to obtain the overall impact of the three types of nuclear reactions on the yield of ²⁶Al, we select several reactions with higher sensitivity in each type of reaction, and show in Fig. 7.

Figure 7.  (color online) Comparison of three types of nuclear reaction sensitivity.

Fig. 7 shows that among all the three types of nuclear reactions, ²⁵Mg(p, $ {\rm{\gamma }} $)²⁶Al is the one that has the greatest impact on the yield of ²⁶Al. Therefore, we recommend that nuclear experimentalists pay more attention to it.

In 3$ M_{\odot} $ AGB stars, the network calculation of ²⁶Al nucleosynthesis and the sensitivity analysis of nuclear reaction data have been given. Firstly, we build a complete reaction network from carbon to silicon combing with the latest nuclear reaction rates. As the nuclear reactions proceed, the nucleosynthesis of ²⁶Al is computed in detail, and the calculation of the reaction flow shows the existence of the MgAl cycle. Furthermore, we also analyze the sensitivity of three reaction types (n, $ {\rm{\gamma }} $), (p, $ {\rm{\gamma }} $) and ($ \alpha,\, {\rm{\gamma }}$) to the production of ²⁶Al. The result indicates: the reaction rates of ²⁵Mg(n, $ {\rm{\gamma }} $)²⁶Mg, ²⁵Mg(p, $ {\rm{\gamma }} $)²⁶Al, ²⁶Mg(p, $ {\rm{\gamma }} $)²⁷Al, ²¹Ne(p, $ {\rm{\gamma }} $)²²Na, ¹⁸O($ \alpha,\, {\rm{\gamma }}$)²²Ne and ²²Ne($ \alpha,\,{\rm{\gamma }} $)²⁶Mg have the greatest impact on the nucleosynthesis of ²⁶Al. In particular, the ²⁵Mg(p, $ {\rm{\gamma }} $)²⁶Al reaction is the most important reaction of all, further experimental studies of this reaction are recommended.

Acknowledgments This work is supported by the Excellent Course Project of Beijing University of Chemical Technology. The author sincerely thanks the anonymous reviewers for their excellent reviews and their valuable comments.