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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).
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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:$$ \begin{split}& \frac{{\rm d}n_{i}}{{\rm d}t}= -\sum_{j}\frac{a_{i}}{1+\delta _{ij}}r_{ij}(m)+\sum_{k,l}\frac{b_{i}}{1+\delta_{kl}}r_{kl}(i),\\ & \delta_{ij}= \left\{\begin{matrix} 0,~i\neq j\\1,~i = j, \end{matrix}\right. \quad \delta_{kl} = \left\{\begin{matrix} 0,~k\neq l\\1,~k = l, \end{matrix}\right. \end{split} $$ (1) 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 Yi is the element abundance, the particles density can be written as:$$ n_{i} = \frac{\rho X_{i}}{A_{i}}N_{A} = \rho N_{A}Y_{i},$$ (2) 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:$$ r_{ij}(m) = n_{i}n_{j}\langle\sigma\nu\rangle_{ij},\quad r_{kl}(i) = n_{k}n_{l}\langle\sigma\nu\rangle_{kl}, $$ (3) 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:$$ \begin{split} \frac{{\rm d}Y_{i}}{{\rm d}t} =& -\rho N_{A}\sum\limits_{j}\frac{a_{i}}{1+\delta _{ij}}Y_{i}Y_{j}\langle\sigma\nu\rangle_{ij}+\\ &\rho N_{A}\sum\limits_{k,l}\frac{b_{i}}{1+\delta_{kl}}Y_{k}Y_{l}\langle \sigma\nu\rangle_{kl}\mathbf{}. \end{split} $$ (4) 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:$$ \begin{split} \frac{{\rm d}Y_{j}}{{\rm d}t} =& \rho Y_{\alpha}\sum(N_{A}\langle\sigma\nu\rangle_{i\alpha}Y_{i}-N_{A}\langle\sigma\nu\rangle_{j\alpha}Y_{j})+\\ & \sum(Y_{i}\lambda_{ij}-Y_{j}\lambda_{ji})+ N_{\rm n}(\langle\sigma\nu\rangle_{in}Y_{i}-\\ &\langle \sigma\nu\rangle_{jn}Y_{j})+ N_{\rm p}(\langle\sigma\nu\rangle_{ip}Y_{i}-\langle\sigma\nu\rangle_{jp}Y_{j}),\end{split} $$ (5) 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]:
$$ \left\{\,\begin{aligned} &y'=f(t,y),\;\;\;a\leqslant t\leqslant b \\& y(a) = \eta \end{aligned}\right., $$ (6) the numercial solution format can be determined by the following forms
$$ \begin{split} y_{n+1} =& y_{n}+w_{1}k_{1}+w_{2}k_{2}\\ k_{1} =& h[1-ha_{1}A(t_{n},y_{n})]^{-1}f(t_{n},y_{n})\\ k_{2} =& h[1-ha_{2}A(t_{n}+c_{2}h,y_{n}+c_{21}k_{1})]^{-1}\times\\& f(t_{n}+b_{2}h,y_{n}+b_{21}k_{1}),\\ \nonumber\end{split} $$ 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$$ \begin{split} a_{1} = &1+\frac{\sqrt{6}}{6} = 1.408\,248\,29,\\ a_{2} =& 1-\frac{\sqrt{6}}{6} = 0.591\,751\,71,\\ b_{21} =& c_{21} = \frac{-6-\sqrt{6}+\sqrt{58+20\sqrt{6}}}{6+2\sqrt{6}} = 0.173\,786\,67,\\ b_{2} =& c_{2} = 0.173\,786\,67,\\ w_{1} =& -0.413\,154\,32,\\w_{2} = &1.413\,154\,32. \end{split}$$ 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.
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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. -
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:
$$ F_{ij}(t) = \int\nolimits_{0}^{t}[Y_{i}(i\rightarrow j)]{\rm d}t, $$ (7) 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.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:
$$ \begin{array}{l} D^{2} = \sum(F-F^{0})^{2}\cdot h,\end{array} $$ (8) 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. -
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. -
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.
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.
The Network Calculation of 26Al Nucleosynthesis in 3M⊙ AGB Stars and the Sensitivity Analysis of Nuclear Reaction Rates
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摘要: 研究了3
$M_{\odot}$ AGB星中26Al核合成的网络计算和核反应率的灵敏度分析。结合最新的核反应率数据,建立了一个从碳到硅完整的核反应网络,计算了26Al的丰度。结果表明,26Al首先在AGB星中有效合成,随着核反应的进行,然后被一系列的核反应消耗。MgAl循环出现在26Al的网络中。我们将核反应网络中的主要核反应分为三类:(n,${\rm{\gamma }}$ ),(p,${\rm{\gamma }}$ )和($\alpha$ ,${\rm{\gamma }}$ ),并对核反应率的灵敏度进行了详细的分析。已经确定了每一类中最有影响的核反应,它们是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和22Ne($\alpha$ ,${\rm{\gamma }}$ )26Mg。在目前网络所涉及的所有核反应中,25Mg(p,${\rm{\gamma }}$ )26Al是对26Al的产量有最大的影响,它值得核实验物理学家的关注。Abstract: The network calculation of 26Al nucleosynthesis in 3$M_{\odot}$ AGB stars and the sensitivity analysis of nuclear reaction rates have been investigated in this article. After establishing a complete nuclear reaction network from carbon to silicon, combined with the latest nuclear reaction rate data, we have calculated the abundance of 26Al. The results show that 26Al is effectively synthesized in the AGB stars at the beginning, but as the reaction proceeding, 26Al is consumed by a series of nuclear reactions. The MgAl cycle appears in the network of 26Al. We divide the main nuclear reactions in the reaction network into three categories (n,${{\gamma }} $ ), (p,${{\gamma }}$ ) and ($\alpha,\,{{\gamma }}$ ), and the sensitivity of nuclear reaction rates has been analyzed in detail. We have identified the most influential reactions in each type of nuclear reactions, they are: 25Mg(n,${{\gamma }}$ )26Mg, 25Mg(p,${{\gamma }}$ )26Al, 26Mg(p,${{\gamma }}$ )27Al, 21Ne(p,${{\gamma }}$ )22Na, 18O($\alpha,\,{\rm{\gamma }}$ )22Ne and 22Ne($\alpha,\,{\rm{\gamma }}$ )26Mg. Among all the nuclear reactions involved in the present network, 25Mg(p,${\rm{\gamma }}$ )26Al is the one that has the greatest impact on the yield of 26Al, which deserves the attention of nuclear experimentalists.-
Key words:
- 26Al abundance /
- AGB stars /
- nuclear reaction network /
- sensitivity analysis
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