Astrophysical $S$ factor of the ${}^{12}\text{C}(\alpha ,\gamma ){}^{16}\text{O}$ reaction calculated with reduced $R\text{-matrix}$ theory

Determination of the accurate astrophysical $S$ factor of ${}^{12}\text{C}(\alpha ,\gamma ){}^{16}\text{O}$ reaction has been regarded as the holy grail of nuclear astrophysics for decades. In current stellar models, a knowledge of that value to better than 10% is desirable. Due to the practical issues, tremendous experimental and theoretical efforts over nearly 50 years are not able to reach this goal, and the published values contradicted with each other strongly and their uncertainties are two times larger than the required precision. To this end we have developed a reduced $R\text{-matrix}$ theory based on the classical $R\text{-matrix}$ theory of Lane and Thomas, which treats primary transitions to the ground state and four bound states as the independent reaction channels in the channel spin representation. With the coordination of covariance statistics and error-propagation theory, a global fitting for almost all available experimental data of ${}^{16}\mathrm{O}$ system has been multi-iteratively analyzed by our powerful code. A reliable, accurate, and self-consistent astrophysical $S$ factor of ${}^{12}\text{C}(\alpha ,\gamma ){}^{16}\text{O}$ was obtained with a recommended value $S_{\mathrm{tot}}\left(0.3\mathrm{MeV}\right)=162.7\pm 7.3$ keV b (4.5%) which could meet the required precision.

Figure 1

The electric multipole transition processes to ground state $\left(0^{+}\right)$ described by channel spin scheme and $\mathcal{L}$ representation for ${}^{16}\text{O}$ system.

Figure 2

Level scheme of the ${}^{16}\text{O}$ nucleus [60]. All states relevant for the analysis are indicated.

Figure 3

The flow chart for the procedure of the iterative fit.

Figure 4

Results of the best $R\text{-matrix}$ fit for the $S_{\mathrm{tot}}$ data from Schürmann 2005 [7], Schürmann 2011 [8], Plag 2012 [9], and Fujita 2013 [10], together with the decomposition into different energy-level contributions. For comparison, the results of Kunz 2002 [65], Hammer 2005 [43], Schürmann 2012 [40], and Xu 2013 [66] are also shown, respectively.

Figure 5

Results of the best $R\text{-matrix}$ fits (black line) for the ${}^{16}\text{N}\alpha$ spectra from Tang 2010 [26], Azuma 1994 [27], and Zhao 1993 [28], together with the decomposition into $p\text{−}$ (dashed line) and $f\text{-wave}$ (dotted line) contributions. For comparison, the best fits of Tang 2010 [26], Azuma 1994 [27], and Schürmann 2012 [40] are shown, scaled by the corresponding coefficients.

Figure 6

Results of the best $R\text{-matrix}$ fits for the ground-state transitions $S_{\text{g.s.}}$ data from Brochard 1972 [67], Ophel 1976 [20], Kettner 1982 [23], Redder 1987 [12], Ouellet 1996 [13], Kunz 2001 [15], Assunção 2006 [16], Makii 2009 [18], Schürmann 2011 [8], and Plag 2012 [9], together with the decomposition into different energy-level contributions. For comparison, the results of Brune 1999 [29], Kunz 2002 [65], Hammer 2005 [43], Schürmann 2012 [40], Oulebsir 2012 [31], and Xu 2013 [66] are shown.

Figure 7

Fits to the ${}^{12}\text{C}(\alpha ,{\gamma }_0){}^{16}\text{O}_0$ angular distributions of Ouellet 1996 [13] at (a) $E_{\mathrm{c}.\mathrm{m}.}=1.362$, (b) 1.370, (c) 1.382, (d) 1.576), (e) 1.578, and (f) 1.580 MeV.

Figure 8

Fits to the ${}^{12}\text{C}(\alpha ,{\gamma }_0){}^{16}\text{O}_0$ angular distributions of Ouellet 1996 [13] at (a) $E_{\mathrm{c}.\mathrm{m}.}=1.777$, (b) 1.979, (c) 2.172, (d) 2.383, (e) 2.390, and (f) 2.570 MeV.

Figure 9

Fits to the ${}^{12}\text{C}(\alpha ,{\gamma }_0){}^{16}\text{O}_0$ angular distributions of Ouellet 1996: [13] at (a) $E_{\mathrm{c}.\mathrm{m}.}=2.590$, (b) 2.835, (c) 2.982, and (d) 2.985 MeV.

Figure 10

Fits to the ${}^{12}\text{C}(\alpha ,{\gamma }_0){}^{16}\text{O}_0$ angular distributions of Makii 2009 [18] at (a) $E_{\mathrm{c}.\mathrm{m}.}=1.225$, (b) 1.470, (c) 1.467, and (d) 1.591 MeV.

Figure 11

Fits to the ${}^{12}\text{C}(\alpha ,{\gamma }_0){}^{16}\text{O}_0$ angular distributions of Dyer 1974 [11] at (a) $E_{\mathrm{c}.\mathrm{m}.}=2.180$, (b) 2.420, (c) 2.560, and (d) 2.831 MeV.

Figure 12

Fits to the ${}^{12}\text{C}(\alpha ,{\gamma }_0){}^{16}\text{O}_0$ angular distributions of Redder 1987 [12] at (a) $E_{\mathrm{c}.\mathrm{m}.}=1.710$, (b) 2.360, (c) 2.684, and (d) 2.830 MeV.

Figure 13

Fits to the ${}^{12}\text{C}(\alpha ,{\gamma }_0){}^{16}\text{O}_0$ angular distributions of Assunção 2006 [16] at (a) $E_{\mathrm{c}.\mathrm{m}.}=1.310$, (b) 1.340, (c) 2.268, (d) 2.660, (e) 2.677, and (f) 2.684 MeV.

Figure 14

Fits to the ${}^{12}\text{C}(\alpha ,{\gamma }_0){}^{16}\text{O}_0$ angular distributions of Plag 2012 [9] at (a) $E_{\mathrm{c}.\mathrm{m}.}=1.002$, (b) 1.308, (c) 1.416, and (d) 1.510 MeV.

Figure 15

Fits to the ${}^{12}\text{C}(\alpha ,{\gamma }_0){}^{16}\text{O}_0$ angular distributions of Fey 2004 [17] at (a) $E_{\mathrm{c}.\mathrm{m}.}=1.666$, (b) 1.965, (c) 2.040, (d) 2.116, (e) 2.192, and (f) 2.230 MeV.

Figure 16

Fits to the ${}^{12}\text{C}(\alpha ,{\gamma }_0){}^{16}\text{O}_0$ angular distributions of Fey 2004 [17] at (a) $E_{\mathrm{c}.\mathrm{m}.}=2.343$, (b) 2.455, (c) 2.578, (d) 2.607, (e) 2.652, and (f) 2.682 MeV.

Figure 17

Fits to the ${}^{12}\text{C}(\alpha ,{\gamma }_0){}^{16}\text{O}_0$ angular distributions of Fey 2004 [17] at (a) $E_{\mathrm{c}.\mathrm{m}.}=2.684$, (b) 2.687, (c) 2.757, and (d) 2.780 MeV.

Figure 18

Fits to the ${}^{12}\text{C}(\alpha ,{\gamma }_0){}^{16}\text{O}_0$ angular distributions of Kunz 1997 [14] at (a) $E_{\mathrm{c}.\mathrm{m}.}=2.250$, (b) 2.400, (c) 2.480, and (d) 2.684 MeV.

Figure 19

Fits to the ${}^{12}\text{C}(\alpha ,{\gamma }_0){}^{16}\text{O}_0$ angular distributions of Larson 1964 [19] and Kernel 1971 [21] at (a) $E_{\mathrm{c}.\mathrm{m}.}=5.295$, (b) 5.565, (c) 5.910, (d) 6.000, (e) 5.775, and (f) 5.910 MeV.

Figure 20

Fits to the ${}^{12}\text{C}(\alpha ,{\gamma }_0){}^{16}\text{O}_0$ differential cross-section data at (a) ${45}^{\circ }$, (b) ${61}^{\circ }$, (c) ${90}^{\circ }$, and (d) ${135}^{\circ }$ from Larson 1964 [19], Kernel 1971 [21], Ophel 1976 [20], and Mitchell 1964 [22].

Figure 21

Calculation of $S_{E10}$ with the best $R\text{-matrix}$ fit. For comparison, previous fits of Azuma 1994 [27], Ouellet 1996 [13], Brune 1999 [29], Gialanella 2001 [70], Kunz 2002 [65], Hammer 2005 [43], Schürmann 2012 [40], Oulebsir 2012 [31], and Xu 2013 [66] are shown. Data points shown are taken from Dyer 1974 [11], Redder 1987 [12], Ouellet 1996 [13], Roters 1999 [69], Kunz 2001 [15], Gialanella 2001 [70], Assunção 2006 [16], Makii 2009 [18], Schürmann 2011 [8], and Plag 2012 [9].

Figure 22

Calculation of $S_{E20}$ with the best $R\text{-matrix}$ fit. For comparison, previous fits of Ouellet 1996 [13], Brune 1999 [29], Kunz 2002 [65], Hammer 2005 [43], Dufour 2008 [71], Oulebsir 2012 [31], Schürmann 2012 [40], Sayre 2012 [41] and Xu 2013 [66] are shown. Data points shown are taken from Redder 1987 [12], Ouellet 1996 [13], Kunz 2001 [15], Assunção 2006 [16], Makii 2009 [18], Schürmann 2011 [8], and Plag 2012 [9].

Figure 23

Results of best $R\text{-matrix}$ fit for the cascade transitions $S_{6.05}$ from Schürmann 2011 [8] and Matei 2006 [24], together with the decomposition into different energy-level contributions. For comparison, the results of Schürmann 2011 [8], Matei 2006 [24], and Xu 2013 [66] are shown in this figure.

Figure 24

Results of the best $R\text{-matrix}$ fit for the cascade transitions $S_{6.13}$ from Schürmann 2011 [8] together with the decomposition into different energy-level contributions. For comparison, the result of Xu 2013 [66] is shown in this figure.

Figure 25

Results of the best $R\text{-matrix}$ fit for the cascade transitions $S_{6.92}$ from Redder 1987 [12], Kettner 1982 [23], Kunz 2001 [15], and Schürmann 2011 [8], together with the decomposition into different energy-level contributions. For comparison, the results of Schürmann 2012 [40] and Xu 2013 [66] are shown in this figure.

Figure 26

Results of the best $R\text{-matrix}$ fit for the cascade transitions $S_{7.12}$ from Redder 1987 [12], Kunz 2001 [15], and Schürmann 2011 [8], together with the decomposition into different energy-level contributions. For comparison, the results of Schürmann 2012 [40] and Xu 2013 [66] are shown in this figure.

Figure 27

Fits to the ${}^{12}\text{C}(\alpha ,\alpha ){}^{12}\text{C}$ angular distributions of Plaga 1987 [32] at (a) $E_{\mathrm{c}.\mathrm{m}.}=1.100$, (b) 1.480, (c) 1.550, (d) 1.555, (e) 1.704, and (f) 1.870 MeV.

Figure 28

Fits to the ${}^{12}\text{C}(\alpha ,\alpha ){}^{12}\text{C}$ angular distributions of Plaga 1987 [32] at (a) $E_{\mathrm{c}.\mathrm{m}.}=2.003$, (b) 2.153, (c) 2.228, (d) 2.302, (e) 2.303, and (f) 2.378 MeV.

Figure 29

Fits to the ${}^{12}\text{C}(\alpha ,\alpha ){}^{12}\text{C}$ angular distributions of Plaga 1987 [32] at (a) $E_{\mathrm{c}.\mathrm{m}.}=2.452$, (b) 2.457, (c) 2.528, (d) 2.588, (e) 2.678, and (f) 2.738 MeV.

Figure 30

Fits to the ${}^{12}\text{C}(\alpha ,\alpha ){}^{12}\text{C}$ angular distributions of Plaga 1987 [32] at (a) $E_{\mathrm{c}.\mathrm{m}.}=2.829$, (b) 2.888, (c) 3.038, (d) 3.131, (e) 3.196, and (f) 3.284 MeV.

Figure 31

Fits to the ${}^{12}\text{C}(\alpha ,\alpha ){}^{12}\text{C}$ angular distributions of Plaga 1987 [32] at (a) $E_{\mathrm{c}.\mathrm{m}.}=3.338$, (b) 3.488, (c) 3.638, (d) 3.788, (e) 3.944, and (f) 4.088 MeV.

Figure 32

Fits to the ${}^{12}\text{C}(\alpha ,\alpha ){}^{12}\text{C}$ angular distributions of Plaga 1987 [32] at (a) $E_{\mathrm{c}.\mathrm{m}.}=4.125$, (b) 4.163, (c) 4.200, (d) 4.238, (e) 4.275, and (f) 4.290 MeV.

Figure 33

Fits to the ${}^{12}\text{C}(\alpha ,\alpha ){}^{12}\text{C}$ angular distributions of Plaga 1987 [32] at (a) $E_{\mathrm{c}.\mathrm{m}.}=4.352$, (b) 4.364, (c) 4.387, (d) 4.424, (e) 4.462, and (f) 4.499 MeV.

Figure 34

Fits to the ${}^{12}\text{C}(\alpha ,\alpha ){}^{12}\text{C}$ angular distributions of Plaga 1987 [32] at (a) $E_{\mathrm{c}.\mathrm{m}.}=4.507$, (b) 4.542, (c) 4.582, (d) 4.619, (e) 4.657, and (f) 4.694 MeV.

Figure 35

Fits to the ${}^{12}\text{C}(\alpha ,\alpha ){}^{12}\text{C}$ angular distributions of Plaga 1987 [32] at (a) $E_{\mathrm{c}.\mathrm{m}.}=4.769$, (b) 4.844, and (c) 4.919 MeV.

Figure 36

Fits to the ${}^{12}\text{C}(\alpha ,\alpha ){}^{12}\text{C}$ angular distributions of Tischhauser 2009 [34] at (a) $E_{\mathrm{c}.\mathrm{m}.}=2.291$, (b) 3.192, (c) 3.913, and (d) 4.902 MeV.

Figure 37

Fits to the ${}^{12}\text{C}(\alpha ,\alpha ){}^{12}\text{C}$ angular distributions of Morris 1968 [35] at (a) $E_{\mathrm{c}.\mathrm{m}.}=4.950$, (b) 5.163, (c) 5.224, (d) 5.243, (e) 5.298, and (f) 5.320  MeV.

Figure 38

Fits to the ${}^{12}\text{C}(\alpha ,\alpha ){}^{12}\text{C}$ angular distributions of Morris 1968 [35] at (a) $E_{\mathrm{c}.\mathrm{m}.}=5.375$, (b) 5.550, (c) 5.625, (d) 5.775, (e) 5.825, and (f) 5.850 MeV.

Figure 39

Fits to the ${}^{12}\text{C}(\alpha ,\alpha {}_1){}^{12}\text{C}$ cross section of Mitchell 1965 [37] and deBoer 2012 [38].

Figure 40

Fits to the ${}^{12}\text{C}(\alpha ,\alpha {}_1){}^{12}\text{C}$ angular distributions of Mitchell 1965 [37] at (a) $E_{\mathrm{c}.\mathrm{m}.}=5.963$, (b) 6.015, and (c) 6.105 MeV.

Figure 41

Fits to the ${}^{12}\text{C}(\alpha ,\alpha {}_1){}^{12}\text{C}$ differential-cross-section data of Mitchell 1965 [37].

Figure 42

Fits to the ${}^{12}\text{C}(\alpha ,\alpha {}_1){}^{12}\text{C}$ angular distributions of deBoer 2012 [38] at (a) $E_{\mathrm{c}.\mathrm{m}.}=5.964$, (b) 5.984, (c) 6.001, (d) 6.020, (e) 6.039, and (f) 6.059 MeV.

Figure 43

Fits to the ${}^{12}\text{C}(\alpha ,\alpha {}_1){}^{12}\text{C}$ angular distributions of deBoer 2012 [38] at (a) $E_{\mathrm{c}.\mathrm{m}.}=6.096$ MeV, (b) 6.113, (c) 6.133, (d) 6.153, and (e) 6.169 MeV.

Figure 44

Fits to the ${}^{12}\text{C}(\alpha ,p){}^{15}\text{N}$ differential-cross-section data of Mitchell 1965 [37].