NEW ASTROPHYSICAL REACTION RATE FOR THE ¹²C(α, γ)¹⁶O REACTION

The S-factors of ¹²C(α, γ)¹⁶O are constituted by several resonant peaks with strong interference patterns. Furthermore, the complicated mechanism of this reaction results in unpredictable interference effects from the first principles (Kunz et al. 2002). The global fitting for the ¹⁶O system with a multilevel, multichannel R-matrix allows for the simultaneous analysis of differential cross-section data and the corresponding angle-integrated cross section of an ¹⁶O compound nucleus (An et al. 2015). A multichannel R-matrix analysis provides the possibility of reducing uncertainties in the extrapolated total and partial S-factors of the ¹²C(α, γ)¹⁶O reaction, and the interpretation of the interference mechanism via the additional constraint offered by the simultaneous analysis of multiple reaction channels (Azuma et al. 2010).

The error propagation formulae (Smith 1991) are adopted to determine the uncertainty of the S factor in the whole energy region. Our extrapolation value is S_tot(0.3 MeV) = 162.7 ± 7.3 keV b, which is composed of S_E10(0.3 MeV) = 98.0 ± 7.0 keV b, S_E20(0.3 MeV) = 56.0 ± 4.1 keV b ground-state captures and of cascade captures, S_casc(0.3 MeV) = 8.7 ± 1.8 keV b. Furthermore, the cascade transitions of S_6.05 and S_6.13 at 0.3 MeV are 4.91 ± 1.11 keV.b and 0.16 ± 0.26 keV b, respectively. They are quite consistent with the constructive interference result of S_6.05 (0.3 MeV) = 4.36 ± 0.45 keV.b and destructive interference of S_6.13 (0.3 MeV) = 0.12 ± 0.04 keV b of Avila et al. (2015), which constrained the contribution of the values by measuring the asymptotic normalization coefficients (ANCs) for these states using the α-transfer reaction ⁶Li(¹²C, d)¹⁶O. We adopt a similar approach for the fitting of cascade transitions. This is one of the reasons that uncertainty of the extrapolated S factor reduces dramatically. The values of the other two cascade transitions are S_7.12 (0.3 MeV) = 0.63 ± 0.22 keV b and S_6.92(0.3 MeV) = 3.00 ± 0.42 keV b, which are in excellent agreement with the recent results of Schürmann et al. (2012), respectively.

The absolute values of reaction rates, ${N}_{{\rm{A}}}\langle \sigma v\rangle$ of ¹²C(α, γ)¹⁶O can be obtained by Equation (1) from the self-consistent total S factor and its uncertainties. Table 1 lists 80 points of reaction rates in the temperature range of 0.04 ≤ T₉ ≤ 10. To be precise, the total reaction rate of ¹²C(α, γ)¹⁶O is achieved after multiplying ${N}_{{\rm{A}}}\langle \sigma v\rangle$ with the probability densities of the reaction partners and integrating over the energy interval. The uncertainties of the reaction rates obtained from our R-matrix model are also tabulated for the high rate and the low rate in Table 1.

Table 1.  The ¹²C(α, γ)¹⁶O Reaction Rates (in cm³ mol⁻¹ s⁻¹)

T9	${N}_{{\rm{A}}}\langle \sigma v\rangle$	High	Low	10n	T9	${N}_{{\rm{A}}}\langle \sigma {\text{}}v\rangle$	High	Low	10n
0.040	9.27	9.75	8.79	−31	0.60	3.28	3.40	3.17	−8
0.042	3.94	4.14	3.74	−30	0.65	8.00	8.27	7.73	−8
0.045	2.92	3.07	2.77	−29	0.70	1.78	1.84	1.73	−7
0.050	5.69	5.98	5.40	−28	0.75	3.70	3.82	3.58	−7
0.055	7.60	7.98	7.21	−27	0.80	7.19	7.42	6.96	−7
0.060	7.51	7.88	7.13	−26	0.85	1.32	1.37	1.28	−6
0.065	5.81	6.10	5.52	−25	0.90	2.33	2.40	2.26	−6
0.070	3.67	3.85	3.49	−24	0.95	3.93	4.05	3.81	−6
0.075	1.95	2.05	1.86	−23	1.00	6.41	6.60	6.22	−6
0.080	9.00	9.44	8.57	−23	1.10	1.56	1.60	1.51	−5
0.085	3.66	3.84	3.49	−22	1.20	3.42	3.52	3.33	−5
0.090	1.34	1.40	1.27	−21	1.30	6.96	7.15	6.78	−5
0.095	4.45	4.67	4.24	−21	1.40	1.33	1.36	1.30	−4
0.100	1.36	1.43	1.30	−20	1.50	2.41	2.47	2.35	−4
0.105	3.88	4.06	3.69	−20	1.60	4.19	4.29	4.09	−4
0.11	1.03	1.08	0.98	−19	1.70	7.02	7.18	6.85	−4
0.12	6.18	6.48	5.89	−19	1.80	1.14	1.16	1.11	−3
0.13	3.06	3.20	2.91	−18	1.90	1.80	1.83	1.76	−3
0.14	1.29	1.35	1.23	−17	2.00	2.76	2.82	2.71	−3
0.15	4.75	4.97	4.53	−17	2.10	4.15	4.23	4.07	−3
0.16	1.56	1.64	1.49	−16	2.20	6.10	6.22	5.98	−3
0.17	4.67	4.88	4.46	−16	2.30	8.79	8.96	8.62	−3
0.18	1.28	1.34	1.23	−15	2.50	1.73	1.76	1.69	−2
0.19	3.27	3.42	3.13	−15	2.75	3.66	3.73	3.60	−2
0.20	7.83	8.18	7.48	−15	3.00	7.13	7.26	7.00	−2
0.21	1.77	1.85	1.69	−14	3.25	1.29	1.32	1.27	−1
0.22	3.79	3.96	3.63	−14	3.50	2.20	2.24	2.16	−1
0.24	1.53	1.59	1.46	−13	3.75	3.58	3.64	3.51	−1
0.26	5.29	5.52	5.07	−13	4.00	5.57	5.67	5.46	−1
0.28	1.62	1.69	1.55	−12	4.25	8.37	8.54	8.21	−1
0.30	4.47	4.66	4.29	−12	4.50	1.22	1.25	1.20	0
0.32	1.13	1.18	1.08	−11	5.00	2.42	2.48	2.36	0
0.34	2.65	2.75	2.54	−11	5.50	4.43	4.56	4.31	0
0.36	5.81	6.04	5.57	−11	6.00	7.60	7.87	7.34	0
0.38	1.20	1.25	1.15	−10	6.50	1.23	1.28	1.18	1
0.40	2.37	2.46	2.27	−10	7.00	1.90	1.99	1.81	1
0.42	4.46	4.63	4.28	−10	7.50	2.79	2.94	2.64	1
0.45	1.07	1.11	1.03	−9	8.00	3.95	4.18	3.71	1
0.50	3.91	4.05	3.76	−9	9.00	7.13	7.63	6.64	1
0.55	1.21	1.25	1.17	−8	10.0	1.15	1.24	1.06	2

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According to the Gamow theory (Rolfs & Rodney 1988), for the nonresonant cross section, the Gamow window (significant integral interval) of each T₉ is selected in [E₀ − ΔE₀/2, E₀ + ΔE₀/2], with E₀ = (${E}_{{\rm{G}}}^{1/2}{k}_{{\rm{B}}}$T/2)^2/3 and ΔE₀ = (16E₀k_BT/3)^1/2. Considering the typical T₉ temperatures involved, we have E₀ = 0.3 MeV of helium burning starting at T₉ = 0.2. Furthermore, for the S factor data, the chief center-of-mass energy range is 0.1–0.5 MeV. As E₀ and ΔE₀ increase with stellar temperature, the S(E) factor data in the higher energy range gradually play a leading role for the reaction rate.

To understand the influence of the S(E) factor of ¹²C(α, γ)¹⁶O on the reaction rate at different temperatures, probability density functions of the total reaction rates at T₉ = 0.2, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 8.0, and 10.0 are shown in Figure 1. At T₉ = 0.2 and 1.0, the probability density functions are located almost in the extrapolated S factor, without resonance peaks, so the Gamow window can be well approximated by a Gaussian distribution with the most effective energy E₀ = 0.3 MeV and E₀ = 0.9 MeV. As the T₉ increases from 1 to 10, however, the influence of resonances of the S(E) factor becomes more and more remarkable, and the probability density functions can no longer be approximated by the Gaussian distribution. So, the ¹²C(α, γ)¹⁶O reaction rate at these temperatures can be obtained just by the S factor measurements at energies as wide as possible.

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The S_tot measurements of Schürmann et al. (2005, 2011), in reverse kinematics using the recoil mass separator, allowed us to acquire data with a high degree of accuracy (<3%) in a wide energy scope of E_c.m. = 1.5–4.9 MeV. These data would provide good restrictions on the probability density functions of T₉ = 2.0, 3.0, and 4.0 (Figures 1(C)–(E)), and the uncertainties of the ¹²C(α, γ)¹⁶O reaction rate are smaller than 3%, from 2.0 ≤ T₉ ≤ 4.0.

Kunz et al. (2002) studied the S factor in the ground state of ¹²C(α, γ)¹⁶O at higher energies, covering the ${1}_{3}^{-}$ (E_c.m. = 5.28 MeV) and ${1}_{4}^{-}$ (E_c.m. = 5.93 MeV), using resonance parameters of Tilley et al. (1993) in the calculation. The published data in two independent experiments  (Brochard et al. 1973; Ophel et al. 1976) of the ground-state transition were neglected. Possible interference effects were included in the calculation of S_E10 and S_E20 by applying the R-matrix fitting procedures, but they were somewhat speculative, and the results of S_g.s were about 2 ∼ 5 times away from experimental data. Therefore, from the probability density functions of T₉ = 5.0, 6.0, 8.0, and 10.0 (Figures 1(F)–(I)), it can be inferred that the rate calculation of Kunz et al. (2002) is significantly higher.

The S_tot can be indicated according to different types of J^π (0⁺, 1⁻, 2⁺, 3⁻, and 4⁺). Figure 2(A) shows the fractional contributions of different values of J^π compared to the total reaction rates in 0.04 ≤ T₉ ≤ 10.0. It can be seen that J^π = 1⁻ and 2⁺ dominate the reaction rate up to T₉ = 2.0. From the probability density functions of T₉ = 0.2, 1.0, and 2.0 (Figures 1(A)–(C)), we note that the contribution stems mainly from J^π = 1⁻ (E_c.m. = 2.42 MeV) and J^π = 2⁺ (E_c.m. = 2.58 MeV) levels in ¹⁶O. The rate above T₉ = 2.0, the fraction from 3⁻ gradually increases with temperature. This comes from the contribution of J^π = 3⁻ (E_c.m. = 4.44, 5.97, and 6.10 MeV) resonances. The contribution of J^π = 4⁺ increases with T₉ first, and then decreases, having two 4⁺ resonances just at E_c.m. = 3.20 and 3.93 MeV in the integral interval. Thus, the fractional contributions of different values of J^π reconfirmed the validity of the probability density functions in Figure 1.

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The contributions of ground-state capture and cascade captures to the reaction rate of ¹²C(α, γ)¹⁶O are illustrated in Figure 2(B). Ground state capture (S_E10 and S_E20) dominates the reaction rate up to T₉ = 0.1. Furthermore, the contributions from the cascade transitions increase with T₉. Besides at the important He-burning temperature T₉ = 0.2, the rate is still important all the way up to T₉ = 5.0, because the inverse reaction of ¹²C(α, γ)¹⁶O plays an important role in silicon burning (S. E. Woosley 2013, private communication). Thus, the cascade transition is necessary for the precise calculation of the reaction rate.

Figure 3 shows comparisons between our new reaction rate and previous estimates. In each panel, the dashed line shows the ratio of a previous determination to our new rate. The gray bands are the uncertainty of the published rate, e.g., in Figure 3(A), the edges of the gray zone are reaction rate ratios of NARCE II's limits to the principal value of our rates. The blue bands estimate the uncertainty of our rate. Below T₉ ≈ 4.0, our recommended results are within the uncertainties of Buchmann (1996), Kunz et al. (2002), and NACRE II. Above T₉ = 4, the results agree with the analysis of NACRE.

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By comparing all of the published rates with our present results, the principal values of each rate are shown in Figure 4. At an astrophysical temperature of T₉ = 0.2, the new rate is about 10% larger than the rate of NACRE II (S_tot (0.3 MeV) = 148 ± 27 keV b) and Buchmann (1996) (S_tot (0.3 MeV) = 146 keV b), about 16% lower than the rate of the NACRE (S_tot(0.3 MeV) = 199 ± 64 keV b), and it is quite consistent with the adopted value of Kunz et al. (2002) (S_tot (0.3 MeV) = 165 ± 50 keV b).

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In the intermediate range of 0.5 ≤ T₉ ≤ 3, our recommended rate is in good agreement with NACRE II. The temperature dependence of our recommended value differs significantly from the rates of Katsuma (2012), which stems from the higher total S factor at ${1}_{2}^{-}$ (E_c.m. = 2.42 MeV) resonance-peak, overestimating the cross section of Schürmann et al. (2005) in their calculations (Katsuma 2008). In the same temperature range, the deviation from Kunz et al. (2002) mainly originates from the lower calculation values of the total S factor from E_c.m. = 0.5 MeV to E_c.m. = 2.0 MeV. The lower value of NACRE is a direct consequence of the considered cascade transitions for the total S-factors.

For the rates above T₉ = 3, our reaction rate increases with T₉, but has lower values than Kunz et al. (2002) and NACRE II, because the high-energy data covering the ${1}_{3}^{-}$ and ${1}_{4}^{-}$ resonance are apparently overestimated in their calculations.

A common form of reaction rate is an analytical formula with an appropriate parametrization for applications in stellar models. Equation (2) is a usual expression (Buchmann 1996; Kunz et al. 2002).

$$\begin{eqnarray}&&{N}_{{\rm{A}}}\langle \sigma {\text{}}v{\rangle }^{{AF}}\\ &&\quad =\displaystyle \frac{{a}_{0}}{{T}_{9}^{2}{(1+{a}_{1}{T}_{9}^{-2/3})}^{2}}\mathrm{exp}\left[-\displaystyle \frac{{a}_{2}}{{T}_{9}^{1/3}}-{\left(\displaystyle \frac{{T}_{9}}{{a}_{3}}\right)}^{2}\right]\\ &&\quad +\displaystyle \frac{{a}_{4}}{{T}_{9}^{2}{(1+{a}_{5}{T}_{9}^{-2/3})}^{2}}\mathrm{exp}\left(-\displaystyle \frac{{a}_{6}}{{T}_{9}^{1/3}}\right)+\displaystyle \frac{{a}_{7}}{{T}_{9}^{3/2}}\\ &&\quad \times \mathrm{exp}\left(-\displaystyle \frac{{a}_{8}}{{T}_{9}}\right)+\displaystyle \frac{{a}_{9}}{{T}_{9}^{2/3}}(1+{a}_{10}{T}_{9}^{1/3})\mathrm{exp}\left(-\displaystyle \frac{{a}_{11}}{{T}_{9}^{1/3}}\right).\end{eqnarray} \tag{ 2 }$$

The difference between the fitting formula to the tabulated rate is shown in Figure 3(F). The blue bands indicate the uncertainty of the adopted rate in Table 1. The dotted line shows the ratio of the adopted values of the analytical expression normalized to the adopted tabulated ones. It is applicable in the temperature range of 0.04 ≤ T₉ ≤ 10 with a maximum deviation of 4% to the recommended rate in Table 1. For the most important range of T₉ = 0.1–0.3 the maximal deviation is 1%. And the parameters ${a}_{0}-{a}_{11}$ are a₀ = 4.70 × 10⁸; a₁ = 0.312; a₂ = 31.8; a₃ = 400; a₄ = 1.08 × 10¹⁵; a₅ = 23.6; a₆ = 41.3; a₇ = 2.49 × 10³; a₈ = 28.5; a₉ = 1.19 × 10¹¹; a₁₀ = −98.0; a₁₁ = 36.5.