THEORETICAL REACTION RATES OF ¹²C(α, γ)¹⁶O BELOW T₉ = 3

The ¹²C(α, γ)¹⁶O reaction is considered as the key reaction for the carbon–oxygen ratio in the universe, and it influences not only the production of all elements heavier than A = 16 nuclei, but also stellar evolution from the He-burning phase to the explosive phase (Rolfs & Rodney 1988). The knowledge of ¹²C(α, γ)¹⁶O, however, has not been established yet, because the energy near the Gamow window corresponding to He-burning temperatures (E_c.m. = 0.3 MeV) is too low to measure the cross section in present laboratories. In the sequence of reactions in He burning, 3α → ¹²C(α, γ)¹⁶O(α, γ)²⁰Ne, the reaction rates of the triple-α process is comparatively well determined experimentally through the study of the resonance at the excitation energy E_x = 7.65 MeV in ¹²C. The ¹⁶O nuclei cannot be consumed significantly, because there is no resonance in ²⁰Ne at the astrophysically interesting energies. At present, the low-energy ¹²C(α, γ)¹⁶O cross section, from which the reaction rate is derived, is a remaining unsolved problem. Current theoretical discussions about the α-capture reactions are available at, e.g., Katsuma (2008), Mohr (2005), Ogata et al. (2009), and Dufour & Descouvemont (2008).

Recently, precise measurements of γ-ray angular distributions (Assunção et al. 2006; Makii et al. 2009; Kunz et al. 2001; Ouellet et al. 1996; Redder et al. 1987), cascade transitions (Kettner et al. 1982; Redder et al. 1987; Matei et al. 2006), and β-delayed α spectra for ¹⁶N (Buchmann & Barnes 2006; Tang et al. 2010 and references therein) have been performed to understand the low-energy ¹²C(α, γ)¹⁶O reaction more thoroughly. The phase shifts of α + ¹²C elastic scattering at low energies have been investigated to describe accurately the initial scattering channel (Plaga et al. 1987; Tischhauser et al. 2009). Through these experimental studies, the extension of the cross section to low energies has been extrapolated. Other experimental information are available in Dyer & Barnes (1974), Kremer et al. (1988), Barker & Kajino (1991), Buchmann et al. (1993, 1996), Azuma et al. (1994), Ouellet et al. (1996), Trautvetter et al. (1997), Angulo & Descouvemont (2000), Kunz et al. (2001), Tischhauser et al. (2002), Kettner et al. (1982), Barker (1987), Roters et al. (1999), Buchmann (2001), Hammer et al. (2005), Schürmann et al. (2005), Redder et al. (1987), and Matei et al. (2006). The extrapolation has also been predicted by the studies of the indirect measurements (⁶Li,d) and (⁷Li,t), e.g., Belhout et al. (2007) and Brune et al. (1999). Despite these efforts, uncertainties still remain in the astrophysical reaction rates at the important He-burning temperatures, T₉ ≈ 0.1–0.2 (T₉ is defined by T = T₉ × 10⁹ K).

Representative results for the reaction rate have been provided by Kunz et al. (2002, 2001), Buchmann (1996), Buchmann et al. (1996), Azuma et al. (1994), and Buchmann et al. (1993), based on the R-matrix analyses of the radiative capture data, the β-delayed α spectrum of ¹⁶N, and the phase shifts for elastic scattering. In this article, their results are referred to as KU02 (Kunz et al. 2002, 2001) and BU96 (Buchmann 1996; Buchmann et al. 1996; Azuma et al. 1994; Buchmann et al. 1993). For the extrapolated astrophysical S factors at E_c.m. = 0.3 MeV, KU02 estimate S_E1 = 76 ± 20 keV b, S_E2 = 85 ± 30 keV b, and S_casc = 4 ± 4 keV b for the E1, E2, and cascade transitions. BU96 predict S_E1 = 79 ± 21 keV b, S_E2 = 70 ± 70 keV b, and S_casc = 16 ± 16 keV b. The derived reaction rates give different temperature dependences from that in the Fowler's pioneering compilation of the reaction rates (Caughlan & Fowler 1988, hereafter CF88).

The NACRE compilation (Angulo et al. 1999) is the milestone work after CF88, providing evaluated reaction rates for the astrophysical community. For ¹²C(α, γ)¹⁶O, the NACRE reaction rate is based on the results from the R-matrix method of Buchmann et al. (1993) and Azuma et al. (1994) for E1 and Barker & Kajino (1991) for E2. Their extrapolated values at E_c.m. = 0.3 MeV are S_E1 = 79 ± 21 keV b and S_E2 = 120 ± 60 keV b. It is assumed that the contribution from the cascade transition through the excited states in ¹⁶O is negligible. The contribution from resonances up to E_c.m. = 6 MeV is added separately. The NACRE reaction rates of the ¹²C(α, γ)¹⁶O reaction now seem somewhat out-of-date compared with the recommended rates from the recent experimental and theoretical works.

The low-energy α+¹²C cross sections have been investigated with microscopic models (e.g., Dufour & Descouvemont 2008; Descouvemont & Baye 1987; Descouvemont et al. 1984). Hybrid models have also been performed in, e.g., Langanke & Koonin (1985), Funck et al. (1985), Langanke & Koonin (1983), and Koonin et al. (1974). The qualitative analyses by the models may seem successful. The effects of coupling to other channels were taken into account theoretically. But they might not have described ¹²C(α,γ)¹⁶O with sufficient accuracy, because of the limitations of their models. Besides, the coupling included explicitly might be redundant for the low-energy α+¹²C reaction because elastic scattering can be described by the single channel calculation of Katsuma (2010a).

For the evaluation of the reaction rates, the E1 contribution from the 1⁻₁ state at E_x = 7.12 MeV, just below the α-particle threshold of ¹⁶O, is assumed to play a crucial role in the reproduction of the available data at low energies. Our recent theoretical study (Katsuma 2008) using the potential model, however, shows that the low-energy cross section is dominated by the E2 component. The additive contribution from the 1⁻₁ state does not seem to be necessarily required. The derived S factors at E_c.m. = 0.3 MeV are S_E1 = 3 keV b, S_E2 = 150 keV b, and S_casc ≈ 20 keV b. Moreover, Katsuma (2008) illustrates that the derived reaction rate is concordant with published rates at the He-burning temperatures, even if the low-energy E1 cross section is not enhanced by the subthreshold 1⁻₁ state. Our potential model is constructed so as to satisfy the semi-classical limit of α-clustering aspect deduced by the optical potential describing rainbow phenomena at high energies (Katsuma 2008). Additional information from comments and replies is available in Descouvemont et al. (2010) and Katsuma (2010b).

The α+¹²C elastic scattering provides the initial channel for the direct capture potential model. The result from Katsuma (2010a), using the empirical potential (Katsuma 2008), gives a successful reproduction of the elastic differential cross sections, especially the enhancement of backscattering. It also shows survival of the single-particle motion of α-particle in the excitation function below E_c.m. = 5 MeV. Katsuma (2010a) endorses the applicability of the potential model with a simple α+¹²C configuration.

In this article, the numerical values of the ¹²C(α, γ)¹⁶O reaction rate below T₉ = 3 are listed in a tabular form. By performing a polynomial expansion of the reaction rates, a simple analytic expression is proposed. To estimate the theoretical uncertainties, we examine the sensitivity to the model parameters. The phase shifts of α + ¹²C elastic scattering are used to guide the variation of the potential parameters. The main purpose of this article is to distribute the theoretical reaction rates for assessment in astrophysical applications. The difference between our reaction rate and others (KU02; BU96; NACRE; CF88) originates from the assumed reaction mechanism in the analysis of the ¹²C(α, γ)¹⁶O cross section. We assume a direct capture mechanism and adopt the simple potential model. We take account of the contribution from the cascade transition through the excited states of ¹⁶O.

In the following section, we describe the procedure for estimating the reaction rates. In Section 3, the sensitivity of the ¹²C(α, γ)¹⁶O cross sections to the model parameters is investigated. The reaction rates are compared with KU02, BU96, NACRE, and CF88 and are given in tabular form and by an analytic expression. We examine the reaction rates deduced from the new phase-shift data of Tischhauser et al. (2009). A summary is given in Section 4.

The reaction rates of the ¹²C(α, γ)¹⁶O reaction are estimated from the potential model. The model used in this article is the same as that in the previous work (Katsuma 2008). We describe here the theoretical reaction rates, the procedures for estimating uncertainties, and analytic expressions. The details of the model can be found in Katsuma (2008).

The Maxwellian averaged thermonuclear reaction rate (Rolfs & Rodney 1988; Angulo et al. 1999) is defined by

$$\begin{eqnarray} N_A \langle \sigma v \rangle &=& N_A \frac{(8/\pi)^{1/2}}{\mu ^{1/2}(k_BT)^{3/2}} \int \!\! \sigma (E_{\rm c.m.}) E_{\rm c.m.}\nonumber\\ &&\times \exp \!\left(-\frac{E_{\rm c.m.}}{k_BT}\right) dE_{\rm c.m.}, \end{eqnarray} \tag{ 1 }$$

where N_A, k_B, μ, and T are the Avogadro number, Boltzmann constant, reduced mass, and temperature, respectively. E_c.m. is the center-of-mass energy. The σ(E_c.m.) is the capture cross section obtained from the potential model with a simple α + ¹²C configuration, and it is given as

$$\begin{eqnarray} \sigma (E_{\rm c.m.}) &=& 4 \pi \mathop {\sum }_{\lambda f i} d_0 \frac{k_\gamma ^{2\lambda +1}}{\hbar v} |\langle \phi _f || M^E_\lambda || \phi _i \rangle |^2, \end{eqnarray} \tag{ 2 }$$

where ϕ_f and ϕ_i are the wave functions for the final bound state and initial scattering channel, respectively. M^E_λ is the electromagnetic operator of the Eλ transition with the effective charge. The λ is the multipolarity of the transition; k_γ is the wave number of γ-rays; v is the relative velocity; d₀ is the factor depending on the orbital angular momentum and λ.

The effective charge in the electromagnetic operator is expressed with $\tilde{e}^{E\lambda }= e[Z_1(A_2/A)^\lambda + Z_2(-A_1/A)^\lambda ]$ (Barker & Kajino 1991; Descouvemont 2003), where A₁(Z₁) and A₂(Z₂) are mass number (charge) of nuclei in the initial channel and A = A₁ + A₂. However, the effective charge is treated as an adjustable parameter in our model (Katsuma 2008), including corrections in normalization of the transition amplitude. For example, the effective charge for the E1 transition is $\tilde{e}^{E1}=0$. This may express the isospin selection rule. However, $\tilde{e}^{E\lambda }$ can be a small value due to violation of the isospin selection rule. The normalization also originates from the simplicity of the model. It may depend on the assumption of the direct capture. The final bound state may have the complicated nuclear structure, which reflects to a spectroscopic factor deviating from unity. The effective charge is basically a part of the electromagnetic operator leading to the transition from the scattering channel to the bound state. It is, however, expressed as a value depending on the final bound state $\tilde{e}^{E\lambda }_f$ because of the inclusion of the spectroscopic factor.

To generate the initial waves ϕ_i, we use the parity-dependent potential,

$$\begin{eqnarray} V(r) &=& -\frac{V_\xi }{1+\exp \,[(r-R_\xi)/a_\xi ]}, \end{eqnarray} \tag{ 3 }$$

where ξ denotes the "+" (even) and "−" (odd) parities of the scattering channel. For the final bound state, the same functional form of the potential is used, of which the strength parameter is adjusted to reproduce the α-particle separation energy. In the case of the final bound state, ξ is used as a label of the state, assigned in order of the excitation energy. The method generating the bound state with the experimental particle separation energy is generally implemented in the direct reaction calculations (Satchler 1983) using, for example, the distorted-wave Born approximation. For the Coulomb potential, we use the uniform charge sphere with a radius R_C = 3.5 fm.

In our potential model (Katsuma 2008), the given potentials generate both the initial and final wave functions. The effective charges $\tilde{e}_\xi ^{E\lambda }$ of the bound state are adjusted so as to minimize the difference between the experimental and theoretical cross sections (Assunção et al. 2006; Kunz et al. 2001; Kettner et al. 1982; Redder et al. 1987; Matei et al. 2006). The reaction rates are obtained from the numerical integrals of the calculated cross section, including the extrapolated values. Thus, it is a difficult task to find reasonable parameters of the potential.

The astrophysical S factor is used to compensate for the rapid variation of the cross section below the Coulomb barrier and is defined by

$$\begin{eqnarray} &&S(E_{\rm c.m.}) = E_{\rm c.m.} \exp (2\pi \eta) \,\sigma (E_{\rm c.m.}), \end{eqnarray} \tag{ 4 }$$

where η is the Sommerfeld parameter, η = Z₁Z₂e²/(ℏv); Z₁ and Z₂ are the charges of interacting nuclei.

Before describing the procedure for estimating the uncertainties, let us recall the results from Katsuma (2008).

• 1.  
  At low energies, the S factor is dominated by the E2 transition to the ¹⁶O ground state, whereas the E1 component is not enhanced by the subthreshold 1⁻₁ state. The E2 component originates from the 2⁺₁ state (E_x = 6.92 MeV).
• 2.  
  The cascade transitions through the excited states below the α-particle threshold are not very important below E_c.m. = 1 MeV.
• 3.  
  The reaction rates above T₉ = 1 cannot be determined without reference to the cascade transition of the 3⁻₁ state (E_x = 6.13 MeV).

These imply the following.

• (A)  
  The variation of the even-parity potential leads to the different energy dependence of the E2 cross section.
• (B)  
  Above T₉ = 1, the unknown cascade transition to the 3⁻₁ state of ¹⁶O produces a large part of the uncertainties.

We estimate the uncertainties of the S factor and reaction rates by variation of the model parameters, i.e., the even-parity potential and the 3⁻₁ effective charge.

To examine (A), we use the even-parity potentials (Katsuma 2008) labeled as P4a–P4g in Table 1, which are obtained from a χ² fit to the phase-shift data (Plaga et al. 1987). The χ²_ps is defined by

$$\begin{eqnarray} \chi ^2_{\rm ps} &=& \mathop {\sum }_l\mathop {\sum }_j \,\left[\,\frac{\delta _l^{\rm ex}(E_j) -\delta _l^{\rm th}(E_j)}{\Delta \delta _l(E_j)} \,\right]^2, \end{eqnarray} \tag{ 5 }$$

where δ^ex_l and δ^th_l are the experimental and theoretical phase shifts, respectively. Δδ_l is the uncertainty of the data, assumed to be Δδ_l = 1°. The angular momentum has values of l = 0, 2, 4. Figure 1 shows the differences in the reproduction of the phase shifts. The solid curves are calculated from the best-fit parameters (P4d). The dotted and dashed curves are obtained from P4b and P4f, respectively. The phase shifts vary smoothly with the listed potential. In this article, we presume that the parameter sets of P4b–P4f are within the theoretical uncertainty. The odd-parity potential is fixed to V₋ = 168.1 MeV, R₋ = 2.76 fm, and a₋ = 0.567 fm.

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For (B), the E2 effective charge of the 3⁻₁ state is varied within the range of $\tilde{e}^{E2}_2= 1.4\pm 1.4\,e$. In Katsuma (2008), it was assumed that $\tilde{e}^{E2}_2= 1.4\,e$.

An analytic expression of the reaction rate is useful not only to implement the astrophysical applications but also to understand the contributions to the reaction rates. Several types of the analytic expression have been considered (Rolfs & Rodney 1988; Caughlan & Fowler 1988; Angulo et al. 1999). We try to express the reaction rate by using only a few parameters, as one of the forms:

$$\begin{eqnarray} N_A \langle \sigma v\rangle &=& h_0 T_9^{-4/3} \exp \left[-\frac{g_1}{T_9^{1/3}} -\left(\frac{T_9}{g_2}\right)^2\right]\nonumber\\ &&\times\left(1+ h_1 T_9 + h_2 T_9^2 +h_3 T_9^3 \right), \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} N_A \langle \sigma v\rangle &=& \tilde{h}_0 T_9^{-2} \exp \left[-\frac{g_1}{T_9^{1/3}} -\left(\frac{T_9}{g_2}\right)^2\right]\nonumber\\ &&\times \big(1+ \tilde{h}_1 T_9 + \tilde{h}_2 T_9^2 +\tilde{h}_3 T_9^3 \big), \end{eqnarray} \tag{ 7 }$$

where h_n, $\tilde{h}_n$ (n = 0–3), and g_n (n = 1, 2) are the coefficients of the expansion, which are adjusted to make the appropriate reaction rates. Although g₁ is given by g₁ = 3(π²μ/(2k_B))^1/3(e²Z₁Z₂/ℏ)^2/3 ≈ 32.1, it is adjusted to make a good approximation. The expression of Equation (7) is adopted in the NACRE compilation. The leading term at low temperatures in both equations appears to originate from the subthreshold state, as described in Section 3.3. The other terms do not have any specific mechanisms, but are used for the polynomial expansion, so the negative values of the coefficient do not have a special interpretation. The deviation between the analytic expression and tabular rate is defined by

$$\begin{eqnarray} &&\Delta = \left|\,1- \frac{N_A \langle \sigma v\rangle _{\rm Eq}}{N_A \langle \sigma v\rangle _{\rm Table}}\right| \times 100\%, \end{eqnarray} \tag{ 8 }$$

where N_A〈σv〉_Eq and N_A〈σv〉_Table are the analytic expression and the tabular value, respectively. The accuracy of the expansion is characterized by the maximum value of Δ.

The theoretical reaction rates in the temperature T₉ = 0.012–3.0 region are calculated with the potential model. The values of T₉ are the same in the table of CF88 (Caughlan & Fowler 1988). However, the reaction rate at T₉ = 0.012 is too small; the order of magnitude is approximately 10⁻⁵⁰ cm³ mol⁻¹ s⁻¹. In the NACRE compilation, the reaction rates below 10⁻²⁵ cm³ mol⁻¹ s⁻¹ are assumed to be practically negligible for all astrophysical conditions. Thus, the tabular rates for T₉ = 0.04–3.0 are provided in this article. This choice of the lowest temperature is the same as that in KU02 (Kunz et al. 2002). When making an analytic expression, the reaction rates in the entire region are taken into account to reproduce the theoretical model. Above T₉ = 3, the uncertainties of the reaction rates are expected to increase because of the contribution from the experimental data concerning narrow resonances. Moreover, the unknown 3⁻₁ cascade transition leads to the uncertainties (Katsuma 2008). We truncate the reaction rates at T₉ = 3 because the rates above T₉ = 3 do not correspond to the range of the considered cross section data.

In this section, we indicate the sensitivity of the S factor to the variation of the model parameters and compare the theoretical rate with the published rates (KU02; BU96; NACRE; CF88). The uncertainties of the reaction rates are also discussed. After listing the numerical values of the reaction rates, we present polynomial expansions with Equations (6) and (7), and provide a recommended expression. We finally discuss the results from the new phase-shift data of Tischhauser et al. (2009).

3.1. S Factors

The astrophysical S factors of the E2 transition for the ¹²C(α, γ₀)¹⁶O reaction are shown in Figure 2. The solid curve is the result calculated with the parameter set of P4d, which is the same curve as in Figure 5 of Katsuma (2008). The dotted and dashed curves are obtained from P4b and P4f, respectively. The E1 and E2 effective charges of the ground state, listed in Table 2, are adjusted to minimize the difference between the experimental and theoretical cross sections. The E2/E1 ratio seems reasonable from the angular distribution of γ-rays (Katsuma 2008).

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Table 2. Sensitivity of the S Factors at E_c.m. = 0.3 MeV and Derived Effective Charges of the Ground State

Potentials	$\tilde{e}_0^{E1}$	$\tilde{e}_0^{E2}$	SE1	SE2
	(10−3 e)	(e)	(keV b)	(keV b)
P4a	8.33	1.16	3.0	121
P4b	8.80	1.29	2.9	133
P4c	9.46	1.54	2.9	140
P4d	9.96	1.69	2.9	150
P4e	10.38	1.86	2.9	156
P4f	11.97	2.37	2.9	191
P4g	15.20	3.57	3.0	257

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In Table 2, the obtained S factors are listed. The E2 S factor varies smoothly, and it is found to be S_E2 = 150⁺⁴¹₋₁₇ keV b at E_c.m. = 0.3 MeV, if the potentials of P4b, P4c, P4d, P4e, and P4f with the optimized effective charges are adopted. In contrast, S_E1 ≈ 3 keV b is estimated. Although we examined the sensitivity of the S factor by varying the odd-parity potential keeping the l = 1, 3 phase shifts, the resultant E1 S factor was hardly affected.

To examine the sensitivity of the total S factor to the variation of the parameters, the cascade transitions must be taken into account. The effective charges of the excited states are listed in Table 3, as well as those of the ground state. The $\tilde{e}_\xi ^{E\lambda }$ of 0⁺₂, 2⁺₁, and 1⁻₁ are obtained from the fits to the cross section data (Kettner et al. 1982; Redder et al. 1987; Matei et al. 2006). The values $\tilde{e}_2^{E2}=0$, 1.4, and 2.8 e are used for the 3⁻₁ cascade transition. At E_c.m. = 0.3 MeV, the resultant S factor of the cascade transitions is S^casc_{E1 + E2} = 18 ± 4.5 keV b. The total S factor is, therefore, found to be S^tot_{E1 + E2} = 171⁺⁴⁶₋₂₂ keV b.

Table 3. E1 and E2 Effective Charges of the Bound States

Potentials	$\tilde{e}_0^{E1}$	$\tilde{e}_0^{E2}$	$\tilde{e}_1^{E1}$	$\tilde{e}_1^{E2}$	$\tilde{e}_2^{E1}$	$\tilde{e}_2^{E2}$	$\tilde{e}_3^{E1}$	$\tilde{e}_3^{E2}$	$\tilde{e}_4^{E1}$	$\tilde{e}_4^{E2}$
	(10−3 e)	(e)	(10−3 e)	(e)	(10−3 e)	(e)	(10−3 e)	(e)	(10−3 e)	(e)
P4a	8.33	1.16	0.	1.77	0.	0., 1.4, 2.8	4.50	2.03	0.	1.38
P4b	8.80	1.29	0.	1.86	0.	0., 1.4, 2.8	5.54	2.07	0.	1.38
P4c	9.46	1.54	0.	1.97	0.	0., 1.4, 2.8	4.96	2.37	0.	1.38
P4d	9.96	1.69	0.	2.05	0.	0., 1.4, 2.8	5.33	2.48	0.	1.38
P4e	10.38	1.86	0.	2.11	0.	0., 1.4, 2.8	6.17	2.49	0.	1.38
P4f	11.97	2.37	0.	2.32	0.	0., 1.4, 2.8	9.63	2.21	0.	1.38
P4g	15.20	3.57	0.	2.63	0.	0., 1.4, 2.8	14.43	1.27	0.	1.38

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We now discuss what is known about the asymptotic normalization constants (ANC; Mukhamedzhanov & Tribble 1999; Mukhamedzhanov et al. 1995). The α-particle ANC is defined by $C\stackrel{r\rightarrow \infty }{=} S^{1/2}_\alpha \phi _f(r)/W(r)$, where W(r) is the Whittaker function. S_α is the α-particle spectroscopic factor, given by $S_\alpha = (\tilde{e}^{E2}_\xi /\tilde{e}^{E2}_3)^2$ in our model (Katsuma 2008). The ANCs of the bound states used in the calculation of the recommended reaction rates shown in the next subsection are C² = 5.01 × 10²⁸ fm⁻¹, 2.03 × 10¹⁰ fm⁻¹, 1.18 × 10⁴ fm⁻¹, 2.76 × 10⁶ fm⁻¹, and 3.00 × 10⁵ fm⁻¹ for 1⁻₁, 2⁺₁, 3⁻₁, 0⁺₂ (E_x = 6.05 MeV), and 0⁺₁ (ground state). The ANC and S_α seem to be consistent with previous analyses (Belhout et al. 2007; Brune et al. 1999; Sparenberg 2004).

The calculated lifetime and B(E2) of the 2⁺₁ state allow the model to be assessed. The lifetime t = 5.98^+0.72_−3.33 fs and B(E2; 2⁺₁ → 0⁺₁) = 8.62^+10.8_−0.93 e² fm⁴ values are obtained from the parameters of Tables 1 and 3. The density distribution of α-particle (Satchler & Love 1979) and ¹²C (Kamimura 1981) are used. The α-particle spectroscopic factor of 2⁺₁ is assumed to be S_a = 1. The evaluated value is t = 6.78 ± 0.19 fs (Tilley et al. 1993).

3.2. Reaction Rates

In Figure 3, the reaction rates obtained from the potential model are compared with the published rates: (a) CF88, (b) NACRE, (c) KU02, and (d) BU96. In each panel, the theoretical reaction rates are indicated with the ratio to the respective reaction rates. The solid curves are the ratio of our recommended reaction rates. For reference, the ratio of the upper (lower) limit evaluated by NACRE, KU02, and BU86 is indicated with the dotted (dashed) curve. There is no indication of the dotted and dashed curves in Figure 3(a) because CF88 do not provide the uncertainties. Below T₉ ≈ 0.4, our reaction rates are within the uncertainties of NACRE, KU02, and BU96. From Figure 3 one sees that our reaction rates resemble BU96 in the temperature dependence, rather than CF88 and NACRE. Our reaction rate has a different trend with temperature compared with CF88 and NACRE, and shows a steep slope with increasing T₉. At high temperatures, the deviation from NACRE and CF88 mainly originates from the contribution of the cascade transition. The contribution from the experimental narrow resonances stemming from the compound nucleus mechanism is not analyzed in this paper.

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The uncertainties of the reaction rates obtained from the variation of the parameters are illustrated by the shaded areas in Figure 3. The pale shade is the uncertainty estimated from the variation of the potential parameters and $\tilde{e}^{E2}_2$. The dense shade is with fixed $\tilde{e}^{E2}_2= 0$. We confirm from this figure that the uncertainties at low temperatures stem from the potential variation. In contrast, the large part of the uncertainties above T₉ = 1 originates from the cascade transition through the 3⁻₁ state. To put the reaction rates on a firm foundation, measurement of the 3⁻₁ cascade cross section would be required. Conceivably, the 3⁻₁ cascade cross section may be as important as contributions from narrow resonances in the 1 < T₉ < 3 temperature region. The recommended values have been calculated by assuming that the effective charge of the 3⁻₁ state is approximately equal to that of the 1⁻₁ state (Katsuma 2008). The uncertainties estimated here appear to be reduced at low temperatures compared with other evaluations.

The numerical values of the present reaction rate are listed in Table 4, including the upper and lower limits. The reaction rates are given in units of cm³ mol⁻¹ s⁻¹. The values in each row should be multiplied by 10ⁿ, where n is listed in the sixth row.

Table 4. Theoretical ¹²C(α,γ)¹⁶O Reaction Rates

T9	Low	Recommend	High	NACREa	10n
0.04	6.14	6.84	8.96	⋅⋅⋅	−31
0.05	4.03	4.50	5.86	⋅⋅⋅	−28
0.06	5.59	6.25	8.11	10.2	−26
0.07	2.84	3.18	4.11	4.98	−24
0.08	7.20	8.07	10.4	12.2	−23
0.09	1.10	1.23	1.58	1.80	−21
0.10	1.15	1.29	1.64	1.81	−20
0.11	8.83	9.91	12.6	13.5	−20
0.12	5.37	6.02	7.66	7.98	−19
0.13	2.69	3.02	3.83	3.89	−18
0.14	1.15	1.29	1.63	1.61	−17
0.15	4.27	4.80	6.05	5.86	−17
0.16	1.42	1.60	2.01	1.91	−16
0.18	1.19	1.32	1.67	1.53	−15
0.20	7.30	8.20	10.2	9.11	−15
0.25	2.76	3.09	3.81	3.21	−13
0.30	4.32	4.84	5.93	4.75	−12
0.35	3.84	4.31	5.24	4.03	−11
0.40	2.31	2.60	3.14	2.31	−10
0.45	1.06	1.18	1.42	1.00	−9
0.50	3.86	4.32	5.16	3.52	−9
0.60	3.27	3.66	4.33	2.75	−8
0.70	1.79	2.01	2.37	1.40	−7
0.80	7.25	8.14	9.61	5.36	−7
0.90	2.36	2.66	3.16	1.66	−6
1.00	6.56	7.42	8.87	4.41	−6
1.25	5.15	5.96	7.57	3.19	−5
1.50	2.60	3.13	4.40	1.53	−4
1.75	9.79	12.6	19.8	5.76	−4
2.00	3.04	4.19	7.15	1.84	−3
2.50	1.92	2.85	5.26	1.27	−2
3.00	7.71	11.9	22.0	5.81	−2

Notes. The reaction rates are in units of cm³ mol⁻¹ s⁻¹. ^aNACRE reaction rates (Angulo et al. 1999).

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3.3. Analytic Expression of the Reaction Rates

We find a polynomial expansion of the tabular form of the reaction rates using Equation (6). The coefficients of the expansion are listed in Table 5. The maximum deviations, defined by Equation (8), are 6%, 8%, and 7% for the recommended, the upper limit, and the lower limit of the rates. The tabular reaction rates are satisfactorily expressed by Equation (6). The coefficients are |h_n| < 1 (n = 1, 2, 3). The reaction rate is thus found to be dominated by the first term at low temperatures.

Table 5. Coefficients of the Expansion of the Theoretical Reaction Rates, Defined by Equation (6)

Tabular Form	h0	h1	h2	h3	g1	g2	Max. Δa
	(cm3 mol−1 s−1)
Recommend	1.16 × 109	−0.371	−0.106	0.264	32.369	3.17	6%
High	1.32 × 109	−0.0686	−0.845	0.711	32.330	2.80	8%
Low	1.02 × 109	−0.422	0.124	0.0894	32.363	4.04	7%

Note.^a The maximum deviation is defined by Equation (8).

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A better expression could not be found with Equation (7) adopted in NACRE. The coefficients for the recommended reaction rates are listed in Table 6. $\tilde{h}_1$ becomes large, $\tilde{h}_1> 10$. Taking T₉ = 0.1, for example, the reaction rate is found to be dominated by the second term.

Table 6. The Same as Table 5, but the Coefficients Are Defined by Equation (7)

Tabular Form	$\tilde{h}_0$	$\tilde{h}_1$	$\tilde{h}_2$	$\tilde{h}_3$	g1	g2	Max. Δ
	(cm3 mol−1 s−1)
Recommend	9.04 × 106	186.3	−189.8	88.6	32.189	3.45	16%

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Figure 4 makes a comparison between the analytic expression and the tabular form. For the recommended reaction rates, the deviations 1 − N_A〈σv〉_Eq/N_A〈σv〉_Table are indicated with the circles (Equation (6)) and triangles (Equation (7)). Around T₉ = 0.05 and 0.4, the analytic expressions deviate from the recommended tabular rates.

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Let us consider the reaction mechanism associated with Equation (6). Using Equation (14) of Angulo et al. (1999) with ωγ = 0, the contribution from the subthreshold state, N_A〈σv〉_SR, is expressed in units of cm³ mol⁻¹ s⁻¹ as follows:

$$\begin{eqnarray} N_A \langle \sigma v\rangle _{\rm SR} &=& 12.4\!\times \!10^9 S(E_0) \, T_9^{-2/3} \!\exp \!\left(\!-\frac{32.1}{T_9^{1/3}}\!\right), \end{eqnarray} \tag{ 9 }$$

where S(E₀) is the S factor in MeV b units at the most effective energy E₀ (Rolfs & Rodney 1988; Angulo et al. 1999) approximately given as E₀ ≈ 0.922T^2/3₉ MeV for the considered reaction (Angulo et al. 1999). Suppose the slope of the S factor is simplified with S∝1/E_c.m., then T^−2/3₉ dependence is found in S(E₀). The corresponding reaction rate implicitly depends on T^−4/3₉. The T⁻²₉ dependence is found if S∝1/E²_c.m. is assumed. It thus seems that the first term of Equation (6) is the so-called tail contribution from the subthreshold state, having a weak energy dependence.

If S(E₀) = 0.17 MeV b is substituted into Equation (9), the coefficient becomes 2 × 10⁹. This appears to be in the same order of magnitude as h₀ (Table 5). For the expression of Equation (7), $\tilde{h}_0$ is small compared with 10⁹, while $\tilde{h}_0\cdot \tilde{h}_1 = 1.68\times 10^9$ becomes comparable (Table 6). We thus consider the analytic expression of Equation (6) to be better than that of Equation (7) for the present reaction rates because it is easier to understand the reaction rates at low temperatures. Besides, the accuracy of Equation (6) is good.

At high temperatures, the analytic expression of Equation (6) consists of the contributions from the total S factor and includes the cascade transitions. The resonant contributions of the states belonging to the α + ¹²C rotational band in ¹⁶O are also included. Therefore, the polynomial expansion does not have any specific interpretation.

3.4. Discussion about the New Phase Shift Data

The theoretical ¹²C(α, γ)¹⁶O reaction rates below T₉ = 3 have been considered with the potential model, and they have been provided in tabular form and by an analytic expression. The uncertainties of the reaction rates and cross sections have been estimated from the variation of the model parameters. The even-parity potential and 3⁻₁ effective charge are varied. As a guide to the variation of the potential parameters, the sensitivity of the phase shifts for α + ¹²C elastic scattering was examined.

At E_c.m. = 0.3 MeV, the S factor is estimated to be S_E2 = 150⁺⁴¹₋₁₇ keV b for the E2 component. The uncertainties of the E1 and cascade transitions are not so important as that of S_E2. The E1 S factor is S_E1 ≈ 3 keV b. The S factor of the cascade transition is S^casc_{E1 + E2} = 18 ± 4.5 keV b. Therefore, the total S factor is S^tot_{E1 + E2} = 171⁺⁴⁶₋₂₂ keV b.

The reaction rates are listed in Table 4, as well as the upper and lower limits, and they are compared with KU02, BU96, NACRE, and CF88. Although the assumed reaction mechanism is different, the reaction rate seems to be concordant with the published rates at low temperatures. The temperature dependence of our reaction rates is similar to that of BU96.

The recommended reaction rates for T₉ ⩽ 3 are expressed as

$$\begin{eqnarray} N_A \langle \sigma v\rangle &=& 1.16\!\times \!10^9 \,T_9^{-4/3} \exp \left[-\frac{32.369}{T_9^{1/3}}-\left(\frac{T_9}{3.17}\right)^2\,\right] \nonumber \\ &&\times \left(\, 1 -0.371 \,T_9 -0.106 \,T_9^2 +0.264 \,T_9^3 \right), \end{eqnarray} \tag{ 10 }$$

in units of cm³ mol⁻¹ s⁻¹. The maximum deviation from the tabular reaction rate is 6%. The first term appears to represent the tail contribution from the subthreshold 2⁺₁ state and dominates the reaction rate at low temperatures. For the present rates, the expression beginning with T^−4/3₉ (Equation (6)) is superior to that with T⁻²₉ (Equation (7)). The upper and lower limits, N_A〈σv〉_H and N_A〈σv〉_L, are given by

$$\begin{eqnarray} N_A \langle \sigma v\rangle _{\rm H} &=& 1.32\!\times \!10^9 \,T_9^{-4/3} \exp \left[-\frac{32.330}{T_9^{1/3}}-\left(\frac{T_9}{2.80}\right)^2\,\right] \nonumber \\ &&\times \big( 1 -0.0686 \,T_9 -0.845 \,T_9^2 +0.711 \,T_9^3 \big),\nonumber\\ \end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray} N_A \langle \sigma v\rangle _{\rm L} &=& 1.02\!\times \!10^9 \,T_9^{-4/3} \exp \left[-\frac{32.363}{T_9^{1/3}}-\left(\frac{T_9}{4.04}\right)^2\,\right] \nonumber \\ &&\times \big(1 -0.422 \,T_9 +0.124 \,T_9^2 +0.0894 \,T_9^3 \big).\nonumber\\ \end{eqnarray} \tag{ 12 }$$

The maximum deviations are 8% and 7%, respectively.

The author thanks Professors Y. Sakuragi, A. Kawauchi, and Y. Ohnita for their comments and encouragement, and thanks Professor I. J. Thompson for a careful reading of the manuscript. He also thanks Professors P. Descouvemont, M. Dufour, J.-M. Sparenberg, and A.E. Champagne for calling his attention to the present problem. He is grateful to Professors M. Arnould, A. Jorissen, K. Takahashi, and H. Utsunomiya for their hospitality and encouragement during his stay at Université Libre de Bruxelles (ULB). He thanks anonymous referee for comments. This work has been done in part by the Interuniversity Attraction Pole IAP 5/07 of the Belgian Federal Science Policy (Konan University—ULB convention), and was partially supported by the JSPS institutional program for young researcher overseas visits, "Promoting International Young Researchers in Mathematics and Mathematical Sciences led by Osaka City University Advanced Mathematical Institute."