Impact of ¹⁶O($\gamma$,$\alpha$)¹²C measurements on the ¹²C($\alpha,\gamma$)¹⁶O astrophysical reaction rate

The ¹²C($\alpha,\gamma$)¹⁶O reaction is believed to be one of the most important reactions in nuclear astrophysicsFowler (1984); Woosley et al. (2003). A recent reviewdeBoer et al. (2017) highlights the key role played by this reaction in both the evolution of and nucleo-synthetic yields from massive stars. The purpose of this study is to explore the role that forthcoming measurements of the inverse reaction - ¹⁶O($\gamma,\alpha$)¹²C (OSGA) - could have on reducing the overall uncertainty in the cross section for the ¹²C($\alpha,\gamma$)¹⁶O reaction at helium burning temperatures. To do this we perform fits to the existing data using the $R$-matrix approachLane and Thomas (1958) and study the impact of including new data on the inverse reaction. This is achieved by starting with a reasonable $R$-matrix fit that can be used as a basis for comparison to fits with and without projected ¹⁶O($\gamma,\alpha$)¹²C data. For the inverse capture data we start with a proposed JLab experimentSuleiman et al. (2013) in order to assess the possible role of new measurements in reducing the overall uncertainty in the cross sectionHolt et al. (2018). A detailed $R$-matrix analysis of this reaction and and excellent review of the subject is given in Ref.deBoer et al. (2017).

In the present work, we employed the $R$-matrix approach to calculate the total cross section, $\sigma(E)$, for alpha-capture to the ground state. Considering only ground state capture is sufficient for this study since the capture to excited states is believeddeBoer et al. (2017) to contribute only about 5$\%$ to the total capture rate at 300 keV. The cross section is then used to calculate the astrophysical $S$ factor given by

where $E$ is the energy in the center of mass, $\eta$ is the Sommerfeld parameter, $\sqrt{\frac{\mu}{2E}}Z_{1}Z_{2}\frac{e^{2}}{\hbar}$, and $\mu$ is the reduced mass of the carbon ion and alpha particle. Measurements of the $S$ factor as a function of energy are often reported in the literature. For the ¹²C($\alpha,\gamma$)¹⁶O reaction, the value of $S$ at $E=300\ keV$ is typically quoted as the most probable energy for stellar helium burning. Of course, the cross section is so small at 300 keV that it cannot be directly measured. Thus, extrapolations to 300 keV must be performed to study the impact of data on the extrapolation. Of course, efforts aimed at improving the data and extrapolation are underwaySuleiman et al. (2013); Gai (2018); Balabanski et al. (2017); Costantini et al. (2009); Robertson et al. (2016); Liu (2017); Bemmerer et al. (2018); Xu et al. (2007); Friščić et al. (2019) at a number of laboratories worldwide. The new inverse reaction (OSGA) experimentsSuleiman et al. (2013); Xu et al. (2007); Gai (2018); Balabanski et al. (2017); Friščić et al. (2019) bring a different set of systematic errors than previous experiments and thus provide an additional check on systematics.

The collision matrix for the OSGA reaction will be given in terms of the Hamiltonian $H^{\mathcal{L}}$ which electromagnetically couples the photon of multipolarity $\mathcal{L}$ to the nucleus. We introduce the wave function $\Psi_{E(J)}$ that describes the alpha-¹²C system in total spin state $J$ and an initial state wave function $\psi_{i(J_{i})}$ which describes the nucleus (¹⁶O) in its ground state. Then the collision matrix is given by

where $k_{\gamma}=E_{\gamma}/\hbar c$ is the photon wave number and the subscript c refers to the final $\alpha-^{12}$C channel with quantum numbers $slJ$. Here $s$ is the channel spin (zero in this case), $l$ is the orbital angular momentum, and $J=l+s$ is the total angular momentum. In principle, we would perform the radial integration in Eq. 2 from the origin to the channel radius (internal piece) and from the channel radius to infinity (external piece). According to the R-matrix theoryLane and Thomas (1958) inside the channel radius $a$, the final state wave function, $\Psi_{E(J)}$, can be expanded in terms of a complete set of states, $X_{\lambda(J)}$

where $\phi_{c}$ is a Coulomb phase shift, $\Gamma_{\mu c}$ is the width of level $\mu$ in channel $c$, and $A_{\lambda\mu}$ is the matrix that relates the internal wave function and the observed resonances. Here

where $E_{\lambda}$ is a level energy, $\delta_{\lambda\mu}$ is the Kronecker $\delta$ and $\xi$ is given in terms of the Coulomb shift factor, $S_{c}$, the boundary condition constant, $b_{c}$, and the Coulomb penetration factor, $P_{c}$

where here c refers to essentially the $\alpha$ channel in this case and the $\gamma_{\lambda c}$ are the $\alpha$ reduced width amplitudes. The $\alpha$ channel is the only open channel and closed channels are neglected.

The internal part of the collision matrix for radiative capture to the ground state is given by

where $\phi_{l}$ is the Coulomb phase shift for orbital angular momentum $l$, $\Gamma_{\lambda\alpha lJ}$ and $\Gamma_{\mu\gamma lJ}$ are the formal ground state $\alpha$ and radiative widths, respectively. For a given level, the observed width can be relatedLane and Thomas (1958) to the reduced width by

while the reduced widths for the bound states are given by

where $S_{bl}$ is the bound state shift factor for orbital angular momentum $l$. For the photon radiative width, we have

where $\Gamma_{\lambda\gamma lJ\circ}$ is the observed radiative width and

where $Q$ is the $Q$-value for the reaction and $E_{r\lambda}$ are the physical resonance energies as given in the equation $E_{r\lambda}=E_{\lambda}+(b_{c}-S_{c})\gamma_{\lambda\alpha}^{2}$.

We then calculated the $E\mathcal{L}$ ground state radiative cross sectionCarr and Baglin (1971) for the ¹²C($\alpha$,$\gamma$)¹⁶O reaction from the collision matrix for spin-zero nuclei:

We only considered ground state transitions and statistical errors in this study. We initially chose a channel radius of 5.43 fm to be consistent with a previous analysisdeBoer et al. (2017), but later consider a larger channel radius to be consistent with other analysesKirsebom et al. (2018); Shen et al. (2019). We employed five $E1$ resonance levels and four $E2$ resonance levels in the internal part of the the R-matrix analysis as shown in Table  1. This analysis is similar to that of refs.Azuma et al. (2010) and Holt et al. (1978), and the details comport with the results of Lane and ThomasLane and Thomas (1958). In order to speed up computations, we turned off the external part for this study. This external contribution is most sensitive to the $E2$ part of the cross section since the $E1$ external part is greatly reduced by isospin symmetry. In fact, the external E1 part would vanish under perfect isospin conservation. We performed the fit for data less than 3 MeV, where the external part is small. As a check, we turned on the external piece for several fits, but it did not significantly change the results.

We used a SIMPLEX fitterNelder and Mead (1965) for the present work. Our best $R$-matrix fit of the existing $E1$ and $E2$ $S$-factor data, shown in Fig. 1, was taken as the most probable description of the $S$-factor data. In order to explore the statistical variation in the S-factor extrapolations, we created $S$-factor pseudo-data by random variation according to a Gaussian probability distribution about the best fit $S$-factor values at the measured energies. In the randomizations, we multiplied the individual pseudo-data uncertainties by the square root of the ratio of the original best fit values to the original measured uncertainties. We further multiplied these uncertainties by the square root of the $E1$ and $E2$ reduced chi squares, the Birge factorBirge (1932), for the $E1$ and $E2$ fits, respectively. This procedure should give a conservative estimate for statistical uncertainties. For the subtheshold states, we fixed the radiative widths of the subthreshold states at the measured values and varied the reduced alpha widths. We allowed the reduced alpha and radiative width of the first $E1$ state above threshold to vary in the fit, while we allowed the radiative width of the fifth $E1$ state to vary. We also allowed the radiative width of the fourth $E2$ R-matrix level to vary. The first $E2$ state above threshold is very narrow and we fixed the parameters of this level at those of ref.deBoer et al. (2017). The radiative width of the third $E2$ resonance was treated separately. We observed that using the value in ref.deBoer et al. (2017) resulted in a cross section that was significantly smaller than the data of ref.Schurmann et al. (2011). Rather, we made a fit to $E2$ data that included the data of ref.Schurmann et al. (2011). We then fixed the third $E2$ radiative width at -0.65 eV found from the fit and used it in subsequent fits to the data below 3 MeV. Indeed, we fixed all other parameters except the third $E2$ radiative width and those marked with an asterisk in Table 1 at the values of ref.deBoer et al. (2017). The parameters allowed to vary are denoted by an asterisk in table 1.

Also, following ref. deBoer et al. (2017), we performed the fits by maximizing L rather than minimizing $\chi^{2}$, where L is givenSivia and Skilling (2006) by

and $R_{i}=(f(x_{i})-d_{i})^{2}/\sigma_{i}^{2}$ is the usual quantity used in $\chi^{2}$ minimizations. Here $f(x_{i})$ is the function to be fitted to data, $d_{i}$, with statistical error $\sigma_{i}$. The L maximization has the feature that it reduces the impact of large error bar data on the fit and generally gives larger S-factor uncertainties in projected values of $S(300\ keV)$ than that of a $\chi^{2}$ minimization. In this work $L_{tot}$ is maximized and defined by

where $L_{E1(2)}$ is $L$ for $E1(2)$ data and $L_{OSGA}$ represents $L$ for the inverse reaction data or JLab data in this case.

The parameters of the bound levels are very important for the projection to 300 keV. The resonance energies were fixed, but the parameters, $E_{\lambda}$, depend on the reduced width of the levels. We allowed the reduced widths of the bound states to vary, so the $E_{\lambda}$ varies. We chose the $R$-matrix boundary condition constants to cancel out this effect for the second levels so that $E_{\lambda}=E_{r\lambda}$ for these levels. For the third and higher levels, the reduced widths were not varied because alpha elastic scattering determined these widths and allowing them to vary did not make a significant difference. We used the $S$-factor data sets given in refs. Dyer and Barnes (1974); Kremer et al. (1988); Redder et al. (1987); Ouellet et al. (1992); Roters et al. (1999); Gialanella et al. (2001); Kunz et al. (2001); Assuncao et al. (2006); Makii et al. (2009); Plag et al. (2012) and show the E1 and E2 ground state S factors in Fig.1.

From this study it appears that inverse reaction data can have a significant impact on the projection of $S$(300 keV) based on the projected OSGA data. We took the projected JLab data to represent $E1$ + $E2$ data since only total cross sections to the ground state will be measured. The projected standard deviation for the 1000 fits to the $E1$ and $E2$ data with the proposed JLab data is generally smaller than that without JLab data. The JLab data constrain the total $E1$ + $E2$ cross section in the fit. This leads to smaller standard deviations than fitting $E1$ and $E2$ separately. Fitting only data above 1.6 MeV leads to a significant shift upward in the projected $S$-factors at 300 keV. This illustrates the importance of lower energy data in the extrapolation to 300 keV. Since the expected OSGA data will be less than 1.6 MeV and even lower than existing data, we can infer that the proposed OSGA data will have a significant impact on the value of the low energy extrapolation. The significant difference between $S$(300 keV) for the fits with channel radii of 5.43 and 6.5 fm indicates model uncertainty. The lower energy OSGA data may help resolve this ambiguity. For example, if the uncertainties on the projected 7 JLab data points are reduced by a factor of two, the $S$(300 keV) from a fit with a 5.43 fm channel radius is brought into agreement with that from a 6.5 fm fit. This level of accuracy at low energies would represent an interesting goal not only for the upcoming JLab experiment, but also for the other future experiments.