New approach to determining radiative capture reaction rates at astrophysical energies

Radiative capture reactions play a crucial role in stellar nucleosynthesis but have proved challenging to determine experimentally. In particular, the large uncertainty ($\approx 100\%$) in the measured rate of the ${}^{12}\mathrm{C}(\alpha ,\gamma ){}^{16}\mathrm{O}$ reaction is the largest source of uncertainty in any stellar evolution model. With development of high-current energy-recovery linear accelerators (ERLs) and high-density gas targets, measurement of the ${}^{16}\mathrm{O}(e,e^{\prime }\alpha ){}^{12}\mathrm{C}$ reaction close to threshold using detailed balance allows a new approach to determine the ${}^{12}\mathrm{C}(\alpha ,\gamma ){}^{16}\mathrm{O}$ reaction rate with significantly increased precision ($<20\%$). We present the formalism to relate photo- and electrodisintegration reactions and consider the design of an optimal experiment to deliver increased precision. Once the new ERLs come online, an experiment to validate the approach we propose should be carried out. This approach has broad applicability to radiative capture reactions in astrophysics.

Figure 1

Measured astrophysical $S_{E1}$ and $S_{E2}$ factors for $E_{\alpha }<1.7\mathrm{MeV}$ from Dyer and Barnes [5], Redder [6], Kremer [7], Ouellet [8], Roters [9], Kunz [11, 12], Gialanella [10], Fey (a) turning table measurement and (b) EUROGAM measurement [13], Assunção (a) two-parameter fit and (b) three-parameter fit [15], Makii [16], and Plag [18]. An $R$-matrix fit to the data, represented by the solid line, was performed using the azure2 code [24].

Figure 2

Feynman diagram for the photodisintegration of ${}^{16}\mathrm{O}$ involving a real photon, $\gamma$, which requires that $q=\omega =E_{\gamma }$. The kinematic variables here will be discussed in more detail in Sec. 3.

Figure 3

First-order Feynman diagram for the electrodisintegration of ${}^{16}\mathrm{O}$ involving one virtual photon $\gamma \text{*}$ exchange to be compared with Fig. 2. Again, the kinematic variables here will be discussed in more detail in Sec. 3.

Figure 4

Transverse response function $R_T$ as function of the photon energy $\omega$ and the three-momentum transfer $q$, for the real photon case $q=\omega$ (solid line) and virtual photon case $q>\omega$ (surface plot), where ${\omega }_{\mathrm{th}}$ denotes the value of the threshold photon energy for the reaction.

Figure 5

Kinematics of the exclusive ${}^{16}\mathrm{O}(e,e^{\prime }\alpha ){}^{12}\mathrm{C}$ reaction.

Figure 6

The dependences of the invariant mass above threshold $W-W_{\mathrm{th}}$ (a), of the transferred three-momentum $q$ (b), of the ratio of the transferred energy to the transferred three-momentum $\omega /q$ (c), and of the virtuality of the exchanged photon $\rho \equiv |Q^2/q^2|$ (d) on the scattered electron energy $E_e^{\prime }$ for kinematics corresponding to within 2 MeV above threshold.

Figure 7

Leading-order coefficients $a_{E1}^{\prime }$ and $a_{E2}^{\prime }$ as functions of electron scattering angle ${\theta }_e$ at a beam energy $E_e$ of 114 MeV.

Figure 8

The semi-inclusive electrodisintegration differential cross section as a function of ${\theta }_{\alpha }^{\text{c.m}.}$ at a beam energy $E_e$ of 114 MeV and an electron scattering angle ${\theta }_e$ of ${15}^{\circ }$, for $E_{\alpha }^{\text{c.m.}}=0.7\mathrm{MeV}$ (a) and $E_{\alpha }^{\text{c.m}.}=1.7\mathrm{MeV}$ (b). There are 16 curves on each plot corresponding to $+$ and $-$ sign choices for each of the four next-to-leading-order coefficients $b_{C1}^{\prime }, b_{C2}^{\prime }, b_{E1}^{\prime }$, and $b_{E2}^{\prime }$. The difference introduced by the change of the sign is so small that the most of the lines are overlapping. At each of the local maxima, it is possible to distinguish two groups of lines which correspond to $+b_{C2}^{\prime }$ and $-b_{C2}^{\prime }$ contributions. Here all interferences involving the $C0$ multipole have been taken to have a plus sign together with the choices of phase differences discussed in the text. Alternatively, all such inteferences could enter with a minus sign.

Figure 9

Electric ($t_{E1}, t_{E2}$) and Coulomb ($|t_{C1}|, |t_{C2}\left|\right)$ multipole matrix elements as functions of the electron scattering angle ${\theta }_e$ at a beam energy $E_e$ of 114 MeV.

Figure 10

Angular distribution of the ${}^{16}\mathrm{O}(e,e^{\prime }\alpha ){}^{12}\mathrm{C}$ differential cross section for beam energy of $E_e=114\mathrm{MeV}$, electron scattering angle ${\theta }_e={15}^{\circ }$, and $\alpha$-particle c.m. kinetic energies $E_{\alpha }^{\text{c.m}.}=1.3$, 1.4, and 1.5 MeV. The electron beam lies on a ray from ${180}^{\circ }$ to $0^{\circ }$, the direction of the scattered electron is represented by a dashed line, and the direction of the virtual photon is shown by a solid line. The results were calculated as a function of $\alpha$-particle c.m. production angle ${\theta }_{\alpha }^{\text{c.m.}}$, but plotted with respect to the direction that the virtual photon has in the laboratory system. Panel (a) shows $t_{C0}$ case A and (b) case B.

Figure 11

Angular distribution of the ${}^{16}\mathrm{O}(e,e^{\prime }\alpha ){}^{12}\mathrm{C}$ differential cross section multiplied by $sin{\theta }_e$ for beam energy of $E_e=114\mathrm{MeV}, E_{\alpha }^{\text{c.m}.}=1.5\mathrm{MeV}$, for different electron scattering angles ${\theta }_e=5^{\circ },7.5^{\circ },{10}^{\circ }\left(\mathrm{a}\right);{15}^{\circ },{25}^{\circ },{35}^{\circ }\left(\mathrm{b}\right)$. The electron beam lies on a ray from ${180}^{\circ }$ to $0^{\circ }$, the directions of the scattered electrons are represented by dashed lines, and the other type of lines on the positive angle side represents the direction of the virtual photon in the laboratory system.

Figure 12

Angular distribution of the ${}^{16}\mathrm{O}(e,e^{\prime }\alpha ){}^{12}\mathrm{C}$ differential cross section multiplied by $sin{\theta }_{e}sin{\theta }_{\alpha }^{\text{c.m}.}$ for beam energy of $E_e=114\mathrm{MeV}, E_{\alpha }^{\text{c.m}.}=1.5\mathrm{MeV}$, for different electron scattering angles ${\theta }_e=5^{\circ },7.5^{\circ },{10}^{\circ }\left(\mathrm{a}\right);{15}^{\circ },{25}^{\circ },{35}^{\circ }\left(\mathrm{b}\right)$. The electron beam lies on a ray from ${180}^{\circ }$ to $0^{\circ }$, the directions of the scattered electrons are represented by dashed lines, and the other type of lines on the positive angle side represents the direction of the virtual photon in the laboratory system.

Figure 13

Angular distribution of the ${}^{16}\mathrm{O}(e,e^{\prime }\alpha ){}^{12}\mathrm{C}$ differential cross section multiplied by $sin{\theta }_e$ (a) and $sin{\theta }_{e}sin{\theta }_{\alpha }^{\text{c.m.}}$, (b) for beam energies of $E_e=78$, 114, and 150 MeV, $E_{\alpha }^{\text{c.m.}}=1.5\mathrm{MeV}$, and electron scattering angle of ${\theta }_e={15}^{\circ }$. The electron beam lies on a ray from ${180}^{\circ }$ to $0^{\circ }$, the directions of the scattered electrons are represented by dashed lines, and the other type of lines on the positive angle side represents the direction of the virtual photon in the laboratory system.

Figure 14

(a) The theoretical photo-nuclear cross section ${\sigma }_{\gamma N}$ as a function of the $\gamma$ energy $E_{\gamma }$ for the signal reaction ${}^{16}\mathrm{O}(\gamma ,\alpha ){}^{12}\mathrm{C}$, and background reactions ${}^{17}\mathrm{O}(\gamma ,\alpha ){}^{13}\mathrm{C}, {}^{18}\mathrm{O}(\gamma ,\alpha ){}^{14}\mathrm{C}$, and ${}^{14}\mathrm{N}(\gamma ,p){}^{13}\mathrm{C}$, from Ref. [52]. The same curves are shown on panel (b), but now the cross sections of the oxygen isotopes were normalized under the assumption that the natural abundances of ${}^{17}\mathrm{O}$ and ${}^{18}\mathrm{O}$ were depleted by a factor of 1000, and that oxygen gas is contaminated with 5 ppmv of ${}^{14}\mathrm{N}$.

Figure 15

Energy-loss corrected kinetic energy $E_k^{\text{Lab}}$ and time-of-flight ToF as functions of laboratory ion production angle ${\theta }_{\text{ion}}^{\text{Lab}}$ assuming that the ions were produced by electrons involved in the electrodisintegration of ${}^{16}\mathrm{O}$, at $E_e=114\mathrm{MeV}$ and ${\theta }_e={15}^{\circ }$: panels (a) and (b) cut on $0.7\le E_{\alpha }^{\text{c.m.}}\le 0.8\mathrm{MeV}$ and panels (c) and (d) cut on $1.0\le E_{\alpha }^{\text{c.m}.}\le 1.1\mathrm{MeV}$.

Figure 16

Standard deviation of the $\alpha$-production angle $\mathrm{\Delta }{\theta }_{\alpha }^{\text{Lab}}$ as a function of $\alpha$-particle kinetic energy $E_{\alpha }^{\text{Lab}}$ for an $\alpha$ particle passing through a 2-mm-wide ${}^{16}\mathrm{O}$ gas jet with a density of $6.65\times {10}^{-4}\mathrm{g}/{\mathrm{cm}}^3$.

Figure 17

Energy-loss-corrected kinetic energy of $\alpha$ particles $E_{\alpha }^{\text{Lab}}$ as functions of laboratory $\alpha$ production angle ${\theta }_{\alpha }^{\text{Lab}}$ for two values of transferred 3-momentum of the virtual photon $q$. The (a) panel shows $\alpha$ particles in the range of $0.7\le E_{\alpha }^{\text{c.m}.}\le 0.8\mathrm{MeV}$, and panel (b) shows them in the range of $1.0\le E_{\alpha }^{\text{c.m.}}\le 1.1\mathrm{MeV}$.

Figure 18

Schematic layout of our proposed ${}^{16}\mathrm{O}(e,e^{\prime }\alpha ){}^{12}\mathrm{C}$ experiment: ${}^{16}\mathrm{O}$, inside a gas cluster-jet target, is disintegrated by the electron beam into $\alpha$ particles and ${}^{12}\mathrm{C}$ nuclei. The scattered electron is detected in an electron spectrometer and the produced $\alpha$ particle in the ion detectors.

Figure 19

Top-view layout of proposed ${}^{16}\mathrm{O}(e,e^{\prime }\alpha ){}^{12}\mathrm{C}$ experiment, showing the in-plane angular acceptance of the ion detectors.

Figure 20

Coincidence rate per day for $t_{C0}$ cases A and B, for electron beam energy of $E_e=114\mathrm{MeV}$, central electron scattering angle of ${\theta }_e={15}^{\circ }$, electron spectrometer acceptance of 10.5 msr, $\alpha$-particle detector acceptance of 3.14 sr, and luminosity of $1.25\times {10}^{36}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$.

Figure 21

Number of events as a function of ${\theta }_{\alpha }^{\text{c.m}.}$ assuming 100 days of data collection at the luminosity of $2.5\times {10}^{36}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$. The left column represents $t_{C0}$ case A and the right one case B. Horizontal bars denote the width of the ${\theta }_{\alpha }^{\text{c.m}.}$ bin, which here is equal to ${10}^{\circ }$. Horizontal placement of the data point within a bin was done according to the procedure recommended in Ref. [64]. The $q$ bins are 1.91 MeV/c wide and $E_{\alpha }^{\text{c.m.}}$ bins are 100 keV. For all events inside a particular $q$ and $E_{\alpha }^{\text{c.m}.}$ bin, the specified $q$ values represent the average value, and the stated $E_{\alpha }^{\text{c.m}.}$ values represent the average of the expected and averaged $E_{\alpha }^{\text{c.m.}}$ value.

Figure 22

Differential cross section as function of ${\theta }_{\alpha }^{\text{c.m.}}$. The left column represents $t_{C0}$ case A and right column is case B. Vertical bars correspond to statistical uncertainties assuming 100 days of data collection at the luminosity of $2.5\times {10}^{36}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$. The same binning procedure was performed as described in caption of Fig. 21.

Figure 23

Reconstructed astrophysical $S_{E1}$ and $S_{E2}$ factors, showing their absolute (left column) and relative uncertainties (right column) for $t_{C0}$ cases A and B, $E_e=114\mathrm{MeV}$ and ${\theta }_e={15}^{\circ }$. $S_{aC0}$ does not have an astrophysical counterpart and it just a conversion of the third fitting parameter into an $S$ factor and corresponding uncertainty in order to put it in perspective with $S_{E1}$ and $S_{E2}$.

Figure 24

Reconstructed astrophysical $S_{E1}$ and $S_{E2}$ factors with statistical error bars (represented by solid circles) from our calculation for $E_e=$ 114 MeV, ${\theta }_e={15}^{\circ }, t_{C0}$ case A, together with experimental data from Refs. [5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 18] and azure2 R-matrix fit [24].

Figure 25

Relative uncertainties of reconstructed the astrophysical $S_{E1}$ and $S_{E2}$ factors (represented by solid circles) from our calculation for $E_e=114\mathrm{MeV}, {\theta }_e={15}^{\circ }, t_{C0}$ case A and relative errors from Refs. [5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 18] experiments. Data points with uncertainties larger then 140% are not shown.

Figure 26

Relative uncertainties of the $S_{E1}, S_{E2}$, and $S_{aC0}$ factors for several values of beam energies $E_e$, electron scattering angles ${\theta }_e$, and for $t_{C0}$ case A. The $S_{aC0}$ does not have an astrophysical counterpart and it just a conversion of the third fitting parameter into an $S$ factor and corresponding uncertainty in order to put it in a perspective with $S_{E1}$ and $S_{E2}$.

Figure 27

Same as in the caption of Fig. 26, but for $t_{C0}$ case B.