Thermonuclear fusion rates for tritium + deuterium using Bayesian methods

The ${}^3\mathrm{H}(d,n){}^4\mathrm{He}$ reaction has a large low-energy cross section and will likely be utilized in future commercial fusion reactors. This reaction also takes place during Big Bang nucleosynthesis. Studies of both scenarios require accurate and precise fusion rates. To this end, we implement a one-level, two-channel $R$-matrix approximation into a Bayesian model. Our main goals are to predict reliable astrophysical $S$-factors and to estimate $R$-matrix parameters using the Bayesian approach. All relevant parameters are sampled in our study, including the channel radii, boundary condition parameters, and data set normalization factors. In addition, we take uncertainties in both measured bombarding energies and $S$-factors rigorously into account. Thermonuclear rates and reactivities of the ${}^3\mathrm{H}(d,n){}^4\mathrm{He}$ reaction are derived by numerically integrating the Bayesian $S$-factor samples. The present reaction rate uncertainties at temperatures between 1.0 MK and 1.0 GK are in the range of 0.2% to 0.6%. Our reaction rates differ from previous results by 2.9% near 1.0 GK. Our reactivities are smaller than previous results, with a maximum deviation of 2.9% near a thermal energy of 4 keV. The present rate or reactivity uncertainties are more reliable compared to previous studies that did not include the channel radii, boundary condition parameters, and data set normalization factors in the fitting. Finally, we investigate previous claims of electron screening effects in the published ${}^3\mathrm{H}(d,n){}^4\mathrm{He}$ data. No such effects are evident and only an upper limit for the electron screening potential can be obtained.

Figure 1

The data used in our analysis: (red circles) Jarmie et al. [11]; (black diamonds) Brown et al. [12]; (green squares) Kobzev et al. [21]; (blue triangles) Arnold et al. [22]; and (purple triangles) Conner et al. [6]. Absolute cross sections were not determined in Ref. [12] and their data were normalized to those of Ref. [11]. Only statistical uncertainties are shown, but for many of the data points they are smaller than the symbol sizes. Details regarding the data evaluation are given in the Appendix. The energy ranges important for fusion reactors and Big Bang nucleosynthesis are $4–120\mathrm{keV}$ and $13–252\mathrm{keV}$, respectively.

Figure 2

Astrophysical $S$-factors computed using the single-level, two-channel approximation [see Eqs. (1, 2, 3)]. The data are the same as in Fig. 1. The blue curve is computed with the best-fit parameter values of Barker [4]. The red curve is obtained by arbitrarily multiplying Barker's reduced widths by a factor of 10 and adjusting the eigenenergy and boundary condition parameters slightly. The red curve does not represent any best-fit result and serves for illustrative purposes only.

Figure 3

Astrophysical $S$-factors obtained from the Bayesian $R$-matrix fit. The data are the same as in Fig. 1. The red lines represent credible $S$-factors computed using 500 sampled parameter sets that were chosen at random from the complete set of samples. The inset shows a magnified view of the credible $S$-factor samples.

Figure 4

Marginalized posterior of the $S$-factor at a representative center-of-mass energy of 40 keV. Percentiles of the distribution are listed in Table 1.

Figure 5

Marginalized posterior densities of the eigenenergy ($E_0$), the energy at which the level shift is set to zero ($E_B$), the deuteron and neutron reduced widths (${\gamma }_d^2,{\gamma }_n^2$), and the deuteron and neutron channel radii ($a_d,a_n$). Percentiles of the distributions are listed in Table 1.

Figure 6

Marginalized posterior density for the electron screening potential, $U_e$. No evidence for electron screening in the ${}^3\mathrm{H}(d,n){}^4\mathrm{He}$ reaction can be extracted from the available data, contrary to the claims of Langanke and Rolfs [45], and only an upper limit, $U_e\le 14.7\mathrm{eV}$, can be obtained (Table 1).

Figure 7

Marginalized posteriors of the $S$-factor normalization factors, $f_S$. The labels refer to the same data sets as shown in Fig. 1. Percentiles of the distributions are listed in Table 2.

Figure 8

(Top) Present ${}^3\mathrm{H}(d,n){}^4\mathrm{He}$ thermonuclear rates (gray) compared to the evaluation of Descouvemont et al. [13] (purple). (Bottom) Present ${}^3\mathrm{H}(d,n){}^4\mathrm{He}$ reactivities (gray) compared with the results of Bosch and Hale [16] (green). The gray bands signify 68% coverage probabilities. For a better comparison, all rates or reactivities are normalized to our new recommended (i.e., median) values (see Tables 3 and 4). The solid lines shows the ratio of previous and present recommended results.