Cross sections of $\alpha$-induced reactions slightly below doubly magic ${}^{40}\mathrm{Ca}$ from the statistical model

New experimental data for the ${}^{38}\mathrm{Ar}(\alpha ,n){}^{41}\mathrm{Ca}$ and ${}^{38}\mathrm{Ar}(\alpha ,p){}^{41}\mathrm{K}$ reactions were used to find a best-fit set of parameters for statistical model calculations. The very good agreement between the experimental data and the best-fit calculation confirms the applicability of the statistical model for nuclei in the vicinity of the doubly magic ${}^{40}\mathrm{Ca}$ despite their relatively low level densities. The present study investigates the sensitivities and finds that the $\alpha$-nucleus and the nucleon-nucleus potentials are the most important ingredients for the calculation of ($\alpha ,n$) and ($\alpha ,p$) reactions in the statistical model. Furthermore, the width fluctuation correction plays an essential role in the peculiar case of ${}^{38}\mathrm{Ar}$. The best-fit parameters from ${}^{38}\mathrm{Ar}$ are applied to the mirror nucleus ${}^{38}\mathrm{Ca}$ and the neighboring ${}^{36}\mathrm{Ar}$ and ${}^{40}\mathrm{Ar}$ nuclei. For ${}^{38}\mathrm{Ca}$ this results in an astrophysical reaction rate of the ${}^{38}\mathrm{Ca}(\alpha ,p){}^{41}\mathrm{Sc}$ reaction, which has a flatter temperature dependence compared to all previous calculations. For ${}^{40}\mathrm{Ar}$ a better reproduction of ${}^{40}\mathrm{Ar}(\alpha ,p){}^{43}\mathrm{K}$ data from literature is obtained. The disagreement between calculation and an early experimental data point for the ${}^{36}\mathrm{Ar}(\alpha ,p){}^{39}\mathrm{K}$ reaction persists.

Figure 1

Total cross section ${\sigma }_{\mathrm{reac}}$ and compound-elastic ${\sigma }_{\mathrm{compound}}$($\alpha ,\alpha$) for (a) ${}^{38}\mathrm{Ar}$, calculated without WFCF (dotted red) and with WFCF (dashed blue). Because of the increased ${\sigma }_{\mathrm{compound}}$($\alpha ,\alpha$), the (b) ($\alpha ,n$), and (c) ($\alpha ,p$) cross sections are reduced by the WFCF. For further discussion see text.

Figure 2

Sensitivity of the ${}^{38}\mathrm{Ar}$($\alpha ,n$)${}^{41}\mathrm{Ca}$ and ${}^{38}\mathrm{Ar}$($\alpha ,p$)${}^{41}\mathrm{K}$ reactions for different choices for the width fluctuation correction. All cross sections are normalized to the reference calculation (see text) which uses the Moldauer approach for the WFCFs. The deviation of the Hofmann approach (red dashed) is practically negligible, and also the GOE approach (blue points) does not show a major deviation from the reference calculation. However, the cross sections without width fluctuation correction are significantly higher below 10 MeV (green dash-dotted), in particular for the ($\alpha ,n$) channel.

Figure 3

Same as Fig. 2 for the sensitivity of the ${}^{38}\mathrm{Ar}$($\alpha ,n$)${}^{41}\mathrm{Ca}$ and ${}^{38}\mathrm{Ar}$($\alpha ,p$)${}^{41}\mathrm{K}$ reactions on the chosen A-OMP. Except the early potentials by Nolte et al. [35] and Avrigenau et al. [36], the reproduction of the experimental data is quite good. But clearly the best description is obtained from the McFadden/Satchler potential [11].

Figure 4

Same as Fig. 2 for the sensitivity of the ${}^{38}\mathrm{Ar}$($\alpha ,n$)${}^{41}\mathrm{Ca}$ and ${}^{38}\mathrm{Ar}$($\alpha ,p$)${}^{41}\mathrm{K}$ reactions on the chosen N-OMP. The JLM-type potentials from [41, 42, 43] show a trend to overestimate the ($\alpha ,p$) data and underestimate the ($\alpha ,n$) data.

Figure 5

Same as Fig. 2 for the sensitivity of the ${}^{38}\mathrm{Ar}$($\alpha ,n$)${}^{41}\mathrm{Ca}$ and ${}^{38}\mathrm{Ar}$($\alpha ,p$)${}^{41}\mathrm{K}$ reactions on the chosen LD. The best fit is obtained from the generalized superfluid model. At higher energies above 10 MeV, the mHFB-G LD, which is based on Gogny forces, shows a much larger ($\alpha ,p$) cross section and lower ($\alpha ,n$) cross section than all other LDs. As expected, at low energies the role of the LD is very minor (see text).

Figure 6

Total cross section ${\sigma }_{\mathrm{reac}}$ (full black line), compound-elastic ${\sigma }_{\mathrm{compound}}$($\alpha ,\alpha$) (green dashed and dotted), and ${}^{38}\mathrm{Ca}$($\alpha ,p$)${}^{41}\mathrm{Sc}$ (blue dash-dotted) and ${}^{38}\mathrm{Ca}$($\alpha ,2p$)${}^{40}\mathrm{Ca}$ (red long-dashed) reaction cross sections. The uncertainty of the ${}^{38}\mathrm{Ca}$($\alpha ,p$)${}^{41}\mathrm{Sc}$ cross section from the choice of different LDs is indicated by the blue shaded area. The dotted magenta line indicates the production cross section of ${}^{41}\mathrm{Sc}$, i.e., without contributions from higher-lying levels in the residual ${}^{41}\mathrm{Sc}$, which decay preferentially by proton emission. The full orange line for the ($\alpha ,p$) cross section is calculated from the back-shifted Fermi gas LD and will be discussed later in Sec. 4c.

Figure 7

Astrophysical reaction rate $N_A〈\sigma v〉$ of the ${}^{41}\mathrm{Sc}$ production from the ${}^{38}\mathrm{Ca}$($\alpha ,p$)${}^{41}\mathrm{Sc}$ reaction: comparison of the new reference rate from the ${}^{38}\mathrm{Ar}$ reference parameters to various previous calculations [24, 51, 52, 54, 55, 56, 57]. For better visualization, all rates are normalized to the new reference rate $N_A{〈\sigma v〉}_{\mathrm{ref}}$ from this work. For further discussion see text.

Figure 8

Total cross section ${\sigma }_{\mathrm{reac}}$ and compound-elastic ${\sigma }_{\mathrm{compound}}$($\alpha ,\alpha$) for (a) ${}^{36}\mathrm{Ar}$, calculated from the reference parameters without WFCF (dotted red) and with WFCF (dashed blue). Because of the increased ${\sigma }_{\mathrm{compound}}$($\alpha ,\alpha$), the (b) ($\alpha ,n$) and (c) ($\alpha ,p$) cross sections are slightly reduced by the WFCF. The minor sensitivity to the choice of the LD is illustrated by the blue shaded areas. Further discussion see text.

Figure 9

Total cross section ${\sigma }_{\mathrm{reac}}$ and compound-elastic ${\sigma }_{\mathrm{compound}}$($\alpha ,\alpha$) for (a) ${}^{40}\mathrm{Ar}$, calculated from the reference parameters without WFCF (dotted red) and with WFCF (dashed blue). The ($\alpha ,n$) and ($\alpha ,p$) cross sections are shown in (b) and (c). The range of calculations from different parametrizations of the LD is shown as light-blue shaded area. Whereas the ($\alpha ,n$) data are relatively insensitive to the choice of the LD, the description of the ($\alpha ,p$) data can be improved using a different LD. For further discussion see text.

Figure 10

Same as Fig. 8, but with isospin correction: Total cross section ${\sigma }_{\mathrm{reac}}$ and compound-elastic ${\sigma }_{\mathrm{compound}}$($\alpha ,\alpha$) for (a) ${}^{36}\mathrm{Ar}$, calculated from the reference parameters with width fluctuation correction (dashed blue) and with additional isospin correction (dash-dotted green) from Eq. (5). The isospin correction leads to an increased ${\sigma }_{\mathrm{compound}}$($\alpha ,\alpha$), and it reduces the (b) ($\alpha ,n$) and (c) ($\alpha ,p$) cross sections by about 30%.