Measurement of the 10 keV resonance in the ${}^{10}\mathrm{B}(p,{\alpha }_0){}^7\mathrm{Be}$ reaction via the Trojan Horse method

The ${}^{10}\mathrm{B}\left(p,{\alpha }_0\right){}^7\mathrm{Be}$ bare nucleus astrophysical $S\left(E\right)$ factor has been measured for the first time at energies from about 100 keV down to about 5 keV by means of the Trojan Horse method (THM). In this energy region, the $S\left(E\right)$ factor is strongly dominated by the 8.699 MeV ${}^{11}\mathrm{C}$ level ($J^{\pi }={\frac{5}{2}}^{+}$), producing an $s$-wave resonance centered at about 10 keV in the entrance channel. Up to now, only the high–energy tail of this resonance has been measured, while the low-energy trend is extrapolated from the available direct data. The THM has been applied to the quasifree ${}^2\mathrm{H}({}^{10}\mathrm{B},{\alpha }_0$ ${}^7\mathrm{Be}$)n reaction induced at a boron-beam energy of 24.5 MeV. An accurate analysis leads to the determination of the ${}^{10}\mathrm{B}\left(p,{\alpha }_0\right){}^7\mathrm{Be}$ $S\left(E\right)$ factor and of the corresponding electron screening potential $U_e$, thus giving for the first time an independent evaluation of it.

Figure 1

(a) Diagram representing the quasifree process $A+B\to C+D+S$. The upper vertex describes the virtual decay of the THM nucleus $A$ into the clusters $x$ (participant) and $S$ (spectator); the cluster $S$ is considered to be spectator to the $x+B\to C+D$ reaction that takes place in the lower vertex. (b) Schematic diagram for the quasifree reaction ${}^2\mathrm{H}$ + ${}^{10}\mathrm{B}$ $\to \alpha$ + ${}^7\mathrm{Be}$ + $n$.

Figure 2

Schematic drawing of the adopted experimental setup, showing the $\Delta E-E$ system, made up of an ionization chamber (I.C.) and a position sensitive detector (${\mathrm{PSD}}_A$), devoted to ${}^7\mathrm{Be}$ detection, and ${\mathrm{PSD}}_B$ and ${\mathrm{PSD}}_C$, devoted to alpha-particle detection.

Figure 3

The two-dimensional $\Delta E-E$ plot, showing the energy loss in the ionization chamber ($\Delta E$) as a function of the residual energy detected in ${\mathrm{PSD}}_A$.

Figure 4

Experimental $Q$-value spectrum. The vertical arrow marks the position of the theoretical Q value of the ${}^2\mathrm{H}({}^{10}\mathrm{B},{\alpha }^7\mathrm{Be}$)n reaction. No reactions besides ${}^{10}\mathrm{B}$ + $d\to {\alpha }_0+{}^7\mathrm{Be}+n$ contribute, but a small background (not larger than $\sim 4$%).

Figure 5

The experimental kinematical locus $E_{\text{Be}}$ vs $E_{\alpha }$ for the ${}^2\mathrm{H}({}^{10}\mathrm{B},{\alpha }_0$ ${}^7\mathrm{Be})n$ reaction (black points) compared with the theoretical one (red points). The comparison has been made for a fixed detection angular pair.

Figure 6

Possible simplified diagrams for the ${}^2\mathrm{H}({}^{10}\mathrm{B},{\alpha }_0$ ${}^7\mathrm{Be})n$ reaction. Diagrams (a), (b), and (c) represent two-step processes, proceeding through the formation of the compound nuclei ${}^{11}\mathrm{C}$, ${}^8\mathrm{Be}$, and ${}^5\mathrm{He}$, respectively. Diagram (d) represents a direct breakup mechanism.

Figure 7

Two-dimensional plots of the $E_{\alpha \text{−}n}$ and $E_{\text{Be-}n}$ relative energies as a function of $E_{\alpha \text{-Be}}$. The arrows mark the positions of the 8.104 MeV (1), 8.420 MeV (2), 8.654 MeV, and 8.699 MeV [unresolved levels (3) and (4)] in ${}^{11}\mathrm{C}$. No evidence of horizontal loci, due to the population of ${}^5\mathrm{He}$ and ${}^8\mathrm{Be}$ excited levels, respectively, is present.

Figure 8

Two-dimensional plots of the $E_{\alpha \text{-Be}}$ relative energy as a function of the experimental momenta values $p_n$ of the undetected neutron. The labels maintain the same meaning as those of Fig. 7. It should be noticed that the detected ${}^{11}\mathrm{C}$ excited levels populate the low neutron momentum window, corresponding to the kinematical region where the bulk of the QF mechanism is expected.

Figure 9

Experimental momentum distribution (black points) compared with the theoretical one given by the squared Hulthén wave function in momentum space (black line) and the one given in terms of a DWBA calculation performed via the fresco code (red dashed line). The error bars include only statistical errors. The vertical blue line delimits the momentum region $p_n\le 30$ MeV/$c$ selected for the further analysis.

Figure 10

Events corresponding to the kinematical condition $0\le p_n\le 30$ MeV/$c$ (as discussed in the text). Panel (a) shows the events corresponding to the 8.420 MeV ${}^{11}\mathrm{C}$ level (2), while in panel (b) the events corresponding to the two unresolved 8.654 and 8.699 MeV levels [(3) and (4)] in ${}^{11}\mathrm{C}$ are displayed.

Figure 11

Selected events corresponding to the 10 keV resonance for ${\mathrm{the}}^{10}B\left(p,{\alpha }_0\right){}^7\mathrm{Be}$ reaction after removing the subthreshold contribution due to the 8654 keV ${}^{11}\mathrm{C}$ resonant level, as discussed in the text.

Figure 12

Direct astrophysical $S\left(E\right)$ factor of the ${}^{10}\mathrm{B}\left(p,{\alpha }_0\right){}^7\mathrm{Be}$ reaction [6, 7, 10]. The lines represent the $R$-matrix calculation with the resonance parameters from the literature [9, 64], for bare (dashed line) and screened (full line) nuclei. Red symbols are used to mark the data of [7], corrected for the factor 1.83 as done in [10]; blue symbols refer to the measurement performed by [10]; and purple symbols refer to the thick-target measurements of [67]. All these data are included in the NACRE compilation of [6].

Figure 13

The experimental TH S(E)-factor (black dots) together with its fit (solid line). The error bars include the sources of uncertainty described in the text.

Figure 14

The experimental TH $S\left(E\right)$ factor (black points with the corresponding uncertainties of Table 5) together with its fit. The blue line represents the same TH $S\left(E\right)$ factor after removing the energy resolution effects, as discussed in the text.

Figure 15

The TH ${}^{10}\mathrm{B}\left(p,{\alpha }_0\right){}^7\mathrm{Be}$ $S\left(E\right)$ factor at infinite resolution, together with its allowed upper and lower limits as given by the corresponding uncertainties, is compared with the low-energy direct data of [10]. While at energies lower than 30 keV direct data are strongly influenced by electron screening effects, the TH $S\left(E\right)$ factor describes the typical bare-nucleus behavior.

Figure 16

The THM $S\left(E\right)$ factor (black stars), as given in Table 4, compared with the $R$-matrix calculation discussed in Sec. 5b (dashed line) and with the one including electron screening (full line). Red symbols mark the direct data from [7], corrected for the factor 1.83 as recommended in [10], blue symbols the data from [10], and purple symbols the thick-target data in [67].

Figure 17

The bare-nucleus THM $S\left(E\right)$ factor at infinite resolution (blue line) together its upper and lower values (dashed blue line). The low-energy data of [10] have been fitted leaving the screening potential $U_e$ as the only free parameter, leading to $U_e=240\pm 200$ eV. The result is shown as the full red line, together with its upper and lower values.