Trojan horse measurement of the ${}^{10}\mathrm{B}(p,{\alpha }_0){}^7\mathrm{Be}$ cross section in the energy range from 3 keV to 2.2 MeV

The ${}^{10}\mathrm{B}(p,{\alpha }_0){}^7\mathrm{Be}$ excitation function has been studied in a wide energy range, from 2.2 MeV down to astrophysical energies, reproducing the cross section above and below the Coulomb barrier in a single experiment. An optimized experimental setup ensured good energy resolution and for the first time a clear separation of ${\alpha }_0$ and ${\alpha }_1$ channels of the ${}^{10}\mathrm{B}+{}^2\mathrm{H}$ interaction has been achieved by applying the Trojan Horse method. An improved normalization of the Trojan Horse bare-nucleus astrophysical $S$($E$)-factor to direct data was performed and a value of $U_e=391\pm 74\mathrm{eV}$ was obtained for the electron screening potential.

Figure 1

Polar diagram representing the quasifree $a+A\to C+c+s$ reaction.

Figure 2

A schematic view of the experimental setup.

Figure 3

The particle energy loss ($\mathrm{\Delta }E$) measured in the ionization chamber ${\mathrm{IC}}_2$ as a function of the residual energy ($E$) detected by the ${\mathrm{PSD}}_4$.

Figure 4

An example of the $E_{{}^7\mathrm{Be}}-E_{\alpha }$ correlation plot obtained from the coincidence system ${\mathrm{C}}_4$. The kinematical locus for the ${}^2\mathrm{H}({}^{10}\mathrm{B},\alpha {}^7\mathrm{Be}$)$n$ reaction is clearly visible [ellipsoidal shape marked with (A)], but contributions from the two-body ${}^1\mathrm{H}({}^{10}\mathrm{B},{\alpha }_0){}^7\mathrm{Be}$ and ${}^1\mathrm{H}({}^{10}\mathrm{B},{\alpha }_1){}^7\mathrm{Be}$ reactions [two straight lines marked with (B)] and spurious events have to be rejected.

Figure 5

Three-body $Q$-value spectrum obtained for the ${}^7\mathrm{Be}-\alpha {\mathrm{C}}_4$ coincidence events. Two peaks with negative $Q$-values [(a) and (b)] correspond to the ${}^2\mathrm{H}({}^{10}\mathrm{B},{\alpha }_1{}^7\mathrm{Be})n$ and ${}^2\mathrm{H}({}^{10}\mathrm{B},{\alpha }_0{}^7\mathrm{Be}$)$n$ reactions, while two peaks with positive $Q$-values [(c) and (d)] correspond to the ${}^1\mathrm{H}({}^{10}\mathrm{B},{\alpha }_1){}^7\mathrm{Be}$ and ${}^1\mathrm{H}({}^{10}\mathrm{B},{\alpha }_0){}^7\mathrm{Be}$ reactions. The theoretical values $Q_{{\alpha }_0}^{\mathrm{th}}=-1.079$ MeV and $Q_{{\alpha }_1}^{\mathrm{th}}=-1.54$ MeV are marked by arrows in the lower panel. Error bars include only the statistical errors. Parameters obtained from the least-squares fit using the sum of two incoherent Voigt functions are given in Table 3.

Figure 6

Upper panel: the experimental $E_{{}^7\mathrm{Be}}-E_{\alpha }$ kinematical loci for the ${}^2\mathrm{H}({}^{10}\mathrm{B},{\alpha }_0{}^7\mathrm{Be})n$ (black points) and ${}^2\mathrm{H}({}^{10}\mathrm{B},{\alpha }_1{}^7\mathrm{Be})n$ (blue points) reactions for the coincidences ${\mathrm{C}}_4$. Lower panel: the experimental $E_{{}^7\mathrm{Be}}-E_{{\alpha }_0}$ locus (blue points) obtained at angles ${\theta }_{{\alpha }_0}={28}^{\circ }\pm 1^{\circ }$ and ${\theta }_{{}^7\mathrm{Be}}=14.5^{\circ }\pm 1^{\circ }$ for coincidences ${\mathrm{C}}_4$. The simulation for the corresponding angular conditions is shown by black dots.

Figure 7

(a) Polar diagram for the ${}^2\mathrm{H}({}^{10}\mathrm{B},{\alpha }_0{}^7\mathrm{Be})n$ quasi-free process. Sequential decay mechanisms proceeding through the formation of the excited intermediate nuclei ${}^{11}\mathrm{C}^{*}, {}^8\mathrm{Be}^{*}$ and ${}^5\mathrm{He}^{*}$ are represented with polar diagrams (b), (c) and (d), respectively, while the direct breakup is represented with polar diagram (e).

Figure 8

The two-dimensional plots of the ${}^7\mathrm{Be}-n$ and ${\alpha }_0-n$ relative energies as a function of the ${}^7\mathrm{Be}-{\alpha }_0$ relative energy for coincidences detected by the symmetrical pair of PSD's (${\mathrm{C}}_2$) (a) and (b) and for coincidences ${\mathrm{C}}_4$ (c). The arrows mark the positions of the excited ${}^{11}\mathrm{C}$ levels at energies of 8.420 (2), 8.654 (3), 8.699 (4), 9.780 (8), 9.970 (9), and 10.679 MeV (11) and position of the 19.235 MeV ${}^8\mathrm{Be}$ excited level.

Figure 9

An example of coincidence spectra projected onto the $E_{{\alpha }_0}$ axis for a fixed ${\theta }_{{\alpha }_0}={15}^{\circ }$ and for three different ${\theta }_{{}^7\mathrm{Be}}$ in an angular range of $\pm 1^{\circ }$. The condition where the neutron momentum has a minimum value is marked by an arrow. The error bars include only statistical errors.

Figure 10

The two-dimensional plot of the ${}^7\mathrm{Be}-{\alpha }_0$ relative energy vs the neutron momentum $p_s$, reconstructed for coincidence events ${\mathrm{C}}_2$ in the whole angular range covered by the detectors. Arrows mark the positions of the ${}^{11}\mathrm{C}$ 8.420 (2), 8.654 (3), 8.699 (4), 9.780 (8), 9.970 (9) and 10.679 MeV (11) excited levels.

Figure 11

The experimental momentum distribution as a function of the neutron momentum $p_s$ (black points) compared with the theoretical distribution given by the squared Hultén wave function in momentum space (black line). The vertical blue dashed line indicates the momentum region $p_s\le 40\mathrm{MeV}$/c considered in the future analysis. Error bars include only statistical errors.

Figure 12

The ${}^{10}\mathrm{B}(p,{\alpha }_0){}^7\mathrm{Be}$ angular distribution (weighted average for coincidence ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$) obtained in the energy range $E_{\text{c.m.}}=10\pm 5\mathrm{keV}$. The solid line represents the least-squares linear fit to the data assuming a dominant $s$-wave contribution. Error bars include only statistical errors.

Figure 13

Obtained yields for the kinematical condition $|p_s|\le 40\mathrm{MeV}$/c as a function of the ${}^{11}\mathrm{C}$ excitation energy as measured by the symmetric pair of PSD's (upper panel) and by the asymmetric pair of PSD's (lower panel). Arrows mark positions of known ${}^{11}\mathrm{C}$ excited states.

Figure 14

The weighted average data for coincidence systems ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$ corresponding to the ${}^{11}\mathrm{C}$ resonances at energies of 8.654 (3) and 8.699 MeV (4). The least-squares fit using Eq. (10) is displayed as a full black line, while the separate level contributions are displayed as dotted red (3), dashed green (4) and dash-dotted purple (5) lines. All the parameters obtained from the least-squares fit are listed in the Table 5. Error bars include only the statistical errors. Blue dash-dotted lines represent the upper and lower 95% confidence limits to the fit. Arrows mark positions of the ${}^{11}\mathrm{C}$ excited states as found in literature [48].

Figure 15

The least-squares fit to the experimental data corresponding to the 8.699 MeV ($E_{\text{c.m.}}=10$ keV) ${}^{11}\mathrm{C}$ resonance after removing the contribution of the subthreshold level at energy of 8.654 MeV (3). Error bars include the statistical errors and uncertainty due to the subthreshold level subtraction.

Figure 16

Weighted average of all four coincidences data of the experimental ${}^{11}\mathrm{C}$ excitation function. Arrows mark positions of excited states found in the literature [48] and correspond to the resonances at energies of 9.200 (5), 9.645 (7), 9.780 (8), 9.970 (9), 10.083 (10) and 10.679 MeV (11). Since the experimental data do not show evidence for the 9.360 MeV resonance (6), this level was not included in the fit. The least-squares fit using the sum of six incoherent Breit-Wigner functions is displayed as a full black line, while the separate level contributions are displayed as a dashed green lines. Blue dash-dotted lines represent the upper and lower $95\%$ confidence limits to the fit. All the parameters obtained from the least-squares fit are listed in the Table 6.

Figure 17

Bare-nucleus astrophysical $S\left(E_{\text{c.m.}}\right)$ factor obtained by the THM (present work, black points) compared with data from Ref. [22] and data from Ref. [21] renormalized as suggested in Ref. [22]. $R$-matrix fit from Ref. [11] (red line) is also given.