Asymptotic normalization coefficients for ${}^7\mathrm{Be}+p\to {}^8\mathrm{B}$ from the peripheral ${}^7\mathrm{Be}(d,n){}^8\mathrm{B}$ reaction and their astrophysical application

The proton transfer ${}^7\mathrm{Be}\left(d,n\right){}^8\mathrm{B}$ reaction at the energy of 4.5 MeV (center of mass) has been analyzed within the modified distorted-wave Born approximation. New estimates and their uncertainties are obtained for values of the asymptotic normalization coefficients for $p+{}^7\mathrm{Be}\to {}^8\mathrm{B}$, the astrophysical $S_{17}\left(0\right)$ factor, and the $s$-wave $p+{}^7\mathrm{Be}$ scattering lengths.

Figure 1

Diagrams describing transfer of the proton $p$ and taking into account possible subsequent Coulomb-nuclear rescattering of particles (${}^7\mathrm{Be},p$, and $n$) in the intermediate state.

Figure 2

Fit of the elastic $d+{}^7\mathrm{Li}$ and $d+{}^7\mathrm{Be}$ scattering cross sections by using different sets of optical potentials. In (a), the solid, dashed, and dotted lines correspond to sets 4, 3, and 2, whereas the dash-dotted line corresponds to set S1 from Ref. [23]; experimental data for the elastic $d+{}^7\mathrm{Li}$ scattering at $E_d=6\mathrm{MeV}$ (closed triangles) and for the $d+{}^7\mathrm{Be}$ scattering $E_d=5.79\mathrm{MeV}$ (open circles) are taken from Refs. [39] and [23], respectively. In (b), the solid and dashed lines correspond to sets $1a$ and $2a$ of the present work, and the dash-dotted and dotted lines correspond to sets 1 and 2 recommended in Ref. [22]; experimental data (open and close points) for the elastic $d+{}^7\mathrm{Li}$ scattering [39] at $E_d=7.0$ and 8.0 MeV, respectively.

Figure 3

The dependence of ${\mathcal{R}}_{3/21/2}(E_i,\theta ;b_{B;3/2},b_{d;1/2})$ on the single-particle $b_{B;3/2}$ at $b_{d;1/2}=0.942{\mathrm{fm}}^{-1/2}$ and the different angles $\theta$ for the energy of $E_i=4.5\mathrm{MeV}$ and with potential parameters of set 3 in Table 1. In (a), (b), and (c), the calculated ${\mathcal{R}}_{3/21/2}(E_i,\theta ;b_{B;3/2},b_{d;1/2})$ function is plotted for $\theta =16.{58}^{\circ },25.2^{\circ }$, and $34.{23}^{\circ }$, respectively. The width of the bands for fixed values of $b_{B;3/2}$ corresponds to variation of the parameters $r_o$ and $a$ of the adopted Woods-Saxon potential within the intervals from $r_o=1.10$ to 1.40 fm and $a=0.50$ to 0.75 fm.

Figure 4

Angular distributions depending on different cutoff radii $\left(R_{\mathrm{cut}}\right)$ for $E_i=4.5\mathrm{MeV}$ (a) and $E_i=5.8\mathrm{MeV}$ (b) calculated in DWBA with potential parameters of sets 3 and $1a$ (Table 1), respectively. The solid, dotted, and dashed lines are the differential cross sections calculated for the cutoff radius $R_{\mathrm{cut}}=0.0$, 4.0, and 5.0 fm, respectively.

Figure 5

The radial behavior of the single-particle ${}^8\mathrm{B}[=({}^7\mathrm{Be}+p)$] bound-state wave function $r{\phi }_{B;j_B}\left(r\right)$ with $j_B=3/2$ calculated for the Woods-Saxon potential with different sets of $\left(r_o,a\right)$ pairs and $b_{B,3/2}$: (1.00 fm; 0.50 fm) and $0.6200{\mathrm{fm}}^{-1/2}$ (the dashed line), (1.25 fm; 0.65 fm) and $0.7679{\mathrm{fm}}^{-1/2}$ (the solid line), and (1.40 fm; 0.80 fm) and $0.7960{\mathrm{fm}}^{-1/2}$ (the dashed-dotted line). The Coulomb radius $r_C=1.30\mathrm{fm}$. The dotted line is the tail $b_{B,3/2}{W}_{-{\eta }_B;3/2}\left(2\kappa r\right)$ of the bound-state wave function $r{\phi }_{l_B{j}_B}\left(r\right)$ with $b_{B,3/2}=0.7679{\mathrm{fm}}^{-1/2}$ ($r_o=1.25\mathrm{fm}$ and $a=0.65\mathrm{fm}$).

Figure 6

Values of the spectroscopic factors $Z_B$ (a) and the squared ANC $C_B^2$ (b) as a function of the single-particle $b_{B;3/2}$ obtained using the modified DWBA ${}^7\mathrm{Be}(d,n){}^8\mathrm{B}$ analysis at the energy $E_i=4.5\mathrm{MeV}$ and $b_{d,1/2}=0.942{\mathrm{fm}}^{-1/2}$ for the different fixed angles $\theta$ and the sets of the optical potentials. Data denoted by $\circ$ and $▵ (•$ and $▴)$ correspond to the experimental points $\theta$ of $16.6^{\circ }$ and $29.6^{\circ }$, respectively, for set 3(2).

Figure 7

The coupling scheme used in calculations of the cross section for the ${}^7\mathrm{Be}(d,n){}^7\mathrm{Be}$ reaction by the coupled reaction channels method.

Figure 8

The values of the squared ANC $C_B^2$ for ${}^7\mathrm{Be}+p\to {}^8\mathrm{B}$ for each of the experimental $\theta$. Data in (a)–(d) are obtained from the analysis of the experimental differential cross sections of Refs. [23] at $E_i=4.5\mathrm{MeV}$ and sets 1–4 of the optical potentials, respectively. The solid lines present the results for the weighted mean values. The widths of the bands are the corresponding weighed uncertainties.

Figure 9

Angular distribution of the differential cross section for the ${}^7\mathrm{Be}(d,n){}^8\mathrm{B}$ reaction at $E_i=4.5\mathrm{MeV}$ together with the theoretical calculations performed for the different sets of the optical parameters from Table 1 using the corresponding weighted mean of the ANC value from Table 2. The experimental data are taken from Ref. [23].