Transfer reactions with the Lagrange-mesh method

We apply the $R$-matrix method in distorted wave Born approximation (DWBA) calculations. The internal wave functions are expanded over a Lagrange mesh, which provides an efficient and fast technique to compute matrix elements. We first present an outline of the theory, by emphasizing the $R$-matrix aspects. The model is applied to the ${}^{16}\mathrm{O}(d,p){}^{17}\mathrm{O}$ and ${}^{12}\mathrm{C}{(}^7\mathrm{Li},t){}^{16}\mathrm{O}$ reactions, typical of nucleon and of $\alpha$ transfer, respectively. We illustrate the sensitivity of the cross sections with respect to the $R$-matrix parameters and show that an excellent convergence can be achieved with relatively small bases. We also discuss the effects of the remnant term in DWBA calculations and address the question of the peripherality in transfer reactions. We suggest that uncertainties on spectroscopic factors could be underestimated in the literature.

Figure 1

Schematic diagram of process (1). The valence cluster $v$ is transferred from the projectile $A$ to the target $b$. The angles $\mathrm{\Omega }$ and ${\mathrm{\Omega }}^{\prime }$ are associated with the coordinates $\mathit{R}$ and ${\mathit{R}}^{\prime }$.

Figure 2

${}^{16}\mathrm{O}(d,p){}^{17}\mathrm{O}$ angular distributions for the ground ($5/2^{+}$) and first excited ($1/2^{+}$) states of ${}^{17}\mathrm{O}$ at two deuteron energies. Dashed and solid lines correspond to the calculations with and without the inclusion of the remnant term in the potential. The dotted lines correspond to the fresco calculations [5]. Experimental data are taken from Ref. [35].

Figure 3

Coefficients $T_J$ (17) for the ${}^{16}\mathrm{O}(d,p){}^{17}\mathrm{O}$ reaction at $E_d=25.4$ MeV as a function of $J$. The lines are guide to the eye.

Figure 4

${}^{16}\mathrm{O}(d,p){}^{17}\mathrm{O}$ transfer cross section to the $1/2^{+}$ state at $E_d=25.4$ MeV for various channel radii.

Figure 5

${}^{16}\mathrm{O}(d,p){}^{17}\mathrm{O}$ transfer cross section at $E_d=25.4$ MeV and $\theta =2^{\circ }$ as a function of the number of basis functions $N$ (a) and of the channel radius (b).

Figure 6

${}^{12}\mathrm{C}{(}^7\mathrm{Li},t){}^{16}\mathrm{O}$ angular distributions at two different ${}^7\mathrm{Li}$ energies. The solid lines correspond to the calculations without the remnant terms. Dashed and dotted lines are obtained by including the remnant terms with the $t-{}^{12}\mathrm{C}$ potentials from Refs. [37, 38], respectively. The dash-dotted lines are obtained with the alternative $\alpha +{}^{12}\mathrm{C}$ potentials (see text). Experimental data (solid dots) are taken from Ref. [36].

Figure 7

Coefficients $(2J+1)T_J$ (17) for the ${}^{12}\mathrm{C}{(}^7\mathrm{Li},t){}^{16}\mathrm{O}$ reaction at $E_{Li}=28$ MeV as a function of $J$ and for two ${}^{16}\mathrm{O}$ states. The lines are guide to the eye.

Figure 8

Modulus of the modified scattering matrix ${\tilde{U}}^{J\pi }$ (43) for the ${}^{16}\mathrm{O}(d,p){}^{17}\mathrm{O}$ (g.s.) reaction at $E_d=25.4$ MeV, and for different $J$ values. The minimum distance $r_{\min }$ corresponds to the $p+n$ coordinate in panel (a) and to the ${}^{16}\mathrm{O} +$ n coordinate in panel (b).

Figure 9

Modified scattering cross sections $d\tilde{\sigma }\left(r_{\min }\right)/d\mathrm{\Omega }$ for the ${}^{16}\mathrm{O}(d,p){}^{17}\mathrm{O}$(g.s.) reaction at $E_d=25.4$ MeV, and for different scattering angles.

Figure 10

Modulus of the modified scattering matrix ${\tilde{U}}^{J\pi }$ (43) for the ${}^{12}\mathrm{C}{(}^7\mathrm{Li},t){}^{16}\mathrm{O}\left(2_1^{+}\right)$ reaction at $E_{\mathrm{Li}}=28$ MeV. The minimum distance $r_{\min }$ corresponds to the $\alpha +t$ coordinate in panel (a) and to the $\alpha +{}^{12}\mathrm{C}$ coordinate in panel (b).

Figure 11

Modified scattering cross sections $d\tilde{\sigma }\left(r_{\min }\right)/d\mathrm{\Omega }$ for the ${}^{12}\mathrm{C}{(}^7\mathrm{Li},t){}^{16}\mathrm{O}\left(2_1^{+}\right)$ reaction at $E_{\mathrm{Li}}=28$ MeV, and for different scattering angles.