Rapid convergence of the Weinberg expansion of the deuteron stripping amplitude

Theories of $(d,p)$ reactions frequently use a formalism based on a transition amplitude that is dominated by the components of the total three-body scattering wave function where the spatial separation between the incoming neutron and proton is confined by the range of the $n$-$p$ interaction, $V_{np}$. By comparison with calculations based on the continuum discretized coupled channels method we show that the $(d,p)$ transition amplitude is dominated by the first term of the expansion of the three-body wave function in a complete set of Weinberg states. We use the ${}^{132}$Sn($d,p$)${}^{133}$Sn reaction at 30 and 100 MeV as examples of contemporary interest. The generality of this observed dominance and its implications for future theoretical developments are discussed.

Figure 1

CDCC bin state to Weinberg state transformation coefficients $C_{ij}$, of Eq. (9), for Weinberg states $i=1,2,...,5$ and CDCC bin states $j=1,2,...,14$. The deuteron ground state is denoted by $j=0$. The CDCC bins were calculated up to $n$-$p$ relative momenta $k_{\max }=1.4$ fm${}^{-1}$ in steps $\Delta k_i=0.1$ fm${}^{-1}$. See Sec. 3 for full details.

Figure 2

Convergence of selected partial waves of the Weinberg components ${\chi }_i^W$ with respect to the maximum $n$-$p$ continuum energy included in Eq. (9). Results are for (a) ${\chi }_1^W$ and $E_d=100$ MeV, (b) ${\chi }_2^W$ and $E_d=100$ MeV, and (c) ${\chi }_1^W$ and $E_d=30$ MeV. The partial wave values, $L$, associated with each ${\chi }_i^W$ are indicated in each panel.

Figure 3

Calculated Weinberg state distorted waves ${\chi }_i^W$ demonstrating the dominance of ${\chi }_1^W$. Curves compare the moduli of ${\chi }_1^W$, ${\chi }_2^W$, and ${\chi }_3^W$ for the ${}^{132}$Sn(${d,p)}^{133}$Sn reaction for (a) $E_d=100$ MeV and partial wave $L=18$ and (b) $E_d=30$ MeV and partial wave $L=12$.

Figure 4

Comparisons of the calculated differential cross sections for the ${}^{132}$Sn($d,p$)${}^{133}$Sn reaction at (a) 100 MeV and (b) 30 MeV, using Weinberg distorted wave components ${\chi }_1^W$, ${\chi }_2^W$, and ${\chi }_3^W$, showing the dominance of the first Weinberg component ${\chi }_1^W$; see the text for details.