Surface-integral formalism of deuteron stripping

The purpose of this paper is to develop an alternative theory of deuteron stripping to resonance states based on the surface-integral formalism of Kadyrov et al. [Ann. Phys. 324, 1516 (2009)] and continuum-discretized coupled channels (CDCC). First we demonstrate how the surface-integral formalism works in the three-body model and then we consider a more realistic problem in which a composite structure of target nuclei is taken via optical potentials. We explore different choices of channel wave functions and transition operators and show that a conventional CDCC volume matrix element can be written in terms of a surface-integral matrix element, which is peripheral, and an auxiliary matrix element, which determines the contribution of the nuclear interior over the variable $r_{nA}$. This auxiliary matrix element appears because of the inconsistency in treating of the $n\text{−}A$ potential: This potential should be real in the final state to support bound states or resonance scattering and complex in the initial state to describe $n\text{−}A$ scattering. Our main result is formulation of the theory of the stripping to resonance states using the prior form of the surface-integral formalism and CDCC method. It is demonstrated that the conventional CDCC volume matrix element coincides with the surface matrix element, which converges for the stripping to the resonance state. Also the surface representation (over the variable ${\mathbf{r}}_{nA}$) of the stripping matrix element enhances the peripheral part of the amplitude although the internal contribution does not disappear and increases with an increase of the deuteron energy. We present calculations corroborating our findings for both stripping to the bound state and the resonance.

Figure 1

Prior DWBA differential cross sections for the ${}^{14}\mathrm{C}(d,p){}^{15}\mathrm{C}(2s_{1/2},E_x=0.0\mathrm{MeV})$ at $E_d=23.4$ MeV. The solid red line is obtained using the optical potential $U_{nA}$ when calculating $U_{dA}$; the blue dotted line is obtained with $U_{nA}=V_{nA}^{\mathrm{sp}}$ in $U_{dA}$.

Figure 2

Prior CDCC differential cross sections for the ${}^{14}\mathrm{C}(d,p){}^{15}\mathrm{C}(2s_{1/2},E_x=0.0\mathrm{MeV})$ at $E_d=23.4$ MeV. Notations are the same as in Fig. 1.

Figure 3

Prior DWBA differential cross sections for the ${}^{14}\mathrm{C}(d,p){}^{15}\mathrm{C}(2s_{1/2},E_x=0.0\mathrm{MeV})$ at $E_d=60$ MeV. Notations are the same as in Fig. 1.

Figure 4

Post (a) and prior (b) CDCC differential cross sections for the ${}^{14}\mathrm{C}(d,p){}^{15}\mathrm{C}(2s_{1/2},E_x=0.0\mathrm{MeV})$ at $E_d=23.4$ MeV. In the post form (a) the cutoff is introduced over the $p\text{−}n$ partial waves in the continuum component of the initial CDCC scattering wave function: $l_{pn}^{\max }=0$, red solid line; $l_{pn}^{\max }=2$, black dashed line; $l_{pn}^{\max }=4$, blue short dashed line; $l_{pn}=6$, dots. In the prior form (b) the cutoff is introduced over the $n\text{−}A$ partial waves in the continuum component of the final CDCC scattering wave function: $l_{nA}^{\max }=1$, red solid line; $l_{nA}^{\max }=2$, black dashed line; $l_{nA}^{\max }=3$, blue short dashed line; $l_{nA}=4$, dots.

Figure 5

Convergence of the post (a) and prior (b) CDCC differential cross sections for the ${}^{14}\mathrm{C}(d,p){}^{15}\mathrm{C}(2s_{1/2},E_x=0.0\mathrm{MeV})$ at $E_d=23.4$ MeV. $R_{\text{match}}=20$ fm, red solid line; $R_{\text{match}}=30$ fm, black dashed line; $R_{\text{match}}=40$ fm, blue short dashed line.

Figure 6

Dependence of the normalized post CDCC differential cross sections $R_x$ on $r_{nA}^{\max }$ for the ${}^{14}\mathrm{C}(d,p){}^{15}\mathrm{C}(2s_{1/2},E_x=0.0\mathrm{MeV})$ at $E_d=23.4$ MeV. $R_x$ is the ratio of the peak CDCC differential cross section, in which the integral over $r_{nA}$ is calculated up to $r_{nA}^{\max }$, to the full peak CDCC differential cross section calculated at $r_{nA}^{\max }\to \infty$. The solid red line is the normalized post CDCC form; the blue dotted line is the normalized prior CDCC fiorm.

Figure 7

Dependence of the normalized ADWA and CCBA differential cross sections $R_x$ on $r_{nA}$ for the deuteron stripping to resonance ${}^{16}\mathrm{O}(d,p){}^{17}\mathrm{O}\left(1d_{3/2}\right)$ at $E_d=36$ MeV. Blue short and long dash-dotted lines, the ratios $R_X$ of the peak prior ADWA and CCBA differential cross sections, correspondingly, in which the radial integral over $r_{nA}$ is calculated for $r_{nA}\ge r_{nA}^{\text{min}}$, to the full differential cross section. Similarly, magenta dotted and green dashed lines are the ratios $R_X$ of the peak prior ADWA and CCBA differential cross sections, correspondingly, in which the radial integral over $r_{nA}$ is calculated in the interval $0\le r_{nA}\le r_{nA}^{\max }$, to the full differential cross section. The red solid line is the $R_X$ dependence on $r_{nA}^{\max }$ calculated for the post ADWA form. Hence, $r_{nA}$ on the abscissa is $r_{nA}^{\text{min}}$ for the blue short and long dashed lines and $r_{nA}^{\max }$ for the dotted magenta, dashed green, and solid red lines.

Figure 8

Solid red line, dependence on $r_{nA}^{\max }$ of the normalized post ADWA differential cross section $R_x$ for the stripping to resonance ${}^{16}\mathrm{O}(d,p){}^{17}\mathrm{O}\left(1d_{3/2}\right)$ at $E_d=36$ MeV. $R_X$ is calculated as the ratio of the ADWA differential cross section, in which the radial integral over $r_{nA}$ is calculated for $0\le r_{nA}\le r_{nA}^{\max }$, to the full differential cross section. At each $r_{nA}^{\max }$ the peak value of the differential cross section is used. Blue dotted line, dependence on $r_{nA}$ of the binned radial resonant scattering wave function ${\overline{\psi }}_{k_{nA}s=1/2l_{nA}=2J_F=3/2}^{\left(\text{res}\right)}\left(r_{nA}\right)$.

Figure 9

Angular distributions for the deuteron stripping to resonance ${}^{16}\mathrm{O}(d,p){}^{17}\mathrm{O}\left(1d_{3/2}\right)$ at $E_d=36$ MeV. The red solid line is the DWBA, the blue short dashed line is the ADWA, and the green dashed line is the CCBA. All the angular distributions are normalized in the region of the forward peak to the experimental one, red dots [40].

Figure 10

Angular distributions for the deuteron stripping to resonance ${}^{16}\mathrm{O}(d,p){}^{17}\mathrm{O}\left(1d_{3/2}\right)$ at $E_d=36$ MeV calculated using prior CCBA for three different bins: 1 MeV, red solid line; $0.8$ MeV, dashed black line; $0.6$ MeV, short dashed blue line.

Figure 11

Solid red line- dependence on $r_0$ of the neutron resonance width extracted from the CCBA calculations of the ${}^{16}\mathrm{O}(d,p){}^{17}\mathrm{O}\left(1d_{3/2}\right)$ reaction at $E_d=36$ MeV. The blue dashed line is the experimental neutron resonance width of the $1d_{3/2}$ resonance in ${}^{17}\mathrm{O}$ and the blue strip is the resonance width's experimental uncertainty.

Figure 12

Solid red line- dependence on $r_0$ of the spectroscopic factor extracted from the CCBA calculations of the ${}^{16}\mathrm{O}(d,p){}^{17}\mathrm{O}\left(1d_{3/2}\right)$ reaction at $E_d=36$ MeV.