Examination of the adiabatic approximation for $(d,p)$ reactions

Background: Deuteron-induced one-neutron transfer reactions have been used to extract single-particle properties of nuclei, and the adiabatic (AD) approximation is often used to simply treat the deuteron breakup states.

Purpose: The primary goal is to examine the validity of the AD approximation for the $(d,p)$ reaction systematically. We clarify also the role of the closed channels often ignored in the description of breakup reactions.

Methods: We calculate the $(d,p)$ cross sections with the continuum-discretized coupled-channels method (CDCC) for 128 reaction systems and compare the results with those obtained by the CDCC calculation with the AD approximation. Effect of the closed channels are investigated by ignoring them in CDCC.

Results: The AD approximation affects in general the $(d,p)$ cross section by less than 20%, but some exceptional (nonadiabatic) cases for which the AD approximation breaks down are found. The closed channels turn out to have significant effects on the cross section at deuteron energies less than about 10 MeV.

Conclusions: The use of the AD approximation in the description of the $(d,p)$ reaction can be justified in many cases, with the uncertainty of less than about 20%. The existence of some nonadiabatic cases nevertheless should be realized. The neglect of the closed channels without confirming the convergence of the CDCC result is not recommended.

Figure 1

Illustration of the three-body system.

Figure 2

Angular distributions of the $(d,p)$ cross sections calculated with CDCC (solid lines) and CDCC-AD (dashed lines) for (a) ${}^{\text{100}}\text{Zr}(d,p){}^{101}\text{Zr}\left(2s_{1/2}\right)$ at $E_d=5$ MeV with $S_n=0.1$ MeV and (b) ${}^{40}\text{Ca}(d,p){}^{41}\text{Ca}\left(2s_{1/2}\right)$ at $E_d=40$ MeV with $S_n=8$ MeV.

Figure 3

(a) Same as in Fig. 2 but for $S_n=8$ MeV; (b) ET cross section and (c) BT cross section.

Figure 4

(a) TMDs for the ET part of the cross section at ${40}^{\circ }$ for ${}^{100}\text{Zr}(d,p){}^{101}\text{Zr}\left(2s_{1/2}\right)$ at $E_d=5$ MeV. The solid (dashed) and dotted (dash-dotted) lines show the results with CDCC (CDCC-AD) for $S_n=8$ and 0.1 MeV, respectively. (b) Enlarged view of panel (a) for $r\le 10$ fm; the dotted and dash-dotted lines are multiplied by 100.

Figure 5

Same as in Fig. 3 but for ${}^{40}\text{Ca}(d,p){}^{41}\text{Ca}\left(2s_{1/2}\right)$ at $E_d=40$ MeV with $S_n=0.1$ MeV.

Figure 6

TMD for the BT cross section in Fig. 5 at $0^{\circ }$.

Figure 7

Same as in Fig. 3 but for ${}^{40}\text{Ca}(d,p){}^{41}\text{Ca}\left(2s_{1/2}\right)$ at $E_d=5$ MeV with $S_n=0.1$ MeV.

Figure 8

Angular distributions of the $(d,p)$ cross sections for ${}^{20}\text{Ne}(d,p){}^{21}\text{Ne}\left(0f_{7/2}\right)$ at $E_d=5$ MeV with $S_n=8$ MeV. The solid and dashed lines show the results of CDCC with and without the closed channels, respectively.

Figure 9

Same as in Fig. 8 but for (a) ${}^{20}\text{Ne}(d,p){}^{21}\text{Ne}\left(0f_{7/2}\right)$, (b) ${}^{40}\text{Ca}(d,p){}^{41}\text{Ca}\left(2s_{1/2}\right)$, and (c) ${}^{100}\text{Zr}(d,p){}^{101}\text{Zr}\left(2s_{1/2}\right)$ at $E_d=5$ MeV with $S_n=0.1$ MeV.