  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 reactions 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 the closed channels
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
give 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.
