Systematic theoretical study of the finite-range effects in ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:math> ) transfer reactions

Background: The adiabatic wave approximation (ADWA) is widely applied in systematic studies of ($d,p$) and ($p,d$) single-nucleon transfer reactions. To simplify the calculations, a zero-range approximation (ZRA), usually corrected with a local energy approximation (LEA) to take into account finite-range effects (FREs), is often employed in calculating the $T$ matrix of ($p,d$) and ($d,p$) reactions within the ADWA framework. In addition to the ZRA in the reaction $T$ matrix, a zero-range version of the effective deuteron optical model potential proposed by Johnson and Soper (${U}_{d}^{\text{JS}}$) is also much more widely used in ADWA calculations than its finite-range version proposed by Johnson and Tandy (${U}_{d}^{\text{JT}}$). Therefore, it is necessary to assess the influence of FREs and the effectiveness of finite-range corrections (FRCs) with the LEA in ADWA calculations. Previous systematic studies about this subject were mainly made at relatively low incident energies below about 40 MeV/nucleon and on target nuclei whose mass numbers are mostly larger than 40.Purpose: To study the FREs in ADWA calculations of the ($p,d$) reactions within a wider energy range on a finer distribution of target masses, so that the systematics of the FREs with respect to the ZRA calculations, and the accuracy of the FRC with the LEA, can be better assessed.Methods: This study analyzes 48 sets of experimental data of ($p,d$) reactions on 10 targets ($^{12}\mathrm{C}, ^{16}\mathrm{O}, ^{28}\mathrm{Si}, ^{40}\mathrm{Ca}, ^{58}\mathrm{Ni}, ^{90}\mathrm{Zr}, ^{102}\mathrm{Ru}, ^{118}\mathrm{Sn}, ^{142}\mathrm{Nd}$, and $^{208}\mathrm{Pb}$) within a wide range of incident energies from 18 to 200 MeV/nucleon. Three types systematic nucleon-nucleus optical model potentials (OMPs), namely, the phenomenological KD02 [A. J. Koning, et al., Nucl. Phys. A 713, 231 (2003)], the semimicroscopic JLMB (Jeukenne-Lejeune-Mahaux-Bruy\`eres) [E. Bauge, et al., Phys. Rev. C 63, 024607 (2001)], and the microscopic CTOM (Optical Model by co-operation between China Nuclear Data Center Tuebingen University) [R. R. Xu, et al., Phys. Rev. C 94, 034606 (2016)], are used. They allow us to investigate how the FREs depend on the incident energies, on the target masses, and on the choice of systematic OMPs.Results: Our results demonstrate that (i) the FREs due to ${U}_{d}^{\text{JT}}$ do not show clear dependence on incident energies for all three systematic OMPs used; (ii) the FREs in the $T$ matrix depend strongly on the incident energy, rising from around $0%$ to about $150%$ within the range of incident energies studied; (iii) the finite-range correction of the $T$ matrix by using the LEA effectively reduced the discrepancy between zero-range and finite-range (FR) calculations, particularly on medium and heavy targets ($A\ensuremath{\geqslant}90$); and (iv) on average, the $\mathrm{LEA}+\mathrm{JT}$ calculations overestimate the $\mathrm{FR}+\mathrm{JT}$ calculations by around $7.5%$ over the whole range of incident energies studied in this work.Conclusions: These results suggest that although the LEA can greatly improve the zero-range ADWA for ($p,d$) reactions, exact finite-range calculations ($\mathrm{FR}+\mathrm{JT}$) are still recommended in the analysis of ($p,d$) [and ($d,p$)] reactions as much as possible.