Introduction

Single-nucleon transfer reactions such as the (d, p) and (p, d) reactions are valuable tools for studying the single-particle structure of atomic nuclei. In recent years, many studies on transfer reactions with radioactive beams have contributed significantly to our understanding of the evolution of nuclear structures and nuclear astrophysics (see, for instance, the review papers of Refs. [2-4]). Nuclear reaction theories are essential for mediating nuclear structure information and measuring nuclear reaction data. In nearly all the theories for transfer reactions, for instance, the distorted wave Born approximation (DWBA) [5, 6], adiabatic wave approximation (ADWA) [7, 8], continuum discretized coupled-channel method (CDCC) [9-11], and Faddeev equation-based models [12-14], optical model potentials (OMPs) are indispensable model inputs. These OMPs affect the theoretical cross sections, thus affecting the informal nuclear structure obtained from experimental data [15]. Therefore, reliable OMPs are crucial for the nuclear reaction calculations.

Phenomenological global OMPs such as CH89 [16], KD02 [17], Becchetti–Greenless [18], the more recent WP [19] for nucleons, and the OMPs cited in refs. [20-25] for deuterons or other nuclei are deduced by fitting the elastic scattering data of a particle within a wide range of incident energies and target masses. Phenomenological global OMPs are widely used for nuclear reaction calculations [26-34]. However, most experimental data concern stable nuclei. Therefore, caution is necessary when applying these phenomenological OMPs to reactions involving exotic isotopes. Microscopic optical potentials, which are based on the more fundamental principles of nuclear many-body interactions, can be more appropriate for describing elastic scattering and transfer reactions simultaneously for unstable nuclei. Microscopic OMPs for nucleons, such as the Bruyères Jeukenne–Lejeune–Mahaux model potential (the JLMB potential) [35] and the global potential based on the Dirac–Brueckner–Hartree–Fock approach developed by R. R. Xu et al. (CTOM [36, 37]), and those for deuteron [38], triton [39], α[40], and heavy-ion [41], among others, are used with increasing frequency in applications to direct nuclear reactions. In addition, microscopic OMPs have been found to be more reliable for extracting nuclear structure information from transfer reaction data [42]. However, many microscopic OMPs are strictly semi-microscopic because some parameters constrained by nucleon scattering data are still present.

It has been found that nucleon elastic scattering and transfer reactions are sensitive to different regions of the OMPs [42]. It is possible that phenomenological OMPs that were constrained only by elastic scattering data (thus, only a certain radial range was well constrained by the experimental data) are not the best for theoretical calculations of transfer reactions. Recently, Whitehead, Lim, and Holt constructed a microscopic global nucleon–nucleus optical potential based on an analysis of 1800 isotopes in the framework of many-body perturbation theory with state-of-the-art nuclear interactions from chiral effective field theory (EFT) [1]. An attractive feature of the WLH potential is that none of its parameters are fitted to the nucleon–nucleus scattering data. One might expect that being derived fully microscopically, the WLH potential might be more suitable for probing nuclear structure information via transfer reactions. Thus, it is necessary to test the WLH potential with (p,d) transfer reactions to compare the results with experimental data and with the same calculations using phenomenological global nucleon–nucleus potentials.

In this study, tests were conducted using 28 sets of (p,d) reaction angular distributions on 15 nuclei, including both stable and unstable nuclei, for incident energies below 200 MeV/nucleon. The global phenomenological nucleon–nucleus potential of Koning and Delaroche (KD02 potential) [17] was used for comparison. The Johnson–Soper adiabatic wave approximation (ADWA) was adapted for the (p,d) and (d,p) reaction calculations. The ADWA, which is essentially a three-body nuclear reaction model, considers the deuteron breakup effect simply but effectively [7]. It is widely used in the analysis of (p,d) and (d,p) reactions (see Refs. [43, 44]). More importantly, it does not involve the deuteron-target OMPs; instead, it considers the deuteron as a p+n system and calculates the distorted waves of the (p+n)+target three-body system with proton- and neutron-target OMPs. Thus, only nucleon-target OMPs are required in the ADWA model, which suits our need to estimate the microscopic WLH potential with (p,d) and (d,p) reactions.

The remainder of this paper is organized as follows: in Sec. 2, we briefly introduce the theoretical model and procedures of data analysis. An analysis of the transfer reactions is presented in Sec. 3.1. The results for incident energies higher than 150 MeV are presented in Sec. 3.2 and for unstable nuclei are given in Sec. 3.3. A discussion of the spectroscopic factors (SFs) is presented in Sec. 3.4. Finally, we summarize our conclusions in Sec. 4.

Methodology

To study the effects of microscopic WLH potential on the cross section of the transfer reaction, we investigated the (p, d) transfer reactions with 15 target nuclei, which included ²⁸Si, ^34,46Ar, ⁴⁰Ca, ⁵⁴Fe, ^56,58,60Ni, ⁹⁰Zr, ¹²⁰Sn, ¹⁰²Ru, ¹⁴⁰Ce, ^142,144Nd, and ²⁰⁸Pb. The incident energies varied between 18 and 200 MeV/nucleon. The ranges of the target nuclei and incident energies were mainly limited by the upper bounds of the ranges of incident energies of the WLH and KD02 potentials, the lower bound of the target mass of KD02, and the availability of experimental data. All experimental data were obtained from the nuclear reaction database EXFOR/CSISRS [45] or digitized from original papers, as shown in Tables 1 and 2.

We adopted the three-body model reaction methodology (TBMRM) proposed by Lee et al. to analyze (p,d) reactions [43, 44, 72]. This methodology uses the Johnson–Soper ADWA model for the (p,d) and (d,p) reactions [7], where the amplitude of the A(p,d)B reaction is given by [73] $T_{pd}=S{F}_{nlj}^{1/2}\langle {\chi }_{dB}^{(-)}{\varphi }_{np}\left|V_{np}\right|{\chi }_{pA}^{(+)}{\varphi }_{nlj}\rangle \mathrm{,}$ (1) where SFnlj is the spectroscopic with n, l, and j being the node number, angular momentum, and total angular momentum, respectively, of the single neutron wave function ϕnlj in the nucleus A (A=B+n); χpA and χdB are the entrance- and exit-channel distorted waves, respectively; and Vnp is the neutron–proton interaction that supports the bound state of the n–p pair ϕnp (the deuteron wave function).

In this work, Vnp is the Gaussian potential with a depth of 72.15 MeV and a radius of 1.484 fm, which is taken from Ref. [74]. With this potential, only the s-wave was considered in ϕnp. Using the ADWA model, exit-channel distorted waves are generated with the following effective “deuteron” (as a subsystem composed of neutrons and protons) potential [7, 8]: $U_{dB}\left(R\right)=\frac{\langle {\varphi }_{np}\left|V_{np}\left[U_{nB}\left(R+\frac{r}{2}\right)+U_{pB}\left(R-\frac{r}{2}\right)\right]\right|{\varphi }_{np}\rangle }{\langle {\varphi }_{np}\left(r\right)\left|V_{np}\left(r\right)\right|{\varphi }_{np}\left(r\right)\rangle }\mathrm{,}$ (2) where UnB and UpB are the neutron and proton OMPs on the target nucleus B evaluated at half of the deuteron incident energies (the “E_d/2 rule”). In the zero-range version of ADWA, the effective deuteron potential becomes $U_{dB}\left(R\right)=U_{nB}\left(R\right)+U_{pB}\left(R\right)\mathrm{.}$ (3) Clearly, with the three-body ADWA model, only the neutron and proton OMPs are required. This allows the calculation of the (p,d) and (d,p) reactions using only one set of systematic OMPs. Zero-range ADWA was found to be satisfactory for describing the deuteron pickup and stripping reactions [43, 44]. Therefore, although the finite-range version of the ADWA model and more sophisticated yet complicated continuum-discretized coupled channel models are available for the analysis of (d,p) reactions [8, 73], we adopted the zero-range ADWA model in this study. To examine the microscopic WLH potential and compare its results with the phenomenological KD02 potential, zero-range ADWA should be sufficient.

The single-particle wave functions were calculated using the separation energy prescription with the Woods–Saxon form of the single-particle potentials. The depths of these potentials were adjusted to reproduce the neutron separation energies in the ground state of the target nuclei. The radius and diffuseness parameters of these potentials, r₀ and a₀, are also important for the nuclear transfer reactions. Their empirical values were r₀ = 1.25 fm and a₀ = 0.65 fm. However, these empirical values do not represent the specific structure of a single nucleus. A better solution is to confine the r₀ and a₀ values with a reliable nuclear structure theory. The TBMRM constrains the r₀ and a₀ values using modern Hartree–Fock (HF) calculations [43, 72, 72, 75-83]. Using this procedure, the diffuseness parameter was fixed at a₀=0.7 fm. The radius parameter r₀ was determined by requiring the root mean square (rms) radius of the single neutron wave function, $\sqrt{\langle r^2\rangle }$, to be related with the rms radius of the corresponding single particle orbital from HF calculations, $\sqrt{{\langle r^2\rangle }_{HF}}$, by $\langle r^2\rangle =\left[A/(A-1)\right]{\langle r^2\rangle }_{\text{HF}}$. The factor [A/(A-1)] was used to correct the fixed potential center assumption used in the HF calculations, where A is the mass number of the composite nucleus. All the HF calculations performed in this work were based on the SkX interaction [84]. We adopted the same procedure as that used in our previous studies [42, 81]. All transfer differential cross sections were calculated using the TWOFNR code [85].

Results and Discussions

3.1

Transfer Reactions on Stable Nuclei at Low Energies

The reliability of the WLH potential for the (p,d) transfer reactions was first checked with stable nuclei at incident energies Ep≤ 150 MeV, which fall within the range of incident energies of the WLH potential. The target nuclei included ²⁸Si, ⁴⁰Ca, ⁵⁴Fe, ^58,60Ni, ⁹⁰Zr, ¹²⁰Sn, ¹⁰²Ru, ¹⁴⁰Ce, ^142,144Nd, and ²⁰⁸Pb. The results are shown in Figs. 1-3 together with the experimental data. For comparison, the results calculated using the global phenomenological KD02 potential are also presented. All the calculated results were normalized to the experimental data at the first peaks of the angular distributions or at the angles where the maximum measured differential cross sections occurred. Neutron SFs were obtained using this procedure. All experimental data correspond to neutron transfer from the ground states of the target nuclei to the ground states of the residual nuclei; thus, the results obtained in this work are single-neutron SFs in the ground states of the target nuclei.

Fig. 1Full-screen View

(color online) Comparisons between the two optical model calculations and experimental data of (p, d) reactions on ²⁸Si, ⁴⁰Ca, ⁵⁴Fe, and ⁵⁸Ni at the incident energies, up to 150 MeV, indicated in the figures. The solid and dashed curves are calculated with the WLH and KD02 potentials, respectively. The values of the differential cross sections are multiplied by the corresponding SF as labeled

Fig. 2Full-screen View

(color online) Same comparison as that in Fig. 1 but on ⁶⁰Ni, ⁹⁰Zr, ¹⁰²Ru, and ¹²⁰Sn

Fig. 3Full-screen View

(color online) Same comparison as that in Fig. 1 but on ¹²⁰Sn, ¹⁴⁰Ce, ^142,144Nd, and ²⁰⁸Pb

As shown in Fig.1, 2, and 3, generally, the experimental data can be satisfactorily reproduced by the WLH potential at forward angles as well as by the KD02 potential. The WLH potential was even better than the KD02 potential in some cases, such as ¹²⁰Sn_g.s.(p, d)¹¹⁹Sn_g.s. at 18 MeV. In some individual cases, the experimental angular distributions were not reproduced satisfactorily using either OMPs, for example, ¹²⁰Sn_g.s.(p, d)¹¹⁹Sn_g.s. at 26.3 MeV. Problems other than the OMP may exist in these cases.

3.2

Extrapolation to Higher Energies

The WLH potential is expected to work for incident energies below 150 MeV. Above this energy range, the theoretical uncertainties may become uncontrolled. However, it is interesting to observe how this operates when extrapolated to higher energies. When the incident energies are higher than approximately 150 MeV, the randomly generated WLH potential may generate unphysical positive potential. In these cases, we set the number of random pulls to 1500, which allowed us to obtain the least 1000 negative-valued potentials from which we could obtain reasonable averaged potentials. In Fig. 4, we show results for (p,d) reactions on ²⁸Si and ⁴⁰Ca targets at incident energies between 150 and 200 MeV. Again, it can be observed that the WLH potential reasonably reproduced these higher-energy data and yielded results very close to those of the KD02 potential at forward angles.

Fig. 4Full-screen View

(color online) Comparisons between two optical model calculations and experimental data of (p, d) reactions on ²⁸Si and ⁴⁰Ca at the incident energies >150 MeV indicated in the figures. The solid and dashed curves are calculated with the WLH and KD02 potentials, respectively. The values of the differential cross sections are multiplied by the corresponding SF as labeled

3.3

Transfer Reactions on Unstable Nuclei

There is a pressing need for high-quality optical potentials for nuclei that are far from stable, which represent a frontier in nuclear physics [86-88]. Phenomenological potentials are most suitable for reactions where the masses of the target nuclei are within the mass range of the experimental database from which the potential parameters are constrained. Because most elastic scattering data used to obtain phenomenological OMPs were measured on stable targets, the resulting OMP parameters may be biased by the limited range of target masses in the database. Therefore, caution is always advised when extrapolating these potentials to unstable nuclei, particularly when they are far from the β stability line. Meanwhile, microscopic OMPs, which are derived from effective nucleon–nucleon interactions with reliable nuclear structure models, are expected to be more sophisticated when applied to reactions with unstable nuclei. It is interesting to examine how the microscopic WLH potential works on unstable nuclei, and how it compares with the phenomenological KD02 potential. The results are presented in Fig. 5 for (p,d) reactions on unstable targets ^34,46Ar and ⁵⁶Ni. As can be observed, again, the WLH potential reproduced the experimental data reasonably well. In the ⁵⁶Ni case, the WLH potential was even slightly better than the KD02 potential. Unfortunately, the experimental data for (p,d) reactions on unstable nuclei are limited. More measurements of the (p,d) reaction data on targets further away from the β-stability line are required to examine the applicability of the WLH potential.

Fig. 5Full-screen View

(color online) Comparisons between two optical model calculations and experimental data of (p, d) reactions on unstable nuclei, including ³⁴Ar, ⁴⁶Ar, and ⁵⁶Ni, at the incident energies indicated in the figures. The solid and dashed curves are calculated with the WLH and KD02 potentials, respectively. The values of the differential cross sections are multiplied by the corresponding SF as labeled

3.4

Spectroscopic Factors

The OMPs affect not only the angular distributions but also the magnitudes of the (p,d) reactions. Both aspects of experimental data are important in nuclear reaction studies. Comparisons between the theoretical and experimental angular distributions determine whether the assumed reaction mechanism in the theoretical model and their parameters are appropriate, whereas the magnitudes of (p,d) reactions determine the SFs that provide important information about the nuclear single-particle structure. As stated above, the SFs are extracted experimentally by matching the theoretical and experimental angular distributions at the maximum cross sections, where the uncertainties of the experimental data and the extracted SFs are at their minimum [70] (see Figs. 1, 2, 3, 4, and 5). The SFs from all reactions analyzed in this study are presented in Tables 1 and 2.

Comparisons between the SFs obtained using the microscopic WLH potential and the phenomenological KD02 potential are shown in Fig. 6 for their ratios, SF_WLH/SF_KD02, at different incident energies. The target nuclei included are the doubly-magic isotopes ⁴⁰Ca and ²⁰⁸Pb, stable nuclei ²⁸Si, ^58,60Ni, ⁹⁰Zr, ¹²⁰Sn, ¹⁰²Ru, ¹⁴⁰Ce, and ^142,144Nd, and unstable nuclei ^34,46Ar and ⁵⁶Ni. Interestingly, when the incident energy was less than approximately 120 MeV, the two systematic potentials gave, on average, very similar SFs for all types of nuclei. However, when the range of incident energies increased to approximately 200 MeV, the ratios showed an increasing trend. This is especially apparent for the doubly magic nucleus ⁴⁰Ca (there is a lack of experimental data for ²⁰⁸Pb above 100 MeV). When the results in the tables are examined, it can be observed that the changes in SFs of ⁴⁰Ca with the KD02 potential are much smaller over the ranges of incident energies than those with the WLH potential.

Fig. 6Full-screen View

(color online) Ratios of experimental ground-state neutron SF values between the WLH and KD02 calculations as a function of mass number. The dashed line indicates a perfect agreement between data and theory. Open diamonds, open circles, and open triangles denote SF ratios extracted from (p, d) reactions on doubly magic nuclei, stable nuclei(not doubly magic), and unstable nuclei, respectively

It is well known that OMPs with doubly magic nuclei do not follow the systematics of OMPs established for other nuclei because of the relatively larger excitation energies of their first few excited states [16, 22, 94]. The increased disagreement between the SFs and the WLH and KD02 potentials at higher energies may indicate deficiencies in these potentials. Therefore, we compared the descriptions of proton elastic scattering from ⁴⁰Ca at higher energies. The results are presented in Fig. 7 for the ⁴⁰Ca target. It can be observed that, at lower energies, the two potentials describe the experimental data similarly. However, at higher energies, the phenomenological KD02 potential appears to be better than the microscopic WLH potential. However, neither potential reproduced the higher-energy data nor the low-energy data. Because all other parameters are the same in the transfer reaction calculations, this could be the reason for the differences among the SFs obtained at both potentials. Note that these incident energies are outside the range of WLH, which is for Ep<150 MeV. This suggests that caution is needed when extrapolating the WLH potentials at higher incident energies. Meanwhile, improvement of the systematic potential to lighter targets and/or higher-energy regions would be interesting and useful, especially for doubly magic nuclei.

Fig. 7Full-screen View

(color online) Comparison of the elastic scattering reaction cross section of the WLH calculation (solid lines), KD calculation (dashed lines), and experimental data (points) from Refs. [89-93] for p + ⁴⁰Ca

Summary

We verified the performance of the new microscopic optical WLH potential based on EFT theory in (p, d) reactions. In the present study, we performed zero-range adiabatic calculations for 15 nuclei, covering a wide range of 18–-200 MeV/nucleon. The phenomenological KD02 potential parameters were used for comparison. This pure microscopic optical potential effectively reproduced the angular distributions of both stable and unstable nuclei as well as the phenomenological KD02 potential. Our results suggest that the microscopic WLH potential can be used as an advanced approach for the prediction of transfer reactions on unstable nuclei and for extracting nuclear structure information of exotic nuclei.

Furthermore, we studied the amplitudes of the transfer cross sections with the WLH potential. The SFs extracted using the WLH potential were close to those obtained using the KD02 potential for all types of nuclei below approximately 120 MeV. However, when the range of incident energies was increased to approximately 200 MeV, the ratios showed a significant increasing trend with an increase in incident energy, especially for the doubly magic nucleus ⁴⁰Ca case. Comparisons between the WLH and KD02 potentials in their descriptions of proton elastic scattering from ⁴⁰Ca were performed up to 200 MeV. The results showed that some imperfections existed at higher energies for both potentials. An improvement in the WLH potential at higher incident energies for doubly magic nuclei is expected. Nevertheless, in general, the successful application of the WLH potential in (p,d) transfer reactions seems encouraging.