Application of a microscopic optical potential of chiral effective field theory in (p, d) transfer reactions

Yi‑Ping Xu · Si‑Lu Chen · Dan‑Yang Pang

Abstract The microscopic global nucleon–nucleus optical model potential (OMP) proposed by Whitehead, Lim, and Holt, the WLH potential (Whitehead et al., Phys Rev Lett 127:182502, 2021), which was constructed in the framework of many-body perturbation theory with state-of-the-art nuclear interactions from chiral effective field theory (EFT), was tested with (p, d)transfer reactions calculated using adiabatic wave approximation.The target nuclei included both stable and unstable nuclei,and the incident energies reached 200 MeV.The results were compared with experimental data and predictions using the phenomenological global optical potential of Koning and Delaroche, the KD02 potential.Overall, we found that the microscopic WLH potential described the (p, d) reaction angular distributions similarly to the phenomenological KD02 potential;however, the former was slightly better than the latter for radioactive targets.On average, the obtained spectroscopic factors(SFs) using both microscopic and phenomenological potentials were similar when the incident energies were below approximately 120 MeV.However, their difference tended to increase at higher incident energies, which was particularly apparent for the doubly magic target nucleus 40Ca.

Keywords Microscopic optical model potential · (p, d) transfer · Spectroscopic factors

1 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 manybody 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 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 ap+nsystem and calculates the distorted waves of the(p+n)+target three-body system with proton- and neutrontarget 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 Sect.2, we briefly introduce the theoretical model and procedures of data analysis.An analysis of the transfer reactions is presented in Sect.3.1.The results for incident energies higher than 150 MeV are presented in Sect.3.2 and for unstable nuclei are given in Sect.3.3.A discussion of the spectroscopic factors (SFs) is presented in Sect.3.4.Finally,we summarize our conclusions in Sect.4.

2 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 included28Si,34,46Ar,40Ca,54Fe,56,58,60Ni,90Zr,120Sn,102Ru,140Ce,142,144Nd, and208Pb.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 theA(p,d)Breaction is given by [73]

whereSFnljis the spectroscopic withn,l, andjbeing the node number, angular momentum, and total angular momentum, respectively, of the single-neutron wave functionφnljin the nucleusA(A=B+n);χpAandχdBare the entranceand exit-channel distorted waves, respectively; andVnpis the neutron–proton interaction that supports the bound state of then–ppairφnp(the deuteron wave function).

In this work,Vnpis 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 thes-wave was considered inφnp.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]:

whereUnBandUpBare the neutron and proton OMPs on the target nucleusBevaluated at half of the deuteron incident energies (the “Ed∕2 rule”).In the zero-range version of ADWA, the effective deuteron potential becomes

Table 1 Spectroscopic factors (SFs) obtained with the WLH and KD02 potentials for stable isotopes at different energies

Table 2 SFs obtained with the WLH and KD02 potentials for unstable isotopes at different energies

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.

3 Results and discussion

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 energiesEp≤ 150 MeV, which fall within the range of incident energies of the WLH potential.The target nuclei included28Si,40Ca,54Fe,58,60Ni,90Zr,120Sn,102Ru,140Ce,142,144Nd, and208Pb.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.1 (Color online) Comparisons between the two optical model calculations and experimental data of (p, d) reactions on 28Si, 40Ca,54Fe, and 58 Ni 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

As shown in Figs.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 as120Sng.s.(p,d)119Sng.s.at 18 MeV.In some individual cases, the experimental angular distributions were not reproduced satisfactorily using either OMPs, for example,120Sng.s.(p,d)119Sng.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 on28Si and40Ca 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.

3.3 Transfer reactions on unstable nuclei

Fig.2 (Color online) Same comparison as that in Fig.1 but on 60Ni, 90Zr, 102Ru, and 120Sn

Fig.3 (Color online) Same comparison as that in Fig.1 but on 120Sn, 140Ce, 142,144Nd, and 208Pb

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 theβstability line.Meanwhile, microscopic OMPs, which are derived from effective nucleon–nucleon interactions with reliablenuclear 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 targets34,46Ar and56Ni.As can be observed, again, the WLH potential reproduced the experimental data reasonably well.In the56Ni 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 theβ-stability line are required to examine the applicability of the WLH potential.

Fig.4 (Color online) Comparisons between two optical model calculations and experimental data of (p, d) reactions on 28 Si and 40 Ca 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

3.4 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.

Fig.5 (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

Fig.6 (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

Comparisons between the SFs obtained using the microscopic WLH potential and the phenomenological KD02 potential are shown in Fig.6 for their ratios, SFWLH/SFKD02,at different incident energies.The target nuclei included are the doubly magic isotopes40Ca and208Pb, stable nuclei28Si,58,60Ni,90Zr,120Sn,102Ru,140Ce, and142,144Nd, and unstable nuclei34,46Ar and56Ni.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 nucleus40Ca (there is a lack of experimental data for208Pb above 100 MeV).When the results in the tables are examined, it can be observed that the changes in SFs of40Ca with the KD02 potential are much smaller over the ranges of incident energies than those with the WLH potential.

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 from40Ca at higher energies.The results are presented in Fig.7 for the40Ca 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 lowenergy 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 forEp＜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.7 (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

4 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 nucleus40Ca case.Comparisons between the WLH and KD02 potentials in their descriptions of proton elastic scattering from40Ca 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.

Author ContributionsAll authors contributed to the study conception and design.Material preparation, data collection, and analysis were performed by Yi-Ping Xu, Si-Lu Chen, and Dan-Yang Pang.The first draft of the manuscript was written by Yi-Ping Xu, and all authors commented on the previous versions of the manuscript.All authors read and approved the final manuscript.

Data availabilityThe data that support the findings of this study are openly available in Science Data Bank at https:// cstr.cn/ 31253.11.scien cedb.14619 and https:// www.doi.org/ 10.57760/ scien cedb.14619.

Declarations

Conflict of interestThe authors declare that they have no competing interests.