Introduction

Proton transfer reactions, such as (d,³He) and (³He,d) reactions, are important in nuclear physics. They provide not only valuable nuclear reaction data for various applications but also important tools for studying the single-particle structure of atomic nuclei, such as the spectroscopic and asymptotic normalization factors, which are of fundamental importance to nuclear physics and nuclear astrophysics [1-8]. Reaction theories are necessary for extracting such structural information from nuclear reaction measurements [9, 10]. Reliable nuclear structure information relies not only on the precision of measurements but also on the reaction theories used. Experimentalists have always endeavored to make measurements more precise; at the same time, it is also important to quantify the uncertainties of the theoretical results of the reaction cross-sections [11, 12].

In most cases, A(d,³He)B and A(³He,d)B reactions can be well described with the distorted wave Born approximation (DWBA), in which the amplitude is expressed as $\begin{array}{c}{T}_{\beta \alpha }^{DW}={\displaystyle \sum _{nlj}{a}_{nlj}}{\displaystyle \int }\text{d}{r}_{\alpha }{\displaystyle \int }\text{d}{r}_{\beta }{\chi }_{\beta }^{(-)\ast }\left(k_{\beta }\mathrm{,}{r}_{\beta }\right){\varphi }_{nlj}(R)\\ V_{tr}f(r){\chi }_{\alpha }^{(+)}\left(k_{\alpha }\mathrm{,}{r}_{\alpha }\right)\mathrm{,}\end{array}$ (1) where ${\chi }_{\alpha }\left(k_{\alpha }\mathrm{,}{r}_{\alpha }\right)$ and ${\chi }_{\beta }\left(k_{\beta }\mathrm{,}{r}_{\beta }\right)$ are distorted waves describing the relative motions between the two particles in the entrance and exit channels, which are separated by vectors $r_{\alpha }$ and $r_{\beta }$ with wave numbers $k_{\alpha }$ and $k_{\beta }$, respectively. ${\varphi }_{nlj}(R)$ is the normalized single-particle wave function of the transferred proton in the target nucleus, whose associated principal, orbital, and total angular momenta are n, l, and j; anlj is the associated spectroscopic amplitude, which is the square root of the spectroscopic factor Snlj. $f(r)=\langle {\psi }_{{}^3\text{He}}\left({\xi }_{\text{d}}\mathrm{,}r\right)|{\Psi }_{\text{d}}\left({\xi }_{\text{d}}\right)\rangle$ is the overlap between the internal wave functions of ³He and deuteron. The post- and prior-forms of the interaction V_tr are $U_{\text{dB}}+V_{\text{dp}}-U_{{}^3\text{HeB}}$ and $V_{\text{dp}}+U_{\text{dB}}-U_{\text{dA}}$, respectively, for an A(d,³He)B reaction, and $V_{\text{dp}}+U_{\text{dA}}-U_{\text{dB}}$ and $U_{\text{dA}}+V_{\text{pA}}-U_{{}^3\text{HeA}}$, respectively, for an A(³He,d)B reaction. Here, Upq and Vpq are the interactions between particles p and q, with Upq for optical model potentials (OMPs), which are complex valued, and Vpq being single-particle binding potentials, which are real. Within the framework of DWBA, the post- and prior-forms are equivalent. For transfer reactions with only one single-particle wave function involved, the cross-section is proportional to the spectroscopic factor, that is, ${\left(\frac{\text{d}\sigma }{\text{d}\Omega }\right)}^{DW}\propto {\left|T_{\beta \epsilon }^{DW}\right|}^2=S_{nlj}{\left(\frac{\text{d}\sigma }{\text{d}\Omega }\right)}_{a_{nlj}=1},$ where ${\left(\frac{\text{d}\sigma }{\text{d}\Omega }\right)}_{a_{nlj}=1}$ is the cross-section calculated assuming the spectroscopic amplitude to be unity.

Experimentally, the spectroscopic factors are obtained by normalizing ${\left(\frac{\text{d}\sigma }{\text{d}\Omega }\right)}_{a_{nlj}=1}$ to the experimental data: $S_{nlj}={\left(\frac{\text{d}\sigma }{\text{d}\Omega }\right)}_{\text{exp}}/{\left(\frac{\text{d}\sigma }{\text{d}\Omega }\right)}_{a_{nlj}=1}\mathrm{.}$ (2) It is now clear that the spectroscopic factors obtained in this way inherit uncertainties from the uncertainties in both the experimental data and theoretical calculations.

Uncertainties in the theoretical calculations of the transfer reaction cross-sections are rooted mainly in the uncertainties of the following: i) the OMPs, which are responsible for the entrance- and exit-channel distorted waves [χα and χβ in Eq. (1)], ii) single-particle potential parameters, which are responsible for the single-particle wave functions [ϕnlj in Eq. (1)], and iii) the reaction model used, for instance, whether the f(r) term in the transition amplitude is treated exactly or treated with zero-range approximation, and whether the nucleon is assumed to be transferred from one nucleus to another through a direct one-step process only or through higher-order processes such as channel-couplings with nuclear excitations. For a given choice of reaction model, the uncertainties of the theoretical calculations are mainly from uncertainties in the OMPs and single-particle potential parameters.

The parameters of single-particle potentials are usually determined with the separation energy prescription, with which the potential depths are determined by the separation energies of the transferred nucleon when the shape parameters, e.g., the radius parameter r₀ and diffuseness parameter a of a Woods-Saxon (WS) potential, are preselected. For WS potentials, the empirical values r₀=1.25 fm and a=0.65 fm are frequently used [13], although attempts have been made to determine r₀ with, for instance, (e,e’p) measurements [14] and Hartree-Fock calculations [15-18], with a fixed choice of a.

Optical model potentials are usually phenomenologically determined by requiring them to describe elastic scattering angular distributions [19]. It is well known that potential parameters determined in this way suffer rather serious uncertainties, especially when only a limited number of experimental data are used to confine these parameters [20, 21]. This situation is better for systematic or global optical model potentials, whose parameters are constrained by experimental data, which cover large ranges of incident energies and target masses. It has been shown that the usage of systematic OMPs helps to reduce the uncertainties of the spectroscopic factors obtained from transfer reactions [22, 23, 5]. Owing to their importance in the study of nuclear reactions and related subjects, such as nuclear structure, nuclear astrophysics, and nuclear applications, much effort has been devoted to the study of systematic optical model potentials [24-26]. However, most references to existing systematic OMPs do not report the uncertainties of their parameters. Fortunately, for the OMPs needed in the study of (³He,d) and (d,³He) reactions, the recently proposed systematic potentials of ³He and deuteron are given with uncertainties [27-29]. These uncertainties were obtained using the bootstrap statistical method. The bootstrap method simulates many repeated measurements of the elastic scattering data by creating new datasets of the same size as the original using random sampling with replacement. This procedure was repeated many times, generating the distributions of the OMP parameters, from which the uncertainties of these parameters were obtained. The details of this method can be found in Refs. [24, 27]. These systematic OMPs allow us to quantify the uncertainties of the (³He,d) and (d,³He) reaction cross-sections due to the uncertainties of the optical model potentials.

The reactions analyzed in this study are ³⁰Si(³He,d)³¹P [30, 31], ¹³B(d,³He)¹²Be [7], and ³⁴S(³He,d)³⁵Cl [30, 31]. The angular momentum transfers, which are the same as the orbital angular momentum l of the corresponding single proton wave functions with these three reactions, are $l=0\hslash$, 1ℏ, and 2ℏ, respectively. The incident energies of these reactions were 25, 46, and 25 MeV, respectively. These reactions are analyzed using exact finite-range DWBA, considering the full complex remnant term. The single proton wave function in the ground state of ^3He 3He is calculated with the p+d single-particle potential provided in Ref. [32], which reproduces the $\langle 3\text{He}|\text{d}\rangle$ overlap function calculated with the Green’s function Monte-Carlo method [32]. The single proton wave functions in the ground states of target nuclei in the exit channels are determined using the usual separation energy prescription with WS potentials whose radii and diffuseness parameters are r₀=1.25 fm and a₀=0.65 fm, respectively. The calculations were performed using the computer code FRESCO [33].

This paper is organized as follows: the forms of OMPs and their parameters are introduced in Sect. 2, the results of our theoretical calculations and the discussion of uncertainties of the differential cross-sections at different scattering angles are discussed in Sect. 3, and the conclusions of this paper are summarized in Sect. 4.

Optical model potential parameters

The phenomenological OMPs used in this work are defined as $\begin{array}{c}U(r)=-V_{\text{r}}{f}_{\text{ws}}(r)-i{W}_v{f}_{\text{ws}}(r)\\ -i{W}_{\text{s}}\left(-4a_w\right)\frac{\text{d}}{\text{d}r}{f}_{\text{ws}}(r)+V_{\text{C}}\mathrm{,}\end{array}$ (3) where V_r, W_v, and W_s are the depths of the real, volume-imaginary, and surface-imaginary parts of the central potential, respectively. V_C is the Coulomb potential: $V_{\text{C}}(r)=\{\begin{array}{ll}\frac{Z_{\text{P}}{Z}_{\text{T}}{e}^2}{r}\mathrm{,}\hfill & \left(r>R_{\text{C}}\right)\hfill \\ \frac{Z_{\text{P}}{Z}_{\text{T}}{e}^2}{2R_{\text{C}}}\left(3-\frac{r^2}{R_{\text{C}}^2}\right),\hfill & \left(r⩽R_{\text{C}}\right),\hfill \end{array}$ (4) where Z_P and Z_T are the charge numbers of the projectile and target nuclei, respectively, and R_C is the Coulomb radius of the target nuclei. f_ws is the WS form factor: $f_{\text{ws}}(r)=\frac{1}{1+\exp \left(r-R_i\right)/a_i}\mathrm{,}$ (5) where Ri and ai are the radius and diffuseness parameters, respectively, with $i=r$, v, and s labeling the real, volume-imaginary, and surface-imaginary terms in Eq.3, respectively. For the OMPs of ³He and deuteron needed by the analysis of (d,³He) and (³He,d) reactions, Ri is calculated with $r_i{A}_{\text{T}}^{1/3}$, where ri is the reduced radius parameter and A_T is the mass number of the target nucleus. The OMPs of ³He and deuteron projectiles both have a total set of seven OMP parameters {V_r, r_v, a_v, W_v, W_s, r_w, a_w}. The uncertainties in these parameters can be obtained from Refs. [27, 29].

Uncertainties of the transfer reaction cross-sections

The uncertainties in the OMP parameters for ³He and deuteron for the three reactions studied in this work are listed as Δp in Table 1. Shown together are the uncertainties in the differential cross-sections caused by varying each parameter within their ranges of validity while keeping all other parameters fixed, Δ_σ, and the uncertainties in the differential cross-sections caused by varying all of the parameters simultaneously, ${\Delta }_{\sigma }\text{(total)}$. Note that these OMP parameter uncertainties are given in percentages. For a parameter P, whose mean value is $\overline{P}$ and absolute uncertainty is Δ P, its uncertainty, in percentage, is ${\Delta }_{\text{p}}=\frac{\Delta P}{\overline{P}}\times 100\%.$ The absolute uncertainties of these parameters can be found in Ref. [27] for ³He and Ref. [29] for deuteron. The uncertainties of the calculated differential cross-sections are defined in the same way. The uncertainties are evaluated at the center-of-mass angles θ_c.m.=0°, 9°, and 13° for the ³⁰Si(³He,d)³¹P, ¹³B(d,³He)¹²Be, and ³⁴S(³He,d)³⁵Cl reactions, respectively, where their maximum differential cross-sections occur. Within the ranges of these parameters, the cross-sections depend nearly linearly on the values of each individual parameter. Thus, when one parameter, p, is changed within the range $\left[\left(1-{\Delta }_p\right)\times p\mathrm{,}\left(1+{\Delta }_p\right)\times p\right]$ while keeping all other parameters fixed, the differential cross-section will vary between its upper and lower limits, σ_max and σ_min, respectively. Δ_σ (in %) is defined as $100\%\times \left({\sigma }_{\text{max}}-{\sigma }_{\text{min}}\right)/\left({\sigma }_{\text{max}}+{\sigma }_{\text{min}}\right)$.

In contrast to how the differential cross-sections depend almost linearly on each of the parameters within their uncertainties (when all other parameters are fixed), they tend to follow a Gaussian distribution when all the OMP parameters are varied simultaneously. As an example, Fig. 1 shows the distributions of the differential cross-sections for the three reactions at their first three peak angles. The cross-sections are normalized to their mean values: ${\delta }_{\sigma }=(\sigma -\overline{\sigma })/\overline{\sigma }$, where $\overline{\sigma }$ are the mean values. Fig. 1 shows the results of 20000 calculations, each with the parameter sets {V_r, r_v, a_v, W_v, W_s, r_w, a_w} randomly sampled within their individual uncertainties. Uniform random numbers were used for this procedure. This number of samples is sufficient for our analysis because the results are nearly the same as those with 10000 samples. Uncertainties of differential cross-sections when all the OMP parameters vary simultaneously, Δ_σ,total, are taken to be the standard deviations of these distributions, approximated by the standard deviations of the Gaussian function that best fits these distributions in Fig. 1. The Δ_σ,total values for the ³⁰Si(³He,d)³¹P, ¹³B(d,³He)¹²Be, and ³⁴S(³He,d)³⁵Cl reactions at scattering angles of 0°, 9°, and 13°, respectively, are listed in Table. 1.

Fig. 1Full-screen View

(Color online) Distributions of the normalized cross-sections as results of calculations, randomly sampled 20000 times, of the OMP parameters at the three peaks of the angular distributions for the (a) ³⁰Si(³He,d)³¹P, (b) ¹³B(d,³He)¹²Be, and (c) ³⁴S(³He,d)³⁵Cl reactions at incident energies of 25, 46, and 25 MeV, respectively. The scattering angles indicated in these figures are in center-of-mass systems. The curves represent the Gaussian functions whose heights and width parameters are found to best fit these distributions. The Y-axis is the number of cases whose normalized cross-sections δ_σ fall in the intervals of $\left[{\delta }_{\sigma }-{\Delta }_{{\delta }_{\sigma }}/\mathrm{2,}{\delta }_{\sigma }+{\Delta }_{{\delta }_{\sigma }}/2\right]$ with ${\Delta }_{{\delta }_{\sigma }}=0.01$. See the text for details.

From Table. 1, the following observations are made: 1) the uncertainties of OMPs in both exit- and entrance-channels result in an uncertainty of around 5% in the (³He,d) and (d,³He) reactions, which seems to be independent of the angular momentum transfer, the target masses, and the incident energies; 2) Δ_σ,total are smaller than the sums of Δ_σ for all the three reactions, suggesting that some correlations exist among the effects of uncertainties induced by different parameters; and 3) among all these parameters, the radius and diffuseness parameters of the imaginary potentials r_w and a_w of the ³He potential are the most sensitive to the transfer reactions. The uncertainties in the cross-sections caused by these two parameters are approximately twice the uncertainties in the parameters themselves. This suggests that these two parameters should be the focus of future systematic OMP studies to further reduce the theoretical uncertainties of the (³He,d) and (d,³He) reactions.

Experimentally, spectroscopic factors studied with transfer reactions are usually obtained by matching the theoretical cross-sections to the experimental ones at the peaks of the angular distributions. This is a reasonable choice because at these angles, the differential cross-sections are at their maximum values, which means that their experimental uncertainties, at least the statistical uncertainties, are the smallest. The spectroscopic factors obtained at these angles have minimum experimental uncertainties. However, cross-sections at the first peaks may not always be available experimentally. In those cases, the spectroscopic factors must be obtained using experimental data at the second or even the third peaks. It is interesting to determine the extent to which the uncertainties of the spectroscopic factors will increase in such cases. This information is shown in Fig. 1 where the distributions of cross-sections at the second and the third peaks are given for the three reactions analyzed in this study, which have angular momentum transfers l of 0 ℏ, 1 ℏ, and 2 ℏ, respectively. For the ³⁰Si(³He,d)³¹P reaction, which has $l=0\hslash$, the uncertainties of cross-sections at the second and third peaks are nearly the same, which are 7.5% on average, larger than that with the first peak by a factor of 68%. However, for the ¹³B(d,³He)¹²Be and ³⁴S(³He,d)³⁵Cl reactions that have $l=1\hslash$ and $2\hslash$, respectively, the uncertainties of the cross-sections at their first and second peaks are rather close, 6.0% and 5.8%, respectively, much smaller than those of their third peaks, which are around 11.0%. Whether these results apply generally for all reactions at all incident energies might be an interesting subject for further study.

The uncertainties of the theoretical cross-sections at different scattering angles are more clearly seen in Fig. 2, in which the uncertainties caused by the uncertainties in the OMP parameters are depicted as shaded bands for the three reactions analyzed in this study. The widths of these bands represent the upper and lower bounds of the cross-sections at each scattering angle. The corresponding uncertainties in percentage are shown by the brown curves with values shown by the vertical ordinate on the right. The following conclusions can be drawn from this figure: 1) within their ranges of uncertainties, as shown in Table 1, variations of the OMP parameters mainly affect the amplitudes of the differential cross-sections and do not change the angular distributions, especially at smaller angles where theoretical cross-sections are compared with experimental ones to get the spectroscopic factors; and 2) the uncertainties of differential cross-sections increase with increasing scattering angles. Further, the theoretical uncertainties at the shoulders of the peaks may be even smaller than those at the peak angles. However, these shoulders are where the differential cross-sections change most abruptly with respect to the scattering angles. Spectroscopic factors obtained by matching the theoretical and experimental cross-sections at these angles will have the disadvantage that they will need to have much higher angular resolutions than those measured at the peaks.

Fig. 2Full-screen View

(Color online) Uncertainty of the ³⁰Si(³He,d)³¹P, ¹³B(d,³He)¹²Be, and ³⁴S(³He,d)³⁵Cl reactions due to the uncertainties in the OMP parameters of both the entrance- and exit-channels. The widths of the bands represent the upper and lower bounds of the cross-sections (in mb/sr), which are the results of randomly sampling the OMP parameters 20000 times. The brown curves are their corresponding uncertainties in percentage, whose values are given by the right y-axis.

Summary

In summary, (³He,d) and (d,³He) reactions are important tools for studying the single-particle structure of atomic nuclei. Knowledge of the uncertainties in the theoretical calculations for these reactions is important for nuclear structure information, such as spectroscopic factors, obtained from these reactions. This study analyzed the uncertainties of theoretical cross-sections, owing to the uncertainties in the entrance- and exit-channel optical model potentials, for these reactions. The systematic potential of ³He and deuteron projectiles were used, whose parameter uncertainties were available in Refs. [27] and [29], respectively. Three reactions, ³⁰Si(³He,d)³¹P, ¹³B(d,³He)¹²Be, and ³⁴S(³He,d)³⁵Cl, at incident energies of 25, 46, and 25 MeV, respectively, were analyzed within the framework of an exact finite-range DWBA. The momentum transfers of these chosen reactions were 0ℏ, 1ℏ, and 2ℏ, respectively. The analysis was performed through 20000 calculations, for each reaction, randomly sampling the entrance- and exit-channel parameters simultaneously. It was found that the uncertainties of the theoretical cross-sections of these reactions caused by the uncertainties in the OMP parameters were around 5% at scattering angles where the reactions have the largest cross-sections. Uncertainties in the single proton spectroscopic factors with these reactions were concluded to be the same, owing to the uncertainties in the OMP parameters. Such uncertainties seem to be independent of the angular momentum transfer and the target masses for the range of incident energies analyzed in this study.