Adiabatic Distorted Wave Approximation Research Articles

Astrophysical N15(n,γ)N16 reaction rate from precision measurement of the N15(d,p)N16 angular distributions

Background: Asymptotic giant branch stars are one of the possible sites of fluorine production as confirmed by astronomical observation. The $^{15}\mathrm{N}(n,\ensuremath{\gamma})^{16}\mathrm{N}$ reaction competes with $^{15}\mathrm{N}(\ensuremath{\alpha},\ensuremath{\gamma})^{19}\mathrm{F}$ on the reaction chain and thus influences the production of the only stable fluorine isotope $^{19}\mathrm{F}$. The $^{15}\mathrm{N}(n,\ensuremath{\gamma})^{16}\mathrm{N}$ reaction rate at low temperatures of astrophysical interest depends on the neutron spectroscopic factors of the four low-lying states in $^{16}\mathrm{N}$.Purpose: The $^{16}\mathrm{N}$ neutron spectroscopic factors from two previous measurements of $(d,p)$ reaction differ by a factor of $\ensuremath{\approx}2$. This work was intended to investigate these spectroscopic factors via a precision measurement of the $^{15}\mathrm{N}(d,p)^{16}\mathrm{N}$ angular distributions using a Q3D magnetic spectrograph.Methods: The high resolution of the Q3D magnetic spectrograph led to clear separation of these four closely spaced states. The neutron spectroscopic factors were extracted from the present angular distributions with the distorted wave Born approximation and adiabatic distorted wave approximation methods.Results: The neutron spectroscopic factors of the $^{16}\mathrm{N}$ ground, 0.120, 0.298, and 0.397 MeV states were extracted to be $1.32\ifmmode\pm\else\textpm\fi{}0.12, 1.33\ifmmode\pm\else\textpm\fi{}0.18, 1.10\ifmmode\pm\else\textpm\fi{}0.10$, and $1.30\ifmmode\pm\else\textpm\fi{}0.14$, respectively. The new results support the conclusion that these four states in $^{16}\mathrm{N}$ are all good single-particle levels. The cross section and reaction rate of $^{15}\mathrm{N}(n,\ensuremath{\gamma})^{16}\mathrm{N}$ were calculated based on the present spectroscopic factors.Conclusions: The present work provides a precision measurement of the spectroscopic factors of the $^{16}\mathrm{N}$ states and the resulting $^{15}\mathrm{N}(n,\ensuremath{\gamma})^{16}\mathrm{N}$ reaction rate. The present higher reaction rate of the $^{15}\mathrm{N}(n,\ensuremath{\gamma})^{16}\mathrm{N}$ reaction suggests higher flow through $^{15}\mathrm{N}(n,\ensuremath{\gamma})^{16}\mathrm{N}$ compared with previous expectation.

Application of R-Matrix and Lagrange-Mesh Methods to Nuclear Transfer Reactions

Background: Nuclear transfer reactions are a useful tool to study the structure of a nucleus. For reactions involving weekly bound nuclei, breakup effects can play significant role and theoretical calculations can be computational expensive in such cases.
 Purpose: To utilize the Lagrange-mesh and R-matrix methods for nuclear transfer reactions.
 Methods: We use the adiabatic distorted wave approximation (ADWA) method which can approximately treats the breakup effects in a simpler manner. In our approach, we apply the R-matrix method combining it with the Lagrange-mesh method, which is known to provide the fast and accurate computations.
 Results: As a test case, we calculate the angular distribution of the cross sections for the 56Fe(d, p)57Fe reaction, where deuteron breakup effects play important role.
 Conclusions: We show that these methods work well in the ADWA framework, and we look forward to applying these methods in coupled channel calculations.

Considering nonlocality in the optical potentials within eikonal models

Background: For its simplicity, the eikonal method is the tool of choice to analyze nuclear reactions at high energies ($E>100$ MeV/nucleon), including knockout reactions. However, so far, the effective interactions used in this method are assumed to be fully local.Purpose: Given the recent studies on nonlocal optical potentials, in this work we assess whether nonlocality in the optical potentials is expected to impact reactions at high energies and then explore different avenues for extending the eikonal method to include nonlocal interactions.Method: We compare angular distributions obtained for nonlocal interactions (using the exact R-matrix approach for elastic scattering and the adiabatic distorted wave approximation for transfer) with those obtained using their local-equivalent interactions.Results: Our results show that transfer observables are significantly impacted by nonlocality in the high-energy regime. Because knockout reactions are dominated by stripping (transfer to inelastic channels), nonlocality is expected to have a large effect on knockout observables too. Three approaches are explored for extending the eikonal method to nonlocal interactions, including an iterative method and a perturbation theory.Conclusions: None of the derived extensions of the eikonal model provide a good description of elastic scattering. This paper suggests that nonlocality removes the formal simplicity associated with the eikonal model.

Three-body optical potentials in (d,p) reactions and their influence on indirect study of stellar nucleosynthesis

Model uncertainties arising due to suppression of target excitations in the description of deuteron scattering and resulting in a modification of the two-body interactions in a three-body system are investigated for several $(d,p)$ reactions serving as indirect tools for studying the astrophysical $(p,\ensuremath{\gamma})$ reactions relevant to $rp$ process. The three-body nature of the deuteron-target potential is treated within the adiabatic distorted-wave approximation (ADWA) which relies on a dominant contribution from the components of the three-body deuteron-target wave function with small $n\text{\ensuremath{-}}p$ separations. This results in a simple prescription for treating the explicit energy dependence of two-body optical potentials in a three-body system requiring nucleon optical potentials to be evaluated at a shifted energy with respect to the standard value of half the deuteron incident energy. In addition, the ADWA allows for leading-order multiple-scattering effects to be estimated, which leads to a simple renormalization of the adiabatic potential's imaginary part by a factor of two. These effects are assessed using both nonlocal and local optical potential systematics for $^{26}\mathrm{Al}$, $^{30}\mathrm{P}$, $^{34}\mathrm{Cl}$, and $^{56}\mathrm{Ni}$ targets at a deuteron incident energy of 12 MeV, which is typical for experiments with radioactive beams in inverse kinematics. The model uncertainties induced by the three-body nature of deuteron-target scattering are found to be within $40%$ both in the main peak of angular distributions and in total $(d,p)$ cross sections. At higher deuteron energies, around 60 MeV, model uncertainties can reach 100% in the total cross sections. A few examples of application to astrophysically interesting proton resonances in $^{27}\mathrm{Si}$ and $^{57}\mathrm{Cu}$ obtained using $(d,p)$ reactions and mirror symmetry are given.

Perturbative correction to the adiabatic approximation for (d, p) reactions

The adiabatic distorted wave approximation (ADWA) is widely used by the nuclear community to analyse deuteron stripping (d, p) experiments. It provides a quick way to take into account an important property of the reaction mechanism: deuteron breakup. In this work we provide a numerical quantification of a perturbative correction to this theory, recently proposed in Johnson (2014 J. Phys. G: Nucl. Part. Phys. 41 094005) for separable rank-one nucleon–proton potentials. The correction involves an additional, nonlocal, term in the effective deuteron–target ADWA potential in the entrance channel. We test the calculations with perturbative corrections against continuum-discretized coupled channel predictions which treat deuteron breakup exactly.

Three-nucleon force contribution to the deuteron channel in (d,p) reactions

The contribution of a three-nucleon (3N) force, acting between the neutron and proton in the incoming deuteron with a target nucleon, to the deuteron-target potential in the entrance channel of the $A(d,p)B$ reaction has been calculated within the adiabatic distorted wave approximation (ADWA). Four different 3N interaction sets from local chiral effective field theory ($\chi$EFT) at next-to-next-to-leading order (N2LO) were used. Strong sensitivity of the adiabatic deuteron-target potential to the choice of the 3N force format has been found, which originates from the enhanced sensitivity to the short-range physics of nucleon-nucleon (NN) and 3N interactions in the ADWA. Such a sensitivity is reduced when a Watanabe folding model is used to generate $d$-$A$ potential instead of ADWA. The impact of the 3N force contribution on $(d,p)$ cross sections depends on assumptions made about the $p$-$A$ and $n$-$A$ optical potentials used to calculate the distorted $d$-$A$ potential in the entrance channel. It is different for local and nonlocal optical potentials and depends on whether the induced three-body force arising due to neglect of target excitations is included or not.

From Halo Effective Field Theory to the study of breakup and transfer reactions: reliably probing the halo structure of 11Be and 15C

In this work we study one-neutron halo nuclei, and in particular 11Be and 15C, which can be seen as an inert core of 10Be or 14C plus a loosely bound neutron. During the last decades several transfer and breakup reactions involving these systems have been measured on different targets and energies. We study these processes using one single structure model for each nucleus applying the halo effective field theory (Halo EFT) at next-to-leading order NLO. The main parameters of this EFT are adjusted on nuclear-structure data and/or ab initio predictions. We model the transfer reaction within the Adiabatic Distorted Wave Approximation (ADWA) and the breakup process applying an eikonal model with a consistent treatment of nuclear and Coulomb interactions at all orders. At high energy, our model includes a proper treatment of special relativity Our theoretical calculations are in good agreement with experiment for a variety of reaction observables, thus assessing the robustness of the structure model provided for these nuclei. This new idea enables us also to reliably estimate the nuclear-structure observables that actually affect the reaction process, and hence that can be inferred from such measurements.

Astrophysical 22Mg(p, γ)23Al reaction rates from asymptotic normalization coefficient of 23Ne→22Ne+n * * Supported by the National Key Research and Development Program of China (2016YFA0400502), the National Natural Science Foundation of China (11975316, 11490561, 11535004, 11775013), the Continuous Basic Scientific Research Project (WDJC-2019-13) and the 973 Program (2013CB834406)

The radionuclide 22Na generates the emission of a characteristic 1.275 MeV -ray. This is a potential astronomical observable, whose occurrence is suspected in classical novae. The reaction is relevant to the nucleosynthesis of 22Na in Ne-rich novae. In this study, employing the adiabatic distorted wave approximation and continuum discretized coupled channel methods, the squared neutron asymptotic normalization coefficients (ANCs) for the virtual decay of + n were extracted, and determined as fm−1 and fm−1 for the ground state and the first excited state from the experimental angular distributions of populating the ground state and the first excited state of , respectively. Then, the squared proton ANC of was obtained as fm−1 according to the charge symmetry of the strong interaction. The astrophysical S-factors and reaction rates for the direct capture contribution in were also presented. Furthermore, the proton width of the first excited state of was derived to be eV from the neutron ANC of its mirror state in and used to compute the contribution from the first resonance of . This result demonstrates that the direct capture dominates the reaction at most temperatures of astrophysical relevance for .

Precision measurement of the angular distribution for the 16O(d, p)17O transfer reaction to the ground state of 17O

The mechanism of the (d, p) reaction, which has been proven a powerful spectroscopic tool, is believed to be reasonably well understood. However, the 16O(d, p)17O reaction leading to the ground state in 17O seems to be a possible exception. It was found in the previous experiments that the 16O(d, p)17O angular distribution exhibits a rapid decline below the peak at the angle of 15 degrees. To date this abnormal behavior has not been reproduced by theoretical models. In this work we present a re-measurement of the 16O(d, p)Og.s.17 angular distribution by using a high-precision magnetic spectrograph. Our new angular distribution shows a slower decline than the previous data do. A comparative analysis with the distorted Born approximation (DWBA), the adiabatic distorted wave approximation (ADWA) and the continuum discretized coupled channels (CDCC) method suggests that the drop of the differential cross sections at the forward angles is due to the deuteron breakup coupling effects. In addition, we extracted the spectroscopic factor (SF) and the asymptotic normalization coefficient (ANC) for the 17O ground state.

Systematic analysis of the peripherality of the Be10(d,p)Be11 transfer reaction and extraction of the asymptotic normalization coefficient of Be11 bound states

We reanalyze the experiment of Schmitt $\textit{et al.}$ on the $^{10}$Be$(d,p)^{11}$Be transfer reaction [Phys. Rev. Lett. $\textbf{108}$, 192701 (2012)] by exploring the beam-energy and angular ranges at which the reaction is strictly peripheral. We consider the adiabatic distorted wave approximation (ADWA) to model the reaction and use a Halo-EFT description of $^{11}$Be to systematically explore the sensitivity of our calculations to the short-range physics of the $^{10}$Be-$n$ wave function. We find that by selecting the data at low beam energy and forward scattering angle the calculated cross sections scale nearly perfectly with the asymptotic normalization coefficient (ANC) of the $^{11}$Be bound states. Following these results, a comparison of our calculations with the experimental data gives a value of $C_{1s1/2}=0.785\pm0.03$ fm$^{-1/2}$ for the $\frac{1}{2}^+$ ground-state ANC and $C_{0p1/2}=0.135\pm0.005$ fm$^{-1/2}$ for the $\frac{1}{2}^-$ excited-state, which are in perfect agreement with the $\textit{ab initio}$ calculations of Calci $\textit{et al.}$, who obtain $C^{\textit{ab initio}}_{1/2^+}=0.786$ fm$^{-1/2}$ and $C^{\textit{ab initio}}_{1/2^-}=0.129$ fm$^{-1/2}$ [Phys. Rev. Lett. $\textbf{117}$, 242501 (2016)].

Uncertainty Quantification of the Experimental Spectroscopic Factor from Transfer Reaction Models

We study the uncertainty stemming from different theoretical models in the spectroscopic factors extracted from experiments. We use three theoretical approaches, the distorted wave Born approximation (DWBA), the adiabatic distorted wave approximation (ADWA) and the continuum discretized coupled-channels method (CDCC), and analyze the 12C(d, p)13C, 14C(d, p)15C reactions. We find that the uncertainty associated with the adopted theoretical models is less than 20%. We also investigate the contribution from the remnant term and observe that it gives less than 10% uncertainty. We finally make an attempt to explain the discrepancy in the spectroscopic factors of $$^{17}C(\frac{3}{2}^+)$$ between the ones extracted from experiments and from shell model calculations by analyzing the 16C(d, p)17C reaction.

Nuclear structure of Cd112 studied through the Cd111(d⃗,p) reaction

The nuclear structure of $^{112}\mathrm{Cd}$ has been investigated with the $^{111}\mathrm{Cd}(\stackrel{P\vec}{d},p)^{112}\mathrm{Cd}$ reaction. Isotopically enriched targets of $^{111}\mathrm{Cd}$ were bombarded with 22 MeV polarized deuterons, and reaction products were analyzed with a magnetic spectrograph. Angular distributions and analyzing powers were determined for 129 states, 49 of which are newly observed, up to approximately 4.2 MeV in excitation energy. The observed angular distributions were compared with distorted wave Born approximation (DWBA) and adiabatic distorted wave approximation (ADWA) calculations to extract the spectroscopic factors. Two-quasineutron configurations involving coupling to the ${s}_{\frac{1}{2}}$ orbital are suggested. The sum of spectroscopic strengths extracted by using the ADWA for the individual $l$ transfers are combined with previous results from the $^{111}\mathrm{Cd}(\stackrel{P\vec}{d},t)$ reaction and show good agreement with the $2j+1$ sum rule, whereas those extracted with the DWBA calculations are significantly less.

Weak sensitivity of three-body ( d,p ) reactions to np force models

Adiabatic distorted-wave approximation (ADWA) study of three-body $(d,p)$ transfer reactions [G.W. Bailey, N.K. Timofeyuk, and J.A. Tostevin, Phys. Rev. Lett. 117, 162502 (2016)] reported strong sensitivity of cross sections to the neutron-proton $(np)$ interaction model when the nucleon-nucleus optical potential is nonlocal. The verification of this unusual finding using more reliable methods is aimed for in the present work. A rigorous Faddeev-type three-body scattering theory is applied to the study of $(d,p)$ transfer reactions. The equations for transition operators are solved in the momentum-space partial-wave framework. Differential cross sections for $^{26}$Al$(d,p)^{27}$Al reactions are calculated using nonlocal nuclear optical potentials and a number of realistic $np$ potentials. Only a weak dependence on the $np$ force model is observed, typically one order of magnitude lower than in the previous ADWA study. The shape of the angular distribution of the experimental data is well reproduced. Cross sections of $(d,p)$ transfer reactions calculated using a rigorous three-body method show little sensitivity to the $np$ interaction model. This indicates a failure of the ADWA in the context of nonlocal potentials. Some evident shortcomings of the ADWA are pointed out.

A New Measurement of 11Be(p,d) Transfer Reaction**Supported by the National Key Research and Development Program of China under Grant No 2018YFA0404403, and the National Natural Science Foundation of China under Grant Nos 11775004 and 11775036.

A new 11Be(p,d) transfer reaction experiment is performed in inverse kinematics with a radioactive 11Be beam at 26.9A MeV. Three low-lying states, namely the 0+ ground state, the 2+ state at Ex = 3.37 MeV, and the multiplet at around 6 MeV in 10Be, are populated by this one-neutron transfer reaction. These three states in 10Be are clearly discriminated from the Q-value spectrum, which is rebuilt from energies and angles of the recoil deuterons in coincidence with 10Be. A spectroscopic factor for each state is extracted by comparing the experimental differential cross sections to the theoretical calculation results using the finite range adiabatic distorted wave approximation method with different global nucleon-nucleus potentials. It is found that the newly extracted spectroscopic factors for the 0+ and 2+ states are consistent with the previous ones, but the factor for the multiplet is smaller than the value in the reference, and the possible reason is discussed.

Spectroscopy of Fe61 via the neutron transfer reaction H2(Fe60,p)Fe*61

The direct component of the 60Fe(n,γ)61Fe cross section was investigated by populating the bound states of the 61Fe nucleus through the (d,pγ) transfer reaction in inverse kinematics using a radioactive beam of 60Fe at 27A MeV. The experiment was performed at GANIL using the MUST2 array and an annular double-sided silicon strip detector for the detection of the light charged particle in coincidence with the photons measured in the EXOGAM γ-ray detectors. For the first time, the spectroscopic factors of the first 3/2−, 5/2−, 1/2−, and 9/2+ states of 61Fe were deduced experimentally from an adiabatic distorted wave approximation analysis of the data. The obtained results show a very good agreement with the shell-model predictions. The calculated direct component of the (n,γ) cross section was found negligible and of about 2% of the total, indicating a dominant resonant component.

Transfer reaction code with nonlocal interactions

We present a suite of codes (NLAT for nonlocal adiabatic transfer) to calculate the transfer cross section for single-nucleon transfer reactions, (d,N) or (N,d), including nonlocal nucleon–target interactions, within the adiabatic distorted wave approximation. For this purpose, we implement an iterative method for solving the second order nonlocal differential equation, for both scattering and bound states. The final observables that can be obtained with NLAT are differential angular distributions for the cross sections of A(d,N)B or B(N,d)A. Details on the implementation of the T-matrix to obtain the final cross sections within the adiabatic distorted wave approximation method are also provided. This code is suitable to be applied for deuteron induced reactions in the range of Ed=10–70MeV, and provides cross sections with 4% accuracy. Program summaryProgram title: NLATCatalogue identifier: AFAY_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AFAY_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public License, version 3No. of lines in distributed program, including test data, etc.: 662136No. of bytes in distributed program, including test data, etc.: 15253066Distribution format: tar.gzProgramming language: Fortran 90.Computer: Dell Poweredge R620 (Intel XEON E5-2650v2) 2.9 GHz Intel Core i5.Operating system: Linux (Debian 7) Mac OSX.RAM: Around 3 GBClassification: 17.8, 17.9, 17.11, 17.16.Nature of problem:Calculates cross sections for deuteron induced single-nucleon transfer reactions using nonlocal potentials within the adiabatic distorted wave approximation.Solution method:See ‘Nature of problem’.Running time:Less than 2 hours. See sample output files without extension for indication of time taken.

Explicit inclusion of nonlocality in(d,p)transfer reactions

Background: Traditionally, nucleon-nucleus optical potentials are made local for convenience. In recent work we studied the effects of including nonlocal interactions explicitly in the final state for (d,p) reactions, within the distorted wave Born approximation. Purpose: Our goal in this work is to develop an improved formalism for nonlocal interactions that includes deuteron breakup and to use it to study the effects of including nonlocal interactions in transfer (d,p) reactions, in both the deuteron and the proton channel. Method: We extend the finite-range adiabatic distorted wave approximation to include nonlocal nucleon optical potentials. We apply our method to (d,p) reactions on 16O, 40Ca, 48Ca, 126Sn, 132Sn, and 208Pb at 10, 20 and 50 MeV. Results: We find that nonlocality in the deuteron scattering state reduces the amplitude of the wave function in the nuclear interior, and shifts the wave function outward. In many cases, this has the effect of increasing the transfer cross section at the first peak of the angular distributions. This increase was most significant for heavy targets and for reactions at high energies. Conclusions: Our systematic study shows that, if only local optical potentials are used in the analysis of experimental (d,p) transfer cross sections, the extracted spectroscopic factors may be incorrect by up to 40% due to the local approximation.

Spin dependence of the incident channel distorted wave in the theory of theA(d,p)Breaction

The adiabatic distorted wave approximation (ADWA) model of $A(d,p)B$ reactions is generalized to include the effects of nucleon-nucleus spin-orbit potentials and the neutron-proton tensor force and associated $D$-state terms in the interaction between the neutron and proton in the incident deuteron. The ADWA predicts that the incident channel distorting potential must include momentum-dependent rank-2 spin tensor force terms with novel properties not normally considered in the deuteron-nucleus scattering.

Determination of astrophysical 7Be(p, γ)8B reaction rates from the 7Li(d, p)8Li reaction

The 7Be(p, γ)8B reaction plays a central role not only in the evaluation of solar neutrino fluxes but also in the evolution of the first stars. Study of this reaction requires the asymptotic normalization coefficient (ANC) for the virtual decay 8B g.s. → 7Be + p. By using the charge symmetry relation, we obtain this proton ANC with the single neutron ANC of 8Li g.s. →7Li + n, which is determined with the distorted wave Born approximation (DWBA) and adiabatic distorted wave approximation (ADWA) analysis of the 7Li(d, p)8Li angular distribution. The astrophysical S-factors and reaction rates of the direct capture process in the 7Be(p, γ)8B reaction are further deduced at energies of astrophysical relevance. The astrophysical S-factor at zero energy for direct capture, S 17(0), is derived to be (19.9 ± 3.5) eV b in good agreement with the most recent recommended value. The contributions of the 1+ and 3+ resonances to the S-factor and reaction rate are also evaluated. The present result demonstrates that the direct capture dominates the 7Be(p, γ)8B reaction in the whole temperature range. This work provides an independent examination to the current results of the 7Be(p, γ)8B reaction.

Nonlocality in the adiabatic model ofA(d,p)Breactions

In a previous publication [Phys. Rev. Lett. 110, 112501 (2013)] we have proposed a generalization of the adiabatic model of $(d,p)$ reactions that allows the nonlocality of the nucleon optical potential to be included in a consistent way together with the deuteron breakup. In this model an effective local $d\text{\ensuremath{-}}A$ potential is constructed from local nucleon optical potentials taken at an energy shifted by $\ensuremath{\sim}40$ MeV with respect to the widely used ${E}_{d}/2$ value, where ${E}_{d}$ is the deuteron incident energy. The effective $d\text{\ensuremath{-}}A$ potential is shallower than that traditionally used in the analysis of $(d,p)$ reactions within the adiabatic distorted wave approximation and this affects the calculated cross sections and the nuclear structure quantities obtained from their comparison with experimental data. In the present paper we give full derivation of the deuteron effective potential, consider its leading-order term within the local-energy approximation and discuss corrections to the leading-order term. The new method is applied to $(d,p)$ reactions on ${}^{16}$O, ${}^{36}$Ar, and ${}^{40}$Ca targets and the influence of the deviation from the ${E}_{d}/2$ rule on the calculated cross sections is quantified.