We have developed an approximate way of dealing with explicit energy dependence of nonlocal nucleon optical potentials as used to predict the (d,p) cross sections within the adiabatic theory. Within this approximation, the nonlocal optical potentials have to be evaluated at an energy shifted from half the incident deuteron energy by the n−p kinetic energy averaged over the range of the n−p interaction and then treated as an energy-independent nonlocal potential. Thus, the evaluation of the distorting potential in the incident channel is reduced to a problem solved in our previous work [N. K. Timofeyuk and R. C. Johnson, Phys. Rev. Lett. 110, 112501 (2013); Phys. Rev. C 87, 064610 (2013)]. We have demonstrated how our new model works for the case of 16O(d,p)17O, 36Ar(d,p)37Ar, and 40Ca(d,p)41Ca reactions and highlighted the need for a detailed understanding of the energy dependence of nonlocal potentials. We have also suggested a simple way of correcting the d−A effective potentials for nonlocality when the underlying energy-dependent nonlocal nucleon potentials are unknown but energy-dependent local phenomenological nucleon potentials are available.
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 [R.C. Johnson, J. Phys. G: Nucl. Part. Phys. 41 (2014) 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.
A widely accepted practice for treating deuteron breakup in A(d,p)B reactions relies on solving a three-body A+n+p Schrödinger equation with pairwise A−n, A−p and n−p interactions. However, it was shown in Phys. Rev. C 89, 024605 (2014) that projection of the many-body A+2 wave function into the three-body A+n+p channel results in a complicated three-body operator that cannot be reduced to a sum of pairwise potentials. It contains explicit contributions from terms that include interactions between the neutron and proton via excitation of the target A. Such terms are normally neglected. We estimate the first-order contribution of these induced three-body terms and show that applying the adiabatic approximation to solving the A+n+p model results in a simple modification of the two-body nucleon optical potentials. We illustrate the role of these terms for the case of 40Ca(d,p)41Ca transfer reactions at incident deuteron energies of 11.8, 20, and 56 MeV, using several parametrizations of nonlocal optical potentials.
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−A potential is constructed from local nucleon optical potentials taken at an energy shifted by ∼40 MeV with respect to the widely used Ed/2 value, where Ed is the deuteron incident energy. The effective d−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 16O, 36Ar, and 40Ca targets and the influence of the deviation from the Ed/2 rule on the calculated cross sections is quantified.
Deuteron stripping and pick-up experiments - (d; p) and (p; d) - have been used for a long time to study the structure of nuclei. Today these experiments are often carried out in inverse kinematics in state-of-the-art radioactive beams facilities around the world, extending the boundaries of our knowledge of the nuclear chart. The nuclear structure information obtained from these experiments relies entirely on transfer reaction theory. We review the theory of (d; p) and (p; d) reactions starting from early formulations and ending with the most recent developments. In particular, we describe the recent progress made in the understanding of the three-body dynamics associated with the deuteron breakup degrees of freedom, including effects of nonlocality, and discuss the role of many-body degrees of freedom within the three-body context. We also review advances in structure model calculations of one-nucleon overlap functions - an important structure input to (d; p) and (p; d) reaction calculations. We emphasize the physics missing in widely-used standard approaches available to experimentalists and review ideas and efforts aimed at including this physics, formulating the crucial tasks for further development of deuteron stripping and pickup reaction theory
It has been realised recently that charge symmetry of the nucleon-nucleon interaction leads to a certain relation between Asymptotic Normalization Coefficients (ANCs) in mirror-conjugated one-nucleon overlap integrals. This relation can be approximated by a simple analytical formula that involves mirror neutron and proton separation energies, the core charge and the range of the strong nucleon-core interaction. We perform detailed microscopic multi-channel cluster model calculations and compare their predictions to the simple analytical formula as well as to calculations within a single-particle model in which mirror symmetry in potential wells and spectroscopic factors are assumed. The validity of the latter assumptions is verified on the basis of microscopic cluster model calculations. For mirror pairs in which one of the states is above the proton decay threshold, a link exists between the proton partial width and the ANC of the mirror neutron. This link is given by an approximate analytical formula similar to that for a bound-bound mirror pair. We compare predictions of this formula to the results of microscopic cluster model calculations. Mirror symmetry in ANCs can be used to predict cross sections for proton capture at stellar energies using neutron ANCs measured with stable or “less radioactive” beams.
Theories of (d,p) reactions frequently use a formalism based on a transition amplitude that is dominated by the components of the total three-body scattering wave function where the spatial separation between the incoming neutron and proton is confined by the range of the n-p interaction, Vnp. By comparison with calculations based on the continuum discretized coupled channels method we show that the (d,p) transition amplitude is dominated by the first term of the expansion of the three-body wave function in a complete set of Weinberg states. We use the 132Sn(d,p)133Sn reaction at 30 and 100 MeV as examples of contemporary interest. The generality of this observed dominance and its implications for future theoretical developments are discussed.
We propose a new method for the analysis of deuteron stripping reactions, A(d,p)B, in which the nonlocality of nucleon-nucleus interactions and three-body degrees of freedom are accounted for in a consistent way. The model deals with equivalent local nucleon potentials taken at an energy shifted by ∼40 MeV from the “Ed/2” value frequently used in the analysis of experimental data, where Ed is the incident deuteron energy. The “Ed/2” rule lies at the heart of all three-body analyses of (d, p) reactions performed so far with the aim of obtaining nuclear structure properties such as spectroscopic factors and asymptotic normalization coefficients that are crucial for our understanding of nuclear shell evolution in neutron- and proton-rich regions of the nuclear periodic table and for predicting the cross sections of stellar reactions. The large predicted shift arises from the large relative kinetic energy of the neutron and proton in the incident deuteron in those components of the n+p+A wave function that dominate the (d, p) reaction amplitude. The large shift reduces the effective d−A potentials and leads to a change in predicted (d, p) cross sections, thus affecting the interpretation of these reactions in terms of nuclear structure.