TORUS: Theory of Reactions for Unstable iSotopes A Topical Collaboration for Nuclear Theory

# Research Papers

Ch. Elster, L. Hlophe, V. Eremenko, F.M. Nunes, I.J. Thompson, G. Arbanas, J.E. Escher, Separable Potentials for (d,p) Reaction Calculations [pdf] (12 Nov 2015)

An important ingredient for applications of nuclear physics to e.g. astrophysics or nuclear energy are the cross sections for reactions of neutrons with rare isotopes. Since direct measurements are often not possible, indirect methods like (d,p) reactions must be used instead. Those (d,p) reactions may be viewed as effective three-body reactions and described with Faddeev techniques. An additional challenge posed by (d,p)reactions involving heavier nuclei is the treatment of the Coulomb force. To avoid numerical complications in dealing with the screening of the Coulomb force, recently a new approach using the Coulomb distorted basis in momentum space was suggested. In order to implement this suggestion, one needs to derive a separable representation of neutron- and proton-nucleus optical potentials and compute their matrix elements in this basis.

I.J. Thompson, J.E. Escher, G. Arbanas, Ch. Elster, V. Eremenko, L. Hlophe, and F.M. Nunes, One-nucleon transfers to resonance states [pdf] (18 June 2015)

We examine the contributions from interior, surface and exterior parts of the matrix elements for (d,p) neutron-transfer matrix elements, and show how their sum may be written as interior-post, exterior-prior terms along with a surface term. If we locate our surface according to the distance of the transferred neutron to the target, then the three terms depend on the neutron wave function at specific radii, and the surface and exterior terms depend only on that part of the neutron wave function determined by R-matrix parameters for neutron-target scattering.

G. Arbanas, I. J. Thompson, J. E. Escher, Ch. Elster, F. M. Nunes and S.-S. Zhang, A model of neutron capture and deuteron stripping on deformed nuclei [pdf] (16 June 2015)

A coupled channel model of direct neutron capture and deuteron stripping reactions, which consistently accounts for effects of nuclear deformations in both reactions, is constructed by coupling all incoming and outgoing partitions of both reactions to the same set of collective states. This model is demonstrated using the FRESCO coupled-channels code [I. J. Thompson, Comp. Phys. Rep. 7, 167 (1988)], and it is applied to capture and stripping reactions on even-mass calcium isotopes 40,42,44,46,48Ca. All incoming and outgoing partitions in capture and stripping reactions were coupled to 2+, 4+, and 3- collective states using a consistent set of deformation lengths. Coupling to these collective states significantly decreases the direct capture cross section relative to the capture in a spherical model for the nuclides considered. Similarly, deuteron stripping is approximately cut in half for the same nuclides. These results suggest that single-particle spectroscopic factors used in this model of direct capture ought to be refitted by computing deuteron stripping with coupling to the same collective states.

Shi-Sheng Zhang, Jin-Peng Peng, M. S. Smith, G. Arbanas, and R. L. Kozub, Exploration of direct neutron capture with covariant density functional theory inputs [pdf], Phys. Rev. C 91, 045802 (10 April 2015)

Predictions of direct neutron capture are of vital importance for simulations of nucleosynthesis in supernovae, merging neutron stars, and other astrophysical environments. We calculated direct capture cross sections using nuclear structure information obtained from a covariant density functional theory as input for the fresco coupled reaction channels code. We investigated the impact of pairing, spectroscopic factors, and optical potentials on our results to determine a robust method to calculate cross sections of direct neutron capture on exotic nuclei. Our predictions agree reasonably well with experimental cross section data for the closed shell nuclei 16O and 48Ca, and for the exotic nucleus 36S. We then used this approach to calculate the direct neutron capture cross section on the doubly magic unstable nucleus 132Sn which is of interest for the astrophysical r-process.

V. Eremenko, N. J. Upadhyay, I.J. Thompson, Ch. Elster, F.M. Nunes, G. Arbanas, J.E. Escher and L. Hlophe, Coulomb wave functions in momentum space [pdf], Computer Physics Communications, 187, 195 (2015)

An algorithm to calculate non-relativistic partial-wave Coulomb functions in momentum space is presented. The arguments are the Sommerfeld parameter η, the angular momentum l, the asymptotic momentum q and the ‘running’ momentum p, where both momenta are real. Since the partial-wave Coulomb functions exhibit singular behavior when p → q, different representations of the Legendre functions of the 2nd kind need to be implemented in computing the functions for the values of p close to the singularity and far away from it. The code for the momentum-space Coulomb wave functions is applicable for values of |η| in the range of 10−1 to 10, and thus is particularly suited for momentum space calculations of nuclear reactions.

L. Hlophe, V. Eremenko, Ch. Elster, F.M. Nunes, G. Arbanas, J.E. Escher, and I.J. Thompson, Separable Representation of Proton-Nucleus Optical Potentials [pdf], Phys. Rev C. 90, 061602(R)  (9 Dec 2014)

Recently, a new approach for solving the three-body problem for (d,p) reactions in the Coulomb-distorted basis in momentum space was proposed. Important input quantities for such calculations are the scattering matrix elements for proton- and neutron-nucleus scattering. We present a generalization of the Ernst-Shakin-Thaler scheme in which a momentum space separable representation of proton-nucleus scattering matrix elements can be calculated in the Coulomb basis. The viability of this method is demonstrated by comparing S-matrix elements obtained for p+48Ca and p+208Pb for a phenomenological optical potential with corresponding coordinate space calculations.

N. J. Upadhyay, V. Eremenko, L. Hlophe, F. M. Nunes, Ch. Elster, G. Arbanas, J. E. Escher, and I. J. Thompson (TORUS Collaboration), Coulomb problem in momentum space without screening [pdf], Phys. Rev C. 90, 014615 (28 July 2014)

Background: The repulsive Coulomb force poses severe challenges when solving the three-body problem for (d,p) reactions on intermediate mass and heavy nuclei. Recently, a new approach based on the Coulomb-distorted basis in momentum space was proposed.
Purpose:
In this work, we demonstrate the feasibility of using the Coulomb-distorted basis in momentum space for calculating matrix elements expected in a wide range of nuclear reactions.
Method:
We discuss the analytic forms of the Coulomb wave function in momentum space. We analyze the singularities in the Coulomb-distorted form factors and the required regularization techniques. Employing a separable interaction derived from a realistic nucleon-nucleus optical potential, we compute and study the Coulomb-distorted form factors for a wide range of cases, including charge, angular momentum, and energy dependence. We also investigate in detail the precision of our calculations.
Results:
The Coulomb-distorted form factors differ significantly from the nuclear form factors except for the very highest momenta. Typically, the structure of the form factor is shifted away from zero momentum due to the Coulomb interaction. Unlike the Yamaguchi forms typically used in three-body methods, our realistic form factors have a short high-momentum tail, which allows for a safe and efficient truncation of the momentum grid.
Conclusions:
Our results show that the Coulomb-distorted basis can be effectively implemented.

I.J. Thompson and G. Arbanas, Coupled-channel Treatment of Isobaric Analog Resonances in (p,p′) Capture Processes [pdf], Nuclear Data Sheets 118, 298 (2014)

With the advent of nuclear reactions on unstable isotopes, there has been a renewed interest in using isobaric analogue resonances (IAR) as a tool for probing the nuclear structure. The position and width of isobaric analogue resonances in nucleon-nucleus scattering are accurate and detailed indicators of the positions of resonances and bound states with good single-particle characters. We report on implementation within our coupled-channels code FRESCO of the charge-exchange interaction term that transforms an incident proton into a neutron. Isobaric analog resonances are seen as peaks in g-ray spectrum when the proton is transformed into a neutron at an energy near a neutron bound state. The Lane coupled-channels formalism was extended to follow the non-orthogonality of this neutron channel with that configuration of an inelastic outgoing proton, and the target being left in a particle-hole excited state. This is tested for 208Pb, for which good (p,p'g) coincidence data exists.

B.V. Carlson, J.E. Escher, and M.S. Hussein, Theoretical descriptions of compound-nuclear reactions: open problems & challenges [pdf], J. Phys. G focus issue on Open Problems in Reaction Theory, 41, 094003 (15 July 2014)

Compound-nuclear processes play an important role for nuclear physics applications and are crucial for our understanding of the nuclear many-body problem. Despite intensive interest in this area, some of the available theoretical developments have not yet been fully tested and implemented. We revisit the general theory of compound-nuclear reactions, discuss descriptions of pre-equilibrium reactions, and consider extensions that are needed in order to get cross section information from indirect measurements.

I.J. Thompson, Computational challenges to the development of modern theories of nuclear reactions [pdf], J. Phys. G focus issue on Open Problems in Reaction Theory, 41, 094009 (15 July 2014)

Detailed theories of nuclear reactions now lead to and require extensive computations. Only then can their results be used to make verifiable predictions and to contribute to the development of nuclear physics. I focus on low-energy reactions of nucleons and light clusters on heavier nuclei, and discuss the computational challenges in the evaluation of coupled-channel theories of those reactions.

I.J. Thompson, J.E. Escher and G. Arbanas, Coupled-Channel Models of Direct-Semidirect Capture via Giant-Dipole Resonances [pdf], Nuclear Data Sheets 118, 292 (2014)

Semidirect capture, a two-step process that excites a giant-dipole resonance followed by its radiative de-excitation, is a dominant process near giant-dipole resonances, that is, for incoming neutron energies within 5–20 MeV. At lower energies such processes may affect neutron capture rates that are relevant to astrophysical nucleosynthesis models. We implement a semidirect capture model in the coupled-channel reaction code Fresco and validate it by comparing the cross section for direct-semidirect capture 208Pb(n,g)209Pb to experimental data. We also investigate the effect of low-energy electric dipole strength in the pygmy resonance. We use a conventional single-particle direct-semidirect capture code Cupido for comparison. Furthermore, we present and discuss our results for direct-semidirect capture reaction 130Sn(n,g)131Sn, the cross section of which is known to have a significant effect on nucleosynthesis models.

J.E. Escher, I.J. Thompson, G. Arbanas, Ch. Elster, V. Eremenko, L. Hlophe, F.M. Nunes, Reexamining surface-integral formulations for one-nucleon transfers to bound and resonance states [pdf], Phys. Rev C. 89, 054605 (8 May 2014)

One-nucleon transfer reactions, in particular (d,p) reactions, have played a central role in nuclear structure studies for many decades. Present theoretical descriptions of the underlying reaction mechanisms are insufficient for addressing the challenges and opportunities that are opening up with new radioactive beam facilities. We investigate a theoretical approach that was proposed recently to address shortcomings in the description of transfers to resonance states. The method builds on ideas from the very successful R-matrix theory; in particular it uses a similar separation of the parameter space into interior and exterior regions, and introduces a parameterization that can be related to physical observables, which, in principle, makes it possible to extract meaningful spectroscopic information from experiments. We carry out calculations, for a selection of isotopes and energies, to test the usefulness of the new approach.

L. Hlophe, Ch. Elster, R.C. Johnson, N.J. Upadhyay, F.M. Nunes, G. Arbanas, V. Eremenko, J.E. Escher, and I.J. Thompson, Separable Representation of Phenomenological Optical Potentials of Woods-Saxon Type [pdf], Phys. Rev C 88, 064608 (11 Dec 2013)

Background: One important ingredient for many applications of nuclear physics to astrophysics, nuclear energy, and stockpile stewardship are cross sections for reactions of neutrons with rare isotopes. Since direct measurements are often not feasible, indirect methods, e.g. (d,p) reactions, should be used.} Those (d,p) reactions may be viewed as three-body reactions and described with Faddeev techniques.
Purpose: Faddeev equations in momentum space have a long tradition of utilizing separable interactions in order to arrive at sets of coupled integral equations in one variable. While there exist several separable representations for the nucleon-nucleon interaction, the optical potential between a neutron (proton) and a nucleus is not readily available in separable form. The purpose of this paper is to introduce a separable representation for complex phenomenological optical potentials of Woods-Saxon type.
Results: Starting from a global optical potential, a separable representation thereof is introduced based on the Ernst-Shakin-Thaler (EST) scheme. This scheme is generalized to non-hermitian potentials. Applications to n+48Ca, n+132Sn and n+208Pb are investigated for energies from 0 to 50 MeV and the quality of the representation is examined.
Conclusions: We find a good description of the on-shell t-matrix for all systems with rank up to 5. The required rank depends inversely on the angular momentum. The resulting separable interaction exhibits a different off-shell behavior compared to the original potential, reducing the high momentum contributions.

A. Orazbayev, Ch. Elster, S.P. Weppner, Open Shell Effects in a Microscopic Optical Potential for Elastic Scattering of 6(8)He, [pdf]  Phys. Rev C 88,034610 (2013)

Elastic scattering observables (differential cross section and analyzing power) are calculated for the reaction 6He(p,p)6He at projectile energies starting at 71 MeV/nucleon. The optical potential needed to describe the reaction is based on a microscopic Watson first-order folding potential, which explicitly takes into account that the two neutrons outside the 4He-core occupy an open p-shell. The folding of the single-particle harmonic oscillator density matrix with the nucleon-nucleon t-matrix leads for this case to new terms not present in traditional folding optical potentials for closed shell nuclei. The effect of those new terms on the elastic scattering observables is investigated. Furthermore, the influence of an exponential tail of the p-shell wave functions on the scattering observables is studied, as well as the sensitivity of the observables to variations of matter and charge radius. Finally elastic scattering observables for the reaction 8He(p,p)8He are presented at selected projectile energies.

T.J. Ross, R.O. Hughes, C.W. Beausang, J.M. Allmond, C.T. Angell, M.S. Basunia, D.L. Bleuel, J.T. Burke, R.J. Casperson, J.E. Escher, P. Fallon, R. Hatarik, J. Munson, S. Paschalis, M. Petri, L.W. Phair, J.J. Ressler, and N. D. Scielzo, Remnants of spherical shell structures in deformed nuclei: The impact of an N = 64 neutron subshell closure on the structure of N ≈ 90 gadolinium nuclei, [pdf] Phys. Rev C 88, 031301(R) (2013)

Odd-mass gadolinium isotopes around N=90 were populated by the (p,d) reaction, utilizing 25-MeV protons, resulting in population of low-spin quasineutron states at energies near and below the Fermi surface. Systematics of the single quasineutron levels populated are presented. A large excitation energy gap is observed between levels originating from the 2d3/2, 1h11/2, and 3s1/2 spherical parents (above theN=64 gap), and the 2d5/2 (below the gap), indicating that the spherical shell model level spacing is maintained at least to moderate deformations.

Ch. Elster, A. Orazbayev, S. P. Weppner, Microscopic Optical Potentials for Helium-6 Scattering off Protons, [pdf] Few Body Systems, 54, 1399 (2013) (13 Jan 2013)

The differential cross section and the analyzing power are calculated for elastic scattering of 6He from a proton target using a microscopic folding optical potential, in which the 6He nucleus is described in terms of a 4He-core with two additional neutrons in the valence p-shell. In contrast to previous work of that nature, all contributions from the interaction of the valence neutrons with the target protons are taken into account.

M. Gulino, C. Spitaleri, X. D. Tang, G. L. Guardo, L. Lamia, S. Cherubini, B. Bucher, V. Burjan, M. Couder, P. Davies, R. deBoer, X. Fang, V. Z. Goldberg, Z. Hons, V. Kroha, L. Lamm, M. La Cognata, C. Li, C. Ma, J. Mrazek, A. M. Mukhamedzhanov, M. Notani, S. O’Brien, R. G. Pizzone, G. G. Rapisarda, D. Roberson, M. L. Sergi, W. Tan, I. J. Thompson, and M. Wiescher, Suppression of the centrifugal barrier effects in the off-energy-shell neutron + 17O interaction [pdf], PRC 87, 012801(R) (4 Jan 2013)

The reaction 17O(n,α)14C was studied at energies from Ec.m.=0 to Ec.m.=350 keV by using the quasifree deuteron breakup in the three-body reaction 17O+d→α+14C+p, which extended the Trojan Horse indirect method (THM) to neutron-induced reactions. It is found that the 18O excited state at E*=8.125±0.002 MeV, observed in THM experiments, is absent in the direct measurement because of its high centrifugal barrier. The angular distributions of the populated resonances have been measured by using this method. The results unambiguously indicate the ability of the THM to overcome the centrifugal barrier suppression effect and to pick out the contribution of the bare nuclear interaction.

T.J. Ross, C.W. Beausang, R.O. Hughes, N.D. Scielzo, J.T. Burke, J.M. Allmond, C.T. Angell, M.S. Basunia, D.L. Bleuel, R.J. Casperson, J.E. Escher, P. Fallon, R. Hatarik, J. Munson, S. Paschalis, M. Petri, L. Phair, and J. J. Ressler, Spectroscopy of 88Y by the (p,dγ) reaction, [pdf] PRC 86, 067301 (4 Dec 2012)

Low-spin, high-excitation energy states in 88Y have been studied using the 89Y(p,dγ) reaction. For this experiment a 25 MeV proton beam was incident upon a monoisotopic 89Y target. A silicon telescope array was used to detect deuterons, and coincident γ rays were detected using a germanium clover array. Most of the known low-excitation-energy low-spin states populated strongly via the (p,d) reaction mechanism are confirmed. Two states are seen for the first time and seven new transitions, including one which bypasses the two low-lying isomeric states, are observed.

Ch. Elster, L. Hlophe, Nuclear Reactions: A Challenge for Few- and Many-Body Theory, Proceedings of the HITES 2012 Conference, J. Phys.: Conf. Ser. 403, 012025, arxiv.org/abs/1209.0838

A current interest in nuclear reactions, specifically with rare isotopes concentrates on their reaction with neutrons, in particular neutron capture. In order to facilitate reactions with neutrons one must use indirect methods using deuterons as beam or target of choice. For adding neutrons, the most common reaction is the (d,p) reaction, in which the deuteron breaks up and the neutron is captured by the nucleus. Those (d,p) reactions may be viewed as a three-body problem in a many-body context. This contribution reports on a feasibility study for describing phenomenological nucleon-nucleus optical potentials in momentum space in a separable form, so that they may be used for Faddeev calculations of (d,p) reactions.

J E Escher, A M Mukhamedzhanov and I J Thompson, Exploring R-matrix ideas for the description of one-nucleon transfer reactions, Proceedings of the HITES 2012 Conference , J. Phys.: Conf. Ser. 403 012026

Deuteron-induced reactions, in particular (d,p) one-neutron transfer reactions, have been used for decades to investigate the structure of nuclei. These reactions, carried out in inverse kinematics, are expected to play a central role in the study of weakly-bound systems at modern radioactive beam facilities. While the theoretical framework and its computational implementation for describing (d,p) reactions have seen much progress over the decades, open questions remain and need to be addressed, including the proper treatment of transfers to resonance states. Recently, a new formalism was proposed [1] that, in principle, describes transfers to both bound and resonance states. The new formalism is summarized here and illustrated; implications are discussed.

F.M. Nunes and N.J. Upadhyay, Status of reaction theory for studying rare isotopes, Proceedings of the HITES 2012 Conference, J. Phys.: Conf. Ser. 403, 012029

Reactions are an important tool to study nuclear structure and for extracting reactions relevant for astrophysics. In this paper we focus on deuteron induced reactions which can provide information on neutron shell evolution as well as neutron capture cross sections. We review recent work on the systematic comparison of the continuum discretized coupled channel method, the adiabatic wave approximation and the Faddeev momentum-space approach. We also explore other aspects of the reaction mechanism and discuss in detail difficulties encountered in the calculations.

M. La Cognata, C. Spitaleri, O. Trippella, G. G. Kiss, G. V. Rogachev, A. M. Mukhamedzhanov, M. Avila, G. L. Guardo, E. Koshchiy, A. Kuchera, L. Lamia, S. M. R. Puglia, S. Romano, D. Santiago, and R. Spartà, Measurement of the -3 keV resonance in the reaction 13C(a,n)16O of importance in the s-process [pdf], Phys. Rev. Lett. 109, 232701 (4 Dec 2012)

The 13C(α,n)16O reaction is the neutron source for the main component of the s-process, responsible for the production of most nuclei in the mass range 90<A<204. It is active inside the helium-burning shell in asymptotic giant branch stars, at temperatures <=108  K, corresponding to an energy interval where the 13C(α,n)16O is effective from 140 to 230 keV. In this region, the astrophysical S(E)-factor is dominated by the -3 keV subthreshold resonance due to the 6.356 MeV level in 17+O, giving rise to a steep increase of the S(E)-factor. Notwithstanding that it plays a crucial role in astrophysics, no direct measurements exist inside the s-process energy window. The magnitude of its contribution is still controversial as extrapolations, e.g., through the R matrix and indirect techniques, such as the asymptotic normalization coefficient (ANC), yield inconsistent results. The discrepancy amounts to a factor of 3 or more right at astrophysical energies. Therefore, we have applied the Trojan horse method to the 13C(6Li,n16O)d quasifree reaction to achieve an experimental estimate of such contribution. For the first time, the ANC for the 6.356 MeNV level has been deduced through the Trojan horse method as well as the n-partial width, allowing to attain an unprecedented accuracy in the 13C(α,n)16O study. Though a larger ANC for the 6.356 MeV level is measured, our experimental S(E)-factor agrees with the most recent extrapolation in the literature in the 140–230 keV energy interval, the accuracy being greatly enhanced thanks to this innovative approach.

A.M. Mukhamedzhanov, Coulomb renormalization and ratio of proton and neutron asymptotic normalization coefficients for mirror nuclei [pdf] PRC 86, 044615 (15 Oct 2012)

Asymptotic normalization coefficients (ANCs) are fundamental nuclear constants playing important role in nuclear reactions, nuclear structure and nuclear astrophysics. In this paper the physical reasons of the Coulomb renormalization of the ANC are addressed. Using Pinkston-Satchler equation the ratio for the proton and neutron ANCs of mirror nuclei is obtained in terms of the Wronskians from the radial overlap functions and regular solutions of the two-body Schrödinger equation with the short-range interaction excluded. This ratio allows one to use microscopic overlap functions for mirror nuclei in the internal region, where they are the most accurate, to correctly predict the ratio of the ANCs for mirror nuclei, which determine the amplitudes of the tails of the overlap functions. Calculations presented for different nuclei demonstrate the Coulomb renormalization effects and independence of the ratio of the nucleon ANCs for mirror nuclei on the channel radius. This ratio is valid both for bound states and resonances. One of the goals of this paper is to draw attention on the possibility to use the Coulomb renormalized ANCs rather than the standard ones especially when the standard ANCs are too large.

R. L. Kozub, G. Arbanas, A. S. Adekola, D. W. Bardayan, J. C. Blackmon, K.Y. Chae, K. A. Chipps, J. A. Cizewski, L. Erikson, R. Hatarik, W. R. Hix, K. L. Jones, W. Krolas, J. F. Liang, Z. Ma, C. Matei, B. H. Moazen, C. D. Nesaraja, S. D. Pain, D. Shapira, J. F. Shriner, Jr., M. S. Smith, and T. P. Swan, Neutron single particle structure in 131Sn and direct neutron capture cross sections [pdf], PRL 109, 172501 (2012).

Recent calculations suggest that the rate of neutron capture by 130Sn has a significant impact on late-time nucleosynthesis in the r-process. Direct capture into low-lying bound states is expected to be significant in neutron capture near the N=82 closed shell, so r- process reaction rates may be strongly impacted by the properties of neutron single particle states in this region. In order to investigate these properties, the (d,p) reaction has been studied in inverse kinematics using a 630 MeV beam of 130Sn (4.8 MeV/u) and a (CD2)n target. An array of Si strip detectors, including the SIDAR and an early implementation of the ORRUBA, was used to detect reaction products. Results for the 130Sn(d,p)131Sn reaction are found to be very similar to those from the previously reported 132Sn(d, p)133Sn reaction. Direct-semidirect (n, γ) cross section calculations, based for the first time on experimental data, are presented. The uncertainties in these cross sections are thus reduced by orders of magnitude from previous estimates.

Shi-Sheng Zhang, M. S. Smith, G. Arbanas, and R. L. Kozub, Structures of exotic 131,133Sn isotopes and effect on r-process nucleosynthesis [pdf], PRC 86, 032802 (2012)

Background: Four strong single-particle bound levels with strikingly similar level spacings have recently been measured in 131Sn and 133Sn. This similarity has not yet been addressed by a theoretical nuclear structure model. Information on these single-particle bound levels, as well as on resonant levels above the neutron capture threshold, is also needed to determine neutron capture cross sections—and corresponding capture reaction rates—on 130,132Sn. The 130Sn(n,γ) rate was shown in a recent sensitivity study to significantly impact the synthesis of heavy elements in the r-process in supernovae.

Purpose: Understand the structure of bound and resonant levels in 131,133Sn, and determine if the densities of unbound resonant levels are sufficiently high to warrant statistical model treatments of neutron capture on 130,132Sn.

Method: Single-particle bound and resonant levels for 131,133Sn are self-consistently calculated by the analytical continuation of the coupling constant (ACCC) method based on a relativistic mean field (RMF) theory with BCS approximation.

Results:We obtain four strong single-particle bound levels in both 131,133Sn with an ordering that agrees with experiments and spacings that, while differing from experiment, are consistent between the Sn isotopes. We also find at most one single-particle level in the effective energy range for neutron captures in the r-process.

Conclusions:Our RMF+ACCC+BCS model successfully reproduces observed single-particle bound levels in 131,133Sn and self-consistently predicts single-particle resonant levels with densities too low for widely used traditional statistical model treatments of neutron capture cross sections on 130,132Sn employing Fermi gas level density formulations.

A.M. Mukhamedzhanov, V. Eremenko, A.I. Sattarov, Generalized Faddeev equations in the AGS form for deuteron stripping with explicit inclusion of target excitations and Coulomb interaction [pdf], PRC 86, 034001 (2012)

Theoretical description of reactions in general, and the theory for (d,p) reactions, in particular, needs to advance into the new century. Here deuteron stripping processes off a target nucleus consisting of A nucleons are treated within the framework of the few-body integral equations theory. The generalized Faddeev equations in the AGS form, which take into account the target excitations, with realistic optical potentials provide the most advanced and complete description of the deuteron stripping. The main problem in practical application of such equations is the screening of the Coulomb potential, which works only for light nuclei. In this paper we present a new formulation of the Faddeev equations in the AGS form taking into account the target excitations with explicit inclusion of the Coulomb interaction. By projecting the (A + 2)-body operators onto target states, matrix three-body integral equations are derived which allow for the incorporation of the excited states of the target nucleons. Using the explicit equations for the partial Coulomb scattering wave functions in the momentum space we present the AGS equations in the Coulomb distorted wave representation without screening procedure. We also use the explicit expression for the off-shell two-body Coulomb scattering T-matrix which is needed to calculate the effective potentials in the AGS equations. The integrals containing the off-shell Coulomb T-matrix are regularized to make the obtained equations suitable for calculations. For NN and nucleon-target nuclear interactions we assume the separable potentials what significantly simplifies solution of the AGS equations.

K. T. Schmitt, K. L. Jones, A. Bey, S. H. Ahn, D.W. Bardayan, J.C. Blackmon, S.M. Brown, K.Y. Chae, K. A. Chipps, J.A. Cizewski, K.I. Hahn, J.J. Kolata, R.L. Kozub, J.F. Liang, C.Matei, M. Matoš, D. Matyas, B. Moazen, C. Nesaraja, F.M. Nunes, P.D. O’Malley, S.D. Pain, W.A. Peters, S.T. Pittman, A. Roberts, D. Shapira, J.F. Shriner, Jr., M.S. Smith, I. Spassova, D.W. Stracener, A.N. Villano, and G.L. Wilson, Halo Nucleus 11Be: A Spectroscopic Study via Neutron Transfer [pdf]. (Phys. Rev. Letts, 108, 192701 (2012))

The best examples of halo nuclei, exotic systems with a diffuse nuclear cloud surrounding a tightly bound core, are found in the light, neutron-rich region, where the halo neutrons experience only weak binding and a weak, or no, potential barrier. Modern direct-reaction measurement techniques provide powerful probes of the structure of exotic nuclei. Despite more than four decades of these studies on the benchmark one-neutron halo nucleus  11Be, the spectroscopic factors for the two bound states remain poorly constrained. In the present work, the 10Be(d,p) reaction has been used in inverse kinematics at four beam energies to study the structure of 11Be. The spectroscopic factors extracted using the adiabatic model were found to be consistent across the four measurements and were largely insensitive to the optical potential used. The extracted spectroscopic factor for a neutron in an nℓj=2s1/2 state coupled to the ground state of 10Beis 0.71(5). For the fist excited state at 0.32 MeV, a spectroscopic factor of 0.62(4) is found for the halo neutron in a 1p1/2 state.

N. J. Upadhyay, A. Deltuva, F. M. Nunes, Testing the continuum discretized coupled channel method for deuteron induced reactions [pdf] (PRC 85, 054621, 2012)

The continuum-discretized coupled channels (CDCC) method is a well established theory for direct nuclear reactions which includes breakup to all orders. Alternatively, the three-body problem can be solved exactly within the Faddeev formalism which explicitly includes breakup and transfer channels to all orders. With the aim to understand how CDCC compares with the exact three-body Faddeev formulation, we study deuteron- induced reactions on (i) 10Be at Ed = 21.4, 40.9 and 71 MeV; (ii) 12C at Ed = 12 and 56 MeV; and (iii) 48Ca at Ed = 56 MeV. We calculate elastic, transfer, and breakup cross sections. Overall, the discrepancies found for elastic scattering are small with the exception of very backward angles. For transfer cross sections at low energy, ∼10 MeV/u, CDCC is in good agreement with the Faddeev-type results and the discrepancy increases with beam energy. In contrast, breakup observables obtained with CDCC are in good agreement with Faddeev-type results for all but the lower energies considered here.

T. J. Ross, C. W. Beausang, R. O. Hughes, J. M. Allmond, C. T. Angell. M. S. Basunia, D. L. Bleuel, J. T. Burke, R. J. Casperson, J. E. Escher, P. Fallon, R. Hatarik, J. Munson, S. Paschalis, M. Petri, L. Phair, J. J. Ressler, N. D. Scielzo, and I. J. Thompson, Measurement of the entry-spin distribution imparted to the high excitation continuum region of gadolinium nuclei via (p,d) and (p,t) reactions [pdf], Phys. Rev. C 85, 051304(R), (2012)

Over the last several years, the surrogate reaction technique has been successfully employed to extract (n,f ) and (n,γ ) cross sections in the actinide region to a precision of 5% and 20%, respectively. However, attempts to apply the technique in the rare earth region have shown large (factors of 2–3) discrepancies between the directly measured (n,γ ) and extracted surrogate cross sections. One possible origin of this discrepancy lies in differences between the initial spin-parity population distribution in the neutron induced and surrogate reactions. To address this issue, the angular momentum transfer to the high excitation energy quasicontinuum region in Gd nuclei has been investigated. The (p,d) and (p,t) reactions on 154,158Gd at a beam energy of 25 MeV were utilized. Assuming a single dominant angular momentum transfer component, the measured angular distribution for the (p,d) reactions is well reproduced by distorted-wave Born approximation (DWBA) calculations for 􏰀L = 4 h ̄ transfer, whereas the (p,t ) reactions are better characterized by 􏰀L = 5 h ̄ . A linear combination of DWBA calculations, weighted according to a distribution of L transfers (peaking around 􏰀L = 4–5 h ̄ ), is in excellent agreement with the experimental angular distributions.

S.P. Weppner, Ch. Elster, Elastic Scattering of 6He based on a Cluster Description [pdf]. (Phys. Rev. C 85, 044617 (2012))

Elastic scattering observables (differential cross section and analyzing power) are calculated for the reaction 6He(p,p)6He at projectile energies starting at 71 MeV/nucleon. The optical potential needed to describe the reaction is derived describing 6He in terms of a 4He-core and two neutrons. The Watson first order multiple scattering ansatz is extended to accommodate the internal dynamics of a composite cluster model for the 6He nucleus scattering from a nucleon projectile. The calculations are compared with the recent experiments at the projectile energy of 71 MeV/nucleon. In addition, differential cross sections and analyzing powers are calculated at selected higher energies.

P. Capel, H. Esbensen and F.M. Nunes, Comparing nonperturbative models of the breakup of neutron-halo nuclei [pdf], Phys. Rev. C 85, 044605 (2012)

Breakup reactions of loosely bound nuclei are often used to extract structure and/or astrophysical information. Here we compare three nonperturbative reaction theories often used when analyzing breakup experiments, namely the continuum discretized coupled channel model, the time-dependent approach relying on a semiclassical approximation, and the dynamical eikonal approximation. Our test case consists of the breakup of 15C on Pb at 68 MeV/nucleon and 20 MeV/nucleon.

R. O. Hughes, C. W. Beausang, T. J. Ross, J. T. Burke, N. D. Scielzo, M. S. Basunia, C. M. Campbell, R. J. Casperson, H. L. Crawford, J. E. Escher, J. Munson, L. W. Phair, and J. J. Ressler, Utilizing (p,d) and (p,t) reactions to obtain (n,f) cross sections in uranium nuclei via the surrogate-ratio method [pdf], Phys. Rev. C 84, 024613 (2012)

The surrogate ratio method has been tested for (p,d) and (p,t) reactions on uranium nuclei. 236U and 238U targets were bombarded with 28-MeV protons and the light ion recoils and fission fragments were detected using the Silicon Telescope Array for Reaction Studies detector array at the 88-Inch Cyclotron at Lawrence Berkeley Na- tional Laboratory. The (p,df ) reaction channels on 236U and 238U targets were used as a surrogate to determine the σ [236 U(n,f )]/σ [234 U(n,f )] cross-section ratio. The (p,tf ) reaction channels were also measured with the same targetsasasurrogatefortheσ[235U(n,f)]/σ[(233U(n,f)]ratio.Forthe(p,df)and(p,tf)surrogatemeasurements, there is good agreement with accepted (n,f ) values over equivalent neutron energy ranges of En = 0–7 MeV and En = 0–5.5 MeV, respectively. An internal surrogate ratio method comparing the (p,d) and (p,t) reaction channels on a single target is also discussed. The σ[234U(n,f)]/σ[233U(n,f)] and σ[236U(n,f)]/σ[235U(n,f)] cross-section ratios are extracted using this method for the 236U and 238U targets, respectively. The resulting fission cross-section ratios show relatively good agreement with accepted values up to En ∼ 5 MeV.

A.M. Mukhamedzhanov, Theory of deuteron stripping. From surface integrals to generalized R-matrix approach [pdf-v4] (Phys. Rev. C 84, 044616 (2011))

There are two main reasons for absence of the practical theory of stripping to resonance states which could be used by experimental groups: numerical problem of the convergence of the DWBA matrix element when the full transition operator is included and it is unclear what spectroscopic information can be extracted from the analysis of transfer reactions populating the resonance states. The purpose of this paper is to address both questions. The theory of the deuteron stripping is developed, which is based on the post continuum discretized coupled channels (CDCC) formalism going beyond of the DWBA and surface integral formulation of the reaction theory [A. S. Kadyrov et al., Ann. Phys. 324, 1516 (2009)]. First, the formalism is developed for the DWBA and then extended to the CDCC formalism, which is ultimate goal of this work. The CDCC wave function takes into account not only the initial elastic $d+ A$ channel but also its coupling to the deuteron breakup channel $p + n + A$ missing in the DWBA. Stripping to both bound states and resonances are included. The convergence problem for stripping to resonance states is solved in the post CDCC formalism. The reaction amplitude is parametrized in terms of the reduced width amplitudes (ANCs), inverse level matrix, boundary condition and channel radius, that is the same parameters which are used in the conventional $R$-matrix method. For stripping to resonance states many-level, one and two-channel cases are considered. The theory provides a consistent tool to analyze both binary resonant reactions and deuteron stripping in terms of the same parameters.

N.B.Nguyen, S.J.Waldecker, F.M.Nunes, R.J.Charity, W.H.Dickhoff, Transfer reactions and the dispersive optical-model [pdf] (Phys. Rev. C 84, 044611 (2011)).

The dispersive optical-model is applied to transfer reactions. A systematic study of (d,p) reactions on closed-shell nuclei using the finite-range adiabatic reaction model is performed at several beam energies and results are compared to data as well as to predictions using a standard global optical-potential. Overall, we find that the dispersive optical-model is able to describe the angular distributions as well as or better than the global parameterization. In addition, it also constrains the overlap function. Spectroscopic factors extracted using the dispersive optical-model are generally lower than those using standard parameters, exhibit a reduced dependence on beam energy, and are more in line with results obtained from (e,e'p) measurements.

L.J. Titus, P. Capel, F.M. Nunes, Asymptotic normalization of mirror states and the effect of couplings [pdf] (PRC 85, 035805, 2011)

Assuming that the ratio between asymptotic normalization coefficients of mirror states is model independent, charge symmetry can be used to indirectly extract astrophysically relevant proton capture reactions on proton-rich nuclei based on information on stable isotopes. The assumption has been tested for light nuclei within the microscopic cluster model. In this work we explore the Hamiltonian independence of the ratio between asymptotic normalization coefficients of mirror states when deformation and core excitation is introduced in the system. For this purpose we consider a phenomenological rotor + N model where the valence nucleon is subject to a deformed mean field and the core is allowed to excite. We apply the model to 8Li/8B, 13C/13N, 17O/17F, 23Ne/23Al, and 27Mg/27P. Our results show that for most studied cases, the ratio between asymptotic normalization coefficients of mirror states is independent of the strength and multipolarity of the couplings induced. The exception is for cases in which there is an s-wave coupled to the ground state of the core, the proton system is loosely bound, and the states have large admixture with other configurations. We discuss the implications of our results for novae.

F.M. Nunes and A. Deltuva, Adiabatic versus Faddeev for (d,p) and (p,d) reactions [pdf] (Phys. Rev. C 84, 034607 (2011))

The finite range adiabatic wave approximation provides a practical method to analyze (d,p) or (p,d) reactions, however until now the level of accuracy obtained in the description of the reaction dynamics has not been determined. In this work, we perform a systematic comparison between the finite range adiabatic wave approximation and the exact Faddeev method. We include studies of $^{11}$Be(p,d)$^{10}$Be(g.s.) at $E_p=$5, 10 and 35 MeV; $^{12}$C(d,p)$^{13}$C(g.s.) at $E_d=$7, 12 and 56 MeV and $^{48}$Ca(d,p)$^{49}$Ca(g.s.) at $E_d=$19, 56 and 100 MeV. Results show that the two methods agree within $\approx 5%$ for a range of beam energies ($E_d \approx 20-40$ MeV) but differences increase significantly for very low energies and for the highest energies. Our tests show that ADWA agrees best with Faddeev when the angular momentum transfer is small $\Delta l=0$ and when the neutron-nucleus system is loosely bound.

M. La Cognata, A.M. Mukhamedzhanov, C. Spitaleri, et al., The Fluorine Destruction In Stars: First Experimental Study Of The 19F(p,a0)16O Reaction at Astrophysical Energies. (Ap.J.Letts, 738, L54, 2011)

The 19F(p,a)16O reaction is an important fluorine destruction channel in the proton-rich outer layers of asymptotic giant branch (AGB) stars and it might also play a role in hydrogen-deficient post-AGB star nucleosynthesis. So far, available direct measurements do not reach the energy region of astrophysical interest (Ecm <~ 300 keV), because of the hindrance effect of the Coulomb barrier. The Trojan Horse (TH) method was thus used to access this energy region, by extracting the quasi-free contribution to the 2H(19F, α16O)n and the 19F(3He, α16O)d reactions. The TH measurement of the α0 channel shows the presence of resonant structures not observed before, which cause an increase of the reaction rate at astrophysical temperatures up to a factor of 1.7, with potential consequences for stellar nucleosynthesis.

A.M. Mukhamedzhanov, L. D. Blokhintsev, B.F. Irgaziev, Reexamination of the astrophysical S factor for the α+ d -> 6Li +g reaction (arxiv, PRC 83, 055805, 2011)

Recently, a new measurement of the 6Li (150 A MeV) dissociation in the field of 208Pb has been reported [Hammache et al., Phys. Rev. C 82, 065803 (2010)] to study the radiative capture α+d6Li+γ process. However, the dominance of the nuclear breakup over the Coulomb one prevented the information about the α+d6Li+γ process from being obtained from the breakup data. The astrophysical S24(E) factor has been calculated within the α-d two-body potential model with potentials determined from the fits to the α-d elastic scattering phase shifts. However, the scattering phase shift, according to the theorem of the inverse scattering problem, does not provide a unique α-d bound-state potential, which is the most crucial input when calculating the S24(E) astrophysical factor at astrophysical energies. In this work, we emphasize the important role of the asymptotic normalization coefficient (ANC) for 6Li→α+d, which controls the overall normalization of the peripheral α+d6Li+γ process and is determined by the adopted α-d bound-state potential. Since the potential determined from the elastic scattering data fit is not unique, the same is true for the ANC generated by the adopted potential. However, a unique ANC can be found directly from the elastic scattering phase shift, without invoking intermediate potential, by extrapolation the scattering phase shift to the bound-state pole [Blokhintsev et al., Phys. Rev. C 48, 2390 (1993)]. We demonstrate that the ANC previously determined from the α-d elastic scattering s-wave phase shift [Blokhintsev et al., Phys. Rev. C 48, 2390 (1993)], confirmed by ab initio calculations, gives S24(E), which at low energies is about 38% less than the other one reported [Hammache et al., Phys. Rev. C 82, 065803 (2010)]. We recalculate also the reaction rates, which are lower than those obtained in that same study [Hammache et al., Phys. Rev. C 82, 065803 (2010)].

F.M. Nunes, A. Deltuva, and June Hong, Improved description of 34;36;46Ar(p,d) transfer reactions (Phys. Rev. C 83, 034610 (2011)).

An improved description of single neutron stripping from 34;36;46Ar beams at 33 MeV/nucleon by a hydrogen target is presented and the dependence on the neutron-proton asymmetry of the spectroscopic factors is further investigated. A finite range adiabatic model is used in the analysis and compared to previous zero range and local energy approximations. Full three-body Faddeev calculations are performed to estimate the error in the reaction theory. In addition, errors from the optical potentials are also evaluated. From our new spectroscopic factors extracted from transfer, it is possible to corroborate the neutron-proton asymmetry dependence reported from knockout measurements.

A.M. Mukhamedzhanov and A.S. Kadyrov, Unitary correlation in nuclear reaction theory: divorce of reaction theory and spectroscopic factors (PRC 82, 051601(R), 2011)

Future exact many-body theory will allow us to calculate nuclear reactions based on the adopted NN and many-body nuclear potentials. But NN potentials are not observable and there are infinite number of the phase-equivalent NN potentials related via finite-range unitary transformations. We show that asymptotic normalization coefficients, which are the amplitudes of the asymptotic tails of the overlap functions are invariant under finite-range unitary transformations but spectroscopic factors are not. We prove also that the exact amplitudes for the (d,p), (d,pn) and (e, e′p) reactions determining the asymptotic behavior of the exact scattering wave functions in the corresponding channels, in contrast to spectroscopic factors, are invariant under finite-range unitary transformations. Moreover, the exact reaction amplitudes are not parametrized in terms of the spectroscopic factors and nuclear reactions in the exact approach cannot provide a tool to determine spectroscopic factors which are not observable.

## Preprints

Ch. Elster, L. Hlophe, V. Eremenko, F.M. Nunes, G. Arbanas, J.E. Escher, I.J. Thompson, Separable Optical Potentials for (d,p) Reactions [pdf], Contribution to the Proceedings of the International Conference on Nuclear Theory in the Supercomputing Era (NTSE 2014)

An important ingredient for applications of nuclear physics to e.g. astrophysics or nuclear energy are the cross sections for reactions of neutrons with rare isotopes. Since direct measurements are often not possible, indirect methods like (d,p) reactions must be used instead. Those (d,p) reactions may be viewed as effective three-body reactions and described with Faddeev techniques. An additional challenge posed by (d,p) reactions involving heavier nuclei is the treatment of the Coulomb force. To avoid numerical complications in dealing with the screening of the Coulomb force, recently a new approach using the Coulomb distorted basis in momentum space was suggested. In order to implement this suggestion, one needs not only to derive a separable representation of neutron- and proton-nucleus optical potentials, but also compute the Coulomb distorted form factors in this basis..

### Review Papers

F. Nunes, Continuum-Discretised Coupled Channels methods (Scholarpedia)

One of the most common methods to study rare isotopes with large proton/neutron asymmetry, is through breakup reactions. These exotic nuclei are so loosely bound they can break up very easily while leaving the target in its ground state. While the first efforts to study breakup assumed the process could be treated perturbatively, it was soon realized that multi-step effects are important. Also, it is not simple to isolate Coulomb effects from nuclear effects. A non perturbative method that treats breakup to all orders, and includes Coulomb and nuclear effects on equal footing is the Continuum discretized coupled channel method (CDCC). Over the last two decades this method has become increasingly popular. Here we present the method as it was originally introduced, address implementation aspects and provide some applications. We also discuss limitations of the method as well as new developments that allow to solve more complicated problems.

### Related Research Papers

P. Capel and  P. Danielewicz and F. M. Nunes, Deducing spectroscopic factors from wave-function asymptotics

In a coupled-channel model, we explore the effects of coupling between configurations on the radial behavior of the wave function and, in particular, on the spectroscopic factor (SF) and the asymptotic normalization coefficient (ANC). We evaluate the extraction of a SF from the ratio of the ANC of the coupled-channel model to that of a single-particle approximation of the wave function. We perform this study within a core+n collective model, which includes two states of the core that connect by a rotational coupling. To get additional insights, we also use a simplified model that takes a δ function for the coupling potential. Calculations are performed for11Be. Fair agreement is obtained between the SF inferred from the single-particle approximation and the one obtained within the coupled-channel models. Significant discrepancies are observed only for large coupling strength and/or large admixture, that is, a small SF. This suggests that reliable SFs can be deduced from the wave-function asymptotics when the structure is dominated by one configuration, that is, for a large SF.