Proton-Nucleus Elastic Scattering: Comparison between Phenomenological and Microscopic Optical Potentials

Proton-Nucleus Elastic Scattering: Comparison between Phenomenological and Microscopic Optical Potentials

Matteo Vorabbi    Paolo Finelli    Carlotta Giusti TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia, V6T 2A3, Canada Dipartimento di Fisica e Astronomia, Università degli Studi di Bologna and
INFN, Sezione di Bologna, Via Irnerio 46, I-40126 Bologna, Italy
Dipartimento di Fisica, Università degli Studi di Pavia and
INFN, Sezione di Pavia, Via A. Bassi 6, I-27100 Pavia, Italy
July 25, 2019
Abstract

Background: Elastic scattering is a very important process to understand nuclear interactions in finite nuclei. Despite decades of efforts, the goal of reaching a coherent description of this physical process in terms of microscopic forces is still far from being completed.

Purpose: In previous papers Vorabbi:2015nra; Vorabbi:2017rvk we derived a nonrelativistic theoretical optical potential from nucleon-nucleon chiral potentials at fourth (NLO) and fifth order (NLO). We checked convergence patterns and established theoretical error bands. With this work we study the performances of our optical potential in comparison with those of a successful nonrelativistic phenomenological optical potential in the description of elastic proton scattering data on several isotopic chains at energies around and above 200 MeV.

Methods: We use the same framework and the same approximations as in Refs. Vorabbi:2015nra; Vorabbi:2017rvk, where the nonrelativistic optical potential is derived at the first-order term within the spectator expansion of the multiple scattering theory and adopting the impulse approximation and the optimum factorization approximation.

Results: The cross sections and analyzing powers for elastic proton scattering off calcium, nickel, tin, and lead isotopes are presented for several incident proton energies, exploring the range MeV, where experimental data are available. In addition, we provide theoretical predictions for Ni at 400 MeV, which is of interest for the future experiments at EXL.

Conclusions: Our results indicate that microscopic optical potentials derived from nucleon-nucleon chiral potentials at NLO can provide reliable predictions for the cross section and the analyzing power both of stable and exotic nuclei, even at energies where the reliability of the chiral expansion starts to be questionable.

pacs:
24.10.-i; 24.10.Ht; 24.70.+s; 25.40.Cm

I Introduction

The scattering process of an incident nucleon off a target nucleus is a widespread experimental tool for investigating, with specific nuclear reactions, the different properties of a nuclear system. Elastic scattering is probably the main event occurring in the nucleon-nucleus (NA) scattering and measurements of cross sections and polarization observables in elastic proton-nucleus (pA) scattering have provided a lot of detailed information on nuclear properties PaetzSchieck; Glendenning.

A huge amount of experimental data has been collected over the last years concerning stable nuclei (usually with proton or neutron numbers corresponding to some magic configurations) but nowadays one of the most active areas of research in nuclear physics is to understand the properties of nuclei far from the beta-stability line. A number of radioactive ion beam facilities will be used in next years for this purpose. In particular, we would like to mention the FAIR project, with the section dedicated to electromagnetic and light hadronic probes (EXL) fair; exl, where the structure of unstable exotic nuclei in light-ion scattering experiments at intermediate energies will be extensively studied. Some preliminary measurements have already been performed by investigating the reaction Ni(p,p)Ni at an energy of MeV in inverse kinematics refId0. The authors of Ref. refId0 claim that the preliminary results are very promising and demonstrate the feasibility of the intended program of EXL. This result strongly supports the need of a reliable description of the interaction of a nucleon with stable and unstable nuclei. Unfortunately, such processes are characterized by many-body effects that make their theoretical description an extremely hard task.

A very useful framework to achieve this goal is provided by the theoretical concept of the Optical Potential (OP), where the complicated nature of the NA interaction is described introducing a complex effective potential whose real part describes the average interaction between the projectile and the target, and the imaginary part the effect of all inelastic processes which tend to deplete the flux in the elastic channel FESHBACH1958357. The OP was originally employed to analyze the NA elastic scattering data, but its use has been afterwards extended to inelastic scattering and to a wide variety of nuclear reactions.

Different OPs for elastic NA scattering have been derived either by phenomenological analyses of experimental data or by a more fundamental microscopic calculation. Phenomenological OPs are obtained assuming an analytical form of the potential which depends on some free parameters specifying the well and the geometry of the system and that are determined by a fitting procedure over a set of available experimental data of elastic pA scattering. This approach provides good OPs, which perform very well in many regions of the nuclear chart and for several energy ranges where data are available, but which may lack predictive power when applied to situations where experimental data are not yet available. Instead, microscopic OPs are derived starting from the Nucleon-Nucleon (NN) interaction; they can be obtained using different NN potentials and different methods depending on the mass of the target and on the energy of the reaction of interest, and do not contain free adjustable parameters. A recent list of the different approaches can be found in Ref. dickhoff. Being the result of a model and not of a fitting procedure, microscopic OPs should have more theoretical content and might have a more general predictive power than phenomenological OPs, but the approximations which are needed to reduce the complexity of the original many-body problem, whose exact solution is for complex nuclei beyond our present capabilities, might give a poorer agreement with available empirical data.

In Ref. Vorabbi:2015nra we constructed a microscopic OP for elastic pA scattering starting from NN chiral potentials derived up to NLO in the chiral expansion and we studied the chiral convergence of the NN potential in reproducing the pA scattering observables. The OP was obtained at the first-order term within the spectator expansion of the nonrelativistic multiple scattering theory and adopting the impulse approximation and the optimum factorization approximation. In a subsequent work Vorabbi:2017rvk we adopted the same model to obtain the OP and we studied the chiral convergence of a new generation of NN chiral interactions derived up to NLO. Our conclusion was that the convergence has been reached at NLO.

In this work we perform a systematic investigation of the predictive power of our microscopic OP derived in Ref. Vorabbi:2017rvk from different chiral potentials at NLO and of the successful phenomenological OP of Refs. Koning:2003zz; KONING2014187 in comparison with available data for the observables of elastic proton scattering on different isotopic chains, located in different areas of the nuclear chart. Results are presented for several proton energies around and above 200 MeV, with the aim to test the upper energy limit of applicability of our OP before the chiral expansion scheme breaks down.

The paper is organized as follows: In Sec. II we summarize the main features of both microscopic and phenomenological approaches to the OP. In Sec. III we show and discuss the results for the observables of elastic pA scattering. Finally, in Sec IV we draw our conclusions.

Ii Theoretical Models

The underlying assumption on which the OP is based is that the interaction between the projectile and the target nucleus can be modelled by a complex mean field potential. The numerical solution of the Schrödinger equation (or the Lippman-Schwinger equation in the momentum space representation) with this complex potential brings valuable information, i.e. the differential cross section , the analyzing power , and the spin rotation among the others PaetzSchieck; Glendenning.

In our approach we limit ourselves to the study of the elastic channel, defined as the case in which the target and the projectile remain in the stationary state, at energies where a non-relativistic description can be conveniently applied. Given a state which describes the relative motion of both collision partners, after some manipulations, a Schrödinger equation can be derived for the projectile and the target in the relative motion that reads as follows

(1)

is the kinetic energy operator, is the available energy in the elastic channel, and is the so-called generalized OP FESHBACH1958357 defined as

(2)

where is a two-body potential, and and are idempotent projection operators introduced to isolate the contribution of the elastic channel: projects onto the elastic channel and onto the complementary space, i.e. onto all the non-elastic channels, for which the following relations hold: , and .

The first term in Eq. (2) is the contribution of the static two-body interaction, while the second term takes into account the effect of non-elastic channels in the space . This term is the dynamic part of the OP and depends on the energy of the elastic channel. As a consequence, the Feshbach approach naturally leads to an energy-dependent OP .

The general structure of is extremely complicated and can be simplified for some specific applications. We refer the reader to Refs. PaetzSchieck; Glendenning; Jeukenne:1976uy; Mahaux:1985zz for exhaustive discussions about the approximations which are necessary to deal with the treatment of . Here we will restrict our considerations to the bare essentials.

Generally speaking, there are two available methods for the construction of an OP.

In the microscopic approach, one starts from a realistic NN interaction (i.e. able to reproduce the experimental NN phase shifts with a per datum very close to one Machleidt:1989tm; Machleidt:2011zz; Epelbaum:2008ga) and an educated guess for the radial density of the target PaetzSchieck; Glendenning; Jeukenne:1976uy. A suitable combination of these two terms (a procedure usually called ”folding”) produces the optical potential . The main features of the microscopic OP are the independence from phenomenological inputs and, in particular with the most recent NN microscopic potentials, the ability to assess reliable error estimates (see Ref. Epelbaum:2014efa for extensive discussions about this topic). In the ideal case where no approximations are made to derive the microscopic OP, the absence of phenomenology would lead to accurate predictions that are probably better than those obtained with a phenomenological OP, but in practice the full calculation of the OP turns out to be too complicated and the approximations that must necessarily be introduced reduce the accuracy and the reliability of these predictions.

On the other hand, a more pragmatic phenomenological approach can be pursued with the adoption of an analytical form of the potential, i.e. like a Woods-Saxon shape, where the adjustable parameters are fitted to a set of available experimental data Varner:1991zz.

Since a lot of efforts has been put over the last years on both methods, we believe that it can be useful to make a comparison between the above mentioned approaches. For this purpose, we decided to use our recent microscopic OPs derived Vorabbi:2017rvk from NN chiral interactions at NLO Epelbaum:2014sza; Epelbaum:2014efa; Entem:2014msa; Entem:2017gor and the most recent analysis by Koning et al. Koning:2003zz who developed a very successful nonrelativistic phenomenological OP (KD) for energies below 200 MeV but also with an extension up to 1 GeV KONING2014187.

ii.1 Microscopic optical potentials at

The theoretical justification for the description of the NA optical potential in terms of the microscopical NN interaction has been addressed for the first time by Watson et al. Riesenfeld:1956zza and then formalized by Kerman et al. (KMT) Kerman:1959fr, where the so-called multiple scattering approach to the NA optical potential is expressed by a series expansion of the free NN scattering amplitudes. Over the last decades several authors made important contributions to this approach. Just to mention the most relevant ones, we would like to remind the works with the KMT optimum factorized OP PhysRevC.30.1861; PhysRevC.40.881, the calculation of the full-folding OP with harmonic oscillator densities PhysRevC.41.814; PhysRevLett.63.605; PhysRevC.41.2188; PhysRevC.41.2257, the calculation of the second-order OP in the multiple scattering theory PhysRevC.46.279, the calculation of the medium contributions to the first-order OP PhysRevC.48.2956; PhysRevC.51.1418; PhysRevC.52.1992, and the calculation of the full-folding OP with realistic densities PhysRevC.56.2080. Concerning the inclusion of medium effects we also want to mention the works based on the matrix of Amos et al. Amos2002 and Arellano et al. PhysRevC.52.301; PhysRevC.84.034606.

In Refs. Vorabbi:2015nra; Vorabbi:2017rvk a microscopic OP was obtained at the first-order term within the spectator expansion of the nonrelativistic multiple scattering theory, corresponding to the single-scattering approximation. The impulse approximation was adopted, where nuclear binding forces on the interacting target nucleon are neglected, as well as the optimum factorization approximation, where the two basic ingredients of the calculations, i.e. the nuclear density and the NN matrix, are factorized. We refer the reader to Refs. Vorabbi:2015nra; Vorabbi:2017rvk for all relevant details and an exhaustive bibliography. In the momentum space, the factorized is obtained as

(3)

where represents the proton-proton (pp) and proton-neutron (pn) free matrix evaluated at a fixed energy , the neutron and proton profile density, and the momentum variables and are conveniently expressed by the variables and (see Sect. II of Ref. Vorabbi:2015nra).

For the neutron and proton densities of the target nucleus we use as in Refs. Vorabbi:2015nra; Vorabbi:2017rvk a Relativistic Mean-Field (RMF) description Niksic20141808, which has been quite successful in the description of ground state and excited state properties of finite nuclei, in particular in a Density Dependent Meson Exchange (DDME) version, where the couplings between mesonic and baryonic fields are assumed as functions of the density itself PhysRevC.66.024306. We are aware that a phenomenological description of the target is not fully consistent with the goal of a microscopic description of elastic NA scattering. In a very recent paper Gennari:2017yez a microscopic OP has been derived using ab initio translationally invariant nonlocal one-body nuclear densities computed within the No-Core Shell Model (NCSM) approach BARRETT2013131, which is a technique particularly well suited for the description of light nuclei. Indeed the use of a nonlocal ab initio density improves significantly the agreement with data of elastic proton scattering off He and C, while for O no significant improvement is obtained in comparison with the RMF results. The work reported in Ref. Gennari:2017yez represents a great leap forward towards the construction of a microscopic OP for light nuclei, but the aim of our present work is to investigate the predictive power of a microscopic OP over a wide range of nuclei and isotopic chains in different regions of the nuclear chart.

For the NN interaction we use here two different versions of the chiral potentials at fifth order (NLO) recently derived by Epelbaum, Krebs, and Meißner (EKM) Epelbaum:2014sza; Epelbaum:2014efa and Entem, Machleidt, and Nosyk (EMN) Entem:2014msa; Entem:2017gor. As explained in Ref. Vorabbi:2017rvk, the two versions of the chiral NLO potentials have significant differences concerning the renormalization procedures and we follow the same prescriptions adopted there. The strategy followed for the EKM potentials Epelbaum:2014sza; Epelbaum:2014efa consists in a coordinate space regularization for the long-range contributions , by the introduction of , and a conventional momentum space regularization for the contact (short-range) terms, with a cutoff . Five choices of are available: , and fm, leading to five different potentials.

On the other hand, for the EMN potentials, a slightly more conventional approach was pursued Entem:2014msa; Entem:2017gor. A spectral function regularization, with a cutoff MeV, was employed to regularize the loop contributions and a conventional regulator function, with , and MeV, to deal with divergences in the Lippman-Schwinger equation. For all details we refer the reader to Refs. Entem:2014msa; Entem:2017gor; Vorabbi:2017rvk.

The aim of the present work is to test the predictive power of our microscopic OP in comparison with available experimental data and it can be useful to show the uncertainties on the predictions produced by NN chiral potentials obtained with different values of the regularization parameters. For this purpose, all calculations have been performed with three of the EKM Epelbaum:2014sza; Epelbaum:2014efa potentials, corresponding to , and fm, and with two of the EMN Entem:2014msa; Entem:2017gor potentials, corresponding to and MeV. In all the figures presented in Sec. III the bands give the differences produced by changing for EKM (red bands) and for EMN (green bands). Thus the bands have here a different meaning than in Ref. Vorabbi:2017rvk, where the EKM and EMN NN chiral potentials at NLO were also used. The aim of Ref. Vorabbi:2017rvk was to investigate the convergence and to assess the theoretical errors associated with the truncation of the chiral expansion and the bands were given to investigate these issues. We also showed in Ref. Vorabbi:2017rvk that EKM calculations based on different values of are quite close and consistent with each other (although, as remarked in Ref. Epelbaum:2014sza, larger values of R are probably less accurate due to a larger influence of cutoff artifacts). The same assumption can be made about the EMN potentials: changing the cutoffs does not lead to sizeable differences in the datum (see Tab.VIII in Ref. Entem:2017gor) and it is safe to perform calculations with only two potentials. Because we want to explore elastic scattering at energies around and above 200 MeV, we exclude the EKM potentials with and fm and the EMN potential with MeV. We are confident that for our present purposes showing results with only a limited set of NN chiral potentials will not affect our conclusions in any way.

ii.2 Phenomenological potentials

One of the most recent and successful phenomenological OP has been developed by Koning et al. Koning:2003zz. As quoted in the original paper, the authors provided a phenomenological OP able to challenge the best microscopic approaches in terms of predictive power.

The phenomenological OP can be separated into a real () and an imaginary () part. Both contributions can be expressed as follows

(4)

in terms of central (), surface dependent (), and spin-orbit () components. All the components can be separated in energy-dependent well depths and energy-independent shape functions as , where the radial functions usually resemble a Wood-Saxon shape.

The potential of Ref. Koning:2003zz is a so-called ”global” OP, which means that the free adjustable parameters are fitted for a wide range of nuclei () and of incident energies ( keV MeV) with some parametric dependence of the coefficients in terms of the target mass number and of the incident energy . An alternative choice, not adopted in Ref. Koning:2003zz, would be to produce an OP for each single target nucleus. We refer the reader to Ref. Koning:2003zz for more details. Recently, an extension of the OP of Ref. Koning:2003zz up to 1 GeV has been proposed KONING2014187. It is generally believed that above MeV the Schrödinger picture of the phenomenological OP should be taken over by a Dirac approach talys, but the extension was done just with the aim to test at which energy the validity of the predictions of the nonrelativistic OP fail. We are aware that above 200 MeV an approach based on the Dirac equation would probably be a more consistent choice, but since we are interested in testing the limit of applicability of our (nonrelativistic) microscopic OP we will use such an extension to perform some benchmark calculations at center-of-mass energies close to 300 MeV. All the calculations have been performed by ECIS-06 ecis as a subroutine in the TALYS software talys; talys2.

Iii Results

The aim of the present paper is to investigate and compare the predictive power of our microscopic OP derived from the EKM Epelbaum:2014sza; Epelbaum:2014efa and EMN Entem:2014msa; Entem:2017gor chiral potentials at NLO and of the phenomenological global OP KD derived by Koning et al. Koning:2003zz; KONING2014187 in comparison with available data of elastic pA scattering. To this aim, in this section we present and discuss the predictions of the different OPs for the differential cross section , presented as ratio to the Rutherford cross section, , and analyzing power of proton elastic scattering over a wide range of nuclei and isotopes chains, from oxygen to lead, and for proton energies between 156 and 333 MeV, for which experimental data are available.

The energy range considered for our investigation was chosen on the basis of the assumptions and approximations adopted in the derivation of the theoretical OP. In particular, the impulse approximation does not allow us to use our microscopic OP with enough confidence at much lower energies, where we can expect that the phenomenological KD potential is able to give a better agreement with the experimental data. The upper energy limit is determined by the fact that the EKM and EMN chiral potentials are able to describe NN scattering observables up to 300 MeV Epelbaum:2014sza; Epelbaum:2014efa; Entem:2014msa; Entem:2017gor. The phenomenological global KD potential was originally constructed for energies up to 200 MeV Koning:2003zz and it was then extended up to 1 GeV KONING2014187. It can therefore be interesting to test and compare the validity of the predictions of both microscopic and phenomenological OPs up to about 300 MeV.

In Ref. Vorabbi:2017rvk we compared the results obtained with different versions of EKM and EMN chiral potentials at for the pp and pn Wolfenstein amplitudes and for the scattering observables of elastic proton scattering from O, C, and Ca nuclei at an incident proton energy MeV. For the sake of comparison with our previous work, we show in Fig. 1 the ratio of the differential cross section to the Rutherford cross section for elastic proton scattering off O at MeV. The results obtained with the EKM and EMN potentials and with the KD optical potential are compared with the experimental data taken from Refs. kelly; exfor. The EKM and EMN results correspond to the results shown in Fig. 2 of Ref. Vorabbi:2017rvk for the differential cross section (of course with a different meaning of the bands) and give a reasonable, although not perfect, agreement with data. The experimental ratio is slightly overestimated at lower angles and somewhat underestimated for . The differences between the EKM and EMN results are small and not crucial, EKM gives a smaller cross section around the maxima and therefore a somewhat better agreement with the data in this region. The bands, representing the uncertainties on the regularization of the NN chiral potentials, are generally small and not influential for the comparison with data. The KD result gives a good description of the experimental cross section for and underpredicts the data for larger angles. We point out, to be honest, that KD was obtained for nuclei in the mass range while O is below this range. We present the result only for the sake of comparison.

The ratios of the differential cross sections to the Rutherford cross sections for elastic proton scattering off calcium, nichel, tin, and lead isotopes are shown in Figs. 2, 3, 4, and 5. The results are compared with the experimental data taken from Refs. kelly; exfor.

All the results for Ca isotopes in Fig. 2 are for an incident proton energy of 200 MeV. The experimental database used to generate the KD potential includes Ca at MeV. In Fig. 2 KD gives indeed an excellent agreement with Ca data, and a good agreement also for the other isotopes. The results with the EKM and EMN potentials are very close to each other, the uncertainty bands are narrow, and the agreement with data, which is reasonable and of about the same quality for all the isotopes, is however somewhat worse than with KD, in particular at larger angles. At lower angles the EKM and EMN results well reproduce the behaviour of the experimental cross section, which is sometimes a bit overestimated by the calculations. A better agreement with data would presumably be obtained improving or reducing the approximations adopted in the calculation of the microscopic OP.

In Fig. 3 we show the results for Ni at and 295 MeV, Ni at MeV, and Ni at MeV. The experimental database used to generate the KD potential includes Ni up to 200 MeV and Ni up to 65 MeV. For Ni KD gives a good description of the data at 192 MeV, while a much worse agreement is obtained at the higher energy of 295 MeV, where only the overall behavior of the experimental cross section is reproduced by the phenomenological OP. The EKM and EMN results give a better and reasonable description of the data at 295 MeV, up to . At 192 MeV the microscopic OP can roughly describe the shape of the experimental cross section, but the size is somewhat overestimated. KD gives only a poor description of the data for Ni at 178 MeV and a very good agreement for Ni at 156 MeV. The microscopic OP gives a better and reasonable agreement with the Ni data, over all the angular distribution, while for Ni the results are a bit larger than those of the KD potential. The EKM and EMN results are always very close to each other and the bands are generally narrow.

The results for Sn isotopes at 295 MeV and for Sn at 200 MeV are displayed in Fig. 4. In this case all the OPs give qualitatively similar results and a reasonable agreement with data, in particular, for . The agreement generally declines for larger angles. KD gives a better description of Sn data at 200 MeV, where the EKM and EMN results are a bit larger than the data at the maxima and a bit lower at the minima. We note that Sn is included in the experimental database for the KD potential for proton energies up to 160 MeV. At 295 MeV, the microscopic OP gives, in general, a slightly better agreement with the data than KD for all the tin isotopes shown in the figure.

The results for Pb isotopes at 295 MeV and for Pb data at 200 MeV are displayed in Fig. 5. Also in this case the experimental cross section at 200 MeV is well described by KD, the agreement is better than with the microscopic OP. The experimental database for KD includes Pb up to 200 MeV. At 295 MeV a better agreement with data is generally given by the EKM and EMN results, in particular by EMN: for all the three isotopes considered, the two results practically overlap for , where they are also very close to the KD result, then they start to separate and the EMN result is a bit larger than the EKM one and in better agreement with data. We point out that the uncertainty bands, that are generally narrow, in this case become larger increasing the scattering angle, when also the agreement with data declines.

The results that we have shown till now indicate that, in comparison with the phenomenological KD potential, our microscopic OP, in spite of the approximations made to derived it, has a comparable and in some cases even better predictive power in the description of the cross sections on the isotopic chains and energy range here considered. KD is able to give a better and excellent description of data in specific situations, in particular, in the case of nuclei included in the experimental database used to generate the original KD potential and at the lower energies considered. For energies above 200 MeV our microscopic OP gives, in general, a better agreement with data. This conclusion is confirmed by the results shown in Fig. 6, where the ratios of the differential cross sections to the Rutherford cross sections are displayed for elastic proton scattering off O and Ca at MeV and Ni at MeV in comparison with the data taken from Refs. kelly; exfor. The differences between the phenomenological and microscopic OPs increase with the increasing scattering angle and proton energy. For Ni at 333 MeV both EKM and EMN give a much better and very good description of data. In the other cases KD is able to describe data only at the lowest angles. The EKM and EMN results are in general very close to each other. In both cases the width of the uncertainty bands increases at larger scattering angles but the uncertainties are not crucial for the comparison with data.

As mentioned in the Introduction (Sec. I), in the near future new experimental data will be available for exotic nuclei fair; exl. For this purpose, in Fig. 7 we show theoretical predictions for EKM and EMN potentials for the test case Ni(p,p)Ni at an energy of MeV in inverse kinematics. Even if the energy scale involved is beyond the supposed range of validity of our approach, it is interesting to see if microscopic potentials and KD look reasonable and if error bands are acceptable. We only included a selection of the potentials because of the energy: and fm for EKM potentials and and MeV for the EMN ones. At 400 MeV KD and EKM give similar predictions with reasonable shapes as a function of the angle . The EMN potentials seem under control only for small angles (). Unfortunately, experimental data are still under scrutiny and not yet published datani56. From this figure we can thus conclude that, contrary to the EMN potentials, the EKM potentials have not yet reached the limit beyond which the chiral expansion scheme breaks down. Of course, this limit is not unique and depends on the regularization scheme adopted to derive the NN interaction.

In Figs. 8, 9, and 10 we show the analyzing power for some of the same nuclei and at the same proton energies presented in Figs. 1, 2, 3, 4, and 5.

The analyzing power for calcium isotopes is shown Fig. 8, which corresponds to Fig. 2 for the ratio of the differential cross section to the Rutherford cross section. Polarization observables are usually more difficult to reproduce and also in this case the agreement with data is far from perfect, but all the results are able to describe the overall behavior of the experimental , in particular at lower angles. A better result in comparison with data is given in this case by the phenomenological KD potential. The differences between the EKM and EMN results are small, but the error bands get large at the largest angles considered.

The results for tin isotopes in Fig. 9 correspond to the results of Fig. 4 for the ratio . Also in this case the agreement with data is worse than in Fig. 4 and it is difficult to judge which OP gives the better description of the experimental : KD is somewhat better at 200 MeV and the microscopic OP at 295 MeV. It must be emphasized that the extension to 1 GeV was performed to have the total reaction cross section under control up to this energy with no concerns about polarization observables. The bad performance in the description of is anyhow not surprising if we consider that Woods-Saxon like form factors are not supposed to work properly above 200 MeV koning.

The analyzing power for O and Pb at 200 MeV, Ni at 192 MeV, and Ni at 178 MeV are displayed in Fig. 10. In the case of O, the results basically confirm what was already found for the ratio in Fig. 1: KD gives a good description of data for lower angles, in particular for , EKM and EMN give a less good description of data at lower angles but a reasonable agreement up to . The experimental analyzing powers of Ni and Ni are well described by KD at the lowest angles, but over all the angular distribution the general agreement (or disagreement) with data of the microscopic and phenomenological OPs is of about the same quality. Also in the case of Pb KD describes the experimental well for , better than the EKM and EMN results, which, on the other hand, are in a somewhat better (although not perfect) agreement with data for larger angles.

Iv Conclusions

In recent papers Vorabbi:2015nra; Vorabbi:2017rvk we derived a microscopic optical potential for elastic pA scattering from NN chiral potentials at fourth order (NLO) and fifth order (NLO), with the purpose to study the domain of applicability of NN chiral potentials to the construction of an optical potential, to investigate convergence patterns, and to assess the theoretical errors associated with the truncation of the chiral expansion. Numerical examples for the cross section and polarization observables of elastic proton scattering on C, O, and Ca nuclei were presented and compared with available experimental data. Our results indicated that building an optical potential within the chiral perturbation theory is a promising approach for describing elastic proton-nucleus scattering and they allowed us to conclude that convergence has satisfactorily been achieved at NLO.

In the present work we have extended our previous investigation to isotopic chains exploring the mass number dependence and the energy range of applicability of our microscopic optical potential. As a benchmark, we have tested our calculations against one of the best phenomenological parametrizations developed by Koning and Delaroche Koning:2003zz and, of course, experimental data where available.

Our main goal was to check the robustness of our approach and the capability of our optical potential to be applied to exotic nuclei, i.e. nuclei with values of proton-to-neutron ratio far from the stability, where phenomenological models might be unreliable.

Numerical results have been presented for the unpolarized differential cross section and the analyzing power of elastic proton scattering off calcium, nickel, tin, and lead isotopes in a proton energy range between 156 and 333 MeV. A theoretical prediction for the cross section of elastic proton scattering off Ni at 400 MeV has also been presented, which is of interest for the future EXL experiment on exotic nuclei at FAIR fair; exl, although the energy is beyond the supposed range of validity of the chiral potentials.

Because of the renormalization procedure, NN chiral potentials with almost the same level of accuracy (i.e. the associated to the reproduction of NN phase shifts) come with different values of the cut-off parameters. We have restricted our calculations to a limited set of the aforementioned parameters and we have associated theoretical error bands to the uncertainties produced by the different parameters.

The agreement of our present results with empirical data is sometimes worse and sometimes better but overall comparable to the agreement given by the phenomenological OP, in particular for energies close to 200 MeV and above 200 MeV. The example shown at 400 MeV suggests that at this energy the EKM potentials have not yet reached the limit after which the chiral expansion scheme breaks down.

The microscopic optical potential generally provides a qualitatively similar agreement with data for all the nuclei of an isotopic chain. This clearly shows that changing the values of does not affect the predictive power of our optical potential. The agreement is worse for the analyzing power than for the cross section and in general declines for larger values of the scattering angle.

A better description of empirical data requires a more sophisticated model for the microscopic optical potential. Work is in progress about three major improvements: calculation of the full-folding integral (to go beyond the optimum factorization), treatment of nuclear-medium effects (to go beyond the impulse approximation), and the inclusion of three-body forces.

V Acknowledgements

The authors are deeply grateful to E. Epelbaum (Institut für Theoretische Physik II, Ruhr-Universität Bochum, Germany) for providing the chiral potential of Ref. Epelbaum:2014sza; Epelbaum:2014efa and R. Machleidt (Department of Physics, University of Moscow, Idaho, USA) for the chiral potential of Ref. Entem:2014msa; Entem:2017gor. We also would like to acknowledge useful communications with A. Koning (Nuclear Research and Consultancy Group NRG, Petten, The Netherlands) and T. Kröll (Institut für Kernphysik, Technische Universität Darmstadt, Germany).

References

Figure 1: (Color online) Ratio of the differential cross section to the Rutherford cross section as a function of the center-of-mass scattering angle for elastic proton scattering off O. Calculations are performed at MeV (laboratory energy) with the microscopic OPs derived from the EKM Epelbaum:2014sza; Epelbaum:2014efa(EKM, red band) and EMN Entem:2014msa; Entem:2017gor (EMN, green band) NN chiral potentials at NLO and with the phenomenological global OP of Ref. talys (KD, violet line). The interpretation of the bands is explained in the text. Experimental data from Refs. kelly; exfor.
Figure 2: (Color online) The same as in Fig. 1 for Ca isotopes at 200 MeV. Experimental data from Refs. kelly; exfor.
Figure 3: (Color online) The same as in Fig. 1 for Ni isotopes: Ni at and MeV, Ni at MeV, and Ni at MeV. Experimental data from Refs. kelly; exfor.
Figure 4: (Color online) The same as in Fig. 1 for Sn isotopes: Sn at MeV and Sn at MeV. Experimental data from Refs. kelly; exfor.
Figure 5: (Color online) The same as in Fig. 1 for Pb isotopes: Pb at MeV and Pb at MeV. Experimental data from Refs. kelly; exfor.
Figure 6: (Color online) The same as in Fig. 1 for O and Ca at MeV and Ni at MeV. Experimental data from Refs. kelly; exfor.
Figure 7: (Color online) The same as in Fig. 1 for Ni at MeV.
Figure 8: (Color online) The same as in Fig. 2 but for the analyzing power . Experimental data from Refs. kelly; exfor.
Figure 9: (Color online) The same as in Fig. 4 but for the analyzing power . Experimental data from Refs. kelly; exfor.
Figure 10: (Color online) Analyzing power as a function of the angle for elastic proton scattering on O, and Pb at MeV, Ni at MeV, and Ni at MeV. Experimental data from Refs. kelly; exfor.
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