for the NUMEN collaboration

# Multi-channel experimental and theoretical constraints
for the 116Cd(20Ne,20F)116In charge exchange reaction at 306 MeV

S. Burrello  Technische Universität Darmstadt, Fachbereich Physik, Institut
für Kernphysik, Darmstadt, Germany  Departamento de Física Atomica, Molecular
y Nuclear, Facultad de Física, Universidad de Sevilla, Sevilla, Spain  S.
Calabrese  Istituto Nazionale di Fisica Nucleare, Laboratori Nazionali del
Sud, Catania, Italy  Dipartimento di Fisica e Astronomia “Ettore Majorana”,
Università di Catania, Catania, Italy  F. Cappuzzello  Istituto Nazionale di
Fisica Nucleare, Laboratori Nazionali del Sud, Catania, Italy  Dipartimento di
Fisica e Astronomia “Ettore Majorana”, Università di Catania, Catania, Italy
D. Carbone  Istituto Nazionale di Fisica Nucleare, Laboratori Nazionali del
Sud, Catania, Italy  M. Cavallaro  Istituto Nazionale di Fisica Nucleare,
Laboratori Nazionali del Sud, Catania, Italy  M. Colonna  Istituto Nazionale
di Fisica Nucleare, Laboratori Nazionali del Sud, Catania, Italy  J. A. Lay
Departamento de Física Atomica, Molecular y Nuclear, Facultad de Física,
Universidad de Sevilla, Sevilla, Spain  Instituto Carlos I de Física Teórica y
Computacional, Universidad de Sevilla, Sevilla, Spain  H. Lenske  Institut für
Theoretische Physik, Justus-Liebig-Universität Giessen, Giessen, Germany  C.
Agodi  Istituto Nazionale di Fisica Nucleare, Laboratori Nazionali del Sud,
Catania, Italy  J. L. Ferreira  Instituto de Física, Universidade Federal
Fluminense, Niterói, Brazil  S. Firat  Department of Physics, Akdeniz
University, Antalya, Turkey  A. Hacisalihoglu  Institute of Natural Sciences,
Karadeniz Teknik Universitesi, Trabzon, Turkey  L. La Fauci  Istituto
Nazionale di Fisica Nucleare, Laboratori Nazionali del Sud, Catania, Italy
Dipartimento di Fisica e Astronomia “Ettore Majorana”, Università di Catania,
Catania, Italy  A. Spatafora  Istituto Nazionale di Fisica Nucleare,
Laboratori Nazionali del Sud, Catania, Italy  Dipartimento di Fisica e
Astronomia “Ettore Majorana”, Università di Catania, Catania, Italy  L. Acosta
Instituto de Física, Universidad Nacional Autónoma de México, Mexico City,
Mexico  INFN, Sezione di Catania, Catania, Italy  J. I. Bellone  Istituto
Nazionale di Fisica Nucleare, Laboratori Nazionali del Sud, Catania, Italy  T.
Borello-Lewin  Instituto de Fisica, Universidade de São Paulo, São Paulo,
Brazil  I. Boztosun  Department of Physics, Akdeniz University, Antalya,
Turkey  G. A. Brischetto  Istituto Nazionale di Fisica Nucleare, Laboratori
Nazionali del Sud, Catania, Italy  Dipartimento di Fisica e Astronomia “Ettore
Majorana”, Università di Catania, Catania, Italy  D. Calvo  INFN, Sezione di
Torino, Torino, Italy  E. R. Chávez-Lomelí  Instituto de Física, Universidad
Nacional Autónoma de México, Mexico City, Mexico  I. Ciraldo  Istituto
Nazionale di Fisica Nucleare, Laboratori Nazionali del Sud, Catania, Italy
Dipartimento di Fisica e Astronomia “Ettore Majorana”, Università di Catania,
Catania, Italy  M. Cutuli  Istituto Nazionale di Fisica Nucleare, Laboratori
Nazionali del Sud, Catania, Italy  Dipartimento di Fisica e Astronomia “Ettore
Majorana”, Università di Catania, Catania, Italy  F. Delaunay  Istituto
Nazionale di Fisica Nucleare, Laboratori Nazionali del Sud, Catania, Italy
Dipartimento di Fisica e Astronomia “Ettore Majorana”, Università di Catania,
Catania, Italy  LPC Caen, Normandie Université, ENSICAEN, UNICAEN, CNRS/IN2P3,
Caen, France  P. Finocchiaro  Istituto Nazionale di Fisica Nucleare,
Laboratori Nazionali del Sud, Catania, Italy  M. Fisichella  Istituto
Nazionale di Fisica Nucleare, Laboratori Nazionali del Sud, Catania, Italy  A.
Foti  INFN, Sezione di Catania, Catania, Italy  F. Iazzi  INFN, Sezione di
Torino, Torino, Italy  DISAT, Politecnico di Torino, Torino, Italy  G.
Lanzalone  Istituto Nazionale di Fisica Nucleare, Laboratori Nazionali del
Sud, Catania, Italy  Facoltà di Ingegneria e Architettura, Università di Enna
“Kore”, Enna, Italy  R. Linares  Instituto de Física, Universidade Federal
Fluminense, Niterói, Brazil  J. Lubian  Instituto de Física, Universidade
Federal Fluminense, Niterói, Brazil  M. Moralles  Instituto de Pesquisas
Energeticas e Nucleares IPEN/CNEN, São Paulo, Brazil  J. R. B. Oliveira
Instituto de Fisica, Universidade de São Paulo, São Paulo, Brazil  A. Pakou
Department of Physics, University of Ioannina and Hellenic Institute of
Nuclear Physics, Ioannina, Greece  L. Pandola  Istituto Nazionale di Fisica
Nucleare, Laboratori Nazionali del Sud, Catania, Italy  H. Petrascu  IFIN-HH,
Bucarest, Romania  F. Pinna  INFN, Sezione di Torino, Torino, Italy  DISAT,
Politecnico di Torino, Torino, Italy  G. Russo  INFN, Sezione di Catania,
Catania, Italy  O. Sgouros  Istituto Nazionale di Fisica Nucleare, Laboratori
Nazionali del Sud, Catania, Italy  S. O. Solakci  Department of Physics,
Akdeniz University, Antalya, Turkey  V. Soukeras  Istituto Nazionale di Fisica
Nucleare, Laboratori Nazionali del Sud, Catania, Italy  G. Souliotis
Department of Chemistry, University of Athens and Hellenic Institute of
Nuclear Physics, Athens, Greece  D. Torresi  Istituto Nazionale di Fisica
Nucleare, Laboratori Nazionali del Sud, Catania, Italy  S. Tudisco  Istituto
Nazionale di Fisica Nucleare, Laboratori Nazionali del Sud, Catania, Italy  A.
Yildirin  Department of Physics, Akdeniz University, Antalya, Turkey  V. A. B.
Zagatto  Instituto de Física, Universidade Federal Fluminense, Niterói, Brazil

(August 28, 2024; August 28, 2024)

###### Abstract

Background

Charge exchange (CE) reactions offer a major opportunity to excite nuclear
isovector modes, providing important clues about the nuclear interaction in
the medium. Moreover, double charge exchange (DCE) reactions are proving to be
a tempting tool to access nuclear transition matrix elements (NME) related to
double beta-decay processes. The latter are also of crucial importance to
extract neutrino properties from the half-life of the hypothetical
neutrinoless double beta decay and to search for physics beyond the standard
model.

Purpose

Through a multi-channel experimental analysis and a consistent theoretical
approach of the 116Cd(20Ne,20F)116In single charge exchange (SCE) reaction at
306 MeV, we aim at disentangling from the experimental cross section the
contribution of the competing mechanisms, associated with second or higher
order sequential transfer and/or inelastic processes.

Method

We measured excitation energy spectra and absolute cross sections for elastic
+ inelastic, one-proton transfer and SCE channels, using the MAGNEX large
acceptance magnetic spectrometer to detect the ejectiles. For the first two
channels, we also extracted the experimental cross section angular
distributions. The experimental data are compared with theoretical predictions
obtained by performing two-step distorted wave Born approximation and coupled
reaction channel calculations. We employ spectroscopic amplitudes for single-
particle transitions derived within a large-scale shell model approach and
different optical potentials for modeling the initial and the final state
interactions.

Results

The present study significantly mitigates the possible model dependence
existing in the description of these complex reaction mechanisms, thanks to
the satisfactory reproduction of several channels at once. In particular, our
work demonstrates that the two-step transfer mechanisms produce a non
negligible contribution to the total cross section of the 116Cd(20Ne,20F)116In
reaction channel, although a relevant fraction is still missing, being
ascribable to the direct SCE mechanism, which is not addressed here.

Conclusions

Our analysis provides a careful estimation of the sequential transfer
processes which are competing with the direct SCE mechanism for the heavy ion
reaction under investigation. The study suggests that the direct SCE should
play an important role among the mechanisms populating the final channel.
Nevertheless, the analysis of the higher order processes considered here is
mandatory to isolate the direct SCE process contribution and approach
structure information on the corresponding NME from the reaction cross
section. The description of the latter process and the competition between the
two mechanisms deserves further investigation.

##  I Introduction

Charge exchange (CE) transitions induced by the nuclear interaction have
raised a widespread theoretical and experimental interest over the last
decades, being regarded as a major source of information on nuclear isovector
excitations Osterfeld ; Ichimura06 ; Frekers13 . On the other hand, CE
reactions offer a quite appealing opportunity to excite these modes
Brendel:1988mwl ; AndersonPRC1991 ; blomgren ; YakoPLB2005 ; FrekersPPNP2006 ;
FujitaPRC2007 ; FujitaPPNP2011 ; Douma2020 , providing crucial clues about the
nucleon-nucleon (NN) interaction in the medium, whose behavior in the spin-
isospin channels is still not fully understood. Indeed, CE reactions are
nowadays the object of recent intensive experimental campaigns EPJA2018 ;
EPJA2015 ; Agodi1 ; Agodi2 ; sasanoPRC2012 ; Kisamori2016 ; Miki2017 .

In the case of high-energy reactions with light projectiles, where the
reaction mechanism is rather well known, single CE (SCE) reactions have been
widely investigated to probe spin-isospin properties of nuclei. Moreover,
$\beta$ decay was used as calibration whenever possible, in view of the close
relationship between $\beta$-decay strengths and CE cross sections at low
momentum transfer Taddeucci87 ; Alford .

A renewed interest has also recently emerged in studying CE reactions with
heavy ions, which are used to address a wide spectrum of research topics, from
work on quasi–elastic SCE excitations with low and intermediate energy ion
beams, to investigations of the subnuclear sector by excitations of nucleon
resonance. e.g. $\Delta_{33}(1232)$, at relativistic beam energies, as
discussed in the recent review article Ref.LenPPNP2019 .

Over the years, a plethora of heavy ion SCE data has been collected and used
for spectroscopic work. The study of the charge-converting components of the
projectile-target nucleon–nucleon (NN) interactions has been in many cases the
driving force behind experimental and theoretical research. In this context,
some progress on the theoretical description of the complex reaction dynamics
has been made over the past years Lenske18 ; LenPPNP2019 ; CavFRO2021 . Not to
the least, the interest in that kind of interaction is motivated by the
similarity of the nuclear transition operators to those acting in nuclear
beta–decay, as elucidated in Refs. Lenske18 ; LenPPNP2019 ; BelPLB2020 ;
LenskeUniverse .

Even the large research potential of heavy ion double charge exchange (DCE)
reactions was recognized quite early in a first (unsuccessful) attempt to
study the double Gamow–Teller (DGT) mode blomgren . More recently,
experimental studies are also investigating the DCE reactions between heavy
ions to identify the connections of these processes not only with DGT or
double Fermi transitions but also with double-$\beta$ ($2\beta$) decay
SanPRC2018 ; BelPLB2020 . In this context, the NUMEN and NURE projects at LNS-
INFN have proposed an innovative way to exploit DCE reactions to access
nuclear transition matrix elements (NME) related to those involved in $2\beta$
decay EPJA2018 . A very accurate knowledge of the NME is indeed instrumental
to extract the properties of neutrinos from the half-life of the hypothetical
neutrinoless double beta decay (0$\nu$2$\beta$), that is, to search for
physics beyond the standard model EngRPP2017 ; ShimizuPRL2018 . Besides,
extraction of nuclear structure information from the total DCE reaction cross
section is helpful to add new and more stringent constraints on the nuclear
models EngRPP2017 ; GamPRL2020 . These goals pose an exciting challenge on the
study of DCE reactions that are stimulating recent developments, aimed to
describe the (virtually unexplored) underlying reaction mechanism SanPRC2018 ;
BelPLB2020 . Before being able to address those spectroscopic issues on a
quantitative level, the experimental and theoretical research program of the
NUMEN and NURE projects demands thus a full understanding of the reaction
mechanism of the underlying SCE and DCE reactions at beam energies in the
region of 10 to 20 AMeV, which is located well above the Coulomb barrier in
practical experiments.

Heavy ion collisions above the Coulomb barrier are however generally
characterized by a large number of different populated channels. For example,
it is well known that nuclear charge exchange processes proceed in principle
by two distinct, but interfering mechanisms. Direct CE is a collisional
process mediated by the isovector nucleon–nucleon (NN) interactions, acting
between projectile and target. However, an unwanted but unavoidable
complication in the heavy ion induced CE reactions is the admixture of multi-
step reaction mechanisms, typically given by sequential transfer of nucleons
or inelastic processes of second or higher order CapLenske04 . The two-step
transfer mechanism is indeed sensitive to the nucleon-nucleus mean-field
potential and cannot probe the NN interaction responsible for Fermi and Gamow-
Teller isospin-flip transitions, which are observed in a CE reaction
LayBurrello . For SCE reactions, the competition of one-step direct charge
exchange and two-step sequential processes was discussed in detail, e.g., in
Refs. Oertzen ; Brendel:1988mwl ; Winfield ; Len89 . The probably first fully
microscopic theoretical investigations of heavy ion SCE reaction showed that,
at energies close to the Coulomb barrier, transfer CE is by far the dominant
reaction mechanism Brendel:1988mwl , but with increasing beam energy the
strength of direct SCE rapidly increases and finally dominates the SCE cross
sections Len89 .

In Ref. Len89  it was also shown that direct CE calculations give differential
cross section angular distributions with a steeper decrease at backward angles
than the experimental data, so indicating the increased importance of higher-
order processes at large momentum transfer. Other pioneering studies
DasPRC1986  showed that, under suitable conditions, the sequential transfer of
proton and neutron pairs could even constitute the dominant process populating
the final channel of a DCE reaction, at least when not very forward angles
were considered.

The second order character and the dominance of mean–field dynamics lead to a
quite pronounced dependence of transfer SCE on incident energies, quantum
numbers and matching conditions of the involved single particle orbitals, and
the structure and multipolarities of the initial, intermediate, and final
states, see e.g. Ref. Len89 . In other words, although the reaction mechanism
follows general rules, the transfer SCE yield may depend critically on the
reacting nuclei and the kinematical conditions and a full understanding of the
competition between the different mechanisms calls for further investigation.
To complicate matters, one should take into account that also projectile and
target inelastic excitations could play a relevant role in the description of
the reaction mechanism, producing strong coupling effects among the various
reaction channels. Therefore, if one wants to extract information on the NME,
it is mandatory to disentangle from the experimental CE cross section the
contribution of each competing process.

On the other hand, the theoretical description of the complex mechanisms
involved requires several ingredients, whose reliability cannot be generally
guaranteed when applied beyond the domain in which they have been determined.
For instance, the nucleus-nucleus potential should be in principle
specifically modeled for the projectile-target system under investigation.
However, recent studies have shown that, in the energy region of our interest,
a satisfactory reproduction of the scattering cross sections is achieved for a
wide range of masses, under suitable general prescriptions SpaPRC19 ; FauPRC21
; CavFRO2021 ; Akis ; CarUNI2021 . It is nonetheless important to validate
this strategy for each system under investigation and, in this context, the
experimental measurement and analysis of the elastic and inelastic scattering
reveals thus crucial to constrain also the SCE and DCE calculations.

Because of the complexity in diagonalizing the Hamiltonian of systems
involving open shell medium and heavy nuclei, other requirements are then also
needed to perform the nuclear structure calculations. In particular, further
constraints are required for the model space and interaction combination which
is frequently adopted to deduce the single-particle transitions characterizing
one- and two-nucleon transfer processes Diana ; Jonas ; Akis . A comprehensive
analysis should take the limitations of the model spaces into account and
assess whether the global interaction parameters are suitably applicable to
the system under investigation.

A possible gateway to overcome this issue, which has revealed successfull in
some recent works CavFRO2021 ; Akis , is trying to perform measurements of
several reaction mechanisms under the same experimental conditions and to
describe them within an unified theoretical reaction scheme. In the present
work, we propose thus to perform a multi-channel experimental analysis, to be
compared with a combined structure and reaction theoretical study, with the
aim to assess the reliability of the ingredients adopted in the description of
the involved mechanisms.

As part of the intensive experimental campaigns proposed by the NUMEN project,
we focus on the 116Cd(20Ne,20F)116In SCE reaction at 306 MeV. 116Cd was chosen
as a target not to the least because that nucleus is one of the candidates for
the pursued $0\nu 2\beta$ decay sasanoPRC2012 . Keeping that goal in mind,
there are other, lighter nuclei under scrutiny as $2\beta$ decay–candidates.
However, once reactions on 116Cd are understood, then SCE – and finally DCE –
reactions on lighter target nuclei can be analyzed accordingly by the same
methods.

The goal of this paper is twofold: first of all, the work serves to show the
experimental feasibility of such a demanding reaction for spectroscopic
purposes. Secondly, on the theory side we aim at the evaluation of the various
competing transfer contributions feeding the outgoing SCE channel by a
comparison with a more comprehensive set of experimental observables beyond
what had been done so far.

The paper is organized as follows. In Sect. II we illustrate the theoretical
framework adopted, trying to underline the main differences existing among the
various reaction schemes considered. In Sect. III the experimental set-up and
the data reduction are described. In Sect. IV we discuss the results
concerning the analysis of the elastic scattering, the one-proton transfer
reaction and the sequential-nucleon transfer processes. In the last section,
conclusions are drawn and future perspectives are indicated.

##  II Theoretical framework

We consider an ion-ion SCE reaction according to the scheme

${}^{a}_{z}x+^{A}_{Z}X\to^{a}_{z\pm 1}y+^{A}_{Z\mp 1}Y$ (1)

leading from the entrance channel $\alpha=\\{x;X\\}$ to the exit channel
$\beta=\\{y;Y\\}$. In the outgoing nuclei, states of multipolarity $J^{\pi}$
are excited, including Gamow-Teller modes of unnatural parity $\pi=(-)^{J+1}$
and Fermi–modes of natural parity, $\pi=(-)^{J}$, respectively. During the
reaction, the mass partition is retained but the charge partition is changed,
either by a balanced redistribution of protons and neutrons in a sequence of
single particle transfer reactions or by direct single charge exchange.

Direct SCE is mediated by the exchange of (virtual) isovector mesons between
nucleons in projectile and target. It is a process of first order in the
projectile-target isovector interaction, described well by Distorted Wave Born
Approximation (DWBA) methods. Hence, in direct SCE the initial and final
states are connected in an especially simple manner by transition operators of
the same spin–isospin structure as known from nuclear beta–decay Lenske18 ;
LenPPNP2019 . From the point of view of nuclear structure physics, the direct
SCE reaction mechanism is the by far preferred mechanism because of the
unfiltered access to nuclear isovector spectroscopy.

However, the same exit channels may also be populated in a considerably more
complicated manner by transfer SCE. That alternative process requires a
sequence of proton–neutron exchange processes between projectile and target
nuclei, finally merging into a configuration which is part of the wave
function of the final state. Obviously, that process requires the appropriate
stepwise rearrangement of neutrons and protons. In a reaction like the one
here investigated, we encounter the neutron pick up–proton stripping sequence

${}^{a}_{z}x+^{A}_{Z}X\to^{a+1}_{z}x^{\prime}+^{A-1}_{Z}X^{\prime}\to^{a}_{z-1}y+^{A}_{Z+1}Y$
(2)

and the complementary proton stripping–neutron pickup sequence

${}^{a}_{z}x+^{A}_{Z}X\to^{a-1}_{z-1}y^{\prime}+^{A+1}_{Z+1}Y^{\prime}\to^{a}_{z-1}y+^{A}_{Z+1}Y,$
(3)

both being at least second order reactions in the respective mean–field
potentials defining the transfer interactions. From a nuclear structure point
of view, transfer SCE is probing the overlap of the final projectile and
target configurations with specific proton and neutron configurations. Since
these configurations are populated stepwise in a sequence of pick up and
stripping reactions, also the single particle spectroscopy in the
intermediately reached $A\pm 1$ and $a\mp 1$ nuclei, respectively, plays an
important role. Moreover, in order to exhaust the full single particle
strengths, the transfer steps should in principle scan over the whole spectral
distributions for each single particle spin and parity.

Also admixtures of core–excited configurations may affect significantly the
spectroscopic distributions especially in the intermediately populated
odd–even and even–odd nuclei. A complete theoretical analysis might require to
take into account also the coupling of the various channels to inelastically
excited states in projectile– and target–like nuclei. The proper handling of
these complexities in the reaction mechanism requires methods going beyond
standard DWBA.

At this stage we do not consider direct SCE processes further, mainly due to
yet to be resolved conceptual uncertainties with respect to the consistency of
the theoretical models. The widely used standard approach to the nuclear
response function of direct SCE is the Quasiparticle Random Phase
Approximation (QRPA). QRPA by definition focuses on the nuclear 1 particle-1
hole response in a quite complete manner, even over very large energy
intervals as entering e.g. as intermediate states into the theory of DCE
reactions BelPLB2020 ; LenskeUniverse . The spectroscopy of the transfer
channels, however, is described by many–body shell model methods, which
emphasize the multi–configuration dynamics in a limited number of the nuclear
valence shells. Thus, in order to exclude arbitrary effects due to possibly
incommensurable theoretical input, at this point we refrain from combining the
two distinct sectors, leaving that demanding task to later work.

It is worthwhile to notice that the strong absorption in the overlap region
enforces the localization of the interaction regions mostly to the surface or
even tail regions of the density distributions of the colliding ions.
Different direct reaction frameworks are available to tackle such processes.
Firstly, the DWBA framework has proven in many cases to describe successfully
direct reactions as single nucleon transfer reactionsSat83 ; TimoPPNP2019 ;
MontPRC2011 ; GasPRC2018 ; Akis . For a reaction $\alpha\to\beta$ the DWBA
amplitude is given as a matrix element of distorted waves
$\chi^{(\pm)}_{\alpha\beta}$ and a form factor or reaction kernel

$M_{\alpha\beta}=\langle\chi^{(-)}_{\beta}|F_{\alpha\beta}|\chi^{(+)}_{\alpha}\rangle,$
(4)

where the nuclear structure information is contained in the (non–local)
reaction kernel

$F_{\alpha\beta}(\mathbf{r}_{\alpha},\mathbf{r}_{\beta})=\langle
yY|U_{\alpha\beta}|xX\rangle$ (5)

depending on the channel coordinates $\mathbf{r}_{\alpha,\beta}$. The
transition potential $U_{\alpha\beta}$ accounts for the nuclear dynamics of
the reaction. For single nucleon transfer reactions, $U_{\alpha\beta}$ is
defined in terms of the binding potentials of the transferred nucleon plus
eventually necessary non–orthogonality terms, for details see e.g. Tamura:1974
; Sat83 . In the case of a second order reaction like transfer SCE,
$U_{\alpha\beta}$ is by itself a second order operator, accounting for the
population of the intermediate states and the corresponding distorted wave
channel propagators, see BelPLB2020 ; LenskeUniverse .

A widely used successful approach is to separate the transition potentials
into (products of) spectroscopic amplitudes, determined either by nuclear
theory or phenomenologically by fits to data, and reduced form factors
carrying the information on single particle wave functions and potentials. The
calculations discussed below are performed with realistic single particle wave
functions and potentials describing separation energies and nuclear radii.
Spectroscopic amplitudes are taken from many–body shell model calculations.

The DWBA approximation assumes, however, that the probability of the process
under study is small with respect to the elastic scattering and requires that
the final state is reached directly, i.e. without any intermediate excitations
of excited states lying in the same mass partitions. Such requirement is not
always fulfilled, especially in the case of strongly excited rotational or
vibrational modes. As a result, when coupling effects among the various
reaction channels are strong, the DWBA approach fails to reproduce the
experimental data. For such cases, the reaction should be described by solving
a set of coupled wave equations. For the direct solution of such coupled
channels (CC) problems standard numerical methods are available, but
restricted to couplings within the same mass partition. Iterative methods are
used if transfer channels are to be included. A widely used approach is the
Coupled Channels Born Approximation (CCBA). On top of the CCBA solutions the
coupling of initial and final SCE channels via sequential transfer is added
iteratively by the Coupled Reaction Channel (CRC) formalism. For details we
refer again to the literature Tamura:1974 ; Sat83 ; KeePRC2020 . The various
formal approaches are realized numerically in the Exact Finite Range (EFR)
computer code FRESCO Thompson  which treats properly the afore mentions
non–localities of transfer form factors. FRESCO was used for all reaction
calculations.

In the following sections, we first study elastic scattering. As discussed in
Ref. LenPPNP2019 , the order of magnitude of the quasi–elastic CE and transfer
cross sections is determined by the distorted waves. Their properties are
fixed by the optical potentials used to describe elastic scattering. Hence, a
proper description of elastic cross sections and total reaction cross sections
is essential for a realistic transfer yield. Therefore, below some space is
devoted to elastic scattering and the derivation of optical potentials.
Turning to the transfer SCE, we start with studying the population of several
low-lying excited states reached in the first step of the full SCE sequence.
All of them are relevant intermediate channels of the nucleon transfer
processes which coherently contribute to the full SCE reaction. Finally, the
transitions to the final SCE states are considered by investigating the
combined sequential one neutron/one proton transfer rearrangements. Further
details concerning each reaction framework, the description of the ion-ion
interaction as well as about the structure ingredients required within our
scheme, will be provided especially in section IV.

##  III Experiment and data reduction

The experiment was performed at the INFN-LNS laboratory in Catania. A 20Ne4+
beam was accelerated at 306 MeV incident energy by the K800 Superconducting
Cyclotron, fully stripped by crossing a thin carbon foil located at the
accelerator exit and then transported to the scattering chamber. The targets
were 96% isotopically enriched 116Cd foils produced by rolling at the LNS
target laboratory of thickness (1370 $\pm$ 70), (1080 $\pm$ 60) and (1330
$\pm$ 70) $\mu$g/cm2 in the case of one-proton transfer, SCE and scattering
measurements, respectively. The first two were coupled to natural C foils with
thicknesses of (990 $\pm$ 50) and (900 $\pm$ 45) $\mu$g/cm2, while the third
one to a CH2 (950 $\pm$ 45) $\mu$g/cm2 thick. Such post-strippers foils were
introduced CAVALLARO2020334 ; shima1992  to conveniently readjust the charge
state distribution of the ions emerging from the cadmium targets cavallaro2019
. However, due to the kinematics of the explored reactions, the data shown in
the following are not affected, in the analyzed regions, by the contributions
due to reactions on post-stripper materials.
A copper Faraday cup of 0.8 cm entrance diameter and 3 cm depth was used in
order to stop the beam and collect its charge. It was mounted 15 cm downstream
of the target and equipped with an electron suppressor polarized at -200 V to
ensure, even with beam currents of few enA as the ones typically available in
the described experimental conditions EPJA2018 , a charge collection accuracy
better than 10%.
The reaction ejectiles were momentum analyzed by the MAGNEX spectrometer
magnex_review , whose experimental setup was optimized to select, in separated
runs, specific reaction channel:

  * •

For the 116Cd(20Ne,20Ne)116Cd elastic and inelastic scattering measurement,
the spectrometer optical axis was set at three different angular settings,
$\theta_{opt}$ = 8°, 13° and 20° in the laboratory frame, spanning the 3°
$<\theta_{lab}<$ 26° angular range. All the runs were performed adopting the
full angular acceptance ($\Omega\sim$ 50 msr).

  * •

For the 116Cd(20Ne,19F)117In one-proton transfer measurement the spectrometer
optical axis was placed at $\theta_{opt}$ = 8°, corresponding to the
3°$<\theta_{lab}<$ 14° angular range. The MAGNEX angular acceptance was
slightly decreased, determining a solid angle of $\sim$ 45 msr.

  * •

For the 116Cd(20Ne,20F)116In single charge exchange the spectrometer optical
axis was oriented at $\theta_{opt}$= 9°, thus exploring the angular range 4°
$<\theta_{lab}<$ 15°. In this run the vertical angular acceptance was
considerably reduced in order to limit the overall event-rate at the focal
plane detector, decreasing the covered solid angle to $\sim$ 1.3 msr.

The ejectile identification as well as the data reduction techniques are
described in details in Refs. cappuzzello2010 ; CAVALLARO2020334 . They
require the accurate measurements of the horizontal and vertical positions and
angles provided by the MAGNEX focal plane detector cavallaro2012 ;
TORRESI2021164918  and the knowledge of the high-order transport map of the
particles throughout the spectrometer magnex_review .
The differential cross sections were extracted for all the measured reaction
channels according to the technique described in Ref. magnex_review . The
error bars reported in the corresponding energy and angular distributions
account for the statistical contribution, the uncertainty due to the solid
angle evaluation and the fit procedure when performed. A systematic
uncertainty of $\sim$10%, not shown in the plots, is common to all the
experimental data, originating from target thickness measurement and Faraday
cup charge collection uncertainties.

##  IV Results

###  IV.1 Scattering channel

A crucial issue for a quantitative understanding of heavy ion reactions is the
proper treatment of projectile-target interaction, thus motivating elastic and
inelastic scattering studies.

Figure 1: Excitation energy spectra of the scattering channel in two different
angular regions. Panel (a): Energy spectrum in the angular range 8.7°
$<\theta_{lab}<$ 9.0°. The magenta curve represents the energetically
unresolved ground and first excited states of the target. The cyan and blue
Gaussian functions are introduced to reproduce the small bump associated to
the unresolved excitation of the first low-lying states of 116Cd and 20Ne (see
the text). The red curve reproduces the continuos spectrum in the excitation
energy region above $\sim$2.5 MeV. The sum of all the mentioned contributions
is shown as the green line. Panel (b): Energy spectrum in the angular range
16.0° $<\theta_{lab}<$ 16.8°. The ground and the first 116Cd excited states
are fitted separately as the grey and yellow curves. The brown curve
represents the additional Gaussian functions peaked around 2.2 MeV, not
present in the forward angle spectrum of panel (a). For all the other
contributions as well as for the total sum function the same color code as in
panel (a) is adopted.

The excitation energy spectrum of the (20Ne, 20Ne) scattering channel is shown
in Fig. 1, being $E_{x}$ = $Q_{0}$ \- $Q$ with $Q_{0}$ the ground-state-to-
ground state Q-value. The limited experimental energy resolution ($\delta
E\sim$ 1.1 MeV full width half maximum, mainly due to straggling effects in
the used target) and the available statistics do not allow to discriminate the
elastic transition and the first excited state of the system, the
($2^{+}_{1}$) state of the 116Cd expected at 0.513 MeV, except for a
restricted angular region (15° $\lesssim\theta_{lab}\lesssim$ 18°). In Fig. 1
(a), the sum of the contributions of such states is represented by the
dominant Gaussian function, whereas in Fig. 1(b) the two transitions are
considered separately. Additional Gaussian functions peaked around 1.24 and
1.64 MeV, introduced to better reproduce the bump visible in such energy
region, are shown in Figs. 1(a) and (b). These distributions are expected to
correspond to the sum of the cadmium two-phonon quadrupole states ($2^{+}$,
$4^{+}$, $0^{+}$) at 1.213, 1.219, 1.282 MeV, respectively, and the (2+) 20Ne
and 116Cd states at 1.634 and 1.642 MeV. However, they can also collect the
contributions from other possible state like the 116Cd (0+) predicted at 1.382
MeV and, partially, from the cadmium ones foreseen around 2 MeV. The
excitation energy spectrum of Fig. 1(b), corresponding to the backward angular
range, presents a further Gaussian distribution introduced to better model the
counting shape observed around 2.2 MeV, which is attributable to the latest
quoted excited states of the target as well as to the simultaneous excitation
of the first target and projectile transitions. Due to the low yields, the
high number of the involved possible states and the limited energy resolution
available which does not allow to distinguish among the different possible
transitions, the distributions lying beyond 1 MeV will not be further
considered in the following. Finally, the experimental spectrum beyond the
commented structures resulting in a continuous and suppressed shape is fitted
by a smooth curve. The experimental differential cross section of the quasi-
elastic transition, which includes the ground state (g.s.) and the
(2${}^{+}_{1}$) state of 116Cd at 0.513 MeV, is shown in Fig. 2 (a) and
compared to the results of the theoretical calculations, The experimental
elastic and (2${}^{+}_{1}$) 116Cd inelastic differential cross sections, as
extracted in the restricted angular range indicated above, with the
corresponding calculations, are reported separately in Fig. 2 (b). Figures 2
(a) and (b) also report the scale of transferred linear momentum $q$. As seen
from the upper abscissa of Fig. 2 (a), the data cover a large range of linear
momentum transfers, extending from $q\sim 0.6$ fm-1 to $q\sim 6$ fm-1. Thus,
heavy ion reactions of this kind are probing nuclear properties in much
detail.

In the present work, the ion-ion interaction is described by a complex optical
potential $U$, whose radial dependence is microscopically derived by doubly
folding the one-body g.s. projectile and target nuclear densities, which are
parameterized by two-parameters Fermi-Dirac distributions, with an effective
NN interaction. In the numerical calculations of this work, two different
double folding optical potentials will be tested.

On the one hand, we consider the São Paulo Potential (SPP) PerPLB09 , which
has already demonstrated to be successful in describing the elastic scattering
and peripheral reaction channels for several heavy-ion reactions involving
light PerPLB2012 ; FonPRC2019 , medium SpaPRC19 ; PerNPA2009 ; ZagPRC2018  and
heavy AlvNPA2003  mass targets in a wide energy region.

The general SPP parameterization, being based on theoretical calculations with
the Dirac-Hartree-Bogoliubov model and also on experimental results for charge
distributions, has no adjustable parameter PerPLB09 . However, small
deviations around the average values might be expected due to the effects of
the structure of the nuclei.

On the other hand, another optical potential, which was used in some other
works CapLenske04 ; Lenske18 ; LenskeUniverse  and labeled as DFOL, is
employed. The latter makes use of the complex NN T-matrix derived by Franey
and Love FraneyLove , including both the isoscalar and isovector components
and it is extrapolated down to the present energy region. Differently from the
SPP case, where the imaginary part is obtained from a simple scaling of the
real one, for the DFOL optical potential the imaginary potential is directly
available from the folding procedure.

As briefly addressed in a recent paper LenskeUniverse , we are fully aware of
the changes of the T-matrix to be expected in the low-energy region. Results
of a recent investigation show that below $T_{lab}=$ 50 MeV the strength of
the real parts of the (anti-symmetrized) spin-scalar, isoscalar and isovector
components of the NN T-matrix remains almost constant, while the imaginary
parts increase on a moderate level. A comparison of the total neutron-proton
cross section to data and to the one obtained with the CD Bonn potential in
the full Lippmann-Schwinger formalism shows that the strong increase of the
cross section towards low energies is mainly driven by kinematical effects.
The NN T-matrix parameters, as obtained for the central interactions at
$T_{lab}=15$ MeV in the relevant channels for the ion-ion potentials, i.e. the
spin - scalar ($S=0$) and the isospin $T=0,1$ channels, are given in Table 1.

Table 1:  Nucleon-nucleon T-matrix interaction strengths at $T_{lab}=15$ MeV in the spin - scalar ($S=0$) and the isospin $T=0,1$ channels. The ranges are expressed in fm, all other quantities are in MeV. Range | SE | TE | SO | TO
---|---|---|---|---
Real
0.25 | 6048.76 | 7972.38 | -51218.10 | 2091.19
0.40 | -1754.49 | -2235.67 | 5541.75 | -641.32
1.40 | -10.50 | -10.50 | 31.50 | 3.50
Imaginary
0.25 | -1328.78 | 20805.12 | 31381.92 | 16175.65
0.40 | -382.85 | -6968.75 | -4100.90 | -2485.94

Although the folding procedure is easily extended to cover deformed matter
distributions of projectile and target, in the present case by numerical
reasons we use spherical density profiles in the calculations of the DFOL and
SPP optical potentials. In a more extended approach, reaction observables need
to be averaged anyway over the orientations of the deformed ions. To a good
approximation the resulting net effects can be simulated by slightly changing
the geometrical parameters of the density distributions SpaPRC19 . In
particular, since the g.s. quadrupole moment of projectile and target are not
negligible Pri2016 ; Ram2001 , one may infer that, especially in the case of
20Ne, the nucleus is significantly deformed, as also confirmed by the large
experimental values of the quadrupole deformations ($\beta_{2}^{C}=0.721$ for
20Ne, $\beta_{2}^{C}=0.194$ for 116Cd). An effective way to take this aspect
into account in building the optical potential is thus a change of the density
profiles, as also described in Ref. SpaPRC19 , with the requirement to keep
constant the volume integral of the nuclear densities to fix the number of
nucleons. In Ref. SpaPRC19 , where a different reaction involving the same
projectile nucleus was analyzed, an effective geometrical modification in the
nuclear densities was introduced, increasing by 5% the radius of the density
profiles, while renormalizing the central density parameter. For sake of
consistency, the same prescription is here adopted for the projectile 20Ne,
while a smaller modification (3%) is used for the target, to achieve the best
possible agreement between data and calculations. The root mean square radius
and volume integral per nucleon, as obtained for the real and the imaginary
parts of both potentials adopted here, are listed in Tab. 2. One observes
that, even though the root mean square radii of the real parts practically
coincide, a significant difference exists among the volume integrals of the
two optical potentials, although both values are still compatible with the
typical ones expected from the systematics SatcPR79 .

It is worthwhile to remind that, for the imaginary part, a scaling factor has
to be then introduced, depending on the reaction framework adopted, to account
for missing couplings to states not explicitly considered here.

Although it is not obvious that the same scaling factors should be assumed for
different nuclear systems, the most reasonable choice would be to adopt a
consistent approach with the analysis of some previous works CarPRC17 ;
PaePRC17 ; ErmPRC17 ; ErmPRC16 . For both DFOL and SPP optical potentials,
such a factor is then set to 0.78 in case of optical model (OM) calculations
and 0.6 whenever CC calculations are performed.

Table 2:  Real $J_{V}$ and imaginary $J_{W}$ part of the volume integral per nucleon (in MeV fm3) and root mean square radii (in fm) for the real $\sqrt{\langle R^{2}\rangle_{V}}$ and imaginary $\sqrt{\langle R^{2}\rangle_{W}}$ part of DFOL and SPP potentials adopted. | $J_{V}$ | $J_{W}$ | $\sqrt{\langle R^{2}\rangle_{V}}$ | $\sqrt{\langle R^{2}\rangle_{W}}$
---|---|---|---|---
DFOL | -437.56 | -408.94 | 5.884 | 5.770
SPP | -331.68 | -331.68 | 5.883 | 5.883

A comparison with the measured quasi-elastic scattering data in terms of the
corresponding Rutherford cross section $\sigma_{Ruth}$ is shown in Fig. 2 (a).
Actually, as already explained in the previous section, the experimental
values for the quasi-elastic channel include also the inelastic scattering to
the (2${}_{1}^{+}$) target state at 513 keV. Therefore, the OM (for the
elastic) + DWBA (for the inelastic) calculations plotted in Fig. 2 (a) take
into account the incoherent sum of the two different contributions, namely the
genuine elastic cross section and the DWBA cross section of inelastic
scattering to the first excited state of 116Cd.

Figure 2: Panel (a): Experimental angular distribution of differential quasi-
elastic scattering cross section in the $\sigma/\sigma_{\textup{Ruth}}$
representation. Theoretical results obtained for the DFOL (red lines) and SPP
(blue lines) potentials, within two different reaction schemes, namely OM
(elastic) + DWBA (inelastic), in dashed lines, and CC, in full lines. Both
experimental values and theoretical calculations also include the contribution
of the inelastic scattering to the (2${}_{1}^{+}$) target state. The dotted
and dashed-dotted lines show the results obtained with SPP by alternatively
switching on the contributions of the (2${}_{1}^{+}$) target and projectile
states, respectively. Panel (b): Experimental angular distribution of
differential cross section for elastic and (2${}_{1}^{+}$) 116Cd inelastic
excitation, as separately considered and compared with OM/DWBA or CC
calculations, with SPP potential. Panel (c): Radial dependence of the real V
(red full line) and imaginary W (red dashed line) part for DFOL and the real
part V for SPP (blue full line). The inset shows also a zoom on the peripheral
region.

Despite the differences existing in the internal region of the two optical
potentials, there is no significant difference between the cross sections
obtained by the two calculations. This fact confirms that the theoretical
description of elastic scattering is not strongly dependent on the choice of
these optical potentials since the strong absorption confines the reaction
source on the surface of the colliding systems, where the two potentials have
a similar behavior.

Heavy ion reactions and therefore elastic scattering and peripheral inelastic
reactions are indeed mainly sensitive to the nuclear surface regions of the
interacting nuclei and, as a result, the incoming flux is mostly absorbed into
a multitude of reaction channels, as demonstrated also by the large total
reaction cross sections.

One observes that theoretical OM+DWBA calculations fail to reproduce the slope
exhibited by the experimental data beyond the grazing angle, whatever optical
potentials is employed, including the two considered here. Since the behavior
around the grazing angle is properly reproduced, the present case is similar
to the one discussed in SpaPRC19 , where the observed discrepancies with
respect to the experimental data were attributed to the coupling of the
elastic channel with possible inelastic excitations to the 2${}_{1}^{+}$
states of both projectile and target. In Fig. 2 (a) CC calculations performed
with both DFOL and SPP optical potentials are therefore also included.

Both one-step DWBA and CC calculations are performed within the rotational
model, following the same prescription adopted in some recent works CarUNI2021
; CavFRO2021 , despite the low-lying states in 116Cd nuclei might appear to be
of vibrational nature. Reduced transition probabilities B(E2;0${}^{+}\to
2^{+}$) = 0.0333 $e^{2}b^{2}$ for 20Ne and B(E2;0${}^{+}\to 2^{+}$)= 0.58
$e^{2}b^{2}$ for 116Cd are taken from Pri2016 ; Pri2017 ; Kib2002  and used to
describe the strength of Coulomb deformation of both projectile and target.

Nuclear deformations are described in terms of first-order derivative of the
OM potential U(r)

$V(r)=-\frac{\delta_{2}}{\sqrt{4\pi}}\frac{dU(r)}{dr}$ (6)

where the strength of the deformation is embedded in the deformation length
$\delta_{2}$ Sat83 :

$\delta_{2}=\beta_{2}R=\frac{4\pi}{3Z}\frac{\sqrt{B(E2;0^{+}\to
2^{+})}}{R_{V}}$ (7)

In Eq. (7), $\beta_{2}$ is the deformation parameter characterizing the
transition of the given nucleus of charge Z, $R=1.2A^{1/3}$ is the radius and
$R_{V}$ is the root mean square radius of the real part of the adopted optical
potential (see Tab. 2). The deformation lenghts obtained following such a
prescription are $\delta_{2}=1.299$ fm and $\delta_{2}=1.130$ fm, for
projectile and target, respectively. Exploratory calculations changing the
method for the determination of $R_{V}$ give however similar results.
Moreover, the same radial form factors are assumed also for the imaginary
coupling potentials.

One observes that the CC calculations well reproduce the slope exhibited by
the quasi-elastic experimental data also at larger angles. A better agreement
is thus observed with respect to OM+DWBA, especially in the region of larger
momentum transfer, confirming the relevant role played by these rotational
(collective) 2${}_{1}^{+}$ states, whose couplings with the g.s. have to be
explicitly included. By alternatively switching off the contributions of the
various inelastic excitations, we found moreover a leading role played by the
projectile $2_{1}^{+}$ state, which is strongly coupled to the elastic channel
because of its large deformation parameter. The best agreement is however
achieved only when the combined effect with the target inelastic excitation is
included.

The overall outcome is moreover not significantly influenced by the small
changes introduced in the target and projectile nuclear density radii. Indeed,
we checked that, when ignoring such modifications, the total cross section in
the elastic channel differ at most of 3% with respect to the results shown
before, both in the OM and in the CC approach, in the case of SPP potential.

A further validation of the ingredients adopted in the description of the
scattering channel analysis is reported in Fig. 2 (b). Indeed, a nice
agreement is obtained between experimental data and theoretical predictions
when disentangling the elastic and the first inelastic transitions in the
angular range considered, especially when the CC calulations are performed. In
Fig. 2 (b), only the SPP potential is considered, although an analogous result
is obtained in the case of DFOL potential. Despite the limitedness of the
angular range considered, such a result strengthens all the conclusions drawn
from the quasi-elastic analysis. In particular, the matching between the
experimental and theoretical elastic distribution confirms the reliability of
the prescriptions adopted for the optical potentials, whereas the first
inelastic transition provides another proof of the model assumed to describe
the coupling with the inelastic excitations considered. In the following
sections, we will explicitly take into account the effect of these inelastic
excitations, looking also at their influence on the transfer channels
investigated within our study.

###  IV.2 One-proton transfer 116Cd(20Ne,19F)117In reaction

Figure 3: Excitation energy spectrum of the one-proton transfer channel in the
8° $<\theta_{lab}<$ 14° angular range. In the inset, the spectrum up to $\sim$
4.5 MeV for the interval 8° $<\theta_{lab}<$ 9° is reported. The red hatched
area corresponds to the analyzed region as described in the text.

The excitation energy spectrum measured for the (20Ne,19F) one-proton transfer
reaction is presented in Fig. 3. In agreement with Brink’s transfer matching
conditions BRINK197237 , the spectrum is peaked around $E_{x}^{opt}\sim$5 MeV.
In order to extract the differential cross section angular distribution in the
low excitation energy region, the spectra have been analyzed in one-degree
angular step. The inset of Fig. 3 shows the experimental energy distribution
in the 8° $<\theta_{lab}<$ 9° interval up to $\sim$ 4.5 MeV. Countings at low
excitation energy are smoothly distributed due to the 117In and 19F high
nuclear level densities without exhibiting any clear structure in such region.
In fact, even if the ground-state-to-ground-state transition as well as the
first excited states cannot be distinguished due to the limited energy
resolution, none of them dominates over the others thus resulting in a
continuous yield trend. Beyond 1 MeV, instead, some bumps are visible which
are expected to be the unresolved sum of several different transitions.
Examining the inset of Fig. 3, however, a local minimum is observed around 2
MeV. Then the differential cross section angular distribution in the [-1, 2]
MeV excitation energy range was extracted and it is shown in Fig. 4. We
limited to the lowest excitation energy region, where the reproduction of the
energy spectrum in terms of single-particle configurations adopted in the
following calculations is expected to be more reliable. It is interesting to
note that the angular distribution shape results clearly peaked close to the
grazing angle ($\theta_{gr}\sim$ 15°) region. This “bell-shaped” behaviour is
the typical feature expected in absorptive reactions involving heavy nuclei
PaePRC17 ; KahANP1977 ; Diana .

Figure 4: Experimental angular distribution of the differential cross section
for the one-proton transfer reaction 116Cd(20Ne,19F)117In, as obtained in the
[-1, 2] MeV excitation energy range. Theoretical results refer to three
different reaction schemes: DWBA, CCBA and CRC. Both DFOL and SPP optical
potentials are employed.

Once the initial state interaction has been properly addressed, it would be
important to check whether the theoretical framework adopted in our work
allows to get a reliable reproduction also of the transfer channels. Direct
reaction calculations for the transfer mechanism have been performed by
employing EFR transfer coupling within, at first, the DWBA approximation.
Following the analysis performed in some recent works Diana , it is
constructive to look also at the effect of the coupling with the inelastic
excitations of low-lying states in the incoming partition, within CCBA or CRC
formalism.

To perform microscopic transfer reaction calculations, one needs to also know
the spectroscopic amplitudes associated to the wave function single-particle
overlaps. Such amplitudes are derived through large scale shell model
calculations performed by using the NuShellX code NuShellX .

For projectile-like isotopes, the effective Zuker-Buck-McGrory ($zbm$)
interaction zbm  is considered, within a model space which assumes 12C as a
closed core and valence protons and neutrons in the $1p_{1/2}$, $1d_{5/2}$ and
$2s_{1/2}$ orbits. We also checked that similar results would be obtained when
employing the $psdmod$ psdmod  interaction, which was also adopted in some
recent works CarPRC17 ; cardozo .

For the target-like isotopes involved here, discrepancies are generally
observed in reproducing the single particle spectrum. Nevertheless, the
recently introduced effective interaction $88Sr45$ coraggio2016 , allows for a
quite satisfying determination of the excitation energy of the low-lying
states in this mass region Diana . In that case, 88Sr is assumed as a closed
core within a model space which considers valence protons in the $2d_{5/2}$,
$2p_{1/2}$, $1g_{9/2}$ and $1g_{7/2}$ orbits and neutrons in the $2d_{5/2}$,
$3s_{1/2}$, $1g_{7/2}$ and $2d_{3/2}$ orbits.

The couplings and level schemes obtained considering $zbm$ and $88Sr45$
interactions for projectile-like and target-like nuclei, respectively, are
sketched in Fig. 5, while the spectroscopic amplitudes of the relevant
overlaps involved are listed in Table 3.

The single-particle wave functions representing the overlaps are generated by
Woods-Saxon binding potentials, whose depths are varied to reproduce the
experimental one-proton separation energies. The same procedure is adopted in
the following for the one-neutron transfer, where the depths are varied in
order to reproduce the experimental one-neutron separation energies. The
calculations have been performed by setting the reduced radii and diffuseness
parameters of the binding potentials to 1.23 fm and 0.65 fm for 116Cd + p and
to 1.26 fm and 0.7 fm for 19F + p. Such values are compatible with the
systematics usually adopted and allow for the reproduction of the experimental
data Diana .

The angular distribution of the differential cross section for the
116Cd(20Ne,19F)117In reaction is shown in Fig. 4, as obtained within three
different theoretical approaches: DWBA, CCBA and CRC. In all cases, the post
representation is adopted and full complex remnant terms are considered.
Moreover, the same prescriptions used in OM calculations are employed to
determine the corresponding core-core potential. Both DFOL and SPP optical
potentials are employed and the corresponding results are compared with the
experimental data discussed above.

Regardless of the reaction framework considered, one may observe that, while
the DFOL potential tends to overestimate the contribution at smaller angles,
the SPP potential returns a satisfactory agreement of the typical bell-shape
exhibited by the experimental data around the grazing angle. As shown by the
figure, the shape of the angular distribution in the angular range covered by
the experimental data is significantly dependent on the details of the optical
potentials, especially in the surface region. From a qualitative point of
view, we verified that the same shape would be preserved indeed even when
performing the calculations by assuming unitary values for the spectroscopic
amplitudes for all the states considered in our coupling scheme of Fig. 5 or
when assuming a different effective interaction in our shell model
calculations. So, the result of Fig. 4 confirms the reliability of the
potentials adopted to describe the projectile-target interaction, especially
for the SPP parameterization. Nevertheless, it provides some hints to
distinguish among different options for the optical potentials, which equally
well reproduce the experimental results in the elastic channel, supporting the
preferable choice of the SPP potential.

Moreover, not only the shape but also the reproduction of the order of
magnitude of the differential cross section is observed in Fig. 4. Such
result, reached without the need of any arbitrary scaling factor or any
significant modification of the structure parameters of the binding
potentials, demonstrates the trustworthiness of the model space and of the
effective interaction employed in our shell model calculations, despite some
existing and hard to quantify uncertainties.

Figure 5: Coupling schemes of the projectile and target overlaps used in the calculations of the 116Cd(20Ne,19F)117In one-proton transfer reaction, within the considered excitation energy range. Red dashed arrows concern the coupling considered within the DWBA approximation, while the blue full arrows indicate the additional couplings introduced within the CCBA reaction scheme. In the CRC framework, not shown in the figure, all arrows have to be considered bidirectional (see text). Table 3: One-proton spectroscopic amplitudes (S.A.) adopted in DWBA and CCBA calculation of the 116Cd(20Ne,19F)117In one-proton transfer reaction. These amplitudes are obtained through shell model calculations by employing $zbm$ interaction for projectile-like nuclei and $88Sr45$ interaction for target-like ones. The column nlj indicates the principal quantum number, the orbital and the total angular momentum of the single valence proton, respectively, through the usual spectroscopic notation. Initial state | Final state | nlj | S.A.
---|---|---|---
20Ne${}_{\textup{g.s.}}$ (0+) |  19F${}_{\textup{g.s.}}$ (1/2+) | (2s1/2) | -0.8584
19F0.110 (1/2-) | (1p1/2) | -1.2702
19F0.197 (5/2+) | (1d5/2) | 1.1741
20Ne1.634 (2+) |  19F${}_{\textup{g.s.}}$ (1/2+) | (1d5/2) | 0.6712
19F0.197 (5/2+) | (2s1/2) | -0.6922
(1d5/2) | -0.6416
116Cd${}_{\textup{g.s.}}$ (0+) |  117In${}_{\textup{g.s.}}$ (9/2+) | (1g9/2) | -0.4066
117In0.315 (1/2-) | (2p1/2) | -0.3847
117In0.748 (7/2+) | (1g7/2) | 0.0098
117In0.881 (5/2+) | (2d5/2) | -0.1593
117In1.052 (5/2+) | (2d5/2) | -0.2670
117In1.360 (5/2+) | (2d5/2) | -0.7192
116Cd0.513 (2+) | 117In${}_{\textup{g.s.}}$ (9/2+) | (1g9/2) | -0.6092
(1g7/2) | 0.0073
(2d5/2) | 0.1182
117In0.589 (3/2-) | (2p3/2) | -0.3319
117In0.660 (3/2+) | (1g7/2) | -0.0157
(2d5/2) | -0.1502
117In0.748 (7/2+) | (1g9/2) | -0.2232
(1g7/2) | 0.0170
(2d5/2) | 0.0674
117In0.749 (1/2+) | (2d5/2) | -0.2796
117In0.881 (5/2+) | (1g9/2) | 0.4308
(1g7/2) | 0.0201
(2d5/2) | -0.1781
117In1.028 (5/2-) | (2p1/2) | 0.3306
117In1.052 (5/2+) | (2d5/2) | 0.5693
(1g9/2) | -0.1326
117In1.360 (5/2+) | (1g9/2) | -0.1813
117In1.366 (9/2+) | (2d5/2) | -0.4463
117In1.376 (3/2+) | (2d5/2) | -0.2946
117In1.713 (3/2+) | (2d5/2) | -0.1521
117In1.785 (3/2+) | (2d5/2) | 0.1214
117In1.892 (1/2+) | (2d5/2) | -0.6646
(1g7/2) | 0.1382

In Fig. 4, the effect of the coupling with inelastic states in the entrance
partition can be inferred from the comparison between the different reaction
schemes adopted. Regardless of the optical potential employed, some general
features may be discussed. Indeed, similarly to what was observed in our
recent work Diana , the inclusion of couplings with low-lying collective
inelastic excitations moderately changes the shape of the angular
distribution. In particular, it slightly modifies the oscillating behavior of
the transfer cross section at small angles, owing to the different orbital
angular momentum transferred, when passing through the intermediate states
involved in the inelastic excitations. Moreover, a tiny shift is introduced on
the bell-shaped peak, due to the larger spatial distribution of the collective
wave functions adopted in the coupled channel calculations. As would be
expected, a larger cross section above the grazing angle is observed in CCBA
compared to DWBA. Such behavior is attributed to couplings with the target
2${}_{1}^{+}$ state, as one observes selectively switching off the
contributions of the various inelastic excitations. As far as the coupling
with the projectile $2_{1}^{+}$ state is concerned, its inclusion does not
imply any change in the shape of the differential cross section, except only
for a small global reduction, which suggests a destructive interference with
the 0+ state contribution. In Fig. 4, CRC results are also displayed. Whatever
the optical potential considered, the curve related to CRC calculation is
always practically superimposed on the one corresponding to the CCBA results,
in the explored angular region. We remind here that, in CRC calculations, the
couplings related to the single-nucleon transfer transitions are implemented
iteratively, till the absolute difference between successive S-matrix elements
becomes less than 0.01%, so even including the back-coupling. Therefore, our
result demonstrates that accounting for the effect of the latter is safely
negligible and this holds not only for the transfer but also for both elastic
and inelastic channels. Indeed, in all cases the difference between CRC and
CCBA calculations on the total cross section is smaller than 0.2%.

On the other hand, the small differences found between DWBA and CCBA
calculations demonstrate a mild influence of the high-order couplings between
transfer and scattering channels at least in the present angular window. The
role of the inelastic excitations might be thus disregarded at a first level
of accuracy or effectively embedded in the dynamic polarization potential.
This is especially true when looking at other quantities, such as the
(angular-integrated) total cross section, which is obviously less sensitive to
the details of the diffraction pattern.

To summarize, Figs. 2 and 4 show that with a unified choice of the optical
potential and with a rather simple coupling scheme, we are able to reproduce
at once the angular distribution of both the quasi-elastic and one-proton
transfer channel. Of course, one might in principle consider more
sophisticated reaction frameworks or approximations and more complex coupling
schemes. Nevertheless, since this simplified approach already demonstrates an
acceptable agreement with the experimental data, resorting to more complex
reaction schemes reveals unnecessary, as long as one does not want to describe
more channels. Moreover, it allows to test the basic ingredients of our
calculations, validating our choice of the optical potential adopted, as well
as the model space and interaction combination considered in the shell model
calculations.

###  IV.3 Contribution of sequential neutron-proton transfer on the SCE

The (20Ne,20F) SCE reaction absolute cross section energy spectrum is shown in
Fig. 6. In the inset, a zoom for the excitation energy region up to 1.5 MeV is
reported. The continuum shape is a combined effect of the high level density
of the involved ejectile and residual nuclei and the limited energy
resolution. Such conditions, together with the poor statistics, do not allow
further experimental analysis. Therefore, cross section values were extracted
only by integrating on selected energy ranges, thus including the
contributions from transitions to several states of the final partition. In
order to compare the experimental results with the theoretical ones a small
excitation energy interval has been considered, namely the one between -0.35
and 0.35 MeV. The list of the expected states in the selected energy interval
for the ejectile and the residual nuclei and the corresponding experimental
integrated cross section value are listed in Tab. 4.

Figure 6: Excitation energy cross section spectrum of the single charge
exchange channel in the 7.5° $<\theta_{lab}<$ 12.5° angular range. In the
inset, the spectrum up to 1.5 MeV in the full angular interval 4.5°
$<\theta_{lab}<$ 14.5° is reported. The dashed lines and the red hatched area
highlight the selected integration energy region discussed in the text.

In the present work, we focus on the competing multi-nucleon transfer
mechanisms which end up in the same exit channel as the one reached in the
direct SCE process. Hence, in our approach, all excited states in either the
projectile- or the target-like outgoing nuclei is thus populated only through
the proper rearrangement of neutrons and protons.

A schematic representation of the two different paths considered is sketched
in Fig. 7: the same final states may then be reached by performing either a
one-neutron pick-up reaction followed by a one-proton stripping (hereafter
labeled as path1) or viceversa (hereafter labeled as path2).

Figure 7: Schematic representation of the sequential neutron-proton transfer
processes, discussed before in Section II. The one-neutron pick-up + one-
proton stripping (path1, Eq. (2)) and the one-proton stripping + one-neutron
pick-up (path2, Eq. (3)) routes are depicted on the left and right,
respectively. For completeness, also the direct SCE channel is indicated.

In the following, the two different reaction paths will be first considered
separately and thereafter they will be included within the same scheme, in
order to evaluate the interference between the two processes. The prior-post
representation is used and full complex remnant terms are considered.

Before embarking in the full calculations, we also checked the equivalence
between the different (prior/post) representations adopted, once the non-
orthogonality terms are included ThomBCS2013 . We compare thus the prior-post
calculation for which non-orthogonality terms identically vanish Sat83 , with
two other possible combinations, where non-orthogonality corrections are
included. As a result, a very nice agreement is generally observed. Such an
agreement, even though is never worse than 3%, slightly deteriorates for
larger angles (beyond 15 degrees), where however the differential cross
section has a steep decrease and its contribution to the total integrated
value practically vanishes. Such a result confirms in turn the good numerical
convergence of our calculations. In the second transfer steps of path1 and
path2, contributions of spectroscopic strength less than 0.1 are neglected.

#### Path1.

Let us consider the path1, whose couplings and level schemes of the involved
nuclei are sketched in Fig. 8. The values of the spectroscopic amplitudes
extracted by the shell model calculation, when employing $zbm$ and $88Sr45$
interaction, are listed in Table 5 for projectile-like overlaps and in Table 6
and 7 for target-like overlaps involved in the first and in the second
transfer step, respectively.

20Neg.s. (0+)1.634 (2+)21Neg.s. (3/2+)0.351 (5/2+)1.746 (7/2+)2.794
(1/2+)3.736 (5/2+)20Fg.s. (2+)116Cdg.s. (0+)0.513 (2+)115Cdg.s. (1/2+)0.229
(3/2+)0.361 (5/2+)0.389 (7/2+)0.389 (9/2+)0.473 (3/2+)0.473 (5/2+)0.507
(3/2+)0.507 (5/2+)0.649 (1/2+)1.062 (7/2+)116Ing.s. (1+)0.128 (5+)0.223
(4+)0.273 (2+)0.313 (4+) Figure 8: Coupling schemes of the projectile and
target overlaps used in the sequential two-nucleon transfer reaction
calculations, along path 1. The $zbm$ and $88Sr45$ interactions are employed
in shell model calculations, for projectile-like and target-like nuclei,
respectively. Red arrows concern the coupling considered within the DWBA
approximation, while the blue arrows indicate the additional couplings
introduced within the CCBA reaction scheme.

As one may observe in Fig. 8, there is a large number of states in the
intermediate partition which may be coupled, especially when target-like
excitation levels are considered. Moreover, the coupling between two different
states may involve even more overlaps, as indicated in Table 6 and 7. In
principle, one should consider in fact all the possible states of the
intermediate partition and, therefore, extend as much as possible the number
of couplings included. However, at least when taking into account transitions
which involve low-lying states of projectile and target in the initial
partition as in this case, the coupling with states having larger excitation
energy should be generally reduced.

Moreover, in view of the comparison with the experimental data, which may
isolate the contribution to the low-lying states also of the final partition,
a further reduction of the weight of high-energy contributions to the total
cross section is expected. Nevertheless, these states may play a role within
CCBA formalism, whereas the coupling with the $2_{1}^{+}$ states of the
initial partition is explicitly taken into account. However, in that case,
their contribution is expected to be suppressed by the larger number of steps
involved in the whole process. In any case, complete CCBA calculations have
been also performed to get a quantitative estimation of these contributions.

#### Path2.

Analogously to the analysis of the previous paragraph, a two-step transfer
reaction calculation through the intermediate partition (19F + 117In) has been
performed, at first within the DWBA approximation. Couplings and level schemes
of the involved nuclei are sketched in Fig. 9. The values of the spectroscopic
amplitudes extracted by the shell model calculation, when employing $zbm$ and
$88Sr45$ interactions, for projectile-like and target-like overlaps, are
listed in Table 8 and 9, respectively.

20Neg.s. (0+)1.634 (2+)19Fg.s. (1/2+)0.197 (5/2+)1.346 (5/2-)1.554 (3/2+)2.780
(9/2+)4.378 (7/2+)20Fg.s. (2+)116Cdg.s. (0+)0.513 (2+)117Ing.s. (9/2+)0.748
(7/2+)0.881 (5/2+)1.030 (5/2+)1.052 (5/2+)1.365 (7/2+)1.365 (9/2+)116Ing.s.
(1+)0.128 (5+)0.223 (4+)0.273 (2+)0.313 (4+) Figure 9: Coupling schemes of the
projectile and target overlaps used in the sequential two-nucleon transfer
reaction calculations, along path 2. The $zbm$ and $88Sr45$ interactions are
employed in shell model calculations, for projectile-like and target-like
nuclei, respectively. Red arrows concern the coupling considered within the
DWBA approximation, while the blue arrows indicate the additional couplings
introduced within the CCBA reaction scheme.

#### Coherent sum of two paths.

With the aim to properly evaluate the role of the two-step transfer mechanisms
in determining the total cross section of the 116Cd(20Ne,20F)116In SCE
reaction, the contribution of the two different paths illustrated above should
be included in a coherent way within the same calculation. The resulting
angular behavior of the differential angular distribution cross section, as
obtained within the DWBA approximation along path1 or path2 or when including
the coherent sum of the two paths, is shown in Fig. 10, for the different
target-like states lying within the excitation energy range considered. For
the projectile-like states, only the g.s. is taken into account. Only the SPP
parameterization is moreover shown for the optical potential, for sake of
simplicity.

Figure 10: Angular distribution of the differential cross section for the
different final states of the 116Cd(20Ne,20F${}_{\textup{g.s.}}$ (2+))116In
SCE reaction, lying within the considered excitation energy range, as obtained
when considering only the two-step transfer mechanisms. The two-step DWBA
approximation is adopted and the SPP optical potential is employed. The two
paths are considered separetely and with their coherent sum.

As a general feature, one observes that the bell-shape exhibited by the
angular distribution in the one-proton transfer case is modified in the
sequential two-step transfer mechanism. The result is, up to a large extent, a
quite flat angular behavior of the differential cross section, at least at
intermediate angles. Moreover, for a given combination of states in the exit
channel, the diffraction pattern seems to be practically independent from the
considered path while it depends on the angular momentum of the final states
involved, i.e. on the total angular momentum transferred in the process. In
particular, at least from a qualitative point of view, one notices that the
angular position of the first maximum shifts to larger angles when considering
target states in the outgoing partition with larger J values. However, this
correspondence which is easily recovered in the one-nucleon transfer case, is
blended and less clear in the two-step process where an intermediate partition
is crossed. As a consequence, some differences arise in the diffraction
pattern at small angles even when the angular momentum of the target states in
the final partition is the same (see for example the differences existing in
the diffraction patterns for the two different 4+ states of 116In shown in
Fig. 10).

A remarkable feature to be noted is that the contribution provided along path1
is systematically larger than the corresponding one obtained along path2. Such
a result might be mainly attributed to the larger level density predicted in
the low excitation energy region for the system 115Cd, with respect to the
117In, within the model space here adopted.

In addition, the role of the coherent interference between the two path turns
out to be rather important. The coherent sum of the two contributions may
considerably alter indeed the diffraction pattern of the general process,
especially when the two paths have a comparable magnitude. As a result, the
angular distribution of the differential cross section might appear rather
different from the single path components, whereas the absolute cross section
is only partially affected.

Similar conclusions might be drawn when more complicated coupling schemes are
investigated. For this purpose, it could be helpful to compare the DWBA
results with the CCBA ones, in order to shed light on the role of the
inelastic excitations and on the transfer mechanisms which they are involved
in. The corresponding results are displayed in Fig. 11, where the differential
cross section is plotted as a function of the scattering angle in the center
of mass system. Only the total cross section, given by the coherent sum of the
two path contributions and the incoherent sum on combinations of projectile
and target states lying within the excitation energy range considered is
reported. As in the one-nucleon transfer case, the CCBA results only slightly
impact the diffraction pattern at forward angles, in light of the different
combinations of states and of angular momentum transfer involved when
exploring the intermediate partition. The results are then slightly sensitive
also to the choice of the optical potential, in view of the delicate interplay
between the spatial extension of the transition form factor and the surface
details of the ion-ion interaction. Indeed, analogously to what was observed
also in Fig. 4 for the one-proton transfer case, DFOL results tend to be
larger at small angles but predict a steeper decrease than SPP with increasing
$\theta_{c.m.}$ scattering angle.

Figure 11: Angular distribution of the differential cross section for all the
final states of the 116Cd(20Ne,20F)116In SCE reaction, lying within the
considered excitation energy range, obtained from two-step DWBA (dashed lines)
or CCBA (full lines) calculations. Calculation with DFOL and SPP ion-ion
potentials are indicated by red and blue lines, respectively.

The results discussed above allow thus to finally achieve a quantitative
estimation of the total SCE cross section, at least when describing this
process as only given by the sequence of two-nucleon transfer mechanisms and
inelastic excitations in the initial partition. Although the genuine direct
SCE transition driven by the direct isovector nuclear interaction is not
included in such a scheme, it is thus certainly worthwhile to provide a first
comparison with the experimental results. In Table 4, we list the theoretical
cross section, integrated in the angular range [4.5°, 14.5°] of the laboratory
system, for all the states lying within the considered excitation energy
range. In Table 4, we also reported the total sum, to be compared with the
corresponding experimental value.

Table 4: SCE cross section (in $\mu b$) as obtained with the SPP and the DFOL potential through two-step transfer mechanisms, when integrating in the angular range [4.5°, 14.5°] of the laboratory system and for all the combination of projectile and target states lying within the considered excitation energy range. The integrated experimental value is also reported.  |  |  | DFOL |  | SPP |
---|---|---|---|---|---|---
$J^{\pi}_{p}$ | $J^{\pi}_{t}$ | $E_{t}^{*}$ | $\sigma_{\textup{DWBA}}$ | $\sigma_{\textup{CCBA}}$ |  | $\sigma_{\textup{DWBA}}$ | $\sigma_{\textup{CCBA}}$ | $\sigma_{\textup{exp}}$
2${}^{+}_{\textup{g.s.}}$ | 1+ | 0.000 | 0.015 | 0.024 |  | 0.020 | 0.033 |
2${}^{+}_{\textup{g.s.}}$ | 5+ | 0.128 | 0.017 | 0.043 |  | 0.019 | 0.049 |
2${}^{+}_{\textup{g.s.}}$ | 4+ | 0.223 | 0.014 | 0.021 |  | 0.018 | 0.028 |
2${}^{+}_{\textup{g.s.}}$ | 2+ | 0.273 | 0.078 | 0.078 |  | 0.099 | 0.100 |
2${}^{+}_{\textup{g.s.}}$ | 4+ | 0.313 | 0.128 | 0.129 |  | 0.153 | 0.156 |
total |  |  | 0.252 | 0.295 |  | 0.309 | 0.366 | 0.7 $\pm$ 0.3

In light of the results of Table 4, one can assess that, when describing the
116Cd(20Ne,20F)116In reaction only in terms of two successive transfer
processes, the order of magnitude of the cross section is nicely recovered in
our theoretical calculations. Moreover, this is achieved already within the
simplified DWBA framework, although the role of the coupling with the
inelastic excitations of the states which belong to the initial partition
seems to be not completely negligible.

It is also worth noting that, due to the finite experimental energy
resolution, the set interval upper limit rules out part of the high-energy
contributions of the considered transitions. On the other hand, the present
theoretical predictions of the sequential proton-neutron transfer processes
fully account for such contributions, leading to an expected overestimation of
the relative amount in the total reaction cross section attributed to the
multi-step mechanism. Nonetheless, this does not affect the validity of this
first study, since the order of magnitude of the transfer processess would be
in any case preserved.

We would like to remind here that our result has been obtained by employing as
theoretical ingredients an optical potential, which has demonstrated to
provide a reliable reproduction of the elastic channel, and a well-defined
model space with its corresponding shell model effective interaction, which
returns a satisfying spectroscopic description of the nuclear systems involved
here and allows for reasonable agreement with experimental results of the one-
proton transfer process. We notice, by the way, that the same ingredients have
been already successfully applied also in the description of two-neutron and
two-proton transfer reactions, having in common the same initial partition
Diana .

Of course, our result might still be considered affected by a certain degree
of model dependence, as suggested by the small sensitivity here highlighted
with respect to the optical potential adopted and to the fine details of the
structure ingredients. This unavoidable model dependence, which might be
ascribable to the variety of different processes involved and to the
uncertainties existing in the description of the considered nuclei, is however
strongly mitigated in our work by the multi-channel experimental constraints
and the fully consistent theoretical approach employed. Moreover, our study
suggests that SPP might be globally considered a preferrable choice with
respect to DFOL. Although they both give a good reproduction of the scattering
data, SPP gives indeed a better description of the transfer channels, in one-
proton transfer as well as in the two-neutron and two-proton channels
considered in Ref. Diana .

The analysis performed here does not prevent us thus to draw important
conclusions. On the one hand, our work demonstrates that the two-step transfer
mechanisms provide a non negligible contribution to the total cross section of
the 116Cd(20Ne,20F)116In reaction channel. Such a result, in the energy/mass
range investigated in the present work, was not so obvious. Although in some
previous works (Refs. Oertzen ; Winfield ; Len89 ; DasPRC1986 ), which mainly
refer to lighter systems, a primary role of the multi-step transfer
contribution was already assessed, in some other light-ion induced studies
(Ref. CapLenske04 ) a different behaviour was observed, thus highlighting a
not universal scenario. Moreover, in this work, different kinematical
conditions, which are of crucial importance in determining the magnitude of
the transfer mechanisms, have been generally explored. On the other hand, our
finding suggests that a relevant fraction of this total cross section is still
missing within our picture. Such a fraction should be provided by the direct
mechanism, namely from the direct SCE process, which has not been addressed
here,owing to unavoidable uncertainties which would affect the results for a
such heavy system, both from the experimental and theoretical perspectives.
Such a statement might be then definitely validated only when confirmed by a
consistent calculation taking into account the direct charge-exchange
contribution. Nevertheless, for the time being, we highlight that this result
seems to be in line with recent findings discussed in Ref. CavFRO2021 , where
a significant contribution from direct isovector exchange mechanism is
highlighted from the analysis of the 40Ca(18O,18F)40K SCE reaction, at the
same bombarding energy per nucleon.

The competition between the two processes is however anything but trivial,
since its fully consistent analysis would require all the structure
ingredients needed for describing both mechanisms being extracted within the
same coherent theoretical framework. Such a competition is currently under
investigation and will be the topic of a forthcoming paper.

However, since the transfer mechanisms do not seem to be suppressed, the
present study already suggests that in order to extract structure information
on the NME from the SCE cross section, a careful estimation of the competing
transfer processes is needed. These higher order processes are instead
expected to be suppressed when further increasing the number of steps
involved, for example when investigating DCE reactions, and preliminary
attempts along this direction seem to confirm such an expectation LayBurrello
; Diana .

##  V Conclusions and outlooks

In the present work, the 116Cd(20Ne,20F)116In reaction has been studied at an
incident energy of 306 MeV, through a multi-channel analysis, involving also
the elastic scattering and the one-proton transfer 116Cd(20Ne,19F)117In
reaction. Such a wide collection of experimental data was indeed instrumental
to put more stringent constraints on the nuclear models that are also adopted
for the calculations of the SCE direct process, the corresponding NME and the
related beta-decay ones. In perspective, studying CE reactions on 116Cd will
contribute to encircle and to understand the expected $0\nu 2\beta$-decay
properties of that nucleus.

The experimental data for multi-nucleon transfer have been then compared with
the theoretical predictions obtained by performing (sequential) two-step
distorted wave Born approximation and higher-order coupled reaction channel
calculations. Two different optical potentials were employed for modeling the
initial and final state interactions and a large-scale shell model approach
was adopted to evaluate the spectroscopic amplitudes for the single-particle
transitions characterizing one-nucleon transfer processes.

Through a combination of structure and reaction studies and a multi-channel
comparison with the experimental data, we validated the choice of the optical
potentials adopted and we checked the reliability of the model space and
interactions assumed in the shell model calculations. Moreover, we shed light
on the influence of the high-order couplings between transfer and scattering
channels, bringing to light a bland influence of the inelastic excitations in
the initial partition both on the one-proton transfer process and on the SCE
reaction.

Finally, within the same unified scheme, our calculations were able to
reproduce also the order of magnitude of the experimental cross section, by
describing the 116Cd(20Ne,20F)116In reaction only in terms of successive
transfer processes. This result was obtained already within the simplified
two-step DWBA framework, proving that resorting to more complex reaction
schemes, at a first level of accuracy, reveals unnecessary. It is worth noting
that the robustness of our finding is corroborated by the fact that no
parameter has been pratically adjusted with respect to the systematics.

However, the sequence of second or higher order transfer or inelastic
processes considered in our study does not entirely exhaust the total cross
section. The origin of the remaining difference between the theoretical and
experimental results is not easy to access at this stage, although being for a
large extent ascribable to the direct SCE mechanism, which is not included
within the present picture. Our conclusion is thus that room is left for a
possible contribution of the direct isovector exchange mechanism in the
explored SCE transitions.

The genuine contribution of the latter mechanism might be isolated when a
fully consistent analysis, taking into account the competition among all the
possible processes involved, including the direct SCE mechanism, is carried
out within the same theoretical framework. As a perspective, the study of the
competition among the different charge exchange reaction mechanisms demands
the exploration of new set of data. To this aim, the analysis of simple
systems, in which transitions to individual states can be isolated, is
particularly helpful for untangling direct and multi-step contributions and
comparing with the predictions of theoretical models. Studies of heavy-ion
induced charge exchange transitions on 12C target are already in progress
spatafora , taking advantage of the low level density in the light reaction
products and of the well known spectroscopic properties of the involved
nuclei. In this view, the analysis of new charge exchange data at different
beam energies is also interesting and it is among the research lines of the
NUMEN and NURE projects EPJA2018 . Work is in progress along this direction,
with the aim of extending this analysis also to the DCE channel, of particular
interest for the possibility to extract information on the NME of
2$\beta$-decay.

Table 5:  Spectroscopic amplitudes (S.A.) obtained through shell model $zbm$ interaction and adopted in CCBA calculations for all the projectile-like nuclei involved along the path1 (see text) of the 116Cd(20Ne,20F)116In SCE reaction, whose final states lie within the excitation energy range here considered. The column nlj indicates the principal quantum number, the orbital and the total angular momentum of the single valence proton, respectively, through the usual spectroscopic notation. Initial state | Final state | nlj | S.A.
---|---|---|---
20Ne${}_{\textup{g.s.}}$ (0+) |  21Ne0.351 (5/2+) | (1d5/2) | 0.7696
21Ne2.794 (1/2+) | (2s1/2) | -0.7355
21Ne3.736 (5/2+) | (1d5/2) | -0.0891
20Ne${}_{\textup{1.634}}$ (2+) | 21Ne${}_{\textup{g.s.}}$ (3/2+) | (2s1/2) | 0.2077
(1d5/2) | -1.0262
21Ne0.351 (5/2+) | (2s1/2) | 0.3181
(1d5/2) | 0.1650
21Ne1.746 (7/2+) | (1d5/2) | -0.9131
21Ne2.794 (1/2+) | (1d5/2) | -0.6516
21Ne3.736 (5/2+) | (2s1/2) | -0.7344
(1d5/2) | 0.5534
21Ne${}_{\textup{g.s.}}$ (3/2+) | 20F${}_{\textup{g.s.}}$ (2+) | (2s1/2) | -0.6366
(1d5/2) | 0.5209
21Ne0.351 (5/2+) | (2s1/2) | 0.0656
(1d5/2) | -0.6412
21Ne1.746 (7/2+) | (1d5/2) | -0.4159
21Ne2.794 (1/2+) | (1d5/2) | -0.0431
21Ne3.736 (5/2+) | (2s1/2) | -0.0331
(1d5/2) | -0.2296
Table 6:  Spectroscopic amplitudes (S.A.) obtained through shell model $88Sr45$ interaction and adopted in CCBA calculations for all the target-like nuclei involved along the first step of path1 (see text) of the 116Cd(20Ne,20F)116In SCE reaction. The column nlj indicates the principal quantum number, the orbital and the total angular momentum of the single valence proton, respectively, through the usual spectroscopic notation. Initial state | Final state | nlj | S.A.
---|---|---|---
116Cd${}_{\textup{g.s.}}$ (0+) |  115Cd${}_{\textup{g.s.}}$ (1/2+) | (3s1/2) | 0.8913
115Cd0.229 (3/2+) | (2d3/2) | -1.2814
115Cd0.361 (5/2+) | (2d5/2) | -0.0927
115Cd0.389 (7/2+) | (1g7/2) | 1.8526
115Cd0.473 (3/2+) | (2d3/2) | 0.0356
115Cd0.473 (5/2+) | (2d5/2) | 0.3192
115Cd0.507 (3/2+) | (2d3/2) | -0.0219
115Cd0.507 (5/2+) | (2d5/2) | 0.0950
115Cd0.649 (1/2+) | (3s1/2) | 0.9347
116Cd${}_{\textup{0.513}}$ (2+) | 115Cd0.229 (3/2+) | (3s1/2) | 0.1381
(2d3/2) | 0.9889
(1g7/2) | -0.1855
115Cd0.361 (5/2+) | (3s1/2) | -0.5688
(2d3/2) | 0.5895
(1g7/2) | 0.3823
115Cd0.389 (7/2+) | (2d1/2) | -0.1917
(1g3/2) | 0.5723
115Cd0.389 (9/2+) | (2d5/2) | 0.1209
(1g3/2) | -0.5996
115Cd0.473 (3/2+) | (3s1/2) | -0.4137
(1g7/2) | -0.2472
115Cd0.473 (5/2+) | (3s1/2) | 0.1866
(1g7/2) | 0.5306
115Cd0.507 (3/2+) | (3s1/2) | 0.1228
(2d5/2) | 0.1089
(1g7/2) | -0.4713
115Cd0.507 (5/2+) | (3s1/2) | -0.1367
(2d3/2) | 0.2446
(1g7/2) | -0.2707
115Cd0.649 (1/2+) | (2d3/2) | -0.2863
| (2d5/2) | 0.1247
115Cd1.062 (7/2+) | (2d3/2) | 0.5040
(1g7/2) | 0.4286
Table 7:  Spectroscopic amplitudes (S.A.) obtained through shell model $88Sr45$ interaction and adopted in CCBA calculations for all the target-like nuclei involved along the second step of path1 (see text) of the 116Cd(20Ne,20F)116In SCE reaction and whose final states lie within the excitation energy range here considered. The column nlj indicates the principal quantum number, the orbital and the total angular momentum of the single valence proton, respectively, through the usual spectroscopic notation. Initial state | Final state | nlj | S.A.
---|---|---|---
115Cd0.229 (3/2+) | 116In${}_{\textup{g.s.}}$ (1+) | (2d5/2) | -0.1978
115Cd0.389 (7/2+) | (1g9/2) | -0.9798
115Cd0.389 (9/2+) | (1g9/2) | 0.3486
115Cd0.473 (3/2+) | (2d5/2) | -0.1457
115Cd0.473 (5/2+) | (2d5/2) | 0.1498
115Cd0.507 (3/2+) | (2d5/2) | -0.1540
115Cd1.062 (7/2+) | (2d5/2) | 0.1505
(1g9/2) | 0.2514
115Cd${}_{\textup{g.s.}}$ (1/2+) | 116In0.128 (5+) | (1g9/2) | -0.3145
115Cd0.229 (3/2+) | (1g9/2) | 0.5385
115Cd0.361 (5/2+) | (1g9/2) | 0.1772
115Cd0.389 (7/2+) | (1g9/2) | -0.2004
115Cd0.389 (9/2+) | (2d5/2) | -0.1084
115Cd0.473 (3/2+) | (1g9/2) | -0.2877
115Cd0.473 (5/2+) | (1g9/2) | -0.2605
115Cd0.507 (3/2+) | (1g9/2) | 0.1585
115Cd0.507 (5/2+) | (1g9/2) | -0.2633
115Cd0.649 (1/2+) | (1g9/2) | 0.1831
115Cd1.062 (7/2+) | (1g9/2) | -0.8241
115Cd${}_{\textup{g.s.}}$ (1/2+) | 116In0.223 (4+) | (1g9/2) | -0.4179
115Cd0.229 (3/2+) | (1g9/2) | -0.4774
115Cd0.361 (5/2+) | (2d5/2) | -0.1412
(1g9/2) | 0.5945
115Cd0.389 (7/2+) | (1g9/2) | 0.1464
115Cd0.389 (9/2+) | (2d5/2) | 0.1171
(1g9/2) | -0.8536
115Cd0.473 (3/2+) | (1g9/2) | 0.1176
115Cd0.473 (5/2+) | (1g9/2) | -0.4535
115Cd0.507 (5/2+) | (1g9/2) | -0.2989
115Cd0.649 (1/2+) | (1g9/2) | 0.2337
115Cd1.062 (7/2+) | (1g9/2) | 0.5216
115Cd0.361 (5/2+) | 116In0.273 (2+) | (2d5/2) | -0.1858
115Cd0.389 (7/2+) | (2d5/2) | -0.1401
(1g9/2) | 0.5919
115Cd0.389 (7/2+) | (1g9/2) | -0.4539
115Cd0.473 (3/2+) | (2d5/2) | 0.1242
115Cd0.473 (5/2+) | (1g9/2) | -0.1142
115Cd0.507 (5/2+) | (2d5/2) | 0.1230
115Cd1.062 (7/2+) | (2d5/2) | -0.1611
(1g9/2) | -0.1623
115Cd${}_{\textup{g.s.}}$ (1/2+) | 116In0.313 (4+) | (1g9/2) | -0.1678
115Cd0.473 (3/2+) | (1g9/2) | 0.1308
115Cd0.649 (1/2+) | (1g9/2) | -0.3518
Table 8:  Spectroscopic amplitudes (S.A.) obtained through shell model $zbm$ interaction and adopted in CCBA calculations for all the projectile-like nuclei involved along the path2 (see text) of the 116Cd(20Ne,20F)116In SCE reaction and whose final states lie within the excitation energy range here considered. The column nlj indicates the principal quantum number, the orbital and the total angular momentum of the single valence proton, respectively, through the usual spectroscopic notation. Initial state | Final state | nlj | S.A.
---|---|---|---
20Ne${}_{\textup{g.s.}}$ (0+) |  19F${}_{\textup{g.s.}}$ (1/2+) | (2s1/2) | -0.8584
19F0.197 (5/2+) | (1d5/2) | 1.1741
20Ne${}_{\textup{1.634}}$ (2+) |  19F${}_{\textup{g.s.}}$ (1/2+) | (1d5/2) | 0.6712
19F0.197 (5/2+) | (2s1/2) | -0.6922
(1d5/2) | -0.6416
19F1.346 (5/2-) | (1p1/2) | -0.9779
19F1.554 (3/2+) | (2s1/2) | 0.3767
(1d5/2) | 0.2923
19F2.780 (9/2+) | (1d5/2) | -0.6248
19F4.378 (7/2+) | (1d5/2) | -0.1995
19F${}_{\textup{g.s.}}$ (1/2+) | 20F${}_{\textup{g.s.}}$ (2+) | (1d5/2) | -0.2291
19F0.197 (5/2+) | (2s1/2) | 0.1215
(1d5/2) | -0.8891
19F${}_{\textup{1.346}}$ (5/2-) | (1p1/2) | -0.1475
19F${}_{\textup{1.554}}$ (3/2+) | (1d5/2) | -0.4445
19F${}_{\textup{2.780}}$ (9/2+) | (1d5/2) | -0.5137
19F${}_{\textup{4.378}}$ (7/2+) | (1d5/2) | -0.5312
Table 9:  Spectroscopic amplitudes (S.A.) obtained through shell model $88Sr45$ interaction and adopted in CCBA calculations for all the target-like nuclei involved along the path2 (see text) of the 116Cd(20Ne,20F)116In SCE reaction and whose final states lie within the excitation energy range here considered. The column nlj indicates the principal quantum number, the orbital and the total angular momentum of the single valence proton, respectively, through the usual spectroscopic notation. Initial state | Final state | nlj | S.A.
---|---|---|---
116Cd${}_{\textup{g.s.}}$ (0+) |  117In${}_{\textup{g.s.}}$ (9/2+) | (1g9/2) | -0.4066
117In0.748 (7/2+) | (1g7/2) | 0.0098
117In0.881 (5/2+) | (2d5/2) | -0.1593
117In1.030 (5/2+) | (2d5/2) | -0.2670
117In1.052 (5/2+) | (2d5/2) | -0.7192
117In1.365 (7/2+) | (1g7/2) | -0.0027
|  117In1.365 (9/2+) | (1g9/2) | 0.0383
116Cd${}_{\textup{0.513}}$ (2+) | 117In${}_{\textup{g.s.}}$ (9/2+) | (2d5/2) | 0.1182
(1g9/2) | -0.6092
117In0.748 (7/2+) | (1g9/2) | -0.2232
117In0.881 (5/2+) | (2d5/2) | -0.1781
(1g9/2) | 0.4308
117In1.052 (5/2+) | (2d5/2) | 0.5693
(1g9/2) | -0.1326
117In1.365 (7/2+) | (2d5/2) | 0.1889
117In1.365 (9/2+) | (1g9/2) | -0.1813
117In${}_{\textup{g.s.}}$ (9/2+) | 116In${}_{\textup{g.s.}}$ (1+) | (1g7/2) | -0.5008
117In0.881 (5/2+) | (2d3/2) | -0.2335
117In1.030 (5/2+) | (2d3/2) | -0.4548
117In1.052 (5/2+) | (2d3/2) | 0.2233
117In${}_{\textup{g.s.}}$ (9/2+) | 116In${}_{\textup{0.128}}$ (5+) | (1g7/2) | -0.1534
(2d3/2) | -0.9107
(3s1/2) | -0.1969
117In0.748 (7/2+) | (1g7/2) | 0.1283
(2d3/2) | 0.7751
117In0.881 (5/2+) | (2d5/2) | 0.1249
117In1.030 (5/2+) | (1g7/2) | 0.1178
117In1.365 (7/2+) | (2d3/2) | -0.4105
117In1.365 (9/2+) | (1g7/2) | -0.1519
(2d3/2) | -0.4675
117In${}_{\textup{g.s.}}$ (9/2+) | 116In${}_{\textup{0.223}}$ (4+) | (2d3/2) | 0.6852
(3s1/2) | -0.2581
117In0.748 (7/2+) | (1g7/2) | 0.1143
(2d3/2) | 0.2821
117In0.881 (5/2+) | (2d3/2) | 0.9649
117In1.030 (5/2+) | (2d3/2) | -0.4573
117In1.365 (9/2+) | (3s1/2) | -0.1604
117In${}_{\textup{g.s.}}$ (9/2+) | 116In${}_{\textup{0.273}}$ (2+) | (1g7/2) | 0.6212
117In0.748 (7/2+) | (2d3/2) | -0.1605
117In0.881 (5/2+) | (2d3/2) | 0.1612
117In1.030 (5/2+) | (2d3/2) | 0.3029
117In1.365 (7/2+) | (2d3/2) | -0.3473
117In${}_{\textup{g.s.}}$ (9/2+) | 116In${}_{\textup{0.313}}$ (4+) | (2d5/2) | -0.1060
(2d3/2) | -0.2653
(3s1/2) | -0.8248
117In0.748 (7/2+) | (2d3/2) | -0.1569
117In0.881 (5/2+) | (2d3/2) | 0.1433
117In1.030 (5/2+) | (2d3/2) | 0.2034
117In1.052 (5/2+) | (2d3/2) | -0.1049
117In1.365 (7/2+) | (2d3/2) | 0.2134

## Acknowledgments

The authors warmly acknowledge the operators of INFN-LNS Accelerator Division
for the production and delivery of the 20Ne beam and their support throughout
the experiments. This project has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 Research and Innovation
Program (Grant Agreement No. 714625). S. Burrello and J. A. Lay acknowledge
support from the Spanish Ministry of Economy and Competitiveness and the
European Regional Development Fund (FEDER) under Project Nº FIS2017-88410-P
and FIS2014-53448-C2-1-P. E. R. Chávez-Lomelí acknowledges DGAPA-UNAM
IN107820, AG101120 and CONACyT 314857. J.R.B. Oliveira acknowledges "FAPESP
proc. 2019/07767-1 and INCT-FNA, project n. 464898/2014-5), Brazil, by JRBO".
DGAPA-UNAM IN107820, AG101120 and CONACyT 314857.

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