A Constrained Analysis of the 40Ca(18O,18F)40K Direct Charge Exchange Reaction Mechanism at 275 MeV

Introduction

Single charge exchange (SCE) reactions are considered to be the best probe to explore the isospin and spin–isospin nuclear response to the strong interaction. Light-ion-induced SCE reactions have been widely investigated (Taddeucci et al., 1987; Osterfeld, 1992; Alford and Spicer, 1998) especially thanks to the high energy resolution (30 keV) achieved in (³He,t) experiments (Fujita et al., 1997, 2007; Douma et al., 2020) which have allowed to study the details of the populated energy spectra, including their relationships with beta decay transition strengths (Fujita et al., 2011; Frekers et al., 2013; Diel et al., 2019). The use of light-ion-induced SCE reactions for the spectroscopic studies of Fermi (J^π = 0⁺; L = 0, T = 1, S = 0) and Gamow–Teller (J^π = 1⁺; L = 0, T = 1, S = 1) excitations is well-established. Recent studies have also shown a sensitivity of (³He,t) reactions to higher multipoles, such as the spin dipole (J^π = 2⁻; L = 1, T = 1, S = 1) (Ejiri et al., 2014; Akimune et al., 2020). The isospin response at high multipolarity has attracted interest in the last few years in view of its possible connection to neutrinoless double beta decay (0νββ) nuclear matrix elements (NMEs), where indeed higher multipoles are expected to play a relevant role in an intermediate virtual state (Ejiri et al., 2019). Ordinary muon capture (OMC) techniques, despite the moderate energy resolution of the order of few MeV, have been recently developed to explore nuclear high multipoles stimulated by isospin weak interaction operators (Ejiri, 2005; Jokiniemi et al., 2019).

In this perspective, heavy-ion-induced SCE reactions are particularly interesting because of their enhanced probability to populate high-spin states, due to the large momentum transferred from the heavy projectile to the target even at forward scattering angles. Hence, heavy-ion SCE reactions are expected to provide a relevant contribution to our understanding of the matrix elements of nuclear multipole transitions (Lenske et al., 2018, 2019). In this case, the achievable energy resolution is a few hundred keV, mainly limited by energy straggling due to the projectile–target interaction, which is often enough to allow the separation of low-lying states in the inclusive spectra. Different heavy-ion probes have been proposed in the years, mainly induced by ^6,7Li, ^12,13C, ⁴⁸Ti beams at energies above the Coulomb barrier (Lenske et al., 2019). The main findings concern the competition between direct meson exchange and sequential nucleonic transfer mechanisms depending on the specific dynamical conditions. When the direct mechanism is dominant, a proportionality of SCE cross sections with beta decay strengths is found and a general trend to favor the population of unnatural parity transitions is observed (Cappuzzello et al., 2004a). Not much is known about the (¹⁸O,¹⁸F) SCE reaction, which is discussed in the present paper. Such a probe has been studied on the ²⁸Si and ³⁶S targets from 3 to 19.6 A MeV (Kim et al., 1979; Horen et al., 1986; Fifield et al., 1993). In all cases, a significant role of the direct mechanism has been deduced, especially at the highest beam energies, but uncertainties in the adopted models for the nuclear structure and reaction inputs did not allow to draw any firm conclusion.

The study of such SCE reactions is of crucial importance in the NUMEN and NURE projects (Cappuzzello et al., 2015a, 2018; Cavallaro et al., 2017) at the INFN-LNS laboratory, which aims at investigating double-charge exchange (DCE) reactions (¹⁸O,¹⁸Ne) and (²⁰Ne,²⁰O) to an unprecedented level of accuracy, both from the experiment and theory side, in view of their connections with 0νββ decay. The key aspects of this research program are the similarities between the two classes of processes, DCE and 0νββ, both characterized by the exchange of two units of isospin between the initial state and the final state, although mediated by the strong interaction and the weak interaction, respectively (Cappuzzello and Cavallaro, 2020).

A theory for heavy-ion SCE and DCE reactions, describing the two kinds of reaction in a unified manner, was missing up to few years ago and is presently under development within the NUMEN activities (Lenske et al., 2018, 2019; Santopinto et al., 2018; Bellone et al., 2020; Carbone et al., 2020; Ferreira et al., 2021; Burrello et al., in preparation). The competition between the direct mechanisms, mediated by the exchange of one (SCE) or two (DCE) charged mesons between the projectile and target and probing the isospin structure of the ions, and the sequential transfer of protons and neutrons, probing the mean field structure of the involved nuclei, is one of the topics of this research. In particular, Bellone et al. (2020) demonstrated that the (¹⁸O,¹⁸F) cross section is fundamental to constrain the DCE sequential meson exchange mechanism along the ¹⁸O → ¹⁸F → ¹⁸Ne transition. Moreover, the experimental measurement and analysis of the elastic scattering and inelastic scattering cross sections for the same projectile–target system is crucial to constrain the SCE and DCE calculations. Indeed, the projectile–target nucleus-nucleus potential needs to be accurately modeled both in the entrance [initial state interaction (ISI)] and exit [final state interaction (FSI)] channels (Spatafora et al., 2019; Carbone et al., 2021; La Fauci et al., submitted).

A case of interest for a description of charge exchange cross sections is the ⁴⁰Ca(¹⁸O,¹⁸F)⁴⁰K reaction at an incident energy of 275 MeV for which the theoretical formalism and the details of a microscopic quantum-mechanical calculation were reported by Lenske et al. (2018). Such an approach includes a nuclear structure part modeled by quasi-particle random phase approximation- (QRPA-) based transition densities, and a reaction part via distorted wave Born approximation (DWBA) cross section calculations. Here, we provide for the first time the experimental cross section angular distribution data and discuss their comparison with the theoretical calculations. Contextually, the elastic and inelastic scattering of ¹⁸O beam on ⁴⁰Ca target at the same bombarding energy were measured in a wide momentum transfer range and studied to extract the optical potential to be used in the SCE calculations. The importance of experimental constraints coming from the elastic scattering data to the determination of the ion–ion interaction was already stressed by Lenske et al. (2018). However, such information was missing in that work and is available here for the first time. The effect of the couplings with inelastic excitations is also explored here.

Experiment and Data Reduction

The ⁴⁰Ca(¹⁸O,¹⁸F)⁴⁰K charge exchange reaction was measured at INFN-Laboratori Nazionali del Sud in Catania (Italy). A beam of ¹⁸O⁴⁺ ions, extracted by the K800 Superconducting Cyclotron accelerator, bombarded a 280 ± 14 μg/cm² natural calcium (96.9% ⁴⁰Ca) target at an incident energy of 275 MeV. The calcium material for the target was evaporated on a carbon backing of 25 μg/cm² thick, and a further carbon layer of 15 μg/cm² was evaporated on the top of the calcium layer to reduce oxidation processes. The ¹⁸O + ⁴⁰Ca elastic scattering and inelastic scattering were measured in a dedicated run using a 250 ± 12 μg/cm² thick calcium target evaporated on 47 μg/cm² carbon backing. A Faraday cup located inside a scattering chamber and 15 cm downstream of the target was used to collect the beam charge. An electron suppressor polarized at −200 V and a low-noise charge integrator allowed a charge collection accuracy better than 10%. The ejectiles produced in the collisions were selected by the MAGNEX large acceptance spectrometer (Cappuzzello et al., 2016) and identified by its focal plane detector (Cavallaro et al., 2012; Torresi et al., 2021). For the charge exchange measurement, the optical axis of the spectrometer was located at θ_opt = +7° with respect to the beam direction. Thanks to the MAGNEX angular acceptance, an angular range of 3° < θ_lab < 9° in the laboratory reference frame was explored in a single angular setting. For the measurement of elastic scattering and inelastic scattering, the optical axis of MAGNEX was set at θ_opt = +8°, +14°, and +18° in three different acquisition runs, exploring an overall angular range of 3° < θ_lab < 19°. The procedure to reduce the collected data and extract the energy spectra and the cross section angular distributions for the measured transitions is described in detail by Cappuzzello et al. (2010), Cappuzzello et al. (2011), Cappuzzello et al. (2014), Calabrese et al. (2020), Cavallaro et al. (2011), and Carbone (2015).

In Figure 1, an example of a measured spectrum is shown as a function of excitation energy E_x = Q₀ −Q (where Q is the reaction Q-value, Q₀ = 0 for elastic scattering) for the ¹⁸O+⁴⁰Ca scattering in the angular region 8.2° < θ_lab < 9.0°. The peaks related to the transition to the 0^{+ 40}Ca ground state and to the first 2^{+ 18}O excited state at E_x = 1.982 MeV are clearly visible. The structures above E_x = 3 MeV are due to unresolved excitations of both ejectile and residual nuclei. The contributions arising due to the carbon and oxygen impurities in the target are expected at E_x > 6 and E_x > 4 MeV, respectively, so not present in the range of the explored transitions. An energy resolution of about 500 keV (full width at half maximum) is measured.

FIGURE 1

Figure 1. Excitation energy spectrum of ⁴⁰Ca obtained in the ¹⁸O + ⁴⁰Ca elastic scattering and inelastic scattering at an incident energy of 275 MeV and 8.2° < θ_lab < 9.0°.

The cross section angular distribution for the ¹⁸O + ⁴⁰Ca elastic scattering, expressed in terms of its ratio to the Rutherford cross section, is shown in Figure 2A. The corresponding scale of transferred linear momentum q is also given. The angular distribution for the inelastic scattering to the first excited state of ¹⁸O at 1.982 MeV is shown in Figure 2B.

FIGURE 2

Figure 2. (A) Cross section angular distribution of the ¹⁸O + ⁴⁰Ca elastic scattering at 275 MeV in terms of its ratio to the Rutherford cross section σ_Ruth. (B) Angular distribution of the inelastic channel ⁴⁰Ca(¹⁸O,¹⁸O_1.982)⁴⁰Ca_g.s. In both plots, the blue dotted line shows the optical model (OM; for elastic) and distorted wave Born approximation (DWBA; for inelastic) calculations with São Paulo potential (SPP), the orange dashed line shows the coupled channel (CC) calculations with SPP and N_W = 0.6, the green dashed line shows the CC calculations with SPP and N_W = 0.78, and the red solid line shows the OM and DWBA calculations with coupled channel equivalent polarization potential (CCEP).

The ⁴⁰K excitation energy spectrum E_x = Q₀ − Q (where Q is the reaction Q-value and Q₀ the ground state to ground state Q-value) extracted from the ⁴⁰Ca(¹⁸O,¹⁸F)⁴⁰K SCE measurement in the angular region of 3° < θ_lab < 5° is shown in Figure 3. Both counts and energy differential cross section dσ/dE in the absolute value (mb/MeV) are indicated in two scales. Some structures are observed in the low excitation energy region; however, the limited resolution and the high-level density do not allow to isolate single transitions. The groups of levels belonging to the three energy regions indicated in Figure 3 as well as the corresponding integrated cross sections in the angular interval of 4.8° < θ_CM < 11.3° are listed in Table 1.

FIGURE 3

Figure 3. Excitation energy spectrum of the ⁴⁰Ca(¹⁸O, ¹⁸F)⁴⁰K charge exchange reaction at 275 MeV and 3° < θ_lab < 5°. The three hatched areas indicate the regions of interest for the study of the angular distributions.

TABLE 1

Table 1. Transitions expected for the three regions of interest of the ⁴⁰Ca(¹⁸O,¹⁸F)⁴⁰K energy spectrum as indicated in Figure 3.

The angular distributions for the SCE cross section extracted in the three regions of interest of the ¹⁸F+⁴⁰K excitation energy sketched in Figure 3 are shown in Figure 4. In particular, the ground state region (including the unresolved transitions to the ground state and the first excited state of ⁴⁰K at 0.029 MeV), the region at energies of 0.5 MeV < E_x < 1.2 MeV and the region at energies of 1.7 MeV < E_x < 2.6 MeV are explored. For the extraction of the cross section in the ground state region, a Gaussian fitting procedure was performed for each angular bin.

FIGURE 4

Figure 4. Angular distributions for the ⁴⁰Ca(¹⁸O,¹⁸F)⁴⁰K transitions corresponding to the three selected regions in the single charge exchange (SCE) spectrum sketched in Figure 3. The red curves are the sum of the cross sections associated to the different transitions belonging to the three regions resulting from the quasi-particle random phase approximation (QRPA) DWBA calculations.

The error bars reported in the experimental data in Figures 2, 4 include the statistical contribution and the uncertainties coming from the determination of the solid angle intervals. An overall uncertainty of about 10%, due to the determination of charge collection and target thickness, is not shown in Figures 2, 4 as it is common to all data points in the angular distributions.

Analysis of the Elastic and Inelastic Scattering and Determination of the Optical Potential

The measured angular distributions of the elastic and inelastic experimental cross section are compared with the quantum-mechanical optical model (OM), DWBA, and coupled channel (CC) calculations, using the FRESCO code (Thompson, 1988). The double-folding São Paulo potential (V_SPP) (Chamon et al., 2002) is adopted as the real part of a complex optical potential U(r) = V_SPP(r) + i N_W V_SPP(r). The imaginary part is assumed with the same shape of the real one scaled by a factor N_W = 0.78, as typically done, for example, in Alvarez et al. (2003), Spatafora et al. (2019), and Oliveira et al. (2013). For the CC calculations, the results obtained by using a smaller scaling factor, N_W = 0.6 (Pereira et al., 2009), are also shown and commented in the following. In the double-folding procedure, two-parameter Fermi distributions are adopted to describe the projectile and target matter densities, with radius and diffuseness obtained from systematic analyses (Chamon et al., 2002). In particular, the diffuseness of the matter density of the projectile is set to a_p = 0.61 fm, which is increased by 10% with respect to SPP systematics (a_p = 0.56 fm), as commonly done for ¹⁸O projectile (Crema et al., 2011; Cavallaro et al., 2013; Cappuzzello et al., 2015b; Ermamatov et al., 2016, 2017; Carbone et al., 2017; Paes et al., 2017; Cardozo et al., 2018; Linares et al., 2018; Fonseca et al., 2019).

The couplings introduced in the DWBA and CC calculations account for the collective low-lying quadrupole and octupole transitions. Namely, we consider the first ¹⁸O(2⁺) excited state at E_x = 1.982 MeV, and the first ⁴⁰Ca(3⁻) and ⁴⁰Ca(2⁺) excited states, at 3.737 MeV and 3.904 MeV, respectively. The adopted coupling scheme is sketched in Figure 5.

FIGURE 5

Figure 5. Coupling scheme adopted in the DWBA (blue dotted single arrows) and CC (green dashed double arrows) calculations in the present work. In DWBA, the single arrows sketch that back coupling effects are not included.

Both DWBA and CC calculations are performed within a rotational model. Calculations within a vibrational model do not give relevant differences in the results. Reduced transition probabilities B(E2; 0⁺ → 2⁺) = 0.0043 e²b² for ¹⁸O, B(E3 0⁺ → 3⁻) = 0.0184 e²b³ and B(E2; 0⁺ → 2⁺) = 0.00924 e²b² for ⁴⁰Ca are taken from Pritychenko et al. (2016), Pritychenko et al. (2017), and Kibedi and Spear (2002) and used to describe the strength of Coulomb deformation of both the projectile and target (Satchler, 1983; Khoa and Satchler, 2000).

Nuclear deformations (for the multipolarity λ) are described in terms of a first-order derivative of the potential U(r), which is used to describe the corresponding elastic channel:

$\begin{array}{ll}{V}_{\lambda }(r)=-\frac{{\delta }_{\lambda }}{\sqrt{4\pi }}\frac{dU(r)}{dr}& (1)\end{array}$

The strength of the deformation is embedded in the deformation length δ_λ (Satchler, 1983):

$\begin{array}{ll}{\delta }_{\lambda }={\beta }_{\lambda }{R}_V=\frac{4\pi }{3Ze}\frac{\sqrt{(2I+1)B(E\lambda ;I\to I^{\prime })}}{R_V^{\lambda -1}}& (2)\end{array}$

where β_λ is the deformation parameter characterizing the transition (of multipole λ) of the given nucleus of charge Ze and is deduced from the electric reduced transition probability B(Eλ; I→I′) from a state of spin I to a state of spin I′, and R_V = 3.73 fm is the radius of the real part of the adopted optical potential U(r), extracted from its fit with a Woods–Saxon shape (Satchler, 1983). For the imaginary coupling potentials, the same radial form factors are assumed. Exploratory calculations changing the method for the determination of R_V or introducing a N/Z correction factor in Equation (2) to account for possible differences in the density profiles for neutrons and protons (Satchler, 1983), give very similar results, within the quoted uncertainty of B(E_λ; I→I′) (Kibedi and Spear, 2002; Pritychenko et al., 2016, 2017).

The results of the calculations are compared with the experimental data in Figure 2, where all the theoretical curves are folded with the experimental angular resolution (~0.9° in the center of mass reference frame). The OM and DWBA calculations including São Paulo-derived optical potential describe reasonably well the experimental cross sections at small angles, showing an oscillatory pattern, which is more pronounced than the experimental data and a general overestimation of the data especially at larger angles and momentum transfer (q > 4 fm⁻¹). This behavior resembles the one found in the similar analyses (Zagatto et al., 2018; Spatafora et al., 2019; Carbone et al., 2021; La Fauci et al., submitted; Burrello et al., in preparation) and is due to the lack of important couplings of elastic channel with inelastic channels.

The results of the CC calculations, also reported in Figure 2, show that the explored cross sections are affected by the couplings among the selected channels. The main effect is to reduce the cross section, especially at a large momentum transfer. Moreover, the couplings damp the oscillations and slightly shift the positions of the minima toward larger scattering angles, thus improving the description of the data. The coupling to the first 2⁺ excited state of the ¹⁸O projectile, keeping the target in its ground state, gives a main contribution to the elastic channel. Nonetheless, the explicit inclusion of the coupling to the first 3⁻ state of ⁴⁰Ca, keeping the projectile in its ground state, improves the description of the slope of the inelastic cross section (Ohkubo and Hirabayashi, 2014; Cappuzzello et al., 2016; Zagatto et al., 2018). We note that the calculations with N_W = 0.78 better describe the experimental data both in absolute cross section and shape of the angular distributions compared to those obtained assuming N_W = 0.6, which is often used in large-scale CC calculations (Pereira et al., 2009). The larger value for the N_W scaling factor, also reported by Spatafora et al. (2019) and Zagatto et al. (2018), may be explained due to the few states included in the present coupling scheme (see Figure 5) and it is compatible with uncertainties in the adopted B(E2) and B(E3) strengths.

As mentioned above, a proper description of the experimental elastic scattering and inelastic scattering data in the fully explored transferred momentum range is achieved, thanks to the introduction of couplings to inelastic channels. This result indicates that also the optical potential necessary for the description of the other quasi-elastic reaction channels induced by the same projectile and target at the same incident energy and angular momentum transfer should account for these couplings. However, it is evident that higher-order effects as introduced by CC calculations are most important for strong channel couplings as, e.g., the coupling of low-lying vibrational or rotational inelastic excitations to the elastic ground state channel. Since the SCE channels are rather weakly coupled to the other configurations, they are well-described already in a first-order DWBA approach. Moreover, in a CC description, the form factors describing each considered reaction channel should come from consistent nuclear structure models. Developments in that direction are the subject of ongoing theory projects and will be discussed elsewhere.

In this view, following the approach of Thompson et al. (1989), we have implicitly incorporated the effect of channel couplings in the elastic optical potential by means of an effective polarization potential term. In general, the formal theory gives a polarization potential, which is not only complex and energy dependent but also non-local and angular momentum dependent. The approach of Thompson et al. (1989), instead, produces an average local and L-independent polarization potential named trivially equivalent local potential (TELP). Adding the TELP to the São Paulo bare optical potential (V_SPP) used in CC calculations, we get a CCEP (Thompson et al., 1985; Rangel et al., 2016). Such CCEP has been used herein in the OM and DWBA calculations for the elastic scattering and inelastic scattering, respectively, and in the DWBA calculations for the SCE channel described in Section The Charge Exchange Reaction Channel. We emphasize that, in the TELP approach, the effect of couplings is derived from a CC solution of the scattering problem, guaranteeing that the physics of coupling on average is accounted for. Thus, this method is more accurate than the common procedure to scale the imaginary part or slightly modify the geometry of the optical potential (Lubian and Nunes, 2007).

The elastic scattering and inelastic scattering experimental data are compared with the OM and DWBA calculations, respectively, using the CCEP optical potential in Figure 2. It is evident that the introduction of the polarization potential is essential to reproduce the experimental cross section even in an OM and a DWBA approach. We stress that in the present work the use of a CCEP has been introduced in view of its application to the SCE and DCE analyses. For the elastic scattering and inelastic scattering studies, it would not be necessary, since CC calculations are feasible and, indeed, have been performed.

The volume integrals and root mean square radii obtained for the SPP and CCEP potentials here adopted are listed in Table 2. The results agree with the values known from systematic studies (Satchler, 1983), confirming a very satisfactory description of the optical potential properties.

TABLE 2

Table 2. Volume integrals per interacting pair and root mean square radii for the real (V) part and imaginary (W) part of the São Paulo potential (SPP) and coupled channel equivalent polarization potential (CCEP) optical potentials adopted in the calculations.

The Charge Exchange Reaction Channel

As discussed in the introduction, the ⁴⁰Ca(¹⁸O,¹⁸F)⁴⁰K SCE reaction could proceed via two mechanisms: one mediated by charged meson exchange (direct SCE) and one governed by mean field interaction among the interacting nucleons (sequential multi-nucleon transfer). In the present work, we show the calculations performed for the former mechanism, which is the one of main interest in view of DCE studies. Calculations are performed within a framework of DWBA, where the cross section in a center of mass rest frame is given by the expression:

$\begin{array}{l}\frac{d^2\sigma }{d{E}_{x}d\Omega }=\frac{E_{\alpha }{E}_{\beta }}{{\left(2\pi {\hslash }^2{c}^2\right)}^2}\frac{k_{\beta }}{k_{\alpha }}\frac{1}{(2J_A+1)(2J_a+1)}\\ \begin{array}{l}{{\displaystyle \sum }}_{M_A,M_{a,}{M}_B,M_b}|{\displaystyle \int }{d}^{3}r{\chi }_{\beta ,{\text{k}}_{\beta }}^{*}(r)F_{\alpha \beta }^{(SCE)}(r){\chi }_{\alpha ,{\text{k}}_{\alpha }}(r){|}^2\end{array}& (3)\end{array}$

where E_α (E_β) and k_α (k_β) denote the energy and relative momentum in the entrance α = a + A (exit β = b + B) channel; χ_α (χ_β) is the distorted wave, accounting for the initial (final) state interaction, i.e., it is the solution of the Schroedinger equation for the Hamiltonian including Coulomb and optical potential (CCEP in the present study) of the entrance (exit) channel. $F_{\alpha \beta }^{(SCE)}(\text{r})$ is the SCE form factor, given by the folding of the nucleon–nucleon effective local interaction potential V_NN(r) and the one-body transition densities of projectile and target nuclei ρ(r).

The effective local nucleon–nucleon direct and exchange interaction potential V_NN(r) contains both central and tensor components. It is parameterized by the sum of three Yukawa functions, with parameters coming from a proper interpolation procedure (Lenske, 1988) between two different parameterizations: (i) the G-matrix calculated by solving the Bethe–Salpeter equation with Paris nucleon–nucleon potential for E/A ≤ 10 MeV (Anantaraman et al., 1983) and (ii) the Franey–Love parameterization of effective nucleon–nucleon T-matrix for E/A ≥ 50 MeV (Franey and Love, 1985). The one-body transition densities ρ(r) are the matrix elements of the one-body operator describing SCE transitions in the projectile and target nuclei (Lenske et al., 2018). These transition densities are calculated within an extended QRPA framework, going beyond the standard approach by accounting for the coupling of the two-quasi-particle configurations to core excitations by polarization self-energies. Residual interactions are determined self-consistently in the context of Fermi-liquid theory. The single quasi-particle states are derived from Hartree–Fock–Bogoliubov (HFB) calculations with state-dependent pairing gaps, describing the ground state properties of the involved nuclei very accurately (Lenske and Tsoneva, 2019). The HFB wave functions and the QRPA configuration amplitudes are used to construct the radial transitions densities, entering finally into the charge exchange transition form factors.

Calculations are performed in terms of the partial wave decomposition of distorted waves and the multipole expansion of the form factor, which is a quite standard procedure (Satchler, 1983). Detailed information on the adopted theoretical modeling is given by Lenske et al. (2018). This approach has been successfully used to also study other heavy-ion-induced SCE reactions (Cappuzzello et al., 2004a,b; Nociforo et al., 2006; Cavallaro, 2011).

On the reaction side, we performed the DWBA calculations using the CCEP optical potential defined in Section 3 for the three excitation energy ranges for which experimental angular distributions have been extracted. The cross sections corresponding to transitions to different excited states are calculated by using the HIDEX code (Lenske, private communication). The calculations shown in Figure 4 represent the sum of the angular distributions of the SCE reactions leading to all the possible excited states experimentally populated both by the ejectile and residual nucleus with total excitation energy within the above-mentioned three ranges. These exit channels are listed in Table 1, together with the corresponding values of the calculated cross section integrated in the same angular range explored by the data.

Table 1 shows that in the first region of interest, the transitions to the 4⁻ ground state and 3⁻ excited state of ⁴⁰K at 0.029 MeV, with the ¹⁸F ejectile at its 1⁺ ground state, give comparable contributions. In the [0.5, 1.7] MeV region, the transition to the first 2⁻ state of ⁴⁰K at 0.800 MeV and to ¹⁸F_g.s.(1⁺) dominates over the other transitions. Conversely, in the [1.7, 2.6] MeV energy range, several multipolarities are comparable in magnitude, even if the cross sections of the transitions to the 2⁻ states of ⁴⁰K at 2.047 and 2.419 MeV, with ejectile ¹⁸F_g.s.(1⁺), are somewhat larger than the others.

An interesting result comes out from a comparison of the calculated cross sections for the 2⁻,3⁻,4⁻, and 5⁻ multiplet of states at 0.800, 0.029, 0.0, and 0.891 MeV, respectively. Although these states refer to the same particle-hole structure with one neutron in the 1f_7/2 orbital coupled to a 1d_3/2 proton hole, the calculated cross sections strongly enhance the 2⁻. Indeed, the 2⁻ cross section is larger than the 3⁻, 4⁻, and 5⁻, by a factor of 21, 17, and 185, respectively. A close inspection of the calculated form factors reveals that the (L = 1, S = 1)-component of the 2⁻ transition carries a certain amount of collectivity, reminiscent of the spin-dipole collectivity seen in other reactions, see Lenske et al. (2019) and Austin et al. (2001). The L = 3 multipoles entering in the 2⁻, 3⁻, 4⁻, and 5⁻ transitions are comparable, while the L = 5 components, contributing to the excitations of the 4⁻ and 5⁻ states, are the smallest. Thus, an interesting selectivity of the smallest L transfer for this SCE reaction is found. Similar behavior is observed in (³He,t) (Ajzenberg-Selove et al., 1985) and (¹²C,¹²N) reactions (von Oertzen, 1988) on the same target, where again the 2⁻ state is the most populated, thus suggesting a universal property of the nuclear response to the nucleon–nucleon isovector interaction.

The plots in Figure 4 tell us that the direct SCE mechanism allows the reproduction of the shape of the measured angular distribution. Regarding the absolute cross sections, the data are slightly underestimated in the regions of low excitation, while a good agreement is found in the excitation energy region between 1.7 and 2.6 MeV.

Simple semiclassical estimates for the energy and angular momentum matching for the transfer processes (Brink, 1972) suggest that a cross section of a relevant magnitude may be expected in the region of the ground state, progressively decreasing at higher excitation energies. Thus, the multi-nucleon transfer mechanism is expected to play a major role in the low excitation energy region of the spectrum explaining the larger experimental cross section with respect to the direct mechanism calculations.

Conclusions

The ⁴⁰Ca(¹⁸O,¹⁸F)⁴⁰K charge exchange reaction, together with the elastic scattering and inelastic scattering of ¹⁸O on ⁴⁰Ca, has been studied at an incident energy of 275 MeV and at forward angles using the MAGNEX spectrometer. The availability of this complete set of data, where the absolute cross section at different angles has been measured with high precision, has allowed a constrained and reliable description of the direct reaction mechanism for the charge exchange process. Charge exchange cross section calculations have been performed in DWBA using a CCEP tested against the elastic and inelastic scattering data and form factors extracted from double folding of a nucleon–nucleon isovector interaction with QRPA transition densities. Such full quantum-mechanical calculations with microscopic nuclear structure inputs describe the order of magnitude and shape of the observed cross sections. Selectivity of this reaction to the different angular momentum transfer components is revealed, making this probe an interesting tool to explore the nuclear response to isovector operators with high momentum as those entering the 0νββ NME.

It is worthwhile to emphasize that the above-mentioned calculations are based on a quite involved approach, starting from microscopic nuclear response functions, combined with free-space nucleon–nucleon interactions to derive the transition form factors and fixing ion–ion ISI and FSI by elastic scattering and inelastic scattering. The advantage of such a demanding program is evident from the fact that the data are quite well-described without the need to introduce additional scaling factors, as it was found necessary in the former heavy-ion-induced SCE reactions on the same target such as the ⁴⁰Ca(⁷Li,⁷Be)⁴⁰K reaction at 35 MeV (Williams-Norton et al., 1979) or the pioneering studies on (¹⁸O,¹⁸F) on different targets (Kim et al., 1979; Horen et al., 1986; Fifield et al., 1993).

The origin of the remaining difference between the theoretical and experimental results is not easy to access. On a nuclear structure side, there is some room left for refinements of the nuclear response functions by going deeper into the dynamics of polarization effects, e.g., in the multi-phonon approach of Lenske and Tsoneva (2019) or by second RPA methods of Gambacurta et al. (2020). On the reaction side, the competing sequential nucleon transfer mechanism should be studied, mainly for reasons of completeness. Based on the presently available results, a rather successful description of the higher excitation energy region by the direct SCE mechanism strongly hints at a rather insignificant role of nucleon transfer, at least in that spectral region, as indeed it is expected from the matching conditions for transfer processes (Brink, 1972). Nevertheless, the contribution of the transfer mechanism to the SCE process would not prevent any extrapolation to DCE reactions, where transfer processes are expected to be safely negligible as experimentally checked (Cappuzzello et al., 2015a; Ferreira et al., in preparation) and because they are processes of a higher order than a direct DCE.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Author Contributions

MCa, JB, SC, CA, SB, FC, DCar, and MCo contributed to conception and design of the study. MCa, SC, CA, FC, DCar, AS, LA, PA-V, TB-L, GB, DCal, VC, EC, IC, FD, HD, CE, PF, SF, MF, AF, MG, AH, FI, RL, NM, MM, JO, AP, LP, HP, FP, GR, OS, SS, VS, GS, DT, ST, AY, and VZ participated to the data taking. SC, ND, and DCar performed the data reduction. JB, SC, HL, JL, and AS performed the calculations. MCa wrote the first draft of the manuscript. JB wrote a section of the manuscript. All authors contributed to manuscript revision, read, and approved the submitted version.

Funding

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). Partial funding by the European Union's Horizon 2020 research and innovation program under Grant No. 654002 is acknowledged. The grants DGAPA-UNAM IN107820, AG101120, and CONACyT 314857, Mexico, FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo, proc. 2019/07767-1), and INCT: Instituto Nacional de Ciência e Tecnologia-Física Nuclear e Aplicações (INCT-FNA, research project 464898/2014-5), Brazil are also acknowledged. HL acknowledges support by INFN and DFG, Grant Le439/16.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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