Optical model potentials for deuteron scattering off ²⁴Mg, ²⁸Si, ⁵⁸Ni, ⁹⁰Zr, ¹¹⁶Sn, and ²⁰⁸Pb at $\sim$100 MeV/nucleon

With the increasing availability of radioactive ion beams of reasonable intensities, there has been enhanced recent interest in the investigation of giant resonances (GR), which are highly collective oscillations of atomic nuclei. In particular, the isoscalar giant monopole resonance (ISGMR) in nuclei far from the stability line has the potential to make significant strides in our understanding of nuclear incompressibility, $K_{\infty}$, and especially the asymmetry term, $K_{\tau}$ [1]. Such measurements have to be performed in inverse kinematics, with deuterium and helium gases being the best targets available so far. While inelastic scattering of $\alpha$ particles has been the mainstay of such studies for a long time now [1], there has not been much work with deuterons since the mid-70’s primarily because of various experimental constraints [2, 3]. On the other hand, most of the measurements of GR with radioactive ion beams so far have employed active-target time projection chambers (AT-TPC) with deuterium as the component gas [4, 5, 6, 7]. Because of paucity of GR data with deuterons, it was important to validate in known cases the results obtained in $(d,d^{\prime})$ measurements via direct comparison with results of inelastic $\alpha$ scattering. Such a detailed investigation was carried out by measuring small-angle inelastic deuteron scattering off ¹¹⁶Sn and ²⁰⁸Pb, and it was established that the extracted ISGMR strength distributions, using a multipole decomposition analysis similar to that done for inelastic $\alpha$ scattering, agree very well with those deduced from the $(\alpha,\alpha^{\prime})$ data [8].

Such analyses hinge upon obtaining good-quality optical model (OM) parameters from elastic scattering data. The OM potential is widely used to generate wave functions for elastic scattering. These wave functions serve, in turn, as the door waves, widely known as the distorted waves, for the analysis of other direct reaction channels within the distorted-wave Born approximation (DWBA) or coupled-channel formalism. The inelastic scattering leading to the excitation of collective states of the target nucleus is essential for the determination of the underlying nuclear structure properties from the measured angular distributions. The nuclear interaction potential is, however, inherently complicated. Therefore, the phenomenological optical potential (OP) of a simple functional form is often employed in the OM and DWBA calculations to describe direct nuclear reactions and to extract nuclear properties from the angular distribution data. It represents a simple “effective” interaction used in the collective model of nuclear scattering [9] to describe both the elastic and inelastic scattering channels.

For elastic deuteron-nucleus scattering, the existing OM studies of the deuteron OP are mainly based on the phenomenological potential model using the Woods-Saxon (WS) functional form as described, for example, in Ref. [10]. This simple OP was successfully used to describe the elastic as well as inelastic angular distributions of deuteron scattering off heavy nuclei (A $\gtrsim$ 40), based on the collective model of nuclear scattering [9]. However, the collective model approach seems to overestimate the (one-step) transition probabilities from the ground state to the low-lying excited states of some light nuclei. For example, as noted in Ref. [11], the OM parameter set that gives the best fit of the elastic angular distribution requires 40% lower value of the deformation parameter $\beta_{2}$, as compared to the adopted value, to reproduce the experimentally observed 2${}^{+}_{1}$ angular distribution in the nucleus ¹⁶O.

In this work, we report on the elastic and inelastic deuteron-nucleus scattering measurements at the beam energy of 98 MeV/nucleon for several targets, ranging from light- to medium- and heavy-mass nuclei. The measured $(d,d)$ and $(d,d^{\prime})$ scattering data have been analyzed within the OM and DWBA, respectively, using both the WS phenomenological potential model as well as a semi-microscopic deuteron-nucleus potential obtained in the double-folding model [12, 13, 14]. The extracted OM parameters are expected to find use in analyzing the giant resonance data with radioactive ion beams.

The experiments were performed at the ring cyclotron facility of the Research Center for Nuclear Physics (RCNP), Osaka University, Japan. The $E_{d}=196$ MeV deuteron beam was scattered off six highly enriched (more than 90%) self-supporting targets listed in Table 1. Elastic and inelastic deuteron-nucleus scattering measurements were made over an angular range of $\theta_{\rm lab}\sim 3.5^{\circ}-32^{\circ}$. Each experimental angular opening ($\sim 2^{\circ}$-wide) was subdivided into three parts for the analysis, each corresponding to a solid angle of 0.42 msr.

The scattered particles were momentum analyzed by the magnetic spectrometer Grand Raiden and focused onto the focal-plane detector system [17] consisting of two multi-wire drift chambers (MWDC) and two plastic scintillators [18]. The time-of-flight and energy-loss techniques were used for the identification of the scattered particles. Grand Raiden was used in the double-focusing mode in order to identify and eliminate practically all instrumental background from the final spectra [8]. Particle tracks were reconstructed using the ray-tracing technique described in Refs. [17, 19]. This, in turn, allowed for the reconstruction of the scattering angle. The experimental angular resolution was $\sim$ 0.15^∘, including the nominal broadening of the scattering angle due to the emittance of the beam and the multiple Coulomb scattering effects. Further details of the experimental and data analysis procedures have been provided in Ref. [8, 20].

The two available global deuteron-nucleus optical potentials were developed by Daehnick et al. [10] and Bojowald et al. [21] in the 1980s using the phenomenological WS form for both the real and imaginary OP, covering the mass range of 27 $\leq$ A $\leq$ 238 and energy range of $E_{d}\sim 10$ MeV–90 MeV. The deuteron carries one unit of spin ($\mbox{\boldmath$s$}=\mbox{\boldmath$1$}$) in its ground state, and this requires the inclusion of a real spin-orbit term $V_{\ell s}$ into the OP. For the imaginary OP, in addition to the volume absorption $W$, an imaginary surface term $W_{D}$ is also included to account for the surface absorption which is significant due to the deuteron breakup [22, 23]. Thus, the total OP is determined explicitly as

and $V_{C}(r)$ is the Coulomb potential of a uniformly charged sphere of radius $R_{C}=1.3~{}A^{1/3}$ fm. The functional form f is chosen in the WS form for all the terms

The phenomenological OM potential given by Eqs. (1)-(6) is used in the OM analysis of the measured elastic $(d,d)$ scattering data, and to generate the distorted waves for the DWBA description of the inelastic $(d,d^{\prime})$ scattering data. The associated OM parameters were obtained from a $\chi^{2}$-minimization fit to the elastic $(d,d)$ data using the code ECIS97 [24], and they are given explicitly in Table 2. The global deuteron OPs [10, 21] were used for the parameter initialization in the $\chi^{2}$ search.

The OM results obtained using the best-fit OP parameter set are shown as dashed lines in Figs. 1 and 2, in comparison with elastic $(d,d)$ scattering data measured at $E_{d}=196$ MeV for the targets under study.

The DWBA analysis of inelastic deuteron-nucleus scattering data is restricted to the direct one-step excitation of a collective state of the target, neglecting the contributions of indirect two- or three-step transitions and the channel coupling effects. The $(d,d^{\prime})$ scattering cross section is calculated using the following DWBA inelastic scattering amplitude [25]:

where the distorted waves $\chi^{\pm}$ are generated by the OP (1)-(6). The transition matrix element of the deuteron-nucleus interaction $\langle dA^{\prime}|V_{d-A}|dA\rangle$ is dubbed as the inelastic scattering form factor (FF). A widely adopted method is to obtain the nuclear inelastic scattering FF by radially deforming nuclear part of the OP, based on the collective vibrational (or rotational) model of nuclear scattering [9]. For the $(d,d^{\prime})$ scattering to a $2^{\lambda}$-pole collective excited state of target, the inelastic scattering FF is determined as

Prescription (9)-(10) is known as the deformed optical potential (DOP) method to generate the inelastic nuclear scattering FF. The spin-orbit contribution (10) to the total $(d,d^{\prime})$ cross section is minor, but it helps to better reproduce the observed oscillation pattern of inelastic $(d,d^{\prime})$ cross section for medium mass targets. For an isoscalar excitation, the nuclear deformation length $\delta^{(N)}_{\lambda}$ is usually assumed to be the same as the proton deformation length $\delta^{(p)}_{\lambda}$, determined from the reduced transition rate $B(E\lambda\uparrow)$ [25] as

The Coulomb inelastic FF in Eq. (8) can be approximately obtained in a “model independent” form [25]:

Thus, the OP parameters (1)-(6) and the reduced transition rate $B(E\lambda\uparrow)$ of the target excitation are the main inputs for the DWBA analysis of inelastic $(d,d^{\prime})$ scattering using the collective model of nuclear scattering [9] associated with the phenomenological OP). The DWBA calculation of the $(d,d^{\prime})$ scattering cross section has been done using two choices of the $B(E\lambda)$:

i) $B(E\lambda\uparrow)$ is fixed to the experimentally adopted transition rates [26, 27, 28], denoted as $B(E\lambda)_{\rm adopted}$ in Table 3;

ii) $B(E\lambda\uparrow)$ is deduced from the best DWBA fit of the calculated $(d,d^{\prime})$ cross section to the $(d,d^{\prime})$ scattering data and denoted as $B(E\lambda)_{\rm DOP}$ or $B(E\lambda)_{\rm DFM}$ in Table 3.

The DWBA description of inelastic $(d,d^{\prime})$ scattering data measured for the excitation of 2${}^{+}_{1}$ states of ²⁴Mg, ²⁸Si, ⁵⁸Ni targets, and 3${}^{-}_{1}$ state of ²⁰⁸Pb target given by the collective-model inelastic FF (8)-(12) is shown in Fig. 3. As can be seen, the 3${}^{-}_{1}$ angular distribution measured for ²⁰⁸Pb target is reproduced well with the adopted $B(E3)_{\rm adopted}$ value [28]. However, the DWBA description of the measured 2${}^{+}_{1}$ cross section using the adopted $B(E2)$ value, $B(E2)_{\rm adopted}$, seems to get worse for light ²⁴Mg and ²⁸Si nuclei, where the calculated $(d,d^{\prime})$ cross section overestimates data at small angles, and then becomes non-oscillatory for angles above $\sim$ 25^∘. The best DWBA fit to inelastic $(d,d^{\prime})$ data given by the collective-model FF requires $B(E2)_{\rm DOP}$ value of around 40% lower than $B(E2)_{\rm adopted}$ value. This likely indicates a deficiency of the collective model of nuclear scattering [9] based on the phenomenological OP (1)-(6) for light-mass nuclei, as also found earlier for ¹⁶O in Ref. [11].

The DWBA has been proven to be a reliable approximation for the direct reaction studies at energies around or above 100 MeV/nucleon. Therefore, a good agreement of the $(d,d^{\prime})$ cross section calculated in the DWBA with $(d,d^{\prime})$ data measured for 2${}^{+}_{1}$ and 3${}^{-}_{1}$ states of target must be a prerequisite for the validation of the phenomenological OP (1)-(6). In the present work, the DWBA analysis of inelastic deuteron scattering off ⁹⁰Zr and ¹¹⁶Sn targets was chosen as test ground for the phenomenological deuteron-nucleus OP. The DWBA results obtained for ¹¹⁶Sn and ⁹⁰Zr targets are shown in Figs. 4 and 5, respectively. At variance with light targets, the DWBA cross sections given by the collective model inelastic FF based on the best-fit OP and adopted $B(E2)_{\rm adopted}$ values [26, 27] agree well with the measured 2${}^{+}_{1}$ angular distribution for both ⁹⁰Zr and ¹¹⁶Sn targets. The best DWBA fit yields $B(E2)_{\rm DOP}$ values around 5% and 15% larger than $B(E2)_{\rm adopted}$ values for ⁹⁰Zr and ¹¹⁶Sn, respectively (see Table 3).

The situation is different, on the other hand, for the 3${}^{-}_{1}$ angular distributions measured for these targets. While the DWBA calculation using the collective model inelastic FF based on the adopted $B(E3)_{\rm adopted}$ value [28] describes well the $(d,d^{\prime})$ data measured for the 3${}^{-}_{1}$ state of ¹¹⁶Sn, the same prescription overestimates the 3${}^{-}_{1}$ angular distribution measured for the ⁹⁰Zr target over the whole angular range. Given $B(E3)_{\rm adopted}\approx 0.098\ e^{2}$b³ adopted from $(e,e^{\prime})$ data, and the empirical $B(E3)$ values of 0.037 to 0.079 $e^{2}$b³ deduced from inelastic nucleon- and light-ion scattering data (see Table V in Ref. [28]), such a disagreement with the 3${}^{-}_{1}$ angular distribution for ⁹⁰Zr would not be unexpected. Indeed, the best DWBA fit to the measured 3${}^{-}_{1}$ cross sections yields a lower value of $B(E3)_{\rm DOP}\approx 0.051\ e^{2}$b³.

The microscopic description of the nucleus-nucleus interaction is usually based on a chosen effective pairwise nucleon-nucleon (NN) interaction $v$ between projectile nucleon and bound nucleon of target. The density dependence of $v$ presumably takes into account the three-nucleon interaction and higher-order NN correlations. Among such models, the double-folding model (DFM) has been used widely to calculate the $\alpha$-nucleus and heavy-ion OP [12, 13, 14]. The established success of the DFM in describing the observed elastic scattering for many nucleus-nucleus systems indicates that it produces the dominant part of the nucleus-nucleus OP. Within the DFM, the deuteron-nucleus scattering potential can be formally determined [14] as

where the diagonal ($A^{\prime}=A$ ) and nondiagonal ($A^{\prime}\neq A$) matrix elements describe elastic and inelastic deuteron-nucleus scattering, respectively. The antisymmetrization $\mathcal{A}$ accounts for all single-nucleon exchanges between deuteron and target, giving rise to a nonlocal exchange term of the potential (13)

where $v_{\rm D(EX)}$ is the direct (exchange) part of $v$. Given the nonlocal exchange potential, one has to solve an integro-differential OM equation involving a nonlocal kernel of the deuteron-nucleus OP, which has not been done so far due the complexity of the nonlocal DFM computation. In fact, the exact solution of the OM equation with the nonlocal folded OP was obtained only for elastic nucleon scattering (see, e.g., Ref. [16] and references therein). Like previous DFM calculations [29, 14], we have used in the present work the well-proven local approximation for the exchange potential, based on the WKB approximation for the change in the deuteron-nucleus relative motion wave function induced by the exchange of spatial coordinates of each interacting nucleon pair (see more details in Ref. [15]). In this case, both the direct and exchange terms of the deuteron-nucleus potential (14) are evaluated [14] by folding the deuteron and target densities of with the chosen effective NN interaction $v$. The accuracy of such a WKB approximation was shown again in a recent OM study of elastic nucleon-nucleus scattering [16]. With a localized exchange potential, not only the OM calculation is much simpler, but also the comparison with the (local) phenomenological OP (1)-(6) is more direct which is essential for the present study.

Elastic and inelastic deuteron scattering have been measured off ²⁴Mg, ²⁸Si, ⁵⁸Ni, ⁹⁰Zr, ¹¹⁶Sn, and ²⁰⁸Pb at an energy of 98 MeV/nucleon. The measured $(d,d)$ and $(d,d^{\prime})$ angular distributions were analyzed within the OM and DWBA using the phenomenological deuteron OP associated with the collective model of nuclear scattering, and the hybrid potential model for the deuteron OP and inelastic FF based on the microscopic DFM calculation. The $E2$ and $E3$ transition rates of the $2^{+}_{1}$ and $3^{-}_{1}$ excitations of these target nuclei have been deduced from the best DWBA fits to the $(d,d^{\prime})$ scattering data, which agree reasonably with the adopted $B(E\lambda)$ values [26, 28].

The results of the OM analysis of elastic $(d,d)$ data using the hybrid folded OP show that the deuteron breakup does not affect significantly the volume part of the deuteron OP given by the DFM calculation. However, the breakup effect seems to imply an enhanced absorption at the surface, which can be taken into account effectively by a surface WS potential added to the imaginary deuteron OP.

While both potential models describe equally well the $(d,d)$ and $(d,d^{\prime})$ angular distributions measured for medium- and heavy-mass target nuclei, the DWBA calculation using the collective-model inelastic FF (DOP) gives a poorer description of the $2^{+}_{1}$ angular distribution measured for the light-mass ²⁴Mg and ²⁸Si nuclei, in comparison with the DWBA description of the same data using the semi-microscopic folded inelastic FF (DFM).

The DWBA analysis of the $3^{-}_{1}$ angular distribution measured for ⁹⁰Zr target indicates that $B(E3)_{\rm adopted}$ value from $(e,e^{\prime})$ data [28] is too high, and the DWBA results given by the two potential models agree well with $(d,d^{\prime})$ scattering data, with the best-fit $B(E3)$ values close to those deduced from inelastic nucleon- and light-ion scattering data.

A hindrance of the $E3$ transition rate determined by the collective-model inelastic FF compared to that determined by the inelastic folded FF was also found, which illustrates the inconsistency between the DWBA description of nuclear excitation with $\lambda\gtrsim 3$ based on the DOP approach and that based on the DFM approach discussed earlier by Beene et al. [40].

We acknowledge the efforts of the RCNP staff in providing high-quality deuteron beams required for these measurements, and thank A. Okamoto (Konan), T. Sako (Kyoto) and K. Schlax (Notre Dame) for their assistance with the experiment. This work has been supported in part by the National Science Foundation (Grants No. PHY-1068192, No. PHY-2011890, and No. PHY-2310059). Two of us (D.C.C. and D.T.K.) were supported by the National Foundation for Science and Technology Development of Vietnam (NAFOSTED Project No. 103.04-2021.74).