Coulomb-nuclear dynamics in the weakly-bound ⁸Li breakup

As mentioned in the introduction, in our numerical approach we use the CDCC formalism, in which we model the projectile ${}^{8}{\rm Li}$ as ${}^{7}{\rm Li}$ core nucleus, to which a neutron is loosely bound with ground-state energy $\varepsilon_{b}=2.03\,{\rm MeV}$. This state is defined in the core-neutron centre-of-mass (c.m.) by $n=1$, $\ell_{0}=$1, $\tilde{\j}_{0}^{\pi}=2^{+}$ quantum numbers, where $n$ stands for the radial state, $\ell_{0}$ the orbital angular momentum and $\tilde{\j}_{0}^{\pi}$ the projectile total angular momentum with parity $\pi$. It is obtained by applying the usual spin-orbit coupling ${\bm{j}}_{0}={\bm{\ell}}_{0}+{\bf 1/2}$; $\tilde{\bm{\j}}_{0}={\bm{j}}_{0}+{\bm{I}}_{c}$, with the core spin $I_{c}={3}/{2}$. In addition to the ground state, an excited bound state with energy $\varepsilon_{\rm ex}=0.98\,{\rm MeV}$ (located in the $\tilde{\j}_{0}^{\pi}=1^{+}$ state Nut10 ) was also considered in our coupling scheme. We would like to emphasize that we are not considering possible core excitations in our calculations.

In this formalism, we first consider the expansion of the three-body wave function on the projectile internal states. After that, by introducing the three-body expansion into the corresponding Schrödinger equation, a one-dimensional radial set of coupled differential equations can be derived for the radial wave-function components $\chi_{\alpha}^{LJ}(R)$, in terms of the projectile-target c.m. coordinate $R$, which is given by

where $L$ is the orbital angular momentum associated with $R$, $J$ is the total angular momentum, and $\mu_{pt}$ the projectile-target ($pt$) reduced mass. The total energy is given by $E$, with $\varepsilon_{\alpha}$ being the projectile bin energies. The index $\alpha$ appearing in the equation is representing a set of quantum numbers describing the projectile states, as given by $\alpha\equiv(i,\ell,s,j,I_{c},\tilde{\j})$, $i=0,1,2,\ldots,N_{b}$ ($N_{b}=$ number of bins).

With the projectile-target potential given as a sum of the core-target ($ct$) and neutron-target ($nt$) terms, i.e., $U_{pt}({\bm{r}},{\bm{R}})=U_{ct}({\bm{R}}_{ct})+U_{nt}({\bm{R}}_{nt})$, where ${\bm{R}}_{ct}\equiv{\bm{R}}+\frac{1}{8}{\bm{r}}$ and ${\bm{R}}_{nt}\equiv{\bm{R}}-\frac{7}{8}{\bm{r}}$ (with ${\bm{r}}$ being the projectile internal coordinate), the potential matrix elements $U_{\alpha\alpha^{\prime}}^{LL^{\prime}J}(R)$ in (II.1) are given by its Coulomb and nuclear parts, such that

where $\mathcal{Y}_{\alpha L}({\bm{r}},\Omega_{R})\equiv[\hat{\Phi}_{\alpha}({\bm{r}})\otimes{\rm i}^{L}Y_{L}^{\Lambda}(\Omega_{R})]_{JM}$ is the direct product of the angular part of ${\bm{R}}$ with the projectile channel wave function, $\hat{\Phi}_{\alpha}({\bm{r}})$, which contains the square integrable discretized bin wave functions. The nuclear terms express the sums of real and imaginary parts. The former are responsible for the nuclear dissociation, whereas the latter accounts for the nuclear absorption. These nuclear terms are, respectively, given by $U_{ct}^{nucl}({\bm{R}}_{ct})=V_{ct}^{nucl}({\bm{R}}_{ct})+{\rm i}W_{ct}^{nucl}({\bm{R}}_{ct})$ and $U_{nt}^{nucl}({\bm{R}}_{nt})=V_{nt}^{nucl}({\bm{R}}_{nt})$+ ${\rm i}W_{nt}^{nucl}({\bm{R}}_{nt})$, with the Woods-Saxon shape being adopted for both components. The diagonal coupling matrix elements $U_{\alpha\alpha}^{LLJ}(R)$, contain the monopole nuclear term in the projectile-target c.m., which we denote by $V_{\beta_{0}\beta_{0}}^{LJ}(R)=\langle\Phi_{\beta_{0}}({\bm{r}})|U_{ct}^{nucl}+U_{nt}^{nucl}|\Phi_{\beta_{0}}({\bm{r}})\rangle$, where $\beta_{0}$ represents the set of ground-state projectile quantum numbers, $\beta_{0}\equiv(k_{0},\ell_{0},s,j_{0},I_{c},\tilde{\j}_{0})$. The imaginary part accounts for the absorption in the projectile-target c.m. motion.

The separation of the Coulomb and nuclear interactions to obtain the Coulomb and nuclear breakup cross sections ($\sigma_{Coul}$ and $\sigma_{nucl}$, respectively) remains a challenge in nowadays theories, making an accurate description of the Coulomb-nuclear interference a more tricky task. For that, in this work we resort to an approximate approach, as follows: The nuclear breakup cross sections, defined as $\sigma_{nucl}$, are obtained by including in the coupling matrix elements, the nuclear components of $U_{ct}$ and $U_{vt}$ potentials, plus the diagonal monopole Coulomb potential. On the other hand, the Coulomb breakup cross sections, defined as $\sigma_{Coul}$, are obtained by including in the matrix elements the Coulomb component of the projectile-target potential, i.e., $V_{ct}^{Coul}(R_{ct})$ (as $V_{nt}^{Coul}=0$), plus the monopole nuclear potential. The total breakup cross sections $\sigma_{tot}$ are obtained by including the full $U_{pt}$ potential in the calculations.

Since the early works on Coulomb and nuclear breakup studies Thomp20 ; 1999Th , this approach has been widely adopted to study Coulomb and nuclear breakup cross sections, as one can follow from the review 2015Canto (and references therein). In Ref. Pierre100 , where different methods are considered in order to decompose the total breakup into its Coulomb and nuclear components, this approach is also referred as weak-coupling approximation. Two methods emerged from their discussion, which they refer to as method 1 and method 2. The weak-coupling approximation is very close to method 1 for nuclear breakup, and close to method 2 for Coulomb breakup. While this approximate procedure will not completely eliminate the ambiguities surrounding the separation of the total breakup cross section into its Coulomb and nuclear components (as also outlined in Ref.Pierre100 ), we believe that it is particularly justified in the present work, since by using the ¹²C target, the breakup is naturally dominated by nuclear dissociation, whereas by using the ²⁰⁸Pb target the breakup is dominated by Coulomb dissociation.

Once the matrix elements (II.1) are computed, the coupled Eq. (II.1) is solved with the usual asymptotic conditions, which for $k_{\alpha}\equiv\sqrt{{(2\mu_{pt}/\hbar^{2})(E-\varepsilon_{\alpha})}}$ is given by

where $H_{\alpha}^{\mp}(k_{\alpha}R)$ are the usual incoming (-) and outgoing (+) Coulomb Hankel functions Abramo100 , with $S_{\alpha\alpha^{\prime}}(k_{\alpha})$ being the scattering S-matrix elements. Due to the short-range nature of nuclear forces, the matrix elements corresponding to the nuclear interaction in Eq. (II.1) will vanish at large distances, ${R\gg R_{n}}$, where

determines the range of the nuclear forces ($r_{0}$ being the nucleon size, with $r_{0}A_{p}^{1/3}$ and $r_{0}A_{t}^{1/3}$ the projectile and target sizes, respectively). The function $\delta_{R}(\varepsilon_{b})$ is introduced to take into account the well-known effect which occurs in weakly-bound systems (low breakup thresholds), as in halo nuclei, in which the nuclear forces can be stretched beyond $R_{0}=r_{0}(A_{p}^{1/3}+A_{t}^{1/3})$. The various breakup cross sections are obtained by using the relevant S-matrix, as outlined for example in Ref. Thompson100 .

At large distance ($R\to\infty$), Eq.(II.1) contains only the Coulomb interaction, which can be expanded as Hussein50

where $Z_{t}e$ is the target charge, with $\lambda$ the multipole order truncated by $\lambda_{\rm max}$. $\mathcal{O}_{\lambda}^{\epsilon}({\bm{r}})$ is the projectile electric operator, given by

where $Z_{c}e$ is the charge of the projectile core, with $Z_{\lambda}$ being defined as the effective charge. The projectile electric transition probability for the transition from the projectile ground-state to the continuum states can be obtained through $\mathcal{O}_{\lambda}^{\epsilon}({\bm{r}})$ Bertulani50 . For excitation energies $\varepsilon$, the corresponding variation of the electric transition probability $B(E\lambda)$ can be written as

where ($\beta$) refers to the set of quantum numbers in the continuum states $\beta\equiv(k,\ell,s,j,I_{c},\tilde{\j})$], $k=\sqrt{2\mu_{cn}\varepsilon/\hbar^{2}}$, $k_{0}=\sqrt{2\mu_{cn}|\varepsilon_{0}|/\hbar^{2}}$, with $\mu_{cn}$ the core-neutron reduced mass. By defining $\hat{l}\equiv\sqrt{2l+1}$ for general angular quantum numbers, from the above we obtain

where $u_{k_{0}\ell_{0}}^{\tilde{\j}_{0}}(r)$, and $u_{k\ell}^{\tilde{\j}}(r)$ are the ground-state and continuum radial wave functions. The Eqs. (5)-(8) are indicating how the Coulomb breakup is being affected by the projectile structure.

The energies and corresponding wave functions which appear in the set of coupled differential equations (II.1), for the bound and continuum states of the ⁷Li+n system, are obtained by considering a two-body Woods-Saxon potential as input, whose parameters are the same as in Ref. Moro200 . The depth $V_{0}$ of the central part of the potential was adjusted to reproduce the ground and excited bound-state energies. These parameters are summarized in Table 1.

Similarly, the other binding energies considered in this work are obtained by adjusting $V_{0}$. The same ground-state potential parameters are adopted to calculate the corresponding continuum wave functions. With these potential parameters, we first calculate the electric transition probability $B(E1)$ variation with the excitation energy $\varepsilon$, given by Eq.(8), corresponding to the transition from the ground-state to continuum $s$- plus $d$-states, for the binding energies $\varepsilon_{b}=0.01\,{\rm MeV},1.0\,{\rm MeV}$ and $2.03\,{\rm MeV}$. The results are shown in the upper panel of Fig.1. One notices that $B(E1)$ varies substantially for $\varepsilon_{b}=0.01\,{\rm MeV}$ as compared with values obtained for larger $\varepsilon_{b}$. These results highlight the strong dependence of the Coulomb breakup on the projectile internal structure, particularly in the asymptotic region. In this regard, it is also instructive to verify how the projectile root-mean-square radii $\sqrt{\langle r^{2}\rangle}$ vary with the projectile ground-state binding energies. For that, we add the lower panel of Fig. 1, with the corresponding root-mean-square radii, obtained for the projectile ground-state $\Phi_{\beta_{0}}(\bm{r})$. As expected, the root-mean-square radii behavior is reflecting the large increasing of the wave function as the binding energy comes close to zero. Also, for $\varepsilon_{b}=2.033$ MeV, we note that we obtain $\sqrt{\langle r^{2}\rangle}=2.39$ fm, in very close agreement with the corresponding values reported in Refs. 2015Fan and 1988Tanihata (respectively, $\sqrt{\langle r^{2}\rangle}=2.39\pm 0.05$ fm and $\sqrt{\langle r^{2}\rangle}=2.37\pm 0.02$ fm).

In order to evaluate the coupling matrix elements of Eq. (II.1), fragments-target optical potentials are needed. The ⁷Li$+^{12}$C optical potential parameters were taken from Ref. Barioni100 , whereas the ⁷Li$+^{208}$Pb optical potential parameters were obtained from the ⁷Li global potential of Ref. Cook300 , with the depth of the real part slightly modified to fit the elastic scattering experimental data. For the $n-$target optical potentials, we adopted the global potential of Ref. 1969Green . The CDCC limiting values of the model space parameters, used for the numerical solution of Eq.(II.1), are listed in Table 2, for the two targets we are considering, ¹²C and ²⁰⁸Pb, where $\ell_{\rm max}$ is the maximum angular momentum between ${}^{7}{\rm Li}$ and the neutron, $\lambda_{\rm max}$ is the maximum order of the potential multipole expansion, $\varepsilon_{\rm max}$ is the maximum bin energies, $r_{\rm max}$ is the maximum matching radius for bin potential integration, $L_{\rm max}$ is the maximum angular momentum of the relative c.m. motion, and $R_{\rm max}$ is the maximum matching radius of the integration for the coupled differential equations, with $\Delta R$ the corresponding $R-$step size. The reported main values are found to give enough converged results for $\varepsilon_{b}\geq$0.4 MeV. However, as we decrease the projectile binding energy, for $\varepsilon_{b}\leq 0.08$MeV, to guarantee enough good convergence and precision of the results we found necessary to increase the maximum values for the projectile matching radius $r_{\rm max}$, for the matching radius $R_{\rm max}$, and for the relative angular momentum of the c.m. motion, $L_{\rm max}$, correspondingly to each of the target. These values for smaller $\varepsilon_{b}$ are shown within parenthesis, below the respective values obtained for larger $\varepsilon_{b}$. The adopted bin widths were, $\Delta\varepsilon=0.5\,{\rm MeV}$, for $s$- and $p$-states, $\Delta\varepsilon=1.0\,{\rm MeV}$, for $f$- and $d$-states and $\Delta\varepsilon=1.5\,{\rm MeV}$ for ${\rm g}$-states.

We start the first part of this section by analyzing the dependence of the elastic scattering cross sections on the projectile ground-state binding energy. These cross sections are displayed in Fig. 2 for ${}^{12}{\rm C}$ target; and in Fig. 3 for ${}^{208}{\rm Pb}$ target, considering two incident energies. In both the cases, we assume different values of $\varepsilon_{b}$, from the experimental one down to 0.01 MeV. In the case of ${}^{12}{\rm C}$ target, which is a nuclear-dominated reaction, from the results shown in Fig. 2 one can observe a weak dependence on $\varepsilon_{b}$ in the range $0.4\,{\rm MeV}\leq\varepsilon_{b}\leq 2.03\,{\rm MeV}$, for both incident energies, $E_{lab}=$14 MeV [panel (a)] and 24 MeV [panel (b)]. However, it becomes relatively significant for $\varepsilon_{b}\leq 0.08\,{\rm MeV}$ [see panel (a)]. Also shown in Fig. 2 is that the experimental data are well reproduced by the model for both incident energies.

For the Coulomb-dominated reaction with ${}^{208}{\rm Pb}$, the results given in Fig. 3 for $E_{lab}=$36 MeV (a) and 60 MeV (b)] are indicating strong dependence of the elastic scattering cross sections on all binding energies at forward angles (asymptotic region), where the Coulomb breakup is particularly dominant. However, at backward angles (short distance), where the nuclear breakup is expected to provide meaningful effects, the elastic cross sections become almost independent of the binding energy.

These results lead to a conclusion that, when the nuclear breakup is dominant or relatively significant, the effect of the binding energy on the elastic scattering cross section is rather small, whereas it is more pronounced when the Coulomb breakup is dominant. Therefore, since a relatively significant effect for the ${}^{8}{\rm Li}+{}^{12}{\rm C}$ reaction is observed when $\varepsilon_{b}\leq 0.08\,{\rm MeV}$, it is possible that the ${}^{8}{\rm Li}+{}^{12}{\rm C}$ reaction is already dominated by the Coulomb breakup for $\varepsilon_{b}\leq 0.08\,{\rm MeV}$. As the binding energy decreases, the Coulomb breakup becomes dominant over its nuclear counterpart, as anticipated. It also follows that the probability of the projectile to fly on the outgoing trajectory unbroken decreases, diminishing the corresponding elastic scattering cross section. In the next section, we will look into this observation in more detail.

The differential total, Coulomb and nuclear breakup cross sections, for the ${}^{12}{\rm C}$ target, are depicted in Fig. 4, for $E_{lab}=$14 MeV [(a)-(e) panels] and $E_{lab}=$24 MeV [(f)-(j) panels]. As anticipated, in the case of nuclear-dominated reactions, for both incident energies, $d\sigma_{nucl}/d\Omega$ $\simeq d\sigma_{tot}/d\Omega$ $\gg d\sigma_{Coul}/d\Omega$ as $\varepsilon_{b}\to 2.03\,{\rm MeV}$, with $d\sigma_{Coul}/d\Omega\to 0$. However, it is interesting to notice that as $\varepsilon_{b}$ decreases, the Coulomb breakup increases rapidly, such that for $\varepsilon_{b}\to 0.01\,{\rm MeV}$, $d\sigma_{nucl}/d\Omega$ $\ll d\sigma_{Coul}/d\Omega$ $\simeq d\sigma_{tot}/d\Omega$ at forward angles, for both incident energies. On the light of these results it follows that as the binding energy further decreases, the Coulomb breakup becomes more relevant, and comparable with the total breakup even in such a naturally nuclear-dominated reaction. This can be attributed to the fact that the breakup becomes more peripheral as $\varepsilon_{b}$ decreases, where only Coulomb forces are available. Hence, the importance of the Coulomb breakup in this case relies mainly on the long-range behavior of the Coulomb forces, and on its direct dependence on the electromagnetic transition matrix elements, in agreement with our assessment in Sect. III.1. Furthermore, these results show that the “nuclear-dominated reaction” concept may be relative to the projectile binding energy.

As the projectile binding energy varies from 2.03 MeV down to 0.01 MeV, one may wonder how relevant higher-order partial-waves ($\ell$) are in the breakup process for such very low binding energy, particularly for heavy targets. In order to verify the importance of higher-order partial-waves in this case, we performed a convergence test of the total, Coulomb and nuclear differential breakup cross sections for ²⁰⁸Pb target at $E_{lab}=36$ MeV. The different breakup cross sections are shown in Fig. 5, as functions of the c.m. angle $\theta$, for different maximum projectile internal angular momenta $\ell_{\rm max}$, and only for $\varepsilon_{b}=0.01$ MeV and 2.03 MeV binding energies. As evidenced by the results in this figure, there is no meaningful difference between $\ell_{\rm max}=4$ and $\ell_{\rm max}=7$, regardless the binding energy. This implies that, by reducing the ground-state binding energy, the convergence of the breakup cross sections is not affected, in respect to the maximum core-neutron orbital angular momentum $\ell_{\rm max}$.

In Fig.6, displays, the total, Coulomb and nuclear breakup angular cross-section distributions as functions of the c.m. angle $\theta$, for the different binding energies $\varepsilon_{b}$, for ⁸Li$+^{208}$Pb reaction. We first observe that as $\varepsilon_{b}$ decreases, the peaks of $d\sigma_{tot}/d\Omega$ and $d\sigma_{Coul}/d\Omega$ are shifted to forward angles. In fact, for $\varepsilon_{b}\leq$ 0.08 MeV, the peaks are located close to zero degree. This is a clear display of the peripheral nature of the breakup process as $\varepsilon_{b}$ decreases. A careful look at this figure also indicates that as $\varepsilon_{b}$ decreases, even the peak of $d\sigma_{nucl}/d\Omega$ is shifted to forward angles, which may suggest that even the nuclear breakup process becomes peripheral as $\varepsilon_{b}\to$ 0.01 MeV. The peripherality of the nuclear breakup in this case, can be understood by considering the function $\delta_{R}(\varepsilon_{b})$, which appears in Eq.(4). The nuclear breakup dynamics require that $\delta_{R}(\varepsilon_{b})\to 0$, as $\varepsilon_{b}$ increases, implying that $R_{n}\to R_{0}$, due to the short-range nature of nuclear forces. However, as $\varepsilon_{b}\to 0$, $\delta_{R}(\varepsilon_{b})$ increases and so does $R_{n}$, leading to a significant nuclear effect in the peripheral region. Therefore, the function $\delta_{R}(\varepsilon_{b})$ is introduced to take into account the well-known effect which occurs in weakly-bound systems, as in halo nuclei, in which the nuclear forces are stretched beyond the usual range.

Quantitatively, since this ⁸Li$+^{208}$Pb reaction is Coulomb-dominated, we observe that at forward angles both $d\sigma_{tot}/d\Omega$ and $d\sigma_{Coul}/d\Omega$ are substantially larger than $d\sigma_{nucl}/d\Omega$ (about three orders of magnitude as $\varepsilon_{b}$ decreases). A further inspection of this figure shows that for $E_{lab}=60$ MeV, we notice that the total and Coulomb breakup cross sections are more similar compared to $E_{lab}=36$ MeV, with the difference coming from the competition between the nuclear and Coulomb interactions above the barrier (for a discussion on the role of the diagonal Coulomb interaction, see also Ref. MukeruPRC2020 ).

In order to better elucidate the importance of the nuclear absorption in the breakup process, we present in Fig. 7, for the ⁸Li+²⁰⁸Pb reaction, the integrated total breakup cross section as well as the total fusion cross sections as functions of $\varepsilon_{b}$. In this regard, we are extending a previous analysis done for this reaction in Ref. 2020Mukeru , in which the total fusion cross sections are shown as functions of the incident energy for different projectile binding energies. The breakup cross section obtained in the presence of nuclear absorption (i.e., $W_{ct}^{nucl}\neq 0$, $W_{nt}^{nucl}\neq 0$), are indicated by the label “BU”. The breakup cross section obtained in the absence of nuclear absorption (i.e, $W_{ct}=W_{nt}=0$), are indicated by “NA”. The total fusion cross section is labeled as “TF”. By observing this figure, it follows that, as $\varepsilon_{b}\to 2.03$ MeV, the nuclear absorption contributes to largely reduce the breakup cross section about one order magnitude in the log-scale. However, we observe that the nuclear absorption plays a minor role on the breakup cross section for smaller binding energies, being negligible for $\varepsilon_{b}\to 0.01$ MeV, in particular at $E_{lab}=60$ MeV. In this case, $\sigma_{\rm NA}\simeq\sigma_{\rm BU}\gg\sigma_{\rm TF}$, ( where $\sigma_{\rm BU}$ is the breakup cross section followed by fragments absorption, and $\sigma_{\rm NA}$ is the breakup cross section without fragments absorption after breakup). A larger breakup cross section over the total fusion cross section can be understood as due to the fact that, when the breakup occurs where classically the trajectory is far away from the target, the projectile fragments have no easy access to the absorption region, thus significantly reducing the flux that contributes to the fusion cross section. However, as expected, as $\varepsilon_{b}\to 2.03$ MeV, where the breakup process occurs closer to the target, where the probability for the projectile fragments to survive absorption is significantly reduced, we observe that $\sigma_{\rm BU}\ll\sigma_{\rm NA}<\sigma_{\rm TF}$. A weak dependence of the total fusion cross section on the binding energy compared to the breakup cross section is also observed. The energy region well above the Coulomb barrier is particularly dominated by the complete fusion process. As shown in Ref. Lei100 , the complete fusion cross section is insignificantly dependent on the projectile $\varepsilon_{b}$ for ⁷Li$+^{209}$Bi reaction. We believe that these observations would be valid for any loosely bound projectile, and hence there is nothing unusual in the breakup of the ${}^{8}{\rm Li}$ nucleus.

Concerning our approach to total fusion (TF) and absorption, let us clarify that: In the standard CDCC method, the optical potentials are chosen to describe the elastic scattering of the fragments by the target. So, their imaginary parts account for the absorption to fusion and other direct channels (surface reactions). Nevertheless, as direct reaction cross sections are expected to be small for the interactions between fragments and targets selected in this work, the TF cross section provides the major contribution to this absorption.

For a better quantitative assessment of these results, we consider the integrated total ($\sigma_{tot}$), Coulomb ($\sigma_{Coul}$), and nuclear ($\sigma_{nucl}$) breakup cross sections, which are displayed as functions of $\varepsilon_{b}$ in Fig. 8 (for the ${}^{12}{\rm C}$ target), and in Fig. 9 (for the ${}^{208}{\rm Pb}$ target). The results in both figures confirm the conclusions already drawn from Figs.4 and 6. For example, both panels of Fig.8, show that as $\varepsilon_{b}\to$ 0.01 MeV, $\sigma_{Coul}>\sigma_{nucl}$ ($\sigma_{Coul}\simeq\sigma_{tot}$), whereas $\sigma_{Coul}<\sigma_{nucl}\simeq\sigma_{tot}$ ($\sigma_{nucl}\simeq\sigma_{tot}$) as $\varepsilon_{b}\to 2.03$ MeV. For ${}^{208}{\rm Pb}$ target, the results are shown in the presence of nuclear absorption. When the nuclear absorption is taken into account [panels (a) and (b)], we notice that $\sigma_{coul}\simeq\sigma_{tot}\gg\sigma_{nucl}$ and this is independent of $\varepsilon_{b}$. In the absence of the nuclear absorption [panels (c) and (d)], while $\sigma_{coul}\simeq\sigma_{tot}\gg\sigma_{nucl}$ remains valid for $\varepsilon_{b}\to 0.01$ MeV, it is noticed that $\sigma_{tot}\simeq\sigma_{nucl}>\sigma_{Coul}$, for $\varepsilon_{b}\to 2.03$ MeV, which further highlights the importance of the nuclear absorption for large binding energies. The results in this figure further support the fact that strong nuclear absorption in the inner region is the main factor that dictates the importance of the Coulomb breakup cross section over its nuclear counterparts.

In Table 3, we provide more quantitative results, given as fractions from $\sigma_{tot}$ and $\sigma_{nucl}$, reflecting the competition between the different cross sections, by selecting the two limiting binding energies we are studying, i.e., $\varepsilon_{b}=$0.01 MeV and $\varepsilon_{b}=$2.03 MeV. We are also including $\sigma_{int}$, as defined by

which we naively regard as the Coulomb-nuclear interference and that will be discussed in the next subsection. From this table, it becomes evident that, when $\varepsilon_{b}$ decreases, $\sigma_{Coul}$ (approaching to $\sigma_{tot}$) becomes substantially larger than $\sigma_{nucl}$. Also, for the light ¹²C target, at $E_{lab}=14$ MeV and 24 MeV, we note that $\sigma_{Coul}/\sigma_{nucl}$ rapidly grows, when varying $\varepsilon_{b}$ from 2.03 MeV down to 0.01 MeV. As shown, in this energy interval, $\sigma_{Coul}/\sigma_{nucl}$ increases from 0.03 to 3.12 for 14 MeV, and from 0.06 to 2.10 for 24 MeV. This indicates that, as the binding energy decreases, the ⁸Li$+^{12}$C reaction becomes like a “Coulomb-dominated reaction”, with the emergence of a long-range behavior. Moreover, with the heavy target at $E_{lab}=$36 MeV, in the presence of nuclear absorption, for $\varepsilon_{b}=$2.03 MeV, $\sigma_{coul}/\sigma_{nucl}=12$, whereas $\sigma_{coul}/\sigma_{nucl}\simeq 90$ for $\varepsilon_{b}=$ 0.01 MeV. It is noticed in this case that this ratio is substantially affected in the absence of nuclear absorption (NA), becoming $\sigma_{coul}/\sigma_{nucl}\simeq 0.06$ ($\varepsilon_{b}=2.03$ MeV), and $\sigma_{coul}/\sigma_{nucl}\simeq 34$ ($\varepsilon_{b}=0.01$ MeV).

It is well-known that the incoherent sum of the Coulomb and nuclear breakup cross section $(\sigma_{Coul}+\sigma_{nucl})$ is always different from their coherent sum, $\sigma_{tot}$, due to the Coulomb-nuclear interference effect. To assess this effect in the context of very weak ground-state binding energy, we consider $\sigma_{int}$ as defined in Eq. (19) to estimate the Coulomb-nuclear interference. Given that, for the two limiting binding energies, the quantitative results for $\sigma_{int}$ are already furnished in Table 3 as ratios with respect to $\sigma_{tot}$ and $\sigma_{nucl}$. In Figs. 10 and 11 (respectively, for ¹²C and ²⁰⁸Pb targets), we provide the exact $\sigma_{int}$ behaviors, together with their respective ratios $\sigma_{int}/\sigma_{tot}$, as functions of $\varepsilon_{b}$, in a way to clarify that the differences between $\sigma_{tot}$ and $(\sigma_{Coul}+\sigma_{nucl})$ are quite large in both the cases, with the amount varying with $E_{lab}$ ($|\sigma_{int}|$ decreasing with increasing $E_{lab}$). The Coulomb-nuclear interference is strongly dependent on $\varepsilon_{b}$. As one can notice, it appears to increase as $\varepsilon_{b}$ decreases, and becomes quite small as $\varepsilon_{b}\to 2.03$ MeV.

For the ⁸Li+²⁰⁸Pb reaction, nuclear absorption which is already shown to reduce the breakup cross section (Fig.7), is expected to be more relevant on the Coulomb-nuclear interference. The Coulomb-nuclear interference obtained when the breakup is followed by nuclear absorption (i.e., $W_{ct}^{nucl}\neq 0,W_{nt}^{nucl}\neq 0$), is denoted by $\sigma_{int}^{\rm WA}$ (WA standing for “with absorption”), and by $\sigma_{int}^{\rm NA}$ the Coulomb-nuclear interference obtained when $W_{ct}^{nucl}=W_{nt}^{nucl}=0$. Therefore, in order to assess the relevance of the nuclear absorption on this interference, we compare $\sigma_{int}^{\rm WA}$ with $\sigma_{int}^{\rm NA}$. The results are presented in Fig.11. In this figure, the panels (a) and (b) are for the exact $\sigma_{int}$ results, whereas in panels (c) and (d) we have the respective ratios $\sigma_{int}/\sigma_{tot}$. The upper panels are for $E_{lab}=36$ MeV, and the lower panels for $E_{lab}=60$ MeV. The absorption contribution to $\sigma_{int}$ is verified by the observed difference $|\sigma_{int}^{\rm NA}-\sigma_{int}^{\rm WA}|$, which are clearly shown for both $E_{lab}$ energies, as $\varepsilon_{b}$ varies.

Besides the fact that the Coulomb-nuclear interference is shown to be larger in the very small binding energy limits, such larger values may also be influenced by the large magnitudes of the total and Coulomb breakup cross sections, which are shown in Figs. 9 and 10. However, as verified from Table 3, the ratios $\sigma_{Coul}/\sigma_{tot}$ for the smaller binding are enough deviating from one (when full absorption is considered, in the ²⁰⁸Pb case). Consistently, we also noticed from the results given in Table 3, that $\sigma_{int}/\sigma_{tot}$ is larger for $\varepsilon_{b}=2.03$ MeV when we have the usual cross section values with absorption. Further investigation may be required to clarify the $\varepsilon_{b}$ dependence of Coulomb-nuclear interference, in support to the actual results that are shown an overall significant effect of nuclear absorption.

In the case of such weakly-bound projectiles, a better understanding of the function $\delta_{R}(\varepsilon_{b})$, which appears in Eq.(4), could shed more light on the complexity of the Coulomb-nuclear interference. In such cases, $R_{n}$ can significantly deviate from $R_{0}$, since the nuclear breakup dynamics requires that $\delta_{R}(\varepsilon_{b})\to 0$ for larger values of $\varepsilon_{b}$. Particularly, the main characteristics of this function could show up in a study with charged projectiles, considering that strong Coulomb/nuclear interference has been observed for the reaction of proton halo ⁸B with ⁵⁸Ni target 2001Tostevin ; 2002Margueron ; Tarutina10 ; 2009Lubian , in which we have a very weakly-bound projectile with breakup threshold of 0.137 MeV.