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

B. Mukeru  Department of Physics, University of South Africa, P.O. Box 392,
Pretoria 0003, South Africa  J. Lubian  Instituto de Física, Universidade
Federal Fluminense, Niterói, RJ, 24210-340, Brazil  Lauro Tomio  Instituto de
Física Téorica, Universidade Estadual Paulista, 01140-070 São Paulo, SP,
Brazil

###### Abstract

A detailed study of total, Coulomb and nuclear breakup cross sections
dependence on the projectile ground-state binding energy $\varepsilon_{b}$ is
presented, by considering the 8Li+12C and 8Li+208Pb breakup reactions. To this
end, apart from the experimental one-neutron separation energy of 8Li nucleus
($\varepsilon_{b}=2.03$ MeV), lower values of $\varepsilon_{b}$ down to
$\varepsilon_{b}=0.01$ MeV, are also being considered. It is shown that all
breakup processes become peripheral as $\varepsilon_{b}\to 0.01$ MeV, which is
understood as due to the well-known large spacial extension of ground-state
wave functions associated to weakly-bound projectiles. The Coulomb breakup
cross section is found to be more strongly dependent on $\varepsilon_{b}$ than
the nuclear breakup cross section, such that the Coulomb breakup becomes more
significant as $\varepsilon_{b}$ decreases, even in a naturally nuclear-
dominated reaction. This is mainly due to the long-range nature of the Coulomb
forces, leading to a direct dependence of the Coulomb breakup on the
electromagnetic transition matrix. It is also highlighted the fact that the
nuclear absorption plays a minor role for small binding when the breakup
becomes more peripheral.

Keywords: Nuclear fusion reactions; Cross sections; halo nuclei; 8Li; 208Pb;
12C

###### pacs:

24.10.Eq,24.10.-i

##  I Introduction

In the breakup of weakly-bound projectiles against heavy nuclei targets, a
relevant phenomenon which has been investigated is the Coulomb-nuclear
dynamics, such that considerable efforts have been made to understand the role
of Coulomb-nuclear interference and the dynamics of fragments absorption in
the breakup process. The established studies in this matter with the
corresponding most relevant works can be found in Refs. 2003Suzuki ;
Thompson100 ; Chat10 . For other complementary studies done in past two
decades on the Coulomb-nuclear dynamics involving weakly-bound projectiles,
with particular interest to our present investigation, we select the Refs.
Thomp20 ; 1999Th ; 2002Margueron ; 2003Capel ; Tarutina10 ; Hussein20 ;
2006Canto ; 2009Lubian ; 2009Canto , as well as more recent works (among which
we include contributions by some of us) in Refs. Kucuk10 ; 2014Capel ;
Hussein200 ; 2015Lubian ; Mukeru10 ; Mukeru20 ; 2016Manjeet ; Pierre100 ;
2017Mukeru ; MukeruPRC2020 . Despite the advances verified by these studies,
the question on how both Coulomb and nuclear forces interfere to produce a
total breakup remains far from being fully established. Some of the challenges
emanate from the fact that, in a Coulomb-dominated reaction, small
contribution of the nuclear breakup does not automatically imply insignificant
Coulomb-nuclear interference Nakamura10 ; Noc10 ; Aum10 ; Fukuda10 ; Abu10 .
It could be interesting to verify what happens in nuclear-dominated reactions.

In view of the long-range nature of Coulomb forces, a low breakup threshold is
expected to lead to peripheral collisions, where the Coulomb breakup prevails
over the nuclear breakup. In this peripheral region, the Coulomb breakup cross
section depends on the projectile structure through the electromagnetic matrix
elements of the projectile. Although not a general rule, according to the
Coulomb dissociation method Bertulani10 ; Winther10 ; Baur10 ; Baur20 , the
breakup cross section is simply the product of the reaction parameters and the
projectile dipole electric transition probability. As the binding energy
decreases, the reaction becomes more peripheral, with the ratio between the
Coulomb breakup cross section to the nuclear counterpart being expected to
rise significantly, regardless the target mass. Intuitively, in this case, one
would expect that the total breakup cross section becomes comparable to the
Coulomb one, owing to both dynamic and static breakup effects. From the fact
that lower is the ground-state binding, longer is the tail of the associated
wave function, the nuclear forces are fairly stretched beyond the projectile-
target radius. Therefore, for a projectile with very weak binding energy, even
the nuclear breakup can be assumed to be a peripheral phenomenon, with the
Coulomb-nuclear interference becoming stronger in the peripheral region.

The dependence of various reaction observables on the projectile ground-state
binding energy has being studied recently in Refs. Wang10 ; Rath100 ;
2016Rangel ; Lei100 ; Mukeru15 ; 2020Mukeru , in which different projectiles
with different binding energies have been considered. One of the drawbacks
being that all the projectiles do not have the same ground-state structure,
mass and charge. Among other ways to circumvent such shortcomings, at least
theoretically, one could artificially consider different binding energies for
the same projectile (i.e., with nucleon-number $A$ and charge $Z$ unchanged),
within an approach that has been adopted for instance in Refs. 2016Rangel ;
Lei100 ; Mukeru15 ; 2020Mukeru . Even though a given nucleus has fixed ground-
state energy, this is a convenient theoretical approach to unambiguously
establish the dependence of the reaction observables on the projectile binding
energy.

Another important aspect in breakup dynamics, relies on possible effects on
other reaction observables, such as on fusion cross sections. While it is
widely understood that the complete fusion suppression strongly depends on the
projectile breakup threshold (see Ref. 2020Jha , for recent related studies),
strong charge clustering has recently been identified as the main factor
responsible for such suppression, in the breakup of 8Li on a heavy target
Cook20 . Some behavior in the breakup of this nucleus, has also been reported
in Refs. Pak20 ; Gum20 . Unlike several other loosely bound nuclei (such as
8B, 6,7Li, and 11Be), not much has been reported on the breakup dynamics of
the ${}^{8}{\rm Li}$ nucleus.

In view of the above discussion, we are motivated to study the breakup of
${}^{8}{\rm Li}$ nucleus, within a model in which a valence neutron (n) is
loosely bound to the ${}^{7}{\rm Li}$ nucleus by a binding energy
$\varepsilon_{b}=2.03$ MeV Nut10 , by considering the light and heavy targets
12C and 208Pb. The present study on the 8Li$+^{208}$Pb breakup reaction is
also extending a previous recent analysis for this reaction done in Ref.
2020Mukeru , where a critical angular momentum for complete fusion was also
considered. We are particularly interested in analyzing the dependence of the
resulting total, Coulomb and nuclear breakup cross sections, as well as the
Coulomb-nuclear interference, on the projectile ground-state binding energy,
in order to test the validity of the assumptions presented in the previous
paragraphs. Within a more detailed investigation, we expect to show that for a
much weaker projectile binding energy, the Coulomb breakup becomes dominant
regardless the target mass, and the nuclear breakup becomes relatively
peripheral, leading to a peripheral Coulomb-nuclear interference. Since both
Coulomb and nuclear breakup cross sections increase with the decrease of the
binding energy, a clear separation of their effects is not a simple task. The
choice of 12C and 208Pb as the targets is motivated by the fact that, in the
former case, the reaction should be dominated by the nuclear breakup, whereas
it is dominated by Coulomb breakup in the latter case. If fact, 12C was also
used in Ref. Fukuda10 , as a reference target when studying the 11Be Coulomb
dissociation on 208Pb target. In our approach to obtain the corresponding
total, Coulomb and nuclear breakup cross sections, we adopt the Continuum
Discretized Coupled Channels (CDCC) formalism Aust100 , with the Fresco code
1988Thompson  being used for the numerical solutions.

The next sections are organized as follows: Sect. II provides some details on
the model approach, with a summary of the CDCC formalism. Sect. III contains
the main results for elastic and breakup cross sections, together with our
analysis on the Coulomb-nuclear interference and possible absorption
contributions. Finally, the Sect. IV presents a summary with our conclusions.

##  II Formalism and computational approach

###  II.1 Brief description of the CDCC formalism

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

$\displaystyle\left[-\frac{\hbar^{2}}{2\mu_{pt}}\bigg{(}\frac{d^{2}}{dR^{2}}-\frac{L(L+1)}{R^{2}}\bigg{)}+U_{\alpha\alpha}^{LLJ}(R)\right]\chi_{\alpha}^{LJ}(R)$
$\displaystyle+\sum_{\alpha^{\prime}L^{\prime}(\alpha^{\prime}\neq\alpha)}U_{\alpha\alpha^{\prime}}^{LL^{\prime}J}(R)\chi_{\alpha^{\prime}}^{L^{\prime}J}=(E-\varepsilon_{\alpha})\chi_{\alpha}^{LJ},$
(1)

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

$\displaystyle U_{\alpha\alpha^{\prime}}^{LL^{\prime}J}(R)$ $\displaystyle=$
$\displaystyle\langle\mathcal{Y}_{\alpha
L}({\bm{r}},\Omega_{R})|V_{ct}^{Coul}({\bm{R}}_{ct})|\mathcal{Y}_{\alpha^{\prime}L^{\prime}}({\bm{r}},\Omega_{R})\rangle$
$\displaystyle+$ $\displaystyle\langle\mathcal{Y}_{\alpha
L}({\bm{r}},\Omega_{R})|U_{ct}^{nucl}({\bm{R}}_{ct})|\mathcal{Y}_{\alpha^{\prime}L^{\prime}}({\bm{r}},\Omega_{R})\rangle$
$\displaystyle+$ $\displaystyle\langle\mathcal{Y}_{\alpha
L}({\bm{r}},\Omega_{R})|U_{nt}^{nucl}({\bm{R}}_{nt})|\mathcal{Y}_{\alpha^{\prime}L^{\prime}}({\bm{r}},\Omega_{R})\rangle,$

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

$\displaystyle\chi_{\alpha}^{LJ}(R)\stackrel{{\scriptstyle
R\to\infty}}{{\longrightarrow}}\frac{\rm
i}{2}\left[H_{\alpha}^{-}(k_{\alpha}R)\delta_{\alpha\alpha^{\prime}}-H_{\alpha}^{+}(k_{\alpha}R)S_{\alpha\alpha^{\prime}}^{LL^{\prime}J}\right],$
(3)

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

$R_{n}\equiv r_{0}(A_{p}^{1/3}+A_{t}^{1/3})+\delta_{R}(\varepsilon_{b})\equiv
R_{0}+\delta_{R}(\varepsilon_{b})$ (4)

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

$\displaystyle V^{Coul}({\bm{r}},{\bm{R}})\stackrel{{\scriptstyle
R\to\infty}}{{\longrightarrow}}4\pi Z_{t}e\sum_{\lambda=0}^{\lambda_{\rm
max}}\frac{\sqrt{2\lambda+1}}{R^{\lambda+1}}\left[\mathcal{O}_{\lambda}^{\epsilon}({\bm{r}})\otimes
Y_{\lambda}(\Omega_{R})\right]^{0},$ (5)

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

$\displaystyle\mathcal{O}_{\lambda\mu}^{\epsilon}({\bm{r}})$ $\displaystyle=$
$\displaystyle\left[Z_{c}e\left(-\frac{A_{n}}{A_{p}}\right)^{\lambda}\right]r^{\lambda}Y_{\lambda}^{\mu}(\Omega_{r})=Z_{\lambda}r^{\lambda}Y_{\lambda}^{\mu}(\Omega_{r}),$
(6)

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

$\displaystyle\frac{dB(E\lambda)}{d\varepsilon}$ $\displaystyle=$
$\displaystyle\frac{\mu_{cn}}{\hbar^{2}k}\sum_{\tilde{\j}}(2{\tilde{\j}}+1)\left|\langle\Phi_{\beta_{0}}({\bm{r}})|\mathcal{O}_{\lambda}^{\epsilon}({\bm{r}})|\Phi_{\beta}({\bm{r}})\rangle\right|^{2},$
(7)

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

$\displaystyle\frac{dB(E\lambda)}{d\varepsilon}$ $\displaystyle=$
$\displaystyle\frac{\mu_{cn}}{\hbar^{2}k}\sum_{\tilde{\j}}(2{\tilde{\j}}+1)|\mathcal{F}_{\lambda,{\tilde{\j}}}|^{2},\;\;{\rm
with}$ (8) $\displaystyle\mathcal{F}_{\lambda,j}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{4\pi}Z_{\lambda}\hat{\ell}_{0}\hat{\ell}\hat{\lambda}^{2}\hat{j}_{0}\hat{j}(-1)^{\ell_{0}+\ell+s+j+j_{0}+I_{c}+{\tilde{\j}}}$
(15) $\displaystyle\times$
$\displaystyle\left(\begin{array}[]{ccccc}\ell&\lambda&\ell_{0}\\\
0&0&0\end{array}\right)\left(\begin{array}[]{ccccc}j&\lambda&j_{0}\\\
0&0&0\end{array}\right)\left\\{\ \begin{array}[]{cccc}s&\ell_{0}&j_{0}\\\
\lambda&j&\ell\end{array}\right\\}\ $ $\displaystyle\times$
$\displaystyle\left\\{\ \begin{array}[]{cccc}I_{c}&j_{0}&\tilde{\j}_{0}\\\
\lambda&\tilde{\j}&j\end{array}\right\\}\
\int_{0}^{\infty}dr\,u_{k_{0}\ell_{0}}^{\tilde{\j}_{0}}(r)r^{\lambda}u_{k\ell}^{\tilde{\j}}(r),$
(18)

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.

###  II.2 Computational details

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.

Table 1:  Woods-Saxon potential parameters for the projectile (n$-^{7}$Li) ground and excited bound-state energies. $\tilde{\j}^{\pi}$ | $V_{0}$ | $r_{0}$ | $a_{0}$ | $V_{\rm SO}$ | $r_{\rm SO}$ | $a_{\rm SO}$
---|---|---|---|---|---|---
| (MeV) | (fm) | (fm) | (MeV/fm2) | (fm) | (fm)
$2^{+}$ | 37.22 | 1.25 | 0.52 | 4.89 | 1.25 | 0.52
$1^{+}$ | 46.65 | 1.25 | 0.52 | 4.89 | 1.25 | 0.52

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).

Figure 1:  In panel (a), considering three different 7Li-n ground-state
binding energies $\varepsilon_{b}$, it is shown how the derivative of the
electric transition probability, given by (8), varies with the excitation
energy $\varepsilon$, for transitions from ground to continuum $s$\- plus
$d$-states. In panel (b), the root-mean-square radii is shown as a function of
the binding energy $\varepsilon_{b}$.

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.

Table 2:  Maximum model space parameters, for optimal numerical convergence of Eq. (II.1) for both 12C and 208Pb targets. The main reported values are for $\varepsilon_{b}\geq 0.4$MeV, with the corresponding ones within parenthesis for $\varepsilon_{b}\leq 0.08$MeV. Target | $\ell_{\rm max}$ | $\lambda_{\rm max}$ | $\varepsilon_{\rm max}$ | $r_{\rm max}$ | $L_{\rm max}$ | $R_{\rm max}$ | $\Delta R$
---|---|---|---|---|---|---|---
|  ($\hbar$) |  | (MeV) | (fm) |  ($\hbar$) | (fm) | (fm)
${}^{12}{\rm C}$ | 3 | 3 | 6 | 80 | 300 | 300 | 0.08
|  |  |  | (100) | (1000) | (500) |
${}^{208}{\rm Pb}$ | 4 | 4 | 10 | 80 | 1000 | 600 | 0.03
|  |  |  | (100) | (10000) | (1000) |

Figure 2:  8Li+12C elastic scattering cross sections for the incident
energies, $E_{lab}=$14 MeV and $E_{lab}=$23.9 MeV. The model results are for
different 8Li binding energies $\varepsilon_{b}$ (in MeV units), as indicated
inside panel (a) for both panels. The available experimental data, converted
to Rutherford $\sigma_{R}$ units, are from Refs. 1993Becchetti  [panel (a)]
and Barioni100  [panel (b)], as indicated in the database reported in Ref.
Jinr20 .

Figure 3:  8Li+208Pb elastic scattering cross sections (in units of the
Rutherford $\sigma_{R}$), obtained for the incident energies $E_{lab}=$36 MeV
[panel (a)] and $E_{lab}=$60 MeV [panel (b)]. As in Fig. 2, the results are
for the same set of $\varepsilon_{b}$ (in MeV). From Ref. 2002Kolata , we
included in (a) the closest available experimental data, which are for
$E_{lab}=$30.6 MeV, as indicated in the database reported in Ref. Jinr20 .

##  III Results and Discussion

###  III.1 Elastic scattering cross sections

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.

###  III.2 Breakup cross sections

Figure 4:  For incident energies $E_{lab}=14\,{\rm MeV}$ (left column) and
$E_{lab}=24\,{\rm MeV}$ (right column), with fixed different $\varepsilon_{b}$
(shown inside the panels), the 8Li$+^{12}$C angular distributions for the
total, Coulomb and nuclear differential breakup cross sections
$d\sigma/d\Omega$ (identified inside the upper panels) are shown as functions
of the c.m. angle $\theta$.

Figure 5:  Convergence sample results for the 8Li$+^{208}$Pb, total (upper
frames), Coulomb (middle frames) and nuclear (bottom frames) breakup angular
distributions, $d\sigma/d\Omega$, at $E_{lab}=36\,{\rm MeV}$, considering
different maximum projectile internal angular momenta $\ell_{\rm max}$
(indicated in the upper-left frame). The left set [(a)-(c)] is for
$\varepsilon_{b}=0.01$MeV, with the right set [(d)-(f)] for
$\varepsilon_{b}=2.03$MeV.

Figure 6:  For incident energies $E_{lab}=36\,{\rm MeV}$ (left column) and
$E_{lab}=60\,{\rm MeV}$ (right column), with fixed different $\varepsilon_{b}$
(shown inside the panels), the ${}^{8}{\rm Li}+{}^{208}{\rm Pb}$ angular
distributions for the total, Coulomb and nuclear $d\sigma/d\Omega$ (identified
in the upper panels) are shown as functions of the c.m. angle $\theta$.

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 ).

Figure 7:  For the 8Li$+^{208}$Pb reaction, by considering $E_{lab}=$36 MeV
(a) and 60 MeV (b), it is shown the integrated breakup cross sections (BU)
(when both $W_{ct}^{nucl}$ and $W_{nt}^{nucl}$ are contributing), the breakup
cross section without absorption (NA) (when $W_{ct}^{nucl}=W_{nt}^{nucl}=0$),
and the total fusion cross section (TF), as functions of the projectile
binding energy $\varepsilon_{b}$.

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.

Figure 8:  For the 8Li+12C breakup reaction, the angular-integrated total,
Coulomb and nuclear breakup cross sections are given as functions of the
projectile binding energy $\varepsilon_{b}$, for the incident energies
$E_{lab}=$14 MeV (a) and 24 MeV (b).

Figure 9:  The angular-integrated total, Coulomb and nuclear breakup cross
sections are given for the 8Li+208Pb breakup reaction as functions of
$\varepsilon_{b}$, with nuclear absorption in the panels (a) and (b); and
without absorption in the panels (c) and (d). As indicated, the incident
energies are $E_{lab}=$36 MeV [panels (a) and (c)] and 60 MeV [panels (b) and
(d)].

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.

Table 3:  Coulomb, nuclear and interference cross-sections for the ${}^{8}{\rm Li}+{}^{12}{\rm C}$ and 8Li$+^{208}$Pb, considering n$-^{7}$Li binding energies $\varepsilon_{b}=$0.01 MeV and 2.03 MeV. For each target, we present our results, in terms of ratios, for two colliding energies. For 208Pb target, with no-nuclear absorption (NA) the results are shown within parenthesis below the ones with absorption.  Target | ${E_{lab}}$ |  | $\varepsilon_{b}=$ | 2.03 | MeV |  |  | $\varepsilon_{b}=$ | 0.01 | MeV |
---|---|---|---|---|---|---|---|---|---|---|---
| (MeV) | $\frac{\sigma_{Coul}}{\sigma_{tot}}$ | $\frac{\sigma_{nucl}}{\sigma_{tot}}$ | $\frac{\sigma_{Coul}}{\sigma_{nucl}}$ | $\frac{\sigma_{int}}{\sigma_{nucl}}$ | $\frac{\sigma_{int}}{\sigma_{tot}}$ | $\frac{\sigma_{Coul}}{\sigma_{tot}}$ | $\frac{\sigma_{nucl}}{\sigma_{tot}}$ | $\frac{\sigma_{Coul}}{\sigma_{nucl}}$ | $\frac{\sigma_{int}}{\sigma_{nucl}}$ | $\frac{\sigma_{int}}{\sigma_{tot}}$
${}^{12}{\rm C}$ | 14 | 0.024 | 0.824 | 0.029 | 0.186 | 0.153 | 0.836 | 0.268 | 3.123 | -0.387 | -0.104
| 24 | 0.048 | 0.808 | 0.059 | 0.190 | 0.154 | 0.701 | 0.339 | 2.069 | -0.118 | -0.040
${}^{208}{\rm Pb}$ | 36 | 1.800 | 0.150 | 12.00 | -6.333 | -0.950 | 1.033 | 0.012 | 90.16 | -3.850 | -0.044
|  | (0.344) | (1.481) | (0.232) | (-0.557) | (-0.825) | (1.032) | (0.029) | (33.97) | (-2.061) | (-0.063)
| 60 | 1.326 | 0.087 | 15.25 | -4.750 | -0.413 | 1.015 | 0.010 | 104.6 | -2.540 | -0.025
|  | (0.198) | (0.783) | (0.253) | (0.025) | (0.019) | (1.000) | (0.059) | (17.03) | (-0.991) | (-0.058)

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

$\displaystyle\sigma_{int}=\sigma_{tot}-(\sigma_{Coul}+\sigma_{nucl}),$ (19)

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).

Figure 10:  The 8Li+12C integrated Coulomb-nuclear interference $\sigma_{int}$
[panel (a)], given by (19), with the respective ratio
$\sigma_{int}/\sigma_{tot}$ [panel (b)], are shown as functions of
$\varepsilon_{b}$, for the colliding energies $E_{\rm lab}=$14 and 24 MeV.

Figure 11:  The ${}^{8}{\rm Li}+{}^{208}{\rm Pb}$ integrated Coulomb-nuclear
interference $\sigma_{int}$ [panels (a) and (b)], given by (19), with their
ratios $\sigma_{int}/\sigma_{tot}$ [panels (c) and (d)], are shown as
functions of $\varepsilon_{b}$, for $E_{\rm lab}=$36 and 60 MeV (upper and
lower frames, respectively). $\sigma_{int}^{\rm WA}$ (solid lines) denotes the
interference when the breakup is followed by nuclear absorption, with
$\sigma_{int}^{\rm NA}$ (dot-dashed lines) denoting interference with no
nuclear absorption.

###  III.3 Coulomb-nuclear interference

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.

##  IV Conclusion

We have presented a study on the breakup of the weakly-bound 8Li (n$-^{7}$Li)
projectile on light and heavy target masses, namely, 12C and 208Pb. Our main
objective was to investigate the dependence of the total, Coulomb and nuclear
breakup cross sections, on the 8Li ground-state binding energy
$\varepsilon_{b}$, in order to study the peripherality of the total, Coulomb
and nuclear breakup processes, which are associated to the weaker binding
energy of the projectile. To this end, apart from the experimentally-known
ground-state binding energy of the n$-^{7}$Li system, we artificially
considered four other binding energies, below the experimental value, down to
$\varepsilon_{b}=0.01$ MeV, which is much smaller than the experimental value,
$\varepsilon_{b}=2.03$ MeV. From our analysis it is shown that the total,
Coulomb and nuclear breakup processes become peripheral as $\varepsilon_{b}\to
0.01$ MeV, regardless the target mass. We argue that the peripherality of the
nuclear breakup in this case is primarily related to the spacial extension of
the corresponding ground-state wave function, which is related to weaker
binding energy. The peripheral region is determined by the range of the
nuclear forces $R_{0}$, and the corresponding extension of the ground-state
wave function, which is associated to a function
$\delta_{R}(\varepsilon_{b})$, expressed by $R_{n}$ defined in Eq. (4). By
taking into account the fact that close to the $n-$core $\varepsilon_{b}\to 0$
limit, a long-range interaction is expected to emerge between projectile and
target (similar as for three-body halo-nuclei systems 2012Frederico ), the
size of the associated wave function will increase significantly in this
limit. So, a detailed investigation of this function
$\delta_{R}(\varepsilon_{b})$ (which should go to zero by increasing
$\varepsilon_{b}$) can shed more light into the dynamics of nuclear breakups
induced by loosely bound projectiles. It is also noticed that the variation of
$\varepsilon_{b}$ strongly affects the Coulomb breakup, as compared to the
nuclear breakup, such that as $\varepsilon_{b}\to$ 0.01 MeV, the Coulomb
breakup becomes dominant even for the ${}^{12}{\rm C}$ target, which is known
to be naturally dominated by nuclear breakup. Therefore, in view of this
binding-energy dependence, one may infer that the expression “naturally-
dominated by nuclear breakup” may be relative to the projectile binding
energy. It is also verified that the nuclear absorption has an insignificant
effect on the total and nuclear breakup cross sections when decreasing the
binding energy to small binding such as $\varepsilon_{b}\to$ 0.01 MeV. In this
small binding energy region, we found that the total breakup cross section is
larger than the calculated total fusion cross section, while as expected, the
opposite is observed as $\varepsilon_{b}\to$ 2.03 MeV.

## Acknowledgements

We thank T. Frederico, B.V. Carlson and L. F. Canto for useful discussions.
B.M. is also grateful to the South American Institute of Fundamental Research
(ICTP-SAIFR) for local facilities. For partial support, we also thank Conselho
Nacional de Desenvolvimento Científico e Tecnológico [INCT-FNA
Proc.464898/2014-5 (LT and JL), Proc. 304469/2019-0(LT) and Proc.
306652/2017-0(JL)], and Fundação de Amparo à Pesquisa do Estado de São Paulo
[Projs. 2017/05660-0(LT)].

## References

  * (1) Y. Suzuki, K. Yabana, R.G. Lovas, K. Varga, Structure and Reactions of Light Exotic Nuclei (Taylor & Francis Group, London and New York, 2003).
  * (2) I. J. Thompson and F. M. Nunes, Nuclear Reactions for Astrophysics (Cambridge University Press, New York, 2009).
  * (3) R. Chatterjee and R. Shyam, Prog. Part. Nucl. Phys. 103, 67 (2018).
  * (4) F. M. Nunes and I. J. Thompson, Phys. Rev. C 57, R2818 (1998).
  * (5) F. M. Nunes and I. J. Thompson, Phys. Rev. C 59, 2652 (1999).
  * (6) J. Margueron, A. Bonaccorso, and D. M. Brink, Nucl. Phys. A 703, 105 (2002).
  * (7) P. Capel, D. Baye and V. S. Melezhik, Phys. Rev. C 68, 014612 (2003).
  * (8) T. Tarutina and M. S. Hussein, Phys. Rev. C 70, 034603 (2004).
  * (9) M. S. Hussein, R. Lichtenthaler, F. M. Nunes and I. J. Thompson, Phys. Lett. B 640, 91 (2006).
  * (10) L. F. Canto, P. R. S. Gomes, R. Donangelo, and M. S. Hussein, Phys. Rep. 424, 1 (2006).
  * (11) J. Lubian, T. Correa, E. F. Aguilera, L. F. Canto, A. Gomez-Camacho, E. M. Quiroz, and P. R. S. Gomes, Phys. Rev. C 79, 064605 (2009).
  * (12) L. F. Canto, J. Lubian, P. R. S. Gomes, and M. S. Hussein, Phys. Rev. C 80, 047601 (2009).
  * (13) Y. Kucuk and A. M. Moro, Phys. Rev. C 86, 034601 (2012).
  * (14) P. Capel, J. Phys. G: Nucl. Part. Phys. 41, 094002 (2014).
  * (15) D. R. Otomar, P. R. S. Gomes, J. Lubian, L. F. Canto and M. S. Hussein, Phys. Rev. C 87, 014615 (2013).
  * (16) D. R. Otomar, P. R. S. Gomes, J. Lubian, L. F. Canto and M. S. Hussein, Phys. Rev. C 92, 064609 (2015).
  * (17) B. Mukeru, M. L. Lekala and A. S. Denikin, J. Phys. G: Nucl. Part. Phys. 42, 015109 (2015).
  * (18) B. Mukeru and M. L. Lekala, Phys. Rev. C 91, 064609 (2015).
  * (19) S. G. Manjeet, Chin. Phys. C 40, 054101 (2016).
  * (20) P. Descouvemont, L.F. Canto, M.S. Hussein, Phys. Rev. C 95, 014604 (2017).
  * (21) B. Mukeru and M.L. Lekala, Int. J. Mod. Phys. E 26,1750075 (2017).
  * (22) B. Mukeru, T. Frederico, and L. Tomio, Phys. Rev. C 102, 064623 (2020).
  * (23) T. Nakamura, N. Kobayashi, Y. Kondo,Y. Satou, N. Aoi, H. Baba, et al., Phys. Rev. Lett. 103, 262501 (2009).
  * (24) C. Nociforo et al., Phys. Lett. B 605, 79 (2005).
  * (25) T. Aumann and T. Nakamura, Phys. Scr. T 152, 014012 (2013).
  * (26) N. Fukuda, T. Nakamura,N. Aoi,N. Imai, M. Ishihara,T. Kobayashi, H. Iwasaki, T. Kubo, A. Mengoni, M. Notani, H. Otsu,H. Sakurai, S. Shimoura, T. Teranishi,Y. X. Watanabe,K. Yoneda, Phys. Rev. C 70, 054606 (2004).
  * (27) B. Abu-Ibrahim and Y. Suzuki, Progr. Theor. Phys. 112, 1013 (2004).
  * (28) C. A. Bertulani and G. Baur, Phys. Rep. 163, 299 (1988).
  * (29) A. Winther and K. Alder, Nucl. Phys. A 319, 518 (1979).
  * (30) G. Baur, K. Hencken, and D. Trautmann, Prog. Part. Nucl. Phys. 51, 487 (2003).
  * (31) G. Baur and H. Rebel, Annu. Rev. Nucl. Part. Sci. 46, 321 (1996).
  * (32) B. Wang, W. J. Zhao, P. R. S. Gomes, E. G. Zhao, and S.-G. Zhou, Phys. Rev. C 90, 034612 (2014).
  * (33) P. K. Rath, S. Santra, N. L. Singh, R. Tripathi, V. V. Parkar, B. K. Nayak, K. Mahata, R. Palit, S. Kumar, S. Mukherjee, S. Appannababu, R. K. Choudhury, Phys. Rev. C 79, 051601(R) (2009).
  * (34) J. Rangel, J. Lubian, L. F. Canto, and P. R. S. Gomes, Phys. Rev. C 93, 054610 (2016).
  * (35) J. Lei A. M. Moro, Phys. Rev. Lett. 122, 042503 (2019).
  * (36) B. Mukeru, J. Phys. G: Nucl. Part. Phys. 45, 065201 (2018).
  * (37) B. Mukeru, M.L. Lekala, J. Lubian and L. Tomio, Nucl. Phys. A 996, 121700 (2020).
  * (38) V. Jha, V. V. Parkar and S. Kailas, Phys. Rep. 845 1 (2020).
  * (39) K. J. Cook, I. P. Carter, E. C. Simpson, M. Dasgupta, D. J. Hinde,L. T. Bezzina, S. Kalkal, C. Sengupta, C. Simenel, B. M. A. Swinton-Bland, K. Vo-Phuoc,E. Williams, Phys. Rev. C 97 021601(R) (2018).
  * (40) A. Pakou et al., Eur. Phys. J. A 51, 55 (2015).
  * (41) V. Guimarães, J. Lubian, J. J. Kolata, E. F. Aguilera, M. Assunção, V. Morcelle, Eur. Phys. J. A 54, 223 (2018).
  * (42) M. Wang, et al., Chin. Phys. C 41, 030003 (2017). [See also at https://www.nndc.bnl.gov/nudat2/].
  * (43) N. Austern  et al., Phys. Rep. 154,125 (1987).
  * (44) I. J. Thompson, Comput. Phys. Rep. 7, 167 (1988)
  * (45) L. F. Canto, P. R. S. Gomes, R. Donangelo, J. Lubian and M. S. Hussein, Phys. Rep. 596, 1 (2015).
  * (46) M. Abramowitz and I. Stegun,  Handbook of Mathematical Functions: with Formulas,Graphs, and Mathematical Tables (Dover Publications, National Bureau of Standards, New York, 1964).
  * (47) L. F. Canto and M. S. Hussein, Scattering theory of molecules, atoms and nuclei (World Scientific Publishing Co. Pte. Ltd, Singapore, 2013)
  * (48) C. A. Bertulani, Comput. Phys. Commun. 156, 123 (2003).
  * (49) A. M. Moro, R. Crespo, H. Garcia-Martinez, E. F. Aguilera, E. Martinez-Quiroz, J. Gomez-Camacho, F. M. Nunes, Phys. Rev. C 68, 034614 (2003).
  * (50) G. W. Fan, M. Fukuda, D. Nishimura, X. L. Cai, S. Fukuda, I. Hachiuma, et al., Phys. Rev. C 91, 014614 (2015).
  * (51) I. Tanihata, T. Kobayashi, O. Yamakawa, S. Shimoura, K. Ekuni, K. Sugimoto, N. Takahashi, T. Shimoda, H. Sato, Phys. Lett. B206, 592 (1988).
  * (52) A. Barioni, V. Guimarães, A. Lépine-Szily, R. Lichtenthäler, D. R. Mendes, E. Crema, K. C. C. Pires, M.C. Morais, V. Morcelle, P. N. deFaria, R. P. Condori., A. M. Moro., D. S. Monteiro, J. M. B. Shorto, J. Lubian, M. Assunção, Phys. Rev. C 80, 034617 (2009).
  * (53) J. Cook, Nucl. Phys. A 388, 153 (1982).
  * (54) A. J. Koning and J. P. Delaroche, Nucl. Phys.A 713, 231 (2003).
  * (55) F. D. Becchetti, W. Z. Liu, K. Ashktorab, J. F. Bajema, J. A. Brown, J. W. Janecke, D. A. Roberts, J. J. Kolata, K. L. Lamkin, A. Morsad, R. J. Smith, X. J. Kong, R. E. Warner, Phys. Rev. C 48, 308 (1993).
  * (56) A. V. Karpov, A. S. Denikin, A. P. Alekseev, V. I. Zagrebaev, V. A. Rachkov, M. A. Naumenko, and V. V. Saiko, Phys. Atom. Nucl. 79,749 (2016) and http://nrv.jinr.ru/nrv/webnrv/expdata database.
  * (57) J. J. Kolata, V. Z. Goldberg, L. O. Lamm, M. G. Marino, C. J. OKeeffe, G. Rogachev, et al., Phys. Rev. C 65, 054616 (2002).
  * (58) J. A. Tostevin, F. M. Nunes, and I. J. Thompson, Phys. Rev. C 63, 024617 (2001).
  * (59) T. Frederico, A. Delfino, L. Tomio and M.T. Yamashita, Prog. Part. Nucl. Phys. 67, 939 (2012).