distorted wave with momentum ${\mbox{\boldmath$k$}}_{b}$, obtained as a
solution of the single-channel equation

$\left[T_{bB}+U_{bB}^{\dagger}-E_{b}\right]\chi_{b}^{(-)}(\mbox{\boldmath$k$}_{b},{\mbox{\boldmath$r$}}_{b})=0,$
(97)

where $E_{b}$ is the kinetic energy associated with the $b+B$ relative motion.
The wave function describing the motion of $x$ after the breakup, that will be
denoted as $Z_{x}(\xi,{\mbox{\boldmath$r$}}_{x})$, can be obtained projecting
the total wave function [Eq. (95)] onto this particular state of the $b$
particle, i.e.,

$\displaystyle Z_{x}(\mbox{\boldmath$k$}_{b},\xi,{\mbox{\boldmath$r$}}_{x})$
$\displaystyle\equiv\langle{\mbox{\boldmath$r$}}_{x}\chi_{b}^{(-)}|\Psi\rangle=\left[E^{+}-E_{b}-H_{B}\right]^{-1}\langle{\mbox{\boldmath$r$}}_{x}\chi_{b}^{(-)}|V_{\mathrm{post}}|\Psi\rangle,$
(98)

which can be also written in differential form as

$\left[E^{+}-E_{b}-H_{B}\right]Z_{x}(\mbox{\boldmath$k$}_{b},\xi,{\mbox{\boldmath$r$}}_{x})=\langle{\mbox{\boldmath$r$}}_{x}\chi_{b}^{(-)}|V_{\mathrm{post}}|\Psi\rangle.$
(99)

The source term of this equation involves the exact and hence unknown wave
function $\Psi$ which, in actual calculations, must be approximated by some
calculable form. IAV adopted the DWBA factorized form

$\Psi(\xi,{\mbox{\boldmath$r$}}_{x},{\mbox{\boldmath$r$}}_{b})\approx\phi^{0}_{A}(\xi)\phi_{a}(\mbox{\boldmath$r$}_{bx})\chi^{(+)}_{a}(\mbox{\boldmath$k$}_{a},\mbox{\boldmath$r$}_{a}),$
(100)

where $\phi_{a}(\mbox{\boldmath$r$}_{bx})$ is the projectile ground-state wave
function and $\chi^{(+)}_{a}(\mbox{\boldmath$k$}_{a},\mbox{\boldmath$r$}_{a})$
is a distorted wave describing the $a+A$ motion in the incident channel. In
practice, the latter is commonly replaced by the solution of some optical
potential describing $a+A$ elastic scattering.

Using the Feshbach projection formalism, the
$Z_{x}(\mbox{\boldmath$k$}_{b},\xi,{\mbox{\boldmath$r$}}_{x})$ function is
decomposed as

$Z_{x}(\mbox{\boldmath$k$}_{b},\xi,{\mbox{\boldmath$r$}}_{x})={\cal
P}Z_{x}+{\cal Q}Z_{x},$ (101)

where ${\cal P}$ is the projector operator onto the target ground state and
${\cal Q}=1-{\cal P}$. Consequently, we can write ${\cal
P}Z_{x}=\varphi_{x}(\mbox{\boldmath$k$}_{b},{\mbox{\boldmath$r$}}_{x})\phi^{0}_{A}(\xi)$,
where the function
$\varphi_{x}(\mbox{\boldmath$k$}_{b},{\mbox{\boldmath$r$}}_{x})$ is the seeked
$x$-channel wavefunction that describes the $x+A$ relative motion when the
target is in the ground state. It verifies the equation

$(E^{+}_{x}-T_{x}-{\cal
U}_{x})\varphi_{x}(\mbox{\boldmath$k$}_{b},{\mbox{\boldmath$r$}}_{x})=\langle{\mbox{\boldmath$r$}}_{x}\chi_{b}^{(-)}|V_{bx}|\Psi\rangle$
(102)

with $E_{x}=E-E_{b}$, and ${\cal U}_{x}$ the formal optical model potential
describing $x+A$ elastic scattering, i.e.,

${\cal
U}_{x}=\langle\phi^{0}_{A}|V_{xA}+V_{xA}Q[E^{+}-E_{b}-H_{QQ}]^{-1}V_{xA}|\phi^{0}_{A}\rangle,$
(103)

where $H_{QQ}\equiv{\cal Q}H_{B}{\cal Q}$. As usual, the formal potential
${\cal U}_{x}$ can be approximated by a simpler, energy-averaged and possibly
local potential with parameters adjusted to describe $a+A$ elastic scattering
(denoted $U_{x}$ hereafter).

To obtain the expression for the NEB component one makes use of the coupled-
channels optical theorem, as formulated by Cotanch [286], which can be
regarded as a generalization of the usual optical theorem to the multichannel
case. If $\chi_{i}$ is the channel wave function and $W_{i}$ the diagonal
imaginary part for this channel, the contribution to the absorption in this
particular channel is given by [286]

$\sigma^{i}_{\mathrm{abs}}=-\frac{2}{\hbar
v_{el}}\langle\chi_{i}|W_{i}|\chi_{i}\rangle,$ (104)

where $v_{el}$ is the projectile–target relative velocity in the incident
(elastic) channel.

We may use this result to calculate the NEB contribution by noting that the
latter is nothing but the absorption occurring in the $x+A$ channel. The
channel wave function is given by
$\varphi_{x}(\mbox{\boldmath$k$}_{b},{\mbox{\boldmath$r$}}_{x})$, which is a
solution of Eq. (102). Since this equation corresponds to a definite energy
and direction of the $b$ particle, we consider the differential cross section
corresponding to a range of the outgoing momenta of $b$,

$d^{2}\sigma=-\frac{2}{\hbar
v_{i}}\langle\varphi_{x}|W_{x}|\varphi_{x}\rangle\,\frac{d\mbox{\boldmath$k$}_{b}}{2\pi^{3}}.$
(105)

Transforming the element of momentum into energy and solid angle elements, we
get the double differential cross section

$\left.\frac{d^{2}\sigma}{dE_{b}d\Omega_{b}}\right|^{\mathrm{IAV}}_{\mathrm{NEB}}=-\frac{2}{\hbar
v_{i}}\rho_{b}(E_{b})\langle\varphi_{x}|W_{x}|\varphi_{x}\rangle.$ (106)

which, in combination with with Eq. (102), provides the differential NEB cross
section with respect to the angle and energy of the detected fragment $b$.

###  4.2 The three-body model of Austern et al.

Austern et al. [22] proposed also the three-body approximation

$\Psi(\xi,{\mbox{\boldmath$r$}}_{x},{\mbox{\boldmath$r$}}_{b})\approx\phi^{0}_{A}(\xi)\Psi^{\mathrm{3b}}({\mbox{\boldmath$r$}}_{x},{\mbox{\boldmath$r$}}_{b}),$
(107)

where $\Psi^{\mathrm{3b}}$ is the solution of the three-body equation:

$[\hat{T}_{aA}+\hat{T}_{bx}+V_{bx}+U_{bA}+U_{xA}-E]|\Psi^{\mathrm{3b(+)}}\rangle=0.$
(108)

Within this three-body approximation, the $x$-channel wavefunction is simply
given by

$\varphi^{\mathrm{3b}}_{x}({\mbox{\boldmath$r$}}_{x})=\langle{\mbox{\boldmath$r$}}_{x}\chi_{b}^{(-)}|\Psi^{\mathrm{3b(+)}}\rangle.$
(109)

In spite of its apparent simplicity, this equation has the difficulty of
requiring an accurate representation of the three-body wavefunction
$\Psi^{\mathrm{3b(+)}}$ in the full configuration space, since there is no
natural cutoff in the integration variable ${\mbox{\boldmath$r$}}_{b}$.

Alternatively, the three-body function
$\varphi^{\mathrm{3b}}_{x}({\mbox{\boldmath$r$}}_{x})$ can be obtained from an
inhomogeneous equation similar to (102),

$(E^{+}_{x}-K_{x}-{\cal
U}_{x})\varphi^{\mathrm{3b}}_{x}=\langle{\mbox{\boldmath$r$}}_{x}\chi_{b}^{(-)}|V_{\mathrm{post}}|\Psi^{\mathrm{3b}^{(+)}}\rangle.$
(110)

Although this equation is a priori more difficult to implement than Eq. (109),
it has the advantage that the $\Psi^{\mathrm{3b(+)}}$ function appears
multiplied by $V_{\mathrm{post}}$, that will tend to emphasize small $b-x$
separations and hence one requires only an approximate three-body wavefunction
accurate within that range. This can be achieved, for instance, expanding
$\Psi^{\mathrm{3b(+)}}$ in terms of $b-x$ eigenstates, as done in the CDCC
method, or in terms of Weinberg states [287]. The implementation of the method
with CDCC wavefunctions is numerically challenging and the first calculation
of this kind was reported only recently [288].

Figure 20: Angular distribution of $\alpha$ particles produced in the reaction
118Sn(6Li,$\alpha$)$X$ at the incident energies indicated by the labels. The
dashed and dotted lines are the EBU and NEB contributions, and the solid line
is their incoherent sum (TBU).

The IAV model has been recently revisited by several groups [289, 285, 290]
and its accuracy assessed against experimental data with rather encouraging
results [291, 283]. As an example, in Fig. 20 we show the application to the
evaluation of the $\alpha$ production in the 6Li+118Sn reaction at the
incident energies indicated by the labels (adapted from Ref. [291]). The EBU
contribution (dashed line) was evaluated with the CDCC method whereas the NEB
part (dotted line) was obtained with the IAV method. Interestingly, one can
see that the inclusive $\alpha$-yield is largely dominated by the NEB
mechanism. The EBU is only important at small scattering angles (distant
collisions, in a classical picture). This dominance of the NEB over the EBU
mechanism has been observed in other 6,7Li-induced reactions on a variety of
energies and targets. The situation is qualitatively different for more weakly
bound nuclei, such as the neutron halo nuclei 11Be, 11Li and the proton-halo
8B. In all these cases, inclusive breakup cross sections corresponding to the
production of the heavier projectile fragment (the core) shows a dominance of
the EBU mechanism.

Figure 21:  Angular (a) and energy(b) distribution of 10Be fragments produced
in the reaction 208Pb(11Be,10Be)$X$ at the incident energy of 140 MeV.
Experimental data are compared with the calculations for the EBU and NEB
contributions, and their sum. Figure adapted from Ref. [248].

An example is shown in Fig. 21, which shows the angular and energy
distributions of 10Be fragments measured in the collision 11Be+208Pb at 140
MeV [248]. The EBU and NEB contributions were calculated with the CDCC and IAV
methods, respectively. It is seen that the EBU mechanism dominates for most
angles and the NEB is only non-negligible in an angular region around the
grazing angle. This dominance of the EBU in the case of halo nuclei can be
interpreted as a consequence of the fact that, due to its large probability of
being at large distances from the core, the halo particle(s) can be more
easily removed from the projectile without distorting or exciting the core or
the target, as compared to the case of a well bound nucleus. The orange arrow
in the energy distributions correspond to $(10/11)E(\mathrm{{}^{11}Be})$,
where $E(\mathrm{{}^{11}Be})$ is the expected 11Be kinetic energy calculated
assuming a binary-like collision. The shift with the observed position of the
energy peak is due to a Coulomb post-acceleration effect, as explained in Ref.
[248].

##  5 Heavy-ion fusion reactions

###  5.1 Light systems and nuclear astrophysics

In this section, we discuss fusion reactions. Fusion of light nuclei, such as
$p+^{7}$Be and $\alpha+^{12}$C reactions, at deep subbarrier energies play an
important role in nucleosynthesis [5, 6]. For these reactions, even if the
Coulomb barrier is overcome, it is unlikely that there exists a level of a
compound nucleus at the same energy as the incident energy. Therefore, in
order for fusion to take place, a di-nucleus system has to emit gamma rays to
populate levels in the compound nucleus. In such radiative capture process,
fusion reaction thus depends both on the tunnel dynamics of the Coulomb
barrier and on the dynamics of electromagnetic transitions. This is the case
also for the triple-alpha process to form the 12C nucleus [178, 179, 292, 293,
294, 295, 296, 297, 298]. Since the operators of the electromagnetic
transitions are long-ranged, fusion reactions may take place even outside the
Coulomb barrier. Moreover, a few specific resonance states often play a
decisive role in fusion of light nuclei.

As the atomic number of a compound nucleus increases, the level density of the
compound nucleus becomes large, and the compound nucleus may be formed even
without emitting gamma-rays. We here discuss particularly the 12C+12C fusion
reaction, which plays an important role in several astrophysical phenomena,
such as the carbon burning in stellar evolution, type Ia supernovae, and the
X-ray superburst of an accreting neutron star. It has been known that the
fusion cross sections for the 12C+12C reaction show many fine structures [299,
300, 301, 302, 303]. This is in contrast to the cross sections of the
neighboring systems, 12C+13C and 13C+13C, which are more smooth functions of
energy. The difference in the fusion cross sections for these three systems
may be understood in terms of the level density of the compound nuclei, 24Mg,
25Mg and 26Mg[304]. An interesting observation is that the fusion cross
sections for the 12C+12C system match with the cross sections for the 12C+13C
and 13C+13C systems at a few resonance energies, while the cross sections for
the 12C+12C system are hindered as compared to the cross sections for the
other systems at off-resonance energies [305, 306].

Figure 22:  The modified astrophysical $S$-factor for the 12C+12C reaction.
The solid line is obtained by solving the coupled-channels equations with
excitations to the first 2+ state in both the projectile and the target
nuclei, and their mutual excitations. The contribution of $J=2$ is denoted by
the thin dashed line, where $J$ denotes the spin of the compound nucleus. The
dashed line shows the result of the single-channel calculation. The parameters
for the coupled-channels calculation are given in Ref. [77].

It has been known that the excitation of 12C to the first 2+ state plays an
important role in the 12C+12C fusion reaction at subbarrier energies [77, 307,
308, 309, 310]. Fig. 22 shows the result of the coupled-channels calculation
which corresponds to that in Ref. [77]. Here, $S^{*}(E)$ is the modified
astrophysical $S$-factor, defined as
$S^{*}(E)=\sigma(E)E\,\exp(87.21/\sqrt{E}+0.46E)$, where $\sigma(E)$ is the
fusion cross section and the energy $E$ is measured in units of MeV. The
coupling to the first 2+ state both in the projectile and in the target, as
well as their mutual excitations, are taken into account. The solid and dashed
lines show the result of the coupled-channels and the single-channel
calculations, respectively. For the coupled-channels calculation, the
contribution of $J=2$, $J$ being the spin of the compound nucleus, is also
shown by the thin dashed line. One can clearly see that the number of
resonance peaks dramatically increases by taking account the channel coupling
effects. This effect can be explained in terms of the double resonance
mechanism [309, 310]. That is, the number of resonance peak increases when
both the entrance channel (at the energy $E$ and the angular momentum $J$) and
an excited channel (at the energy $E-\epsilon_{2}$ and the angular momentum
$J-2$) have resonance states. This is, in a sense, similar to the Feshbach
resonance, which has been intensively discussed in physics of cold atoms [311,
312].

See also Refs. [313, 314, 315] for other recent coupled-channels calculations
for this system.

###  5.2 Medium-heavy systems

####  5.2.1 Subbarrier enhancement of fusion cross sections

Figure 23: Fusion cross sections for the 16O+154Sm (the left panel) and the
58Ni+58Ni (the right panel) systems. The dashed lines show the results of
single-channel calculations, while the solid lines take into account the
channel coupling effects. For the 16O+154Sm system, the excitation of the
ground state rotational band in 154Sm is included. On the other hand, for the
58Ni+58Ni system, the quadrupole excitations up to the double phonon states
are taken into account. The experimental data are taken from Refs. [316, 317].

In fusion of medium-heavy systems, the absorption inside the Coulomb barrier
is strong and it is a good assumption that the compound nucleus is
automatically formed once two nuclei touch with each other. In this
circumstance, resonance structure in fusion cross sections cease to appear and
the fusion dynamics is entirely governed by the penetration of the Coulomb
barrier [43]. We refer to Refs. [43, 318, 319, 320, 321, 322, 323] for
previous reviews on heavy-ion fusion reactions in medium-heavy systems.

A characteristic feature of fusion reactions in medium-heavy systems at
energies around the Coulomb barrier is that fusion cross sections are
significantly enhanced by the channel coupling effects. Figure 23 shows the
results of the coupled-channels calculations for the 16O+154Sm (the left
panel) [43] and for the 58Ni+58Ni (the right panel) [324, 325] as typical
examples. For the 16O+154Sm systems, the excitations of the ground state
rotational band in 154Sm are taken into account while 16O is assumed to be
inert. On the other hand, for the 58Ni+58Ni system, the excitations of the one
and two quadrupole phonon states in both of the 58Ni nuclei are taken into
account. In both systems, it is clearly seen that the fusion cross sections
are significantly enhanced by several orders of magnitude at energies below
the Coulomb barrier as compared to the prediction of a single-channel
calculation (the dashed lines). The coupled-channels calculations well account
for the enhancement of the fusion cross sections.

Figure 24:  The dependence of the internucleus potential for the 16O+154Sm
system on the orientation angle $\theta_{d}$ of the deformed 154Sm target. The
solid and the dashed lines show the potentials for $\theta_{d}$= 0 and 90
degrees, respectively, for the quadrupole deformation parameter of
$\beta_{2}=0.306$ and the hexadecapole deformation parameter of
$\beta_{4}=0.05$. The dotted line shows the potential in the spherical limit.

The subbarrier enhancement of fusion cross sections can be easily understood
for the 16O+154Sm system. Notice that the energy of the first 2+ state,
$\epsilon_{2}=89$ keV in the deformed 154Sm nucleus, is much smaller than the
typical energy scale of fusion reactions characterized by the curvature of the
Coulomb barrier [43], which is typically around 3.5 MeV. Neglecting the
excitation energies of the ground state rotational band in 154Sm, the fusion
cross sections can be calculated with Eq. (51) as a weighted average of
contributions of different orientation angles. The solid and the dashed lines
in Fig. 24 show the internucleus potential for $\theta_{d}=0$ and $\pi/2$,
respectively, with the deformation parameters of $\beta_{2}=0.306$ and
$\beta_{4}=0.05$. For prolate nuclei with $\theta_{d}=0$, the projectile
nucleus approaches the target nucleus from the longer axis of the target, and
thus the Coulomb barrier is lower than the potential in the spherical limit
(see the dotted line), since the nuclear attraction acts from larger distances
For $\theta_{d}=\pi/2$, the opposite happens and the Coulomb barrier is higher
than that for the spherical case. The tunneling probability, and thus the
fusion cross section, increases for $\theta_{d}=0$ and decreases for
$\theta_{d}=\pi/2$. However, since the tunneling probability has an
exponential dependence on the incident energy, the enhancement for
$\theta_{d}\sim 0$ leads to the main contribution to the fusion cross section,
leading to the subbarrier enhancement of fusion cross sections. This also
explains why the single-channel and the coupled-channels calculations lead to
similar fusion cross sections at high energies, since the tunneling
probability becomes close to unity irrespective of the deformation effect.

For the 58Ni + 58Ni system, the interpretation is more complicated, since the
geometrical interpretation cannot be applied as in the 16O+154Sm system.
However, as we have demonstrated in Fig. 1, the potential is dynamically
altered by the channel coupling effects, which in general results in an
enhancement of the penetrability at energies below the barrier. The subbarrier
enhancement of fusion cross sections for this system reflects this effect.

####  5.2.2 Barrier distributions

Equation (51) for fusion cross sections with a deformed target can be
symbolically expressed as a weighted sum of the contribution of eigen-channels
$\alpha$ as

$\sigma_{\rm fus}(E)=\sum_{\alpha}w_{\alpha}\sigma_{\rm
fus}^{(0)}(E;V_{\alpha}(R)),$ (111)

where $w_{\alpha}$ are the weight factors and $\sigma_{\rm
fus}^{(0)}(E;V_{\alpha}(R))$ is the single-channel fusion cross section with
the potential $V_{\alpha}(R)$. For Eq. (51), the eigen-channel $\alpha$
corresponds to the orientation angle $\theta$ with the weight factor
$w_{\theta}=2\pi\sin\theta$. Within the isocentrifugal approximation, this
formula is exact when the excitation energy of the intrinsic state is zero,
but the same formula holds to a good approximation even when the excitation
energy is finite [43, 326]. In the latter case, the eigen-channel $\alpha$ can
be defined as the eigenstates of $S^{\dagger}S$ [43], for which the potential
$V_{\alpha}$ is obtained by diagonalizing $H_{0}(\xi)+V_{\rm
coup}(\mbox{\boldmath$R$},\xi)$ in eq. (10) at each $r$.

For a single-channel problem, one can derive a compact expression for fusion
cross sections by approximating the Coulomb barrier by an inverse parabola
[43, 320, 327, 328, 329, 330]. The resultant Wong formula [327] reads

$\sigma_{\rm
fus}^{(0)}(E;V_{\alpha}(R))=\frac{\hbar\Omega_{\alpha}}{2E}\,R_{\alpha}^{2}\,\ln\left[1+\exp\left(\frac{2\pi}{\hbar\Omega_{\alpha}}(E-B_{\alpha})\right)\right],$
(112)

where $B_{\alpha}$, $R_{\alpha}$, and $\hbar\Omega_{\alpha}$ are the height,
the position, and the curvature of the Coulomb barrier for the potential
$V_{\alpha}(R)$, respectively. Notice that the first energy derivative of
$E\sigma_{\rm fus}^{(0)}$ from this formula is proportional to the $s$-wave
penetrability of the Coulomb barrier with a parameter set of
$(B,R_{b},\hbar\Omega)$,

$\frac{d}{dE}[E\sigma_{\rm fus}^{(0)}(E)]=\pi R_{b}^{2}\,P(E),$ (113)

with

$P(E)=\frac{1}{1+\exp\left[-\frac{2\pi}{\hbar\Omega}(E-B)\right]}.$ (114)

Since the first derivative of the penetrability $P(E)$ is a Gaussian-like
function peaked at $E=B$, the second energy derivative of $E\sigma_{\rm
fus}(E)$ shows in general multiple of peaks at $E=B_{\alpha}$ due to the
channel coupling effects. This function thus provides a distribution of the
barrier heights, and is referred to as the fusion barrier distribution $D_{\rm
fus}$ [321, 331],

$D_{\rm fus}(E)=\frac{d^{2}(E\sigma_{\rm fus}(E))}{dE^{2}}.$ (115)

The fusion barrier distribution has been extracted experimentally for several
systems [321, 316]. These barrier distribution have shown that the fusion
barrier distribution is in general sensitive to the details of the channel
couplings and thus provides a powerful tool to visualize the underlying
dynamics of subbarrier fusion reactions. Fig. 25 shows the barrier
distribution for the 16O+154Sm system as an example. The figure also shows the
barrier distribution obtained with the coupled-channels calculation (the solid
line), for which the contribution of each orientation angle is shown by the
dashed lines. One can see that the barrier distribution is structured, and the
coupled-channels calculation well accounts for the shape of the barrier
distribution. By analyzing the barrier distribution, one can infer the
energies of the barrier height for the side ($\theta_{d}=\pi/2$) and for the
tip ($\theta_{d}=0$) collisions. This has been utilized in fusion for
superheavy elements, which we shall discuss in Sec. 5.4. We also mention that
the Bayesian analysis may be used to analyze fusion barrier distributions as
well [332].

Figure 25:  The fusion barrier distribution for the 16O+154Sm system. The
dashed lines show the contributions of different orientation angles of the
deformed 154Sm, while the solid line denotes the sum of all the contributions.
The experimental data are taken from Ref. [316].

A similar barrier distribution can be extracted using an excitation function
of quasi-elastic scattering at backward angles. Here, the quasi-elastic
barrier distribution is defined as the first energy derivative of the
Rutherford ratio of quasi-elastic scattering [333],

$D_{\rm qel}(E)=-\frac{d}{dE}\left(\frac{d\sigma_{\rm
qel}(E)}{d\sigma_{R}(E)}\right),$ (116)

where $d\sigma_{\rm qel}/d\Omega$ and $d\sigma_{R}/d\Omega$ are quasi-elastic
and the Rutherford cross sections, respectively. The quasi-elastic barrier
distribution is defined at backward angles, corresponding to low partial waves
in the semi-classical approximation, so that the quasi-elastic barrier
distribution has a good correspondence to the fusion barrier distribution. A
mapping onto the barrier distribution for the $s$-wave scattering can be
achieved by introducing the effective energy defined by [333, 334]

$E_{\rm eff}=2E\frac{\sin(\theta/2)}{1+\sin(\theta/2)},$ (117)

that is derived by subtracting from the total energy $E$ the centrifugal
energy for the classical trajectory for the scattering angle $\theta$. The
quasi-elastic barrier distribution so obtained corresponds to the first energy
derivative of the reflection probability of the Coulomb barrier, and is thus
complementary to the fusion barrier distribution. In fact, it has been shown
that the shape of the quasi-barrier distribution is similar to that of the
fusion barrier distribution, even though the former is slightly less sensitive
to the details of the couplings because of a low-energy tail in the
distribution originated from the tail of the nuclear potential [334].

Figure 26: The quasi-elastic barrier distribution for the 24Mg+90Zr system and
its comparison to the coupled-channels calculations with several values of the
deformation parameters of the 24Mg nucleus. The right panel shows the
posterior probability distribution from the Bayesian analysis. Taken from Ref.
[335].

Recently, the experimental quasi-elastic barrier distribution for the
24Mg+90Zr system has been analyzed to extract the quadrupole and the
hexadecapole deformation parameters of the 24Mg nucleus [335]. To this end,
the authors of Ref. [335] employed the Bayesian statistics and determined the
values and the sign of the deformation parameters with the Markov-chain Monte-
Carlo method. This is shown in Fig. 26. From this analysis, the hexadecapole
deformation parameter of 24Mg, $\beta_{4}=-0.11\pm 0.02$, has been determined
precisely for the first time.

Incidentally, the shape of the measured quasi-elastic barrier distribution has
been shown significantly different between the 20Ne+90Zr and 20Ne+92Zr systems
[336]. While the barrier distribution for the 20Ne+90Zr can be well reproduced
by taking into account the rotational excitations of 20Ne and the vibrational
excitations of 90Zr, similar coupled-channels calculations do not reproduce
the data for the 20Ne+92Zr system, despite the fact that the couplings are
well dominated by the rotational excitations of the strongly deformed 20Ne
nucleus. See also Refs. [337, 338, 339, 340]. The difference has been
attributed to non-collective excitations in the target nuclei [341, 342, 343,
344], originated from the two extra neutrons in 92Zr.

####  5.2.3 Deep subbarrier fusion hindrance

At deep subbarrier energies, it has been systematically observed that
experimental fusion cross sections fall off much steeper as compared to
theoretical fusion cross sections based on the coupled-channels calculations
[345, 322, 346]. Two theoretical models have been proposed to interpret this
phenomenon, based either on the sudden approximation [347, 348] or on the
adiabatic approximation [349, 350, 351]. In both models, the deep subbarrier
fusion hindrance is attributed to the dynamics after two colliding nuclei
touch with each other [352]. In the sudden model, one considers a shallow
potential, for which the Pauli principle leads to a repulsive contribution at
short distances [353]. Fusion cross sections are then hindered at deep
subbarrier energies as the contribution of high partial waves is suppressed.
In contrast to the sudden model, the adiabatic model assumes that the
internuclear potential is smoothly connected to a mono-nucleus potential,
which is often described by the liquid drop model. After two nuclei touch with
each other, it is not reasonable to expand the total wave function with the
eigenstates of isolated nuclei, or at least it is not practical to do so as
one would have to include a huge number of states. To take this into account,
the adiabatic model assumes that the coupling is effectively damped after the
touching [349, 350, 351], which has been justified microscopically by the
random phase approximation (RPA) with two-center shell model wave functions
[354, 355]. In the adiabatic model, the deep subbarrier hindrance is
originated from a quenching of an enhancement of fusion cross sections due to
the channel coupling effects.

Figure 27: The fusion cross sections for the 64Ni+64Ni system obtained with
the sudden model (the left panel) and the adiabatic model (the right panel).
In both models, the standard coupled-channels calculations (the dotted line in
the light panel and the dashed and the dot-dashed lines in the right panel)
overestimate fusion cross sections at deep subbarrier energies. The arrow in
the right panel indicates the threshold energy for the deep subbarrier fusion
hindrance. Taken from Refs. [347, 351].

Figure 27 shows a comparison between the sudden (the left panel) and the
adiabatic (the right panel) models for the fusion cross sections for the
64Ni+64Ni system. The results of the standard coupled-channels calculations
are indicated by the dotted line in the left panel and by the dashed and the
dot-dashed lines in the right panel. Those calculations well account for the
experimental fusion cross sections at energies around the Coulomb barrier, but
they significantly overestimate the cross sections at deep subbarrier
energies. By either introducing a repulsive potential in the sudden model or
quenching the channel coupling effects in the adiabatic model, the deep
subbarrier hindrance is well reproduced (see the solid lines). One can clearly
see that both models work equally well. This indicates that fusion cross
sections themselves do not distinguish the two models and other observables
are needed to judge which model is more reasonable. For instance, average
angular momenta of a compound nucleus could be used for this purpose [356],
even though it would be challenging to measure them directly at deep
subbarrier energies.

###  5.3 Semi-microscopic approach

Figure 28: The fusion cross sections for the 16O+208Pb system. The dashed and
the solid lines are obtained with single-channel calculations with the double
folding potential and with a Woods-Saxon potential, respectively. For the
Woods-Saxon potential, the diffuseness parameter of $a=1.12$ fm is employed.
The experimental data are taken from Ref. [357].

In Sec. 3, we have discussed the microscopic coupled-channels method and its
application to inelastic scattering. Unfortunately, a similar approach does
not work for heavy-ion fusion reactions. This is demonstrated in Fig. 28 (see
also Refs. [358, 359]). In the figure, the experimental fusion cross sections
for the 16O+208Pb system are compared with a single-channel calculation based
on the double folding model with the density-dependent M3Y interaction (with
the zero-range approximation for the exchange term) [360, 361]. One can see
that the double folding model significantly overestimates the fusion cross
sections at energies above the Coulomb barrier. This feature remains the same
even if the channel coupling effects are taken into account [359]. The double
folding potential can be well fitted with a Woods-Saxon potential with the
surface diffuseness parameter of $a\sim$ 0.63 fm. It has been known that the
surface diffuseness parameter has to be phenomenologically increased to fit
the experimental fusion cross sections (see the solid line in the figure)
[324, 316, 359, 362, 363], even though scattering data can be fitted with a
surface diffuseness parameter that is consistent with a double folding
potential [364, 365, 366, 367, 368].

The failure of the double folding model for fusion reactions may not be
surprising, given that the model takes into account only a simple exchange
term, that is, the knock-on exchange term in which the exchange operator is
applied only to two interacting nucleons. From the point of view of many-body
theories, such as the resonating group method (RGM), there are many other
types in the exchange terms, which may play an important role especially in
short distances relevant to fusion reactions [369, 370, 371, 372]. Moreover,
couplings to high-lying collective and non-collective states may also play a
role in fusion reactions at energies above the barrier [363, 373, 374].

Given this situation, a semi-microscopic coupled-channels approach has been
developed [325, 375]. In this approach, the operator $\alpha_{\lambda\mu}$ in
Eq. (26) is replaced by (see Eq. (29)),

$\alpha_{\lambda\mu}=\frac{3eZ_{T}R_{T}^{\lambda}}{4\pi}\,Q_{\lambda\mu},$
(118)

where $Q_{\lambda\mu}$ is a many-body multipole operator

$Q_{\lambda\mu}=\sum_{i}r_{i}^{\lambda}Y_{\lambda\mu}(\mbox{\boldmath$r$}_{i}).$
(119)

For the Coulomb coupling, $Q_{\lambda\mu}$ in Eq.(27) is replaced by Eq.
(119). Using many-body wave functions from microscopic nuclear structure
calculations for $\varphi_{\alpha Im_{I}}(\xi)$ in Eq. (11), one can then
construct the coupled-channels equations microscopically. This approach still
uses a phenomenological potential $V_{N}(R)$, thus it is called the semi-
microscopic approach. In this way, the difference in the coupling strengths
between the nuclear and the Coulomb couplings can be automatically taken into
account. In practical calculations, one may scale the coupling strengths to an
empirical value known for the transition between the ground state and the
first excited state for a given nucleus. Also one may scale the excitation
energies to the experimental energy of the first excited state.

Figure 29: The fusion cross sections (the left panel) and the fusion barrier
distribution (the right panel) for the 16O+208Pb system. The dotted lines show
the result of the single-channel calculation. The dashed and the solid lines
are obtained with the coupled-channels method with the vibrational couplings
up to the double quadrupole and octupole phonon states in 208Pb. The dashed
lines use the harmonic oscillator (HO) model, while the solid lines take into
account the effects of anharmonicity with the generator coordinate method
(GCM) based on the covariant density functional theory. The experimental data
are taken from Ref. [357].

Fig. 29 shows the results of the semi-microscopic approach for the 16O+208Pb
system[375]. To this end, the intrinsic states are constructed with the multi-
reference covariant density functional theory, in which many Slater
determinants are linearly superposed after the angular momentum and the
particle number projections are applied. The left and the right panels show
the fusion cross sections and the corresponding fusion barrier distribution,
respectively. The dotted lines show the result of the single-channel
calculation. On the other hand, the dashed lines are obtained by taking into
account up to two octupole and quadrupole phonon states in the harmonic
oscillator approximation. In this approximation, the octupole and the
quadrupole phonons are treated as independent phonons. Such calculation
underestimates the fusion cross sections, and overestimates the height of the
main peak in the fusion barrier distribution [357]. This problem is resolved
by taking into account the anharmonicities of the octupole and the quadrupole
phonons with the semi-microscopic coupled-channels approach, as is shown by
the solid lines. It has been pointed out that the couplings between the
octupole and the quadrupole phonons plays an important role in reproducing the
experimental data [375].

The semi-microscopic coupled-channels approach can be combined with any
microscopic nuclear structure calculation, such as shell model and (quasi-
particle) random phase approximation. We mention that earlier studies with the
interacting boson model (IBM) [376, 377, 378, 379, 380, 381, 382] also belong
to this approach.

###  5.4 Heavy and superheavy systems

Superheavy elements, defined as the transactinide elements with $Z\geq 104$,
have attracted a lot of attention in recent years [383, 384, 385, 386, 387,
388, 389]. A standard way to synthesize those superheavy elements is to use
heavy-ion fusion reactions at energies around the Coulomb barrier [383, 384].
A characteristic in fusion of such massive systems is that the touching point
appears outside the fission barrier, and thus the formation of a compound
nucleus is not yet achieved even if the Coulomb barrier is overcome [390]. To
form a compound nucleus, the inner fission barrier has to be overcome, but
before that two nuclei may reseparate with a large probability after
exchanging a few nucleons and energies. Such process is referred to as the
quasi-fission process [386]. Formation cross sections of a compound nucleus,
that is, capture cross sections are then given as (see Eq. (22)),

$\sigma_{\rm cap}(E)=\frac{\pi}{K^{2}}\sum_{J_{T}}(2J_{T}+1)T_{J_{T}}(E)P_{\rm
CN}(E,J_{T}).$ (120)

In this equation,
$T_{J_{T}}(E)=\left(1-\sum_{c}|S^{J_{T}}_{cc_{0}}|^{2}\right)$ is the
penetrability of the Coulomb barrier, while $P_{\rm CN}(E,J_{T})$ is the
probability to overcome the inner barrier to form a compound nucleus. The
Langevin approach has often been used to describe $P_{\rm CN}(E,J_{T})$ [390,
391, 392, 393, 394, 395]. $P_{\rm CN}(E,J)$ is almost one in light and medium-
heavy systems, since the touching point is inside the inner barrier and thus
it is a good approximation to assume that the compound nucleus is
automatically formed once the touching configuration is reached. A transition
to the massive regime is considered to occur when the charge product of the
projectile and the target nuclei, $Z_{P}Z_{T}$, is around 1600-1800 [396,
397].

The compound nucleus in the superheavy region predominantly decays by fission.
A complication is that the final products of the fusion-fission process often
overlap with those of the quasi-fission process. It is thus difficult to
separate them unambiguously, and a detection of fission fragments alone does
not guarantee a formation of a compound nucleus. For this reason, usually a
formation of a compound nucleus is identified by measuring evaporation
residues. Cross sections for the formation of evaporation residues are then
given by,

$\sigma_{\rm ER}(E)=\frac{\pi}{K^{2}}\sum_{J_{T}}(2J_{T}+1)T_{J_{T}}(E)P_{\rm
CN}(E,J_{T})W_{\rm suv}(E^{*},J_{T}),$ (121)

where $E^{*}$ is the excitation energy of the compound nucleus and $W_{\rm
suv}(E^{*},J_{T})$ is the survival probability of the compound nucleus against
fission, which can be estimated using a statistical model.

Figure 30: (Upper panel) The barrier distribution for the capture process for
the 48Ca+248Cm system. The solid line is obtained with the coupled-channels
calculation which includes the rotational excitation of 248Cm, the octuple
phonon excitation of 48Ca, and a one-neutron transfer process. The dashed line
shows the contribution of the side collision with $\theta_{d}=\pi/2$. The
experimental data are taken from Ref. [398]. (Lower panel) The evaporation
residue cross sections for the same system. The experimental data are taken
from Ref. [399].

Recently, the quasi-elastic barrier distribution has been successfully
measured for several systems which are relevant to syntheses of superheavy
elements [400, 401, 398, 402]. Such barrier distribution is related to the
capture process, whose probability is given by $T_{J_{T}}$ in Eq. (121). As an
example, the top panel of Fig. 30 shows the experimental barrier distribution
for the 48Ca+248Cm system [398] and its comparison to the coupled-channels
calculations. In the coupled-channels calculation, the excitations of the
rotational band in the deformed 248Cm nucleus, the octupole phonon excitation
of 48Ca, as well as a one-neutron transfer reaction are taken into account.
The coupled-channels calculation provides information on the energy region
corresponding to the side collision, $\theta_{d}=\pi/2$ with respect to the
beam direction, which is indicated by the dashed line in the figure. The
measured evaporation residue cross sections $\sigma_{\rm ER}$ for this system
[399] are shown in the lower panel of the figure. One can clearly see that the
peak of $\sigma_{\rm ER}$ appears in the energy region corresponding to the
side collision. This measurement of the capture barrier distribution thus
provided the first experimental confirmation of the notion of compactness,
that is, the side collision provides the dominant contribution to the compound
nucleus formation since the touching configuration is compact such that the
height of the inner barrier is effectively lowered [403].

Figure 31:  Evaporation residue cross sections for the 48Ca+248Cm system
obtained with the extended fusion-by-diffusion model taking into account the
orientation angles of the deformed target nucleus (the solid lines). 3$n$ and
4$n$ indicate the 3 neutron and 4 neutron evaporation channels, respectively.
The dotted and the dashed lines show the contribution of the tip
($\theta_{d}=0$) and the side ($\theta_{d}=\pi/2$) collisions, respectively.
The experimental data are taken from Ref. [399].

Figure 31 shows the role of orientation angle of 248Cm in the evaporation
formation cross sections[404]. To this end, the evaporation residue cross
sections are computed as

$\sigma_{\rm
ER}(E)=\frac{\pi}{K^{2}}\int^{1}_{0}d(\cos\theta_{d})\sum_{J_{T}}(2J_{T}+1)T_{J_{T}}(E;\theta_{d})P_{\rm
CN}(E,J_{T};\theta_{d})W_{\rm suv}(E^{*},J_{T}),$ (122)

where $T_{J_{T}}(E;\theta_{d})$ and $P_{\rm CN}(E,J_{T};\theta_{d})$ are the
penetrability of the Coulomb barrier and the formation probability of the
compound nucleus for a given orientation angle $\theta_{d}$, respectively. See
also Ref. [405]. $P_{\rm CN}$ is estimated with the fusion-by-diffusion model
[390, 391, 393, 406] with an extension to introduce the angle dependence of
the injection point. The injection point $s_{\rm inj}$ is the initial position
to solve the diffusion equation to overcome the inner barrier, and is taken to
be [404],

$s_{\rm inj}(\theta_{d})=s_{\rm inj}^{(0)}+R_{T}\sum_{\lambda}\beta_{\lambda
T}Y_{\lambda 0}(\theta_{d}),$ (123)

where $s_{\rm inj}^{(0)}$ is the injection point in the spherical limit and
$\beta_{\lambda T}$ is the deformation parameter. With $\beta_{2T}>0$, the
injection point for $\theta_{d}=\pi/2$ is smaller than that for
$\theta_{d}=0$. Since the effective barrier height is given by $B_{\rm
eff}=V_{\rm fis}(s_{\rm sd})-V_{\rm fis}(s_{\rm inj})$, where $V_{\rm fis}$ is
a fission barrier and $s_{\rm sd}$ is the saddle point, $B_{\rm eff}$ is lower
for $\theta_{d}=\pi/2$ as compared to $B_{\rm eff}$ for $\theta_{d}=0$. In the
figure, the dashed and the dotted lines show the contributions of the side
($\theta_{d}=\pi/2$) and the tip ($\theta_{d}=0$) collisions, respectively.
For the 4 neutron evaporation channel ($4n$), which provides the most
important evaporation channel in this system, one can clearly see that the
side collision indeed provides the dominant contribution.

The injection point has subsequently been determined more microscopically by
using the time-dependent Hartree-Fock (TDHF) method [407]. Such method
provides a hybrid approach which combines the TDHF with the Langevin
approaches, which is suitable for applications to the region in which no
experimental data is available. In fact, this approach has been applied to
formation reactions of the superheavy element $Z=120$, revealing that the
capture cross sections are not sensitive to a choice of the reaction systems
[407].

###  5.5 Fusion of weakly-bound nuclei

Fusion of neutron-rich nuclei is important for nuclear astrophysics [408, 409]
as well as for superheavy elements [410]. In addition, the reaction mechanism
of fusion of neutron-rich nuclei is in itself intriguing, as several effects
interplay with each other. Those effects include: i) the extended density
distribution due to the weakly bound nature of neutron-rich nuclei, which
results in a lowering of the Coulomb barrier [411], ii) the breakup process,
which may hinder fusion cross sections since the lowering of the Coulomb
barrier disappears; at the same time, it may also enhance fusion cross
sections if couplings to a breakup channel dynamically lowers the Coulomb
barrier [412, 413, 414], and iii) the transfer processes, for which the
$Q$-value is positive for neutron-rich nuclei and fusion reaction are then
significantly affected [415, 416]. In the case of reactions with weakly-bound
nuclei, in addition to the complete fusion process, in which the whole
projectile is absorbed by the target, an important contribution to the fusion
comes from the so-called incomplete fusion (ICF), in which only a part of the
projectile is absorbed. The understanding and quantitative evaluation of ICF
has revealed as a very challenging theoretical problem. We refer the reader to
Refs. [24, 25, 417] for reviews as well as Refs. [418, 419] for earlier works
based on the semi-classical approximation.

Figure 32: Fusion cross sections for the 11Be+208Pb system obtained in Ref.
[413]. The dashed and the solid lines shows the total and the complete fusion
cross sections, respectively. The dotted line denotes the result of a single-
channel calculation without breakup of the projectile nucleus.

####  5.5.1 The CF suppression phenomenon

Experiments with light weakly-bound stable nuclei (such as 6,7Li or 9Be) have
shown that the CF (conveniently defined in these experiments as capture of the
complete charge of the projectile) are suppressed by $\sim$20-30% compared to
the case of tightly bound nuclei [420, 421, 422, 423, 424, 425, 426]. Early
analyses of these experiments tried to explain the phenomenon using coupled-
channels calculations, including the coupling to low-lying excited states of
the projectile and target [420, 422, 427, 428, 429]. Yet, these calculations
systematically failed to reproduce such suppression. This failure has been
attributed to the omission in these calculations of the breakup of the
projectile; a scenario was suggested in which the weakly bound projectile
breaks up prior to reaching the fusion barrier, with the subsequent reduction
of probability of complete capture. This interpretation is supported by the
presence of large $\alpha$ yields as well as target-like residues which are
consistent with the capture of one of the fragment constituents of the
projectile, that is, ICF. To account for these observations, some authors have
proposed a two-step scenario [422, 430] consisting on the elastic dissociation
of the projectile followed by the capture of one of the fragments by the
target. However, calculations based on a three-dimensional classical dynamical
model [430], which incorporates this two-step breakup-fusion mechanism, can
only explain a small fraction of the observed CF suppression for 9Be [431] and
8Li [432] reactions. More encouraging results have been obtained with
different methods based on the CDCC formalism, as described in the following
subsections.

####  5.5.2 Computation of fusion cross sections with the CDCC method

Although the CDCC method was originally devised to evaluate the elastic and
breakup observables, some attempts have been made to use this method to obtain
complete and incomplete fusion cross sections. In Refs. [413, 414], the CDCC
approach was applied to fusion of a weakly-bound projectile. In these
calculations, the CF was identified with the absorption from the entrance
channel while the ICF with the absorption from the breakup channels. Figure 32
shows the result of such calculation for the 11Be+208Pb reaction [413]. The
solid line shows the CF cross sections, while the dashed line shows the total
fusion cross sections (that is, a sum of complete and incomplete fusion cross
sections). For comparison, the figure also shows the result of the single-
channel calculation (dotted line). By comparing the solid and the dotted
lines, one can see that the CF cross sections may be enhanced as compared to
the fusion cross sections without the breakup process at energies below the
Coulomb barrier, while they are suppressed above the barrier.

A limitation of this method is that it can only be applied to projectiles
composed by a heavy charged fragment and a light uncharged one (such as 11Be),
as it relies on the assumption that the center of mass of the projectile is
close to that of the heavy fragment, and far from the light one. Thus, it
cannot be used for projectiles like 6,7Li, that break up into two fragments of
comparable masses. To overcome this difficulty, Hashimoto et al. [433]
proposed an alternative approach based also on the CDCC method, and applied it
to the case of deuteron scattering. Their idea is to transform the CDCC
wavefunction from its natural coordinates {$r$, $R$} to the coordinates
{${\mbox{\boldmath$r$}}_{p}$, ${\mbox{\boldmath$r$}}_{n}$},

$|\Psi({\mbox{\boldmath$r$}},{\mbox{\boldmath$R$}})|^{2}\,d{\mbox{\boldmath$r$}}d{\mbox{\boldmath$R$}}=|\tilde{\Psi}({\mbox{\boldmath$r$}}_{p},{\mbox{\boldmath$r$}}_{n})|^{2}d{\mbox{\boldmath$r$}}_{p}d{\mbox{\boldmath$r$}}_{n}\,,$
(124)

and then associate the CF and ICF cross sections with the absorption taking
place in different regions of the $\\{r_{p},r_{n}\\}$ space, as illustrated in
the left panel of Fig. 33. The distances $r^{\mathrm{ab}}_{p}$ and
$r^{\mathrm{ab}}_{n}$ denote the absorption radii for the proton and neutron,
such that for $r_{p}>r^{\mathrm{ab}}_{p}$ ($r_{n}>r^{\mathrm{ab}}_{n}$) the
proton (neutron) absorption becomes negligible. The CF is identified with the
absorption taking place when both the proton and neutron are inside their
respective absorption radii, i.e.,

$\displaystyle\sigma_{\rm
CF}=\frac{1}{|N|^{2}}\frac{K}{E}\int_{r_{p}<r_{p}^{\rm
ab}}d{\mbox{\boldmath$r$}}_{p}\int_{r_{n}<r_{n}^{\rm
ab}}d{\mbox{\boldmath$r$}}_{n}|\tilde{\Psi}({\mbox{\boldmath$r$}}_{p},{\mbox{\boldmath$r$}}_{n})|^{2}\\{W_{p}({\mbox{\boldmath$r$}}_{p})+W_{n}({\mbox{\boldmath$r$}}_{n})\\},$
(125)

where $K$ is the incident wave number, $E$ the energy and $N$ is a
normalization constant. Similarly, the ICF cross section is obtained from the
absorption occurring in the region where one of the two fragments is inside
its absorption radius while the other is outside it, i.e.,

$\displaystyle\sigma_{\rm
ICF}^{(p)}=\frac{1}{|N|^{2}}\frac{K}{E}\int_{r_{p}<r_{p}^{\rm
ab}}d{\mbox{\boldmath$r$}}_{p}\int_{r_{n}>r_{n}^{\rm
ab}}d{\mbox{\boldmath$r$}}_{n}|\tilde{\Psi}({\mbox{\boldmath$r$}}_{p},{\mbox{\boldmath$r$}}_{n})|^{2}W_{p}({\mbox{\boldmath$r$}}_{p}),$
(126) $\displaystyle\sigma_{\rm
ICF}^{(n)}=\frac{1}{|N|^{2}}\frac{K}{E}\int_{r_{p}>r_{p}^{\rm
ab}}d{\mbox{\boldmath$r$}}_{p}\int_{r_{n}<r_{n}^{\rm
ab}}d{\mbox{\boldmath$r$}}_{n}|\tilde{\Psi}({\mbox{\boldmath$r$}}_{p},{\mbox{\boldmath$r$}}_{n})|^{2}W_{n}({\mbox{\boldmath$r$}}_{n}).$
(127)

Figure 33:  Left: Schematic illustration of the four integration regions
employed in the method of Hashimoto et al. [433]. Right: a) Comparison of
$\sigma_{\rm ICF}^{(p)}$ (short-dashed line) and $\sigma_{\rm ICF}^{(n)}$
(dash-dotted line) obtained in [433] with $\sigma_{\rm STR}^{(p)}$ (squares)
and $\sigma_{\rm STR}^{(n)}$ (triangles) given by the Glauber model [434] b)
Complete fusion cross sections calculated with the Glauber model (dots) and
with the method of [433] (solid line).

The method was applied to the reaction $d$+7Li [433]. Although no comparison
with experimental data was attempted, the authors compared their results with
those computed with the Glauber calculations of Ref. [434]. This comparison is
shown in the right panel of Fig. 33 with the upper and lower panels
corresponding respectively to the ICF and CF cross sections as a function of
the deuteron incident energy. The agreement for the ICF is very satisfactory
for energies as low as $E_{d}$=10 MeV. Since the Glauber model is a high-
energy approximation, the agreement at these relatively low energies is
somewhat unexpected. By contrast, for the CF part the Glauber method gives
significantly smaller cross sections. A discussion of these results is
provided in Ref. [434].

Another procedure to extract the CF and ICF cross section from the CDCC method
was proposed by Parkar and co-workers [435]. The central idea of their method
is to perform a series of CDCC calculations with different choices of the
fragment-target potentials. In particular, to obtain the ICF cross section for
the capture of a given fragment, they perform a CDCC calculation using short-
range imaginary potential for the interaction of that fragment and the target,
and a real potential for the other fragment. Using this method, these authors
have been able to obtain a reasonable account of the CF, ICF and TF cross
sections of reactions induced by 6,7Li projectiles [435, 436], although with
some arbitrary choice of the short-range of the fusion potential.

More recently, the authors of Refs. [437, 438] have proposed an alternative
method which requires a single CDCC calculation to compute the TF, CF and ICF
cross sections. In this CDCC calculation the fragment-target interactions are
also modeled with optical potentials with a short-ranged imaginary part. For a
two-body projectile, the total fusion cross section is then computed as:

$\sigma_{\mathrm{TF}}=\frac{1}{|N|^{2}}\ \frac{K}{E}\ \ \left\langle\
\Psi^{(+)}\,\left|\,W^{(1)}+W^{(2)}\,\right|\Psi^{(+)}\ \right\rangle\,,$
(128)

where $W^{(1,2)}$ represent the imaginary parts of the fragment-target
interactions. This CDCC wavefunction can be split into bound ($\Psi^{B}$) and
continuum ($\Psi^{C}$) components:

${\Psi}^{(+)}({{\mbox{\boldmath$R$}}},{\mbox{\boldmath$r$}})={\rm\Psi}^{B}({{\mbox{\boldmath$R$}}},{{\mbox{\boldmath$r$}}})+{\Psi}^{C}({{\mbox{\boldmath$R$}}},{{\mbox{\boldmath$r$}}}),$
(129)

where ${\Psi}^{B}$ and ${\Psi}^{C}$ are respectively its components in the
bound and bin subspaces. They are given by the channel expansions

$\displaystyle{\Psi}^{B}({{\mbox{\boldmath$R$}}},{{\mbox{\boldmath$r$}}})$
$\displaystyle=$ $\displaystyle\sum_{\beta\,\in\,B}\ \chi_{\beta}({\bf
R})\phi_{\beta}({\bf r})$ (130)
$\displaystyle{\Psi}^{C}({{\mbox{\boldmath$R$}}},{\bf r})$ $\displaystyle=$
$\displaystyle\sum_{\gamma\,\in\,C}\
\chi_{\gamma}({{\mbox{\boldmath$R$}}})\phi_{\gamma}({{\mbox{\boldmath$r$}}}),$
(131)

where $\phi_{\beta}$ and $\phi_{\gamma}$ are respectively bound and unbound
states of the projectile, and $\chi_{\beta}$ and $\chi_{\gamma}$ are the
corresponding wave function describing the projectile-target relative motion.

Assuming that matrix-elements of the imaginary potentials connecting bound
channels to bins are negligible, Eq. (128) can be put in the form

$\sigma_{\mathrm{TF}}=\sigma_{\mathrm{TF}}^{B}\ +\ \sigma_{\mathrm{TF}}^{C},$
(132)

with

$\displaystyle\sigma_{\mathrm{TF}}^{B}$ $\displaystyle=$
$\displaystyle\frac{1}{|N|^{2}}\frac{K}{E}\sum_{\beta,\beta^{\prime}\,\in\,{B}}\
\left\langle\chi_{\beta}\left|W_{\beta\beta^{\prime}}^{(1)}+W_{\beta\beta^{\prime}}^{(2)}\,\right|\chi_{\beta^{\prime}}\right\rangle$
(133) $\displaystyle\sigma_{TF}^{C}$ $\displaystyle=$
$\displaystyle\frac{1}{|N|^{2}}\frac{K}{E}\sum_{\gamma,\gamma^{\prime}\in\,{C}}\
\left\langle\chi_{\gamma}\left|W_{\gamma\gamma^{\prime}}^{(1)}+W_{\gamma\gamma^{\prime}}^{(2)}\,\right|\chi_{\gamma^{\prime}}\right\rangle.$
(134)

where
$W_{\alpha\alpha^{\prime}}^{(i)}=\left(\phi_{\alpha}\left|{W}^{(i)}\right|\phi_{\alpha^{\prime}}\right),$
with $\alpha,\alpha^{\prime}$ standing for either $\beta,\beta^{\prime}$ or
$\gamma,\gamma^{\prime}$, are the matrix-elements of the imaginary potentials.

Then, by performing an angular momentum expansion of the wave functions and
the imaginary potentials, Eqs. (133) and (134) become

$\displaystyle\sigma_{TF}^{B}$ $\displaystyle=$
$\displaystyle\frac{\pi}{K^{2}}\,\sum_{J_{T}}(2J_{T}+1)\
\mathcal{P}_{B}^{TF}(J_{T})$ (135) $\displaystyle\sigma_{TF}^{C}$
$\displaystyle=$ $\displaystyle\frac{\pi}{K^{2}}\,\sum_{J_{T}}(2J_{T}+1)\
\mathcal{P}_{C}^{TF}(J_{T}),$ (136)

with

$\displaystyle\mathcal{P}_{B}^{TF}(J_{T})$ $\displaystyle=$
$\displaystyle\mathcal{P}_{B}^{(1)}(J_{T})+\mathcal{P}_{B}^{(2)}(J_{T})$ (137)
$\displaystyle\mathcal{P}_{C}^{TF}(J_{T})$ $\displaystyle=$
$\displaystyle\mathcal{P}_{C}^{(1)}(J_{T})+\mathcal{P}_{C}^{(2)}(J_{T}).$
(138)

where $\mathcal{P}_{B}^{(i)}(J_{T})$ and $\mathcal{P}_{C}^{(i)}(J_{T})$ are
the probabilities of absorption of fragment $c_{i}$ in bound channels and in
the continuum, respectively, resulting from the contributions of ${W}^{(i)}$
to the TF cross section.

In terms of these probabilities, the authors of Ref. [437, 438] introduce the
ICF probabilities

$\displaystyle{\mathcal{P}}^{ICF1}(J_{T})$ $\displaystyle=$
$\displaystyle{\mathcal{P}}^{(1)}_{C}(J_{T})\times\left[\
1-{\mathcal{P}}^{(2)}_{C}(J_{T})\ \right]$ (139)
$\displaystyle{\mathcal{P}}^{ICF2}(J_{T})$ $\displaystyle=$
$\displaystyle{\mathcal{P}}^{(2)}_{C}(J_{T})\times\left[\
1-{\mathcal{P}}^{(1)}_{C}(J_{T})\ \right],$ (140)

and the sequential complete fusion probability

${\mathcal{P}}^{\mathrm{SCF}}(J_{T})=2\
{\mathcal{P}}^{(1)}_{C}(J_{T})\times{\mathcal{P}}^{(2)}_{C}(J_{T}).$ (141)

In terms of the introduced probabilities, the following fusion cross sections
are defined:

  * •

Direct complete fusion (CF):

$\sigma_{DCF}=\sigma_{TF}^{B},$ (142)

which describes the simultaneous capture of the two fragments.

  * •

Sequential complete fusion (SCF):

$\sigma_{SCF}=\frac{\pi}{K^{2}}\,\sum_{J_{T}}(2J_{T}+1)\
\mathcal{P}^{SCF}(J_{T}).$ (143)

  * •

ICF of fragment $c_{i}$ (ICFi)

$\sigma_{ICFi}=\frac{\pi}{K^{2}}\,\sum_{J_{T}}(2J_{T}+1)\
\mathcal{P}^{ICFi}(J_{T}).$ (144)

In this formalism, the CF, ICF and TF cross sections are then given by

$\displaystyle\sigma_{\mathrm{CF}}$ $\displaystyle=$
$\displaystyle\sigma_{DCF}\,+\,\sigma_{SCF},$ (145)
$\displaystyle\sigma_{\mathrm{ICF}}$ $\displaystyle=$
$\displaystyle\sigma_{ICF1}\,+\,\sigma_{ICF2},$ (146)
$\displaystyle\sigma_{TF}$ $\displaystyle=$
$\displaystyle\sigma_{CF}+\sigma_{ICF}.$ (147)

An application of this model is shown in Fig. 34, where CF (left panel) and
ICF (right panel) data for the reaction 7Li+209Bi [422, 423] are compared with
the predictions of the model. The upper and lower panels display the same
results in logarithmic and linear scale, respectively. For the CF, the
agreement with the data is very satisfactory, both above and below the barrier
(indicated by the arrow). For the ICF, the separate contributions for triton
capture and $\alpha$ capture are shown, with the former giving a much larger
cross section. The calculations are found to reproduce well the data up to
$E_{\mathrm{c.m.}}\approx 34$ MeV, but overestimate them at higher energies. A
detailed discussion of these results can be found in Ref. [438].

Figure 34:  CF (left) and ICF (right) cross sections for 7Li+209Bi.
Experimental data from Refs. [422, 423] are compared with the calculations of
Ref. [438]. Figure reproduced from this reference with permission from APS.

####  5.5.3 Evaluation of CF and ICF cross sections with the IAV model

As discussed in Sec. 4, the IAV model is intended to provide the total
inclusive cross section corresponding to the detection of the $b$ fragment in
reactions of the form $A(a,b)X$. This results from the fact that the imaginary
part appearing in the expectation value of Eq. (106) accounts in principle for
all processes in which the participant fragment $x$ interacts nonelastically
with the target nucleus. This will include the ICF cross section, but also
other NEB processes not associated with the formation of a compound nucleus of
the $x+A$ system, such as target excitation. The isolation of the ICF cross
section from the total NEB cross section is indeed not a non-trivial problem.
An intuitive approach consists in identifying the ICF with the absorption due
to a short-ranged imaginary potential. This strategy has been however barely
explored so far within the IAV model.

A tentative application of this idea is shown in the right panel of Fig. 35,
corresponding to the reaction 7Li+209Bi. The symbols correspond to the ICF
data of Dasgupta et al. [422, 423]. For the calculations we show the
individual contributions to the ICF cross section, namely, $\alpha$-ICF (i.e.,
$\alpha$ absorbed) and $t$-ICF ($t$-absorbed) as well as their sum. To compute
the $\alpha$-ICF ($t$-ICF), the imaginary part of the $\alpha$+209Bi
($t$+209Bi) system was replaced by a short-range imaginary potential of Woods-
Saxon form and parameters $W_{0}=50$ MeV, $r_{i}=1.0$ fm, $a=0.2$ fm. The
results are very similar to those reported in Ref. [438] and shown in Fig. 34,
in which the authors made use of the absorption and survival probabilities
extracted from the CDCC calculations. Further calculations are needed to
elucidate the usefulness and applicability of the IAV model to evaluate ICF
cross sections.

Figure 35:  CF (left) and ICF (right) cross sections for 7Li+209Bi.
Experimental data from [422, 423] are compared with the calculations based on
the IAV model. In the CF plot, we include also the reaction cross section
(from a CDCC calculation) and the fusion computed with the barrier penetration
model (BPM).

The IAV model has been also used to infer CF cross sections of weakly bound
nuclei [439, 288]. The idea of this method is to decompose the reaction cross
section as follows

$\sigma_{R}\approx\sigma_{\mathrm{CF}}+\sigma_{\mathrm{inel}}+\sigma_{\mathrm{EBU}}+\sigma^{(b)}_{\mathrm{NEB}}+\sigma^{(x)}_{\mathrm{NEB}}.$
(148)

In this expression, $\sigma_{\mathrm{inel}}$ corresponds to the excitation of
the projectile and/or target without dissociation (i.e., inelastic
scattering). The terms $\sigma_{\mathrm{EBU}}$ and
$\sigma^{(b,x)}_{\mathrm{NEB}}$ correspond to the EBU and NEB contributions.
In the latter, one distinguishes the cases in which either the fragment $x$ or
$b$ interacts nonelastically with the target whereas the other scatters
elastically. A successful determination of the CF cross section from the
decomposition (148) requires that all other quantities involved in this
formula can be evaluated accurately. The pure inelastic scattering cross
sections ($\sigma_{\mathrm{inel}}$) are standardly computed by means of
coupled-channels calculations including low-lying collective excitations of
the projectile and target (see Sec. 2.3). The EBU part can be accurately
calculated, for instance, using the continuum-discretized coupled-channels
(CDCC) method discussed in Sec. 3. Finally, the NEB contributions can be
evaluated with the IAV model, at least for projectiles with a developed two-
boy structure, such as 6,7Li.

An application of the method to the 7Li+209Bi reaction is shown in the left
panel of Fig. 35, adapted from Ref. [439]. The circles are the CF data from
Ref. [423], the solid green line is the reaction cross section obtained from a
CDCC calculation and the solid red line the calculated CF cross section
inferred from Eq. (148) assuming a two-body model ($\alpha+t$) for 7Li. For
comparison, a single-channel barrier penetration model (BPM) calculation
(dashed lined) is shown. As is seen, the data are largely suppressed with
respect to the BPM. By contrast, the CF extracted from Eq. (148) explains very
well the data. As discussed in Ref. [439], the reduction with respect to the
BPM is found to be mostly due to the competition due to the 209Bi(7Li,
$\alpha$)X channel, which includes, among others, the $t$-ICF channel (see the
right panel).

####  5.5.4 Origin of the ICF large cross sections

As shown in Refs. [285, 291, 439], the elastic dissociation (EBU) mechanism
yields a comparatively small contribution to the total inclusive breakup cross
section in reactions induced by weakly bound nuclei such as deuterons or
6,7Li. Consequently, the widely accepted hypothesis that the CF suppression
arises as due to the two-step mechanism, consisting on the elastic
dissociation of the projectile followed by the capture of one of the produced
fragments does not seem to be supported by these calculations. In fact, IAV
calculations performed for the 7Li+209Bi reaction using the three-body version
of the IAV method, Eq. (110), in which the 7Li continuum is explicitly
included, did not find significant differences with respect to the DWBA IAV
calculations [288] for the calculated $\alpha$ production yields. In fact, the
calculations of Ref. [439] suggest that the large $\alpha$ yields observed in
6,7Li reactions can be explained in terms of a Trojan horse mechanism. The
idea is that, for a three-body reaction of the form $a+A$, with $a=b+x$, a
particular channel of the form $a+A\to b+c+C$ will be enhanced with respect to
the free, two-body reaction $x+A\to c+C$ due to the fact that the $a+A$ system
is above its Coulomb barrier. Loosely speaking, the $x$ particle is brought
inside its Coulomb barrier by the heavier particle $a$. The Trojan Horse
method has become a standard tool in nuclear astrophysics as an indirect way
of obtaining information of low-energy charged-particle induced reactions by
means of three-body reactions (see e.g. [440]) and its formal aspects can be
found elsewhere [441]. We illustrate in Fig. 36 the phenomenon for the
7Li+209Bi at hand. For that we compare the reaction cross sections for the
two-body reaction $t$+209Bi, as a function of the center-of-mass energy for
each system, with the three-body cross sections 209Bi(7Li, $\alpha$$X$) for
two different 7Li incident energies. The vertical arrow indicates the position
of the Coulomb barrier for the $t$+209Bi system. As expected, the reaction
cross section for the two-body reaction drops very quickly as the energy
decreases and approaches the Coulomb barrier. By contrast, the three-body
cross sections remain very large, even at energies well below their nominal
barrier. These results provide a natural explanation of the large $\alpha$
yields observed experimentally and confirmed by the IAV model.

The picture that emerges from these calculations is the following. The weakly-
bound projectile $a$ overcomes the $a+A$ Coulomb barrier, bringing also the
$x$ fragment inside its Coulomb barrier via the just described Trojan Horse
mechanism. This triggers the non-elastic processes between $x$ and $A$ which
give rise to the large variety of emerging fragments observed experimentally
and, in turn, to the suppression of CF. The present results add numerical
support to the suggestion put forward by Cook et al. [432].

Figure 36:  Illustration of the Trojan Horse mechanism for the 209Bi(7Li,
$\alpha$$X$) reaction. The broken lines are the three-body cross sections
(7Li, $\alpha$$X$) at the incident energies indicated by the labels. The black
solid lines are the two-body reaction cross sections (in arbitrary units). The
nominal position of the barrier for the $t$+209Bi reactions is indicated by
the arrow.

####  5.5.5 Application to surrogate reactions

The gradual improvement in the models oriented to the computation of ICF cross
sections has driven their applicability of the so-called surrogate method
(SRM). This method provides an indirect way of evaluating compound-nucleus
cross sections in reactions for which the direct measurement is difficult or
even not possible. An example is the extraction of neutron-induced cross
sections of the form $(n,\chi)$, where $\chi$ is a given decay channel product
($\gamma$, fission fragment, etc). Following the Bohr hypothesis, it is
customary assumed that in these reactions the formation and decay of a
compound nucleus take place independently of each other. In order to obtain
information on the decay of the compound nucleus ($B^{*}$) occurring in the
reaction of interest ($a+A\to B^{*}\to c+C$), one uses the alternative
(surrogate) reaction $a+A\to b+B^{*}$ that involves a projectile-target
combination ($a+A$) that is experimentally more accessible.

We consider for definiteness the extraction of $(n,\chi)$ cross sections from
a $(d,pX)$ surrogate reaction. Compound-nuclear reactions are properly
described in the Hauser-Feshbach formalism, which considers the conservation
of angular momentum $J$ and parity $\pi$. The cross section for the “desired”
reaction $A(n,\chi)$ is given by

$\displaystyle\sigma_{(n,\chi)}(E_{n})$ $\displaystyle=$
$\displaystyle\sum_{J_{T},\pi}\sigma_{J_{T},\pi}^{CN}(E_{ex},J_{T},\pi)\;\;G_{\chi}^{CN}(E_{ex},J_{T},\pi)\;,$
(149)

where $\sigma_{J_{T},\pi}^{CN}(E_{ex},J_{T},\pi)$ is the cross section for the
CN formation and $G_{\chi}^{CN}(E_{ex},J_{T},\pi)$ the branching ratio for the
decay in the channel $\chi$. Note that the excitation energy $E_{ex}$ of the
compound nucleus $B^{*}$ is related to the center-of-mass energy $E_{a}$ in
the entrance channel via the energy needed for separating $a$ from $B$:
$E_{a}=E-S_{a}(B)$. The objective of the surrogate method is to experimentally
determine or constrain the decay probabilities
$G_{\chi}^{CN}(E_{ex},J_{T},\pi)$, which are often difficult to calculate
accurately.

In the surrogate reaction, the same CN nucleus $B^{*}$ is formed and the decay
product of interest ($\chi$) is measured in coincidence with the outgoing
particle $b$. The probability for this process can be written as:

$P_{S,\chi}(E_{ex})=\sum_{J_{T},\pi}F_{S}^{CN}(E_{ex},J_{T},\pi)\;G_{\chi}^{CN}(E_{ex},J_{T},\pi)\;,$
(150)

where the subscript $S$ denotes the specific surrogate reaction (in this case,
$A(d,p)B^{*}$), $F_{S}^{CN}(E_{ex},J_{T},\pi)$ is the probability of forming
$B^{*}$ in this surrogate reaction (with specific values of $E_{ex}$, $J$ and
$\pi$) and $G_{\chi}^{CN}(E_{ex},J_{T},\pi)$ is the same branching ratio
appearing in the direct reaction (149). The probability $P_{S,\chi}(E_{ex})$
can be obtained experimentally as the ratio between the number of coincidences
between the $b$ particle and the decay particle $\chi$, $N_{S,\chi}$, and the
total number of surrogate events, $N_{S}$, i.e.:

$P^{exp}_{S,\chi}(E_{ex})=\frac{N_{S,\chi}}{N_{S}\,\epsilon_{\chi}}\;,$ (151)

where $\epsilon_{\chi}$ is the efficiency of detecting the exit-channel $\chi$
for the reactions in which $b$ is detected.

Ideally, if a reliable prediction of $F_{S}^{CN}(E_{ex},J_{T},\pi)$ is
possible, with an accurate determination of $P^{exp}_{S,\chi}(E_{ex})$ for a
range of energies and angles of $b$, one might be able to extract
$G_{\chi}^{CN}(E_{ex},J_{T},\pi)$ which can be then used to calculate the
desired cross section by means of (149). In practice, this approach is not
always feasible due to the lack of some of this required information and the
approach has relied on additional approximations. In particular, most
practical applications have made use of the so-called “Weisskopf-Ewing
approximation”, which assumes that the branching ratios
$G_{\chi}^{CN}(E_{ex},J_{T},\pi)$ are independent of the angular momentum and
spin, giving rise to the simplified cross section:

$\displaystyle\sigma_{(n,\chi)}(E_{n})$ $\displaystyle=$
$\displaystyle\sigma^{CN}(E_{ex})\;G_{\chi}^{CN}(E_{ex})\;,$ (152)

where $\sigma^{CN}(E_{ex})$ is to be understood as the CN cross section summed
over all possible $J_{T},\pi$ values. Applying the same approximation to the
surrogate reaction, and using $\sum_{J_{T},\pi}F_{S}^{CN}(E_{ex},J_{T},\pi)=1$
we have

$P_{S,\chi}(E_{ex})=G_{\chi}^{CN}(E_{ex})\;,$ (153)

allowing the determination of the desired cross section as

$\displaystyle\sigma_{(n,\chi)}(E_{n})$ $\displaystyle=$
$\displaystyle\sigma^{CN}(E_{ex})\;P_{S,\chi}(E_{ex})\;,$ (154)

which avoids the need of the probabilities $F_{S}^{CN}(E_{ex},J_{T},\pi)$.

One of the few cases in which the idealized SRM approach has been successfully
applied is shown in Fig. 37, taken from Ref. [442]. In this case, the
important required probability $F_{S}^{CN}(E_{ex},J_{T},\pi)$ was provided by
the IAV model. The extracted $(n,\gamma$) cross section is in excellent
agreement with the direct measurement (black and red squares). As shown also
in this figure, the result using Weisskopf-Ewing approximation departs
significantly from the direct measurement.

Figure 37:  Application of the surrogate method (SRM) to the 95Mo$(n,\gamma)$
reaction. The solid blue region is the $(n,\gamma)$ cross section obtained
from the SRM, the red circles and black squares are the direct measurements
[443, 444]. The uncertainty due to experimental data and fitting error is
indicated by the shaded band. The result obtained using the Weisskopf-Ewing
approximation is also shown (gold diamonds). Quoted from Ref. [442].

##  6 Summary

The coupled-channels method has been a standard tool to analyze and interpret
experimental data for nuclear reactions. Unlike the distorted wave Born
approximation (DWBA), it treats couplings to excited states in the colliding
nuclei to all orders in the coupling strengths, and thus it automatically
describes the interplay among several nuclear reaction processes. Moreover, a
microscopic optical potential can also be formulated via the coupled-channels
approach. In this paper, we have reviewed the current status and the recent
developments of the coupled-channels approach to nuclear reactions.

It has been known well that the channel coupling effects play an important
role in heavy-ion fusion reactions of medium-heavy systems at energies around
the Coulomb barrier. There, the fusion cross sections are largely enhanced due
to the channel coupling effects as compared to the prediction of single-
channel calculations. An interesting observation is that the standard coupled-
channels calculations, which well account for fusion cross sections around the
Coulomb barrier, overestimates fusion cross sections at deep subbarrier
energies. To explain this, a model has been proposed in which the channel
coupling effects are effectively quenched after two nuclei touch with each
other.

The channel coupling effects also play an important role in nuclear
astrophysics and in fusion for superheavy elements. For instance, it has been
shown that the number of resonance peaks increase significantly in the 12C+12C
fusion reaction due to the coupling to the first 2+ state in 12C. For
superheavy elements, it has been pointed out that the channel coupling effects
play a crucial role in determining the optimum energy for fusion with a
deformed target nucleus.

Conventionally, the coupled-channels approach has been formulated with the
macroscopic collective models. Such treatment has limitations, since one often
has to assume the harmonic oscillator for vibrations and the rigid rotor for
rotational motions. In this regard, the (semi-)microscopic coupled-channels
approach has been developed. In this approach, excited states of the colliding
nuclei are described with microscopic nuclear structure calculations. The
internucleus potentials may also be constructed microscopically based on the
double folding model. This method has been applied both to inelastic
scattering and to subbarrier fusion reactions. In particular, for inelastic
scattering, this approach has been successfully applied to extract information
on cluster structure in light nuclei.

In weakly-bound nuclei, the breakup process plays an important role. To
describe such process, the coupled-channels approach has to be extended to
include excitations to the continuum spectrum. For this purpose, the
continuum-discretized coupled-channels (CDCC) method has been widely used.
Although the method was originally devised to describe deuteron elastic and
breakup reactions, it was soon extended to other weakly-bound systems and
observables, including reactions induced by other weakly bound stable (e.g.
6,7Li) as well as two- and three-body halo nuclei (such as 11Be, 19C, 6He and
11Li) impinging on protons and composite target nuclei. Another advance has
been the inclusion of core degrees of freedom and the use of microscopic
fragment-target interactions. Additionally, the method has been successfully
employed in transfer (e.g. $(d,p)$), proton-induced knockout reactions of the
form $(p,pn)$ and $(p,2p)$ and fusion reactions of weakly bound nuclei at
near-barrier energies, which have been found to be affected by the breakup
channels. In recent years, several models based on the CDCC method have been
proposed to the evaluation of incomplete fusion cross sections, which has
remained a challenging and elusive problem for many years. The revival of the
inclusive breakup models first proposed in the 1980s, provides also a
promising tool to the evaluation of incomplete fusion cross sections.

In this way, the coupled-channels approach has played a central role in
nuclear reaction physics. However, at the same time, there still remain many
theoretical challenges. For instance, it has still been difficult to include
breakup, transfer, and inelastic excitations simultaneously. This is
particularly the case with pair transfer and multi-nucleon transfer channels,
which is relevant to reactions of exotic nuclei with neutron skins. Moreover,
it has also been a theoretical challenge to develop a many-body theory to
describe reaction processes involving high excitations, in which the concepts
of diffusion and friction are relevant. This is particularly important for
heavy-ion deep inelastic collisions and fusion for superheavy nuclei. Even
though some of these issues may need to wait for a development of powerful
computers, several works are now in progress towards these goals.

Another important remark is that some of the developments described in this
paper, such as CDCC with core and target excitations, the four-body CDCC
calculations, nonelastic breakup (IAV), and multiple scattering methods, are
not currently available in a form of publicly available codes. It would be
advisable that these codes become available in future once the formalisms will
have been consolidated and tested, so future generations can employ these
formalisms and developments.

Acknowledgements

We thank all our collaborators for discussions and fruitful collaborations.
This work was supported in part by JSPS KAKENHI Grant Numbers JP19K03861,
JP21H00120, JP17740148, JP22740169, JP22560820, JP25400255, JP16K05352, and in
part by the COREnet program of RCNP, Osaka University. A.M.M. is supported by
the Spanish Ministerio de Ciencia e Innovación y Universidades under project
PID2020-114687GB-I00 and by the Consejería de Economía, Conocimiento, Empresas
y Universidad (Junta de Andalucía) under project P20_01247.

##  Appendix A Details of the microscopic coupled-channels method

In this appendix, we describe the MCC framework for direct reactions, elastic
and inelastic scattering in particular, based on the MST. Some formulae
appearing below are essentially the same as those in Sec. 2. Nevertheless, we
intend to reformulate them with explicit indices to show unambiguously the
representation of the structural input, i.e., transition densities of Eq.
(179), and its implementation in the MCC reaction framework.

###  A.1 Multiple scattering theory

The basis of the MCC method discussed in this article is the MST developed by
Foldy [108], Watson [109], and Kerman, Mc-Manus, and Thaler [110]. Below we
briefly recapitulate the MST. Let us consider an NA scattering process. The
Schödinger equation is written by

$\left(T_{\mathrm{NA}}+\sum_{i}v_{i}+H_{\mathrm{A}}-E\right)\Psi^{\left(+\right)}=0,$
(155)

where $T_{\mathrm{NA}}$ is the kinetic energy operator regarding the NA
relative motion, $v_{i}$ is the NN bare interaction between the incident
nucleon and the $i$th nucleon in A, $H_{\mathrm{A}}$ is the internal
Hamiltonian of A, $E$ is the total energy of the system. For simplicity, we
omit many-nucleon forces in Eq. (155), and the Coulomb interaction is
disregarded in the formulae shown below. The total wave function is denoted by
$\Psi^{\left(+\right)}$, which contains the complete information on the
reaction system. Although it is represented only implicitly,
$\Psi^{\left(+\right)}$ involves components corresponding to all possible
boundary conditions.

As the starting point of the MST, we restrict ourselves to consider the
process whose boundary condition is expressed by the following LS equation.

$\bar{\Psi}^{\left(+\right)}=\phi+G_{0}^{\left(+\right)}\sum_{i}v_{i}\bar{\Psi}^{\left(+\right)},$
(156)

where $\phi$ is the plane wave and $G_{0}^{(+)}$ is the Green function given
by

$G_{0}^{\left(+\right)}=\frac{1}{E-T_{\mathrm{NA}}-H_{\mathrm{A}}+i\varepsilon}.$
(157)

It is then understood that one can impose the boundary condition

$\bar{\Psi}^{\left(+\right)}\rightarrow\phi+\sum_{c}f_{c}\frac{e^{iKR}}{R}$
(158)

when the displacement $R$ of the incident nucleon from the c.m. of A is
sufficiently larger than the range $R_{\mathrm{N}}$ of the strong interaction.
The scattering amplitude is denoted by $f_{c}$ with $c$ being the channel
index. The important point is that $\bar{\Psi}^{\left(+\right)}$ is a subset
of $\Psi^{\left(+\right)}$ because of the restriction of the boundary
condition. For instance, $\bar{\Psi}^{\left(+\right)}$ does not explicitly
include the rearrangement channels.

The transition matrix corresponding to Eq. (156) is given by

$T=\sum_{i}\Lambda_{i},$ (159)

with

$\Lambda_{i}\equiv
v_{i}+v_{i}G_{0}^{\left(+\right)}\sum_{j}v_{j}+v_{i}G_{0}^{\left(+\right)}\sum_{j}v_{j}G_{0}^{\left(+\right)}\sum_{k}v_{k}+...$
(160)

where $i,$ $j,$ $k,$ $...$ are the indices of the nucleons in A. The essence
of the MST is the resummation of the series in Eq. (160) as follows. First, we
rewrite Eq. (160) as

$\Lambda_{i}=v_{i}+v_{i}G_{0}^{\left(+\right)}\Lambda_{i}+v_{i}G_{0}^{\left(+\right)}\sum_{j\neq
i}\Lambda_{j},$ (161)

which reads

$\Lambda_{i}=\frac{1}{1-v_{i}G_{0}^{\left(+\right)}}v_{i}+\frac{1}{1-v_{i}G_{0}^{\left(+\right)}}v_{i}G_{0}^{\left(+\right)}\sum_{j\neq
i}\Lambda_{j}.$ (162)

By introducing the effective interaction $\tau_{i}$ defined through

$\tau_{i}=v_{i}+v_{i}G_{0}^{\left(+\right)}\tau_{i},$ (163)

$\Lambda_{i}$ is written by

$\Lambda_{i}=\tau_{i}+\tau_{i}G_{0}^{\left(+\right)}\sum_{j\neq
i}\Lambda_{j}.$ (164)

One may find that Eq. (164) is a series of multiple scattering with respect to
$\tau_{i}$, whereas Eq. (160) to $v_{i}$. Because $\tau_{i}$ takes into
account infinite number of interacting processes between the incident nucleon
and the $i$th nucleon in A, it is much easier to handle than $v_{i}$ as
$\tau_{i}$ has no repulsive core.

The transition matrix with the resummation reads

$T=\sum_{i}\tau_{i}+\sum_{i}\tau_{i}G_{0}^{\left(+\right)}\sum_{j\neq
i}\tau_{j}+\sum_{i}\tau_{i}G_{0}^{\left(+\right)}\sum_{j\neq
i}\tau_{j}G_{0}^{\left(+\right)}\sum_{k\neq j}\tau_{k}+...$ (165)

If the wave function of A to be operated by this $T$ is antisymmetrized
regarding the exchange of each nucleon-pair inside A, we obtain

$T=\frac{A}{A-1}\bar{T},$ (166)
$\bar{T}=\bar{\tau}+\bar{\tau}G_{0}^{\left(+\right)}\bar{T},$ (167)
$\bar{\tau}=\frac{A-1}{A}\tau,$ (168)

where $A$ is the number of nucleons in A; we have dropped the index $i$ of
$\tau$. One can interpret this result of the MST as a rewriting of Eq. (155)
to

$\left[T_{\mathrm{NA}}+A\bar{\tau}+H_{\mathrm{A}}-E\right]\bar{\Psi}^{\left(+\right)}=0$
(169)

with the modification of Eq. (166) to the resulting transition matrix
$\bar{T}$. Equation (169) makes sense, however, only for the scattering
process governed by the asymptotic condition of Eq. (158).

###  A.2 Coupled-channel equations based on the MST

Let us concentrate for a while on the NA elastic and inelastic processes. For
simplicity, we do not take the intrinsic spin of the incident nucleon into
account. We first expand the total wave function $\bar{\Psi}^{\left(+\right)}$
in terms of the eigenstates $\Phi_{nIm_{I}}$ of $H_{\mathrm{A}}$, where $n$,
$I$, $m_{I}$ are the energy index, the total spin, its third component,
respectively, of A; the parity of A is understood to be specified but not
shown in the subscript of $\Phi$. Here, it is assumed that $\Phi_{nIm_{I}}$
are either bound states or resonant states with narrow widths. When a broad
resonance comes into play, it is not isolated well from continuum states near
the resonant energy, and we need to consider both resonant and nonresonant
continuum states on the same footing. Such cases are discussed in Sec. 3.3.

To derive CC equations to be solved, we make the partial-wave expansion of
$\bar{\Psi}^{\left(+\right)}$. The partial scattering wave characterized by
the total angular momentum $J_{T}$ and its third component $M_{T}$ is defined
by

$\displaystyle\bar{\Psi}^{\left(J_{T}M_{T}\right)}_{n_{0}I_{0}L_{0}}\left(\boldsymbol{R},\xi\right)$
$\displaystyle=\sum_{nIL}\frac{\chi_{nIL,n_{0}I_{0}L_{0}}^{\left(J_{T}\right)}\left(K_{nI},R\right)}{KR}\sum_{m_{I},m_{L}}\langle
Im_{I}Lm_{L}|J_{T}M_{T}\rangle
i^{P}\Phi_{nIm_{I}}\left(\xi\right)i^{L}Y_{Lm_{L}}(\boldsymbol{\hat{R}})$
$\displaystyle=\sum_{nIL}\frac{\chi_{nIL,n_{0}I_{0}L_{0}}^{\left(J_{T}\right)}\left(K_{nI},R\right)}{KR}\left[i^{P}\Phi_{nI}\left(\xi\right)\otimes
i^{L}Y_{L}(\boldsymbol{\hat{R}})\right]_{J_{T}M_{T}}.$ (170)

Here, $\boldsymbol{R}$ and $\xi$ are the NA relative coordinate and all
internal coordinates of A, respectively, $P=0$ (1) for the positive (negative)
parity state of A. $\Phi_{nIm_{I}}$ satisfies

$H_{\mathrm{A}}\Phi_{nIm_{I}}\left(\xi\right)=\epsilon_{nI}\Phi_{nIm_{I}}\left(\xi\right),$
(171)

where $\epsilon_{nI}$ is the eigenenergy. The NA scattering wave function is
denoted by $\chi_{nIL,n_{0}I_{0}L_{0}}^{\left(J_{T}\right)}$, where $L$ is the
relative orbital angular momentum. The NA relative wave number $K_{nI}$ is
defined by

$K_{nI}=\frac{\sqrt{2\mu E_{nI}}}{\hbar},$ (172)

where $\mu$ is the NA reduced mass and

$E_{nI}=E-\epsilon_{nI}.$ (173)

In the present case, the channel is characterized by $n$, $I$, $L$; when
convenient, these are abbreviated by $c$ below. Similarly, the incident
channel is represented by $c_{0}\equiv\left(n_{0},I_{0},L_{0}\right)$. For
simplicity, $K_{nI}$ in the incident channel, $K_{n_{0}I_{0}}$, is denoted by
$K$ in Eq. (170).

Multiplying the Schrödinger equation for
$\bar{\Psi}^{\left(J_{T}M_{T}\right)}_{n_{0}I_{0}L_{0}}$

$\left[-\frac{\hbar^{2}}{2\mu}\frac{1}{R}\frac{d^{2}}{dR^{2}}R+\frac{\boldsymbol{\hat{L}}^{2}}{2\mu
R^{2}}+A\bar{\tau}+H_{\mathrm{A}}-E\right]\bar{\Psi}^{\left(J_{T}M_{T}\right)}_{n_{0}I_{0}L_{0}}\left(\boldsymbol{R},\xi\right)=0$
(174)

by $\left[i^{P^{\prime}}\Phi_{n^{\prime}I^{\prime}}\left(\xi\right)\otimes
i^{L^{\prime}}Y_{L^{\prime}}(\boldsymbol{\hat{R}})\right]_{J_{T}M_{T}}^{\ast}$
from the left, performing the integration over $\xi$ and
$\boldsymbol{\hat{R}}$, one gets

$\left[-\frac{\hbar^{2}}{2\mu}\frac{d^{2}}{dR^{2}}+\frac{\hbar^{2}}{2\mu}\frac{\left(L^{\prime}\right)^{2}}{R^{2}}-E_{n^{\prime}I^{\prime}}\right]\chi_{c^{\prime}c_{0}}^{\left(J_{T}\right)}\left(k_{n^{\prime}I^{\prime}},R\right)=-\sum_{c}U^{(J_{T})}_{c^{\prime}c}\left(R\right)\chi_{cc_{0}}^{\left(J_{T}\right)}\left(k_{nI},R\right),$
(175)

where the coupling potential is given by

$U^{(J_{T})}_{c^{\prime}c}\left(R\right)=\left.\left\langle\left[i^{P^{\prime}}\Phi_{n^{\prime}I^{\prime}}\left(\xi\right)\otimes
i^{L^{\prime}}Y_{l^{\prime}}(\boldsymbol{\hat{R}})\right]_{J_{T}M_{T}}\right|A\bar{\tau}\left|\left[i^{P}\Phi_{nI}\left(\xi\right)\otimes
i^{L}Y_{L}(\boldsymbol{\hat{R}})\right]_{J_{T}M_{T}}\right\rangle_{\xi,\boldsymbol{\hat{R}}}.\right.$
(176)

Note that $U^{(J_{T})}_{c^{\prime}c}$ is independent of $M_{T}$ as seen below
(see also Sec. 2.2). The CC equations (175) are solved under the boundary
condition

$\chi_{c^{\prime}c_{0}}^{\left(J_{T}\right)}\left(K_{n^{\prime}I^{\prime}},R\right)\rightarrow\frac{i}{2}\left[KRh_{L^{\prime}}^{\left(-\right)}\left(KR\right)\delta_{c^{\prime}c_{0}}-\sqrt{\frac{K}{K_{n^{\prime}I^{\prime}}}}S_{c^{\prime}}^{\left(J_{T}\right)}K_{n^{\prime}I^{\prime}}Rh_{L^{\prime}}^{\left(+\right)}\left(K_{n^{\prime}I^{\prime}}R\right)\right],\quad\left(\text{for
}R>R_{\mathrm{N}}\right),$ (177)

where $h_{L^{\prime}}^{\left(-\right)}$ and $h_{L^{\prime}}^{\left(+\right)}$
are the incoming and outgoing Hankel functions, respectively. The scattering
matrix to the $c^{\prime}$ channel is denoted by
$S_{c^{\prime}}^{\left(J_{T}\right)}$, with which scattering observables can
be evaluated in the standard manner. It should be noted that we only consider
the open channels, for which $E_{nI}>0$. It is known that at low energies, the
coupling with the closed channels corresponding to $E_{nI}<0$ becomes crucial
[22, 174]. We do not apply the MCC framework to that case because the MST
itself will be questionable there.

###  A.3 Coupling potentials

Now we describe the form of the coupling potential $U_{c^{\prime}c}$. Let us
consider using a G-matrix interaction $g$ as $\tau$. Similarly to
$\bar{\tau}$, we use

$\bar{g}\equiv\frac{A-1}{A}g=\bar{g}^{\mathrm{dr}}+\bar{g}^{\mathrm{ex}}\hat{P}^{\mathrm{ex}},$
(178)

where $\bar{g}^{\mathrm{dr}}$ and $\bar{g}^{\mathrm{ex}}$ are the direct and
exchange parts, respectively, $\hat{P}^{\mathrm{ex}}$ is the operator that
exchanges the incident nucleon and a nucleon in A. We concentrate on the
central part of $U_{c^{\prime}c}$ and do not differentiate proton and neutron
in the following formulation. We here introduce the one-body transition
density of A defined by

$\rho_{n^{\prime}I^{\prime}m_{I}^{\prime}nIm_{I}}^{\mathrm{tr}}\left(\boldsymbol{r}_{\mathrm{A}}\right)\equiv\int\Phi_{n^{\prime}I^{\prime}m_{I}^{\prime}}^{\ast}\left(\xi\right)\sum_{i=1}^{A}\delta\left(\boldsymbol{r}_{i}-\boldsymbol{r}_{\mathrm{A}}\right)\Phi_{nIm_{I}}\left(\xi\right)d\xi=\sum_{\lambda\mu}\langle
Im_{I}\lambda\mu|I^{\prime}m_{I}^{\prime}\rangle\rho_{n^{\prime}I^{\prime}nI,\lambda}^{\mathrm{tr}}\left(r_{\mathrm{A}}\right)Y_{\lambda\mu}^{\ast}\left(\boldsymbol{\hat{r}}_{\mathrm{A}}\right),$
(179)

where $\boldsymbol{r}_{i}$ is the coordinate of the $i$th nucleon from the
c.m. of A. It is understood that the c.m. motion of the nucleus A has been
eliminated in $\Phi_{Im_{I}}$ and the integration over $\xi$ is carried out
with the constraint of

$\sum_{i=1}^{A}\boldsymbol{r}_{i}=0.$ (180)

The transition density is orthonormalized as

$\sqrt{4\pi}\int\rho_{n^{\prime}I^{\prime}nI,\lambda}^{\mathrm{tr}}\left(r_{\mathrm{A}}\right)r_{\mathrm{A}}^{2}dr_{\mathrm{A}}=\delta_{\lambda
0}\delta_{n^{\prime}n}\delta_{I^{\prime}I}A.$ (181)

The one-body nuclear density of A is thus given by

$\rho_{nI}\left(r_{\mathrm{A}}\right)=\frac{1}{\sqrt{4\pi}}\rho_{nInI,0}^{\mathrm{tr}}\left(r_{\mathrm{A}}\right).$
(182)

It should be noted that the transition density has the following property

$\rho_{nIn^{\prime}I^{\prime},\lambda}^{\mathrm{tr}}\left(r_{\mathrm{A}}\right)=\frac{\hat{I}^{\prime}}{\hat{I}}\left(-1\right)^{I^{\prime}-I}\rho_{n^{\prime}I^{\prime}nI,\lambda}^{\mathrm{tr}}\left(r_{\mathrm{A}}\right).$
(183)

Let us first consider the direct part of $U_{c^{\prime}c}$, which is written
with the transition density by

$\displaystyle U_{c^{\prime}c,J_{T}}^{\mathrm{dr}}\left(R\right)$
$\displaystyle=\int
d\boldsymbol{\hat{R}}\,\sum_{m_{I}^{\prime}m_{L}^{\prime}}\langle
I^{\prime}m_{I}^{\prime}L^{\prime}m_{L}^{\prime}|J_{T}M_{T}\rangle\,i^{-P^{\prime}}i^{-L^{\prime}}Y_{L^{\prime}m_{L}^{\prime}}^{\ast}(\boldsymbol{\hat{R}})$
$\displaystyle\times\sum_{m_{I}m_{L}}\langle
Im_{I}lm_{L}|J_{T}M_{T}\rangle\,i^{P}i^{L}Y_{Lm_{L}}(\boldsymbol{\hat{R}})\sum_{\lambda\mu}\langle
Im_{I}\lambda\mu|I^{\prime}m_{I}^{\prime}\rangle\,\mathcal{U}_{n^{\prime}I^{\prime}nI,\lambda}^{\mathrm{dr}}\left(\boldsymbol{R}\right),$
(184)
$\mathcal{U}_{n^{\prime}I^{\prime}nI,\lambda}^{\mathrm{dr}}\left(\boldsymbol{R}\right)\equiv\int\bar{g}^{\mathrm{dr}}\left(s,k_{\mathrm{F};n^{\prime}I^{\prime}nI}\left(\left|\boldsymbol{R}+\boldsymbol{s}/2\right|\right)\right)\rho_{n^{\prime}I^{\prime}nI,\lambda}^{\mathrm{tr}}\left(r_{\mathrm{A}}\right)Y_{\lambda\mu}^{\ast}\left(\boldsymbol{\hat{r}}_{\mathrm{A}}\right)d\boldsymbol{r}_{\mathrm{A}},$
(185)

where

$\boldsymbol{s}\equiv\boldsymbol{r}_{\mathrm{A}}-\boldsymbol{R}$ (186)

and $k_{\mathrm{F};n^{\prime}I^{\prime}nI}$ stands for the Fermi momentum
defined by

$k_{\mathrm{F};n^{\prime}I^{\prime}nI}\left(r\right)=\left(\frac{3\pi^{2}}{2}\frac{\rho_{n^{\prime}I^{\prime}}\left(r\right)+\rho_{nI}\left(r\right)}{2}\right)^{1/3}.$
(187)

After some manipulation, one finds

$U_{c^{\prime}c,J_{T}}^{\mathrm{dr}}\left(R\right)=i^{-P^{\prime}-L^{\prime}+P+L}\sum_{\lambda}X_{I^{\prime}L^{\prime}IL,J_{T}\lambda}\mathcal{\bar{U}}_{n^{\prime}I^{\prime}nI,\lambda}^{\mathrm{dr}}\left(R\right),$
(188)

with

$X_{I^{\prime}L^{\prime}IL,J_{T}\lambda}\equiv\sqrt{2I^{\prime}+1}\sqrt{2L+1}\sqrt{2\lambda+1}\frac{\left(-\right)^{I^{\prime}+L-J_{T}}}{\sqrt{4\pi}}W\left(ILI^{\prime}L^{\prime};J_{T}\lambda\right)\langle
L0\lambda 0|L^{\prime}0\rangle,$ (189)
$\mathcal{\bar{U}}_{n^{\prime}I^{\prime}nI,\lambda}^{\mathrm{dr}}\left(R\right)\equiv
4\pi\sum_{l=0}^{\lambda}\frac{\sqrt{2\left(\lambda-l\right)+1}}{\left(2l+1\right)\sqrt{2\lambda+1}}\sqrt{{}_{2\lambda+1}C_{2l}}R^{\lambda-l}\langle
l0,\lambda-l,0|\lambda
0\rangle\,\int\mathcal{F}_{n^{\prime}I^{\prime}nI,\lambda
l}^{\mathrm{dr}}\left(R,s\right)s^{l+2}ds,$ (190)
$\mathcal{F}_{n^{\prime}I^{\prime}nI,\lambda
l}^{\mathrm{dr}}\left(R,s\right)\equiv\frac{2l+1}{2}\int_{-1}^{1}\bar{\rho}_{n^{\prime}I^{\prime}nI,\lambda}^{\mathrm{tr}}\left(\left|\boldsymbol{R}+\boldsymbol{s}\right|\right)\bar{g}^{\mathrm{dr}}\left(s,k_{\mathrm{F};n^{\prime}I^{\prime}nI}\left(\left|\boldsymbol{R}+\boldsymbol{s}/2\right|\right)\right)P_{l}\left(x\right)dx,$
(191)
$\bar{\rho}_{n^{\prime}I^{\prime}nI,\lambda}^{\mathrm{tr}}\left(r_{\mathrm{A}}\right)\equiv
r_{\mathrm{A}}^{-\lambda}\rho_{n^{\prime}I^{\prime}nI,\lambda}^{\mathrm{tr}}\left(r_{\mathrm{A}}\right),$
(192)

where $W\left(abcd;ef\right)$ is the Racah coefficient,
${}_{2\lambda+1}C_{2l}$ is the binomial coefficient, and

$x\equiv\frac{\boldsymbol{R}\cdot\boldsymbol{s}}{Rs}.$ (193)

To calculate the exchange part of $U_{c^{\prime}c}$, we follow the
localization method proposed by Brieva and Rook (BR) [445] for NA elastic
scattering. The resulting form is

$U_{c^{\prime}c,J_{T}}^{\mathrm{ex}}\left(R\right)=i^{-P^{\prime}-L^{\prime}+P+L}\sum_{\lambda}X_{I^{\prime}L^{\prime}IL,J_{T}\lambda}\mathcal{\bar{U}}_{n^{\prime}I^{\prime}nI,\lambda}^{\mathrm{ex}}\left(R\right),$
(194)
$\mathcal{\bar{U}}_{n^{\prime}I^{\prime}nI,\lambda}^{\mathrm{ex}}\left(R\right)=4\pi\sum_{l=0}^{\lambda}\frac{\sqrt{2\left(\lambda-l\right)+1}}{\left(2l+1\right)\sqrt{2\lambda+1}}\sqrt{{}_{2\lambda+1}C_{2l}}R^{\lambda-l}\langle
l0,\lambda-l,0|\lambda 0\rangle\,\int
j_{0}\left(K\left(R\right)s\right)\mathcal{F}_{n^{\prime}I^{\prime}nI,\lambda
l}^{\mathrm{ex}}\left(R,s\right)s^{l+2}ds,$ (195)
$\displaystyle\mathcal{F}_{n^{\prime}I^{\prime}nI,\lambda
l}^{\mathrm{ex}}\left(R,s\right)$
$\displaystyle=\frac{2l+1}{2}\int_{-1}^{1}\bar{\rho}_{n^{\prime}I^{\prime}nI,\lambda}^{\mathrm{tr}}\left(\left|\boldsymbol{R}+\boldsymbol{s}/2\right|\right)\bar{g}^{\mathrm{ex}}\left(s;k_{\mathrm{F};n^{\prime}I^{\prime}nI}\left(\left|\boldsymbol{R}+\boldsymbol{s}/2\right|\right)\right)$
$\displaystyle\times\frac{3j_{1}\left(k_{\mathrm{F};n^{\prime}I^{\prime}nI}\left(\left|\boldsymbol{R}+\boldsymbol{s}/2\right|\right)s\right)}{k_{\mathrm{F};n^{\prime}I^{\prime}nI}\left(\left|\boldsymbol{R}+\boldsymbol{s}/2\right|\right)s}P_{l}\left(x\right)dx,$
(196)

where $j_{0}$ ($j_{1}$) is the spherical Bessel function of the first kind for
order 0 (1). The NA local wave number $K\left(R\right)$ is self-consistently
determined to satisfy

$\frac{\hbar^{2}}{2\mu}K^{2}\left(R\right)+\mathcal{\bar{U}}_{n_{0}I_{0}n_{0}I_{0},0}^{\mathrm{dr}}\left(R\right)+\mathcal{\bar{U}}_{n_{0}I_{0}n_{0}I_{0},0}^{\mathrm{ex}}\left(R\right)=E_{n_{0}I_{0}}.$
(197)

It should be noted that in Ref. [115], the nonlocality in the exchange term
was explicitly treated, without the BR localization.

Thus, the coupling potential of central type is given by

$U_{c^{\prime}c,J_{T}}\left(R\right)=i^{-P^{\prime}-L^{\prime}+P+L}\sum_{\lambda}X_{I^{\prime}L^{\prime}IL,J_{T}\lambda}\mathcal{\bar{U}}_{n^{\prime}I^{\prime}nI,\lambda}\left(R\right),$
(198)
$\mathcal{\bar{U}}_{n^{\prime}I^{\prime}nI,\lambda}\left(R\right)\equiv\mathcal{\bar{U}}_{n^{\prime}I^{\prime}nI,\lambda}^{\mathrm{dr}}\left(R\right)+\mathcal{\bar{U}}_{n^{\prime}I^{\prime}nI,\lambda}^{\mathrm{ex}}\left(R\right).$
(199)

Although it is rather trivial, only the channels having the same parity are
coupled. When this condition is satisfied, $U_{c^{\prime}c}$ is symmetric
under interchange of $c^{\prime}$ and $c$. In numerical calculations of
reaction processes,
$\bar{\rho}_{n^{\prime}I^{\prime}nI,\lambda}^{\mathrm{tr}}$ of Eq. (192) is
the primary input to be prepared with nuclear structure calculations. It will
be worth pointing out that $U_{c^{\prime}c}$ is an energy-dependent and
complex potential. It is sometimes argued that the physical origin of the
imaginary part of the transition potential, $U_{c^{\prime}c}$ for
$c^{\prime}\neq c$, is not clear and it is naively disregarded in some cases.
As shown above, the MST clearly gives the origin of the imaginary part of
$U_{c^{\prime}c}$. There is no justification to assume $U_{c^{\prime}c}$ to be
real when $c^{\prime}\neq c$, at least from the viewpoint of the MST.

The spin-orbit part of the coupling potential can be formulated in a similar
manner. The explicit form of the diagonal spin-orbit potential for the elastic
channel is given in Refs. [164, 446]. There are a few applications of the MCC
framework including both central and spin-orbit interactions to inelastic
scattering [447, 139].

###  A.4 Extended NAF model for alpha-nucleus scattering

In the extended NAF model, the $\alpha$A coupling potential
$U_{c^{\prime}c,J}^{\alpha\mathrm{A}}$ is calculated by folding the NA
coupling potential with the one-body density $\rho^{\left(\alpha\right)}$ of
$\alpha$; explicit couplings through excited (breakup) channels of $\alpha$
are disregarded because of its large binding energy. It is given with
$\mathcal{\bar{U}}_{n^{\prime}I^{\prime}nI,\lambda}$ of Eq. (199) by

$U_{c^{\prime}c,J_{T}}^{\alpha\mathrm{A}}\left(R\right)=i^{-P^{\prime}-L^{\prime}+P+L}\sum_{\lambda}X_{I^{\prime}L^{\prime}IL,J_{T}\lambda}\sum_{\Lambda=0}^{\lambda}U_{n^{\prime}I^{\prime}nI,\lambda\Lambda}^{\alpha\mathrm{A}}\left(R\right)\frac{4\pi}{\hat{\Lambda}^{2}}\sqrt{{}_{2\lambda+1}C_{2\Lambda}}\left(-\right)^{\Lambda}\langle\Lambda
0\lambda 0|\lambda-\Lambda,0\rangle,$ (200)
$U_{n^{\prime}I^{\prime}nI,\lambda\Lambda}^{\alpha\mathrm{A}}\left(R\right)\equiv\int\rho^{\left(\alpha\right)}\left(s_{\alpha}\right)\mathfrak{\bar{U}}_{n^{\prime}I^{\prime}nI,\lambda\Lambda}\left(s_{\alpha},R\right)s_{\alpha}^{\Lambda}R^{\lambda-\Lambda}s_{\alpha}^{2}ds_{\alpha},$
(201)
$\mathfrak{\bar{U}}_{n^{\prime}I^{\prime}nI,\lambda\Lambda}\left(s_{\alpha},R\right)=\frac{\hat{\Lambda}^{2}}{2}\int_{-1}^{1}\frac{\mathcal{\bar{U}}_{n^{\prime}I^{\prime}nI,\lambda}\left(\left|\boldsymbol{s}_{\alpha}+\boldsymbol{R}\right|\right)}{\left|\boldsymbol{s}_{\alpha}+\boldsymbol{R}\right|^{\lambda}}P_{\Lambda}\left(y\right)dy,$
(202) $y\equiv\frac{\boldsymbol{R}\cdot\boldsymbol{s}_{\alpha}}{Rs_{\alpha}},$
(203)

where $\boldsymbol{s}_{\alpha}$ is the coordinate of each nucleon in $\alpha$
with respect to the c.m of $\alpha$. Because of its simplicity, this extended
NAF model has been applied to various $\alpha$ inelastic scattering processes
[32, 33, 35, 127, 128, 36, 37, 129]. The CC equations (175) can be used as
they are because $\alpha$ is a spinless particle.

###  A.5 Nucleus-nucleus scattering

There are several applications of the MCC framework to nucleus-nucleus (AA)
scattering [26, 27, 28, 165]; here we consider target nuclei heavier than
$\alpha$. For a complete formulation of the MCC for AA scattering, we need to
change the form of the channel wave function to include the spins of both
particles. To avoid complexity, we do not show the explicit form of the wave
function and coupling potential in this article. Readers are referred to the
literature, for instance, Refs. [27, 28].

## References

  * [1] T. Tamura, Rev. Mod. Phys. 37 (1965) 679.
  * [2] G. R. Satchler, Direct Nuclear Reactions, Clarendn Press, Oxford, 1983.
  * [3] P. Fröbrich, R. Lipperheide, Theory of Nuclear Reactions, Clarendn Press, Oxford, 1996.
  * [4] R. A. Broglia, A. Winther, Heavy-ion Reactions, Westview Press, Cambridge, MA, 2004\.
  * [5] C. A. Bertulani, P. Danielewicz, Introduction to Nuclear Reactions, IOP Publishing, Bristol, UK, 2004.
  * [6] I. J. Thompson, F. M. Nunes, Nuclear Reactions for Astrophysics, Cambridge University Press, Cambridge, 2009.
  * [7] L. F. Canto, M. S. Hussein, Scattering Theory of Molecules, Atoms and Nuclei, World Scientific Publishing Co. Pte. Ltd., Singapore, 2013.
  * [8] J. Raynal, Saclay Report No. DPh-T 69/42 (unpublished).
  * [9] A. Lépine-Szily, R. Lichtenthäler, Euro. Phys. J. A 57 (2021) 99.
  * [10] I. J. Thompson, Comput. Phys. Rep. 7 (1988) 167.
  * [11] K. Hagino, N. Rowley, A. T. Kruppa, Comput. Phys. Comm. 123 (1999) 143.
  * [12] E. Rost, Phys. Rev. 154 (1967) 994.
  * [13] K. Hagino, H. Sagawa, S. Kanaya, A. Odahara, Prog. Theor. Exp. Phys. 2020 (2020) 023D01.
  * [14] A. T. Kruppa, B. Barmore, W. Nazarewicz, T. Vertse, Phys. Rev. Lett. 84 (2000) 4549\.
  * [15] H. Esbensen, C. N. Davids, Phys. Rev. C 63 (2000) 014315.
  * [16] C. N. Davids, H. Esbensen, Phys. Rev. C 64 (2001) 034317.
  * [17] K. Hagino, Phys. Rev. C 64 (2001) 041304(R).
  * [18] K. Hagino, N. V. Giai, Nucl. Phys. A 735 (2004) 55.
  * [19] S. M. Wang, W. Nazarewicz, F. R. Xu, Phys. Rev. C 96 (2017) 044307.
  * [20] S. M. Wang, W. Nazarewicz, Phys. Rev. Lett. 120 (2018) 212502.
  * [21] M. Kamimura, M. Yahiro, Y. Iseri, Y. Sakuragi, H. Kameyama, M. Kawai, Prog. Theo. Phys. Suppl. 89 (1986) 1.
  * [22] N. Austern, Y. Iseri, M. Kamimura, M. Kawai, G. Rawitscher, M. Yahiro, Phys. Rep. 154 (1987) 125.
  * [23] M. Yahiro, K. Ogata, T. Matsumoto, K. Minomo, Prog. Theo. Exp. Phys. 2012 (2012) 01A206.
  * [24] L. F. Canto, P. R. S. Gomes, R. Donangelo, M. S. Hussein, Phys. Rep. 424 (2006) 1\.
  * [25] L. F. Canto, P. R. S. Gomes, R. Donangelo, J. Lubian, M. S. Hussein, Phys. Rep. 596 (2015) 1.
  * [26] M. Takashina, T. Furumoto, Y. Sakuragi, Phys. Rev. C 81 (2010) 047605.
  * [27] T. Furumoto, Y. Sakuragi, Phys. Rev. C 87 (2013) 014618.
  * [28] K. Minomo, M. Kohno, K. Ogata, Phys. Rev. C 93 (2016) 014607.
  * [29] Y. Kim, K. Amos, S. Karataglidis, W. Richter, Nucl. Phys. A 807 (2008) 119–144.
  * [30] P. Descouvemont, M. S. Hussein, Phys. Rev. Lett. 111 (2013) 082701.
  * [31] G. R. Satchler, W. G. Love, Phys. Rep. 55 (1979) 183.
  * [32] Y. Kanada-En’yo, K. Ogata, Phys. Rev. C 99 (2019) 064601.
  * [33] Y. Kanada-En’yo, K. Ogata, Phys. Rev. C 99 (2019) 064608.
  * [34] Y. Kanada-En’yo, K. Ogata, Phys. Rev. C 100 (2019) 064616.
  * [35] Y. Kanada-En’yo, K. Ogata, Phys. Rev. C 101 (2020) 014317.
  * [36] Y. Kanada-En’yo, Y. Shikata, Y. Chiba, K. Ogata, Phys. Rev. C 102 (2020) 014607\.
  * [37] Y. Kanada-En’yo, K. Ogata, Phys. Rev. C 103 (2021) 024603.
  * [38] K. Hagino, A. B. Balantekin, Phys. Rev. A 70 (2004) 032106.
  * [39] C. H. Dasso, S. Landowne, A. Winther, Nucl. Phys. A 405 (1983) 381.
  * [40] C. H. Dasso, S. Landowne, A. Winther, Nucl. Phys. A 407 (1983) 221.
  * [41] A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, Princeton, New Jersey, 1960.
  * [42] S. Landowne, S. C. Pieper, Phys. Rev. C 29 (1984) 1352.
  * [43] K. Hagino, N. Takigawa, Prog. Theor. Phys. 128 (2012) 1061.
  * [44] P. W. Wen, C. J. Lin, R. G. Nazmitdinov, S. I. Vinitsky, O. Chuluunbaatar, A. A. Gusev, A. K. Nasirov, H. M. Jia, A. Gozdz (2020).  arXiv:2011.01806.
  * [45] H. Esbensen, S. Landowne, C. Price, Phys. Rev. C 36 (1987) 2359.
  * [46] P. Ring, P. Schuck, The Nuclear Many-Body Problem, Springer-Verlag, New York, 1980\.
  * [47] K. Hagino, N. Takigawa, M. Dasgupta, D. J. Hinde, J. R. Leigh, Phys. Rev. C 55 (1997) 276.
  * [48] H. Esbensen, S. Landowne, Phys. Rev. C 35 (1987) 2090.
  * [49] A. Bohr, B. R. Mottelson, Nuclear Structure, Vol. 2, Benjaming W.A., Reading, MA, 1975.
  * [50] H. Esbensen, S. Landowne, C. Price, Phys. Rev. C 36 (1987) 1216.
  * [51] K. Hagino, N. Rowley, Phys. Rev. C 69 (2004) 054610.
  * [52] O. Tanimura, Phys. Rev. C 35 (1987) 1600.
  * [53] M. V. Andres, N. Rowley, M. A. Nagarajan, Phys. Lett. B 202 (1988) 292.
  * [54] M. A. Nagarajan, A. B. Balantekin, N. Takigawa, Phys. Rev. C 34 (1986) 894.
  * [55] N. Takigawa, F. Michel, A. B. Balantekin, G. Reidemeister, Phys. Rev. C 44 (1991) 477.
  * [56] J. A. Christley, R. Johnson, I. J. Thompson, J. of Phys. G 20 (1994) 169.
  * [57] D. J. Hinde, N. Rowley, M. Dasgupta, R. D. Butt, C. R. Morton, A. Mukherjee, Phys. Rev. C 68 (2003) 044606.
  * [58] K. Hagino, S. Sakaguchi, Phys. Rev. C 100 (2019) 064614.
  * [59] S. E. Koonin, D. C. Meredith, Computational Physics Fortran Version, Addison-Wesley, 1990.
  * [60] M. A. Melkanoff, T. Sawada, J. Raynal, Methods in Computational Physics 6 (1966) 1.
  * [61] Z. H. Levine, Phys. Rev. A 30 (1984) 1120.
  * [62] T. N. Rescigno, A. E. Orel, Phys. Rev. A 25 (1982) 2402.
  * [63] W. E. Baylis, S. J. Peel, Comput. Phys. Comm. 25 (1982) 7–19.
  * [64] W. Brenig, T. Brunner, A. Gross, R. Russ, Z. Phys. B 93 (1993) 91.
  * [65] W. Brenig, R. Russ, Surf. Sci. 315 (1994) 195.
  * [66] B. R. Johnson, J. of Comp. Phys. 13 (1973) 445.
  * [67] B. R. Johnson, Phys. Rev. A 32 (1985) 1241.
  * [68] M. Rhoades-Brown, M. H. Macfarlane, S. C. Pieper, Phys. Rev. C 21 (1980) 2436.
  * [69] M. Ichimura, M. Igarashi, S. Landowne, C. H. Dasso, B. S. Nilsson, R. A. Broglia, A. Winther, Phys. Lett. B 67 (1977) 129.
  * [70] K. Alder, F. Roesel, R. Morf, Nucl. Phys. A 284 (1977) 145.
  * [71] S. N. Ershov, J. S. Vaagen, M. V. Zhukov, Phys. Rev. C 84 (2011) 064308.
  * [72] M. Kamimura, Prog. Theor. Phys. Supp. 62 (1977) 236.
  * [73] T. Matsuse, Y. Abe, Y. Kondo, Prog. Theor. Phys. 59 (1978) 1904.
  * [74] W. Kohn, Phys. Rev. 74 (1948) 1763.
  * [75] L. Hulthén, Ark. Mat. Astr. Fys. 35A (1948) 25.
  * [76] T. Kato, Prog. Theor. Phys. 6 (1951) 394.
  * [77] Y. Kondo, T. Matsuse, Y. Abe, Prog. Theor. Phys. 59 (1978) 465.
  * [78] M. Hesse, J.-M. Sparenberg, F. Van Raemdonck, D. Baye, Nucl. Phys. A 640 (1998) 37.
  * [79] P. Descouvemont, Comput. Phys. Comm. 200 (2016) 199.
  * [80] D. Baye, P.-H. Heenen, J. Phys. A 19 (1986) 2041.
  * [81] D. Baye, M. Hesse, J.-M. Sparenberg, M. Vincke, J. Phys. B 31 (1998) 3439.
  * [82] M. Hesse, J. Roland, D. Baye, Nucl. Phys. A 709 (2002) 184.
  * [83] D. Baye, J. Goldbeter, J.-M. Sparenberg, Phys. Rev.A 65 (2002) 052710.
  * [84] D. M. Brink, Semi-classical Methods for Nucleus-Nucleus Scattering, Cambridge University Press, Cambridge, England, 1985.
  * [85] P. Pechukas, Phys. Rev. 181 (1969) 174.
  * [86] A. B. Balantekin, N. Takigawa, Ann. Phys. (NY) 160 (1985) 441.
  * [87] N. Takigawa, Y. Alhassid, A. B. Balantekin, Phys. Rev. C 45 (1992) 1850.
  * [88] N. Takigawa, K. Hagino, M. Abe, Phys. Rev. C 51 (1995) 187.
  * [89] K. Alder, A. Winther, Electromagnetic Excitations, North-Holland, Amsterdam, 1975\.
  * [90] K. Alder, A. Bohr, T. Huus, B. Mottelson, A. Winther, Rev. Mod. Phys. 28 (1956) 432\.
  * [91] C. A. Bertulani, L. F. Canto, M. S. Hussein, A. F. R. de Toledo Piza, Phys. Rev. C 53 (1996) 334.
  * [92] E. Maglione, G. Pollarolo, A. Vitturi, R. A. Broglia, A. Winther, Phys. Lett. B 162 (1985) 59.
  * [93] W. von Oertzen, A. Vitturi, Rep. Prog. Phys. 64 (2001) 1247.
  * [94] J. A. Lay, L. Fortunato, A. Vitturi, Phys. Rev. C 89 (2014) 034618.
  * [95] G. Scamps, K. Hagino, Phys. Rev. C 92 (2015) 054614.
  * [96] A. Winther, Nucl. Phys. A 572 (1994) 191.
  * [97] C. H. Dasso, G. Pollarolo, A. Winther, Phys. Rev. Lett. 73 (1994) 1907.
  * [98] A. Winther, Nucl. Phys. A 594 (1995) 203.
  * [99] G. Pollarolo, A. Winther, Phys. Rev. C 62 (2000) 054611.
  * [100] H. D. Marta, L. F. Canto, R. Donangelo, P. Lotti, Phys. Rev. C 66 (2002) 024605\.
  * [101] H. D. Marta, L. F. Canto, R. Donangelo, Phys. Rev. C 78 (2008) 034612.
  * [102] L. F. Canto, R. Donangelo, H. D. Marta, Phys. Rev. C 73 (2006) 034608.
  * [103] H. D. Marta, L. F. Canto, R. Donangelo, Phys. Rev. C 89 (2014) 034625.
  * [104] G. D. Kolinger, L. F. Canto, R. Donangelo, S. R. Souza, Phys. Rev. C 98 (2018) 044604\.
  * [105] T. Vockerodt, A. Diaz-Torres, Phys. Rev. C 100 (2019) 034606.
  * [106] T. Vockerodt, A. Diaz-Torres, Phys. Rev. C 104 (2021) 064601.
  * [107] N. Yamanaka, Y. Kino, Phys. Rev. A 64 (2001) 042715.
  * [108] L. L. Foldy, Phys. Rev. 67 (1945) 107.
  * [109] K. M. Watson, Phys. Rev. 89 (1953) 115.
  * [110] A. K. Kerman, H. McManus, R. M. Thaler, Ann. Phys. (NY) 8 (1959) 551.
  * [111] R. A. Arndt, W. J. Briscoe, I. I. Strakovsky, R. L. Workman, Phys. Rev. C 76 (2007) 025209.
  * [112] R. L. Workman, W. J. Briscoe, I. I. Strakovsky, Phys. Rev. C 94 (2016) 065203.
  * [113] M. A. Franey, W. G. Love, Phys. Rev. C 31 (1985) 488.
  * [114] C. J. Horowitz, Phys. Rev. C 31 (1985) 1340.
  * [115] K. Amos, P. J. Dortmans, H. V. von Geramb, S. Karataglidis, J. Raynal, Advances in nuclear physics, Vol. 25, Plenum, New York, 1973.
  * [116] M. Yahiro, K. Minomo, K. Ogata, M. Kawai, Prog. Theor. Phys. 120 (2008) 767.
  * [117] R. Crespo, R. C. Johnson, J. A. Tostevin, Phys. Rev. C 46 (1992) 279–297.
  * [118] R. Crespo, R. C. Johnson, Phys. Rev. C 60 (1999) 034007.
  * [119] R. Crespo, I. J. Thompson, A. A. Korsheninnikov, Phys. Rev. C 66 (2002) 021002.
  * [120] H. F. Arellano, F. A. Brieva, W. G. Love, Phys. Rev. C 52 (1995) 301–315.
  * [121] H. F. Arellano, W. G. Love, Phys. Rev. C 76 (2007) 014616.
  * [122] M. Gennari, M. Vorabbi, A. Calci, P. Navrátil, Phys. Rev. C 97 (2018) 034619.
  * [123] M. Burrows, R. B. Baker, C. Elster, S. P. Weppner, K. D. Launey, P. Maris, G. Popa, Phys. Rev. C 102 (2020) 034606.
  * [124] T. Furumoto, Y. Sakuragi, Y. Yamamoto, Phys. Rev. C 79 (2009) 011601(R).
  * [125] T. Furumoto, Y. Sakuragi, Y. Yamamoto, Phys. Rev. C 80 (2009) 044614.
  * [126] M. Toyokawa, M. Yahiro, T. Matsumoto, K. Minomo, K. Ogata, M. Kohno, Phys. Rev. C 92 (2015) 024618.
  * [127] Y. Kanada-En’yo, K. Ogata, Phys. Rev. C 101 (2020) 064607.
  * [128] Y. Kanada-En’yo, K. Ogata, Phys. Rev. C 101 (2020) 064308.
  * [129] Y. Kanada-En’yo, K. Ogata, Prog. Theor. Exp. Phys. 2021 (2021) 043D01.
  * [130] K. Egashira, K. Minomo, M. Toyokawa, T. Matsumoto, M. Yahiro, Phys. Rev. C 89 (2014) 064611.
  * [131] K. Minomo, K. Washiyama, K. Ogata, arXiv:1712.10121.
  * [132] Y. Kanada-En’yo, H. Horiuchi, A. Ono, Phys. Rev. C 52 (1995) 628.
  * [133] Y. Kanada-En’yo, H. Horiuchi, Phys. Rev. C 52 (1995) 647.
  * [134] Y. Kanada-En’yo, M. Kimura, A. Ono, Prog. Theor. Exp. Phys. 2012 (2012) 01A202.
  * [135] F. Maréchal, T. Suomijärvi, Y. Blumenfeld, A. Azhari, E. Bauge, D. Bazin, J. A. Brown, P. D. Cottle, J. P. Delaroche, M. Fauerbach, M. Girod, T. Glasmacher, S. E. Hirzebruch, J. K. Jewell, J. H. Kelley, K. W. Kemper, P. F. Mantica, D. J. Morrissey, L. A. Riley, J. A. Scarpaci, H. Scheit, M. Steiner, Phys. Rev. C 60 (1999) 034615.
  * [136] H. Scheit, F. Maréchal, T. Glasmacher, E. Bauge, Y. Blumenfeld, J. P. Delaroche, M. Girod, R. W. Ibbotson, K. W. Kemper, J. Libert, B. Pritychenko, T. Suomijärvi, Phys. Rev. C 63 (2000) 014604.
  * [137] M. Takashina, Y. Kanada-En’yo, Phys. Rev. C 77 (2008) 014604.
  * [138] J.-P. Jeukenne, A. Lejeune, C. Mahaux, Phys. Rev. C 16 (1977) 80.
  * [139] T. Furumoto, M. Takashina, Phys. Rev. C 103 (2021) 044602.
  * [140] Y. Yamamoto, T. Furumoto, N. Yasutake, T. A. Rijken, Phys. Rev. C 88 (2013) 022801(R).
  * [141] Y. Yamamoto, T. Furumoto, N. Yasutake, T. A. Rijken, Phys. Rev. C 90 (2014) 045805\.
  * [142] T. A. Rijken, M. M. Nagels, Y. Yamamoto, Prog. Theor. Phys. Suppl. 185 (2010) 14\.
  * [143] K. Sato, T. Furumoto, Y. Kikuchi, K. Ogata, Y. Sakuragi, Prog. Theor. Exp. Phys. 2019 (2019) 101D01.
  * [144] D. H. Youngblood, Y.-W. Lui, H. L. Clark, Phys. Rev. C 55 (1997) 2811.
  * [145] D. H. Youngblood, Y.-W. Lui, H. L. Clark, Phys. Rev. C 60 (1999) 014304.
  * [146] D. H. Youngblood, H. L. Clark, Y.-W. Lui, Phys. Rev. Lett. 82 (1999) 691.
  * [147] M. Itoh, H. Sakaguchi, M. Uchida, T. Ishikawa, T. Kawabata, T. Murakami, H. Takeda, T. Taki, S. Terashima, N. Tsukahara, Y. Yasuda, M. Yosoi, U. Garg, M. Hedden, B. Kharraja, M. Koss, B. K. Nayak, S. Zhu, H. Fujimura, M. Fujiwara, K. Hara, H. P. Yoshida, H. Akimune, M. N. Harakeh, M. Volkerts, Phys. Lett. B 549 (2002) 58.
  * [148] M. Itoh, H. Sakaguchi, M. Uchida, T. Ishikawa, T. Kawabata, T. Murakami, H. Takeda, T. Taki, S. Terashima, N. Tsukahara, Y. Yasuda, M. Yosoi, U. Garg, M. Hedden, B. Kharraja, M. Koss, B. K. Nayak, S. Zhu, H. Fujimura, M. Fujiwara, K. Hara, H. P. Yoshida, H. Akimune, M. N. Harakeh, M. Volkerts, Phys. Rev. C 68 (2003) 064602.
  * [149] M. Uchida, H. Sakaguchi, M. Itoh, M. Yosoi, T. Kawabata, Y. Yasuda, H. Takeda, T. Murakami, S. Terashima, S. Kishi, U. Garg, P. Boutachkov, M. Hedden, B. Kharraja, M. Koss, B. K. Nayak, S. Zhu, M. Fujiwara, H. Fujimura, H. P. Yoshida, K. Hara, H. Akimune, M. N. Harakeh, Phys. Rev. C 69 (2004) 051301(R).
  * [150] T. Li, U. Garg, Y. Liu, R. Marks, B. K. Nayak, P. V. R. Madhusudhana, M. Fujiwara, H. Hashimoto, K. Kawase, K. Nakanishi, S. Okumura, M. Yosoi, M. Itoh, M. Ichikawa, R. Matsuo, T. Terazono, M. Uchida, T. Kawabata, H. Akimune, Y. Iwao, T. Murakami, H. Sakaguchi, S. Terashima, Y. Yasuda, J. Zenihiro, M. N. Harakeh, Phys. Rev. Lett. 99 (2007) 162503.
  * [151] M. Itoh, S. Kishi, H. Sakaguchi, H. Akimune, M. Fujiwara, U. Garg, K. Hara, H. Hashimoto, J. Hoffman, T. Kawabata, K. Kawase, T. Murakami, K. Nakanishi, B. K. Nayak, S. Terashima, M. Uchida, Y. Yasuda, M. Yosoi, Phys. Rev. C 88 (2013) 064313.
  * [152] S. Bagchi, J. Gibelin, M. N. Harakeh, N. Kalantar-Nayestanaki, N. L. Achouri, H. Akimune, B. Bastin, K. Boretzky, H. Bouzomita, M. Caama$\tilde{\mathrm{n}}$o, L. Càceres, S. Damoy, F. Delaunay, B. Fernández-Domínguez, M. Fujiwara, U. Garg, G. F. Grinyer, O. Kamalou, E. Khan, A. Krasznahorkay, G. Lhoutellier, J. F. Libin, S. Lukyanov, K. Mazurek, M. A. Najafi, J. Pancin, Y. Penionzhkevich, L. Perrot, R. Raabe, C. Rigollet, T. Roger, S. Sambi, H. Savajols, M. Senoville, C. Stodel, L. Suen, J. C. Thomas, M. Vandebrouck, J. V. de Walle, Phys. Lett. B 751 (2015) 371.
  * [153] M. Vandebrouck, J. Gibelin, E. Khan, N. L. Achouri, H. Baba, D. Beaumel, Y. Blumenfeld, M. Caama$\tilde{\mathrm{n}}$o, L. Càceres, G. Colò, F. Delaunay, B. Fernandez-Dominguez, U. Garg, G. F. Grinyer, M. N. Harakeh, N. Kalantar-Nayestanaki, N. Keeley, W. Mittig, J. Pancin, R. Raabe, T. Roger, P. Roussel-Chomaz, H. Savajols, O. Sorlin, C. Stodel, D. Suzuki, J. C. Thomas, Phys. Rev. C 92 (2015) 024316.
  * [154] Y. K. Gupta, U. Garg, J. Hoffman, J. Matta, P. V. M. Rao, D. Patel, T. Peach, K. Yoshida, M. Itoh, M. Fujiwara, K. Hara, H. Hashimoto, K. Nakanishi, M. Yosoi, H. Sakaguchi, S. Terashima, S. Kishi, T. Murakami, M. Uchida, Y. Yasuda, H. Akimune, T. Kawabata, M. N. Harakeh, Phys. Rev. C 93 (2016) 044324\.
  * [155] T. Peach, U. Garg, Y. K. Gupta, J. Hoffman, J. T. Matta, D. Patel, P. V. M. Rao, K. Yoshida, M. Itoh, M. Fujiwara, K. Hara, H. Hashimoto, K. Nakanishi, M. Yosoi, H. Sakaguchi, S. Terashima, S. Kishi, T. Murakami, M. Uchida, Y. Yasuda, H. Akimune, T. Kawabata, M. N. Harakeh, G. Colò, Phys. Rev. C 93 (2016) 064325.
  * [156] S. Adachi, T. Kawabata, K. Minomo, T. Kadoya, N. Yokota, H. Akimune, T. Baba, H. Fujimura, M. Fujiwara, Y. Funaki, T. Furuno, T. Hashimoto, K. Hatanaka, K. Inaba, Y. Ishii, M. Itoh, C. Iwamoto, K. Kawase, Y. Maeda, H. Matsubara, Y. Matsuda, H. Matsuno, T. Morimoto, H. Morita, M. Murata, T. Nanamura, I. Ou, S. Sakaguchi, Y. Sasamoto, R. Sawada, Y. Shimizu, K. Suda, A. Tamii, Y. Tameshige, M. Tsumura, M. Uchida, T. Uesaka, H. P. Yoshida, S. Yoshida, Phys. Rev. C 97 (2018) 014601.
  * [157] Y. Suzuki, Prog. Theor. Phys. 56 (1976) 111.
  * [158] T. Yamada, Y. Funaki, H. Horiuchi, K. Ikeda, A. Tohsaki, Prog. Theor. Phys. 120 (2008) 1139.
  * [159] D. T. Khoa, D. C. Cuong, Phys. Lett. B 660 (2008) 331.
  * [160] K. Minomo, K. Ogata, Phys. Rev. C 93 (2016) 051601(R).
  * [161] M. Kamimura, Nucl. Phys. A 351 (1981) 456.
  * [162] K. Ogata, Y. Chiba, Y. Sakuragi, Prog. Theor. Exp. Phys. 2021 (2021) 103D01.
  * [163] R. A. Miskimen, A. M. Bernstein, G. Bernhardt, C. F. Williamson, B. A. Brown, R. Alarcon, Phys. Rev. C 37 (1988) 1600–1608.
  * [164] T. Furumoto, Y. Sakuragi, Y. Yamamoto, Phys. Rev. C 78 (2008) 044610.
  * [165] T. Furumoto, T. Suhara, Y. Sakuragi, Phys. Rev. C 97 (2018) 044602.
  * [166] L. D. Faddeev, Zh. Eksp. Theor. Fiz. 39 (1960) 1459.
  * [167] N. Austern, M. Yahiro, M. Kawai, Phys. Rev. Lett. 63 (1989) 2649.
  * [168] N. Austern, M. Kawai, M. Yahiro, Phys. Rev. C 53 (1996) 314.
  * [169] M. C. Birse, E. F. Redish, Nucl. Phys. A 406 (1982) 149.
  * [170] E. Hiyama, Y. Kino, M. Kamimura, Prog. Part. Nucl. Phys. 51 (2003) 223.
  * [171] F. Pérez-Bernal, I. Martel, J. M. Arias, J. Gómez-Camacho, Phys. Rev. A 63 (2001) 052111.
  * [172] A. M. Moro, F. Pérez-Bernal, J. M. Arias, J. Gómez-Camacho, Phys. Rev. C 73 (2006) 044612.
  * [173] N. J. Upadhyay, A. Deltuva, F. M. Nunes, Phys. Rev. C 85 (2012) 054621.
  * [174] K. Ogata, K. Yoshida, Phys. Rev. C 94 (2016) 051603(R).
  * [175] E. O. Alt, P. Grassberger, W. Sandhas, Nucl. Phys. B 2 (1967) 167.
  * [176] A. Deltuva, A. M. Moro, E. Cravo, F. M. Nunes, A. C. Fonseca, Phys. Rev. C 76 (2007) 064602.
  * [177] A. Deltuva, A. C. Fonseca, P. U. Sauer, Phys. Rev. C 71 (2005) 054005.
  * [178] K. Ogata, M. Kan, M. Kamimura, Prog. Theor. Phys. 122 (2009) 1055.
  * [179] T. Akahori, Y. Funaki, K. Yabana, Phys. Rev. C 92 (2015) 022801(R).
  * [180] Y. Kikuchi, T. Myo, M. Takashina, K. Katō, K. Ikeda, Prog. Theor. Phys. 122 (2009) 499.
  * [181] Y. Kikuchi, T. Matsumoto, K. Minomo, K. Ogata, Phys. Rev. C 88 (2013) 021602(R).
  * [182] Y. Kikuchi, K. Katō, T. Myo, M. Takashina, K. Ikeda, Phys. Rev. C 81 (2010) 044308\.
  * [183] A. Saito, Ph.D. thesis, Tokyo Institute of Technology (2021).
  * [184] Y. Kikuchi, N. Kurihara, A. Wano, K. Katō, T. Myo, M. Takashina, Phys. Rev. C 84 (2011) 064610.
  * [185] S. Watanabe, K. Ogata, T. Matsumoto, Phys. Rev. C 103 (2021) L031601.
  * [186] S. Watanabe, T. Matsumoto, K. Ogata, M. Yahiro, Phys. Rev. C 92 (2015) 044611.
  * [187] K. Ogata, T. Myo, T. Furumoto, T. Matsumoto, M. Yahiro, Phys. Rev. C 88 (2013) 024616\.
  * [188] Y. Suzuki, K. Ikeda, Phys. Rev. C 38 (1988) 410.
  * [189] T. Matsumoto, K. Katō, M. Yahiro, Phys. Rev. C 82 (2010) 051602(R).
  * [190] U. Fano, Phys. Rev. 124 (1961) 1866.
  * [191] T. Suzuki, T. Otsuka, C. Yuan, N. Alahari, Phys. Lett. B 753 (2016) 199.
  * [192] T. Matsumoto, J. Tanaka, K. Ogata, Prog. Theor. Exp. Phys. 2019 (2019) 123D02.
  * [193] R. Kanungo, A. Sanetullaev, J. Tanaka, S. Ishimoto, G. Hagen, T. Myo, T. Suzuki, C. Andreoiu, P. Bender, A. A. Chen, B. Davids, J. Fallis, J. P. Fortin, N. Galinski, A. T. Gallant, P. E. Garrett, G. Hackman, B. Hadinia, G. Jansen, M. Keefe, R. Krücken, J. Lighthall, E. McNeice, D. Miller, T. Otsuka, J. Purcell, J. S. Randhawa, T. Roger, A. Rojas, H. Savajols, A. Shotter, I. Tanihata, I. J. Thompson, C. Unsworth, P. Voss, Z. Wang, Phys. Rev. Lett. 114 (2015) 192502.
  * [194] J. Tanaka, R. Kanungo, M. Alcorta, N. Aoi, H. Bidaman, C. Burbadge, G. Christian, S. Cruz, B. Davids, A. Diaz Varela, J. Even, G. Hackman, M. Harakeh, J. Henderson, S. Ishimoto, S. Kaur, M. Keefe, R. Krücken, K. Leach, J. Lighthall, E. Padilla Rodal, J. Randhawa, P. Ruotsalainen, A. Sanetullaev, J. Smith, O. Workman, I. Tanihata, Physics Letters B 774 (2017) 268–272.
  * [195] J. Aguilar, J. M. Combes, Commun. Math. Phys. 22 (1971) 269.
  * [196] E. Balslev, J. M. Combes, Commun. Math. Phys. 22 (1971) 280.
  * [197] H. Feshbach, Ann. Phys. 5 (1958) 357.
  * [198] H. Feshbach, Ann. Phys. 19 (1962) 287.
  * [199] E. C. Pinilla, P. Descouvemont, D. Baye, Phys. Rev. C 85 (2012) 064610.
  * [200] M. Cubero, J. P. Fernández-García, M. Rodríguez-Gallardo, L. Acosta, M. Alcorta, M. A. G. Alvarez, M. J. G. Borge, L. Buchmann, C. A. Diget, H. A. Falou, B. R. Fulton, H. O. U. Fynbo, D. Galaviz, J. Gómez-Camacho, R. Kanungo, J. A. Lay, M. Madurga, I. Martel, A. M. Moro, I. Mukha, T. Nilsson, A. M. Sánchez-Benítez, A. Shotter, O. Tengblad, P. Walden, Phys. Rev. Lett. 109 (2012) 262701.
  * [201] S. Ogawa, T. Matsumoto, Y. Kanada-En’yo, K. Ogata, Phys. Rev. C 104 (2021) 044608\.
  * [202] J. Casal, J. Gómez-Camacho, Phys. Rev. C 99 (2019) 014604.
  * [203] S. Ogawa, T. Matsumoto, Phys. Rev. C 102 (2020) 021602(R).
  * [204] Y. Kanada-En’yo, K. Ogata, arXiv:2004.14597.
  * [205] T. Matsumoto, E. Hiyama, K. Ogata, Y. Iseri, M. Kamimura, S. Chiba, M. Yahiro, Phys. Rev. C 70 (2004) 061601.
  * [206] M. Rodriguez-Gallardo, J. M. Arias, J. Gomez-Camacho, A. M. Moro, I. J. Thompson, J. A. Tostevin, Four-body continuum-discretized coupled-channels calculations, Phys.Rev. C 80 (2009) 051601.
  * [207] J. Casal, M. Rodriguez-Gallardo, J. M. Arias, Analytical transformed harmonic oscillator basis for three-body nuclei of astrophysical interest: Application to 6he, Phys.Rev. C 88 (2013) 014327.
  * [208] J. Casal, M. Rodriguez-Gallardo, J. M. Arias, I. J. Thompson, Astrophysical reaction rate for 9be formation within a three-body approach, Phys.Rev. C 90 (2014) 044304.
  * [209] J. Casal, M. Rodriguez-Gallardo, J. M. Arias, 9be elastic scattering on 208pb and 27al within a four-body reaction framework, Phys.Rev. C 92 (2015) 054611.
  * [210] J. Singh, T. Matsumoto, T. Fukui, K. Ogata, Phys. Rev. C 104 (2021) 034612.
  * [211] P. Descouvemont, Phys. Lett. B 772 (2017) 1.
  * [212] P. Descouvemont, Phys. Rev. C 97 (2018) 064607.
  * [213] C. A. Bertulani, Phys. Rev. Lett. 94 (2005) 072701.
  * [214] K. Ogata, C. A. Bertulani, Prog. Theor. Phys. 121 (2009) 1399.
  * [215] K. Ogata, C. A. Bertulani, Prog. Theor. Phys. 123 (2010) 701.
  * [216] K. Ogata, M. Yahiro, Y. Iseri, T. Matsumoto, M. Kamimura, Phys. Rev. C 68 (2003) 064609.
  * [217] L. Moschini, P. Capel, Phys. Lett. B 790 (2019) 367.
  * [218] G. Goldstein, D. Baye, P. Capel, Phys. Rev. C 73 (2006) 024602.
  * [219] T. Fukui, K. Ogata, P. Capel, Phys. Rev. C 90 (2014) 034617.
  * [220] A. M. Moro, Phys. Rev. C 92 (2015) 044605.
  * [221] G. Jacob, T. A. J. Maris, Rev. Modern Phys. 38 (1966) 121.
  * [222] N. S. Chant, P. G. Roos, Phys. Rev. C 27 (1983) 1060.
  * [223] T. Wakasa, K. Ogata, T. Noro, Prog. Part. Nucl. Phys. 96 (2017) 32.
  * [224] K. Yoshida, M. Gómez-Ramos, K. Ogata, A. M. Moro, Phys. Rev. C 97 (2018) 024608\.
  * [225] M. Gómez-Ramos, A. Deltuva, A. M. Moro, Phys. Rev. C 102 (2020) 064613.
  * [226] M. Gómez-Ramos, J. Casal, A. M. Moro, Phys. Lett. B 772 (2017) 115.
  * [227] M. Gómez-Ramos, A. M. Moro, Phys. Lett. B 785 (2018) 511.
  * [228] A. Bonaccorso, D. M. Brink, Phys. Rev. C 38 (1988) 1776–1786.
  * [229] A. Bonaccorso, D. M. Brink, Phys. Rev. C 44 (1991) 1559–1568.
  * [230] A. Bonaccorso, G. F. Bertsch, Phys. Rev. C 63 (2001) 044604.
  * [231] F. Flavigny, A. Obertelli, A. Bonaccorso, G. F. Grinyer, C. Louchart, L. Nalpas, A. Signoracci, Phys. Rev. Lett. 108 (2012) 252501.
  * [232] K. Minomo, K. Washiyama, K. Ogata, J Nucl. Sci. Tech. 54 (2017) 127.
  * [233] T. Sato, K. Niita, N, Matsuda, S. Hashimoto, Y. Iwamoto, S. Noda, T. Ogawa, H. Iwase, H. Nakashima, T. Fukahori, K. Okumura, T. Kai, S. Chiba, T. Furuta, L. Sihver, J Nucl. Sci. Tech. 50 (2013) 913.
  * [234] P. Garin, M. Sugimoto, J Nucl. Mater. 417 (2011) 1262.
  * [235] S. Nakayama, H. Kouno, Y. Watanabe, O. Iwamoto, K. Ogata, Phys. Rev. C 94 (2016) 014618.
  * [236] S. Nakayama, O. Iwamoto, Y. Watanabe, K. Ogata, J Nucl. Sci. Tech. 58 (2021) 805\.
  * [237] K. Hencken, G. Bertsch, H. Esbensen, Phys. Rev. C 54 (1996) 3043.
  * [238] R. J. Glauber, Lectures in Theoretical Physics, Vol. I, Interscience, New York, 1959\.
  * [239] K. Ogata, T. Fukui, Y. Kamiya, A. Ohnishi, Phys. Rev. C 103 (2021) 065205.
  * [240] R. Crespo, A. Deltuva, A. M. Moro, Phys. Rev. C 83 (2011) 044622.
  * [241] A. M. Moro, R. Crespo, Phys. Rev. C 85 (2012) 054613.
  * [242] N. C. Summers, et al., Phys. Rev. C 74 (2006) 014606.
  * [243] R. de Diego, J. M. Arias, J. A. Lay, A. M. Moro, Phys. Rev. C 89 (2014) 064609\.
  * [244] V. Pesudo, et al., Phys. Rev. Lett. 118 (2017) 152502.
  * [245] A. Di Pietro, A. M. Moro, J. Lei, R. de Diego, Phys.Lett. B 798 (2019) 134954\.
  * [246] J. A. Lay, R. de Diego, R. Crespo, A. M. Moro, J. M. Arias, R. C. Johnson, Phys.Rev. C 94 (2016) 021602.
  * [247] J. Chen, J. L. Lou, Y. L. Ye, Z. H. Li, Y. C. Ge, Q. T. Li, J. Li, W. Jiang, Y. L. Sun, H. L. Zang, N. Aoi, E. Ideguchi, H. J. Ong, Y. Ayyad, K. Hatanaka, D. T. Tran, T. Yamamoto, M. Tanaka, T. Suzuki, N. T. Tho, J. Rangel, A. M. Moro, D. Y. Pang, J. Lee, J. Wu, H. N. Liu, C. Wen, Phys. Rev. C 93 (2016) 034623\.
  * [248] F. Duan, Y. Yang, K. Wang, A. M. Moro, V. Guimarães, D. Pang, J. Wang, Z. Sun, J. Lei, A. Di Pietro, X. Liu, G. Yang, J. Ma, P. Ma, S. Xu, Z. Bai, X. Sun, Q. Hu, J. Lou, X. Xu, H. Li, S. Jin, H. Ong, Q. Liu, J. Yao, H. Qi, C. Lin, H. Jia, N. Ma, L. Sun, D. Wang, Y. Zhang, X. Zhou, Z. Hu, H. Xu, Phys. Lett. B 811 (2020) 135942.
  * [249] A. M. Moro, J. A. Lay, J. Gomez Camacho, Phys.Lett. B 811 (2020) 135959.
  * [250] N. C. Summers, F. M. Nunes, Phys. Rev. C 76 (2007) 014611.
  * [251] N. C. Summers, F. M. Nunes, I. J. Thompson, Phys. Rev. C 89 (2014) 069901.
  * [252] A. Shrivastava, et al., Phys. Lett. B 596 (2004) 54.
  * [253] J. A. Lay, A. M. Moro, J. M. Arias, J. Gómez-Camacho, Phys. Rev. C 85 (2012) 054618\.
  * [254] T. Tarutina, L. C. Chamon, M. S. Hussein, Phys. Rev. C 67 (2003) 044605.
  * [255] A. M. Moro, J. A. Lay, Phys. Rev. Lett. 109 (2012) 232502.
  * [256] Y. Iseri, M. Yahiro, M. Kamimura, Progress of Theoretical Physics Supplement 89 (1986) 84.
  * [257] J. Tostevin, Nucl. Phys. A 682 (2001) 320.
  * [258] R. de Diego, R. Crespo, A. M. Moro, Phys. Rev. C 95 (2017) 044611.
  * [259] H. Fuchs, Nucl. Instrum. Methods Phys. Res. 200 (1982) 361.
  * [260] M. Yahiro, Y. Iseri, H. Kameyama, M. Kamimura, M. Kawai, Progress of Theoretical Physics Supplement 89 (1986) 32–83.
  * [261] A. Deltuva, Nucl. Phys. A 947 (2016) 173.
  * [262] A. Kiss, O. Aspelund, G. Hrehuss, K. Kn Nucl. Phys. A 262 (1976) 1 – 18.
  * [263] R. Varner, W. Thompson, T. McAbee, E. Ludwig, T. Clegg, Physics Reports 201 (1991) 57.
  * [264] M. Gómez-Ramos, A. M. Moro, Phys. Rev. C 95 (2017) 034609.
  * [265] P. Chau Huu-Tai, Jour. of Phys.: Conf. Series 312 (2011) 082018.
  * [266] L. D. Faddeev, Zh. Eksp. Teor. Fiz. 39 (1960) 1459.
  * [267] A. Deltuva, Phys. Rev. C 79 (2009) 054603.
  * [268] G. Baur, R. Shyam, F. Rösel, D. Trautmann, Phys. Rev. C 28 (1983) 946.
  * [269] S. Typel, G. Baur, Phys. Rev. C 50 (1994) 2104.
  * [270] H. Esbensen, G. F. Bertsch, Nucl. Phys. A 600 (1996) 37.
  * [271] T. Kido, K. Yabana, Y. Suzuki, Phys. Rev. C50, R1276.
  * [272] P. Capel, G. Goldstein, D. Baye, Phys. Rev. C 70 (2004) 064605.
  * [273] J. E. Escher, J. T. Burke, F. S. Dietrich, N. D. Scielzo, I. J. Thompson, W. Younes, Rev. Mod. Phys. 84 (2012) 353.
  * [274] J. Pampus, J. Bisplinghoff, J. Ernst, T. Mayer-Kuckuk, J. Rama Rao, G. Baur, F. Rosel, D. Trautmann, Nucl. Phys. A311 (1978) 141.
  * [275] G. Baur, R. Shyam, F. Rosel, D. Trautmann, Phys. Rev. C 21 (1980) 2668.
  * [276] R. Shyam, G. Baur, F. Rosel, D. Trautmann, Phys. Rev. C 22 (1980) 1401.
  * [277] T. Udagawa, T. Tamura, Phys. Rev. C 24 (1981) 1348.
  * [278] T. Udagawa, X. H. Li, T. Tamura, Phys. Lett. 135B (1984) 333.
  * [279] N. Austern, C. M. Vincent, Phys. Rev.C 23 (1981) 1847.
  * [280] M. Ichimura, N. Austern, C. M. Vincent, Phys. Rev. C 32 (1985) 431.
  * [281] M. Ichimura, N. Austern, C. M. Vincent, Phys. Rev.C 32 (1985) 431.
  * [282] M. Ichimura, Phys. Rev. C 41 (1990) 834.
  * [283] G. Potel, G. Perdikakis, B. V. Carlson, M. C. Atkinson, W. H. Dickhoff, J. E. Escher, M. S. Hussein, J. Lei, W. Li, A. O. Macchiavelli, A. M. Moro, F. M. Nunes, S. D. Pain, J. Rotureau, The European Physical Journal A 53 (2017) 178\.
  * [284] M. Gómez-Ramos, J. Gómez-Camacho, J. Lei, A. M. Moro, The European Physical Journal A 57 (2021) 1.
  * [285] J. Lei, A. M. Moro, Phys. Rev. C 92 (2015) 044616.
  * [286] S. R. Cotanch, Nucl. Phys. A842 (2010) 48.
  * [287] D. Y. Pang, N. K. Timofeyuk, R. C. Johnson, J. A. Tostevin, Phys. Rev. C 87 (2013) 064613.
  * [288] J. Lei, A. M. Moro, Phys. Rev. Lett. 122 (2019) 042503.