2.1 The single-channel description

The semiclassical theory is used to estimate the fusion cross section in the onedimensional potential model which assumes that one can describe the degree of freedom only of the relative motion between the colliding heavy ions [19, 20]. The semiclassical theory deals with the Schrödinger equation assuming independent energy and angular momentum and the potential energy for the radial part of the relative motion through quantum tunneling:

$$\left[-\hbar^2\nabla^2/2\mu + V(r) - E\right]\Psi(\mathbf{r}) = \mathbf{0} \tag{1}$$

where μ is the reduced mass of the system and V rð Þ is the total potential energy of the system. Semiclassical reaction amplitudes can be evaluated as a function of time, assuming the particle trajectory to be determined by classical dynamics, including Coulomb and real nuclear and centrifugal potentials, which can be written in the form

$$V(r) = V\_C(r) + V\_N(r) + V\_l(r) \tag{2}$$

In coupled-channel effects on the elastic channel, the imaginary part should be added to the nuclear potential, represented by complex potential as

$$V\_N(r) = U\_N(r) - iW(r) \tag{3}$$

Fusion Reaction of Weakly Bound Nuclei DOI: http://dx.doi.org/10.5772/intechopen.80582

4 He+<sup>7</sup>

systems <sup>6</sup>

Li with separation energy Sα = 8.664 MeV and 12C breakup into three α

Be.

The semiclassical theory is used to estimate the fusion cross section in the onedimensional potential model which assumes that one can describe the degree of freedom only of the relative motion between the colliding heavy ions [19, 20]. The semiclassical theory deals with the Schrödinger equation assuming independent energy and angular momentum and the potential energy for the radial part of the

where μ is the reduced mass of the system and V rð Þ is the total potential energy of the system. Semiclassical reaction amplitudes can be evaluated as a function of time, assuming the particle trajectory to be determined by classical dynamics, including Coulomb and real nuclear and centrifugal potentials, which can be

In coupled-channel effects on the elastic channel, the imaginary part should be

added to the nuclear potential, represented by complex potential as

<sup>=</sup>2<sup>μ</sup> <sup>þ</sup> V rð Þ� <sup>E</sup> <sup>Ψ</sup>ð Þ¼ <sup>r</sup> <sup>0</sup> (1)

V rð Þ¼ VCð Þþ r VNð Þþ r Vlð Þr (2)

VNð Þ¼ r UNð Þ� r iW rð Þ (3)

channel is described by the continuum discretized coupled-channel (CDCC) method. The continuum that describes the breakup channel is discretized into bins [13, 14]. To study the coupled-channel problem, this requires a profound truncation of the continuum into discrete bin of energy into equally spaced states. The CDCC method is totally based on this concept. Surrey group extended the discretization procedure discussed in [14] for the deuteron case to study the breakup and fusion reactions of systems involving weakly bound nuclei [15, 16]. Recently, Majeed and Abdul-Hussien [17] utilized the semiclassical approach based on the theory of Alder and Winther. They carried out their calculations to investigate the role of the breakup channel on the fusion cross section σfus and fusion barrier distribution Dfus for 6,8H halo [17]. Semiclassical coupled-channel calculations in heavy-ion fusion reactions for the systems 40Ar + 110Pd and 132Sn + 48Ca were carried out by Majeed et al. [18]. They argued that including the channel coupling between the elastic channel and the continuum enhances the fusion reaction cross section σfus and the fusion barrier distribution Dfus calculations quite well below and above the Coulomb barrier for medium and heavy systems. This study aims to employ a semiclassical approach by adopting Alder and Winther (AW) [19] theory originally used to treat the Coulomb excitation of nuclei. The semiclassical approach has been implemented and coded using FORTRAN programming codenamed (SCF) which is written and developed by Canto et al. [20]. The fusion cross section σfus and fusion barrier distribution Dfus are calculated here utilizing the semiclassical approach. The results from the present study are compared with the quantum mechanical calculations using the FORTRAN code (CC) [21] and with the experimental data for the three

Nuclear Fusion - One Noble Goal and a Variety of Scientific and Technological Challenges

) 3α [3, 11, 12]. The breakup

particles induced by neutrons or protons by 12C (p, p<sup>0</sup>

Li + 64Ni, 11B + 159Tb, and 12C + <sup>9</sup>

2. The semiclassical theory

written in the form

22

2.1 The single-channel description

relative motion through quantum tunneling:

�ℏ<sup>2</sup> ∇2

The method can be extended to describe interference of different l waves due to strong nuclear attraction and absorption caused by the imaginary nuclear potential [6, 20, 22]. When the two nuclei come across the potential barrier into the inner region, the fusion occurs according to the semiclassical theory, and the penetrability probability below barrier can be evaluated usingWKB approximation [5, 6, 19, 23, 24]:

$$P\_{fus}^{WKB}(l,E) = \frac{1}{\mathbf{1} + \exp\left[2\int\_{r\_b^{(l)}}^{r\_s^{(l)}} \kappa\_l(r) dr\right]}\tag{4}$$

Then, the latter can be rewritten as follows:

$$P\_{fus}^{\rm WKB}(l, E) = \frac{1}{1 + \exp\left[\frac{2\pi}{\hbar\Omega\_l}(V\_b(l) - E)\right]}\tag{5}$$

where κlð Þr is the local wave number and r ð Þl <sup>b</sup> and r ð Þl <sup>a</sup> are the inner and outer classical turning points at the fusion barrier potential. If one approximates the fusion barrier by a parabolic function, then the penetrability probability above barrier is given by the Hill-Wheeler formula [3, 19]:

$$P\_{\rm fus}^{\rm WH}(l, E) = \frac{1}{\mathbf{1} + \exp\left[\frac{2\pi}{\hbar \Omega\_l} (E - \mathbf{V}\_b(l))\right]} \tag{6}$$

where Vbð Þl and Ω<sup>l</sup> are the height and the curvature parameter of the fusion barrier for the partial wave, respectively, and E is the bombarding energy. Ignoring the l dependence of ω and of the barrier position Rb and assuming that the l dependence of Vbð Þl is given only by the difference of the centrifugal potential energy, one can obtain Wong's formula which is given in Section 5. The fusion cross sections can be estimated by the one-dimensional WKB approximation by the following relations [21, 24]:

$$
\sigma\_{\rm fus}(E) = \frac{\pi}{\kappa^2} \sum (2l+1) P\_{\rm fus}^{\rm MVKB}(l, E) \tag{7}
$$

$$P\_{fus}^{\prime}(l,E) = \frac{4k}{E} \int dr \left| u\_{\eta l}(k\_{\gamma}, r) \right|^2 \mathcal{W}\_{fus}^{\prime}(r) \tag{8}$$

where u<sup>γ</sup><sup>l</sup> k<sup>γ</sup> ;r � � represents the radial wave function for the partial wave l in channel γ and W<sup>γ</sup> fusð Þr is the absolute value of the imaginary part of the potential associated to fusion in that channel.

The complete fusion cross section in heavy ions evaluated using semiclassical theory is based on the classical trajectory approximation r. And the relevant intrinsic degrees of freedom of the projectile, represented by ξ with applying the continuum discretized coupled-channel (CDCC) approximation of Alder and Winther (AW) theory [16], have been proposed [17]. The projectile Hamiltonian is then given by

$$h = h\_0(\xi) + V(\xi, r) \tag{9}$$

where h0ð Þξ is the intrinsic Hamiltonian and Vð Þ¼ ξ;r VNð Þþ ξ;r VCð Þ ξ;r , Vð Þ ξ;r is the interaction between the projectile and target nuclei. The Rutherford trajectory depends on the collision energy, E, and the angular momentum, l. In this case the trajectory is the solution of the classical motion equations with the potential

V rð Þ¼ h i Ψ0jV rð Þj ; ξ Ψ<sup>0</sup> , where Ψ<sup>0</sup> is the ground state (g.s.) of the projectile. In this way, the interaction becomes time-dependent in the <sup>ξ</sup>-space Vlð Þ¼ <sup>ξ</sup>:<sup>t</sup> V rl tð Þ; <sup>ξ</sup> , and the eigenstates of the intrinsic Hamiltonian jΨ<sup>γ</sup> satisfy the Schrödinger equation [25, 26]:

$$h|\Psi\_{\mathcal{I}}\rangle = \varepsilon|\Psi\_{\mathcal{I}}\rangle\tag{10}$$

ψ ξð Þ¼ ; t ∑

α

σβ <sup>¼</sup> <sup>π</sup> <sup>k</sup><sup>2</sup> <sup>∑</sup> ℓ

σ<sup>F</sup> ¼ ∑ β

PF <sup>ℓ</sup>ð Þ¼ β

tial associated to fusion in that channel.

approach on the classical trajectory and Tð Þ <sup>β</sup>

π <sup>k</sup><sup>2</sup> <sup>∑</sup> ℓ

4k E ð W<sup>F</sup>

PF <sup>ℓ</sup>ð Þ β ≃P

obtains the AW equations [27]

Fusion Reaction of Weakly Bound Nuclei DOI: http://dx.doi.org/10.5772/intechopen.80582

<sup>ℓ</sup> <sup>¼</sup> <sup>a</sup>βð Þ <sup>ℓ</sup>; <sup>t</sup> ! þ<sup>∞</sup> � � �

expansions, we get [28]

with

channel β, and W<sup>F</sup>

approximation [27]

ð Þ β

tial barrier in channel β [19].

where P

That is [19],

25

Pð Þ <sup>β</sup>

iℏa\_ <sup>β</sup>ð Þ¼ ℓ; t ∑

�

β

and inserting this expansion into the Schrödinger equation for ψ ξð Þ ; t , one

a<sup>γ</sup> ð Þ ℓ; t φβjVℓð Þj ξ; t φβ � �e

<sup>2</sup> and the angle-integrated cross section is [19]

ð Þ <sup>2</sup><sup>ℓ</sup> <sup>þ</sup> <sup>1</sup> <sup>P</sup>ð Þ <sup>β</sup>

ð Þ <sup>2</sup><sup>ℓ</sup> <sup>þ</sup> <sup>1</sup> PF

<sup>β</sup> ð Þ<sup>r</sup> <sup>u</sup>β<sup>ℓ</sup> <sup>k</sup>β;<sup>r</sup> � � � � �

<sup>ℓ</sup> is the probability that the system is in channel β at the point of closest

<sup>ℓ</sup> E<sup>β</sup>

<sup>β</sup> is the absolute value of the imaginary part of the optical poten-

Above, uβ<sup>ℓ</sup> kβ;r � � represents the radial wave function for the ℓth partial wave in

To use the AW method to evaluate the fusion cross section, we make the

ð Þ β <sup>ℓ</sup> <sup>T</sup>ð Þ <sup>β</sup> <sup>ℓ</sup> E<sup>β</sup>

with energy E<sup>β</sup> ¼ E � εβ and reduced mass μ ¼ MPMT=ð Þ MP þ MT , where MP, MT are the masses of the projectile and target, respectively, tunnels through the poten-

We now proceed to study the CF cross sections in reactions induced by weakly bound projectiles. For simplicity, we assume that the g.s. is the only bound state of the projectile and that the breakup process produces only two fragments, F<sup>1</sup> and F2. In this way, the labels β ¼ 0 and β 6¼ 0 correspond, respectively, to the g.s. and the breakup states represented by two unbound fragments. Neglecting any sequential contribution, the CF can only arise from the elastic channel. In this way, the cross section σCF can be obtained from Eq. (20), dropping contributions from β 6¼ 0.

<sup>ℓ</sup>ð Þ β

� 2

� � (18)

To extend this method to fusion reactions, we start with the quantum mechanical calculation of the fusion cross section in a coupled-channel problem. For simplicity, we assume that all channels are bound and have zero spin. The fusion cross section is a sum of contributions from each channel. Carrying out partial-wave

These equations are solved with the initial conditions aβð Þ¼ ℓ; t ! �∞ δβ0, which means that before the collision ð Þ t ! �∞ , the projectile was in its ground state. The final population of channel β in a collision with angular momentum ℓ is

aβð Þ ℓ; t φβð Þξ e

�iεβ <sup>t</sup>=<sup>ℏ</sup> (15)

�ið Þ εβ�εγ <sup>t</sup>=<sup>ℏ</sup> (16)

<sup>ℓ</sup> (17)

dr (19)

� � (20)

� � is the probability that a particle

After expanding the wave function in the basis of intrinsic eigenstates

$$\Psi(\xi, t) = \sum a\_{\gamma}(l, t)\Psi\_{\gamma}(\xi)e^{-ic\_{\gamma}t/\hbar} \tag{11}$$

and inserting Eq. (11) into the Schrödinger equation for Ψð Þ ξ; t , the AW equations can be obtained:

$$i\hbar \,\dot{a}\_{\gamma}(l,t) = \sum\_{\varepsilon} \left< \Psi\_{\gamma} | V(\xi, t) | \Psi\_{\gamma} \right> e^{i(\varepsilon\_{\gamma} - \varepsilon\_{\varepsilon})t/\hbar} \gamma e(l, t) \tag{12}$$

These equations should be solved with initial conditions a<sup>γ</sup> ð Þ¼ l; t ! �∞ δγ<sup>0</sup> which mean that before the collision ð Þ t ! �∞ , the projectile was in its ground state. The final population of the channel γ in a collision with angular momentum l is P<sup>γ</sup> fusð Þ¼ <sup>l</sup>; <sup>E</sup> <sup>a</sup><sup>γ</sup> ð Þ <sup>l</sup>; <sup>t</sup> ! �<sup>∞</sup> 2 . Eq. (8) gives the general expression for the fusion cross section in multichannel scattering [26].

#### 2.2 The coupled-channel description

The variables employed to describe the collision are the projectile-target separation vector r ! and the relevant intrinsic degrees of freedom of the projectile ξ. For simplicity, we neglect the internal structure of the target. The Hamiltonian is then given by [27]

$$H = H\_0(\xi) + V(\overrightarrow{r}, \xi) \tag{13}$$

where H0ð Þξ is the intrinsic Hamiltonian of the projectile and Vðr ! , ξÞ represents the projectile-target interaction. Since the main purpose of the present work is to test the semiclassical model in calculations of sub-barrier fusion, the nuclear coupling is neglected. Furthermore, for the present theory-theory comparison, only the Coulomb dipole term is taken into account. The eigenvectors of H0ð Þξ are given by the equation [27]

$$H\_0|\rho\_\beta\rangle = \varepsilon\_\beta|\rho\_\beta\rangle\tag{14}$$

where εβ is energy of internal motion. The AW method is implemented in two steps. First, one employs classical mechanics for the time evolution of the variable r !. The ensuing trajectory depends on the collision energy, E, and the angular momentum, l. In its original version, an energy symmetrized Rutherford trajectory rℓ ! ð Þt was used. In our case, the trajectory is the solution of the classical equations of motion with the potential Vðr !Þ ¼ ⟨φ0|Vð<sup>r</sup> !, <sup>ξ</sup>Þφ0⟩, where <sup>j</sup>φ0<sup>i</sup> is the ground state of the projectile. In this way, the coupling interaction becomes a time-dependent interaction in the ξ-space, Vℓð Þ� ξ; t Vðr<sup>ℓ</sup> ! ð Þ<sup>t</sup> , <sup>ξ</sup>Þ. The second step consists in treating the dynamics in the intrinsic space as a time-dependent quantum mechanics problem. Expanding the wave function in the basis of intrinsic eigenstates [19]

Fusion Reaction of Weakly Bound Nuclei DOI: http://dx.doi.org/10.5772/intechopen.80582

$$\Psi(\xi, t) = \sum\_{\beta} a\_{\beta}(\ell, t) \rho\_{\beta}(\xi) e^{-i\epsilon\_{\beta}t/\hbar} \tag{15}$$

and inserting this expansion into the Schrödinger equation for ψ ξð Þ ; t , one obtains the AW equations [27]

$$i\hbar \dot{\mathbf{a}}\_{\beta}(\ell, \mathbf{t}) = \sum\_{a} \mathbf{a}\_{\gamma}(\ell, \mathbf{t}) \langle \boldsymbol{\uprho}\_{\beta} | \mathbf{V}\_{\ell}(\xi, \mathbf{t}) | \boldsymbol{\uprho}\_{\beta} \rangle e^{-i \left(\boldsymbol{\uprho} - \boldsymbol{\uprho}\_{\gamma}\right) \mathbf{t}/\hbar} \tag{16}$$

These equations are solved with the initial conditions aβð Þ¼ ℓ; t ! �∞ δβ0, which means that before the collision ð Þ t ! �∞ , the projectile was in its ground state. The final population of channel β in a collision with angular momentum ℓ is Pð Þ <sup>β</sup> <sup>ℓ</sup> <sup>¼</sup> <sup>a</sup>βð Þ <sup>ℓ</sup>; <sup>t</sup> ! þ<sup>∞</sup> � � � � <sup>2</sup> and the angle-integrated cross section is [19]

$$
\sigma\_{\beta} = \frac{\pi}{k^2} \sum\_{\ell} (2\ell + 1) P\_{\ell}^{(\beta)} \tag{17}
$$

To extend this method to fusion reactions, we start with the quantum mechanical calculation of the fusion cross section in a coupled-channel problem. For simplicity, we assume that all channels are bound and have zero spin. The fusion cross section is a sum of contributions from each channel. Carrying out partial-wave expansions, we get [28]

$$\sigma\_{\boldsymbol{F}} = \sum\_{\boldsymbol{\beta}} \left[ \frac{\boldsymbol{\pi}}{k^2} \sum\_{\ell} (2\ell + \mathbf{1}) P\_{\ell}^{\boldsymbol{F}}(\boldsymbol{\beta}) \right] \tag{18}$$

with

V rð Þ¼ h i Ψ0jV rð Þj ; ξ Ψ<sup>0</sup> , where Ψ<sup>0</sup> is the ground state (g.s.) of the projectile. In this way, the interaction becomes time-dependent in the <sup>ξ</sup>-space Vlð Þ¼ <sup>ξ</sup>:<sup>t</sup> V rl tð Þ; <sup>ξ</sup> ,

Nuclear Fusion - One Noble Goal and a Variety of Scientific and Technological Challenges

hjΨ<sup>γ</sup> ¼ εjΨ<sup>γ</sup>

After expanding the wave function in the basis of intrinsic eigenstates

Ψð Þ¼ ξ; t ∑a<sup>γ</sup> ð Þ l; t Ψ<sup>γ</sup> ð Þξ e

and inserting Eq. (11) into the Schrödinger equation for Ψð Þ ξ; t , the AW

Ψ<sup>γ</sup> jVð Þj ξ; t Ψ<sup>γ</sup> e

These equations should be solved with initial conditions a<sup>γ</sup> ð Þ¼ l; t ! �∞ δγ<sup>0</sup> which mean that before the collision ð Þ t ! �∞ , the projectile was in its ground state. The final population of the channel γ in a collision with angular momentum l

The variables employed to describe the collision are the projectile-target separa-

simplicity, we neglect the internal structure of the target. The Hamiltonian is then

H ¼ H0ð Þþ ξ V r!; ξ

the projectile-target interaction. Since the main purpose of the present work is to test the semiclassical model in calculations of sub-barrier fusion, the nuclear coupling is neglected. Furthermore, for the present theory-theory comparison, only the Coulomb dipole term is taken into account. The eigenvectors of H0ð Þξ are given by

¼ εβjφβ

where εβ is energy of internal motion. The AW method is implemented in two steps. First, one employs classical mechanics for the time evolution of the variable

! ð Þt was used. In our case, the trajectory is the solution of the classical equations of

treating the dynamics in the intrinsic space as a time-dependent quantum mechanics problem. Expanding the wave function in the basis of intrinsic eigenstates [19]

where H0ð Þξ is the intrinsic Hamiltonian of the projectile and Vðr

H0jφβ 

!. The ensuing trajectory depends on the collision energy, E, and the angular momentum, l. In its original version, an energy symmetrized Rutherford trajectory

the projectile. In this way, the coupling interaction becomes a time-dependent

!Þ ¼ ⟨φ0|Vð<sup>r</sup>

! and the relevant intrinsic degrees of freedom of the projectile ξ. For

<sup>i</sup>ð Þ εγ�εϵ <sup>t</sup>

. Eq. (8) gives the general expression for the fusion

satisfy the Schrödinger

(10)

�iεγ <sup>t</sup>=<sup>ℏ</sup> (11)

<sup>=</sup><sup>ℏ</sup> γϵð Þ <sup>l</sup>; <sup>t</sup> (12)

(13)

! , ξÞ represents

(14)

!, <sup>ξ</sup>Þφ0⟩, where <sup>j</sup>φ0<sup>i</sup> is the ground state of

! ð Þ<sup>t</sup> , <sup>ξ</sup>Þ. The second step consists in

and the eigenstates of the intrinsic Hamiltonian jΨ<sup>γ</sup>

iℏ a\_ <sup>γ</sup> ð Þ¼ l; t ∑

 2 ϵ

equation [25, 26]:

equations can be obtained:

fusð Þ¼ <sup>l</sup>; <sup>E</sup> <sup>a</sup><sup>γ</sup> ð Þ <sup>l</sup>; <sup>t</sup> ! �<sup>∞</sup>

2.2 The coupled-channel description

cross section in multichannel scattering [26].

is P<sup>γ</sup>

tion vector r

given by [27]

the equation [27]

motion with the potential Vðr

interaction in the ξ-space, Vℓð Þ� ξ; t Vðr<sup>ℓ</sup>

r

rℓ

24

$$P\_{\ell}^{F}(\beta) = \frac{4k}{E} \int \mathcal{W}\_{\beta}^{F}(r) |u\_{\beta \ell}(k\_{\beta}, r)|^{2} dr \tag{19}$$

Above, uβ<sup>ℓ</sup> kβ;r � � represents the radial wave function for the ℓth partial wave in channel β, and W<sup>F</sup> <sup>β</sup> is the absolute value of the imaginary part of the optical potential associated to fusion in that channel.

To use the AW method to evaluate the fusion cross section, we make the approximation [27]

$$P\_{\ell}^{\mathrm{F}}(\boldsymbol{\beta}) \simeq \overline{\mathbf{P}}\_{\ell}^{(\boldsymbol{\beta})} T\_{\ell}^{(\boldsymbol{\beta})} \left( \mathbf{E}\_{\boldsymbol{\beta}} \right) \tag{20}$$

where P ð Þ β <sup>ℓ</sup> is the probability that the system is in channel β at the point of closest approach on the classical trajectory and Tð Þ <sup>β</sup> <sup>ℓ</sup> E<sup>β</sup> � � is the probability that a particle with energy E<sup>β</sup> ¼ E � εβ and reduced mass μ ¼ MPMT=ð Þ MP þ MT , where MP, MT are the masses of the projectile and target, respectively, tunnels through the potential barrier in channel β [19].

We now proceed to study the CF cross sections in reactions induced by weakly bound projectiles. For simplicity, we assume that the g.s. is the only bound state of the projectile and that the breakup process produces only two fragments, F<sup>1</sup> and F2. In this way, the labels β ¼ 0 and β 6¼ 0 correspond, respectively, to the g.s. and the breakup states represented by two unbound fragments. Neglecting any sequential contribution, the CF can only arise from the elastic channel. In this way, the cross section σCF can be obtained from Eq. (20), dropping contributions from β 6¼ 0. That is [19],

Nuclear Fusion - One Noble Goal and a Variety of Scientific and Technological Challenges

$$
\sigma\_{\rm CF} = \frac{\pi}{k^2} \sum\_{\ell} (2\ell + \mathbf{1}) P\_{\ell}^{\rm Surv} T\_{\ell}^{(0)}(E) \tag{21}
$$

4. Results and discussion

Fusion Reaction of Weakly Bound Nuclei DOI: http://dx.doi.org/10.5772/intechopen.80582

barrier are displayed in Table 1.

4.1 The reaction <sup>6</sup>

12C + <sup>9</sup>

6

6

Table 1.

Figure 1.

fusion reaction barrier distribution Dfus (mb/MeV).

6

27

12C + <sup>9</sup>

In this section, the theoretical calculations are obtained for total fusion reaction σfus, and the fusion barrier distribution Dfus using the semiclassical theory adopted the continuum discretized coupled channel (CDCC) to describe the effect of the breakup channel on fusion processes. The semiclassical calculations are carried out

Be. The values of the height Vc, radius Rc, and curvature ℏω for the fusion

The calculations of the fusion cross section σfus and fusion barrier distribution Dfus are presented in Figure 1 panel (a) and panel (b), respectively, for the system

The experimental data for this system are obtained from Ref. [32]. The real and

Li + 64Ni. The dashed blue and red curves represent the semiclassical and full quantum mechanical calculations without coupling, respectively. The solid blue and red curves are the calculations including the coupling effects for the semiclassical and full quantum mechanical calculations, respectively. Panel (a) shows the comparison between our semiclassical and full quantum mechanical calculations with

imaginary Akyüz-Winther potential parameters obtained by using chi-square

System Vc Rc ℏω Refs

Li + 64Ni 12:41 9:1 3:9 [32] 11B + 159Tb 40:34 10:89 4:42 [33]

The fusion barrier parameters are height Vc ð Þ MeV , radius Rcð Þ fm , and curvature ℏω ð Þ MeV :

Be 4.28 7.43 2.61 [34]

The comparison of the coupled-channel calculations of semiclassical treatment (red curves) and full quantum mechanical (blue curves) with the experimental data of complete fusion (black-filled circles) [32] for

Li + 64Ni system. Panel (a) refers to the total fusion reaction cross section σfus (mb), and panel (b) provides the

Li + 64Ni, 11B + 159Tb and

using the (SCF) code, while the full quantum mechanical calculations are

performed by using the code (CC) for the systems <sup>6</sup>

Li + 64Ni

the respective experimental data (solid circles).

where

$$P\_{\ell}^{\text{Surv}} \equiv \overline{P}\_{\ell}^{(0)} = |a\_0(\ell, t\_{ct})|^2 \tag{22}$$

is usually called survival (to breakup) probability [19].
