4. Results and discussion

<sup>σ</sup>CF <sup>¼</sup> <sup>π</sup>

PSurv <sup>ℓ</sup> � P

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

where

3. Fusion barrier distribution

well as numerical uncertainties [29–31]:

δDstat fus ð Þ E ≈

26

q

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

ð Þ 0

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

Nuclear fusion is related to the transmission of the incident wave through a potential barrier, resulting from nuclear attraction plus Coulomb repulsion. However, the meaning of the fusion barrier depends on the description of the collision. Coupled-channel calculations include static barriers, corresponding to frozen den-

sities of the projectile and the target. Its most dramatic consequence is the enhancement of the total fusion reaction cross section σfus at Coulomb barrier energies Vb, in some cases by several orders of magnitude. One of the possible ways to describe the effect of coupling channels is a division of the fusion barrier into several, the so-called fusion barrier distribution Dfus and given by [20, 29]

Dfusð Þ¼ <sup>E</sup> <sup>d</sup><sup>2</sup>

where F Eð Þ is related to the total fusion reaction cross section through [29]

The experimental determination of the fusion reaction barrier distribution has led to significant progress in the understanding of fusion reaction. This comes about because, as already mentioned, the fusion reaction barrier distribution gives information on the coupling channels appearing in the collision. However, we note from Eq. (24) that, since fusion reaction barrier distribution should be extracted from the values of the total fusion reaction cross section, it is the subject to experimental as

Dfusð Þ <sup>E</sup> <sup>≈</sup>F Eð Þþ <sup>þ</sup> <sup>Δ</sup><sup>E</sup> F Eð Þ� � <sup>Δ</sup><sup>E</sup> <sup>2</sup>F Eð Þ

where δF Eð Þ means the uncertainty in the measurement of the product of the energy by the total fusion reaction cross section at a given value of the collision

> ffiffiffi 6 <sup>p</sup> <sup>δ</sup>F Eð Þ

energy. Then the uncertainties can approximately be written as [30]

δDstat fus ð Þ E ≈

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ � <sup>δ</sup>F Eð Þ <sup>þ</sup> <sup>Δ</sup><sup>E</sup> <sup>2</sup> <sup>þ</sup> ½ � <sup>δ</sup>F Eð Þ � <sup>Δ</sup><sup>E</sup> <sup>2</sup> <sup>þ</sup> <sup>4</sup>½ � <sup>δ</sup>F Eð Þ <sup>2</sup>

where ΔE is the energy interval between measurements of the total fusion reaction cross section. From Eq. (25), one finds that the statistical error associated with the fusion reaction barrier distribution is approximately given by [30]

F Eð Þ

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

<sup>ℓ</sup> <sup>T</sup>ð Þ <sup>0</sup>

<sup>ℓ</sup> ð Þ E (21)

<sup>ℓ</sup> <sup>¼</sup> j j <sup>a</sup>0ð Þ <sup>ℓ</sup>; tca <sup>2</sup> (22)

dE<sup>2</sup> (23)

<sup>Δ</sup>E<sup>2</sup> (25)

ð Þ <sup>Δ</sup><sup>E</sup> <sup>2</sup> (26)

ð Þ <sup>Δ</sup><sup>E</sup> <sup>2</sup> (27)

F Eð Þ¼ Eσfusð Þ E (24)

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 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, 11B + 159Tb and 12C + <sup>9</sup> Be. The values of the height Vc, radius Rc, and curvature ℏω for the fusion barrier are displayed in Table 1.

#### 4.1 The reaction <sup>6</sup> Li + 64Ni

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 6 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 the respective experimental data (solid circles).

The experimental data for this system are obtained from Ref. [32]. The real and imaginary Akyüz-Winther potential parameters obtained by using chi-square


Table 1.

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

#### Figure 1.

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

method are the strength W<sup>0</sup> ¼ 50 MeV, radius ri ¼ 1:0 fm, and diffuseness ai ¼ 0:25 fm, and for the real part, the depth is V<sup>0</sup> ¼ 35:0 MeV, radius is <sup>r</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup>:<sup>1</sup> fm, and diffuseness is <sup>a</sup><sup>0</sup> <sup>¼</sup> <sup>0</sup>:<sup>8</sup> fm. The <sup>χ</sup><sup>2</sup> values obtained for the total fusion cross section σfus are 1.5057 and 1.1286 in the case of no coupling for semiclassical and quantum mechanical calculations, respectively. The χ<sup>2</sup> values obtained for the case of coupling effects included are 0.2431 and 0.3115 for semiclassical and quantum mechanical calculations, respectively. The χ<sup>2</sup> values show clearly that semiclassical calculations including coupling effects are more consistent with the experimental data than full quantum mechanical including coupling effects. The χ<sup>2</sup> values obtained using single-channel calculations for the fusion reaction barrier distribution Dfus are 0.1823 and 1.1914 for semiclassical and quantum mechanical calculations, respectively. The χ<sup>2</sup> values obtained when coupled channels are included are 0.1827 and 0.1321 for semiclassical and quantum mechanical calculations, respectively; the fusion barrier distribution Dfus has been extracted from the experimental data using Wong fit model along with the three-point difference method. The comparison with the experimental data for Dfus shows that the quantum mechanical calculations are in better agreement than the semiclassical calculations including the coupling effects.
