4.2 The reaction 11B + 159Tb

In similar analysis we compare our theoretical calculations of the fusion cross section σfus and fusion barrier distribution Dfus with the corresponding experimental data in panels (a) and (b) of Figure 2, respectively, for the system 11B + 159Tb. The experimental data for this system is obtained from Ref. [33]. The real and imaginary Akyüz-Winther potential parameters are obtained by using chi-square method: V0 ¼ 126:1 MeV, r0 ¼ 1:2 fm, and a0 ¼ 0:5 fm and W0 <sup>¼</sup> <sup>55</sup>:9 MeV, ri <sup>¼</sup> <sup>0</sup>:986 fm, and ai <sup>¼</sup> <sup>0</sup>:614 fm. The <sup>χ</sup><sup>2</sup> values 0.9473 and 0.2486 are obtained for σfus using semiclassical and quantum mechanical distribution calculations without including the coupling, respectively, while the χ<sup>2</sup> for σfus using the semiclassical and quantum mechanical distribution calculations including the

coupling effects are 0.2681 and 0.1657, respectively. The χ<sup>2</sup> for the fusion barrier distribution Dfus using the semiclassical and quantum calculations are 0.5828 and 1.2329 for no coupling and 4.5969 and 0.0616 including coupling effects, respectively. The χ<sup>2</sup> values for σfus and Dfus give clear evidence that the quantum mechanical calculations are in better agreement than the semiclassical calculations as

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) [34] for

Be system. Panel (a) shows the total fusion reaction cross section σfus (mb), and panel (b) gives the fusion

Figure 3 (panels (a) and (b)) presents the comparison between our theoretical calculations for σfus and Dfus using both semiclassical and quantum mechanical calculations with the corresponding experimental data for the system 12C + <sup>9</sup>

experimental data for this system are obtained from Ref. [34]. The real and imaginary Akyüz-Winther potential parameters are obtained by using chi-square method: V<sup>0</sup> ¼ 40:3 MeV, r<sup>0</sup> ¼ 1:11 fm, a<sup>0</sup> ¼ 0:590 fm, W<sup>0</sup> ¼ 0 MeV, ri ¼ 1:1 fm, and ai <sup>¼</sup> <sup>0</sup>:<sup>50</sup> fm. The <sup>χ</sup><sup>2</sup> values obtained from the comparison between the results and experimental data for σfus are 1.0633 and 1.1447 without coupling, and 0.4924 and 0.2072 with coupling, for semiclassical and quantum mechanical calculations, respectively. The obtained χ<sup>2</sup> values for Dfus using semiclassical and quantum mechanical calculations are 1.2383 and 0.6185 without coupling and 0.9875 and

The semiclassical and quantum mechanical calculations for the total fusion reaction σfus and the fusion barrier distribution Dfus calculations below and around

consideration to describe the total fusion reaction σfus and the fusion barrier distribution Dfus for reaction of light projectiles. The full quantum mechanical calculations are closer to the experimental data than the semiclassical calculations;

Be. We conclude that the breakup channel is very important to be taken into

Li + 64Ni, 11B + 159Tb, and

Be. The

compared with experimental data.

reaction barrier distribution Dfus (mb/MeV).

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

Be

0.1868 with account for coupling, respectively.

Coulomb barrier were discussed for the systems <sup>6</sup>

4.3 The reaction 12C + <sup>9</sup>

Figure 3.

12C + <sup>9</sup>

5. Conclusion

12C + <sup>9</sup>

29

#### Figure 2.

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) [33] for 11B + 159Tb system. Panel (a) shows the total fusion reaction cross section σfus (mb), and panel (b) gives the fusion reaction barrier distribution Dfus (mb/MeV).

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

Figure 3.

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

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

In similar analysis we compare our theoretical calculations of the fusion cross

experimental data in panels (a) and (b) of Figure 2, respectively, for the system 11B + 159Tb. The experimental data for this system is obtained from Ref. [33]. The real and imaginary Akyüz-Winther potential parameters are obtained by using chi-square method: V0 ¼ 126:1 MeV, r0 ¼ 1:2 fm, and a0 ¼ 0:5 fm and W0 <sup>¼</sup> <sup>55</sup>:9 MeV, ri <sup>¼</sup> <sup>0</sup>:986 fm, and ai <sup>¼</sup> <sup>0</sup>:614 fm. The <sup>χ</sup><sup>2</sup> values 0.9473 and 0.2486 are obtained for σfus using semiclassical and quantum mechanical distribution calculations without including the coupling, respectively, while the χ<sup>2</sup> for σfus using the semiclassical and quantum mechanical distribution calculations including the

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) [33] for 11B + 159Tb system. Panel (a) shows the total fusion reaction cross section σfus (mb), and panel (b) gives the

section σfus and fusion barrier distribution Dfus with the corresponding

calculations including the coupling effects.

4.2 The reaction 11B + 159Tb

Figure 2.

28

fusion reaction barrier distribution Dfus (mb/MeV).

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) [34] for 12C + <sup>9</sup> Be system. Panel (a) shows the total fusion reaction cross section σfus (mb), and panel (b) gives the fusion reaction barrier distribution Dfus (mb/MeV).

coupling effects are 0.2681 and 0.1657, respectively. The χ<sup>2</sup> for the fusion barrier distribution Dfus using the semiclassical and quantum calculations are 0.5828 and 1.2329 for no coupling and 4.5969 and 0.0616 including coupling effects, respectively. The χ<sup>2</sup> values for σfus and Dfus give clear evidence that the quantum mechanical calculations are in better agreement than the semiclassical calculations as compared with experimental data.

#### 4.3 The reaction 12C + <sup>9</sup> Be

Figure 3 (panels (a) and (b)) presents the comparison between our theoretical calculations for σfus and Dfus using both semiclassical and quantum mechanical calculations with the corresponding experimental data for the system 12C + <sup>9</sup> Be. The experimental data for this system are obtained from Ref. [34]. The real and imaginary Akyüz-Winther potential parameters are obtained by using chi-square method: V<sup>0</sup> ¼ 40:3 MeV, r<sup>0</sup> ¼ 1:11 fm, a<sup>0</sup> ¼ 0:590 fm, W<sup>0</sup> ¼ 0 MeV, ri ¼ 1:1 fm, and ai <sup>¼</sup> <sup>0</sup>:<sup>50</sup> fm. The <sup>χ</sup><sup>2</sup> values obtained from the comparison between the results and experimental data for σfus are 1.0633 and 1.1447 without coupling, and 0.4924 and 0.2072 with coupling, for semiclassical and quantum mechanical calculations, respectively. The obtained χ<sup>2</sup> values for Dfus using semiclassical and quantum mechanical calculations are 1.2383 and 0.6185 without coupling and 0.9875 and 0.1868 with account for coupling, respectively.

### 5. Conclusion

The semiclassical and quantum mechanical calculations for the total fusion reaction σfus and the fusion barrier distribution Dfus calculations below and around Coulomb barrier were discussed for the systems <sup>6</sup> Li + 64Ni, 11B + 159Tb, and 12C + <sup>9</sup> Be. We conclude that the breakup channel is very important to be taken into consideration to describe the total fusion reaction σfus and the fusion barrier distribution Dfus for reaction of light projectiles. The full quantum mechanical calculations are closer to the experimental data than the semiclassical calculations;

however, semiclassical ones can be considered a successful tool for studying fusion reaction of systems involving light projectiles.

References

6 Li, <sup>7</sup>

377-379

9

044608

31

mechanisms for the <sup>9</sup>

Li, and <sup>9</sup>

[1] Gomes PRS, Padron I, Crema E, Capurro OA, Fernández Niello JO, et al. Comprehensive study of reaction

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

at near- and sub-barrier energies. Physical Review C. 2006;73:064606

[2] Gomes PRS, Rios JL, Borges JR, Otomar DR. Fusion, breakup and scattering of weakly bound nuclei at near barrier energies. The Open Nuclear & Particle Physics Journal. 2013;6:10-15

[3] Dasgupta M, Gomes PRS, Hinde DJ, Moraes SB, Anjos RM, Berriman AC, et al. Effect of breakup on the fusion of

Physical Review C. 2004;70:024606

[5] Hussein MS, Pato MP, Canto LF, Donangelo R. Near-barrier fusion of 11Li with heavy spherical and deformed targets. Physical Review C. 1992;46:

[6] Dasso CH, Vitturi A. Does the presence of 11Li breakup channels reduce the cross section for fusion processes? Physical Review C. 1994;50:R12-R14

[7] Canto LF, Donangelo R, Lotti P. Effect of Coulomb dipole polarizability of halo nuclei on their near-barrier fusion with heavy targets. Physical Review C. 1995;52:R2848-R2850

[8] Martí GV, Gomes PRS, Rodríguez MD, Fernández Niello JO. Fusion, reaction, and breakup cross sections of

Be on a light mass target. Physical

[9] Padron I, Gomes PRS, Anjos RM, Lubian J, Muri C, Alves JJ. Fusion of stable weakly bound nuclei with 27Al and 64Zn. Physical Review C. 2002;66:

Review C. 2005;71:027602

[4] Takigawa N, Sagawa H. Interaction potential and fusion of a halo nucleus. Physics Letters B. 1991;265:23-28

Be with heavy nuclei.

Be+144Sm system

[10] Wang B, Zhao W, Gomes PRS, Zhao E, Zhou S. Systematic study of breakup effects on complete fusion at energies above the Coulomb barrier. Physical

[11] Wolfs FLH, White CA, Bryan DC, Freeman CG, Herrick DM, et al.

Breakup of 82 MeV 11B. Physical Review

[12] Bogatin VI, Novak Z, Ostroumov VI. Breakup of 12C into three alpha particles accompanying inelastic

scattering of 80-MeV π<sup>þ</sup> mesons. Soviet Physics - JETP. 1963;16:1116-1121

[13] Matsumoto T, Hiyama E, Ogata K, Iseri Y, Kamimura M, Chiba S, et al. Continuum-discretized coupledchannels method for four-body nuclear

Review C. 2004;0616061(R):70

[14] Austern N, Iseri Y, Kamimura M, Kawai M, Rawitscher G, Yahiro M. Continuum-discretized coupledchannels calculations for three-body models of deuteron-nucleus reactions. Physics Reports. 1987;154:125-204

[15] Nunes FM, Thompson IJ. Nuclear

breakup. Physical Review C. 1998;57:

[16] Tostevin JA, Nunes FM, Thompson

[17] Majeed FA, Abdul-Hussien YA. Semiclassical treatment of fusion and breakup processes of 6,8He halo nuclei. Journal of Theoretical and Applied

[18] Majeed FA, Hamodi RSh, Hussian FM. Journal of Computational and Theoretical

Nanoscience. 2017;14:2242-2247

B breakup. Physical

IJ. Calculations of three-body

Review C. 2001;63:024617

Physics. 2016;107:1-6

interference effects in <sup>8</sup>

R2818-R2820

observable in <sup>8</sup>

He+12C scattering. Physical

B sub-Coulomb

Review C. 2014;90:034612

C. 1994;49:2538-2548

breakup in <sup>6</sup>
