Open access peer-reviewed chapter

Fusion Reaction of Weakly Bound Nuclei

Written By

Fouad A. Majeed, Yousif A. Abdul-Hussien and Fatima M. Hussian

Submitted: June 9th, 2018 Reviewed: July 29th, 2018 Published: January 26th, 2019

DOI: 10.5772/intechopen.80582

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Abstract

Semiclassical and full quantum mechanical approaches are used to study the effect of channel coupling on the calculations of the total fusion reaction cross section σfus and the fusion barrier distribution Dfus for the systems 6Li + 64Ni, 11B + 159Tb, and 12C + 9Be. The semiclassical approach used in the present work is based on the method of the Alder and Winther for Coulomb excitation. Full quantum coupled-channel calculations are carried out using CCFULL code with all order coupling in comparison with our semiclassical approach. The semiclassical calculations agree remarkably with the full quantum mechanical calculations. The results obtained from our semiclassical calculations are compared with the available experimental data and with full quantum coupled-channel calculations. The comparison with the experimental data shows that the full quantum coupled channels are better than semiclassical approach in the calculations of the total fusion cross section σfus and the fusion barrier distribution Dfus.

Keywords

  • fusion reaction
  • breakup channel
  • weakly bound nuclei
  • fusion barrier

1. Introduction

In recent years, big theoretical and experimental efforts had been dedicated to expertise the effect of breakup of weakly bound nuclei on fusion cross sections [1, 2]. This subject attracts special interests for researchers and scholars, because the fusion of very weakly bound nuclei and exotic radioactive nuclei is reactions that have special interests in astrophysics which play a very vital role in formation of superheavy isotopes for future applications [3, 4, 5, 6, 7, 8]. Since the breakup is very important to be considered in the fusion reaction of weakly bound nuclei, the following should be considered: the elastic breakup (EBU) in which neither of the fragments is captured by the target; incomplete fusion reaction (ICF), which happens when one of the fragments, is captured by the target; and complete fusion following BU (CFBU), which happens in all breakup fragments that are captured by the target, is called the sequential complete fusion (SCF). Therefore, the total breakup cross section is the sum of three contributions: EBU, ICF, and CFBU, whereas the sum of complete fusion (including two body fusions and CFBU) and incomplete fusion is called total fusion (TF) [1, 8, 9, 10]. Fusion reactions with high-intensity stable beams which have a significant breakup probability are good references for testing the models of breakup and fusion currently being developed. The light nuclei such as 6Li breakup into 4He+2H, with separation energy Sα = 1.48 MeV; 11B breakup into 4He+7Li with separation energy Sα = 8.664 MeV and 12C breakup into three α particles induced by neutrons or protons by 12C (p, p′) 3α [3, 11, 12]. The breakup 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 systems 6Li + 64Ni, 11B + 159Tb, and 12C + 9Be.

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2. The semiclassical theory

2.1 The single-channel description

The semiclassical theory is used to estimate the fusion cross section in the one-dimensional 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:

22/2μ+VrEΨr=0E1

where μ is the reduced mass of the system and Vr 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

Vr=VCr+VNr+VlrE2

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

VNr=UNriWrE3

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 using WKB approximation [5, 6, 19, 23, 24]:

PfusWKBlE=11+exp2rblralκlrdrE4

Then, the latter can be rewritten as follows:

PfusWKBlE=11+exp2πΩlVblEE5

where κlr is the local wave number and rbl and ral 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]:

PfusWHlE=11+exp2πΩlEVblE6

where Vbl and Ωl 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 Vbl 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]:

σfusE=πκ22l+1PfusWKBlEE7
PfusγlE=4kEdruγlkγr2WfusγrE8

where uγlkγr represents the radial wave function for the partial wave l in channel γ and Wfusγr 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=h0ξ+VξrE9

where h0ξ is the intrinsic Hamiltonian and Vξr=VNξr+VCξr, Vξr 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 Vr=Ψ0VrξΨ0, where Ψ0 is the ground state (g.s.) of the projectile. In this way, the interaction becomes time-dependent in the ξ-space Vlξ.t=Vrltξ, and the eigenstates of the intrinsic Hamiltonian Ψγ satisfy the Schrödinger equation [25, 26]:

hΨγ=εΨγE10

After expanding the wave function in the basis of intrinsic eigenstates

Ψξt=aγltΨγξeiεγt/E11

and inserting Eq. (11) into the Schrödinger equation for Ψξt, the AW equations can be obtained:

iȧγlt=ϵΨγVξtΨγeiεγεϵtγϵltE12

These equations should be solved with initial conditions aγlt=δγ0 which mean that before the collision t, the projectile was in its ground state. The final population of the channel γ in a collision with angular momentum l is PfusγlE=aγlt2. 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=H0ξ+VrξE13

where H0ξ is the intrinsic Hamiltonian of the projectile and V(r,ξ) 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 H0ξ are given by the equation [27]

H0φβ=εβφβE14

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,ℏl. In its original version, an energy symmetrized Rutherford trajectory rt was used. In our case, the trajectory is the solution of the classical equations of motion with the potential V(r)=φ0|V(r,ξ)φ0, where φ0 is the ground state of the projectile. In this way, the coupling interaction becomes a time-dependent interaction in the ξ-space, VξtV(rt,ξ). 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]

ψξt=βaβtφβξeiεβt/E15

and inserting this expansion into the Schrödinger equation for ψξt, one obtains the AW equations [27]

iȧβt=αaγtφβVξtφβeiεβεγt/E16

These equations are solved with the initial conditions aβt=δβ0, which means that before the collision t, the projectile was in its ground state. The final population of channel β in a collision with angular momentum is Pβ=aβt+2 and the angle-integrated cross section is [19]

σβ=πk22+1PβE17

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]

σF=βπk22+1PFβE18

with

PFβ=4kEWβFruβkβr2drE19

Above, uβkβr represents the radial wave function for the th partial wave in channel β, and WβF 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]

PFβP¯βTβEβE20

where P¯β is the probability that the system is in channel β at the point of closest approach on the classical trajectory and TβEβ is the probability that a particle with energy Eβ=Eεβ and reduced mass μ=MPMT/MP+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, F1 and F2. In this way, the labels β=0 and β0 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 β0. That is [19],

σCF=πk22+1PSurvT0EE21

where

PSurvP¯0=a0tca2E22

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

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3. Fusion barrier distribution

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 densities 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]

DfusE=d2FEdE2E23

where FE is related to the total fusion reaction cross section through [29]

FE=EσfusEE24

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 well as numerical uncertainties [29, 30, 31]:

DfusEFE+E+FEE2FEE2E25

where E 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]

δDfusstatEδFE+E2+δFEE2+4δFE2E2E26

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

δDfusstatE6δFEE2E27
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4. Results and discussion

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 6Li + 64Ni, 11B + 159Tb and 12C + 9Be. The values of the height Vc, radius Rc, and curvature ω for the fusion barrier are displayed in Table 1.

SystemVcRcωRefs
6Li + 64Ni12.419.13.9[32]
11B + 159Tb40.3410.894.42[33]
12C + 9Be4.287.432.61[34]

Table 1.

The fusion barrier parameters are height VcMeV, radius Rc fm, and curvature ωMeV.

4.1 The reaction 6Li + 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 6Li + 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).

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 6Li + 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).

The experimental data for this system are obtained from Ref. [32]. The real and imaginary Akyüz-Winther potential parameters obtained by using chi-square method are the strength W0=50MeV, radius ri=1.0fm, and diffuseness ai=0.25fm, and for the real part, the depth is V0=35.0MeV, radius is r0=1.1fm, and diffuseness is a0=0.8fm. The χ2 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 χ2 values obtained for the case of coupling effects included are 0.2431 and 0.3115 for semiclassical and quantum mechanical calculations, respectively. The χ2 values show clearly that semiclassical calculations including coupling effects are more consistent with the experimental data than full quantum mechanical including coupling effects. The χ2 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 χ2 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.

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=126.1MeV, r0=1.2fm,anda0=0.5fm and W0=55.9MeV, ri=0.986fm, and ai=0.614fm. The χ2 values 0.9473 and 0.2486 are obtained for σfus using semiclassical and quantum mechanical distribution calculations without including the coupling, respectively, while the χ2 for σfus using the semiclassical and quantum mechanical distribution calculations including the coupling effects are 0.2681 and 0.1657, respectively. The χ2 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 χ2 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.

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).

4.3 The reaction 12C + 9Be

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 + 9Be. 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: V0=40.3MeV, r0=1.11fm,a0=0.590fm, W0=0MeV, ri=1.1fm, and ai=0.50fm. The χ2 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 χ2 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.

Figure 3.

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 + 9Be system. Panel (a) shows the total fusion reaction cross section σfus (mb), and panel (b) gives the fusion reaction barrier distribution Dfus (mb/MeV).

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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 6Li + 64Ni, 11B + 159Tb, and 12C + 9Be. 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.

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Acknowledgments

The author FA Majeed gratefully acknowledges financial assistance from Conselho Nacional de Desenvolvimento Científico e Tecnolόgico (CNPq) (Brazil) and is especially indebted to the World Academy of Sciences for the advancement of science in developing countries (TWAS) (Italy) for a 1-year grant under the scheme (TWAS-CNPq exchange programs for postdoctoral researchers).

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Written By

Fouad A. Majeed, Yousif A. Abdul-Hussien and Fatima M. Hussian

Submitted: June 9th, 2018 Reviewed: July 29th, 2018 Published: January 26th, 2019