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May 26, 2015 - Abstract. We study the role of the projectile breakup in the fusion process by example of the 6Li reactions with the 59Co, 144Sm and 209Bi ...
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Eur. Phys. J. A (2015) 51: 62

DOI 10.1140/epja/i2015-15062-7

6

Li breakup and suppression of complete fusion above the Coulomb barrier N.A. Elmahdy, A.S. Denikin, M. Ismail and A.Y. Ellithi

Eur. Phys. J. A (2015) 51: 62 DOI 10.1140/epja/i2015-15062-7

THE EUROPEAN PHYSICAL JOURNAL A

Regular Article – Theoretical Physics

6

Li breakup and suppression of complete fusion above the Coulomb barrier N.A. Elmahdy1 , A.S. Denikin2,3,a , M. Ismail4 , and A.Y. Ellithi4 1 2 3 4

Modern Academy for Engineering and Technology, Cairo, Egypt Dubna International University, Dubna, Russia Flerov Laboratory of Nuclear Reactions, JINR, Dubna, Russia Cairo University, Cairo, Egypt Received: 17 December 2014 / Revised: 18 March 2015 c Societ` Published online: 26 May 2015 –  a Italiana di Fisica / Springer-Verlag 2015 Communicated by F. Gulminelli Abstract. We study the role of the projectile breakup in the fusion process by example of the 6 Li reactions with the 59 Co, 144 Sm and 209 Bi targets in vicinity of the Coulomb barrier. The coupled channel and distorted wave approaches are employed in order to calculate the complete fusion and the breakup cross sections, respectively. The partial cross sections in both the channels are compared in order to estimate the breakup fraction responsible for the suppression of complete fusion. The calculations are compared with available experimental data. The conclusions and recommendations are made.

1 Introduction Numerous experimental and theoretical studies have been devoted to the analysis of the breakup processes and their influence on fusion reactions involving light and weakly bound nuclei having the pronounced cluster structure [1– 10]. The fusion reactions induced by weakly bound projectiles at the near-barrier energies demonstrate the strong coupling with the collective inelastic excitations of the target nucleus [11–17] as well as with the neutron rearrangement channels [18–20], that may result in the significant enhancement of the fusion probability at deep subbarrier energies. Coupling to the breakup channels of the weakly bound projectiles plays a noticeable role at the energies near and above the Coulomb barrier [21–28] where the nuclear interaction contributes to the breakup more and more. In particular, it was shown [29, 9, 30] that the theoretical fusion cross sections in the case of light projectiles (like the 6 Li ones) overestimate the experimental data above the barrier up to 20–40% of their values. Corresponding example are shown in fig. 1. It shows good agreement between the predicted cross section (solid line) and the data on the 16 O + 154 Sm reaction (solid circles) at all the collision energies. On the other hand, the experimental data on the 6 Li + 144 Sm complete fusion (open squares) show the significant reduction (up to 40%) of the cross section at over barrier energies in comparison with the calculated one (dashed line). It is attributed usually to a

e-mail: [email protected]

Fig. 1. The complete fusion cross sections for the reactions 6 Li + 144 Sm and 16 O + 154 Sm are shown in dependence on the center-of-mass energy in ratio to the Coulomb barrier height VC . The experimental data [31, 32] are shown by the dots. Curves represents the coupled channel model calculations [16]. The discrepancy between theoretical prediction and experimental data is shown at the over barrier energies in the case of light projectile reaction.

the influence of the projectile breakup channels, although no direct numerical estimations were made.

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Eur. Phys. J. A (2015) 51: 62

Presently there are no theoretical approaches allowing one to take into account the collective excitations of colliding nuclei along with the rearrangement channels like the transfer and breakup ones. The fusion cross sections can be calculated quite reliably within the quantum coupled channel (CC) approaches taking into account the nuclear collective excitation only. The breakup processes are described usually within either continuum discretized coupled channel (CDCC) method [33–35] or the distorted wave Born approximation (DWBA) [36–39]. In the present work we perform the coupled channel calculations of the complete fusion cross section and the DWBA analysis of the breakup of light projectile at near barrier energies for the 6 Li induced reactions with 59 Co, 144 Sm, and 209 Bi targets. The goal of this work is to estimate the breakup fraction in the complete fusion cross section in order to answer the question: may the breakup channels alone be responsible to the fusion suppression at the over barrier energies?

2 Theoretical models 2.1 Complete fusion The complete fusion of the atomic nuclei in vicinity of the Coulomb barrier is interpreted well within the coupled channel approach [12, 14]. The fusion cross section is usually decomposed over partial waves, 2

σfus (E) =

π¯ h 2µE

∞ 

(2l + 1)Tl (E),

(1)

l=0

where µ is the reduced mass of the system, E is the centreof-mass energy. The coefficient Tl contains all the information about the fusion dynamics and defines the quantummechanical probability to penetrate through the potential barrier. Calculation of the Tl values is the complicated mathematical problem since the relative nuclear motion is accompanied by the excitation of many internal degrees of freedom. Within the most developed CC models the coupling to the collective vibration and/or rotational modes is considered. The details of the CC method in application to the complete fusion may be found in refs. [12, 14]. Here we just mention that the calculation of the complete fusion cross sections requires the information on the properties of the lowest excited states, i.e. excitation energy ελ , transition multipolarity λ and corresponding zero-point amplitude β0,λ for the spherical nuclei or the deformation parameters βλ in the case of deformed ones. The total wave function is decomposed over the complete set of the basis functions describing either rotational or vibrational states. Obtained set of coupled Schr¨ odinger equations with proper boundary conditions are solved by an appropriate numerical method [14]. The coefficients Tl are defined by the ratio of the flux passed over the potential barriers in all reaction channels to the incoming flux. Thus the probability to fuse is considered as probability for nuclei to come into touch, supposing the probability of the compound nuclei to survive is unit.

Table 1. Nuclear properties and Woods-Saxon potential parameters for coupled channel calculations. The reduced Coulomb radius rC = 1.1 fm is used for all the reactions. Target

λπ

ελ

β0,λ

(MeV) 59 144

V0

rV

aV

(MeV)

(fm)

(fm)

Co

2+

1.10

0.14

100.0

1.12

0.70

Sm

2+

1.66

0.09

100.0

1.10

0.60

100.0

1.12

0.70

209

Bi

3+

1.81

0.15

3−

2.62

0.15

5

3.09

0.11



In our work we employ the computer code available on the web site of the NRV low-energy nuclear knowledge base [16, 14] in order to describe the complete fusion of the 6 Li projectile with the 59 Co, 144 Sm, and 209 Bi targets in vicinity to the Coulomb barrier. The parameters of the collective excitation of the targets as well as the potential parameters are listed in table 1. In the case of the 59 Co target the parameters were defined following to the recommendations of ref. [25]. The calculated complete fusion cross sections σfus are shown in fig. 2 by the short-dashed curves in comparison with the experimental data [40, 32, 41]. 2.2 Projectile breakup In the present work we applied the standard threebody DWBA [36–39] to the analysis of the 6 Li projectile breakup in the collisions with the 59 Co, 144 Sm and 209 Bi targets. The calculations were performed assuming the (α + d) cluster structure for the 6 Li nucleus. The differential cross section for break-up is [37] d3 σDW µ = (2π)−5 ρphase |Tbu |2 , dEd dΩd dΩα k0

(2)

where k0 are the projectile wave number, Tbu is the breakup amplitude, the phase space factor ρ defines the density of the final states. In the case of three-body exit channel the phase space factor is defined by (see ref. [37] for details) ρphase =

m A mα m d p α p d , ¯ 6 (mA + md + md (pα − Ptot )pd /p2d ) h

(3)

where mA is the target mass, mα,d are the masses of projectile clusters while the pα,d are the corresponding momenta, Ptot is the total momentum. The integration of eq. (2) over the available phase space provides the total breakup cross section σDW . The prior-form of the DWBA breakup amplitude [36, 38] reads       (−) (−) Tbu = ψp′ (R) φk (r) VR,r φg.s. (r) ψp(+) (R) , (4)

where r is the α + d relative distance, the wave functions (±) φg.s. (r) and φk (r) describe the ground and excited states

Eur. Phys. J. A (2015) 51: 62

Page 3 of 6 Table 2. Parameters for α + d interaction potential V (r) = V0 exp(−r2 /Rv2 ) + VC (r).

Fig. 2. The fusion σfus and the DWBA breakup σDW cross sections for the reactions 6 Li + 59 Co, 144 Sm, 209 Bi are shown by the short-dashed and long-dashed curves, respectively. The solid curves are the modified fusion cross sections taking into account the influence of the projectile breakup (see the text for details). The arrows show the Coulomb barrier heights. The experimental data [40, 32, 41] are shown by the dots.

of the projectile, R is the distance between the target (±) nucleus and c.m. of the projectile and ψp (R) are the distorted waves characterising the projectile-target relative motion in the entrance and exit channels. The potential V (R, r) defines the interaction of the projectile constituents with the target nucleus. Note that leaving in the V (R, r) only the Coulomb parts one obtains so called Coulomb breakup cross section, while the contribution from the nuclear potentials is called as nuclear breakup.

State

l, ¯ h

V0 , MeV

Rv , fm

RC , fm

Bound

0

75.06

2.236

2.24

Unbound

0

76.12

2.236

2.24

1

79.00

2.132

2.24

2

81.00

2.294

2.24

≥3

76.12

2.236

2.24

The α + d potential proposed in ref. [54] is adapted in order to describe the bound and low-lying continuum states. The corresponding parameters are listed in table 2. It allows, in particular, to reproduce the binding energy and r.m.s. radius for the ground state. Jenny et al. [55] carried out the analysis of α + d elastic scattering, and extracted the corresponding phase shifts. Table 2 contains the potential parameters providing reasonable description of these data at energies Eαd ≤ 15 MeV. In our calculations the 6 Li continuum is truncated at 15 MeV excitation including the (α + d) relative angular momentum l ≤ 4. The calculation of the breakup amplitude (4) requires also the α-target and deuteron-target optical potentials appearing in the V (R, r) interaction, and the 6 Li-target interaction required for the distorted waves calculations. In order to simplify the calculations we adapt the available global optical potentials proposed in refs. [42–44], describing the deuteron-, α- and 6 Li-induced elastic scattering in wide energy domain. Figure 3 shows the optical model cross sections in comparison with the measured data on d, α and 6 Li elastic scattering with different targets at the energies within the domain of our interest. The calculations in fig. 3 approve the validity of the optical potentials. Note also that these optical potentials provide the heights of the Coulomb barrier close to those shown in fig. 2. The 6 Li spectrum does not include any bound excited states. Thus all the projectile excitations are located in the α + d continuum and are treated therefore as the breakup processes. This approximation was already used in order to describe the data on the coincidence experiments, in particular, the deuteron elastic breakup by the different target nuclei. In spite of the failure of the first application [36] it was shown later [38, 39, 56, 57] that the prior-form DWBA provides adequate description of the breakup in specific kinematic conditions. It is due to the cross section (2) includes actually the contributions both from direct breakup σbu and from incomplete fusion σicf when one of projectile’s constituent after breakup is captured by the target. Moreover including the imaginary parts of the cluster-target potentials to the V (R, r) operator one takes implicitly into account the inelastic channels in the cluster-target subsystems. Corresponding cross section is denoted as σinel . Thus the integrated cross section σDW (2) effectively involves the contribution from the different reaction channels.

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Eur. Phys. J. A (2015) 51: 62

Fig. 3. The calculated cross sections as well as the available experimental data on the deuteron (a), α (b) and 6 Li (c) elastic scattering reactions with different targets. The curves show the optical model calculations with the deuteron [42], α [43] and 6 Li [44] global optical potentials. The following data are used: d(20 MeV) + 58 Ni [45], d(14 MeV) + 141 Pr [46], d(22 MeV) + 208 Pb [47], α(25 MeV) + 59 Co [48], α(31 MeV) + 141 Pr [49, 50], α(35 MeV) + 209 Bi [51], 6 Li(30 MeV) + 56 Co [21], 6 Li(32 MeV) + 144 Sm [52], 6 Li(36 MeV) + 209 Bi [53].

The matrix elements (4) have the same form as the coupling matrix elements in the CDCC approach which is widely used in studying the interplay between the breakup and elastic channels. It takes into account first-order processes and does not include the second-order ones like continuum-to-continuum coupling. The latter type of coupling results in the repulsive contribution to the real part of the dynamical polarization potential that leads to the reduction of the breakup cross section to some extent. The comparison of the experimental data on the deuteron and 3 He breakup with the DWBA and CDCC calculations [33] shows that the experimental data are in between the models predictions. The CDCC calculations performed in ref. [22] for the 6,7 Li-induced reactions also confirm these conclusions. Thus, one employs the DWBA method to the reactions considered here keeping in mind that this model somewhat overestimates the breakup cross section. It is necessary to mention another limitation of the approach applied to the description of projectile breakup. At the energies near and below the Coulomb barrier the low partial waves in the projectile-target relative motion start to dominate in the breakup. The exit channel is treated as the unbound two-body projectile, whose center of mass moves in the field of the target nucleus. When the projectile is broken up the motion of its center of mass is treated unrealistically, since projectile’s clusters are unbound and move separately. For higher beam energies this approximation is rather reasonable because of the peripheral character of the breakup, the low excitation energy (slow internal motion), and the high nucleus-nucleus relative velocity. At the same time for low energies the results can be inaccurate. It should be noted that the same limitations are valid for the CDCC analysis since the coupling matrix has the same properties as the breakup DWBA amplitude (4). The projectile breakup cross sections σDW calculated within the aforesaid scheme are shown in fig. 2 by the long-dashed curves. These reactions have been analyzed

experimentally and within the CDCC method in refs. [58, 22, 25, 26, 59]. We remind the reader that the cross section σDW represents the contribution from the direct breakup, from the incomplete fusion and the inelastic processes σDW = σbu + σicf + σinel . The calculation scheme does not allow to separate the contributions σbu and σicf . In particular the 6 Li(25.5 MeV)+ 59 Co reaction the cross section σDW = 703 mb, whereas the breakup and incomplete fusion contributions σbu + σicf = 508 mb. One may compare this value with the data on the α and deuteron yields estimated in ref. [26] for the same reaction excluding the compound nuclei decay contribution. Corresponding cross sections were found to be equal σα = 332 ± 33 mb and σd = 126 ± 15 mb. One may claim that the data are in consistency, since σbu + σicf ≈ σα + σd . The breakup cross sections calculated for the reactions considered here are also reported in ref. [25]. They are apparently less than those shown in fig. 2. There are two reasons for this difference. First, it was already mentioned above that the CDCC approach provides the breakup cross sections reduced in comparison with the DWBA. However, as it was shown early in ref. [33], the reduction is expected to be weaker. The second reason lies in the different projectile-target optical potentials applied here and in ref. [25]. The real part was chosen in ref. [25] in the form of the folding potential while the imaginary part had the constant depths W0 = −50 MeV which does not depend on energy and mass and it is more than twice deeper than the one used here. The optical potential having deep imaginary part leads to the distorted waves suppressed in the interaction domain. Thus such strong absorptive part actually allows to exclude the incomplete fusion contribution and estimate the direct breakup one only. It may explain the difference in the cross sections. Concluding one may expect that the estimations performed here correspond to the upper limit of the projectile breakup cross sections taking into account indirectly additional channels that may follow and accompany the breakup.

Eur. Phys. J. A (2015) 51: 62

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Fig. 4. Partial fusion (solid curves) and breakup (dashed curves) cross sections for the 6 Li-induced reactions with 59 Co, 144 Sm and 209 Bi targets at above-barrier energies. The shadowed area corresponds to the σ efus cross section (see the text). The arrows show the grazing orbital momenta Lg .

3 Discussion and conclusions The comparison of the complete fusion cross sections calculated within the CC approach with the experimental data are shown in fig. 2. It is obvious that the calculated cross sections σfus (E) systematically overestimate the data at over barrier energies for all the considered reactions as it was already mentioned in the introduction. The σDW cross sections demonstrate dependence on the Coulomb charge of the target nucleus. The heavier the target nucleus the larger the breakup cross section at nearbarrier energies. Indeed, there are two factors contributing the breakup. They are the nuclear and the Coulomb forces. The latter reveals itself by the extensive tails in the partial cross sections at large L values which are evidently seen in fig. 4 for the targets with larger Coulomb charges. The nuclear breakup contributions are localized near the partial waves Lg corresponding to the grazing collisions. The competition between the complete fusion and the breakup do not involve the high angular momenta, where the fusion cross section vanishes. Fusion takes place only for central collisions. Therefore, in order to estimate the maximal possible suppression of the complete fusion by the projectile breakup we calculate the difference σ fus =

Lm 

(σfus,L − σDW,L ),

(5)

L=0

where Lm is the maximal orbital momentum at which the difference (σfus,L − σDW,L ) is still positive. This procedure

is illustrated in fig. 4. The shadowed area in the figure represents the modified cross section σ fus . The residual part of the fusion cross section can be treated as the part provided by the projectile breakup which is not taken explicitly into account within the CC approach. In fig. 2 the modified fusion cross sections σ fus calculated according to eq. (5) are shown by the solid curves. One may see the significant improvement of the agreement between the calculated and measured cross sections at over barrier energies. The estimation of the fusion suppression is quite simplified. Nevertheless it provides surprisingly good agreement with the data at over-barrier energies. Remembering that the cross section σDW includes actually the contributions from the different reaction channels one may conclude the following. The projectile breakup strongly affects on the fusion in peripheral collisions. However the breakup itself cannot provide enough suppression of the fusion cross section in this energy domain. Particularly it is necessary to take into account the transfer channels significantly populated at the barrier energies especially in the case of reactions with weakly bound nuclei. It supports the conclusions formulated in ref. [22]. The applied approximation does not take also into account the coupling between inelastic and breakup channels. The second order of perturbation (like the continuum-to-continuum couplings) are not included as well. All these couplings may lead to the interference effects that requires additional studies within a more sophisticated model. One may follow the recommendation given in ref. [33]. The authors concluded that the best way of considering the breakup is the post-form CCBA formalism, in which the initial channel wave function is obtained by the CDCC method, and the transition to the final channels is calculated with a post-form perturbation. This idea has to be revised since it was formulated for relatively high energies. However it may improve the CDCC approach reclaiming shortcomings mentioned above. This work is supported in part by the grant under Arab Republic of Egypt and JINR Agreement and the Russian Foundation for Basic Research (project numbers: 13-07-00714).

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