Effects of the valence neutron of the lithium isotopes in 6,7,8Li+208Pb systems

Using the measured experimental data for 6,7,8Li+208Pb system, we investigated the relation between the interaction distance and the lowest threshold separation energy of the projectiles. Although 6Li can be considered more weakly bound projectile than 8Li, the lithium isotope with the largest interaction distance is not 6Li but 8Li. Also, we can infer that the difference of the semiexperimental direct reaction (DR) cross sections and the total breakup / transfer reaction cross sections comes from α and 7Li capture. Finally, the barrier height, VB, of 8Li + 208Pb system compared with 7Li + 208Pb system are reduced about 3MeV. This is the rise of flux from the elastic into the reaction channel, which tends to enhance σexp R . As a result, we can deduce that the valence neutron play a role as the lowering of the barrier height.


Introduction
In our recent study [1], we carried out analyses based on the extended optical model [2,3], in which the optical potential consists of the energy independent Hartree-Fock part and the energy dependent complex polarization potential for the 6 Li + 208 Pb and 7 Li + 208 Pb systems.In Ref. [1], using such an extended optical potential, we performed the simultaneous χ 2 analyses of the elastic scattering and fusion cross section data, determining the two components of the polarization potentials as functions of the incident energy.
In the present work, we will investigate the characteristic features of the heavy ion reaction for the lithium isotopes including 8 Li.The unstable radioactive nucleus 8 Li is a loosely bound nuclei, with a threshold separation energy S n =2.033MeV ( 8 Li → 7 Li+n).The threshold separation energy of 8 Li is between those for 6 Li [S α =1.475MeV ( 6 Li → α+d)] and 7 Li [S α =2.468MeV ( 7 Li → α+t)].This primary breakup channels, 8 Li → 7 Li+n, includes 1n-transfer and breakup processes producing 7 Li nuclei and one neutron.It is similar to the breakup process for neutron rich nuclei such as 6 He and 11 Be.From consecutive breakup process of 7 Li nuclei ( 7 Li → α+t), however, an additional α-particle yield can be observed [4].At near and below Coulomb-barrier energies, therefore, the large transfer and breakup cross sections for 8 Li+ 208 Pb system have been measured in Ref. [4].As shown in Table .1, indeed, the total reaction cross section is dominated by the DR cross section including the transfer and the breakup cross sections.According to the dominant DR cross section, therefore, we can expect that the complete fusion (CF) suppression takes place at above Coulomb-barrier [5].In order to a e-mail: wyso@kangwon.ac.kr investigate this suppression, we study precisely the effect of the breakup on the fusion for 8 Li+ 208 Pb system.
In Sec. 2, we will first explain the so-called semiexperimental total reaction cross sections, σ semi-exp R , from the elastic scattering, and then will obtain the semiexperimental DR cross sections, σ semi-exp D from Eq. (2) using the fusion cross section data.In Sec. 3, we will analyses the experimental cross section data for lithium isotopes.In addition, we discuss the relation between the interaction distance and the lowest threshold separation energy of the projectiles and the distinction of the total reaction cross section for lithium isotopes.Also, we will examine the role of the valence neutron(s) on the the barrier height.Section 4 concludes the paper.

Extracting semiexperimental DR and total reaction cross sections
To carry out the present study, we have first to extract the semiexperimental total reaction cross sections, σ semi-exp which can be extracted from the measured elastic scattering cross sections, dσ E /dΩ, in heavy-ion collisions [6, 7].
Here σ C and σ E are the angle-integrated total Rutherford and elastic scattering cross sections, respectively.Note that the semiexperimental total reaction cross sections are defined as the sum of all the non elastic integrated cross sections including the inelastic cross sections.Equation ( 1) is based on the optical theorem for charged particles [6], which reads σ semi-exp The measured complete fusion cross sections and the extracted DR and total reaction cross sections for 6,7,8 Li + 208 Pb systems are listed in Table 1.For the 6 Li + 208 Pb system, since the cross sections σ exp F obtained from Ref. [8]  fluctuate with respect to the incident energies, we extracted their experimental cross sections using Wong's formula, as the measured fusion cross sections.Here, V B , ω, and R B are the Coulomb barrier energy, the barrier curvature, and the interaction radius, respectively [10].Table 2 shows these parameters for the complete fusion cross sections of lithium isotopes incident on 208 Pb and 209 Bi. Figure 1 shows the complete fusion are shown as functions of (a) E c.m. and (b) E c.m. −V B for 6,8,9 Li + 208 Pb and 6,7 Li + 209 Bi systems.We can see that the complete fusion data are nearly the same shape.As seen in Fig. 1 (b),

Analyses of the experimental cross section data for lithium isotopes
In this paper, one of the characteristic features for the experimental cross section data is the relation between the interaction distance and the lowest threshold separation energy of the projectiles.In order to discuss this relation, we define the ratio, P E , given as  are the elastic and Coulomb scattering cross section, respectively [1].Generally, the collision between the heavy nuclei near coulomb barrier energy, V B , is governed by the Coulomb and nuclear forces.In the asymptotic region, where the nuclear forces do not reach, the projectile nuclei mainly move along the classical Coulomb scattering trajectory.Thus, P E remains close to unity until two ions approach each other within interaction distance, d I .The interaction distance is determined as the place at which the ratio P E drops to 0.99.However, if the two nuclei come close on feeling sufficient nuclear force, the projectile particles due to the strong attractive nuclear force transfer direct reaction such as neutron transfer reaction, breakup reaction, and incomplete fusion, or complete fusion rather than the elastic scattering moving along the Coulomb scattering trajectory.Within d I , as a result, P E begins to fall off.Actually, the experimental values of P E as a function of the reduced distance d for 6,7,8 Li+ 208 Pb systems are plotted on Fig. 2 and the resultant value of the interaction distance and the lowest α-particle or neutron threshold separation energy, S i (i = n, or p, or α) are listed in table 3. From the point of view of the threshold separation energy, as seen in table 3, we can  2).Also shown is the total transfer/breakup cross section (diamonds) as well as the 7 Li production cross sections (stars) for 8 Li+ 208 Pb system.
consider that 6 Li is the most characteristic weakly bound projectile, 8 Li is the next, 7 Li is the least among these three under consideration.Thus, we can predict that 6 Li has the larger interaction distance than 7,8 Li because it is more easily broken by the nuclear forces.As seen in Table .3, however, lithium isotope with the largest interaction distance is not 6 Li but 8 Li.Although the threshold separation energy is lower, thus, the interaction distance is not always far.Another characteristic feature of the experimental data for lithium isotopes can be seen the distinction of the total reaction cross section including the fusion and DR cross sections.We found that the complete fusion cross sections are essentially independent for the considered lithium isotopes in Sec. 2. However, it is found that the DR and total reaction cross sections for lithium isotopes are related considerably to the projectiles from Fig. 3.As seen in Fig. 3 (a), the DR cross sections of 6 Li and 7 Li are slightly different.Thus, the total reaction cross sections of them are also not significantly different.But, the DR and total reaction cross sections of 6,7 Li and 8 Li are considerably distinct.
In order to explain the difference of the DR and total reaction cross sections for 8 Li with respect to 6,7 Li, we will investigate to the contribution for the constituent of the DR cross sections and the Coulomb barrier for the total reaction cross sections.Figure 3 (a) shows the total breakup / transfer cross sections as well as the 7 Li production cross sections.There are two processes which product 7 Li cluster in 8 Li+ 208 Pb reaction.one is the elastic breakup (EBU) process which any particles are not captured by the target.One 7 Li par-16015-p.3ticles are measured at the EBU process, 8 Li → 7 Li + n.Another is the 1n-transfer process which a neutron is captured by the target after EBU process, so that one 7 Li particles are also measured.Thus, the 7 Li production cross sections denoted as the diamonds mean the sum of the elastic breakup cross (EBU) sections and 1n-transfer cross sections, σ breakup + σ 1n-transfer = σ 7 Li−production and the stars in Fig. 3 indicate the sum of the 7 Li and α production cross sections.In the present experiment, 7 Li, α, and tritium resulting from 8 Li breakup can be generated.
According to the Ref. [5], most of tritium seem likely to fuse into the target in the present incident energy region and 7 Li are possible to capture into the target in the upper half of the experimental region.On the other hand, most of α seem unlikely to fuse into the target in the present incident energy region.Therefore, we can infer that the difference of the semiexperimental DR cross sections and the total breakup / transfer reaction cross sections in Fig. 3 (a) comes from α and 7 Li capture.
Figure 3 (b) shows the semiexperimental total reaction cross sections for 6,7,8 Li + 208 Pb systems.Using Eq. ( 3), as done in Sec. 2, we extracted the barrier parameter, V B , with respect to the semiexperimental total reaction cross sections.One of the most interesting things in Table .4 is about 3MeV reduction of V B for 8 Li + 208 Pb system compared with 7 Li + 208 Pb system and about 4MeV reduction of V B for the semiexperimental total reaction cross sections compared with that of the complete fusion cross sections for 8 Li + 208 Pb system.This is somewhat similar to 4,6 He + 209 Bi systems reported in Ref. [13].Actually, the reported in Ref. [13] reduction of V B for 6 He + 209 Bi system compared with 4 He + 209 Bi system is about 6MeV.One special feature is that 6 He or 8 Li consists of α particle and two valence neutrons or 7 Li and one valence neutrons, respectively.If the projectile and target nuclei come close on feeling sufficient nuclear force, the valence neutron(s) of the projectile particles react faster than the core nuclei on target nuclei.As a result, we can consider that the reaction processes of 6 He and 8 Li compared with those of 4 He and 7 Li take place from farther.This is the rise of flux from the elastic into the reaction channel, which increases σ exp R and also corresponds to the lowering of the barrier height, which tends to enhance σ exp R .Thus, we can deduce that the valence neutron(s) play a role as the lowering of the barrier height, V B .

Conclusions
We investigated the relation between the interaction distance and the lowest threshold separation energy of the projectiles.From the point of view of the threshold separation energy, 6 Li can be considered more weakly bound projectile than 8 Li.But the lithium isotope with the largest interaction distance is not 6 Li but 8 Li.Although the threshold separation energy is lower, as a result, the interaction distance is not always far.
In the present experiment, 7 Li, α, and tritium resulting from 8 Li breakup can be generated.We can infer that the difference of the semiexperimental DR cross sections and the total breakup / transfer reaction cross sections comes from α and 7 Li capture.
The Coulomb barrier for 8 Li + 208 Pb system compared with 7 Li + 208 Pb system are reduced about 3MeV.This is the rise of flux from the elastic into the reaction channel, which tends to enhance σ exp R .As a result, we can deduce that the valence neutron(s) play a role as the lowering of the barrier height.

2 FUSION11
) as a function of the distance of the closest approach D (or the reduced distance d), where dσ el /dσ Ω and dσ C /dσ Ω 16015-p.

Fig. 3 .
Fig. 3. (Color online) The (a) DR and (b) total reaction cross section, σ D and σ R , are shown as functions of E c.m. for 6,7,8 Li+ 208 Pb systems.These values are taken from Eq. (2).Also shown is the total transfer/breakup cross section (diamonds) as well as the7 Li production cross sections (stars) for 8 Li+ 208 Pb system.

Table 1 .
The measured complete fusion cross sections and the ex- † These values are extrapolated from Eq. (3).∆σ is given in terms of the nuclear part of the forward scattering amplitude, f N (θ) =0, as ∆σ = 4π/k Im [ f N (θ)].For heavy ion collisions, ∆σ is generally small, justifying the use of Eq. (4) for generating σ semi-exp R .Once σ semi-exp R is extracted, and if the experimental fusion cross section, σ exp F , is available, one can further generate a semiexperimental DR cross section, σ semi-exp D , as σ semi-exp

Table 2 .
The Coulomb barrier parameters for the complete fusion cross sections of lithium isotopes incident on208Pb and 209 Bi.The Coulomb barrier parameters for the complete fusion for are extracted by using Wong's formula, Eq. (3).
7articulary, such feature can best be seen as functions of E c.m. − V B .At high energy E c.m. ≈ 37.7MeV, however, the values of σ exp F for6Li + 208 Pb system deviate from those of σ exp F for7Li + 208 Pb system by about 30%.

Table 4 .
The same as in Table2, but for the total reaction cross sections.