DIFFUSENESS OF WOODS SAXON POTENTIAL AND SUB-BARRIER FUSION

Modern Physics Letters A Vol. 26, No. 28 (20) 229 234 c World Scientific Publishing Company DOI: 0.42/S0277303654 DIFFUSENESS OF WOODS SAXON POTENTIAL AND SUB-BARRIER FUSION MANJEET SINGH, SUKHVINDER S. DUHAN and RAJESH KHARAB Department of Physics, Kurukshetra University, Kurukshetra 369, India kharabrajesh@rediffmail.com Received 8 July 20 We have proposed an energy-dependent parametrization scheme for determining the diffuseness parameter of Woods Saxon potential which when used in conjunction with the coupled channel code CCFULL explains very well the fusion excitation function data around the barrier for various systems. Keywords: Diffuseness; heavy-ion sub-barrier fusion; coupled channel analysis. PACS Nos.: 25.60.Pj, 2.60.Ev, 24.0.Eq In the analysis of any nuclear reaction data the shape of nucleus nucleus potential, which consists of Coulomb repulsive interaction, centrifugal term and the attractive short range nuclear potential, is one of the most important input. The Coulomb and centrifugal terms are well understood whereas there are large ambiguities in the nuclear potential which is generally parametrized by Woods Saxon form. As the elastic and inelastic scattering is described by the tail region of Coulomb barrier so the elastic and inelastic scattering are sensitive mainly to the surface region of the nuclear potential and these processes basically provide the relevant information about the surface properties of nuclear potential. On the other hand, the nuclear fusion reactions are highly sensitive to the inner region of nuclear potential and the relevant information about fusion dynamics depends upon both sides of barrier position. Therefore the study of elastic and inelastic scattering and the fusion process provides the complementary information about the nuclear potential. 3 In the Woods Saxon parametrization of nuclear potential, the diffuseness parameter is one ofthe important parameterwhich defines the slope ofthe nuclear potential in the tail region of Coulomb barrier, where fusion starts to take place. A surface diffuseness a = 0.65 fm of Woods Saxon nuclear potential has been deduced from the elastic scattering data. In the last decade, it has been recognized that large value of Corresponding author 229

230 M. Singh, S. S. Duhan & R. Kharab diffuseness parameter is required in order to reproduce the sub-barrier fusion excitation function. The range of diffuseness parameter a = 0.75 fm to a =.5 fm, which is much larger than the expected value a = 0.65 fm deduced from the elastic scattering data, is required to reproduce the systematic of sub-barrier fusion cross-section. 4 0 The correct underlying reasons for this diffuseness anomaly are still not clear. For sub-barrier fusion reactions, the barrier curvature is found to be roughly inversely proportional to the square root of the diffuseness parameter,5,0 so the change in the diffuseness parameter ultimately produces the variation in the tunneling region between the colliding nuclei. The Woods Saxon nuclear potential which fits the elastic scattering data overpredicts the fusion cross-section in below and above barrier energies. Thus the inconsistency of Woods Saxon nuclear potential to fit simultaneously the elastic scattering data and fusion data is one of the most challenging problems in heavy ion reactions. Recently, it was realized that the energy dependence of the heavy ion bare potential is to be properly taken into account. 2 The energy dependence of the potential is also shown in the double folding model wherein it originates from the energy dependence of underlying nucleon nucleon interaction and from nonlocal quantum effects. Thus the energy dependence, which may mocks up some dynamical effects which are important in the analysis of the sub-barrier fusion process, may provide better description of these processes. In Refs. 2, 4 and 7 the energy dependence of the nuclear potential is included through the energy dependence of its radius. Here we take into account the energy dependence of the potential through the diffuseness parameter. We propose an energy-dependent form of the diffuseness parameter such that when the resulting Woods Saxon parametrization for nuclear potential is used in the coupled channel code CCFULL, the sub-barrier fusion excitation function data for various systems could be reproduced reasonably well. The computer code CCFULL is a highly versatile code developed by Hagino et al., wherein the coupled differential equations are solved by using the modified Numerov method. This code employs the isocentrifugal approximation where the angular momentum in a particular channel is replaced by the total angular momentum and the angular momentum in the entrance channel is equal to that of exit channel so as to reduce the number of coupled channel equations. The nuclear coupling matrix elements appearing in the coupled channel equations are evaluated through the matrix diagonalization method. The fusion cross-section and the mean angular momentum of the compound nucleus at different incident energies are the output of the program CCFULL. The surface diffuseness parameter a of nuclear potential which is to be given as an input to CCFULL is of immense importance in connection with diffuseness anomaly. As mentioned earlier, a wide range (a = 0.75 fm to a =.5 fm) of values of diffuseness parameter has been used to explain the sub-barrier fusion data in the literature. The need for large diffuseness may be attributed to the fact that larger diffuseness leads to smaller barrier position and smaller barrier curvature which in turn results in large tunneling region. Sometimes the following parametrization

Diffuseness of Woods Saxon Potential and Sub-Barrier Fusion 23 schemes, wherein the projectile and target mass number dependence is taken into account,duetowinther 2,3 andakyüz Winther 4,5 areusedtoevaluatethevalue of diffuseness parameter. a AW =.7 [ +0.53(A /3 a AKZ W =.6 [ +0.48(A /3 P +A /3 T P +A /3 T ) ] fm, ) ] fm. However, by using the so obtained values of diffuseness parameter, the experimental fusion excitation functions of various heavy ion fusion reactions could not be reproduced. Thus we have proposed the following energy-dependent parametrization scheme for evaluating the diffuseness parameter and use it in the calculations, performed with code CCFULL, of fusion excitation functions. a(e) = 0.85 + r 0 3.75(A /3 P +A /3 T ) ( +exp ( E V B 0.96 0.03 )) fm. In order to obtain this expression firstly we have reproduced the fusion excitation function data by varying diffuseness parameter of Woods Saxon nuclear potential for wide range of projectile and target combinations and have found that different values of diffuseness are needed to fit the data in different energy regions. Then we have fitted the diffuseness parameter as a function of energy through the sigmoidal fitting which leads to the above expression. In addition, the fact that the process becomes sharp with increasing energy was an intuitive guide to incorporate the energy dependence. In order to understand the behavior of sub-barrier fusion excitation function with respect to coupling to low lying vibrational states it is useful to consider the fusion of nuclei having closed shell configurations and near closed shell configurations. Here we have considered the fusion of,36 6 S+ 90,96 40 Zr systems in near barrier energy regions. The 36 6S nucleus consists of 20 neutrons, which is a neutron magic number, whereas the 6S nucleus has the nuclear configuration which is close to magic nuclei. The 90 40Zr nucleus is doubly magic nucleus which consists of both proton and neutron magic number, whereas the 96 40Zr nucleus lies close to doubly magic configuration. For these Sulfur and Zirconium isotopes, it is sufficient to consider only low lying surface vibrational states about their equilibrium shape. The values of the deformation parameter (β λ ) and the excitation energies (E λ ) corresponding to quadrupole (λ = 2) and octupole (λ = 3) vibrational states for these isotopes, which are needed as inputs to the code CCFULL, are listed in Table. The other important ingredients needed in the calculations are the depth (V 0 ), range (r 0 ) and the heights of Coulomb barrier for different projectile target combination. The potential depth is fixed at V 0 = 00 MeV and r 0 =. fm throughout while the Coulomb barrier heights for different systems are listed in Table 2.

2 M. Singh, S. S. Duhan & R. Kharab Table. The deformation parameter (β λ ) and the excitation energy (E λ ) of the quadrupole and octupole vibrational states of various isotopes considered here. Nucleus β 2 E 2 β 3 E 3 Reference 6S 0. 2.230 0.40 5.006 6 36 6 S 0.6 3.29 0.38 4.92 7 90 40Zr 0.09 2.86 0.22 2.748 6, 7 96 40Zr 0.08.75 0.27.897 6, 7 This value is extracted from the B(E3) = 0.0083e 2 b 3 given in Ref. 8 using the expression B(E3) =.702 0 7 (ZAβ 3 ) 2 e 2 b 3. Table 2. Heights of Coulomb barriers, V B, for different projectile target combination used here. System V B Reference 6 S+90 40Zr 8.2 6 6 S+96 40Zr 80. 6 36 6 S+90 40Zr 79.0 7 36 6 S+90 40Zr 77.2 7 In Fig., we compare the fusion excitation functions of,36 6 S + 90 40Zr systems obtained with the CCFULL code in conjunction with our new prescription for potential diffuseness with the corresponding data. We have considered three phonon quadrupole and octupole vibration in target and one phonon quadrupole and octupole vibration in projectile. The results are also compared with those obtained by using Winther and Akyüz Winther parametrization for potential diffuseness. It may be clearly observed in this figure that the experimental data is substantially underestimated by the calculation performed by using the Winther and Akyüz Winther parametrization schemes while the agreement between the data and the present calculations is reasonably good. It is worth mentioning here that our energy-dependent diffuseness parameter varies from a = 0.97 fm to a = 0.85 fm as energy changes from E = 70 MeV to E = 00 MeV while the diffuseness parameter remains a = 0.66 fm and a = 0.68 fm for Winther and Akyüz Winther scheme respectively for all energies. Our results favor the fact that larger diffuseness is needed to explain the sub-barrier fusion data. In the case of,36 6 S+96 40Zr systems, the situation is somewhat different as the strength of quadrupole vibrational state of 96 40Zr nucleus is comparable to that of 90 40Zr nucleus (Table ), but the corresponding excitation energyforthe 96 40 Zrnucleusissignificantlysmall.Further,astrongoctupolevibrational state of 96 40Zr nucleus occurs at energy smaller than the energy of the corresponding

Diffuseness of Woods Saxon Potential and Sub-Barrier Fusion 233 000 000 00 00 0 S + 90 Zr System 0 36 S + 90 Zr System 70 75 80 85 90 95 00 70 75 80 85 90 95 00 (a) (b) Fig.. Fusion excitation functions of,36 6 S+ 90 40Zr system corresponding to the calculations performed by using different parametrization schemes for potential diffuseness parameter are compared with the experimental data ( ). The experimental data are taken from Refs. 6 and 7 for 6 S+90 40Zr and 36 6 S+90 40Zr systems respectively. 000 000 00 00 0 S + 96 Zr System 0 36 S + 96 Zr System 70 75 80 85 90 95 00 70 75 80 85 90 95 00 (a) (b) Fig. 2. Same as Fig. but for,36 6 S+96 40Zr systems. state in 90 40Zr nucleus. Both of these facts lead to the larger enhancement in the fusion cross-section when coupling to low lying states of 96 40Zr nucleus is considered in comparison to 90 40Zr target. In Fig. 2, the fusion excitation function data for,36 6 S+96 40Zr systems are compared with the corresponding calculations performed by using different diffuseness schemes. Again, we find the agreement between data

234 M. Singh, S. S. Duhan & R. Kharab and prediction is very good in the case of our scheme in comparison to the other schemes. To summarize, we have proposed a new parametrization scheme for evaluating the value of diffuseness parameter of Woods Saxon potential, used to describe nuclear fusion, wherein the energy dependence is properly taken into account. When it is used in conjugation with the code CCFULL, considering the coupling to low lying 2 + and 3 surface vibrational states of the projectile and target, reproduces the fusion excitation data very well for various systems. References. K. Hagino et al., Phys. Rev. C 67, 054603 (2003). 2. W. M. Seif, Nucl. Phys. A 767, 92 (2006). 3. K. Hagino and Y. Watanabe, Phys. Rev. C 76, 0260(R) (2007). 4. J. O. Newton et al., Phys. Rev. C 70, 024605 (2004). 5. K. Hagino et al., Phys. Rev. C 7, 04462 (2005). 6. I. I. Gontchar et al., Phys. Rev. C 69, 02460 (2004). 7. J. O. Newton et al., Phys. Lett. B 586, 29 (2004). 8. J. R. Leigh et al., Phys. Rev. C 52, 35 (995). 9. K. Washiyama et al., Phys. Rev. C 73, 034607 (2006). 0. A. M. Stefanini et al., Phys. Rev. C 73, 034606 (2006).. K. Hagino, N. Rowley and A. T. Kruppa, Comput. Phys. Commun. 23, 43 (999). 2. Ishwar Dutt and Rajeev K. Puri, arxiv:005.523v2. 3. A. Winther, Nucl. Phys. A 594, 203 (995). 4. R. N. Sagaidak et al., Phys. Rev. C 76, 034605 (2007). 5. Ö. Akyüz and A. Winther, in Nuclear Structure and Heavy-Ion Collisions, Proc. Int. School of Physics Enrico Fermi, Course LXXVII, Varenna, Italy, 979, eds. R. A. Broglia, C. H. Dasso and R. Richi (North-Holland, 98), p. 492. 6. H. Q. Zhang et al., arxiv:005.0727v. 7. A. M. Stefanini et al., Phys. Rev. C 62, 0460 (2000). 8. T. Kibedi and R. H. Spear, Atomic Data and Nuclear Data Tables 80, 35 (2002).