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Coulomb reorientation effect on near-barrier heavy-ion fusion reactions in rigid-body rotational dynamics calculation
Nuclear Physics A 834 (2010) 195c–197c www.elsevier.com/locate/nuclphysa
Coulomb reorientation effect on near-barrier heavy-ion fusion reactions in rigid-body rotational dynamics calculation S. S Godrea and P. R. Desaib a
Department of Physics, Veer Narmad South Gujarat University, Surat - 395007, India
b
Navyug Science College, Rander road, Surat - 395009, India
A classical rigid-body dynamics model is used for study of Coulomb reorientation effect on the fusion dynamics. Model calculations show that barrier parameters depend on the initial orientations and also on the collision energy. Calculated fusion cross sections for 24 Mg+208 Pb and 16 O+154 Sm reactions show that reorientation effect is more pronounced in the case of 24 Mg+208 Pb reaction which involves a lighter deformed nucleus. 1. INTRODUCTION Heavy-ion collisions at energies near the Coulomb barrier are strongly affected by the internal structure of the colliding nuclei [1]. For nuclei with a significant static deformation the reorientation of the deformed nucleus under the influence of the torque produced by the long-range Coulomb force plays a notable role in the sub-barrier collisions [2–6]. This is especially important in the case of sub-barrier energy collisions where the interaction time between the two nuclei is sufficiently large to cause reorientation of the colliding nuclei which can be crucial in determining the approach state of the two nuclei for fusion. It is possible to include all the degrees of freedom in a classical microscopic calculation for heavy-ion collision such as in Classical Molecular Dynamics model (CMD-model) [7]. However, the contribution of an individual degree of freedom to fusion gets obscured in the presence of all the other degrees of freedom in this model. On the other hand a microscopic Static Barrier Penetration Model (SBPM) [8] has also been used to calculate fusion cross sections. In SBPM static deformation is taken care of but all the dynamical effects are explicitly neglected. Therefore, a Classical Rigid-Body Dynamics model (CRBD-model) is developed [5] to specifically study the effects of Coulomb reorientation. Brief description of the CRBD-model calculations is given in section 2. Comparison of fusion cross sections calculated in SBPM and CRBD-model can bring out the contribution of the reorientation effect. Effect of reorientation on the reaction dynamics and fusion cross sections for 24 Mg+208 Pb and 16 O+154 Sm systems are presented in section 3. 2. CALCULATIONAL DETAILS Nucleons are assumed to be classical point particles. The NN potential used is a phenomenological soft-core Gaussian potential [8]. The individual nuclei are first generated using a STATIC energy minimization procedure [7] in which a random distribution of 0375-9474/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2009.12.038
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S.S. Godre, P.R. Desai / Nuclear Physics A 834 (2010) 195c–197c
all the nucleon positions is generated and then the total potential energy of this nucleon configuration is cyclically minimized with respect to small displacements of the individual nucleon coordinates. Potential parameter set P4 [8] (V0 = 1155 MeV, C = 2.07 fm and r0 = 1.2 fm) is used which approximately reproduces the ground-state binding energy, rms radius and deformation β2 of the nuclei used in the collision calculations. The simulation process is initiated by placing the two nuclei with a given relative orientation and collision energy along their Rutherford trajectories with their centre of masses separated by Rin =2500 fm. In CRBD-model the two nuclei are assumed to be rigid and for given initial conditions motion of the centre-of-mass and the orientations of the principal-axes of the two nuclei are found by solving the classical equations of motion for rigid-bodies. These equations take into account the total force and torque experienced by each nucleus due to the other which are calculated from the nuclear and the Coulomb potential experienced by individual particles of the two interacting nuclei. Barrier parameters for a given initial orientation of the two nuclei for l =0 collision are found from the ion-ion potential generated from the dynamical simulation for a given collision energy. Using the barrier parameters corresponding to the given collision energy, fusion cross section for this energy is calculated using the Wong’s formula [9], 2 RB ECM − VB h ¯ ω0 ]ln{1 + exp(2π )} (1) 2ECM h ¯ ω0 Large numbers of initially random orientations are considered at every collision energies for calculating the orientation-averaged fusion cross-section.
σ(ECM ) = [
3. RESULTS AND DISCUSSION As the two colliding nuclei evolve from their initial separation Rin , the torque produced by the long range Coulomb force rotates and reorients the deformed nucleus. The extent of reorientation from the initial orientation depends on the magnitude of the torque and the interaction time which in turn depends on the initial separation Rin and the collision energy. Therefore, a large value of Rin is essential to account for the full extent of the Coulomb torque. It is reported in TDHF calculations [2,3], which are actually started at smaller distances, that the barrier parameters are independent of the collision energy. However, in the CRBD-model study of 24 Mg+208 Pb [6] it is shown that the extent of reorientation and barrier parameters not only depend on the initial orientation but also on the collision energy. The reorientation effect is small at higher energies and becomes significant at collision energies close to the average barrier height. It is also observed that heavy-spherical nucleus 208 Pb does not reorient much while the lighter deformed nucleus 24 Mg reorients significantly giving rise to non-uniform distribution of relative orientations at a distance near the barrier, even though the distribution of orientations initially at large distance is uniform. Fusion cross sections for 24 Mg+208 Pb system calculated in the CRBD-model and SBPM model are shown in the figure 1. Since both the calculations use the same NN potential and the same nuclei in ground state, a comparison of the two calculations shows the effect of reorientation of the deformed nucleus on fusion cross sections in CRBD-model. Fusion cross sections calculated in CRBD-model are suppressed at below barrier energies as compared to SBPM calculation in which rotational dynamics is neglected.
S.S. Godre, P.R. Desai / Nuclear Physics A 834 (2010) 195c–197c
Figure 1. Fusion cross-sections for 24 Mg+208 Pb reaction
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Figure 2. Fusion cross-sections for 16 O+154 Sm reaction
Fusion cross sections for light-spherical and heavy-deformed 16 O+154 Sm system are also calculated in the CRBD-model and are shown in the figure 2. Comparison of CRBD-model and SBPM calculations for this reaction shows that reorientation of the deformed nucleus does have some effect on fusion cross section but since the deformed nucleus in this case is heavy, this effect is much less as compared to that in the case of 24 Mg+208 Pb reaction in which the deformed nucleus is lighter. 4. CONCLUSIONS Heavy-ion fusion dynamics studied in a CRBD-model shows that barrier parameters are not only dependent on the initial orientation but also on the collision energy, contrary to that reported in some TDHF calculations. Fusion cross sections for 24 Mg+208 Pb reactions show suppression at below barrier energies due to reorientation effect. However 16 O+154 Sm reaction involving a heavy deformed nucleus shows much less effect of reorientation. REFERENCES 1. M. Dasgupta, D. J. Hinde, N. Rowley, A. M. Stefanini, Annu. Rev. Nucl. Part. Sci. 48 (1998) 401. 2. C. Simenel, Ph. Chomaz and G. de France, Phys. Rev. Lett. 93 (2004) 102701. 3. A. S. Umar and V. E. Oberacker,Phys. Rev. C 76 (2007) 014614. 4. B. K. Nayak et al. Phys. Rev. C 75 (2007) 054615. 5. P. R. Desai and S. S. Godre, Proc. Symp. on Nucl. Phys.(Baroda) 49 B (2006) 419. 6. P. R. Desai and S. S. Godre, in preparation. 7. S. S. Godre and Y. R. Waghmare, Phys. Rev. C 36 (1987) 1632. 8. S. S. Godre, Nucl. Phys. A 734 (2004) E17. 9. C. Y. Wong, Phys. Rev. Lett. 31 (1973) 766. 10. B. B. Back, Phys. Rev. C 31(1985) 6. 11. J. R. Leigh et al. Phys. Rev. C 47(1993) R437.
Coulomb reorientation effect on near-barrier heavy-ion fusion reactions in rigid-body rotational dynamics calculation S. S Godrea and P. R. Desaib a
Department of Physics, Veer Narmad South Gujarat University, Surat - 395007, India
b
Navyug Science College, Rander road, Surat - 395009, India
A classical rigid-body dynamics model is used for study of Coulomb reorientation effect on the fusion dynamics. Model calculations show that barrier parameters depend on the initial orientations and also on the collision energy. Calculated fusion cross sections for 24 Mg+208 Pb and 16 O+154 Sm reactions show that reorientation effect is more pronounced in the case of 24 Mg+208 Pb reaction which involves a lighter deformed nucleus. 1. INTRODUCTION Heavy-ion collisions at energies near the Coulomb barrier are strongly affected by the internal structure of the colliding nuclei [1]. For nuclei with a significant static deformation the reorientation of the deformed nucleus under the influence of the torque produced by the long-range Coulomb force plays a notable role in the sub-barrier collisions [2–6]. This is especially important in the case of sub-barrier energy collisions where the interaction time between the two nuclei is sufficiently large to cause reorientation of the colliding nuclei which can be crucial in determining the approach state of the two nuclei for fusion. It is possible to include all the degrees of freedom in a classical microscopic calculation for heavy-ion collision such as in Classical Molecular Dynamics model (CMD-model) [7]. However, the contribution of an individual degree of freedom to fusion gets obscured in the presence of all the other degrees of freedom in this model. On the other hand a microscopic Static Barrier Penetration Model (SBPM) [8] has also been used to calculate fusion cross sections. In SBPM static deformation is taken care of but all the dynamical effects are explicitly neglected. Therefore, a Classical Rigid-Body Dynamics model (CRBD-model) is developed [5] to specifically study the effects of Coulomb reorientation. Brief description of the CRBD-model calculations is given in section 2. Comparison of fusion cross sections calculated in SBPM and CRBD-model can bring out the contribution of the reorientation effect. Effect of reorientation on the reaction dynamics and fusion cross sections for 24 Mg+208 Pb and 16 O+154 Sm systems are presented in section 3. 2. CALCULATIONAL DETAILS Nucleons are assumed to be classical point particles. The NN potential used is a phenomenological soft-core Gaussian potential [8]. The individual nuclei are first generated using a STATIC energy minimization procedure [7] in which a random distribution of 0375-9474/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2009.12.038
196c
S.S. Godre, P.R. Desai / Nuclear Physics A 834 (2010) 195c–197c
all the nucleon positions is generated and then the total potential energy of this nucleon configuration is cyclically minimized with respect to small displacements of the individual nucleon coordinates. Potential parameter set P4 [8] (V0 = 1155 MeV, C = 2.07 fm and r0 = 1.2 fm) is used which approximately reproduces the ground-state binding energy, rms radius and deformation β2 of the nuclei used in the collision calculations. The simulation process is initiated by placing the two nuclei with a given relative orientation and collision energy along their Rutherford trajectories with their centre of masses separated by Rin =2500 fm. In CRBD-model the two nuclei are assumed to be rigid and for given initial conditions motion of the centre-of-mass and the orientations of the principal-axes of the two nuclei are found by solving the classical equations of motion for rigid-bodies. These equations take into account the total force and torque experienced by each nucleus due to the other which are calculated from the nuclear and the Coulomb potential experienced by individual particles of the two interacting nuclei. Barrier parameters for a given initial orientation of the two nuclei for l =0 collision are found from the ion-ion potential generated from the dynamical simulation for a given collision energy. Using the barrier parameters corresponding to the given collision energy, fusion cross section for this energy is calculated using the Wong’s formula [9], 2 RB ECM − VB h ¯ ω0 ]ln{1 + exp(2π )} (1) 2ECM h ¯ ω0 Large numbers of initially random orientations are considered at every collision energies for calculating the orientation-averaged fusion cross-section.
σ(ECM ) = [
3. RESULTS AND DISCUSSION As the two colliding nuclei evolve from their initial separation Rin , the torque produced by the long range Coulomb force rotates and reorients the deformed nucleus. The extent of reorientation from the initial orientation depends on the magnitude of the torque and the interaction time which in turn depends on the initial separation Rin and the collision energy. Therefore, a large value of Rin is essential to account for the full extent of the Coulomb torque. It is reported in TDHF calculations [2,3], which are actually started at smaller distances, that the barrier parameters are independent of the collision energy. However, in the CRBD-model study of 24 Mg+208 Pb [6] it is shown that the extent of reorientation and barrier parameters not only depend on the initial orientation but also on the collision energy. The reorientation effect is small at higher energies and becomes significant at collision energies close to the average barrier height. It is also observed that heavy-spherical nucleus 208 Pb does not reorient much while the lighter deformed nucleus 24 Mg reorients significantly giving rise to non-uniform distribution of relative orientations at a distance near the barrier, even though the distribution of orientations initially at large distance is uniform. Fusion cross sections for 24 Mg+208 Pb system calculated in the CRBD-model and SBPM model are shown in the figure 1. Since both the calculations use the same NN potential and the same nuclei in ground state, a comparison of the two calculations shows the effect of reorientation of the deformed nucleus on fusion cross sections in CRBD-model. Fusion cross sections calculated in CRBD-model are suppressed at below barrier energies as compared to SBPM calculation in which rotational dynamics is neglected.
S.S. Godre, P.R. Desai / Nuclear Physics A 834 (2010) 195c–197c
Figure 1. Fusion cross-sections for 24 Mg+208 Pb reaction
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Figure 2. Fusion cross-sections for 16 O+154 Sm reaction
Fusion cross sections for light-spherical and heavy-deformed 16 O+154 Sm system are also calculated in the CRBD-model and are shown in the figure 2. Comparison of CRBD-model and SBPM calculations for this reaction shows that reorientation of the deformed nucleus does have some effect on fusion cross section but since the deformed nucleus in this case is heavy, this effect is much less as compared to that in the case of 24 Mg+208 Pb reaction in which the deformed nucleus is lighter. 4. CONCLUSIONS Heavy-ion fusion dynamics studied in a CRBD-model shows that barrier parameters are not only dependent on the initial orientation but also on the collision energy, contrary to that reported in some TDHF calculations. Fusion cross sections for 24 Mg+208 Pb reactions show suppression at below barrier energies due to reorientation effect. However 16 O+154 Sm reaction involving a heavy deformed nucleus shows much less effect of reorientation. REFERENCES 1. M. Dasgupta, D. J. Hinde, N. Rowley, A. M. Stefanini, Annu. Rev. Nucl. Part. Sci. 48 (1998) 401. 2. C. Simenel, Ph. Chomaz and G. de France, Phys. Rev. Lett. 93 (2004) 102701. 3. A. S. Umar and V. E. Oberacker,Phys. Rev. C 76 (2007) 014614. 4. B. K. Nayak et al. Phys. Rev. C 75 (2007) 054615. 5. P. R. Desai and S. S. Godre, Proc. Symp. on Nucl. Phys.(Baroda) 49 B (2006) 419. 6. P. R. Desai and S. S. Godre, in preparation. 7. S. S. Godre and Y. R. Waghmare, Phys. Rev. C 36 (1987) 1632. 8. S. S. Godre, Nucl. Phys. A 734 (2004) E17. 9. C. Y. Wong, Phys. Rev. Lett. 31 (1973) 766. 10. B. B. Back, Phys. Rev. C 31(1985) 6. 11. J. R. Leigh et al. Phys. Rev. C 47(1993) R437.