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Heavy-Ion Fusion Cross Sections in Microscopic Barrier Penetration Model
Nuclear Physics A 734 (2004) E17–E20 www.elsevier.com/locate/npe
Heavy-Ion Fusion Cross Sections in Microscopic Barrier Penetration Model S. S. Godre Department of Physics, South Gujarat University, Surat 395007, India Heavy ion fusion barrier parameters are calculated by making use of configuration of classical nucleon positions of the colliding nuclei. Calculated fusion cross sections and barrier distributions of 154 Sm + 16 O, 12 C + 232 Th, and 86 Kr + 208 Pb reactions agree well with the experiments. PACS Nos: 25.70.Jj, 24.10-i. Key Words: Heavy-ion reactions; sub-barrier fusion; barrier penetration model. 1. INTRODUCTION The possibility of forming Super Heavy Elements in cold fusion reactions [1-3] has renewed much interest in various heavy-ion reactions at sub-barrier energy. Fusion cross sections for heavy-ion reactions at sub-barrier energies have been calculated in various classical and semiclassical barrier-penetration models or coupled channel calculations [4]. All such calculations are essentially one-dimensional macroscopic calculations that make use of ion-ion potential that is calculated macroscopically [5]. In the present study, we consider a microscopic barrier-penetration model. This model calculation makes use of ion-ion potential that is obtained by using classical microscopic configurations of nuclei. Fusion cross sections and barrier-distribution calculation of 154 Sm + 16 O, 12 C + 232 Th and 86 Kr + 208 Pb reactions are presented here. 2. CALCULATIONAL DETAILS Heavy-ion fusion cross-sections in one-dimensional fusion barrier model, in the classical limit, is given by [6] 2 [1 − VB /Ecm ] σf us = πRB
(1)
where VB (barrier-height) and RB (barrier-radius) are for a head-on collision (b = 0). In this approach fusion cross-section vanishes at energies below the barrier. However because of distribution of barriers one can find non-vanishing fusion cross sections below the average barrier also. Taking into account of the barrier penetrability one can calculate fusion cross sections at energies below the barrier also. For a quantum mechanical system for a single barrier fusion cross section is approximately given by the Wong’s formula [7]
σf us =
h ¯ω 2π 2 (Ecm − VB ) RB ln 1 + exp 2Ecm h ¯ω
0375-9474/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2004.03.009
(2)
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S.S. Godre / Nuclear Physics A 734 (2004) E17–E20
where ω is the oscillator frequency corresponding to the barrier top. Fusion cross section and barrier parameter calculations in the present model requires microscopic configuration for the two nuclei in ground state and a suitable NN potential for interaction. In the present calculation, nucleons are considered as classical point particles, which interact via two-body forces. Nucleon spin is explicitly neglected. The Coulomb potential between protons has the usual form VijC (rij ) = 1.44/rij (MeV )
(3)
and the NN potential chosen is a soft-core gaussian potential of the form [5] VijN (rij ) = −V0 (1 − C/rij )exp(−rij2 /r02 )
(4)
The NN potential between like particles is taken to be about 20% weaker than that between unlike particle [8]. The distribution of nucleons in each ions are first obtained by cyclically minimizing the total potential energy of an initially random distribution of nucleons, with respect to small spatial displacements of individual nucleon coordinates [5]. In the present approach, the zero-point energy in the ground state is explicitly neglected. However, a purely phenomenological NN potential is chosen with its parameters adjusted to reproduce binding energy and rms radius, for the cluster of particles in the ground state. For the reactions studied in this presentation a parameter set called potential P4 of ref. [5] is used (V0 = 1155 MeV, C= 2.07 fm, r0 = 1.2 fm). The ion-ion potential is calculated as a function of c.m. separation (Rcm ) of the two nuclei in the sudden-approximation (i.e., keeping the configuration of the nucleons in the two nuclei frozen at all times) [9]. The ion-ion potential is the sum of the nuclear and Coulomb potential between all the nucleons of the two ions. The barrier parameters VB and RB correspond to the outer maximum of the ion-ion potential and correspond to the second derivative of this peak. This gives barrier parameters (VB , RB , ω) for head-on collisions for a given orientation of the two nuclei. Large numbers of such randomly chosen orientations of the two nuclei are considered in this study. 3. RESULTS & DISCUSSION 154
Sm + 16 O Reaction : Fusion cross sections for this reaction calculated with eqs.(1) and (2) and averaged over 25 orientations are shown in Fig.-1. Calculations with eq.(1) agree well with the experiment [10] except at very low energies. The lowest value of VB (for 25 orientations) is 55.1 MeV. Therefore, fusion cross section is zero below this energy. The average value of the barrier-parameters are VB =61.53 MeV, and RB =11.34 fm. Static semiclassical microscopic calculations with Wong’s model eq.(2), which takes into account of the barrier penetrability, compares very well with the experiment [10]. The barrier-distributions (BD) are obtained from the averaged fusion cross sections as the second derivative of (Ecm , σf us ) with respect to Ecm . Fusion cross sections were
S.S. Godre / Nuclear Physics A 734 (2004) E17–E20
154
16
Fig.1: Fusion cross sections for Sm+ O reaction.
E19
154
16
Fig.2: Barrier distribution for Sm+ O reaction.
calculated with 0.5 MeV interval and the second derivatives were evaluated with a point difference formula with ∆Ecm = 2 MeV as in ref. [10,11]. Calculated BD with both eq.(1) and eq.(2) are shown in Fig. 2 and compared with experiment [10]. They agree very well with the experiment in the location, width, and overall shape of the main lobe. 12
C + 232 Th Reaction : Fusion cross sections are obtained for the 50 sets of barrier parameters using eq.(1) or eq.(2) and the barrier-distributions BD are obtained from the averaged fusion cross sections as described above. Calculated BD for this reaction is shown in Fig. 3. Calculated BD with eq.(2) are smooth compared to that with eq.(1) and agree well with the experimental BD of Mein [12] except at Ecm =67.5 MeV. Also shown in Fig. 3 are experimental results of Varma [13] obtained from quasi-elastic scattering. Comparing the BD calculated with eq.(1) with experimental result of [13] it is notable that it reproduces the location of the peak of the main lobe as well as a small bump at around Ecm = 66.5 MeV and the overall wide asymmetric shape. Fig.3 also shows BD obtained from the experimental data of [14] for comparison. 86 Kr + 208 Pb Reaction: Recently much interest was generated in 86 Kr + 208 Pb reaction. The possibility of using this reaction for production of Super Heavy Element at Z = 118 was reported by Smolanczuk [1] and an experiment performed by Ninov et al [2] reported formation of Super Heavy Element at Z = 118 using this reaction with projectile energy of 449 MeV. Ninov et al have, however, withdrawn their claim later [2]. Therefore it is of interest to use the present model to find the fusion probability or fusion cross section for this reaction. Fusion cross sections are obtained for 25 sets of barrier parameters. The averaged fusion cross sections are shown in Fig.4. Fusion cross section at Ecm = 317 MeV (Elab = 449 MeV) is about 25 mb which is very large as compared to that reported in [2].
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12
232
Fig.3: Barrier distributions for C+ Th reaction.
86
208
Fig.4: Fusion cross section for Kr+ Pb reaction.
4. CONCLUSION The simple model presented here improves on the barrier penetration model used in the literature by calculating the ion-ion potential, which makes use of an NN potential and configuration of nucleons in each nucleus. Calculated results show good agreements for fusion cross sections and barrier distribution for the reactions studied here. REFERENCES 1. R. Smolanczuk, Phy. Rev. C 59 (1999) 2634. 2. V. Ninov et al, Phy. Rev. Lett. 83 (1999) 1104; V. Ninov et al, Phy. Rev. Lett. 89 (2002) 039901. 3. Yu. Ts. Oganessian et al, Nature 400 (1999) 242. 4. M. Dasgupta, D. J. Hinde, N. Rowley and A. M. Stefanini, Ann. Rev. of Nucl. and Part. Sci., 48 (1998) 401. 5. S. S. Godre and Y. R. Waghmare, Phys.Rev.C36 (1987) 1632. 6. C. Ngo, IL Nuovo Cimento, 81A (1984) 47. 7. C. Y. Wong, Phys. Rev. Lett., 31 (1973) 766. 8. W. D. Myers and W. Swiatecki, Nucl. Phys. 81 (1966) 1. 9. S. S. Godre, Symp.on Nucl. Phys. (Book of Abstracts), 36B (1996) 194. 10. J. R. Leigh et al, Phys. Rev. C52 (1995) 31511. 11. N. Rowley et al, Phys. Lett. B254 (1991) 25. 12. J. C. Mein et al, Phys. Rev. C55 (1997) R995. 13. R. Varma et al, Phys. Rev. C57 (1998) 3462. 14. N. Majumdar et al, Phy. Rev. Lett. 77 (1996) 5027.
Heavy-Ion Fusion Cross Sections in Microscopic Barrier Penetration Model S. S. Godre Department of Physics, South Gujarat University, Surat 395007, India Heavy ion fusion barrier parameters are calculated by making use of configuration of classical nucleon positions of the colliding nuclei. Calculated fusion cross sections and barrier distributions of 154 Sm + 16 O, 12 C + 232 Th, and 86 Kr + 208 Pb reactions agree well with the experiments. PACS Nos: 25.70.Jj, 24.10-i. Key Words: Heavy-ion reactions; sub-barrier fusion; barrier penetration model. 1. INTRODUCTION The possibility of forming Super Heavy Elements in cold fusion reactions [1-3] has renewed much interest in various heavy-ion reactions at sub-barrier energy. Fusion cross sections for heavy-ion reactions at sub-barrier energies have been calculated in various classical and semiclassical barrier-penetration models or coupled channel calculations [4]. All such calculations are essentially one-dimensional macroscopic calculations that make use of ion-ion potential that is calculated macroscopically [5]. In the present study, we consider a microscopic barrier-penetration model. This model calculation makes use of ion-ion potential that is obtained by using classical microscopic configurations of nuclei. Fusion cross sections and barrier-distribution calculation of 154 Sm + 16 O, 12 C + 232 Th and 86 Kr + 208 Pb reactions are presented here. 2. CALCULATIONAL DETAILS Heavy-ion fusion cross-sections in one-dimensional fusion barrier model, in the classical limit, is given by [6] 2 [1 − VB /Ecm ] σf us = πRB
(1)
where VB (barrier-height) and RB (barrier-radius) are for a head-on collision (b = 0). In this approach fusion cross-section vanishes at energies below the barrier. However because of distribution of barriers one can find non-vanishing fusion cross sections below the average barrier also. Taking into account of the barrier penetrability one can calculate fusion cross sections at energies below the barrier also. For a quantum mechanical system for a single barrier fusion cross section is approximately given by the Wong’s formula [7]
σf us =
h ¯ω 2π 2 (Ecm − VB ) RB ln 1 + exp 2Ecm h ¯ω
0375-9474/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2004.03.009
(2)
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S.S. Godre / Nuclear Physics A 734 (2004) E17–E20
where ω is the oscillator frequency corresponding to the barrier top. Fusion cross section and barrier parameter calculations in the present model requires microscopic configuration for the two nuclei in ground state and a suitable NN potential for interaction. In the present calculation, nucleons are considered as classical point particles, which interact via two-body forces. Nucleon spin is explicitly neglected. The Coulomb potential between protons has the usual form VijC (rij ) = 1.44/rij (MeV )
(3)
and the NN potential chosen is a soft-core gaussian potential of the form [5] VijN (rij ) = −V0 (1 − C/rij )exp(−rij2 /r02 )
(4)
The NN potential between like particles is taken to be about 20% weaker than that between unlike particle [8]. The distribution of nucleons in each ions are first obtained by cyclically minimizing the total potential energy of an initially random distribution of nucleons, with respect to small spatial displacements of individual nucleon coordinates [5]. In the present approach, the zero-point energy in the ground state is explicitly neglected. However, a purely phenomenological NN potential is chosen with its parameters adjusted to reproduce binding energy and rms radius, for the cluster of particles in the ground state. For the reactions studied in this presentation a parameter set called potential P4 of ref. [5] is used (V0 = 1155 MeV, C= 2.07 fm, r0 = 1.2 fm). The ion-ion potential is calculated as a function of c.m. separation (Rcm ) of the two nuclei in the sudden-approximation (i.e., keeping the configuration of the nucleons in the two nuclei frozen at all times) [9]. The ion-ion potential is the sum of the nuclear and Coulomb potential between all the nucleons of the two ions. The barrier parameters VB and RB correspond to the outer maximum of the ion-ion potential and correspond to the second derivative of this peak. This gives barrier parameters (VB , RB , ω) for head-on collisions for a given orientation of the two nuclei. Large numbers of such randomly chosen orientations of the two nuclei are considered in this study. 3. RESULTS & DISCUSSION 154
Sm + 16 O Reaction : Fusion cross sections for this reaction calculated with eqs.(1) and (2) and averaged over 25 orientations are shown in Fig.-1. Calculations with eq.(1) agree well with the experiment [10] except at very low energies. The lowest value of VB (for 25 orientations) is 55.1 MeV. Therefore, fusion cross section is zero below this energy. The average value of the barrier-parameters are VB =61.53 MeV, and RB =11.34 fm. Static semiclassical microscopic calculations with Wong’s model eq.(2), which takes into account of the barrier penetrability, compares very well with the experiment [10]. The barrier-distributions (BD) are obtained from the averaged fusion cross sections as the second derivative of (Ecm , σf us ) with respect to Ecm . Fusion cross sections were
S.S. Godre / Nuclear Physics A 734 (2004) E17–E20
154
16
Fig.1: Fusion cross sections for Sm+ O reaction.
E19
154
16
Fig.2: Barrier distribution for Sm+ O reaction.
calculated with 0.5 MeV interval and the second derivatives were evaluated with a point difference formula with ∆Ecm = 2 MeV as in ref. [10,11]. Calculated BD with both eq.(1) and eq.(2) are shown in Fig. 2 and compared with experiment [10]. They agree very well with the experiment in the location, width, and overall shape of the main lobe. 12
C + 232 Th Reaction : Fusion cross sections are obtained for the 50 sets of barrier parameters using eq.(1) or eq.(2) and the barrier-distributions BD are obtained from the averaged fusion cross sections as described above. Calculated BD for this reaction is shown in Fig. 3. Calculated BD with eq.(2) are smooth compared to that with eq.(1) and agree well with the experimental BD of Mein [12] except at Ecm =67.5 MeV. Also shown in Fig. 3 are experimental results of Varma [13] obtained from quasi-elastic scattering. Comparing the BD calculated with eq.(1) with experimental result of [13] it is notable that it reproduces the location of the peak of the main lobe as well as a small bump at around Ecm = 66.5 MeV and the overall wide asymmetric shape. Fig.3 also shows BD obtained from the experimental data of [14] for comparison. 86 Kr + 208 Pb Reaction: Recently much interest was generated in 86 Kr + 208 Pb reaction. The possibility of using this reaction for production of Super Heavy Element at Z = 118 was reported by Smolanczuk [1] and an experiment performed by Ninov et al [2] reported formation of Super Heavy Element at Z = 118 using this reaction with projectile energy of 449 MeV. Ninov et al have, however, withdrawn their claim later [2]. Therefore it is of interest to use the present model to find the fusion probability or fusion cross section for this reaction. Fusion cross sections are obtained for 25 sets of barrier parameters. The averaged fusion cross sections are shown in Fig.4. Fusion cross section at Ecm = 317 MeV (Elab = 449 MeV) is about 25 mb which is very large as compared to that reported in [2].
E20
S.S. Godre / Nuclear Physics A 734 (2004) E17–E20
12
232
Fig.3: Barrier distributions for C+ Th reaction.
86
208
Fig.4: Fusion cross section for Kr+ Pb reaction.
4. CONCLUSION The simple model presented here improves on the barrier penetration model used in the literature by calculating the ion-ion potential, which makes use of an NN potential and configuration of nucleons in each nucleus. Calculated results show good agreements for fusion cross sections and barrier distribution for the reactions studied here. REFERENCES 1. R. Smolanczuk, Phy. Rev. C 59 (1999) 2634. 2. V. Ninov et al, Phy. Rev. Lett. 83 (1999) 1104; V. Ninov et al, Phy. Rev. Lett. 89 (2002) 039901. 3. Yu. Ts. Oganessian et al, Nature 400 (1999) 242. 4. M. Dasgupta, D. J. Hinde, N. Rowley and A. M. Stefanini, Ann. Rev. of Nucl. and Part. Sci., 48 (1998) 401. 5. S. S. Godre and Y. R. Waghmare, Phys.Rev.C36 (1987) 1632. 6. C. Ngo, IL Nuovo Cimento, 81A (1984) 47. 7. C. Y. Wong, Phys. Rev. Lett., 31 (1973) 766. 8. W. D. Myers and W. Swiatecki, Nucl. Phys. 81 (1966) 1. 9. S. S. Godre, Symp.on Nucl. Phys. (Book of Abstracts), 36B (1996) 194. 10. J. R. Leigh et al, Phys. Rev. C52 (1995) 31511. 11. N. Rowley et al, Phys. Lett. B254 (1991) 25. 12. J. C. Mein et al, Phys. Rev. C55 (1997) R995. 13. R. Varma et al, Phys. Rev. C57 (1998) 3462. 14. N. Majumdar et al, Phy. Rev. Lett. 77 (1996) 5027.