- Email: [email protected]
New challenges in understanding heavy ion fusion
Nuclear Physics A 787 (2007) 144c–149c
New challenges in understanding heavy ion fusion M. Dasguptaa∗ , D.J. Hindea , A. Mukherjeea† and J.O. Newtona a
Department of Nuclear Physics, RSPhysSE, Australian National University, Canberra ACT 0200, AUSTRALIA
Advances enabling precision measurements of fusion cross-sections, and the development of realistic theoretical models, have together resulted in considerable progress being made towards understanding fusion involving heavy nuclei (Z1 Z2 ≥ 400). However, discrepancies between measurements and predictions are becoming evident when a consistent description of the various reaction processes is sought. Even the process of fusion spanning energies well below to well above the barrier cannot be consistently explained by the commonly used coupled channels model. The inadequacy of our current understanding, and the need for a re-examination of the assumptions in the current models of fusion is discussed. 1. Introduction Measurements [1,2] of fusion of heavy nuclei clearly demonstrated the inadequacy of the single barrier penetration model in describing fusion dynamics [3]. The need to include internal degrees of freedom, such as rotational and vibrational states of the interacting nuclei, became apparent as differences were seen amongst isotopes [1,2]. The coupled channels formalism, which includes the coupling between the relative motion and the internal degrees of freedom of the colliding nuclei, was successfully used to explain fusion observables [4,5]. Experimentally, the role of collective degrees of freedom was vividly demonstrated by the experimental barrier distributions [6] which were extracted from high precision measurements [7,8] of fusion excitation functions. This led to a flurry of activity in near barrier fusion reaction studies, and a large body of high precision data was obtained which demonstrated the role of complex vibrations and transfer reactions [5]. The high precision data necessitated further developments in the coupled reaction channels formalism [9]. Currently, this model is the most accepted and used model of low energy nuclear reactions, and in principle has the potential to describe all reaction processes within a single framework. Inadequacies in the coupled channels model as currently implemented, start becoming evident when attempting to obtain a simultaneous description of a new generation of high precision measurements of different reaction observables e.g. elastic, quasi-elastic, fusion. This paper discusses some of these aspects. ∗ †
Research Supported by an Australian Research Council Discovery Grant Permanent Address: Saha Institute of Nuclear Physics, 1/AF Bidhan Nagar, Kolkata 700 064, India.
0375-9474/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2006.12.025
M. Dasgupta et al. / Nuclear Physics A 787 (2007) 144c–149c
145c
2. The nuclear potential The internuclear potential is one of the main ingredients of any model, and one which is often questioned where there are discrepancies between predictions and measurements. The most commonly used form is the phenomenological Woods-Saxon potential defined by the depth V0 , the radius r0 , and the diffuseness parameter a, and is given by: 1/3
1/3
Vnuc (r) = −V0 {1 + exp[(r − r0 (A1 + A2 ))/a]}−1 .
(1)
Here r denotes the center-of-mass distance between the two colliding nuclei with mass numbers A1 and A2 . There are many parameterizations in the literature for V0 , r0 and a. Commonly used are those due to the Woods-Saxon parametrization [10] of the Aky¨ uzWinther potential. Another commonly used potential is the double folding [10] potential obtained from a more microscopic basis. The double folding potential is valid when nuclei are well separated and is thought to be reasonable for separations where there is a small overlap between the nuclei. It has been shown that the double folding potential can be very well described by a Woods-Saxon form [11] with a diffuseness parameter, which characterizes the fall-off of the nuclear potential, of 0.65 fm for internuclear separations down to 1 fm inside the barrier radius. 3. Model predictions and observations It is well documented that the elastic scattering cross-sections are well described by the coupled channels (CC) model using a Woods-Saxon potential with a diffuseness parameter of 0.65 fm (or equivalently a double folding potential, since as discussed above, it closely follows a Woods-Saxon potential near the barrier). Fusion cross-sections and barrier distributions at energies near the barrier are also well reproduced [5] with a CC model, but description of fusion cross-sections at energies well above the barrier is found to require larger values of the diffuseness parameter [12], with “apparent” diffuseness of up to 1.2 fm. Fusion cross-sections at energies well below the barrier also show large deviations [13] from coupled channels calculations using the standard nuclear potential. A nuclear potential with a very small depth has been proposed [14] to explain the experimental data. Thus, whilst individual processes over a limited energy range can be fitted by the coupled channels model either by changing the couplings or the internuclear potential shape, it has not been possible to obtain a simultaneous description of the different processes with a consistent parameter set. This inconsistency was not clear prior to the availability of high precision fusion crosssection data, and indeed elastic scattering and fusion were thought to be described reasonably consistently [15,16] within the coupled channels approach. High precision fusion cross-section data determines the energy of the average barrier very well, usually better than ± 1.5%, and constraint the input potential parameters in a CC calculation. Thus the consistency between elastic, inelastic and fusion (uncertainties ∼ 5 - 10%) previously found for example for the 12 C + 208 Pb reaction, is destroyed when the barrier energy is constrained by the new high precision fusion data; the elastic scattering and fusion appear to require different nuclear potential shape, characterized by different diffuseness parameters, as shown in Fig. 1. It is interesting that signs of failure of the CC model,
146c
M. Dasgupta et al. / Nuclear Physics A 787 (2007) 144c–149c 10
1400
12C
+ 208Pb
12C
+ 208Pb
1200
1
σfus (mb)
dσelas/dσRuth
1000
0.1
800
600
400
0.01
new data from Ref. [16]
from Ref. [16] 200
a = 0.56 fm
a = 0.56 fm
a = 0.95 fm
a = 0.95 fm 0
0.001 20
70
120
θC.M. (deg.)
50
70
90
EC.M. (MeV)
Figure 1. The elastic scattering angular distribution at a laboratory energy of 69.9 MeV (left panel) and fusion cross-sections as a function of centre-of-mass energy (right panel) for the 12 C + 208 Pb system. The lines are the results of a coupled channels calculation using FRESCO with two different values of the diffuseness parameter for the Woods-Saxon potential; couplings are the same as used in Ref. [16].
as currently implemented in FRESCO [17] and CCFULL [18], is evident even for reactions induced by light projectiles such as 12 C. An explanation put forward to understand the different potential shapes required to fit the elastic scattering and fusion data is that the nuclear potential deviates from a Woods-Saxon shape, and that the discrepancy arises as elastic scattering and fusion probe different parts of the nuclear potential. Elastic scattering probes the tail region, whilst fusion the internal region. Changing the shape of the nuclear potential is a possibility, and has indeed been proposed [19]. The ‘new’ potential would need to be constrained by (i) the average barrier determined from fusion and (ii) the outer region determined from elastic scattering. Calculations using an error function, with these constraints, have been tried [20] and preliminary results suggests that a simultaneous reproduction of elastic and fusion data may not be achievable simply by changing the form of the potential. 4. Above-barrier and deep sub-barrier fusion The lack of knowledge about the exact form of the potential may give rise to controversies in simultaneous descriptions of elastic scattering and fusion. However, one can ask does the CC model fare better in describing fusion alone? In the framework of the current model, fusion at energies well below the lowest barrier is tunnelling dominated, with the slope of the cross-sections as a function of energy determined by the width of the lowest barrier, which in turn is characterized by the diffuseness of the nuclear potential. Fusion at energies around the barrier is coupling dominated, giving rise to the observed distribution of barriers. At energies well above the average barrier fusion the cross-sections are
M. Dasgupta et al. / Nuclear Physics A 787 (2007) 144c–149c
147c
150
r 100
V (MeV)
High l
50
Low l
0 5
10
15
20
r (fm)
Figure 2. The calculated total internuclear potential as a function of the centre-of-mass separation between 16 O and 208 Pb. The distance of separation at the inner turning point at sub-barrier energies is similar to the barrier radius for high angular momenta.
potential dominated again, being determined mainly by the high barrier parameters. As shown in Fig. 2, the inner turning point at sub-barrier energy occurs at a similar separation distance as the barrier peak for higher -values. Thus intuitively, fusion well below the barrier and well above the barrier probes similar separation distances. In the CC approach one might then expect both regions to be explained by the same nuclear potential (with or without couplings, as they do not affect the shape of the excitation function in these energy regimes). However, as shown in Fig. 3 for the reaction of 16 O with 208 Pb, the fusion data at energies well below the barrier (Fig. 3 (a)) appear to be well reproduced using a diffuseness of 1.65 f m, in contrast the above barrier data [21] appears to require a diffuseness of 1.18 f m. Note that both these values are much higher than obtained from fits to elastic scattering data. The deep sub-barrier, and the above-barrier data taken together, represent a new challenge to theoretical models, as calculations with a nuclear potential which explains the above barrier-data do not explain the deep sub-barrier data, and vice-versa. Simultaneous explanation of the above-barrier and deep sub-barrier data using a single nuclear potential of any form may not be possible. This failure of the commonly used model appears to indicate that some physical aspects of the collision are not being modeled correctly. What might these be? In the coupled channels calculations, the fusion process is simulated by a mathematical “trick”; either using a imaginary potential or by applying an incoming wave boundary condition. Imposing a boundary condition at a critical distance means that if a shallow potential is used, as proposed [14] for example in explaining the deep sub-barrier fusion data, an un-physical scenario can result where fusion continues to occur even though no trapping potential pocket exists for higher angular momenta. The physical process of energy damping is not included explicitly in any of the models, although it is known that experimentally deep inelastic products are observed. If, as
148c
M. Dasgupta et al. / Nuclear Physics A 787 (2007) 144c–149c 1500
10
101
(a) 10
(b)
01
16O
+ 208Pb
16O
204Pb
16O
+ 208Pb
0.1 -1
+
1000
σfus (mb)
σfus (mb)
10
0.01 -2
10
10-3 0.001
500 0.0001 -4
10
a = 0.66 fm a = 1.18 fm
0.00001 -5
10
a = 0.66 fm a = 1.18 fm a = 1.65 fm
a = 1.65 fm -6 0.000001
10
-12
0
-10
-8
-6
-4
-2
0
EC.M. - B (MeV)
-10
0
10
20
30
40
EC.M. - B (MeV)
Figure 3. The measured fusion cross-sections as a function of the centre of mass energy with respect to the barrier energy B at deep sub-barrier energies (a) and all energies (b). The calculations using three different values of the diffuseness of the Woods-Saxon nuclear potential are shown. The deep sub-barrier data and above barrier data are not consistently described using the same value of the parameter.
argued above, both deep sub-barrier and above barrier data probe similar separation distances, then they are both sensitive to regions inside the average barrier radius where dissipative effect may be significant. Including such effects in the theoretical formalism may help in obtaining a simultaneous description of the above barrier and below barrier data. A related question that must be addressed is whether it is correct to treat the tunnelling of finite sized nuclei on the same footing as that of point particles. This and other aspects, such as multi-nucleon transfer [22], may need to be investigated in order to obtain a consistent explanation of the fusion process over a range of energies. 5. Fusion with light weakly-bound nuclei Fusion with weakly-bound light nuclei, both stable and radioactive, is of interest as attested by the large number of contributions to this conference. The challenge, again, is to be able to relate the breakup, transfer, complete- and incomplete-fusion processes, and obtain a consistent description of all these processes in a single framework. The data are usually described in the framework of the Continuum Discretized Coupled Channels (CDCC) framework [23]. This model is able to predict breakup cross-sections if none of the fragments are captured by the target, but a major drawback is that it cannot distinguish complete fusion from fusion of one of the breakup fragments (incomplete fusion). This is a major failing which cannot be rectified; and one which critically affects interpretation of experimental data. The distinction between complete and incompletefusion is possible if the trajectories of the breakup fragments are followed. A 3-dimensional classical trajectory model, which can relate breakup well below the barrier to complete
M. Dasgupta et al. / Nuclear Physics A 787 (2007) 144c–149c
149c
and incomplete fusion above the barrier is currently being developed [24] and verified using asymptotic observables from CDCC. 6. Discussion The past decade has seen enormous advances in experiments, with innovative techniques enabling precision measurements. Beautiful examples of exclusive measurements, even with low intensity radioactive beams, were presented at this conference. Experimentally, complete data sets for a few well chosen reactions would be very useful in furthering theoretical developments and our understanding of reaction mechanisms. The most commonly used model of low energy reactions (the coupled channels model), as currently implemented, is unable to provide a consistent description of new high quality experimental data. Using different reaction processes, and fusion data over a wide range of energies, it is argued here that physical features not currently implement in the “standard” model need to be included to make the model more realistic. Alternately, other models aiming to give a consistent description may need to be pursued. For weakly-bound systems, the most commonly used CDCC model can not distinguish complete and incomplete fusion, giving impetus to the development of a model based on classical trajectories. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.
R.G. Stokstad et al., Phys. Rev. C 21 (1980) 2427. M. Beckerman et al., Phys. Rev. Lett. 45 (1980) 1472. A.B. Balantekin et al., Phys. Rev. C 28 (1983) 1565. C.H. Dasso et al., Nucl. Phys. A 405 (1983) 381; 407 (1983) 221. M. Dasgupta et. al., Annu. Rev. Nuc. Part. Sci. 48 (1998) 401, and references therein. N. Rowley et al., Phys. Lett. B 254 (1991) 25. J.X. Wei et al., Phys. Rev. Lett. 67 (1991) 3368–3371. J.R. Leigh et al., Phys. Rev. C 52 (1995) 3151–3166. K. Hagino et. al., Phys. Rev. C 55 (1997) 276; Phys. Rev. Lett. 79 (1997) 2014. R.A. Broglia and A. Winther, Heavy Ion Reactions Lecture Notes, Volume I: Elastic and Inelastic Reactions, Benjamin/Cummings, Reading, 1981. I.I. Gontchar et al., Phys. Rev. C 69 (2004) 024610. J.O. Newton et al., Phys. Lett. B 586 (2004) 219. C.L. Jiang et al., Phys. Rev. C 73 (2006) 014613 and references therein. S¸. Mi¸sicu and H. Esbensen, Phys. Rev. Lett. 96 (2006) 112701. I.J. Thompson et al., Nucl. Phys. A 505 (1989) 84. S. Santra et al., Phys. Rev. C 64 (2001) 024602. I.J. Thompson, Comp. Phys. Reports 7 (1988), 167. K. Hagino, N. Rowley, and A.T. Kruppa, Comput. Phys. Commun. 123 (1999) 143. J.O. Newton et al., Phys. Rev. C 64 (2001) 064608. M. Zamrun and K. Hagino, University of Sendai, Private Communication. C.R. Morton et al., Phys. Rev. C 60 (1999) 044608. G. Pollarolo, AIP Conference Proccedings, eds. L. Corradi et al., 853 (2006) 29. Y. Sakuragi, M. Yahiro, and M. Kamimura, Prog. Theo. Phys. Suppl. 89 (1986) 136 A. Diaz-Torres et al., submitted to Phys. Rev. Lett. (2006).
New challenges in understanding heavy ion fusion M. Dasguptaa∗ , D.J. Hindea , A. Mukherjeea† and J.O. Newtona a
Department of Nuclear Physics, RSPhysSE, Australian National University, Canberra ACT 0200, AUSTRALIA
Advances enabling precision measurements of fusion cross-sections, and the development of realistic theoretical models, have together resulted in considerable progress being made towards understanding fusion involving heavy nuclei (Z1 Z2 ≥ 400). However, discrepancies between measurements and predictions are becoming evident when a consistent description of the various reaction processes is sought. Even the process of fusion spanning energies well below to well above the barrier cannot be consistently explained by the commonly used coupled channels model. The inadequacy of our current understanding, and the need for a re-examination of the assumptions in the current models of fusion is discussed. 1. Introduction Measurements [1,2] of fusion of heavy nuclei clearly demonstrated the inadequacy of the single barrier penetration model in describing fusion dynamics [3]. The need to include internal degrees of freedom, such as rotational and vibrational states of the interacting nuclei, became apparent as differences were seen amongst isotopes [1,2]. The coupled channels formalism, which includes the coupling between the relative motion and the internal degrees of freedom of the colliding nuclei, was successfully used to explain fusion observables [4,5]. Experimentally, the role of collective degrees of freedom was vividly demonstrated by the experimental barrier distributions [6] which were extracted from high precision measurements [7,8] of fusion excitation functions. This led to a flurry of activity in near barrier fusion reaction studies, and a large body of high precision data was obtained which demonstrated the role of complex vibrations and transfer reactions [5]. The high precision data necessitated further developments in the coupled reaction channels formalism [9]. Currently, this model is the most accepted and used model of low energy nuclear reactions, and in principle has the potential to describe all reaction processes within a single framework. Inadequacies in the coupled channels model as currently implemented, start becoming evident when attempting to obtain a simultaneous description of a new generation of high precision measurements of different reaction observables e.g. elastic, quasi-elastic, fusion. This paper discusses some of these aspects. ∗ †
Research Supported by an Australian Research Council Discovery Grant Permanent Address: Saha Institute of Nuclear Physics, 1/AF Bidhan Nagar, Kolkata 700 064, India.
0375-9474/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2006.12.025
M. Dasgupta et al. / Nuclear Physics A 787 (2007) 144c–149c
145c
2. The nuclear potential The internuclear potential is one of the main ingredients of any model, and one which is often questioned where there are discrepancies between predictions and measurements. The most commonly used form is the phenomenological Woods-Saxon potential defined by the depth V0 , the radius r0 , and the diffuseness parameter a, and is given by: 1/3
1/3
Vnuc (r) = −V0 {1 + exp[(r − r0 (A1 + A2 ))/a]}−1 .
(1)
Here r denotes the center-of-mass distance between the two colliding nuclei with mass numbers A1 and A2 . There are many parameterizations in the literature for V0 , r0 and a. Commonly used are those due to the Woods-Saxon parametrization [10] of the Aky¨ uzWinther potential. Another commonly used potential is the double folding [10] potential obtained from a more microscopic basis. The double folding potential is valid when nuclei are well separated and is thought to be reasonable for separations where there is a small overlap between the nuclei. It has been shown that the double folding potential can be very well described by a Woods-Saxon form [11] with a diffuseness parameter, which characterizes the fall-off of the nuclear potential, of 0.65 fm for internuclear separations down to 1 fm inside the barrier radius. 3. Model predictions and observations It is well documented that the elastic scattering cross-sections are well described by the coupled channels (CC) model using a Woods-Saxon potential with a diffuseness parameter of 0.65 fm (or equivalently a double folding potential, since as discussed above, it closely follows a Woods-Saxon potential near the barrier). Fusion cross-sections and barrier distributions at energies near the barrier are also well reproduced [5] with a CC model, but description of fusion cross-sections at energies well above the barrier is found to require larger values of the diffuseness parameter [12], with “apparent” diffuseness of up to 1.2 fm. Fusion cross-sections at energies well below the barrier also show large deviations [13] from coupled channels calculations using the standard nuclear potential. A nuclear potential with a very small depth has been proposed [14] to explain the experimental data. Thus, whilst individual processes over a limited energy range can be fitted by the coupled channels model either by changing the couplings or the internuclear potential shape, it has not been possible to obtain a simultaneous description of the different processes with a consistent parameter set. This inconsistency was not clear prior to the availability of high precision fusion crosssection data, and indeed elastic scattering and fusion were thought to be described reasonably consistently [15,16] within the coupled channels approach. High precision fusion cross-section data determines the energy of the average barrier very well, usually better than ± 1.5%, and constraint the input potential parameters in a CC calculation. Thus the consistency between elastic, inelastic and fusion (uncertainties ∼ 5 - 10%) previously found for example for the 12 C + 208 Pb reaction, is destroyed when the barrier energy is constrained by the new high precision fusion data; the elastic scattering and fusion appear to require different nuclear potential shape, characterized by different diffuseness parameters, as shown in Fig. 1. It is interesting that signs of failure of the CC model,
146c
M. Dasgupta et al. / Nuclear Physics A 787 (2007) 144c–149c 10
1400
12C
+ 208Pb
12C
+ 208Pb
1200
1
σfus (mb)
dσelas/dσRuth
1000
0.1
800
600
400
0.01
new data from Ref. [16]
from Ref. [16] 200
a = 0.56 fm
a = 0.56 fm
a = 0.95 fm
a = 0.95 fm 0
0.001 20
70
120
θC.M. (deg.)
50
70
90
EC.M. (MeV)
Figure 1. The elastic scattering angular distribution at a laboratory energy of 69.9 MeV (left panel) and fusion cross-sections as a function of centre-of-mass energy (right panel) for the 12 C + 208 Pb system. The lines are the results of a coupled channels calculation using FRESCO with two different values of the diffuseness parameter for the Woods-Saxon potential; couplings are the same as used in Ref. [16].
as currently implemented in FRESCO [17] and CCFULL [18], is evident even for reactions induced by light projectiles such as 12 C. An explanation put forward to understand the different potential shapes required to fit the elastic scattering and fusion data is that the nuclear potential deviates from a Woods-Saxon shape, and that the discrepancy arises as elastic scattering and fusion probe different parts of the nuclear potential. Elastic scattering probes the tail region, whilst fusion the internal region. Changing the shape of the nuclear potential is a possibility, and has indeed been proposed [19]. The ‘new’ potential would need to be constrained by (i) the average barrier determined from fusion and (ii) the outer region determined from elastic scattering. Calculations using an error function, with these constraints, have been tried [20] and preliminary results suggests that a simultaneous reproduction of elastic and fusion data may not be achievable simply by changing the form of the potential. 4. Above-barrier and deep sub-barrier fusion The lack of knowledge about the exact form of the potential may give rise to controversies in simultaneous descriptions of elastic scattering and fusion. However, one can ask does the CC model fare better in describing fusion alone? In the framework of the current model, fusion at energies well below the lowest barrier is tunnelling dominated, with the slope of the cross-sections as a function of energy determined by the width of the lowest barrier, which in turn is characterized by the diffuseness of the nuclear potential. Fusion at energies around the barrier is coupling dominated, giving rise to the observed distribution of barriers. At energies well above the average barrier fusion the cross-sections are
M. Dasgupta et al. / Nuclear Physics A 787 (2007) 144c–149c
147c
150
r 100
V (MeV)
High l
50
Low l
0 5
10
15
20
r (fm)
Figure 2. The calculated total internuclear potential as a function of the centre-of-mass separation between 16 O and 208 Pb. The distance of separation at the inner turning point at sub-barrier energies is similar to the barrier radius for high angular momenta.
potential dominated again, being determined mainly by the high barrier parameters. As shown in Fig. 2, the inner turning point at sub-barrier energy occurs at a similar separation distance as the barrier peak for higher -values. Thus intuitively, fusion well below the barrier and well above the barrier probes similar separation distances. In the CC approach one might then expect both regions to be explained by the same nuclear potential (with or without couplings, as they do not affect the shape of the excitation function in these energy regimes). However, as shown in Fig. 3 for the reaction of 16 O with 208 Pb, the fusion data at energies well below the barrier (Fig. 3 (a)) appear to be well reproduced using a diffuseness of 1.65 f m, in contrast the above barrier data [21] appears to require a diffuseness of 1.18 f m. Note that both these values are much higher than obtained from fits to elastic scattering data. The deep sub-barrier, and the above-barrier data taken together, represent a new challenge to theoretical models, as calculations with a nuclear potential which explains the above barrier-data do not explain the deep sub-barrier data, and vice-versa. Simultaneous explanation of the above-barrier and deep sub-barrier data using a single nuclear potential of any form may not be possible. This failure of the commonly used model appears to indicate that some physical aspects of the collision are not being modeled correctly. What might these be? In the coupled channels calculations, the fusion process is simulated by a mathematical “trick”; either using a imaginary potential or by applying an incoming wave boundary condition. Imposing a boundary condition at a critical distance means that if a shallow potential is used, as proposed [14] for example in explaining the deep sub-barrier fusion data, an un-physical scenario can result where fusion continues to occur even though no trapping potential pocket exists for higher angular momenta. The physical process of energy damping is not included explicitly in any of the models, although it is known that experimentally deep inelastic products are observed. If, as
148c
M. Dasgupta et al. / Nuclear Physics A 787 (2007) 144c–149c 1500
10
101
(a) 10
(b)
01
16O
+ 208Pb
16O
204Pb
16O
+ 208Pb
0.1 -1
+
1000
σfus (mb)
σfus (mb)
10
0.01 -2
10
10-3 0.001
500 0.0001 -4
10
a = 0.66 fm a = 1.18 fm
0.00001 -5
10
a = 0.66 fm a = 1.18 fm a = 1.65 fm
a = 1.65 fm -6 0.000001
10
-12
0
-10
-8
-6
-4
-2
0
EC.M. - B (MeV)
-10
0
10
20
30
40
EC.M. - B (MeV)
Figure 3. The measured fusion cross-sections as a function of the centre of mass energy with respect to the barrier energy B at deep sub-barrier energies (a) and all energies (b). The calculations using three different values of the diffuseness of the Woods-Saxon nuclear potential are shown. The deep sub-barrier data and above barrier data are not consistently described using the same value of the parameter.
argued above, both deep sub-barrier and above barrier data probe similar separation distances, then they are both sensitive to regions inside the average barrier radius where dissipative effect may be significant. Including such effects in the theoretical formalism may help in obtaining a simultaneous description of the above barrier and below barrier data. A related question that must be addressed is whether it is correct to treat the tunnelling of finite sized nuclei on the same footing as that of point particles. This and other aspects, such as multi-nucleon transfer [22], may need to be investigated in order to obtain a consistent explanation of the fusion process over a range of energies. 5. Fusion with light weakly-bound nuclei Fusion with weakly-bound light nuclei, both stable and radioactive, is of interest as attested by the large number of contributions to this conference. The challenge, again, is to be able to relate the breakup, transfer, complete- and incomplete-fusion processes, and obtain a consistent description of all these processes in a single framework. The data are usually described in the framework of the Continuum Discretized Coupled Channels (CDCC) framework [23]. This model is able to predict breakup cross-sections if none of the fragments are captured by the target, but a major drawback is that it cannot distinguish complete fusion from fusion of one of the breakup fragments (incomplete fusion). This is a major failing which cannot be rectified; and one which critically affects interpretation of experimental data. The distinction between complete and incompletefusion is possible if the trajectories of the breakup fragments are followed. A 3-dimensional classical trajectory model, which can relate breakup well below the barrier to complete
M. Dasgupta et al. / Nuclear Physics A 787 (2007) 144c–149c
149c
and incomplete fusion above the barrier is currently being developed [24] and verified using asymptotic observables from CDCC. 6. Discussion The past decade has seen enormous advances in experiments, with innovative techniques enabling precision measurements. Beautiful examples of exclusive measurements, even with low intensity radioactive beams, were presented at this conference. Experimentally, complete data sets for a few well chosen reactions would be very useful in furthering theoretical developments and our understanding of reaction mechanisms. The most commonly used model of low energy reactions (the coupled channels model), as currently implemented, is unable to provide a consistent description of new high quality experimental data. Using different reaction processes, and fusion data over a wide range of energies, it is argued here that physical features not currently implement in the “standard” model need to be included to make the model more realistic. Alternately, other models aiming to give a consistent description may need to be pursued. For weakly-bound systems, the most commonly used CDCC model can not distinguish complete and incomplete fusion, giving impetus to the development of a model based on classical trajectories. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.
R.G. Stokstad et al., Phys. Rev. C 21 (1980) 2427. M. Beckerman et al., Phys. Rev. Lett. 45 (1980) 1472. A.B. Balantekin et al., Phys. Rev. C 28 (1983) 1565. C.H. Dasso et al., Nucl. Phys. A 405 (1983) 381; 407 (1983) 221. M. Dasgupta et. al., Annu. Rev. Nuc. Part. Sci. 48 (1998) 401, and references therein. N. Rowley et al., Phys. Lett. B 254 (1991) 25. J.X. Wei et al., Phys. Rev. Lett. 67 (1991) 3368–3371. J.R. Leigh et al., Phys. Rev. C 52 (1995) 3151–3166. K. Hagino et. al., Phys. Rev. C 55 (1997) 276; Phys. Rev. Lett. 79 (1997) 2014. R.A. Broglia and A. Winther, Heavy Ion Reactions Lecture Notes, Volume I: Elastic and Inelastic Reactions, Benjamin/Cummings, Reading, 1981. I.I. Gontchar et al., Phys. Rev. C 69 (2004) 024610. J.O. Newton et al., Phys. Lett. B 586 (2004) 219. C.L. Jiang et al., Phys. Rev. C 73 (2006) 014613 and references therein. S¸. Mi¸sicu and H. Esbensen, Phys. Rev. Lett. 96 (2006) 112701. I.J. Thompson et al., Nucl. Phys. A 505 (1989) 84. S. Santra et al., Phys. Rev. C 64 (2001) 024602. I.J. Thompson, Comp. Phys. Reports 7 (1988), 167. K. Hagino, N. Rowley, and A.T. Kruppa, Comput. Phys. Commun. 123 (1999) 143. J.O. Newton et al., Phys. Rev. C 64 (2001) 064608. M. Zamrun and K. Hagino, University of Sendai, Private Communication. C.R. Morton et al., Phys. Rev. C 60 (1999) 044608. G. Pollarolo, AIP Conference Proccedings, eds. L. Corradi et al., 853 (2006) 29. Y. Sakuragi, M. Yahiro, and M. Kamimura, Prog. Theo. Phys. Suppl. 89 (1986) 136 A. Diaz-Torres et al., submitted to Phys. Rev. Lett. (2006).