- Email: [email protected]
Search for experimental evidence of chaotic behavior in nuclear scattering
4 February 1999
Physics Letters B 447 Ž1999. 41–45
Search for experimental evidence of chaotic behavior in nuclear scattering G.V. Martı´ 1, A.J. Pacheco, J.E. Testoni, D. Abriola, O.A. Capurro, D.E. Di Gregorio, ´ J.O. Fernandez Niello, E. Achterberg, D.E. Alvarez ´ Laboratorio TANDAR, Departamento de Fısica, Comision AÕ. del Libertador 8250, 1429 Buenos Aires, ´ ´ Nacional de Energıa ´ Atomica, ´ Argentina Received 21 October 1998; revised 14 December 1998 Editor: R.H. Siemssen
Abstract Angular distributions for the elastic and inelastic scattering in the 16 O q 28 Si system have been measured in two energy regions, one close to the Coulomb barrier and the other well above. Fine steps in both bombarding energy and scattering angle make it possible to compare the data with the theoretical calculations that predict, for each of these regions, distinctive cross-section patterns in correspondence with the classical occurrence of either regular or chaotic regimes. The experimental results show specific differences between the two explored energy ranges in qualitative agreement with the theoretical predictions. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 24.60.Lz; 25.70.Bc
Since the earliest stages of heavy-ion research, the study of elastic and inelastic scattering in low-absorption reaction systems has revealed peculiar behaviors. Several theoretical approaches have successfully accounted for different aspects of the experimental results, although a comprehensive picture has failed to emerge from these models, as was pointed out in the review article of Ref. w1x. More recently, it has been suggested that the observed irregular fluctuations of the cross sections as a function of both the bombarding energy and the scattering angle could have as their basic mechanism the chaotic regime that, under certain circumstances, underlies classical
1
E-mail: [email protected]
scattering w2–8x. According to these studies, the chaotic regime in scattering situations becomes prevalent whenever there is a significant coupling between the relative motion of the nuclei and some internal degrees of freedom. This situation is more likely to occur at near barrier energies in the presence of a surface-transparent optical potential which makes it possible to explore the region inside the barrier for a large number of trajectories w3–7x. The system 16 O q 28 Si exhibits the above mentioned features associated with chaotic motion, namely a low-absorption optical potential and an adequate degree of freedom related to the rotational character of the 28 Si nucleus. In a recent paper w8x the connection between chaos in classical scattering and its quantum-mechanical manifestations has been investigated in the
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 1 5 6 2 - 7
42
G.V. Martı´ et al.r Physics Letters B 447 (1999) 41–45
context of low energy nuclear reactions. Detailed calculations performed with the coupled-channel code FRESCO w9x for the 16 O q 28 Si system showed that the differences between the regular and the chaotic regimes predicted by the classical models have a clear quantal correspondence that should be accessible to experimental verification. In particular, it was shown that the mere observation of fluctuations in an angular distribution or in an excitation function is not in itself enough to distinguish between the two regimes. Rather, the signature of either regular or erratic behavior is found in a bombarding energyscattering angle plot of the cross sections. This is illustrated in the three dimensional plot shown in Fig. 1, constructed using the data of Ref. w8,10x, which shows the results of the coupled-channel calculation of the cross section for the elastic and inelastic scattering of 16 O leading to the first 2q excited state of 28 Si in the regular and chaotic regions pointed out by the classical analysis. It should be stressed that these are ordinary coupled-channel calculations performed with standard codes. The main point of Ref. w8x is associated with the observation of two different cross-section patterns, as shown in Fig. 1, and its interpretation in terms of either chaos or regularity in the corresponding classical picture. The purpose of the calculation was not to describe specific angular distributions in detail, but rather to distinguish globally between these different types of behavior. The experimental verification of whether these predicted differences in the cross-section patterns are indeed present requires more detailed measurements in the relevant bombarding energy-scattering angle regions than those reported in the literature w1,11x. In this letter, we present the results of an experimental search for the predicted signature of the regular and chaotic behaviors in the 16 O q 28 Si system. For that purpose differential cross sections for the elastic scattering and for the inelastic scattering leading to the first excited 2q state of 28 Si have been measured. These measurements were carried out over two Elab y uc.m . ranges Ž39 MeV F Elab F 45 MeV, 908 F uc.m .F 1208, and 70 MeV F Elab F 77 MeV, 848 F uc.m .F 1108. which approximately correspond to the predicted chaotic and regular regimes shown in Fig. 1. The data were taken using angular and energy steps of 0.88 and 0.5 or 1 MeV, respec-
Fig. 1. Differential cross sections generated by the code FRESCO Žusing the data from Ref. w8,10x. for the inelastic scattering leading to the 2q Ž1.78 MeV. state of 28 Si in the reaction 16 O q 28 Si. The three-dimensional plots of the angular distributions as a function of the laboratory bombarding energy and the center-ofmass scattering-angle are shown in the two regions which, according to the classical predictions, are characterized by regular or chaotic behavior Žsee Fig. 11 Žtop. of Ref. w8x..
tively, small enough to enable an adequate comparison with the typical structures predicted by the theory. The experiments were performed using the 20 UD accelerator TANDAR in Buenos Aires. Beams of 16 O were directed onto a 50 mgrcm2-thick silicon target evaporated on a 10 mgrcm2 carbon backing. The reaction products were detected in coincidence using two solid-state position-sensitive detectors. The data were recorded in event-by-event mode, each event being defined by four parameters: the energy and the position of each of the two reaction products.
G.V. Martı´ et al.r Physics Letters B 447 (1999) 41–45
By using simple two-body kinematic relationships the off-line analysis allows one to unambiguously identify the relevant reaction channels and to separate them from other channels stemming from the carbon backing. Absolute cross sections were obtained with the aid of two solid-state detectors placed at forward angles to monitor Rutherford scattering. The measured elastic scattering angular distribution taken at Elab s 41 MeV could be directly compared with the data of Ref. w12x, with which it was found to be in good agreement. Further experimental details and results will be given in a future paper. A sample of the measured data is shown in Fig. 2, which shows the angular distributions for the inelastic 2q Ž1.78 MeV. state, measured at E lab s 72 MeV ŽFig. 2a. and at Elab s 44 MeV ŽFig. 2b.. According to the theoretical calculations, these two energies correspond to the regular and chaotic regimes, respectively. The solid curves in this figure are intended to guide the eye through the most relevant angular structures defined by the data points, disregarding less significant fluctuations. As previously mentioned, Fig. 2 shows that the analysis of an angular
Fig. 2. Measured angular distribution for the inelastic scattering leading to the 2q state Ž1.78 MeV. of 28 Si in the reaction 16 Oq28 Si at: Ža. Elab s 72 MeV and Žb. Elab s 44 MeV. The solid curves are intended to guide the eye through the relevant structure defined by the data points.
43
distribution at a single energy in each of the two regions does not provide in itself a clear signature of either type of behavior. The experimental cross sections as a function of both scattering angle and bombarding energy for the elastic and the 2q Ž1.78 MeV. inelastic channel are shown in Fig. 3. The dark curves correspond to the experimental angular distributions. For the sake of clarity, these are representative curves such as those illustrated in Fig. 2, rather than the actual data points. The white curves are linear interpolations between adjacent experimental angular distributions. For both elastic and inelastic scattering the results exhibit a pronounced difference in the qualitative behavior when comparing the low-energy regime with the high-energy regime. In the high-energy region the valleys and ridges follow a markedly regular pattern which is very well defined by all the measured angular distributions. In turn, such regularities are not apparent in the low-energy region which shows rather nonregular shifts of the positions of the maxima and the minima when going from one energy to the next, and also a much less regular trend in the evolution of the absolute values of the cross section. The qualitative difference in the behavior of the experimental cross-section pattern between the predicted regular and chaotic energy regimes appears to be in correspondence with that illustrated in Fig. 1. In order to make the comparison between the theoretical calculations and the experimental results somewhat more quantitative, the different cross-section patterns have been analyzed using a procedure that is intended to be particularly sensitive to the relevant qualitative features discussed above. The following is an outline of the procedure: for a given region in the E lab y uc.m . plane an arbitrarily oriented straight segment Ždefined by an orientation angle c and limited by the boundaries of the region. is taken. Next, the one dimensional function that arises from the corresponding cut of the cross-section surface along that segment is considered, and the total number of relative maxima and minima is counted. These steps are then repeated for a representative set of parallel segments that cover the region under consideration. For this angle c the total number of maxima and minima per unit length over all the parallel segments, nŽ c ., is computed. By
G.V. Martı´ et al.r Physics Letters B 447 (1999) 41–45
44
Fig. 3. Differential cross sections for elastic and inelastic scattering in the 16 O q28 Si system, in the two regions of the bombarding energy–scattering angle plane that are predicted to present regular or chaotic behavior Žhigh-energy and low-energy regimes, respectively. according to Ref. w8x. The black curves represent the experimental angular distributions as was illustrated in Fig. 2. The white curves are linear interpolations to guide the eye.
plotting nŽ c . vs c , one should obtain two possible extreme results. For a very regular pattern of the cross-sections, the function nŽ c . is expected to show a pronounced structure. Irregular patterns Žsuch as those obtained as a consequence of the theoretical calculations for the chaotic region. should produce more or less flat nŽ c . functions. The degree of regularity may be finally characterized by a single number e obtained as follows:
e;
)
Ý n Ž c . y ² n: ² n:
2
Ž 1.
with ² n: the average value of nŽ c .. Additional details, related to the normalizations and the choice of the relative scales between the energy and the angle variables, will be given in a future publication. In order to test the usefulness of e as an estimator, this procedure was first applied to paradigmatic patterns. For this purpose, regular patterns consisting
of parallel ridges and valleys were constructed using appropriate two-variable analytic functions. Paradigmatic irregular patterns were obtained from arrays of gaussian peaks whose centroids and widths were randomly distributed. In a subsequent stage, the theoretical patterns resulting from the coupled-channel calculations of Ref. w8x and the present experimental data for inelastic scattering were tested using the same procedure. The results obtained are summarized in Table 1. The first row shows, as a reference, the values of e obtained for two representative
Table 1 Values of the parameter e obtained from the application of the procedure described in the text for the different cases Case
e
Case
e
Regular paradigmatic 31 Irregular paradigmatic 7.8 Regular theoretical 33 Chaotic theoretical 14 High-energy experimental 27 Low-energy experimental 5.8
G.V. Martı´ et al.r Physics Letters B 447 (1999) 41–45
paradigmatic regular and irregular cases. The second row corresponds to the theoretical coupled-channel results for the Elab y uc.m . windows shown in Fig. 1. The third row presents the resulting values of e for the experimental inelastic cross-section patterns shown in the right-hand side of Fig. 3, corresponding to the same E lab y uc.m . windows. The error bars of the experimental cross sections do not contribute significantly to the uncertainties in the values of e and therefore these values were quoted without error in Table 1. Several conclusions can be drawn from these results. In the first place, the estimator e is able to quantitatively differentiate between regular and irregular patterns, as is illustrated with the specific paradigmatic examples presented here. Secondly, it is shown that according to these criteria the theoretical patterns for the high-energy and low-energy regimes can be characterized as regular and irregular, respectively, in accordance with the regular and chaotic character discussed in Ref. w8x. Finally, the experimental values for the same regions are also in very reasonable agreement with both the theory and the paradigmatic cases. In summary, we have searched for manifestations of regular and chaotic behavior in nuclear scattering which, according to a classical model, can be associated with definite regions in the bombarding energy-scattering angle plane. Standard quantal coupled-channel calculations predict a characteristic two-dimensional cross section pattern for each type of regime. The present experimental results for both elastic and inelastic scattering were found to be in reasonable agreement with the theoretical predictions, both from a qualitative comparison between the cross-section patterns and from a quantitative estimate of their regular or irregular character. Therefore, the present data lend support to the interpretation that under certain circumstances chaotic
45
behavior in nuclear scattering may manifest itself in the bombarding energy-scattering angle cross-section patterns. Acknowledgements The technical support of M.Ramırez is greatly ´ acknowledged. Financial supports from the Consejo Nacional de Investigaciones Cientıficas y Tecnicas ´ ´ ŽCONICET. and Fundacion ´ Antorchas are acknowledged. Some of us ŽA.J.P., J.E.T., D.E.D.G. and J.O.F.N.. are fellows of the Carrera del Investigador Cientıfico of the CONICET. ´ References w1x P. Braun-Munzinger, J. Barrette, Phys. Rep. 87 Ž1982. 209. w2x A. Glaesner, W. Dunweber, M. Bantel, W. Hering, D. Kon¨ nerth, R. Ritzka, W. Trautmann, W. Trombik, W. Zipper, Nucl. Phys. A 509 Ž1990. 331. w3x A. Rapisarda, M. Baldo, Phys. Rev. Lett. 66 Ž1991. 2581. w4x R. Blumel, U. Smilansky, Phys. Rev. Lett. 60 Ž1988. 477. ¨ w5x M. Baldo, E.G. Lanza, A. Rapisarda, Nucl. Phys. A 545 Ž1992. 467C. w6x C.H. Dasso, M. Gallardo, M. Saraceno, Nucl. Phys. A 549 Ž1992. 265. w7x C.H. Dasso, M. Gallardo, M. Saraceno, Nucl. Phys. A 587 Ž1995. 339. w8x C.H. Dasso, G. Pollarolo, M. Saraceno, Nucl. Phys. A 602 Ž1996. 77. w9x I.J. Thompson, Comp. Phys. Rep. 7 Ž1988. 167. w10x G. Pollarolo, private communication. w11x J.M. Oliveira Jr., A. Lepine-Szily, A.C.C. Villari, R. Licht´ enhaller, L.C. Gomes, W. Sciani, P. Fachini, G.L. Lima, ¨ A.C.G. Martins, V. Chyiste, J.M. Casandijian, A.J. Pacheco, J.E. Testoni, D. Abriola, A.O. Macchiavelli, F. Hasenbalg, O.A. Capurro, D. Tomasi, J.O. Fernandez Niello, D. Bran´ dizzi, Phys. Rev. C 53 Ž1996. 2926. w12x P. Braun-Munzinger, G.M. Berkowitz, M. Gai, C.M. Jachcinski, T.R. Renner, C.D. Uhlhorn, J. Barrette, M.J. LeVine, Phys. Rev. C 24 Ž1981. 1010.
Physics Letters B 447 Ž1999. 41–45
Search for experimental evidence of chaotic behavior in nuclear scattering G.V. Martı´ 1, A.J. Pacheco, J.E. Testoni, D. Abriola, O.A. Capurro, D.E. Di Gregorio, ´ J.O. Fernandez Niello, E. Achterberg, D.E. Alvarez ´ Laboratorio TANDAR, Departamento de Fısica, Comision AÕ. del Libertador 8250, 1429 Buenos Aires, ´ ´ Nacional de Energıa ´ Atomica, ´ Argentina Received 21 October 1998; revised 14 December 1998 Editor: R.H. Siemssen
Abstract Angular distributions for the elastic and inelastic scattering in the 16 O q 28 Si system have been measured in two energy regions, one close to the Coulomb barrier and the other well above. Fine steps in both bombarding energy and scattering angle make it possible to compare the data with the theoretical calculations that predict, for each of these regions, distinctive cross-section patterns in correspondence with the classical occurrence of either regular or chaotic regimes. The experimental results show specific differences between the two explored energy ranges in qualitative agreement with the theoretical predictions. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 24.60.Lz; 25.70.Bc
Since the earliest stages of heavy-ion research, the study of elastic and inelastic scattering in low-absorption reaction systems has revealed peculiar behaviors. Several theoretical approaches have successfully accounted for different aspects of the experimental results, although a comprehensive picture has failed to emerge from these models, as was pointed out in the review article of Ref. w1x. More recently, it has been suggested that the observed irregular fluctuations of the cross sections as a function of both the bombarding energy and the scattering angle could have as their basic mechanism the chaotic regime that, under certain circumstances, underlies classical
1
E-mail: [email protected]
scattering w2–8x. According to these studies, the chaotic regime in scattering situations becomes prevalent whenever there is a significant coupling between the relative motion of the nuclei and some internal degrees of freedom. This situation is more likely to occur at near barrier energies in the presence of a surface-transparent optical potential which makes it possible to explore the region inside the barrier for a large number of trajectories w3–7x. The system 16 O q 28 Si exhibits the above mentioned features associated with chaotic motion, namely a low-absorption optical potential and an adequate degree of freedom related to the rotational character of the 28 Si nucleus. In a recent paper w8x the connection between chaos in classical scattering and its quantum-mechanical manifestations has been investigated in the
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 1 5 6 2 - 7
42
G.V. Martı´ et al.r Physics Letters B 447 (1999) 41–45
context of low energy nuclear reactions. Detailed calculations performed with the coupled-channel code FRESCO w9x for the 16 O q 28 Si system showed that the differences between the regular and the chaotic regimes predicted by the classical models have a clear quantal correspondence that should be accessible to experimental verification. In particular, it was shown that the mere observation of fluctuations in an angular distribution or in an excitation function is not in itself enough to distinguish between the two regimes. Rather, the signature of either regular or erratic behavior is found in a bombarding energyscattering angle plot of the cross sections. This is illustrated in the three dimensional plot shown in Fig. 1, constructed using the data of Ref. w8,10x, which shows the results of the coupled-channel calculation of the cross section for the elastic and inelastic scattering of 16 O leading to the first 2q excited state of 28 Si in the regular and chaotic regions pointed out by the classical analysis. It should be stressed that these are ordinary coupled-channel calculations performed with standard codes. The main point of Ref. w8x is associated with the observation of two different cross-section patterns, as shown in Fig. 1, and its interpretation in terms of either chaos or regularity in the corresponding classical picture. The purpose of the calculation was not to describe specific angular distributions in detail, but rather to distinguish globally between these different types of behavior. The experimental verification of whether these predicted differences in the cross-section patterns are indeed present requires more detailed measurements in the relevant bombarding energy-scattering angle regions than those reported in the literature w1,11x. In this letter, we present the results of an experimental search for the predicted signature of the regular and chaotic behaviors in the 16 O q 28 Si system. For that purpose differential cross sections for the elastic scattering and for the inelastic scattering leading to the first excited 2q state of 28 Si have been measured. These measurements were carried out over two Elab y uc.m . ranges Ž39 MeV F Elab F 45 MeV, 908 F uc.m .F 1208, and 70 MeV F Elab F 77 MeV, 848 F uc.m .F 1108. which approximately correspond to the predicted chaotic and regular regimes shown in Fig. 1. The data were taken using angular and energy steps of 0.88 and 0.5 or 1 MeV, respec-
Fig. 1. Differential cross sections generated by the code FRESCO Žusing the data from Ref. w8,10x. for the inelastic scattering leading to the 2q Ž1.78 MeV. state of 28 Si in the reaction 16 O q 28 Si. The three-dimensional plots of the angular distributions as a function of the laboratory bombarding energy and the center-ofmass scattering-angle are shown in the two regions which, according to the classical predictions, are characterized by regular or chaotic behavior Žsee Fig. 11 Žtop. of Ref. w8x..
tively, small enough to enable an adequate comparison with the typical structures predicted by the theory. The experiments were performed using the 20 UD accelerator TANDAR in Buenos Aires. Beams of 16 O were directed onto a 50 mgrcm2-thick silicon target evaporated on a 10 mgrcm2 carbon backing. The reaction products were detected in coincidence using two solid-state position-sensitive detectors. The data were recorded in event-by-event mode, each event being defined by four parameters: the energy and the position of each of the two reaction products.
G.V. Martı´ et al.r Physics Letters B 447 (1999) 41–45
By using simple two-body kinematic relationships the off-line analysis allows one to unambiguously identify the relevant reaction channels and to separate them from other channels stemming from the carbon backing. Absolute cross sections were obtained with the aid of two solid-state detectors placed at forward angles to monitor Rutherford scattering. The measured elastic scattering angular distribution taken at Elab s 41 MeV could be directly compared with the data of Ref. w12x, with which it was found to be in good agreement. Further experimental details and results will be given in a future paper. A sample of the measured data is shown in Fig. 2, which shows the angular distributions for the inelastic 2q Ž1.78 MeV. state, measured at E lab s 72 MeV ŽFig. 2a. and at Elab s 44 MeV ŽFig. 2b.. According to the theoretical calculations, these two energies correspond to the regular and chaotic regimes, respectively. The solid curves in this figure are intended to guide the eye through the most relevant angular structures defined by the data points, disregarding less significant fluctuations. As previously mentioned, Fig. 2 shows that the analysis of an angular
Fig. 2. Measured angular distribution for the inelastic scattering leading to the 2q state Ž1.78 MeV. of 28 Si in the reaction 16 Oq28 Si at: Ža. Elab s 72 MeV and Žb. Elab s 44 MeV. The solid curves are intended to guide the eye through the relevant structure defined by the data points.
43
distribution at a single energy in each of the two regions does not provide in itself a clear signature of either type of behavior. The experimental cross sections as a function of both scattering angle and bombarding energy for the elastic and the 2q Ž1.78 MeV. inelastic channel are shown in Fig. 3. The dark curves correspond to the experimental angular distributions. For the sake of clarity, these are representative curves such as those illustrated in Fig. 2, rather than the actual data points. The white curves are linear interpolations between adjacent experimental angular distributions. For both elastic and inelastic scattering the results exhibit a pronounced difference in the qualitative behavior when comparing the low-energy regime with the high-energy regime. In the high-energy region the valleys and ridges follow a markedly regular pattern which is very well defined by all the measured angular distributions. In turn, such regularities are not apparent in the low-energy region which shows rather nonregular shifts of the positions of the maxima and the minima when going from one energy to the next, and also a much less regular trend in the evolution of the absolute values of the cross section. The qualitative difference in the behavior of the experimental cross-section pattern between the predicted regular and chaotic energy regimes appears to be in correspondence with that illustrated in Fig. 1. In order to make the comparison between the theoretical calculations and the experimental results somewhat more quantitative, the different cross-section patterns have been analyzed using a procedure that is intended to be particularly sensitive to the relevant qualitative features discussed above. The following is an outline of the procedure: for a given region in the E lab y uc.m . plane an arbitrarily oriented straight segment Ždefined by an orientation angle c and limited by the boundaries of the region. is taken. Next, the one dimensional function that arises from the corresponding cut of the cross-section surface along that segment is considered, and the total number of relative maxima and minima is counted. These steps are then repeated for a representative set of parallel segments that cover the region under consideration. For this angle c the total number of maxima and minima per unit length over all the parallel segments, nŽ c ., is computed. By
G.V. Martı´ et al.r Physics Letters B 447 (1999) 41–45
44
Fig. 3. Differential cross sections for elastic and inelastic scattering in the 16 O q28 Si system, in the two regions of the bombarding energy–scattering angle plane that are predicted to present regular or chaotic behavior Žhigh-energy and low-energy regimes, respectively. according to Ref. w8x. The black curves represent the experimental angular distributions as was illustrated in Fig. 2. The white curves are linear interpolations to guide the eye.
plotting nŽ c . vs c , one should obtain two possible extreme results. For a very regular pattern of the cross-sections, the function nŽ c . is expected to show a pronounced structure. Irregular patterns Žsuch as those obtained as a consequence of the theoretical calculations for the chaotic region. should produce more or less flat nŽ c . functions. The degree of regularity may be finally characterized by a single number e obtained as follows:
e;
)
Ý n Ž c . y ² n: ² n:
2
Ž 1.
with ² n: the average value of nŽ c .. Additional details, related to the normalizations and the choice of the relative scales between the energy and the angle variables, will be given in a future publication. In order to test the usefulness of e as an estimator, this procedure was first applied to paradigmatic patterns. For this purpose, regular patterns consisting
of parallel ridges and valleys were constructed using appropriate two-variable analytic functions. Paradigmatic irregular patterns were obtained from arrays of gaussian peaks whose centroids and widths were randomly distributed. In a subsequent stage, the theoretical patterns resulting from the coupled-channel calculations of Ref. w8x and the present experimental data for inelastic scattering were tested using the same procedure. The results obtained are summarized in Table 1. The first row shows, as a reference, the values of e obtained for two representative
Table 1 Values of the parameter e obtained from the application of the procedure described in the text for the different cases Case
e
Case
e
Regular paradigmatic 31 Irregular paradigmatic 7.8 Regular theoretical 33 Chaotic theoretical 14 High-energy experimental 27 Low-energy experimental 5.8
G.V. Martı´ et al.r Physics Letters B 447 (1999) 41–45
paradigmatic regular and irregular cases. The second row corresponds to the theoretical coupled-channel results for the Elab y uc.m . windows shown in Fig. 1. The third row presents the resulting values of e for the experimental inelastic cross-section patterns shown in the right-hand side of Fig. 3, corresponding to the same E lab y uc.m . windows. The error bars of the experimental cross sections do not contribute significantly to the uncertainties in the values of e and therefore these values were quoted without error in Table 1. Several conclusions can be drawn from these results. In the first place, the estimator e is able to quantitatively differentiate between regular and irregular patterns, as is illustrated with the specific paradigmatic examples presented here. Secondly, it is shown that according to these criteria the theoretical patterns for the high-energy and low-energy regimes can be characterized as regular and irregular, respectively, in accordance with the regular and chaotic character discussed in Ref. w8x. Finally, the experimental values for the same regions are also in very reasonable agreement with both the theory and the paradigmatic cases. In summary, we have searched for manifestations of regular and chaotic behavior in nuclear scattering which, according to a classical model, can be associated with definite regions in the bombarding energy-scattering angle plane. Standard quantal coupled-channel calculations predict a characteristic two-dimensional cross section pattern for each type of regime. The present experimental results for both elastic and inelastic scattering were found to be in reasonable agreement with the theoretical predictions, both from a qualitative comparison between the cross-section patterns and from a quantitative estimate of their regular or irregular character. Therefore, the present data lend support to the interpretation that under certain circumstances chaotic
45
behavior in nuclear scattering may manifest itself in the bombarding energy-scattering angle cross-section patterns. Acknowledgements The technical support of M.Ramırez is greatly ´ acknowledged. Financial supports from the Consejo Nacional de Investigaciones Cientıficas y Tecnicas ´ ´ ŽCONICET. and Fundacion ´ Antorchas are acknowledged. Some of us ŽA.J.P., J.E.T., D.E.D.G. and J.O.F.N.. are fellows of the Carrera del Investigador Cientıfico of the CONICET. ´ References w1x P. Braun-Munzinger, J. Barrette, Phys. Rep. 87 Ž1982. 209. w2x A. Glaesner, W. Dunweber, M. Bantel, W. Hering, D. Kon¨ nerth, R. Ritzka, W. Trautmann, W. Trombik, W. Zipper, Nucl. Phys. A 509 Ž1990. 331. w3x A. Rapisarda, M. Baldo, Phys. Rev. Lett. 66 Ž1991. 2581. w4x R. Blumel, U. Smilansky, Phys. Rev. Lett. 60 Ž1988. 477. ¨ w5x M. Baldo, E.G. Lanza, A. Rapisarda, Nucl. Phys. A 545 Ž1992. 467C. w6x C.H. Dasso, M. Gallardo, M. Saraceno, Nucl. Phys. A 549 Ž1992. 265. w7x C.H. Dasso, M. Gallardo, M. Saraceno, Nucl. Phys. A 587 Ž1995. 339. w8x C.H. Dasso, G. Pollarolo, M. Saraceno, Nucl. Phys. A 602 Ž1996. 77. w9x I.J. Thompson, Comp. Phys. Rep. 7 Ž1988. 167. w10x G. Pollarolo, private communication. w11x J.M. Oliveira Jr., A. Lepine-Szily, A.C.C. Villari, R. Licht´ enhaller, L.C. Gomes, W. Sciani, P. Fachini, G.L. Lima, ¨ A.C.G. Martins, V. Chyiste, J.M. Casandijian, A.J. Pacheco, J.E. Testoni, D. Abriola, A.O. Macchiavelli, F. Hasenbalg, O.A. Capurro, D. Tomasi, J.O. Fernandez Niello, D. Bran´ dizzi, Phys. Rev. C 53 Ž1996. 2926. w12x P. Braun-Munzinger, G.M. Berkowitz, M. Gai, C.M. Jachcinski, T.R. Renner, C.D. Uhlhorn, J. Barrette, M.J. LeVine, Phys. Rev. C 24 Ž1981. 1010.