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On the limits to fusion in light heavy-ion systems
Volume 95B, number 2
PHYSICS LETTERS
22 September 1980
ON THE LIMITS TO FUSION IN LIGHT HEAVY-ION SYSTEMS J.J. KOLATA
University of Notre Dame, Notre Dame, IN, USA Received 28 June 1980
Certain striking features of the interaction of 12C and 160 ions wxth each other are shown to be related to the limits to fusion m these light systems. It appears that the limiting mechanism is m some way dependent on properties of the compound system, and that its onset is more violent than previously suspected.
The reaction cross section for the 12C + 12C system has recently been shown [1] to display some striking phenomena which are not easily explained on the basis of current understanding of the i o n - i o n potential. In particular, the trajectory of L c, the critical angular momentum for fusion, shows discontinuities in slope at successive even values of l, as is appropriate for systems of identical bosons. However, the transitions are much more rapid than predicted on the basis of optical-model calculations using potentials available in the literature. Secondly, at high energies the trajectory of L c appears to follow the extended ground-state band of 24Mg, despite the fact that this band is not likely to be yrast [2]. Finally, the transition between the low- and high-energy regimes is also astonishingly abrupt. In this letter, we demonstrate that these phenomena are not peculiar to the 12C + 12C system, but rather are common features of the interaction of 12C and 160 ions with each other. In addition, we point out a characteristic signature in certain reaction cross sections which occurs when the limits to fusion are reached. Consider the partial-wave expansion of a reaction cross section:
a~i) = rr~2 ~ ( 2 l + 1)TIP(I i) , l where ~ is the reduced wavelength, T l the optical model transmission coefficient, and Pl(i) the proba-
(1)
billty that the system will emerge in the ith channel. In particular, the optical-model total reaction cross section results if P/(i) is set equal to unity for all partial waves. The critical angular momentum for fusion, L c, is then defined in the sharp-cutoff approach
TIPI(f) = 1,
for l < L c ,
= 0,
for l > L c ,
(2)
and the series (1) is summed to yield o~f) = 7r7~2 (L c + 1) 2
(distinguishable particles),
o~f)= rrX2(L c + 1)(L c + 2)
(identical bosons).
(3)
The critical angular momentum ~s computed from the experimental fusion cross section using eq. (3), where the orbital angular momentum is treated as a continuous variable in order to approximately compensate for use of the sharp-cutoff model. The experimental values o f L c for the 12C + 12C system are shown m fig. 1, where we plot excitation energy in the compound system versus Lc(L c + 1) in order to bring out the quasi-rotational nature of the data. Also shown in fig. 1 are the trajectories of the grazing angular momentum lg (defined as the angular momentum for which Tt = 0.5 at a given center-ofmass energy) and the critical angular momentum for the total reaction cross section l c (computed from eq. (3) using the predicted total reaction cross section), as calculated from an optical-model potential [3] which fits the elastic-scattering data. Note first that
215
Volume 95B, number 2 O
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PHYSICS LETTERS 12
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1(~*1) Fig. ], Critical-angular-momentum trajectories for the 12C + 12C reaction. The curves labeled !g and l c trace the loci of the grazing angular momentum and the critical angular momentum for the total reaction cross section calculated with the optical-model potential of ref, [3] (see text for details).
The curve labeled Expt is calculated from the experimental total fusion cross section of ref. [1 ]. The straight line represents the extension of the 24Mg gsb (having moment-of-inertia parameter ~2/20 = 205 keV). the trajectory lg shows distinct discontinuities in slope at successive integral values of angular momentum, which are related to "shape resonances" in the rather transparent optical-model potential necessary to describe the 12C + 12C elastic data. They occur in both even and odd partial waves because symmetrization of the reaction amplitudes has not yet been invoked. In contrast, the trajectory of l c shows these discontinuities in slope at successive even/-values only since properly symmetrized amplitudes were used to calculate the theoretical reaction cross section. Note also that 1c and lg, whde they are approximately parallel, do not coincide. This difference (which disappears below the Coulomb barrier) implies that reaction cross sections calculated from eq. (3) using as input models for the trajectory of the grazing angular momentum will be systematically too high. Comparing now with the experimental result, it appears that the critical angular momentum for fusion at low energies is approximately two units tess than that computed for the total reaction cross section at the same c.m. energy. This is consistent with a model in which the last available partial wave does not fuse, but instead is distributed over several direct reaction channels. The experimental trajectory at 216
22 September 1980
c.m. energies less than 25 MeV also shows discontinuities in slope at consecutive even/-values, but the transitions are far more abrupt than expected from the optical-model calculation. Next, note the really dramatic change In the behavior of the trajectory which occurs just as it crosses the extended groundstate band (gsb) of 24Mg (having moment-of-inertia parameter ~ 2 / 2 0 = 205 keV). From this point, the critical angular momentum appears to track the gsb to at least 110 MeV [4]. The crossover at E x = 40 MeV is correlated with a striking resonance-hke structure which appears in several reaction channels but is strongest in the 160 yield [1]. We now demonstrate that all of the features discussed above appear in the critical-angular-momentum trajectory for the 160 + 160 system (fig. 2). In this case, the transition to the regime of limited fusion occurs at I = 16 and is correlated with a strong resonance [5,6] in the 20Ne yield. We have been able to follow the trajectory to quite high energy, and find that discontinuities in slope continue well beyond the crossover. This is somewhat surprising since it appears to suggest that partial waves four tO six units below grazing are sensitive to changes in the most peripheral partial waves. On the other hand, we cannot be certain that the mechanism which produces these features is the same at high energies as it is below the crossover where optical-potential effects dominate. 1
0
6
8
.
.
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.
14 .
16
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Fig. 2. Critical-angular-momentum trajectories for the 160 + 160 system. In this case, the straight line is a least-squares fit to the high-energy portion of the experimental data. It is observed to have h2/20 = 165 keV, and to pass through the ground state of 32S. See also fig. 1 caption.
Volume 95B, number 2
22 September 1980
PHYSICS LETTERS
The average behavior of the trajectory of L c is also of some interest, since no experimental information presently exists on high-spin states in 32S. On the basis of the known positions of the 2 + and 4 + levels [7], it would appear that the 32S gsb is vibrational rather than quasi-rotational. Nevertheless, we performed a least-squares fit to the data in fig. 2 (above the crossover point) and found that a straight line having/t2/20 = 165 keV and passing through the origin (the ground state of 32S) gave the best description of the average high-energy behavior of L c. This result is remarkable for two reasons. First of all, it seems unlikely that dynamic models for fusion limitation could explain the observation that the criticalangular-momentum trajectory in the high-energy regime extrapolates back to the ground state of the compound system for both 12C + 12C and 160 + 160. Secondly, a shell-model calculation due to Wddenthal [8] predicts that the 32S gsb becomes quasi-rotational just above the 4 + state, with average moment-of-inertia parameter h2/20 = 175 keV. The low-energy data on the trajectory o f L c provide an opportumty for comparison with predictions using several optical-model potentials. The calculation shown, which uses a rather extreme surface-transparent potential [9], graphically demonstrates the extent to which optical-model predictions underestimate the suddeness of the observed transitions. A calculation using an explicitly angular-momentum-dependent potential [10] does slightly better, but still cannot achieve a good representation of the experimental trajectory. The final system we wall consider is 12C + 160, which is somewhat more complicated by virtue of the fact that all partial waves can contribute to the reaction. As one result of this, the predicted reaction cross section using a standard optical-model potential [11 ] is essentially smooth even though the grazing-angularmomentum trajectory shows resonance-like structure (fig. 3). The experimental data [12] (which have been supplemented at low energies by results from other laboratories [13] and at high energies by recent resuits at the University of Notre Dame) exhibit much structure. The crossover point occurs at 1 = 14 and E x = 37 MeV and is accompanied by a strong resonance in the 2°Ne channel [12]. In this case, one might also expect a crossover with an odd-spin band having a different band-head energy and moment of
0
8
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14
16
18
20
22
24
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6
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Fig. 3. Critical-angular-momentum tralectories for 12C + 16O. The straight line is the extension of the 2SSi gsb. See also fig. 1 caption. inertia. Possible signatures of such an event occur at l = 15 and E x = 42 MeV but also at 1 = 11 and E x = 33 MeV, suggesting the need for further work in the energy region from 1 0 - 2 0 MeV c.m. The experimental trajectory does, however, provide a simple explanation for the observation that the structures at E x = 37 and 42 MeV appear to have an energy spacing characteristic of a change of two units in angular momentum [14]. The moment-of-inertia parameter for the 28Si gsb is much greater than that corresponding to the optical-model grazing angular momentum, leading to a larger separation between consecutive/-values above the crossover. In conclusion, critical-angular-momentum trajectories for the interactions of 12C and 160 ions with each other have been shown to display several features in common. The most striking of these are the association of the limits to fusion in the high energy regime with the extension of the ground-state band in the three systems revolved, and the correlation of the crossing point with a strong resonance-like structure in certain reaction channels. The former result implies that the fusion-limiting mechanism depends upon properties of the compound system. The latter behavior suggests that this mechanism, whatever its nature, does more violence to the i o n - i o n interaction than previously suspected. Some clue to the reaction mechanism involved might be found in the fact that, in each case, it is the alpha-transfer channel which most strongly displays structure. 217
Volume 95B, number 2
PHYSICS LETTERS
References [1 ] J.J. Kolata et al., Phys. Rev. C21 (1980) 579. [2] D. Glas and U. Mosel, Phys. Lett. 78B (1978) 9 [3] W. Reilly et al., in. Proc. 5th Intern. Conf. on Nuclear reactions induced by heavy ions (Heidelberg, Germany, 1969), eds. R. Bock and W.R. Hering (North-Holland, Amsterdam, 1970) p. 95. [4] M.N. Namboodiri, E.T. Chulick and J.B. Natowitz, Nucl. Phys. A263 (1976) 491. [5] P.P. Singh et al., Phys. Rev. Lett. 28 (1972) 1714. [6] J.J. Kolata et al., Phys, Rev. C16 (1977) 891 ; C19 (1979) 2235.
218
22 September 1980
[7] P.M. Endt and C. van der Leun, Nucl. Phys. A310 (1978) 1. [8] B.H. Wildenthal, private communication. [9] A. Gobbi et al., Phys. Rev. C7 (1973) 30. [10] R.A. Chatwm et al., Phys. Rev. C1 (1970) 795. [11 ] R.E. Malmin, Argonne Physics Division Informal Report No. Phy-1972 F, unpubhshed [12] J.J. Kolata et al., Phys. Lett. 65B (1976) 333; Phys. Rev. C19 (1979) 408 [13] D. Branford, B.N. Nagorka and J.O. Newton, J. Phys. G3 (1977) 1565; Y.D. Chan et al., Nucl. Phys. A303 (1978) 500, B. Cujec and C.A. Barnes, Nucl. Phys. A266 (1976) 461. [14] P. Sperr et al., Phys. Rev. Lett. 36 (1976) 405.
PHYSICS LETTERS
22 September 1980
ON THE LIMITS TO FUSION IN LIGHT HEAVY-ION SYSTEMS J.J. KOLATA
University of Notre Dame, Notre Dame, IN, USA Received 28 June 1980
Certain striking features of the interaction of 12C and 160 ions wxth each other are shown to be related to the limits to fusion m these light systems. It appears that the limiting mechanism is m some way dependent on properties of the compound system, and that its onset is more violent than previously suspected.
The reaction cross section for the 12C + 12C system has recently been shown [1] to display some striking phenomena which are not easily explained on the basis of current understanding of the i o n - i o n potential. In particular, the trajectory of L c, the critical angular momentum for fusion, shows discontinuities in slope at successive even values of l, as is appropriate for systems of identical bosons. However, the transitions are much more rapid than predicted on the basis of optical-model calculations using potentials available in the literature. Secondly, at high energies the trajectory of L c appears to follow the extended ground-state band of 24Mg, despite the fact that this band is not likely to be yrast [2]. Finally, the transition between the low- and high-energy regimes is also astonishingly abrupt. In this letter, we demonstrate that these phenomena are not peculiar to the 12C + 12C system, but rather are common features of the interaction of 12C and 160 ions with each other. In addition, we point out a characteristic signature in certain reaction cross sections which occurs when the limits to fusion are reached. Consider the partial-wave expansion of a reaction cross section:
a~i) = rr~2 ~ ( 2 l + 1)TIP(I i) , l where ~ is the reduced wavelength, T l the optical model transmission coefficient, and Pl(i) the proba-
(1)
billty that the system will emerge in the ith channel. In particular, the optical-model total reaction cross section results if P/(i) is set equal to unity for all partial waves. The critical angular momentum for fusion, L c, is then defined in the sharp-cutoff approach
TIPI(f) = 1,
for l < L c ,
= 0,
for l > L c ,
(2)
and the series (1) is summed to yield o~f) = 7r7~2 (L c + 1) 2
(distinguishable particles),
o~f)= rrX2(L c + 1)(L c + 2)
(identical bosons).
(3)
The critical angular momentum ~s computed from the experimental fusion cross section using eq. (3), where the orbital angular momentum is treated as a continuous variable in order to approximately compensate for use of the sharp-cutoff model. The experimental values o f L c for the 12C + 12C system are shown m fig. 1, where we plot excitation energy in the compound system versus Lc(L c + 1) in order to bring out the quasi-rotational nature of the data. Also shown in fig. 1 are the trajectories of the grazing angular momentum lg (defined as the angular momentum for which Tt = 0.5 at a given center-ofmass energy) and the critical angular momentum for the total reaction cross section l c (computed from eq. (3) using the predicted total reaction cross section), as calculated from an optical-model potential [3] which fits the elastic-scattering data. Note first that
215
Volume 95B, number 2 O
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PHYSICS LETTERS 12
14
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1(~*1) Fig. ], Critical-angular-momentum trajectories for the 12C + 12C reaction. The curves labeled !g and l c trace the loci of the grazing angular momentum and the critical angular momentum for the total reaction cross section calculated with the optical-model potential of ref, [3] (see text for details).
The curve labeled Expt is calculated from the experimental total fusion cross section of ref. [1 ]. The straight line represents the extension of the 24Mg gsb (having moment-of-inertia parameter ~2/20 = 205 keV). the trajectory lg shows distinct discontinuities in slope at successive integral values of angular momentum, which are related to "shape resonances" in the rather transparent optical-model potential necessary to describe the 12C + 12C elastic data. They occur in both even and odd partial waves because symmetrization of the reaction amplitudes has not yet been invoked. In contrast, the trajectory of l c shows these discontinuities in slope at successive even/-values only since properly symmetrized amplitudes were used to calculate the theoretical reaction cross section. Note also that 1c and lg, whde they are approximately parallel, do not coincide. This difference (which disappears below the Coulomb barrier) implies that reaction cross sections calculated from eq. (3) using as input models for the trajectory of the grazing angular momentum will be systematically too high. Comparing now with the experimental result, it appears that the critical angular momentum for fusion at low energies is approximately two units tess than that computed for the total reaction cross section at the same c.m. energy. This is consistent with a model in which the last available partial wave does not fuse, but instead is distributed over several direct reaction channels. The experimental trajectory at 216
22 September 1980
c.m. energies less than 25 MeV also shows discontinuities in slope at consecutive even/-values, but the transitions are far more abrupt than expected from the optical-model calculation. Next, note the really dramatic change In the behavior of the trajectory which occurs just as it crosses the extended groundstate band (gsb) of 24Mg (having moment-of-inertia parameter ~ 2 / 2 0 = 205 keV). From this point, the critical angular momentum appears to track the gsb to at least 110 MeV [4]. The crossover at E x = 40 MeV is correlated with a striking resonance-hke structure which appears in several reaction channels but is strongest in the 160 yield [1]. We now demonstrate that all of the features discussed above appear in the critical-angular-momentum trajectory for the 160 + 160 system (fig. 2). In this case, the transition to the regime of limited fusion occurs at I = 16 and is correlated with a strong resonance [5,6] in the 20Ne yield. We have been able to follow the trajectory to quite high energy, and find that discontinuities in slope continue well beyond the crossover. This is somewhat surprising since it appears to suggest that partial waves four tO six units below grazing are sensitive to changes in the most peripheral partial waves. On the other hand, we cannot be certain that the mechanism which produces these features is the same at high energies as it is below the crossover where optical-potential effects dominate. 1
0
6
8
.
.
I0 .
12 .
.
14 .
16
(, 18
20
,."
,o
22
24
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20
Go 30
tial
• I0
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/ I 150
I :500
I 450
I 600
1(~*1)
Fig. 2. Critical-angular-momentum trajectories for the 160 + 160 system. In this case, the straight line is a least-squares fit to the high-energy portion of the experimental data. It is observed to have h2/20 = 165 keV, and to pass through the ground state of 32S. See also fig. 1 caption.
Volume 95B, number 2
22 September 1980
PHYSICS LETTERS
The average behavior of the trajectory of L c is also of some interest, since no experimental information presently exists on high-spin states in 32S. On the basis of the known positions of the 2 + and 4 + levels [7], it would appear that the 32S gsb is vibrational rather than quasi-rotational. Nevertheless, we performed a least-squares fit to the data in fig. 2 (above the crossover point) and found that a straight line having/t2/20 = 165 keV and passing through the origin (the ground state of 32S) gave the best description of the average high-energy behavior of L c. This result is remarkable for two reasons. First of all, it seems unlikely that dynamic models for fusion limitation could explain the observation that the criticalangular-momentum trajectory in the high-energy regime extrapolates back to the ground state of the compound system for both 12C + 12C and 160 + 160. Secondly, a shell-model calculation due to Wddenthal [8] predicts that the 32S gsb becomes quasi-rotational just above the 4 + state, with average moment-of-inertia parameter h2/20 = 175 keV. The low-energy data on the trajectory o f L c provide an opportumty for comparison with predictions using several optical-model potentials. The calculation shown, which uses a rather extreme surface-transparent potential [9], graphically demonstrates the extent to which optical-model predictions underestimate the suddeness of the observed transitions. A calculation using an explicitly angular-momentum-dependent potential [10] does slightly better, but still cannot achieve a good representation of the experimental trajectory. The final system we wall consider is 12C + 160, which is somewhat more complicated by virtue of the fact that all partial waves can contribute to the reaction. As one result of this, the predicted reaction cross section using a standard optical-model potential [11 ] is essentially smooth even though the grazing-angularmomentum trajectory shows resonance-like structure (fig. 3). The experimental data [12] (which have been supplemented at low energies by results from other laboratories [13] and at high energies by recent resuits at the University of Notre Dame) exhibit much structure. The crossover point occurs at 1 = 14 and E x = 37 MeV and is accompanied by a strong resonance in the 2°Ne channel [12]. In this case, one might also expect a crossover with an odd-spin band having a different band-head energy and moment of
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18
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22
24
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Fig. 3. Critical-angular-momentum tralectories for 12C + 16O. The straight line is the extension of the 2SSi gsb. See also fig. 1 caption. inertia. Possible signatures of such an event occur at l = 15 and E x = 42 MeV but also at 1 = 11 and E x = 33 MeV, suggesting the need for further work in the energy region from 1 0 - 2 0 MeV c.m. The experimental trajectory does, however, provide a simple explanation for the observation that the structures at E x = 37 and 42 MeV appear to have an energy spacing characteristic of a change of two units in angular momentum [14]. The moment-of-inertia parameter for the 28Si gsb is much greater than that corresponding to the optical-model grazing angular momentum, leading to a larger separation between consecutive/-values above the crossover. In conclusion, critical-angular-momentum trajectories for the interactions of 12C and 160 ions with each other have been shown to display several features in common. The most striking of these are the association of the limits to fusion in the high energy regime with the extension of the ground-state band in the three systems revolved, and the correlation of the crossing point with a strong resonance-like structure in certain reaction channels. The former result implies that the fusion-limiting mechanism depends upon properties of the compound system. The latter behavior suggests that this mechanism, whatever its nature, does more violence to the i o n - i o n interaction than previously suspected. Some clue to the reaction mechanism involved might be found in the fact that, in each case, it is the alpha-transfer channel which most strongly displays structure. 217
Volume 95B, number 2
PHYSICS LETTERS
References [1 ] J.J. Kolata et al., Phys. Rev. C21 (1980) 579. [2] D. Glas and U. Mosel, Phys. Lett. 78B (1978) 9 [3] W. Reilly et al., in. Proc. 5th Intern. Conf. on Nuclear reactions induced by heavy ions (Heidelberg, Germany, 1969), eds. R. Bock and W.R. Hering (North-Holland, Amsterdam, 1970) p. 95. [4] M.N. Namboodiri, E.T. Chulick and J.B. Natowitz, Nucl. Phys. A263 (1976) 491. [5] P.P. Singh et al., Phys. Rev. Lett. 28 (1972) 1714. [6] J.J. Kolata et al., Phys, Rev. C16 (1977) 891 ; C19 (1979) 2235.
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[7] P.M. Endt and C. van der Leun, Nucl. Phys. A310 (1978) 1. [8] B.H. Wildenthal, private communication. [9] A. Gobbi et al., Phys. Rev. C7 (1973) 30. [10] R.A. Chatwm et al., Phys. Rev. C1 (1970) 795. [11 ] R.E. Malmin, Argonne Physics Division Informal Report No. Phy-1972 F, unpubhshed [12] J.J. Kolata et al., Phys. Lett. 65B (1976) 333; Phys. Rev. C19 (1979) 408 [13] D. Branford, B.N. Nagorka and J.O. Newton, J. Phys. G3 (1977) 1565; Y.D. Chan et al., Nucl. Phys. A303 (1978) 500, B. Cujec and C.A. Barnes, Nucl. Phys. A266 (1976) 461. [14] P. Sperr et al., Phys. Rev. Lett. 36 (1976) 405.