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A resonant interpretation of gross structure in 12C + 12C and16O + 16O inelastic scattering
Volume 87B, number 3
PHYSICS LETTERS
5 November 1979
A RESONANT INTERPRETATION OF GROSS STRUCTURE IN 12C + 12C AND 160 + 160 INELASTIC SCATTERING ~r
W.A. FRIEDMAN, K.W. McVOY and M.C. NEMES Physics Department, University of Wisconsin-Madison, Madison, W153706, USA Received 23 August 1979 A barrier-top-resonancemodel calculation is found adequate to describe the gross structure behavior thus far observed in 12C + 12C and 160 + 160 inelastic scattering excitation functions. The excitation functions for 12C + 12C and 160 + 160 inelastic scattering to certain low-lying states exhibit gross structure maxima, with widths of a few MeV, in the energy range of two to four times the Coulomb barrier. Although single-particle or pocket resonances ("quasi-molecular states") have often been suggested as an explanation of these excitation function oscillations, Phillips et al [1], noting that the amount of absorption normally assumed for optical potentials will badly damp these pocket resonances, have recently shown that the Blair-Austern [2] strong absorption model is also moderately successful in accounting for the observed structure. This success stems from the fact that the model provides/-windows for entrance and exit channels through factors of the form d~t/dl. As the authors of ref. [1] point out, these /-windows, separated a few units in I by the Q-value of the reaction, slide upward in l as the bombarding energy is increased and consequently produce structure in the excitation function. Although we agree with the authors of ref. [1 ] that a Blair-Austern model can provide a fit, we wish to show here that the presence of strong internal absorption alone by no means excludes all resonant effects, In particular, we explicitly display a model, based on resonances which survive in the presence of realistic absorption, that fits the data as well as the BlairAustern description does. The absorption assumed in ref. [1 ] provides just ¢' Supported in part by the National Science Foundation, the University of Wisconsin-Madison, and FAPESP of Brazil.
the conditions necessary for barrier top resonances [3]. These orbiting-type resonances occur as a general feature of internally absorptive potentials. Like pocket resonances, they manifest themselves as poles in either the energy or/-dependence of optical wavefunctions, but unlike pocket resonances they are always sufficiently overlapping that they do not cause any individual features in the elastic S-functions, r~I. Because the barrier top resonances occur in the optical wavefunctions, they influence the/-dependence of the radial integrals I(l, l') of a DWBA analysis. Studies of well-matched heavy-ion transfer reactions have provided strong evidence for the existence of two dominant poles [3,4], one associated with each channel (entrance and exit). These poles have been found to be the one barrier top resonance in each channel which lies closest to the real l-axis. While the poles are not easily identified in elastic scattering amplitudes because of the strong overlapping, the pole nearest the real axis attains enhanced importance for the transfer reactions, which are peripheral due to the short ranged form factor [3]. Our purpose here is to test whether in inelastic scattering, as in transfer reactions, these two barrier top poles may also dominate the reaction amplitudes. If so, these poles may provide /-windows having the required properties for explainhag the inelastic gross structure. Specifically we assume, following Fuller and Dragun [4], that the inelastic radial integral I(l, l'), introduced in ref. [1 ], is given by two moving Regge poles 1(1, l') = c/(l - £)(1 - £') ,
(1) 179
Volume 87B, number 3 t00)
+160
PHYSICS LETTERS
5 November 1979
+ t2c (2+)
(3-) i50
~" 80]
/,, 125 .Q vE
:oL. i
--
~
i
t ~i--
",., iI/ ..v"
26
2'I
32
b
JsternBlair (Ref.I)
36
75 50
•
: ~--q~,,,, ~ ResonoTe Model
I s~ IS"
4o
ENERGY
I0
Fig. 1. Measured and calculated angle-integrated inelastic scattering excitation functions for t~O + I60. The data are represented by the solid curve. The dashed curve is calculated from the present barrier-top-resonance model, and the dotted curve from the Austern-Blair model of ref. [1 ].
where £ ( E ) and £ ' ( E ) are the dominant barrier t o p poles o f the entrance and the exit channels. The angle integrated cross section to a state o f spin I, given b y
(Kf/Ki) + 1)l(l'OIOllO)121I(l,/)12 ,
100
15
20
25
30
Ec. ( MeVl Fig. 2. Measured and calculated angleqntegrated inelastic scattering excitation functions for 12C + 12C. The data from ref. [ 1 ] are represented by the solid curve. The dashed curve is calculated from the present barrier-top-resonance model, and the dotted curve uses the barrier top model for £r(E) but with Fl taken as a constant (2.2). The nature o f the interplay between these rotational bands is shown b y fig. 3, which displays the two
Oft(L7~ = constant × ~(2l'
II
Q,
(2)
can be evaluated with the insertion o f eq. (1). Writing £ = £r + iFl/2, we have determined the trajectory £r(E) for the barrier t o p resonance from folded potentials for b o t h the 160 + 160 and the i2C + 12C systems. These potentials will described below. We have also used these potentials to calculate FI(E), a function which exerts a critical influence on the observed gross structure. Using the calculated barrier-top values for £(E) we obtain the results shown in figs. 1 and 2. Each curve contains an arbitrary normalization which is not determined b y our model. Because the moving/-windows which we calculate are similar to those employed in ref. [1], our calculat e d excitation functions are similar to the ones obtained there. The interpretation is very different, however. Our excitation functions result directly from resonances, in spite o f the presence o f strong absorption. Specifically, they arise from two rotational bands o f resonances which have been obtained with essentially no free parameters from entrance and exit channel optical potentials. 180
3c
"\x...
/
1 ¢
'- .... 7
I0
I
12 or #'
14
16
Fig. 3. Rotational trajectories employed for the 12C + I:C system. The solid line trajectory is for the entrance channel and employs E o = 2.6 MeV, ~2/29 = 107 keV, as given in ref. [ 1]. The dashed-line trajectory is for the exit channel, and is simply the entrance-channel curve shifted up by the Q-value of the reaction. The lorentzian curve along the/-axis is the Iwindow II - £r(E) 1-2, whose motion, as a function of E, generates the lorentzian curves shown along the E-axis, as the Regge pole £r(E) moves past even integers.
Volume 87B, number 3
PHYSICS LETTERS
trajectories (displaced in energy by the relative Qvalue of the two channels), for the l and l' = I - 2 cases of the 12C + 12C reaction. This indicates how a moving/-pole produces a peak in the factor f(E) = It - £(E)1-2 from eq. (1), which for each even/-value is shown projected onto the energy axis, producing just the pattern of overlapping peaks which was pointed out and discussed in ref. [1]. It also makes clear how the Q-displacement gives the two bands slightly different slopes, dE/d/, at a given energy, thus causing the two sequences of energy peaks to cross as a function of increasing energy or I. This happens to occur at E ~ 19 MeV, near the even integers l = 10, l' = 12. Although this produces no visible effect on the excitation function shown, it could have done so if the widths had been narrower. The values of PI (width in/) found for both reactions were about 2 in their energy ranges of interest. Calculations show that had PI been a factor of 2 smaller, the crossing would have caused a substantial enhancement of the peak at E = 19 MeV. On the other hand, had the widths been a factor of 2 larger, successive states would have overlapped enough to almost wash out all structure in the excitation function. The fact that F t 2 indicates that it is essential that the entrance channels consist of two identical particles, for only if the integer l's occurring in the sums of eq. (2) are separated by a A/larger than F l can the excitation function exhibit clearly separated maxima. If the two entrance channel particles had not been identical, both even and odd rs would have been permitted, making Pt > A/and substantially washing out the oscillations in the excitation function. This suggests that the oscillations which are seen in certain reactions with nonidentical entrance channels [e.g., in 24Mg(160, 12C) 28Si] [5] are likely to be of a different nature. To obtain the barrier top resonance parameters without fttUy searching on a complex scattering amplitude [6], we calculated the real and imaginary parts by a semiclassical procedure outlined in ref. [3]. For such a procedure to be valid, we require an optical potential whose pocket remains open over the range of energy required. The potential of Gobbi et al. [7] discussed in ref. [1 ] does, however, fill in at an energy of about 28 MeV, and is thus not convenient for a semiclassical analysis. Consequently we have used the following folded potentials for the two reactions:
5 November 1979
VO_ O = - ( 4 0 0 + i 25) exp(-r2/11.48 fm2),
(3)
VC_ C = - ( 2 2 0 + i 25) exp(-r2/9.46 fm2).
(4)
These were constructed following the general prescription discussed by Dover and Vary [8], taking
V(r) = f PA(rl)PB(r2)G(r + q -- r2) drldr 2,
(5)
with
G(r) = fiN exp(-r2/r2) , and N chosen so that
fG(r) d3r = --2T~f~h2Jmp
.
We used r 0 = 1.4 fro, f = 1.3 for 160 + 160, a n d f = 1.0 for 12C + 12C, consistent w i t h f = 1.7 used by Golin to obtain fits for 120 + 12C [9]. Gaussian functions with (r 2) given by electron scattering were used for the density functions. We have used an imaginary strength of 25 MeV, consistent with the strength required to fit 160 + 12C scattering [9]. The £r(E) was determined from the real part of the optical potential by the method discussed in ref. [3]. The numerical result obtained from the folded potentials was found to agree closely with the trajectories used in ref. [1 ]. The width, FI, was also calculated according to the procedures of ref. [3], i.e., we considered two contributions, Ps and Pw, one determined from the curvature of the real part of the effective potential at the barrier top (Fs) , the other from the value of the imaginary potential at the barrier radius (['w)" The values of F t are extremely critical for determining the structure in the excitation function. We note that the energy-widths obtained for the barrier top resonances are slowly varying with energy, a property of barrier top resonances that distinguishes them from pocket resonances, whose widths tend to increase .rapidly with energy. The trajectory for the final channel was obtained from that for the initial channel by assuming the same potential for both, so that £f(E) = £i(E + Q), where Q is the Q-value of the reaction. Hence all trajectories were obtained from potentials which were constructed via the Dover-Vary prescription without free parameters.
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Volume 87B, number 3
PHYSICS LETTERS
Examining fig. 2 we dearly see that the fit to the 12C + 12C excitation function is poor. But the sharpness of the peaks suggests that the structure most prominent here is not that associated with the gross structure. It would appear, however, that an energy average over this structure will provide oscillations more in the nature of that presented for 160 + 160, and that the present analysis would be in qualitative agreement with that as far as energy spacing is concerned. At the top o f the energy range for 12C + 12C, the potential pocket is filling in, making less valid our semiclassical determination of the barrier top poles and presumably drastically increasing the width above that calculated. We have allowed the width to grow quadratically with I above l = 15 (the point where our approximation should fail). The qualitative effect of this rapid increase in width seems reflected in the data, and the decline o f the cross section at higher energies displayed by Cormier et al. [10], may be related to the loss of the barrier top resonances. In summary we have presented a barrier-top-resonance model to account for the gross structure in two inelastic reactions. We have used reasonable optical (folded) potentials with no free parameters. The agreement found seems reasonable for 160 + 160 but poor for 12C + 12C where, however, energy averaging must be done before a meaningful comparison is attempted.
182
5 November 1979
The barrier-top-resonance model can be viewed as describing the interplay of two rotational bands o f i o n ion resonances. We emphasize that the resonances we considered, as distinct from the pocket resonances, exist even in the presence of strong absorption and they have energy widths relatively independent of energy. It would appear that these resonances, which have been useful in understanding transfer reactions, may play an important role in inelastic reactions as well.
References [1] R.L. Phillips, K.A. Erb, D.A. Bromley and J. Weneser, Phys. Rev. Lett. 42 (1979) 566. [2] N. Austern and J.S. Blair, Ann. Phys. (NY) 33 (1965) 15. [3] W.A. Friedman and C.J. Goebel, Ann. Phys. (NY) 104 (1977) 145. [4] R.C. Fuller and O. Dragun, Phys. Rev. Lett. 32 (1974) 617; B.V. Carlson and K.W. McVoy, Nucl. Phys. A292 (1977) 310. [5] M. Paul et al., Phys. Rev. Lett. 40 (1978) 1310. [6] T. Tamura and H.H. Wolter, Phys. Rev. C6 (1972) 1976. [7] A. Gobbi et al., Phys. Rev. C7 (1973) 30; W. Reilly et al., Nuovo Cimento A13 (1973) 897. [8] J.P. Vary and C.B. Dover, Phys. Rev. Lett 31 (1973) 1510. [9] M. Colin, Phys. Lett. 74B (1978) 23. [10] T.M. Cormier et al., Phys. Rev. Lett. 40 (1978) 924.
PHYSICS LETTERS
5 November 1979
A RESONANT INTERPRETATION OF GROSS STRUCTURE IN 12C + 12C AND 160 + 160 INELASTIC SCATTERING ~r
W.A. FRIEDMAN, K.W. McVOY and M.C. NEMES Physics Department, University of Wisconsin-Madison, Madison, W153706, USA Received 23 August 1979 A barrier-top-resonancemodel calculation is found adequate to describe the gross structure behavior thus far observed in 12C + 12C and 160 + 160 inelastic scattering excitation functions. The excitation functions for 12C + 12C and 160 + 160 inelastic scattering to certain low-lying states exhibit gross structure maxima, with widths of a few MeV, in the energy range of two to four times the Coulomb barrier. Although single-particle or pocket resonances ("quasi-molecular states") have often been suggested as an explanation of these excitation function oscillations, Phillips et al [1], noting that the amount of absorption normally assumed for optical potentials will badly damp these pocket resonances, have recently shown that the Blair-Austern [2] strong absorption model is also moderately successful in accounting for the observed structure. This success stems from the fact that the model provides/-windows for entrance and exit channels through factors of the form d~t/dl. As the authors of ref. [1] point out, these /-windows, separated a few units in I by the Q-value of the reaction, slide upward in l as the bombarding energy is increased and consequently produce structure in the excitation function. Although we agree with the authors of ref. [1 ] that a Blair-Austern model can provide a fit, we wish to show here that the presence of strong internal absorption alone by no means excludes all resonant effects, In particular, we explicitly display a model, based on resonances which survive in the presence of realistic absorption, that fits the data as well as the BlairAustern description does. The absorption assumed in ref. [1 ] provides just ¢' Supported in part by the National Science Foundation, the University of Wisconsin-Madison, and FAPESP of Brazil.
the conditions necessary for barrier top resonances [3]. These orbiting-type resonances occur as a general feature of internally absorptive potentials. Like pocket resonances, they manifest themselves as poles in either the energy or/-dependence of optical wavefunctions, but unlike pocket resonances they are always sufficiently overlapping that they do not cause any individual features in the elastic S-functions, r~I. Because the barrier top resonances occur in the optical wavefunctions, they influence the/-dependence of the radial integrals I(l, l') of a DWBA analysis. Studies of well-matched heavy-ion transfer reactions have provided strong evidence for the existence of two dominant poles [3,4], one associated with each channel (entrance and exit). These poles have been found to be the one barrier top resonance in each channel which lies closest to the real l-axis. While the poles are not easily identified in elastic scattering amplitudes because of the strong overlapping, the pole nearest the real axis attains enhanced importance for the transfer reactions, which are peripheral due to the short ranged form factor [3]. Our purpose here is to test whether in inelastic scattering, as in transfer reactions, these two barrier top poles may also dominate the reaction amplitudes. If so, these poles may provide /-windows having the required properties for explainhag the inelastic gross structure. Specifically we assume, following Fuller and Dragun [4], that the inelastic radial integral I(l, l'), introduced in ref. [1 ], is given by two moving Regge poles 1(1, l') = c/(l - £)(1 - £') ,
(1) 179
Volume 87B, number 3 t00)
+160
PHYSICS LETTERS
5 November 1979
+ t2c (2+)
(3-) i50
~" 80]
/,, 125 .Q vE
:oL. i
--
~
i
t ~i--
",., iI/ ..v"
26
2'I
32
b
JsternBlair (Ref.I)
36
75 50
•
: ~--q~,,,, ~ ResonoTe Model
I s~ IS"
4o
ENERGY
I0
Fig. 1. Measured and calculated angle-integrated inelastic scattering excitation functions for t~O + I60. The data are represented by the solid curve. The dashed curve is calculated from the present barrier-top-resonance model, and the dotted curve from the Austern-Blair model of ref. [1 ].
where £ ( E ) and £ ' ( E ) are the dominant barrier t o p poles o f the entrance and the exit channels. The angle integrated cross section to a state o f spin I, given b y
(Kf/Ki) + 1)l(l'OIOllO)121I(l,/)12 ,
100
15
20
25
30
Ec. ( MeVl Fig. 2. Measured and calculated angleqntegrated inelastic scattering excitation functions for 12C + 12C. The data from ref. [ 1 ] are represented by the solid curve. The dashed curve is calculated from the present barrier-top-resonance model, and the dotted curve uses the barrier top model for £r(E) but with Fl taken as a constant (2.2). The nature o f the interplay between these rotational bands is shown b y fig. 3, which displays the two
Oft(L7~ = constant × ~(2l'
II
Q,
(2)
can be evaluated with the insertion o f eq. (1). Writing £ = £r + iFl/2, we have determined the trajectory £r(E) for the barrier t o p resonance from folded potentials for b o t h the 160 + 160 and the i2C + 12C systems. These potentials will described below. We have also used these potentials to calculate FI(E), a function which exerts a critical influence on the observed gross structure. Using the calculated barrier-top values for £(E) we obtain the results shown in figs. 1 and 2. Each curve contains an arbitrary normalization which is not determined b y our model. Because the moving/-windows which we calculate are similar to those employed in ref. [1], our calculat e d excitation functions are similar to the ones obtained there. The interpretation is very different, however. Our excitation functions result directly from resonances, in spite o f the presence o f strong absorption. Specifically, they arise from two rotational bands o f resonances which have been obtained with essentially no free parameters from entrance and exit channel optical potentials. 180
3c
"\x...
/
1 ¢
'- .... 7
I0
I
12 or #'
14
16
Fig. 3. Rotational trajectories employed for the 12C + I:C system. The solid line trajectory is for the entrance channel and employs E o = 2.6 MeV, ~2/29 = 107 keV, as given in ref. [ 1]. The dashed-line trajectory is for the exit channel, and is simply the entrance-channel curve shifted up by the Q-value of the reaction. The lorentzian curve along the/-axis is the Iwindow II - £r(E) 1-2, whose motion, as a function of E, generates the lorentzian curves shown along the E-axis, as the Regge pole £r(E) moves past even integers.
Volume 87B, number 3
PHYSICS LETTERS
trajectories (displaced in energy by the relative Qvalue of the two channels), for the l and l' = I - 2 cases of the 12C + 12C reaction. This indicates how a moving/-pole produces a peak in the factor f(E) = It - £(E)1-2 from eq. (1), which for each even/-value is shown projected onto the energy axis, producing just the pattern of overlapping peaks which was pointed out and discussed in ref. [1]. It also makes clear how the Q-displacement gives the two bands slightly different slopes, dE/d/, at a given energy, thus causing the two sequences of energy peaks to cross as a function of increasing energy or I. This happens to occur at E ~ 19 MeV, near the even integers l = 10, l' = 12. Although this produces no visible effect on the excitation function shown, it could have done so if the widths had been narrower. The values of PI (width in/) found for both reactions were about 2 in their energy ranges of interest. Calculations show that had PI been a factor of 2 smaller, the crossing would have caused a substantial enhancement of the peak at E = 19 MeV. On the other hand, had the widths been a factor of 2 larger, successive states would have overlapped enough to almost wash out all structure in the excitation function. The fact that F t 2 indicates that it is essential that the entrance channels consist of two identical particles, for only if the integer l's occurring in the sums of eq. (2) are separated by a A/larger than F l can the excitation function exhibit clearly separated maxima. If the two entrance channel particles had not been identical, both even and odd rs would have been permitted, making Pt > A/and substantially washing out the oscillations in the excitation function. This suggests that the oscillations which are seen in certain reactions with nonidentical entrance channels [e.g., in 24Mg(160, 12C) 28Si] [5] are likely to be of a different nature. To obtain the barrier top resonance parameters without fttUy searching on a complex scattering amplitude [6], we calculated the real and imaginary parts by a semiclassical procedure outlined in ref. [3]. For such a procedure to be valid, we require an optical potential whose pocket remains open over the range of energy required. The potential of Gobbi et al. [7] discussed in ref. [1 ] does, however, fill in at an energy of about 28 MeV, and is thus not convenient for a semiclassical analysis. Consequently we have used the following folded potentials for the two reactions:
5 November 1979
VO_ O = - ( 4 0 0 + i 25) exp(-r2/11.48 fm2),
(3)
VC_ C = - ( 2 2 0 + i 25) exp(-r2/9.46 fm2).
(4)
These were constructed following the general prescription discussed by Dover and Vary [8], taking
V(r) = f PA(rl)PB(r2)G(r + q -- r2) drldr 2,
(5)
with
G(r) = fiN exp(-r2/r2) , and N chosen so that
fG(r) d3r = --2T~f~h2Jmp
.
We used r 0 = 1.4 fro, f = 1.3 for 160 + 160, a n d f = 1.0 for 12C + 12C, consistent w i t h f = 1.7 used by Golin to obtain fits for 120 + 12C [9]. Gaussian functions with (r 2) given by electron scattering were used for the density functions. We have used an imaginary strength of 25 MeV, consistent with the strength required to fit 160 + 12C scattering [9]. The £r(E) was determined from the real part of the optical potential by the method discussed in ref. [3]. The numerical result obtained from the folded potentials was found to agree closely with the trajectories used in ref. [1 ]. The width, FI, was also calculated according to the procedures of ref. [3], i.e., we considered two contributions, Ps and Pw, one determined from the curvature of the real part of the effective potential at the barrier top (Fs) , the other from the value of the imaginary potential at the barrier radius (['w)" The values of F t are extremely critical for determining the structure in the excitation function. We note that the energy-widths obtained for the barrier top resonances are slowly varying with energy, a property of barrier top resonances that distinguishes them from pocket resonances, whose widths tend to increase .rapidly with energy. The trajectory for the final channel was obtained from that for the initial channel by assuming the same potential for both, so that £f(E) = £i(E + Q), where Q is the Q-value of the reaction. Hence all trajectories were obtained from potentials which were constructed via the Dover-Vary prescription without free parameters.
181
Volume 87B, number 3
PHYSICS LETTERS
Examining fig. 2 we dearly see that the fit to the 12C + 12C excitation function is poor. But the sharpness of the peaks suggests that the structure most prominent here is not that associated with the gross structure. It would appear, however, that an energy average over this structure will provide oscillations more in the nature of that presented for 160 + 160, and that the present analysis would be in qualitative agreement with that as far as energy spacing is concerned. At the top o f the energy range for 12C + 12C, the potential pocket is filling in, making less valid our semiclassical determination of the barrier top poles and presumably drastically increasing the width above that calculated. We have allowed the width to grow quadratically with I above l = 15 (the point where our approximation should fail). The qualitative effect of this rapid increase in width seems reflected in the data, and the decline o f the cross section at higher energies displayed by Cormier et al. [10], may be related to the loss of the barrier top resonances. In summary we have presented a barrier-top-resonance model to account for the gross structure in two inelastic reactions. We have used reasonable optical (folded) potentials with no free parameters. The agreement found seems reasonable for 160 + 160 but poor for 12C + 12C where, however, energy averaging must be done before a meaningful comparison is attempted.
182
5 November 1979
The barrier-top-resonance model can be viewed as describing the interplay of two rotational bands o f i o n ion resonances. We emphasize that the resonances we considered, as distinct from the pocket resonances, exist even in the presence of strong absorption and they have energy widths relatively independent of energy. It would appear that these resonances, which have been useful in understanding transfer reactions, may play an important role in inelastic reactions as well.
References [1] R.L. Phillips, K.A. Erb, D.A. Bromley and J. Weneser, Phys. Rev. Lett. 42 (1979) 566. [2] N. Austern and J.S. Blair, Ann. Phys. (NY) 33 (1965) 15. [3] W.A. Friedman and C.J. Goebel, Ann. Phys. (NY) 104 (1977) 145. [4] R.C. Fuller and O. Dragun, Phys. Rev. Lett. 32 (1974) 617; B.V. Carlson and K.W. McVoy, Nucl. Phys. A292 (1977) 310. [5] M. Paul et al., Phys. Rev. Lett. 40 (1978) 1310. [6] T. Tamura and H.H. Wolter, Phys. Rev. C6 (1972) 1976. [7] A. Gobbi et al., Phys. Rev. C7 (1973) 30; W. Reilly et al., Nuovo Cimento A13 (1973) 897. [8] J.P. Vary and C.B. Dover, Phys. Rev. Lett 31 (1973) 1510. [9] M. Colin, Phys. Lett. 74B (1978) 23. [10] T.M. Cormier et al., Phys. Rev. Lett. 40 (1978) 924.