On the variations in fusion cross sections for different light heavy-ion systems

Nuclear Physics A339 (1980) 167-174; © North-Holbtd Publishinp Co., Muterrlatn xot to be raproduced by ahotopriat or micro8lm wühout

parmiwion l}om táe publishar

ON THE VARIATIONS IN FUSION CROSS SECTIONS FOR DIFFERENT LIGHT HEAVY-ION SYSTEMS R. VANDENBOSCH

Nuclear Physics Laboratory, Unioasity of Washington, Seattle, WA 98195, USA Received 22 October 1979

Aórtraet : Fusion cross sections are calculated with a classical trajectory model incorporating the nuclear

proximity potential and ono-body proximity friction with radius and diffuseness parameters for each system taken from electron scattering results . Qualitative agreement with experiment is achieved, although significant discrepancies exist for several systems . Comparisons with recent TDHF calculations are also made .

1. Introdacdon

In the last few years precision measurements 1- t~ of fusion cross sections for and t 6 0 projectiles on p and sd shell nuclei have revealed some significant variations from system to system . Although there was at first some indication of an approximate 200 mb discontinuity between p and s-d shell nuclei, counter examples to this trend have also been reported 3' 4). It seems important to understand to what extent these variations can be understood in terms of known properties of the nuclei, such as their geometrical properties, so that the possible dependence on other properties, such as the availability of high-spin states to carry the angular momentum, can be isolated. To this end we have attempted to incorporate the known geometrical properties in a particular dynamical model forthe fusion process. We consider two heavy ions interacting by both conservative and dissipative forces. The conservative forces are derived from the proximity potential t'), and the dissipatioe forces from proximity friction ts) . The microscopic origin of the latter is nucleon exchange between the moving nuclei. We integrate the classical equations of motion to determine the trajectory of the ions. ü the reactants get trapped in the internuclear potential by energy dissipation we assume that the nuclei will eventually fuse to form an amalgamated system. This model has been fairly successful in accounting for some of the qualitative features of fusion cross-section excitation functions t 3). 'ZC

2. Para~et~rizatioo of the model The proximity potential depends on the radii and diffuseness of the nuclei involved. For not too lightnuclei it has bcen suggested that one might calculate the half-density 167

168

R . VANDENBOSCH

radius C from C = R(1-b2/R) with and b = 1 . For lighter nuclei . the diliuseness is smaller and the half-central~ensity radius is larger than given by these "standard" parameters . We have therefore gone back to the electron scattering results to-te) for charge density distributions and extracted half-central~ensity radii and diffuseness for self-conjugate nuclei . The diffuseness b is related to the skin thickness t to _ 9o bY ") Since most matter density radial dependences are not well known, we have assumed the neutron and proton densities are the same. for light self-conjugate nuclei, and have interpolated according to an A} dependence for non-selfconjugate nuclei . The values_ used, an average over several determinations in many cases, are listed in table 1 . T~atE l Half-central-density radius C and diffuseness b values deduced from electron scattering results Nucleus

'°B

' 2C ' aN "N ' 60 "O ' s0 '9 F ~°Ne ~`Mg

C

b

2 .15 2 .26 2 .44 (2 .48) 2 .65 (2 .69) (2 .72) (2 .75) 2 .79 2 .92

0 .88 0 .83 0 .78 (0.84) 0 .82 (0.88) (0.93) (0.99) 1 .04 1 .05

The values in parentheses were obtained by interpolation.

We have used these parameters to calculate fusion cross sections for a variety of light heavy-ion systems. The calculations were performed treating the orbital angular momentum (defining the impact parameter) as a continuous variable . The highest !-value for which fusion occurs, lm, was determined to an accuracy of 0.2 units, and the cross section was calculated from the relation Qr~. = a~t~lm(lm+ 1). 3. Comprieoa wiü experiment Most of the systems under consideration exhibit experimental maximum fusion cross sections at energies between 20 and 26 MeV (c.m.), and we compare the calculated and experimental values at the energy corresponding to the maximum of the fusion cross section in this energy range in fig. 1 . There are no adjustable

FUS10N CROSS SECTIONS

169

a E p b`

26 27 28 29 30 31 32 COMPOUND NUCLEUS MASS NUMBER Fig. 1 . Comparison of calcLlated (histogram) and experimental (points) fusion cross sections.

Fig . 2 . Energy dependence of the fusion cross sections for the two entrance channels leading to the' 6A1 . wmpound nucleus. Data from refs . _-`).

parameters in these calculations . (It should be remarked that the uacertainties in the charge density radii, as evidénced by variations between the results of different measurements, leads to an uacertainty in the calculated cross sections of 3-8 %.) The calculations reproduce the general trend of the data, but fail to fully reproduce some of the "structure" in the data, such as the difference in the moderate energy fusion cross sections for the 1. 2 C+ 1aN and 10B+ 160 systems, and the somewhat larger thaw average fusion cross sections for nuclei with a few sd-shell nucleons.

170

R. VANDENBOSCH

The fusion cross section for the 1 °B + 160 system has recently been shown ~) to exhibit an anomalous behavior relative to nearby systems. The fusion cross section continues to rise to nearly the highest energy investigated, with a maximum cross section approximately 35 ~ larger than that for the 1ZC+ 1aN system leading to the same compound nucleus. These results, together with our calculations for these systems, are shown in fig. 2. The calculations fail to account for the dramatic difference between the two systems. What differences there are in the calculation arise primarily from the larger average diffuseness for the 10B + 16 0 system, with a smaller difference in the same direction arising from a few percent difference in the sum of the radii. The sharp decrease in the calculated fusion cross section as 1/E decreases occurs when the maximum initial l-value for fusion stops increasing with increasing bombarding energy . In the present model this is determined by whether sufficient angular momentum is dissipated to reach an !-value for which the internuclear potential exhibits a pocket . It is important to recognize that an essential ingredient of this model is the dissipation of angular momentum as well as energy . This is illustrated in fig. 3, where we see that the potential for l = 19 does not exhibit a

1i=20 ~

r c~

W 2 W

lf " 15 .2 -

0~ 2

i 3

i 4

i 5

i 6

i 7

i .i

i

8 2 3 r (fm)

i

4

~

5

i

6

i

7

i

8

Fig . 3 . The dashed curves are the total internuclear potentials for selected 1-values. The full curves give the trajectories in ~r space. The trajectory in (a) is for h = 19, which leads to fusion . The trajectory in (b) is for h = 20 at the same initial energy .

pocket but that suûtcient angular momentum is transferred from relative orbital motion to intrinsic degrees of freedom during the collision to reach an !-value exhibiting a pocket . In the example in fig. 3a sufitcient energy is lost for the system to become trapped, whereas the next higher partial wave does not folow a trajectory with sufficient damping to be trapped (fig . 3b).

FUSION CROSS SECTIONS

171

We have examined the sensitivity ofthecalculation to the strength of the dissipative forces . Previous comparisons ts.'~ of the proximity friction model with energy losses associated with deeply inelastic collisions between much heavier nuclei have shown that this "frozen spheres" model underestimates the energy dissipation. For energies below about 40 MeV (c.m .) the calculated fusion cross sections are quite insensitive to the strength of the dissipative force, varying by only a few percent for changes of a factor of two in the strength of the friction term from that given by the model. At high energies, above the break in the calculated fusion cross sections, the results become more sensitive to the rate of energy dissipation. This is illustrated in fig. 4, where we show the fusion cross section calculated with the standard proximity friction (full curve), and with a friction twice as strong as the standard proximity friction . With the increased friction angular momentum dissipation is sufficiently complete to nearly approach the "rolling" limit of ? for transfer of orbital angular momentum into intrinsic rotation, and the fusion cross section approaches zero as 1/E goes to zero. 1400 1200 1000 a _E .~ 800 b` s00 400 200 0

0 .1

0.2

0 .3

0.4 Oá I/EcM

0 .6

0 .7

0 .8

Fig . 4. Dependence of the calculated fusion cross section on the strength of the friction . The full curve is calculated with the standard proximity friction and the dashed curve with a frictional force twig as strong . The vertical dashed line indicates the energy at which the dirtanae of closest approach for the lowest noo-tbsion l--value equals the sum of the gaff-donsity radii .

The reason for the failure of the calculations to reproduce the energy at which the cross section starts to decrease rapidly is not clear. It may simply reflect the inadequacies of the model, particularly the frozen shapes assumed. The vertical dashed line in fig. 4 indicates the energy above which the distance of closest approach for non-fusing trajectories becomes less than tha sum of the half-density radii. For the more penetrating trajectories associated with higher energies the

172

R . VANDENBOSCH

sudden approximation of frozen densities is not likely to be valid The origin of the failure of the model to account for the different behavior exhibited by the ' 2C+'4N and' °B+"O systems is also not apparent. One might wonder whether the neglect of rolling friction in the present model might be playing a role, as rolling friction would be relatively more important in a more asymmetric system . A first attemptZ°) to include rolling friction in the framework of the present model indicates its effect is relatively small. Other factors which may play a role in determining the differences between the two systems is the fact that the excitation energy of the compound system for a given E~ .m. is larger for '°B+'60 than for 'ZC+'4N. The channel spins are also larger for the former system . 4. Comparison w3tó timo-depeadent Hartree-Fork calculations There have been several time-dependent Hartree-Fuck (TDHF) calculations of fusion cross sections for light heavy-ion systems. We compare the proximity friction calculations with two of these. The first system to be considered is the' 6 0+'60, for which a number zt-2a) of TDHF calculations have been reported . We compare here with the excitation function calculations of Bonche et al. zz) . Their results together with the proximity friction model results and experimental s . e . ~ fusion cross sections are displayed in fig. 5. At low energies both calculations are in reasonable aceord with experiment. The decrease in the TDHF cross section at higher energy is primarily a result of the low-l non-fusion region predicted by these

~ 600 0

e soo aoo 200 0

0.1

0 .2

0.3

0.4

0 .5 I/E~

0.6

0.7

0.8

0.9

L0

Fig. 5. Cimparisofi of results of present calculation with TDHF results'=) and with experiment . The range of values oonsiatent with the TDHF calculation corresponds to the region between the dashed lines . She squares and triangles sine representative points from the experimental results given in refs . ° " s . 6) respectively.

FUSION CROSS SECTIONS

173

~C+ ~0 FUSION CROSS SECTIONS

Fig. 6. Comparison of results of present calculation with TDHF results s') and with e~pcriment . The circles, rectangles and triangles are from the work of refs . ~ " ~e " ") iespectivdy .

calculations. The proximity friction trajectory calculation does not exhibit a fusion window at the energies we have investigated. Krieger and Davies zs) have performed calculations for the asymmetric systems ' zC+ 160 and 1 zC+ 1s0. We compare our results with theirs and with experiment in fig. 6. The proximity model underestimates the cross sections, and the TDHF calculation overestimates the cross sections somewhat. The underestimation of the fusion cross sections at low energy in the proximity model calculations implies that this model gives too large a barrier. The TDHF calculations are expected to be a little high, since the Hartree-Fork radius comes out a little larger thaw experiment in this calculation. If the TDHF calculation scales with radius as does the proximity model calculation this would result in about a 5 % reduction in the calculated cross section . S. Co~clodae The simplé proximity friction model with a priori parameters predict low-energy fusion cross sections in . qualitative agreement with experiment. Some but not all of thé variations in maximum cross sections observed between diBerent systems are

reproduced i. The model is less successful in accotmting for the energy dependence offusion cross sections at higher energy, failing to account for the energy at which the break in the dependence of the fusion cross section with 1 /E occurs, and failing to account for the differences in behavior for related systems. Thus further work is required to elucidate all the factors determining whether fusion will occur in a particular collision. Comparison with TDHF calculations also shows qualitative but not quantitative agreement. The proximity friction model gives lower cross sections at low energies in some cases, and does not exhibit the low-1 non-fusion region indicated in some TDHF calculations . It is a pleasure to acknowledge helpful discussion with H. Doubre, A. Lazzarini and E. Plagnol. This work has been supported in part by the US Department of Energy. t Horn and Ferguson 3° ) have successfully parameterized a number of fusion exitadon functions using an empirical dependence of the cross section on the tail of the charge distribution .

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