Systematics of precise nuclear fusion cross sections: the need for a new dynamical treatment of fusion?

Physics Letters B 586 (2004) 219–224 www.elsevier.com/locate/physletb

Systematics of precise nuclear fusion cross sections: the need for a new dynamical treatment of fusion? J.O. Newton a , R.D. Butt a , M. Dasgupta a , D.J. Hinde a,∗ , I.I. Gontchar a,b , C.R. Morton a , K. Hagino c a Department of Nuclear Physics, Research School of Physical Sciences and Engineering, Australian National University,

Canberra, ACT 0200, Australia b Omsk State Transport University, pr. Marksa 35, Omsk RU-64406, Russia c Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan

Received 17 December 2003; received in revised form 12 February 2004; accepted 23 February 2004 Editor: J.P. Schiffer

Abstract A large number of precision fusion excitation functions, at energies above the average fusion barriers, have been fitted using the Woods–Saxon form for the nuclear potential. Values of the empirical diffuseness parameter greatly exceed those which generally reproduce elastic scattering data and tend to increase strongly with the reaction charge product Z1 Z2 . Possible reasons for this may lie in the Woods–Saxon form being inappropriate, or in the reduction of fusion cross sections by processes such as deep-inelastic collisions. These results point to a need for renewed efforts in dynamical calculations of fusion.  2004 Published by Elsevier B.V. PACS: 25.70.J; 24.10.Eq; 21.60.Ev; 27.60.+j Keywords: Heavy-ion fusion; Nuclear potential; Deep-inelastic scattering

Many measurements have now been made of precise fusion excitation functions [1–3] for mass-asymmetric reactions, with the aim of determining fusion barrier distributions. In interpreting these data, both simplified [4–6] and realistic [7,8] coupled channels codes have been used. An energy-independent Woods–Saxon (WS) form for the real nuclear poten-

* Corresponding author.

E-mail address: [email protected] (D.J. Hinde). 0370-2693/$ – see front matter  2004 Published by Elsevier B.V. doi:10.1016/j.physletb.2004.02.052

tial: VN (r) = −

V0 1/3 
r−r0 A1/3 1 −r0 A2 1 + exp a

(1)

is used in these codes. Here A1 and A2 are the mass numbers of the projectile and target nuclei, V0 is the depth, r0 is the radius parameter, and a is the diffuseness parameter. The slope of the fusion cross sections at energies well above the fusion barrier VB is relatively insensitive to the couplings that change the shape of the fu-

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sion barrier distribution itself [9]. Above-barrier cross sections have thus often [2] been fitted by varying only the parameters of the WS potential. The most sensitive parameter is the diffuseness a, as long as the calculated barrier is constrained to match the experimentally determined average barrier energy (by adjusting either the depth or radius parameter). Many fits reported in the literature [2,3] have been carried out with the approximate coupled channels code CCMOD [6]. Above-barrier excitation functions have been reproduced to better than 1% simply by optimizing a, whilst constraining the calculated average VB to reproduce the experimental value. Thus the empirical diffuseness parameters may be thought of as characterising the behaviour of the above-barrier fusion excitation functions. However, the question arises as to whether these values (a ∼ 1 fm) relate to the actual shape of the nuclear potentials, since they are much larger than the values of a ∼ 0.65 fm extracted by fitting elastic scattering data [10,11]. Furthermore, double-folding model calculations [12,13] can be well approximated by the WS form, down to separation distances somewhat smaller than the barrier radius RB , typically with a < 0.7 fm. Thus it may be that the empirical diffuseness required to fit the fusion excitation functions is large because of other physical effects not included in commonly used [4–8] fusion models. Only now have precise fusion excitation functions been measured for a sufficient number of reactions to allow a broadly-based investigation. We report here the first systematic study, focussing on the physical interpretation of the large values of the empirical diffuseness that have previously been reported only for some individual reactions [2]. In calculations of above-barrier fusion cross sections, changing the diffuseness of the WS potential (whilst maintaining VB ) scales the above-barrier cross sections by a factor which is essentially energyindependent [14]. This is illustrated in Fig. 1, where experimental data for the reaction 16 O + 208 Pb have been divided by a factor = 0.9 and 1.1, and refitted by optimising the diffuseness and radius parameter of the WS potential. Fits of equal quality are obtained, but the values of a required are very different. This highlights a problem in trying to determine empirical values of a, namely, that the experimental data should not only have high precision in the relative cross sections, but also high accuracy in the absolute cross

Fig. 1. Excitation functions for the reaction 16 O + 208 Pb, assuming

has the values 0.9, 1.0 and 1.1. The experimental points are indicated by filled circles [38]. The open points are the latter divided by the indicated values of ; the corresponding values of a required to fit these data with V0 = 100 MeV are also shown. The dashed and dotted lines are the fits to the points giving the values of a shown. The dot-dash line, labelled X–P, derives from a calculation with the extra-push model adjusted to give the experimental VB .

sections σfus (abs). A measurement whose efficiency was actually lower than had been assumed, would give a value of a which is too large, as shown in Fig. 1. It would thus be valuable to have data for a given reaction from more than one laboratory; this is rarely the case, however a procedure has been developed to identify measurements where the efficiency may be in error. Fusion cross sections for 46 reactions were fitted with a χ 2 minimisation code based on CCMOD [6]. For reactions forming heavy compound nuclei, where a substantial part of the cross section comprises fission, the cross sections are more correctly described as capture cross sections, as no distinction could be made between compound-nucleus fission and quasifission in the measurements [15]. Only values of σfus exceeding 200 mb were included, to essentially eliminate sensitivity to the couplings that may be present. This was confirmed by comparisons of fits with and without couplings for a number of cases. The diffuseness determined from fitting a given data set depends somewhat on the other parameters of the

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WS potential. Thus in order to carry out a systematic study, a consistent approach is required in fitting the experimental data, as described below. The values of r0 and a, giving the best fit to the fusion data were found for a fixed value of V0 . An “efficiency factor”

could be introduced in the data fitting procedure to account for identified errors in the σfus (abs). Thus

= 1.0 would mean that the σfus (abs) were taken as correct, whereas = 0.8 was applied to cross sections believed to be (for example) 20% low. Further details of the procedures used will be published elsewhere [9]. The approximate coupled channels code CCFUS [4] calculates fusion cross sections based on the Wong formula [16]:  RB2 h¯ ω  ln 1 + exp[2π(E − VB )/h¯ ω] . (2) 2E However, the Wong formula does not take into account the l-dependence of the barrier curvature h¯ ω or radius. It has been shown [14] that neglect of the latter, for E > VB , leads to fitted values of a that are too large. Thus the shift of the barrier to smaller separation distances with increasing l has to be taken into account in order to obtain a meaningful fit to the above barrier cross sections. This is achieved in CCMOD, following Ref. [14], by replacing RB in Eq. (2) by: σ (E) =

RE = RB − a ln[1 + 2(E − VB )/VB ]

(3)

for E > VB . The use of the modified Wong formula in CCMOD results in fast calculations. However, it must be recognised that a result of using the Wong formula is that the loss of the potential pocket at high angular momentum does not limit fusion, unlike in the code CCFULL [7], which solves the Schrödinger equation and uses the ingoing wave boundary condition at the minimum in the pocket. However, until the pocket is lost, CCFULL and CCMOD give essentially identical results for zerocoupling calculations. The experimental data themselves show no evidence for limitation of fusion due to loss of the potential pocket at high angular momenta. They can be extremely well fitted by CCMOD up to the highest measured energies, for example up to 1.7VB for 19 F + 208 Pb. Capture may be expected to occur even where there is no pocket, because of tangential friction, as for example in the extra-push model due to Swiatecki [17,18].

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Using the code CCFULL, the data at high energies can only be reproduced with very deep nuclear potentials, for which CCMOD and CCFULL agree well. It can be argued that these deep potentials are unrealistic [19]. Consequently, for this systematic study, the fusion data were fitted with the code CCMOD, and a fixed potential depth of 100 MeV. This is relatively close to the potential depths of the Woods–Saxon parametrization of the exponential Akyüz–Winther potential [10], as described in Ref. [11], which reproduces elastic scattering data. This will be referred to as the AW potential. Fixing the depth when fitting the data avoids introducing the (relatively weak) dependence of a on V0 . To identify measurements with inaccuracies in , i.e., in σfus (abs), the experimental values of r0 obtained from fitting each excitation function were compared with the radius parameter of the AW potential [11], for consistency also modified to have V0 = 100 MeV (MAW) [9]. A smooth trend in the ratios of the radius parameters might be expected if plotted against the charge product Z1 Z2 . Any large departure from this trend should indicate an error in σfus (abs). The parameter could then be adjusted to bring r0 into accordance with the trend values by rescaling the measured cross sections. In most cases the ratios r0 (exp)/r0 (MAW) were distributed evenly about a constant mean value of 0.932 ± 0.021, independent of Z1 Z2 . However, for 10 cases out of 46, the value of

had to be significantly changed from unity. Having established those data showing consistent behaviour, and those suspected of having errors in the absolute cross sections, the interpretation of the parameters resulting from fitting the data could proceed. Because of the similar effect of changing a and on the calculated cross sections, in principle the data can be reproduced either by varying a whilst keeping

unchanged, or by taking a prescription for a, and varying . The former will be discussed first. The deduced values of a, for V0 = 100 MeV, are shown in Fig. 2(a) as a function of Z1 Z2 for the whole data set. The experimental errors shown arise from statistical errors in the data together with, in some cases, errors arising when is not unity [9]. A linear least squares fit to the data is shown by the dashed line. Although there is considerable scatter about this trend line, all points are well above the values of a for the MAW potential, shown by the full line, and

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Fig. 2. Values of the empirical diffuseness parameter a as a function of Z1 Z2 : (a) for all cases; (b) for 16 O induced reactions only. The dashed lines show a linear least-squares fit to all of the experimental data and the solid lines indicate the values of a for the MAW potential. The symbols indicate values calculated with the double-folding model [13] (small open circles) and target nuclei with closed shells (stars), of vibrational character (circles) and of rotational character (squares); the open symbols indicate cases where = 1.0. The following systems were analysed: 9 Be + 208 Pb; 12 C + 92 Zr, 204 Pb; 16 O + 58, 62 Ni, 92 Zr, 112, 116 Sn, 144, 148, 154 Sm, 182, 186 W, 194, 198 Pt, 208 Pb, 238 U; 17 O + 144 Sm; 19 F + 197 Au, 208 Pb, 232 Th; 28 Si + 92 Zr, 144 Sm, 178 Hf, 208 Pb; 29 Si + 178 Hf; 30 Si + 186 W; 31 P + 175 Lu; 32 S + 89 Y, 208 Pb, 232 Th; 34 S + 89 Y, 168 Er; 35 Cl + 92 Zr; 40 Ca + 48 Ca, 46, 48, 50 Ti, 90, 96 Zr, 124 Sn, 192 Os, 194 Pt; 58 Ni + 60 Ni.

values calculated with the double-folding model [13] (small open circles). For the whole data set there is no significant difference between the average values for closed shell, vibrational or deformed nuclei, indicated by stars, circles and squares, respectively.

To investigate whether the scatter about the dashed line results only from experimental uncertainties, or could have a physical origin, various sub-sets of the data are considered. Data from a single projectile (16 O) are shown in Fig. 2(b), of which 8 were measured by the same group. Comparisons between them should therefore be more reliable than comparisons between members of the whole data set. The 16 O + A Sm reactions (Z Z = 496) show an increase in a 1 2 going from 144 Sm through 148 Sm to 154 Sm (a = 0.75, 0.99 and 1.06 fm, respectively). These isotopes show a systematic increase in β2 going from the spherical closed shell nucleus 144 Sm to the strongly deformed 154 Sm, hence the increase in a might be attributed to this change. It could also be attributed to increasing neutron-richness, as the 36 S and 40 Ca induced reactions on 90, 96 Zr [21,22] show a similar large increase in a (0.95 to 1.22 fm and 1.05 to 1.38 fm, respectively) in going from 90 Zr to 96 Zr, but all are spherical nuclei. This suggests that the increase in a may be correlated with neutron-richness of the target, rather than with increasing β2 . Some support is given for this conclusion by the neutron-rich reaction 40 Ca + 124 Sn [20] (Z1 Z2 = 1000), which has a ≈ 1.5 fm, well above the dashed line, whilst the neutron-poor reaction 28 Si + 144Sm [23] ((Z1Z2 = 868), similarly to 16 O + 144 Sm case, has a ≈ 0.89 fm, well below the line. A reaction between a very neutron-rich target and neutron-deficient projectile will necessarily involve multi-neutron transfer with positive Q-values. Examples are the 40 Ca + 96 Zr reaction (6 neutrons) and the 40 Ca + 124 Sn reaction (8 neutrons). However, it is found that the 40 Ca induced reactions on 194Pt and the more neutron-rich 192 Os [24] (both allowing positive-Q transfer of up to 8 neutrons), and also those on 46, 48, 50 Ti [25], show no significant variation in a, so a firm conclusion on this aspect cannot yet be given. There seems no doubt about the overall systematic increase in the effective a as a function of Z1 Z2 . However, the physical reasons for this behaviour are not yet clear. It might be that the WS form for the nuclear potential inside RB is not appropriate [26]; it has no physical justification. There is considerable scatter of the points about the trend line, and although there appears to be some correlation with the characteristics of the interacting nuclei, it would be well worth while if the high-energy cross sections were re-

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measured by a single group using detectors with near 100% efficiency. It should then be possible to reduce fluctuations due to experimental uncertainties and derive more definitive information. We now discuss the second way to fit the data, by varying whilst constraining a to the MAW value (≈ 0.67 fm). This also has a physical interpretation, as even if the measurement were correct, it might be that the fusion cross section is depleted by some mechanism. This certainly happens in the reaction 9 Be + 208 Pb [27], where the loosely bound 9 Be has a significant chance of being broken up by the nuclear field before the fusion barrier is reached [28]. Here one can say that complete fusion is suppressed by a factor S ≈ 0.68 [27]. Provided that S is independent of bombarding energy it can therefore be taken as equivalent to . Processes such as quasi-fission or deep-inelastic collisions (DIC) may also deplete fusion in reactions with other projectiles. However, in order to produce a suppression, it would be necessary that in the measurement these products were excluded from the events identified as fusion. For heavier projectiles, it is likely that in most measurements incomplete fusion, as seen for 9 Be, and quasi-fission, would be included in the fusion (capture) cross sections. Assuming that the large empirical diffuseness values arise from fusion depletion, values of S were determined, by taking the ratios of the measured cross sections to those calculated using the MAW potential, (slightly adjusted to give the experimental value for VB ). They are shown in Fig. 3 as a function of Z1 Z2 . The data show a consistent downward trend with increasing Z1 Z2 , displaying considerable scatter, as did the empirical a values. The deviation of S from unity should exactly mirror the deviation of a from the values of the MAW potential (solid line in Fig. 2), since a linear scaling of cross sections is part of both fitting procedures. The trend of S with Z1 Z2 could be consistent with known physical effects. Fig. 3 shows, by the large filled hexagons, two points at Z1 Z2 = 1400 for the reactions 58 Ni + 112, 124Sn [29,30] and one at Z1 Z2 = 1184 for the reaction 32 S + 182W [31]. Both show very significant contributions from DIC, at energies close to the fusion barrier. Wolfs [29,30] showed that this was so even at energies below the barrier. Here, DIC was identified with reaction products having a small mass change from the incoming nuclei, but with kinetic

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Fig. 3. Calculated suppression factors S with respect to the MAW potential (see text). Symbols are as in Fig. 2, except that the large filled hexagons refer to the reactions 58 Ni + 112, 124 Sn [29] and 32 S + 182 W [31].

energy loss larger than expected from peripheral excitation processes. The value of S has here been taken as σfus /[σfus + σDIC ], where σDIC is the cross section for DIC. The cross sections were summed over the measured energies with σfus > 200 mb. Wolfs’ values of σfus + σDIC were well reproduced [32] by a barrier passing calculation, suggesting that DIC is taking flux away from fusion. This is also supported by a number of calculations, including friction and quantal fluctuations [33], and Langevin calculations not including quantal fluctuations [34–36], which gave fair agreement with Wolfs’ results. The three points do not seem inconsistent with the other points with large Z1 Z2 in Fig. 3. Strongly damped reactions similar to DIC have also been reported to occur in lighter systems, for example, 32 S + 64 Ni [37] (Z1 Z2 = 448). It would be interesting to know the detailed behaviour of the DIC cross sections for reactions covering a range of Z1 Z2 , at a number of energies around and above the barrier. The DIC might make a large contribution to the trends shown in Figs. 2 and 3. However, the extra-push model in its present form does not resolve the problem, as the dash-dot line in Fig. 1 shows. It gives a value for a of 0.67 fm for 16 O + 208 Pb, when analysed by our procedure. Its

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results would give an approximately constant value for a, similar to that for the MAW potential, for lower values of Z1 Z2 and then a rather sudden and drastic increase to values > 2 fm when DIC became significant. The discrepancies between calculated and measured fusion cross sections are still far from being resolved. Therefore it seems timely to revisit the DIC process, and the effects of friction on fusion, through higher precision measurements, comparisons with existing models and development of new models. To summarise, 46 above-barrier fusion excitation functions have been fitted using the Woods–Saxon form of the nuclear potential. The data are extremely well fitted by adjusting the diffuseness a, which is found to increase on average with Z1 Z2 , and to be much larger than the value of ≈ 0.65 fm needed to reproduce elastic scattering data. This may indicate that the WS form of potential is not appropriate in the barrier region. It could also result from dynamical effects not included in the model used to fit the data. Indeed, the data can equally well be fitted by assuming that fusion is reduced by an energy-independent factor S. This factor decreases with increasing Z1 Z2 , and is not inconsistent with experimental results at high Z1 Z2 showing significant deep-inelastic cross sections. This first systematic survey of high precision above-barrier fusion excitation functions reveals the need to incorporate physical effects which have not been accounted for in recent coupled-channels models of fusion.

Acknowledgements M.D. and D.J.H. acknowledge the support of an ARC Discovery Grant.

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