A unified homographic law for fusion excitation functions above the barrier

We report on a systematics of fusion cross section data at energies above the reaction threshold to those of disappearance of fusion process. By an appropriate scaling of both cross sections and energy, a fusion excitation function common to all the data points is established. A universal description of the fusion excitation function relying on basic nuclear concepts is proposed and its dependence on the reaction cross section used for the cross section normalization is discussed.


Introduction
The evolution of the fusion reaction mechanism above the barrier to the Fermi-energy region is closely related to the existing competition between (nuclear) mean field and two-body dissipation. Thus, the study of both the complete (CF) and incomplete (IF) fusion excitation functions may be a proper way to understand and to constrain the ingredients entering in theoretical models, such as Sky3D [1], TDHF3D [2] or DYWAN [3], used to describe heavyion reactions at that energy range. Examples of burning issues are the choice of effective interactions [4] and the modeling of nucleon-nucleon collisions.
In two recent papers [5,6] we have published a systematic study of both CF and IF fusion cross sections σ fus for incident energies E lab /nucleon = E in higher than about 3A -4A MeV. From the works published during the past 40 years we have collected 382 CF and CF+IF σ fus data points belonging to 81 reaction systems with a vast variety of projectile-target pairs. By properly reducing σ fus values with the (total) reaction cross section σ reac and by applying an original energy scaling we have demonstrated that, irrespectively of the details of a given reaction system, the fusion excitation function follows a simple universal functional law. On that way we have been able to identify those data points for which, the most likely, the reported σ fus values suffer from a non-fusion contribution [5,6] and those for which other reaction mechanism has been erroneously identified as fusion [7]. Discarding these data one ends with 76 systems and 316 CF+IF σ fus data points plus 57 CF data points. Besides on experimental uncertainties of σ fus the exact parameter values of the universal functional law are somewhat dependent on the values of σ reac used for the σ fus normalization. In this contribution we focus on the data normalization problem and the impact of the implemented σ reac . a e-mail: eudes@subatech.in2p3.fr b e-mail: basrak@irb.hr 2 Scaling of fusion cross section Figure 1 displays the 316 CF+IF kept σ fus data points as a function of E in . Clearly, most of these non-scaled raw σ fus data points gather in a narrow domain of the σ fus vs E in plane although lighter systems (blue, cyan, and green symbols) gather along an arclike structure decreasing with E in while heavier ones (pink, red, and orange symbols) follow a line which sharply rises with E in .
The fact that systems of low and high mass do not fall together is natural due to the known dependence of σ fus on the system size. In order to compare so many different systems it is not enough to scale only the cross sections. We have shown [5,6] that energy has to be scaled too. In order to express on the same footing the mass asymmetric systems with those which are mass symmetric the system energy should be expressed in units of the so-called system available energy that is nothing but the center-of-mass energy per nucleon where A tot = A t + A p is the system mass and A p and A t are projectile and target mass, respectively. One accounts for the proportionality of cross section to the size of a reaction system by the renormalizing σ fus by (total) reaction cross section σ reac at the same E in . Accurate measurement of the total σ reac is rather hard so that these data are scarce and it would be inappropriate to apply them to the ensemble of our σ fus data. Therefore, one commonly resorts to phenomenological approaches to calculate σ reac , a solution which suffers for its own uncertainties and ambiguities. Thus, the uncertainty arising from the use of a particular parameterization of σ reac has to be investigated. Among a number of phenomenological parameterizations we have investigated five of them [8][9][10][11][12] including the pioneer one due to Bass [8] and the most recent one by Tripathi [12]. In order to infer reliability of   )   14N+12C  20Ne+16O  24Mg+12C  12C+27Al  16O+24Mg  20Ne+20Ne  28Si+12C  16O+26Mg  18O+24Mg  16O+27Al  32S+12C  20Ne+26Mg  20Ne+27Al 20Ne+27Al  35Cl+12C  16O+32S  16O+40Ca  28Si+28Si 28Si+28Si  32S+24Mg  19F+40Ca  32S+27Al  35Cl+24Mg  12C+48Ti  23Na+40Ca  16O+48Ti  40Ar+27Al   28Si+40Ca  12C+58Ni  36Ar+KCl  12C+63Cu  40Ar+40Ca  40Ca+40Ca  58Ni+27Al  24Mg+63Cu  48Ti+45Sc  36Ar+58Ni  16O+92Mo  32S+76Ge  40Ar+68Zn  52Cr+56Fe  19F+93Nb  64Zn+48Ti  58Ni+58Ni  18O+100Mo  78Kr+40Ca  82Kr+40Ca  12C+124Sn  14N+124Sn  20Ne+124Sn  40Ar+109Ag  84Kr+65Cu   40Ar+116Sn  40Ar+121Sb  16O+146Nd  132Xe+30Si  40Ar+124Sn  14N+154Sm  14N+159Tb  16O+159Tb  20Ne+159Tb  58Ni+124Sn  20Ne+169Tm  12C+182W  19F+175Lu  40Ar+154Sm  14N+181Ta  16O+181Ta  40Ar+164Dy  24Mg+181Ta  40Ar+165Ho  12C+197Au  14N+197Au 16O+197Au 16O+197Au 20Ne+197Au 40Ar+197Au 14N+232Th 40Ar+238U Figure 1. Raw fusion cross sections σ fus plotted as a function of E in . The inventoried systems are distinguished among them by symbols and a color code. The same symbols and the color code are used in Ref. [6] where an interested reader may find detailed information on energies, σ values, and references to original works. each of the models in figure 2 we compare them with 134 experimental σ reac measured for 46 systems. All these approaches rely on the strong absorption picture of nuclear processes and differ among themselves in the way the basic relation is parameterized. In Eq. (2) the cross section depends on the inverse of the (center-of-mass) energy E while the radius R and the potential depth V may be, in a first approximation, considered constant for a given system. Various models differ in the treatment of R and V by introducing or not a certain dependence on energy and/or on some other system property, usually A p or A tot , to the one or both of them. From figure 2 one may conclude that the Bass approach overpredicts the measured cross sections at all E avail , that the one of Kox underpredicts them at low energies (E avail 2 MeV/nucleon) while that of Gupta at higher energies (E avail 2 MeV/nucleon). These conclusions are plausible from the histograms obtained by projecting those ratios on the ordinate of each of panels in figure 2.   Figure 2. Ratios of experimental reaction cross sections and theoretical predictions for five models as a function of E avail . Panels display σ reac ratios with the models of a) Bass [8], b) Gupta and Kailas [9], c) Kox et al. [10], d) Shen et al. [11], and e) Tripathi et al. [12]. An interested reader may find detailed information on experimental σ reac and references to original works in Ref. [6].
mental and model reaction cross sections we heuristically define the most appropriate combination of model predictions. So, the Mixed σ reac is taken as that of Tripathi [12] for E avail < 2 MeV/nucleon and A tot < 86, as that of Gupta [9] for E avail < 2 MeV/nucleon and A tot ≥ 86, while for E avail ≥ 2 MeV/nucleon mixed σ reac is the average of the Shen [11] and Tripathi [12] predictions. It is interesting to note that out of the 316 data points in the case of Gupta and that of Kox 2 and 41 points, respectively violate the physically allowed range for the normalized cross section values σ fus /σ reac = σ red , namely between 0 and 1. All these points are due to σ red overflow that occurs at low energies.
Assuming the applicability of the strong absorption concept in the first approximation both σ fus and σ reac may be expressed by the same functional form given by Eq.
ures 3 and 4 have been fitted with the three-parameters homographic function (3). The obtained best fit result is displayed by the red full curve for each σ red . Interestingly enough, independently of the σ reac chosen for the normalization procedure, the best fit curve gives for the energy of disappearance of the CF+IF fusion process the same value:

Complete fusion cross section
Only twelve experiments have explicitly been designed to measure both complete and incomplete fusion components  figure 4) and for the complete fusion data (main inset) as a function of E avail . The full violet curve in the inset and the main panel is due to the best fit of the complete fusion data by the homographic function (3). The full red curve in the main panel is due to the same kind of fit to CF+IF data. The orange and light-violet background bands around the best fit curve in the main panel and in the inset, respectively are due to the errors on the fit parameters. The dashed blue curve is the difference of both fusion excitation functions. The inset in the main inset displays the ratio of CF and CF+IF best-fit excitation functions as a function of E avail . [5,6]. These 57 CF σ red data points belonging to the 14 reaction systems and obtained with the heuristic σ reac discussed above are displayed in the main inset of figure 5 as a function of E avail . The same homographic law (3) used in fitting the CF+IF data is here used to obtain the best fit result to the CF data. It is shown by the full violet curve. The fitting code which is used provides an uncertainty on the fit parameters a, b, and c [13]. These uncertainties define the light-violet background drawn around the best fit curve. Owing to the rather large experimental error bars and the relatively small number of data points the energy of CF disappearance is not very accurately defined. It reads E avail = 6.2 +1.3 −1.1 MeV/nucleon. Similarly, CF data display a stronger dependence on the normalization σ reac used. However, the deduced energy of CF disappearance for each of the σ reac used lays well within the above stated uncertainty limits.

Discussion and conclusions
In the main panel of figure 5 is repeated panel c) of figure 4, namely σ red for the CF+IF data points obtained with the heuristic σ reac displayed as a function of E avail . The orange background drawn around the full red best fit curve is due to the uncertainties of the fit parameters. Much smaller errors of the fit parameters are in accordance with the already discussed stability of the CF+IF σ red data. In the main panel is repeated the full violet best fit curve to the CF data. Making difference of both allows to infer the main properties of the incomplete fusion excitation function: IF process opens around E avail ≈ 1.5 MeV/nucleon, reaches maximal value at the energy of CF disappearance (E avail ≈ 6 MeV/nucleon), and vanishes at E avail ≈ 13 MeV/nucleon. An inset in the main inset displays how the best-fit CF excitation function decreases relatively to the CF+IF one, an observable which has been investigated a long time ago by Morgenstern et al. [14].
To summarize, the scrutiny of the existing fusion cross sections well above the reaction barrier allowed us to establish a universal dependence of these data on energy. The established homographic functional description of both the complete and the complete plus incomplete fusion excitation functions is rather stable and the inferred global features of these excitation functions quite weakly depend on the details of the data normalization. Nevertheless, the normalization may be improved if additional high quality measurements on both fusion and reaction cross section would be available.