Assessment of saddle-point-mass predictions for astrophysical applications

Using available experimental data on fission barriers and ground-state masses, a detailed study on the predictions of different models concerning the isospin dependence of saddle-point masses is performed. Evidence is found that several macroscopic models yield unrealistic saddle-point masses for very neutron-rich nuclei, which are relevant for the r-process nucleosynthesis.


INTRODUCTION
In order to have a full understanding of r process nucleosynthesis it is indispensable to have proper knowledge of the fission process. In the r process, fission can have decisive influence on the termination of the r-process as well as on the yields of transuranium elements and, consequently, on the determination of the age of the Galaxy and the Universe [1]. In cases where high neutron densities exist over long periods, fission will also influence the abundances of nuclei in the region A ~ 90 and 130 due to fission cycling [2,3].
First studies on the role of fission in the r process began forty years ago [2].
Meanwhile, extensive investigations on beta-delayed and neutron-induced fission have been performed; see e.g. [4,3,5,6]. Recently, first studies on the role of neutrino-induced fission in the r-process have also been done [7,8]. One of the common conclusions from all this work is that the influence of fission on the r process is very sensitive to the fission-barrier heights of heavy r-process nuclei with A > 190 and Z > 84, since they determine the calculated fission probabilities of these nuclei. Unfortunately, experimental information on the height of the fission barrier is only available for nuclei in a rather limited region of the chart of the nuclides. Therefore, for heavy r-process nuclei one has to rely on theoretically calculated barriers. Due to the limited number of available experimental barriers, in any theoretical model, constraints on the parameters defining the dependence of the fission barrier on neutron excess are rather weak. This leads to large uncertainties in estimating the heights of the fission barriers of heavy nuclei involved in the r process. For example, it was shown in Ref. [5] that predictions on the beta-delayed fission probabilities for nuclei in the region A ~ 250 -290 and Z ~ 92 -98 can vary between 0% and 100 % depending on the mass model used (see e.g. Table 2 of Ref. [5]), thus strongly influencing the r-process termination point. Moreover, the uncertainties within the nuclear models used to calculate the fission barriers can have important consequences on the r process. Meyer et al. have shown that a change of 1 MeV in the fission-barrier height can have strong consequences on the production of the progenitors (A ~ 250) of the actinide cosmochronometers, and thus on the nuclear cosmochronological age of the Galaxy [9].
Recently, important progress has been made in developing full microscopic approaches to nuclear masses (see e.g. [10]). Nevertheless, due to the complexity of the problem, this type of calculations is difficult to apply to heavy nuclei, where one has still to deal with semi-empirical models. Often used models are of the macroscopicmicroscopic type, where the macroscopic contribution to the masses is based either on some liquid-drop, droplet or Thomas-Fermi model, while microscopic corrections are calculated separately, mostly using the Strutinsky method [11]. The free parameters of these models are fixed using the nuclear ground-state properties and, in some cases, the height of fission barriers when available. Some examples of such calculations are shown in Fig. 1 (upper part), where the fission-barrier heights given by the results of the Howard-Möller fission-barrier calculations [12], the finite-range liquid drop model (FRLDM) [13], the Thomas-Fermi model (TF) [14], and the extended Thomas-Fermi model with Strutinsky integral (ETFSI) [15] are plotted as a function of the mass number for several uranium isotopes (A = 200-305). In case of the FRLDM and the TF model, the calculated ground-state shell corrections of Ref. [16] were added as done in Ref. [17]. In cases where the fission barriers were measured, the experimental values are also shown. Howard-Möller tables [12] (full grey line). In case of FRLDM and TF the ground-state shell corrections were taken from Ref. [16]. The macroscopic part of the Howard-Möller results is based on the droplet model [18]. The existing experimental data shown in the upper part of the figure are taken from the compilation in Ref. [19]. The small insert in the upper left part represents a zoom of the region where experimental data are available.
From the figure it is clear that as soon as one enters the experimentally unexplored region there is a severe divergence between the predictions of different models. Of course, these differences can be caused by both -macroscopic and microscopic -parts of the models, but in the present work we will concentrate only on macroscopic models. For this, we have two reasons: Firstly, different models show large discrepancies in the isotopic trend of macroscopic fission barriers * as can be seen in the lower part of Fig. 1.
Secondly, we want to avoid uncertainties and difficulties in calculating the shell corrections at large deformations corresponding to saddle-point configurations.
* The macroscopic part of the Howard-Möller calculations is based on the droplet model [18]. Therefore, in this paper, we consider the macroscopic part of the above-mentioned models and study the behaviour of the macroscopic contribution to the fission barriers when extrapolating to very neutron-rich nuclei. This study is based on the approach of Dahlinger et al. [19], where the predictions of the theoretical models are examined by means of a detailed analysis of the isotopic trends of ground-state and saddle-point masses. It is not our intention to develop a new model for calculating fission barriers or to suggest possible improvements in already existing model. The goal of this paper is to test the existing models and to suggest those which are the most reliable to be used in astrophysical applications.

METHOD
Usually, when one tests the predicted fission barriers of a theoretical model, one compares the heights of experimentally determined and calculated fission barriers. In doing so, one is obliged to use the theoretically calculated ground-state and saddle-point shell corrections, which can introduce an additional important uncertainty in the model predictions. To avoid this problem, we compare the measured saddle-point masses and the model-calculated macroscopic saddle-point masses, as was already suggested in [19].
In order to test the consistency of these models, we study the difference between the Eq. (1) represents the most direct test of the macroscopic model, as it does not refer to empirical or calculated ground-state shell corrections. If a certain macroscopic model is realistic, then the difference between the experimental saddle-point mass and the calculated macroscopic saddle-point massequal to δU sad in Eq. 1 -should correspond to the shell-correction energy at the saddle point. It is well known that the shell-correction energy oscillates with deformation and neutron/proton number. If we consider deformations corresponding to the saddle-point configuration, then the oscillations in the microscopic corrections for heavy-nuclei region we are interested in have a period of between about 10 ~ 30 neutrons depending on the single-particle potential used, see e.g. [21,22,23,24]. This means that if we follow the isotopic trend of the quantity δU sad over a large enough region of neutron numbers, in case of a realistic macroscopic model this quantity should show only local variations as shell corrections have local character. Moreover, according to the topographic arguments * of Myers and Swiatecki [17], these local variations should be very small -on the 1 -2 MeV level [14]. Of course, shell effects will change the deformation corresponding to the * According to the topographic theorem of Myers and Swiatecki [17] for nuclei with non-vanishing macroscopic fission barriers, the measured saddle-point masses should be very close to the values calculated by the macroscopic theory, i.e. the saddle-point shell-saddle point, but we are here interested in the mass at the saddle point and not in its position in the potential-energy landscape.
On the other hand, if the macroscopic part of a model does not describe realistically the isotopic trend, the quantity δU sad as defined by Eq. 1 will not correspond to the shellcorrection energy at the saddle point, and, consequently, this quantity will show global tendencies (e.g. increase or decrease) with respect to the neutron content. This mean that a general trend in δU sad with respect to the neutron content resulting from our analysis would indicate severe shortcomings of a given macroscopic model in extrapolating to nuclei far from stability. compilation [25], while the experimental fission barriers for these nuclei are taken from the compilation of Dahlinger et al. [19]. We have taken into account only the highest experimental fission barrier for two reasons: Firstly, this barrier is determined experimentally with less ambiguity than the lower barrier and, secondly, according to the topographic theorem [17], it should also be closer to the macroscopic barrier. Due to the large uncertainties in the measured heights of fission barriers of lighter nuclei, we have considered only the nuclei with atomic number Z ≥ 90. The wide span of the available data over more than 20 neutrons, see Fig. 3, guarantees the global aspect of the study. correction energy should be very small of the order of 1-2 MeV [14]. The validity of the topographic theorem was shown on the empirical basis by Myers and Swiatecki in Ref. [14], as a direct consequence of the smoothness of the measure saddle-point masses.

RESULTS AND DISCUSSION
For the case of uranium isotopes, the variable δU sad as defined by Eq. (1) is shown in Fig.   4 as a function of the neutron number N. The FRLDM and the Thomas-Fermi model result in a quite similar behaviour of δU sad (N) with slopes close to zero. On the contrary, the results from the droplet model (DM) show that δU sad increases strongly with the neutron number, while the ETF model predicts a decrease. For this analysis, we did not have the macroscopic ETF ground-state masses available. Therefore, we have used the Thomas-Fermi masses from Ref. [17]. This is justified by the fact that the macroscopic part of the ground-state masses in the different models is adjusted to the large body of existing data, and different models predict very similar values and tendencies as a function of neutron number for the macroscopic ground-state masses (at least in the region of masses where the present analysis is performed). This was checked by comparing the results from the FRLDM, the DM and the TF model. If we would extrapolate the behaviour of δU sad from Fig. 4 to the case of e.g. 300 U, which could be encountered on the r-process path [5], in the case of the ETF model one would get an increase in the macroscopic barrier relative to 238   For all studied nuclei the droplet model predicts an increase in δU sad as a function of neutron excess, and, thus, positive values of A 1 . This would imply that the macroscopic fission barriers decrease too strongly with increasing neutron number for all studied elements. The value of the mean slope averaged over the studied Z range is 0.16 ± 0.01 MeV. This value of slope adds up to a variation in δU sad of about 3.5 MeV over the range of 22 neutron numbers studied. All this indicates that the description of the isospin dependence of saddle-point masses within the droplet model is not consistent. The same conclusion was obtained in Refs. [19,26]. This result sheds a doubt on the applicability of the Howard-Möller tables of fission barriers [12] in regions far from stability. This finding is consistent with the analysis of the abundances produced in nuclear explosions performed by Hoff in 1987 [27], which also gave evidence that the Howard-Möller fission barriers of neutron-rich nuclei are too low.
In case of the ETF model we had available the macroscopic barriers for the uranium isotopes only. Already in this case we see, Fig. 4, that over the range of 10 studied   Fig. 5 supports the assumption that the shell-correction energy at the saddle point is small, and lends additional credence to the topologic theorem.
One can, of course, raise the question of the origin of these differences between the predictions of the different models. Any of the mass models has a certain set (depending on the physical assumptions) of parameters -macroscopic and microscopic -whose values are obtained through a least-squares adjustment to the experimental ground-state masses and, in some cases, to the fission barriers. As the ground-state masses are influenced by strong shell effects, the values of the macroscopic parameters cannot be determined completely independently from the microscopic part of the model. Moreover, in the least-squares adjustment, due to the much larger number of available experimental ground-state masses as compared to the small number of measured fission barrier heights, the weight of the former in the determination of the model parameters is much larger, and, consequently, the model parameters are mostly determined by the ground-state masses. This kind of parameter adjustment can cause problems in calculating the fission barrier, as the barriers can be much more sensitive to specific parameters (e.g. curvature coefficient) than masses [15]. For example, the DM and the TF model use for the curvature coefficient a cv values of 0 MeV [18] and 12.1 MeV [17], respectively, which could be one source for the different predictions of these two models. Another parameter that has decisive influence on the fission-barrier heights is the symmetry-energy coefficient J. Mamdouh et al [15] discuss that if they use J = 32 MeV instead of their default value of 27 MeV they obtain a decrease in the height of the ETFSI fission barrier for 276 U of 6 MeV, while for 238 U the barrier is, in this case, decreased by only 1 MeV. This problem can be overcome, if the method proposed in the present work is applied, where the differences between experimental and calculated saddle-point masses would be used to fix the model parameters, thus avoiding any influence of ground-state shell corrections, which are difficult to model with the necessary precision. We expect that this powerful method will give much better constraints on a realistic description of the saddlepoint masses and fission barriers for very neutron-rich nuclei.

CONCLUSIONS
We have studied four different macroscopic models in order to check whether they are adapted for predicting realistic values for the saddle-point masses of nuclei far from stability, in particular in model calculations of r-process nucleosynthesis. The results of this study show that the most realistic predictions are expected from the Thomas-Fermi model [14]. A similar conclusion can be made for the finite-range liquid-drop model [13] while inconsistencies in the saddle-point mass predictions of the droplet model [18] and the extended Thomas-Fermi model [15] are seen. This result raises severe doubts on the applicability of the Howard-Möllers fission-barrier tables [12] and the predictions of the ETFSI model [15] in modeling r-process nucleosynthesis.