Chaos Weighted Wall Formula Research Articles

Measurement of fusion evaporation residue cross sections in the Ti48+Ba138 reaction

Evaporation residue cross sections are measured for the reaction $^{48}\mathrm{Ti}+^{138}\mathrm{Ba}$ which forms the compound nucleus ${}^{186}{\mathrm{Pt}}^{*}$. The cross sections are measured at beam energies in the range of 189.3--234.4 MeV. The experimental evaporation residue cross sections are compared with the dynamical model which employs one-dimensional Langevin dynamical calculations. The dissipation strength of the Langevin equation is calculated using both chaos weighted wall formula and a constant reduced dissipation function. The measured ER cross sections are found to be much less than the theoretical predictions. Further, the measured ER cross sections for the system $^{48}\mathrm{Ti}+^{138}\mathrm{Ba}$ are compared with those of $^{32}\mathrm{S}+^{154}\mathrm{Sm}$ forming the same compound nucleus. The suppression in the evaporation residue cross sections of the former reaction may be attributed to the increasing competition from quasifission. The quasifission reaction is found to have superseded any effect of neutron shell closure $(N=82)$ of the target in the present study.

Dissipation of the tilting degree of freedom in heavy-ion-induced fission from four-dimensional Langevin dynamics

A stochastic approach based on four-dimensional Langevin fission dynamics is applied to the calculation of a wide set of experimental observables of excited compound nuclei from 199Pb to 248Cf formed in reactions induced by heavy ions. In the model under investigation, the tilting degree of freedom (K coordinate) representing the projection of the total angular momentum onto the symmetry axis of the nucleus is taken into account in addition to three collective shape coordinates introduced on the basis of \(\{c,h,\alpha\}\) parametrization. The evolution of the K coordinate is described by means of the Langevin equation in the overdamped regime. The friction tensor for the shape collective coordinates is calculated under the assumption of the modified version of the one-body dissipation mechanism, where the reduction coefficient ks of the contribution from the “wall” formula is introduced. The calculations are performed both for the constant values of the coefficient \(k_{s}\) and for the coordinate-dependent reduction coefficient \(k_{s}(\mathbf{q})\) which is found on the basis of the “chaos-weighted wall formula”. Different possibilities of the deformation-dependent dissipation coefficient \((\gamma_{K})\) for the K coordinate are investigated. The presented results demonstrate that an impact of the \(k_{s}\) and \(\gamma_{K}\) parameters on the calculated observable fission characteristics can be selectively probed. It was found that it is possible to describe the experimental data consistently with the deformation-dependent \(\gamma_{K}(\mathbf{q})\) coefficient for shapes featuring a neck, which predicts quite small values of \(\gamma_{K}=0.0077\) (MeV zs)-1/2 and constant \(\gamma_{K}=0.1\)-0.4 (MeV zs)-1/2 for compact shapes featuring no neck.

Incorporation of a tilting coordinate into the multidimensional Langevin dynamics of heavy-ion-induced fission: Analysis of experimental data from fusion-fission reactions

A four-dimensional dynamical model was developed and applied to study fission characteristics in a wide range of a fissility parameter. Three collective shape coordinates and the $K$ coordinate were considered dynamically from the ground-state deformation to the scission into fission fragments. A modified one-body mechanism for nuclear dissipation with a reduction coefficient ${k}_{s}$ of the contribution from a ``wall'' formula has been used in the study. The inclusion of the $K$ coordinate in the dynamical consideration and use of the ``chaos-weighted wall formula'' with a deformation-dependent scaling factor ${k}_{s}({q}_{1})$ lead to fairly good reproduction of the variances of the fission-fragment mass distribution and the prescission neutron multiplicity for a number of fissioning compound nuclei in a wide fissility range. The four-dimensional dynamical calculations describe better experimental prescission neutron multiplicity and variances of fission-fragment mass distribution for heaviest nuclei with respect to a three-dimensional dynamical model, where the $K$ coordinate is assumed to be equal to zero. The estimate of a dissipation coefficient for the orientation degree of freedom, ${\ensuremath{\gamma}}_{K}\ensuremath{\simeq}0.077$ ${(\text{MeV}\phantom{\rule{0.16em}{0ex}}\text{zs})}^{\ensuremath{-}1/2}$, is good for heavy nuclei and a larger value of ${\ensuremath{\gamma}}_{K}\ensuremath{\simeq}0.2$ ${(\text{MeV}\phantom{\rule{0.16em}{0ex}}\text{zs})}^{\ensuremath{-}1/2}$ is needed for nuclei with mass ${A}_{\mathrm{CN}}$ $\ensuremath{\simeq}$ 200.

The Dependency on the Dissipation Tensor of Multi-modal Nuclear Fission

In the study of the fission of actinide nuclei at low excitation energies including the spontaneous fission, it was found that the fragment mass distribution and the total kinetic energy (TKE) distribution consist of more than one component, in contrast to the simple single peak structure that is found in the fission at high excitation energies. 1– 6 This phenomenon is attributed to the existence of more than one fission path and is called the multi-modal fission. The mass and TKE distributions depend sensitively on the excitation energy and the position of the peaks of the mass distribution suggests the influence of the closed shell structure of the fragments. Therefore, it is supposed that the microscopic energy plays an important role for the manifestation of this phenomenon. It is a great challenge for us to understand this phenomenon in terms of nuclear many-body dynamics. Several authors studied the potential energy surface (PES) including the microscopic energy in a multi-dimensional parameter space that describes various nuclear shapes; one can deduce the possible fission paths by studying the location of the saddle points and the fission valleys in multi-dimensional parameter space. 7 With this method, they could explain the general trend of the position of the peaks of the mass distribution. The dynamical point of view is necessary to progress the study of the fission mode. We have applied the Langevin approach to the study of the fission modes in uranium nuclei and in fermium nuclei. 8–10 We studied the mass and TKE distributions and demonstrated that we can decompose the fission events into several components by tracing the Langevin trajectories. We also studied the isotope dependence and the excitation energy dependence of the fission mode. 8–10 In the previous studies, we adopted the wall-and-window type one-body friction as the dissipation mechanism of the nuclear fission dynamics. The validity of this dissipation mechanism has been demonstrated by one of the authors (T.W.) who studied the dissipation tensor dependence of the pre-scission neutron multiplicity and the mean TKE. 11–14 From the comparison of the results of the dynamical calculation with experimental data, they excluded the possibility of the two-body type dissipation to be the dominant mechanism by showing that it cannot reproduce the pre-scission neutron data and the TKE data simultaneously. On the other hand, they showed that the wall-and-window type one-body friction can reproduce both data reasonably well and concluded that it is a reasonable model for the dissipation mechanism of nuclear fission. There are other models that are of one-body nature, e.g. surface-plus-window formula, modified wall-and-window formula and chaos weighted wall formula. 15–17 Though there were no free parameters in the original derivation of the one-body friction, 18, 19 the strength has been modified frequently in order to reproduce some experimental data. For example, in the study of the light particle evaporation and the mass distribution, Schmitt et al. used the strength as a free parameter. 17 The modification itself should be acceptable when we take account of the simplicity of the model; it is a macroscopic model without any microscopic effect and it has no dependence on the temperature. However, when one modifies the strength of the nuclear dissipation just to reproduce only one physical quantity, it might be inappropriate to conclude that the deduced strength has definite physical meanings. It may reflect the other effects completely different from the dissipation, like the insufficiency of the model space. It is very important to compare many (at least more than one) physical quantities at the same time. Among the physical quantities that are measured in nuclear fission, the TKE and mass distributions are well investigated experimentally in many cases. In this study, we use these quantities to discuss the dissipation dependence of the fission modes. It is shown that the TKE distribution is, as was expected, directly connected to the strength of the dissipation and we can put some constraints on the strength of the dissipation. Furthermore, it is shown that the mass distribution changes rather drastically when one uses different models for the dissipation mechanism. These results demonstrate the importance and the usefulness of the dynamical approach to the study of the fission mode. Section 2 gives a concise description of our framework. Results are shown in Sec. 3 concerning the fission of 264 Fm nucleus at Ex = 20 MeV. Summary is given in Sec. 4. 2. Methods

Evaporation residue cross-sections as a probe for nuclear dissipation in the fission channel of a hot rotating nucleus

Evaporation residue cross-sections are calculated in a dynamical description of nuclear fission in the framework of the Langevin equation coupled with statistical evporation of light particles. A theoretical model of one-body nuclear friction which was developed earlier, namely the chaos-weighted wall formula, is used in this calculation for the 224Th nucleus. The evaporation residue cross-section is found to be very sensitive to the choice of nuclear friction. The present results indicate that the chaotic nature of the single-particle dynamics within the nuclear volume may provide an explanation for the strong shape-dependence of nuclear friction which is usually required to fit experimental data.

Prescission neutron multiplicity and fission probability from Langevin dynamics of nuclear fission

A theoretical model of one-body nuclear friction which was developed earlier, namely the chaos-weighted wall formula, is applied to a dynamical description of compound nuclear decay in the framework of the Langevin equation coupled with statistical evaporation of light particles and photons. We have used both the usual wall formula friction and its chaos-weighted version in the Langevin equation to calculate the fission probability and prescission neutron multiplicity for the compound nuclei $^{178}$W, $^{188}$Pt, $^{200}$Pb, $^{213}$Fr, $^{224}$Th, and $^{251}$Es. We have also obtained the contributions of the presaddle and postsaddle neutrons to the total prescission multiplicity. A detailed analysis of our results leads us to conclude that the chaos-weighted wall formula friction can adequately describe the fission dynamics in the presaddle region. This friction, however, turns out to be too weak to describe the postsaddle dynamics properly. This points to the need for a suitable explanation for the enhanced neutron emission in the postsaddle stage of nuclear fission.

Fission widths of hot nuclei from Langevin dynamics

The fission dynamics of excited nuclei is studied in the framework of the Langevin equation. One-body wall-and-window friction is used as the dissipative force in the Langevin equation. In addition to the usual wall formula friction, the chaos-weighted wall formula developed earlier is also considered here. The fission rate calculated with the chaos-weighted wall formula is found to be larger by about a factor of 2 compared to that obtained with the usual wall friction. The systematic dependence of the calculated fission width on temperature and spin of the fissioning nucleus is investigated and a simple parametric form of the fission width is obtained.

Chaos modified wall formula damping of the surface motion of a cavity undergoing fissionlike shape evolutions

The chaos weighted wall formula developed earlier for systems with partially chaotic single particle motion is applied to large amplitude collective motions similar to those in nuclear fission. Considering an ideal gas in a cavity undergoing fission-like shape evolutions, the irreversible energy transfer to the gas is dynamically calculated and compared with the prediction of the chaos weighted wall formula. We conclude that the chaos weighted wall formula provides a fairly accurate description of one body dissipation in dynamical systems similar to fissioning nuclei. We also find a qualitative similarity between the phenomenological friction in nuclear fission and the chaos weighted wall formula. This provides further evidence for one body nature of the dissipative force acting in a fissioning nucleus.

Chaos in single particle motion and one body dissipation

The validity of the chaos weighted wall formula developed for systems in which the particle motion is not fully randomized is tested. We consider an ideal gas in a cavity undergoing volume conserving shape oscillations and compare the energy transferred from the wall to the gas with the prediction of the chaos weighted wall formula. We find that the chaos weighted wall formula provides a fairly reliable description of one body dissipation for nearly integrable to highly chaotic systems. {copyright} {ital 1997} {ital The American Physical Society}