Semianalytical expressions are suggested for the temperature dependence of those combinations of transport coefficients that govern the fission process. This is based on experience with numerical calculations within the linear response approach and the locally harmonic approximation. A reduced version of the latter is seen to comply with Kramers's simplified picture of fission. It is argued that for variable inertia his formula has to be generalized, as already required by the need that for overdamped motion the inertia must not appear at all. This situation may already occur above TϷ2 MeV, where the rate is determined by the Smoluchowski equation. Consequently, comparison with experimental results does not give information on the effective damping rate, as often claimed, but on a special combination of local stiffnesses and the friction coefficient calculated at the barrier.

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Previous computations of the potential landscape with the shapes parameterized in terms of Cassini ovaloids are extended to collective dynamics at finite excitations. Taking fission as the most demanding example of large scale collective motion, transport coefficients are evaluated along a fission path. We concentrate on those for average motion, namely stiffness C, friction \gamma and inertia M. Their expressions are formulated within a locally harmonic approximation and the help of linear response theory. Different approximations are examined and comparisons are made both with previous studies, which involved different descriptions of single particle dynamics, as well as with macroscopic models. Special attention is paid to an appropriate definition of the deformation of the nuclear density and its relation to that of the single particle potential. For temperatures above 3 MeV the inertia agrees with that of irrotational flow to less than a factor of two, but shows larger deviations below, in particular in its dependence on the shape. Also friction exhibits large fluctuations along the fission path for small excitations. They get smoothed out above 3 - 4 MeV where \gamma attains values in the range of the wall formula. For T > (or=) 2 MeV the inverse relaxation time \beta = \gamma /M turns out to be rather insensitive to the shape and increases with T.Comment: 30 pages, Latex, 15 Postscript figures; to appear in PRC; e-mail: hhofmann@physik.tu-muenchen.de www home page: http://www.physik.tu-muenchen.de/tumphy/e/T36/hofmann.htm

The mass asymmetry in the fission of 236 U at low excitation energy is clarified by the analysis of the trajectories obtained by solving the Langevin equations for the shape degrees of freedom. It is demonstrated that the position of the peaks in the mass distribution of fission fragments is determined mainly by the saddle point configuration originating from the shell correction energy. The width of the peaks, on the other hand, results from the shape fluctuations close to the scission point caused by the random force in the Langevin equation. We have found out that the fluctuations between elongated and compact shapes are essential for the fission process. According to our results the fission does not occur with continuous stretching in the prolate direction, similarly to that observed in starch syrup, but is accompanied by the fluctuations between elongated and compact shapes. This picture presents a new viewpoint of fission dynamics and the splitting mechanism.

1 4 Strutinsky smoothing and collective dynamics in the independent particle model 4.1 Averaged response functions for nucleonic motion 4.2 Friction from averaged response functions 4.2.1 The innitely deep square well 4.2.2 The oscillator potential 5 Discussion of numerical results for friction 5.1 Friction from averaging procedures 5.2 Friction and collisional damping 5.3 Hydrodynamic behavior 6 Summary, conclusions and outlook A Appendices A.1 Single particle models A.1.1 The innitely deep square well A.1.2 The oscillator potential A.2 The wall formulaWe study slow collective motion of isoscalar type at nite excitation. The collective variable is parameterized as a shape degree of freedom and the mean eld is approximated by a deformed shell model potential. We concentrate on situations of slow motion, as guaranteed, for instance, by the presence of a strong friction force, which allows us to apply linear response theory. The prediction for nuclear dissipation of some models of internal motion are contrasted. They encompass such opposing cases as that of pure independent particle motion and the one of "collisional dominance". For the former the wall formula appears as the macroscopic limit, which is here simulated through Strutinsky smoothing procedures. It is argued that this limit hardly applies to the actual nuclear situation. The reason is found in large collisional damping present for nucleonic dynamics at nite temperature T . The level structure of the mean eld as well as the T -dependence of collisional damping determine the T -dependence of friction. Two contributions are isolated, one coming from real transitions, the other being associated to what for innite matter is called the "heat pole". The importance of the latter depends strongly on the level spectrum of internal motion, and thus is very dierent for "adiabatic" and "diabatic" situations, both belonging to dierent degrees of "ergodicity".2

We developed a four-dimensional Langevin model which can treat the deformation of each fragment independently and applied it to low energy fission of 236 U, the compound system of the reaction n+ 235 U. The potential energy is calculated with the deformed two-center Woods-Saxon (TCWS) and the Nilsson type potential with the microscopic energy corrections following the Strutinsky method and BCS pairing. The transport coefficients are calculated by macroscopic prescriptions. It turned out that the deformation for the light and heavy fragments behaves differently, showing a sawtooth structure similar to that of the neutron multiplicities of the individual fragments ν(A). Furthermore, the measured total kinetic energy T KE(A) and its standard deviation are reproduced fairly well by the 4D Langevin model based on the TCWS potential in addition to the fission fragment mass distributions. The developed model allows a multi-parametric correlation analysis among, e.g., the three key fission observables, mass, TKE, and neutron multiplicity, which should be essential to elucidate several long-standing open problems in fission such as the sharing of the excitation energy between the fragments.

We have developed new Langevin-model codes to calculate fission observables as a contract with MEXT. Developed are 1) a Langevin code which can take account of microscopic transport coefficients (the mass and friction tensors) calculated by linear response theory, and 2) a 4-dimensional Langevin code in which deformations of 2 fission fragments are treated to be independent. Calculated results by them will be presented placing emphasis on those with microscopic transport coefficients and their effects on fission observables.

Abstract. We provide a semiclassical description of the shell structure in large prolate cavities. Level densities and shell-correction energies are obtained from periodic orbit theory, using a version of Gutzwiller's trace formula that takes into account continuous symmetries. The semiclassical results are compared to their quantum-mechanical counterparts.