Mass-energy distribution of fragments within Langevin dynamics of fission induced by heavy ions

A stochastic approach based on four-dimensional Langevin fission dynamics is applied to calculating mass-energy distributions of fragments originating from the fission of excited compound nuclei. In the model under investigation, the coordinate K 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 the {c, h, ??} parametrization. The evolution of the orientation degree of freedom (K mode) is described by means of the Langevin equation in the overdamped regime. The tensor of friction is calculated under the assumption of the reducedmechanismof one-body dissipation in the wall-plus-window model. The calculations are performed for two values of the coefficient that takes into account the reduction of the contribution from the wall formula: k _s = 0.25 and k _s = 1.0. Calculations with a modified wall-plus-window formula are also performed, and the quantity measuring the degree to which the single-particle motion of nucleons within the nuclear system being considered is chaotic is used for k _s in this calculation. Fusion-fission reactions leading to the production of compound nuclei are considered for values of the parameter Z ²/A in the range between 21 and 44. So wide a range is chosen in order to perform a comparative analysis not only for heavy but also for light compound nuclei in the vicinity of the Businaro-Gallone point. For all of the reactions considered in the present study, the calculations performed within four-dimensional Langevin dynamics faithfully reproduce mass-energy and mass distributions obtained experimentally. The inclusion of the K mode in the Langevin equation leads to an increase in the variances of mass and energy distributions in relation to what one obtains from three-dimensional Langevin calculations. The results of the calculations where one associates k _s with the measure of chaoticity in the single-particle motion of nucleons within the nuclear system under study are in good agreement for variances of mass distributions. The results of calculations for the correlations between the prescission neutron multiplicity and the fission-fragment mass, ??n _pre(M)??, and between, this multiplicity and the kinetic energy of fission fragments, ??n _pre(E _k )??, are also presented.     

Allowance for the shell structure of colliding nuclei in the fusion-fission process

The motion of two nuclei toward each other in fusion-fission reactions is considered. The state of the system of interacting nuclei is specified in terms of three collective coordinates (parameters). These are the distance between the centers of mass of the nuclei and the deformation parameter for each of them (the nose-to-nose orientation of the nuclei is assumed). The evolution of collective degrees of freedom of the system is described by Langevin equations. The energies of the Coulomb and nuclear (Gross-Kalinovsky potential) interactions of nuclei are taken into account in the potential energy of the system along with the deformation energy of each nucleus with allowance for shell effects. The motion of nuclei toward each other are calculated for two reaction types: reactions involving nuclei that are deformed (₄₂¹⁰⁰Mo + ₄₂¹⁰⁰Mo → ₈₄²⁰⁰Po) and those that are spherical (₈₂²⁰⁸Pb + ₈¹⁸O → ₉₀²²⁶Th) in the ground state. It is shown that the shell structure of interacting nuclei affects not only the fusion process as a whole (fusionbarrier height and initial-reaction-energy dependence of the probability that the nuclei involved touch each other) but also the processes occurring in each nucleus individually (shape of the nuclei and their excitation energies at the point of touching).     

Cold events in thermal-neutron-induced fission of heavy nuclei

An interpretation of the cold fission events in thermal-neutron-induced fission of heavy nuclei is given. The descent from the saddle point is considered as a dynamical process with reversible coupling between collective and intrinsic degrees of freedom. The distribution function for the collective variables is expressed as a product of two terms: the adiabatical and the dynamical factors. A simple model for symmetric fission to study the mass distribution is presented. As example, the calculations are performed for the nucleus ²⁶⁴Fm. Gross features of the cold fission are discussed as well as the dependence of the theoretical mass distribution on the parameters of the model. Received: 29 April 1998 / Revised version: 11 September 1998     

Collective mass parameters in the nuclear fission process

The theory of nuclear fission process is considered. The effective mass parameters are calculated as a function of the following collective coordinates: the separation between the centres of harmonic oscillators, the mass asymmetry parameter and the necking parameter. Numerical calculations are carried out for the fission of²³⁶U and²³⁸U nuclei. Symmetric and asymmetric cases are considered. From the present calculation we see the importance of taking into account the necking parameter as a dynamical collective coordinate.     

Description of the two-humped mass distribution of fission fragments of mercury isotopes on the basis of the multidimensional stochastic model

The dynamical model proposed earlier for describing fusion-fission reactions is applied to describing the two-humped mass distribution of fission fragments of mercury isotopes. In this model, the calculation of the time evolution of collective coordinates of the system is broken down into two stages. The first stage is that within which the projectile approaches the target nucleus, while the second is that of the evolution of the system formed after the touching of the projectile and target nuclei. The dynamical evolution of the system within both stages of the calculation is described on the basis of Langevin equations. The shell structure of colliding nuclei is taken into account at either stage of the calculation. Mass distributions are calculated for fragments originating from the fission of the mercury isotopes ^{190, 184}Hg formed in the fusion-fission reactions ⁴⁸Ca + ¹⁴²Nd → ¹⁹⁰Hg and ⁴⁰Ar + ¹⁴⁴Sm → ¹⁸⁴Hg. The process in which the isotope ¹⁸⁰Hg undergoes fission from the ground state is also calculated. The results obtained in this way are compared with the results of previous theoretical calculations and with available experimental data.     

Allowance for the shell structure of the 42100 Mo and 46110 Pd nuclei in the synthesis of 84200 Po, 88210 Ra, and 92220 U

The effect of the shell structure of colliding nuclei in calculating the entrance channel on the ensuing evolution of the product system is investigated. The entrance channel is calculated under the assumption of the nose-to-nose orientation of colliding nuclei. The following three reactions involving nuclei that are deformed in the ground state are considered: ₄₂¹⁰⁰Mo + ₄₂¹⁰⁰Mo → ₈₄¹⁰⁰Po, ₄₂¹⁰⁰Mo + ₄₆¹⁰⁰Pd → ₈₈²¹⁰Ra, and ₄₆¹¹⁰Pd + ₄₆¹¹⁰Pd → ₉₂²²⁰U. The state of the system at the point of touching is determined by the results obtained by calculating the entrance reaction channel. The shape of the system is specified by three collective coordinates (deformation parameters). The evolution of collective coordinates of the system is described in terms of Langevin equations. The potential energy of the system of colliding nuclei is calculated with allowance for their shell structure. It is shown that allowance for individual features of interacting nuclei in the entrance channel of the fusion-fission reactions makes it possible to obtain, for the reactions being considered, cross sections for evaporation-residue formation that are closer to available experimental data than their liquid-drop counterparts.     

Langevin fission dynamics of hot rotating nuclei: Systematic application to Z 2/A=34–42 heavy nuclei

A stochastic approach that treats fission dynamics on the basis of three-dimensional Langevin equations is used to calculate the mass-energy distributions of fragments originating from the fission of compound nuclei whose fissility parameter lies in the range Z ²/A=34–42. In these calculations, use was made of the liquid-drop model allowing for finite-range nuclear forces and the diffuseness of the nuclear surface in calculating the potential energy and a modified one-body mechanism of viscosity in describing dissipation. The emission of light prescission particles is taken into account on the basis of the statistical model. The calculations performed within three-dimensional Langevin dynamics reproduce well all parameters of the experimental mass-energy distributions of fission fragments and all parameters of the prefission-neutron multiplicity for various parameters of the compound nucleus. The inclusion of the third collective coordinate in the Langevin equations leads to a considerable increase (by up to 40–50%) in the variances of mass-energy distributions in relation to what was previously obtained from two-dimensional Langevin calculations. For the parameters of the mass-energy distributions of fission fragments and the parameters of the prefission-neutron multiplicity to be reproduced simultaneously, the reduction coefficient K _s must be diminished at least by a factor of 2(0.2≤K _s ≤0.5) in relation to that in the case of total one-body viscosity (K _s =1).     

Stochastic model of angular distributions of fragments originating from the fission of excited compound nuclei

The anisotropy of angular distributions of fission fragments and the average multiplicity of prescission neutrons were calculated within a stochastic approach to fission dynamics on the basis of three-dimensional Langevin equations. This approach was combined with a Monte Carlo algorithm for the degree of freedom K (projection of the total angular momentum I onto the fission axis). The relaxation time τ _K in the coordinate K was considered as a free parameter of the model; it was estimated on the basis of a fit to experimental data on the anisotropy of angular distributions. Specifically, the relaxation time τ _K was estimated at 2 × 10^?21 s for the compound nuclei ²²⁴Th and ²²⁵Pa and at 4 × 10^?21 s for the heavier nuclei ²⁴⁸Cf, ²⁵⁴Fm, and ²⁶⁴Rf. The potential energy was calculated on the basis of the liquid-drop model with allowance for finiteness of the range of nuclear forces and for the diffuseness of the nuclear surface. A modified one-body viscosity mechanism featuring a coefficient k _s that takes into account the reduction of the contribution from the wall formula was used to describe collective-energy dissipation. The coefficient k _s was also treated as a free parameter and was estimated at 0.5 on the basis of a fit to experimental data on the average prescission multiplicity of neutrons.     

Nonuniform character of the population of spin projections K for a fissile nucleus at the scission point and anisotropies in the angular distributions of fragments originating from the induced fission of nuclei

It is shown that the emergence of anisotropies in the angular distributions of fragments originating from the spontaneous and induced fission of oriented actinide nuclei is possible only if nonuniformities in the population of the projectionsM (K) of the fissile-nucleus spin onto the z axis of the laboratory frame (fissile-nucleus symmetry axis) appear simultaneously in the vicinity of the scission point but not in the vicinity of the outer saddle point of the deformation potential. The possibilities for creating the orientation of fissile nuclei for spontaneous and induced fission and the effect of these orientations on the anisotropies under analysis are considered. The role of Coriolis interaction as a unique source of the mixing of different-K fissile-nucleus states at all stages of the fission process is studied with allowance for the dynamical enhancement of this interaction for excited thermalized states of the nucleus involved that is characterized by a high energy density. It is shown that the absence of thermalization of excited states of the fissile nucleus that appear because of the effect of nonadiabaticity of its collective deformation motion in the vicinity of the scission point is a condition of conservation of the influence that transition fission states formed at the inner and outer fission barriers exerts on the distribution of the spin projections K for lowenergy spontaneous nuclear fission. It is confirmed that anisotropies observed in the angular distributions of fragments originating from the fission of nuclei that is induced by fast light particles (multiply charged ions) are due to the appearance of strongly excited equilibrium(nonequilibrium) states of the fissile nucleus in the vicinity of its scission point that have a Gibbs (non-Gibbs) distribution of projections K.     

Rate of excited-nucleus fission within a multidimensional stochastic approach

A stochastic approach to describing the dynamics of the nuclear-fission process is applied to study the effect of the number of dimensions on the fission rate within the dynamical model used. The time dependence of the fission rate is calculated on the basis of a multidimensional Langevin equation without taking into account particle evaporation. The one-, two-, and three-dimensional cases are considered for the example of the “c, h, α” parametrization of nuclear-surface shapes. The calculations are performed for a large number of compound nuclei whose parameter Z ²/A falls within the range 20 ≤ Z ²/A ≤ 40. The stationary level of the fission rate is found to increase considerably upon going over from the one- to the three-dimensional case. This increase is especially pronounced for light fissile nuclei in the vicinity of the Businaro-Gallone point. Also, the stationary fission-rate level obtained from our dynamical simulation is compared with its counterpart calculated by the Kramers formula generalized to the multidimensional case.     

Potential energy surfaces and fission fragment mass yields of even-even superheavy nuclei

Potential energy surfaces and fission barriers of superheavy nuclei are analyzed in a macroscopic-microscopic model. The Lublin-Strasbourg Drop (LSD) model is used to obtain the macroscopic part of the energy, whereas the shell and pairing energy corrections are evaluated using the Yukawa-folded potential; a standard flooding technique is utilized to determine barrier heights. A Fourier shape parametrization containing only three deformation parameters is shown to effectively reproduce the nuclear shapes of nuclei approaching fission. In addition, a non-axial degree of freedom is taken into account to better describe the structure of nuclei around the ground state and in the saddle region. In addition to the symmetric fission valley, a new highly asymmetric fission mode is predicted in most superheavy nuclei. The fission fragment mass distributions of the considered nuclei are obtained by solving 3D Langevin equations.     

Angular distribution of fission fragments in reactions of complete fusion of deformed nuclei

Formation of angular distributions of fission fragments for the ¹⁶O + ²³²Th and ¹²C + ^235,236U reactions has been analyzed within a dynamic approach. In this approach, the component of the total angular momentum along the fission axis K is considered as a fluctuating quantity and the corresponding relaxation time is assumed to be the main parameter controlling the evolution of this mode. Particular attention is paid to the analysis of the effect of initial distributions over K (formed during fusion) on the angular distribution of fission fragments of nuclei having fission barriers comparable with the nuclear temperatures.     

Realistic Langevin Treatment of Fission-Fragment Energy Distribution

The average kinetic energy and its variance of fission fragments for the compound nucleus ²¹³At are calculated by four-dimensional Langevin equation. Two collective coordinates are taken into account: the distance p between the centers of the nascent fragments and the neck parameter h using the {c, h, a} parametrization. The model allows for a realistic coordinate dependence of all coeficien ts appearing in the Langevin equation as computed by the Werner-Wheeler method. The diffusion process for thc fissioning nuclear system from the ground state to the scission line on the realistic energy surface is investigated. The results are in good agreement with the experiments.     

Study of fission dynamics with the three-dimensional Langevin equations

The dynamics of fission has been studied by solving one- and three-dimensional Langevin equations with dissipation generated through the chaos weighted wall and window friction formula. The average prescission neutron multiplicities, fission probabilities and the mean fission times have been calculated in a broad range of the excitation energy for compound nuclei ²¹⁰Po and ²²⁴Th formed in the fusion-fission reactions ⁴He + ²⁰⁶Pb , ¹⁶O + ²⁰⁸Pb and results compared with the experimental data. The analysis of the results shows that the average prescission neutron multiplicities, fission probabilities and the mean fission times calculated by one- and three-dimensional Langevin equations are different from each other, and also the results obtained based on three-dimensional Langevin equations are in better agreement with the experimental data.     

Stochastic model of tilting mode in nuclear fission

A model of induced nuclear fission was developed with consideration of thermodynamically fluctuating orientation degree of freedom (tilting) of deformed nuclei. This model was applied to analysis of the experimental angular anisotropy of fission fragments in the ¹⁶O + ²³²Th, ²³⁸U, ²⁴⁸Cm, ²⁰⁸Pb, ²⁰⁹Bi; ¹²C + ²³⁶U; ¹⁹F + ²⁰⁸Pb; and ¹¹B + ²³⁷Np reactions. Information on the equilibrating time of the tilting mode was obtained. The text was submitted by the authors in English.     

Dynamical Behaviour of the Mass Asymmetry Mass Parameter in Nuclear Fission

The theory of nuclear fission is reconsidered. We study the behaviour of the mass parameter as a dynamical quantity of the mass asymmetry. The dependence of the mass asymmetry mass parameter is studied as a function of the five collective coordinates. These parameters are reconsidered by including the temperature to show the temperature dependence of the mass parameter. The cranking model is used in developing all the mathematical and theoretical expressions. Numerical calculations of the obtained analytical expressions are carried out for the two fissioning nuclei ²³⁶U and ²³⁸U. The mass asymmetry mass parameters are calculated including the temperature as a function of the different five collective coordinates. The present study shows that the values of this mass asymmetry mass parameters are stable against the change of the temperature for temperature values greater than 1 MeV for all the different five collective coordinates.     

CALCULATION OF THE KINETIC-ENERGY DISTRIBUTIONS OF FISSION-FRAGMENTS FROM MONTE-CARLO SIMULATION TO THE FOUR-DIMENSIONAL LANGEVIN EQUATION

The four-dimensional Langevin equation for two collective coordinates (the distance bet ween the centers of mass of the dascent fragments and the neck parameter) and their conjugate momenta is used as a dynamical equation to describe the descent of Brownian particles from the saddle-point to the scission points Monte-carlo method is used to slove the Langevin equation.The variances of the kinetic-energy distributions of nuclear fission-fragments in the range 322/A<40 have been calculated.The results of calculation are in good agreement with the experimental data.