Three-dimensional Langevin Equations Research Articles

DYNAMICAL CALCULATION FOR FUSION-FISSION PROCESS IN SUPERHEAVY MASS REGION

The fusion-fission process for the synthesis of superheavy elements is discussed on the basis of fluctuation-dissipation model. We employ three-dimensional Langevin equation. We investigate the fusion-fission process by analyzing not only the mass distribution of fission fragments but also neutron multiplicity on fusion-fission process.

Reduction coefficient in surface-plus-window dissipation: Analysis of experimental data from fusion-fission reactions within a stochastic approach

A stochastic approach based on three-dimensional Langevin equations was applied to detailed study of fission dynamics in fusion-fission reactions. The reduction coefficient in surface-plus-window dissipation is deduced by analyzing the available experimental data on various observable quantities in fusion-fission reactions. Analysis of the results shows that not only characteristics of the mass-energy distribution of fission fragments but also the mass and kinetic-energy dependence of the prescission and postscission neutron multiplicities, the angular anisotropy, and fission probability can be reproduced using surface-plus-window dissipation with the reduction coefficient from a wall formula ks =0 .25-0.5 for com- pound nuclei lighter than 224 Th. The analysis performed reveals that the coordinate-independent reduction coefficient ks is not compatible with the simultaneous description of the main fission characteristics for fissioning systems heavier than 224 Th. c � 2003 MAIK "Nauka/Interperiodica".

More detailed study of fission dynamics in fusion-fission reactions within a stochastic approach

A stochastic approach based on three-dimensional Langevin equations was applied to a more detailed study of fission dynamics in fusion-fission reactions. The dynamical model has been developed and extended to investigate fission characteristics of light fissioning nuclei at low excitation energies. The energy dependences of an anisotropy of the fission fragment angular distribution, an evaporation residue cross section, a fission cross section, mean prescission neutron, and giant dipole $\ensuremath{\gamma}$ multiplicities have been analyzed for the ${}^{16}\mathrm{O}{+}^{208}\mathrm{Pb}$-induced fission of ${}^{224}\mathrm{Th}.$ Also, dependence of the prescission neutron multiplicity on the fragment mass asymmetry and total kinetic energy have been calculated. Analysis of the results shows that not only characteristics of the mass-energy distribution of fission fragments, but also the mass and kinetic-energy dependence of the prescission neutron multiplicity, the angular anisotropy, and fission probability can be reproduced using a modified one-body mechanism for nuclear dissipation with a reduction coefficient of the contribution from a wall formula ${k}_{s}=0.25--0.5$ for compound nuclei ${}^{172}\mathrm{Yb},$ ${}^{205}\mathrm{Fr},$ ${}^{215}\mathrm{Fr},$ and ${}^{224}\mathrm{Th}.$ Decrease of the prescission neutron multiplicities with fragment mass asymmetry is due to a decrease of the fission time. The results obtained show that prescission neutrons are evaporated predominantly from the nearly spherical compound nucleus at an early stage of fission process before the saddle point is reached. From performed analysis one can conclude that coordinate-independent reduction coefficient ${k}_{s}$ is not compatible with simultaneous description of the main fission characteristics for heavy fissioning systems ${}^{256}\mathrm{Fm}$ and ${}^{252}\mathrm{Fm}.$

Langevin description of charge fluctuations in fission of highly excited nuclei

A stochastic approach to fission dynamics based on three-dimensional Langevin equations is used to study charge fluctuations in fission of highly excited nuclei. Elongation, neck and charge asymmetry parameters are chosen as relevant collective coordinates. Using as an example 4He + 232Th induced fission of 236U in a broad range of excitation energies (from 60 to 160 MeV), the isobaric charge variance is investigated to obtain information on nuclear dissipation. The friction parameter of the charge mode is calculated under the assumption of both the one-body and two-body mechanisms of nuclear viscosity. The results obtained for the variances of the charge distribution within the applied approach reveal that the optimal reproduction of the available data is achieved with the value of the two-body viscosity coefficient (0.6≤ν≤1.8)×10-23MeV s fm-3 that is close to those deduced earlier from the description of the fission fragment mass-energy distribution. The expression for the friction parameter of the charge mode is derived within the one-body mechanism of nuclear dissipation. The one-body mechanism of nuclear dissipation also provides a good reproduction of data on the isobaric charge variance without any adjustable strength parameter.

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 =3 4-42. 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% )i n 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 ks must be diminished at least by a factor of 2 (0.2 ≤ ks ≤ 0.5) in relation to that in the case of total one- body viscosity (ks =1 ). c � 2002 MAIK "Nauka/Interperiodica".

Three-dimensional Langevin calculations of fission fragment mass-energy distribution from excited compound nuclei

A stochastic approach to fission dynamics based on three-dimensional Langevin equations was applied to calculate fission fragment mass-energy distribution from a number of excited compound nuclei formed in reactions induced by heavy ions. Evaporation of prescission light particles along Langevin fission trajectories from the ground state of the compound nucleus to its scission has been taken into account using a Monte Carlo simulation technique. Inclusion of the third collective coordinate in Langevin dynamics leads to a considerable increase of the variance of the mass and the kinetic-energy distributions of fission fragments as compared with two-dimensional Langevin calculations. A liquid-drop model with finite range of nuclear forces and a modified one-body mechanism for nuclear dissipation have been used in the calculations. The results of the calculations are compared with the available experimental data. The calculations performed using the three-dimensional Langevin dynamics reproduce sufficiently well all the parameters of the two-dimensional fission fragment mass-energy distribution and their dependence on various parameters of the compound nucleus. The mean prescission neutron multiplicities are also reproduced with good accuracy. In order to reproduce simultaneously the measured prescission neutron multiplicities and the variance of the fission fragment mass-energy distribution, the reduction coefficient of the contribution from a wall formula has to be decreased at least by half of the one-body dissipation strength $(0.25<~{k}_{s}<~0.5).$

Dynamics of the Synthesis of Superheavy Elements

Present status of the study of superheavy elements is reviewed. The dynamical process of synthesizing superheavy elements is studied on the basis of the fluctuation-dissipation dynamics. The whole process is divided into three stages, i.e., the approaching stage, the formation stage, and the surviving stage. For the study of the formation stage, a three-dimensional Langevin equation is used after the contact of two nuclei to calculate the competition between the complete fusion and the quasi-fission. For the description of the surviving stage, a statistical model is used to estimate the competition between the fission and the evaporation process. In this stage, it is very important to include the temperature-dependent shell correction to the fission barrier and the collective enhancement of the level density. From the study of the isotope dependence of the production cross section, the survival probability is found to be very sensitive to the separation energy of neutron. The results show the importance of the use of neutron-rich beams and targets.

Stochastic dynamics of slow gas–surface collisions

The dynamics of slow gas atoms at surfaces of solids is studied. A slow motion implies that the particle velocity is lower than the sound speed in the solid. A specific property of such systems is that the energy exchange with the long-wave phonons makes the dominant contribution to the dynamic gas–solid interaction. On this basis the solid is modelled by an elastic continuum. The many-body problem is reduced to the three-dimensional Langevin equation describing the particle motion by excluding the phonon variables. Both bulk and surface phonons are taken into account. The friction matrix is presented as a product of two matrices. One of them is space dependent and the other one depends on the elastic constants of the solid. The latter is calculated for isotropic media and for (001), (011) and (111) faces of some cubic crystals.

On the exact and phenomenological Langevin equations for a harmonic oscillator in a fluid

We have derived, starting from the Liouville equation and using projection operators, the three-dimensional Langevin and Fokker-Planck equations for the time dependence of the momenta and position of two Brownian particles of mass M interacting with a harmonic potential (i.e., a harmonic oscillator) in a fluid of particles with mass m, with M ⪢ m. The resulting Langevin equation has a very complicated structure in that the friction coefficients, x, due to the hydrodynamic interaction between the oscillator particles, are functions of R(t), the time-dependent seperation of the oscillator particles, and the noise terms, though Gaussian, are non-stationary and of a generalized form, i.e., neither “additive” nor “multiplicative”. In addition, there is a term involving the mean force exerted by the fluid on the oscillating Brownian particle. We then investigated the various approximations which must be made to reduce this “exact” Langevin equation to the frequently used one-dimensional, phenomenological Langevin equations (and corresponding Fokker-Planck equations) with purely “additive” and “multiplicative” noise. These approximations are of three types: (a) one must neglect the term arising from the rotational motion of the oscillator in the fluid; (b) one must neglect the R(t) dependence of X[R(t)], leading to purely “additive” noise, or approximate X[R(t)] by a Taylor series expansion to quadratic order in R(t) - R0 (where R0 is the equilibrium separation of the oscillator particles), leading to “additive” and “multiplicative” noise; and (c) one must either neglect the mean force term or approximate it by a term linear in R(t) - R0. We conclude from these results that one-dimensional phenomenological Langevin equations with simple noise structure are of doubtful validity for the dynamical description of the relative momenta (or vibrational energy) of oscillating molecules in a fluid.