Multidimensional Langevin Equation Research Articles

Four-dimensional Langevin approach to fission with Cassini shape parameterization

We apply the Cassini shape parameterization to the dynamical study of actinide fission using the multi-dimensional Langevin equation. The fragment mass distributions for fission of 236U and 256, 258 Fm are calculated with the 4D Langevin equation. The 4D collective space is formed by adding either α3 or α6 to the basic 3D {α, α1, α4} space. We investigate the role of α3 and α6 by comparing the results of 4D (α3 or α6) with that of 3D. For 236U and 256Fm, where asymmetric fission is predominant, the fragment mass distributions calculated in the 4D (α3) space are in good agreement with experimental data. On the other hand, for 258Fm, which is characterized by symmetric fission, 4D (α6) is effective for describing the shape of compact symmetric fragments. We demonstrate the advantages of the Cassini shape parameterization for the dynamical study of actinide fission.

Fusion cross section and total kinetic energy of fission fragments by the dynamical dissipative surface-friction model

The capture cross section, the fusion cross section, and the quasi-fission yield producing symmetric fragments ( ACN/2 ± 20u ) in the 48 Ca+ 238 U reaction are analyzed by the multidimensional Langevin equation taking account of the surface friction effect. From the experimental data, the strength of the tangential friction has been determined. It is presented that tangential friction increases in proportional to the power of the relative velocity of the projectile and the target.

Covariant formulation of nonlinear Langevin theory with multiplicative Gaussian white noises

The multi-dimensional non-linear Langevin equation with multiplicative Gaussian white noises in Ito's sense is made covariant with respect to non-linear transform of variables. The formalism involves no metric or affine connection, works for systems with or without detailed balance, and is substantially simpler than previous theories. Its relation with deterministic theory is clarified. The unitary limit and Hermitian limit of the theory are examined. Some implications on the choices of stochastic calculus are also discussed.

On multidimensional fractional Langevin equations in terms of Caputo derivatives

On multidimensional fractional Langevin equations in terms of Caputo derivatives

Influence of the shell effects on evaporation residue cross section of superheavy nuclei

In order to study the influence of the shell effects on the formation and fission of superheavy elements, we applied multidimensional Langevin equations. The evaporation residue cross sections have been calculated for 3n, 4n, and 5n evaporation channels using three ( $$K=0$$ )- and four ( $$K \ne 0$$ )-dimensional Langevin equations. Calculations were done for 48Ca + 238U and 48Ca + 244Pu hot fusion reactions with 3n, 4n evaporation channels and 70Zn + 208Pb, and 54Cr + 209Bi cold fusion reactions with 1n and 2n evaporation channels. The calculations were performed for 4n and 5n evaporation channels of the 26Mg + 238U reaction, as well. Our results show that with increasing dimension of Langevin equations the residue cross section increases, whereas the fission cross section decreases. The obtained results with four-dimensional Langevin and considering shell effects are in better agreement with experimental data in comparison with three- and four-dimensional Langevin equations without shell effects.

Formation of Superheavy Elements: Study Based on Dynamical Approach

Using multi-dimensional Langevin equations for the probability distribution of the distance between the surfaces of two approaching nuclei, we have studied the formation of superheavy elements via calculation of evaporation and fission cross sections of these elements. Evaporation residue cross sections have been calculated for the 1n, 2n, 3n, 4n, and 5n evaporation channels using one and four dimensional Langevin equations for the 48Ca+226Ra, 232Th, 238U, 237Np, 239,240,242,244Pu, 243Am, 245,248Cm, 249Bk, and 249Cf reactions. Our results show that with increasing dimension of Langevin equations the evaporation residue cross section is increased. Also, obtained results based on fourdimensional Langevin are in better agreement with experimental data in comparison with one-dimensional model.

Dynamical evolution of spectator systems produced in ultrarelativistic heavy-ion collisions

In peripheral heavy-ion collisions at ultra-relativistic energies, usually only parts of the colliding nuclei effectively interact with each other. In the overlapping zone, a fireball or quark-gluon plasma is produced. The excitation energy of the heavy remnant can range from a few tens to several hundreds of MeV, depending on the impact parameter. The decay of these excited spectators is investigated in this work for the first time within a dynamical approach based on the multi-dimensional stochastic Langevin equation. The potential of this exploratory work to understand the connection between electromagnetic fields generated by the heavy spectators and measured pion distributions is discussed.

Dissipation strength of the tilting degree of freedom in fusion-fission reactions

The four-dimensional Langevin model was applied to calculate a wide set of experimental observables for compound nuclei, formed in heavy-ion fusion-fission reactions. A modified one-body mechanism for nuclear dissipation with a reduction coefficient ks of the contribution from a ”wall” formula was used for shapes parameters. Different possibilities of deformation-dependent dissipation coefficient for the K coordinate (γK) were investigated. Presented results demonstrate that the influence of the ks and γK parameters on the calculated quantities can be selectively probed. It was found that it is possible to describe experimental data with the deformation-dependent γK coefficient. One of the possibility is to use large values of γK � 0.2 (MeV zs) −1/2 for compact shapes featuring no neck and small values of γK � 0.0077 (MeV zs) −1/2 for elongated shapes. Fission still is one of the most interesting and challenging topics in nuclear physics providing a perfect opportunity to investigate the large scale evolution of initial compound nucleus into fission products. During the past two decades stochastic approach based on multidimensional Langevin equations has been extensively and rather successfully used to elucidate many problems of collective nuclear dynamics in fusion-fission reactions at high excitation energies [1, 2]. A reasonable choice of collective degrees of freedom for modeling shape evolution and considering particle evaporation allow modeling the complex interplay between static and dynamical effects in fission and succeeding in explaining a wide range of experimental data [3, 4].

Dynamical features of nuclear fission

It is now established that the transition-state theory of nuclear fission due to Bohr and Wheeler underestimates several observables in heavy-ion-induced fusion–fission reactions. Dissipative dynamical models employing either the Langevin equation or equivalently the Fokker–Planck equation have been developed for fission of heavy nuclei at high excitations (T∼1 MeV or higher). Here, we first present the physical picture underlying the dissipative fission dynamics. We mainly concentrate upon the Kramers’ prescription for including dissipation in fission dynamics. We discuss, in some detail, the results of a statistical model analysis of the pre-scission neutron multiplicity data from the reactions 19F+194,196,198Pt using Kramers’ fission width. We also discuss the multidimensional Langevin equation in the context of kinetic energy and mass distribution of the fission fragments.

Theoretical investigation of shape parametrization’s effects on characteristics of fission fragments in 18O-induced fission of 154Sm, 197Au, 238U and 246Cm

The multi-dimensional Langevin equations are employed to investigate mass and energy distributions of fission fragments. The calculations have been performed with two different shape classes of parametrization, namely Funny Hills and Cassinian Ovaloids. It was shown that inclusion of the Funny Hills parameterizations in the dynamical model produced considerable increase in neutron multiplicity as compared with available experimental data for 18O-induced fission of 154Sm, 197Au, 238U and 246Cm. The proposed shape-dependent multi-dimensional dynamical model reproduces well experimental data for mass distribution, neutron multiplicity and average kinetic energy of fission fragments over a wide range of mass–energy regions.

Role of asymmetry parameter and dissipation coefficient ofKcoordinate in angular distribution of fission fragments

Using multidimensional Langevin equations, we have calculated the angular distribution of fission fragments with nonconstant values for the dissipation coefficient of the $K$ coordinate (the projection of the total angular momentum $I$ onto the symmetry axis of the fissioning nuclear system) for symmetry and asymmetry fission. We investigated the relaxation time of the $K$ coordinate, ${\ensuremath{\tau}}_{K}$, as a function of scission criteria and the asymmetry parameter. Calculations are done for ${}^{16}\mathrm{O}+{}^{232}\mathrm{Th}\ensuremath{\rightarrow}{}^{248}\mathrm{Cf}$, ${}^{16}\mathrm{O}+{}^{238}\mathrm{U}\ensuremath{\rightarrow}{}^{254}\mathrm{Fm}$, and ${}^{16}\mathrm{O}+{}^{248}\mathrm{Cm}\ensuremath{\rightarrow}{}^{264}\mathrm{Rf}$ reactions and obtained results are compared with available experimental data. Our calculations with nonconstant values of the $K$ coordinate show that the agreement between theoretical calculations and experimental data for asymmetry fission is better than that for symmetry fission.

Towards Multi-Dimensional Nonlinear Langevin Equation

Towards Multi-Dimensional Nonlinear Langevin Equation

Dynamical model of surrogate reactions

A new dynamical model is developed to describe the whole process of surrogate reactions; transfer of several nucleons at an initial stage, thermal equilibration of residues leading to washing out of shell effects and decay of populated compound nuclei are treated in a unified framework. Multi-dimensional Langevin equations are employed to describe time-evolution of collective coordinates with a time-dependent potential energy surface corresponding to different stages of surrogate reactions. The new model is capable of calculating spin distributions of the compound nuclei, one of the most important quantity in the surrogate technique. Furthermore, various observables of surrogate reactions can be calculated, e.g., energy and angular distribution of ejectile, and mass distributions of fission fragments. These features are important to assess validity of the proposed model itself, to understand mechanisms of the surrogate reactions and to determine unknown parameters of the model. It is found that spin distributions of compound nuclei produced in $^{18}$O+$^{238}$U $\rightarrow ^{16}$O+$^{240*}$U and $^{18}$O+$^{236}$U $\rightarrow ^{16}$O+$^{238*}$U reactions are equivalent and much less than 10$\hbar$, therefore satisfy conditions proposed by Chiba and Iwamoto (PRC 81, 044604(2010)) if they are used as a pair in the surrogate ratio method.

DYNAMICAL SIMULATION OF EMISSION ANGLE DEPENDENCE OF FRAGMENTS SPIN IN HEAVY ION FUSION–FISSION REACTIONS

In this paper, a formalism consisting of multi-dimensional Langevin equation (LE) and Scission point Transition State Model (SCTSM) have been applied to study the angle dependence of fragment's average spin for 16 O + 209 Bi and 16 O + 232 Th systems. The influence of asymmetry and deformation parameters, neck thickness of compound nucleus and pre-scission neutron multiplicity on the fragment's average spin is discussed. Results obtained using this approach are in better agreement with the experimental data as compared with the SCTSM.

Dynamic reaction coordinate in thermally fluctuating environment in the framework of the multidimensional generalized Langevin equations

A framework recently developed for the extraction of a dynamic reaction coordinate to mediate reactions buried in a multidimensional Langevin equation is extended to the generalized Langevin equations without a priori assumption of the forms of the potential (in general, nonlinearly coupled systems) and the friction kernel. The equation of motion with memory effect can be transformed into an equation without memory at the cost of an increase in the dimensionality of the system, and hence the theoretical framework developed for the (nonlinear) Langevin formulation can be generalized to the non-Markovian process with colored noise. It is found that the increased dimension can be physically interpreted as effective modes of the fluctuating environment. As an illustrative example, we apply this theory to a multidimensional generalized Langevin equation for motion on the Müller-Brown potential surface with an exponential friction kernel. Numerical simulations find a boundary between the highly reactive region and the less reactive region in the space of initial conditions. The location of the boundary is found to depend significantly on both the memory kernel and the nonlinear couplings. The theory extracts a reaction coordinate whose sign determines the fate of the reaction taking into account thermally fluctuating environments, memory effect, and nonlinearities. It is found that the location of the boundary of reactivity is satisfactorily reproduced as the zero of the statistical average of the new reaction coordinate, which is an analytical functional of both the original position coordinates and velocities of the system, and of the properties of the environment.

Nonlinear dynamical effects on reaction rates in thermally fluctuating environments

A framework to calculate the rate constants of condensed phase chemical reactions of manybody systems is presented without relying on the concept of transition state. The theory is based on a framework we developed recently adopting a multidimensional underdamped Langevin equation in the region of a rank-one saddle. The theory provides a reaction coordinate expressed as an analytical nonlinear functional of the position coordinates and velocities of the system (solute), the friction constants, and the random force of the environment (solvent). Up to moderately high temperature, the sign of the reaction coordinate can determine the final destination of the reaction in a thermally fluctuating media, irrespective of what values the other (nonreactive) coordinates may take. In this paper, it is shown that the reaction probability is analytically derived as the probability of the reaction coordinate being positive, and that the integration with the Boltzmann distribution of the initial conditions leads to the exact reaction rate constant when the local equilibrium holds and the quantum effect is negligible. Because of analytical nature of the theory taking into account all nonlinear effects and their combination with fluctuation and dissipation, the theory naturally provides us with the firm mathematical foundation of the origin of the reactivity of the reaction in a fluctuating media.

Hierarchy of reaction dynamics in a thermally fluctuating environment

Nonlinear dynamics in the passage over a rank-one saddle is investigated as a function of temperature in the presence of stochastic, thermal fluctuation. The analyses are based on a framework we developed recently adopting a multidimensional underdamped Langevin equation (without any assumption for the form of the potential of mean force). The framework can in principle provide a single coordinate to enable us to predict the final destination of the reaction in a thermally fluctuating media. At each temperature, the preciseness or the error of the reaction coordinate is evaluated in capturing the true reaction dynamics at different levels of approximations. By using the Müller-Brown potential as an illustrative example, it is found that a hierarchy of dynamical structure exists in the region of a rank-one saddle, in which the crossing dynamics qualitatively changes as the temperature increases. We discuss the mechanism of how the reaction coordinate persists, which provides a boundary of the reaction to divide the phase space into the reactive and the nonreactive regions, even in the presence of thermal fluctuation.

Dynamic pathways to mediate reactions buried in thermal fluctuations. I. Time-dependent normal form theory for multidimensional Langevin equation

We present a novel theory which enables us to explore the mechanism of reaction selectivity and robust functions in complex systems persisting under thermal fluctuation. The theory constructs a nonlinear coordinate transformation so that the equation of motion for the new reaction coordinate is independent of the other nonreactive coordinates in the presence of thermal fluctuation. In this article we suppose that reacting systems subject to thermal noise are described by a multidimensional Langevin equation without a priori assumption for the form of potential. The reaction coordinate is composed not only of all the coordinates and velocities associated with the system (solute) but also of the random force exerted by the environment (solvent) with friction constants. The sign of the reaction coordinate at any instantaneous moment in the region of a saddle determines the fate of the reaction, i.e., whether the reaction will proceed through to the products or go back to the reactants. By assuming the statistical properties of the random force, one can know a priori a well-defined boundary of the reaction which separates the full position-velocity space in the saddle region into mainly reactive and mainly nonreactive regions even under thermal fluctuation. The analytical expression of the reaction coordinate provides the firm foundation on the mechanism of how and why reaction proceeds in thermal fluctuating environments.

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 Z2/A falls within the range 20 ≤ Z2/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.

Analytical solution of a multidimensional Langevin equation at high friction limits and probability passing over a two-dimensional saddle

The analytical solution of a multidimensional Langevin equation at the overdamping limit is obtained and the probability of particles passing over a two-dimensional saddle point is discussed. These results may break a path for studying further the fusion in superheavy elements synthesis.