Mass-energy distribution of fragments from the fission of excited nuclei within three-dimensional Langevin dynamics

Two-dimensional mass-energy distributions of fission fragments are calculated for the first time on the basis of three-dimensional stochastic Langevin equations. In these calculations, the emission of light prescission particles is taken into account within the statistical model. The results demonstrate that calculations within three-dimensional Langevin dynamics make it possible to describe most compre-hensively the properties of the mass-energy distribution of fission fragments and the mean multiplicity of prescission neutrons.     

A note on the examination of isospin effects in multi-dimensional Langevin fission dynamics

In [W. Ye, F. Wu, H.W. Yang, Phys. Lett. B 647 (2007) 118] prescission protons and α particles of high-isospin ²⁰⁶Pb were shown to be almost independent of the dissipation strength ks$k_s$. Subsequently, in [P.N. Nadtochy, et al., Phys. Lett. B 685 (2010) 258] prescission light charged particles (LCPs) were shown to have approximately the same sensitivity as neutrons to ks$k_s$ for ²⁰⁶Pb and ²⁰⁴Hg nuclei. In this Letter we point out that the reason for the apparent contradictory conclusions is that the authors in the latter did not compute the changes in the absolute yields of prescission LCPs multiplicities with increasing ks$k_s$ and compare them with typical experimental uncertainties. It is shown that the expected changes are very small in the case of neutron-rich ²⁰⁶Pb and ²⁰⁴Hg systems, which are within experimental error bars. This indicates that, from the viewpoint of experiment, LCPs emission of ²⁰⁶Pb and ²⁰⁴Hg is insensitive to dissipation.     

Analysis of angular momentum dependence of fission fragment mass-energy distribution within Langevin dynamics

Dependence of fission fragment mass-energy distribution on the angular momentum is studied within Langevin dynamics. The calculations are performed in the framework of the generalized temperature-dependent finite-range liquid drop model. The analysis is done for five compound nuclear systems representing heavy fissioning nuclei, medium fissioning nuclei, and a light fissioning one with the angular momentum varied in a wide range from l = 0 to 70?. The coefficients d〈E _K 〉/dl ² and $d\sigma _{{\rm E}_{\rm K} }^2 /dl^2 $ are extracted. Previous analysis of the dσ _M ² /dl ² coefficient is generalized. Excitation energy dependence of the fission fragment mass-energy distribution is also found. The qualitative comparison of the extracted values with the experimental data reveals good agreement for all the cases. The calculated values of the coefficients dσ _M ² /dl ² and $d\sigma _{{\rm E}_{\rm K} }^2 /dl^2 $ are functions of the angular momentum, in contrast to the experimental estimations.     

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.     

Generalized Langevin equations with time-dependent temperature

A generalized Langevin equation describing the evolution of a particle in a heat bath with a time-dependent temperature is derived for a simple model. The temperature is controlled by introducing dissipative terms in the dynamical equations of the heat bath particles. The Langevin equation contains a term that is specifically associated with the variation of the temperature.     

Nonlinear generalized Langevin equations

Exact generalized Langevin equations are derived for arbitrarily nonlinear systems interacting with specially chosen heat baths. An example is displayed in which the Langevin equation is nonlinear but approximately Markovian.Research supported by NSF grant GP-29534.     

Temperature dependent fission fragment distribution in the Langevin equation

The temperature dependent width of the fission fragment distributions was simulated in the Langevin equation by taking two-parameter exponential form of the fission fragment mass variance at scission point for each fission event. The result can reproduce experimental data well, and it permits to make reliable estimate for unmeasured product yields near symmetry fission.     

Temperature dependent fission fragment distribution in the Langevin equation

The temperature dependent width of the fission fragment distributions was simulated in the Langevin equation by taking two-parameter exponential form of the fission fragment mass variance at scission point for each fission event. The result can reproduce experimental data well, and it permits to make reliable estimate for unmeasured product yields near symmetry fission.     

Langevin equations and stochastic integrals

The ambiguity of stochastic integrals involved in Langevin equations is removed by the postulate of invariance with respect to nonlinear transformations of the coordinates. The Stratonovich sense of the integrals, which is imposed thereby, is also strongly suggested by stability considerations requiring small changes of the solutions whenever the perturbations are changed by a small amount. The associated Fokker-Planck equation must include the spurious drift which arises from the transition from the Stratonovich to the Itô sense of the Langevin equations and describes one aspect of the systematic motion due to nonconstant fluctuations.     

Delay and memory-like Langevin equations

By using an exact functional formalism we characterize general stationary properties of linear memory-like Langevin equations, including the case of delay equations driven by arbitrary noises. Spectral properties and stationary distributions are analyzed in detail.     

The validity of nonlinear Langevin equations

In the presence of internal noise the variables describing a system are intrinsically stochastic. If they constitute a Markov process the expansion enables one to extract a deterministic macroscopic equation and to compute the fluctuations in successive approximations. In the lowest or linear noise approximation the fluctuations can be represented by a Langevin equation, provided it is handled appropriately. Higher orders cannot be described by any white noise Langevin equation. The question whether the equation has to be interpreted according to Itô or Stratonovich concerns these higher orders, for which the equation is not valid anyway.     

Langevin analysis of fission excitation functions induced by protons

The stochastic Langevin approach to fission is applied to analyze fission excitation functions measured in p+²06Pb and p+²09Bi systems. A presaddle friction strength of（3–5）×10²¹s^-1is extracted by comparing theoretical predictions with experimental data. Furthermore, the small distortion of the formed compound nuclei with respect to the spherical shape under the condition of low angular momentum suggests that experimentally, populating an excited compound system via light-ion induced reactions favors a more accurate determination of presaddle friction with a fission cross section.     

On fractional Langevin differential equations with anti-periodic boundary conditions

In this paper, we provide existence criteria for the solutions of p-Laplacian fractional Langevin differential equations with anti-periodic boundary conditions. The Caputo fractional as well as Caputo q-fractional operators are used to address the derivatives. The main results are verified by the help of Leray–Schaefer’s fixed point theorem. We construct an example to illustrate the feasibility of the main theorems. Our results are new and provide extensions to some known theorems in the literature.     

Effective potential for complex Langevin equations

We construct an effective potential for the complex Langevin equation on a lattice. We show that the minimum of this effective potential gives the space–time and Langevin time average of the complex Langevin field. The loop expansion of the effective potential is matched with the derivative expansion of the associated Schwinger–Dyson equation to predict the stationary distribution to which the complex Langevin equation converges.