Nuclear Fission Dynamics Research Articles

Recent progress in microscopic nuclear fission dynamics

原子核裂变是一个复杂的非平衡非绝热的大幅度量子多体动力学过程。近年来，随着计算能力的发展，微观裂变动力学取得了很大的进展。本文介绍了非绝热的时间相关的密度泛函理论和绝热的时间相关的生成坐标法等微观裂变动力学模型。裂变位能面的并行约束计算和能量相关性对绝热裂变模型很重要。最近我们扩展了有限温度加动力学涨落的时间相关的密度泛函理论。 基于时间相关的密度泛函理论对裂变过程中的对关联效应，涨落效应，耗散效应，非绝热效应，能量相关性，碎片之间的纠缠等核裂变机制获得了新的认识。此外微观裂变研究对理解天体过程中的r-process也很重要。 最终我们期待发展出更综合更精确的核裂变理论支撑核裂变的应用创新。

Investigating the fission dynamics of the following neutron shell closed nuclei within a stochastic dynamical approach: 210Po, 212Rn, and 213Fr* *D.A. acknowledges the financial support from the University Grants Commission (UGC), Government of India, received in the form of a research fellowship (Ref. No. 19/06/2016/(i) EU-V, Sr. No. 2061651303)

The dissipative dynamics of nuclear fission is a well confirmed phenomenon that can be either described by a Kramers-modified statistical model or by a dynamical model employing the Langevin equation. Although dynamical models as well as statistical models incorporating fission delays have been found to explain the measured fission observables in several studies, they present conflicting results for shell closed nuclei in the mass region of 200. Notably, an analysis of the recent data on neutron shell closed nuclei in the excitation energy range of 4080 MeV failed to provide a satisfactory description of the data, which was attributed to a mismatch with shell effects and/or entrance channel effects, without reaching a definite conclusion. In the present study, we demonstrate that a well established stochastic dynamical code can simultaneously reproduce the available data for pre-scission neutron multiplicities and fission and evaporation residue excitation functions for the following neutron shell closed nuclei Po and Rn and their isotopes Po and Rn without the need for including any extra shell or entrance channel effects. The relevant calculations are performed by using a phenomenological universal friction form factor with no ad-hoc adjustment of the model parameters. However, we note a significant deviation, beyond experimental errors, for some Fr isotopes.

Microscopic analysis of induced nuclear fission dynamics

The dynamics of low-energy induced fission is explored using a consistent microscopic framework that combines the time-dependent generator coordinate method (TDGCM) and time-dependent nuclear density functional theory (TDDFT). While the former presents a fully quantum mechanical approach that describes the entire fission process as an adiabatic evolution of collective degrees of freedom, the latter models the dissipative dynamics of the final stage of fission by propagating the nucleons independently toward scission and beyond. By combining the two methods, based on the same nuclear energy density functional and pairing interaction, we perform an illustrative calculation of the charge distribution of yields and total kinetic energy for induced fission of $^{240}$Pu. For the saddle-to-scission phase a set of initial points for the TDDFT evolution is selected along an iso-energy curve beyond the outer fission barrier on the deformation energy surface, and the TDGCM is used to calculate the probability that the collective wave function reaches these points at different times. Fission observables are computed with both methods and compared with available data. The relative merits of including quantum fluctuations (TDGCM) and the one-body dissipation mechanism (TDDFT) are discussed.

Impact of nuclear dissipation on the fission dynamics within the Langevin approach

The impact of nuclear dissipation on the dynamics of nuclear fission at low excitation energy is studied with a three-dimensional Langevin approach in which the two-center shell model is adopted to describe the nuclear shape and the single-particle potential. With three types of formulas for the friction tensor, such as the wall formula, the wall-and-window formula with and without the term related to the mass asymmetry change, the fragment mass and the total kinetic energy (TKE) distributions in low-energy fission are calculated. The calculated fragment mass distributions are almost identical for both with the wall formula and with the wall-and-window model without the term related to the mass asymmetry change, and both the results are consistent with the evaluated mass yields and the calculated results with the GEF model. Our study found that the term related to the mass asymmetry change in the window formula leads to an unreasonable shift of the peak position of the mass distribution. The TKE values calculated with the wall model are slightly larger than those with the wall-and-window model due to the more compact configuration obtained with the pure wall model. Moreover, the influences of the strength of friction tensor on the fragment distribution, the scission configuration, and the fission time distribution are investigated. With the present model the fission fragment mass distributions of U, Np, Pu, and Am isotopes are well reproduced and the systematic dependence of the averaged TKE on the Coulomb parameter is also well reproduced.

Optimization of basis functions for multiconfiguration mixing using the replica exchange Monte Carlo method and its application to C12

Background: To calculate excited states in quantum many-body systems, multiconfiguration mixing has often been employed. However, it remains unclear how to choose important Slater determinants from a huge model space.Purpose: We propose an efficient method as the replica exchange Monte Carlo (RXMC) method to sample important Slater determinants and optimize and analyze the results obtained. As an application, we investigate the ground and excited states of $^{12}\mathrm{C}$ based on the Bloch-Brink $\ensuremath{\alpha}$ cluster model and show the detailed structure of the obtained states.Methods: The RXMC method enables us to efficiently sample Slater determinants following the Boltzmann distribution on the multidimensional potential-energy surface (PES) under a given model space. To analyze the excited states obtained, we embed sampled basis functions onto the PES calculated with the $\ensuremath{\beta}\text{\ensuremath{-}}\ensuremath{\gamma}$ constraint method and discuss the main component in each state.Results: The RXMC method efficiently performs the samplings with a temperature parameter of ${T}_{L}=2.5$ MeV in $^{12}\mathrm{C}$. We obtain the gas-like state with a wide density distribution in the tail part in the second ${0}^{+}$ state. We also obtain the linear-chain-like states with the bending and stretching vibrational modes in the third and fourth ${0}^{+}$ states, respectively. In the fifth ${0}^{+}$ state, the main component of the basis functions contains expanded equilateral-triangle configurations.Conclusions: The second ${0}^{+}$ gas-like state emerges at the local minimum in the PES, which is the beginning of the valley structure connected to the linear-chain breakup channel. The third and fourth linear-chain-like states emerge in this valley structure. We conclude that the RXMC method is a powerful method to calculate the excited states of nuclei, which would be applied to much more complicated nuclear fission dynamics in heavier nuclei.

3D TDHFB theory with Gogny force and fission dynamicsof <sup>240</sup>Pu

Nuclear fission is one of the most complex processes in nuclear physics, involving a strong interplay between nuclear structure and dynamics, as well as various many-body correlations and fluctuation effects. Systematically theoretical and experimental research on the fission problems is of great significance to clarify the fission mechanism, reveal the complexity of low-energy nuclear dynamics, and test the nuclear model theory based on the nuclear many-body theory. However, despite the huge theoretical effort pursued by several generations of physicists, many crucial facets of this process are still not understood at a fundamental level, warranting further exploration of its dynamics. This is more so when viewed from the microscopic theoretical point of view. In the subject of study on nuclear fission, scission dynamics represents one of the most challenging problems in many-body theory as it plays a crucial role for determining the main properties of fission fragments and the emitted light particles. In the scission and post-scission processes, the fragments can experience a rapid change in shape and thus trigger the non-adiabatic effects coming from couplings between collective and intrinsic degrees of freedom. Therefore, from microscopic point of view, a reasonable description of the entire fission process could be achieved with an adiabatic model describing the slow evolution across the barrier, followed by a non-adiabatic treatment of the scission and post-scission dynamics. In this paper, with the aid of the nuclear mean-field model and Gogny effective nucleon-nucleon interaction, a three-dimensional Cartesian mesh code of time-dependent Hartree-Fock-Bogoliubov theory (Gogny-TDHFB Code for Fission) has been developed for solving the static and/or dynamic equations with isolated or periodic boundary conditions and no further symmetry assumptions. In contrast with the Skyrme HFB, the particle-hole channel and the particle-particle channel are treated on an equal footing in the HFB calculations with the Gogny interaction (Gogny HFB). The code consists of two parts: One is the constrained HFB for calculating the adiabatic fission potential energy surface in the multi-dimensional collective phase space, the other is TDHFB for studying the dynamical process of nuclear fission. In order to describe the extremely large deformation of nuclei in the direction of fission, the basis functions in the spatial grid points of the Lagrange mesh are introduced with respect to the direction of the z -axis, and the deformation harmonic oscillator eigenfunctions are used as the basis functions in the ( x , y ) directions. In the calculation of multi-dimensional potential energy surfaces, the constraints are imposed on expectation values of the multi-pole moments Q lm and the augmented Lagrangian method (ALM) is exploited for multi-constrained calculations. A parallel interface using the message passing interface (MPI) library has been implemented for large-scale calculations on massively parallel computers. Under the help of Gogny-TDHFB Code for Fission and by mainly focusing on the scission process, various aspects including the nuclear fission processes, the multi-dimensional potential energy surface, the configuration of scission point, more importantly, the non-adiabatic dynamics in the course of deformation from saddle to scission and also in post-scission are microscopically investigated in order to make clarification on the fission mechanisms, the nature of scission and some other basic problems. As an example, the spontaneous fission process of 240Pu has been studied. The results on the non-adiabatic dynamics in the course of deformation from saddle to scission and also in post-scission show that the pairing correlation plays crucial roles in the scission dynamics of nuclear fission.

Nuclear Fission Dynamics: Past, Present, Needs, and Future

Recent developments in theoretical modeling and in computational power have allowed us to make significant progress on a goal not achieved yet in nuclear theory: a fully microscopic theory of nuclear fission. The complete microscopic description remains a computationally demanding task, but the information that can be provided by current calculations can be extremely useful to guide and constrain phenomenological approaches. First, a truly microscopic framework that can describe the real-time dynamics of the fissioning system can justify or rule out assumptions and approximations incompatible with an accurate quantum treatment or with our understanding of the inter nucleon interactions. Second, the microscopic approach can be used to obtain trends such as: the excitation energy sharing mechanism between fission fragments (FFs) with increasing excitation energy of the fissioning system, the angular momentum content of the FFs, or even to compute observables that cannot be otherwise calculated in phenomenological approaches or even measured, as in the case of astronomical environments. Merely the characterization of the trends would be of great importance for various application. We present here arguments that a truly microscopic approach to fission does not support the assumption of adiabaticity of the large amplitude collective motion in fission, particularly starting from the outer saddle down to the scission configuration.

Into the Fission Valley of 'magic Nucleus' Polonium

The word radioactive was first coined by Marie Curie when she, along with her husband Pierre Curie, discovered the element Polonium. The nucleus 210Po is a testing ground for many theoretical and experimental aspects of nuclear structure as well as nuclear fission dynamics as it is a magic nucleus with neutron number N=126. At Variable Energy Cyclotron Centre, Kolkata the fission of Polonium nuclei is being studied in order to understand the survival of nuclear shell effects that is known to be the key for the stability of super heavy elements (SHE).

Nuclear Fission: from more phenomenology and adjusted parameters to more fundamental theory and increased predictive power

Two major recent developments in theory and computational resources created the favorable conditions for achieving a microscopic description of nuclear fission almost eighty years after its discovery in 1939 by Hahn and Strassmann (1930). The first major development was in theory, the extension of the Time-Dependent Density Functional Theory (TDDFT) to superfluid fermion systems. The second development was in computing, the emergence of powerful enough supercomputers capable of solving the complex systems of equations describing the time evolution in three dimensions without any restrictions of hundreds of strongly interacting nucleons. Even though the available nuclear energy density functionals (NEDFs) are phenomenological still, their accuracy is improving steadily and the prospects of being able to perform calculations of the nuclear fission dynamics and to predict many properties of the fission fragments, otherwise not possible to extract from experiments, are within reach, all without making recourse anymore to uncontrollable assumptions and simplified phenomenological models.

Fission barriers and probabilities of spontaneous fission for elements with Z ≥ 100

This paper briefly reviews recent progress in theoretical studies on fission barriers and fission half-lives of even–even superheavy nuclei. We compare and discuss results obtained in the semi-classical macroscopic–microscopic approach, the self-consistent mean-field models with the Skyrme and Gogny energy density functionals and in the relativistic mean-field theory. A short part of the paper is devoted to the calculation of the mass parameters and nuclear fission dynamics. We also discuss the predictive power of Skyrme energy density functionals applied to key properties of the fission path of 266Hs. Standard techniques of error estimates in the framework of a χ2 analysis are applied.

The Businaro--Gallone Region: A Playground for Dynamical Models of Fission?

Nuclei in the Businaro–Gallone region are proposed as a relevant tool to study the dynamics of nuclear fission at high temperature. The influence of the macroscopic contribution to the driving potential energy landscape, and its strong dependence on angular momentum, is addressed in this contribution as a first step. An advanced model based on a four-dimensional Langevin approach is used to calculate fission-fragment mass and charge distributions. A good agreement with experiment is observed. The work is in progress, with the goal to explore the major assets of nuclei in this region for probing the modeling of inertia, friction and fluctuations, and particle evaporation.

Nuclear fission dynamics within a generalized Langevin approach

Within the generalized (non-Markovian) multidimensional Langevin approach, the time and energy characteristics of symmetric fission of highly excited heavy nuclei are studied. In two-dimensional space of the collective deformation parameters, it is considered a nuclear descent from the top of the fission barrier to the scission point. The distributions of descent times and total kinetic energy of fission fragments are calculated as functions of memory time, measuring the relative size of memory effects in the collective dynamics. We found that the peculiarities of the non-Markovian dynamics at fairly large values of the memory time are reflected in the saturation of the mean time of motion from the saddle to scission with the growth of the strength of memory effects in the system.

The role of neck degree of freedom in nuclear fission

The dynamics of nuclear fission is investigated using Langevin equations in order to study the role of the neck degree of freedom in fission. The time-dependent fission widths are calculated in both one (with only elongation coordinate) and two (with elongation plus neck) dimensions. A quantitative estimate is also made of the distribution of the fission trajectories in two dimensions. Comparing the results from one and two-dimensional calculations, we find that the stationary fission widths are nearly the same in both the calculations. However, the most probable fission path beyond the saddle ridge in two dimensions deviates considerably from that in one dimension.

Experimental studies of nuclear fission dynamics near the scission point

Conditions for formation of angular and energy distributions of light particles emitted in ternary fission of 233, 235U, 239Pu, and 245Cm nuclei induced by cold polarized neutrons have been studied in the course of investigating T-odd asymmetry in emission of these particles with respect to the plane formed by the fission axis and the polarization axis of the fissioning nucleus. The results obtained lead to the conclusion that in ternary fission charged particles are emitted by the fissioning nuclear system rotating around the polarization direction.

The Dependency on the Dissipation Tensor of Multi-modal Nuclear Fission

In the study of the fission of actinide nuclei at low excitation energies including the spontaneous fission, it was found that the fragment mass distribution and the total kinetic energy (TKE) distribution consist of more than one component, in contrast to the simple single peak structure that is found in the fission at high excitation energies. 1– 6 This phenomenon is attributed to the existence of more than one fission path and is called the multi-modal fission. The mass and TKE distributions depend sensitively on the excitation energy and the position of the peaks of the mass distribution suggests the influence of the closed shell structure of the fragments. Therefore, it is supposed that the microscopic energy plays an important role for the manifestation of this phenomenon. It is a great challenge for us to understand this phenomenon in terms of nuclear many-body dynamics. Several authors studied the potential energy surface (PES) including the microscopic energy in a multi-dimensional parameter space that describes various nuclear shapes; one can deduce the possible fission paths by studying the location of the saddle points and the fission valleys in multi-dimensional parameter space. 7 With this method, they could explain the general trend of the position of the peaks of the mass distribution. The dynamical point of view is necessary to progress the study of the fission mode. We have applied the Langevin approach to the study of the fission modes in uranium nuclei and in fermium nuclei. 8–10 We studied the mass and TKE distributions and demonstrated that we can decompose the fission events into several components by tracing the Langevin trajectories. We also studied the isotope dependence and the excitation energy dependence of the fission mode. 8–10 In the previous studies, we adopted the wall-and-window type one-body friction as the dissipation mechanism of the nuclear fission dynamics. The validity of this dissipation mechanism has been demonstrated by one of the authors (T.W.) who studied the dissipation tensor dependence of the pre-scission neutron multiplicity and the mean TKE. 11–14 From the comparison of the results of the dynamical calculation with experimental data, they excluded the possibility of the two-body type dissipation to be the dominant mechanism by showing that it cannot reproduce the pre-scission neutron data and the TKE data simultaneously. On the other hand, they showed that the wall-and-window type one-body friction can reproduce both data reasonably well and concluded that it is a reasonable model for the dissipation mechanism of nuclear fission. There are other models that are of one-body nature, e.g. surface-plus-window formula, modified wall-and-window formula and chaos weighted wall formula. 15–17 Though there were no free parameters in the original derivation of the one-body friction, 18, 19 the strength has been modified frequently in order to reproduce some experimental data. For example, in the study of the light particle evaporation and the mass distribution, Schmitt et al. used the strength as a free parameter. 17 The modification itself should be acceptable when we take account of the simplicity of the model; it is a macroscopic model without any microscopic effect and it has no dependence on the temperature. However, when one modifies the strength of the nuclear dissipation just to reproduce only one physical quantity, it might be inappropriate to conclude that the deduced strength has definite physical meanings. It may reflect the other effects completely different from the dissipation, like the insufficiency of the model space. It is very important to compare many (at least more than one) physical quantities at the same time. Among the physical quantities that are measured in nuclear fission, the TKE and mass distributions are well investigated experimentally in many cases. In this study, we use these quantities to discuss the dissipation dependence of the fission modes. It is shown that the TKE distribution is, as was expected, directly connected to the strength of the dissipation and we can put some constraints on the strength of the dissipation. Furthermore, it is shown that the mass distribution changes rather drastically when one uses different models for the dissipation mechanism. These results demonstrate the importance and the usefulness of the dynamical approach to the study of the fission mode. Section 2 gives a concise description of our framework. Results are shown in Sec. 3 concerning the fission of 264 Fm nucleus at Ex = 20 MeV. Summary is given in Sec. 4. 2. Methods

Prescission neutron multiplicity and fission probability from Langevin dynamics of nuclear fission

A theoretical model of one-body nuclear friction which was developed earlier, namely the chaos-weighted wall formula, is applied to a dynamical description of compound nuclear decay in the framework of the Langevin equation coupled with statistical evaporation of light particles and photons. We have used both the usual wall formula friction and its chaos-weighted version in the Langevin equation to calculate the fission probability and prescission neutron multiplicity for the compound nuclei $^{178}$W, $^{188}$Pt, $^{200}$Pb, $^{213}$Fr, $^{224}$Th, and $^{251}$Es. We have also obtained the contributions of the presaddle and postsaddle neutrons to the total prescission multiplicity. A detailed analysis of our results leads us to conclude that the chaos-weighted wall formula friction can adequately describe the fission dynamics in the presaddle region. This friction, however, turns out to be too weak to describe the postsaddle dynamics properly. This points to the need for a suitable explanation for the enhanced neutron emission in the postsaddle stage of nuclear fission.

Effective one-dimensional equation of motion for nuclear fission

An approach for describing the dynamics of nuclear fission in the framework of generalized quantum mechanics is discussed. The collective kinetic energy is assumed to be two dimensional, and the reduced mass is allowed to vary with the coordinates. The generalized calculus of variation is employed for minimizing the action after being properly quantized as required by Hamilton's principle, employing a curvilinear coordinate system. The corresponding Euler Lagrange equation is identified as the required generalized equation of motion. The proposed generalized two-dimensional equation of motion is separated into a vibrational eigenvalue equation and a set of coupled-channel one-dimensional equations which describe the translational motion, by exploiting the completeness of the vibrational eigenfunctions. Such a system of coupled equations can be decoupled by replacing the coupling matrix elements by a nonlocal interaction, which can be rendered local after employing the effective mass approximation. As a consequence this differential equation is provided with an effective mass, an effective potential barrier, and a differential boundary term which is responsible for restoring the self-adjointness of the kinetic energy differential operator.

Nuclear fission with diffusive dynamics.

We investigate the dynamics of nuclear fission, assuming purely diffusive motion up to the saddle point. The resulting Smoluchowski equation is solved for conditions appropriate to the $^{16}\mathrm{O}$${+}^{142}$Nd${\ensuremath{\rightarrow}}^{158}$Er reaction at 207 MeV. The solution is characterized by an equilibration time ${\mathrm{\ensuremath{\tau}}}_{0}$ for the system to reach steady state, and the fission decay rate in steady state, \ensuremath{\Lambda}. We find that the equilibration time ${\mathrm{\ensuremath{\tau}}}_{0}$ plays a very small role in determining the number of prescission neutrons. The diffusion coefficient extracted from the experimental data is larger than the theoretical in the work of Bush, Bertsch, and Brown by a factor of 5--11.

Two-dimensional Langevin approach to nuclear fission dynamics

A phenomenological model for large-scale nuclear collective motion in the exit channel of fusion-fission reactions on the basis of a two-dimensional Langevin approach is discussed. For the symmetric fission of the 215Fr compound system the influence of the nuclear temperature T and of the two-body viscosity coefficient μ 0 on the distribution of the total kinetic energy has been studied. Calculated time scales are compared with reaction times extracted from pre-fission multiplicities of light particles.

Dynamics of nuclear fission

The most interesting process of collective flow of nuclear matter is the fission process. The time-scale of this process at moderate to high excitation energies is reviewed. Various methods to measure the time associated with large scale collective flow of nuclei in the range of 10^(-21) to 10^(-17) s are compared. Pre- and post-scission multiplicities of light particles in particular neutrons are shown to be a sensitive probe of the dynamics of nuclear fission. A systematics of pre- and post-scission neutron multiplicities associated with nuclear fission is presented. The magnitude of nuclear dissipation deduced from pre- and post-scission neutron multiplicities is compared with theoretical estimates.