On the propagation of a fissioning system across the barrier towards scission

We describe the motion of a fissioning nuclear system from the potential minimum across the saddle region to scission by means of a Fokker-Planck type equation. We pay particular attention to the problems associated with the motion of broad distributions in phase space. We introduce propagators to obtain global solutions. The results are compared with Kramers' stationary solution for all friction and with simplified dynamical calculations which start behind or on top of the barriers. We are able to find stationary solutions numerically for all finite friction values. We compare the corresponding decay rates with Kramers' formulae for large and small friction.     

Fokker-Planck方程的数值解

采用扩散模型研究核裂变,需要求解Fokker-Planck方程。本文提出一个数值计算方法-平均隐式差分方法。对具有粘滞性的核体系的有关裂变动力学量,如几率分布、裂变率、断点处的平均动能以及鞍点到断点的平均扩散时间等一系列物理量做了计算,并与适合大粘滞性的Kramers的解析解做了比较。通过与解析解的比较及对归一常数的检验,证明计算结果精确可靠。     

平均最后通过时间和热核裂变速率

提出利用平均最后通过鞍点时间计算热核裂变速率,结果表明平均最后通过鞍点时间的倒数比平均首次通过断点时间的倒数更接近朗之万数值模拟值. The mean last passage time is introduced to instead of the mean first passage time for studying the decay of an induced-fissioning system. The stationary fission rate determined by the inverse of mean last passage time across the saddle point is agreement with the resulting rate of Langevin simulation and better than that of mean first passage time arriving at the scission point.     

Systematic studies of fission fragment kinetic energy distributions by Langevin simulations

Fission fluctuation-dissipation dynamics of heavy nuclei has been studied using Langevin Monte Carlo simulations. The covariant form of the fission transport equation and the coefficients related to it are investigated. To learn about the influence of the dynamics from the ground state to the saddle point on the kinetic energy distributions we have studied various systems and compared the calculations both starting from the ground state and from the saddle point. Both the mean total kinetic energy of the fission fragments and its variances can fit with the experimental values in terms of a finite neck radius as scission condition.This work was supported by the National Natural Science Foundation of China.     

Thermal Fission Rate Calculated Numerically by Particles Multi-passing over Saddle Point

Langevin simulation of the particles multi-passing over the saddle point is proposed to calculate thermal fission rate. Due to finite friction and the corresponding thermal fluctuation, a backstreaming exists in the process of the particle descent from the saddle to the scission. This leads to that the diffusion behind the saddle point has influence upon the stationary flow across the saddle point. A dynamical correction factor, as a ratio of the flows of multi- and first-overpassing the saddle point, is evaluated analytically. The results show that the fission rate calculated by the particles multi-passing over the saddle point is lower than the one calculated by the particle firstly passing over the saddle point, and the former approaches the results at the scission point.     

Macroscopic potential-energy surfaces for symmetric fission and heavy-ion reactions

We calculate the macroscopic potential energy of deformation for symmetric configurations of interest in fission and heavy-ion reactions. The shape of the system is characterized in terms of two moments of the matter distribution. These moments correspond to the distance between the centers of mass of the two halves of the system and to the elongation of each half about its center of mass. The configurations studied include a continuous sequence of shapes from the sphere to two-, three-, and four-fragment scission lines. Beyond the scission lines and prior to the line of first contact in heavy-ion reactions we represent the system in terms of separated oblate and prolate spheroids. The macroscopic energy is calculated as the sum of a Coulomb energy and a nuclear macroscopic energy that takes into account the finite range of the nuclear force. For systems throughout the periodic table we display the calculated energy as a function of the two moments in the form of contour maps. Some important features of the contour maps are the binary, ternary, and quaternary saddle points, the fission and fusion (or two-fragment) valleys, and the three- and four-fragment valleys. The maps illustrate how the topography of the potential energy changes as a function of the nuclear system considered. For example, as we move from lighter to heavier nuclear systems the binary saddle point moves from outside the point of first contact in heavy-ion reactions to inside the contact point. Because of this, the formation of a heavy compound nucleus requires additional energy relative to the maximum in a one-dimensional interaction barrier. The maps also illustrate for moderately heavy systems the presence of separate valleys for binary fission and fusion. For still heavier systems the ternary and quaternary saddle points are no longer present. This means that the ternary and quaternary valleys are accessible by paths that decrease monotonically in energy beyond the binary saddle point. Finally, for nuclear systems heavier than about ³⁰⁰120, the binary saddle point itself disappears, which in the absence of single-particle effects precludes altogether the formation of a compound system.     

Investigation of the reaction 208Pb(18O, f): Fragment spins and phenomenological analysis of the angular anisotropy of fission fragments

The average multiplicity of gamma rays emitted by fragments originating from the fission of ²²⁶Th nuclei formed via a complete fusion of ¹⁸O and ²⁰⁸Pb nuclei at laboratory energies of ¹⁸O projectile ions in the range E _lab = 78–198.5 MeV is measured and analyzed. The total spins of fission fragments are found and used in an empirical analysis of the energy dependence of the anisotropy of these fragments under the assumption that their angular distributions are formed in the vicinity of the scission point. The average temperature of compound nuclei at the scission point and their average angular momenta in the entrance channel are found for this analysis. Also, the moments of inertia are calculated for this purpose for the chain of fissile thorium nuclei at the scission point. All of these parameters are determined at the scission point by means of three-dimensional dynamical calculations based on Langevin equations. A strong alignment of fragment spins is assumed in analyzing the anisotropy in question. In that case, the energy dependence of the anisotropy of fission fragments is faithfully reproduced at energies in excess of the Coulomb barrier (E _c.m. ? E _B ≥ 30 MeV). It is assumed that, as the excitation energy and the angular momentum of a fissile nucleus are increased, the region where the angular distributions of fragments are formed is gradually shifted from the region of nuclear deformations in the vicinity of the saddle point to the region of nuclear deformations in the vicinity of the scission point, the total angular momentum of the nucleus undergoing fission being split into the orbital component, which is responsible for the anisotropy of fragments, and the spin component. This conclusion can be qualitatively explained on the basis of linear-response theory.     

Cold deformed fission in232U(n,f) and239Pu(n,f)

Masses, charges and kinetic energies of light fission fragments from the reactions²³²U(n, f) and²³⁹Pu(n, f) induced by thermal neutrons have been measured on the Cosi fan tutte spectrometer of the Institut Laue-Langevin in Grenoble. Both at very high and very low kinetic energies marked fine structures in the mass yields and odd-even staggerings in the charge yields are observed. In the framework of a scission point model the results are shown to point to compact and deformed scission configurations, respectively, where at scission the fragments carry no intrinsic excitation energy. The two limiting processes may, therefore, be called cold compact fission (usually known as cold fission) and cold deformed fission. The latter process as a general phenomenon of low energy fission has come into focus only recently.     

Fission modes in charged-particle induced fission

The population of the three fission modes predicted by Brosa's multi-channel fission model for the uranium region was studied in different fissioning systems. They were produced bombarding²³²Th and²³⁸U targets by light charged particles with energies slightly above the Coulomb barrier. Though the maximum excitation energy of the compound nucleus amounted to about 22 MeV, the influences of various spherical and deformed nuclear shells on the mass and total kinetic energy distributions of fission fragments are still pronounced. The larger variances of the total kinetic energy distributions compared to those of thermal neutron induced fission were explained by temperature dependent fluctuations of the amount and velocity of alteration of the scission point elongation of the fissioning system. From the ratio of these variances the portion of the potential energy dissipated among intrinsic degrees of freedom before scission was deduced for the different fission channels. It was found that the excitation remaining after pre-scission neutron emission is mainly transferred into intrinsic heat and less into pre-scission kinetic energy.     

Analysis of fragment mass distribution in asymmetric area at fission of <Superscript>235</Superscript>U induced by thermal neutrons

The fragment mass yields in fission of ²³⁵U induced by thermal neutrons for A = 145–160 and E_K = 50–75 MeV were measured using a mass spectrometer. The fine structure is observed at A = 153, 154 and E_K = 50–60 MeV. The obtained results were described in the framework of a model based on the dinuclear system concept. The analyzed correlation between the total kinetic energy and mass distribution of fission fragments is connected with the shell structure of the formed fragments of fission. From this correlation and the time dependence of the calculated mass distribution of the binary reaction products, one can conclude that the descent time from a saddle point to a scission point for the more deformed fragments is longer than that for fragments of more compact shape.     

Energy partition in nuclear fission

A scission point model (two spheroid model TSM) including semi-empirical, temperature-dependent shell correction energies for deformed fragments at scission is presented. It has been used to describe the mass-asymmetry-dependent partition of the total energy release on both fragments from spontaneous and induced fission. Characteristic trends of experimental fragment energy and neutron multiplicity data as function of incidence energy in the Th — Cf region of fissioning nuclei are well reproduced. Based on model applications, information on the energy dissipated during the descent from second saddle of fission barrier to scission point have been deduced.     

Étude de la dissipation d'énergie dans la fission de 234u à l'aide de la réaction 233U(d,pf)

The energy balance in the fission of ²³⁴U has been investigated on the basis of experimental results from the ²³³U(d, pf) reaction. Taking into account the neutron evaporation we have deduced the total kinetic energy and excitation energy distributions of the primary fragments as functions of the excitation energy of the fissioning nucleus. The neutron evaporation temperatures have been adjusted so as to reproduce the average value and width of the measured kinetic energy distributions for each fragmentation. Excitation energy distributions of the fragments have been deduced. The data are discussed in the framework of the liquid-drop model with shell corrections. Evidence for energy dissipation in the fission of ²³⁴U, involving drastic changes in the scission configuration, is shown for some fragmentation modes.     

Potential energy surfaces and lifetimes for spontaneous fission of heavy and superheavy elements from a variable density dependent mass formula

The potential energy surface for spontaneous fission is calculated using realistic density distributions for finite nuclei. Particular emphasis is placed on the region of the potential between the saddle and scission point. The method involves computing the energy of the system using an energy density functional consistent with varible density distributions and nuclear masses and obtained from a statistical many body theory. The results show that there exists an external or scission barrier to the fission process. Lifetimes and mass distributions which are computed using these potential energy surfaces are found to be in adequate agreement with observations for ²³⁴U, ²³⁶U, ²⁴⁰Pu, ²⁴⁴Cm, ²⁴⁸Cf, and ²⁵²Cf. Our predicted upper limit for the spontaneous fission half-lives of elements 112 and 114 is one year but the calculation indicates that these could be considerably shorter than a year.     

Investigation of fragment yields from <Superscript>239</Superscript>U fission at anomalously high values of the total kinetic energy

The yields of fragments originating from ²³⁸U fission induced by 5-MeV neutrons are investigated. Accumulated statistics—2.5×10⁶ events of binary fission—make it possible to study fission-fragment yields at anomalously high values of the total kinetic energy. The spectra of the cold fragmentation of ²³⁹U are obtained. Events characterized by the total kinetic energy that is equal to the total reaction energy are found for some fragment masses. Methods of digital signal processing permit a highly reliable identification of these rare events. An interpretation of this phenomenon on the basis of the liquid-drop model of the fission process is proposed.     

CALCULATION OF THE NUCLEAR FISSION BY NUMERICAL SOLUTION OF F-P EQUATION

In this paper, the two-dimensional Fokker-Planck Equation is exactly solved by means of the numerical method. The velocity distribution at the saddle point, the second moments of the coordinate and velocity, and the time development of the nuclear fis-sion rate are studied. The maximum fission rate at a certain viscosity is exhibited by studying the dependence of the fission rate on the nuclear viscosity.     

A combinatorial analysis of pair breaking in fission

We develop a model for pair-breaking in order to explain the behavior of the even-odd effects displayed both by the fragment yields and the fragments' kinetic energies in low energy nuclear fission. Neutron and proton pair breaking are taken into acount. Two pair-breaking mechanisms are considered. In the first, pairs are broken at the saddle point and the individual members of the broken pairs are assumed to localise independently in the fragments. In the second, pairs are broken at scission and individual members of the broken pairs are assumed to end in different fission fragments. With this simple model all existing experimental data can be explained, including results of cold-fragmentation.     

Prompt γ-rays emitted in the thermal-neutron induced fission of 233U and 239Pu

The average number and average energy of γ-rays emitted within ≈ 5 nsec after fission have been determined as functions of fragment mass and as functions of total kinetic energy. They were obtained from a four-parameter experiment that recorded, event-by-event, correlated of γ-rays and of fission-fragment pairs and the time, relative to fission, at which a γ-ray was detected. For ²³³U(n_th, f) the average total number and energy emitted per fission were found to be 6.31 ± 0.3 and 6.69 ± 0.3 MeV, respectively, giving an average photon energy of 1.06 ± 0.07 MeV. The results for ²³⁹Pu(n_th, f) given in the same order, are 6.88 ± 0.35,6.73 ± 0.35 MeV, and 0.98 ± 0.07 MeV.     

Excitation during collective deformation: How simple it is

In the frame of the time dependent Hartree-Fock-Bogolyubov method a simple model for the excitation of a nucleus during a collective deformation is presented. The Hilbert space consists of all single and multiple pair excitations the nucleons of each pair being in time reversed states. The non-adiabatic transitions within this space are treated correctly. Treating the selfconsistency ofλ andΔ only in an average way reduces the multichannel calculation to a set of Landau-Zener problems. These give the excitation probability of each pair. Introducing average quantities yields an analytic solution for the excitation energy in the fission process, the collective kinetic energy at the scission point and the mean number of pair excitations for any deformation.     

Post-scission dynamics in fission

A dynamical model for fission from the classical turning point to scission and beyond is presented. We consider the fissioning nucleus as well as the fission fragments as incompressible irrotational deformable charged liquid drops. We focus on the post-scission time evolution of the neck, stretching of the fragments, kinetic energy and excitation energy.