Deformation parameter changes in fission mass yields within the systematic statistical scission-point model

The fission fragment mass-yields are evaluated for pre-actinide and actinide isotopes using a systematic statistical scission point model. The total potential energy of the fissioning systems at the scission point is presented in approximate relations as functions of mass numbers,deformation parameters and the temperature of complementary fission fragments. The collective temperature, Tcoll, and the temperature of fission fragments, Ti, are separated and the effect of collective temperature on mass yields results is investigated. The fragment temperature has been calculated with the generalized superfluid model. The sum of deformation parameters of complementary fission fragments has been obtained by fitting the calculated results with the experimental data. To investigate the transitions between symmetric and asymmetric modes mass yields for pre-actinide and heavy actinides are calculated with this model. The transition from asymmetric to symmetric fission is well reproduced using this systematic statistical scission point model. The calculated results are in good agreement with the experimental data with Tcoll= 2 Me V at intermediate excitation energy and with T_(coll)= 1MeV for spontaneous fission.Despite the Langevin model, in the scission point model, a constraint on the deformation parameters of fission fragments has little effect on the results of the mass yield.     

A MODEL OF FISSION

Mass and kinetic energy distribution of fragments in thermal-neutron in-duced fission of ²³⁵U are calculated by a α decay-like model. Besides Coulomb andnuclear interactions, the deformation and intrinsic excitation energies of fragments arealso taken into account.     

THE SHAPE OF FISSION SCISSION POINT

A shape, rotating cassin's oval body with symmetry deformation parameter of 1.0, in nuclear fission scission point is suggested. By use of the shape, simple calculations on average total kinetic energy of fission fragments in both symmetry and asymmetry cases show that the results are in agreement with semiempirical formula due to V.E. Viola.     

Odd-even effect in fragment angular momentum in low-energy fission of actinides

Quantitative explanation for the odd-even effect on fragment angular momenta in the low-energy fission of actinides have been provided by taking into account the single particle spin of the odd proton at the fragment’s scission point deformation in the case of odd-Z fragments along with the contribution from the population of angular momentum bearing collective vibrations of the fissioning nucleus at scission point. The calculated fragment angular momenta have been found to be in very good agreement with the experimental data for fragments in the mass number region of 130–140. The odd-even effect observed in the fragment angular momenta in the low-energy fission of actinides has been explained quantitatively for the first time.      

Realistic Langevin Treatment of Fission-Fragment Energy Distribution

The average kinetic energy and its variance of fission fragments for the compound nucleus ²¹³At are calculated by four-dimensional Langevin equation. Two collective coordinates are taken into account: the distance p between the centers of the nascent fragments and the neck parameter h using the {c, h, a} parametrization. The model allows for a realistic coordinate dependence of all coeficien ts appearing in the Langevin equation as computed by the Werner-Wheeler method. The diffusion process for thc fissioning nuclear system from the ground state to the scission line on the realistic energy surface is investigated. The results are in good agreement with the experiments.     

Isomeric yield ratios of fission products in the system of 24 MeV proton-induced fission of238U

Isomeric yield ratios of 30 fission products in 24 MeV proton-induced fission of²³⁸U were measured by the use of the ion-guide isotope separator on-line. The obtained isomeric yield ratios were converted to the angular momenta of primary fission fragments based on the statistical model. The deduced angular momenta were examined from various aspects. It is found that in general the angular momentum continuously increases with the fragment mass number including the region of symmetric mass division. However, there are some exceptions. For Sn isotopes the deduced angular momenta are quite small due to the spherical shape of the nuclear shell configuration. It is also concluded from the consideration of the charge distribution that the angular momentum of fission product scatters considerably within the narrow range of mass division. The dependence of the angular momentum on the available energy of fragments at scission point indicates that the individual fragment possesses a characteristic deformation at scission and/or the deduced angular momentum is seriously affected by the particle excitation after scission.     

Recent studies in heavy ion induced fission reactions

Nuclear fission process involves large scale shape changes of the nucleus, while it evolves from a nearly spherical configuration to two separated fission fragments. The dynamics of these shape changes in the nuclear many body system is governed by a strong interplay of the collective and single particle degrees of freedom. With the availability of heavy ion accelerators, there has been an impetus to study the nuclear dynamics through the investigations of nucleus-nucleus collisions involving fusion and fission process. From the various investigations carried out in the past years, it is now well recognized that there is large scale damping of collective modes in heavy ion induced fission reactions, which in other words implies that nuclear motion is highly viscous. In recent years, there have been many experimental observations in heavy ion induced fission reactions at medium bombarding energies, which suggest possible occurrence of various non-equilibrium modes of fission such as quasi-fission, fast fission and pre-equilibrium fission, where some of the internal degrees of freedom of the nucleus is not fully equilibrated. We have carried out extensive investigations on the fission fragment angular distributions at near barrier bombarding energies using heavy fissile targets. The measured fragment anisotropies when compared with the standard saddle point model (SSPM) calculations show that for projectile-target systems having zero or low ground state spins, the angular anisotropy exhibits a peak-like behaviour at the sub barrier energies, which cannot be explained by the SSPM calculations. For projectiles or targets with large ground state spins, the anomalous peaking gets washed out due to smearing of the K-distribution by the intrinsic entrance channel spins. Recently studies have been carried out on the spin distributions of fission fragments through the gamma ray multiplicity measurements. The fission fragments acquire spin mainly from two sources: (i) due to rigid rotation of the nascent fragments at scission and (ii) due to statistical excitation of the spin bearing collective modes in the fissioning nucleus. One of the collective modes — the tilting mode depends on the K quantum number and is responsible for the emission angle dependence of fragment spin. In our studies, we have shown conclusively that the collective statistical spin modes get strongly suppressed for high K values corresponding to large rotational frequencies along the fission axis. These results bring out the importance of the dynamical effects in the heavy ion induced fusion-fission reactions. The present article will review the work carried out on the above aspects in heavy ion fission reactions as well as on the fission time scales, and some of the recent studies on the mass-energy correlations of fission fragments at near-barrier bombarding energies.     

Quantum and thermodynamic properties of spontaneous and low-energy induced fission of nuclei

It is shown that A. Bohr’s concept of transition fission states can be matched with the properties of Coriolis interaction if an axisymmetric fissile nucleus near the scission point remains cold despite a nonadiabatic character of nuclear collective deformation motion. The quantum and thermodynamic properties of various stages of binary and ternary fission after the descent of a fissile nucleus fromt he outer saddle point are studied within quantum-mechanical fission theory. It is shown that two-particle nucleon-nucleon correlations—in particular, superfluid correlations— play an important role in the formation of fission products and in the classification of fission transitions. The distributions of thermalized primary fission fragments with respect to spins and their projections onto the symmetry axis of the fissile nucleus and fission fragments are constructed, these distributions determining the properties of prompt neutrons and gamma rays emitted by these fragments. A new nonevaporation mechanism of third-particle production in ternary fission is proposed. This mechanism involves transitions of third particles from the cluster states of the fissile-nucleus neck to high-energy states under effects of the shake-off type that are due to the nonadiabatic character of nuclear collective deformation motion.     

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。     

Semi-microscopical description of the scission configuration in the neutronless fission of 252Cf

The neutronless fission of ²⁵²Cf is studied in the frame of a molecular model in which the scission configuration is described by two aligned fragments interacting by means of Coulomb (+ nuclear) forces. The study is carried out for different distances between the fragments tips and excitation energies. For a given deformation, the fragment's total energy is computed via the constrained Hartree-Fock + BCS formalism. The total excitation energy present in the fragments is supposed to contribute only to the fragments deformation and the asymptotic value of the kinetic energy is equated to the inter-fragment potential at scission. These two constraints are yielding a few fission channels for a fixed tip distance and excitation energy. Discarding those fission channels corresponding to a disequilibrium in the sharing of the excitation energy between the two fragments, we establish the most likely scission configurations for a specified excitation energy. Received: 24 September 1999     

MEASUREMENT OF FISSION FRAGMENT ANGULAR DISTRIBUTION IN HEAVY-IONS INDUCED REACTION

The angular distributions of fission fragments for a series of compound nuclei formed in the bombardment of ¹⁸¹Ta,¹⁹⁷Au Pb and ²⁰⁹Bi by ¹²C ions and in the bombardment of natural lead by ¹⁴N ions were measu ed with mica track detectors and gold surface barrier detectors.It is shown that all experimental data for the fission fragment angular distributions can be fitted by the statistical scission model of fission fragment angular distribution.     

Angular Anisotropy of the Fission Fragments in the Dinuclear System Mo del

A theoretical evaluation of the collective excitation spectra of nucleus at large deformations is possible within the framework of the dinuclear system (DNS) model, which treats the wave function of the fissioning nucleus as a superposition of a mononucleus configuration and two-cluster configurations in a dynamical way, permitting exchange of nucleons between clusters. In this work the method of calculation of the potential energy and the collective spectrum of fissioning nucleus at scission point is presented. Combining the DNS model calculations and the statistical model of fission we calculate the angular distribution of fission fragments for the neutron–induced fission of 239Pu.     

Nonuniform character of the population of spin projections K for a fissile nucleus at the scission point and anisotropies in the angular distributions of fragments originating from the induced fission of nuclei

It is shown that the emergence of anisotropies in the angular distributions of fragments originating from the spontaneous and induced fission of oriented actinide nuclei is possible only if nonuniformities in the population of the projectionsM (K) of the fissile-nucleus spin onto the z axis of the laboratory frame (fissile-nucleus symmetry axis) appear simultaneously in the vicinity of the scission point but not in the vicinity of the outer saddle point of the deformation potential. The possibilities for creating the orientation of fissile nuclei for spontaneous and induced fission and the effect of these orientations on the anisotropies under analysis are considered. The role of Coriolis interaction as a unique source of the mixing of different-K fissile-nucleus states at all stages of the fission process is studied with allowance for the dynamical enhancement of this interaction for excited thermalized states of the nucleus involved that is characterized by a high energy density. It is shown that the absence of thermalization of excited states of the fissile nucleus that appear because of the effect of nonadiabaticity of its collective deformation motion in the vicinity of the scission point is a condition of conservation of the influence that transition fission states formed at the inner and outer fission barriers exerts on the distribution of the spin projections K for lowenergy spontaneous nuclear fission. It is confirmed that anisotropies observed in the angular distributions of fragments originating from the fission of nuclei that is induced by fast light particles (multiply charged ions) are due to the appearance of strongly excited equilibrium(nonequilibrium) states of the fissile nucleus in the vicinity of its scission point that have a Gibbs (non-Gibbs) distribution of projections K.     

EFFECTIVE MOMENT OF INERTIA OF MEDIUM MASS NUCLEI AT THE SADDLE POINT

The angular distributions of fission fragments are measured for systems of 120MeV ¹⁴N+¹¹⁸Sn with mica track detectors and gold-surfacebarrier silicon detectors. The measured angular distributions can be fitted by the standard theory of fission angular distributions on the basis of statistical model. The effective moments of inertia of fissioning nuclei at saddle point are derived from angular distribution anisotropies. The shapes of saddle point of fissioning nuclei in the region of Z²/A<30 are discussed.     

The determination of the Coulomb energy of a nuclear system at the scission point upon fission

The dependence of the Coulomb energy of deformed nuclei on the shape and the nuclear matter density distribution is considered. For an ellipsoidal shape of the nucleus and a homogeneous distribution of nuclear matter, the result was obtained in a simple analytical form convenient for further use. Separate attention was paid to the consideration of a Fermi-like distribution of nuclear matter using different shapes of the nucleus that reflect several different kinds of collective motions. After these considerations, the dependence of the Coulomb energy of the fission products at the scission point of binary fission reactions on configurations, shapes and nuclear matter distributions was investigated. Calculation of the dependence of the shape of the nuclei deformation parameters at the scission point from their total kinetic energies was made in detail. Finally, the influence of shell effects on the mass yield of fission products is discussed. The text was submitted by the authors in English.     

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.     

CALCULATION OF THE KINETIC-ENERGY DISTRIBUTIONS OF FISSION-FRAGMENTS FROM MONTE-CARLO SIMULATION TO THE FOUR-DIMENSIONAL LANGEVIN EQUATION

The four-dimensional Langevin equation for two collective coordinates (the distance bet ween the centers of mass of the dascent fragments and the neck parameter) and their conjugate momenta is used as a dynamical equation to describe the descent of Brownian particles from the saddle-point to the scission points Monte-carlo method is used to slove the Langevin equation.The variances of the kinetic-energy distributions of nuclear fission-fragments in the range 322/A<40 have been calculated.The results of calculation are in good agreement with the experimental data.     

Remarks on a quantum mechanical treatment of internal excitation in low energy nuclear fission

Internal excitations of the fissioning nucleus are usually described phenomenologically by friction terms. In the present paper an approach is discussed which is in principle based on a correct quantum mechanical treatment taking the projection form of the Schrödinger equation as a starting point. Considering nuclear fission as an almost adiabatic process an estimate for the friction energy is made. In this very crude estimate only 10–15% of the collective energy gain in going from the saddle to the scission point is transformed into internal excitation energy. This is in agreement with experimental data showing pronounced substructure effects which would be destroyed in the presence of a larger friction. As compared to other microscopic calculations, in the present work the total Hamiltonian is split in such a way that the only perturbation term being responsible for the deformation is essentially the Coulomb energy. By this assumption the calculation of transition probabilities to intrinsically excited states becomes rather insensitive to the exact excitation energy spectrum of the compound nucleus.     

A simplified two-dimensional diffusion model for calculating the fission-fragment kinetic-energy distribution

The variance \(\sigma _{E_k }^2\) of the fission-fragment kinetic-energy distribution is calculated in describing the dynamics of fission of an excited nucleus by a Fokker-Planck equation for the quantum distribution function of two collective coordinates (the distance between the centers of mass of the nascent fragments and the neck parameter) and their conjugate momenta. In the calculations fluctuations in collective coordinates near scission and prescission kinetic energy have been taken into account simultaneously. The results of statistical model calculations for the case of high friction in fission mode and those of the calculations in the framework of a dynamical model for zero nuclear viscosity are the limiting cases of the proposed model. The dependence of \(\sigma _{E_k }^2\) upon the fissility parameter is studied.     

Investigation of fission fragments average spin based on four dimensional Langevin dynamical model

We applied the four dimensional Langevin dynamical model to investigate the average spin of fission fragments. Elongation, neck thickness, asymmetry parameter, and the orientation degree of freedom(K coordinate)are the four dimensions of the dynamical model. We assume that the collective modes depend on the emission angle of the fragments, then different parameters related to the average spin of fission fragments are calculated dynamically.The angle dependence of average spin of fission fragments is investigated by calculating the spin at angles 90?and165?. Also, the obtained results based on the transition state model at scission point are presented. One can obtain better agreement between the results of the dynamical model and experimental data in comparison with the results of the transition state model.