Dynamical Behaviour of the Mass Asymmetry Mass Parameter in Nuclear Fission

The theory of nuclear fission is reconsidered. We study the behaviour of the mass parameter as a dynamical quantity of the mass asymmetry. The dependence of the mass asymmetry mass parameter is studied as a function of the five collective coordinates. These parameters are reconsidered by including the temperature to show the temperature dependence of the mass parameter. The cranking model is used in developing all the mathematical and theoretical expressions. Numerical calculations of the obtained analytical expressions are carried out for the two fissioning nuclei ²³⁶U and ²³⁸U. The mass asymmetry mass parameters are calculated including the temperature as a function of the different five collective coordinates. The present study shows that the values of this mass asymmetry mass parameters are stable against the change of the temperature for temperature values greater than 1 MeV for all the different five collective coordinates.     

Mass Distributions and Charge Dispersion for the Nuclear Fission

The theory of nuclear fission is reconsidered by introducing the charge asymmetry in the asymmetric two center shell model as a dynamical collective coordinate. The quantum mechanical fluctuations, which are accompanied with the collective motion as a function of the mass asymmetry, are responsible for the mass distributions in nuclear fission. Numerical calculations are carried out for the mass distributions and charge dispersions for the nuclear fission of the ²³⁶U and ²³⁸U nuclei. The present obtained theoretical calculations are in good agreement with the experimental measurements.     

Cold events in thermal-neutron-induced fission of heavy nuclei

An interpretation of the cold fission events in thermal-neutron-induced fission of heavy nuclei is given. The descent from the saddle point is considered as a dynamical process with reversible coupling between collective and intrinsic degrees of freedom. The distribution function for the collective variables is expressed as a product of two terms: the adiabatical and the dynamical factors. A simple model for symmetric fission to study the mass distribution is presented. As example, the calculations are performed for the nucleus ²⁶⁴Fm. Gross features of the cold fission are discussed as well as the dependence of the theoretical mass distribution on the parameters of the model. Received: 29 April 1998 / Revised version: 11 September 1998     

Effect of the orientation degree of freedom on the rate and time of fission of highly excited nuclei

The effect of the dimensionality of the dynamical model used on the fission rate and mean time is studied within a multidimensional stochastic approach to fission dynamics. These features of fission of excited compound nuclei are calculated within four-dimensional Langevin dynamics, where the coordinate K, which is the projection of the total angular momentum onto the symmetry axis of the nucleus being considered, is taken into account in addition to three collective shape coordinates introduced on the basis of the {c, h, a} parametrization. The evolution of this orientation degree of freedom (K mode) is described in terms of the Langevin equation in the overdamped regime. The effect of the orientation degree of freedom on the rate and mean time of fission of compound nuclei is studied. The introduction of the orientation degree of freedom is shown to lead to a substantial decrease in the fission rate and, accordingly, to an increase in the mean fission time upon going over from the three- to the four-dimensional model. The reactions induced by the interaction of ¹⁴N and ¹⁶O projectile ions with ¹⁹⁷Au, ²⁰⁸Pb, ²³²Th, and ²³⁸U nuclei at energies above the Coulomb fusion barrier are considered. The effect of the increase in the fission time because of the introduction of the K mode is so strong that it compensates almost completely for an opposite effect from introducing, in the one-dimensional model, the second and third collective coordinates that take into account, respectively, the evolution of the neck in the nuclear shape and the mass asymmetry. Ultimately, the difference between the results in the four- and one dimensional problems is not more than 5 to 25% for the reactions considered here.     

THE MICROSCOPIC CALCULATION OF THE STATISTICAL THEORY OF NUCLEAR FISSION

Based on the statistical fission theory, we have calculated the mass and kinetic energy distributions as well as other physical quantities of ²³⁵U fission induced by thermal neutron using microscopical method. That is, the quantum state densities of the fragments at scission point are calculated by means of BCS hamiltonian. The contribution of the collective deformation of fragments to the quantum state density has been taken into consideration. The potential energy of fragment is calculated by means of Strutinski procedure, and the shelling effect, pairing correlation as well as collective deformation have been taken into consideration in calculating the excitation energy at scission point. The scission point distance is treated as an adjustable parameter.Comparing with other statistical fission theories, our results give better agreement with existing experiments.     

Effect of entrance channel parameters on fission fragment angular momentum in medium energy fission

Independent isomeric yield ratios of¹³²I were radiochemically determined in alpha particle induced fission of²³⁸U in the energy range 25–44 MeV. Fission fragment angular momenta were deduced from the measured isomeric yield ratios using spin dependent statistical model analysis. It was seen that angular momentum of¹³²I increases with increase of excitation energy and angular momentum of the fissioning nucleus. Comparison of the present data on¹³²I in²³⁸U(α,f) with the literature data for the same product in²³⁸U(p, f) and²³⁸U(γ, f) at various excitation energies show that fragment angular momentum strongly depends on the input angular momentum in the range of excitation energy considered. Experimental fragment angular momentum at all excitation energies were seen to be in agreement with the theoretical values calculated based on thermal equilibration of the various collective rotational degrees after considering the occurence of multichance fission. Thus, strong effect of input angular momentum as well as the statistical equilibration among the various collective rotational degrees of freedom in medium energy fission is corroborated.     

Medium energy proton induced fission in Tb,La and Ag

Coincidence studies with silicon surface barrier detectors have been used to determine fragment kinetic energies, angular correlations and fission cross sections in the fission of Ag,¹³⁹La,¹⁵⁹Tb and U nuclei induced by 600 MeV protons. Symmetric mass distributions are deduced for Ag and Tb, whereas La shows an indication of a stable asymmetric mass distribution. We find no indication of the Businaro-Gallone point. Fission-spallation competition calculations are used to deduce values of macroscopic fission barrier heights and nuclear level density parameter values at deformations corresponding to the saddle point shapes. We find macroscopic fission barriers lower than those predicted by macroscopic theories. The total kinetic energies at symmetric mass divisions follow closely the Viola prediction.     

Some dynamical aspects of the fission process

The properties of the nuclear mass parameter and its influence on the fission process are discussed. In particular, correlations between dynamical and statical paths to fission are considered. The least action trajectories idea is used for finding the dynamical path to fission. Ritz method for the action integral minimalization is applied. An example of the nucleus ²⁵²Fm is considered.     

The dynamical description of the mass distribution of fission fragments

The variances of the mass distribution of fission fragments for nuclei in the fissility parameter range 30 < Z²/A < 40 have been calculated. The Fokker-Planck equation for the distribution function of collective variables is used as a dynamical equation to describe the descent of the fissioning nucleus from the saddle to scission. The calculations reproduce the experimentally observed increase of the mass variances with a growth of the parameter Z²/A. The increase of the calculated variances is due to a considerable decrease in the stiffness coefficients corresponding to the saddle points for heavy fissioning nuclei and to the non-equilibrium character of the descent.     

Investigation of polar emission in 252Cf and 235U + nth fission

The energy distributions and relative intensities of protons, deuterons, tritons and α-particles emitted along the fission axis during spontaneous fission of ²⁵²Cf were measured simultaneously with both fission fragment energies. The absolute intensity of particles, the mass distribution of fragments, the total kinetic energy and total excitation energy of both fragments were subsequently deduced from the experimental data. Statistical model calculations based on a hypothesis that the polar particles are evaporated from fission fragments have been performed for ²⁵²Cf and ²³⁶U fission. Although some experimental results agree remarkably well with the evaporation hypothesis, the considered model cannot describe many features of the polar emission phenomenon.     

Angular distribution of fission fragments in reactions of complete fusion of deformed nuclei

Formation of angular distributions of fission fragments for the ¹⁶O + ²³²Th and ¹²C + ^235,236U reactions has been analyzed within a dynamic approach. In this approach, the component of the total angular momentum along the fission axis K is considered as a fluctuating quantity and the corresponding relaxation time is assumed to be the main parameter controlling the evolution of this mode. Particular attention is paid to the analysis of the effect of initial distributions over K (formed during fusion) on the angular distribution of fission fragments of nuclei having fission barriers comparable with the nuclear temperatures.     

Fission mass asymmetry as a dynamic process described by a collective coordinate

A collective coordinate for describing reflection-asymmetric shapes of fissioning nuclei is introduced. The potential energy and mass parameter determining motion in this degree of freedom are calculated in a two-centre shell model. In quantized treatment the probability for a given mass asymmetry is determined and compared to the experimental mass yield curve of ²³⁶U.     

Langevin description of the charge distribution of fragments originating from the fission of excited nuclei

The charge distribution of fragments originatingfrom the fission of the ²³⁶U compound nucleus is calculated within a stochastic approach based on Langevin equations. The elongation coordinate, the neck-thickness coordinate, and the charge-asymmetry coordinate are chosen as collective variables. The friction parameter of the charge mode is calculated on the basis of two nuclear-viscosity mechanisms, that of one-body and that of two-body dissipation. It is shown that the Langevin approach is applicable to studying isobaric distributions. In addition, the charge distribution in question is studied as a function of the excitation energy of the compound nucleus and as a function of the coefficient of two-body viscosity.     

Correlation between fragment mass-distribution fine structure,charge distribution and nuclear structure for thermal-neutron-induced fission of 233U and 235U

Masses corresponding to observed fine-structure peaks in the fragment mass distributions for thermal-neutron-induced fission of ²³³U and ²³⁵U are shown to correspond to average measured masses for even-even nuclear charge splits. Evidence is presented that the yield enhancement for even-Z fragments is not restricted just to fission events with higher-than-average total kinetic energy. The anomalously high yield of fragments with mass 134 in ²³⁵U(n, f) as opposed to ²³³(n, f) is tentatively correlated with rapidly changing nuclear structure properties as a function of the mass of the complementary light (Z = 40) fragments.     

Multimode approximation for 238U photofission at intermediate energies

The yields of products originating from ²³⁸U photofission are measured at the bremsstrahlung endpoint energies of 50 and 3500 MeV. Charge and mass distributions of fission fragments are obtained. Symmetric and asymmetric channels in ²³⁸U photofission are singled out on the basis of the model of multimode fission. This decomposition makes it possible to estimate the contributions of various fission components and to calculate the fissilities of ²³⁸U in the photon-energy regions under study.     

True ternary fission of <Superscript>236</Superscript>U: A sequential mechanism?

A mechanism is proposed and calculations are performed to explain the true ternary fission of ²³⁶U nuclei experimentally observed via the missing mass method. The yield of the first excited fragment ⁷²Ni after the emission of neutrons corresponds to the most probable ⁶⁸Ni yield obtained in the experiment. The good agreement between the theoretical results and the experimental data means that the assumption of the sequential mechanism for true ternary fission is reasonable and that shell effects should be considered in the theoretical model.     

Large-scale collective motion in heavy-ion collisions

Heavy-ion fusion and deep inelastic reactions have been studied for symmetric systems in a classical dynamical model with deformation and necking as the collective shape coordinates. The calculated fusion excitation functions (for compound nucleus formation) are in good agreement with the experimental results from the evaporation residue measurements. It is observed that “nuclear molecules” are formed for not too heavy systems. The calculated reaction time for collisions of very heavy ions like ²³⁸U + ²³⁸U is found to be ~10⁻²¹ sec only and thus the width of the positron spectra observed in these reactions can not be explained in the light of quantum electrodynamics. The extra-extra push energies for fusion of heavy nuclei have also been studied.