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.     

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.     

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.     

Influence of nuclear dissipation on fission dynamics of the excited nucleus <Superscript>248</Superscript>Cf within a stochastic approach

A stochastic approach to fission dynamics based on two-dimensional Langevin equations was applied to calculate the anisotropy of the fission fragments angular distribution and average pre-scission neutron multiplicities for the compound nucleus ²⁴⁸Cf formed in the ¹⁶O+²³²Th reactions. Postsaddle nuclear dissipation strength of (12–14) × 10²¹ s^?1 was extracted for Cf nucleus by fitting the results of calculations with the experimental data. Furthermore, it was found that the results of calculations for the anisotropy of the fission fragments angular distribution and pre-scission neutron multiplicities are very sensitive to the magnitude of post-saddle nuclear dissipation.     

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.     

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.     

Dynamical approach to calculating angular distributions of fission fragments

A dynamical approach is proposed for calculating the angular distributions of fission fragments. The relaxation time for the degree of freedom associated with the projection of the total angular momentum of the nuclear system onto the symmetry axis and the coefficient of damping of the fission mode are the basic parameters of this approach. Experimental data on the anisotropy of the angular distributions of fission fragments and on the multiplicities of prescission neutrons are analyzed within the proposed model for ¹⁶O+²⁰⁸Pb (E _lab=110–148 MeV), ¹⁶O+²³²Th (120–160 MeV), ¹⁶O+²⁴⁸Cm (110–148 MeV), and ¹⁶O+²³⁸U (96–148 MeV). The relaxation time and the damping coefficient are estimated at τ_K=(5–6)×10^?21 s and β=4×10²¹ s^?1, respectively.     

Dynamic picture of the formation of spins of induced fission fragments

A way of calculating the average spins of induced fission fragments is developed, based on the dynamic model of their angular distributions. The range of relaxation times for the degree of freedom associated with the orientation of the axis of symmetry of a fissioning nucleus relative to its total angular momentum is determined by analyzing experimental data on the energy dependences of average spins and the anisotropy of the angular distributions of fission fragments for the ¹²C, ¹⁶O + ²³²Th reactions at E _сm = 55–150 MeV.     

Dynamical model of fission fragment angular distributions

A dynamical model of fission fragment angular distributions is developed. The experimental data on the angular anisotropy of fission fragments is analyzed for the ¹⁶O + ²⁰⁸Pb, ²³²Th, ²³⁸U, and ²⁴⁸Cm reactions at energies of the incident ¹⁶O ions ranging from 90 to 160 MeV. This analysis allows us to extract the relaxation time for the tilting mode. It was also demonstrated that the angular distributions are sensitive to the deformation dependence of the nuclear friction.

     

Angular distributions of fragments originating from the spontaneous fission of oriented nuclei and problem of the conservation of the spin projection onto the symmetry axis of a fissile nucleus

The concept of transition fission states, which was successfully used to describe the angular distributions of fragments for the spontaneous and low-energy induced fission of axisymmetric nuclei, proves to be correct if the spin projection onto the symmetry axis of a fissile nucleus is an integral of the motion for the external region from the descent of the fissile nucleus from the external fission barrier to the scission point. Upon heating a fissile nucleus in this region to temperatures of T ≈ 1 MeV (this is predicted by many theoretical models of the fission process), the Coriolis interaction uniformly mixes the possible projections of the fissile-nucleus spin for the case of low spin values, this leading to the loss of memory about transition fission states in the asymptotic region where the angular distributions of fragments are formed. Within quantum-mechanical fission theory, which takes into account deviations from A. Bohr’s formula, the angular distributions of fragments are calculated for spontaneously fissile nuclei aligned by an external magnetic field at ultralow temperatures, and it is shown that an analysis of experimental angular distributions of fragments would make it possible to solve the problem of spin-projection conservation for fissile nuclei in the external region.     

<Emphasis Type="Italic">P</Emphasis>-even correlations in binary and ternary nuclear fission induced by polarized neutrons

The evolution of a fissile nucleus from transition fission states specified at the saddle point of the deformation potential to fission states associated with prescission configurations of this nucleus and characterized by a pearlike shape of the nucleus is studied within the quantum-mechanical theory of fission processes that is based on the time-independent formalism. The coefficients of P-even asymmetries in the angular distributions of a light fragment and a third particle are calculated on the basis of the idea of the one-step mechanism of the production of a third particle and two fragments from the ternary fission of nuclei that is induced by polarized thermal neutrons. In order to confirm the developed concepts, it is proposed to repeat, at a higher level of statistical accuracy, experiments devoted to observing left-right asymmetries in the angular distributions of alpha particles from the ternary fission of nuclei.     

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.     

T-odd asymmetries for evaporation neutrons in nuclear fission

T-odd asymmetries in the angular distributions of evaporation neutrons emitted by thermalized fission fragments in the fission of axially symmetric deformed nuclei by cold polarized neutrons are investigated within the quantum theory of fission. The asymmetries in question are due to the anisotropy of angular distributions of evaporation neutrons in the center-of-mass systems of the fission fragments, and this anisotropy arises from the orientation of large-value fission fragment spins in the direction perpendicular to the direction K ₀ of the symmetry axis of the fissioning nucleus at the time of its scission, caused by zero wriggling vibrations of the fissioning nucleus. The angle of rotation of the vector k ₀ with respect to the asymptotic direction k ₀ of the fissioning nucleus symmetry axis is calculated with allowance for the interference of fission amplitudes of neutron resonances excited in a fissioning nucleus as it captures an incident neutron. It is shown that the T-odd asymmetry coefficient for evaporation neutrons is close in structure and value to the analogous coefficient for evaporation γ-rays.     

Langevin fission dynamics of hot rotating nuclei: Systematic application to Z 2/A=34–42 heavy nuclei

A stochastic approach that treats fission dynamics on the basis of three-dimensional Langevin equations is used to calculate the mass-energy distributions of fragments originating from the fission of compound nuclei whose fissility parameter lies in the range Z ²/A=34–42. In these calculations, use was made of the liquid-drop model allowing for finite-range nuclear forces and the diffuseness of the nuclear surface in calculating the potential energy and a modified one-body mechanism of viscosity in describing dissipation. The emission of light prescission particles is taken into account on the basis of the statistical model. The calculations performed within three-dimensional Langevin dynamics reproduce well all parameters of the experimental mass-energy distributions of fission fragments and all parameters of the prefission-neutron multiplicity for various parameters of the compound nucleus. The inclusion of the third collective coordinate in the Langevin equations leads to a considerable increase (by up to 40–50%) in the variances of mass-energy distributions in relation to what was previously obtained from two-dimensional Langevin calculations. For the parameters of the mass-energy distributions of fission fragments and the parameters of the prefission-neutron multiplicity to be reproduced simultaneously, the reduction coefficient K _s must be diminished at least by a factor of 2(0.2≤K _s ≤0.5) in relation to that in the case of total one-body viscosity (K _s =1).     

Unified Mechanism behind the Appearance of <Emphasis Type="Italic">T</Emphasis>-Odd TRI and ROT Asymmetries in Actinide Fission Induced by Cold Polarized Neutrons

Some shortcomings of the approaches that are used to describe T-odd ROT and TRI asymmetries in true ternary fission via reactions involving the emission of prescission alpha particles and which are based on employing the classical method of trajectory calculations are analyzed. These shortcomings are caused by the disregard of the interference between the fission widths of different sJ_s neutron resonance states formed in the first well of the deformation potential of fissile compound nuclei. It is shown that the method used in some studies to determine T-odd TRI-asymmetries for prescission alpha particles is at odds with basic concepts of the generalizedmodel of the nucleus and approaches to constructing collective (for example, bending) vibrations of a fissile compound nucleus. Quantum-mechanical fission theory is generalized via employing a unified mechanism of formation of T-odd TRI and ROT asymmetries for prescission alpha particles and evaporated photons (neutrons). The proposed mechanism takes correctly into account the effect of quantum rotation of a fissile compound nucleus on the angular distributions of fission fragments and alpha particles for true ternary fission, as well as on the angular distribution of prompt photons (neutrons) emitted by fragments originating from the delayed fission of the aforementioned nuclei.     

Description of the two-humped mass distribution of fission fragments of mercury isotopes on the basis of the multidimensional stochastic model

The dynamical model proposed earlier for describing fusion-fission reactions is applied to describing the two-humped mass distribution of fission fragments of mercury isotopes. In this model, the calculation of the time evolution of collective coordinates of the system is broken down into two stages. The first stage is that within which the projectile approaches the target nucleus, while the second is that of the evolution of the system formed after the touching of the projectile and target nuclei. The dynamical evolution of the system within both stages of the calculation is described on the basis of Langevin equations. The shell structure of colliding nuclei is taken into account at either stage of the calculation. Mass distributions are calculated for fragments originating from the fission of the mercury isotopes ^{190, 184}Hg formed in the fusion-fission reactions ⁴⁸Ca + ¹⁴²Nd → ¹⁹⁰Hg and ⁴⁰Ar + ¹⁴⁴Sm → ¹⁸⁴Hg. The process in which the isotope ¹⁸⁰Hg undergoes fission from the ground state is also calculated. The results obtained in this way are compared with the results of previous theoretical calculations and with available experimental data.     

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.     

<Emphasis Type="Italic">T</Emphasis>-odd asymmetries for evaporation γ-rays in nuclear fission

T-odd asymmetries in angular distributions of evaporation γ-rays emitted by thermalized fragments resulting from the fission of axially symmetric deformed nuclei induced by cold polarized neutrons are investigated within the quantum theory of fission. The asymmetries in question are due to the anisotropy of the angular distributions of evaporation γ-rays, caused by zero wriggling vibrations of the fissioning nucleus and associated with the orientation of large fission fragment spins in the direction perpendicular to the direction n ₀ of the symmetry axis of the fissioning nucleus at the time of its separation into fragments. The angle of rotation of the vector n ₀ with respect to the asymptotic direction of the light fission fragment emission is calculated with allowance for the interference of fission amplitudes of neutron resonances excited in the fissioning nucleus as it captures the incident neutron. It is shown that the angular and energy characteristics of the T-odd asymmetry calculated for evaporation γ-rays agree with the characteristics of the experimentally investigated T-odd asymmetry in angular distributions of all γ-rays emitted by a fissioning ²³⁶U nucleus.