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.     

Study of fission dynamics with the three-dimensional Langevin equations

The dynamics of fission has been studied by solving one- and three-dimensional Langevin equations with dissipation generated through the chaos weighted wall and window friction formula. The average prescission neutron multiplicities, fission probabilities and the mean fission times have been calculated in a broad range of the excitation energy for compound nuclei ²¹⁰Po and ²²⁴Th formed in the fusion-fission reactions ⁴He + ²⁰⁶Pb , ¹⁶O + ²⁰⁸Pb and results compared with the experimental data. The analysis of the results shows that the average prescission neutron multiplicities, fission probabilities and the mean fission times calculated by one- and three-dimensional Langevin equations are different from each other, and also the results obtained based on three-dimensional Langevin equations are in better agreement with the experimental data.     

Studies of the nuclear inertia in fission and heavy-ion reactions

On the basis of the non-self-consistent cranking model we study some aspects of the nuclear inertia of interest in fission and heavy-ion reactions. First, we consider in the adiabatic limit the inertia for a doubly closed-shell nucleus in a deformed spheroidal harmonic-oscillator single-particle potential plus a small perturbation. When expressed in terms of a coordinate that describes the deformation of the nuclear matter distribution, the inertia for small oscillations about a spherical shape is exactly equal to the incompressible, irrotational value. For large distortions it deviates from the incompressible, irrotational value by up to about ±1 % away from level crossings. Second, in order to study the dependence of the inertia upon a level crossing, we consider in detail two levels of the above system. This is done both in the adiabatic limit and for large collective velocities. At level crossings the adiabatic inertia relative to the deformation of the matter distribution diverges as 1/|ΔV|, where |ΔV| is the magnitude of the perturbation. However, for large collective velocities the contribution to the inertia from a level crossing is less than 4|ΔV|r²_m, where r_m is the collective velocity of the matter distribution. Although we have not considered the effect of large velocities on the remaining levels of the many-body system or the effect of a statistical ensemble of states, some of our results suggest that for high excitation energies and moderately large collective velocities the nuclear inertia approaches approximately the irrotational value.     

Langevin analysis of fission excitation functions induced by protons

The stochastic Langevin approach to fission is applied to analyze fission excitation functions measured in p+²06Pb and p+²09Bi systems. A presaddle friction strength of（3–5）×10²¹s^-1is extracted by comparing theoretical predictions with experimental data. Furthermore, the small distortion of the formed compound nuclei with respect to the spherical shape under the condition of low angular momentum suggests that experimentally, populating an excited compound system via light-ion induced reactions favors a more accurate determination of presaddle friction with a fission cross section.     

Analysis of angular momentum dependence of fission fragment mass-energy distribution within Langevin dynamics

Dependence of fission fragment mass-energy distribution on the angular momentum is studied within Langevin dynamics. The calculations are performed in the framework of the generalized temperature-dependent finite-range liquid drop model. The analysis is done for five compound nuclear systems representing heavy fissioning nuclei, medium fissioning nuclei, and a light fissioning one with the angular momentum varied in a wide range from l = 0 to 70?. The coefficients d〈E _K 〉/dl ² and $d\sigma _{{\rm E}_{\rm K} }^2 /dl^2 $ are extracted. Previous analysis of the dσ _M ² /dl ² coefficient is generalized. Excitation energy dependence of the fission fragment mass-energy distribution is also found. The qualitative comparison of the extracted values with the experimental data reveals good agreement for all the cases. The calculated values of the coefficients dσ _M ² /dl ² and $d\sigma _{{\rm E}_{\rm K} }^2 /dl^2 $ are functions of the angular momentum, in contrast to the experimental estimations.     

Charge distributions in low-energy nuclear fission and their relevance to fission dynamics

Radiochemical and energy loss techniques have been used to measure fragment charge distributions in the systems ²²⁹Th(n_th, f), ²³⁵U(n_{3 MeV}, f) and ²⁵²Cf(s, f). Together with those previously obtained [^{233, 235}U(n_th, f) and ^{239, 241}Pu(n_th, f] the results show that these charge distributions depend very sensitively on both the fissioning species and the initial excitation energy. Very strong even-odd effects are present, for example in the charge distribution of ²²⁹Th(n_th, f) while they almost disappear in ²³⁵U(n_{3 MeV}, f). In the ²²⁹Th(n_th f) and ²⁵²Cf(s, f) cases, kinetic energy distributions of the fragments have been obtained. Fragments with even charges are more energetic, on average, than fragments with odd charges. The results are discussed in terms of a rather small probability for pair breaking during the transition from saddle-point to scission. The relationship between even-odd effects in charge distributions and fine structures in mass distributions is also investigated.     

A note on the examination of isospin effects in multi-dimensional Langevin fission dynamics

In [W. Ye, F. Wu, H.W. Yang, Phys. Lett. B 647 (2007) 118] prescission protons and α particles of high-isospin ²⁰⁶Pb were shown to be almost independent of the dissipation strength ks$k_s$. Subsequently, in [P.N. Nadtochy, et al., Phys. Lett. B 685 (2010) 258] prescission light charged particles (LCPs) were shown to have approximately the same sensitivity as neutrons to ks$k_s$ for ²⁰⁶Pb and ²⁰⁴Hg nuclei. In this Letter we point out that the reason for the apparent contradictory conclusions is that the authors in the latter did not compute the changes in the absolute yields of prescission LCPs multiplicities with increasing ks$k_s$ and compare them with typical experimental uncertainties. It is shown that the expected changes are very small in the case of neutron-rich ²⁰⁶Pb and ²⁰⁴Hg systems, which are within experimental error bars. This indicates that, from the viewpoint of experiment, LCPs emission of ²⁰⁶Pb and ²⁰⁴Hg is insensitive to dissipation.     

The treatment of heavy-ion collisions by Langevin equations

Fusion and deep-inelastic heavy-ion collisions are treated by Langevin equations within the framework of the surface friction model. Cross sections are calculated by Monte Carlo sampling of trajectories. An important result is that in contrast to what is usually assumed statistical fluctuations play a significant role in analysing heavy ion fusion data. It is demonstrated that in the analysis of deep-inelastic collisions the Langevin-Monte-Carlo method is more accurate than solving the corresponding Fokker-Planck equation by the usual moment expansion with the assumption of a gaussian distribution function.     

Mass-energy distribution of fragments from the fission of excited nuclei within three-dimensional Langevin dynamics

Two-dimensional mass-energy distributions of fission fragments are calculated for the first time on the basis of three-dimensional stochastic Langevin equations. In these calculations, the emission of light prescission particles is taken into account within the statistical model. The results demonstrate that calculations within three-dimensional Langevin dynamics make it possible to describe most compre-hensively the properties of the mass-energy distribution of fission fragments and the mean multiplicity of prescission neutrons.     

Ward Takahashi identities and fluctuation-dissipation theorem in a superspace formulation of the Langevin equation

A class of Langevin equations is formulated as a field theory in superspace. The Ward Takahashi identities associated with the hidden supersymmetry are derived which in turn are shown to lead to the fluctuation dissipation theorems.     

Activated aging dynamics and negative fluctuation-dissipation ratios

In glassy materials, aging proceeds at large times via thermal activation. We show that this can lead to negative dynamical response functions and novel and well-defined violations of the fluctuation-dissipation theorem, in particular, negative fluctuation-dissipation ratios. Our analysis is based on detailed theoretical and numerical results for the activated aging regime of simple kinetically constrained models. The results are relevant to a variety of physical situations, such as aging in glass formers, thermally activated domain growth, and granular compaction.     

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, 235}U, 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.     

Further evidence for nuclear rolling friction acting in heavy-ion induced reactions

Recent experimental results on heavy-ion fusion cross sections for asymmetric systems and on angular momentum transfer in damped collisions are shown to be well described, if rolling friction is explicitly taken into account. Its neglect in various other trajectory calculations leads to a significant underestimate of the fusion cross section as well as of the transferred angular momentum.     

Internal pair creation induced by nuclear coulomb excitation in heavy-ion collisions

Internal electron-positron pair creation caused by nuclear Coulomb excitation in heavy-ion collisions is compared with the spontaneous and induced positron production in overcritical quasimolecules with Z₁ + Z₂$?170. Using the rotation-vibration nuclear model in the calculation of the quadrupole Coulomb excitation and the conversion coefficient for electron-positron pair formation, the total cross section for the system ²³⁸₉₂U-²³⁸₉₂U was found to be ${\sigma }_C^{e^{+},e^{?}}=0.125mb$ at E_kin^c.m. = 797 MeV, which is four times smaller than the cross section ${\sigma }_{ind}^{e^{+},e^{?}}$ for the corresponding vacuum decay process if a K-shell vacancy production of L₀ = 10^? is assumed. Evaluation of the ratio $R=d{\sigma }_{ind}^{e^{+},e^{?}}({?}_{c.m.})/d{\sigma }_C^{e^{+},e^{?}}({?}_{c.m.})$ between the differential cross sections with respect to the ion angle ?_c.m leads to R = 60 at ?_c.m. = 160°. In contrast to the induced positron spectrum, the decay of excited nuclear levels in U is not followed by positrons with kinetic energy above 0.9 MeV. Therefore a unique determination of the decay of the vacuum in overcritical fields is experimentally possible provided that L₀ is not drastically reduced. In addition, similar calculations have been performed for pair creation resulting from nuclear Coulomb excitation for the systems U-Cf, Th-Th, Pb-Pb, Ni-Ni, Xe-Ba, Xe-Nd and Xe-Ce. The occupation of vacant bound states of superheavy electronic molecules by conversion electrons of γ-rays from nuclear transitions with E_γ < 1 MeV is also discussed.     

Combining a langevin description of heavy-ion induced fission including neutron evaporation with the statistical model

A model is developed which combines dynamical (Langevin-) calculations with the Kramers modified statistical model in order to describe heavy-ion induced fission including neutron evaporation. In the example of the¹⁹F+¹⁸¹Ta collision, the energy dependence of fission probabilities, neutron multiplicities and (H.I.,xn)-cross sections is calculated and a fair agreement with the data is achieved with a reduced friction parameter ofβ=3^*10²¹sec^?1. We pay particular attention to the angular momentum dependence of the fusion-fission process.