Potential-energy surfaces for asymmetric heavy-ion reactions

We calculate the macroscopic potential energy of deformation as a function of mass asymmetry and distance between mass centers for shape configurations of interest in heavy-ion reactions. For the system³⁰⁰120 we also study the effect of adding microscopic shell and pairing corrections to the macroscopic potential energy. The shape configurations are generated by bringing together two separated spheres of unequal size. After the spheres touch the shapes are constructed by filling in the neck while keeping constant the radii of the end spheres, the nuclear density and the total nuclear volume. 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 distance between mass centers and mass asymmetry in the form of contour maps. Some important features of the contour maps are the stationary points of the potential energy and how they change in character and location as functions of the nuclear system considered. For example, for light systems there is a maximum in the potential energy for symmetric shapes. As we move to heavier systems this peak in the potential-energy surface splits into two asymmetric peaks that are separated by a symmetric saddle point. This occurs when Z²/A ≈ 30 for the total system. As the systems become still heavier the peaks become more and more asymmetric. In heavy-ion reactions for which the asymmetry of the system is smaller than that corresponding to the peak, the smaller nucleus tends to suck up the larger one. For larger asymmetries the larger nucleus tends to suck up the smaller one. For heavy systems the binary fission saddle point is lower than the maximum in the one-dimensional interaction barrier. The penetrability calculated for the multidimensional potential-energy surface is therefore increased relative to that for the one-dimensional barrier. The microscopic shell and pairing corrections lower the potential energy for configurations in which the target and/or projectile are magic or nearly magic. This effect persists to somewhat inside the point of touching. These corrections also lower the energy near the ground state.     

Quantum description of quaternary nuclear fission

The quantum theory of binary and ternary fission is generalized to the case of recently observed quaternary nuclear fission. Formulas for the amplitudes of partial fission widths and angular and energy distributions of quaternary fission products are derived with allowance for strong channel coupling. The nonevaporation mechanism for formation of light particles is used to explain the experimentally observed decrease in the probability for emission of light particles (α, α), (α, t), and (t, t) as compared with the product of emission probabilities for the same particles in ternary fission. It is concluded that in quaternary fission, as in ternary fission, light particles escape from the neck of the fissioning nucleus much earlier than scission of the nucleus into heavy fragments occurs.     

Single-particle enhancement of heavy-element production

Fusion barriers are calculated in a macroscopic-microscopic model for several cold-fusion heavy-ion reactions leading to heavy and superheavy elements. The results obtained in such a picture are very different from those obtained in a purely macroscopic model. For reactions on ²⁰⁸Pb targets, shell effects in the entrance channel result in fusion-barrier energies at the touching point that are only a few MeV higher than the ground state for compound systems near Z = 110. The entrance-channel fragment-shell effects remain far inside the touching point, almost to configurations only slightly more elongated than the ground-state configuration, where the fusion barrier has risen to about 10 MeV above the ground-state energy. Calculated single-particle level diagrams show that few level crossings occur until the peak in the fusion barrier very close to the ground-state shape is reached, which indicates that dissipation is negligible until very late in the fusion process. Whereas the fission valley in a macroscopic picture is several tens of MeV lower in energy than is the fusion valley, we find in the macroscopic-microscopic picture that the fission valley is only about 5 MeV lower than the fusion valley for cold-fusion reactions leading to compound systems near Z = 110. These results show that no significant “extra-extrapush” energy is needed to bring the system inside the fission saddle point and that the typical reaction energies for maximum cross section in heavy-element synthesis correspond to only a few MeV above the maximum in the fusion barrier.     

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.     

Dynamic treatment of ternary fission

The new dynamic model of light charged particle (LCP) formation in ternary fission is presented. The model is based on the assumption that light particles are formed as a result of two random neck ruptures during the time interval about one single-particle period. The connection of the final stage of ternary fission and of the saddle point descent stage was obtained in the framework of the density moments method. The analysis of LCP formation has shown that LCP mass and charge distributions are strongly governed by statistical nucleon exchange in the LCP-light fragment double system. New semiclassical expressions for the calculations of LCP yields and relative ternary fission probability are given. The results of calculations are in satisfactory agreement with the experimental data.     

曲率能对热力学驱动力的影响

为了研究曲率能对核裂变热力学驱动力（TDF）的影响,首先利用包含曲率能的截断版小液滴模型计算了200Pb和224Th的位垒和熵垒,对比液滴模型的计算结果表明:曲率能未改变224Th的位垒鞍点,却将200Pb的位垒鞍点向后推移。能级密度参数的形变关系越强则两系统的熵垒鞍点越靠近基态。为了进一步探究曲率能如何通过位势和熵势影响TDF,以断前中子多重性（PNM）为探针,通过两种方案进行了模拟,结果表明:曲率能降低了两系统的位势驱动力,而增强了其熵势驱动力。结合PNM的计算表明,前一种效应要比后一种效应明显,因此曲率能总体减弱了200Pb和224Th的TDF,进而延缓了两系统的核裂变进程。In order to study the effect of curvature energy on the thermodynamic driving force (TDF) of nuclear fission, the potential and entropy barrier of 200Pb and 224Th systems are calculated by using the truncated droplet model including curvature energy, respectively. Compared with the liquid drop model, the results show that curvature energy does not affect the saddle point of 224Th, but pushes the saddle point of 200Pb backwards the ground state. The stronger the deformation dependence of the level density parameter is, the closer the saddle point of entropy barrier for these systems is to the ground state. In order to further investigate how curvature energy affects TDF through nuclear potential and entropy, respectively, the prescission neutron multiplicity (PNM) is selected as the probe, some simulations based on two schemes are carried out. The results show that curvature energy reduces the potential driving force of 200Pb and 224Th, and enhances the entropy potential driving force. Combined with the calculations and analyses of PNM, the former effect is more obvious than the latter, so curvature energy weakens TDF of two systems on whole, thus delaying the nuclear fission process of two systems.     

Sequential character of low-energy ternary and quaternary nuclear fission

An analysis of low-energy true ternary (quaternary) nuclear fission leads to the conclusion that these fission modes have a sequential two-step (three-step) character such that the emission of a third particle (third and fourth particles) and the separation of fission fragments occur at distinctly different instants, in contrast to the simultaneous emergence of all fission products in the case of onestep ternary (quaternary) fission. This conclusion relies on the following arguments. First, the emission of a third particle (third and fourth particles) from a fissile nucleus is due to a nonevaporative mechanism associated with a nonadiabatic character of the collective deformation motion of this nucleus at the stages preceding its scission. Second, the axial symmetry of the deformed fissile compound nucleus and the direction of its symmetry axis both remain unchanged at all stages of ternary (quaternary) fission. This circumstancemakes it possible to explain themechanism of the appearance of observed anisotropies and T — odd asymmeries in the angular distributions of products of ternary (quaternary) nuclear fission. Third, the T —odd asymmetry discovered experimentally in ternary nuclear fission induced by cold polarized neutrons obeys the T —invariance condition only in the case of a sequential two-step (three-step) character of true ternary (quaternary) nuclear fission. At the same time, this asymmetry is not a T —invariant quantity in the case of the simultaneous emission of products of true ternary (quaternary) nuclear fission from the fissile compound nucleus.     

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.     

Multiplicity distribution of prompt gamma rays in spontaneous ternary fission of252Cf

The first and the second moments of the multiplicity distribution of prompt gamma rays in spontaneous ternary fission of²⁵²Cf have been measured by the multiple coincidence technique. While both these moments were found to be nearly independent of the energy of the light charged particle accompanying the fission fragments, the width of the multiplicity distribution was larger than that in the case of normal binary fission by about 20%.     

On Ternary Fission Induced by Neutrons

Experiments on measuring the rotational effect of the ²³⁴U fissile nucleus at the scission point showed that the fissile nucleus rotates as a right screw with respect to the longitudinally polarized neutron beam direction in the ternary fission of the ²³³U target nucleus induced by polarized s-neutrons; in the binary fission of the same nuclei it rotates in the opposite direction. Moreover, it was found that ternary fission “prefers” the spin state of J = I +1/2. This phenomenon cannot be explained within the existing concepts of ternary fission as one of the two “final” states after neck rupture. The same “parent” ²³⁴U nucleus cannot rotate in opposite directions in the two different final states. It should be assumed that ternary fission is a special branch of descent from the saddle point to the point of neck rupture. It can also be assumed that this branch is formed at the saddle point in a configuration favorable for cluster formation. Why does it prefer the spin state of J = I + 1/2? This is an interesting question for further studies.

     

Stationary Fokker-Planck equation applied to fission dynamics

By use of both analytical and numerical techniques, we study the relaxation of time-dependent solutions of the Fokker-Planck equation for an inverted oscillator to Kramers' stationary solution. This is done by integrating over all time the time-dependent solutions for given initial conditions at the saddle point to obtain stationary solutions, whose densities and higher velocity moments are compared as functions of the coordinate with the corresponding quantities calculated from Kramers' stationary solution. For large values of the coordinate an a symptotic expansion of the density is obtained, but for general values of the coordinate and for higher velocity moments the time integration must be done numerically. With increasing dissipation the relaxation to Kramers' stationary solution occurs at successively smaller values of the coordinate. By use of Kramers' stationary solution, we derive analytical expressions as functions of nuclear temperature and dissipation strength for several quantities of interest in fission dynamics, including the mean time from the saddle point to scission, the mean fission-fragment kinetic energy at the scission point and the contribution to the variance in the fission-fragment kinetic energy resulting from fluctuations in the fission degree of freedom. We apply these results to some examples that have been studied experimentally, including the mean saddle-to-scission time for the heavy-ion-induced fission of the compound nucleus ¹⁶⁸Yb and the mean fission-fragment kinetic energy at scission and the contribution to its variance for the α-particle-induced fission of the compound nucleus ²¹³At.     

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。     

Comparison between the symmetric fission and fusion paths

The symmetric fission path leading to smooth extended shapes and the fusion or new fission path going through the two tangent sphere configuration are investigated within the liquid-drop model including the nuclear proximity energy. Analytical formulae are given for the various shape-dependent functions which govern the dynamics. The quadrupole moment, the perpendicular moment of inertia and the Coulomb energy are similar in the two paths. In contrast, the neck radius, the rupture point between the fragments, the parallel and effective moments of inertia, the surface energy and the critical angular momentum against fission are quite different in the two valleys. The introduction of the proximity energy strongly lowers the deformation energy in the fusion valley and for the light, medium and very heavy nuclei the barrier heights are nearly equal in the two paths. This flattening of the potential surface by the proximity forces allows to better understand the sudden transition between the one- and two-body configurations.     

Study of Fission Barrier Heights of Uranium Isotopes by the Macroscopic-Microscopic Method

Potential energy surfaces of uranium nuclei in the range of mass numbers 229 through 244 are investigated in the framework of the macroscopic-microscopic model and the heights of static fission barriers are obtained in terms of a double-humped structure. The macroscopic part of the nuclear energy is calculated according to Lublin-Strasbourg-drop (LSD) model. Shell and pairing corrections as the microscopic part are calculated with a folded-Yukawa single-particle potential. The calculation is carried out in a five-dimensional parameter space of the generalized Lawrence shapes. In order to extract saddle points on the potential energy surface, a new algorithm which can effectively find an optimal fission path leading from the ground state to the scission point is developed. The comparison of our results with available experimental data and others' theoretical results confirms the reliability of our calculations.     

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.     

Temperature effects in the liquid drop description of nuclear fission

Using recent results of the dependence of the surface tension and nuclear equilibrium density of temperature from an extended Fermi gas model the influence of temperature (compound nuclear excitation energy) on the liquid drop deformation energy and fission barrier height is studied. It is shown that increasing the temperature results in a lower fission barrier and a less constricted saddle point shape.     

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.     

Measurement of the fission moment of inertia of medium mass nuclides

Measurements have been made, by means of mica nuclear track detectors, of the distributions in tracklength and the angular distributions with respect to the beam direction for the fission fragments from the systems Ag, Te and Au plus 80-MeV alpha particles. Analysis of the tracklength distributions indicated that, in each case, full momentum transfer from the incident projectile characterized those interactions that led to fission. Angular-momentum dependent statistical model calculations for the decay of the respective compound nuclei then provided information on fission-evaporation competition in the de-excitation processes, and in particular the distribution in nuclear temperature and angular momentum at which the fission events took place. This information was then employed in the analysis of the measured fission fragment angular distributions, and to extractK ₀ ² values. From these, the moments of inertia of the fission saddle point shapes were calculated, and the results are in good agreement with theoretical estimates.     

多维Kramers公式的研究

本文研究了鞍点附近多维的位能曲面及鞍点所在位置;并用Werner Wheeler及无旋液体等两种方法计算了多维的质量系数与粘滞系数,然后用多维Kramers公式计算了裂变速率.发现裂变速率随着维数的增加而适当增大.不同的形变参量以及不同的计算质量和粘滞系数方法对计算核裂变速率影响不大.从结果看,采用三维计算裂变几率已足够准确.     

Potential energy surfaces and fission fragment mass yields of even-even superheavy nuclei

Potential energy surfaces and fission barriers of superheavy nuclei are analyzed in a macroscopic-microscopic model. The Lublin-Strasbourg Drop (LSD) model is used to obtain the macroscopic part of the energy, whereas the shell and pairing energy corrections are evaluated using the Yukawa-folded potential; a standard flooding technique is utilized to determine barrier heights. A Fourier shape parametrization containing only three deformation parameters is shown to effectively reproduce the nuclear shapes of nuclei approaching fission. In addition, a non-axial degree of freedom is taken into account to better describe the structure of nuclei around the ground state and in the saddle region. In addition to the symmetric fission valley, a new highly asymmetric fission mode is predicted in most superheavy nuclei. The fission fragment mass distributions of the considered nuclei are obtained by solving 3D Langevin equations.