Dependence of mass fission asymmetry on the compound excitation energy

The influence of the excitation energy of compound fissioning nucleus on the potential deformation energy as a function of the fission and asymmetry coordinates has been investigated. The Fermi distribution of nucleons at a certain temperature is assumed and taken into account in the Strutinsky shell-correction method. It has been found that the structure of the potential energy surface becomes smoother with increasing excitation energy and shell effects disappear at a compound-nuclear temperature of 2.0 to 2.5 MeV.     

Spontaneous Fission Barriers Based on a Generalized Liquid Drop Model

The barrier against the spontaneous fission has been determined within the Generalized Liquid Drop Model （GLDM） including the mass and charge asymmetry, and the proximity energy. The shell correction of the spherical parent nucleus is calculated by using the Strutinsky method, and the empirical shape-dependent shell correction is 6mp10yed during the deformation process. A quasi-molecular shape sequence has been defined to describe the whole process from one-body shape to two-body shape system, and a two-touching-ellipsoid is adopted when the superdeformed one-body system reaches the rupture point. On these bases the spontaneous fission barriers are systematically studied for nuclei from 2a~Th to 249 Cm for different possible exiting channels with the different mass and charge asymmetries. The double, and triple bumps are found in the fission potential energy in this region, which roughly agree with the experimental results. It is found that at around Sn-like fragment the outer fission barriers are lower, while the partner of the Sn-like fragment is in the range near l~SRu where the ground-state mass is lowered by allowing axially symmetric shapes. The preferable fission channels are distinctly pronounced, which should be corresponding to the fragment mass distributions.     

Fusion Barrier of Super-heavy Elements in a Generalized Liquid Drop Model

The macroscopic deformed potential energies for super-heavy elements Z = 110,112,114,116,118 arc determined within a generalized liquid drop model (GLDM). A quasi-molecular mechanism is introduced to describe the deformation of a nucleus in the GLDM and the shell model simultaneously. The macroscopic energy of a twocenter nuclear system in the GLDM includes the volume-, surface-, and Coulomb-energies, the proximity effect at each mass asymmetry, and accurate nuclear radius. The shell correction is calculated by the Strutinsky method and the microscopic single particle energies are derived from a shell model in an axially deformed Woods-Saxon potential with the quasi-molecular shape. The total potential energy of a nucleus can be calculated by the macro-microscopic method as the summation of the liquid-drop energy and the Strutinsky shell correction. The theory is applied to predict the fusion barriers of the cold reactions ^64Ni ^208 spb → ^272 110*, ^70Zn ^208pb → ^278 112*, ^76Ge ^208seb → ^284 114*,^82Se ^208pb → ^29 116*, ^86Kr ^208pb → ^294 118*. It is found that the neck in the quasi-molecular shape is responsible for the deep valley of the fusion barrier. In the cold fusion path, double-hump fusion barriers could be predicted by the shell corrections and complete fusion events may occur.     

Energetics of the fission process

The mass asymmetry of fragments from nuclear fission of heavy nuclei is reviewed. While mass asymmetry is a common and well-known phenomenon for low-energy fission of the lighter actinides, more recent experiments have demonstrated that, for the heaviest actinides, the mass distribution switches to a symmetric one. On the other hand, it has been discovered that, though for fissioning nuclei with mass numbersA225 the mass distribution is basically symmetric, an asymmetric component is clearly to be identified for nuclei down to the Pb-region. In the absence of a generally accepted dynamical theory of fission, the above experimental findings are discussed in terms of static energy considerations. Triggered from the outset by the structure of the potential energy surface at the saddlepoint, the energy balance at the scission point between the available energy (Q-value) of the reaction and the Coulomb and deformation energy of the nascent fragments is shown to steer the characteristics of the fragment mass distributions.     

Strongly deformed nuclei near the magic proton shell 82?

The sudden change of the isotope phase shift in the Hg isotopes is explained by studying the deformation energy with the Strutinsky shell correction method. It is shown that going from the spherical isotope ²⁰⁴Hg to the isotope ¹⁸⁴Hg a small negative deformation (up to β₂ = −0.19) is built up. Between the mass numbers 186 and 182 the deformation is shifted through the γ instability valley to larger positive deformations (β₂ ≈ 0.3). The experimentally observed anomaly can be explained with the theoretical isotope shift constant, if this change of ground state deformations is taken into account.     

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.     

Fission Barrier for ^240Pu in the Quadrupole Constrained Relativistic Mean Field Approach

The fission barrier for ^240pu is investigated beyond the second saddle point in the potentiM energy surface by the constrained relativistic mean field method with the newly proposed parameter set PK1. The microscopic correction for the centre-of-mass motion is essentiM to provide the correct potential energy surface. The shell effects that stabilize the nuclei against the fission is also investigated by the Strutinsky method. The shapes for the ground state, fission isomer and saddle-points, etc, are studied in detail.     

Shell structure and orbit bifurcations in finite fermion systems

We first give an overview of the shell-correction method which was developed by V.M. Strutinsky as a practicable and efficient approximation to the general self-consistent theory of finite fermion systems suggested by A.B. Migdal and collaborators. Then we present in more detail a semiclassical theory of shell effects, also developed by Strutinsky following original ideas of M.C. Gutzwiller. We emphasize, in particular, the influence of orbit bifurcations on shell structure. We first give a short overview of semiclassical trace formulae, which connect the shell oscillations of a quantum system with a sum over periodic orbits of the corresponding classical system, in what is usually called the “periodic orbit theory”. We then present a case study in which the gross features of a typical double-humped nuclear fission barrier, including the effects of mass asymmetry, can be obtained in terms of the shortest periodic orbits of a cavity model with realistic deformations relevant for nuclear fission. Next we investigate shell structures in a spheroidal cavity model which is integrable and allows for far-going analytical computation. We show, in particular, how period-doubling bifurcations are closely connected to the existence of the so-called “superdeformed” energy minimum which corresponds to the fission isomer of actinide nuclei. Finally, we present a general class of radial power-law potentials which approximate well the shape of a Woods-Saxon potential in the bound region, give analytical trace formulae for it and discuss various limits (including the harmonic oscillator and the spherical box potentials).     

Probabilistic scission of a fissile nucleus into fragments

A probabilistic criterion is proposed for the scission of a fissile nucleus into fragments. The probability of the rupture of the neck between would-be fragments is estimated by considering scission as a fluctuation. The energy of the prescission configuration and the energy of the separated-fragment configuration are computed on the basis of a macroscopic model that takes into account a finite range of nuclear forces and the diffuseness of the nuclear surface. The effect of the probabilistic criterion of nuclear scission on fission-process observables, such as the moments of the mass-energy distribution of fission fragments, the mean multiplicity of prescission neutrons, and mean fission times, is demonstrated. It is shown that the Strutinsky criterion, according to which nuclear scission occurs at a finite neck radius of 0.3R₀, is a rather good approximation to the probabilistic scission criterion in Langevin dynamical calculations employing the one-body nuclear-viscosity mechanism modified in such a way that the wall-formula contribution is reduced, the reduction factor satisfying the condition k _s <05.     

Rotation of nuclei around a prolate symmetry axis at very high spin states

Classically one expects that nuclei rotate at very high spins (30≦I≦80) around an oblate symmetry axis. It is shown that strong shell correction energies yield for some nuclei at the end of the rare earth region and in the Pb-region yrast states for a rotation around a prolate symmetry axis. Like for the rotation around an oblate symmetry axis one expects also here yrast traps. The deformation energy surfaces for very high spin states are calculated by the Strutinsky method using a Saxon-Woods potential and by a microscopic method built on constraint Nilsson functions. Both methods agree qualitatively. Yrast traps are studied for these nuclei. It is shown that the MONA (Maximisation of theOverlap ofNuclear wave functions byAlignment) effect prefers at high spin rotation around the symmetry axis of a negative deformed shape at the beginning of the shell and of positive deformation at the end of the shell.     

Inclusion of temperature dependent shell corrections in Landau theory for hot rotating nuclei

Landau theory used for studying hot rotating nuclei usually uses zero temperature Strutinsky smoothed total energy for the temperature dependent shell corrections. This is replaced in this work by the temperature dependent Strutinsky smoothed free energy. Our results show that this replacement has only marginal effect for temperatures greater than 1 MeV but plays significant role at lower temperatures.     

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.     

Fine structures by deformation in the mass energy distribution of fission fragments

It is shown within a seission point model that for a fixed mass and charge fragmentation the potential energy of the scission configuration has several minima as a function of the deformation parameters of the fragments. The scission at these minima leads to a relatively enhanced yield of the fragments with a certain TKE and creates fine structures in the TKE-mass distribution which are different from those produced by the odd-even effect.     

Stability of Strutinsky Shell Correction Energy in Relativistic Mean Field Theory

The single-particle spectrum obtained from the relativistic mean field （RMF） theory is used to extract the shell correction energy with the Strutinsky method. Considering the delicate balance between the plateau condition in the Strutinsky smoothing procedure and the convergence for the total binding energy, the proper space sizes used in solving the RMF equations are investigated in detail by taking ^208 Pb as an example. With the proper space sizes, almost the same shell correction energies are obtained by solving the RMF equations either on basis space or in coordinate space.     

The pairing correlations and nuclear shapes at very high angular momenta

The usual Strutinsky shell corrections include the pairing correlations in the BCS approach. At high-spin states the cranked intrinsic wave functions are not symmetric or antisymmetric under time reversal symmetry for general triaxial shapes. On the basis of the Hartree-Fock-Bogoliubov (HFB) approach two generalizations of the Strutinsky procedure are given to describe pairing correlations also for high-spin states and triaxial shapes. The method is applied to the neutron-deficient rare earth nucleus ¹⁵⁰Gd. It is found that pairing has an important effect on the change of nuclear deformation with increasing angular momentum. The proton pairing persists at least up to I ≈ 40.     

Calculation of charge vibration in fission with Strutinsky shell correction

The Strutinsky shell correction method has been applied to the two spheroid model to study charge vibrations in fission. The investigation is carried out by calculating the potential energy surface with respect to three degrees of freedom: charge vibration from the uniform value and the deformations of the two fragments. The results suggest that the effect of shells at Z = 50 and N = 82 do not cause large deviations from the liquid drop model charge density around mass 132; their effect is much more pronounced in the fragment excitation energy. The results also suggest that the fragment excitation and kinetic energies for a given mass ratio are markedly charge density dependent. Some features inherent to this treatment with respect to characteristic periods of individual degrees of freedom have been discussed.     

Fission fragment mass energy correlation for 13 actinides ranging from Th to Cf

Fission fragment mass-energy correlations have been measured for a large range of fissioning systems. The “cold fragmentations” mass distributions show the strong influence of the potential energy surface around scission (shell and charge effects).     

Angular momentum and energy dependence of fission andn,p,4He emission from194Hg compound nucleus

Spin and temperature dependence of the fission and particle emission is studied for¹⁹⁴Hg. The compound nucleus is described using the Strutinsky shell correction approach extended for finite angular momenta and temperature. The shell corrections to the potential energy, free energy and the angular momentum are calculated using the Woods-Saxon average field. Results are compared with the experimental data and show a good qualitative agreement. It is found that the inclusion of the shell effects is necessary to understand the decay properties of¹⁹⁴Hg even for temperatures as high as 1.5–2.0 MeV.