Shell Correction Method Research Articles

Collective effects on mass asymmetry in fission

The development of the mass asymmetry vibrations in the final stages of the fission process is studied with an approximate treatment of the coupling to relative motion. A parametrized friction is introduced and its effects are studied. Numerical results are presented for $^{236}\mathrm{U}$, together with estimates for the kinetic energy of the fragments.RADIOACTIVITY, FISSION $^{236}\mathrm{U}$; calculated mass distribution, kinetic energy distribution. Collective dynamics, shell correction method, cranking model.

On oblate-prolate transition in the ground state rotational band of light mercury isotopes

The transition from an oblate to a prolate shape in the ground state band of even mass Hg isotopes and in the ground states of the chain of odd mass Hg isotopes are studied. The shape is found by minimizing the deformation energy which is calculated by means of Strutinsky's shell correction method. The rotational energy corresponds to the axialsymmetric rotator. The moment of inertia is calculated with the help of the cranking model. Pairing and hexadecupole deformation are included. The results are in good agreement with the available experimental data.

A microscopic, but not self-consistent approach to nuclear binding and deformation energies

Using various versions of the Skyrme force and Negele's interaction, we calculate deformation energies of nuclei by evaluating the expectation value of the many-body Hamiltonian in wave functions taken to be antisymmetrized products of single-particle functions. These single-particle functions are eigenfunctions of a phenomenological potential, here taken to be a deformed Woods-Saxon well. The method can be thought of as an extension of the Strutinsky shell-correction method, to make the connection with the two-body interaction. The method employed here is checked by comparison with Hartree-Fock (HF) results; our method is, however, much faster than the HF method, and, therefore, suitable for a wide range of problems where one tests the sensitivity of results to changes in the two-body interaction. A fairly good agreement with the HF method is obtained for ground-state energies, radii and deformations, as well as for deformations of shape isomers. The main discrepancy is that our energies tend to increase slightly too rapidly with deformation, indicating that we may not have chosen the best phenomenological well. Two-dimensional energy surfaces, which agree quite well with those from the Strutinsky method, are found for 240Pu.

Deformations and moments of inertia of actinide nuclei in the ground and shape isomeric states

Using a Woods-Saxon potential, equilibrium deformations are obtained by the Strutinsky shell-correction method. Deformation parameters β 2 and β 4 of the ground state and the shape isomeric state are extracted for all actinide nuclei. It is shown that the connection of β 2 and β 4 with the multipole moments Q 2 and tQ 4 is not so trivial as sometimes assumed in the literature. The moments of inertia — taken at the same deformations — are evaluated within the cranking model. Their dependence on deformation and temperature (excitation energy) is discussed; the rigid body values are demonstrated to be reached both for large deformations and large temperatures. Where available, experimental data are compared; the agreement is generally very good.

Asymmetric fission in the shell-correction method with the asymmetric two-center shell model

The asymmetry of the fragment mass distribution of fissioning nuclei is described by suitable parametrization of an asymmetric two-center shell model (ATCSM). The deformation energy surface is calculated by applying a modified shell-correction prescription due to Strutinsky. It is shown that the ATCSM is specially suited for the description of the scission asymmetry which turns out to be strongly dependent on the shell structure of the fragments. It is demonstrated that the mass asymmetry at scission is unaffected by whether or not the Strutinsky renormalization for the shell corrections is employed for the calculation of the deformation energy surface.

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.

How much of a bubble is there in 184Hg?

Recent suggestions of the existence of bubble-shape nuclei are examined for a few selected nuclei in terms of a Strutinsky shell-correction method type of calculation, based on the folded-Yukawa model. The inner surface is treated by a modified version of the liquid-drop model, allowing for the finite range of the nuclear diffuseness and nucleon-nucleon interaction. It appears safe to conclude that 184Hg is not bubble-like. The observed large 〈 r 2 〉 in this region of Hg nuclei is explained as being associated with a change in distortion. This change is shown to be largely an effect of the introduction of quadrupole pairing.

Fission Barriers for Nuclei Lighter than Radium

Potential energy surfaces have been calculated for the mass range $180\ensuremath{\le}A\ensuremath{\le}212$ by using the shell-correction method on the basis of a symmetric two-center shell model. All calculations have been performed both with a constant and with a surface-dependent pairing strength. The results show that the fission barriers calculated with the latter prescription are incompatible with experimental data. The calculations for a constant pairing strength, however, in which the shells at the saddle are calculated on the same basis as those at the ground state, agree reasonably well with experimental data of the Berkeley group. The shell corrections for these nuclei are analyzed separately in detail and are found to exhibit remarkable regular structures which can be explained by an interplay between the shells of the fragments and those of the fissioning nucleus. Finally, the existence of shape-isomeric and the treatment of the zero-point energy in these calculations are discussed.

Shell structure for deformed nuclear shapes

Eigenvalues for the harmonic oscillator without l· s or l 2 terms suggest a deformed shell structure for nuclei with axes ratios 2 : 1 and deformation ϵ = 0.6 with corresponding nucleon “magic numbers” 2, 4, 10, 16, 28, 40, 60, 80 110 and 140, subject to small modifications due to spin-orbit and other correction terms. Experimental evidence of reasonably stable highly deformed structures corresponding to nucleon numbers 16, 20, 28, 40 and 60 (64) is presented. Attempts to calculate the corresponding potential energy surfaces using the Strutinsky shell correction method are described.

A foundation to the shell correction method

A foundation to the shell correction approach in studies of variations of nuclear masses and deformation energies is presented, based on the Hartree-Fock and the Fermi-liquid theories and on the local density approximation of the Brueckner-Bethe theory. In the first two cases the equations for the second order energy shell corrections and a relationship between the irregular component of the spatial density distribution and that of the shell model are obtained. The numerical results prove the rapid convergence of the shell correction expansion. Some technical problems involved in the calculation of first order corrections are considered.

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 β 2 = −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 (β 2 ≈ 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.

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.

Funny Hills: The Shell-Correction Approach to Nuclear Shell Effects and Its Applications to the Fission Process

This paper reviews various results related to the single-particle structure in spherical and deformed nuclei, discussed from the viewpoint of the so-called shell-correction method. This method stresses the importance of large-scale nonuniformities in the energy distribution of the individual particles especially near the Fermi energy. The way in which these nonuniformities affect in an essential way many nuclear properties, such as the shape stiffness, the spatial density distribution, the total mass of the nucleus, the mass and inertia of the nuclear shape variations, etc. is also discussed. Against this background, the behavior of the nuclear deformation energy is described, in particular for larger distortions relevant to the fission process. In this connection, some qualitative singularities of the phenomenological liquid-drop deformation energy at large shape distortions are pointed out, and their possible implications for fission are discussed. As the problems considered cover a wide range of nuclear properties, this paper is not a review in the narrow sense of the word. Comparison with other approaches as well as historic references are given mainly to clarify specific points, because a complete review would be a monumental undertaking.

Shape of the Island of Superheavy Nuclei

On the basis of the Strutinsky shell-correction method applied to realistic diffuse-surface potentials we have calculated the shape of the island of superheavy nuclei. With respect to spontaneous fission, the island is a mountain ridge extending from 114 protons to about 124 protons. The descent from the mountain down to the sea of instability is rather gentle for decreasing neutron numbers below 184, but is more rapid on the other three sides.

Asymmetry in nuclear fission

The deformation energies for fissioning nuclei in the actinide region are calculated using the Strutinsky shell-correction method. Instability of the second barriers against an asymmetric shape degree of freedom is found in agreement with Möller and Nilsson. The inclusion of these asymmetric deformations improves the agreement between calculated and experimental barrier heights. The asymmetries found have the correct magnitude in order to explain the experimental mass ratios of the fission fragments.

Theoretical predictions concerning superheavy elements

The problem of the existence of islands of relative nuclear stability outside of the « stability peninsula » of presently studied elements is reviewed on the basis of various theoretical investigations. Most emphasis is put on those that are based on smooth extrapolations of one-body nuclear potentials to higher A-values but also calculations based on a manybody treatment of various semirealistic or effective two-body interactions are discussed. Different distortion parametrisation schemes are compared. Various aspects of the calculations are reviewed as the application of the Strutinsky shell correction method, and the treatment of pairing in the extrapolations to new mass values and new distortions. The criticality of the estimates of the inertial parameters for the fission half-life evaluations is also considered. The existence of a Z = 114 and N = 184 island of elements of relative longevity appears to receive rather unanimous support from the theoretical investigations reviewed.

“Shells” in deformed nuclei

New calculation results based on the shell correction method are described. The results are presented in such a way as to illustrate more clearly many effects of nucleon shells in deformed and spherical nuclei. Some complementary theoretical arguments related to the shell correction method and details of the calculation are given.