Macroscopic-microscopic Method Research Articles

Spontaneous fission of superheavy nuclei

The macroscopic–microscopic method is extended to calculate the deformation energy and penetrability for binary nuclear configurations typical for fission processes. The deformed two-centre shell model is used to obtain single-particle energy levels for the transition region of two partially overlapped daughter and emitted fragment nuclei. The macroscopic part is obtained using the Yukawa-plus-exponential potential. The microscopic shell and pairing corrections are obtained using the Strutinsky and BCS approaches and the cranking formulae yield the inertia tensor. Finally, the WKB method is used to calculate penetrabilities and spontaneous fission half-lives. Calculations are performed for the decay of 282,292120 nuclei.

Fission decay of 282Cn studied using cranking inertia

Superheavy nuclei produced until now are decaying mainly by α emission and spontaneous fission. Calculated α decay half-lives are in agreement with experimental data to within one order of magnitude. The discrepancy between theory and experiment can be as high as ten orders of magnitude for spontaneous fission. We analyze a way to improve the accuracy by using the action integral based on cranking inertia and a potential barrier computed by the macroscopic-microscopic method with a two-center shell model. Illustrations are given for 282Cn which has a measured fission half-life.

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.

Projected total energy surface description for axial shape asymmetry in 172W

The projected total energy surface (PTES) approach has been developed based on the triaxial projected shell model (TPSM) hybridized with the macroscopic-microscopic method. The total energy of an atomic nucleus is decomposed into macroscopic, microscopic and rotational terms. The macroscopic and microscopic components are described with the liquid drop model and Strutinsky method, respectively, and the rotational energy is given by the TPSM, the term beyond the mean field. To test theory, the PTES calculations have been carried out for the yrast states of the well deformed rare earth nucleus 172W, and the theoretical results are in good agreement with the experimental data. By using the equilibrium quardrupole deformations (ɛ 2 and γ) determined by the PTES, the calculation of the transition quardrupole moment (Q t ) in function of spin also reproduces the experimental data. A comparison between the PTES and TRS methods has been made for theoretical and application uses.

Predictions for α-decay half-lives of heavy and superheavy elements with a generalized liquid drop model

Theoretical estimates for the lifetimes of several isotopes of heavy and superheavy elements with Z = 100–126 are performed in the framework of the generalized liquid drop model including the proximity effects, the mass and charge asymmetry. The α-decay half-lives calculated using the experimental Q-values were found to be in reasonable agreement with experimental data. Half-lives with the Q-values from the finite-range droplet model (FRDM) and the macroscopic–microscopic method (MMM) are presented for comparison. It is found that the half-life calculations are quite sensitive to the choice of Q-values. The results show that the half-lives obtained by the theoretical Q-values from MMM are better in agreement with experimental ones than those with Q-values from FRDM. Finally, some predications on the α-decay half-lives using the two kinds of theoretical Q-values are provided for future experimental assignment and identification.

Low-energy fission within the Lublin–Strasbourg drop and Yukawa folded model

The evolution of fragment mass distribution in low-energy fission is studied as a function of the composition of the fissioning system over a wide range, from Hg to Fm isotopes. The relative importance of symmetric and asymmetric fission modes is investigated within a static picture based on elaborate, four-dimensional potential energy spaces using the modified Funny–Hills deformation coordinates. The potential energy is computed with the macroscopic-microscopic method. The Lublin–Strasbourg drop model is used to calculate the macroscopic part of the potential energy, while the Yukawa folded single-particle potential with an improved Strutinsky method and the Bardeen–Cooper–Schrieffer theory are employed to evaluate the shell and pairing contributions to the macroscopic energy. From the analysis of the potential energy surfaces, equilibrium deformations and saddle-point shapes are determined. A coexistence of symmetric and asymmetric fission valleys is observed. The relative depth of these valleys depends on the fissioning nucleus. The valleys' competition is used to explain the evolution of the fission-fragment mass distribution, from asymmetric fission in Hg to symmetric fission in Fm. Good qualitative agreement is found with the experiment.

Microscopic-macroscopic method for studying single-particle level density of superheavy nuclei

The intrinsic level densities of superheavy nuclei in the a-decay chains of 296,298,300120 nuclei are calculated using the single-particle spectra obtained with the modified two-center shell model. The level density parameters are extracted and compared with their phenomenological values used in the calculations of the survival of excited heavy nuclei. The dependences of the level density parameters on the mass and charge numbers as well as on the ground-state shell corrections are studied.

Nuclear inertia and the decay modes of superheavy nuclei

Superheavy nuclei produced up to now decay mainly by α emission and spontaneous fission. For atomic numbers larger than 121 cluster decay has a good chance to compete. While calculated α decay half-lives are in agreement with experimental data within one order of magnitude and cluster decay experiments are also very well accounted for, the discrepancy between theory and experiment can be as high as ten orders of magnitude for spontaneous fission. We analyze some ways of improving the accuracy: using a semiempirical formula for α decay and changing the parameters of analytical superasymmetric fission and of the universal curve for cluster decay. For spontaneous fission we act on nuclear dynamics based on potential barriers computed by the macroscopic–microscopic method and employing various nuclear inertia variation laws. Applications are illustrated for 284Cn and Z = 118–124 even–even parent nuclei.Communicated by Steffen Bass

Potential energy landscapes of Th isotopes within the Lublin Strasbourg drop + Yukawa-folded model

Advanced calculations of potential energy surfaces in a four-dimensional deformation space were performed for even–even Th isotopes using the macroscopic–microscopic method. The macroscopic part of the total energy was derived from the Lublin Strasbourg drop model. The Yukawa-folded potential was used to generate the single-particle spectra. The asymptotic Strutinsky shell-correction method and the Bardeen–Cooper–Schrieffer theory with approximate particle number projection were used to evaluate the microscopic part of the nuclear energy. Predictions for the ground-state masses and fission-barrier heights agree well with the experimental data. An accurate analysis of the computed multi-dimensional energy maps is in progress. Our preliminary results indicate that the present approach is capable of addressing the evolution of fission modality in the actinide region.

An improved nuclear mass formula: WS3

We introduce a global nuclear mass formula which is based on the macroscopic-microscopic method, the Skyrme energy-density functional and the isospin symmetry in nuclear physics. The rms deviation with respect to 2149 known nuclear masses falls to 336 keV, and the rms deviations from 1988 neutron separation energies and α-decay energies of 46 superheavy nuclei are significantly reduced to 286 and 248 keV, respectively. The predictive power of the mass formula for describing new measured masses in GSI and those in AME2011 is excellent. In addition, we introduce an efficient and powerful systematic method, radial basis function approach, for further improving the accuracy and predictive power of global nuclear mass models.

α decay half-lives for Z = 108, 114, 120, 126 isotopes and N = 162, 184 isotones

Within the generalized liquid drop model (GLDM) including the proximity effects between nuclei in the neck and the gap between the nascent fragments, the α decay half-lives of the 108, 114, 120, 126 isotopes and 162, 184 isotones are investigated in the quasi-molecular shape path. The Qα is obtained from the macroscopic–microscopic method (MMM) or from the experimental data. The calculated results are compared with experimental data, and both reveal that 270Hs is a double magic nucleus. The correct agreement allows us to make predictions for the α decay half-lives of other still-unknown superheavy nuclei(SHN) in the same framework. It is concluded that 298114 may be a double magic nucleus and N = 184 the neutron magic number for Z = 120 and Z = 126, however, further experimental evidence is needed.

Superdeformed Oblate Superheavy Nuclei in the Self-consistent Approach

The HFB self-consistent method has been applied to study the properties of several neutron deficient superheavy nuclei with Z=120-124, N=160-168. Their distinctive feature is the existence of minima of the total HFB energy for strongly deformed, oblate shapes. The self-consistent results agree quite remarkably with those currently obtained by using microscopic-macroscopic method.

Ground State Properties of Ds Isotopes Within the Relativistic Mean Field Theory

The ground state properties of Ds (Z = 110) isotopes (N = 151–195) are studied in the framework of the relativistic mean field (RMF) theory with the effective interaction NL-Z2. The pairing correlation is treated within the conventional BCS approximation. The calculated binding energies are consistent with the results from finite-range droplet model (FRDM) and Macroscopic-microscopic method (MMM). The quadrupole deformation, α-decay energy, α-decay half-live, charge radius, two-neutron separation energy and single-particle spectra are analyzed for Ds isotopes to find new characteristics of superheavy nuclei (SHN). Among the calculated results it is rather distinct that the isotopic shift appears evidently at neutron number N = 184.

Theoretical description of superheavy nuclei

AbstractA short review of the studies of superheavy nuclei (SHN), done recently in our theoretical group of Warsaw, is presented. Main attention is given to description of the properties of SHN. The description is performed by macroscopic–microscopic methods. Such properties as mass,α-decay energy andα-decay half-life are considered. Special attention is devoted to the analysis of the half-life. Although mainly treated in a phenomenological way, the role of the microscopic structure of a nucleus in this quantity is tested. It is found that this structure may significantly change the half-life of nuclei with the odd nucleon (or nucleons).

TEST OF TETRAHEDRAL SYMMETRY FOR HEAVY AND SUPERHEAVY NUCLEI

In the recent years an interesting suggestion has been made that nucleus, like many other quantum objects, can have tetrahedral shape at the ground state. We test this hypothesis within the macroscopic-microscopic method, by using a 12 dimensional manifold of shapes, including: axial, nonaxial and reflection-asymmetric ones. We systematically calculate potential energy surfaces for even-even heavy and superheavy nuclei in the range of proton numbers 82 ≤ Z ≤ 128 and neutron numbers 98 ≤ N ≤ 194. We find that in the whole region of the investigated superheavy nuclei, tetrahedral symmetry does not play a significant role at the ground state.

Th and U fission barriers within the Woods-Saxon two center shell model

Abstract Fission barriers of actinides are calculated in the framework of the macroscopic-microscopic method. The single particle energies are obtained within a new version of the Woods-Saxon two-center shell model. A nuclear shape parametrization characterized by five degrees of freedom is used. The barriers are calculated along the minimal action trajectory in the configuration space and the inertia is evaluated within the cranking formalism. The reliability of the model is tested by comparing the theoretical results with values deduced from experimental data.

Improved microscopic-macroscopic approach incorporating the effects of continuum states

The Woods-Saxon-Strutinsky method (the microscopic-macroscopic method) combined with Kruppa's prescription for positive energy levels, which is necessary to treat neutron rich nuclei, is studied to clarify the reason for its success and to propose improvements for its shortcomings. The reason why the plateau condition is met for the Nilsson model but not for the Woods-Saxon model is understood in a new interpretation of the Strutinsky smoothing procedure as a low-pass filter. Essential features of Kruppa's level density is extracted in terms of the Thomas-Fermi approximation modified to describe spectra obtained from diagonalization in truncated oscillator bases. A method is proposed which weakens the dependence on the smoothing width by applying the Strutinsky smoothing only to the deviations from a reference level density. The BCS equations are modified for the Kruppa's spectrum, which is necessary to treat the pairing correlation properly in the presence of continuum. The potential depth is adjusted for the consistency between the microscopic and macroscopic Fermi energies. It is shown, with these improvements, that the microscopic-macroscopic method is now capable to reliably calculate binding energies of nuclei far from stability.

Spheroidal cap configurations of atomic clusters on planar surfaces

Deposited atomic cluster configurations are investigated under the hypothesis of spheroidal cap shapes being a stable geometry. The macroscopic-microscopic method is employed to calculate the deformation energy. A new specialized single-particle model is developed in order to account for the quantum effects, and the liquid-drop approach is used to calculate the macroscopic part of the energy. The minima within the total deformation energy are interpreted as equilibrium states of atomic clusters on surfaces. Calculations have been performed for the metallic clusters of Na with atom numbers N = 20, 70 and 200.

Modification of nuclear mass formula by considering isospin effects

We propose a semiempirical nuclear mass formula based on the macroscopic-microscopic method in which the isospin and mass dependence of model parameters are investigated with the Skyrme energy density functional. The number of model parameters is considerably reduced compared with the finite range droplet model. The rms deviation with respect to 2149 measured nuclear masses is reduced by 21%, falling to 0.516 MeV. The new magic number $N=16$ in light neutron-rich nuclei and the shape coexistence phenomena for some nuclei have been examined with the model. The shell corrections of superheavy nuclei are also predicted.

ROTATIONAL BANDS IN HEAVY AND SUPERHEAVY NUCLEI WITHIN THE LUBLIN STRASBOURG DROP + YUKAWA FOLDED MODEL

The potential energies of heavy and superheavy even even nuclei were found within the macroscopic–microscopic method with LSD model, Yukawa–folded mean field, Strutinsky shell correction and BCS pairing theory in order to calculate the rotational state with cranking moments of inertia. The calculations were performed for a couple of nuclei: 250 No - 264 Hs on a deformation grid c = 0.8 up to 1.6 with step length 0.05 and h = -0.4 to 0.4 with step length 0.1. The equilibrium deformations were found and the masses compared to the experimental ones. Rotational energies of No isotopes for L/ħ = 2, 4, 6, 8, 10, 12 agree well with the data when a pairing strengths [Formula: see text] and a pairing window of [Formula: see text] levels is used. A prediction of the E2+ rotational states for all the investigated nuclei is given.