Shell Correction Method Research Articles

Nuclear level density in the statistical semiclassical micro-macroscopic approach

Level density ρ is derived for a finite system with strongly interacting nucleons at a given energy E, neutron N, and proton Z particle numbers, projection of the angular momentum M, and other integrals of motion, within the semiclassical periodic-orbit theory (POT) beyond the standard Fermi-gas saddle-point method. For large particle numbers, one obtains an analytical expression for the level density which is extended to low excitation energies U in the statistical micro-macroscopic approach (MMA). The interparticle interaction averaged over particle numbers is taken into account in terms of the extended Thomas - Fermi component of the POT. The shell structure of spherical and deformed nuclei is taken into account in the level density by the Strutinsky shell correction method through the mean-field approach used near the Fermi energy surface. The MMA expressions for the level density ρ reaches the well-known macroscopic Fermi-gas asymptote for large excitation energies U and the finite combinatoric power-expansion limit for low energies U. We compare our MMA results for the averaged level density with the experimental data obtained from the known excitation energy spectra by using the sample method under statistical and plateau conditions. Fitting the MMA ρ to these experimental data on the averaged level density by using only one free physical parameter - inverse level density parameter K - for several nuclei and their long isotope chain at low excitation energies U one obtains the results for K. These values of K might be much larger than those deduced from neutron resonances. The shell, isotopic asymmetry, and pairing effects are significant for low excitation energies.

Fission barriers of superheavy nuclei for emitted fragment isotopes near proton magic numbers

Fission barriers for supposed doubly magic $^{298}\mathrm{Fl}$ are calculated using a specialized binary macroscopic-microscopic method. The microscopic effects are obtained from the developed binary Strutinsky shell correction method. The main input data are the proton and neutron energy levels calculated with the deformed two-center shell model. One is able in this way to provide the microscopic transition via the level schemes, from the parent to two overlapped and finally separated level schemes. The macroscopic part is acquired by the deformed charged liquid drop model with the double Yukawa-plus-exponential finite range potential. The method is applied to the reactions accompanied by isotopes of Sn and Pb within the fission channels, as being favored by strongly negative shell corrections.

Comparative Evaluation on Nuclear Shell Correction Calculation using Regulation Bayesian and Moving Average Methods

Since Strutinsky had proposed a shell correction for nuclear mass calculations in 1967, there has been much further research on this topic. In addition to the development carried out on the Strutinsky method, some studies try to apply the shell correction. This paper is included in the development of the shell correction method. There are two techniques used in this work the first is the average method, while the second is the Bayesian regulation method. The two techniques are very different from those Strutinsky originally developed. He used the hermit polynomial as an extension of delta Dirac. This paper will show a comparative study of the two alternative techniques for calculating shell correction.

On the stability of superheavy nuclei

Potential energy surfaces of even–even superheavy nuclei are evaluated within the macroscopic-microscopic approximation. A very rapidly converging analytical Fourier-type shape parametrization is used to describe nuclear shapes throughout the periodic table, including those of fissioning nuclei. The Lublin Strasbourg Drop and another effective liquid-drop type mass formula are used to determine the macroscopic part of nuclear energy. The Yukawa-folded single-particle potential, the Strutinsky shell-correction method, and the BCS approximation for including pairing correlations are used to obtain microscopic energy corrections. The evaluated nuclear binding energies, fission-barrier heights, and \(Q_\alpha \) energies show a relatively good agreement with the experimental data. A simple one-dimensional WKB model à la Świa̧tecki is used to estimate spontaneous fission lifetimes, while \(\alpha \) decay probabilities are obtained within a Gamow-type model.

A search for neutron magicity in the isotopic series of Z = 122, 128 superheavy nuclei

The superheavy nuclei have been examined systematically in the region 158 ⩽ N ⩽ 218, 162 ⩽ N ⩽ 212 for Z = 122 and 128, respectively. The explicit density-dependent meson-exchange (DD-ME) and point-coupling (DD-PC) models within the framework of covariant density functional theory (CDFT) have been used to study the structural and decay properties of the isotopic series which includes the separable form of a finite range of pairing interaction. From the potential energy curves, the ground state properties of nuclei are predicted. Due to the importance of the shell effect in the superheavy region, the Strutinsky shell correction method has been employed for a better understanding of the extra stability of nuclei. The results from neutron pairing energy, two-neutron separation energy (S 2n ), single-particle energy levels, and total shell-correction energy strongly support N = 168, 174, and 178 as deformed neutron-magic numbers from both the force parameter, in both the isotopic series. N = 172 and 184 are predicted as spherical magic with DD-ME2 interaction in the Z = 122 isotopic series. Using three different semi-empirical approaches named UNIV2, SemFIS2, and ImSahu, the α-decay properties are studied and compared with available experimental data, FRDM2012 and the WS4 mass model. The stability of synthesized superheavy nuclei can be determined by comparing spontaneous fission half-lives with α-decay half-lives.

Polar-Covalent Bonding Beyond the Zintl Picture in Intermetallic Rare-Earth Germanides.

A comparative chemical bonding analysis for the germanides La2 MGe6 (M=Li, Mg, Al, Zn, Cu, Ag, Pd) and Y2 PdGe6 is presented, together with the crystal structure determination for M=Li, Mg, Cu, Ag. The studied compounds adopt the two closely related structure types oS72-Ce2 (Ga0.1 Ge0.9 )7 and mS36-La2 AlGe6 , containing zigzag chains and corrugated layers of Ge atoms bridged by M species, with La/Y atoms located in the biggest cavities. Chemical bonding was studied by means of the quantum chemical position-space techniques QTAIM (quantum theory of atoms in molecules), ELI-D (electron localizability indicator), and their basin intersections. The new penultimate shell correction (PSC0) method was introduced to adapt the ELI-D valence electron count to that expected from the periodic table of the elements. It plays a decisive role to balance the Ge-La polar-covalent interactions against the Ge-M ones. In spite of covalently bonded Ge partial structures formally obeying the Zintl electron count for M=Mg2+ , Zn2+ , all the compounds reveal noticeable deviations from the conceptual 8-N picture due to significant polar-covalent interactions of Ge with La and M ≠ Li, Mg atoms. For M=Li, Mg a formulation as a germanolanthanate M[La2 Ge6 ] is appropriate. Moreover, the relative Laplacian of ELI-D was discovered to reveal a chemically useful fine structure of the ELI-D distribution being related to polyatomic bonding features. With the aid of this new tool, a consistent picture of La/Y-M interactions for the title compounds was extracted.

The microscopic mechanism behind the fission-barrier asymmetry (II): The rare-earth region 50 < Z < 82 and 82 < N < 126

It is well known that most actinides fission into fragments of unequal size. This contradicts liquid-drop-model theory from which symmetric fission is expected. The first attempt to understand this difference suggested that division leading to one of the fragments being near doubly magic 132Sn is favored by gain in binding energy. After the Strutinsky shell-correction method was developed an alternative idea that gained popularity was that the fission saddle might be lower for mass-asymmetric shapes and that this asymmetry was preserved until scission. Recently it was observed [Phys. Rev. Lett. 105 (2010) 252502] that 180Hg preferentially fissions asymmetrically in contradiction to the fragment-magic-shell expectation which suggested symmetric division peaked around 90Zr, with its magic neutron number N=50, so it was presented as a “new type of asymmetric fission”. However, in a paper [Phys. Lett. B 34 (1971) 349] a “simple” microscopic mechanism behind the asymmetry of the actinide fission saddle points was proposed to be related to the coupling between levels of type [40ΛΩ] and [51ΛΩ]. The paper then generalizes this idea and made the remarkable prediction that analogous features could exist in other regions. In particular it was proposed that in the rare-earth region couplings between levels of type [30ΛΩ] and [41ΛΩ] would favor mass-asymmetric outer saddle shapes. In this picture the asymmetry of 180Hg is not a “new type of asymmetric fission” but of analogous origin as the asymmetry of actinide fission. This prediction has never been cited in the discussion of the recently observed fission asymmetries in the “new region of asymmetry”, in nuclear physics also referred to as the rare-earth region. We show by detailed analysis that the mechanism of the saddle asymmetry in the sub-Pb region is indeed the one predicted half a century ago.

Semiclassical approaches to nuclear dynamics

The extended Gutzwiller trajectory approach is presented for the semiclassical description of nuclear collective dynamics, in line with the main topics of the fruitful activity of V.G. Solovjov. Within the Fermi-liquid droplet model, the leptodermous effective surface approximation was applied to calculations of energies, sum rules and transition densities for the neutron-proton asymmetry of the isovector giant-dipole resonance and found to be in good agreement with the experimental data. By using the Strutinsky shell correction method, the semiclassical collective transport coefficients such as nuclear inertia, friction, stiffness, and moments of inertia can be derived beyond the quantum perturbation approximation of the response function theory and the cranking model.The averaged particle-number dependence of the low-lying collective vibrational states are described in good agreement with basic experimental data, mainly due to an enhancement of the collective inertia as compared to its irrotational flow value. Shell components of the moment of inertia are derived in terms of the periodic-orbit free-energy shell corrections. A good agreement between the semiclassical extended Thomas-Fermi moments of inertia with shell corrections and the quantum results is obtained for different nuclear deformations and particle numbers. Shell effects are shown to be exponentially dampted out with increasing temperature in all the transport coefficients.

Fragment Mass Distributions in Low-energy Fission of $^{236}$Pu

The potential energy landscapes of Hg–U nuclei were calculated within the macroscopic–microscopic method with a Fourier nuclear shape parametrisation and the Lublin–Strasbourg Drop for the macroscopic energy. Microscopic corrections based on the Yukawa folded single-particle potential were obtained with the Strutinsky shell-correction method and the BCS approximation. The energy landscapes of even–even isotopes of Hg, Po, Ra and U were analysed in a 4-dimensional deformation space and projected onto the quadrupole–octupole plane. Extrema, ridges and valleys were localized and the electric quadrupole moments in these minima were evaluated. A comparison with the experimental data for the ground state is shown for the Hg isotopes.

Potential Energy Surfaces of Mercury up to Uranium Isotopes in the 4D Fourier Shape Parametrisation

The potential energy landscapes of Hg–U nuclei were calculated within the macroscopic–microscopic method with a Fourier nuclear shape parametrisation and the Lublin–Strasbourg Drop for the macroscopic energy. Microscopic corrections based on the Yukawa folded single-particle potential were obtained with the Strutinsky shell-correction method and the BCS approximation. The energy landscapes of even–even isotopes of Hg, Po, Ra and U were analysed in a 4-dimensional deformation space and projected onto the quadrupole–octupole plane. Extrema, ridges and valleys were localized and the electric quadrupole moments in these minima were evaluated. A comparison with the experimental data for the ground state is shown for the Hg isotopes.

Remarks on the nuclear shell-correction method

It is shown that the nuclear single-particle energy spectra obtained in the selfconsistent way obey similar particle number dependence as the spectrum of a pure 3D harmonic oscillator potential. This effect was used to improve the quality of evaluation of the shell correction energy in weakly bound nuclei. The magnitude of the traditional Strutinsky shell correction energy obtained by the smearing of the single-particle energy spectrum of light nuclei is compared with that obtained by the smoothing of the single-particle energy sums in the particle number space.

Surface corrections to the moment of inertia and shell structure in finite Fermi systems

The moment of inertia for nuclear collective rotations is derived within a semiclassical approach based on the Inglis cranking and Strutinsky shell-correction methods, improved by surface corrections within the nonperturbative periodic-orbit theory. For adiabatic (statistical-equilibrium) rotations it was approximated by the generalized rigid-body moment of inertia accounting for the shell corrections of the particle density. An improved phase-space trace formula allows to express the shell components of the moment of inertia more accurately in terms of the free-energy shell correction. Evaluating their ratio within the extended Thomas-Fermi effective-surface approximation, one finds good agreement with the quantum calculations.

Chiral rotational {models

The chiral rotation in atomic nuclei has attracted significant attention in recent years. This paper introduces the particle rotor model and the titled axis cranking model, which are used to analyze the chiral rotation with multi-quasiparticle configuration. {For the particle rotor model, the basis space and the method to solve the model are discussed in detail: by considering the single-particle states of valence nucleons and the selection rule of angular momentum, one can construct the strong coupling basis which is symmetrized under the point group $D_2$; the dimension of the basis space will be very large when several valence nucleons are included, thus the Lancozos method is adopted to diagonalize the large scale hamiltonian matrix. For the titled axis cranking model, two versions based on the pairing plus quadrupole model and the shell correction method are introduced respectively: the cranking Hartree-Fock-Bogoliubov equations are given starting with the two-body Routhian; the 3-dimensional tilted axis cranking solutions are discussed in the intrinsic principal axes frame, and the vector of the angular velocity is parallel with the vector of the expectation values of the angular momentum components when the self-consistent solution is obtained.} By summarizing and developing further the two models, it will be helpful to clarify the debates in the understanding of the nuclear chiral rotation.

Semiclassical moment of inertia shell-structure within the phase-space approach

The moment of inertia for nuclear collective rotations is derived within a semiclassical approach based on the cranking model and the Strutinsky shell-correction method by using the non-perturbative periodic-orbit theory in the phase-space variables. This moment of inertia for adiabatic (statistical-equilibrium) rotations can be approximated by the generalized rigid-body moment of inertia accounting for the shell corrections of the particle density. A semiclassical phase-space trace formula allows us to express the shell components of the moment of inertia quite accurately in terms of the free-energy shell corrections for integrable and partially chaotic Fermi systems, which is in good agreement with the corresponding quantum calculations.

Fission properties of Po isotopes in different macroscopic–microscopic models

Fission-barrier heights of nuclei in the Po isotopic chain are investigated in several macroscopic–microscopic models. Using the Yukawa-folded single-particle potential, the Lublin–Strasbourg drop (LSD) model, the Strutinsky shell-correction method to yield the shell corrections and the BCS theory for the pairing contributions, fission-barrier heights are calculated and found in quite good agreement with the experimental data. This turns out, however, to be only the case when the underlying macroscopic, liquid-drop (LD) type, theory is well chosen. Together with the LSD approach, different LD parametrizations proposed by Moretto et al are tested. Four deformation parameters describing respectively elongation, neck-formation, reflectional-asymmetric, and non-axiality of the nuclear shape thus defining the so called modified Funny Hills shape parametrization are used in the calculation. The present study clearly demonstrates that nuclear fission-barrier heights constitute a challenging and selective tool to discern between such different macroscopic approaches.

The low-energy quadrupole mode of nuclei

The phenomenological classification of collective quadrupole excitations by means of the Bohr–Hamiltonian (BH) is reviewed with focus on signatures for triaxility. The variants of the microscopic BH derived by means of the Adiabatic Time-Dependent Mean Field theory from the Pairing-plus-quadrupole–quadrupole interaction, the Shell Correction Method, the Skyrme Energy Density Functional, the Relativistic Mean Field Theory and the Gogny interaction are discussed and applications to concrete nuclides reviewed. The Generator Coordinate Method for the five-dimensional quadrupole deformation space and first applications to triaxial nuclei are presented. The phenomenological classification in the framework of the Interacting Boson Model is discussed with a critical view on the boson number counting rule. The recent success in calculating the model parameters by mapping the mean field deformation energy surface on the bosonic one is discussed and the applications listed. A critical assessment of the models is given with focus on the limitations due to the adiabatic approximation. The Tidal Wave approach and the Triaxial Projected Shell Model are presented as practical approaches to calculate spectral properties outside the adiabatic region.

The potential energy surface of 240Pu around scission

The potential energy surface of is analyzed in the scission region within an elaborate macroscopic–microscopic approach, using the Lublin–Strasbourg-drop model, an improved Strutinsky shell-correction method and the BCS (Bardeen–Cooper–Schrieffer) treatment of pairing. The modified funny-hills nuclear shape parameterization, used in the present study, is further improved by introducing new collective coordinates κ and ψ for the elongation and neck-constriction. These variables are shown to be very well suited for the scission region. Within a limited number of dimensions, the model is able to explain fission modes in the actinide region. More specifically, the present work indicates that the peak of the mass distribution at in the low-energy fission of is mainly caused by strong neutron shell corrections.

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.

SHAPE EVOLUTION WITH SPIN IN 158Er AND 162Hf NUCLEI

The nuclear potential energies of neutron deficient even–even rare earth nuclei 158 Er and 162 Hf for the spin range 0–60 are computed within the framework of cranked Nilsson–Strutinsky shell correction method. The potential energy surface diagrams are analyzed in terms of quadrupole deformation and triaxiality parameter. The shape evolution of these isotopes with respect to spin is studied. The spin dependence of nuclear equilibrium potential energy is also verified.

Equilibrium shapes and deformations of some rare earth nuclei in cranked Nilsson-Strutinsky shell correction approach

The nuclear potential energies of neutron-deficient even-even rare earth nuclei ^{150-160}Dy, ^{150-160}Er, ^{150-160}Yb, ^{158,162,166-176}Hf, ^{160,164-178}W, and ^{162,166,170-180}Os are computed within the framework of the cranked Nilsson--Strutinsky shell correction method. The ground state potential energy surface diagrams of these nuclei are analysed in terms of quadrupole deformation and the triaxiality parameter. The nuclear shapes and deformations in the ground state are found to be functions of Z and N. Even though most of the isotopes in this region seem to be prolate in the ground state, oblate and triaxial shapes are also predicted for some isotopes. It is also found that the rare earth nuclei provide a platform for the study of phenomena such as nuclear shape changes and shape coexistence.