Strutinsky 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.

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.

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.

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 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.

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.

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.

Rotational states and masses of heavy and superheavy nuclei

The macroscopic-microscopic model with the Lublin-Strasbourg drop, the Strutinsky shell-correction method, and the BCS approach for pairing correlations is used with the cranking model to describe nuclear masses and rotational bands in even-even Ra to Cn isotopes of actinide and transactinide nuclei. The single-particle levels and potential-energy surfaces are calculated with the Yukawa-folded single-particle potential using the “modified funny hills” (c, h) shape parametrization. A monopole pairing force is used in our calculations. At equilibrium deformation the pairing strength is adjusted for every nucleus so as to reproduce the experimentally known rotational E2+ state within the cranking model. The pairing strength obtained in this way is used to predict the masses and rotational states in superheavy No to Cn nuclei. It is also shown that the rotational states with L 10 in Ra to No nuclei evaluated using a simple rotational formula agree quite well with the data. We propose a simple mechanism which takes into account a dynamical coupling of rotation with the pairing field that then allows one to obtain an excellent agreement with the data up to the states with the largest angular momenta (L 30) measured in this mass region.

ROTATIONAL STATES IN HEAVIEST ISOTOPES

Masses and rotational energies of even-even Fm - Hs isotopes are obtained using the Yukawa-folded single-particle potential and the Lublin Strasbourg Drop . The Strutinsky shell-correction method and the BCS approximation were used to evaluate the shell and pairing corrections. The paring force strength is adjusted to the experimental rotational states of the considered nuclei. The ground-state masses and the lowest rotational states obtained by using the cranking moments of inertia at the equilibrium deformations agree well with the experimental data.

Curvature correction in Strutinsky's method

Mass calculations carried out by Strutinsky's shell correction method are based on the notion of smooth single-particle level density. The smoothing procedure is always performed using curvature correction. In the presence of curvature correction a smooth function remains unchanged if smoothing is applied. Two new curvature correction methods are introduced. We study the performances of the standard and new methods using the harmonic oscillator and realistic potentials.

FISSION POTENTIAL ENERGY SURFACES IN TEN-DIMENSIONAL DEFORMATION SPACE

Various fission processes are described in terms of high-dimensional potential energy surface in the frame of the Strutinsky shell correction method for actinide region. The complete deformation space is necessary to study the potential energy minima responsible for the cluster radioactivity, cold fission and cold multi-fragmentation valleys. The nuclear shape families for the different fission configurations are obtained without any specific change of the parameters. The coordinate system based on the Cassini ovaloids makes it possible to increase the number of independent deformation parameters without divergence. The higher orders of the deformation are shown to play an important role in the description of the potential energy surface structure.

Doubly magic properties in superheavy nuclei

A systematic study of global properties of superheavy nuclei in the framework of the Liquid Drop Model and the Strutinsky shell correction method is performed. The evolution equilibrium deformations, TRS graphs and α-decay energies are calculated using the TRS model. The analysis covers a wide range of even-even superheavy nuclei from Z = 102 to 122. Magic numbers and their observable influence occurring in this region have been investigated. Shell closures appear at proton number Z = 114 and at neutron number N = 184.

Accelerating the convergence of the total energy evaluation in density functional theory calculations

A special feature of the Strutinsky shell correction method (SCM) [D. Ullmo et al., Phys. Rev. B 63, 125339 (2001)] and the recently proposed orbital-corrected orbital-free density functional theory (OO-DFT) [B. Zhou and Y. A. Wang, J. Chem. Phys. 124, 081107 (2006)] is that the second-order corrections are incorporated in the total energy evaluation. In the SCM, the series expansion of the total electronic energy is essentially the Harris functional with its second-order correction. Unfortunately, a serious technical problem for the SCM is the lack of the exact Kohn-Sham (KS) density rho KS(r) required for the evaluation of the second-order correction. To overcome this obstacle, we design a scheme that utilizes the optimal density from a high-quality density mixing scheme to approximate rho KS(r). Recently, we proposed two total energy density functionals, i.e., the Zhou-Wang-lambda (ZW lambda) and the Wang-Zhou-alpha (WZ alpha) functionals, for use in the OO-DFT method. If the two interpolation parameters, lambda and alpha, are chosen to allow the second-order errors of the ZW lambda and the WZ alpha functionals to vanish, these two functionals reduce to the Hohenberg-Kohn-Sham functional with its second-order correction. Again, the optimal density from a high-quality density mixing scheme is used to approximate rho KS(r) in the evaluation of lambda and alpha. This approach is tested in iterative KS-DFT calculations on systems with different chemical environments and can also be generalized for use in other iterative first-principles quantum chemistry methods.