International Journal of Modern Physics E

An overview of present experimental investigation of superheavy elements is given. Using cold fusion reactions which are based on lead and bismuth targets, relatively neutron deficient isotopes of the elements from 107 to 113 were synthesized at GSI in Darmstadt, Germany, and/or at RIKEN in Wako, Japan. In hot fusion reactions of ⁴⁸Ca projectiles with actinide targets more neutron rich isotopes of the elements from 112 to 116 and even 118 were produced at FLNR in Dubna, Russia. Recently, part of these data which represent the first identification of nuclei located on the predicted island of SHEs were confirmed in two independent experiments. The data are compared with theoretical descriptions.

The static fission-barrier height of heaviest nuclei is studied in a multidimensional deformation space. The main attention is given to the effect of the hexadecapole non-axial shapes on . The analysis is performed within a macroscopic-microscopic approach. A 10-dimensional deformation space is used. A large number of about 300 even-even heavy and superheavy nuclei with proton number 98 ≤ Z ≤ 126 and neutron number 134 ≤ N ≤ 192 are considered. It is found that the inclusion of the non-axial hexadecapole shapes lowers the barrier by up to about 1.5 MeV.

Predictions based on the "Fusion by Diffusion" model in a version adapted for calculating xn channels, with the ground-state masses, shell effects and barrier heights as given by Muntian, Patyk and Sobiczewski are presented. Sensitivity of the model to uncertainties in determination of the theoretical fission barriers is discussed. Predictions concerning formation of the element Z = 120 include comparison of fusion of the most asymmetric systems, ⁵⁰Ti + ²⁴⁹Cf and ⁵⁴Cr + ²⁴⁸Cm, with less asymmetric combinations ⁵⁸Fe + ²⁴⁴Pu and ⁶⁴Ni + ²³⁸U.

Energy landscapes of superheavy even-even nuclei with 116 ≤ Z ≤ 126 and 176 ≤ N ≤ 184 are studied within the macroscopic-microscopic method over a 12 dimensional manifold of shapes, including both axial- and mass-asymmetry. In addition to the spherical, prolate and oblate minima we find also oblate-octupole minima, with Y₃₃ symmetry along the oblateness axis, that compete to become ground states. All fission saddle points are triaxial and by 0.5-2.5 MeV lower than the axial ones. The Woods-Saxon model predicts the abrupt lowering of barriers near Z = 126, in a stark contrast to the mostly used self-consistent effective interactions.

We have calculated the liquid drop fission barriers of medium and heavy nuclei within the Lublin-Strasbourg-Drop model. Exploiting in addition the topographical theorem by Myers and Światecki we propose a simple but quite accurate approximation of the fission barrier heights. When comparing the r.m.s. deviation of approximated versus experimental values of fission barrier heights for known nuclei with Z > 70 a value 1.1 MeV is obtained which is comparable with the experimental uncertainties. The Strutinsky optimal shape method is generalized to the left-right asymmetric shapes of nuclei in order to investigate the influence of this degree of freedom on the barrier heights.

A microscopic theory for the decay of a compound nucleus is presented which is formulated in the spirit of transport theories. Its basic physical assumption is the existence of two time scales, a rapid one concerning the creation of intrinsic excitations and a slow one controlling the change of the nuclear shape. This fact is used in the derivation of an equation of motion for the reduced density by applying the so-called Markov approximation. A canonical distribution of the system with regard to the intrinsic excitations at a given shape is assumed as an approximation and a temperature T(q, t) depending on the nuclear shapes and time t is defined so as to render the canonical distribution as realistic as possible.

We consider a realistic description of hot, fast rotating nuclei with the help of the macroscopic nuclear model, LSD, of Refs.^1,2 We examine in particular a competition between Jacobi-type instabilities that preserve the left-right symmetry of nuclear systems and the Poincaré-type ones that break this symmetry. It is shown that in exotic nuclei these two mechanisms should be considered together in order to reach the lowest-energy nuclear configurations. Results from our related experiments are discussed together with the examples of preliminary theoretical predictions.

The generalized Wick theorem for coupled operators is applied for a description of a system of N interacting bosons. The matrix elements of the N-bosons Hamiltonian containing one- and two-body terms are expressed in terms of overlaps between N-boson states spanning a boson space. A possibility of transfering of the presented formalism for the description of 2N-fermion systems is also considered.

A class of relativistic nuclear energy density functionals is explored, in which only nucleon degrees of freedom are explicitly used in the construction of effective interaction terms. Short-distance correlations, as well as intermediate and long-range dynamics, are encoded in the nucleon-density dependence of the strength functionals of an effective interaction Lagrangian. The resulting phenomenological effective interaction, adjusted to experimental binding energies of a large set of axially deformed nuclei, together with a new separable pairing interaction adjusted to reproduce the pairing gap in nuclear matter calculated with the Gogny force, is applied in triaxial relativistic Hartree-Bogoliubov calculations of sequences of heavy nuclei: Th, U, Pu, Cm, Cf, Fm, and No.

The self-consistent separable random-phase approximation (SRPA) with Skyrme forces is extended to the case of magnetic excitations and applied to the description of spin-flip and orbital M1 giant resonances in the isotopic chain ^142-152Nd. The Skyrme forces SkT6, SkM*, SLy6 and SkI3 are used. The calculations show an onset of the scissors mode with increasing deformation. A specific three-peak structure of the spin-flip response is found and explained by particular neutron and proton spin-flip transitions. Although the employed forces provide an acceptable qualitative description, the Skyrme functional still needs further improvement to reproduce quantitatively the experiment for spin modes.

In the framework of the Higher Tamm-Dancoff Approximation which allows for a consistent treatment of pairing and quadrupole vibrational correlations preserving the particle-number symmetry and the Pauli principle, we compare the isospin content of ground and excited K^π = 0⁺ states including either type of the above mentioned correlations using an approximate projection-after-variation technique for isospin. The pairing correlations are described with a delta interaction using the same strength in the T = 0 and T = 1 channels, whereas the vibrational correlations are treated with an isoscalar quadrupole-quadrupole interaction. As a first study, we apply this approach in the N = Z, even-even ¹⁶O nucleus. For the same amount of correlations measured by the diffusivity of Fermi surface or depletion of particle-hole vacuum in the ground-state solution, we find very similar isospin mixing in the ground state for both types of correlations, but different patterns for the distribution in the calculated solutions of the expectation value of the square of the isospin operator.

We discuss the great theoretical insights on nuclear stability that emerged in the late 1960ies, especially the realization that observable nuclei might include a superheavy island 20+ protons beyond uranium with 92 protons. However, we now realize that the early models were not sufficiently quantitative to yield definite conclusions. Further groundwork for such models was laid in the 1970ies. Around 1980 more quantitative, global and universal models appeared and together with experimental advances the mapping of the superheavy island, or as it turns out now, continent started in earnest. We review these early developments and conclude with some examples of our current insights. In particular, 1) some theoretical models do have predictive capabilities for nuclear properties such as ground-state masses, shapes, and half-lives 10 or more neutrons and protons away from previously known regions and 2) the superheavy island may be a continent connected to the actinides by a narrow land bridge; the continent itself may extend from about proton number Z = 110 to Z = 120 or slightly beyond.

The high-spin yrast states in ¹⁶¹Lu are calculated in a Cranked Nilsson-Strutinsky approach keeping track of configurations and their evolution with spin. The outcome from calculations neglecting pairing and including pairing are compared. It is predicted that several yrast region configurations terminate at a very favored energy around I = 50 and that one of the observed bands is two transitions short of termination.

The influence of the reflection asymmetry degrees of freedom on the stability of rotating charged or gravitating liquid objects is investigated. The Modified Funny Hills shape parametrization as well as the optimal shapes developed by the Strutinsky group have been used. It is shown that the Poincaré instability can, indeed, appear in rotating gravitating objects, but is quite unlikely in rotating nuclei.

The addition of an extra nucleon to an even-even core generates, in most cases, a polarization of this core which is essentially due to a spin paramagnetic type of coupling. It is shown here that these results (which have been obtained so far only with the SIII Skyrme force) seems to be robust against the change of the Skyrme force parameters for the charge state corresponding to the added nucleon and somewhat less for the other charge state. The consequences drawn from calculations with the SIII force about the effect of the core polarization on the effective spin gyromagnetic factors, hold also for the two other Skyrme force parametrizations (SkM* and SLy4) considered here. A very good reproduction of the average phenomenological quenching of these factors from their free values, is also obtained. Finally, the rare cases where these rules do not hold are discussed.

Collective excitations involving a hypothetical nuclear tetrahedral symmetry, the latter described with the help of the T_d point group are discussed. A model combining the zero- and one-phonon excitations with the solutions of the "generalised collective rotor" of Ref.¹ is constructed. The solutions belonging to the irreducible representations of the T_d-group are constructed and used to calculate transition probabilities from hypothetical tetrahedral bands to the ground-state bands. An example of ¹⁵⁶Dy is discussed.

The technique of band-interaction calculations has been widely used for many years in the context of nuclear structure studies and a number of approximations have generally been made in this context. In this article we examine the impact of certain of these simplifications, basing our study on the interaction of some model bands. We show that large uncertainties can result when the interacting-band model is used under the assumptions that quite often are not even explicitly mentioned - neither verified.

Spatial symmetries of the densities appearing in the nuclear Density Functional Theory are discussed. General forms of the local densities are derived by using methods of construction of isotropic tensor fields. The spherical and axial cases are considered.

In this article we discuss the problem of a reliable determination of parameters of the nuclear mean-field Hamiltonians in view of obtaining high quality comparisons with the experimental single particle spectra: a. In the case of extrapolations to extreme (exotic) isospin nuclei as well as b. In terms of predictions of the energy levels that have not been so far measured also in 'non-exotic' nuclei. We formulate our discussion in terms of the Inverse Problem, a quickly growing sub-field of Applied Mathematics, and introduce a number of mathematical properties which are illustrated in Ref.¹ We focus on those mechanisms that may significantly limit the predictive power of the resulting Hamiltonians.

Determination of the mean-field Hamiltonian parameters can be seen as gathering information about all the single-particle states out of a very partial information on only a few experimentally known levels. This is exactly what the inverse problem in applied mathematics is about. We illustrate some of the related concepts in view of a preparation of the fully statistically significant parameter adjustment procedures. For this purpose we construct the exactly soluble inverse problems associated with the realistic and phenomenologically powerful nuclear Woods-Saxon Hamiltonian and we analyse a few both physical and mathematical aspects of such procedures. Presented illustrations suggest that to be able to discuss the predictive power of the mean-field Hamiltonians the parameter adjustment procedures must be based on a relatively complex statistical analysis partially addressed in Ref.¹

A new detector system dedicated for very/super heavy elements (VHE/SHE) detection that decay by spontaneous fission is presented. Such a decay mode of nuclei can be found e.g. in the region of Z≈100 (Fermium). Future experiments realized in the frame of French-Polish collaboration at the cyclotrons in GANIL and in the Heavy Ion Laboratory (HIL), University of Warsaw, is described. The results of test measurements made in the HIL for reactions ²⁰Ne(8 A.MeV)+¹²⁰Sn, ¹⁷⁹Au are presented.

A study of fragment production in peripheral nuclear collisions at Fermi energies with a transport model (Boltzmann–Nordheim–Vlasov, BNV) is presented. It is compared to an experiment for the reaction ¹⁸O on ¹⁸¹Ta at E/A = 35 MeV measured at very forward angles. The data are decomposed empirically into a two-component structure, one centered at beam velocity ("direct component") and one at lower velocities ("dissipative component"). It is seen that BNV calculations describe the general features of the dissipative component of the reaction. The direct component of experimental data is interpreted with the Goldhaber model, and the width parameter is extracted. We also include the deexcitation of the excited, primary projectile–like residues due to statistical decay to be able to compare the results of calculations more in detail with experiment. This secondary evaporation is important for both isotope and velocity distributions, but a more consistent estimate of the excitation energy of the residues has to be persued.

We consider many-body E-l transition matrix-elements between two nuclear states of different axially-symmetric deformations characterised by two different (mutually non-orthogonal) sets of single-particle wave-functions. Yet, when varying the deformations of the initial, final, or both these states one notices abrupt changes in the form of vanishing and possibly reappearance of the transition matrix elements calculated between the corresponding Slater determinants. The mechanism is explained in terms of the conservation of the |m| quantum number (absolute value of the projection of individual-nucleonic angular-momenta); consequences for the more general calculations of this type also without axial symmetry are discussed.

We compare different methods of obtaining the neutrinoless double beta decay nuclear matrix elements (NME). On the example of ⁷⁶Ge we use the NME to calculate the Majorana neutrino transition magnetic moments, generated through particle–sparticle R-parity violating loop diagrams whithin the minimal supersymmetric standard 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: ²⁵⁰No - ²⁶⁴Hs 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 and a pairing window of levels is used. A prediction of the E₂₊ rotational states for all the investigated nuclei is given.

Low energy quadrupole excitations of the ^84-110Mo nuclei are studied in the frame of the general Bohr collective model. The potential energy and inertial functions are calculated using the ATDHFB method starting from the Skyrme effective interaction. Obtained energies of the 2₁, 4₁, 2₂ and 0₂ levels and B(E2) values for the 2₁ → 0_g.s transitions are in good agreement with experimental data.

Recently, we reported on the observation of a new reaction mechanism of fast collinear breakup of the ¹⁹⁷Au + ¹⁹⁷Au system into three and four fragments of comparable size. Here we present results of an additional analysis that shows that these ternary and quaternary breakup reactions occur at extremely deep-inelastic collisions corresponding to small values of the impact parameter, while more peripheral collisions lead to well known binary deep-inelastic reactions.

A new ternary decay channel connected with magic constituents from Cf and U has been observed in recent experiments. It was called Collinear Cluster Tripartition (CCT) because the decay products fly apart almost collinearly. It is a decay of the fissioning system into the three fragments of comparable masses from the strongly elongated configuration. A total yield of the CCT events was found at the order of 10^-3 per binary fission. The first theoretical description of the high lying energy minima on the potential energy landscape are presented for U and Cf. They have been obtained by the Strutinsky shell correction calculations in the multidimensional deformation space.

We briefly review a recently developed semiclassical theory¹ for quantum oscillations in the spatial (particle and kinetic energy) densities of finite fermion systems and present some examples of its results. We then discuss the inclusion of correlations (finite temperatures, pairing correlations) in the semiclassical theory.

The collective moment of inertia is derived analytically within the cranking model in the adiabatic mean-field approximation at finite temperature. Using the nonperturbative periodic-orbit theory the semiclassical shell-structure components of the collective moment of inertia are obtained for any potential well. Their relation to the free-energy shell corrections are found semiclassically as being given through the shell-structure components of the rigid-body moment of inertia of the statistically equilibrium rotation in terms of short periodic orbits. Shell effects in the moment of inertia disappear exponentially with increasing temperature. For the case of the harmonic-oscillator potential one observes a perfect agreement between semiclassical and quantum shell-structure components of the free energy and the moment of inertia for several critical bifurcation deformations and several temperatures.

We propose to use the semi-classical Wigner-Kirkwood ħ expansion to calculate shell corrections for spherical and deformed nuclei. The expansion is carried out up to fourth order in ħ. A systematic study of Wigner-Kirkwood averaged energies is presented as a function of the deformation degrees of freedom. The shell corrections, along with the pairing energies obtained by using the Lipkin-Nogami scheme are used in the microscopic-macroscopic approach to calculate binding energies. The macroscopic part is obtained from a liquid drop formula with six adjustable parameters. Considering a set of 367 spherical nuclei, the liquid drop parameters are adjusted to reproduce the experimental binding energies, which yields a rms deviation of 630 keV.

The equation of state (EOS) in a form suitable for calculation in the model of quantum molecular dynamics (QMD) is presented. Using the proposed form of the EOS, different properties of nuclei such as ground state energy, nuclear radii and neutron skin could be calculated. The preliminary results of calculations are compared with the experimental data.

Role of the non-axial octupole deformation a₃₂(Y₃₂ + Y_3-2) on the potential energy of heavy nuclei is studied in a large deformation space. The study is performed within a macroscopic-microscopic approach. A large region of nuclei with proton number 88 ≤ Z ≤ 112 and neutron number 128 ≤ N ≤ 156 is considered. It is found that while the a₃₂ deformation alone lowers the energy of the nuclei by up to about 3 MeV (for nuclei around ²³⁸Fm), it has practically no effect on the global energy minima of considered nuclei, when the analysis is done in a large space.

We show the possible evolution of the nuclear deep inelastic structure function with nuclear density ρ. The nucleon deep inelastic structure function represents distribution of quarks as a function of Björken variable x, which measures the longitudinal fraction of the momentum carried by them during deep inelastic scattering (DIS) of electrons on nuclear targets. The quark localization is proportional to 1/x and this relation introduces the dependence of the nucleon structure function on the nuclear medium. Starting with small density and negative pressure in nuclear matter (NM), we have relatively large inter-nucleon distances and increasing role of nuclear interaction mediated by virtual mesons. When the density approaches the saturation point, ρ = ρ₀, we have no longer separate mesons and nucleons but eventually modified nucleon structure function (SF) in the medium. The ratio of the nuclear to the nucleon SF measured at the saturation point is well known as the "EMC effect". For larger density, ρ > ρ₀, when the localization of quarks is smaller than 0.3 fm, the nucleons overlap. We argue that nucleon mass should start to decrease in order to satisfy the momentum sum rule (MSR) of DIS. These modifications of the nucleon structure function are calculated in the frame of the nuclear relativistic mean field (RMF) convolution model. The correction to the Fermi energy from a term proportional to the pressure is very important and its inclusion modifies the equation of state (EoS) for the nuclear matter.

The relation between the interaction parameters for fermions on the spatial lattice and the two-body T matrix is discussed. The presented method allows determination of the interaction parameters through the relatively simple computational scheme which includes the effect of finite lattice spacing. In particular the relation between the interaction parameters and the effective range expansion parameters is derived in the limit of large lattices.

The ground-state properties of neutron-deficient Hg isotopes have been investigated by the constrained self-consistent Hartree-Fock-Bogoliubov approach with the Skyrme and Gogny effective forces. In the case of the Skyrme interaction we have also applied the Hartree-Fock+BCS model with the state-dependent δ-pairing interaction. Potential energy surfaces and pairing properties have been compared for the both types of forces.

Calculations for infinite nuclear matter with realistic nucleon-nucleon interactions suggest that the isoscalar effective mass (IEM) of a nucleon at the saturation density equals m*/m ~ 0.8 ± 0.1, at variance with empirical data on the nuclear level density in finite nuclei which are consistent with m*/m ≈ 1. This contradicting results might be reconciled by enriching the radial dependence of IEM. In this work four new terms are introduced into the Skyrme-force inspired local energy-density functional: τ(∇ρ)², , τ² and τ∆ρ. The aim is to investigate how they influence the radial dependence of IEM and, in turn, the single-particle spectra.