Nuclear Structure of 12,14Be Within Deformed Quasi-Particle Random Phase Approximation (DQRPA)

We developed a deformed quasi-particle random phase approximation (DQRPA) to describe the Gamow–Teller (GT) transitions on even–even neutron-rich nuclei. To describe deformed nuclei, we exploited the deformed axially symmetric Woods–Saxon potential, the deformed BCS, and the deformed QRPA with realistic two-body interaction calculated by Brueckner G-matrix based on Bonn CD potential. The deformed single particle states are expanded in terms of the spherical harmonic oscillator basis in order to take the realistic G-matrix stored in the spherical basis. We calculated GT strength distributions, B(GT), of two nuclei ^12,14Be for many different deformation parameter β ₂ values as a function of the excitation energy E _ex w.r.t. the ground state of a parent nucleus. Our results for ¹²Be predict to prefer a prolate shape and B(GT) results of ¹⁴Be turn out to be independent of the β ₂ values.     

Deformation and shape coexistence in medium mass nuclei

Emerging evidence for deformed structures in medium mass nuclei is reviewed. Included in this review are both nuclei that are ground state symmetric rotors and vibrational nuclei where there are deformed structures at excited energies (shape coexistence). For the first time. Nilsson configurations in odd-odd nuclei within the region of deformation are identified. Shape coexistence in nuclei that abut the medium mass region of deformation is also examined. Recent establishment of a four-particle, four-hole intruder band in the doublesubshell closure nucleus⁹⁶Zr₅₆ is presented and its relation to the Nuclear Vibron Model is discussed. Special attention is given to the N=59 nuclei where new data have led to the reanalysis of⁹⁷Sr and⁹⁹Zr and the presence of the [404 9/2] hole intruder state as isomers in these nuclei. The low energy levels of the N=59 nuclei from Z=38 to 50 are compared with recent quadrupole-phonon model calculations that can describe their transition from near-rotational to single closed shell nuclei. The odd-odd N=59 nuclei are discussed in the context of coexisting shape isomers based on the (p[303 5/2]n[404 9/2])2^– configuration. Ongoing in-beam (t.p conversion-electron) multiparameter measurements that have led to the determination of monopole matrix elements for even-even₄₂Mo nuclei are presented, and these are compared with initial estimates using lBA-2 calculations that allow mixing of normal and cross subshell excitations. Lastly, evidence for the neutron-proton³S₁ force's influence on the level structure of these nuclei is discussed within the context of recent quadrupole-phonon model calculations.Work supported by USDOE contract Nr.W-7405-Eng-48 and NATO Grant Nr.RGO565/82.     

Exponential convergence and acceleration of Hartree-Fock calculations

We show that we can expect an exponential behaviour for the convergence of the Hartree-Fock solution during the HF iteration procedure. We use this property to extrapolate some collective degrees of freedom, in this case the shape, in order to speed up the self-consistent calculation. For axially deformed nuclei we apply the method to the quadrupole moment which corresponds to a simple scaling transformation on the single particle wave functions. Results are shown for the deformed nuclei ²⁰Ne and ²⁸Si with a Skyrme interaction.     

Nuclear pears: Recent developments and future prospects

We report studies of examples of reflection-asymmetric nuclei which are difficult to access using compound nucleus reactions. The octupole radium isotopes withN>132 and radon isotopes are not accessible by reactions employing stable targets and beams; we have shown that multinucleon transfer reactions can populate these nuclei with sufficient yield for their structure to be determined. We report high-spin studies in^{218, 220, 222}Rn and^{222, 224, 226, 228, 230}Ra: these show that the Ra isotopes withA<228 have the characteristics of octupole deformed nuclei whereas the Rn isotopes behave like octupole vibrators. Measurements of theB(E1)/B(E2) ratios indicate that the electric dipole moment in these nuclei is constant with spin. The most octupole deformed nuclei are predicted to be uranium isotopes withN≈132; measurements of the very fissile nucleus²²⁶U suggest that it is octupole deformed and has a large intrinsic electric dipole moment. Finally, we speculate that the best examples of pear shapes are the hyperdeformed minima predicted to lie low in uranium isotopes withN≈140; their signature of high-multiplicity low-energyE1 photon cascades should be detectable using present-day high-efficiency germanium arrays.     

Potential energy surfaces for<Emphasis Type="Italic">N</Emphasis> =<Emphasis Type="Italic">Z</Emphasis>,<Superscript>20</Superscript>Ne-<Superscript>112</Superscript>Ba nuclei

We have calculated the potential energy surfaces forN = Z,²⁰Ne-¹¹²Ba nuclei in an axially deformed relativistic mean field approach. A quadratic constraint scheme is applied to determine the complete energy surface for a wide range of the quadrupole deformation. The NL3, NL-RA1 and TM1 parameter sets are used. The phenomenon of (multiple) shape coextistence is studied and the calculated ground and excited state binding energies, quadrupole deformation parameters and root mean square (rms) charge radii are compared with the available experimental data and other theoretical predictions.     

Calculated properties of superheavy nuclei

Ground-state properties of the heaviest nuclei are analyzed within a macroscopic-microscopic approach. The main attention is paid to such properties as deformation, deformation energy, energy of the first rotational state 2⁺ of a nucleus, and the branching ratio of α decay to this 2⁺ state with respect to the decay to the ground state 0⁺. The analysis concerns the problem of experimental confirmation of theoretically predicted deformed shapes of superheavy nuclei situated in the region around the nucleus ²⁷⁰Hs. A large region of even-even nuclei with proton, Z=82–128, and neutron, N=126–190, numbers is considered.     

The collective 1+ state in triaxial nuclei

The energy spectra of the relative neutron-proton surface vibrations are discussed within the framework of the neutron-proton deformation model (NPD) in cases of deformed axial-symmetric and triaxial nuclei. It is shown that the energy spacing between the lowest 1⁺ states is of the order of the energy of the mass vibrations in axially symmetric nuclei and approaches values of rotational energies in the case of maximal asymmetry.     

Magnetic moments of K isomers as indicators of octupole collectivity

The relation between the quadrupole-octupole deformation and the structure of high-K isomers in heavy even-even nuclei is studied through a reflection asymmetric deformed shell model including a BCS procedure with constant pairing interaction. Two-quasiparticle states with K ^?? = 4^?, 5^?, 6^?, 6⁺ and 7^? are considered in the region of actinide nuclei (U, Pu and Cm) and rare-earth nuclei (Nd, Sm and Gd). The behaviour of two-quasiparticle energies and magnetic dipole moments of these configurations is examined over a wide range in the plane of quadrupole and octupole deformations (?? ₂ and ?? ₃. In all considered actinide nuclei, the calculations show that there is pronounced sensitivity of the magnetic moments to the octupole deformation. In the rare-earth nuclei, the calculations for ^{154, 156}Gd show stronger sensitivity of the magnetic moment to the octupole deformation than in the other considered cases.     

Fragmentation of giant multipole resonances in deformed nuclei

The fragmentation of the giant monopole resonance in deformed nuclei is first studied by coupling the monopole oscillation with the quadrupole oscillation by means of the variational procedure for resonance frequencies. It is shown that, for non-axial symmetry, the monopole oscillation couples with both m = 0 and 2 modes of the quadrupole oscillation and the giant monopole resonance is split into three components, whereas for axial symmetry, the fragmentation is given by E₀(1 + 0.86δ² ± 1.25δ³) and E₀(0.74 ± 0.22δ ? 0.21δ² ± 0.57δ³), where E₀ is the g monopole resonance energy for spherical nuclei, δ is the deformation parameter, and the upper and lower signs stand for prolate and oblate deformations, respectively. The initial fragmentation of the giant quadrupole resonance is seen to be little modified by the coupling, except for the m = 0 mode which is split into two components. The variational method is extended to general multipoles for an ellipsoid and the fragmentation of giant multipole resonances in deformed nuclei is investigated for both axial and non-axial symmetries. A brief discussion is also made about the meaning of the energy eigenvalue involved in the model wave equation in terms of multipole sum rules. The giant dipole resonance for the static octupole deformation is shortly considered. The giant E0 and E3 resonances for largely deformed nuclei are finally examined by solving the spheroidal eigenvalue equation and they are compared with the results of the giant dipole and quadrupole resonances.     

An investigation of the odd-A rare-earth nuclei using rotational models

Using recently compiled data for band-head energies of 53 odd-A rare-earth nuclei, rotational models for strongly deformed nuclei have been used to determine the variation of deformations, spin-orbit parameter κ and the l² parameter, μ, in these nuclei. The deformation is found to be consistent with experimental deformations within 20 %. The spin-orbit parameter, κ, is found to vary from 0.037 up to 0.070, a 25 % variation from Nilsson's 0.050. The l² parameter μ is found to vary from 0.30 up to 0.71, a 30 % variation from Nilsson's 0.45. The trends observed in the values of spin-orbit strength C indicate a correlation with deformation, δ. Simultaneous shifts of the l² strength D for neutron numbers 95–97 and 101–103 may be interpreted as a sudden shift in the squareness of the potential well possibly caused by shell filling. Inclusion of hexadecapole deformation greatly improves the band-head energies for the mass region 150 ≦ A ≦ 165.     

Rotational bands from shell-model Hamiltonians

Shell-model Hamiltonians with eigenstates forming rotational bands are considered. Such states have eigenvalues proportional to J(J+ 1) and can be projected from a Slater determinant of deformed orbitals. The latter are linear combinations of single-nucleon wave functions of j-orbits in a major shell. Conditions on matrix elements and single-nucleon energies are obtained in terms of the deformation parameters. An actual effective interaction is constructed yielding exact ground-state rotational bands for ²⁰Ne and ²⁴Mg which gives reasonable agreement with energies of other sd shell nuclei. Unlike the case of SU(3) symmetry, spin-orbit interaction and different single-nucleon energies can be accommodated and the procedure is not confined to oscillator major shells. Other welcome departures of our effective interaction from the SU(3) picture are the absence of rotational spectra in oxygen isotopes and that the ²⁴Mg ground-state band is projected from a Nilsson-type deformed state with axial symmetry.     

Deformation effects on isospin mixing and isobar analogue resonance for 74−80Kr isotopes

Pyatov’s method has been applied to investigate Fermi beta transitions in deformed ^74–80Kr isotopes. This self-consistent method, which was used to study the isobar analogue states in the spherical odd-odd nuclei, has to date not been applied for the isobar analogue states in deformed nuclei. The nucleon-nucleon residual interaction has been included so that the broken isospin symmetry in the mean field approximation has been restored and the strength parameter of the effective interaction has been taken out to be a free parameter. The energies and wave functions of the isobaric analogue excitations in ^74–80Rb isotopes have been obtained within the framework of the pnQRPA method. The probability of the isospin mixing in the ground states and the centroid energies of the isobar analogue resonance have been presented and the deformation effects on these quantities have been quantified.     

Allowance for the shell structure of colliding nuclei in the fusion-fission process

The motion of two nuclei toward each other in fusion-fission reactions is considered. The state of the system of interacting nuclei is specified in terms of three collective coordinates (parameters). These are the distance between the centers of mass of the nuclei and the deformation parameter for each of them (the nose-to-nose orientation of the nuclei is assumed). The evolution of collective degrees of freedom of the system is described by Langevin equations. The energies of the Coulomb and nuclear (Gross-Kalinovsky potential) interactions of nuclei are taken into account in the potential energy of the system along with the deformation energy of each nucleus with allowance for shell effects. The motion of nuclei toward each other are calculated for two reaction types: reactions involving nuclei that are deformed (₄₂¹⁰⁰Mo + ₄₂¹⁰⁰Mo → ₈₄²⁰⁰Po) and those that are spherical (₈₂²⁰⁸Pb + ₈¹⁸O → ₉₀²²⁶Th) in the ground state. It is shown that the shell structure of interacting nuclei affects not only the fusion process as a whole (fusionbarrier height and initial-reaction-energy dependence of the probability that the nuclei involved touch each other) but also the processes occurring in each nucleus individually (shape of the nuclei and their excitation energies at the point of touching).     

Nuclear level densities with collective rotations included

The level density is calculated from the single particle energies in a Woods-Saxon potential with pairing included in the BCS approximation. The collective rotations are included by addition of a rotational band on top of each of the intrinsic levels. The nuclei investigated have mass numbers in the region 100 ≦ A ≦ 253. At the ground state deformation and at the neutron separation energy for the nucleus in question we compare calculated and observed level densities. The dependence on the parameters in the model are investigated. Considering the uncertainties in these parameters the calculated results are believed accurate to within a factor of 3. The rotations contribute typically a factor of 40. They must be included for deformed and not for spherical nuclei. We underestimate systematically the level density by a factor of 4 with fluctuations around the average value by a factor of 3. The nuclei lighter than ¹³⁸Ba are an exception. We obtain around a factor 100 too few levels in the calculation.     

α-clustering in Be isotopes

The Be chain of isotopes is investigated within the Sernimicroscopic Algebraic Cluster Model (SACM). For that the SACM is extended to odd-mass p-shell nuclei. We show that the main features of the Be isotopes can be well explained. Though for the neutron halo nucleus ¹¹Be some problems remain, the shell model still can describe the rough structures of these nuclei. In ¹¹Be a shell inversion takesplace reflecting the high deformation of the system. The quadrupole—quadrupole interaction plays a dominant role in p-shell nuclei.     

Role of thermal fluctuations in the damping of the giant dipole resonance of spherical and deformed nuclei: 90Zr and 164Er

We study the damping width of the giant dipole resonance (GDR) built on the ground state and on excited states of ⁹⁰Zr, which is spherical at zero spin and temperature, and ¹⁶⁴Er which is deformed. Neither of these nuclei changes appreciably their equilibrium shape when excited. Nonetheless, fluctuations due to finite temperature produce a marked increase in the width of the GDR of ⁹⁰Zr, leaving essentially unchanged that of ¹⁶⁴Er. This is because in the case of deformed nuclei thermal fluctuations imply the sampling of both larger and smaller deformations around the equilibrium shape, while spherical nuclei can only feel larger distortions. In spite of the stability of the GDR of ¹⁶⁴Er, the angular distribution of the associated γ-rays is strongly affected by the temperature.     

Einzelteilchen-Berechnungen zur Kernspaltung

We use an axial symmetric, energy-, spin-, and isospin-dependent shell model Hamiltonian to describe heavy deformed nuclei, where the equipotential surfaces are taken to be spheroids, corrected towards a more realistic nuclear shape by means of Legendre polynomials of maximal order four, and the potential is assumed to depend on the modified radial coordinate like a Fermi function. The eigenvalue problem is solved by a perturbation treatment which starts with the eigensolutions for purely spheroidal deformations as a first approximation. By analysis of an expression which involves the single particle energies for a given deformation, the sets of deformation parameters for the ground state and the transition point of a fissioning nucleus are found, and from them the total energy at the transition point is obtained. For Th²³¹, Th²³³, U²³⁵, U²³⁶ and U²³⁷ we give our results and compare them with measurements and liquid drop model calculations.     

Shape evolution for Sm isotopes in relativistic mean-field theory

The evolution of shape from the spherical to the axially deformed shapes in the Sm isotopes is investigated microscopically in relativistic mean-field theory. The microscopic and self-consistent quadrupole deformation constrained relativistic mean-field calculations show a clear shape change for the even-even Sm isotopes with N = 82-96. The potential surfaces for ¹⁴⁸Sm, ¹⁵⁰Sm and ¹⁵²Sm are found to be relatively flat, which may be the possible critical-point nuclei. By examining the single-particle spectra and nearest-neighbor spacing distribution of the single-particle levels, one finds that the single-particle levels in ¹⁴⁸Sm , ¹⁵⁰Sm, and ¹⁵²Sm distribute more uniformly.