DIVISION OF NUCLEAR STRUCTURE THEORY
Head: Prof. Jacek Dobaczewski
Academic teachers: Prof. Witold Nazarewicz1, Prof. Stanisław G. Rohoziński, Dr hab. Wojciech Satuła, Dr hab. Tomasz Werner
Postgraduate students: Mgr Rainald Kirchner, Mgr Przemysław Olbratowski
Graduate students: Sebastian Głowacz
In 2000 and 2001, five members of the Division have published 40 papers in refereed periodicals (see the list below), and have presented 40 invited talks and 9 contributions at international conferences. During this period of time the following main research projects have been realized:
Paper  is a continuation of  and discusses breaking of symmetries that belong to the double point group D . Subgroup structure of the D group indicates that there can be as much as 28 physically different, broken-symmetry mean-field schemes -- starting with solutions obeying all the symmetries of the D group, through 26 generic schemes in which only a non-trivial subgroup of D is conserved, down to solutions that break all of the D symmetries. Choices of single-particle bases and the corresponding structures of single-particle hermitian operators are analysed for several subgroups of D .
In Ref.  it was shown how to construct a basis in which two arbitrary complex antisymmetric matrices C and C' acquire simultaneously canonical forms. The construction is not restricted by any conditions on properties of the C+C' matrix. Canonical bases pertaining to the generator-coordinate-method treatment of many-fermion systems were discussed.
Shell corrections of the finite deformed Woods-Saxon potential were calculated  using the Green's function method and the generalized Strutinsky smoothing procedure. They were compared with the results of the standard prescription which are affected by the spurious contribution from the unphysical particle gas. In the new method, the shell correction approaches the exact limit provided that the dimension of the single-particle (harmonic oscillator) basis is sufficiently large. For spherical potentials, the present method is faster than the exact one in which the contribution from the particle continuum states is explicitly calculated. For deformed potentials, the Green's function method offers a practical and reliable way of calculating shell corrections for weakly bound nuclei.
Nuclei with large neutron-to-proton ratios have neutron skins, which manifest themselves in an excess of neutrons at distances greater than the radius of the proton distribution. In addition, some drip-line nuclei develop very extended halo structures. The neutron halo is a threshold effect; it appears when the valence neutrons occupy weakly bound orbits. In Ref. , nuclear skins and halos were analysed within the self-consistent Skyrme-Hartree-Fock-Bogoliubov and relativistic Hartree-Bogoliubov theories for spherical shapes. It was demonstrated that skins, halos, and surface thickness can be analysed in a model-independent way in terms of nucleonic density form factors. Such an analysis allows for defining a quantitative measure of the halo size. The systematic behavior of skins, halos, and surface thickness in even-even nuclei was discussed.
In Ref.  particle and pairing densities in spherical even-even neutron-rich nuclei were studied within the Skyrme-Hartree-Fock-Bogoliubov approach with the density-dependent pairing interaction. The influence of the density dependence of the pairing interaction on asymptotic properties of nucleonic distributions were analysed. It was demonstrated that the size of the neutron halo dramatically depends on the behavior of the pairing interaction at low density.
In Ref.  the high-spin behavior of deformed neutron-rich nuclei was discussed. In particular, quasi-particle Routhian spectra of heavy Er isotopes were discussed within the deformed shell model, and rotational properties and isovector shape deformations of heavy Ne and Mg isotopes were studied with the self-consistent cranked Skyrme-Hartree-Fock theory. What is the response of the neutron drip-line nuclei, the large, diffused, and possibly superfluid many-body systems to rotation? Both the schematic and self-consistent calculations contained in this paper gave interesting insights to this question. On the one hand, the variation of the neutron shell structure with neutron number, mainly influencing the position of the high-j unique-parity shell, is expected to modify the pattern of quasiparticle excitations in the rotating nucleus. On the other hand, since the Coriolis force mainly acts on the high-j orbitals which are strongly localized within the nuclear volume because of the large centrifugal barrier, no strong isovector effects (due to neutron halo or skin) are expected at high spins. For instance, our unpaired calculations indicate that proton and neutron deformations are very similar at high rotational frequencies.
Life in nuclear ``terra incognita'' is different from that around the stability line; the promised access to completely new combinations of proton and neutron numbers offers prospects for new structural phenomena. The main objective of Ref.  was to discuss some of the challenges and opportunities for nuclear structure research with radioactive nuclear beams.
Two applications of mean-field calculations based on 3D coordinate-space techniques were presented in Ref. . The first concerns the structure of odd-N superheavy elements that have been recently observed experimentally and shows the ability of the method to describe, in a self-consistent way, very heavy odd-mass nuclei. Our results are consistent with the experimental data. The second application concerns the introduction of correlations beyond a mean-field approach by means of projection techniques and configuration mixing. Results for Mg isotopes demonstrate that the restoration of rotational symmetry plays a crucial role in the description of 32Mg.
Radii and diffuseness parameters of heavy and superheavy nuclei were in Ref.  analysed for spherical and axially deformed shapes within the Skyrme-Hartree-Fock+BCS theory with zero-range pairing force. The characteristics of self-consistent density distributions have been analysed using the generalized Helm model extended to the case of deformation.
In Ref. , quantum stabilization of superheavy elements was quantified in terms of the shell-correction energy. The shell correction was computed at spherical shape using self-consistent nuclear models: the non-relativistic Skyrme–Hartree–Fock approach and the relativistic mean-field model, for a number of parametrizations. All the forces applied predict a broad valley of shell stabilization around Z=120 and N=17-184. Also two broad regions of shell stabilization in hyperheavy elements with N258 and N308 were predicted. Due to the large single-particle level density, shell corrections in the superheavy elements differ markedly from those in lighter nuclei. With increasing proton and neutron numbers, the regions of nuclei stabilized by shell effects become poorly localized in particle number, and the familiar pattern of shells separated by magic gaps is basically gone.
The newly developed nonadiabatic method based on the coupled-channel Schrödinger equation with Gamow states was used  to study the phenomenon of proton radioactivity. The new method, adopting the weak coupling regime of the particle-plus-rotor model, allows for the inclusion of excitations in the daughter nucleus. This can lead to rather different predictions for lifetimes and branching ratios as compared to the standard adiabatic approximation corresponding to the strong coupling scheme. Calculations were performed for several experimentally seen, nonspherical nuclei beyond the proton dripline. By comparing theory and experiment, we were able to characterize the angular momentum content of the observed narrow resonance. Theoretical approaches to deformed proton emitters were briefly reviewed in Ref. .
During the last years much theoretical and experimental work was devoted to the physics of K-isomers. In several Coulomb-excitation experiments surprisingly strong population of these states were observed in spite of that the transitions linking them to the ground state are forbidden due to the K number. A natural explanation is that the Coulomb excitation goes via some intermediate states, although these states are not yet identified. Paper  reported on Coulomb excitation of Ta, which is a relatively easy case, because the forbidness in question is not due to K, but to the asymptotic Nilsson numbers, and thus not so strong. It was possible, therefore, to observe the mediating states. Reduced transition probabilities were derived from the experimental data and it was calculated, that feeding through intermediate states was about fifty times more efficient than the direct way from the ground state. Comparison of the data with calculations within the Quasiparticle-Plus-Phonon model shows that the newly found states are one-phonon gamma-vibrational excitations.
Structure of eight experimentally known superdeformed bands in the nucleus 151Tb was in Ref.  analysed using the results of the Hartree-Fock and Woods-Saxon cranking approaches. It was demonstrated that far going detailed similarities between the two approaches exist and predictions related to the structure of rotational bands calculated within the two models are nearly parallel. An interpretation scenario for the structure of the superdeformed bands was presented and predictions related to the exit spins were made. Small but systematic discrepancies between experiment and theory, analysed in terms of the dynamical moments, , were shown to exist. These discrepancies could be parametrized in terms of a scaling factor f, such that modifications together with the implied scaling of the frequencies , correspond systematically better with the experimental data ( ) for both the Woods-Saxon and Hartree-Fock with Skyrme SkMinteractions.
In Ref.  we described the new version (v1.75r) of the code HFODD which solves the nuclear Skyrme-Hartree-Fock problem by using the Cartesian deformed harmonic-oscillator basis. Three minor errors that went undetected in the previous version have been corrected. The new version contains an interface to the LAPACK subroutine ZHPEV. Several methods of terminating the Hartree-Fock iteration procedure, and an algorithm that allows to follow the diabatic configurations, have been implemented.
Three superdeformed bands were established in 65Zn using the 40Ca(29Si, 4p)65Zn reaction, and the lifetimes were measured for two of the three bands . The configurations of these bands were assigned based on the Hartree-Fock calculations. One of the three bands exhibits at low a rise in the J(2) dynamic moments of inertia that is similar to the alignment gain observed in 60Zn. A comparison of the superdeformed band configurations and their J(2) dynamic moments of inertia for light Zn isotopes supports the suggestion that the rise in J(2)may be related to the T=0 pair correlations.
In Refs. [15,38], high-spin states in 59Cu and 57Co, respectively, were found by using the Gammasphere array in conjunction with ancillary detector systems that allowed for the identification of superdeformed rotational bands in these nuclei, which were firmly linked to low-spin yrast states. Using directional correlations of oriented states, a spin-parity assignments of the band heads were possible. The average quadrupole moments of the bands were measured. The characteristics of the bands were compared to neighboring nuclei and predictions of different mean-field theories were analysed.
It has been possible, using Gammasphere plus Microball, to extract differential lifetime measurements free from common systematic errors for over 15 different nuclei (various isotopes of Ce, Pr, Nd, Pm and Sm) at high spin within a single experiment. A comprehensive study  established the effective single-particle quadrupole moments in the A135 light rare-earth region. Detailed comparisons were made with calculations using the self-consistent cranked mean-field theory.
In Ref.  rotational structures in NZ40 nuclei, calculated using the standard TRS model, were presented. It was shown, that th model involving only isovector pairing force reveals in N=Z nuclei certain shortcomings. It was argued that inclusion of isoscalar pairing may improve agreement to the data. In particular, the presence of such correlations can shift crossing frequencies [signature conserving isoscalar pairing mode] and enhance rigidity [signature breaking isoscalar pairing mode] of moment of inertia (MoI) at very high-spins.
In Ref.  a model including proton-neutron pairing and extension of Lipkin-Nogami approximate number projection to non-separable proton-neutron systems was presented. It was shown that the number projection allows for mixing of different pairing phases but, simultaneously, acts destructively on the proton-neutron correlations. An impact of isoscalar pairing on the binding energy was also analysed. It was shown that these correlations may provide a natural microscopic explanation of the Wigner energy, lacking in mean-field models. A possibility of phase transition from isovector to isoscalar phase at high angular momenta was discussed as well.
In Ref.  we discussed pairing correlations in weakly bound neutron rich nuclei, by using the coordinate-space Hartree-Fock-Bogolyubov approach which allows to take into account the coupling to particle continuum. We showed that the additional pairing binding energy acts against a development of an infinite rms radius that characterizes standard =0 mean-field eigenfunctions in the limit of vanishing binding energy. As a result, neutron radii of even-N nuclei do not diverge in the limit of vanishing Fermi energy. Only the broken-pair ground states of odd-N nuclei can exhibit diverging neutron radii, provided an =0 (or 1) quasiparticle state appears near the Fermi surface. We also show that the pairing-increased (although not infinite) rms radii of even-Nnuclei result from the coupling to low-lying =0 continuum, which is always available for virtual pair excitations, independently of what are the angular momenta of least-bound single-particle levels.
In Ref.  odd-even staggering of binding energies in finite fermion systems with pairing correlations was discussed. The binding energy indicators which measure the magnitude of pairing correlations and the effective single-particle spacings in a given system were constructed and studied for several exactly solvable many-body Hamiltonians. Analytical formulas were derived that can be applied in the weak and strong pairing limits.
In Ref.  the concept of isospin cranked mean-field was introduced, and the response of various phases of pairing correlations to rotation in isospace was worked out in detail. In particular, it was shown that isoscalar pairing strongly reduces the MoI in isospace in a similar way as the isovector pairing reduces the spatial MoI. Moreover, the MoI undergoes a phase transition similar to the well-known Meissner effect in semiconductors. In turn, the T=2 states in even-even nuclei which were calculated to be beyond the phase transition, were shifted up in energy in agreement with experimental data.
In Ref.  the isospin cranking model was extended to deal with the T=1 excitations in even-even (e-e) N=Z nuclei and T=0 and T=1 states in odd-odd (o-o) N=Z nuclei. It was shown that the T=0 states in o-o nuclei, and the T=1 states in e-e nuclei must be treated as two-quasiparticle (2qp), time-reversal symmetry-breaking excitations to account for their angular momenta I0. On the other hand, the lowest T=2 states in e-e nuclei, and the lowest T=1 states in o-o nuclei are both 0qp type states, but excited in the isospace. This interpretation was found to be consistent with an interpretation of their isobaric analogue states.
In Ref.  various aspects and consequences of application of isoscalar-paired and isospin cranked model were overviewed. Among others such issues like: (i) the Wigner energy problem, (ii) the influence of shell-structure on the excitation energies of the lowest isobaric excitations, (iii) the influence of isovector pairing on the MoI in isospace (analytical discussion), and (iv) structure of the so called terminating states were discussed in detail.
The nuclear quadrupole moment (NQM) of the I=3/2- excited nuclear state of 57Fe at 14.41 keV, important in Mossbauer spectroscopy, was in Ref.  determined from the large-scale nuclear shell-model calculations for 54Fe, 57Fe, and also from the electronic ab initio and density functional theory calculations including solid state and electron correlation effects for the molecules Fe(CO) and Fe(C5H5)2. Both independent methods yield very similar results. The recommended value is 0.15(2)eb. The NQM of the isomeric 10+ in 54Fe has also been calculated. The new NQM values for 54Fe and 57Fe are consistent with the perturbed angular distribution data.