K. Goeke

Pairing correlations and simultaneous projection of particle number and angular momentum in light nuclei

Abstract The Hartree-Bogolubov theory with simultaneous projection on sharp angular momentum quantum number and sharp particle number is presented as a description of the ground state rotational bands of doubly even nuclei. Calculations with effective G-matrix elements of the Hamada-Johnston potential and also of the Yale potential have been performed within this framework for the nuclei 20Ne, 22Ne, 24Ne, 24Mg, 26Mg, 28Si, 30Si, 32S and 34S. Rotational energies, moments of inertia, pairing correlations, quadrupole moments and B(E2) values are discussed and compared with available experimental data. The effect of particle number fluctuation on the ground state rotational spectrum and on the intrinsic wave function is investigated. The particle number projection turns out to be important for the description of the rotational levels, whereas it has only a negligible effect on the properties of the intrinsic wave function. Comparison with Hartree-Fock results and with experimental data shows that the inclusion of pairing correlations provides a significant improvement on the excitation energies, their relative ratios and the angular momentum dependence of the moments of inertia. All solutions show an antistretching effect in the quadrupole moments and especially in the B(E2) values. The rotational bands in 20Ne, 22Ne, 24Ne and 28Si turn out to be well described in the framework presented.

Antipairing and antistretching in 22Ne

Abstract The ground state rotational band of 22 Ne has been investigated with the angular momentum and particle number projected Hartree-Bogoliubov theory. Variation before projections was performed in the framework of the constrained Hartree-Bogoliubov theory with the quadrupole moment and the degree of pairing as constraints. It is shown that there is a clear decrease of the pairing correlations (antipairing) and a decrease of the quadrupole deformation (antistretching) with increasing angular momentum. The antipairing effect appears to be essential to reproduce the experimentally known deviation of the spectrum from the J ( J + 1) rule. The antistretching indicated by the B (E2) values is much stronger than that indicated by the calculated static quadrupole moments. This amounts to a breakdown of the rotational picture which is possibly connected with the low band cut-off values in the sd shell. The antipairing effect decreases the antistretching only slightly. The interaction between the pairing and quadrupole degrees of freedom is found to be too weak to change the earlier conclusions concerning antipairing and antistretching in the whole sd shell. No inert core was assumed in the calculations. The effective G -matrix elements of the Hamada-Johnston potential were used as an interaction.

On number and angular momentum projection from Hartree-Bogoliubov states

Rotational spectra, calculated by angular momentum projection from Hartree-Bogoliubov states may be completely distorted by particle number nonconservation. A simple method for correcting this is presented, which brings them close to the number projected spectra. It is also shown that by a slight modification of the usual number projection operator this projection may then be executed two times faster and with better numerical accuracy.We study the effect ofJ- and/orN projection before the variation in a constrained Hartree-Bogoliubov model. It is demonstrated that only simultaneous projection of both particle number and angular momentum before the variation is meaningful. Then a more gradual antipairing effect is found than known from previous work. We conclude however that the diagonalization of the Hamiltonian in a space of appropriately chosen generator wave functions is preferable to projection before the variation.In all cases the examples are nuclei in thesd-shell, calculated selfconsistently without separating off and inert core. The nucleon-nucleon force is the Hamada-Johnston potential.

The effect of particle number fluctuation on the rotational states of paired light nuclei

Abstract Hartree-Bogoliubov calculations with simultaneous projection on sharp particle number and good angular momentum after the variation are performed for 22 Ne and 26 Mg. Energies, pairing correlations, rms-radii and quadrupole moments are compared with experimental data. The effect of particle number fluctuation on the intrinsic wave function and on the properties of the rotational states is investigated. The particle number fluctuation turns out to be quite important for the description of the rotational levels, whereas it has only negligible effect on the properties of the intrinsic wave function.

Importance of quasiparticle excitations for backbending

Abstract The role of two quasiparticle excitations for the backbending phenomenon in rare earth nuclei is studied in terms of a variational procedure and a dynamical mixing in a trial space spanned simultaneously by BCS ground state configurations and two quasiparticle configurations, both projected accurately on good angular momentum and good particle number. It turns out that in 166Yb the backbending may be caused by the (j, m) = ( 13 2 , ± 5 2 ) two quasiparticle excitation, being closest to the Fermi surface, whereas in 168Yb similar excitations are too high in energy and cause therefore no anomaly of the moment of inertia. In both cases a good agreement between theory and experiment is observed without refitting the P + Q force.

Antipairing effect in the rotational states of light nuclei

Abstract Properties of rotational levels of sd shell nuclei are studied in the self-consistent deformed Hartree-Bogoliubov model with a variation performed approximately after angular momentum projection. This formalism, which consists of a constrained Hartree-Bogoliubov theory with an additional pairing potential, is explained in detail. Excitation energies, moments of inertia, pairing correlations and quadrupole moments of the ground state rotational bands of 22 Ne, 24 Ne and 30 Si are investigated and compared with experimental data. In the calculation all particles are treated alike without the assumption of an inert core, using the effective G -matrix elements of Barrett et al. , the Yale-Shakin interaction and the B1 force of Brink and Boeker. It turns out to be important to perform the projection before the variation and not afterwards, since the pairing correlations change drastically in going to higher J -states. In the nuclei 24 Ne and 30 Si the calculations indicate a phase transition into the unpaired states at J = 4 + and 8 + respectively. The agreement with experiment is satisfactory and the experimental deviation from the J (J+1) law is quite well reproduced. In each case the Barrett G -matrix gives the best results, which are, in the case of 22 Ne, in good agreement with experimental values.

Phase Transition in the Ground State Rotational Band of sd Shell Nuclei

A similar anomaly at high angular momentum states as the back bending effect in rare earth nuclei is found in the sd shell. It can be quantitatively understood by a phase transition due to the Coriolis Antipairing (CAP) effect. The phase transition is calculated by performing a Hartree-Fock-Bogoliubov (HB) calculation with angular momentum projection approximately before variation (PHB). This method determines for each angular momentum the amount of pairing independently. Results obtained utilizing reaction matrix elements derived from the Hamada-Johnston potential agree in 22Ne and 24Ne surprisingly well with the experimental data.

Do electron scattering and muonic x-ray data really require a central depression in the charge distribution of208Pb?

The charge distribution of208pb calculated in the Hartree-Fock (HF) approach using the density dependent nucleon-nucleon interaction of Ehlers and Moszkowski is tested by simultaneous comparison with experimental data from 502 MeV elastic electron scattering and muonic atoms. In both cases the agreement is very good and nearly as good as the best fits with a phenomenological charge distribution of Fermi type, if the effect of the polarization of the nucleus due to the presence of the muon is properly taken into account. In contradiction to the Fermi fits the HF distribution shows a hump at the center of the nucleus.

Selfconsistency and backbending in rare earth nuclei

Abstract Band mixing calculations are performed for the nucleus 166 Yb using angular momentum and particle number projected Hartree-Fock-Bogoliubov wave functions. The calculated spectrum for the yrast band is compared with that obtained from a non-selfconsistent Nilsson + BCS prescription.

High spin states in doubly even pf-shell nuclei

Abstract High spin states of the yrast band in 50 Ti, 52 Cr, 54 and 46 Ti are investigated in a microscopic approach allowing a coupling of rotations, vibrations and quasiparticle excitations. The lowering of the 6 + state in N = 28 nuclei is shown to originate mainly from proton K = 0 two-quasiparticle excitations. Using as a basis the entire pf-shell 8 + and 10 + states can be predicted for 50 Ti and 54 Fe.