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An attempt is made to take into account single particle degrees of freedom in angular momentum projected HFB-calculations via the explicit inclusion of two quasiparticle states. For this purpose a formalism is used which, though being rather general, nevertheless is rather suitable for numerical applications even in realistic model spaces. As a first test the method is applied in thei13/2-shell, where exact shell model solutions are available. Though using only a rather truncated quasiparticle space we obtain almost perfect agreement with these exact solutions.

Recent theoretical investigations revealed two different mechanisms for the backbending in 130Ce and 128Ba. While in 130Ce the well-known alignment of two nucleons at the beginning of the π h 11 2 shell is found, in 128Ba the effect is produced by the particles in the more than half-filled ν h 11 2 orbit and hence is preceded by the scattering of a nucleon pair in the nearby ν g 7 2 shell in order to increase the available alignment energy. The present paper is devoted to the possible experimental fingerprints of these different mechanisms. Using the wave functions already obtained, the total spin of each state is decomposed into the individual contributions from the different shells. Furthermore g-factors, static and dynamic quadrupole and also static hexadecapole moments are calculated. It turns out that the only decisive measurable quantities are the occupations of the relevant shells before and after the backbend, which are hardly accessible experimentally, at least at present.

Backbending due to the rotational alignment of two i13/2 neutrons oscillates as a function of the neutron number. This is explained in a particle number and angular momentum conserving model. The neutrons in the i13/2-shell are described by shell model wave functions and are weakly coupled to the rotation of the core. It is shown that the alignment is arranged by the nucleus by scattering two i13/2 neutrons mainly from the Ω = 1/2 and Ω = 3/2 i13/2 orbits into the core. The aligned band has therefore in average two neutrons less in the i13/2-shell. Interaction between the valence nucleon and the core is due to the pairing force. It is shown that this interaction has j -3/2 zeros. No interaction between the ground band and the aligned band means strong backbending while a strong interaction means no backbending. In the second part of this contribution we explain the behaviour of the second anomaly of the moment of inertia by the alignment of two h11/2 protons corresponding to the same oscillating behaviour as discussed before. In this way we can explain that for the N = 90 isotones 66156Dy shows no backbending, 68158Er shows upbending and 70160Yb shows a strong second anomaly of the moment of inertia.

The Hartree-Fock-Bogoliubov approach with spin and number projection before the variation extended to the description of nonyrast states (EXCITED VAMPIR) is used to describe simultaneously many properties of the 68,70,72,74Ge and 72,74Se nuclei. Low-lying spectra, neutron and proton occupation numbers for the spherical-basis orbitals, E0 and E2 transitions, and quadrupole moments are calculated and a fairly good agreement is obtained with experiment. The analysis of the EXCITED VAMPIR mean fields reveals that a strong mixing of prolate and oblate intrinsic quasiparticle determinants is responsible for the complex behaviour of the nuclei in A=70 region.

The solution of the Hartree-Fock-Bogoliubov problem with restoration of the broken symmetries before the variation, as it has been formulated within the VAMPIR and EXCITED VAMPIR approaches yields a description of each state of the nucleus by essentially one single symmetry-projected quasi-particle determinant. In the present paper we propose a procedure which allows to go beyond this mean-field approximation and to include the most important additional correlations in a systematic way. In this new approach each state is approximated by a linear combination of several symmetry-projected quasi-particle determinants, and the corresponding quasi-particle transformations are determined successively together with the configuration mixing via a chain of variational calculations. In this way the selection of the relevant configurations for each state is entirely left to the dynamics of the considered system. The method is applied to some selected examples within the 1sOd shell (the lowest three I π = 1 + states in 20 Ne and the 0 + , 2 + , 4 + members of the 28 Si ground-state band). The results are compared to the exact as well as to the VAMPIR and EXCITED VAMPIR results. Considerable improvements with respect to the latter approaches are obtained.

We have calculated the neutrinoless double beta decay rate of 76Ge. We take into account for the first time a relativistic correction to the nuclear current including weak magnetism. Its effect is to cancel a considerable part of the decay amplitude and we obtain less stringent upper limits on the neutrino Majorana mass and the right-handed weak leptonic current compared with previous calculations.

Two main shortcomings of the cranking model in band crossing regions, namely the possible occurrence of discontinuities and the strongly varying angular momentum fluctuations, are investigated. It is found that because of these drawbacks the reliability of the cranking approximation in regions of band crossing is more than doubtful, and that even the suggestions to overcome the difficulties by including the so-called physical parts of the 2qp trajectory do not improve the situation, at least as far as the wave functions are considered.

In the first part of the present series of two papers we discussed several nuclear structure models all working in configuration spaces consisting of spin- and number-projected quasiparticle determinants. In the present paper a particular version of the numerically simplest of these models is presented. This model approximates the nuclear wave functions by linear combinations of the angular momentum- and particle number-projected Hartree-Fock-Bogoliubov vacuum and the equally spin- and number-projected two quasiparticle excitations with respect to it. The model allows the use of realistic two body interactions and rather large model spaces. It can hence be applied to a large number of nuclear structure problems in various mass regions. First applications have been performed for the nuclei /sup 20/Ne, /sup 22/Ne, /sup 46/Ti, and /sup 164/Er. In all these cases the results are very encouraging.

In this first of two successive papers we present a microscopic model for nuclear reactions. It allows the description of elastic and inelastic nucleon scattering as well as photonuclear reactions with real and virtual photons. The model is based on a particular classification scheme of the many-body Hilbert space according to the number of continuum nucleons. By its very definition it is restricted to processes with at most one continuum particle in the entrance and/or exit channel. In a large model space, containing all single-particle resonances as bound states, nuclear wave functions are calculated by means of the structure model MONSTER(VAMPIR). Arbitrary general effective two-body forces may be used for the structure calculations as well as for the coupling of the resulting highly correlated wave functions to the continuum. It is the coupling to a modified resonance free continuum in the framework of a general coupled-channel calculation which allows the description of the intermediate structure of the nuclear continuum. In the second paper results obtained from an application to the reaction /sup 29/Si(e,e'x/sub i/)/sup 27/Al will be presented, while in a forthcoming paper the particular role of a fifth response function arising from polarized electron scattering will be discussed. copyrightmore » 1988 Academic Press, Inc.« less

We present a systematic study of low- and medium-spin properties of odd-mass nuclei in the mass A = 130 region using a microscopic spin- and number-conserving nuclear structure model with a realistic two-nucleon interaction. In this model the wave functions are approximated by linear combinations of number- and spin-projected one-quasiparticle determinants obtained from a selfconsistent Hartree-Fock-Bogoliubov mean field. The choice of the model space and the renormalization of the two-nucleon force extracted via a G -matrix description from nucleon-nucleon scattering data are discussed in detail. We demonstrate that the model can account for the main systematic trends of odd-mass nuclei in the region considered and is even able to reproduce rather fine details. It can hence be used to predict e.g. even such complicated level sequences as displayed here by the positive-parity neutron and proton excitations. The calculated spectroscopic amplitudes for singlenucleon transfer are compared with experiments and an extensive set of predictions is made. Throughout the paper we discuss the limitations of the presently used projected one-quasiparticle approach.

Hartree-Fock-Bogoliubov calculations with spin and number projection before the variation (VAMPIR) are performed for the nuclei 128 Ba and 130 Ce using a slightly renormalized Brueckner G -matrix as effective interaction in a rather large single-particle basis. The results are compared to those of Hartree-Fock-Bogoliubov calculations with projection after the variation, those of multiconfiguration calculations (MONSTER) and to experiment. In both nuclei the VAMPIR and the MONSTER approaches turn out to be of about the same quality and agree rather well with the experimental data. Analysis of the VAMPIR mean fields reveals that two somewhat different mechanisms are responsible for the backbending observed in the yrast bands of the two nuclei. While in 130 Ce the well-known alignment of two high- j quasiparticles (proton h 11 2 ) is found, in 128 Ba first a neutron pair is scattered from the h 11 2 to the g 7 2 orbit, and then the larger alignment energy of the less occupied neutron h 11 2 states produces the backbend. This latter effect is in agreement with the predictions of a simple model presented by us some years ago.

In this second of two successive papers we apply the reaction model MERLIN, presented in the first paper, to the recently measured reaction 28 Si (e, e′ x i ) 27 Al . The tremendous amount of calculated information is first analyzed as coming from a “complete experiment.” This allows a discussion of the assumptions which had to be made by the experimentalists in order to extract reasonable information from the measured data. A real comparison with the 28 Si (e, e′ x i ) 27 Al-experiment is possible only in a rather restricted kinimatical region. Hence, all predictions concerning other kinematics as well as polarization degrees of freedom, etc., will be discussed in forthcoming papers.

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.

The oscillating behaviour of the yrast-yrare interaction as a function of the chemical potential found in various calculations using mostly the Hartree-Fock-Bogoliubov cranking approach is reexamined with the help of an exactly solvable angular momentum conserving model. It is shown, that the oscillations of the yrast-yrare coupling can be explained as being due to the mixing of states with different particle numbers inside the valence shell.

Nuclear structure wave functions for the ground and low-lying excited states of 20 Ne, obtained from the angular momentum projected deformed particle-hole model using a “realistic” many-nucleon hamiltonian (kinetic energy plus a Brueckner G -matrix based on the Hamada-Johnston potential), are used as input to microscopic antisymmetrised DWBA analyses of inelastic proton scattering from 20 Ne. This nuclear structure model, which has been previously shown able to describe the essential features of the giant multipole resonances of both 20 Ne and 28 Si, predicts angular distributions for inelastic proton scattering, exciting a number of states below 9 MeV in 20 Ne, in qualitative agreement with the available data; a somewhat surprising result given the nuclear structure model's completely microscopic formulation. Anomalies observed in the assignment of some predicted levels to experimental states suggest some shortcomings in the form adopted for the hamiltonian.

Shell-model and multi-configuration Hartree-Fock (MCHF) calculations in an Od 5 2 -1s 1 2 oscillator basis are compared for 20 Ne, 22 Ne and 24 Mg. The similarity of the calculated results in the two models shows that the MCHF method is a very good approximation to the shell-model calculations.

In a recent paper we have shown that the use of essentially complex mean fields in the Hartree-Fock-Bogoliubov problem with restoration of the broken symmetries before the variation provides a possibility to account for time-odd unnatural parity-pairing correlations without being forced to give up axial and time-reversal symmetry. In this way the applicability of the previously developed VAMPIR and EXCITED VAMPIR approaches could be extended to states with arbitrary angular momentum and parity in both doubly-even and doubly-odd nuclear systems. In the present paper now the numerical feasibility of this idea is proven. Calculations are performed for selected states of several light nuclei using a 1s0d single-particle basis. The results are compared to those of exact shell-model diagonalizations as well as to those obtained by the older, more restricted versions of the above-mentioned methods. Considerable improvements with respect to the latter approaches are obtained indicating the importance of unnatural parity-pairing correlations in the considered nuclei.

A microscopic study of the yrast bands of various doubly even nuclei in the rare earth region is presented. The calculations are based on a variational principle using angular momentum and particle number projected deformed BCS states as trial wave functions and the pairing plus quadrupole force as residual interaction. The influence of particle number projection and force parameters on the spectra and wave functions are studied as well as the interplay between Coriolis antipairing and deformation effects. Successes and shortcomings of the model are discussed.

The recent attempt to describe the low-spin excitations in some even Ge and Se nuclei self-consistently by symmetry-projected Hartree-Fock-Bogoliubov-type quasi-particle determinants (EXCITED VAMPIR model) is extended to the high-spin states in the 68Ge and 72Se isotopes. The shape coexistence explaining the properties of the low-spin states persists and manifests itself in the structure of the investigated high-spin bands. The coexistence of oblate and prolate deformed mean fields, dynamical changes of the hexadecapole deformations, and special alignments of the nucleons occupying the g 9 2 spherical shell lead to the prediction of a rather complex feeding pattern for the yrast band, including strong M1,ΔI = 0 transitions, some anomalous small intraband B(E2) transitions, and a particular trend of the g-factor values for different calculated bands. Though a couple of discrepancies remain, the overall agreement with the available experimental results is fairly good.

A fully microscopic study of the proton radiative capture on 19 F with excitation of the multipole resonances in 20 Ne is presented. The calculation makes use of a version of the Feshbach formalism of nuclear reactions. The bound states are described as linear combinations of angular momentum projected one particle — one hole configurations in a deformed Hartree-Fock basis. The agreement for the 90° yields with the experimental data is excellent. Gross features as well as intermediate structures are nicely reproduced.

Multi-Configuration-Hartree-Fock (MCHF) calculations with angular momentum projection before the variation of the internal degrees of freedom have been performed for the nuclei20Ne and22Ne. This procedure yields different correlated intrinsic states for the different members of a rotational band. Thus the angular momentum dependence of correlations has been studied. Experimentally, the ground state spectra of20Ne and22Ne show properties similar to the phase transitions observed in some rare earth nuclei which have been well reproduced through the present calculations. The calculated spectra andBE2-values show a significant improvement compared to the ones obtained by variation before the angular momentum projection is effected.

Competition of strong and decoupling is studied systematically for odd-odd mass nuclei as a function of the position of the Fermi surfaceseF(π) andeF(υ) and the deformationsβ andγ. As an example we use80Br. For a prolate deformation the odd proton likes to decouple while the odd neutron couples strongly. This favours a ground state angular momentum of 5 to 6. The strong coupling is described by the admixture of ad5/2 nucleon. Only for the neutrons the admixture is so appreciably that it affects the spectrum and improves the agreement between theory and experiment for80Br.

The Hartree-Fock-Bogoliubov approach with spin and number projection before the variation is extended to the description of non-yrast states. For this purpose, projected Hartree-Fock-Bogoliubov determinants, which are Schmidt-orthogonalized with respect to all the solutions already obtained for the angular momentum considered, are used as test wave functions. This procedure allows the successive construction of an optimal configuration space for the A -nucleon problem, in which finally the residual interaction can be diagonalized. Furthermore, it can account for even drastic changes of the internal structure with increasing excitation energy. As a first test, the method is applied to the four lowest I π = 0 + states in 50 Ti. Considerable improvements with respect to less sophisticated calculations are obtained.

We have calculated the 0νββ and 2νββ decay rates of the transition 76 Ge(0 + 1 ) → 76 Se(0 + 1 ). We took into account a relativistic correction to the nuclear current including weak magnetism. The nuclear two-body transition operator for the 0νββ decay originating from this correction acquires a finite range due to the finite extension of the nucleon and the short-range NN correlations reduce its matrix element only moderately. The contribution from this “second-forbidden” transition plays a dramatic role in the 0νββ decay caused by a specific admixture of a right-handed leptonic current because of the high momentum of the virtual neutrino exchanged between nucleons and systematic cancellations in “allowed” and other “second-forbidden” 0νββ transitions. A new limit on the right-handed current coupling strength |〈 η 〉| −8 was obtained, which is more stringent by an order of magnitude than that obtained recently by the Osaka group.

In the present paper a general survey of the mathematical formalism for microscopic nuclear structure calculations in configuration spaces consisting of arbitrary spin- and number-projected Hartree-Fock-Bogoliubov---type quasiparticle determinants is given. On the basis of this formalism, various levels of approximation are then discussed. These lead to a number of microscopic nuclear structure models in between the standard Hartree-Fock-Bogoliubov theory and the complete diagonalization of a given effective many nucleon Hamiltonian. For all these models variational equations are derived and possibilities for their numerical application are estimated. The second part of the present series of two papers will then present initial results of the applications of the simplest of these models to several nuclei in various mass regions.

The intrashell quartet states proposed earlier on the basis of intrinsic multiconfiguration Hartree-Fock calculations are reexamined for /sup 20/Ne and /sup 24/Mg in angular momentum projection formalism and supports in favor of the existence of such states are obtained. Quartet correlation in excited states of /sup 22/Ne has also been investigated. (AIP)

Opposite to the strong-coupling scheme, in which the back bending effect is discussed as an effect of the Coriolis force in the rotating body fixed frame, we have studied the effect in lab coordinates, in which a Coriolis force does not exist. For this purpose exact solutions of the ( i 13 2 ) N ν problem have been weakly coupled to a rigid rotor core spectrum. In addition to the usual quadrupole coupling also a pairing type interaction was included, which allows for the exchange of neutron pairs between the core and the valence system. It was found that this pairing field has drastic effects on the results and should hence not be neglected.

The solution of the Hartree-Fock-Bogoliubov problem with symmetry-projection before the variation as it has been formulated within the VAMPIR approach leads to symmetry-dependent and, in general, essentially complex mean fields are studied. Tamm-Dancoff as well as random-phase-like equations are derived and discussed. It is shown that for the case of time reversal invariant and axially symmetric Hartree-Fock-Bogoliubov transformations the resulting random-phase equations can be brought into a form involving only real matrices. Advantages and shortcomings with respect to other approaches are investigated.

Parameter-free, microscopic predictions are presented for the B (M1)↑ strength distributions in some deformed even-even nuclei of the mass A = 130 region. An extensive fragmentation of the M1 strength, being predominantly of isovector character, is obtained. The major strength is typically split into two regions around 4.0 and 5.0 MeV with the largest transitions having B ( M 1)↑ values of about 1.5–3.0 μ 0 2 . The total strenghts vary from 6.0 to 9.0 μ 0 2 , a value obtained usually for a single collective 1 + state of phenomenological models. The predicted B ( M 1)↓ branching from the 1 + states is sensitively state dependent, the branching to other than the 0 1 + and 2 1 + states being anywhere between 10% and 200% of the joined intensity to these two lowest levels.

We present a systematic study of even-even nuclei in the mass A = 130 region using a microscopic nuclear structure model, in which the nuclear wave functions are approximated by linear combinations of number- and spin-projected zero-quasiparticle and two-quasipartile determinants obtained from a self-consistent Hartree-Fock-Bogoliubov mean field. With a fixed hamiltonian, being a slightly renormalized Brueckner G-matrix based on the Bonn potential the model is able to reproduce the main trends of empirical energy and electromagnetic properties of the transitional, soft nuclei in the region. The empirical nature of the crossing ( πh 11 2 ) 2 and ( vh 11 2 ) 2 superbands in Ce and Ba isotopes, respectively, is confirmed. The calculations suggest a particular interpretation of the negative-parity bands observed for 128Ba and 124Xe. Furthermore a rich variety of empirically unknown excited 2 q.p. bands with positive parity is predicted to exist throughout the region. We suggest an empirical method to deduce the low-end termination of such bands based on comparisons of microscopic electric quadrupole moments to those of the rotation model. The model fails to account for the low-lying quasi-gamma bands in Ce and Ba isotopes but indicates a strong contribution of proton 2 q.p. excitations for this band in Xe isotopes.

The Hartree-Fock-Bogoliuboy problem with restoration of the broken symmetries before the variation as it has been formulated recently within the VAMPIR and EXCITED VAMPIR approaches is studied in more detail. Special emphasis is put on the implications of various symmetry restrictions imposed on the quasiparticle transformations. It is shown that the use of essentially complex transformations provides a possibility to account for time-odd unnatural parity pairing correlations in the wavefunctions without being forced to give up axial and time-reversal symmetry. It is illustrated that together with parity and neutron-proton mixing the use of such complex mean fields extends the applicability of the VAMPIR and EXCITED VAMPIR approaches to states with arbitrary spin and parity in both doubly even and doubly odd nuclear systems. The corresponding variational equations are explicitly evaluated and the numerical feasibility of their solution is investigated.

The Generator-Coordinate-Method is applied to describe yrast-bands and excited vibrational bands of pairing-vibrational orβ-vibrational type in deformed rare earth nuclei. The model wave functions are supplied by the Pairing+Quadrupole-Model with angular momentum — and particle number projection. Results are shown for the nucleus170Yb. The question if backbending may be produced by the intersection of the ground and the neutron pairing vibrational band is discussed.

In addition to the quartet excitations between major oscillator shells (intershell quartet states), we propose quartet excitations within a shell (intrashell quartet states). A phenomenological quartet shell model is devised to predict the energies of these states. Evidence for such excitations is also found in a microscopic multiconfiguration Hartree-Fock calculation.

The multi-configuration Hartree-Fock (MCHF) theory, a variational method for the self-consistent solution of the nuclear many-body problem with the inclusion of correlations, is applied to 12 C. The force is the Yale potential in the form of the Yale-Shakin matrix elements. 40 different configurations are included. The gain in the total binding energy compared to HF in the same single-particle basis and with the same force is 1.20 MeV. The MCHF theory yields the first excited 0 + state and the first excited 2 + state in good agreement with the experimental data.

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.

The Multi Configuration Hartee-Kock (MCHF) theory is applied to the ground state of 28 Si. The single particle basis states are the 0 d 5 2 and 1 s 1 2 orbitals. A comparison is made with the results of a Hartree-Fock calculation performed in the same space.

In the last years a family of microscopic nuclear structure models based on Hartree-Fock-Bogoliubov mean fields has been developed by the authors, which are known as the MONSTER and the VAMPIR approaches1. These models are supposed to be an alternative to the ‘exact’ shell model configuration mixing approach, which is limited to too small single particle basis spaces in order to be useful for realistic applications in heavy nuclei. We will briefly describe here the theoretical background and the numerical implementation of these models.