Study ofpf-shell nuclei with a symmetry conserving generator coordinate method

Bands based on the 0⁺ ground state and the first excited 0⁺ pairing vibrational state of⁴⁸Ti,⁵²Cr and⁵⁶Fe are studied with the generator coordinate method. The generating wave functions for each value of the angular momentumJ are angular momentum and particle number projected selfconsistent Hartree-Fock-Bogoliubov states where the constrained amount of pairing correlations serves as the generator coordinate. The interaction is given by reaction matrix elements derived from the Hamada-Johnston force. The basis includes the four lowest oscillator shells. The excitation energies of the pairing vibrational states can be reproduced fairly well by the present choice of the generating wave functions, whereas the ground band is not much improved compared to projected Hartree-Bogoliubov calculations. We find that the strength of the pairing correlations in the 0⁺ and 2⁺ states of the ground state and the pairing vibrational bands can be related to data of two-particle transfer reactions. The angular momentum dependence of the pairing correlations and of the moments of inertia are studied. The results show that for a strongly paired ground state the ground state band and the pairing vibrational band intersect. This may produce in the yrast band the anomaly of the moment of inertia known from rare earth nuclei.     

Microscopic description of collective motion in pf shell nuclei

Rotational and vibrational excitations in pf shell nuclei are studied by means of the generator coordinate method. The generator coordinates are the pairing energies and the quadrupole moments of constrained Hartree-Bogoliubov states, projected onto good angular momenta and particle numbers. The Kuo interaction and the one modified by McGrory are used. The vibrational character of the yrast energies appears to be produced by mixing prolate and oblate wave functions. Pairing correlations are essential for this mixing. In contrast to the yrast states the excitation energies of the higher states depend strongly on the interaction used. They show good agreement with experiment, particularly in the case of ⁴⁸Ti with the Kuo force. The calculated B(E2) values exhibit a rotational band structure in general, even if the energies look more vibrational. The force dependence of the excitation energies can qualitatively be understood by inspection of the intrinsic energy surface.     

Quartet states in the sd and fp shells

Two-proton-two-neutron correlations are studied in psd and fp shell nuclei in the frame of the aligned scheme approximation. Four-particle rotational states are obtained in terms of linear combinations of projected aligned Slater determinants and the resulting excitation energies and wave functions are compared with those of elaborate shell-model calculations. The close agreement obtained both for single j-shell and complete shell configuration spaces shows that rotational or quasi-rotational states are almost entirely generated by aligned configurations. The lowest four-particle states obtained in one j-shell can be explained as the rotational spectrum of an oblate intrinsic state. The lowest four-particle states of ¹⁶O, ²⁰Ne, ⁴⁴Ti and ⁶⁰Zn, calculated in a major shell, correspond to the rotational spectrum of a prolate intrinsic state and fit the existing data. Higher states are the result of a mixed deformation.     

The isospin mixing and the superallowed Fermi beta decay

In the present work, the isospin admixtures in the nuclear ground states of the parent nuclei and isospin structure of the isobar analog resonance (IAR) states have been investigated by studying the 0^?+???0^?+? superallowed Fermi ?? decays using Pyatov??s restoration method. Within the random phase approximation (RPA), in this method, the effect of isospin breaking due to the Coulomb forces has been evaluated, taking into account the effect of pairing correlations between nucleons.     

Second Coulomb energy differences of isospin triplets in the 2s-1d shell

Second Coulomb energy differences, which in the present case are proportional to the tensor Coulomb energy, are calculated for 0⁺, T = 1 ground states in the region 18 ≦ A ≦ 42 using a shell model that includes a pairing interaction. The calculation is done with a mathematical formalism that includes p-n pairs as well as p-p and n-n pairs. Besides an enhancement of proton-pair Coulomb energies, the pairing interaction is responsible for lowering the Coulomb energy of N = Z members of isospin triplets and also gives rise to an important term in the second energy difference. Using pairing strengths derived from fitting energy levels for mass-18 and mass-42 nuclei, results of the calculation reproduce experimental second energy differences extremely well.     

The role of isospin on the N=Z line

The role played by isospin in nuclear structure phenomena encountered on the N=Z line is discussed. New results on Coulomb energy differences (CED) at high spin for odd-A nuclei in the f _7/2 shell are presented and interpreted in the framework of a simple Cranked Shell Model treatment involving an exact numerical diagonalisation. Results for the CED between the A=46 even-even mirror pairs are also discussed. The CED between the T=1 states in N=Z odd-odd nuclei and their isobaric analogues are suggested as a possible probe of np pairing on the N=Z line. First results from a numerical diagonalization of IBM-4 are cited.     

An investigation of the influence of the pairing correlations on the properties of the isobar analog resonances inA = 208 nuclei

Within the quasi-particle random phase approximation (QRPA), the method of the self-consistent determination of the isovector effective interaction which restores a broken isotopic symmetry for the nuclear part of the Hamiltonian is given. The effect of the pairing correlations between nucleons on the following quantities were investigated for theA = 208 nuclei: energies of the isobar analog 0⁺ states, the isospin admixtures in the ground state of the even-even nuclei, and the differential cross-section for the²⁰⁸Pb(³He,t)²⁰⁸Bi reaction atE(³He)=450 MeV. Both couplings of the excitation branches withT ^z = T⁰ ± 1, and the analog state with isovector monopole resonance (IVMR) in the quasi-particle representation were taken into account in our calculations. As a result of these calculations, it was seen that the pairing correlations between nucleons have no considerable effect on theT = 23 isospin admixture in the ground state of the²⁰⁸Pb nucleus, and they cause partially an increase in the isospin impurity of the isobar analog resonance (IAR). It was also established that these correlations have changed the isospin structure of the IAR states, and shifted the energies of the IVMR states to the higher values.     

Pairing vibrational states and the generator coordinate method

The pairing vibrational states and the two-neutron transfer cross sections between these states are calculated in Ni, Sn and Pb isotopes by the generator coordinate method (GCM). The particle number fluctuation of the BCS functions is handled by projecting in a good approximation on sharp particle numbers. The results agree quite well with the experimental data.     

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.     

Angular-momentum-projected cranked hfb approach to the study of nuclear rotations

Employing a pairing-plus-quadrupole interaction hamiltonian and projecting out good angular momentum states from the cranked Hartree-Fock-Bogoliubov (CHFB) intrinsic wave functions the yrast spectra of ¹⁵⁸Dy and ¹⁶⁸Yb are calculated up to moderately high spins (I_max = 16) as to include the backbending region. Then the variation of pairing correlation, g-factor and rotational alignment of neutron spin as a function of total angular momentum is studied. The effect of particle number projection on the spin-projected CHFB wave functions is also investigated and is found to be unimportant for the calculation of g-factors. On the other hand, corrections of the excitation energies for number fluctuations in the CHFB wave functions are essential. Furthermore, looking at the distribution of the total projection quantum number K in various cranking wave functions we are able to throw some light on the K ≠ 0 nature of the aligned s-band.A variation-after-spin projection calculation strictly for the axial shape, without cranking, is also carried out for both the nuclei considered here. In the low-spin region this numerically “cheaper” scheme produces energy spectra similar to that of the CHFB method, and may thus be used to readjust the interaction parameters.     

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.     

Coupled-channels analysis of energy-dependent isospin violation in the 12C(d, α)10B(1.74 MeV,T = 1) reaction

We have studied the isospin non-conserving ¹²C(d, α)¹⁰B(1.74 MeV, 0⁺, T= 1) reaction at several incident energies of 9 ≦ E_d ≦ 16 MeV in terms of a coupled-channels method. The reaction processes involved in the present analysis are the successive single-nucleon pick-up processes as well as the inelastic scattering of deuterons from ¹²C. It is assumed that the isospin violation should occur in the intermediate mirror cluster states of ³He + ¹¹B and t + ¹¹C, due to the Coulomb interaction. The calculation reproduces fairly well the observed features of the reaction, i.e. the decreasing cross section with increasing incident energy, and the variation of the angular distribution. We also note that the calculation shows the energy-dependent localization of isospin violation in the angular momentum space, i.e. a specifically narrow localization at the lower incident energies studied and its broadening at the higher energies. This fact is associated with the variation of the angular distribution from a forward-backward symmetry at the lower incident energies to a forward peak at the higher energies.     

High spin states and pairing vibrations

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.     

Isospin mixing between T=0 and 1 states in the 1p-shell

The effect of using different proton and neutron wave functions to evaluate matrix elements of a charge independent nucleon-nucleon interaction is examined. It is shown that this is predominantly an isovector effect and in the 1p-shell can give rise to large off diagonal matrix elements (? 100 keV). The magnitude of these matrix elements are extremely sensitive to the detailed structure of the single particle wave functions. Until this effect is satisfactorily taken into account it will be difficult to demonstrate the need for a change symmetry breaking nucleon-nucleon interaction from a measurement of isospin mixing between T=0 and 1 states.     

The strange backbending behaviour in the Yb isotopes

The rotational energies and the backbending behaviour of ^166,168,170Yb are calculated utilizing angular momentum and particle number projected deformed BCS trial wave functions. The anomaly of the moment of inertia can be reproduced quantitatively with the moment of inertia of an inert core as the only free parameter. A possible explanation for the strange behaviour in the Yb isotopes is given.     

High-K multi-particle bands and pairing reduction in 254No

The multi-particle states and rotational properties of the two-particle bands in 254No are investigated by the cranked shell model with pairing correlations treated by the particle number conserving method.The rotational bands on top of the two-particle K^π=3^+,8^− and 10^+ states and the pairing reduction are studied theoretically in 254No for the first time.The experimental excitation energies and moments of inertia of the multi-particle states are reproduced well by the calculations.Better agreement with the data is achieved by including the high-order deformation ε6,which leads to enlarged Z=100 and N=152 deformed shell gaps.An increase of J¹ in these two-particle bands compared with the ground state band is attributed to the pairing reduction due to the Pauli blocking effect.     

Calculated nearly degenerate isospin doublets

It is noted that in several calculations of the odd J states of N = Z nuclei ³⁶Ar, ⁴⁴Ti, and ⁵²Fe, there is a near degeneracy of the lowest T = 0 and lowest T = 1 states, for several J values. Qualitative physical reasons for this behavior are given and the Coulomb mixing is estimated.     

Core-excited states of Fe from the Fe(p, t)Fe reaction at 52 MeV

The perturbation theory for projected states is applied to the two-level pairing force model. Both approximate and exact forms of number projection are considered. The results are compared with the exact ones and with ordinary perturbation theory based on BCS wave functions without projection.     

Isospin mixing observed in the reaction 12C(6Li, α)14N

The reaction ¹²C(¹²Li, α)¹⁴N was studied to investigate the isospin mixing of high-lying levels in ¹⁸F. Excitation functions and angular distributions of the α-transitions to the ground, first and second excited states in ¹⁴N were measured for bombarding energies from 3.2 to 8.0 MeV. The isospin-forbidden cross section for the excitation of the lowest T = 1 state in ¹⁴N at 2.31 MeV was found to lie between 1–2 % of that of the allowed transitions. A partial wave analysis of the α₁ angular distribution data revealed a strong resonance with J^π = 2⁺ at E_x = 15.99 MeV. Arguments are presented which tentatively identify this resonance as being due to two close-lying 2⁺ levels with different isospin.