Are the low-lying isovector 1 + states scissor vibrations?

Abstract

In 1984 Richter and coworkers detected in deformed nuclei in electron scattering low-lying isovector 1 + states. In the same year Dennis Hamilton and coworkers from the University of Sussex found in (n,γ)-reactions low-lying 2 + isovector states in spherical nuclei. Such states have been predicted in several theoretical models. The first prediction was work of the author in 1966 where he extended the Bohr-Mottelson-model to include independent collective motions of the proton and the neutron distributions in nuclei. Both motions are coupled by the symmetry energy which dislikes demixing of the protons and neutrons. Thus one obtains the usual low-lying isoscalar quadrupole vibrations and rotations, but in addition also isovector states in spherical and deformed nuclei. In these states protons and neutrons move out of phase. The 1 + states found by Richter and coworkers in deformed nuclei can then be interpreted as vibrations of the deformed proton and neutron shapes in a scissor like way. In spherical nuclei the data of Dennis Hamilton and coworkers can be explained as harmonic quadrupole vibrations of protons and neutrons out of phase around the spherical equilibrium. We show here that this collective interpretation can very nicely reproduce the energies of the isovector 1 + states in deformed nuclei and of the isovector 2 + states in spherical nuclei. If the parameters are fixed to the low-lying isoscalar quadrupole rotations or vibrations the energies of the isovector states can be predicted without adjusting an additional parameter. Although the energies are reproduced very nicely one finds in deformed nuclei that the B( M1;0 +→1 +)-transition probability is too collective by a factor 5. In this collective description the M1-strength is concentrated in one state. In reality this isovector 1 + state contains only part of the Oπω-two-quasi-particle exitations. The rest of the strength is shifted to higher energies. A microscopic description using the quasi-particle random phase approximation or MONSTER can reproduce the exitation energies and transition probabilities correctly. It turns out to be important that in deformed nuclei the spurious state connected with the rotation of the whole system has to be put to zero energy by adjusting the parameter of the scalar part of the nucleon-nucleon interaction. In addition, one has to take care that the low-lying isovector 1 +-state is orthogonal to the spurious state |S>. If we request that the Hamiltonian is rotational invariant, that means that it commutes with the angular momentum operator, we can determine the nucleon-nucleon interaction from the given one-body part. For the one-body Hamiltonian we use the kinetic energies of the single nucleons, a deformed Saxon-Woods potential with spin-orbit coupling and slightly different shapes for protons and neutrons. By the quasi-particle transformation we include also a pairing interaction into the one-body piece. With the requirement that the Hamiltonian should commute with the total angular momentum, one can determine the many-body Hamiltonian parameter free if the Saxon-Woods potential and the pairing force are given. This symmetry restoring interaction is very nicely reproducing the energies and the magnetic transition probabilities from the ground state to these low-lying isovector 1 + states in the rare earth nuclei. We furthermore apply this formalism to the Ti-isotopes. There we get also similar satisfactory results using MONSTER and VAMPIR. The theoretical results indicate that the low-lying isovector 1 + states in deformed nuclei are in average half of orbital and half of spin character. A comparison of the excitation of these states with electron scattering and with proton scattering suggest that they are of more purer orbital character. A closer analysis of these measurements shows that they are not contradicting to a spin and an orbital character of these states which is roughly half and half. For the MONSTER we compare also M1-form factors with the data and find excellent agreement.