The linked-cluster expansion for the transition amplitude for deuteron stripping reaction: W. F. Junkin and F. Villars. Laboratory for Nuclear Science, M.I.T., Cambridge, Massachusetts

Abstract

At the Technische Hochschule in Darmstadt the group of Richter and coworkers found in 1983/84 in deformed rare earth nuclei low-lying isovector 1+ states. Such states have been predicted in the generalized Bohr-Mottelson model and in the interacting boson model no. 2 (IBA2). In the generalized Bohr-Mottelson model one allows for proton and neutron quadrupole deformations separately. If one includes only static proton and neutron deformations the generalized Bohr-Mottelson model reduces to the two rotor model. It describes the excitation energy of these states in good agreement with the data but overestimates the magnetic dipole transition probabilities by a factor 5. In the interacting boson model (IBA2) where only the outermost nucleons participate in the excitation the magnetic dipole transition probability is only overestimated by a factor 2. The too large collectivity in both models results from the fact that they concentrate the whole strength of the scissors vibrations into one state. A microscopic description is needed to describe the spreading of the scissors strength over several states. For a microscopic determination of these scissors states one uses the Quasi-particle Random Phase Approximation (QRPA). But this approach has a serious difficulty. Since one rotates for the calculation the nucleus into the intrinsic system the state corresponding to the rotation of the whole nucleus is a spurious state. The usual procedure to remove this spuriosity is to use the Thouless theorem which says that a spurious state created by an operator which commutes with the total hamiltonian (here the total angular momentum, corresponding to a rotation of the whole system) produces the spurious state if applied to the ground state. It says further the energy of this spurious state lies at zero excitation energy (it is degenerate with the ground state) and is orthogonal to all physical states. Thus the usual approach is to vary the quadrupole-quadrupole force strength so that a state lies at zero excitation energy and to identify that with the spuríous state. This procedure assumes that a total angular momentum commutes with a total hamiltonian. But this is not the case since the total hamiltonian contains a deformed Saxon-Woods potential. Thus one has to take care explicitly that the spurious state is removed. This we do in our approach by introducing Lagrange multipliers for each excited states and requesting that these states are orthogonal to the spurious state which is explicitly constructed by applying the total angular momentum operator to the ground state. To reduce the number of free parameters in the hamiltonian we take the Saxon-Woods potential for the deformed nuclei from the literature (with minor adjustments) and determine the proton-proton, neutron-neutron and the proton-neutron quadrupole force constant by requesting that the hamiltonian commutes with the total angular momentum in the (QRPA) ground state. This yields equations fixing all three coupling constants for the quadrupole-quadrupole force allowing even for isospin symmetry violation. The spin-spin force is taken from the Reid soft core potential. A possible spin-quadrupole force has been taken from the work of Soloviev but it turns out that this is not important. The calculation shows that the strength of the scissors vibrations are spread over many states. The main 1+ state at around 3 MeV has an overlap of the order of 14 % of the scissors state. 50% of that state are spread over the physical states up to an excitation energy of 6 MeV. The rest is distributed over higher lying states. The expectation value of the many-body hamiltonian in the scissors vibrational state shows roughly an excitation energy of 7 MeV above the ground state. The results also support the experimental findings that these states are mainly orbital excitations. States are not very collective. Normally only a proton and neutron particle-hole pair are with a large amplitude participating in forming these states. But those protons and neutrons which are excited perform scissors type vibrations.