Background: The scissors mode is a rotational mode of isovector character in deformed nuclei. Together with the isoscalar rotation, it is the prominent collective mode at low excitation energies.

Purpose: We use a simple field theory model to investigate the role of the nucleon spin for the magnetic sum rules associated with the low-lying scissors mode. Special emphasis is put on the coupling of the spin part of the $M1$ operator to the scissors mode.

Methods: We apply the mean-field approximation and random phase approximation to a model Hamiltonian based on a simple quadrupole-quadrupole interaction. The effects of the spin-orbit interaction are included in the mean-field Hamiltonian. Ward-Takahashi relations are used to derive the $M1$ sum rules.

Results: The presence of the spin-orbit interaction leads to interference terms in the inverse energy weighted sum rule. It is shown that the low-lying scissors mode, which is generated by the isovector combination of proton and neutron total angular momenta, gives the main contribution to the spectral sum for this case.

Conclusions: The basic concept of the scissors mode as an isovector vibrational rotation remains valid in the presence of the nucleon spin. The inverse energy weighted sum rule, however, receives nontrivial modifications because of interference terms.

• Received 29 December 2013

DOI:https://doi.org/10.1103/PhysRevC.89.024314