Topics Concerning the Quadrupole-Quadrupole Interaction

We address some properties of the quadrupole-quadrupole ($Q \cdot Q$) interaction in nuclear studies. We first consider how to restore $SU(3)$ symmetry even though we use only coordinate and not momentum terms. Using the Hamiltonian $H=\sum_i (p^2/2m + m/2 \omega^2 r_i^2) -\chi \sum_{i < j}Q(i) \cdot Q(j) - \chi/2 \sum_i Q(i) \cdot Q(i)$ with $Q_{\mu}=r^2 Y_{2,\mu}$, we find that only 2/3 of the single-particle splitting ($\epsilon_{0d}-\epsilon_{1s}$) comes from the diagonal term of $Q \cdot Q$ -the remaining 1/3 comes from the interaction of the valence nucleus with the core. On another topic, a previously derived relation, using $Q \cdot Q$, between isovector orbital $B(M1)$ (scissors mode) and the ``difference'' ($B(E2, isoscalar)-B(E2, isovector)$) is discussed. It is shown that one needs the isovector $B(E2)$ in order that one get the correct limit as one goes to nuclei sufficiently far from stability so that one subshell (neutron or proton) is closed.