Surface Delta Interaction and CollectiveE2andE3Transitions in Rare-Earth Nuclei

$\gamma$-ray transition probabilities from the ground state into the $\gamma$-vibrational band and the $K=0\mathrm{}$ octupole vibrational band in even-mass rare-earth nuclei are calculated. The surface delta interaction (SDI) and the pairing-plus-quadrupole (PQF) or -octupole force (POF) are utilized as residual interactions. The quasi-particle random-phase approximation (QRPA) is employed in the calculation. Both residual interactions (SDI and PQF) give good agreement with experiment for electric quadrupole transitions, using an effective charge equal to $\frac{Z}{A}$. This value is required in a self-consistency argument given by Mottelson. For octupole transitions, good results are also obtained using the same effective charge. The Tamm-Dancoff approximation is compared with the RPA. It is shown that it does not yield reasonable results when used in this kind of calculation.