The generator of the low-lying quadrupole vibration

In the framework of the adiabatic cranking model we analyse the generator of the low-lying quadrupole vibration. In a first step, we show the connection between the cranking approach and the equations-of-motion method used in RPA. Then we calculate the matrix elements of the generator numerically, and use them to find an approximate analytic expression for the operator itself. We find that the low-lying vibration, as well as a high-lying state, which we analysed in the same manner for comparison, has a very regular behaviour with varying particle number. An easy approximate expression for the generator can be given finally, which for doubly magic nuclei even becomes exact. (orig.)