The strong variation in energy of the first excited $0^{+}$ state in the even-even Ge isotopes is investigated with pairing type interactions in various model spaces. Although it had been possible in previous work to describe this variation in a BCS–random-phase-approximation (RPA) calculation in the neutron configuration only, the corresponding exact diagonalization disagrees with the BCS-RPA results and experiment. It is furthermore shown that the behavior of the first excited $0^{+}$ state cannot be obtained by a reasonable variation of parameters in the exact neutron pair calculations. This suggests that the model space be enlarged to include proton, neutron, and proton-neutron pairing. The construction of the corresponding basis is then simplified by performing a boson mapping and employing the ideal collective boson basis. This, however, may lead to the occurrence of spurious states, even at fairly low energies, and three separate methods by means of which they can be identified are discussed. Although the enlarged model space gives a desired lowering of excitation energy of the first excited $0^{+}$ states in an exact diagonalization, the strong experimental variation could not be achieved with this simple interaction. Nevertheless, the analysis yields further and new insight into applications of boson mappings, specifically about how and when spurious states may be identified in calculations which are performed in a truncated space not invariant under a given collective algebra.

• Received 22 February 1994

DOI:https://doi.org/10.1103/PhysRevC.50.177