Four nuclei which are proved to be $2\nu \beta \beta$ emitters (${}^{76}\mathrm{Ge}$, ${}^{82}\mathrm{Se}$, ${}^{150}\mathrm{Nd}$, ${}^{238}\mathrm{U}$), and four suspected, due to the corresponding $Q$-values, to have this property (${}^{148}\mathrm{Nd}$, ${}^{154}\mathrm{Sm}$, ${}^{160}\mathrm{Gd}$, ${}^{232}\mathrm{Th}$), were treated within a proton-neutron quasiparticle random phase approximation $\left(\mathrm{pnQRPA}\right)$ with a projected spherical single particle basis. The advantage of the present procedure over the ones using a deformed Woods-Saxon or Nilsson single particle basis is that the actual $\mathrm{pnQRPA}$ states have a definite angular momentum while all the others provide states having only $K$ as a good quantum number. The model Hamiltonian involves a mean field term yielding the projected single particle states, a pairing interaction for alike nucleons and a dipole-dipole proton-neutron interaction in both the particle-hole $\left(ph\right)$ and particle-particle $\left(pp\right)$ channels. The effect of nuclear deformation on the single beta strength distribution as well as on the double beta Gamow-Teller transition amplitude $\left(M_{\mathrm{GT}}\right)$ is analyzed. The results are compared with the existent data and with the results from a different approach, in terms of the process half-life $T_{1∕2}$. The case of different deformations for mother and daughter nuclei is also presented.

• Received 13 January 2004

DOI:https://doi.org/10.1103/PhysRevC.69.064321