In this paper we present a timely application of the proton-neutron deformed quasiparticle random-phase approximation $(pn$-dQRPA), designed to describe in a consistent way the $1^{+}$ Gamow-Teller states in odd-odd deformed nuclei. For this purpose we apply a projection before variation procedure by using a single-particle basis with projected angular momentum, provided by the diagonalization of a spherical mean field plus quadrupole-quadrupole interaction. The residual Hamiltonian contains pairing plus proton-neutron dipole terms in particle-hole and particle-particle channels, with constant strengths. As an example we describe the two-neutrino double-beta $\left(2\nu \beta \beta \right)$ decay of ${}^{150}\mathrm{Nd}$ to the ground state of ${}^{150}\mathrm{Sm}$. The experimental $\left(p,n\right)$ type of strength in ${}^{150}\mathrm{Nd}$ and the $\left(n,p\right)$ type of strength in ${}^{150}\mathrm{Sm}$ are reasonably reproduced and the $2\nu \beta \beta$ decay matrix element depicts a strong dependence upon the particle-particle strength $g_{pp}$. The experimental half-life is reproduced for $g_{pp}=0.05$. It turns out that the measured half-lives for $2\nu \beta \beta$ transitions between other deformed superfluid partners with mass numbers $A=82$,96,100,128,130,238 are reproduced with fairly good accuracy by using this value of $g_{pp}$.

• Received 29 January 2015
• Revised 28 April 2015

DOI:https://doi.org/10.1103/PhysRevC.91.054329