The CSM extension for description of the positive and negative parity bands in even-odd nuclei

A particle-core Hamiltonian is used to describe the lowest parity partner bands $K^{\pi}=1/2^{\pm}$ in $^{219}$Ra, $^{237}$U and $^{239}$Pu, and three parity partner bands, $K^{\pi}=1/2^{\pm}, 3/2^{\pm}, 5/2^{\pm}$, in $^{227}$Ra. The core is described by a quadrupole and octupole boson Hamiltonian which was previously used for the description of four positive and four negative parity bands in the neighboring even-even isotopes. The particle-core Hamiltonian consists of four terms: a quadrupole-quadrupole, an octupole-octupole, a spin-spin and a rotational $\hat{I}^2$ interaction, with $\hat {I}$ denoting the total angular momentum. The single particle space for the odd nucleon consists of three spherical shell model states, two of positive and one of negative parity. The product of these states with a collective deformed ground state and the intrinsic gamma band state generate, through angular momentum projection, the bands with $K^{\pi}=1/2^{\pm},3/2^{\pm},5/2^{\pm}$, respectively. In the space of projected states one calculates the energies of the considered bands. The resulting excitation energies are compared with the corresponding experimental data as well as with those obtained with other approaches. Also, we searched for some signatures for a static octupole deformation in the considered odd isotopes. The calculated branching ratios in $^{219}$Ra agree quite well with the corresponding experimental data.