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Description of electromagnetic and favored $α$-transitions in heavy odd-mass nuclei

We describe electromagnetic and favored α-transitions to rotational bands in odd-mass nuclei built upon a single particle state with angular momentum projection $Ω=\frac{1}{2}$ in the region $88 \le Z \le 98$. We use the particle coupled to an even-even core approach described by the Coherent State Model (CSM) and the coupled channels method to estimate partial $α$-decay widths. We reproduce the energy levels of the rotational band where favored $α$-transitions occur for 26 nuclei and predict B (E2) values for electromagnetic transitions to the bandhead using a deformation parameter and a Hamiltonian strength parameter for each nucleus, together with an effective collective charge depending linearly on the deformation parameter. Where experimental data is available, the contribution of the single particle effective charge to the total B (E2) value is calculated. The Hamiltonian describing the $α$- nucleus interaction contains two terms, a spherically symmetric potential given by the double-folding of the M3Y nucleon-nucleon interaction plus a repulsive core simulating the Pauli principle and a quadrupole-quadrupole (QQ) interaction. The $α$-decaying state is identified as a narrow outgoing resonance in this potential. The intensity of the transition to the first excited state is reproduced by the QQ coupling strength. It depends linearly both on the nuclear deformation and the square of the reduced width for the decay to the bandhead, respectively. Predicted intensities for transitions to higher excited states are in a reasonable agreement with experimental data. This formalism offers a unified description of energy levels, electromagnetic and favored $α$-transitions for known heavy odd-mass $α$-emitters.