The $0^{+} \leftrightarrow 0^{−}$ first-forbidden 𝛽 decay transitions have been investigated for some spherical nuclei. The theoretical framework is based on a proton–neutron quasiparticle random phase approximation (pnQRPA). The Woods–Saxon potential basis has been used in our calculations. The transition probabilities have been calculated within the 𝜉 approximation. The relativistic 𝛽 moment matrix element has been calculated both directly without any assumption and assuming that it is proportional to the non-relativistic one.

The violated supersymmetry property of the pairing interaction between nucleons were restored using the Pyatov method [Pyatov and Salamov, Nucleonica 22, 127 (1977)]. The eigenvalues and eigenfunctions of the restored Hamiltonian with the separable residual Gamow–Teller effective interactions in the particle–hole and particle–particle channels were solved within the framework of proton–neutron quasirandom phase approximation (pnQRPA). The Gamow–Teller resonance energies for ^112-124Sb isotopes and the differential cross-sections for Sn(³He, 𝑡)Sb reactions at 𝐸(³He) = 200 MeV occurring by the excitation of the Gamow–Teller resonance state were calculated. The calculated values were compared with other calculations and the corresponding experimental data.

We try to present the structure of $1^{−}$ excitations in open-shell $^{124,126,128,130}\rm{Te}$ isotopes. Electric dipole states are investigated within a translational and Galilean invariant model. Also, a theoretical description of charge conserving spin-dipole $1^{−}$ excitations is presented for the same isotopes. The energy spectra for both kinds of excitations are analysed in detail. Furthermore, a comparison of the calculated cross-sections and energies with the available experimental data is given.