In this study, for the first time, the Translational and Galileo Invariant (TGI-) Quasiparticle Phonon Nuclear Model (QPNM) has been formulated and a computer code for this model has been written in FORTRAN programming language, in order to theoretically explain the electrical dipole excitations (E1) observed in odd-A deformed nuclei located in different mass regions of the periodic table and to reveal their systematic properties. The broken translational and Galileo symmetry of the QPNM Hamiltonian of odd-mass nuclei due to the mean area approach has been restored by means of Pyatov-Salamov restoration method.

The fine structure and total properties of E1 excitations in 2-4 MeV in ¹³⁹La, ¹⁴¹Pr,¹⁴³Nd, ^151,153Eu, ^155,157Gd, ¹⁵⁹Tb, ^161,163Dy, ¹⁶⁵Ho, ¹⁶⁷Er, ¹⁶⁹Tm, ¹⁸¹Ta, ¹⁷⁵Lu, ²³⁵U, and ²³⁹Pu,whose low energy (2-4 MeV) dipole spectra were experimentally determined, have been investigated theoretically in the framework of the formulated TGI-QPNM and the obtained results have been compared with the experimental data. In addition, the fine structure and total properties of E1 excitations in the energy range of 2-20 MeV have been calculated for the first time by using the TGI-QPNM for selected ^133-141La, ^153-159Eu,^151-161Gd, ^155-165Dy, ^155-169Ho, ^157-167Er and ^167-179Lu odd-mass isotopes in rare earth region and for selected ^229-233Th and ^233-239U odd-mass isotopes in actinide region, in order to determine whether E1 excitations in different energy regions (2-4 MeV, 4-10 MeV and 10-20 MeV) exhibit systematic properties in odd-mass mass nuclei.