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Inspired by the recent experimental data (Phys. Lett. B {\bf 675} (2009)420), we extend the triaxial projected shell model approach to study the$\gamma$-band structure in odd-mass nuclei. As a first application of the newdevelopment, the $\gamma$-vibrational structure of $^{103}$Nb is investigated.It is demonstrated that the model describes the ground-state band andmulti-phonon $\gamma$-vibrations quite satisfactorily, supporting theinterpretation of the data as one of the few experimentally-known examples ofsimultaneous occurrence of one- and two-$\gamma$-phonon vibrational bands. Thisgeneralizes the well-known concept of the surface $\gamma$-oscillation indeformed nuclei built on the ground-state in even-even systems to$\gamma$-bands based on quasiparticle configurations in odd-mass systems.

Multi-phonon $\gamma$-vibrational bands in odd-mass nuclei studied by triaxial projected shell model approach