Description of energy spectrum and signature inversion for \({}^{160}\hbox {Tm}\) and \({}^{161}\hbox {Tm}\) isotopes using the projected shell model

The projected shell model (PSM) was employed to study the signature inversion of \({}^{160}\hbox {Tm}\) and \({}^{161}\hbox {Tm}\) isotopes. The Hamiltonian of these isotopes were constructed and solved to obtain the energy levels and the \(\gamma \)-ray transition energies. The rotational bands of the odd–odd \({}^{160}\hbox {Tm}\) and odd–even \({}^{161}\hbox {Tm}\) isotopes of thulium nucleus were constructed based on the \(\pi h\frac{11}{2} \otimes \upsilon i\frac{13}{2}\) Nilsson configuration to calculate the kinetic and dynamic moments of inertia, \(\gamma \)-ray transition energy (\(\Delta {I}=1\)) and the staggering parameter (S(I)). Yrast band with two signatures for both isotopes have been extracted and compared with the experimental data. The anomalous splitting and signature inversion occurred due to the quadruple deformation in both isotopes. Analysis of level staggering and signature inversion show that signature inversion for the \({}^{160}\hbox {Tm}\) isotope occurs at low spin region, while for the \({}^{161}\hbox {Tm}\) isotope, it occured in the intermediate spin region. As expected, zig-zag behaviour and phase transition, as well as signature of energy splitting are clearly indicated in figures showing variation of S(I) and \(\Delta E\) with spin. This behaviour is more visible for the odd–even \({}^{161}\hbox {Tm}\) isotope than for the odd–odd \({}^{160}\hbox {Tm}\) isotope. Good agreement between the calculated results and experimental data confirms the success of the PSM model.