Role of Valence Nucleons in Quantum Theory of Backbending Phenomenon within the Framework of the Extended Particles-Plus-Rotor Model

The particle rotor model (PRM) is extended to couple several valence neutrons (nN = 2, 4, 8, 10) to an axially symmetric deformed rotor core containing even number of protons and neutrons and provided with variable moments of inertia (VMI). The considered valence neutrons move in deformed Woods-Saxon potential and confined to an i13/2 subshell with projections Ω = 1/2, 3/2, 5/2 and 7/2. We have been performing all complete Coriolis band mixing calculations, we noticed that the Coriolis interaction is more and more attenuated when more neutrons are added. The dimensions of our Hamiltonian matrices are very large, so we used the Davidson method for diagonalization, to extract the eigenvalues (energies) and the eigenvectors. As a particular example for our explicit numerical calculations, we present in the Appendix the case for two valence particles nN = 2 occupied two Nilsson levels with Ω1 = ±1/2 and Ω2 = ±3/2 (Matrix 4 × 4). For nN = 4 in three levels with Ω1 = ±1/2, Ω2 = ±3/2, Ω3 = ±5/2, the matrix is of order 9 × 9, … etc. It is found that for only two valence neutrons a sharp backbending happens, and when increasing the number of valences neutrons, the backbending decreases, that is a smooth-back respective unbending happen. The proposed model has been applied to study the behavior of the yrast positive parity states in even-even Hafnium isotopes 162–172Hf. The calculated energy levels agree excellently with experimental data up to spin Iπ = 36+. The backbending observed in the yrast, positive parity state in Hf nuclei is attributed to band crossing with rotation aligned i13/2 neutrons bands. The isotopes 162,164Hf show strong backbending and then the backbending become less severe with increasing neutron numbers. The behavior of aligned angular momentum which characterize the rotational bands are also discussed.