Level Spectra of Odd Isotopes of Ga, As, Br, and Rb in the Coriolis Coupling Model with a Residual Interaction of the Pairing Type

Low-lying positive- and negative-parity states of odd isotopes of Ga, As, Br, and Rb in the mass region bounded by $30<Z<\mathrm{}38\mathrm{}$ and $40\le N\le 48$ have been calculated in the Coriolis coupling model with the incorporation of a residual interaction of the pairing type in calculating the intrinsic states. The single-particle energy levels and wave functions are first computed in a deformed well with a spin-orbit strength of $-0.26\mathrm{}\hslash {\omega }_0$ and for a well-flattening parameter $D$ between $-0.035\mathrm{}\hslash {\omega }_0$ and $-0.06\mathrm{}\hslash {\omega }_0$. These values are consistent with the observed location of the $1f_{\frac{5}{2}}$, $2p_{\frac{3}{2}}$, and $2p_{\frac{1}{2}}$ states in ${\mathrm{Sc}}^{41}$ and ${\mathrm{Sc}}^{49}$. The deformation $\beta$ has been restricted to a range around $\beta =\pm 0.2$ which is consistent with the observed $B\left(E2\mathrm{}\right)\mathrm{}$ value to the first excited $2^{+}$ state of neighboring even-even nuclei, assuming a rotational character for this state. Using the orbitals in the deformed well as vacuum states, new particle states are calculated for a residual interaction of the pairing type, using a pairing-force strength consistent with the observed odd-even mass difference and the Bardeen-Cooper-Schrieffer (BCS) variational formalism. The complete variational calculation has been carried out for the ground, the particle, and the core excited states with due consideration of the blocking effect. This procedure, which differs from the usual quasiparticle treatment, ensures that the particle and core excited states are orthonormal to each other and to the ground state and that the average particle number is conserved in each configuration. Band-head energies and the unperturbed bands built on them are calculated, using these new renormalized single-particle states and the moment of inertia obtained from the excitation energy of the first excited $2^{+}$ state of neighboring even-even nuclei, assuming a rotational character for this state. Using the same moment of inertia and the computed wave functions of the deformed well, the Coriolis coupling matrix elements between these bands are calculated. The inclusion of the residual interaction introduces multiplicative "overlap factors" into the Coriolis coupling strength which results in a reduction of its magnitude for off-diagonal terms. These overlap factors have been calculated from the BCS wave functions. Thus we have used no arbitrary free parameters in the computation. The final spectra of the negative- and positive-parity states are obtained by a diagonalization procedure, using, respectively, the bands built on the 10 Nilsson states in the $1f-2p$ shell and on the 15 Nilsson states in the $1g-2d-3s$ shell. The model can correctly account for (i) the observed ground-state spins for all nuclei in this region, (ii) the large number of negative-parity states with spins $\le {\frac{7}{2}}^{-}$ below 1-MeV excitation energy, and (iii) the occurrence of several low-lying positive-parity states, in particular a low-lying ${\frac{9}{2}}^{+}$, ${\frac{5}{2}}^{+}$ doublet.