Level Spectra of Odd-Even1f72-Shell Nuclei in the Coriolis Coupling Model

Abstract

The negative parity levels of odd-even nuclei with protons and neutrons in the $1{f}_{\frac{7}{2}}$ shell are calculated using the strong-coupling symmetric-rotator model including the Coriolis coupling between bands. The single-particle energy levels and wave functions in the deformed well are computed for a spin-orbit strength $C=\ensuremath{-}0.26\ensuremath{\hbar}{\ensuremath{\omega}}_{0}$ which is consistent with the observed splitting in ${\mathrm{Ca}}^{41}$. The well-flattening parameter $D$ is taken to be $\ensuremath{-}0.06\ensuremath{\hbar}{\ensuremath{\omega}}_{0}$ in the middle of the $1{f}_{\frac{7}{2}}$ shell and $\ensuremath{-}0.035\ensuremath{\hbar}{\ensuremath{\omega}}_{0}$ otherwise. The band-head energies are calculated from the appropriate summation over the occupied single-particle energy levels using $\ensuremath{\hbar}{\ensuremath{\omega}}_{0}=\frac{41}{{A}^{\frac{1}{3}}}$ MeV for the energy of the oscillator quantum. The moment of inertia is taken from the excitation energy of the first excited ${2}^{+}$ state of neighboring even-even nuclei assuming a rotational character for this state. The same value is used for all bands in any given nucleus. The matrix elements of the Coriolis coupling are computed from the single-particle wave functions. The final excitation spectra are obtained by diagonalizing the Coriolis coupling term with the rotational wave function based on the ten available particle or core excited states in the $1f\ensuremath{-}2p$ shell. Energy levels and wave functions are calculated as a function of the deformation parameter $\ensuremath{\beta}$; the level spectrum for each individual nucleus is given for a tentative choice of $\ensuremath{\beta}$. The effect of a quenching of the Coriolis interaction on the calculated level spectrum is investigated. The computed level spectra for the nuclei (i) ${\mathrm{Sc}}^{43}$, ${\mathrm{Sc}}^{45}$, ${\mathrm{Sc}}^{47}$, (ii) ${\mathrm{Ca}}^{43}$, ${\mathrm{Ti}}^{45}$, ${\mathrm{V}}^{47}$, ${\mathrm{V}}^{49}$, ${\mathrm{V}}^{51}$, (iii) ${\mathrm{Ca}}^{45}$, ${\mathrm{Ti}}^{47}$, ${\mathrm{Cr}}^{49}$, ${\mathrm{Mn}}^{51}$, ${\mathrm{Mn}}^{53}$, and (iv) ${\mathrm{Ti}}^{49}$, ${\mathrm{Cr}}^{51}$ compare favorably with experiment. The model predicts the correct ground-state spin for all nuclei, including the anomalous cases of ${\mathrm{Ti}}^{47}$, ${\mathrm{Cr}}^{49}$, and ${\mathrm{Mn}}^{51}$ which have a ${\frac{5}{2}}^{\ensuremath{-}}$ ground state. Thus the observed ${\frac{7}{2}}^{\ensuremath{-}}$ (and occasionally ${\frac{5}{2}}^{\ensuremath{-}}$) ground-state configuration cannot be adduced as evidence supporting the validity of the spherical-shell model in the $1{f}_{\frac{7}{2}}$ shell. The positions of negative parity states with known spin--- especially the low-lying ${\frac{3}{2}}^{\ensuremath{-}}$ states---are well accounted for in this model, in contrast with difficulties encountered in the shell-model treatment. In particular, the model reproduces the observed ground-state triplets in ${\mathrm{V}}^{47}$ and in ${\mathrm{V}}^{49}$ and the low-lying ${\frac{3}{2}}^{\ensuremath{-}}$ state in ${\mathrm{Cr}}^{51}$. Further, this model predicts the correct number of levels below about 2.5-MeV excitation energy for nuclei in the upper half of the $1{f}_{\frac{7}{2}}$ shell. For nuclei in the lower half of the $1{f}_{\frac{7}{2}}$ shell, more levels are observed than predicted, which suggests the presence of positive parity states arising from core excitation of the $2s\ensuremath{-}1d$ shell. The natural classification of the $1{f}_{\frac{7}{2}}$ nuclei in this model according to their odd-nucleon count is borne out by the similarity of the observed spectra. The natural shell-model classification scheme in terms of the conjugate and equivalent pairs, on the other hand, does not seem to be supported by the experimental data.