Nuclear Moments of Inertia due to Nucleon Motion in a Rotating Well

A model of a deformed nucleus in which the spheroidal collective field is steadily cranked about a fixed axis, as introduced in previous papers, serves as a convenient approximation expected to reproduce some of the dynamic inertial properties of the collective motion. The independent-nucleon behavior in a rotating harmonic oscillator potential, deformed by the presence of the open-shell nucleons, gives the rigid-rotation moment of inertia and is discussed here with an attempt at graphic clarity. This result is much larger than observed and attention is here focused on the shortcomings of the harmonic oscillator approximation, although, as suggested by Bohr and Mottleson, the discrepancy may also be largely due to the internucleon interactions which have not been calculated adequately and are here neglected. The $1d\ensuremath{-}2s$ shell is treated as a tractable illustrative case. The characteristic harmonic-oscillator degeneracy of the undeformed levels within the magic-number groups, such as the $1d$ and $2s$ levels, profoundly affects the perturbation calculation of the moment of inertia through the energy denominators. Removing this degeneracy by lowering the states of high $l$ (as required in heavier nuclei for the magic numbers) has the effect of increasing the calculated moment of inertia above the rigid-rotation value in most cases near the beginning of the shell and reducing it in most cases near the end of the shell. The preponderance of prolate deformations is also discussed.

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