Macroscopic Self-Consistency of the Collective Model of Deformed Nuclei
Abstract
It has been suggested, in effect, that in a deformed nucleus the number of protons (or the amount of charge) which are carried along by the rotational motion may, on the average, be approximated as the product of the atomic number $Z$ and the deformation parameter $\ensuremath{\delta}$. The theoretical justification of this suggestion is discussed, and a more accurate expression is obtained. Assuming that the interaction of the rotational motion with an external magnetic field is entirely due to the current associated with the amount of charge following the rotational motion, on the average, and also in accordance with the cranking approximation, a macroscopic expression for the rotational gyromagnetic ratio ${g}_{R}$ is derived. This expression, supplemented by the usual macroscopic formula for the intrinsic quadrupole moment ${Q}_{0}$, may constitute a macroscopic self-consistency relation among the collective parameters in a rotational band, namely, ${Q}_{0}$, ${g}_{R}$, and the moment of inertia $J$. The values of ${g}_{R}$ calculated from the experimental values of $J$ and ${Q}_{0}$ are tabulated for the ground-state rotational bands of both even-even and odd-mass nuclei. The aforementioned macroscopic self-consistency is then tested by comparing these calculated values of ${g}_{R}$ with both empirical values and previous microscopic calculations. According to the present approach, the well-known lowering of ${g}_{R}$ from the usual fluid-model value is mainly due to the limitation on the number of protons which can follow the rotational motion, on the average. It is not clear, however, whether there is any direct connection between this limitation and the pairing interaction which seems to play a rather essential role in the current microscopic calculations. This puzzling situation is further illustrated by considering the moments of inertia of even-even nuclei.