Electric and dynamic quadrupole moments of even-even nuclei

Abstract

For the ground-state bands of nonmagic even-even nuclei, the variable moment-of-inertia law defines an effective moment of inertia $\mathfrak{g}(J)$, using only the energies of the ${2}^{+}$ and ${4}^{+}$ states. From the definition, $\mathfrak{g}(J)$ is never negative. The average moment of inertia ${\mathfrak{g}}_{02}\ensuremath{\equiv}\frac{1}{2}[\mathfrak{g}(0)+\mathfrak{g}(2)]$ may then be related to the transition quadrupole moment ${Q}_{02}$. $\mathfrak{g}(Q)$ is well described by a continuous function ranging over almost two orders of magnitude in both quantities. This function is linear for small values of $Q(\ensuremath{\lesssim}4 e\mathrm{b})$, quadratic for intermediate values ($4\ensuremath{\lesssim}Q\ensuremath{\lesssim}11$ e b), and essentially constant for larger values ($11\ensuremath{\le}Q\ensuremath{\le}13$ e b). The linear section pertains to all nuclei which deviate by not more than two pairs from being singly magic. In this group are both light deformed nuclei (e.g., $^{12}\mathrm{C}$ and $^{24}\mathrm{Mg}$) and heavier spherical nuclei ranging from $30\ensuremath{\lesssim}A\ensuremath{\lesssim}200$. The values of $\mathfrak{g}$ and $Q$ in this region are (within \ifmmode\pm\else\textpm\fi{} 50%) those that would be obtained if the nucleus were replaced by a dumbbell with length equal to the nuclear diameter and an $\ensuremath{\alpha}$ particle at either end. The quadratic relation has a simple interpretation in a picture of the nucleus as a droplet composed of two fluids, a superfluid which does not contribute to the angular momentum, and an inertial or rigidly rotating component which forms a fixed fraction $f$ of the mass density at each point in the nucleus. The fraction $f$, computed according to a simple rule from the degree of deformation of the droplet, vanishes if the droplet is spherical. The constancy of $\mathfrak{g}$ at the largest $Q$ values (associated with spontaneously fissioning nuclei) may be due to a slight proton excess at the poles of the nucleus.NUCLEAR STRUCTURE Quadrupole moments, VMI model, moments of inertia.