Phenomenological Analysis of Ground-State Bands in Even-Even Nuclei

A variable-moment-of-inertia (VMI) model is proposed which permits an excellent fit of level energies of ground-state bands in even-even nuclei. In this model the energy of a level with angular momentum $I$ is given by the sum of a potential energy term $\ensuremath{\propto}{({\mathfrak{g}}_{I}\ensuremath{-}{\mathfrak{g}}_{0})}^{2}$ (where ${\mathfrak{g}}_{0}$ is the ground-state moment of inertia) and a rotational energy term $\frac{{\ensuremath{\hbar}}^{2}I(I+1)}{2{\mathfrak{g}}_{I}}$. It is required that the equilibrium condition $\frac{\ensuremath{\partial}E}{\ensuremath{\partial}\mathfrak{g}}=0$ be satisfied for each state. Each nucleus is described by two adjustable parameters, ${\mathfrak{g}}_{0}$ and $\ensuremath{\sigma}$ (the softness parameter), which are determined by a least-squares fit of all known levels. The calculated level energies and moments of inertia ${\mathfrak{g}}_{I}$, ${\mathfrak{g}}_{0}$, and $\ensuremath{\sigma}$ are tabulated for 88 bands, ranging from Pd to Pt and from Th to Cm. Projections of three-dimensional arrays of ${\mathfrak{g}}_{0}$ and $\ensuremath{\sigma}$ on the ($N,Z$) plane are shown. These parameters are found to vary smoothly as function of $N$ and $Z$. Breaks occur at $N=98, 104, \mathrm{and} 108$. The osmium nuclei show a pronounced maximum for ${\mathfrak{g}}_{0}$ and an equally pronounced minimum for $\ensuremath{\sigma}$ at 108 neutrons. In Pt, ${\mathfrak{g}}_{0}$ decreases steeply to 110 neutrons and then more slowly, while $\ensuremath{\sigma}$ increases correspondingly. The stable Pt nuclei with $A=190,192, \mathrm{and} 194$ still possess appreciable moments of inertia and large but "finite" softness parameters. Hence they may be characterized as "pseudospherical." For nuclei exhibiting a near-harmonic level pattern (like ${\mathrm{Xe}}^{130}$, ${\mathrm{Sm}}^{150}$, and other neutron-deficient rare-earth isotopes), ${\mathfrak{g}}_{0}$ becomes exceedingly small, but already for the 2+ state $\mathfrak{g}$ is several orders of magnitude larger. The parameters of some $K=2$ bands in even-even nuclei and of bands found in odd-odd nuclei are related to those of appropriate ground-state bands in even-even nuclei. Evidence for a rotational band in ${\mathrm{Ir}}^{194}$ is deduced from recently published experimental results. A plot of $\frac{{E}_{4}}{{E}_{2}}$ versus $A$, presented for the discussion of the region of validity of the model, namely, $2.23l\frac{{E}_{4}}{{E}_{2}}l3.33$, reveals new regularities. The empirical "Mallmann curves" ($\frac{{E}_{I}}{{E}_{2}}$ plotted versus $\frac{{E}_{4}}{{E}_{2}}$) are deduced from the VMI model within its region of validity. Graphs are presented which allow the determination of ${E}_{I}$ (for $I\ensuremath{\le}16$) and of $\ensuremath{\sigma}$ and ${\mathfrak{g}}_{0}$ for each even-even nucleus for which the first 2+ and 4+ states are known. The model suggested by Harris, which includes the next-higher-order correction of the cranking model, is shown to be mathematically equivalent to the VMI model. The recently discovered appreciable quadrupole moments of 2+ states of "spherical nuclei" are compatible with the moments of inertia of these states given by the VMI model. The relation between $\frac{B (E2)({2}^{\ensuremath{'}}\ensuremath{\rightarrow}2)}{B (E2)(2\ensuremath{\rightarrow}0)}$ and $\frac{{E}_{4}}{{E}_{2}}$ is explored.