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

Abstract

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 $\propto {({\mathfrak{g}}_I-{\mathfrak{g}}_0)}^2$ (where ${\mathfrak{g}}_0$ is the ground-state moment of inertia) and a rotational energy term $\frac{{\hslash }^{2}I(I+1\mathrm{})\mathrm{}}{2{\mathfrak{g}}_I}$. It is required that the equilibrium condition $\frac{\partial E}{\partial \mathfrak{g}}=0\mathrm{}$ be satisfied for each state. Each nucleus is described by two adjustable parameters, ${\mathfrak{g}}_0$ and $\sigma$ (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 $\sigma$ 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 $\sigma$ 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 $\sigma$ at 108 neutrons. In Pt, ${\mathfrak{g}}_0$ decreases steeply to 110 neutrons and then more slowly, while $\sigma$ 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 $K=2\mathrm{}$ 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, $2.23<\mathrm{}\frac{E_4}{E_2}<3.33\mathrm{}$, 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 $I\le 16$) and of $\sigma$ 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 $\frac{B \mathrm{}\left(E2\mathrm{}\right)(2^{\prime }\to 2)}{B \mathrm{}\left(E2\mathrm{}\right)(2\mathrm{}\to 0)}$ and $\frac{E_4}{E_2}$ is explored.