Lifetimes of Nuclear Isomers

Abstract

Lifetimes of nuclear isomers for gamma-transitions are calculated theoretically on the basis of various independent particle models; e.g., single proton, single neutron, and states of several particles (Sec. II). The calculations of this paper are essentially restricted to the most common type of transition viz., multipole order equal to spin change. The lifetime is expressed in terms of a matrix element, $M$, whose theoretical value depends on the particular model of the nucleus. Radial integrals are calculated numerically, assuming that the nuclear wave functions are given by single particle wave functions for a spherical square well.Empirical values of ${M}^{2}$ can be deduced from measured isomeric lifetimes, corrected for internal conversion. An analysis of empirical ${M}^{2}$ for some gamma transitions points to a number of regularities which, in general, speak in favor of an independent particle model (Sec. III).The regularities are the following:Empirical values of ${M}^{2}$ for $M4$ transitions are of order unity and show little scattering and no distinction between odd proton and odd neutron nuclei. The lack of scattering within each group of transitions is consistent with predictions of a single particle model. However, according to this model, one would expect odd proton nuclei to have lifetimes about half as large as odd neutron nuclei for the same transition energy, and also would expect lifetimes about $\frac{1}{10}$ as large as found empirically. Empirical values of ${M}^{2}$ for $M4$ transitions appear to be larger for transitions in nuclei with nearly closed shells.According to an independent particle model, ${M}^{2}$ for $E3$ transitions of energy 100 kev should be of order ${10}^{\ensuremath{-}8}$ for single neutron transitions, and vanish for many particle transitions, such as those between ${p}_{\frac{1}{2}}$ and $\frac{7}{2}$+states. The fact that empirical ${M}^{2}$ for $E3$ transitions are small can be interpreted as resulting from small deviations from an independent particle model. In fact, empirical ${M}^{2}$ for transitions between ${p}_{\frac{1}{2}}$ and $\frac{7}{2}$+states in odd-neutron nuclei appear to be smaller the more nearly the nucleus can be represented as a closed shell nucleus.The empirical value of ${M}^{2}$ for an $M1$ isomeric transition in ${\mathrm{Li}}^{7}$ is slightly larger than expected according to an independent particle model.A graph of energy levels for a spherical square well potential is presented (Appendix, Fig. 2)