Decay Rate of the Nuclear Isomer ${}^{229}\mathrm{Th}(3/2^{+},7.8\mathrm{eV})$ in a Dielectric Sphere, Thin Film, and Metal Cavity

The main decay channels of the anomalous low-energy $3/2^{+}(7.8\pm 0.5\mathrm{eV})$ isomeric level of the ${}^{229}\mathrm{Th}$ nucleus, namely the $\gamma$ emission and internal conversion, inside a dielectric sphere, dielectric thin film, and conducting spherical microcavity are investigated theoretically, taking into account the effect of media interfaces. It is shown that (1) the $\gamma$ decay rate of the nuclear isomer inside a dielectric thin film and dielectric microsphere placed in a vacuum or in a metal cavity can decrease (increase) in dozen of times, (2) the $\gamma$ activity of the distributed source as a function of time can be nonexponential, and (3) the metal cavity, whose size is of the order of the radiation wavelength, does not affect the probability of the internal conversion in ${}^{229}\mathrm{Th}$, because the virtual photon attenuates at much shorter distances and the reflected wave is very weak.

Figure 1

The averaged Purcell factor $⟨f_P^{M1}⟩$ (6) for the ${}^{229}\mathrm{Th}$ nuclei uniformly distributed inside a dielectric sphere as a function of $R/\lambda$. The inset (a) shows $f_P^{M1}$ for the ${}^{229}\mathrm{Th}$ nuclei placed at the center of a sphere.

Figure 2

The same as in Fig. 1, but with the dielectric sphere placed in a cavity of metallic Al.

Figure 3

Radioactive decay curves $Q(t)$ from Eq. (5) for the ${}^{229}\mathrm{Th}$ nuclei in the ${\mathrm{SiO}}_2$ thin film: (1) with $f_P^{M1}(z)$ for the ${\mathrm{SiO}}_2/\mathrm{Si}$ sample from Eq. (7), (2) with $f_P^{M1}(z)=1$, and (3) the Purcell factor $f_P^{M1}(z)$.