Optical transition of the $^{229}$Th nucleus in a solid-state environment

We describe a novel approach to directly measure the energy of the narrow, low-lying isomeric state in $^{229}$Th. Since nuclear transitions are far less sensitive to environmental conditions than atomic transitions, we argue that the $^{229}$Th optical nuclear transition may be driven inside a host crystal with a high transition Q. This technique might also allow for the construction of a solid-state optical frequency reference that surpasses the precision of current optical clocks, as well as improved limits on the variability of fundamental constants. Based on analysis of the crystal lattice environment, we argue that a precision of $3\times10^{-17}<\Delta f/f <1\times10^{-15}$ after 1 s of photon collection may be achieved with a systematic-limited accuracy of $\Delta f/f \sim 2 \times 10^{-16}$. Improvement by $10^2-10^3$ of the constraints on the variability of several important fundamental constants also appears possible.