Photoexcitation of the $^{229}$Th nuclear clock transition using twisted light

The $^{229}$Th nucleus has a unique transition at only 8 eV which could be used for a novel nuclear clock. We investigate theoretically the prospects of driving this transition with vortex light beams carrying orbital angular momentum. Numerical results are presented for two experimental configurations which are promising for the design of the planned nuclear clock: a trapped ion setup and a large ensemble of nuclei doped into CaF$_2$ crystals which are transparent in the frequency range of the nuclear transition. We discuss the feasibility of the vortex beam nuclear excitation and compare the excitation features with the case of plane wave beams.