Towards a sensitive search for variation of the fine-structure constant using radio-frequency $E1\mathrm{}$ transitions in atomic dysprosium

It has been proposed that the radio-frequency electric-dipole $\mathrm{}\left(E1\mathrm{}\right)\mathrm{}$ transition between two nearly degenerate opposite-parity states in atomic dysprosium should be highly sensitive to possible temporal variation of the fine-structure constant $\left(\alpha \right)$ [V. A. Dzuba, V. V. Flambaum, and J. K. Webb, Phys. Rev. A 59, 230 (1999)]. We analyze here an experimental realization of the proposed search in progress in our laboratory, which involves monitoring the $E1\mathrm{}$ transition frequency over a period of time using direct frequency counting techniques. We estimate that a statistical sensitivity of $|\dot{\alpha }/\alpha |\sim {10}^{-18}/\mathrm{yr}$ may be achieved and discuss possible systematic effects that may limit such a measurement.