Atomic Clocks and Variations of the Fine Structure Constant

We describe a new test for possible variations of the fine structure constant $\alpha$ by comparisons of rates between clocks based on hyperfine transitions in alkali atoms with different atomic number $Z$. H-maser, Cs, and Hg ${}^{+}$ clocks have a different dependence on $\alpha$ via relativistic contributions of order $(Z\alpha {)}^2$. Recent H-maser vs ${\mathrm{Hg}}^{+}$ clock comparison data improve laboratory limits on a time variation by 100-fold to give $\dot{\alpha }/\alpha \le 3.7\times 10^{-14\mathrm{}}/$yr. Future laser cooled clocks (Be ${}^{+}$, Rb, Cs, Hg ${}^{+}$, etc.), when compared, will yield the most sensitive of all tests for $\dot{\alpha }/\alpha$.