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We have measured the atomic masses of ten species with a typical accuracy of ${10}^{\ensuremath{-}10}$ by comparing the cyclotron frequencies of 20 different pairs of alternately trapped ions. The mass of $^{28}\mathrm{Si}$ provides a possible route to realizing an atomic standard of mass. The neutron separation energies for $^{15}\mathrm{N}$ and $^{13}\mathrm{C}$ determined from these mass measurements suggest the need to recalibrate $\ensuremath{\gamma}$-ray energies in the 2 to 13 MeV range, and may lead to an independent determination of both the fine structure constant $\ensuremath{\alpha}$ and the molar Planck constant ${N}_{A}h$ with a precision approaching ${10}^{\ensuremath{-}7}$.
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