  We present an all-optical mass spectrometry technique to identify trapped
ions. The new method uses laser-cooled ions to determine the mass of a
cotrapped dark ion with a sub-dalton resolution within a few seconds. We apply
the method to identify the first controlled synthesis of cold, trapped
$\mathrm{RaOH}^+$ and ${\mathrm{RaOCH}_3}^+$. These molecules are promising for
their sensitivity to time and parity violations that could constrain sources of
new physics beyond the standard model. The nondestructive nature of the mass
spectrometry technique may help identify molecular ions or highly charged ions
prior to optical spectroscopy. Unlike previous mass spectrometry techniques for
small ion crystals that rely on scanning, the method uses a Fourier transform
which is inherently broadband and comparatively fast. The technique's speed
provides new opportunities for studying state-resolved chemical reactions in
ion traps.
