  We demonstrate a spatial-imaging thermometry technique for ions in a
one-dimensional Coulomb crystal by relating their imaged spatial extent along
the linear radiofrequency ion trap axis to normal modes of vibration of coupled
oscillators in a harmonic potential. We also use the thermal spatial spread of
bright ions in the case of a two-species mixed chain to measure the
center-of-mass (COM) resonance frequency of the entire chain and infer the
molecular composition of the co-trapped dark ions. These non-destructive
techniques create new possibilities for better understanding of sympathetic
cooling in mixed-ion chains, improving few-ion mass spectrometry, and
trapped-ion thermometry without requiring a scan of Doppler cooling parameters.
