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In this article, the dynamics of an unbalanced two-ion crystal comprising the'target' and the 'sensor' ions confined in a Penning trap has been studied.First, the low amplitude regime is addressed. In this regime, the overallpotential including the Coulomb repulsion between the ions can be consideredharmonic and the axial, magnetron and reduced-cyclotron modes split up into theso-called 'stretch' and 'common' modes, that are generalizations of thewell-known 'breathing' and 'center-of-mass' motions of a balanced crystal madeof two ions. By measuring the frequency modes of the crystal and the sensor ioneigenfrequencies using optical detection, it will be possible to determine thetarget ion's free-cyclotron frequency. The measurement scheme is described andthe non-harmonicity of the Coulomb interaction is discussed since this mightcause large systematic effects.

Dynamics of an unbalanced two-ion crystal in a Penning trap for application in optical mass spectrometry