  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 overall
potential including the Coulomb repulsion between the ions can be considered
harmonic and the axial, magnetron and reduced-cyclotron modes split up into the
so-called 'stretch' and 'common' modes, that are generalizations of the
well-known 'breathing' and 'center-of-mass' motions of a balanced crystal made
of two ions. By measuring the frequency modes of the crystal and the sensor ion
eigenfrequencies using optical detection, it will be possible to determine the
target ion's free-cyclotron frequency. The measurement scheme is described and
the non-harmonicity of the Coulomb interaction is discussed since this might
cause large systematic effects.
