We have a plasma chamber where the transparent cathode fed very high negative voltage (nearly 10kV ) then the ion from the plasma tend to oscillate in that negative potential well. So, depending on what theory will those ion oscillate in that well?
Almost by definition, an ion will oscillate in a potential well if its total energy is less than the depth of the well (the potential energy reguired to escape to infinity).
In the case of a plasma experiment, the main challenge is characterizing (and controlling) the electrostatic potential profile that a single ion would experience. In general, the space-charge in the plasma will tend to rearrange itself to modify the vacuum electrostatic potential in some manner specific to the discharge. This falls under the genral topic of plasma sheaths. Sheaths are complicated.
If the spatial distribution of the electrostatic potential V(x,y,z) is known, the trajectory of an ion can be calculated from F = m*a.
For example, a mass on a spring can be thought of as a particle in a parabolic well, and x(t) will of course be sinosoidal. For a generic V(x), the ion trajectories will be solutions to the differential equation F=-e*grad(V(x))=m*d^2(x)/dt^2.
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