Heating rate in a linear quadrupole trap

In radio-frequency trap, the temperature of ion ensembles converges towards a hot equilibrium due to radio-frequency heating. This effect is detrimental to the stability of trapped ensembles and is the justification of cooling. The intensity of this radio-frequency heating increases with the amplitude of the radio-frequency field $q_x$. Using an analytical empirical formula, we show that the lifetime of the ion ensemble $t_0$ under cold condition increases with $q_x$ according to a power law $t_0\propto q_x^A$, and does not vary significantly for the several ion quantities $N$ tested. The temperature of the explosive onset $B$ decreases linearly with $q_x$. We also show that non-linear instabilities due to trapping parameters decreases $t_0$ and $B$, and produce a local increase of heating rate for certain temperature ranges.