Probability distributions in quadrupole ion traps

Abstract The solution to Mathieu's equation was modified and combined to only one single sine function. In this form, the cumulative probability functions for a fixed RF-phase can be determined as arcsine distributions. The probability of a normalized position u / u max or normalized velocity u ˙ / u ˙ max is relative to 1 / 1 − ( u / u max ) 2 , u ˙ / u ˙ max respectively. This corrects reference descriptions present in the literature, which stated the probability to be proportional to 1 − ( u / u max ) 2 . Resulting probability plots show that ions that are heavy on the relative mass scale of quadrupole ion traps, are more likely to be found close to the maximum of their oscillation amplitude. Light ions are more likely to be found at the center of their oscillation. The velocity distributions show that the likeliest velocity converges for low q-values to the mean-square velocity but splits into two likely velocity regions with increasing q-value. It is further emphasized, that these distributions describe the probability of a single ion and in order to describe the behavior of an ensemble of ions, it is inevitably needed to define a distribution of oscillation amplitudes u max .