The pseudopotential for quadrupole fields up to q = 0.9080

Abstract

Pseudopotential models for radio frequency (rf) electric fields are widely used. However, these approximate and sometimes intuitively introduced methods, can sometimes lead to ambiguities and false conclusions. While a classical model works well for quadrupole fields with small q where the Mathieu parameter β≤0.4 there has been some discussion in the literature of the behavior of a pseudopotential, and, in particular, the secular frequency of ions for larger q near the far end of the stability zone (for example, near qmax≈0.908045 for a=0). This paper analyzes carefully the behavior of secular frequencies calculated both numerically and analytically and demonstrates that the classical formula for the secular frequency in the pseudopotential, ω=βΩ/2remains valid up to β=1. This means that both the secular frequency and the pseudopotential function well depth increase monotonically with q inside the stability zone. However, some models state that the well depth, identified as the pseudopotential function calculated at the edge of a quadrupole aperture, should be zero at both the low q and high q boundaries of the stability zone, with a maximum somewhere between. It is shown that the paradox that the acceptance is zero at both ends of the stability zone and hence the pseudopotential well depth and the pseudopotential quadratic coefficient should be zero at both boundaries of the stability zone is resolved by introducing for large q a pseudopotential well depth and a pseudopotential well width as separate pseudopotential objects to characterize the acceptance, while a quadratic pseudopotential function is used to describe the secular motion of ions but not the acceptance for large q.