The quadrupole separation system

Exerting an effect on a single charged ion moving near and parallel to the center line inside the separation system and perpendicular to its movement are the forces 

The mathematical treatment of these equations of motion uses Mathieu s differential equations. It is demonstrated that there are stable and unstable ion paths. With the stable paths, the distance of the ions from the separation system center line always remains less than r (passage condition). With unstable paths, the distance from the axis will grow until the ion ultimately collides with a rod surface. The ion will be discharged (neutralized), thus becoming unavailable to the detector (blocking condition). 

Even without solving the differential equation, it is possible to arrive at a purely phenomenological explanation which leads to an understanding of the most important characteristics of the quadrupole separation system. 

If we imagine that we cut open the separation system and observe the deflection of a singly ionized, positive ion with atomic number M, moving in two planes, which are perpendicular one to the other and each passing through the centers of two opposing rods. We proceed step-by-step and 

Here the relationships are exactly opposite to those for i = i (V) since 

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