Mathieu’s differential equation

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). 

The equation of motion (9.49) is known as Mathieu s differential equation. It has stable solutions only for certain values of the parameters a and b [1221]. Charged particles that enter the trap from outside cannot be trapped. Therefore, the ions have to be produced inside the trap. This is generally achieved by electron-impact ionization of neutral atoms. 

Basically, ions entering the quadrupole move through it in a wavelike trajectory around the z-axis between the quadrupole s electrodes. This motion can be described by the so-called Mathieu differential equations (5.2) and (5.3). 

Mathieu s equation is a linear second-order differential equation with periodic coefficients. It belongs to the family of ///// s equations. The one-dimensional Mathieu s equation is written in standard form as... 

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