# Mathieu Equation and Stability Diagram¶

The **Mathieu differential equation** [1] is a differential equation of the form

One area, of interest to us, where this occurs is the equation of
motion for particles in an ideal quadrupole
*Ion Trap* (see [2-3]), where u is
a coordinate in space (e.g. x) and v is proportional to time t.
We can see this by relating (via the *Lorentz Force Law*)
the particle acceleration (x’‘) to the equations
for the gradient of the quadrupole field potential :

The stability of the particle motion in the trap depends on the constants a and q, which we can see are functions of the trap voltages (U and V), angular frequency (), and particle m/z ratio. [2-3]

Even in ion traps with non-idealized geometries, such as hyperbolic electrodes replaced with circular or flat ones, fields maintain some resemblance to the ideal case, and the formalism of a & q still applies.

A **Mathieu stability diagram** plots regions of the u-v space that lead
to stable trajectories, and they are often depicted in introductions
to ion trap theory (e.g. [2-3]).

## SIMION specific¶

SIMION can solve not just the Mathieu equation but the more general problem
of determining particle trajectories most any type of *Ion Trap*
(not limited to idealized traps) or other type of system.
You can simulate in SIMION various ion traps and observe how
choices of a & q lead to stable or unstable particles trajectories.
In fact, SIMION Example: quad conveniently sets quadrupole mass filter voltages
based on a & q parameters entered.
It would be possible to generate a Mathieu stability diagram from SIMION
data (ways to automate multiple runs, like in SIMION Example: tune, can help too).

- SIMION Example: quad
- SIMION Example: trap
- SIMION Example: octupole
*Time-Dependent Field*- [1] Mathworld: Mathieu Differential Equation
- [2] March, Raymond E.
*“An Introduction to Quadrupole Ion Trap Mass Spectrometry”*Journal of Mass Spectrometry, volume 32, issue 4, year 1997, pp. 351-369, doi:10.1002/(SICI)1096-9888(199704)32:4<351::AID-JMS512>3.0.CO;2-Y link (spectroscopynow.com) - [3] Wong, Philip S.H. Cooks, R. Graham.
*“Ion Trap Mass Spectrometry”*. currentseparations. http://www.currentseparations.com/issues/16-3/cs16-3c.pdf