= Quadrupole stability diagram
A \b[quadrupole stability diagram] plots the parameter combinations for which an ideal <quadrupole mass analyzer> gives bounded transverse <ion> motion. The useful region is the overlap of stability in both transverse directions: confinement along only one direction still allows loss along the other. The physical selection mechanism is stability in combined static and high-frequency quadrupole <electric fields>.<Quadrupole mass analyser in IUPAC Gold Book>[Q04971: selected mass-to-charge ratios have stable trajectories; other trajectories are unstable.]{ref}
Using the potential convention stated in <quadrupole mass analyzer>, introduce dimensionless time $\tau=\Omega t/2$ and parameters
$$
a=\frac{8QU}{mr_0^2\Omega^2},\qquad b=\frac{4QV}{mr_0^2\Omega^2}.
$$
Here $Q$ is <electric charge>, $m$ is <mass>, $r_0$ describes geometry, $\Omega$ is drive <angular frequency>, and $U,V$ specify the static and alternating potentials. The transverse equations become
$$
\frac{d^2x}{d\tau^2}+(a+2b\cos2\tau)x=0,\qquad
\frac{d^2y}{d\tau^2}-(a+2b\cos2\tau)y=0.
$$
These are <Mathieu equations>. In the convention $u''+(a-2q\cos2\tau)u=0$, the first equation has $q=-b$; quoting a diagram requires checking this sign and the potential's numerical convention.
A useful limiting test is $b=0$, $a>0$. The $x$ equation oscillates, but the $y$ equation has exponentially growing solutions. Static focusing alone therefore fails to provide a stable overlap. Periodic focusing changes this conclusion; its stability regions cannot be read off from the force at one instant.
At fixed positive $U/V$, changing <mass-to-charge ratio> moves a point along a ray with $a/b=2U/V$. Doubling $m/Q$ halves both coordinates. Doubling both applied potentials moves the same <ion> twice as far along that ray. These scaling relations explain how scanning the potentials selects successive <mass-to-charge ratios>. A stable region is a property of the ideal differential equations; finite rod length, entrance conditions and field imperfections still determine actual transmission and <mass resolving power>.
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