Quadrupole stability diagram

A 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.^[ref]

Using the potential convention stated in quadrupole mass analyzer, introduce dimensionless time $\tau =\Omega t/2$ and parametersHere $Q$ 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 becomeThese are Mathieu equations. In the convention $u^{\prime \prime }+(a-2q\cos 2\tau )u=0$, 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.