Quadrupole mass analyzer

A quadrupole mass analyzer is a mass analyzer that selects ions through stability of motion in combined static and alternating quadrupole electric fields. Selected mass-to-charge ratios follow bounded transverse trajectories through the analyzer; others grow unstable and fail to reach its exit.^[ref]

For an idealized model, define the electric potentialHere $r_0$ is a geometric scale and $U,V$ follow this stated potential convention. An ion with electric charge $Q$ and mass $m$ then obeysAt a given instant the field focuses one transverse direction and defocuses the other. Alternation can nevertheless yield bounded motion in both directions. Rescaling time reduces these equations to forms of the Mathieu equation; a quadrupole stability diagram summarizes the relevant bounded-motion regions.

The equations also show why the filter selects a ratio. Doubling both $m$ and $Q$ leaves the ideal equations unchanged. With fixed geometry, the dimensionless coefficients scale as $QU/(m{\Omega }^2)$ and $QV/(m{\Omega }^2)$. Doubling the drive angular frequency while keeping the potentials fixed divides both coefficients by four; multiplying both potentials by four restores them in this model.

A stable mathematical trajectory is only part of instrument performance. Initial ion positions and velocities, finite flight length, field imperfections and detector response also influence transmission and measured mass resolving power. The ideal potential is an explicit model, not a guarantee about a particular instrument.

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.