 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.