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Quadrupole Mass Filter

A mass analyzer that uses oscillating electric fields to selectively transmit ions based on their m/z ratio.
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The statement of the theorem

Define the potential applied to the quadrupole rods by the voltages UU (DC) and VV (RF) as the potential function Φ\Phi: \nΦ(x,y,z)=UR2(x2−y2)+VR2(x2−y2)cos⁡(ωt)\Phi(x, y, z) = \frac{U}{R^2} (x^2 - y^2) + \frac{V}{R^2} (x^2 - y^2) \cos(\omega t) \nFor an ion with mass-to-charge ratio m/zm/z to maintain a stable trajectory within the quadrupole field, its motion must satisfy the Mathieu equation for both xx and yy coordinates: \nd2dξ2(ξxξy)+(aκ−κa)(ξxξy)=0\frac{d^2}{d\xi^2} \begin{pmatrix} \xi_x \\ \xi_y \end{pmatrix} + \begin{pmatrix} a & \kappa \\ -\kappa & a \end{pmatrix} \begin{pmatrix} \xi_x \\ \xi_y \end{pmatrix} = 0 \nwhere ξ=ωt\xi = \omega t, and the parameters aa and κ\kappa are functions of the applied voltages U,VU, V and the ion's m/zm/z ratio.
Source: Wikipedia