Matthieu equation

The motion of ions in the quadrupole [x, y) is quite complex and can be described by the Matthieu equations. The solution of the Matthieu equations generate two terms, a and q, which are proportional to the RF and DC potentials, respectively. For a detailed description of Matthieu equations, please see reference [53], The trajectories of ions are stable when the ions never reach the rods of the quadrupole. To reach the detector an ion must have a stable trajectory in the x and y directions. With a quadrupole mass analyzer a mass spectrum is obtained by increasing the magnitude of U (DC) and V (RF) at a constant ratio. In a quadrupole mass analyzer when the DC voltage of a quadrupole is set to zero and... 

P = x + because of the cylindrical symmetry of the trap. The motion of ions in the trap is characterized by secular frequencies, one radial and one axial. As for quadrupoles, the motion of ions can be described by the solutions of Matthieu s equations a and q). Ions can be stored in the trap with the condition that trajectories are stable in r and z directions (Fig. 8.13). Each ion of a specific mjz will be trapped at a specific value. The lower mjz will be located at the higher q values. 

The equation of motion (14.29) is known as Matthieu s differential equation. It has stable solutions only for certain values of the parameters a and b [14.55]. Charged particles which enter the trap from outside cannot be trapped. Therefore, the ions have to be produced inside the trap. This is generally achieved by electron-impact ionization of neutral atoms. 

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