Elucidation of ion motion in quadrupole mass spectrometer by Bloch function

In the optimization of the quadrupole mass spectrometer (QP-MS), the understanding of ion motion in terms of the phase space (the combined representation of the trajectory coordinate and momentum) is useful. The phase space representation gives an “ensemble” behavior of ions inside the filter. Even though each ion trajectory does not have the RF periodicity of the applied field, the phase space evolution does. It is only when appropriate ensemble ions are considered together that a proper QP filter characterization is possible. We here report a new framework for the phase space calculation of the QP-MS. The Mathieu–Hill equation is first solved for “complex eigen-trajectory” that has pseudo RF periodicity (the Bloch function). It is then shown that the acceptance phase space can be derived from the Bloch function without a need to calculate each ion trajectory. The ensemble behavior of ions can be estimated from a single Bloch function by expressing the trajectory phase space point by the complex amplitude (coefficient) of the Bloch function. The application of the Bloch function method to the auxiliary (pre-rod) field reveals that the ion injection efficiency may significantly be improved by optimizing the number of RF periods the ions spend in the pre-rod section.