Quantifying the operation of sinusoidal mass filters

Even though sinusoidal quadrupole mass filters have been around for more than 50 years, the relationships defining resolution, resolving power, and transmission from the applied voltages have not been rigorously quantified or discussed. Traditional quadrupole mass filter theory implies that voltages are scanned at constant direct current (DC) to alternating current (AC) voltage ratios with the scanline passing through the origin of the voltage stability diagram. A prominent feature of constant voltage ratio scans is constant baseline theoretical resolving power (m/Delta m) that is the same for all masses. Commercial quadrupole instruments rarely scan at constant resolving power because ion transmission increases with mass. Instead, they scan at constant resolution, meaning that the mass window width is fixed. Constant resolution mass scans are preferred because ion transmission does not change with mass. Commercial mass filter systems create constant resolution scans by linearly changing the DC and AC voltages at a fixed ratio in the presence of an additional negative DC voltage offset. This manuscript systematically quantifies the effects of the DC and AC voltages on resolution, resolving power, pseudopotential well depth, and transmission. To quantify these properties, recently developed spreadsheet tools that calculate the laboratory frame stability of ions from the matrix solutions of Hill's equation were used. Voltage scanning methods and their effects on theoretically determined transmission and sensitivity will be discussed.