Derivation of mathematical expressions to define resonant ejection from square and sinusoidal wave ion traps

Resonant ejection for mass analysis with ion traps is widely used because it markedly improves the mass range and resolution of ion traps. Unfortunately, an easy-to-use analytical expression that defines the ejection mass as a function of the trapping and excitation frequencies is missing in the literature because the secular frequency of the ions in sinusoidal ion traps is not easily determined for all stable values of qz from the Mathieu equation. However, the ion secular frequency for all stable values of qz in digital ion traps can be readily determined from Hill's equation. We have taken this expression and solved it for qz to produce an analytical expression for the ejection mass as a function of trapping and excitation frequency. We also recognized that the expression for the ion mass during resonant ejection for a square wave driven trap can be converted to an expression for a sinusoidal wave trap merely by multiplication by a factor of 4/π. These new expressions open up the possibility of rapid mass calibration for any method of resonant ejection from square or sinusoidal wave driven ion traps.

Graphical abstractAccurate mathematical expressions for the ion secular frequencies and resonant ejection m/z have been determined for any value of trap frequency and voltage and excitation frequency.Figure optionsDownload full-size imageDownload as PowerPoint slide