Calculating the fragmentation process in quadrupole ion traps☆
Graphical abstract
Introduction
P.H. Dawson [1] and J.F.J. Todd were already early on one of the first and few scientists who saw the large potential of the phase-space ellipses that had been mathematically introduced to the mass spectrometry literature by M. Baril and A. Septier [2,3]. These phase-space ellipses can be derived from G.W. Hill's well-known solution [4,5] to the Mathieu equation [[6], [7], [8]] and had also already been shown by Paul and coworkers [9]. They can be used to exactly characterize the strictly speaking aperiodic trajectories in quadrupole fields. Further, they allow to simplify the complicated mathematics behind the solution to Mathieu's equation to basic geometric calculations of points on ellipses. Therefore, it is no surprise that we chose this same method to systematically investigate the collision-induced resonant excitation process in quadrupole ion traps. Dawson was already in extracts describing the phenomenon that happens at the moment an ion dissociates in the time and position dependent electric field, thereby changing its mass-to-charge ratio and with that its -value [10]. This can result in a drastic change in trajectory and lead to fragment ions hitting the electrodes, although having mass-to-charge ratios laying in the stable area of the stability diagram. In addition to that, we use a recently presented similar method of obtaining these space-space ellipses as a second reference method [11,12].
The exact motivation for this investigation is to determine the observed trends in terms of ion loss ratios and especially reached velocities to allow predictions about possible consecutive excitation events. The collision harshness in the quadrupole ion trap and thus resulting fragmentation mass spectra are depending on the reached oscillation amplitude of the ions during the resonant excitation process. While by changing the way we apply auxiliary resonant excitation voltages, we can vary the way an ion increases its oscillation amplitude until it eventually dissociates, the change calculated here, caused by the change in -value, cannot be influenced. This change is purely based on the change in mass-to-charge ratio at the moment of dissociation in the still identic oscillating electric field.
It will be first shown with the simple pseudopotential well-depth equation [13,14] that the only two important parameter describing the ions' velocity function in quadrupole ion traps is their -stability parameter and the oscillation amplitude ions have. The velocity then determines the collision harshness described in the center-of-mass collision energy, which can further be converted into an effective collisional temperature [11,15]. As the excitation q-parameter may be chosen the same for any precursor ion, allows an ion independent characterization of the collision induced dissociation process. The mathematical consequences of the change in q upon dissociation are calculated in two ways that both build up on the same solution to Mathieu's equation. The first approach is geometric and thereby facilitates the understanding of the calculations. The second approach is very compact, also serving directly as a tool for verification. The results are followed by a short discussion of the made assumptions.
Section snippets
Theory
Before the exact solution to Mathieu's or Hill's equation will be used, very important principles are demonstrated, which also need to be understood to follow the actual calculations. These principles can already be shown with the well-known pseudopotential well approximation [13,14].
According to the literature, the pseudopotential well depth is given by:with being the charge of an electron, the pseudopotential (typically only used one-dimensionally for the -direction, i.e.,
Calculating the fragmentation process
The essence of the following calculations is visualized in Fig. 3 and Fig. 4. It will be investigated now, how the oscillation amplitude is affected, when the stability parameters abruptly change, which represents the ion's dissociation.
Discussion
In order to compare different fragmentation LMCO settings or fragmentation -values, the oscillation amplitude was adjusted for each -value in order to result in the same root-mean-square velocity (here ) or effective collisional temperature (here ). This includes the assumption that these trajectories with slightly different velocity probability distributions [12,38] but equal and almost exactly equal [12,36], result in the same reaction
Conclusion
Two ways have been presented of how to calculate the change in trajectory that happens at the moment an ion dissociates in the electric field of a quadrupole ion trap. During the resonant excitation process we can intentionally change the way we increase the oscillation amplitude to induce dissociation by manipulating excitation waveforms. The change in oscillation amplitude calculated here, however, cannot be influenced. Most important, resulting oscillation amplitudes can only be decreased by
Additional information
The shown calculations have also been uploaded to Github in form of a script in the Python programming language: https://github.com/NeugebauerT/ion-trap.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Based in part on work originally submitted by T. S. Neugebauer as a thesis in partial fulfillment for the doctoral degree at the Friedrich-Alexander-Universität Erlangen-Nürnberg.