A modified homotopy perturbation method and the axial secular frequencies of a non-linear ion trap

In this paper, a modified version of the homotopy perturbation method, which has been applied to non-linear oscillations by V. Marinca, is used for calculation of axial secular frequencies of a non-linear ion trap with hexapole and octopole superpositions. The axial equation of ion motion in a rapidly oscillating field of an ion trap can be transformed to a Duffing-like equation. With only octopole superposition the resulted non-linear equation is symmetric; however, in the presence of hexapole and octopole superpositions, it is asymmetric. This modified homotopy perturbation method is used for solving the resulting non-linear equations. As a result, the ion secular frequencies as a function of non-linear field parameters are obtained. The calculated secular frequencies are compared with the results of the homotopy perturbation method and the exact results. With only hexapole superposition, the results of this paper and the homotopy perturbation method are the same and with hexapole and octopole superpositions, the results of this paper are much more closer to the exact results compared with the results of the homotopy perturbation method.     

Characteristics of stability boundary and frequency in nonlinear ion trap mass spectrometer

In this article, the Poincare-Lighthill-Kuo (PLK) method is used to derive an analytical expression on the stability boundary and the ion trajectory. A multipole superposition model mainly including octopole component is adopted to represent the inhomogeneities of the field. In this method, both the motional displacement and secular frequency of ions have been expanded to asymptotic series by the scale of nonlinear term ε, which represents a weak octopole field. By solving the zero and first-order approximate equations, it is found that a frequency shift exists between the ideal and nonlinear conditions. The motional frequency of ions in nonlinear ion trap depends on not only Mathieu parameters, a and q, but also the percentage of the nonlinear field and the initial amplitude of ions. In the same trap, ions have the same mass-to-charge ratio (m/z) but they have different initial amplitudes or velocities. Consequently, they will be ejected at different time through after a mass-selective instability scan. The influences on the mass resolution in quadrupole ion trap, which is coupled with positive or negative octopole fields, have been discussed respectively.     

Potential Distribution and Transmission Characteristics in a Curved Quadrupole Ion Guide

The potential distribution in the curved quadrupole is exactly characterized by the Laplace equation, and an approximate solution to the Laplace equation is calculated. We represent the Laplace equation under the coordinates named minimal rotation frame (MRF) and derive an expression on the hexapole and octopole superposition. Our conclusion is in agreement with the results by the numerical (SIMION) method. Based on the Poincare-Lighthill-Kuo (PLK) method reported in our previous work, the nonlinear effects of ion motion are investigated in detail. The frequency shift of ion motion can be well eliminated by coupling the hexapole component with a positive octopole component, and the transmission efficiency of ions is found to decrease dramatically with the increase of the ionic kinetic energy in the z-direction. Furthermore, the transmission characteristics of ions are discussed with regards to the phase-space theory. The results show that the centrifugally introduced axis shift is mainly responsible for the ion losses. A modified direct current (dc) voltage supply pattern is hence proposed to compensate for this effect.     

Space Charge Induced Nonlinear Effects in Quadrupole Ion Traps

A theoretical method was proposed in this work to study space charge effects in quadrupole ion traps, including ion trapping, ion motion frequency shift, and nonlinear effects on ion trajectories. The spatial distributions of ion clouds within quadrupole ion traps were first modeled for both 3D and linear ion traps. It is found that the electric field generated by space charge can be expressed as a summation of even-order fields, such as quadrupole field, octopole field, etc. Ion trajectories were then solved using the harmonic balance method. Similar to high-order field effects, space charge will result in an “ocean wave” shape nonlinear resonance curve for an ion under a dipolar excitation. However, the nonlinear resonance curve will be totally shifted to lower frequencies and bend towards ion secular frequency as ion motion amplitude increases, which is just the opposite effect of any even-order field. Based on theoretical derivations, methods to reduce space charge effects were proposed.

Linear quadrupoles with added hexapole fields

Linear quadrupoles with added hexapole fields are described. The shifts in ion oscillation frequency caused by the addition of a hexapole field are calculated within the effective potential model. Methods to construct linear quadrupoles with added hexapole fields with exact electrode geometries and with round rods are discussed. A quadrupole with added hexapole field can be constructed with round rods by rotating two rods (say the y rods) towards an x rod. Computer simulations are used to investigate the possibility of mass analysis with quadrupoles with added hexapole fields. We find that a quadrupole with an added hexapole field in the range 2-12% can provide mass analysis provided the dc is applied with the correct polarity and value. When a rod set is constructed with round rods, other multipoles in the potential degrade the peak shape, resolution and transmission. The largest of these after the quadrupole and hexapole are a dipole and octopole term. With round rod sets, the peak shape can be improved by using different diameters for the x and y rod pairs to minimize the octopole term in the potential and by injecting ions at the field center where the dipole term is zero. Calculations of the boundaries of the stability diagram for this case show the boundaries move out, relative to those of a pure quadrupole field, but remain sharp.     

Study of Nonlinear Resonance Effect in Paul Trap

In this article, we investigated the nonlinear resonance effect in the Paul trap with a superimposed hexapole field, which was assumed as a perturbation to the quadrupole field. On the basis of the Poincare-Lighthill-Kuo (PLK) perturbation method, ion motional equation, known as nonlinear Mathieu equation (NME) was expressed as the addition of approximation equations in terms of perturbation order. We discussed the frequency characteristics of ion axial-radial (z-r) coupled motion in the nonlinear field, derived the expressions of ion trajectories and nonlinear resonance conditions, and found that the mechanism of nonlinear resonance is similar to the normal resonance. The frequency spectrum of ion motion in nonlinear field includes not only the natural frequency series but also nonlinear introduced frequency series, which provide the driving force for the nonlinear resonance. The nonlinear field and the nonlinear effects are inevitable in practical ion trap experiments. Our method provides better understanding of these nonlinear effects and would be helpful for the instrumentation for ion trap mass spectrometers.      

Ion excitation in a linear quadrupole ion trap with an added octopole field

Modeling of ion motion and experimental investigations of ion excitation in a linear quadrupole trap with a 4% added octopole field are described. The results are compared with those obtained with a conventional round rod set. Motion in the effective potential of the rod set can explain many of the observed phenomena. The frequencies of ion oscillation in the x and y directions shift with amplitude in opposite directions as the amplitudes of oscillation increase. Excitation profiles for ion fragmentation become asymmetric and in some cases show bistable behavior where the amplitude of oscillation suddenly jumps between high and low values with very small changes in excitation frequency. Experiments show these effects. Ions are injected into a linear trap, stored, isolated, excited for MS/MS, and then mass analyzed in a time-of-flight mass analyzer. Frequency shifts between the x and y motions are observed, and in some cases asymmetric excitation profiles and bistable behavior are observed. Higher MS/MS efficiencies are expected when an octopole field is added. MS/MS efficiencies (N(2) collision gas) have been measured for a conventional quadrupole rod set and a linear ion trap with a 4% added octopole field. Efficiencies are chemical compound dependent, but when an octopole field is added, efficiencies can be substantially higher than with a conventional rod set, particularly at pressures of 1.4 x 10(-4) torr or less.     

Nonlinear Effects in Paul Traps Operated in the Second Stability Region: Analytical Analysis and Numerical Verification

Paul trap working in the second stability region has long been recognized as a possible approach for achieving high-resolution mass spectrometry (MS), which however is still far away from the experimental implementations because of the narrow working area and inefficient ion trapping. Full understanding of the ion motional behavior is helpful for solving the problem. In this article, the ion motion in a superimposed octopole field, which was characterized by the nonlinear Mathieu equation, was solved analytically using Poincare-Lighthill-Kuo (PLK) method. This method equivalently described the nonlinear disturbance by an effective quadrupole field with perturbed Mathieu parameters, a _u ^′ and q _u ^′ , which would bring huge convenience in the studies of nonlinear ion dynamics and was, therefore, used for rapid evaluation of the nonlinear effects of ion motion. Fourth-order Runge-Kutta method (4th R-K) indicated the error of PLK for characterizing the frequency shift of ion motion was within 15%. Figure

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Double resonance ejection in a micro ion trap mass spectrometer

Ion ejection from a cylindrical micro ion trap by resonance excitation of the secular motion is observed to be strongly dependent on the frequency of the secular motion at resonance. Both the intensity of the ion signal and the mass resolution of the resulting mass spectrum are increased when the ion secular frequency is approximately that of a nonlinear resonance of the trap. The resonances are attributed to electrical as well as geometrical considerations.     

Nonlinear resonance effects during ion storage in a quadrupole ion trap

Contributions of higher-order fields to the quadrupolar storage field produce nonlinear resonances in the quadrupole ion trap. Storing ions with secular frequencies corresponding to these nonlinear resonances allows adsorption of power from the higher-order fields. This results in increased axial and radial amplitudes which can cause ion ejection and collision-induced dissociation (CID). Experiments employing long storage times and/or high ion populations, such as chemical ionization, ion-molecule reaction studies, and resonance excitation CID, can be particularly susceptible to nonlinear resonance effects. The effects of higher-order fields on stored ions are presented and the influence of instrumental parameters such as radiofrequency and direct current voltage (q_z and a_z values), ion population, and storage time are discussed.     

Determination of Ion Frequencies in a Quadrupole Ion Trap By Using a Fast Direct Current Pulse as Pump and a Laser Probe

A new technique has been developed which allows the direct measurement of frequencies of ions trapped in a quadrupole ion trap mass spectrometer. This pump/probe method employs a fast direct current (DC) pulse (pump) to displace a kinetically cooled ion population from the center of the trap, and a laser (probe) which recognizes when ions reappear at the center of the trap by the formation of photodissociation fragments. The translationally excited ions undergo periodic motion within the confines of the ion trap, and this periodic motion can be followed by recording the intensity of the photodissociation fragment as a function of the delay time between the DC pump and the laser probe. The DC pulse has a rise time of 15 ns; data are taken 1 ms after its application to allow stable ion motion to be sampled. Sampling of the ion cloud is done at 50 ns intervals, and fast Fourier transformation of the time-based data yields the ion frequencies and their relative magnitudes. Data are reported for ions derived from acetophenone (m/z 105) and 1,4-cyclohexadiene (m/z 80) under various trapping conditions corresponding to different Mathieu q_z values. The measured fundamental secular frequencies, f_z and f_r, are found to agree well with those predicted. The presence of higher order multipole contributions to the trapping field is evident from such ion frequencies as the drive frequency, f_RF,. The ability to measure ion frequencies under operating conditions provides a new tool for comparing simulated and experimental data. Simulation data from the program ITSIM, modified to account for the effects of collisions, are shown to predict the major frequency components observed in the experimental data.     

Ion motion in the rectangular wave quadrupole field and digital operation mode of a quadrupole ion trap mass spectrometer

A quadrupolar electric field driven by a rectangular wave voltage can be used for mass-selective storage and analysis. The ion motion in such an electric field is derived, and the stability of ions is presented in the a-q diagram that is commonly used for sinusoidal wave quadrupole mass spectrometry in association with the solution of the Mathieu equation. The pseudo-potential well is discussed in an approximation that leads to the relation of secular frequency to operating parameters. A scheme for a digital ion trap mass spectrometer is described, based on this theory. An ion optics simulation was performed to check the theory of resonant ejection, and to prove the feasibility of the mass scan method for a practical ion trap of such geometry.     

Nonlinear Ion Harmonics in the Paul Trap with Added Octopole Field: Theoretical Characterization and New Insight into Nonlinear Resonance Effect

The nonlinear harmonics within the ion motion are the fingerprint of the nonlinear fields. They are exclusively introduced by these nonlinear fields and are responsible to some specific nonlinear effects such as nonlinear resonance effect. In this article, the ion motion in the quadrupole field with a weak superimposed octopole component, described by the nonlinear Mathieu equation (NME), was studied by using the analytical harmonic balance (HB) method. Good accuracy of the HB method, which was comparable with that of the numerical fourth-order Runge-Kutta (4th RK), was achieved in the entire first stability region, except for the points at the stability boundary (i.e., β = 1) and at the nonlinear resonance condition (i.e., β = 0.5). Using the HB method, the nonlinear 3β harmonic series introduced by the octopole component and the resultant nonlinear resonance effect were characterized. At nonlinear resonance, obvious resonant peaks were observed in the nonlinear 3β series of ion motion, but were not found in the natural harmonics. In addition, both resonant excitation and absorption peaks could be observed, simultaneously. These are two unique features of the nonlinear resonance, distinguishing it from the normal resonance. Finally, an approximation equation was given to describe the corresponding working parameter, q _nr , at nonlinear resonance. This equation can help avoid the sensitivity degradation due to the operation of ion traps at the nonlinear resonance condition.

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Experimental Characterization of Secular Frequency Scanning in Ion Trap Mass Spectrometers

Secular frequency scanning is implemented and characterized using both a benchtop linear ion trap and a miniature rectilinear ion trap mass spectrometer. Separation of tetraalkylammonium ions and those from a mass calibration mixture and from a pesticide mixture is demonstrated with peak widths approaching unit resolution for optimized conditions using the benchtop ion trap. The effects on the spectra of ion trap operating parameters, including waveform amplitude, scan direction, scan rate, and pressure are explored, and peaks at black holes corresponding to nonlinear (higher-order field) resonance points are investigated. Reverse frequency sweeps (increasing mass) on the Mini 12 are shown to result in significantly higher ion ejection efficiency and superior resolution than forward frequency sweeps that decrement mass. This result is accounted for by the asymmetry in ion energy absorption profiles as a function of AC frequency and the shift in ion secular frequency at higher amplitudes in the trap due to higher order fields. We also found that use of higher AC amplitudes in forward frequency sweeps biases ions toward ejection at points of higher order parametric resonance, despite using only dipolar excitation. Higher AC amplitudes also increase peak width and decrease sensitivity in both forward and reverse frequency sweeps. Higher sensitivity and resolution were obtained at higher trap pressures in the secular frequency scan, in contrast to conventional resonance ejection scans, which showed the opposite trend in resolution on the Mini 12. Mass range is shown to be naturally extended in secular frequency scanning when ejecting ions by sweeping the AC waveform through low frequencies, a method which is similar, but arguably superior, to the more usual method of mass range extension using low q resonance ejection.

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Nonlinear effects induced in the normalized ion density in a linear trap system for a large ion cloud

In this work, we design, by means of a non‐neutral plasma method, a linear trapping model for large ion clouds, which will become the core of an atomic clock. We first obtain the geometry and electromagnetic characteristics of the ion trap. We then perform a systematic analysis describing the main parameters of the ion cloud such as size, secular frequency, and ion number per unit length of temperature. The most appropriate operation point of the ion trap in a set of these specific parameters is evaluated, and a thorough discussion is performed about how minor perturbations introduced in these parameters affect, in a nonlinear response, the performance of the trapping system in sensibility, heating, and radiofrequency potential.     

Reduction of axial kinetic energy induced perturbations on observed cyclotron frequency

With Fourier transform ion cyclotron resonance (FTICR) mass spectrometry one determines the mass-to-charge ratio of an ion by measuring its cyclotron frequency. However, the need to confine ions to the trapping region of the ion cyclotron resonance (ICR) cell with electric fields induces deviations from the unperturbed cyclotron frequency. Additional perturbations to the observed cyclotron frequency are often attributed to changes in space charge conditions. This study presents a detailed investigation of the observed ion cyclotron frequency as a function of ion z-axis kinetic energy. In a perfect three-dimensional quadrupolar field, cyclotron frequency is independent of position within the trap. However, in most ICR cell designs, this ideality is approximated only near the trap center and deviations arise from this ideal quadrupolar field as the ion moves both radially and axially from the center of the trap. To allow differentiation between deviations in observed cyclotron frequency caused from changes in space charge conditions or differences in oscillation amplitude, ions with identical molecular weights but different axial kinetic energy, and thus amplitude of z-axis motion, were simultaneously trapped within the ICR cell. This allows one to attribute deviations in observed cyclotron frequency to differences in the average force from the radial electric field experienced by ions of different axial amplitude. Experimentally derived magnetron frequency is compared with the magnetron frequency calculated using SIMION 7.0 for ions of different axial amplitude. Electron promoted ion coherence, or EPIC, is used to reduce the differences in radial electric fields at different axial positions. Thus with the application of EPIC, the differences in observed cyclotron frequencies are minimized for ions of different axial oscillation amplitudes.     

Linear ion trap with added octopole field component: the property and method

It is well known that superimposition of some positive octopole field will benefit the performance of ion trap mass analyzer. In the radial‐ejection linear ion trap (LIT), adding some octopole field component to the main quadrupole field is usually accomplished by stretching the ejection rod pair. In this study, the effect of octopole potential and some other higher order potential on the performance of LIT mass analyzer is investigated. A simple and effective method, which is to add some octopole component by building a LIT with a pair of rectangular electrodes and a pair of semi‐circular electrodes, is reported. Its properties were studied by numerical simulations and experiments. The results showed that a certain amount of positive octopole component could be produced by simply adjusting the position and width of the rectangular electrodes. A resolution of over 1200 at m/z 609 (~1600 Da/s) was observed in this type of LIT. They also performed tandem mass spectrometry well. The device with optimum geometry for ion ejection from rectangular electrodes provided comparable performance to that for ion ejection from semi‐circular electrodes. This type of LIT design is easy for fabrication and assembly. Copyright © 2015 John Wiley & Sons, Ltd.     

Linear quadrupoles with added octopole fields

Two methods of adding relatively small octopole fields to the main quadrupole field of quadrupoles and linear ion traps with cylindrical rods are investigated. The first, 'stretching' the quadrupole by moving two rods out from the axis, produces a combination of higher order fields with similar magnitudes in which the octopole field is not necessarily the greatest. The quadrupole field strength is changed significantly and a large potential appears on the axis. The second method uses rod pairs of different diameters. It adds octopole components of up to several percent while all other higher order fields remain small. An axis potential is also added, but it is only a few percent of the radio-frequency (RF) voltage and approximately equal to the strength of the octopole field. The axis potential can be removed by moving the larger rod pair out from the axis or applying unbalanced RF to the electrodes.     

Quadrupole excitation of ions in linear quadrupole ion traps with added octopole fields

Modeling and experimental studies of quadrupole excitation of ions in linear quadrupole traps with added octopole fields are described. An approximate solution to the equations of motion of ions trapped in a quadrupole with added octopole and dodecapole fields, with quadrupole excitation and damping is given. The solutions give the steady-state or stationary amplitudes of oscillation with different excitation frequencies. Trajectory calculations of the oscillation amplitudes are also presented. The calculations show that there can be large changes in the amplitude of ion oscillation with small changes in excitation frequency, on both the low and high-frequency sides of a resonance. Results of experiments with quadrupole excitation of reserpine ions in linear quadrupole traps with 2.0%, 2.6%, and 4.0% added octopole fields are given. It is found that as the excitation frequency is changed, two resonances are generally observed, which are attributed to the motion in the x and y directions. The two resonances can have quite different intensities. Sudden jumps or sharp sided resonances are not observed, although in some cases asymmetric resonances are seen. The calculated frequency differences between the two resonances are in approximate agreement with the experiments.     

Losses of ions during forward and reverse scans in a quadrupole ion trap mass spectrometer and how to reduce them

The higher order fields present in the quadrupole ion trap may have beneficial effects such as increases in mass resolution in the mass-selective instability or resonance ejection modes of operation, but may also result in losses of ions due to nonlinear resonances. In this work, the reduction in ion intensities observed in the mass spectra of polyethylene glycol (PEG 1000) has been utilized to monitor the ion losses resulting from these higher order fields during the rf voltage scans in both the forward and reverse directions. Extensive ion losses were observed in reverse rf voltage scans at q _z=0.64 (a _z=0), which corresponds to octopole resonance at β _z=1/2. The losses depended upon rf voltage scan rate and ion mass being greater for lower scan rates and lower masses. For ions of m/z 877, losses of up to 60% of the stored ions were observed at low scan rates (<1×10⁴ Da/s), but were minimal at higher scan rates. Thus, it is possible to avoid such losses during reverse scans by scanning the region q _z=0.64 at rates in excess of 4×10⁴ Da/s. In forward rf voltage scans, ion storage was considerably more reliable, with significant losses observed only at very high scan rates near the region q _z=0.78 (hexapole resonance at β _z=2/3).