Modeling of a linear ion trap with driving rectangular waveforms.

We consider the operation of a digital linear ion trap with resonant radial ejection. A sequence of rectangular voltage pulses with a dipole resonance signal is applied to the trap electrodes. The periodic waveform is piecewise constant, has zero mean, and is determined by an asymmetry parameter d $$ d $$ : one value is taken on interval 0 dT $$ \left(0, dT\right) $$ and another on dT T $$ \left( dT,T\right) $$ , where T $$ T $$ is the RF period. Ion mass scanning is performed by varying the asymmetry parameter d $$ d $$ and amplitude of the negative pulse part with time. The ion oscillation frequencies and acceptance of the linear trap are calculated. The dependence of the ion mass to charge ratio m / z $$ m/z $$ on the parameter d $$ d $$ is m / z ~ d 2 $$ m/z\sim {d}^2 $$ . The maximum value is about m / z = 30 $$ m/z=30 $$  kDa for typical parameters of the linear trap: frequency 0.5 MHz, rod radius 4 mm, and negative pulse amplitude 1 kV. The dipolar excitation frequency is 0.125 MHz at which the LIT acceptance is maximal.

Dipole and quadrupole resonance excitation in linear quadrupoles.

In the development of commercial quadrupole mass spectrometers, there is an interest in improving the performance characteristics such as transmission, resolution, and mass range. In particular, parametric and dipolar resonance excitation of trapping ions are used for linear quadrupole mass filters. Theoretical methods and numerical simulation of ion trajectories were applied for study of ion-optical properties. The review is devoted to description of different excitation methods to improve QMF performance and consists of three parts. The first part presents the results of a linear ion trap simulation for various operating conditions and excitation methods. The second part considers the effects of dipole excitation (DE) on the performance of the quadrupole mass filter. The last part analyzes the formation of stability islands by different methods of quadrupole excitation. To date conditions of mass separation in quadrupole mass filters with sin wave supply were described for stability islands of the first and third stability regions formed by quadrupole and DE. By complicating the electronics such methods allow to overcome the destructive influence of electric field distortions and obtain a resolving power and ion transmission efficiency comparable with commercial devices. At quadrupole resonance excitation by a two-frequency signal, it is possible to reduce the length of electrodes three times without losses in resolution and transmission, which reduces the cost of rod set production with micrometer accuracy. Dipole resonance excitation allows controlling the shape of the mass peak by changing amplitude and phase of the auxiliary AC signal. The main factors affecting the resolving power of a linear ion trap are described theoretically. The numerical modeling results are confirmed by experiment.

Modelling of a linear ion trap operation in the second stability region.

Ion trajectory numerical simulation is used to find the linear ion trap excitation contour in the second stability region. The effects of initial conditions, the ejection Mathieu parameter, scan speed, dipole excitation voltage and gas damping are studied. Modeling shows that in the stability region center the resolution power is ≈ 200 000 (at full width half height of a peak, FWHM) at pressure 0.1 mTorr and 100 % excitation efficiency (not taking into account the space charge).

Modeling dipolar excitation for quadrupole mass filter.

The results of modeling AC and DC dipole excitation of ion oscillations in a quadrupole mass filter are presented. The simulation is done by numerical integration of the ion motion equations, ions' initial coordinates and velocities are distributed normally. For AC dipole excitation the instability bands on the (a, q) stability diagram follow along the isolines ßx/2 and ßy/2, creating regular dips on the transmission contour. We show that AC excitation at frequency Ω/2 makes it possible to control the resolution of the mass filter by either changing the AC amplitude or phase. Instability bands can be used for mass selective excitation in a linear ion trap. Options for the joint use of DC and AC dipole excitation to form the mass peak shape are considered.

A simulation study of excitation contour of a linear trap with spatial harmonics.

The process of nonlinear resonant excitation of ion oscillations in a linear trap is studied. There is still no detailed simulation of the resonance peak in the literature. We propose to use the excitation contour to describe the collective ion resonance. The excitation contour is a resonant mass peak obtained by the trajectory method with the Gaussian distribution of the initial coordinates and velocities. The following factors are considered: excitation time, low order hexapole and octopole harmonics with amplitudes A3 and A4, the depth of the initial ion cloud position. These multipoles are used for selective ion ejection from linear ion trap. All these factors affect the ion yield and the shape of the contours. Obtained data can be useful for control of such processes as ion fragmentation, ion isolation, ion activation, and ion ejection. Simulated resonance peaks are important for the theoretical description of the ion collective nonlinear resonances.

Stable X-islands of quadrupole mass filter.

Quadrupole mass filters are normally operated as narrow band pass filters by appropriate choices of rf and DC voltages corresponding to Mathieu a and q values near the apex of the first stability region. We add an auxiliary quadrupole excitation potential to the main drive voltage. As a result, stability islands appear on the (a,q) plane. The method of the islands mapping on the (a,q) plane is discussed in detail. The DC electric field's effect responsible for removing "shadowing" islands has been studied using a combination of analytic theory and computer modeling. We call a narrow stability region elongated along an iso-ßx line an X-island. Two such stability islands were found where operation is possible without mass spectra interference when the DC potential is used. Those islands are formed by quadrupole potentials with relative excitation frequencies ν=ß,ν=1±ß and ν=2±ß, where ß << 1. Many other stability X-islands are presented in detail and illustrated by their transmission contours. This data is necessary for experimental testing of the separation mode at those islands. The study demonstrates how to use X-islands to achieve relatively high resolution 4000-12,000 at the transmission of about 28-15%, respectively.

Quadrupole mass filter acceptance in stability island created by double-frequency quadrupole excitation.

Increase in quadrupole mass filter resolution at separation in narrow band stability island (X-band) formed by biharmonic resonance excitation of ion oscillation is discussed. X-band and the normal working quadrupole mass filter modes are compared at theoretical resolution of 10,000 and different separation times. Transmission curves, acceptance ellipses parameters, and acceptance characteristics are obtained by numerical simulation. Transmission coefficients are approximately the same in both modes. Dependence of acceptance ellipses parameters on ion inlet phases has a complicated oscillating form in the X-band mode. Acceptance contours calculated for given transition levels have been compared. At low acceptance level, the combined acceptance in the X-band mode was found to be one order of magnitude higher than in the normal mode.

Simulation of the simultaneous dual-frequency resonance excitation of ions in a linear ion trap.

The process of ion resonance dipolar excitation in a linear ion trap by 2 ejection waveforms with close frequencies is studied. The physical mechanism of increasing the resolving power using the ion excitation is a nonlinearity of the electric radio frequency fields caused by space charge. Using 2 resonance forces with 2 close frequencies leads to the completion of 2 excitation processes. In the case of the perfect quadrupole electric field, the ion motion equations are linear, and as a result, the respondent ion ensemble is also a linear and valid superposition principle. Nevertheless, the resolution increases (20%) in the case of lack of a space charge in an operating mode with a dual-frequency. The numerical simulations show that the mass shift is removed, and the mass resolution is increased via dual-frequency resonance excitation when the frequency difference (approximately 2.5 kHz) is relatively small and the phase difference of 2 harmonic signals is 0-π3 even at a high linear ion density of up to 50 000 ions per radius field r0 .

4D phase-space acceptance of the quadrupole mass filter.

Simulations of the four-dimensional (4D) phase-space acceptance volume of a quadrupole mass filter (QMF) are discussed. The 4D acceptance is considered since the ion trajectories in the X and Y phase planes are dependent via the initial RF phase at ion entry into QMF. The QMF parameters are set up for resolution equal to the ion mass number M. For a wide range of ion masses, the acceptance is characterized by relatively large aperture with about 5% transmission, primarily defined by phase dependent ellipses. Contrary to expectations, the small-aperture central spot with 75% transmission accepts a very small portion, namely less than 1% of the passed through particles.

Mass resolution of linear quadrupole ion traps with round rods.

RATIONALE: Auxiliary dipole excitation is widely used to eject ions from linear radio-frequency quadrupole ion traps for mass analysis. Linear quadrupoles are often constructed with round rod electrodes. The higher multipoles introduced to the electric potential by round rods might be expected to change the ion ejection process. We have therefore investigated the optimum ratio of rod radius, r, to field radius, r0, for excitation and ejection of ions. METHODS: Trajectory calculations are used to determine the excitation contour, S(q), the fraction of ions ejected when trapped at q values close to the ejection (or excitation) q. Initial conditions are randomly selected from Gaussian distributions of the x and y coordinates and a thermal distribution of velocities. The N = 6 (12 pole) and N = 10 (20 pole) multipoles are added to the quadrupole potential. Peak shapes and resolution were calculated for ratios r/r0 from 1.09 to 1.20 with an excitation time of 1000 cycles of the trapping radio-frequency. RESULTS: Ratios r/r0 in the range 1.140 to 1.160 give the highest resolution and peaks with little tailing. Ratios outside this range give lower resolution and peaks with tails on either the low-mass side or the high-mass side of the peaks. This contrasts with the optimum ratio of 1.126-1.130 for a quadrupole mass filter operated conventionally at the tip of the first stability region. With the optimum geometry the resolution is 2.7 times greater than with an ideal quadrupole field. Adding only a 2.0% hexapole field to a quadrupole field increases the resolution by a factor of 1.6 compared with an ideal quadrupole field. Addition of a 2.0% octopole lowers resolution and degrades peak shape. With the optimum value of r/r0 , the resolution increases with the ejection time (measured in cycles of the trapping rf, n) approximately as R0.5 = 6.64n, in contrast to a pure quadrupole field where R0.5 = 1.94n. CONCLUSIONS: Adding weak nonlinear fields to a quadrupole field can improve the resolution with mass-selective ejection of ions by up to a factor of 2.7. The optimum ratio r/r0 is 1.14 to 1.16, which differs from the optimum ratio for a mass filter of 1.128-1.130.

Mass selectivity of dipolar resonant excitation in a linear quadrupole ion trap.

RATIONALE: For mass analysis, linear quadrupole ion traps operate with dipolar excitation of ions for either axial or radial ejection. There have been comparatively few computer simulations of this process. We introduce a new concept, the excitation contour, S(q), the fraction of the excited ions that reach the trap electrodes when trapped at q values near that corresponding to the excitation frequency. METHODS: Ion trajectory calculations are used to calculate S(q). Ions are given Gaussian distributions of initial positions in x and y, and thermal initial velocity distributions. To model gas damping, a drag force is added to the equations of motion. The effects of the initial conditions, ejection Mathieu parameter q, scan speed, excitation voltage and collisional damping, are modeled. RESULTS: We find that, with no buffer gas, the mass resolution is mostly determined by the excitation time and is given by R~dß/dq qn, where ß(q) determines the oscillation frequency, and n is the number of cycles of the trapping radio frequency during the excitation or ejection time. The highest resolution at a given scan speed is reached with the lowest excitation amplitude that gives ejection. The addition of a buffer gas can increase the mass resolution. The simulation results are in broad agreement with experiments. CONCLUSIONS: The excitation contour, S(q), introduced here, is a useful tool for studying the ejection process. The excitation strength, excitation time and buffer gas pressure interact in a complex way but, when set properly, a mass resolution R0.5 of at least 10,000 can be obtained at a mass-to-charge ratio of 609.

Trajectory calculations of space-charge-induced mass shifts in a linear quadrupole ion trap.

RATIONALE: If too many ions are stored in a linear quadrupole ion trap, space charge causes the oscillation frequencies to decrease. Ions therefore appear at higher apparent mass-to-charge ratios in a mass spectrum. To further understand this process, we have used trajectory calculations of ions to determine mass shifts. METHODS: Two models of the ion cloud are used. The first assumes that the acceptance of the quadrupole is uniformly filled with ions. The second assumes that the ions have a thermal distribution trapped in the effective potential. Both give analytical descriptions of the field from space charge. Ion trajectories are calculated with and without space charge. Oscillation frequencies are determined with a Fourier transform. Shifts in oscillation frequency with space charge are then used to calculate mass shifts. RESULTS: Both ion cloud models give similar results. More diffuse ion clouds or ion clouds that have higher temperatures produce lower electric fields near the center of the trap and hence lower mass shifts. Space charge produces a nonlinear field. As a result, the discrete resonant frequencies of ions in a pure quadrupole field become distributions of frequencies. Comparisons with experiments show agreement for reasonable values of the parameters of the two ion cloud models. CONCLUSIONS: This relatively simple method for calculating the effects of space charge shows (i) that the spread of oscillation frequencies reduces mass resolution with axial ejection and (ii) that mass shifts are reduced with ion clouds with greater spatial extents or higher ion temperatures.

Upper stability island of the quadrupole mass filter with amplitude modulation of the applied voltages.

Modulation of the voltages applied to a quadrupole mass filter (QMF), either RF or RF and DC, leads to splitting of the stability region into islands of stability. The ion optical properties, such as transmission, resolving power and peak tails of the first upper stability islands have been investigated by numerical simulation of ion trajectories. The dependence of the location of this island on the amplitude of the modulation and the parameter nu = omega/Omega = Q/P where omega is modulation frequency, Omega is main angular radio frequency, and Q and P are integers, is calculated in detail. Different methods of adjusting the QMF resolution are examined. It is found that operation at the upper and lower tips of the stability islands created by amplitude modulation of the RF voltage is preferred, because of the technical simplicity of this method and a reduction of the required separation time. Amplitude modulation improves the performance of a QMF constructed with round rods, in comparison to perfect quadrupole fields. For example, with amplitude modulation of the RF, to reach a resolution of R(0.1) = 1200 requires only about 75 RF cycles of ion motion in a quadrupole field created by round rods.

Quadrupole mass filters with octopole fields.

The performance of quadrupole mass filters with added octopole fields in the range 2.0-4.0% has been investigated. The added fields are much greater than those normally added to conventional rod sets by mechanical tolerances or construction errors. Quadrupole rod sets with added octopole fields were constructed with round rods by making one pair of rods greater in diameter than the other pair. For positive ions, resolution at half height of only about 200 is possible if the negative direct current (dc) output of the quadrupole power supply is connected to the smaller rods. If the positive dc output of the quadrupole power supply is connected to the smaller rods, the resolution improves dramatically; a resolution at half height of 5800 has been observed with a rod set with 2.6% added octopole field. For negative ions the best resolution is obtained with the polarity of the dc reversed, i.e. with the negative dc applied to the smaller rods. These findings are unexpected in view of the literature that argues that to obtain high mass resolution with quadrupole mass filters, higher order multipoles must be kept as small as possible. Numerical simulations of peak shapes agree qualitatively with experiments. Simulation of the boundaries of the first stability region for positive ions shows that when the positive dc is applied to the smaller rods, the addition of a 2.0% octopole field causes the boundaries to shift slightly but the boundaries are well defined, and the tip of the stability region remains sharp. When the positive dc is applied to the larger rods, the boundaries of the stability region move out and become diffuse. For instruments that require a rod set that can be used both as a linear trap and a mass filter, these rod sets may offer improved trap performance while still being capable of providing conventional mass analysis.

Influence of the 6th and 10th spatial harmonics on the peak shape of a quadrupole mass filter with round rods.

The influence of the ratio of the rod radius, r, to field radius, r(0), on the peak shape for a linear quadrupole mass filter constructed with round rods has been investigated. The expansion of the potential in multipoles, phi(N),Phi(x, y) = sum(infinity)(N=0)A(N)phi(N)/r(N)(0) has been considered, and the peak shape and resolution have been determined by numerical calculation of ion trajectories in quadrupoles with different ratios, r/r(0). Geometries that make the dodecapole term (A(6)) zero (r/r(0) = 1.14511) do not give the best performance because the contribution of the 20-pole term, A(10), must be considered. The optimum ratio is r/r(0) approximately 1.13. With this ratio the dodecapole term (A(6)) is ca. 1 x 10(-3), but its effects are partially compensated by the A(10) term which has similar magnitude, but opposite sign.

Matrix methods for the calculation of stability diagrams in quadrupole mass spectrometry.

The theory of the computer calculation of the stability of ion motion in periodic quadrupole fields is considered. A matrix approach for the numerical solution of the Hill equation and examples of calculations of stability diagrams are described. The advantage of this method is that it can be used for any periodic waveform. The stability diagrams with periodic rectangular waveform voltages are calculated with this approach. Calculations of the conventional stability diagram of the 3-D ion trap and the first six regions of stability of a mass filter with this method are presented. The stability of the ion motion for the case of a trapping voltage with two or more frequencies is also discussed. It is shown that quadrupole excitation with the rational angular frequency omega = Nomega/P (where N, P are integers and omega is the angular frequency of the trapping field) leads to splitting of the stability diagram along iso-beta lines. Each stable region of the unperturbed diagram splits into P stable bands. The widths of the unstable resonance lines depend on the amplitude of the auxiliary voltage and the frequency. With a low auxiliary frequency splitting of the stability diagram is greater near the boundaries of the unperturbed diagram. It is also shown that amplitude modulation of the trapping RF voltage by an auxiliary signal is equivalent to quadrupole excitation with three frequencies. The effect of modulation by a rational frequency is similar to the case of quadrupole excitation, although splitting of the stability diagram differs to some extent. The methods and results of these calculations will be useful for studies of higher stability regions, resonant excitation, and non-sinusoidal trapping voltages.