Collisional activation of ions in RF ion traps and ion guides: The effective ion temperature treatment

Ion transfer and storage using inhomogeneous radio frequency (RF) electric fields in combination with gas-assisted ion cooling and focusing constitutes one of the basic techniques in mass spectrometry today. The RF motion of ions in the bath gas environment involves a large number of ion-neutral collisions that leads to the internal activation of ions and their effective "heating" (when a thermal distribution of internal energies results). The degree of ion activation required in various applications may range from a minimum level (e.g., slightly raising the average internal energy) to an intense level resulting in ion fragmentation. Several research groups proposed using the effective temperature as a measure of ion activation under conditions of multiple ion-neutral collisions. Here we present approximate relationships for the effective ion temperature relevant to typical operation modes of RF multipole devices. We show that RF ion activation results in near-thermal energies for ions occupying an equilibrium position at the center of an RF trap, whereas increased ion activation can be produced by shifting ions off-center, e.g., by means of an external DC electric field. The ion dissociation in the linear quadrupole ion trap using the dipolar DC ion activation has been observed experimentally and interpreted in terms of the effective ion temperature.     

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.     

Dipole excitation of ions in linear radio frequency quadrupole ion traps with added multipole fields

Ion motion with auxiliary dipole excitation and collisional damping in a linear radiofrequency quadrupole ion trap incorporating small amounts of even higher order multipoles is studied analytically. The ion motion is modeled in a pseudopotential that is mostly quadratic with small amounts of higher spatial harmonics. Ion motion along x and y axes is characterized by two uncoupled forced and damped anharmonic oscillator equations. A multiple time scales method is used to solve the equations of motion of ions with a first order perturbation correction. Analytical relations between the oscillation amplitudes at steady state (the stationary amplitudes) and excitation frequency are calculated. The frequency response curves show that in some cases bistable behavior might be obtained, i.e., there are two stable stationary amplitudes for a given excitation frequency.     

Simulated ion trajectory and induced signal in ion cyclotron resonance ion traps

We present a numerical method for computation of electrostatic (trapping) and time-varying (excitation) electric fields and the resulting ion trajectory and detected time-domain-induced voltage signal in a rectangular (or cubic) ion cyclotron resonance (ICR) ion trap. The electric potential is calculated by use of the superposition principle and relaxation method with a large number of grid points (e.g., 100 × 100 × 100 for a cubic trap). Complex ICR experiments and spectra may now be simulated with high accuracy. Ion trajectories may be obtained for any combination of trapping and excitation modes, including quadrupolar or cubic trapping in static or dynamic mode; and dipolar, quadrupolar, or parametric excitation with single-frequency, frequency-sweep (chirp), or stored waveform inverse Fourier transform waveforms. The resulting ion trajectory may be represented either as its three dimensional spatial path or as two-dimensional plots of x-, y-, or z-position, velocity, or kinetic energy versus time in the absence or presence of excitation. Induced current is calculated by use of the reciprocity principle, and simulated ICR mass spectra are generated by Fourier transform of the corresponding time-domain voltage signal.     

Theoretical investigation on confinement of ions in a cube‐shaped ion trap

The confinement of ions in a cube‐shaped ion trap and the mathematical formalism governing the behavior of ions in the trap is investigated theoretically. Afterwards, the stability regions are computed using the fourth‐order Rung–Kutta method. Consequently, the influence of the direction of ions, injected into the trap from its center on the stability region, is numerically discussed. Moreover, the maximum angle of injection with respect to the vertical axis of the cube for which the ions could be confined in the trap without invoking any direct current component of voltage (henceforth referred to as limiting angle) was calculated. Strong linear correlation between the angle of injection and the ratio of the stability region areas is confirmed. A nonlinear feature of a cube‐shaped ion trap is demonstrated with a focus on the equations of motion for an ion confined into the trap. It is worthwhile to note that the stability region of our cubic ion trap, which has its own boundary conditions and electrodynamics, has been theoretically investigated for the first time. Besides, the limiting angle as well as the aforementioned strong linear correlation has not been reported in the literature previously. Copyright © 2015 John Wiley & Sons, Ltd.     

Characteristics of electrical field and ion motion in surface‐electrode ion traps

In this article, we calculated the potential function of the surface‐electrode ion trap (SEIT) by using Green's function method, optimized trap size, obtained the coefficients of the multipoles and analyzed ion trajectories in the RF potential. The optimized SEIT not only increases its trapping well depth by a factor of about 15, but also has relatively good linearity of the field (or large quadrupole component). The current design of SEIT can work well either as the ion guide for ion transmission or as the ion trap for ion confinement. Our research can be used to calculate the potential function in the SEIT with different device parameters, understand ion motions in the traps and optimize instrument performance. The method for calculating potential function can be expanded to planar and halo ion traps. Copyright © 2012 John Wiley & Sons, Ltd.     

Radial stratification of ions as a function of mass to charge ratio in collisional cooling radio frequency multipoles used as ion guides or ion traps

Collisional cooling in radio frequency (RF) ion guides has been used in mass spectrometry as an intermediate step during the transport of ions from high pressure regions of an ion source into high vacuum regions of a mass analyzer. Such collisional cooling devices are also increasingly used as 'linear', two-dimensional (2D) ion traps for ion storage and accumulation to achieve improved sensitivity and dynamic range. We have used the effective potential approach to study m/z dependent distribution of ions in the devices. Relationships obtained for the ideal 2D multipole demonstrate that after cooling the ion cloud forms concentric cylindrical layers, each of them composed of ions having the same m/z ratio; the higher the m/z, the larger is the radial position occupied by the ions. This behavior results from the fact that the effective RF focusing is stronger for ions of lower m/z, pushing these ions closer to the axis. Radial boundaries of the layers are more distinct for multiply charged ions, compared to singly charged ions having the same m/z and charge density. In the case of sufficiently high ion density and low ion kinetic energy, we show that each m/z layer is separated from its nearest neighbor by a radial gap of low ion density. The radial gaps of low ion population between the layers are formed due to the space charge repulsion. Conditions for establishing the m/z stratified structure include sufficiently high charge density and adequate collisional relaxation. These conditions are likely to occur in collisional RF multipoles operated as ion guides or 2D ion traps for external ion accumulation. When linear ion density increases, the maximum ion cloud radius also increases, and outer layers of high m/z ions approach the multipole rods and may be ejected. This 'overfilling' of the multipole capacity results in a strong discrimination against high m/z ions. A relationship is reported for the maximum linear ion density of a multipole that is not overfilled.     

Kinetic theory of radio frequency quadrupole ion traps. I. Trapping of atomic ions in a pure atomic gas

A kinetic theory based on the Boltzmann equation is developed for the trapping of atomic ions in a radio-frequency quadrupole ion trap containing enough neutral atoms that ion-neutral collisions cannot be ignored. The collisions are treated at the same level of sophistication and detail as is used to deal with the time- and space-dependent electric fields in the trap. As a result, microscopic definitions are obtained for the damping and stochastic forces that originate from such collisions. These definitions contrast with corresponding phenomenological terms added ad hoc in previous treatments to create damped Mathieu and Langevin equations, respectively. Furthermore, the theory indicates that either collisional cooling or heating of the ions is possible, depending upon details of the ion-neutral mass ratios and interaction potential. The kinetic theory is not dependent on any special assumptions about the electric field strengths, the ion-neutral interaction potentials, or the ion-neutral mass ratio. It also provides an ab initio way to describe the ion kinetic energies, temperatures, and other properties by a series of successive approximations.     

Computer simulation of single-ion trajectories in paul-type ion traps

The computer simulation of single-ion trajectories using a number of computer programs is described together with associated theory. The programs permit calculation of ion trajectories while the ion is subjected to collisions with buffer gas of variable pressure, resonance excitation in any of three modes, and static or ramped DC and radiofrequency levels. Initially, the programs were designed for the calculation of ion trajectories in a quadrupole ion trap. The programs now permit such calculations for ions confined in traps having electrodes shaped to include percentages of hexapole and octupole components in the electric field as well as electrode surface geometries for which there is no closed-form expression. The Langevin collision theory is reviewed and a theoretical treatment of the multipole trap is presented.     

Numerical observation of preferred directionality in ion ejection from stretched rectilinear ion traps

We report on numerical investigations of directionality of ion ejection in stretched rectilinear ion traps (RIT). Three 4-electrode trap geometries have been investigated. In all cases, one pair of electrodes has slits at their center and the other pair has no slits. The studied traps include the RIT-S, in which the mass analyzer electrodes are symmetrically positioned around the central axis; the RIT-X, in which the mass analyzer has a stretch in the direction of the electrodes which have slits (labeled as x-direction); and the RIT-Y, in which the mass analyzer has a stretch in the direction of the electrodes which have no slits (labeled as y-direction).Our analysis has been carried out on two-dimensional (2D) fields at the centre of an infinitely long mass analyzer. The boundary element method (BEM) has been used for field computations. The trajectory of ion motion has been generated using Runge Kutta fourth order integration.Three sets of simulations have been carried out on each of the RIT-S, the RIT-X and the RIT-Y to check for directionality of ion ejection. In the first, we numerically obtain the stability region on the potential (U_dc– V_rf) axes. In the second we generate an escape velocity plot with U_dc=0 for different values of V_rf. In the third, we simulate the mass selective boundary ejection experiment on a single ion.In the symmetric RIT-S, as expected, all three simulations show that there is an equal probability of ion reaching the trap boundary in either of the x- or y-directions. For the stretched traps, however, the results are dramatically different. For the RIT-X, all three simulations suggest that ion destabilization at the stability boundary occurs in the x-direction. Similarly, for the RIT-Y, ions preferentially get destabilized in the y-direction. That is, ions reaching the trap boundary overwhelmingly prefer the stretch direction.     

Energy landscapes of ion clusters in isotropic quadrupolar and octupolar traps

The energy landscapes of ion clouds confined in isotropic quadrupolar and octupolar traps are characterized for several representative cluster sizes. All clusters exhibit stable multishell structures that belong to separate funnels. Quadrupolar confinement leads to more homogeneous clusters and denser distributions of isomers than octupolar confinement. Statistical analysis of the transition states indicates that the barriers associated with intrashell motion are lower but more asymmetric and more cooperative compared to intershell motion. The relaxation between low-energy funnels with different arrangements of shells mostly exhibits Arrhenius kinetics, with a weak variation of the activation energy at higher temperatures.     

Prediction of collective characteristics for ion ensembles in quadrupole ion traps without trajectory simulations

Fundamental aspects are presented of a two-temperature moment theory for quadrupole ion traps developed via transformation of the Boltzmann equation. Solutions of the moment equations correspond to changes in the ensemble average for any function of ion velocity, because the Boltzmann equation reflects changes to an ion distribution as a whole. The function of primary interest in this paper is the ion effective temperature and its behavior during ion storage and resonance excitation. Calculations suggest that increases in ion effective temperature during resonance excitation are due primarily to power absorption from the main RF trapping field rather than from the dipolar excitation signal. The dipolar excitation signal apparently serves mainly to move ions into regions of the ion trap where the RF electric field, and thus ion RF heating, is greater than near the trap center. Both ideal and non-ideal ion trap configurations are accounted for in the moment equations by incorporating parameterized variables a and q , which are modified versions of the commonly used forms for the DC and AC ring voltages, and b and d , which are new forms that account for the voltages applied to the endcaps. Besides extending the applicability of the moment equations to non-ideal quadrupole ion traps, the modified versions of the parameterized variables can have additional utility. Calculation of the spatial dependence of ion secular oscillation frequencies is demonstrated as an example.     

Theoretical and numerical analysis of the behavior of ions injected into a quadrupole ion trap mass spectrometer

A numerical simulation method has been developed for the analysis of trapping ions injected into an ion trap mass spectrometer. This method was applied to clarify the effects of the following parameters on trapping efficiencies: (1) initial phase of the radio frequency (RF) drive voltage, (2) ion injection energy, and (3) RF peak voltage while injecting ions. The following conclusions were obtained by theoretical and simulation approaches. 1. The second and third dominant oscillations contribute significantly to the trapping mechanism of the injected ions, even for low q values. 2. A formula relating the operating parameters, which gives the maximum trapping efficiency, is derived. 3. Based on the above-mentioned formula, an advanced injection method is proposed, in which the RF peak voltage is decreased while injecting ions. The ability of this method to solve the problem of unequal sensitivity among different ion species is indicated by numerical simulation. Copyright 2000 John Wiley & Sons, Ltd.     

Comparison of equilibrium ion density distribution and trapping force in penning, Paul, and combined ion traps

Starting from the classical Boltzmann distribution, we obtain the ion density distribution in the limit of either high temperature/low density (Coulomb interaction energy much less than ion kinetic energy) or low temperature/high density (kinetic energy much less than Coulomb interaction energy), and the trapping force for an ion cloud in Penning ion cyclotron resonance, Paul (quadrupole), and combined (Paul trap in a uniform axial static magnetic field) traps. At equilibrium (total angular momentum conserved), the ion cloud rotates at a constant frequency in Penning and combined traps. In a Penning trap, the maximum ion density is proportional to B ²/m (B is magnetic field and m is the mass of ions), whereas the maximum ion density in a Paul trap is proportional to (V _rf ² /mΩ² r ₀ ⁴ ), with Mathieu equation axial q value <0.4 to satisfy the pseudopotential approximation. Ion maximum densities in both Penning and Paul ion traps depend on the trapping field (magnetic or electric) and ion mass, but not on ion charge. In a Penning trap at maximum ion density (zero pressure), the radial (but not the axial) trapping potential is mass dependent, whereas both radial and axial potentials in a Paul trap at maximum ion density are mass dependent.     

Theoretical notions of aromaticity and antiaromaticity: phenalenyl ions versus fluorenyl ions

The dications 6, 7, and 8 and dianions 9, 10, and 11 of the bistricyclic aromatic enes bifluorenylidene (1), 1,1'-biphenalenylidene (2), and 9-(9H-fluoren-9-ylidene)-1H-phenalene (4), as well as monocations 12a and 13a and monoanions 14a and 15a of phenalene (3) and fluorene (5), were subjected to a systematic DFT and ab initio study. B3LYP and MP2 methods were employed to estimate the relative aromaticity/antiaromaticity of these ions, using energetic, magnetic, and structural criteria. The couplings of monoions 12a-15a to give the respective diions 6-11 result in a similar destabilization in both the fluorene and phenalene series. The interactions between the C13H8 units in diions 6-11 are weak and are not expected to result in a significant loss of aromaticity/gain of antiaromaticity, as compared with the respective monoions. The antiaromaticity of bifluorenylidene dication (6), relative to that of two fluorenyl cations (12a), is only slightly enhanced as compared with the aromaticity of biphenalenylidene dication ((E)-7)) relative to that of two phenalenyl cations (13a). In particular, the homodesmotic reaction 6 + 2.13a = (E)-7 + 2.12a is only slightly exothermic, DeltaE(Tot) = -6.0 kJ/mol. The energetic effect of the analogous reaction involving anions 9 + 2.15a = (E)-10 + 2.14a is even smaller, DeltaE(Tot) = -3.4 kJ/mol. Bifluorenylidene dianion (9) and 1,1'-biphenalenylidene dianion ((E)-10) are aromatic, but the employed criteria disagree about their relative aromaticity. The electronic and structural properties of heteromerous dication 8 and dianion 11 lie between those of the homomerous diions. Thus, dications 6-8 and dianions 9-11 form a continuum of aromaticity/antiaromaticity.