Fourier Transform Ion Cyclotron Resonance Mass Resolution and Dynamic Range Limits Calculated by Computer Modeling of Ion Cloud Motion

Particle-in-Cell (PIC) ion trajectory calculations provide the most realistic simulation of Fourier transform ion cyclotron resonance (FT-ICR) experiments by efficient and accurate calculation of the forces acting on each ion in an ensemble (cloud), including Coulomb interactions (space charge), the electric field of the ICR trap electrodes, image charges on the trap electrodes, the magnetic field, and collisions with neutral gas molecules. It has been shown recently that ion cloud collective behavior is required to generate an FT-ICR signal and that two main phenomena influence mass resolution and dynamic range. The first is formation of an ellipsoidal ion cloud (termed “condensation”) at a critical ion number (density), which facilitates signal generation in an FT-ICR cell of arbitrary geometry because the condensed cloud behaves as a quasi-ion. The second phenomenon is peak coalescence. Ion resonances that are closely spaced in m/z coalesce into one resonance if the ion number (density) exceeds a threshold that depends on magnetic field strength, ion cyclotron radius, ion masses and mass difference, and ion initial spatial distribution. These two phenomena decrease dynamic range by rapid cloud dephasing at small ion density and by cloud coalescence at high ion density. Here, we use PIC simulations to quantitate the dependence of coalescence on each critical parameter. Transitions between independent and coalesced motion were observed in a series of the experiments that systematically varied ion number, magnetic field strength, ion radius, ion m/z, ion m/z difference, and ion initial spatial distribution (the present simulations begin from elliptically-shaped ion clouds with constant ion density distribution). Our simulations show that mass resolution is constant at a given magnetic field strength with increasing ion number until a critical value (N) is reached. N dependence on magnetic field strength, cyclotron radius, ion mass, and difference between ion masses was determined for two ion ensembles of different m/z, equal abundance, and equal cyclotron radius. We find that N and dynamic range depend quadratically on magnetic field strength in the range 1–21 Tesla. Dependences on cyclotron radius and Δm/z are linear. N depends on m/z as (m/z)^–2. Empirical expressions for mass resolution as a function of each of the experimental parameters are presented. Here, we provide the first exposition of the origin and extent of trade-off between FT-ICR MS dynamic range and mass resolution (defined not as line width, but as the separation between the most closely resolved masses).     

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.      

Peak splitting with a quadrupole mass filter operated in the second stability region

Peak splitting or structure has been studied for a quadrupole mass filter operated in the second stability region with Mathieu parameters (a,q) ≈ (0.02.7.55). Two sources of peak splitting are considered and modeled theoretically: nonlinear resonances caused by field imperfections and ion collection effects caused by the periodic properties of ion motion in the quadrupole field. The conditions for the appearance of the nonlinear resonances and ion collection effects are derived and presented in terms of the β variables which determine the frequencies of ion oscillation. Comparisons of calculated peak structure to experimental peak shapes show that ion collection effects dominate, at least for the experimental conditions reported here. It is also shown that neither nonlinear resonances nor ion collection effects can distort the peak at high resolution.     

Thomas-Fermi theory for the ion cloud in the trap

Using Thomas-Fermi Theory, we deal with the motion of a trapped ion cloud with conserved total angular momentum in thermal equilibrium at zero temperature. It is shown that in the case of high magnetic field (≧600 kG) the motion of an electron cloud trapped in a Penning trap would be modified by quantum effects. Under the present laboratory conditions the quantum effects are negligible for the motion of a Mg⁺²⁴ ion cloud trapped in a Paul trap.     

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.

Off‐center shallow donors in a spherical Si quantum dot with dielectric border

Within the effective mass approximation and using a finite element method, the ground state energy and electron cloud localization of the shallow donors in a Si quantum dot (QD) with dielectric border are calculated. Simultaneous effects of dielectric mismatch (DM) at the core–shell interface, the impurity radial position, and the external electric field on the electronic properties are investigated. We found that (i) for a freestanding QD, the binding energy is strongly enhanced due to the additional interactions of the electron with the polarization charges; (ii) the electron cloud distribution can be easily modulated by varying the impurity position; (iii) the electric field‐induced shift in energy levels increases with the DM. Therefore, the electronic energy levels of the nanocrystal could be tuned by properly tailoring the heterostructure parameters (DM with the surrounding matrix, impurity location) as well as by varying the electric field strength. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

The coupling effects of hexapole and octopole fields in quadrupole ion traps: a theoretical study

A theoretical method, the harmonic balance method, was introduced to study the coupling effects of hexapole and octopole fields on ion motion in a quadrupole ion trap. Ion motion characteristics, such as ion motion center displacement, ion secular frequency shift, nonlinear resonance curve and buffer gas damping effects, have been studied with the presence of both hexapole and octopole fields. It is found that hexapole fields have bigger impacts on ion motion center displacement, while octopole fields dominate ion secular frequency shift. Furthermore, the nonlinear features originated from hexapole and octopole fields could enhance or cancel each other, which provide us more space in a practical ion trap design process. As an example, an ion trap with improved performance was designed using a specific combination of hexapole and octopole fields. In this ion trap, a hexapole field was used to achieve efficient ion directional ejection, while an octopole field was added to correct the chemical mass shift and resolution degradation introduced by the hexapole field. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

Improved isotopic abundance measurements for high resolution fourier transform ion cyclotron resonance mass spectra via time-domain data extraction

The simultaneous high resolution and accurate mass measurements possible with Fourier transform ion cyclotron resonance mass spectrometry coupled with the gentle ionization of electrospray hold attractions for protein, peptide, and oligonucleotide characterization, including multistage-mass spectrometry measurements for assignment of fragment masses and greater confidence in structural measurements. The detection of cyclotron motion over extended periods of time (in some cases for several minutes) allows higher resolution and mass accuracy. Generally, signal duration has been considered to be limited primarily by background pressure, with ion-neutral collisions leading to the reduction and dephasing of cyclotron motion, causing signal loss. However, recent theoretical work has shown that the ion cloud stability that is a prerequisite for high performance measurements is highly dependent on the electric field generated by the ion cloud, thus giving rise to a minimum number of charges or ions required for extended time-domain signals. The effects of ion population on ion cloud stability and signal duration, and the subsequent effects on resolution and measured isotopic abundances are reported. Individual time-domain signals for bovine insulin isotopic peaks were extracted to allow a comparison of the damping rates for each of these ion clouds and the measured time-domain amplitude maxima are shown to provide a better match with the theoretically predicted isotopic abundances for insulin. These results show that different damping rates of ions of very similar mass, but different ion cloud population sizes, can have dramatic effects on the observed isotopic patterns. Additionally, more accurate, high resolution spectra can be produced by correcting for the effects of the different damping rates that are observed for different ion population sizes.     

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 Positions,Velocities, and Kinetic Energies of Resonantly Excited Ions in the Quadrupole Ion Trap Mass Spectrometer by Laser Photodissociation

The effects on ion motion caused by the application of a resonance AC dipole voltage to the end-cap electrodes of the quadrupole ion trap are described. An excimer laser is used to photodissociate benzoyl ions, and its triggering is phase locked to the AC voltage to follow the motion of the ion cloud as a function of the phase angle of the AC signal. Resonantly excited ions maintain a coherent motion in the presence of He buffer gas, which dissipates energy from the ions via collisions. Maximum ion displacements, which depend upon the potential well depth (q _z value), occur twice each AC cycle. Axial components of ion velocities are determined by differentiating the displacements of the distributions with respect to time. The experimental data show that these velocities are maximized when the ion cloud passes through zero axial displacement, and they compare favorably with results calculated using a simple harmonic oscillator model. Axial components of ion kinetic energies are low (<5 eV) under the chosen experimental conditions. At low values of q2 (≈ 0.2), the width of the ion distribution increases as the ion cloud approaches the center of the trap and decreases as it approaches the end-cap electrodes. This effect is created by compaction of the ion trajectories when ion velocities are decreased,     

Nonlinear ion acceleration for improved space focusing in time-of-flight mass spectrometry

The initial spatial distribution of gas-phase ions is one of the primary factors limiting the achievable mass resolving power in time-of-flight mass spectrometry. While the effect of the spatial distribution is minimized along the flight path at a location known as the space-focus plane, the use of a single, linear acceleration field to generate this focus represents only a first-order approximation of ideal space focusing. Alternatively, ideal space focusing is possible through the use of a nonlinear ion acceleration field. A computational model delineates the requirements of the nonlinear potential profile, and suggests that substantial improvements to resolving power can be achieved when using an ion source configuration where a first field approximates the optimal nonlinear field gradient and a second, linear accelerating field imparts additional kinetic energy. Experimental results using a novel, static-field ion source geometry designed to allow selective position-specific ionization through photoinduced dissociation indicate that a 10× improvement in focusing can be achieved using this configuration.     

飞行时间质谱中的空间电荷效应研究

在分子多光子电离实验中, 发现母体离子的飞行时间(TOF)质谱峰随激光强度的提高而展宽, 此现象归因于离子团内离子间库仑排斥作用的空间电荷效应. 为从理论上解释此现象, 尝试了从一个简单的理论模型进行推演, 得到飞行时间质谱峰的半高全宽(FWHM)与激光能量、样品气体的分压比、电极板的总电压、离子质量及所采用的透镜聚焦焦距之间关系的解析表达式, 所得的结果能够很好地说明实验现象.     

Realistic modeling of ion cloud motion in a Fourier transform ion cyclotron resonance cell by use of a particle-in-cell approach

Using a 'Particle-In-Cell' approach taken from plasma physics we have developed a new three-dimensional (3D) parallel computer code that today yields the highest possible accuracy of ion trajectory calculations in electromagnetic fields. This approach incorporates coulombic ion-ion and ion-image charge interactions into the calculation. The accuracy is achieved through the implementation of an improved algorithm (the so-called Boris algorithm) that mathematically eliminates cyclotron motion in a magnetic field from digital equations for ion motion dynamics. It facilitates the calculation of the cyclotron motion without numerical errors. At every time-step in the simulation the electric potential inside the cell is calculated by direct solution of Poisson's equation. Calculations are performed on a computational grid with up to 128 x 128 x 128 nodes using a fast Fourier transform algorithm. The ion populations in these simulations ranged from 1000 up to 1,000,000 ions. A maximum of 3,000,000 time-steps were employed in the ion trajectory calculations. This corresponds to an experimental detection time-scale of seconds. In addition to the ion trajectories integral time-domain signals and mass spectra were calculated. The phenomena observed include phase locking of particular m/z ions (high-resolution regime) inside larger ion clouds. A focus was placed on behavior of a cloud of ions of a single m/z value to understand the nature of Fourier transform ion cyclotron resonance (FTICR) resolution and mass accuracy in selected ion mode detection. The behavior of two and three ion clouds of different but close m/z was investigated as well. Peak coalescence effects were observed in both cases. Very complicated ion cloud dynamics in the case of three ion clouds was demonstrated. It was found that magnetic field does not influence phase locking for a cloud of ions of a single m/z. The ion cloud evolution time-scale is inversely proportional to magnetic field. The number of ions needed for peak coalescence depends quadratically on the magnetic field.     

Anomalous small angle x-ray scattering determination of ion distribution around a polyelectrolyte biopolymer in salt solution

The distribution of counterions in solutions of high molecular mass hyaluronic acid, in near-physiological conditions where mono- and divalent ions are simultaneously present, is studied by small angle neutron scattering and anomalous small angle x-ray scattering. The solutions contain either sodium or rubidium chloride together with varying concentrations of calcium or strontium chloride. The effects of monovalent-divalent ion exchange dominate the amplitude and the form of the counterion cloud. In the absence of divalent ions, the shape of the anomalous scattering signal from the monovalent ions is consistent with the distribution calculated from the Poisson-Boltzmann equation, as found by other workers. In mixtures of monovalent and divalent ions, however, as the divalent ion concentration increases, both the diameter and the amplitude of the monovalent ion cloud decrease. The divalent counterions always occupy the immediate neighborhood of the charged polyanion. Above a given concentration their anomalous scattering signal saturates. Even in a large excess of divalent ions, ion exchange is incomplete.     

New method for the computation of ionic distribution around rod-like polyelectrolytes with the helical distribution of charges. I. General approach and a nonlinearized Poisson–Boltzmann equation

This article is a first step in an attempt to reevaluate the relative role of different contributions to the energetics of DNA in salt solutions. To identify individual terms yielding such contributions a new derivation is given of the generalized Poisson–Boltzmann equation, which includes correlation effects, and explicitly shows terms ignored in the regular Poisson–Boltzmann approach. A general method based on the Boundary Element Technique is discussed, which can be used to evaluate these terms in the next steps of the reevaluation. An implementation of this method for the solution of the nonlinear Poisson–Boltzmann equation is described in detail, and is used to compute the ionic atmosphere around DNAs modeled as cylinders with helical distributions of charges. In the B-type DNA models, it is found that the ion densities in the minor and major grooves near the DNA surface differ by up to threefold. This difference is ca. 10-fold for Z-type DNA models. There are 20–25% differences in the magnitude of the maximum ionic charge density between DNA models of the same type. The addition of excess salt (up to 0.15 M) changes this maximum by only 10–15%. This change is not proportional to the concentration of excess salt. The contributions of different factors to the stabilization of alternative forms of DNA are evaluated. These factors are: (1) interactions between the phosphates, (2) interactions of phosphates with water, (3) interactions of phosphates with the ionic cloud, (4) interactions within the ionic cloud, (5) entropy of the ionic cloud. It is found that regardless of large variations in the counterion distributions around different DNAs, energetic contributions from these distributions are similar (?12.65 ± 0.6 kcal/mol · cell). The calculated change in stabilization per unit cell of models of B and Z-type DNAs due to 0.15 M excess NaCl is only ?0.56 ± 0.02 kcal/mol, indicating no tendency toward B-Z transition in this concentration range. Significantly larger variations of the order of 10 kcal/mol per unit cell can result from factors 1–2. Possible effects of the realistic DNA-solvent boundaries on the energetics of DNA solutions are discussed.     

Laser-induced fluorescence for ion tomography in a Penning trap

Laser-induced ion fluorescence of laser-desorbed Ba⁺ ions provides a measure of the relative number of ions near the center of the Penning trap of a Fourier transform ion cyclotron resonance mass spectrometer. Here, we report the detection of Penning-trapped ions by ion fluorescence, subject to radially outward ion cloud expansion (because of ion-neutral collisions), radially inward ion cloud compression (because of quadrupolar axialization), and the effects of buffer gas pressure and electrostatic trapping potential on those processes. At high pressure and high trapping voltage, radial ejection is far more rapid than axial ejection; quadrupolar axialization increases the number of ions near the center of the trap as well as the length of time that ions may be trapped; higher pressure results in faster magnetron radial expansion; and the choice of azimuthal quadrupolar excitation waveform significantly affects the efficacy of axialization. Based on these results, we suggest that directly detected laser-induced ion fluorescence provides a general new tool for mapping the ion distribution and its time evolution in response to various excitatory and damping effects.     

Results of a transient simulation of a drift tube ion mobility spectrometer considering charge repulsion, ion loss at metallic surfaces and ion generation

With optimized geometry and operating parameters both IMS selectivity and sensitivity can be significantly increased. However, finding these parameters and geometry requires an accurate knowledge of the electrical field and the ion concentration within the IMS at any time of operation. Furthermore, the ion loss at metallic surfaces and space charge effects caused by the moving ion cloud must be considered. This is particularly true when using non-radioactive electron emitters which generate a comparably high space charge density at electron currents similar to radioactive beta-sources due to their smaller ionization volume. This can lead to a reduced IMS resolution mainly caused by coulomb repulsion. In this work a transient model which enables a detailed view on the electric field within the IMS considering ion diffusion and migration as well as ion loss and coulomb repulsion is presented. This finite element model provides excellent agreement between simulated IMS spectra and experimental data especially when considering space charge effects and coulomb repulsion respectively. The model is used to design a short drift tube IMS with significantly improved resolution. Furthermore, this model allows considering ion-ion and ion-neutral reactions, such as ion generation, charge transfer reactions and ion-ion recombination. Moreover, fluid dynamics can be considered as required for modeling aspiration type IMS.     

Simple cylindrical ion mirror with three elements

A new cylindrical ion mirror has been designed to create an electric field that is non-linear or curved along the flight path axis for general-purpose time-of-flight mass spectrometers. The inclusion of one or two grids is found to improve the radial field homogeneity especially around the aperture. Only three cylindrical electrodes are used in the design. Changing the electrode dimensions and voltages affects the electric field distribution. Once the electrode dimensions are fixed, there are only two adjustable parameters for achieving optimum nonlinear electric field shape. Resolving powers of 7,000 and 16,100 have been achieved with kinetic energy variations of 34 and 10.5%, respectively. Simulations show that the electric field homogeneity in the radial direction enables the use of ion beam diameters up to 15 mm with only modest loss of resolving power. Increasing the mirror diameter could further increase the practical ion beam diameter. This article details the electric field distribution within the cylindrical mirror in both axial and radial directions. The voltages of the middle and rear electrodes affect the resolving power and the kinetic energy range over which focus can be achieved. The predicted arrival time spread for a single m/z value is narrower than that caused by the turn-around time of ions in a gas-phase ion source. In this case, the broad energy range over which good focus is achieved enables the use of higher extraction fields for turn-around time reduction.     

On electrophoresis of a pH‐regulated nanogel with ion partitioning effects

The electrophoresis of a polyelectrolyte nanoparticle, whose charge condition depends on the salt concentration and pH of the suspended medium as well as the dielectric permittivity difference, is analyzed. The present nonlinear model for the electrophoresis of this pH‐regulated polyelectrolyte (PE) particle is based on the consideration of full set of governing equations of fluid and ion transport coupled with the equation for electric field. The Born energy of the ions are incorporated to account for the difference in the dielectric permittivity of the PE and the electrolyte. The governing equations are computed numerically through a control volume approach. The nonlinear effects are highlighted by comparing with the existing linear model as well as results based on the first‐order perturbation analysis valid for a weak applied field. The ion partitioning effect arising due to the difference in self energy of ions between the two media, have a strong impact on the mobility of the PE. The ion partitioning effect attenuates the penetration of counterions in the PE, which enhances the electric force and hence, results in a larger mobility of the PE. The nonlinear effects due to the double layer polarization and relaxation are intensified due to the ion partitioning effect. The ion partitioning effect influences the association/dissociation of PE functional group by tuning the hydrogen/hydroxide ions. Present study shows that the ion partitioning effect is profound for higher salt concentration and/or higher volume density of PE functional groups.