Science, Faculty of
Chemistry, Department of
DSpace
UBCV
Gao, Cong
Doctor of Philosophy - PhD
University of British Columbia
Space charge effects in a linear quadrupole ion trap operated with mass selective axial ejection have been investigated. If too many ions are confined in an ion trap, the Coulomb repulsion between ions, or from "space charge", causes ions to shift towards higher apparent mass-to-charge ratio (m/z) in a mass spectrum. This phenomenon, called "mass shifts", can be described by a model with the assumption that the trapped ions have a Boltzmann or thermal distribution of positions in an effective potential. The two parameters in this model, the ion temperature and the effective potential, are determined in this thesis, allowing improved modelling of space charge effects with well determined parameters. Space charge effects were also studied experimentally, including the simplest case (self-space charge) and the most difficult case (space charge from a heavy ion species of high abundance). A new technique to lower mass shifts, dual-frequency excitation, was introduced and optimized. As a result, the instrument performance of linear quadrupole ion trap mass analyzers, including the intensity, limits of detection and the trap capacity, can be improved.
Space Charge Effects in Linear Quadrupole Ion Traps by Cong Gao B.Sc., Peking University, 2010 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate and Postdoctoral Studies (Chemistry) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) February 2015 © Cong Gao, 2015 ii Abstract Space charge effects in a linear quadrupole ion trap operated with mass selective axial ejection have been investigated. If too many ions are confined in an ion trap, the Coulomb repulsion between ions, or from "space charge", causes ions to shift towards higher apparent mass-to-charge ratio ( ) in a mass spectrum. This phenomenon, called "mass shifts", can be described by a model with the assumption that the trapped ions have a Boltzmann or thermal distribution of positions in an effective potential. The two parameters in this model, the ion temperature and the effective potential, are determined in this thesis, allowing improved modelling of space charge effects with well determined parameters. Space charge effects were also studied experimentally, including the simplest case (self-space charge) and the most difficult case (space charge from a heavy ion species of high abundance). A new technique to lower mass shifts, dual-frequency excitation, was introduced and optimized. As a result, the instrument performance of linear quadrupole ion trap mass analyzers, including the intensity, limits of detection and the trap capacity, can be improved. iii Preface A version of Chapter 2 has been published as a part of: Qiao, H.; Gao, C.; Mao, D. M.; Konenkov, N.; Douglas, D. J., Space-charge effects with mass-selective axial ejection from a linear quadrupole ion trap. Rapid Communications In Mass Spectrometry 2011, 25 (23), 3509-3520. The project was initiated and designed by Dr. Douglas. Dr. Qiao contributed to the data acquisition, the data analysis and the preparation of the manuscript. Mao and Konenkov conducted ion trajectory simulations. I was responsible for parts of the data acquisition (ion temperature measurements) and provided the relevant tables. A version of Chapter 3 has been published: Gao, C.; Douglas, D. J., Can the effective potential of a linear quadrupole be extended to values of the Mathieu parameter up to 0.90? Journal of the American Society for Mass Spectrometry 2013, 24 (12), 1848-1852. The project was proposed by Dr. Douglas. Also Dr. Douglas contributed to research details and editing the manuscript. I conducted all the data acquisition and the data analysis, and edited the manuscript. Chapter 4 is based on work by Dr. Douglas and Cong Gao. Dr. Douglas contributed greatly to the design of the project and research details. I conducted all the tests, optimized the iv experimental configuration, suggested new experiments, analyzed the data and discussed the results. v Table of Contents Abstract .................................................................................................................................. Preface .................................................................................................................................... ii iii Table of Contents .................................................................................................................. v List of Tables ......................................................................................................................... vii List of Figures ........................................................................................................................ x List of Symbols ...................................................................................................................... xvii List of Abbreviations ............................................................................................................ xx Acknowledgements ................................................................................................................ xxii 1 Introduction ........................................................................................................................ 1 1.1 Theory of Linear Quadrupoles .................................................................................... 3 1.2 Linear Quadrupole Ion Traps ..................................................................................... 8 1.3 Space Charge Effects in Linear Ion Traps .................................................................. 15 1.4 Outline of This Thesis ................................................................................................. 19 2 Measurements of Ion Temperatures ................................................................................ 20 2.1 Introduction ................................................................................................................. 20 2.2 Effusive Flow .............................................................................................................. 22 2.3 Methods ...................................................................................................................... 24 2.4 Results and Discussion ............................................................................................... 32 2.5 Summary ..................................................................................................................... 41 vi 3 Determination of the Effective Potential of a Linear Quadrupole Ion Trap ............... 43 3.1 Background ................................................................................................................. 43 3.2 Theory and Development of the Effective Potential ................................................... 44 3.3 Methods ....................................................................................................................... 49 3.4 Results ......................................................................................................................... 52 3.5 Discussion ................................................................................................................... 60 3.6. Summary .................................................................................................................... 65 4 Dual-Frequency Excitation to Reduce Space Charge Effects ........................................ 66 4.1 Introduction ................................................................................................................. 66 4.2 Methods ....................................................................................................................... 68 4.3 Results ......................................................................................................................... 73 4.4 Discussion ................................................................................................................... 107 4.5 Summary ..................................................................................................................... 121 5 Summary and Future Work ............................................................................................. 122 References .............................................................................................................................. 125 Appendices ............................................................................................................................. 133 Appendix A Derivation of Equation 2.1............................................................................ 133 Appendix B Optimization of Dual-Frequency Excitation ................................................ 136 vii List of Tables Table 2.1 Instrumentation ...................................................................................................... 26 Table 2.2 Operating voltages (V) for mass selective axial ejection ...................................... 27 Table 2.3 Ion temperature ...................................................................................................... 37 Table 2.4 Ratios of mass shifts at two ion temperatures T1 and T2 with a ratio of 1.30. Δm1/Δm2 is the ratio of mass shifts at T1 and T2, and well depth is given by .......... 41 Table 3.1 Quadrupole parameters used in the ion trajectory simulations ............................... 51 Table 3.2 Ion oscillation frequencies at different . Fundamental secular frequencies are calculated by the Equation 3.1 when ............................................................................... 59 Table 4.1 The main ion species, their , their relative abundances in mass spectra of the mixed solution and the corresponding stopping barriers for ion ejection used for these ion species ..................................................................................................................................... 72 Table 4.2 Mass shift reductions and the percentages of reserpine ions ejected by the heating AC with different operating parameters ..................................................................... 78 Table 4.3 Mass shift reductions of ions of 609.3 with different parameters of the heating AC .............................................................................................................................. 92 Table 4.4 The optimum heating AC and the corresponding mass shift reductions for the four ion species at a scan speed of 250 Th/s ........................................................................... 93 Table 4.5 The optimum heating AC and the corresponding mass shift reductions for the four ion species at a scan speed of 500 Th/s ........................................................................... 98 Table 4.6 The optimum heating AC and the corresponding mass shift reductions for the four ion species at a scan speed of 1000 Th/s ......................................................................... 103 viii Table 4.7 The number of trapped ions that result in a 0.1 Th mass shift of 609.3 ......... 108 Table 4.8 The mass shift reductions of 609.3 from dual-frequency excitation .............. 109 Table 4.9 The optimum frequency of the heating AC, and the corresponding time interval between the ion heating and ion ejection for the four ion species at a scan speed of 250 Th/s .......................................................................................................................................... 111 Table 4.10 The optimum time interval (in ms) between the ion heating and ion ejection for the four ions species at the scan speeds of 250 Th/s, 500 Th/s and 1000 Th/s ....................... 113 Table 4.11 The optimum amplitudes of the heating AC for the four ion species at a scan speed of 500 Th/s .................................................................................................................... 114 Table 4.12 Optimization of the heating amplitudes for the four ion species at scan speeds of 250 Th/s, 500 Th/s and 1000 Th/s ...................................................................................... 116 Table 4.13 The FWHM of ions of 374.4 with and without dual-frequency excitation at scan speeds of 250 Th/s, 500 Th/s and 1000 Th/s in experiments with multiple ion species ..................................................................................................................................... 118 Table 4.14 The FWHM of ions of 474.3 with and without dual-frequency excitation at scan speeds of 250 Th/s, 500 Th/s and 1000 Th/s in experiments with multiple ion species ..................................................................................................................................... 118 Table 4.15 The FWHM of ions of 556.6 with and without dual-frequency excitation at scan speeds of 250 Th/s, 500 Th/s and 1000 Th/s in experiments with multiple ion species ..................................................................................................................................... 119 Table 4.16 The FWHM of ions of 609.3 with and without dual-frequency excitation at scan speeds of 250 Th/s, 500 Th/s and 1000 Th/s in experiments with multiple ion species ..................................................................................................................................... 119 ix Table B.1 Mass shift reductions and the percentage of ions ejected by the heating AC with different operating parameters at a scan speed of 500 Th/s .................................................... 136 Table B.2 Mass shift reductions and the percentages of ions ejected by the heating AC with different operating conditions at a scan speed of 1000 Th/s ........................................... 137 Table B.3 Optimization of the heating AC for ions of 374.4 at 250 Th/s ...................... 138 Table B.4 Optimization of the heating AC for ions of 474.3 at 250 Th/s ...................... 139 Table B.5 Optimization of the heating AC for ions of 556.6 at 250 Th/s ...................... 140 Table B.6 Optimization of the heating AC for ions of 374.4 at 500 Th/s ...................... 142 Table B.7 Optimization of the heating AC for ions of 474.3 at 500 Th/s ...................... 143 Table B.8 Optimization of the heating AC for ions of 556.6 at 500 Th/s ...................... 144 Table B.9 Optimization of the heating AC for ions of 609.3 at 500 Th/s ...................... 145 Table B.10 Optimization of the heating AC for ions of 374.4 at 1000 Th/s .................. 147 Table B.11 Optimization of the heating AC for ions of 474.3 at 1000 Th/s .................. 148 Table B.12 Optimization of the heating AC for ions of 556.6 at 1000 Th/s .................. 149 Table B.13 Optimization of the heating AC for ions of 609.3 at 1000 Th/s .................. 150 x List of Figures Figure.1.1 Conventional round quadrupole rods with applied potentials (given as pole to ground) .................................................................................................................................... 4 Figure 1.2 The first stability region of the linear quadrupole. The first stability region is symmetric about the axis and here only the region with is shown .......................... 8 Figure 1.3 Schematic of a linear ion trap with mass selective radial ejection ........................ 12 Figure 1.4 Schematic of a triple quadrupole mass spectrometer. Mass selective axial ejection from either q2 or Q3 is possible ................................................................................. 14 Figure 2.1 (a) Boltzmann distributions (Equation 1.24) of 600 000 ions of 609.3, , in a radial direction trapped in a quadrupole with 1.0 MHz RF, a trap length =20 cm and a field radius = 4.17 mm at = 0.20 (well depth = 10.8 V) at temperatures of = 1000 K and = 7000 K. (b) the number of ions in a ring of radius , , under the same conditions. (c) electric fields from the space charge of the ions with the distributions in (a) and (b) ....................................................................................................... 21 Figure 2.2 An oblique cylinder containing molecules with the velocity that can pass through an orifice in a time interval ......................................................................... 23 Figure 2.3 Schematic of the mass spectrometer system used in this thesis ............................ 25 Figure 2.4 Schematic of the ion trap set up ............................................................................ 28 Figure 2.5 Schematic of the timing sequence for mass selective axial ejection ..................... 29 Figure 2.6 Schematic of the timing sequence for measurements of the ion temperature ...... 31 xi Figure 2.7 (a) The drain curve of reserpine ions ( 609.3) at 0.84 with a delay time of 20 ms. (b) The drain curve (a) with a logarithmic vertical scale. (c) The fit of curve (a) to the equation , giving the ion temperature 1742 178 K. (d) The curve of (c) with a logarithmic vertical scale .................................................................. 33 Figure 2.8 The ion temperatures with different delay times between ion cooling and draining. The measurement uncertainties are shown by the error bars and the average of the ion temperatures with delay times from 10 ms to 100 ms is shown by the red dashed line ........................................................................................................................................... 34 Figure 2.9 Effects of quadrupole DC voltages on mass shifts caused by space charge. Apparent mass of 609.3 vs. the number of trapped ions with (a) = 0.20, = 0.83 to 0.85, scan speed ca. 50 Th/s, = 0.30 Vp-p. (b) = 0.40, = 0.78 to 0.85, scan speed ca. 50 Th/s, = 0.30 Vp-p (c) = 0.20, = 0.83 to 0.85 (positive), 0.85 to 0.87 (negative), positive or negative here means the polarity of DC applied to the electrodes that the dipole excitation was applied to, scan speed ca. 50 Th/s, = 0.30 Vp-p, switched polarity of the DC. (d) = 0.20, = 0.83 to 0.85, scan speed ca. 1000 Th/s, = 1.0 Vp-p. (e) = 0.40, = 0.78 to 0.85, scan speed ca. 1000 Th/s, = 1.0 Vp-p. (f) = 0.20, = 0.83 to 0.85 (positive), 0.85 to 0.87 (negative), scan speed ca. 1000 Th/s, = 1.0 Vp-p, switched polarity of the DC ............... 38 Figure 3.1 The operating interface of the ion trajectory simulation program. The curves in this figure show the ion trajectories in the and directions ................................................ 51 xii Figure 3.2 Ion trajectories in the direction with 0 and 0.85, (a) without and (b) with 75 V pole to pole dipole DC (dipole field V m-1). Initial conditions , , ............................................................................ 54 Figure 3.3 The effective electric field vs. displacement with , 0.85, and the initial conditions of Figure 3.2. A linear fit to , gives , , the linear regression coefficient .......... 55 Figure 3.4 Well depth vs. with the initial conditions of Figure 3.2 ..................................... 56 Figure 3.5 Well depth vs. with the initial conditions of Figure 3.2. A fit to gives , ................................................................ 57 Figure 3.6 Ion oscillation frequencies vs. ,( ■) frequencies calculated from Equation 3.1 with , (▲) frequencies of the standard effective potential model from Equation 3.3, (●) this work ........................................................................................................................... 58 Figure 3.7 Ion trajectories with and . No dipole DC is applied ( ), and damping of the ion motion is included. Initial conditions , , .............................................................................................. 61 Figure 4.1 The ratio, , of the ion oscillation frequency to the trapping RF frequency vs. the oscillation amplitude (normalized to ) for different linear charge densities with an ion temperature = 2000 K, = 4.17 mm, = 609, trapping = 0.85, trapping frequency = 1 MHz. The ion oscillation frequency without space charge is shown by the dashed line. and are initial velocities measured in units of ....................... 68 xiii Figure 4.2 The structures of the analytes, Ala-Ala-Ala-Ala-Ala (a), N-succinyl-Ala-Ala-Ala-p-nitroanilide (b), leucine enkephalin acetate salt hydrate (c), reserpine (d) and hexakis(1H, 1H, 2H-perfluroethoxy) phosphazene (e) ........................................................... 70 Figure 4.3 A mass spectrum of the mixed solution with a logarithmic vertical scale at a scan speed of 250 Th/s ............................................................................................................ 71 Figure 4.4 The apparent mass of ions 609.3 vs. the number of trapped ions with different heating frequencies applied at a scan speed of 250 Th/s. "Single frequency" in this and other figures in this chapter refers to the results with normal axial ejection. For dual-frequency excitation, three frequencies of the heating AC, 382.0 kHz, 383.0 kHz and 384.0 kHz, were used here with a peak-to-peak amplitude of 100 mV .................................. 75 Figure 4.5 The apparent mass of ions of 609.3 vs. the number of trapped ions with different heating amplitudes applied at a scan speed of 250 Th/s. For dual-frequency excitation, the frequency of the heating AC was 383.0 kHz with peak-to-peak amplitudes of 80 mV and 100 mV in this figure ....................................................................................... 77 Figure 4.6 Spectra without (the left, a, c, e, g, i, k) and with (the right, b, d, f, h, j, l) dual-frequency excitation with different numbers of trapped ions at a scan speed of 250 Th/s. The heating AC has a frequency of 383.0 kHz and an amplitude of 100 mV. The peak positions and FWHM of 609.3 are labelled. Figures 4.6 (j) and (i) show that the heating AC ejects a larger fraction of ions when the number of ions is more than 1,163,000; this can be ascribed to the increased repulsions between ions in these cases ....... 81 Figure 4.7 The apparent mass of reserpine ions 609.3 vs. the number of trapped ions with and without dual-frequency excitation. For dual-frequency excitation, a heating AC with a frequency of 383.0 kHz and an amplitude of 160 mV was used ................................. 82 xiv Figure 4.8 Spectra without (the left, a, c, e, g, i, k) and with (the right, b, d, f, h, j, l) dual-frequency excitation with different numbers of trapped ions at a scan speed of 500 Th/s. For dual-frequency excitation, a heating AC with a frequency of 383.0 kHz and a peak-to-peak amplitude of 160 mV was used ...................................................................................... 85 Figure 4.9 The apparent mass of reserpine ions 609.3 vs. the number of trapped ions with and without dual-frequency excitation at 1000 Th/s scan speed. For dual-frequency excitation, a heating AC with a frequency of 382.0 kHz and a peak-to-peak amplitude of 220 mV was used .................................................................................................................... 86 Figure 4.10 Spectra without (the left, a, c, e, g, i, k) and with (the right, b, d, f, h, j, l) dual-frequency excitation with different numbers of trapped ions at a scan speed of 1000 Th/s. For dual-frequency excitation, heating AC with a frequency of 382.0 kHz and a peak-to-peak amplitude of 220 mV was used ...................................................................................... 89 Figure 4.11 The mass shifts vs. the number of trapped ions for different ion species with normal axial ejection ............................................................................................................... 91 Figure 4.12 The mass shifts vs. the number of trapped ions with and without dual-frequency excitation for different ion species. For the cases with dual-frequency excitation, the optimum heating AC for each ion species (listed in Table 4.4) was applied .................... 94 Figure 4.13 Spectra of reserpine ions without (the left, a, c, e, g, i, k) and with (the right, b, d, f, h, j, l) dual-frequency excitation with different numbers of trapped ions at a scan speed of 250 Th/s. For dual-frequency excitation, heating AC with a frequency of 383.0 kHz and a peak-to-peak amplitude of 90 mV was used .......................................................... 97 xv Figure 4.14 The mass shifts vs. the number of trapped ions with and without dual-frequency excitation for different ion species at a scan speed of 500 Th/s. For dual-frequency excitation, the optimum heating AC for each ion species was applied ................. 99 Figure 4.15 Spectra of reserpine ions without (the left, a, c, e, g, i, k) and with (the right, b, d, f, h, j, l) dual-frequency excitation with different numbers of trapped ions at a scan speed of 500 Th/s. For dual-frequency excitation, a heating AC with a frequency of 383.0 kHz and a peak-to-peak amplitude of 160 mV was used. For Figure 4.15 (a), 1000 scans were added. The intensity in Figure 4.15 (a) was normalized to 150 scans, which were used for all the other spectra ................................................................................................... 102 Figure 4.16 The mass shifts vs. the number of trapped ions with and without dual-frequency excitation for different ion species at a scan speed of 1000 Th/s. For dual-frequency excitation, the optimum heating AC for each ion species (listed in Table 4.6) was applied .............................................................................................................................. 104 Figure 4.17 Spectra of reserpine ions without (the left, a, c, e, g) and with (the right, b, d, f, h) dual-frequency excitation with different numbers of trapped ions at a scan speed of 1000 Th/s. For dual-frequency excitation, a heating AC with a frequency of 384.0 kHz and an amplitude of 200 mV was used. For Figure 4.17 (a), 2000 scans were added, and its intensity was normalized to 200 scans, which were used for all the other spectra ................. 106 Figure 4.18 Ion trajectories of the four ion species with and . Dipole excitation was added in the direction with a frequency of 386.453 kHz and a peak-to-peak amplitude of 100 mV, and damping of the ion motion was also included. Initial conditions , , ......................................................... 112 xvi Figure B.1 The mass shifts of different ion species vs. the number of trapped ions at a scan speed of 500 Th/s. Normal single frequency axial ejection was used .................................... 141 Figure B.2 The mass shifts of different ion species vs. the number of trapped ions at a scan speed of 1000 Th/s with normal axial ejection ....................................................................... 146 xvii List of Symbols Time Field radius Cartesian coordinates Polar coordinates , Spherical coordinates Electric potential of a linear quadrupole Number of charges DC voltage Amplitude of trapping RF Angular frequency of trapping RF Mass Electric field in the and -direction Electron charge Mass-to-charge ratio Mass shift Arbitrary coordinate Mathieu parameters for motion in the -direction Dimensionless time variable in the Mathieu equation Expression for the solution of the Mathieu equation Coefficients of Parameters of the stable solutions of the Mathieu equation xviii Angular secular frequency for a given Effective potential Well depth Excitation angular frequency Permittivity of free space Total ion number Electric field from space charge Number density of singly charged ions Boltzman's constant Ion temperature Gas temperature Parameters of Length of the ion trap Cross-sectional area of an orifice Velocity vector Speed, the magnitude of Number of molecules or ions Number density of gas molecules Velocity probability density function Vex Amplitudes of the excitation voltages Pressure of the gas , Fitting parameters of the ion drain curve xix Ion drain rate constant Correction coefficient for absolute ion temperature Damping constant Drag force on an ion Effective electric field , Fitting parameter of Linear regression coefficient , Fitting parameters of well depth Force constant in Hooke's law Initial RF phase Stopping barrier applied to the exit lens Mass resolution from FWHM Collision cross section of an ion Initial position of an ion Initial velocity of an ion Excitation time of an ion Scan speed Frequency of the trapping RF Dipole DC voltage Electric field from dipole DC Parameters of the excitation frequency with quadrupole excitation xx List of Abbreviations Abbreviation Description 3D Three-Dimensional AC Alternating Current AR Analytical Reagent CID Collision Induced Dissociation DC Direct Current ESI Electrospray Ionization FTICR Fourier Transform Ion Cyclotron Resonance FWHM Full Width at Half Maximum MALDI Matrix Assisted Laser Desorption Ionization MCS Multichannel Scalar MS Mass Spectrometry MS/MS, MS3, MSn Tandem Mass Spectrometry MSAE Mass Selective Axial Ejection Q0 Quadrupole Ion Guide Q1 Quadrupole Down-stream of Q0 q2 Collision Cell Q3 Quadrupole Down-stream of q2 RF Radio Frequency RIT Rectilinear ion trap xxi Th Thomson (the unit of the mass-to-charge ratio) TOF Time-of-Flight UHP Ultra High Purity Vp-p Volt, Peak to Peak xxii Acknowledgements I give thanks to Dr. Don Douglas, my supervisor, for his passion in research, his expertise in quadrupoles, and his patience in guiding me in my graduate studies. I am indebted to Dr. Hui Qiao for sharing his knowledge and skills when I first started my research. I would like to extend my thanks to all other former members of Don's group, Yang, Negar, Dunmin, Chuanfan and Kim. I gratefully acknowledge NSERC and AB SCIEX for financial support of this research through an Industrial Research Chair. I would like to thank the Faculty of Graduate and Postdoctoral Studies of UBC for awarding me fellowships. The assistance from the mechanical (Kenny Bach) and electronic shops (Sajjad Haidar and David Tonkin) is greatly appreciated. Special thanks to Dr. Michael Sudakov for the code of the ion trajectory program. I remain grateful to my parents for their unconditional love and support. 1 Chapter 1 Introduction Mass spectrometry (MS) is an analytical technique that can separate gaseous ions according to their mass-to-charge ratios. Mass spectrometers produce spectra of the relative abundances of detected ions. With the advantages of high sensitivity, selectivity and speed, MS is widely used to identify unknown compounds, determine structures of ions and to quantify the amount of an analyte in a sample. The first mass spectrometer was built by Thomson in the early 20th century [1]. During its development over more than a century, not only has mass spectrometry found numerous applications in many fields, but it has experienced an immense commercial growth, with a market value over $3 billion [2]. Its success can be attributed to a series of remarkable advances. These developments include high performance mass analyzers, tandem mass spectrometry, combinations of chromatography and MS, and new ionization sources. One of the many milestones in the history of mass spectrometry was the invention of soft ionization techniques, used mostly for bio-molecules. The 2002 Nobel Prize in Chemistry was partially awarded to John Fenn and Koichi Tanaka for the development of electrospray ionization (ESI) [3] and matrix assisted laser desorption ionization (MALDI) respectively [4]. ESI can produce multiply charged ions and thus extend the mass range of analytes by multiple times for a 2 given maximum of mass-to-charge ratio of a mass analyzer, enabling MS to measure the mass of large biomolecules with molecule weights over 100,000 Da. MALDI is widely used for the ionization of polar molecules, but it also allows analysis of strongly hydrophobic molecules, broadening the range of molecules that can be ionized. Both techniques greatly extended the applications of MS to life sciences. In the last decade, a group of new ionization techniques, known as ambient mass spectrometry, simplified the ionization process and reduced the need for sample preparation or preseparation, helping to bring MS analysis into the "real world" [5]. The developments of improved mass analyzers are equally significant. In 1919, Aston built a mass spectrometer in which isotopes were separated by successive electric and magnetic fields [6]. In the 1950s, Paul and Raether invented quadrupole mass analyzers [7]. Mass analyzers with improved performance, such as time-of-flight (TOF) [8], Fourier transform ion cyclotron resonance (FTICR) [9] and the orbitrap [10] , were successively developed. These advances have brought higher mass accuracy, higher resolution and greater dynamic range, meeting the demands of bioanalytical chemistry and other fields, especially the analysis of extremely complex mixtures. Tandem mass spectrometry, also known as MS/MS or MSn, is a method that enables trace analysis in the presence of "chemical noise" and provides information on the structure of ions. This method involves multiple stages of mass spectrometry analysis, with ion dissociation occurring in between. Typical operations of tandem mass spectrometry are described below. A first mass analyzer is used to select precursor ions. The precursor ions are dissociated using some method, and then the fragment ions produced from the ion dissociation are mass analyzed by a second mass analyzer. If further dissociation steps are performed on the fragment ions, MS3 or MSn (n means the number of stages of mass analysis) mass spectra can be obtained. Several methods can be used for ion dissociation, among which collision-induced dissociation (CID) is the 3 most widely used. CID takes place, when an ion collides with a neutral molecule or atom and the ion's kinetic energy converts into sufficient internal energy to break chemical bonds [11]. Different types of mass analyzers can be used or hybridized in a tandem mass spectrometer. For example, a triple quadrupole mass spectrometer consists of two linear quadrupole mass analyzers in series, with a pressurized quadrupole (referred to as collision cell) in between to produce fragment ions. Tandem mass spectrometry, often combined with gas chromatography or liquid chromatography, features the powerful separation capability of chromatography and the sensitivity, selectivity and speed of MS, broadening the application areas in both academic and industrial research. 1.1 Theory of Linear Quadrupoles The basic theory of quadrupoles was developed by Paul and his coworkers in the 1950s. Since then, quadrupole mass spectrometry has witnessed huge improvements in performance and many new applications. Much of this is described in books by Dawson [12] and March and Hughes [13]. The use of linear quadrupoles in mass spectrometry has been reviewed by Douglas [14]. The linear quadrupole consists of a parallel array of four rod-like electrodes. Ideally, each electrode should be hyperbolic, so that an exact quadrupole electric field can be produced. For ease of manufacture, cylindrical rods are often used in practice, which approximates an ideal quadrupole electric field by optimizing the ratio of rod radius to the distance from the center of the quadrupole to an electrode [15] . The electric field is produced by connecting opposite pairs of rods together and applying radio frequency (RF) and direct current (DC) potentials between the pairs (shown in Figure 1.1). 4 Figure 1.1 Conventional round quadrupole rods with applied potentials (given as pole to ground). Adapted from [13, pp. 43] with permission. The electric potential of an ideal linear quadrupole, , is given by (1.1) where and are Cartesian coordinates, and are a DC voltage and the amplitude of an RF voltage, respectively, applied between an electrode and ground, is the angular frequency of the applied radio frequency voltage and , the field radius, is the distance from the center of the quadrupole to an electrode. Ion motion is determined by Newton's second law (1.2) (1.3) 5 where is the mass of an ion, is time, and are electric fields in the and directions, respectively, is the number of charges on an ion, is the electron charge and is the mass-to-charge ratio. The right sides of the Equations (1.2) and (1.3) give the force on an ion. Considering (1.4) (1.5) the equation of ion motion for the and directions can be written as (1.6) (1.7) The ion motions in the and directions are independent. Defining the variables (1.8) the equations of ion motion can be written as the Mathieu equation (1.9) The solution of the Mathieu equation may be expressed as [13, pp. 39-41] (1.10) where and are constants that are determined by the initial conditions (initial position, velocity and RF phase), the coefficients are factors that characterize the amplitudes of ion motion 6 and depend only upon and , and expressed as , is also a constant that is determined by and . The solutions are classified as stable (where remains finite as increases) and unstable ( tends to infinity as increases ). The possibilities of the solutions have been summarized by Dawson [12, pp. 67] in the following: 1. is real and not zero; the term or will trend to infinity, and the solution is unstable. 2. is complex and leads to an unstable solution. 3. , where is an integer; the solutions are periodic but unstable. These solutions form the boundaries between stable and unstable regions on the stability diagram. 4. and is not an integer; here, the solutions are periodic and stable. Equation (1.10) becomes (1.11) where and . The value of is determined by a continued-fraction expression in terms of and , given by Ions oscillate with frequencies, known as secular frequencies, 7 (1.13) Ions with stable solutions of the Mathieu equation in both the and directions can travel through or be confined in a linear quadrupole. These stable solutions can be described by a diagram in and , named the stability diagram. In practice, the first stability region, shown in Figure 1.2, is often used for mass analysis; its boundaries consist of the lines corresponding and . The primary working principle of a quadrupole mass filter is to control stability and instability of ion trajectories by the magnitudes of the DC ( ) and RF ( ) potentials. Ions, located near the tip ( ) of the first stability diagram, can travel through a linear quadrupole, while the other ions are outside of the first stability region and will hits the rods and be removed because of their unstable trajectories. Mass spectra are produced by scanning and together with a constant ratio to bring ions with different to the tip of stability diagram. Linear ion trap analyzers use different analysis methods, which are introduced below. 8 Figure 1.2 The first stability region of the linear quadrupole. The first stability region is symmetric about the axis and here only the region with is shown. Reproduced from [12, pp. 20] with permission. 1.2 Linear Quadrupole Ion Traps A linear ion trap can be formed by applying RF only voltages to the quadrupole rods and stopping potentials to electrodes at the ends of a linear quadrupole. Ions are confined radially by the RF fields, and axially by the stopping potentials. For a linear quadrupole ion trap, , and thus ions with wide range of masses ( ) have stable trajectories and can be trapped. Linear traps have several advantages over three-dimensional (3D) ion traps, e.g. higher injection 9 efficiency, higher sensitivity and larger ion trap capacity. Linear ion traps can be used in combinations with other mass analyzers (e.g. 3D Paul traps, TOF mass analyzers, FTICR mass analyzers or Orbitrap mass analyzers). In a combined system, linear ion traps can enrich analyte ions and remove unwanted ions prior to mass analysis. As a result, the performance of the mass analyzers can be improved, with an increased duty cycle. Instrumentation for linear quadrupole ion traps has been reviewed by Douglas et al. [16]. 1.2.1 The Effective Potential Approximation Ion motion is usually described by the Mathieu equation. At low values , ion motion in a linear ion trap can be approximated as that of a charged particle undergoing simple harmonic oscillation confined in an effective potential well with a small perturbation by the trapping RF [17]. The effective potential is given by (1.14) where the well depth, (1.15) The frequencies of ion oscillation are given by the term in Equation 1.13, . Although with the approximation , Equation 1.15 was thought valid only at low values, it is shown in Chapter 3 that it can be extended to higher values [18]. 10 1.2.2 Ion Excitation: Dipole and Quadrupole Excitation Ions in a linear quadrupole ion trap can be resonantly excited, with either dipole excitation or quadrupole excitation. Ion excitation leads to an increase of the amplitude of ion oscillation. An auxiliary alternating current (AC) voltage at a frequency needs to be applied, with an amplitude that is several orders of magnitude lower than the amplitude of the trapping RF voltage. Dipole excitation, can be performed by applying the auxiliary AC between two opposite poles of a quadrupole rod set. Resonance occurs when the auxiliary frequency coincides with the secular frequencies given by Equation 1.13; usually, is used as the excitation frequency. Quadrupole excitation takes place when the auxiliary AC is applied to all four rods in the same way as the trapping RF. In other words, for quadrupole excitation, auxiliary AC waveforms, out-of-phase by , are applied to the two pairs of poles respectively. Sudakov et al. [19] showed that the excitation frequencies of ions with quadrupole excitation are given by (1.16) where and . Resonant excitation has been utilized for ion isolation, ion fragmentation and mass selective ejection. Ions of one mass-to-charge ratio can be isolated by applying a broadband waveform with a notch in frequency space. The notch is at the resonance frequency of the ions that are to be isolated. Other unwanted ions are excited, increase their amplitudes of oscillation, and strike the rods, leaving the analyte ions isolated. Ion dissociation can be induced by collisions of isolated analyte ions that are resonantly excited with background gas. This has been used in a tandem mass spectrometry. Another application of ion excitation is mass selective ejection to allow using linear ion traps as mass analyzers, and is introduced in the following sections. 11 1.2.3 Mass Selective Radial Ejection with Linear Ion Traps A linear ion trap mass spectrometer using mass selective radial ejection was built and evaluated by Schwartz et al. [20]. The idea was proposed earlier by Bier and Syka [21]. A schematic of the instrument is shown in Figure 1.3. Hyperbolic rods are cut in three sections for this linear quadrupole ion trap. The center section has slots cut into the x-rods through which ions are ejected. These slots cause imperfections in the fields, which can be compensated by adjusting the spacing of the x-rods [20]. Dipole excitation with auxiliary AC voltages on the x-rods is used to isolate, excite, and eject ions. Ions are isolated by a broadband waveform with frequencies 5–500 kHz, with a notch at the frequency of the analyte ions, usually at . For MS/MS, ions are activated by resonant excitation at a frequency corresponding to a from 0.25 to 0.35. The enhanced ion capacity and injection efficiency, compared with 3-D traps, result in higher sensitivity and lower limits of detection. A mass resolution of 30,000 at a scan speed of 27 Th/s was obtained and MS4 was demonstrated. (The Th or Thomson is the unit of the mass-to-charge ratio of an ion, with mass measured in Daltons and charge measured in units of the absolute charge of the electron). This method is used in the commercial Thermo linear ion trap mass spectrometers. 12 Figure 1.3 Schematic of a linear ion trap with mass selective radial ejection, reproduced from [20] with permission. Typical operating pressures are given. The detector is located near the slot of the center section. In 2004, Cooks et al. introduced a rectilinear ion trap (RIT) [22]. The basics of operation of the RIT are quite similar to the linear quadrupole ion trap described by Schwartz and his co-workers. The RIT uses two pairs ( ) of rectangular electrodes to replace quadrupole rods, and produces approximately a quadrupole field. Ions are ejected radially from slots in the x-rectangles. A mass resolution of 1000 with a mass range of 650 Th was achieved and tandem mass spectrometry was demonstrated. 1.2.4 Mass Selective Axial Ejection (MSAE) with Linear Ion Traps Mass selective axial ejection from a linear ion trap was first described by Hager [23]. This method is based on the earlier experiments of Brinkmann [24] and Hager [25], in which an RF-only quadrupole mass filter with an energy filter located between the exit aperture and the detector was used. The electric field near the exit aperture, referred to as the fringing field, couples the and motion of ions to the axial motion. Ions with values close to the stability boundary ( 13 ) receive increased energy in the and directions and reach the fringing field region with larger radial oscillation amplitudes; as a result, these ions gain sufficient axial kinetic energy in the fringing field to overcome the energy barrier produced by the energy filter, and then reach the detector. In contrast, the other ions can't pass through the energy filter due to their much lower axial kinetic energies. Ions of increasingly higher ratios can be successively brought to the boundary of the stability diagram by scanning the trapping RF; in this way, mass spectra are produced. Hager further developed this technique [23]. A schematic of the instrument is shown in Figure 1.4. A triple quadrupole mass spectrometer was used and ion trapping can occur either in the collision cell (q2) or the down-stream quadrupole (Q3). A stopping potential was applied to the exit aperture. Either dipole excitation or quadrupole excitation can be used for ion excitation. Excited ions can gain sufficient additional axial kinetic energy in the fringing field to overcome the stopping potential and thus to escape axially from the ion trap. Mass spectra are produced by scanning the trapping RF to bring ions of increasingly higher ratios into resonance with a fixed excitation frequency. With q2 ( ) as the ion trap, the mass resolution at 609.3 was 1000 with a scan speed of 1000 Th/s. A slower scan of 5 Th/s produced a resolution of 6000. Axial ejection from Q3 is also possible and a resolution of 6000 was obtained with a scan speed of 100 Th/s. Due to the low pressure in Q3 ( ), a scan speed 20 times greater than that in q2 with the same resolution was possible. An MS/MS experiment was also demonstrated. The precursor ions were mass selected by Q1 and then were injected into q2 with sufficient energy to induce fragmentation. The fragment ions, generated by CID in the pressurized q2, were trapped and axially ejected from Q3. The sensitivity was improved by a factor of 16, 14 compared with conventional triple quadrupole MS/MS. Mass selective axial ejection is used in the AB Sciex linear ion traps. Figure 1.4 Schematic of a triple quadrupole mass spectrometer, reproduced from [23] with permission. Mass selective axial ejection from either q2 or Q3 is possible. Typical operating pressures are given. Sugiyama et al. reported another method that extracts ions axially from a linear ion trap with a DC extraction field [26]. Trapped ions were mass selectively excited by auxiliary AC applied to the excitation lenses. A resolution of 1300 with a scan speed of 500 Th/s at 609 was achieved, with ca. 20% ejection efficiency. 15 1.3 Space Charge Effects in Linear Ion Traps Space charge effects occur when too many ion are introduced to linear ion traps. These effects appear as changes of the apparent mass of ions ( known as mass shifts) and decreases of mass resolution in mass spectra. Space charge not only affects mass measurement accuracy and calibration, it also plays an important role in determining the sensitivity, limits of detection and trap capacity of ion trap mass analyzers. The sensitivity and limit of detection relate to the ion number or linear ion density that causes an onset of mass shifts (usually 0.1 Th). If mass shifts can be reduced, more ions can be trapped, with a given change of the apparent mass; as a result, higher sensitivity and lower limits of detection can be achieved. Therefore, additional investigations of space charge effects, including improved modelling methods and new techniques to reduce mass shifts, may contribute to further improvements in the performance of linear ion traps. Schwartz et al. investigated space charge effects with mass selective radial ejection from a linear ion trap [20]. With the same number of ions, the linear ion trap demonstrated ca. 20 lower mass shifts than a 3D trap. Hager studied mass shifts with mass selective axial ejection from a linear trap [23]. It was found that about 40,000 singly charged ions ( 609.3) produce a 0.1 Th mass shift at . A lower pressure ion trap ( ) gave lower mass shifts. Qiao et al. investigated the effects of operating parameters on the mass shifts and concluded that a proper choice of operating conditions can reduce, but not eliminate, mass shifts caused by space charge [27]. 16 1.3.1 Equations of Ion Motion with Space Charge Space charge, or Coulombic repulsion between ions, changes the character of ion motion and causes the oscillation frequencies of ions to decrease. Thus, a larger trapping RF amplitude is required to bring ions into resonance with the excitation; ions shift to higher apparent masses in mass spectra. The theory of space charge is discussed here. The interactions between ions need to be considered and the equation of ion motion becomes, (1.17) where is the permittivity of free space, is total ion number in the trap, and are Cartesian coordinates of the test ion, and give the positions of the other ions, and the other factors have been described above. Only the ion motion in the direction is described here, but the same treatment can also be applied to the motion in the direction. With space charge, ion motions in the and directions are no longer independent, but instead depend on the relative displacements of other ions at all time. This makes ion motion difficult to model and to simulate. Douglas and Konenkov simplified ion-ion interactions to a static ion cloud, and calculated ion trajectories and mass shifts directly [28]. Similarly, we can approximate ion-ion interactions as a mean field that doesn't change with time. Equation 1.17 can be written as (1.18) 17 where is the mean electric field in the direction from space charge. By introducing an external mean field, ion motion appears not to relate to other individual ions; therefore, an ensemble of trapped ions can be modelled as a nearly independent particle system. 1.3.2 Ion Spatial Distribution At about the pressure at which a linear ion trap is typically operated, a given ion undergoes frequent collisions with background gas and ion spatial distributions can reach a steady state [29]. For ions of reserpine at of N2, the energy relaxation time is calculated to be about s (the details are described in Chapter 2). Thus, the assumption that the mean electric field from space charge doesn't change with time is reasonable. In statistical mechanics, a Boltzman distribution can be used to describe a nearly independent particle system. Introducing the effective potential model and treating space charge as a small perturbation to the ion motion, the number density of singly charged ions (m-3) at a distance from the centre of the quadrupole is given by (1.19) where is the effective potential, is Boltzman's constant, is a normalization constant and is the ion temperature [27, 30]. Here, the " ion temperature " characterizes the energy distribution. Substituting from Equation 1.14 with (1.20) where (1.21) 18 Normalization of the distribution to the total ion number in the trap, , (1.22) gives (1.23) and (1.24) Here, is the length of the ion trap. From Gauss' law, in a cylindrically symmetric space with radius , we have (1.25) The radial electric field from space charge is given by (1.26) The equation of ion motion with space charge is given by Equation 1.18 and in this equation is the component of in the direction. Note , depends on the ratio of two parameters, the ion temperature and the well depth of the effective potential. Therefore, the determination of these two parameters is significant to model and describe ion motion with space charge. 19 1.4 Outline of This Thesis This work focuses on investigating space charge effects in a linear ion trap and developing new techniques to reduce these effects. Chapter 2 describes a method to measure or compare ion temperatures. It is found that applying quadrupole DC can increase the temperature of trapped ions. This work has been published as a part of [27]. Chapter 3 describes a determination of the effective potential for any q value, published as [18]. As a result, these studies provide a better model to simulate ion motion with space charge. Chapter 4 describes a new technique, dual-frequency excitation, to reduce space charge effects in a linear ion trap. Its operating parameters were optimized to obtain the lowest mass shifts. Chapter 5 is a summary of this thesis with suggestions for future work. 20 Chapter 2 Measurements of Ion Temperatures 2.1 Introduction As discussed in Chapter 1, ion motion with space charge can be described by a model that assumes ions are confined with a Boltzmann distribution in an effective potential. In this model, the ion temperature is a crucial parameter. The effects of the ion temperature on the ion spatial distribution and the electric field from space charge are shown in Figure 2.1. An increase in the ion temperature leads to a more diffuse spatial distribution and a lower electric field from space charge; therefore, space-charge-induced mass shifts can be reduced. 21 Figure 2.1 (a) Boltzmann distributions (Equation 1.24) of 600,000 ions of 609.3, , in a radial direction trapped in a quadrupole with 1.0 MHz RF, a trap length =20 cm and a field radius = 4.17 mm at = 0.20 (well depth = 10.8 V) at temperatures of = 1000 K and = 7000 K. (b) the number of ions in a ring of radius , , under the same conditions. (c) electric fields from the space charge of the ions with the distributions in (a) and (b). Reproduced from [27] with permission. x or y (mm)0.0 1.0 2.0 3.0 4.0N(x) or N(y) m-30.02.0e+124.0e+126.0e+128.0e+121000 K7000 K(a)r (mm)0 1 2 3 4E (V/m)02468101214161000 K7000 K(c)r (mm)0 1 2 3 4N(r) dr0.02.0e+44.0e+46.0e+48.0e+41.0e+51.2e+51.4e+51.6e+51000 K7000 K(b) 22 The "ion temperature" characterizes the width or spread of the energy distribution. It also describes the average energy. Usually, the ion temperature is higher than room temperature because ions are heated by the trapping RF. For a Boltzmann distribution, the ion temperature in a linear ion trap can be given by (2.1) where is the average effective potential energy of the trapped ions (see the mathematical derivation of Equation 2.1 in Appendix A). The effective potential for ions is described by Equations 1.14 and 1.15. The calculation of the average potential energy of trapped ions by ion trajectory simulations is possible and thus the ion temperature can be determined. Here, another method is used to measure the ion temperature from experiments directly, with the approximation that when ions drain from a trap they can be described by an effusive flow model. 2.2 Effusive Flow Effusive flow or effusion describes the process in which a gas flows through an orifice when the mean free path of the molecules or atoms of the gas is considerably larger than the orifice size. The number of gas molecules that escape from the orifice with a cross-sectional area in a time interval can be calculated as follows [31]. These molecules are located in an oblique cylinder as shown in Figure 2.2; the number of molecules is given by (2.2) 23 where , is the density of gas molecules (measured in m-3), is the velocity vector of the molecules, is the speed of molecules (the magnitude of ), and are spherical coordinates, and is the velocity probability density function. The negative sign in Equation 2.2 means a decrease of the number of the molecules due to effusive flow. The velocity distribution of an ideal gas can be described by a Maxwell-Boltzmann distribution and is given by (2.3) where is the mass of the gas molecules and T is the gas temperature. Figure 2.2 An oblique cylinder containing molecules with the velocity that can pass through an orifice in a time interval . The effusive flow rate can be calculated by integrating Equation 2.2 over all possible directions of , 24 (2.4) Another form of Equation 2.4 is more often used, given by (2.5) where , the average speed of a molecule, is written as (2.6) Hence, the effusive flow rate depends on the temperature ; this argument can be turned around to provide a method to determine the temperature by measuring the effusive flow rate. 2.3 Methods 2.3.1 Instrument and Reagents The mass spectrometer system used is shown in Figure 2.3. Details are given in Table 2.1. Ions formed by pneumatically assisted electrospray ionization (ESI) pass through a 1.5 mm diameter aperture in a curtain plate, through a dry N2 curtain gas at atmospheric pressure and through a 125 µm diameter orifice in the tip of a skimmer cone. Ions then travel through an RF-only quadrupole ion guide (Q0) (length, 20 cm; = 4.17 mm; pressure, 8 × 10-3 Torr of N2) where they are 25 collisionally cooled to a nominal energy of 1 eV with energy spreads of ca. 1 eV. Ions leave Q0 and pass into the linear quadrupole trap, Q1(length, 20 cm; = 4.17 mm) in a chamber pumped to 2.7 × 10-5 Torr, through an "entrance lens" with a 3.0 mm diameter aperture. The Q0 rods are capacitively coupled to the Q1 trapping RF voltage (1.0 MHz) with an amplitude approximately half of the trapping RF. Two additional lenses, referred to as the “exit lenses”, are located downstream of Q1. The first, with a mesh-covered 9-mm diameter aperture, is located 3.5 mm from the end of Q1. The second, with an open 9-mm diameter aperture, is spaced 5.0 mm from the first. Ions are detected via a continuous dynode electron multiplier, operated in ion counting mode. Operating voltages are shown in Table 2.2. Figure 2.3 Schematic of the mass spectrometer system used in this thesis. 26 Table 2.1 Instrumentation. Q0 rod set Q1 rod set UBC Chemistry mechanical shop AB SCIEX (Concord, ON) Quadrupole power supply AB SCIEX (Concord, ON) Auxiliary AC arbitrary waveform generator Agilent, model 33220 A (Malaysia) Detector Burle Electro-Optics channeltron electron multiplier, model 4824 (USA) Preamplifier Ortec 9302 fast amplifier (USA) Discriminator Ortec, model 436 (USA) Syringe pump Harvard Apparatus, model 22 (USA) Rotary pumps Leybold, Trivac D16B (Germany), pump speed 165 m3/h Turbo pumps Leybold, Turbovac 361(Germany), pump speed 345 L/s Pirani gauge Leybold, Thermovac TM230 (USA) Ion gauge Granville-Philips, model 274005, 270 gauge controller (USA) BenchTop Lite PCI arbitrary waveform generator PC Instruments, Inc. (USA) Multichannel scalar (MCS) PCI card with MCS-32 operating software Ortec, model A73-B32, software version 2.13 (USA) 27 Table 2.2 Operating voltages (V) for mass selective axial ejection. Sprayer +4400 Curtain plate +500 Orifice +12 Q0 rod offset +4Entrance lens +3 or +130 Q1 rod offset +2Exit lens 1 -15, +130, or variable stopping potentials Exit lens 2 -30 The set up of the linear ion trap is shown in Figure 2.4. The voltages of the entrance and exit lenses were controlled by hardware and software from BenchTop Lite (USA), which provides two-channel arbitrary waveforms, with a maximum zero-to-peak amplitude output of 6 V. The output of BenchTop Lite is increased by homemade amplifiers (UBC Chemistry electronics shop) and then applied to the entrance and exit lenses. A timing diagram for the ion trap is given in Figure 2.5. For mass analysis an operating cycle consists of four steps: ion injection, cooling, scanning and draining. For ion injection, the entrance lens voltage is set to +3.0 V, and the exit lens is set to +130 V. The number of trapped ions can be controlled by the ion injection time, varying from 1 millisecond (ms) to 120 ms. The ions are then cooled for 100 ms, during which the 28 entrance lens is raised to +130 V and the exit lens is kept at +130 V. In the scanning step, axial ejection is used for mass analysis; the exit lens voltage is lowered to a stopping potential. Finally, both the entrance lens and exit lens voltages are lowered to -15 V to drain any residual ions from the trap and Q0 before the next cycle starts. Figure 2.4 Schematic of the ion trap set up. 29 Figure 2.5 Schematic of the timing sequence for mass selective axial ejection. Dipole excitation was applied between the electrodes for axial ejection, with a fixed frequency corresponding to the selected ejection value. The trapping RF was scanned to bring ions with increasingly higher mass-to-charge ratios into resonance for ejection. The auxiliary AC voltage from an arbitrary waveform generator was added to the trapping RF and coupled to the quadrupole rods through a balanced transformer. The auxiliary AC voltage was applied over the whole operating cycle. The amplitudes of the excitation voltages, Vex, are given as peak to peak (Vp-p), pole to pole. The trapping RF was set and scanned by a multichannel scalar (MCS). The MCS, plugged into the PCI bus of a computer, produces a ramp output to control the quadrupole power supply and scan the trapping RF voltage. In this way, the mass scan range, the scan time and the RF voltage Injection Cooling Scanning Drain Trap Entrance Trap Exit Trapping RF 1-120 ms 100 ms 400-1200 ms 50 ms 30 for ion injection can be set and controlled. The ramp output of the MCS was triggered by BenchTop Lite, synchronizing the timing of the voltages applied to the entrance and exit lens with the scan of the trapping RF. Mass spectra were acquired by the MCS. Twenty or more scans were repeated to generate a mass spectrum. Mass calibration was made either by linearly fitting a spectrum of a known sample consisting of multiple ions species, or by calculating apparent mass from the ejection RF voltages. The total number of trapped ions was measured by lowering the exit lens voltage and keeping the entrance lens voltage at +130 V. Ions drain from the trap with an approximately exponential decay with time, and the total number of ions in the decay was measured. The proper exit lens voltage is crucial. A low exit lens voltage causes saturation of the ion detector and thus some ions are not counted correctly. Raising this voltage allows the ions to drain more slowly from the trap and avoids saturation of the detector. Typically the exit lens voltage was set to from +1.0 V to +4.0 V, depending on the experimental conditions. Reserpine was analytical reagent (AR) grade and from Sigma, St. Louis, MO. Methanol (99.9%, AR grade) and acetic acid (99.99%, AR grade) were from Fisher Scientific (Nepean, ON, Canada). Nitrogen was ultra high purity (UHP) grade and from Praxair (Vancouver, BC, Canada). A 50 µM reserpine solution (in 50% methanol, 49.5% water and 0.5% acetic acid) was used as the sample. The solution flow rate for all experiments was 1µL/min. 2.3.2 Ion Temperature Measurements Ion temperatures were measured as follows. Ions were injected at or and cooled for 100 ms. The trapping RF was then jumped up to the value used for ion ejection, with delay time from 1 ms to 100 ms, before the ions were allowed to drain from the trap. A schematic 31 of the timing sequence for this process is shown in Figure 2.6. For the ion drain period, exit lens 1 was set at -15.0 V and exit lens 2 at -30 V. Figure 2.6 Schematic of the timing sequence for measurements of the ion temperature. Ions drain from the trap exponentially and thus the number of ions in the trap at time is (2.7) where is the total number of trapped ions and is a rate constant that characterizes the ion drain rate. Differentiating Equation 2.7, the ion drain rate is given by (2.8) By analogy to effusive gas flow through an orifice, described by Equation 2.4, the ion drain rate can also be related to the ion temperature, (2.9) Injection Cooling Delay Drain Trap Entrance Trap Exit Trapping RF 1-120 ms 100 ms 1-100 ms 50 ms 32 where , the area of the "orifice", is the area of the exit lens aperture in this case. Note that for a linear ion trap, the ion density can be written as (2.10) where is the length of the trap, and combining Equation 2.8, 2.9 and 2.10, the ion temperature is given by (2.11) Therefore, the ion temperature can be determined by measuring the rate constant . 2.4 Results and Discussion 2.4.1 Temperatures of ions trapped at = 0.84 The drain process of reserpine ions ( 609.3) trapped at was measured. Ions were injected and cooled for 100 ms at before being drained from the ion trap. A typical ion drain curve is shown in Figure 2.7 (a), with a 20 ms delay time between ion cooling and draining. In Figure 2.7, the axis is the ion drain time measured in ms, and the axis is the number of ions that drain from the trap in a 0.25 ms time interval, According to Equation 2.8, the ion drain rate constant can be determined by fitting the ion drain curve to the equation , where and are the fitting parameters; the data points of the first 5 ms are usually used to fit, for the ion temperatures decrease with drain time when the delay time is not sufficiently long, shown in Figure 2.7 (b). The ion temperature can be then calculated by Equation 2.11. The uncertainty of the ion temperature is estimated by 33 (2.12) where is the fitting uncertainty of . For this experiment, the ion temperature is determined to be 1742 178 K. 0 5 10 15 20 2520004000600080001000012000dNdNdNt/ms0 5 10 15 20 25100100010000t/ms0 1 2 3 4 5020004000600080001000012000dN=a'exp(-b't) t/ms0 1 2 3 4 5100010000dcb dNt/msaFigure 2.7 (a) The drain curve of reserpine ions ( 609.3) at 0.84 with a delay time of 20 ms. (b) The drain curve (a) with a logarithmic vertical scale. (c) The fit of curve (a) to the equation , giving the ion temperature 1742 178 K. (d) The curve of (c) with a logarithmic vertical scale. 34 The ion temperatures with different delay times are shown in Figure 2.8. It was found that over about the first 10 ms, the ion temperatures decreases and then becomes nearly constant. The ion temperatures measured with delay times of 10 ms, 20 ms, 30 ms,..., 100 ms were averaged and the uncertainty in the mean was calculated. The same experiments were repeated on at least three days and the reproducible results gave the ion temperature = 1725 56 K at . 0 20 40 60 80 10015002000250030003500The ion temperature/KDelay time/ms Figure 2.8 The ion temperatures with different delay times between ion cooling and draining. The measurement uncertainties are shown by the error bars and the average of the ion temperatures with delay times from 10 ms to 100 ms is shown by the red dashed line. 35 Figure 2.8 also suggests that ions returns to a new steady state in ca.10 ms when jumps from 0.20 to 0.84. This supports the assumption that the mean field from space charge is independent of time. The new steady state is achieved by the collisions between ions and background gas. The damping of ion motion by collisions is described in [32, 33]. Ions have an energy relaxation time , where , the damping constant, is given by (2.13) where is the collision cross section between the ion and the background gas, is the gas number density, is the gas mass, is Boltzmann's constant, is the gas temperature, and is the ion mass. At a typical trap pressure, the background gas can be treated as an ideal gas and thus the gas number density measured in m-3 is given by (2.14) where is the pressure of the gas measured in pascal (Pa). In this experiment, reserpine ions ( 280 Å2 [34]) trapped at 2.7 10-5 Torr of N2, the energy relaxation time is calculated to be 6.9 10-3 s. This agrees with the experiment results shown in Figure 2.8. For a pressure 1.0 10-5 Torr (mentioned earlier in Section 1.3.2), the energy relaxation time is calculated to be 1.9 10-2 s. 2.4.2 The Ion Temperature with Quadrupole DC The ion temperature with quadrupole DC was also measured. Quadrupole DC was added to the trapping RF, with positive DC applied to the electrodes to which the dipole excitation was applied, and negative DC to the other pairs of electrodes. If negative DC is applied to the electrodes that 36 have the dipole excitation applied and positive DC to the other electrodes, it is referred to as "switched polarity of the quadrupole DC". At an injection 0.20, the highest DC voltage that could be used was 9.0 V. This corresponds to Mathieu parameter = 1.66 × 10-2 for ions of 609.3, close to the stability boundary during ion injection and cooling. It was found that, with ions injected at , at 0.84, adding quadrupole DC of 9.0 V, increases the ion temperature from 1725 56 K to 1925 72 K. In this case, the ion temperature with quadrupole DC is a factor of 1.12 higher. Similar experiments were carried out with injection at 0.40. In this case, the DC voltage that places ions at the boundary of the stability diagram ( 7.0 × 10-2) is about 38.0 V. With a quadrupole DC of 38.0 V, a trapping of 0.77 was used to match the value (determined by and , given by Equation 1.12) at = 0.84 without DC; hence, in both cases, ions have the same oscillation frequency in the direction of the dipole excitation. The case with 9.0 V DC differs; since 9.0 V DC has little effect on the ion oscillation frequency when ions are trapped at = 0.84, no change of trapping is needed. The ion temperatures were measured as 1630 92 K and 2119 23 K without and with 38 V DC respectively. Therefore, the quadrupole DC increases the ion temperature by a factor of 1.30 in this case. A summary of the ion temperatures with different conditions is given in Table 2.3. These results have been published as a part of [27]. In that paper, it was reported that applying quadrupole DC reduces mass shifts caused from space charge at scan speeds of 50 Th/s and 1000 Th/s and the switched polarity of the quadrupole DC gives almost the same reduction of mass shifts, shown in Figure 2.9. In order to determine the possible causes of these observations, trajectory calculations were carried out to conclude that quadrupole DC by itself does not reduce 37 space-charge-induced mass shifts [30]. However, adding quadrupole DC, may change the shape of the ion spatial distribution, causing an increase in the ion temperature. This hypothesis has been experimentally supported by these measurements of the ion temperatures. Therefore, the reduction of mass shifts from quadrupole DC can be attributed to an increase in the ion temperature to some extent. Table 2.3 Ion temperature. Injection Trapping Quadrupole DC /V The ion temperature /K 0.20 0.84 0.0 1725 56 0.20 0.84 9.0 1925 72 0.40 0.84 0.0 1630 92 0.40 0.77 38.0 2119 23 38 Figure 2.9 Effects of quadrupole DC voltages on mass shifts caused by space charge. Apparent mass of 609.3 vs. the number of trapped ions with (a) = 0.20, = 0.83 to 0.85, scan speed ca. 50 Th/s, = 0.30 Vp-p. (b) = 0.40, = 0.78 to 0.85, scan speed ca. 50 Th/s, = 0.30 Vp-p (c) = 0.20, = 0.83 to 0.85 (positive), 0.85 to 0.87 (negative), 0 200000 400000 600000 800000609.3609.5609.7609.9 qinject = 0.20, 1000 Th/s Apparent mass 0.0 V 5.0 V 9.0 V Total ionsd0 200000 400000 600000 800000609.3609.5609.7609.9 Apparent mass 0.0 V 9.0 V 38.0 Vqinject = 0.40, 1000 Th/s Total ionse0 200000 400000 600000 800000609.3609.5609.7609.9610.1610.3 Apparent mass 0.0 V +9.0 V 0.0 V -9.0 Vqinject = 0.20, 50 Th/s Total ionsc0 200000 400000 600000 800000609.3609.5609.7609.9610.1610.3 0.0 V 9.0 V 38.0 Vqinject = 0.40, 50 Th/sb Apparent mass Total ions0 200000 400000 600000 800000609.3609.5609.7609.9610.1610.3 Apparent mass 0.0 V 5.0 V 9.0 V Total ionsaqinject = 0.20, 50 Th/s0 200000 400000 600000 800000609.3609.5609.7609.9610.1610.3 0.0 V +9.0 V 0.0 V -9.0 V Apparent mass qinject = 0.20, 1000 Th/s Total ionsf 39 positive or negative here means the polarity of DC applied to the electrodes that the dipole excitation was applied to, scan speed ca. 50 Th/s, = 0.30 Vp-p, switched polarity of the DC. (d) = 0.20, = 0.83 to 0.85, scan speed ca. 1000 Th/s, = 1.0 Vp-p. (e) = 0.40, = 0.78 to 0.85, scan speed ca. 1000 Th/s, = 1.0 Vp-p. (f) = 0.20, = 0.83 to 0.85 (positive), 0.85 to 0.87 (negative), scan speed ca. 1000 Th/s, = 1.0 Vp-p, switched polarity of the DC. Reproduced from [27] with permission. 2.4.3 Limitations of the Method A method of measuring ion temperatures was introduced above. In this method, the drain process of ions from the trap is modelled as effusive flow of an ideal gas. However, the behavior of ions in an external electric field differ from that of an ideal gas. The rate constant of ion drain mainly depends on the ion velocity distribution in the axial direction, while the ion temperature is mostly related to the motion of ions in the radial direction which is different from that in the axial direction. Therefore, the ion temperature, determined by Equation 2.11, does not give an accurate absolute ion temperature. Considering that the kinetic energy of ions in the radial direction can be transferred to that in the axial direction in the fringing field, the ion temperature is related to the ion drain rate constant . Thus, a correction coefficient can be introduced in Equation 2.11 and the ion temperature can be given by (2.15) 40 where is a coefficient that corrects the errors caused by the method that treats ions as ideal gases. It is still difficult to calculate the ion temperatures directly from Equation 2.15, for is unknown. However, Equation 2.15 suggests that the ratio of the ion temperatures measured by this method is accurate. Hence, the ion temperatures, determined by this method, can be used to calculate the mass shifts caused by space charge. For example, a quadrupole DC of 38 V increases the ion temperature by a factor of 1.30. Since the absolute temperature is not well determined, a series of temperatures with a ratio of 1.30 were used to calculate the mass shifts by ion trajectory simulations [27]. The calculated ratios of mass shifts at the lower temperature to those at higher temperature are shown in Table 2.4. It was found that a temperature ratio of 1.3 produces about the same ratios of mass shifts regardless of the absolute ion temperature. The average ratio of mass shifts in Table 2.4 is 1.53 0.07, which agrees with the experimental result with a scan speed of 50 Th/s shown in Figure 2.9 (b). 41 Table 2.4 Ratios of mass shifts at two ion temperatures T1 and T2 with a ratio of 1.30. Δm1/Δm2 is the ratio of mass shifts at T1 and T2. The well depth is given by . Adapted from [27] with permission. T1 T2 Δm1/Δm2 500 650 1.56 1000 1300 1.48 0.09 1500 1950 1.56 0.10 2000 2600 1.44 0.11 2500 3250 1.50 0.07 3000 3900 1.54 0.09 3500 4550 1.49 0.15 4000 5200 1.67 0.21 2.5 Summary A method of measuring ion temperatures has been described. This method approximates the drain process of ions from the trap as effusive gas flow and thus ion temperatures can be determined by 42 measuring the rate of the ion drain. Although the ion temperature, determined in this way, does not give an exact absolute temperature, the ratio of the measured ion temperatures with different operating conditions is accurate, and can be used in trajectory calculations of space-charge-induced mass shifts. 43 Chapter 3 Determination of the Effective Potential of a Linear Quadrupole Ion Trap 3.1 Background Ion motion in a linear quadrupole can be described by the Mathieu equation (Equations 1.8 and 1.9). Ions oscillate with frequencies, given by (3.1) where is the angular frequency of the trapping RF, and is a function of the Mathieu parameters and (given by Equation 1.12). The effective potential (or pseudopotential) approximation can be used to simplify the description of ion motion in RF electromagnetic fields under some conditions [17, 35, 36]. This method is widely used for both linear quadrupole ion traps and 3-D traps, in areas such as describing the ion density distribution [37, 38], modeling the resolution of resonant ejection [39, 40] and calculating the effects of space charge [27, 29, 30, 41, 42]. In the effective potential approximation, ion motion is decomposed into a slow averaged motion and a high-frequency oscillation correction. The high-frequency oscillation has a small 44 amplitude relative to the averaged motion; thus, the real ion motion can be approximated as the averaged ion motion, which appears as ions undergoing a simple harmonic oscillation confined in an effective time-independent DC potential that does not satisfy Laplace's equation. This effective potential is used to take the place of the real trapping RF potential and can be interpreted as averaging the effects of the trapping RF over time. The effective potential was applied earlier in other areas of physics [43, 44] and was generalized as a method to describe motion in a rapidly oscillating field [45]. For a linear quadrupole, the effective potential is given by [46] (3.2) where the well depth, , is given by (3.3) Equations 3.1 and 3.2 describe the electric effective potential measured in volts. The mechanical effective potential (in eV) is the ion charge times the electric potential ( for a singly charged ion). 3.2 Theory and Development of the Effective Potential The theory of the standard effective potential model for a linear quadrupole trap is reviewed as follows. The treatment is mainly based on the derivation by Gerlich [46]. In this thesis, this model of the effective potential is referred to as the standard model to distinguish it from our work . 45 The electric potential of an RF-only linear trap, , is given by (3.4) where is the initial RF phase. To simplify the derivation, only the motion in the direction is described here; the effective potential is independent of the initial RF phase and is selected to be zero in the derivation below. The electric field in the direction is given by (3.5) The equation of the ion motion is (3.6) The motion resulting from Equation 3.6 is assumed to be decomposed as (3.7) where is a slow averaged motion, and is a small and rapidly oscillating motion that is superimposed on by the trapping RF. Thus, can be approximated as an oscillation in a homogeneous electric field at , and satisfies (3.8) Assume in a short time interval , or X can be treat as time-independent variables relative to the rapidly changing RF period. Selecting the initial condition , we can obtain a solution of Equation 3.8 (3.9) 46 The electric field can be expressed by a Taylor series, (3.10) Taking a first-order approximation to Equation 3.10, we have (3.11) Note that (3.12) Substituting Equations 3.11 and 3.12 into Equation 3.6, we have (3.13) The parts can be cancelled. Averaging Equation 3.13 over time (one cycle of the trapping RF), gives an averaged value of 1/2 and Equation 3.13 becomes (3.14) where , . This equation shows the slow time-averaged motion of ions can be described as a harmonic oscillator in an effective potential. The same treatment can be applied to the motion in the direction of a linear quadrupole and also to ion motion in a 3D ion trap. The derivation above is valid only when all the following assumptions apply: 47 (1) the high frequency correction has a much smaller amplitude relative to the averaged motion, since Equation 3.8 is based on the premise ; (2) Equation 3.9 assumes that the oscillation frequency of the real ion motion is considerably lower than the RF frequency; (3) the first-order approximation of Equation 3.10 is reasonable, or, in other words, higher order terms can be neglected. As emphasized by Douglas et al. [47], the assumptions above are only valid when values are less than about 0.40. At large , near the stability diagram limit ( ), the oscillation frequency of the ion motion is close to one half of the RF frequency, and the high frequency correction has a similar amplitude to that of the averaged motion . Thus, it has generally been thought that the effective potential approximation is limited to the conditions when [46, 48]. However, Makarov noted that the effective potential surprisingly produced an order-of-magnitude calculated resolution of a 3D ion trap operated with a resonant ejection at that was in quite good agreement with experiments [40]. Sudakov and Apatskaya [49] also noted that the effective potential can produce reasonable results for values substantially greater than 0.4. For high values, Sudakov [50] proposed another model that assumes the ion motion can be described as motion in an effective potential with well depth (3.15) 48 Sudakov noted that the ion motion is a mix of sine and cosine waveforms with frequencies given by Equation 3.1, among which, near , the and terms (in Equation 3.1) dominate. These two terms have similar amplitudes and frequencies and thus ion motion appears as a beat between these two frequencies. The Sudakov model correctly gives the beat frequency of the ion motion, which decreases with and becomes zero at the stability boundary where Nevertheless, the Sudakov model is not consistent with some characteristics of the ion motion. The fundamental secular frequencies of the ions, calculated from Equation 3.1 with , are generally considerably higher than the beat frequency, and increase with and , while the beat frequency decreases as increases. Baranov et al. [51] calculated kinetic energies of ions confined in a linear quadrupole with values from 0.20 to 0.85. The kinetic energies of ions depend on the initial conditions of ions, but generally increase to ca. 100 eV at ( s-1, , mm). The well depth calculated from the Sudakov model (Equation 3.15) for these conditions is 11.5 V, significantly less than the ion energies. The well depth given by the Sudakov model decreases as increases, yet the trapping forces increase with because of the higher RF voltages applied to the quadrupole. Here, a new method of determining the effective potential for any is introduced. A dipole DC (also called dipolar DC) electric field is applied, which causes a displaced ion trajectory. We assume that the dipole DC electric field at the center of the displaced trajectory is countered by the electric field from the effective potential. The center of the displaced trajectory was calculated by ion trajectory simulations. Thus, the effective electric field can be probed by the applied dipole DC electric field to obtain the effective potential at any . 49 3.3 Methods 3.3.1 Dipole DC and Ion Displacements A dipole DC field was introduced to the equation of ion motion in order to determine the effective potential at any value. A dipole DC potential can be added to a linear quadrupole by applying a DC potential difference between the two electrodes or the two electrodes. If the potential is applied between the electrodes, the dipole electric potential is given by (3.16) where is the dimensionless amplitude of the dipole potential, and is the difference in DC voltage between the two poles. The electric field from the dipole DC, , is independent of and is given by (3.17) The value of depends on the electrode geometry. For a quadrupole constructed with round rods, [33]. A dipole DC voltage displaces ions from the center of the quadrupole to cause an increase of ion kinetic energies [52]. This has been used as a method of dissociating ions for tandem mass spectrometry in 3D [53-55] and linear [56-58] quadrupoles as well as rectilinear ion traps [59]. At low , the displacement of a singly charged ion from the dipole DC is calculated by setting the force from the dipole electric field equal to the “force” from the effective potential [56]. This argument can be turned around to provide a new method to determine the effective potential of a linear ion trap. If the displacement caused by a dipole DC field can be 50 calculated (or measured [60]), the value of the force of an effective potential can be determined at any by assuming the force from the effective potential at that point is balanced by the force from the dipole DC electric field. In other words, an effective electric field, or , can be "probed" by the dipole DC electric field at the center of the displaced trajectory, using the equation (3.18) By setting a series of values of the dipole DC field that cause different displacements, we can obtain the effective electric field at different distances . The effective potential can then be determined by integrating the effective electric field with respect to . 3.3.2 The Simulation Program for Ion Trajectories The displacements of ions caused by a dipole DC field were calculated by ion trajectory simulations. A computer program, written by Michael Sudakov, was used in the simulations and its operating interface is shown in Figure 3.1. The operating parameters of the quadrupole modelled here were chosen to match those of quadrupoles used in experiments in this lab [27] and also in calculations of space charge effects [28, 30]. The program was originally used to model resonant excitation by auxiliary RF fields. The dipole DC electric field was added by setting the frequency of the auxiliary dipole excitation to a very low value (0.005 kHz). Ion trajectories were typically calculated for 200 RF cycles or 200 µs. During this time the dipole DC electric field changes by only ca. 1 10-6 of its maximum value at . The program integrates the equations of motion with a step size of 10-3 RF cycles. 51 Figure 3.1 The operating interface of the ion trajectory simulation program. The curves in this figure show ion trajectories in the and directions. Table 3.1 Quadrupole parameters used in the ion trajectory simulations. field radius 4.17 10-3 m rod radius 4.69 10-3 m trapping RF angular frequency s-1 ion 609.3 52 The program includes damping of the ion motion by collisions with background gas, although for a time of 200 RF cycles this is very small. The drag force on an ion from the background gas is given by [33] (3.19) where and are the mass and speed of an ion, and is the damping constant given by Equation 2.13. For the experiments of interest to us [27], reserpine ions trapped at 1.0 10-5 torr of N2, = 13.4 s-1. The energy relaxation rate is . In 200 µs (200 RF cycles) the energy decreases by about 1 %. This relaxation does not change the average displacement. The displacement caused by the dipole DC field was calculated by taking the average of the coordinates of an ion trajectory over about 200 RF cycles. In general, 200 RF cycles do not correspond to an integer number of cycles of the ion oscillation. The number of RF cycles was chosen as the number near 200 which gave a minimum in the average of the coordinates. Typically this average was 10-5 or less. The same number of cycles was then used to calculate the average of the coordinates. 3.4 Results 3.4.1 Well Depths at Different Values Ion trajectories for the motion in an RF only quadrupole with and without an applied dipole DC field with values from 0.20 to 0.90 were calculated. As an example, Figure 3.2 shows ion trajectories in for an ion at without (3.2 (a)) and with (3.2 (b)) a dipole DC field of V m-1 (75 V pole to pole for the quadrupole of Table 3.1). The displacement from the 53 dipole DC is evident and in this case was 0.07563 (0.3154 mm). Trajectories were then run to determine the ion displacements with different applied dipole DC fields. The displacement was found to increase linearly with the applied dipole DC field. This has been observed experimentally by Wang et al. for Ba+ ions trapped at = 0.0755, albeit for small displacements (< 100 µm or 0.0286 ) [60]. Thus the effective electric field, determined in this way, also increases linearly with displacement i.e. where is in V m-2 (shown in Figure 3.3). A linear fit to the data gives (2.28031 0.00004) 107 V m-2. The effective potential at a distance from the axis can be defined and determined as (3.20) The well depth, , is the effective potential at or . In this case 198.3 V. The results above are based on the initial RF phase . The well depth was also determined with other initial phases (i.e. and ) for applied dipole DC voltages of 60, 120, 180 and 240 V pole to pole. It was found that the amplitude of ion oscillation changed significantly with the initial phase; however, the displacement caused by the dipole DC was independent of the initial phase, with differences within 1% or less. 54 0 10 20 30 40 50-0.02-0.010.000.010.020 10 20 30 40 500.000.050.100.150.200.25 bX/r0RF cyclesDipole DC=75 VNo dipole DCa X/r0RF cycles Figure 3.2 Ion trajectories in the direction with and , (a) without and (b) with 75 V pole to pole dipole DC (dipole field V m-1). Initial conditions , , . Note the different vertical scales in (a) and (b). Reproduced from [18] with permission. 55 0.0000 0.0002 0.0004 0.0006 0.0008040008000120001600020000 Eeff/Vm-1x/mq=0.85 Figure 3.3 The effective electric field vs. displacement with , 0.85, and the initial conditions of Figure 3.2. A linear fit to , gives , , the linear regression coefficient . Reproduced from [18] with permission. The same procedure was then used to calculate well depths at values from to . The variation of well depth with is shown in Figure 3.4. We find that the well depth can be fit to (3.21) 56 A fit to Equation 3.21 (Figure 3.5) gives . Thus the well depths determined here are within ca. 1% of those of the standard model (Equation 3.3, which is equivalent to in Equation 3.21). Two other sets of initial conditions were then tested: (i) , and (ii) and the well depths determined again for values from 0.20 to 0.90. These gave and . Thus, within the accuracy of the calculations, the well depths do not depend on the initial conditions. 0.2 0.4 0.6 0.8 1.0050100150200250 Well depth/Vq Figure 3.4 Well depth vs. with the initial conditions of Figure 3.2. Reproduced from [18] with permission. 57 0 200 400 600 800050100150200250 Well depth/Vq*Vrf /V Figure 3.5 Well depth vs. with the initial conditions of Figure 3.2. A fit to gives , . Reproduced from [18] with permission. 3.4.2 Ion Oscillation Frequencies at Different Values The oscillation frequencies in the effective potential can be calculated. The restoring force on a singly charged ion is . The angular frequency of this oscillator is . Oscillation frequencies for different values are shown in Figure 3.6. Also shown are the fundamental secular frequencies calculated from Equation 3.1 with , and the frequencies calculated by the standard effective potential model (although it was thought valid only when ). The standard effective potential model and the method here give almost identical 58 results, for they give the well depths within 1% difference; in both methods, ion frequencies increase linearly with for all values of . The relative differences between the fundamental secular frequencies and frequencies predicted by our method are shown in Table 3.2. When , the frequencies predicted by our method agree well with the fundamental secular frequencies, with differences of 4% or less; this suggests that the effective potential method is a reasonable approximation to ion motion under these conditions. However, at higher values, this is no longer true; this method gives lower frequencies, although the fundamental secular frequencies can't accurately describe the real ion motion at high . 0.2 0.4 0.6 0.8 1.00100200300400500 Oscillation frequency/kHzq Figure 3.6 Ion oscillation frequencies vs. ,( ■) frequencies calculated from Equation 3.1 with , (▲) frequencies of the standard effective potential model from Equation 3.3, (●) this work. Reproduced from [18] with permission. 59 Table 3.2 Ion oscillation frequencies at different . Fundamental secular frequencies are calculated from the Equation 3.1 with . Relative differences are calculated by the differences of the two frequencies divided by the fundamental secular frequencies. values Fundamental secular frequency/ kHz Frequency predicted by this method/ kHz Relative difference/ % 0.2 71.27 70.53 0.79 0.3 108.03 105.97 1.8 0.4 146.28 141.49 3.4 0.5 186.87 177.01 5.7 0.6 231.12 212.58 8.9 0.7 281.53 248.27 14 0.8 344.28 283.90 22 0.9 436.97 320.01 37 60 3.5 Discussion 3.5.1 Ion Motion at High Values The effective potential is less successful at predicting the frequencies of ion motion at high . It gives a value for the frequency of a simple harmonic oscillation, which is close to the value of the ion fundamental secular frequency; however, this model doesn't describe the more complex behaviour caused by higher order terms (Equation 3.1 when ) of the ion motion. Actually, at high , these higher order terms have larger contributions to the ion motion compared to those at low ; it may not be reasonable to neglect the higher order terms for an accurate description of ion motion. Therefore, in these cases, the effective potential model is no longer an accurate approximation for describing ion motion. Ion trajectory simulations were carried out to investigate ion motion at different values. Figure 3.7 shows the ion trajectories at values from 0.20 to 0.90. At low values, ion motion is close to a simple harmonic oscillation with small high-frequency ripples (Figure 3.7 (a) and (b)). The amplitudes of the high-frequency components increase with and the ion motion becomes more complex at high values (Figure 3.7 (c) and (d)). At , as Sudakov [50] described, the and terms in Equation 3.1 have similar amplitudes and frequencies, and thus ion motion appears as a beat between these two frequencies, shown in Figure 3.7 (e). 61 Figure 3.7 Ion trajectories with and . No dipole DC is applied ( ), and damping of the ion motion is included. Initial conditions , , . 0 20 40 60 80 100-0.02-0.010.000.010.02q = 0.40RF cyclesX/r0RF cyclesa0 20 40 60 80 100-0.02-0.010.000.010.02bX/r0q = 0.200 20 40 60 80 100-0.02-0.010.000.010.02q = 0.80RF cyclesX/r0RF cyclesc0 20 40 60 80 100-0.02-0.010.000.010.02dX/r0q = 0.600 20 40 60 80 100-0.02- . 10.000.010.02RF cycleseX/r0q = 0.90 62 3.5.2 Possible Applications of Our Method In this work, we defined an effective potential by the average trapping forces of the RF or the effective electric fields at values of from to . This method doesn't rely on the assumptions of the standard effective potential model. The effective potential determined this way suggests that the standard effective potential model can be extended to high . Although the effective potential method can't account for the high-frequency components, given by the Mathieu equation, the effective potential here is consistent with confining ions with up to their maximum energies for any values within the first stability diagram. For example, ions at high can have kinetic energies of ca. 100 eV [51]. The well depths determined here are consistent with confining these ions (e.g. the well depth of 609 at = 0.85 is ca. 198 V), while the Sudakov model only gives a well depth of 11.5 V, significantly less than the ion energies. Recently, Douglas et al. used a matrix method and calculated the effective potential directly based on the Mathieu equation to obtain values of the well depths that are very close to those of this method [47]. Thus, this method enables determining the well depths of trapped ions at any value; this can be used to describe ion spatial distributions and the issues that relate to ion spatial distributions. Space charge effects are of special interest for this thesis. As discussed in Chapter 1, the electric field from space charge depends on the ratio between the well depth of the effective potential and the ion temperature . The standard model of the effective potential was thought to be limited to the cases when , but ions are usually ejected at high values (e. g. 0.85 in our experiments) in practice. The Sudakov model does not account for the ability to trap ions at with 100 eV kinetic energy. Our work suggest that the standard effective potential model can be extended to high , at least for the estimation of the trapping force of the RF and the maximum 63 energy of ions that can be confined. As a result, it is possible to describe and model space charge effects at high . 3.5.3 The Well Depth at the Stability Boundary Our method suggests that the well depth and the trapping forces increase as approaches the stability boundary at . At first, this seems counter-intuitive for ions can't be trapped at the boundary. However, the trapping efficiency decreases at high because the acceptance decreases to zero at [61] (the acceptance here refers to the combinations of values of ion initial coordinates and velocities that lead to ion confinement; these combinations usually form an ellipse area in a graph of ion initial coordinate and velocity, known as the acceptance ellipse [12, pp. 25-31]). Douglas et al. further found that the "potential well shape" depends on the potential well depth (i.e. the acceptable range of initial kinetic energies of the ions) and the potential well size (i.e. the acceptable range of initial coordinates) [47]. Moreover, the potential well depth is separated from the potential well size. At , ion trajectories are unstable because parametric resonance occurs between the component of ion motion with the fundamental frequency of and the trapping RF with a frequency of ; thus, in this case, the potential well size decreases to zero as while the potential well depth is still given by [47]. This argument agrees with the physical intuition that the trapping force increases with the trapping RF voltage, and the well depths are directly proportional to the integration of the trapping force and thus are not zero at . 64 3.5.4 The Effective Potential in 3-D Ion Traps In a 3D quadrupole trap, the potential that an ion experiences at any point is given by (3.22) where is the coordinate in the radial direction and is the coordinate in the axial direction [62]. Ion motion is described by the Mathieu equation, with the definitions (3.23) (3.24) Here and are a DC voltage and the amplitude of an RF voltage applied between an electrode and ground, while in some literature and are used as the pole-to-pole DC voltage and RF amplitude that differ by a factor of 2. The effective potential well depths for a 3-D ion trap at low are given by and (3.25) so that , where and are the well depths in the axial and radial directions [13, pp. 76-77]. The effective potential determined by our method was determined for a linear ion trap. However, because the equations of the motion in linear traps and 3D traps can both be reduced to the Mathieu equation, the effective potentials for linear traps and 3D traps are similar. Therefore, it is possible that the extension of the effective potential to high for linear quadrupoles also 65 applies to 3D traps, after accounting for the different well depths in the axial and radial directions. Experimental validation is needed to test this assumption. 3.6 Summary An effective potential for quadrupole linear ion traps can be defined and calculated at any value within the first stability diagram. We assume that the applied dipole DC electric field is countered by the electric field from the effective potential at the center of the displaced trajectory. The displacements caused by the dipole DC field were calculated by ion trajectory simulations. It is found that the standard effective potential can be extended up to the stability boundary . This method is less successful at predicting the frequencies of ion motion at high ; however, it gives reasonable results for the trapping forces of the trapping RF and the maximum energy of ions that can be confined. The method here can be used for any values and could also be used with other multipole traps (hexapole, octapole, etc.) and any combinations of multipoles. 66 Chapter 4 Dual-Frequency Excitation to Reduce Space Charge Effects 4.1 Introduction A theory of space charge effects in linear quadrupole ion traps was reviewed in Chapter 1, with the assumption that the trapped ions have a Boltzmann distribution of positions in the effective potential. The two parameters in this theory, the ion temperature and the effective potential, can be determined by the methods described in the last two chapters. These studies allow improved modelling of space charge effects with well determined parameters. Qiao et al reported that a proper choice of operating conditions can reduce mass shifts caused by space charge [27]. In this chapter, a new technique to reduce space charge effects, dual-frequency excitation, is introduced and investigated. The working principle of dual-frequency excitation comes from previous ion trajectory calculations [30]. It was found that, in the presence of space charge, the ion oscillation frequency depends on the ion oscillation amplitude and larger oscillation amplitudes result in lower shifts of the oscillation frequency, as shown in Figure 4.1; lower frequency shifts appear as lower mass shifts in mass spectra [27, 30]. Therefore, increasing ion oscillation amplitudes before ions are 67 ejected provides a possible method to reduce space-charge-induced mass shifts. Dual-frequency excitation is based on this idea. In this technique, an additional dipole AC voltage, referred as to the heating AC in this thesis, is applied to excite ions prior to ion ejection. The process of this ion excitation prior to the ion ejection is called "ion heating" herein, and is different from the heating of ions from the trapping RF. The frequency of the heating AC (also called the heating frequency below) is set close to and lower than the ejection frequency. The amplitude of the heating AC (also referred as to the heating amplitude) is usually considerably lower than the ejection amplitude to avoid ejecting ions from the trap in the ion heating process; the heating amplitude is given as peak to peak and pole to pole in this chapter. 68 Figure 4.1 The ratio, , of the ion oscillation frequency to the trapping RF frequency vs. the oscillation amplitude (normalized to ) for different linear charge densities with an ion temperature = 2000 K, = 4.17 mm, = 609, trapping = 0.85, trapping frequency = 1 MHz. The ion oscillation frequency without space charge is shown by the dashed line. and are initial velocities measured in units of (reproduced from [30]). 4.2 Methods The mass spectrometer system described in Chapter 2 was used in this study with the following changes. A shorter quadrupole (length 12.7 cm; = 4.17 mm) was used as the ion trap Q1. A 75 µm diameter ion sampling orifice was used; this gave a lower pressure (1.0 × 10-5 Torr) in Q1. The heating AC was produced by an arbitrary waveform generator (Agilent, model 33220 A, 69 Malaysia), and was added to the auxiliary AC voltage for dipole ejection by a dual sum and invert amplifier (Ortec, model 533, Oak Ridge, USA). A 10 µM reserpine solution (in 50% methanol, 49.5% water and 0.5% acetic acid) was used as a sample for the single ion species experiments. A solution with a mixture of analytes (in 50% methanol, 49.5% water and 0.5% acetic acid) was used to produce several ion species for experiments with multiple ion species. This mixed solution consisted of 2.2 µM Ala-Ala-Ala-Ala-Ala, 15 µM N-succinyl-Ala-Ala-Ala-p-nitroanilide, 3.1 µM leucine enkephalin acetate salt hydrate, 1.1 µM reserpine, 27 µM hexakis(1H, 1H, 2H-perfluroethoxy) phosphazene. The structures of these compounds are shown in Figure 4.2, and a mass spectrum of the mixed solution is given in Figure 4.3. Table 4.1 shows the main ions, the corresponding and the relative abundances in a mass spectrum of the mix solution. Ala-Ala-Ala-Ala-Ala, N-succinyl-Ala-Ala-Ala-p-nitroanilide, leucine enkephalin acetate salt hydrate and reserpine were from Sigma-Aldrich, St. Louis, MO. Hexakis(1H, 1H, 2H-perfluroethoxy) phosphazene was from SynQuest Laboratories, Inc., Alachua, FL. These reagents are analytical reagent (AR) grade. Nitrogen, acetic acid and methanol were as described in Chapter 2. The solution flow rate for the experiments was 0.5 µL/min or 1.0 µL/min. 70 Figure 4.2 The structures of the analytes, Ala-Ala-Ala-Ala-Ala (a), N-succinyl-Ala-Ala-Ala-p-nitroanilide (b), leucine enkephalin acetate salt hydrate (c), reserpine (d) and hexakis(1H, 1H, 2H-perfluroethoxy) phosphazene (e). 71 300 400 500 600 700100100010000m/z 452.3m/z 374.4m/z 474.3 m/z 556.6m/z 622.0m/z 609.3Intensitym/z Figure 4.3 A mass spectrum of the mixed solution with a logarithmic vertical scale at a scan speed of 250 Th/s. The relative abundances of the ion species are shown in Table 4.1. 72 Table 4.1 The main ion species, their , their relative abundances in mass spectra of the mixed solution and the corresponding stopping barriers for ion ejection used for these ion species. The relative abundances are calculated relative to the intensity (peak area) of 374.4. The MH+ and MNa+ refer to protonated ions and sodiated ions respectively. Note that for N-succinyl-Ala-Ala-Ala-p-nitroanilide, the intensity of its MNa+( 474.3) is about 6 times that of its MH+( 452.3) and its MNa+ is used to study mass shifts in these experiments. As for the other reagents, the protonated ions are most abundant. Reagents Main ions Relative abundance Stopping Barriers/ V Ala-Ala-Ala-Ala-Ala N-succinyl-Ala-Ala-Ala-p-nitroanilide MH+ MNa+ 374.4 474.3 1 1.0 10.2 12.9 Leucine enkephalin acetate salt hydrate MH+ 556.6 1.0 15.2 Reserpine MH+ 609.3 1.7 16.7 Hexakis(1H, 1H, 2H-perfluroethoxy) phosphazene MH+ 622.0 20 17.3 The set up and operation of the linear ion trap with axial ejection are described in Chapter 2. Similar operating parameters were used in this work. Ions were injected at a trapping RF voltage (ca. 228 V, zero to peak, pole to ground) corresponding to = 0.21 for 609.3, and cooled for 100 ms. The trapping RF was then scanned to eject ions at = 0.85 with dipole excitation; this 73 ejection corresponds to a frequency of 386.45 kHz for the ejection AC voltage. The amplitude of the ejection AC voltage was 500 mV (peak to peak, pole to pole) for all ions. The stopping potentials applied to the exit lens were increased with with a homemade ramp amplifier (UBC Chemistry electronics shop), which gave , where is the stopping barrier (the stopping potential minus the offset of Q1, measured in V). The stopping barriers for the analyte ions are listed in Table 4.1. The scan speed can be set by controlling the ramp range of the MCS and the scanning time; in this study three scan speeds, 250 Th/s, 500 Th/s and 1000 Th/s were used. Mass spectra with different numbers of trapped ions were collected; 100 or more scans were added for each mass spectrum. The number of trapped ions was controlled by the injection time and measured with a low stopping potential, as described in [27]. The space charge effects from single ion species and multiple ion species were studied. The mass shift reductions resulting from dual-frequency excitation were investigated. The operating parameters of dual-frequency excitation were optimized for each ion species at the three different scan speeds. The experiments were repeated on three different days to ensure reproducible results. 4.3 Results 4.3.1 Experiments with a Single Ion Species Space charge effects from a single ion species, or self-space charge effects, were studied. A 10 µM reserpine solution was used as a sample and the space charge was from the isotopic ions of reserpine only. Dual-frequency excitation was applied to reduce mass shifts from space charge. 74 Different combinations of the heating amplitudes and the heating frequencies were examined and optimized to obtain the lowest mass shifts at scan speeds of 250 Th/s, 500 Th/s and 1000 Th/s. 4.3.1.1 Scan Speed of 250 Th/s Mass spectra were collected without and with dual-frequency excitation applied at a scan speed of 250 Th/s. With normal single frequency axial ejection, the number of reserpine ions that causes a 0.1 Th mass shift is ca. 270,600. With dual-frequency excitation, a proper selection of the frequency and amplitude of the heating AC is crucial. A suitable frequency of the heating AC should be close to the ejection frequency (386.45 kHz in our experiments); otherwise, the excited ions will cool due to collisions with the background gas. The apparent masses of ions 609.3 vs. different numbers of ions with three frequencies of the heating AC and a 100 mV peak-to-peak heating amplitude are shown in Figure 4.4. In this case, 383.0 kHz gives the lowest mass shifts with a slight improvement over 382.0 kHz and 384.0.0 kHz. 75 0 400000 800000 1200000 1600000609.2609.4609.6609.8610.0 Single frequency382.0 kHz, 100 mV383.0 kHz, 100 mV384.0 kHz, 100 mVApparent massTotal ion number250 Th/s Figure 4.4 The apparent mass of ions 609.3 vs. the number of trapped ions with different heating frequencies applied at a scan speed of 250 Th/s. "Single frequency" in this and other figures in this chapter refers to the results with normal axial ejection. For dual-frequency excitation, three frequencies of the heating AC, 382.0 kHz, 383.0 kHz and 384.0 kHz, were used here with a peak-to-peak amplitude of 100 mV. Larger amplitudes of the heating AC were found to be more effective in reducing mass shifts. The amplitudes of the heating AC are given as peak to peak and pole to pole in this chapter. As shown in Figure 4.5, with the heating frequency 383.0 kHz, 100 mV peak-to-peak heating amplitude reduces mass shifts by ca. 47% while 80 mV only gives a mass shift reduction of 33%. 76 However, if the amplitude of the heating AC is too large, the heating AC may eject ions directly from the ion trap; for example, a 120 mV peak-to-peak heating amplitude ejects more than 10% of ions of 609.3. Ideally, the heating AC is designed to avoid ejecting ions; however, these experiments show that a small portion of the ions can still escape from the ion trap with even gentle excitation by the heating AC. Thus, the amplitude of the heating AC should be as large as possible with the limit that only a small portion of the trapped ions are ejected. The percentage of ions that are ejected by the heating AC also depends slightly on the number of trapped ions. In this work, the heating AC is chosen to eject only 10% or less of the ions in the presence of space charge that causes a mass shift of 0.1 Th. 77 0 400000 800000 1200000 1600000609.2609.4609.6609.8610.0 Single frequency383.0 kHz, 80 mV383.0 kHz, 100 mVApparent massTotal ion number250 Th/s Figure 4.5 The apparent mass of ions of 609.3 vs. the number of trapped ions with different heating amplitudes applied at a scan speed of 250 Th/s. For dual-frequency excitation, the frequency of the heating AC was 383.0 kHz with peak-to-peak amplitudes of 80 mV and 100 mV in this figure. Other combinations of the amplitudes and frequencies of the heating AC were also examined. Table 4.2 summarizes the mass shift reductions and the percentages of ions ejected by the heating AC in these experiments. The mass shift reductions were averaged according to the data collected on three different days, and had about 5% standard deviations or less. The percentage of ions ejected by the heating AC was estimated from the spectra with a mass shift of 0.1 Th. The ratios 78 of the peak area from 606.3 to 608.3 and that of 609.3 with and without dual-frequency excitation were calculated respectively; the differences between the two ratios with and without dual-frequency excitation can be attributed to the "ion ejection" by the heating AC. For example, the heating AC with a frequency 383.0 kHz and an amplitude 100 mV ejects 9.7% of trapped ions; at the same time, it reduces the mass shifts by ca. 44%, which is the greatest reduction at this scan speed. Table 4.2 Mass shift reductions and the percentages of reserpine ions ejected by the heating AC with different operating parameters. Heating AC Mass shift reduction (%) Ions ejected by the heating AC (%) 382.0 kHz (80 mV) 30.7 2.4 4.9 382.0 kHz (100 mV) 38.3 1.8 8.1 383.0 kHz (80 mV) 31.3 1.6 5.5 383.0 kHz (100 mV) 44.0 2.0 9.7 384.0 kHz (80 mV) 24.3 3.1 4.3 384.0 kHz (100 mV) 39.0 1.3 7.1 79 The effects of the dual-frequency excitation on the peak shape were also studied. The spectra without and with dual-frequency excitation in the presence of different numbers of trapped ions are shown in Figure 4.6. The peak positions of 609.3 are labelled in these spectra, which show a clear mass shift reduction due to dual-frequency excitation. The full width at half maximum (FWHM) of 609.3 can be calculated by the ratio of the peak area to the peak height; in our experiments, the FWHM was measured directly by the software Origin 7.0. The resolution can then be calculated by the equation . For example, the resolution at = 609.3 in Figure 4.6 (a) is calculated to be ca. 2400. It was found that dual-frequency excitation causes a little larger FWHM in this condition. 606 608 610 612 6140100200300400500dcbm/z609.28aFWHM=0.254606 608 610 612 6140100200300400500FWHM=0.266609.28IntensityIntensitym/z606 608 610 612 61402000400060008000FWHM=0.229609.42Single frequency421,600 ionsm/zIntensityIntensity606 608 610 612 61402000400060008000FWHM=0.260609.36383.0 kHz (100 mV)421,600 ions383.0 kHz (100 mV)28,200 ionsm/zSingle frequency28,200 ions 80 606 608 610 612 61403000600090001200015000hgfm/z609.53eFWHM=0.206606 608 610 612 61403000600090001200015000FWHM=0.261609.42IntensityIntensitym/z606 608 610 612 614040008000120001600020000FWHM=0.188609.61Single frequency900,100 ionsm/zIntensityIntensity606 608 610 612 614040008000120001600020000FWHM=0.274609.48383.0 kHz (100 mV)900,100 ions383.0 kHz (100 mV)689,200 ionsm/zSingle frequency689,200 ions 81 606 608 610 612 614040008000120001600020000lkjm/z609.71iFWHM=0.190606 608 610 612 614040008000120001600020000FWHM=0.316609.53IntensityIntensitym/z606 608 610 612 614040008000120001600020000FWHM=0.204609.83Single frequency1,478,700 ionsm/zIntensityIntensity606 608 610 612 614040008000120001600020000FWHM=0.320609.57383.0 kHz (100 mV)1,478,700 ions383.0 kHz (100 mV)1,163,800 ionsm/zSingle frequency1,163,800 ions Figure 4.6 Spectra without (the left, a, c, e, g, i, k) and with (the right, b, d, f, h, j, l) dual-frequency excitation with different numbers of trapped ions at a scan speed of 250 Th/s. The heating AC has a frequency of 383.0 kHz and an amplitude of 100 mV. The peak positions and FWHM of 609.3 are labelled. Figures 4.6 (j) and (l) show that the heating AC ejects a larger fraction of ions when the number of ions is more than 1,163,000; this can be ascribed to the increased repulsions between ions in these cases. 82 4.3.1.2 Scan speed of 500 Th/s Similar experiments were carried out at a scan speed of 500 Th/s. The same operating parameters were used except for the frequencies and amplitudes of the heating AC. At this scan speed, with normal axial ejection, ca. 188,300 ions cause a 0.1 Th mass shift. It was found that larger amplitudes of the heating AC could be used because of the greater scan speed. The combinations of the frequencies and the amplitudes of the heating AC were optimized (see Appendix B for details). The heating AC 383.0 kHz (160 mV) gives the best result and reduces mass shifts by ca. 53%, as shown in Figure 4.7. 0 200000 400000 600000 800000609.2609.4609.6609.8Single frequency383.0 kHz, 160 mVApparent massTotal ions500 Th/s Figure 4.7 The apparent mass of reserpine ions 609.3 vs. the number of trapped ions with and without dual-frequency excitation. For dual-frequency excitation, a heating AC with a frequency of 383.0 kHz and an amplitude of 160 mV was used. 83 The spectra with different numbers of trapped ions are shown in Figure 4.8. The resolution at 609.3 with normal axial ejection and without space charge is ca. 2050 at a scan speed of 500 Th/s. The application of the dual-frequency causes a larger FWHM and lower resolution; this is consistent with the results at 250 Th/s. 606 608 610 612 6140100200300400500dcbm/z609.28aFWHM=0.296606 608 610 612 6140100200300400500FWHM=0.308609.28IntensityIntensitym/z606 608 610 612 61405001000150020002500FWHM=0.274609.34Single frequency114,500 ionsm/zIntensityIntensity606 608 610 612 61405001000150020002500FWHM=0.321609.32383.0 kHz (160 mV)114,500 ions383.0 kHz (160 mV)22,100 ionsm/zSingle frequency22,100 ions 84 606 608 610 612 61401000200030004000hgfm/z609.42eFWHM=0.249606 608 610 612 61401000200030004000FWHM=0.353609.36IntensityIntensitym/z606 608 610 612 614010002000300040005000FWHM=0.242609.48Single frequency381,700 ionsm/zIntensityIntensity606 608 610 612 614010002000300040005000FWHM=0.357609.40383.0 kHz (160 mV)381,700 ions383.0 kHz (160 mV)267,200 ionsm/zSingle frequency267,200 ions 85 606 608 610 612 6140100020003000400050006000lkjm/z609.54iFWHM=0.230606 608 610 612 6140100020003000400050006000FWHM=0.358609.44IntensityIntensitym/z606 608 610 612 6140100020003000400050006000FWHM=0.242609.62Single frequency648,800 ionsm/zIntensityIntensity606 608 610 612 6140100020003000400050006000FWHM=0.383609.47383.0 kHz (160 mV)648,800 ions383.0 kHz (160 mV)496,200 ionsm/zSingle frequency496,200 ions Figure 4.8 Spectra without (the left, a, c, e, g, i, k) and with (the right, b, d, f, h, j, l) dual-frequency excitation with different numbers of trapped ions at a scan speed of 500 Th/s. For dual-frequency excitation, a heating AC with a frequency of 383.0 kHz and a peak-to-peak amplitude of 160 mV was used. 4.3.1.3 Scan Speed of 1000 Th/s Space charge effects were studied at a scan speed of 1000 Th/s. At this scan speed, ca. 140,100 ions produce a 0.1 Th mass shift with normal axial ejection. The heating AC was optimized 86 among the combinations of the frequencies 380.0 kHz, 382.0 kHz and 384.0 kHz and the amplitudes 180 mV, 200 mV and 220 mV (the details are shown in Appendix B). In this case, the heating AC 382.0 kHz (220 mV) gives the lowest mass shifts, a ca. 48% mass shift reduction, as shown in Figure 4.9. 0 150000 300000 450000 600000609.2609.4609.6609.8 Single frequency382.0 kHz, 220 mVApparent massTotal ion number1000 Th/s Figure 4.9 The apparent mass of reserpine ions 609.3 vs. the number of trapped ions with and without dual-frequency excitation at 1000 Th/s scan speed. For dual-frequency excitation, a heating AC with a frequency of 382.0 kHz and a peak-to-peak amplitude of 220 mV was used. 87 Spectra without and with dual-frequency excitation are shown in Figure 4.10. At this scan speed, the resolution at 609.3 for normal axial ejection without space charge is ca. 2000. With low numbers (less than 295,100) of ions trapped, dual-frequency excitation results in a larger FWHM and lower resolution, as shown in Figure 4.10 (a)-(h). However, when the number of ions increases to 396,000, dual-frequency excitation improves resolution. In these cases with normal axial ejection, space charge causes peak broadening and tailing, with an appearance of shoulder peaks, as shown in Figure 4.10 (i) and (k); dual-frequency excitation eliminates the shoulder peaks and improves peak shapes, resulting in better resolution, as shown in Figure 4.10 (j) and (l). 606 608 610 612 6140306090120150dcbm/z609.28aFWHM=0.304606 608 610 612 6140306090120150FWHM=0.431609.28IntensityIntensitym/z606 608 610 612 6140400800120016002000FWHM=0.282609.34Single frequency104,800 ionsm/zIntensityIntensity606 608 610 612 6140400800120016002000FWHM=0.434609.33382.0 kHz (220 mV)104,800 ions382.0 kHz (220 mV)25,000 ionsm/zSingle frequency25,000 ions 88 606 608 610 612 61405001000150020002500hgfm/z609.40eFWHM=0.286606 608 610 612 61405001000150020002500FWHM=0.439609.38IntensityIntensitym/z606 608 610 612 61405001000150020002500FWHM=0.361609.48Single frequency295,100 ionsm/zIntensityIntensity606 608 610 612 61405001000150020002500FWHM=0.438609.41382.0 kHz (220 mV)295,100 ions382.0 kHz (220 mV)184,200 ionsm/zSingle frequency184,200 ions 89 606 608 610 612 61406001200180024003000lkjm/z609.56iFWHM=0.589606 608 610 612 61406001200180024003000FWHM=0.434609.45IntensityIntensitym/z606 608 610 612 61406001200180024003000FWHM=0.593609.64Single frequency502,300 ionsm/zIntensityIntensity606 608 610 612 61406001200180024003000FWHM=0.514609.48382.0 kHz (220 mV)502,300ions382.0 kHz (220 mV)396,000 ionsm/zSingle frequency396,000 ions Figure 4.10 Spectra without (the left, a, c, e, g, i, k) and with (the right, b, d, f, h, j, l) dual-frequency excitation with different numbers of trapped ions at a scan speed of 1000 Th/s. For dual-frequency excitation, heating AC with a frequency of 382.0 kHz and a peak-to-peak amplitude of 220 mV was used. 4.3.2 Experiments with Multiple Ion Species Space charge effects from a single ion species have been described above. However, in practice, a more common case is that ions of different are trapped during mass analysis. In this section, 90 space charge effects from multiple ion species are investigated. A mixed solution (described in section 4.2) was used as a sample and it mainly produces ions with 374.4, 474.3, 556.6, 609.3 and 622.0. A typical spectrum is shown in Figure 4.3. The majority of the trapped ions are 622.0 and thus space charge is mainly from ions of 622.0. In other words, the self space charge effects can be neglected for ions except 622.0 in this case. In addition, when heavy ions are being ejected, ions with smaller have been ejected earlier and are no longer present in the ion trap; thus, in principle, the number of trapped ions needs to be corrected for each ion species. However, in this experiment, considering the majority of ions are 622.0, we can approximate that the number of trapped ions is same for each ion species to be ejected. The mass shifts of ions of 374.4, 474.3, 556.6 and 609.3 were studied and dual-frequency excitation was optimized for each ion species at scan speeds of 250 Th/s, 500 Th/s and 1000 Th/s respectively. 4.3.2.1 Scan Speed of 250 Th/s Mass spectra were collected with different numbers of trapped ions at a scan speed of 250 Th/s. With normal axial ejection, the mass shifts of ions depend slightly on their , as shown in Figure 4.11. For example, the number of trapped ions that causes a 0.1 Th mass shift is ca. 131,000 for 374.4 and ca. 108,000 for 609.3. In other words, heavy ions have slightly larger mass shifts. The mass shifts of ions of 622.0 are caused by self space charge and belong to the case of a single ion species that has been described above. The mass shifts of 374.4, 474.3, 556.6 and 609.3 are mainly from the interactions of other ion species (mostly ions of 622.0), which is the main concern in this section. Figure 4.11 also suggests 91 that the mass shifts caused by other ion species (the cases of 374.4, 474.3, 556.6 and 609.3) are more than two times those from self space charge (the case of 622.0). 0 100000 200000 300000 400000 5000000.00.10.20.30.40.5 m/z 374.4 m/z 474.3 m/z 556.6 m/z 609.3 m/z 622.0Mass shiftsTotal ions Figure 4.11 The mass shifts vs. the number of trapped ions for different ion species with normal axial ejection. For dual-frequency excitation, the heating AC was optimized for each ion species. Here, take the case of 609.3 as an example. The combinations of the frequencies 382.0 kHz, 383.0 kHz, 384.0 kHz and the peak-to-peak amplitudes 80 mV, 90 mV of the heating AC were tested. The mass shift reductions were averaged from the results on three different days, and are shown in 92 Table 4.3. Among these conditions, the heating AC with a frequency 383.0 kHz and an amplitude 90 mV gives the lowest mass shift reduction, ca. 24.7%. Similar procedures were then carried out to optimize the heating AC for other ion species (see the details in Appendix B). The optimum heating AC for the four ion species and the corresponding mass shift reduction are shown in Table 4.4 and Figure 4.12. It was found that the dual-frequency excitation reduces mass shifts somewhat for all the ion species. With heavy ions, larger heating amplitudes can be used prior to ion ejection; the best heating frequency also varies slightly with the of ions, e.g. 384.0 kHz for ions 374.4 and 383.0 kHz for the other three ion species. Table 4.3 Mass shift reductions of ions of 609.3 with different parameters of the heating AC. Heating AC Mass shift reduction (%) Ions ejected by the heating AC (%) 382.0 kHz (80mV) 20.7 1.8 2.1 382.0 kHz (90mV) 23.0 2.0 2.2 383.0 kHz (80mV) 23.3 1.1 2.7 383.0 kHz (90mV) 24.7 0.4 6.6 384.0 kHz (80mV) 16.7 1.1 3.9 384.0 kHz (90mV) 18.3 1.1 4.0 93 Table 4.4 The optimum heating AC and the corresponding mass shift reductions for the four ion species at a scan speed of 250 Th/s. Ion species Optimum heating AC Mass shift reduction (%) 374.4 384.0 kHz (50 mV) 25.7 4.2 474.3 383.0 kHz (60 mV) 15.0 2.0 556.6 383.0 kHz (70 mV) 16.3 3.1 609.3 383.0 kHz (90 mV) 24.7 0.4 94 0 100000 200000 300000 400000 500000374.4374.6374.8375.0dcbApparent mass Single frequency 384.0 kHz, 50 mVApparent massTotal ionsa0 100000 200000 300000 400000 500000474.2474.4474.6474.8Apparent massApparent massTotal ions Single frequency 383.0 kHz, 60 mV0 100000 200000 300000 400000 500000556.2556.4556.6556.8Total ionsTotal ions Single frequency 383.0 kHz, 70 mV0 100000 200000 300000 400000 500000609.2609.4609.6609.8 Single frequency 383.0 kHz, 90 mVFigure 4.12 The mass shifts vs. the number of trapped ions with and without dual-frequency excitation for different ion species. For the cases with dual-frequency excitation, the optimum heating AC for each ion species (listed in Table 4.4) was applied. The peak shapes without and with dual-frequency excitation are compared in Figure 4.13, with 609.3 as an example. The resolution at = 609.3 with normal ejection and 38,210 trapped ions (the case of Figure 4.13 (a)) is calculated to be ca. 2340. When more ions are trapped, the peaks broaden and resolution decreases. In addition, an overlapping peak appears between the 95 isotopes 609.3 and 610.3; when the number of ions increases to 472,500, it is almost impossible to locate the peak positions, shown in Figure 4.13 (k). With the heating AC applied, the mass shifts can be reduced slightly. Dual-frequency excitation causes slightly lower resolution when the number of ions is 342,000 or less. If more ions are trapped, the heating AC improves the resolution; for example, Figures 4-13 (i) and (j) and also (k) and (l) show that dual-frequency excitation results in better peak shapes and readable peak positions. 606 608 610 612 614030060090012001500FWHM=0.260m/zm/zm/z606 608 610 612 614030060090012001500FWHM=0.270609.28609.28IntensityIntensityIntensityIntensitym/z606 608 610 612 614010002000300040005000FWHM=0.269609.46Single frequency199,250 ions606 608 610 612 614010002000300040005000FWHM=0.304609.40dcba383.0 kHz (90 mV)199,250 ions383.0 kHz (90 mV)38,210 ionsSingle frequency38,210 ions 96 606 608 610 612 614010002000300040005000FWHM=0.298m/zm/zm/z606 608 610 612 614010002000300040005000FWHM=0.346609.50609.56IntensityIntensityIntensityIntensitym/z606 608 610 612 614010002000300040005000FWHM=0.300609.66Single frequency342,100 ions606 608 610 612 614010002000300040005000FWHM=0.363609.56hgfe383.0 kHz (90 mV)342,100 ions383.0 kHz (90 mV)265,200 ionsSingle frequency265,200 ions 97 606 608 610 612 614010002000300040005000FWHM=0.584m/zm/zm/z606 608 610 612 614010002000300040005000FWHM=0.364609.62609.74IntensityIntensityIntensityIntensitym/z606 608 610 612 614010002000300040005000FWHM=0.609~ 609.84Single frequency472,500 ions606 608 610 612 614010002000300040005000FWHM=0.374609.68lkji383.0 kHz (90 mV)472,500 ions383.0 kHz (90 mV)415,600 ionsSingle frequency415,600 ions Figure 4.13 Spectra of reserpine ions without (the left, a, c, e, g, i, k) and with (the right, b, d, f, h, j, l) dual-frequency excitation with different numbers of trapped ions at a scan speed of 250 Th/s. For dual-frequency excitation, heating AC with a frequency of 383.0 kHz and a peak-to-peak amplitude of 90 mV was used. 4.3.2.2 Scan Speed of 500 Th/s Space charge effects were then investigated with a scan speed of 500 Th/s. With normal axial ejection, the mass shifts of ions still depend slightly on their (shown in Figure B.1 in Appendix B); this is similar to the case of 250 Th/s. The number of trapped ions that causes a 0.1 98 Th mass shift is ca. 106,000 for 374.4 and ca. 94,000 for 609.3 at this scan speed. Similar procedures were carried out to optimize dual-frequency excitation for the four ion species; the results are summarized in Table 4.5 and Figure 4.14 (see details in Appendix B). It was found that greater amplitudes of the heating AC can be used, compared to the case at 250 Th/s; dual-frequency excitation reduces mass shifts for all the ion species, by ca. 30% ~ 36%. Table 4.5 The optimum heating AC and the corresponding mass shift reductions for the four ion species at a scan speed of 500 Th/s. Ion species Optimum heating AC Mass shift reduction (%) 374.4 384.0 kHz (90 mV) 36.3 3.6 474.3 383.0 kHz (120 mV) 29.7 0.9 556.6 383.0 kHz (130 mV) 30.0 0* 609.3 383.0 kHz (160 mV) 35.0 1.5 *Three repeated measurements gave a standard deviation of 0. In practice, mass shifts are reproducible within uncertainties of ca. 5.0%, which would gave (30.0 5.0)%. 99 0 100000 200000 300000 400000 500000374.4374.6374.8375.0dcbApparent mass Single frequency 384.0 kHz, 80 mVApparent massTotal ionsa0 100000 200000 300000 400000474.2474.4474.6474.8Apparent massApparent massTotal ions Single frequency 383.0 kHz, 120 mV0 100000 200000 300000 400000556.2556.4556.6556.8Total ionsTotal ions Single frequency 383.0 kHz, 130 mV0 100000 200000 300000 400000609.2609.4609.6609.8 Single frequency 383.0 kHz, 160 mV Figure 4.14 The mass shifts vs. the number of trapped ions with and without dual-frequency excitation for different ion species at a scan speed of 500 Th/s. For dual-frequency excitation, the optimum heating AC for each ion species was applied. The spectra of reserpine ions without and with dual-frequency excitation are shown in Figure 4.15. With normal ejection, the resolution at = 609.3 with 37,000 trapped ions (the case of Figure 4.15 (a)) is calculated to be ca. 2050. With more ions trapped, space charge effects appear: 100 the apparent mass increases with increasing ion numbers, and the peaks broaden and become asymmetric. In addition, an overlapping peak between the isotopes 609.3 and 610.3 appears, making it difficult to read the peak positions when the number of ions increases to 348,400, as shown in Figure 4.15 (k). The application of dual-frequency excitation reduces the mass shifts and results in more symmetric peak shapes, but sacrifices resolution when the number of ions is low (< 286,900). If more than 214,900 ions are trapped, the heating AC eliminates the overlap peak and improves the resolution, as shown in Figure 4.15 (l). 606 608 610 612 614080160240320400FWHM=0.296m/zm/zm/z606 608 610 612 614080160240320400FWHM=0.312609.28609.28IntensityIntensityIntensityIntensitym/z606 608 610 612 6140120240360480600FWHM=0.276609.36Single frequency92,300 ions606 608 610 612 6140120240360480600FWHM=0.308609.34dcba383.0 kHz (160 mV)92,300 ions383.0 kHz (160 mV)37,000 ionsSingle frequency37,000 ions 101 606 608 610 612 61402004006008001000FWHM=0.265m/zm/zm/z606 608 610 612 61402004006008001000FWHM=0.365609.40609.42IntensityIntensityIntensityIntensitym/z606 608 610 612 61402004006008001000FWHM=0.278609.48Single frequency214,900 ions606 608 610 612 61402004006008001000FWHM=0.354609.44hgfe383.0 kHz (160 mV)214,900 ions383.0 kHz (160 mV)153,400 ionsSingle frequency153,400 ions 102 606 608 610 612 614030060090012001500FWHM=0.364m/zm/zm/z606 608 610 612 614030060090012001500FWHM=0.394609.48609.55IntensityIntensityIntensityIntensitym/z606 608 610 612 614030060090012001500FWHM=0.688609.64Single frequency348,400 ions606 608 610 612 614030060090012001500FWHM=0.382609.51lkji383.0 kHz (160 mV)348,400 ions383.0 kHz (160 mV)286,900 ionsSingle frequency286,900 ions Figure 4.15 Spectra of reserpine ions without (the left, a, c, e, g, i, k) and with (the right, b, d, f, h, j, l) dual-frequency excitation with different numbers of trapped ions at a scan speed of 500 Th/s. For dual-frequency excitation, a heating AC with a frequency of 383.0 kHz and a peak-to-peak amplitude of 160 mV was used. For Figure 4.15 (a), 1000 scans were added. The intensity in Figure 4.15 (a) was normalized to 150 scans, which were used for all the other spectra. 103 4.3.2.3 Scan Speed of 1000 Th/s Similar experiments were carried out at a scan speed of 1000 Th/s. With normal axial ejection, the mass shifts of ions increase with their (shown in Figure B.2 in Appendix B), as with scan speeds of 250 Th/s and 500 Th/s. At 1000 Th/s, the number of trapped ions that causes a 0.1 Th mass shift is ca. 95,800 for 374.4 and ca. 77,800 for 609.3. Dual-frequency excitation was optimized for the four ion species; the results are summarized in Table 4.6 and Figure 4.16 (see details in Appendix B). Dual-frequency excitation reduces mass shifts for each ion species at this scan speed. Table 4.6 The optimum heating AC and the corresponding mass shift reductions for the four ion species at a scan speed of 1000 Th/s. Ion species Optimum heating AC Mass shift reduction (%) 374.4 382.0 kHz (160 mV) 48.3 2.2 474.3 384.0 kHz (180 mV) 48.7 1.8 556.6 384.0 kHz (180 mV) 36.3 5.1 609.3 384.0 kHz (200 mV) 40.7 3.6 104 0 100000 200000 300000 400000 500000374.4374.6374.8375.0dcbApparent mass Single frequency 382.0 kHz, 160 mVApparent massTotal ionsa0 70000 140000 210000 280000 350000474.2474.4474.6474.8Apparent massApparent massTotal ions Single frequency 384.0 kHz, 180 mV0 50000 100000 150000 200000 250000556.2556.4556.6556.8Total ionsTotal ions Single frequency 384.0 kHz, 180 mV0 50000 100000 150000 200000 250000609.2609.4609.6609.8 Single frequency 384.0 kHz, 200 mV Figure 4.16 The mass shifts vs. the number of trapped ions with and without dual-frequency excitation for different ion species at a scan speed of 1000 Th/s. For dual-frequency excitation, the optimum heating AC for each ion species (listed in Table 4.6) was applied. The spectra of reserpine ions without and with dual-frequency excitation are shown in Figure 4.17. With normal ejection, the resolution at 609.3 without space charge effects is ca. 1594 (Figure 4.17(a)). With more ions trapped, the peaks shift to higher apparent mass, and the peaks broaden and have tails. When the number of ions increases to 160,500, the peak broadening 105 makes it difficult to read the peak positions accurately, shown in Figure 4.17 (g). The application of the heating AC lowers resolution slightly when the number of ions is low; however, it reduces the mass shifts and results in more symmetric peak shapes. For example, the peak shape of Figure 4.17 (h) is better than that of Figure 4.17 (g). 606 608 610 612 614080160240320400FWHM=0.382m/zm/zm/z606 608 610 612 614080160240320400FWHM=0.464609.28609.28IntensityIntensityIntensityIntensitym/z606 608 610 612 6140120240360480600FWHM=0.316609.36Single frequency71,900 ions606 608 610 612 6140120240360480600FWHM=0.474609.34dcba384.0 kHz (200 mV)71,900 ions384.0 kHz (200 mV)29,800 ionsSingle frequency29,800 ions 106 606 608 610 612 6140160320480640800FWHM=0.502m/zm/zm/z606 608 610 612 6140160320480640800FWHM=0.481609.40609.42IntensityIntensityIntensityIntensitym/z606 608 610 612 61402004006008001000FWHM=0.651609.50Single frequency160,500 ions606 608 610 612 61402004006008001000FWHM=0.517609.43hgge384.0 kHz (200 mV)160,500 ions384.0 kHz (200 mV)117,700 ionsSingle frequency117,700 ions Figure 4.17 Spectra of reserpine ions without (the left, a, c, e, g) and with (the right, b, d, f, h) dual-frequency excitation with different numbers of trapped ions at a scan speed of 1000 Th/s. For dual-frequency excitation, a heating AC with a frequency of 384.0 kHz and an amplitude of 200 mV was used. For Figure 4.17 (a), 2000 scans were added, and its intensity was normalized to 200 scans, which were used for all the other spectra. 107 4.4 Discussion 4.4.1 Self-Space Charge and Space Charge from Other Ion Species The two cases corresponding to self-space charge and the interactions from other ion species are described above. It was found that mass shifts caused by space charge from other ion species are larger than those resulting from self-space charge. In other words, the number of trapped ions that result in a 0.1 Th mass shift of 609.3 in multiple ion species experiments is less than that with reserpine ions only. As shown in Table 4.7, this is seen at all three scan speeds. Take the case at 500 Th/s scan speed as an example. With a 0.1 Th mass shift introduced, about twice as many ions can be stored for single ion species experiments, relative to the case of multiple ion species. This may be because when only one ion species is trapped, the whole ion spatial distribution becomes diffuse due to ion ejection, and at the same time, the number of trapped ions starts to decrease, resulting in fewer interactions between ions. Therefore, self-space charge effects are the easiest to reduce. In contrast, with multiple ion species, only the ions being ejected are excited and the space charge from the other ions remains the same. In these experiments, space charge is mainly from the interactions with ions of 622, and thus space charge effects in these cases are the most severe. 108 Table 4.7 The number of trapped ions that result in a 0.1 Th mass shift of 609.3. Scan speed/ Th/s Space charge from one ion species Space charge from multiple ion species 250 270,600 108,000 500 188,300 94,000 1000 140,100 77,800 The phenomenon that space charge from other ion species causes greater mass shifts than self-space charge was observed earlier and reported in [27]. In that study, it was also found that applying quadrupole DC can reduce mass shifts caused by self-space charge; however, with other ion species in the trap, quadrupole DC is much less effective in reducing space charge effects [27]. This suggests that space charge effects from other ion species are more difficult to reduce than self-space charge effects. This difficulty can be solved to some extent by the application of dual-frequency excitation. As shown in Table 4.8, the applied heating AC is effective in reducing the mass shifts of 609.3 caused from either self-space charge or the interactions from other ions species at the three scan speeds. In addition, dual-frequency excitation results in greater mass shift reductions with self-space charge than with space charge from other ion species. For example, at 500 Th/s scan speed, the heating AC, 383.0 kHz (160 mV), reduces mass shifts by ca. 53%, when only reserpine ions are trapped; in the multiple ion species experiments, the same heating AC gives ca. 35% mass shift reduction for reserpine ions. Dual-frequency excitation is still effective in reducing mass 109 shifts caused by space charge from other ions; however, with the heating AC applied, space charge effects from other ions species are still more difficult to reduce than self-space charge effects. Table 4.8 The mass shift reductions of 609.3 from dual-frequency excitation. The measurement accuracy of the apparent mass is estimated to be 0.02 Th; for a mass shift of 0.4 Th, the relative uncertainty of mass shift reduction is about 5%. Scan speed/ Th/s Space charge from one ion species Space charge from multiple ion species Mass shift reduction (%) Optimized heating AC Mass shift reduction (%) Optimized heating AC 250 44 383.0 kHz (100 mV) 25 383.0 kHz (90 mV) 500 53 383.0 kHz (160 mV) 35 383.0 kHz (160 mV) 1000 48 382.0 kHz (220 mV) 41 382.0 kHz (200 mV) 4.4.2 Optimization of the Heating AC The operating parameters of the heating AC, its frequency and amplitude, were optimized in this study by examining the possible combinations of the heating frequencies and amplitudes for each ion species. This section describes an attempt to determine the rules for the optimization of the heating AC. To simplify the discussion, the selection of the heating frequency and the heating 110 amplitude are discussed separately, although in practice combinations of both parameters were optimized. The frequency of the heating AC and the scan speed determine the time interval between the ion heating and ion ejection for a fixed ejection frequency; this time interval can be used as a parameter to compare and select the frequency of the heating AC for different ion species at different scan speeds. The ions that are excited by the heating AC lose the increased energy through collisions with the background gas. Thus, the time interval between the ion heating and ion ejection should be smaller than the ion relaxation time. A model to calculate the ion relaxation time was described in Chapter 2. At the pressure of these experiments (1.0 10-5 Torr, N2), the energy relaxation time of reserpine ions ( 609.3 with a collision cross section 280 Å2 [34] ) is calculated to be 1.9 10-2 s; for ions of leucine enkephalin ( 556.6 with a cross section 162 Å2 [63]), the energy relaxation time is calculated to be 3.0 10-2 s. We lack data for the cross sections of ions of 374.4 and 474.3 to calculate their relaxation time directly, but we can assume that their relaxation time is of the same scale of reserpine ions. Therefore, a suitable frequency of the heating AC should result in a time interval between the ion heating and ion ejection that is less than 1.9 10-2 s or 19 ms in our experiments. It was found that the optimum time interval between the ion heating and ion ejection increases with the of ions. Table 4.9 shows the optimum frequency of the heating AC and the corresponding time interval between the ion heating and ion ejection for the four ion species at 250 Th/s scan speed. One possible reason is that the time for ions to be fully excited varies with their . In order to verify this argument, ion trajectories with dipole excitation of these four ion species were simulated (the excitation AC has a frequency of 386.453 kHz and a peak-to-peak amplitude of 100 mV), shown in Figure 4.18. These ion trajectories are the results of only one set 111 of initial conditions, and the trapping RF was set to the value that corresponds to 0.85 for each ion species in this simulation, while the trapping RF in real mass analysis is scanning. Thus, the time calculated from this method may differ from the real value in experiments. However, the relative time still describes the rates of different ions to be excited, suggesting that heavy ions require longer times to be fully excited. Table 4.9 The optimum frequency of the heating AC, and the corresponding time interval between the ion heating and ion ejection for the four ion species at a scan speed of 250 Th/s. (The time interval between the ion heating and ion ejection can be either calculated by the trapping RF voltage or measured directly in experiments. The data in this table are the results of experiments in which the ejection AC first had a frequency equal to the heating frequency and then the frequency used with normal ejection.) Calculations of the relative times are described in the caption of Figure 4.18. Ion species 374.4 474.3 556.6 609.3 Optimum frequency of the heating AC / kHz 384.0 383.0 383.0 383.0 Time interval between the ion heating and the ion ejection / ms 4.2 7.9 9.1 10 Relative times for ions to be excited to 1 1.28 1.49 1.63 112 0 500 1000 1500 2000-1.5-1.0-0.50.00.51.01.5x/r0dcbx/r0RF cyclesam/z 374.40 500 1000 1500 2000-1.5-1.0-0.50.00.51.01.5m/z 474.3x/r0x/r0RF cycles0 500 1000 1500 2000-1.5-1.0-0.50.00.51.01.5m/z 556.6RF cyclesRF cycles0 500 1000 1500 2000-1.5-1.0-0.50.00.51.01.5m/z 609.3 Figure 4.18 Ion trajectories of the four ion species with 0.85 and 0. Dipole excitation was added in the direction with a frequency of 386.453 kHz and a peak-to-peak amplitude of 100 mV, and damping of the ion motion was also included. Initial conditions , , . The numbers of RF cycles required to excite ions of 374.4, 474.3, 556.6 and 609.3 to are 1178, 1504, 1755 and 1918 respectively. The relative times for ions to be excited to for these four ion species can then be calculated to be 1, 1.28, 1.49 and 1.63 (normalized to 374.4). 113 The optimum time interval between the ion heating and the ion ejection also depends on the scan speed. Table 4.10 shows that with the optimum heating AC applied, the corresponding time interval between the ion heating and the ion ejection decreases with increasing scan speeds; this may be because greater amplitudes of the heating AC are usually used at higher scan speeds, resulting in less time to fully excite ions. Table 4.10 The optimum time interval (in ms) between the ion heating and ion ejection for the four ions species at the scan speeds of 250 Th/s, 500 Th/s and 1000 Th/s. Note that at 1000 Th/s the time interval for 374.4 is greater than that of 474.3. This phenomenon disagrees with the other data, and may be because only limited numbers of the heating frequencies were tested to produce the corresponding discrete time intervals. Scan speed (Th/s) 374.4 474.3 556.6 609.3 250 4.2 7.9 9.1 10.0 500 2.1 4.0 4.6 5.0 1000 2.0 1.5 1.7 1.8 The optimization of the amplitude of the heating AC is discussed below. Larger amplitudes of the heating AC are more effective in reducing mass shifts caused by space charge; however, at the same time, larger amplitudes also eject more ions directly from the ion trap. Thus, the maximum amplitude of the heating AC depends on the acceptable fraction of the ions that can be ejected by 114 the heating AC. In this study, the amplitude of the heating AC was selected to eject less than 10% of analyte ions. The optimum amplitude of the heating AC also increases approximately in direct proportion to the of ions. Take the case of 500 Th/s scan speed as an example, shown in Table 4.11. The ratios of the optimum heating amplitude to the of ions for the four ion species are around 0.24. The stopping barriers (the stopping potential of the exit lens minus the offset of Q1) varies linearly with the of ions, given by , where is in mV. Thus, the optimum amplitude of the heating AC is directly proportional to the stopping barriers; the higher stopping barriers allow larger amplitudes of the heating AC, resulting in lower mass shifts. Table 4.11 The optimum amplitudes of the heating AC for the four ion species at a scan speed of 500 Th/s. Analyzed ions 374.4 474.3 556.6 609.3 Optimum peak-to-peak amplitude of the heating AC (mV) 80 120 130 160 Ratio of the optimum heating amplitude to the of ions (mV/Th) 0.21 0.25 0.23 0.26 115 The optimum amplitude of the heating AC is also affected by the scan speed. The optimum amplitudes of the heating AC for ions of 374.4, 474.3, 556.6 and 609.3 at the three scan speeds are shown in Table 4.12. Greater scan speeds allow larger optimum amplitudes of heating AC because the available time to excite ions by the heating AC at greater scan speeds decreases. As a result, dual-frequency excitation gives a greater mass shift reduction at a greater scan speed. The ion heating process involves ion excitation, and thus it may be possible to optimize the amplitude of the heating AC by considering the energy that ions receive in the ion excitation. The increased energy of ions from excitation can be described by the product of the excitation time and excitation amplitude ( ). This excitation amplitude is the heating amplitude in these experiments. The excitation time can be calculated approximately by (4.1) where is the scan speed (in Th/s), and the with normal axial ejection was used. Table 4.12 shows that with the optimum heating amplitude applied, the values of are approximately constant at the three scan speeds for each ion species. Therefore, the values of can be used to choose the optimum heating amplitudes for ions with different at different scan speeds. In addition, it was found that is approximately proportional to the of ions; this agrees with the fact that the stopping barriers increase with the increasing of the ions. 116 Table 4.12 Optimization of the heating amplitudes for the four ion species at scan speeds of 250 Th/s, 500 Th/s and 1000 Th/s. Ion species Scan speed (Th/s) Optimum amplitude of the heating (AC/mV) (s mV) Mass shifts reduction by dual-frequency excitation (%) 374.4 250 50 0.084 26 500 90 0.084 36 1000 140 0.076 48 474.3 250 60 0.12 15 500 120 0.13 30 1000 180 0.12 49 556.6 250 70 0.14 16 500 130 0.15 30 1000 180 0.13 37 609.3 250 90 0.19 25 500 160 0.19 35 1000 200 0.15 41 117 To summarize, several rules can be drawn from the optimization of the heating AC. The frequency of the heating AC can be selected according to the time interval from the ion heating to the ion ejection. This time interval increases with of ions and decreases with the scan speed. Larger amplitudes of the heating AC are preferred to reduce mass shifts, with the limit that only an acceptable portion of ions are ejected. The heating amplitude can be optimized by the product of the excitation time and excitation amplitude, and was found to increases with the stopping barrier applied to the exit lens and also with the scan speed. 4.4.3 Effects of Dual-frequency Excitation on Resolution As described above, the heating AC reduces mass shifts caused either by self-space charge or interactions from other ion species. However, dual-frequency excitation also decreases the resolution slightly when the number of trapped ions is low. Consider the case without space charge effects (less than 40,000 trapped ions) as an example. Tables 4.13-4.16 compare the FWHM of the four ion species with and without dual-frequency excitation at three scan speeds. The resolution loss caused by the heating AC may be because the excitation by the heating AC causes a more diffuse spatial distribution of the ions being ejected. 118 Table 4.13 The FWHM of ions of 374.4 with and without dual-frequency excitation at scan speeds of 250 Th/s, 500 Th/s and 1000 Th/s in experiments with multiple ion species. Scan speed/ Th/s FWHM with normal axial ejection FWHM with the optimum heating AC Relative FWHM increase due to the heating AC (%) 250 0.215 0.004 0.231 0.006 7.7 500 0.234 0.002 0.259 0.004 11 1000 0.285 0.010 0.352 0.008 24 Table 4.14 The FWHM of ions of 474.3 with and without dual-frequency excitation at scan speeds of 250 Th/s, 500 Th/s and 1000 Th/s in experiments with multiple ion species. Scan speed/ Th/s FWHM with normal axial ejection FWHM with the optimum heating AC Relative FWHM increase due to the heating AC (%) 250 0.222 0.007 0.240 0.007 7.9 500 0.239 0.004 0.260 0.006 9.2 1000 0.294 0.009 0.349 0.012 19 119 Table 4.15 The FWHM of ions of 556.6 with and without dual-frequency excitation at scan speeds of 250 Th/s, 500 Th/s and 1000 Th/s in experiments with multiple ion species. Scan speed/ Th/s FWHM with normal axial ejection FWHM with the optimum heating AC Relative FWHM increase due to the heating AC (%) 250 0.233 0.002 0.245 0.006 5.3 500 0.250 0.005 0.281 0.010 12 1000 0.321 0.015 0.381 0.012 20 Table 4.16 The FWHM of ions of 609.3 with and without dual-frequency excitation at scan speeds of 250 Th/s, 500 Th/s and 1000 Th/s in experiments with multiple ion species. Scan speed/ Th/s FWHM with normal axial ejection FWHM with the optimum heating AC Relative FWHM increase due to the heating AC (%) 250 0.255 0.008 0.267 0.006 5.0 500 0.293 0.009 0.325 0.010 11 1000 0.342 0.008 0.426 0.012 24 120 It was also found that the loss of resolution is greater at a higher scan speed. At 250 Th/s, the applied heating AC increases the FWHM of the four ion species by ca. 5~8%; when the scan speed increases to 1000 Th/s, the applied heating AC produces a ca. 20% increase of FWHM. In these experiments, the ejection amplitude and the stopping barriers didn't change with the scan speeds. However, in practice, the ejection amplitudes and the stopping barriers are usually increased with increasing scan speeds [64]; a 500 mV peak-to-peak ejection amplitude and the stopping barriers used may not be optimum at 1000 Th/s. This is one possible reason why at 1000 Th/s dual-frequency excitation causes larger losses of resolution in these experiments. Although dual-frequency excitation may cause some peak broadening, it also results in more symmetric peak shapes. When the peaks are seriously broadened by space charge, dual-frequency excitation can improve the peak shapes and the resolution (an example is shown in Figure 4.15 (k) and (l)). 4.4.4 Effects of the Scan Speed The effects of the scan speed have been described above. The section aims to summarize these effects. With normal axial ejection, an increase of the scan speed causes larger mass shifts and lower resolution. As a result, at greater scan speeds, mass shifts may be relatively easier to reduce. Dual-frequency excitation results in greater mass shift reductions and greater losses of resolution at greater scan speeds. These conclusions are based on the precondition that the same ejection amplitude is used at different scan speeds. However, in practice, ejection amplitudes are normally optimized to increase with increasing scan speeds [64]. In this case, as Qiao et al. reported, the mass shifts can be reduced at higher scan 121 speeds [27]. The work in this chapter suggests that the mass shift reduction related to higher scan speeds is not actually caused by higher scan speeds; instead, greater ejection amplitudes used at higher scan speeds are the real causes for the reduction of mass shifts at higher scan speeds in [27]. 4.5 Summary Space charge effects in a linear ion trap, either from a single ion species or from the other ion species, have been investigated. Self-space charge is the simplest case, and space charge effects on lighter ions from a heavy ion species of high abundance are the most difficult to reduce. It was found that space charge from other ion species causes larger mass shifts than self-space charge, as in [27]. The application of dual-frequency excitation was proposed and investigated. This method is somewhat effective in reducing the mass shifts caused either by self-space charge or the interactions from other ion species. Dual-frequency excitation results in greater reductions of mass shifts caused from self-space charge, compared with those in the presence of multiple ion species. The operating parameters of this technique, the amplitude and frequency of the heating AC, were optimized. Larger amplitudes of the heating AC are preferable in reducing mass shifts, but may also eject more ions directly and also decrease the resolution. The optimum heating amplitude can be selected by the product of the excitation time and the heating amplitude; this optimum amplitude increases with scan speeds and with the stopping barriers applied to the exit lens (or the of ions). The frequency of the heating AC can be chosen to optimize the time interval between the heating and ion ejection. This optimum time interval should be the time to fully excite ions by the heating AC, and was found to increase with the of ions, decrease with scan speed, and be much less than the ion relaxation time. 122 Chapter 5 Summary and Future Work Space charge effects in linear quadrupole ion traps operated with mass selective axial ejection are the main concern of this thesis. When too many ions are trapped in an ion trap, the Coulomb repulsion between ions, or from space charge, causes the oscillation frequencies of ions to decrease; thus, greater RF voltages are required for ion ejection and ions shift towards higher apparent in a mass spectrum. The mass shifts caused by space charge can be described by a model that assumes that the interactions between ions can be approximated by an electric field averaged over time. The trapped ions are assumed to have a Boltzmann distribution of positions in an effective potential, and thus an approximate electric field can be determined from the ion distribution, as reviewed in Chapter 1. The two parameters in this model, the ion temperature and the effective potential, were determined by the methods described in Chapter 2 and Chapter 3 respectively. As a result, these studies allow improved modelling of space charge effects with well determined parameters. Chapter 4 described an experimental investigation of space charge effects in a linear quadrupole ion trap. The simplest case (self-space charge) and the most difficult case (space charge from a heavy ion species of high abundance) were studied. A new technique to reduce space charge effects, dual-frequency excitation, was introduced and investigated. This method applied an 123 additional dipole AC voltage (heating AC) to excite ions prior to ion ejection; different combinations of its frequency and amplitude were examined and optimized. At high scan speed (1000 Th/s), ca. 50% reductions in mass shifts were obtained; as a result, the trap capacity and ion intensities can be doubled. Overall, dual-frequency excitation results in only modest reductions of mass shifts induced by space charge. However, this method might be combined with other methods under development elsewhere to give lower mass shifts. In the future, dual-frequency excitation should be optimized with different ejection amplitudes. In this study, a fixed ejection amplitude was used for all the ion species and scan speeds; our instruments don't allow scanning ejection amplitudes when the trapping RF is scanned, and only scan speeds were changed in these controlled experiments. However, in practice, ejection amplitudes are usually increased with the of ions and scan speeds. Thus, it would be of interest to determine the influence of ejection amplitudes on the optimum parameters of the heating AC. Further, we can build a quantitative relation between the optimum parameters of dual-frequency excitation and the , scan speeds, ejection frequency as well as ejection amplitudes. Thus, it might be possible to write a program that can produce a waveform of the optimum heating AC (the heating frequency and amplitude should be changed with scan time) with the input of mass scan range, scan speed, ejection frequency and amplitude. This waveform of the optimum heating AC can then be added to and synchronized with the trapping RF to reduce space charge effects. A method to measure the ion temperature was described in Chapter 2. This work can be used to measure ratios of ion temperatures, but does not give an accurate absolute ion temperature. A better method to determine ion temperatures is to use ion trajectory simulations. The ion 124 temperature in a linear ion trap can be calculated from the average effective potential energy of the trapped ions, , given by (5.1) as describe in Chapter 2. The effective potential energy of the trapped ions is given by (5.2) where the well depth, , is given by Equation 3.3. Therefore, the ion temperature depends on the mean square positions of ions, , given by (5.3) An ensemble of ions with random initial conditions (including initial positions, initial velocities and initial RF phases) can be produced by ion trajectory simulations. The value of of each ion can be calculated from its ion trajectory directly, and thus the ion temperature can then be determined by the ensemble average of . 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Journal of the American Society for Mass Spectrometry 2009, 20, 1132-1140. 133 Appendices Appendix A Derivation of Equation 2.1 In a linear ion trap, the number density of singly charged ions (m-3) at a distance from the centre of the quadrupole is given by (A.1) where , , (A.2) The average potential energy of ions trapped in the effective potential can be calculated as follows: 134 (A.3) Use integration by parts, Equation A.3 becomes (A.4) Substitute Equation A.2, the first part of Equation A.4 is given by The second part of Equation A.4 becomes 135 Therefore, the average of the effective potential is given by (A.5) Considering a linear ion trap that used in our experiments, = 4.17 mm, = 2 MHz, operated at Mathieu parameter , corresponding to the well depth of 10.8 V for ions of 609.3 at the ion temperature of 2000 K, and . In this condition, and Equation A.5 can be written as (A.6) 136 Appendix B Optimization of Dual-Frequency Excitation B.1 Experiments with a Single Ion Species Dual-frequency excitation was optimized by examining the combinations of frequencies and amplitudes of the heating AC at scan speeds of 500 Th/s and 1000 Th/s, shown in Tables B.1 and B.2 respectively. The results at 250 Th/s were described above (Table 4.2). Table B.1 Mass shift reductions and the percentage of ions ejected by the heating AC with different operating parameters at a scan speed of 500 Th/s. Heating AC Mass shift reduction (%) Ions ejected by the heating AC (%) 382.0 kHz (140 mV) 46.0 2.7 6.6 382.0 kHz (160 mV) 50.0 1.3 9.7 383.0 kHz (140 mV) 48.3 2.4 6.6 383.0 kHz (160 mV) 53.0 1.3 9.9 384.0 kHz (140 mV) 46.7 1.1 4.1 384.0 kHz (160 mV) 50.0 0 8.7 137 Table B.2 Mass shift reductions and the percentages of ions ejected by the heating AC with different operating conditions at a scan speed of 1000 Th/s. Heating AC Mass shift reduction (%) Ions ejected by the heating AC (%) 380.0 kHz (180 mV) 44.0 2.7 4.7 380.0 kHz (200 mV) 44.7 2.2 6.5 380.0 kHz (220 mV) 47.0 3.3 9.1 382.0 kHz (180 mV) 45.7 1.6 4.3 382.0 kHz (200 mV) 47.0 3.3 7.6 382.0 kHz (220 mV) 47.7 2.9 9.6 384.0 kHz (180 mV) 39.0 2.7 3.6 384.0 kHz (200 mV) 42.3 1.6 5.8 384.0 kHz (220 mV) 44.0 2.7 9.0 138 B.2 Experiments with Multiple Ion Species B.2.1 Scan Speed 250 Th/s The optimization data for the heating AC for ions of 374.4, 474.3 and 556.6 are shown in Tables B.3, B.4 and B.5 respectively. The results of 609.3 were described above (Table 4.3). Table B.3 Optimization of the heating AC for ions of 374.4 at 250 Th/s. Heating AC Mass shift reduction (%) Ions ejected by the heating AC (%) 382.0 kHz (40 mV) 11.7 1.6 4.2 382.0 kHz (50 mV) 13.3 2.2 8.8 383.0 kHz (40 mV) 14.7 3.1 4.0 383.0 kHz (50 mV) 21.0 1.3 9.2 384.0 kHz (40 mV) 14.7 3.1 1.5 384.0 kHz (50 mV) 25.7 4.2 4.9 139 Table B.4 Optimization of the heating AC for ions of 474.3 at 250 Th/s. Heating AC Mass shift reduction (%) Ions ejected by the heating AC (%) 382.0 kHz (50 mV) 9.7 3.6 3.1 382.0 kHz (60 mV) 14.0 0 3.8 383.0 kHz (50 mV) 15.0 0 5.0 383.0 kHz (60 mV) 15.0 2.0 8.5 384.0 kHz (50 mV) 10.0 3.3 6.9 384.0 kHz (60 mV) 13.7 2.2 7.2 140 Table B.5 Optimization of the heating AC for ions of 556.6 at 250 Th/s. Heating AC Mass shift reduction (%) Ions ejected by the heating AC (%) 382.0 kHz (60 mV) 15.0 2.7 0.7 382.0 kHz (70 mV) 15.7 2.2 1.4 383.0 kHz (60 mV) 14.3 3.1 0.8 383.0 kHz (70 mV) 16.3 3.1 3.0 384.0 kHz (60 mV) 12.7 1.8 3.1 384.0 kHz (70 mV) 13.3 2.2 5.1 B.2.2 Scan speed 500 Th/s With normal axial ejection, the mass shifts of different ion species vs. the number of trapped ions are shown in Figure B.1. The optimization data for ions of 374.4, 474.3, 556.6 and 609.3 are shown in Tables B.6, B.7, B.8 and B.9 respectively. 141 0 80000 160000 240000 320000 4000000.00.10.20.30.40.5 m/z 374.4 m/z 474.3 m/z 556.6 m/z 609.3 m/z 622.0Mass shiftsTotal ions Figure B.1 The mass shifts of different ion species vs. the number of trapped ions at a scan speed of 500 Th/s. Normal single frequency axial ejection was used. 142 Table B.6 Optimization of the heating AC for ions of 374.4 at 500 Th/s. Heating AC Mass shift reduction (%) Ions ejected by the heating AC (%) 382.0 kHz (70 mV) 18.0 1.3 3.6 382.0 kHz (80 mV) 22.0 2.7 6.3 382.0 kHz (90 mV) 24.3 2.2 8.9 383.0 kHz (70 mV) 27.8 2.3 4.8 383.0 kHz (80 mV) 32.8 1.8 7.2 383.0 kHz (90 mV) 36.2 2.3 9.1 384.0 kHz (70 mV) 29.0 2.0 5.2 384.0 kHz (80 mV) 32.3 1.1 8.2 384.0 kHz (90 mV) 36.3 3.6 9.8 143 Table B.7 Optimization of the heating AC for ions of 474.3 at 500 Th/s. Note that a 130 mV peak-to-peak amplitude of the heating AC ejects more than 10% of ions of 474.3; 383.0 kHz and 120 mV were chosen as the optimum. Heating AC Mass shift reduction (%) Ions ejected by the heating AC (%) 382.0 kHz (110 mV) 24.0 0.7 6.7 382.0 kHz (120 mV) 25.0 2.0 7.8 382.0 kHz (130 mV) 27.7 1.1 12.3 383.0 kHz (110 mV) 27.7 1.8 7.1 383.0 kHz (120 mV) 29.7 0.9 9.7 383.0 kHz (130 mV) 34.7 1.8 14.4 384.0 kHz (110 mV) 26.0 0.7 6.5 384.0 kHz (120 mV) 27.0 1.3 12.5 384.0 kHz (130 mV) 32.0 2.7 14.6 144 Table B.8 Optimization of the heating AC for ions of 556.6 at 500 Th/s. Heating AC Mass shift reduction (%) Ions ejected by the heating AC (%) 382.0 kHz (110 mV) 19.7 0.4 2.1 382.0 kHz (120 mV) 23.7 2.4 3.0 382.0 kHz (130 mV) 27.7 1.8 5.6 383.0 kHz (110 mV) 22.0 2.7 3.7 383.0 kHz (120 mV) 25.3 0.4 5.8 383.0 kHz (130 mV) 30.0 0* 9.0 384.0 kHz (110 mV) 23.7 2.4 8.1 384.0 kHz (120 mV) 25.3 0.4 12.1 384.0 kHz (130 mV) 27.7 1.8 14.8 *Three repeated measurements gave a standard deviation of 0. In practice, mass shifts are reproducible within uncertainties of ca. 5.0%, which would gave (30.0 5.0)%. 145 Table B.9 Optimization of the heating AC for ions of 609.3 at 500 Th/s. Heating AC Mass shift reduction (%) Ions ejected by the heating AC (%) 382.0 kHz (120 mV) 21.0 2.0 2.7 382.0 kHz (140 mV) 26.7 4.4 4.2 382.0 kHz (160 mV) 30.7 3.8 8.7 383.0 kHz (120 mV) 26.0 1.0 3.4 383.0 kHz (140 mV) 29.8 2.3 5.0 383.0 kHz (160 mV) 35.0 1.5 9.8 384.0 kHz (120 mV) 24.0 1.3 4.6 384.0 kHz (140 mV) 27.3 0.4 9.2 384.0 kHz (160 mV) 31.3 3.1 15.7 146 B.2.3 Scan speed 1000 Th/s With normal axial ejection, the mass shifts of different ion species vs. the number of trapped ions are shown in Figure B.2. The data for the optimization of the heating AC for ions of 374.4, 474.3, 556.6 and 609.3 are shown in Tables B.10, B.11 B.12 and B.13 respectively. 0 50000 100000 150000 200000 2500000.00.10.20.30.40.5 m/z 374.4 m/z 474.3 m/z 556.6 m/z 609.3 m/z 622.0Mass shiftsTotal ions Figure B.2 The mass shifts of different ion species vs. the number of trapped ions at a scan speed of 1000 Th/s with normal axial ejection. 147 Table B.10 Optimization of the heating AC for ions of 374.4 at 1000 Th/s. Heating AC Mass shift reduction (%) Ions ejected by the heating AC (%) 380.0 kHz (140 mV) 31.7 2.2 3.2 380.0 kHz (160 mV) 33.3 2.2 9.5 380.0 kHz (180 mV) 37.0 0* 16.7 382.0 kHz (140 mV) 48.3 2.2 3.3 382.0 kHz (160 mV) 48.3 2.2 8.9 382.0 kHz (180 mV) 54.3 0.9 16.9 384.0 kHz (140 mV) 38.3 2.2 2.0 384.0 kHz (160 mV) 38.3 2.2 7.6 384.0 kHz (180 mV) 31.7 2.2 15.5 *Three repeated measurements gave a standard deviation of 0. In practice, mass shifts are reproducible within uncertainties of ca. 5.0%, which would gave (37.0 5.0)%. 148 Table B.11 Optimization of the heating AC for ions of 474.3 at 1000 Th/s. Heating AC Mass shift reduction (%) Ions ejected by the heating AC (%) 380.0 kHz (160 mV) 27.0 2.7 3.9 380.0 kHz (180 mV) 28.3 2.2 6.4 382.0 kHz (160 mV) 35.7 1.8 4.5 382.0 kHz (180 mV) 37.0 3.3 9.1 384.0 kHz (160 mV) 46.3 4.9 5.6 384.0 kHz (180 mV) 48.7 1.8 9.8 149 Table B.12 Optimization of the heating AC for ions of 556.6 at 1000 Th/s. Heating AC Mass shift reduction (%) Ions ejected by the heating AC (%) 380.0 kHz (160 mV) 25.3 3.8 2.0 380.0 kHz (180 mV) 28.7 1.6 3.2 380.0 kHz (200 mV) 32.0 4.0 7.8 382.0 kHz (160 mV) 28.7 3.6 3.0 382.0 kHz (180 mV) 33.0 4.0 6.1 382.0 kHz (200 mV) 33.7 4.4 11.5 384.0 kHz (160 mV) 34.0 6.7 2.5 384.0 kHz (180 mV) 36.3 5.1 9.4 384.0 kHz (200 mV) 42.7 4.9 14.5 150 Table B.13 Optimization of the heating AC for ions of 609.3 at 1000 Th/s. 384.0 kHz and 200 mV were chosen as the best operating parameters although this gives 10.3% ejection of ions. Heating AC Mass shift reduction (%) Ions ejected by the heating AC (%) 380.0 kHz (180 mV) 21.0 0.7 3.7 380.0 kHz (200 mV) 29.3 4.4 7.4 382.0 kHz (180 mV) 31.3 4.2 4.7 382.0 kHz (200 mV) 32.7 2.4 9.1 384.0 kHz (180 mV) 34.0 1.3 5.4 384.0 kHz (200 mV) 40.7 3.6 10.3
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Space charge effects in linear quadrupole ion traps
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