Space Charge Effects in Linear. Quadrupole Ion Traps

Transcription

1 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

2 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-tocharge 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. ii

3 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 massselective axial ejection from a linear quadrupole ion trap. Rapid Communications In Mass Spectrometry 2011, 25 (23), 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), 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 iii

4 experimental configuration, suggested new experiments, analyzed the data and discussed the results. iv

5 Table of Contents Abstract... Preface... Table of Contents... List of Tables... List of Figures... List of Symbols... List of Abbreviations... ii iii v vii x xvii xx Acknowledgements... xxii 1 Introduction Theory of Linear Quadrupoles Linear Quadrupole Ion Traps Space Charge Effects in Linear Ion Traps Outline of This Thesis Measurements of Ion Temperatures Introduction Effusive Flow Methods Results and Discussion Summary v

6 3 Determination of the Effective Potential of a Linear Quadrupole Ion Trap Background Theory and Development of the Effective Potential Methods Results Discussion Summary Dual-Frequency Excitation to Reduce Space Charge Effects Introduction Methods Results Discussion Summary Summary and Future Work References Appendices Appendix A Derivation of Equation Appendix B Optimization of Dual-Frequency Excitation vi

7 List of Tables Table 2.1 Instrumentation Table 2.2 Operating voltages (V) for mass selective axial ejection Table 2.3 Ion temperature Table 2.4 Ratios of mass shifts at two ion temperatures T 1 and T 2 with a ratio of Δm 1 /Δm 2 is the ratio of mass shifts at T 1 and T 2, and well depth is given by Table 3.1 Quadrupole parameters used in the ion trajectory simulations Table 3.2 Ion oscillation frequencies at different. Fundamental secular frequencies are calculated by the Equation 3.1 when 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 Table 4.2 Mass shift reductions and the percentages of reserpine ions ejected by the heating AC with different operating parameters Table 4.3 Mass shift reductions of ions of with different parameters of the heating AC 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 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 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 vii

8 Table 4.7 The number of trapped ions that result in a 0.1 Th mass shift of Table 4.8 The mass shift reductions of from dual-frequency excitation 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 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 Table 4.11 The optimum amplitudes of the heating AC for the four ion species at a scan speed of 500 Th/s 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 Table 4.13 The FWHM of ions of 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 Table 4.14 The FWHM of ions of 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 Table 4.15 The FWHM of ions of 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 Table 4.16 The FWHM of ions of 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 viii

9 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 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 Table B.3 Optimization of the heating AC for ions of at 250 Th/s Table B.4 Optimization of the heating AC for ions of at 250 Th/s Table B.5 Optimization of the heating AC for ions of at 250 Th/s Table B.6 Optimization of the heating AC for ions of at 500 Th/s Table B.7 Optimization of the heating AC for ions of at 500 Th/s Table B.8 Optimization of the heating AC for ions of at 500 Th/s Table B.9 Optimization of the heating AC for ions of at 500 Th/s Table B.10 Optimization of the heating AC for ions of at 1000 Th/s Table B.11 Optimization of the heating AC for ions of at 1000 Th/s Table B.12 Optimization of the heating AC for ions of at 1000 Th/s Table B.13 Optimization of the heating AC for ions of at 1000 Th/s ix

10 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 Figure 1.4 Schematic of a triple quadrupole mass spectrometer. Mass selective axial ejection from either q 2 or Q 3 is possible Figure 2.1 (a) Boltzmann distributions (Equation 1.24) of 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) Figure 2.2 An oblique cylinder containing molecules with the velocity that can pass through an orifice in a time interval Figure 2.3 Schematic of the mass spectrometer system used in this thesis Figure 2.4 Schematic of the ion trap set up Figure 2.5 Schematic of the timing sequence for mass selective axial ejection Figure 2.6 Schematic of the timing sequence for measurements of the ion temperature x

11 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 K. (d) The curve of (c) with a logarithmic vertical scale 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 Figure 2.9 Effects of quadrupole DC voltages on mass shifts caused by space charge. Apparent mass of vs. the number of trapped ions with (a) = 0.20, = 0.83 to 0.85, scan speed ca. 50 Th/s, = 0.30 V p-p. (b) = 0.40, = 0.78 to 0.85, scan speed ca. 50 Th/s, = 0.30 V p-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 V p-p, switched polarity of the DC. (d) = 0.20, = 0.83 to 0.85, scan speed ca Th/s, = 1.0 V p-p. (e) = 0.40, = 0.78 to 0.85, scan speed ca Th/s, = 1.0 V p-p. (f) = 0.20, = 0.83 to 0.85 (positive), 0.85 to 0.87 (negative), scan speed ca Th/s, = 1.0 V p-p, switched polarity of the DC 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 xi

12 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,, 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 Figure 3.4 Well depth vs. with the initial conditions of Figure Figure 3.5 Well depth vs. with the initial conditions of Figure 3.2. A fit to gives, 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 Figure 3.7 Ion trajectories with and. No dipole DC is applied ( ), and damping of the ion motion is included. Initial conditions,, 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 xii

13 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) Figure 4.3 A mass spectrum of the mixed solution with a logarithmic vertical scale at a scan speed of 250 Th/s Figure 4.4 The apparent mass of ions 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, khz, khz and khz, were used here with a peak-to-peak amplitude of 100 mv Figure 4.5 The apparent mass of ions of 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 khz with peak-to-peak amplitudes of 80 mv and 100 mv in this figure Figure 4.6 Spectra without (the left, a, c, e, g, i, k) and with (the right, b, d, f, h, j, l) dualfrequency excitation with different numbers of trapped ions at a scan speed of 250 Th/s. The heating AC has a frequency of khz and an amplitude of 100 mv. The peak positions and FWHM of 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 Figure 4.7 The apparent mass of reserpine ions vs. the number of trapped ions with and without dual-frequency excitation. For dual-frequency excitation, a heating AC with a frequency of khz and an amplitude of 160 mv was used xiii

14 Figure 4.8 Spectra without (the left, a, c, e, g, i, k) and with (the right, b, d, f, h, j, l) dualfrequency 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 khz and a peak-topeak amplitude of 160 mv was used Figure 4.9 The apparent mass of reserpine ions 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 khz and a peak-to-peak amplitude of 220 mv was used Figure 4.10 Spectra without (the left, a, c, e, g, i, k) and with (the right, b, d, f, h, j, l) dualfrequency 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 khz and a peak-topeak amplitude of 220 mv was used Figure 4.11 The mass shifts vs. the number of trapped ions for different ion species with normal axial ejection Figure 4.12 The mass shifts vs. the number of trapped ions with and without dualfrequency 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 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 khz and a peak-to-peak amplitude of 90 mv was used xiv

15 Figure 4.14 The mass shifts vs. the number of trapped ions with and without dualfrequency excitation for different ion species at a scan speed of 500 Th/s. For dualfrequency excitation, the optimum heating AC for each ion species was applied 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 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 Figure 4.16 The mass shifts vs. the number of trapped ions with and without dualfrequency excitation for different ion species at a scan speed of 1000 Th/s. For dualfrequency excitation, the optimum heating AC for each ion species (listed in Table 4.6) was applied 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 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 Figure 4.18 Ion trajectories of the four ion species with and. Dipole excitation was added in the direction with a frequency of khz and a peak-topeak amplitude of 100 mv, and damping of the ion motion was also included. Initial conditions,, xv

16 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 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 xvi

17 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 xvii

18 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 V ex Amplitudes of the excitation voltages Pressure of the gas, Fitting parameters of the ion drain curve xviii

19 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 xix

20 List of Abbreviations Abbreviation Description 3D AC AR CID DC ESI FTICR FWHM MALDI MCS MS MS/MS, MS 3, MS n MSAE Q 0 Three-Dimensional Alternating Current Analytical Reagent Collision Induced Dissociation Direct Current Electrospray Ionization Fourier Transform Ion Cyclotron Resonance Full Width at Half Maximum Matrix Assisted Laser Desorption Ionization Multichannel Scalar Mass Spectrometry Tandem Mass Spectrometry Mass Selective Axial Ejection Quadrupole Ion Guide Q 1 Quadrupole Down-stream of Q 0 q 2 Collision Cell Q 3 Quadrupole Down-stream of q 2 RF RIT Radio Frequency Rectilinear ion trap xx

21 Th TOF UHP V p-p Thomson (the unit of the mass-to-charge ratio) Time-of-Flight Ultra High Purity Volt, Peak to Peak xxi

22 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. xxii

23 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 1

24 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 MS n, 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, MS 3 or MS n (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 2

25 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). 3

26 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) 4

27 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 massto-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 ] (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 5

28 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, 6

29 (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. 7

30 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 8

31 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] 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]. 9

32 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, outof-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. 10

33 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 khz, with a notch at the frequency of the analyte ions, usually at activated by resonant excitation at a frequency corresponding to a. For MS/MS, ions are from 0.25 to 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 MS 4 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. 11

34 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 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 ( 12

35 ) 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 (q 2 ) or the down-stream quadrupole (Q 3 ). 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 q 2 ( ) as the ion trap, the mass resolution at was 1000 with a scan speed of 1000 Th/s. A slower scan of 5 Th/s produced a resolution of Axial ejection from Q 3 is also possible and a resolution of 6000 was obtained with a scan speed of 100 Th/s. Due to the low pressure in Q 3 ( ), a scan speed 20 times greater than that in q 2 with the same resolution was possible. An MS/MS experiment was also demonstrated. The precursor ions were mass selected by Q 1 and then were injected into q 2 with sufficient energy to induce fragmentation. The fragment ions, generated by CID in the pressurized q 2, were trapped and axially ejected from Q 3. The sensitivity was improved by a factor of 16, 13

36 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 q 2 or Q 3 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. 14

37 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]. 15

38 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) 16

39 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 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 N 2, 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) 17

40 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. 18

41 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. 19

42 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. 20

43 8.0e e+5 (a) 1.4e+5 (b) N(x) or N(y) m e e e K N(r) dr 1.2e+5 1.0e+5 8.0e+4 6.0e+4 4.0e K 7000 K 7000 K 2.0e x or y (mm) r (mm) (c) K 10 E (V/m) K r (mm) 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. 21

44 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 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) 22

45 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, 23

46 (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 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 N 2 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 (Q 0 ) (length, 20 cm; = 4.17 mm; pressure, Torr of N 2 ) where they are 24

47 collisionally cooled to a nominal energy of 1 ev with energy spreads of ca. 1 ev. Ions leave Q 0 and pass into the linear quadrupole trap, Q 1 (length, 20 cm; = 4.17 mm) in a chamber pumped to Torr, through an "entrance lens" with a 3.0 mm diameter aperture. The Q 0 rods are capacitively coupled to the Q 1 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 Q 1. The first, with a mesh-covered 9-mm diameter aperture, is located 3.5 mm from the end of Q 1. 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. 25

48 Table 2.1 Instrumentation. Q 0 rod set Q 1 rod set Quadrupole power supply Auxiliary AC arbitrary UBC Chemistry mechanical shop AB SCIEX (Concord, ON) AB SCIEX (Concord, ON) Agilent, model A (Malaysia) waveform generator Detector Burle Electro-Optics channeltron electron multiplier, model 4824 (USA) Preamplifier Discriminator Syringe pump Rotary pumps Ortec 9302 fast amplifier (USA) Ortec, model 436 (USA) Harvard Apparatus, model 22 (USA) Leybold, Trivac D16B (Germany), pump speed 165 m 3 /h Turbo pumps Leybold, Turbovac 361(Germany), pump speed 345 L/s Pirani gauge Ion gauge Leybold, Thermovac TM230 (USA) Granville-Philips, model , 270 gauge controller (USA) BenchTop Lite PCI arbitrary waveform PC Instruments, Inc. (USA) generator Multichannel scalar (MCS) PCI card with MCS-32 operating software Ortec, model A73-B32, software version 2.13 (USA) 26

49 Table 2.2 Operating voltages (V) for mass selective axial ejection. Sprayer Curtain plate +500 Orifice +12 Q 0 rod offset +4 Entrance lens +3 or +130 Q 1 rod offset +2 Exit 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 27

50 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 Q 0 before the next cycle starts. Figure 2.4 Schematic of the ion trap set up. 28

51 Injection Cooling Scanning Drain Trap Entrance Trap Exit Trapping RF ms 100 ms ms 50 ms 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, V ex, are given as peak to peak (V p-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 29

52 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 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 30

53 of the timing sequence for this process is shown in Figure 2.6. For the ion drain period, exit lens 1 was set at V and exit lens 2 at -30 V. Injection Cooling Delay Drain Trap Entrance Trap Exit Trapping RF ms 100 ms ms 50 ms 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) 31

54 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 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 The uncertainty of the ion temperature is estimated by 32

55 dn dn dn dn (2.12) where is the fitting uncertainty of. For this experiment, the ion temperature is determined to be K. a b c t/ms dn=a'exp(-b't) d t/ms t/ms t/ms 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 K. (d) The curve of (c) with a logarithmic vertical scale. 33

56 The ion temperature/k 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 = K at Delay 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. 34

57 Figure 2.8 also suggests that ions returns to a new steady state in ca.10 ms when jumps from 0.20 to 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 Å 2 [34]) trapped at Torr of N 2, the energy relaxation time is calculated to be s. This agrees with the experiment results shown in Figure 2.8. For a pressure Torr (mentioned earlier in Section 1.3.2), the energy relaxation time is calculated to be s 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 35

58 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 = 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 K to 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 that places ions at the boundary of the stability diagram ( In this case, the DC voltage ) 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 K and 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 36

59 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

60 Apparent mass Apparent mass Apparent mass Apparent mass Apparent mass Apparent mass a 0.0 V 5.0 V 9.0 V q inject = 0.20, 50 Th/s d 0.0 V 5.0 V 9.0 V q inject = 0.20, 1000 Th/s b 0.0 V 9.0 V 38.0 V Total ions q inject = 0.40, 50 Th/s e 0.0 V 9.0 V 38.0 V Total ions q inject = 0.40, 1000 Th/s c 0.0 V +9.0 V 0.0 V -9.0 V Total ions q inject = 0.20, 50 Th/s f 0.0 V +9.0 V 0.0 V -9.0 V Total ions q inject = 0.20, 1000 Th/s Total ions Total ions Figure 2.9 Effects of quadrupole DC voltages on mass shifts caused by space charge. Apparent mass of vs. the number of trapped ions with (a) = 0.20, = 0.83 to 0.85, scan speed ca. 50 Th/s, = 0.30 V p-p. (b) = 0.40, = 0.78 to 0.85, scan speed ca. 50 Th/s, = 0.30 V p-p (c) = 0.20, = 0.83 to 0.85 (positive), 0.85 to 0.87 (negative), 38

61 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 V p-p, switched polarity of the DC. (d) = 0.20, = 0.83 to 0.85, scan speed ca Th/s, = 1.0 V p-p. (e) = 0.40, = 0.78 to 0.85, scan speed ca Th/s, = 1.0 V p-p. (f) = 0.20, = 0.83 to 0.85 (positive), 0.85 to 0.87 (negative), scan speed ca Th/s, = 1.0 V p-p, switched polarity of the DC. Reproduced from [27] with permission 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) 39

62 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 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 , which agrees with the experimental result with a scan speed of 50 Th/s shown in Figure 2.9 (b). 40

63 Table 2.4 Ratios of mass shifts at two ion temperatures T 1 and T 2 with a ratio of Δm 1 /Δm 2 is the ratio of mass shifts at T 1 and T 2. The well depth is given by. Adapted from [27] with permission. T 1 T 2 Δm 1 /Δm 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 41

64 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-chargeinduced mass shifts. 42

65 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 43

66 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. 44

67 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) 45

68 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: 46

69 (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 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-ofmagnitude 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) 47

70 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 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. 48

71 3.3 Methods 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 49

72 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 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 of its maximum value at. The program integrates the equations of motion with a step size of 10-3 RF cycles. 50

73 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 m rod radius m trapping RF angular frequency s -1 ion