This thesis presents results of investigations on fields and multipole expansion coefficients in axially symmetric (referred to as 3D)and two dimensional (2D)ion trapmass analysers. 3D mass analysers have a three-electrode geometry with two (electrically shorted) endcap electrodes and one central ring electrode. rf-only or rf/dc potential applied across the electrodes creates a linear trapping field in the central cavity of the mass analyser.2Dmass analysers have four longitudinal electrodes in which the opposite pairs of electrodes are electrically shorted. Here, rf-only or rf/dc potential applied across the pair of electrodes creates a linear trapping field and fragment ions of the analyte gas are trapped along the central axis of the mass analyser. Both these mass analysers have apertures machined on the electrodes (holes in case of 3D traps and slits in case of 2D traps) to permit entry of electrons for ionizing the analyte gas and for collection of destabilized fragment ions. This thesis is concerned with how these apertures influence the fields and multipole expansion coefficients within the traps. This thesis is divided into five chapters. Chapter 1 provides the background information which is required for the thesis. It begins with a description of the geometry of the 3D and the 2D mass analysers used in the present work.These include the quadrupole ion trap (QIT) and cylindrical ion trap (CIT) for 3D structures and the linear ion trap (LIT) and the rectilinear ion trap (RIT) for 2D structures. This is followed by a brief description of the numerical method, the boundary element method (BEM), used in the thesis. Also presented here are the Green’s function for 3D and 2D geometries. In the final section, the scope of the thesis is presented. Chapter 2 presents two approximate analytical expressions for nonlinear electric fields in the principal direction in axially symmetric (3D) and two dimensional (2D) ion trap mass analysers with apertures on the electrodes. Considered together (3D and 2D), we present composite approximations for the principal unidirectional nonlinear electric fields in these ion traps. The composite electric field E has the form E= EnoAperture + EdueToAperture where EnoAperture is the field within an imagined trap which is identical to the practical trap except that the apertures are missing; and where EdueToAperture is the field contribution due to apertures on the two trap electrodes. The field along the principal axis of the trap can in this way be well approximated for any aperture that is not too large. To derive EdueToAperture, classical results of electrostatics have been extended to electrodes with finite thickness and different aperture shapes. EnoAperture is a modified truncated multipole expansion for the imagined trap with no aperture. The first several terms in the multipole expansion are in principle exact (though numerically determined using the BEM), while the last term is chosen to match the field at the electrode. This expansion, once computed, works with any aperture in the practical trap. The composite field approximation for axially symmetric (3D) traps is checked for three geometries: the quadrupole ion trap (QIT), the cylindrical ion trap (CIT) and an arbitrary other trap. The approximation for 2D traps is verified using two geometries: the linear ion trap (LIT)and the rectilinear ion trap (RIT). In each case, for two aperture sizes (10% and 50% of the trap dimension), highly satisfactory fits are obtained. These composite approximations may be used in more detailed nonlinear ion dynamics studies than have been hitherto attempted. In Chapter 3we complement and complete the work presented in Chapter 2 by considering off-axis fields in the axially symmetric (3D) and the two dimensional (2D) ion traps whose electrodes have apertures. Our approximation has two parts. The first, EnoAperture, is the field obtained numerically for the trap under study with no apertures. We have used the boundary element method (BEM) for obtaining this field. The second part, EdueToAperture, is an analytical expression for the field contribution of the aperture. In EdueToAperture, aperture size is a free parameter. A key element in our approximation is the electrostatic field near an infinite thin plate with an aperture, and with different constant valued far field intensities on either side. Compact expressions for this field can be found using separation of variables, wherein the choice of coordinate system is crucial. This field is, in turn, used four times within our trap specific approximation. The off-axis field expressions for the 3D geometries were tested on the quadrupole ion trap (QIT) and the cylindrical ion trap (CIT), and the corresponding expressions for the 2D geometries were tested on the linear ion trap (LIT) and rectilinear ion trap (RIT). For each geometry, we have considered apertures which are 10%, 30% and 50% of the trap dimension. We have found that our analytical correction term EdueToAperture, though based on a classical small-aperture approximation, gives good results even for relatively large apertures. Chapter 4 presents approximate analytical expressions for estimating the variation in multipole expansion coefficients with the size of apertures in axially symmetric (3D) and two dimensional (2D) ion trap mass analysers. Following the approach adopted in Chapter 2 and Chapter 3 which focused on the role of apertures to fields within traps, here too, the analytical expression is a sum of two terms, An,noAperture, the multipole expansion coefficient for a trap with no apertures and An,dueToAperture, the multipole expansion coefficient contributed by the aperture. An,noAperture has been obtained numerically and An,dueToAperture is obtained from the nth derivative of the potential within the trap. The expressions derived have been tested on two 3D geometries and two 2D geometries. These include the quadrupole ion trap (QIT) and the cylindrical ion trap (CIT) for 3D geometries and the linear ion trap (LIT) and the rectilinear ion trap (RIT) for the 2D geometries. Multipole expansion coefficients A2 to A24, estimated by our analytical expressions were compared with the values obtained numerically (using the boundary element method) for aperture sizes varying up to 50% of the trap size. In all the plots presented, it is observed that our analytical expression for the variation of multipole expansion coefficients versus aperture size closely follows the trend of the numerical evaluations for the range of aperture sizes considered. The maximum relative percentage errors, which provide an estimate of the deviation of our values from those obtained numerically for each multipole expansion coefficient, are seen to be in the range of 10% to 15%. The leading multipole expansion coefficient, A2, however, is seen to be estimated very well by our expressions, with most values being within 1% of the numerically determined values, with larger deviations seen for the QIT and LIT only at larger aperture sizes. Chapter 5 presents a few concluding remarks.

This thesis presents axially symmetric equivalents of three-dimensional rf ion traps. Miniaturization in mass spectrometry has focused on miniaturizing mass analyzers. Decrease in mass analyzer size facilitates reduction of the size of other components of a mass spectrometer, especially the radio frequency electronics and vacuum system. Miniaturized mass analyzers are made using advanced microfabrication techniques. Due to micromachining limitations, it is not possible to fabricate ion traps with exact axial symmetry. The motivation for this thesis is to investigate newer three-dimensional geometries which do not possess axial symmetry, but are equivalent in performance to axially symmetric ion traps. We introduce a 3D geometry called square ion trap(SIT) having a ring electrode made off our square shaped planar surfaces and square shaped endcap electrodes resembling a cuboid. Initially, a SIT geometry is taken and it will be investigated if this unknown 3D geometry can be made equivalent to a well characterized, axially symmetric ion trap like the CIT. The purpose of showing equivalence will be to understand the ion dynamics and fields inside the new 3D SIT. This thesis consists of five chapters. In Chapter 1, we present the necessary background information required to understand the operation of a mass spectrometer. The Paul trap geometry is introduced followed by the derivation of equation of ion motion inside the Paul trap. The Mathieu stability plot and the modes of operation of a mass spectrometer are briefly discussed. The chapter ends by outlining scope of the thesis. Chapter 2 describes the computational methods employed by us in the thesis. First, the geometry of square ion trap is introduced. Then the boundary element method(BEM) which is used to compute the charge distribution on the electrode surfaces is discussed. This is followed by the three-dimensional Green’s function which should be employed for non-axially symmetric structures. The method to calculate potential and field inside the ion trap from charge distribution is shown. Calculation of multipole coefficients for non-axially symmetric traps using charge distribution is shown. The methods used to generate ion trajectory and stability plot are discussed. The Nelder-Mead simplex method used for optimization is also presented. To verify our numerical methods of charge calculation, we have taken standard textbook problems and compared our results with those presented therein. The multipoles calculation, field and ion trajectory was verified by comparing the results for the Paul trap and cylindrical ion traps. Chapter 3 presents the results for axially symmetric equivalents of 3D rf ion traps. SIT geometry of dimensions equivalent to the CIT0 are taken and field and multipoles are studied in it. Then optimization is applied to create a CIT geometry equivalent to the SIT under study. Axial field and ion trajectory was compared and observed to be matching. Finally, stability plot was generated for both SIT and its equivalent CIT and was found to present a close match. Chapter 4 presents the numerical results obtained for three-dimensional rf ion trap equivalent of CIT. In this chapter, we have considered two standard geometries, the CIT0 and the CITopt. Optimization was applied to create SIT geometries equivalent to the CIT0 and the CITopt respectively. Comparison of fields and ion trajectory confirmed the fact that non-axially symmetric traps can be created equivalent to any axially symmetric ion trap. We have also considered another case of axially symmetric circular planar ion trap which has an annular ring electrode and two planar endcap electrodes. Square equivalent of circular planar trap was created by the optimizer and its equivalent was verified by ion trajectory comparison. Chapter 5 summarizes the thesis with a few concluding remarks.

Axially symmetric rf ion traps consists of a mass analyser having three electrodes, one of which is a central ring electrode and the other two are endcap electrodes. In the ideal Paul trap mass spectrometer, the electrodes have hyperboloidal shape (March and Hughes, 1989) and in mass analyser with simplified geometry, such as the cylindrical ion trap (Wu et al.,2005) the central electrode is a cylinder and the two endcap electrode and flat plates. rf-only or rf/dc potential is applied across the ring electrode and the grounded endcap electrodes for conducting the basic experiments of the mass spectrometer. In recent times, the miniaturisation of ion trap is one of the research interests in the field of mass spectrometry. The miniaturisation has the advantages of compactness, low power consumption and portability. However, this is achieved at the cost of the overall performance of the mass spectrometer with its deleterious effect on resolution. Research groups study the field distribution in the trap for better understanding of ion dynamics in the direction of achieving improved performance with the miniaturised traps. One aspect which has not received any attention in research associated with quadrupole ion traps is the possible role of the magnetic field in improving performance of these traps. Since in the quadrupole ion trap mass analyser ion is confined by an oscillating (rf) field, magnetic fields have been considered superfluous. The motivation of the thesis is to understand the dynamics of ions in axially symmetric rf ion traps, in the presence of the magnetic field. The axially symmetric rf ion trap geometries considered in this thesis are the Paul trap and the cylindrical ion trap (CIT). The changes incurred to the ion motion and Mathieu stability diagram in the presence of magnetic field is observed in this work. Also, the relation between the magnetic field and the Mathieu parameter is shown. The thesis contains 4 chapters: Chapter 1 provides the basic back ground of mass spectrometry and the operating principles. The equations of ion motion in the Paul trap is derived and also the solution to Mathieu equation is provided. The solution to the Mathieu equation are the Mathieu parameters and , when plotted with on the x-axis and on the y-axis, results in the Mathieu stability plot, the explanation of which is also given in the chapter. A brief description of the secular frequency associated with the ion dynamics is given in this chapter. The popular experiments conducted (i.e. the mass selective boundary ejection and resonance ejection) with a mass spectrometer is described here. Finally at the end of the chapter is the scope of the thesis. Chapter 2 facilitates with the preliminary study required fort he accomplishment of the task. The Paul trap and the CIT are the rf ion traps considered in this work. The geometries of these two traps are described in this chapter. The computational methods used for the analysis of various aspects of mass spectrometer is introduced. The computational methods used involve the methods used for calculating the charge distribution on the electrodes, potentials, multipole co-efficients and trajectory calculations. The boundary element method(BEM), calculation for Potentials and the Runge-Kutta method used for the trajectory calculations are introduced in this chapter. The expressions for calculating the multipole co-efficients are also specified. Chapter 3 presents the results obtained. The equations of ion motion in a quadrupole ion trap in the presence of magnetic field is derived here. Verification of numerical results with and without the magnetic field are presented at the end of this chapter. The chapter also presents various graphs showing the impact of magnetic field on the ion dynamics in the Paul trap and the CIT. The impact of the presence of magnetic field on the micro motion in -, -and -directions of the rf ion traps are shown in this chapter. Also the figures showing the variation in the Mathieu stability plots, with varying magnetic field intensity are presented in the chapter. At the end of this chapter the relation between the magnetic field and the Mathieu parameter is derived and plotted. Chapter 4 explains the various observations made from the results obtained. This chapter also highlights the future scope of the work for making this a more applicable one. References in the text have been given by quoting the author’s name and year of publication. Full references have been provide, in an alphabetic order, at the end of the thesis.