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In this thesis we present a computational study of “ion crystals”, the interesting patterns in which ions arrange themselves in ion traps such as Paul and Penning traps. In ion crystals the ions are in equilibrium due to the balance of the repulsive forces between the ions and the overall tendency of the ion trap to pull ions towards the trap centre. We have carried out a detailed investigation of ion crystals in Paul traps by solving their equations of motion numerically. We also propose a model called the spring–mass model to explain the formation of ion crystals. This model is far more efficient than direct numerical simulation for predicting ion crystal structures. Finally, we demonstrate that there is a power law relating distance of an ion from the trap centre in ion crystals to the applied RF voltage amplitude.

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This thesis presents results of investigations that have been carried out to understand dynamics in nonlinear Paul trap mass spectrometers. Of the three problems that have been taken up for study in this thesis, the first concerns understanding early/delayed ejection of ions in mass selective boundary ejection experiments. The second looks at the differential resolution observed in forward and reverse scan resonance ejection experiments. The third study explores a coupled nonlinear resonance within the nominally stable region of trap operation. The method of multiple scales has been to elucidate dynamics associated with early and delayed ejection of ions in mass selective ejection experiments in Paul traps. We develop a slow flow equation to approximate the solution of a weakly nonlinear Mathieu equation to describe ion dynamics in the neighborhood of the stability boundary of ideal traps (where the Mathieu parameter qz = qz* = 0.908046). For positive even multipoles in the ion trapping field, in the stable region of trap operation, the phase portrait obtained from the slow flow consists of three fixed points, two of which are saddles and the third is a center. As the qz value of an ion approaches qz*, the saddles approach each other, and a point is reached where all nonzero solutions are unbounded, leading to an observation of early ejection. The phase portraits for negative even multipoles and odd multipoles of either sign are qualitatively similar to each other and display bounded solutions even for qz > qz*, resulting in the observation of delayed ejection associated with a more gentle increase in ion motion amplitudes, a mechanism different from the case of the positive even multipoles. The second study investigates constraints on pre-ejection dynamical states which cause differential resolution in resonance ejection experiments using Paul traps with stretched geometry. Both analytical and numerical computations are carried out to elucidate the role of damping and scan rate in influencing coherence in ion motion associated with the forward and reverse scan. It has been shown that in the forward scan experiments, for a given damping, low scan rates result in coherent motion of ions oof a given mass at the jump point. At this point, the amplitude and phase of ions of a given mass, starting at different initial conditions, become effectively identical. As the scan rate is increased, coherence is destroyed. For a given scan rate, increasing damping introduces coherence in ion motion, while decreasing damping destroys this coherence. In reverse scan experiments, for a given damping, very low scan rates will cause coherent ion motion. Increasing the scan rate destroys this coherence. The effect of damping in reverse scan experiments is qualitatively similar to that in the forward scan experiments, but settling times in the forward scan are shorter, leading to improved coherence and resolution. For mass spectrometrically relevant scan rates and damping values, significantly greater coherence is obtained in the forward scan. In the third study we investigate the weakly coupled and nonlinear Mathieu equations governing ion motion in axial and radial directions in a Paul trap in the neighborhood of a nonlinear resonance point at az* = -0.2313850427 and qz* = 0.9193009931$. Using harmonic balance based approximate averaging up to second order; we obtain a slow flow that, we numerically demonstrate, approximates the actual ion dynamics. We find that the slow flow is Hamiltonian. We study the slow flow numerically with the objective of exploring and displaying some of the possible types of interesting ion motions. In particular, we choose specific but arbitrary parameter values; study the stability of the individual radial and axial motion invariant manifolds; examine the rather large times associated with escape of ions; notice regions in the averaged phase space wherein trajectories do not, in fact, escape; observe apparently chaotic dynamics preceding escape for ions that do escape; and note that trajectories that do not escape appear to be confined to 4-tori. We conclude with some comments on the implications for practical operation of the Paul trap near this resonant point.

This thesis presents the results of studies on the problem of subharmonic oscillations in nonlinear Paul trap mass spectrometry. The objective of this thesis is to determine whether the subharmonic oscillations of ions in the trap could in any way affect the quality of mass spectrum in resonance ejection experiments. This is accomplished by studying the existence and stability criteria of these oscillations. This study is done for two casesone in which the auxiliary excitation frequency is close to thrice the ion axial secular frequency and other in which it is close to twice the ion axial secular frequency. Initially, the equations describing the ion motion in the presence of auxiliary excitation are derived. The equations describing the ion motion are then brought into a form easily amenable to analysis by techniques of perturbation theory. The necessary background and definitions to understand the basis for the thesis, along with a survey of results obtained in relevant areas in mass spectrometry and nonlinear dynamics is then developed. The first problem is the study of subharmonic oscillations when the auxiliary excitation frequency is in the vicinity of thrice the ion axial secular frequencies taken up. The application of the multiple time scales technique to the equations describing ion motion gives the slow flow equations, that describe the evolution of amplitudes of axial and radial oscillations. The expression for the steady state amplitudes of these oscillations are then derived. From these expressions the conditions for the existence of the oscillations are obtained in terms of the auxiliary amplitude and the frequency detuning. This is then followed by a detailed stability analysis for the subharmonic oscillation with a given amplitude and phase. The study ends with the discussion of the results obtained, the pertinent numerical studies and the relevance of this study to mass spectrometry. The second study is regarding the problem of subharmonic oscillations when the auxiliary excitation frequency is close to twice the ion axial secular frequency is analyzed. When PoincareLindstedt’s method is applied to the ion motion equations, the amplitude frequency relationship that describes the relation between the steady state subharmonic oscillation amplitude and the frequency detuning is obtained. The variation of the oscillation amplitude with the frequency detuning is then studied. Then follows the analysis of stability. The stability of subharmonic oscillation is analyzed using the results from the standard analysis of Hill’s equation of fourth order. This study ends with the discussion of the results obtained in the context of mass spectrometry. Finally, a summary of the results obtained is discussed.

This thesis considers four nonlinear systems with parametric forcing. The first problem involves an inverted pendulum with asymmetric elastic restraints subjected to harmonic vertical base excitation. On linearizing trigonometric terms the pendulum is governed by an asymmetric Mathieu equation. Solutions to this equation are scaleable. The stability regions in the parameter plane are studied numerically. Periodic solutions at the boundaries of stable regions in the parameter plane are found numerically and then their existence is proved theoretically. The second problem involves use of the method of multiple scales to elucidate the dynamics associated with early and delayed ejection of ions from Paul traps. A slow flow equation is developed to approximate the solution of a weakly nonlinear Mathieu equation to describe ion dynamics in the neighborhood of the nominal stability boundary of ideal traps. Since the solution to the unperturbed equation involves linearly growing terms, some care in identification and elimination of secular terms is needed. Due to analytical difficulties, harmonic balance approximations are used within the formal implementation of the method. The third problem involves the attenuation, caused by weak damping, of harmonic waves through a discrete, periodic structure with wave frequency nominally within the Propagation Zone. Adapting the transfer matrix method and using the harmonic balance for nonlinear terms, a four-dimensional map governing the dynamics is obtained. This map is analyzed by applying the method of multiple scales upto first order. The resulting slow evolution equations give the amplitude decay rate in the structure. The fourth problem involves the dynamic response of a strongly nonlinear single-degree-of-freedom oscillator under a constant amplitude, parametric, periodic, impulsive forcing, e.g., a pendulum with strongly nonlinear torsional spring that is periodically struck in the axial direction. Single-term harmonic balance gives an approximate, but explicit, 2-dimensional map governing the dynamics. The map exhibits many fixed points (both stable and unstable), higher period orbits, transverse intersections of stable and unstable manifolds of unstable fixed points, and chaos.

Multielectron bubbles (MEBs) are micron sized cavities in liquid helium that contain electrons confined within a nanometer thick layer on the inner surface of a bubble. These objects present a rich platform to study the behavior of a two dimensional electron gas (2DES) on a curved surface. Most crucially, the surface electron densities in MEBs can vary over a wide range, making it a suitable candidate for studying classical Wigner crystallization and quantum melting in a single system. So far, there has been only limited experimental study of MEBs, with most of the previous investigation transient in nature. As we discuss in our presentation, we have built a cryogenic system for performing transport and optical measurements of MEBs down to 1.3 K. We have developed a new technique of generating MEBs, and trapping them using two different methods. In the first method, we trapped MEBs using a Paul trap for more than hundreds of milliseconds. This allows the MEBs to be further manipulated with buoyant and electric forces, such as to obtain reliable measurements of their physical properties. As we observe experimentally, the surface charge density of a single MEB can vary by orders of magnitude during the course of one measurement, thereby covering a previously unexplored section of the 2DES phase diagram. In the second method, we trapped MEBs using a dielectric coated metal electrode over many seconds. This also allowed the properties of MEBs to be measured in a non-destructive manner. Since MEBs are charged bubbles, their motion can be controlled by electric fields, which allowed us to measure the drag of MEBs as a function of Reynolds number by analysing the trajectories. Due to the low viscosity and surface tension of helium compared to other liquids, these measurements could be performed at Morton Numbers that have never been explored. We also show that how the shape of a single MEB evolves from spherical to ellipsoidal as their speeds vary. During the course of experiments, we observed number of interesting phenomena, such as coalescence of similarly charged bubbles, as well as their splitting into secondary bubbles at high speeds. Most interestingly, we have imaged their dynamics in the presence of static, as well as oscillating electric fields, which may provide insight into the phase of the electronic system present inside the bubbles.

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.