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.

A Paul trap mass spectrometer has a three-electrode geometry mass analyzer consisting of two identical end cap electrodes and a ring electrode. Traditionally, the two end cap electrodes are electrically grounded and an RF potential is applied to the central ring electrode to generate the "trapping field". Ions of the analyte sample are formed in situ by electron bombardment and mass analysis of the fragment ions is performed by mass selectively destabilizing the ions from the trap. The inhornogeneities present in the trapping field (introduced either by misalignment of the trap geometry or by applying a dipolar auxiliary excitation across the end cap electrodes) give rise to various interesting phenomena including, resonance ejection of the trapped ions This thesis is concerned with taking a look into the experimental aspects associated with resonance ejection of ions caused by the dipolar excitation Additionally, u also reports the work undertaken to develop necessary instrumentation for resonant excitation experiments and my contribution to operational>zc the Paul trap mass spectrometer fabricated in the laboratory. The thesis is divided into 5 chapters. Chapter 1 is an introductory chapter. After discussing the conditions for stability of the trapped ions, it goes on to present a brief survey of a variety of applications in literature, which have used resonant excitation. Towards the end, the motivation of the present effort and the scope of work in the thesis have been spelt out. This includes (a) redesign of the ion detector electronics, (b) design of an auxiliary excitation generator, and (c) studies on resonance ejection. Chapter 2 outlines the design considerations, circuit description and fabrication details for the ion detector electronics. The circuits presented in this chapter include (a) electrometer amplifier and (b) -3 kV DC supply for the electron multiplier detector. The electrometer amplifier amplifies the ion current signal from the electron multiplier detector and it needs to have a high input impedance and a high slew rate. The electron multiplier detector requires -3 kV DC power supply for operation. The -3 kV DC power supply is required to have a regulated output voltage with low ripple in the output. Chapter 3 presents the design considerations, circuit description and fabrication details for the auxiliary excitation generator. The auxiliary excitation generator is a three channel DDS (Direct Digital Synthesis) oscillator with independent control of frequency amplitude, and phase of the output signal. Chapter 3 also discusses the micro controller based control sub-system that allows the user to set above mentioned output parameters. The control sub-system provides a user-friendly keyboard interface and 2-line alphanumeric LCD display per channel. It also provides various bus interfaces (such as I2C and SPI) to interface with DDS oscillator ICs, amplitude control DAC, and LCD displays. The chapter then goes on to describe the implementation details of the software written for the control sub-system. The hardware design is simplified by using a micro controller as heart of the control sub-system and employing the software to handle the complex functions. As an example, the design of the keyboard interface is simplified by directly connecting a matrix keyboard to the input/output port of the micro controller. The software is used to scan the keyboard, detect key press and find out the key pressed. Nonetheless, in order to meet specific performance required for the present work, the software needs to have a sense of time, be portable and scalable. Details of the "layered" architecture adopted by as to meet these specific requirements, the lower level "driver" functions implemented for various interfaces of the control sub-system, and the higher level or the "application" software, are described. The application software uses the driver functions to accomplish various tasks required to be executed by the control sub-system. Finally, the chapter presents the design consideration and fabrication details of the coupling transformer used to couple the output of the auxiliary excitation generator to the Paul trap Chapter 4 describes the resonant excitation experiments performed as part of the present work. First of all the chapter presents the improvement in the performance of the Paul trap mass spectrometer as a result of redesigned ion detector electronics It is seen that the resolution is improved significantly due to the improved response time of the electrometer amplifier. The chapter then describes the effect of the resonant excitation on the ions and also that the frequency of the applied auxiliary excitation should be between 500 kHz to 125 kHz. Next, a number of mass spectra for different frequencies of the applied auxiliary excitation are presented. These mass spectra indicate that the resonant ejection sets in for lower masses even at lower amplitude of the auxiliary excitation where as higher amplitude is required for the resonant ejection of the higher masses. It is seen that the resonant excitation of ions improves resolution of the mass spectrum. Moreover, the auxiliary excitation results in ejection of the ions at lower amplitude of the RF voltage and thus allows extending the mass range of the mass spectrometer. We present the mass spectrum of CCI4 which is not possible to normally record in our instrument. We also present results intended to understand the relation between frequency and amplitude of the auxiliary excitation on the mass spectra of benzene. Finally, results of an interesting experiment are presented which indicates the presence of the non-linear resonance points in the Paul trap. Chapter 5 presents the concluding remarks. References cited in the thesis are attached in their alphabetical order at the end of the thesis.

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