Trapped ion magnetic resonance: concepts and designs

A novel spectroscopy of trapped ions is proposed which will bring single-ion detection sensitivity to the observation of magnetic resonance spectra. The approaches developed here are aimed at resolving one of the fundamental problems of molecular spectroscopy, the apparent incompatibility in existing techniques between high information content (and therefore good species discrimination) and high sensitivity. Methods for studying both electron spin resonance (ESR) and nuclear magnetic resonance (NMR) are designed. They assume established methods for trapping ions in high magnetic field and observing the trapping frequencies with high resolution (<1 Hz) and sensitivity (single ion) by electrical means. The introduction of a magnetic bottle field gradient couples the spin and spatial motions together and leads to a small spin-dependent force on the ion, which has been exploited by Dehmelt to observe directly the perturbation of the ground-state electron's axial frequency by its spin magnetic moment.

A series of fundamental innovations is described m order to extend magnetic resonance to the higher masses of molecular ions (100 amu = 2x 10^5 electron masses) and smaller magnetic moments (nuclear moments = 10^(-3) of the electron moment). First, it is demonstrated how time-domain trapping frequency observations before and after magnetic resonance can be used to make cooling of the particle to its ground state unnecessary. Second, adiabatic cycling of the magnetic bottle off between detection periods is shown to be practical and to allow high-resolution magnetic resonance to be encoded pointwise as the presence or absence of trapping frequency shifts. Third, methods of inducing spindependent work on the ion orbits with magnetic field gradients and Larmor frequency irradiation are proposed which greatly amplify the attainable shifts in trapping frequency.

The dissertation explores the basic concepts behind ion trapping, adopting a variety of classical, semiclassical, numerical, and quantum mechanical approaches to derive spin-dependent effects, design experimental sequences, and corroborate results from one approach with those from another. The first proposal presented builds on Dehmelt's experiment by combining a "before and after" detection sequence with novel signal processing to reveal ESR spectra. A more powerful technique for ESR is then designed which uses axially synchronized spin transitions to perform spin-dependent work in the presence of a magnetic bottle, which also converts axial amplitude changes into cyclotron frequency shifts. A third use of the magnetic bottle is to selectively trap ions with small initial kinetic energy. A dechirping algorithm corrects for undesired frequency shifts associated with damping by the measurement process.

The most general approach presented is spin-locked internally resonant ion cyclotron excitation, a true continuous Stern-Gerlach effect. A magnetic field gradient modulated at both the Larmor and cyclotron frequencies is devised which leads to cyclotron acceleration proportional to the transverse magnetic moment of a coherent state of the particle and radiation field. A preferred method of using this to observe NMR as an axial frequency shift is described in detail. In the course of this derivation, a new quantum mechanical description of ion cyclotron resonance is presented which is easily combined with spin degrees of freedom to provide a full description of the proposals.

Practical, technical, and experimental issues surrounding the feasibility of the proposals are addressed throughout the dissertation. Numerical ion trajectory simulations and analytical models are used to predict the effectiveness of the new designs as well as their sensitivity and resolution. These checks on the methods proposed provide convincing evidence of their promise in extending the wealth of magnetic resonance information to the study of collisionless ions via single-ion spectroscopy.

On concentrated loads and Green's functions in elastostatics

This investigation is concerned with the notion of concentrated loads in classical elastostatics and related issues. Following a limit treatment of problems involving concentrated internal and surface loads, the orders of the ensuing displacements and stress singularities, as well as the stress resultants of the latter, are determined. These conclusions are taken as a basis for an alternative direct formulation of concentrated-load problems, the completeness of which is established through an appropriate uniqueness theorem. In addition, the present work supplies a reciprocal theorem and an integral representation-theorem applicable to singular problems of the type under consideration. Finally, in the course of the analysis presented here, the theory of Green's functions in elastostatics is extended.