Axial symmetry and transverse trace-free tensors in numerical relativity

Transverse trace-free (TT) tensors play an important role in the initial conditions of numerical relativity, containing two of the component freedoms. Expressing a TT tensor entirely, by the choice of two scalar potentials, is not a trivial task however. Assuming the added condition of axial symmetry, expressions are given in both spherical and cylindrical coordinates, for TT tensors in flat space. A coordinate relation is then calculated between the scalar potentials of each coordinate system. This is extended to a non-flat space, though only one potential is found. The remaining equations are reduced to form a second order partial differential equation in two of the tensor components. With the axially symmetric flat space tensors, the choice of potentials giving Bowen-York conformal curvatures, are derived. A restriction is found for the potentials which ensure an axially symmetric TT tensor, which is regular at the origin, and conditions on the potentials, which give an axially symmetric TT tensor with a spherically symmetric scalar product, are also derived. A comparison is made of the extrinsic curvatures of the exact Kerr solution and numerical Bowen-York solution for axially symmetric black hole space-times. The Brill wave, believed to act as the difference between the Kerr and Bowen-York space-times, is also studied, with an approximate numerical solution found for a mass-factor, under different amplitudes of the metric.

Computational studies of the transverse structure of AGN jets

Both the emission properties and the evolution of the radio jets of Active Galactic Nuclei are dependent on the magnetic (B) fields that thread them. A number of observations of AGN jets suggest that the B fields they carry have a significant helical component, at least on parsec scales. This thesis uses a model, first proposed by Laing and then developed by Papageorgiou, to explore how well the observed properties of AGN jets can be reproduced by assuming a helical B field with three parameters; pitch angle, viewing angle and degree of entanglement. This model has been applied to multifrequency Very Long Baseline Interferometry (VLBI) observations of the AGN jets of Markarian 501 and M87, making it possible to derive values for the helical pitch angle, the viewing angle and the degree of entanglement for these jets. Faraday rotation measurements are another important tool for investigating the B fields of AGN jets. A helical B field component should result in a systematic gradient in the observed Faraday rotation across the jet. Real observed radio images have finite resolution; typical beam sizes for cm-wavelength VLBI observations are often comparable to or larger than the intrinsic jet widths, raising questions about how well resolved a jet must be in the transverse direction in order to reliably detect transverse Faraday-rotation structure. This thesis presents results of Monte Carlo simulations of Faraday rotation images designed to directly investigate this question, together with a detailed investigation into the probabilities of observing spurious Faraday Rotation gradients as a result of random noise and finite resolution. These simulations clearly demonstrate the possibility of detecting transverse Faraday-rotation structures even when the intrinsic jet widths are appreciably smaller than the beam width.

Fabrication and characterization of doped and porous silicon nanowires as anodes for lithium ion batteries

By using Si(100) with different dopant type (n++-type (As) or p-type (B)), it is shown how metal-assisted chemically (MAC) etched silicon nanowires (Si NWs) can form with rough outer surfaces around a solid NW core for p-type NWs, and a unique, defined mesoporous structure for highly doped n-type NWs. High resolution electron microscopy techniques were used to define the characteristic roughening and mesoporous structure within the NWs and how such structures can form due to a judicious choice of carrier concentration and dopant type. Control of roughness and internal mesoporosity is demonstrated during the formation of Si NWs from highly doped n-type Si(100) during electroless etching through a systematic investigation of etching parameters (etching time, AgNO3 concentration, %HF and temperature). Raman scattering measurements of the transverse optical phonon confirm quantum size effects and phonon scattering in mesoporous wires associated with the etching condition, including quantum confinement effects for the nanocrystallites of Si comprising the internal structure of the mesoporous NWs. Laser power heating of NWs confirms phonon confinement and scattering from internal mesoporosity causing reduced thermal conductivity. The Li+ insertion and extraction characteristics at n-type and p-type Si(100) electrodes with different carrier density and doping type are investigated by cyclic voltammetry and constant current measurements. The insertion and extraction potentials are demonstrated to vary with cycling and the occurrence of an activation effect is shown in n-type electrodes where the charge capacity and voltammetric currents are found to be much higher than p-type electrodes. X-ray photo-electron spectroscopy (XPS) and Raman scattering demonstrate that highly doped n-type Si(100) retains Li as a silicide and converts to an amorphous phase as a two-step phase conversion process. The findings show the succinct dependence of Li insertion and extraction processes for uniformly doped Si(100) single crystals and how the doping type and its effect on the semiconductor-solution interface dominate Li insertion and extraction, composition, crystallinity changes and charge capacity. The effect of dopant, doping density and porosity of MAC etched Si NWs are investigated. The CV response is shown to change in area (current density) with increasing NW length and in profile shape with a changing porosity of the Si NWs. The CV response also changes with scan rate indicative of a transition from intercalation or alloying reactions, to pseudocapactive charge storage at higher scan rates and for p-type NWs. SEM and TEM show a change in structure of the NWs after Li insertion and extraction due to expansion and contraction of the Si NWs. Galvanostatic measurements show the cycling behavior and the Coulombic efficiency of the Si NWs in comparison to their bulk counterparts.