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An Analogue of the LaplaceRungeLenz Vector for Timelike Geodesics in Schwarzschild SpacetimeIn Schwarzschild spacetime, the timelike geodesics are the trajectories of free, massive particles, orbiting a singularity at the origin r = 0. In this work we derive four scalar first integrals of timelike geodesics in Schwarzschild spacetime. Two of the first integrals, corresponding to energy and angular momentum, are wellknown. The other two first integrals, an angular quantity and a temporal quantity, are not as wellknown. Using the freedom to shift first integrals by a constant value we set a ‘zeropoint’ for each of the four first integrals. By choosing a natural point on a noncircular trajectory such as a turning point or inertial point to set the zeropoint value, the angular and temporal first integrals will correspond respectively to the angle and time of the chosen zeropoint. We then take the Newtonian limit of the angular and temporal first integrals, and show that using a natural choice of zeropoint they provide a generalization of the classical LaplaceRungeLenz (LRL) vector. We then evaluate the angular first integral for each type of timelike geodesic in Schwarzschild spacetime. In most cases we are able to choose a turning or inertial point to set a zeropoint. For an unbound or asymptotic trajectory which falls into the singularity of the metric at r = 0, however, we find that we must take a different point, such as the point where the trajectory crosses the horizon at r = 2M, which we call the ‘horizon point.’ For the case of a precessing elliptic orbit we find that the angular first integral is multivalued, with the zeropoint jumping each time the trajectory crosses an apoapsis. It is found that the angular and temporal first integrals provide a relativistic generalization of the classical LRL vector, where we the first integrals correspond to a larger class of physically meaningful points compared to Newtonian orbits and where the LRL vector and angular and temporal first integrals may always correspond to the periapsis.

An analysis of decoupling procedures in generating thermal Green's functions of O([lambda]â ´) by the Zubarev equation of motion methodThe Zubarev equation of motion method has been applied to an anharmonic crystal of O( ,,4). All possible decoupling schemes have been interpreted in order to determine finite temperature expressions for the one phonon Green's function (and self energy) to 0()\4) for a crystal in which every atom is on a site of inversion symmetry. In order to provide a check of these results, the Helmholtz free energy expressions derived from the self energy expressions, have been shown to agree in the high temperature limit with the results obtained from the diagrammatic method. Expressions for the correlation functions that are related to the mean square displacement have been derived to 0(1\4) in the high temperature limit.

Application of [Lambda] to the fourth perturbation theory in calculating the equation of state of rare gas solids and fcc metalsWe have calculated the thermodynamic properties of monatomic fcc crystals from the high temperature limit of the Helmholtz free energy. This equation of state included the static and vibrational energy components. The latter contribution was calculated to order A4 of perturbation theory, for a range of crystal volumes, in which a nearest neighbour central force model was used. We have calculated the lattice constant, the coefficient of volume expansion, the specific heat at constant volume and at constant pressure, the adiabatic and the isothermal bulk modulus, and the Gruneisen parameter, for two of the rare gas solids, Xe and Kr, and for the fcc metals Cu, Ag, Au, Al, and Pb. The LennardJones and the Morse potential were each used to represent the atomic interactions for the rare gas solids, and only the Morse potential was used for the fcc metals. The thermodynamic properties obtained from the A4 equation of state with the LennardJones potential, seem to be in reasonable agreement with experiment for temperatures up to about threequarters of the melting temperature. However, for the higher temperatures, the results are less than satisfactory. For Xe and Kr, the thermodynamic properties calculated from the A2 equation of state with the Morse potential, are qualitatively similar to the A 2 results obtained with the LennardJones potential, however, the properties obtained from the A4 equation of state are in good agreement with experiment, since the contribution from the A4 terms seem to be small. The lattice contribution to the thermal properties of the fcc metals was calculated from the A4 equation of state, and these results produced a slight improvement over the properties calculated from the A2 equation of state. In order to compare the calculated specific heats and bulk moduli results with experiment~ the electronic contribution to thermal properties was taken into account~ by using the free electron model. We found that the results varied significantly with the value chosen for the number of free electrons per atom.

An application of cellcluster theory to a rare gas crystal /n[by] K. Westera.  260 St. Catharines [Ont.] : Dept. of Physics, Brock University,The LennardJones Devonshire 1 (LJD) single particle theory for liquids is extended and applied to the anharmonic solid in a high temperature limit. The exact free energy for the crystal is expressed as a convergent series of terms involving larger and larger sets of contiguous particles called cellclusters. The motions of all the particles within cellclusters are correlated to each other and lead to nontrivial integrals of orders 3, 6, 9, ... 3N. For the first time the six dimensional integral has been calculated to high accuracy using a LennardJones (612) pair interaction between nearest neighbours only for the f.c.c. lattice. The thermodynamic properties predicted by this model agree well with experimental results for solid Xenon.

Application of Reptation Quantum Monte Carlo and Related Methods to LiHThis work investigates mathematical details and computational aspects of MetropolisHastings reptation quantum Monte Carlo and its variants, in addition to the Bounce method and its variants. The issues that concern us include the sensitivity of these algorithms' target densities to the position of the trial electron density along the reptile, timereversal symmetry of the propagators, and the length of the reptile. We calculate the groundstate energy and oneelectron properties of LiH at its equilibrium geometry for all these algorithms. The importance sampling is performed with a singledeterminant large Slatertype orbitals (STO) basis set. The computer codes were written to exploit the efficiencies engineered into modern, highperformance computing software. Using the Bounce method in the calculation of nonenergyrelated properties, those represented by operators that do not commute with the Hamiltonian, is a novel work. We found that the unmodified Bounce gives good ground state energy and very good oneelectron properties. We attribute this to its favourable timereversal symmetry in its target density's Green's functions. Breaking this symmetry gives poorer results. Use of a short reptile in the Bounce method does not alter the quality of the results. This suggests that in future applications one can use a shorter reptile to cut down the computational time dramatically.

The band structure of Cd0The Augmented Pl ane Wave Method has been used to calculate the oneelectron energy band structure of CdO. Energy eigenva l ues were calculated along three symmetry lines and for some other general wavevectors of interest.

Brillouin spectroscopy : measurement of elastic and photoelastic constants of some alkali holide crystalsThe assembly and testing of apparatus for the measurement of elastic and photoelastic constants by Brillouin scattering, using a FabryPerot interferometer and with argon ion laser excitation is described. Such measurements are performed on NaCI, KBr and LiF using the A = 488.0 nm laser line. The elastic constants obtained here are in very good agreement with the ultrasonic data for all three materials. The discrepancy between ultrasonic and hypersonic sound velocities which was reported by some authors for KBr and LiF is not confirmed, and the elastic constants obtained for LiF are the most accurate to date. Also, the present photoelastic constants are in good agreement with the data obtained by ultrasonic techniques for all three crystals. The results for the KBr and LiF crystals constitute the first set of photoelastic constants obtained for these materials by Brillouin spectroscopy. Our results for LiF are the best available to date.

Calculation of nondifferential properties for atomic ground states /A new method for sampling the exact (within the nodal error) ground state distribution and nondiflPerential properties of multielectron systems is developed and applied to firstrow atoms. Calculated properties are the distribution moments and the electronic density at the nucleus (the 6 operator). For this purpose, new simple trial functions are developed and optimized. First, using Hydrogen as a test case, we demonstrate the accuracy of our algorithm and its sensitivity to error in the trial function. Applications to first row atoms are then described. We obtain results which are more satisfactory than the ones obtained previously using Monte Carlo methods, despite the relative crudeness of our trial functions. Also, a comparison is made with results of highly accurate postHartree Fock calculations, thereby illuminating the nodal error in our estimates. Taking into account the CPU time spent, our results, particularly for the 8 operator, have a relatively large variance. Several ways of improving the eflSciency together with some extensions of the algorithm are suggested.

Calculation of the magnetic field penetration depth for highTc cuprate superconductors based on the Interlayer Pair Tunneling model /In this work, the magnetic field penetration depth for highTc cuprate superconductors is calculated using a recent Interlayer Pair Tunneling (ILPT) model proposed by Chakravarty, Sudb0, Anderson, and Strong [1] to explain high temperature superconductivity. This model involves a "hopping" of Cooper pairs between layers of the unit cell which acts to amplify the pairing mechanism within the planes themselves. Recent work has shown that this model can account reasonably well for the isotope effect and the dependence of Tc on nonmagnetic inplane impurities [2] , as well as the Knight shift curves [3] and the presence of a magnetic peak in the neutron scattering intensity [4]. In the latter case, Yin et al. emphasize that the pair tunneling must be the dominant pairing mechanism in the highTc cuprates in order to capture the features found in experiments. The goal of this work is to determine whether or not the ILPT model can account for the experimental observations of the magnetic field penetration depth in YBa2Cu307_a7. Calculations are performed in the weak and strong coupling limits, and the efi"ects of both small and large strengths of interlayer pair tunneling are investigated. Furthermore, as a follow up to the penetration depth calculations, both the neutron scattering intensity and the Knight shift are calculated within the ILPT formalism. The aim is to determine if the ILPT model can yield results consistent with experiments performed for these properties. The results for all three thermodynamic properties considered are not consistent with the notion that the interlayer pair tunneling must be the dominate pairing mechanism in these highTc cuprate superconductors. Instead, it is found that reasonable agreement with experiments is obtained for small strengths of pair tunneling, and that large pair tunneling yields results which do not resemble those of the experiments.

Calculation of the optical conductivity for the Anderson impurity model using the slave boson techniqueThe optical conductivity of the Anderson impurity mode l has been calculated by emp l oying the slave boson technique and an expansion in powers of l i N, where N is the d egeneracy o f the f electron level . This method has been used to find the effective mass of the conduction electrons for temperatures above and below the Kondo tempera ture. For low temperatures, the mass enhancement is f ound to be large while a t high t emperatures, the mass enhancement is sma ll. The conductivity i s f ound to be Drude like with frequency dependent effective mass and scattering time for low independent effective mass and temperatures and scattering time f requency for high t emperatures. The behavior of both the effective mass and the conductivity is in qualitative agreement with experimental r esul t s .

Composition analysis of highTc superconducting thin films by quantitative xray fluorescenceA method is presented for determining the composition of thin films containing the elements Bi, Sr, Br, Cu, and Ca. Quantitative xray fluorescence (XRF) consisting of radioactive sources (secondary foil excitor 241AmMo source and 55Pe source), a Si(Li) detector, and a multichannel analyzer were employed. The XRF system was calibrated by using sol gel thin films of known element composition and also by sputtered thin films analyzed by the conventional Rutherford Back Scattering (RBS). The XRF system has been used to assist and optimize the sputter target composition required to produce highTc BiSrCaCuO films with the desired metal composition.

Confocal and twophoton microscopy : image enhancement /Confocal and twophoton microcopy have become essential tools in biological research and today many investigations are not possible without their help. The valuable advantage that these two techniques offer is the ability of optical sectioning. Optical sectioning makes it possible to obtain 3D visuahzation of the structiues, and hence, valuable information of the structural relationships, the geometrical, and the morphological aspects of the specimen. The achievable lateral and axial resolutions by confocal and twophoton microscopy, similar to other optical imaging systems, are both defined by the diffraction theorem. Any aberration and imperfection present during the imaging results in broadening of the calculated theoretical resolution, blurring, geometrical distortions in the acquired images that interfere with the analysis of the structures, and lower the collected fluorescence from the specimen. The aberrations may have different causes and they can be classified by their sources such as specimeninduced aberrations, opticsinduced aberrations, illumination aberrations, and misalignment aberrations. This thesis presents an investigation and study of image enhancement. The goal of this thesis was approached in two different directions. Initially, we investigated the sources of the imperfections. We propose methods to eliminate or minimize aberrations introduced during the image acquisition by optimizing the acquisition conditions. The impact on the resolution as a result of using a coverslip the thickness of which is mismatched with the one that the objective lens is designed for was shown and a novel technique was introduced in order to define the proper value on the correction collar of the lens. The amoimt of spherical aberration with regard to t he numerical aperture of the objective lens was investigated and it was shown that, based on the purpose of our imaging tasks, different numerical apertures must be used. The deformed beam cross section of the singlephoton excitation source was corrected and the enhancement of the resolution and image quaUty was shown. Furthermore, the dependency of the scattered light on the excitation wavelength was shown empirically. In the second part, we continued the study of the image enhancement process by deconvolution techniques. Although deconvolution algorithms are used widely to improve the quality of the images, how well a deconvolution algorithm responds highly depends on the point spread function (PSF) of the imaging system applied to the algorithm and the level of its accuracy. We investigated approaches that can be done in order to obtain more precise PSF. Novel methods to improve the pattern of the PSF and reduce the noise are proposed. Furthermore, multiple soiu'ces to extract the PSFs of the imaging system are introduced and the empirical deconvolution results by using each of these PSFs are compared together. The results confirm that a greater improvement attained by applying the in situ PSF during the deconvolution process.

Coriolis effects in symmetric top spectra /The "xy Coriolis Coupling Theory" as presented by Dilauro and Mills (1966) is reformulated and extended to the determination of Raman intensities. Theoretical Raman and Infrared spectra are computed in order to understand the effects due to this coupling in both types of spectra. Both the Infrared and Raman spectra obtained indicate very real effects due to Coriolis coupling. In some of the cases chosen the computed spectra are grossly different from the normal spectra where coupling is absent. Such large effects can greatly impede the interpretation of experimental results. Theoretical spectra therefore aids in the interpretation of experimental results, as is clearly demonstrated in the results of this work.

A critical analysis of the 2H NMR data in chlorhexidinecontaining model membranesDeuterium nuclear magnetic resonance spectroscopy was used to study the in uence of chlorhexidine on the internal molecular motions of phospholipid model membranes. Mixtures of dimyristoylphosphocholine (DMPC) and chlorhexidine (CHX) were investigated at several DMPC:CHX molar ratios (1:0,10:1,3:1). Extensive numerical analysis of previously acquired data identi ed the di erences in the temperaturedependence of the order parameters characterizing the rapid molecular motions (on the NMR scale) in both the fatty acid chains of DMPC and in the saturated methylene bridge of CHX. The results are consistent with the known localization of CHX in the membrane determined by neutron scattering [1] and con rmed by molecular dynamics simulations reported earlier [2]. The NMR results indicate that chlorhexidine undergoes di erent motions than those of the bulk lipids in the membrane. The study used a Tikhonovregularizationbased numerical deconvolution technique (dePakeing) that allowed simultaneous determination of the order parameter and of the orientational distribution of domains in the powder sample, partially oriented by the external magnetic eld. Both appeared to suggest an anomalous result in a narrow temperature region for the 3:1 sample, perhaps the existence of a new reentrant phase, but this conclusion could not be made from the limited data available, and requires a further investigation.

Crystal Growth and Characterization of Li0.9Mo6O17Purple bronze Li0.9Mo6O17 has attracted researchers for its low dimensionality and corresponding properties. Although it has been studied for nearly two decades, there are still some unsolved puzzles with this unique material. Single crystals of Li0.9Mo6O17 were grown using the temperature gradient flux technique in this research. The crystal growth was optimized by experimenting different conditions and good quality crystals were obtained. Xray diffraction results have confirmed the right phase of the crystals. Resistivity measurements and magnetic susceptibility measurements were carried out, and anomalous electronic behaviors were found. All of the samples showed the metalinsulator transition near 20K, followed by behavior that differs from sample to sample: either superconducting, metallic or insulating behavior was observed below 2K. Li0.9Mo6O17 was considered as a quasionedimensional crystal and also a superconducting crystal, which implies a dimensional crossover may occur at the metalinsulator transition. A twoband scenario of the Luttinger liquid model was used to fit the resistivity data and excellent results were achieved, suggesting that the Luttinger theory is a very good candidate for the explanation of the anomalous behavior of Li0.9Mo6O17. In addition, the susceptibility measurements showed Curie paramagnetism and some temperature independent paramagnetism at low temperature. The absence of any anomalous magnetic feature near 20K where the resistivity upturn takes place, suggests that a charge density wave mechanism, which has been proposed by some researchers, is not responsible for the unique properties of Li0.9Mo6O17.

Crystal growth, characterization and point contact spectroscopy on CuxTiSe₂Various s e t s of single c rys t a l s and poly c rys t a l s of Cux TiSe2 were grown. X r ay diffraction and ene rgy dispersive spe c t ros copy r e sul t s verified tha t the c rys t a l s were the cor r e c t compos i t ion and c rys t a l s t ruc tur e . Re s i s t ivi ty me a sur ement s and magne t i c sus c ept ibi l i ty me a sur ement s de t e rmined the supe r conduc t ing t r ans i t ion t empe r a tur e s for the c rys t a l s . The c rys t a l s in each growth had various supe r conduc t ing t r ans i t ion t empe r a tur e s . Also, the me a sur ement s indi c a t ed tha t the c rys t a l s were inhomogeneous. Point cont a c t spe c t ros copy expe r iment s were employed on various single c rys t a l s . Inspe c t ion of the da t a indi c a t ed tha t the ma t e r i a l ha s a single ene rgy gap . A progr am was bui l t ut i l i z ing the Levenbe rgMa rqua rdt me thod and theory on point cont a c t spe c t ros copy to de t e rmine the supe r conduc t ing ene rgy gap. Plot s of the supe r conduc t ing ene rgy gap a t various t empe r a tur e s were in di s agr e ement wi th wha t was expe c t ed for a convent iona l supe r conduc tor .

Crystal growth, Raman scattering and optical properties of the superconductor Cd2Re2O7(16O, 18O)/Using the Physical Vapor Transport method, single crystals of Cd2Re207 have been grown, and crystals of dimensions up to 8x6x2 mm have been achieved. Xray diffraction from a single crystal of Cd2Re207 has showed the crystal growth in the (111) plane. Powder Xray diffraction measurements were performed on ^^O and ^^O samples, however no difference was observed. Assigning the space group Fd3m to Cd2Re207 at room temperature and using structure factor analysis, the powder Xray diffraction pattern of the sample was explained through systematic reflection absences. The temperatiure dependence of the resistivity measurement of ^^O has revealed two structural phase transitions at 120 and 200 K, and the superconducting transition at 1.0 K. Using Factor Group Analysis on three different structiures of Cd2Re207, the number of IR and Raman active phonon modes close to the Brillouin zone centre have been determined and the results have been compared to the temperaturedependence of the Raman shifts of ^^O and ^*0 samples. After scaling (via removing BoseEinstein and Rayleigh scattering factors from the scattered light) all spectra, each spectrum was fitted with a number of Lorentzian peaks. The temperaturedependence of the FWHM and Raman shift of mode Eg, shows the effects of the two structurjil phase transitions above Tc. The absolute reflectance of Cd2Re207  '^O single crystals in the farinfrared spectral region (7700 cm~^) has been measured in the superconducting state (0.5 K), right above the superconducting state (1.5 K), and in the normal state (4.2 K). Thermal reflectance of the sample at 0.5 K and 1.5 K indicates a strong absorption feature close to 10 cm~^ in the superconducting state with a reference temperature of 4.2 K. By means of KramersKronig analysis, the absolute reflectance was used to calculate the optical conductivity and dielectric function. The real part of optical conductivity shows five distinct active phonon modes at 44, 200, 300, 375, and 575 cm~' at all temperatures including a Drudelike behavior at low frequencies. The imaginary part of the calculated dielectric function indicates a mode softening of the mode 44 cm~' below Tc.

Crystallinity, Magnetic and Electrical Properties of Bi doped LaVO3We report the results of crystal structure, magnetization and resistivity measurements of Bi doped LaVO3. Xray diffraction (XRD) shows that if doping Bi in the La site is less than ten percent, the crystal structure of La1xBixVO3 remains unchanged and its symmetry is orthorhombic. However, for higher Bi doping (>10%) composite compounds are found where the XRD patterns are characterized by two phases: LaVO3+V2O3. Energydispersive analysis of the xray spectroscopy (EDAX) results are used to find a proper atomic percentage of all samples. The temperature dependence of the mass magnetization of pure and single phase doped samples have transition temperatures from paramagnetic to antiferromagnetic region at TN=140 K. This measurement for biphasic samples indicates two transition temperatures, at TN=140 K (LaVO3) and TN=170 K (V2O3). The temperature dependence of resistivity reveals semiconducting behavior for all samples. Activation energy values for pure and doped samples are extracted by fitting resistivity versus temperature data in the framework of thermal activation process.

Dechanneling of 2 Mev Heâ ½ in goldThick gold single crystals of high quality were prepared for Rutherford Backscattering Dechanneling studies by electropolishing and annealing. The variation.; with temperature of the Random Fraction versus Depth spectrtnn for 2 MeV He+ on < 110> gold was extracted from measured Aligned and Random (Energy) Spectra. The measured dechanneling rate showed a sixfold increase in going from 4loK to 293°K and is in reasonable agreement with calculations made using the Steady Increase in Transverse Energy (SITE) approximation.

Deformations of Maxwell gauge field theoryDeformations of Maxwell gauge theory are studied in 2+1 dimensions. Unlike in previous work in the literature, no Lagrangian structure is assumed for possible deformations, and instead the requirement of gauge invariance of the deformed field equations under the deformed gauge symmetry is used. The results yield a new nonlinear generalization of Maxwell gauge theory. A nonabelian extension of this theory is also obtained.