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NMR characterization of chlorhexidine in lipidbased formulations /A mixture of Chlorhexidine digluconate (CHG) with glycerophospholipid 1,2dimyristoyl <^54glycero3phospocholine (DMPCrf54) was analysed using ^H nuclear magnetic resonance. To analyze powder spectra, the dePakeing technique was used. The method is able to extract simultaneously both the orientation distribution function and the anisotropy distribution function. The spectral moments, average order parameter profiles, and longitudinal and transverse relaxation times were used to explore the structural phase behaviour of various DMPC/CHG mixtures in the temperature range 560°C.

Numerical Solutions of Laplace's Equation for Various Physical SituationsThere are two projects in this thesis. In the first project, a general method is introduced to numerically calculate the resistance of truncated resistors in cylindrical coordinates, with nonconstant crosssectional area. The problem of finding the resistance of a truncated conical resistor is given in some introductory textbooks as a simple problem. The textbook method is flawed however, and leads to the wrong answer. The textbook method assumes that the electric potential distribution inside the truncated cone is approximately equivalent to a cylindrical resistor. This assumption ignores the constricting affect that the boundary of the truncated conical resistor has on the electric potential inside. The deformation of the electric field is not accounted for by excess charge or changing magnetic fields, instead it is the result of a derivative operation called the shear of the field. Numerical solutions for the resistance of truncated conical, ellipsoidal, and hyperboloidal resistors are presented as a function of a/b, where a is the radius of the smallest crosssectional area and b is radius of the largest. It was found that the textbook solution always underestimates the numerical value of the resistance. In the second project, dielectric breakdown clusters were grown with a stochastic two dimensional Dielectric Breakdown Model (DBM) on a honeycomb, square, and triangle lattice, as well as on a random distribution of nodes. On the regular lattices the number of nearest neighbours was a constant at all lattice sites. For a random distribution of nodes there was variation in the number of nearest neighbours at different nodes. Some percentage of the nodes were isolated from the rest of the distribution, because they had 0 nearest neighbours. Distributions of nodes in which many of the nodes had 0 nearest neighbours indicated a medium with high density fluctuations. The motivation for this work was to study the relationship between the fractal dimension of the dielectric breakdown clusters and the number of nearest neighbours, and the density variation of the medium. The singularity spectra were calculated for the clusters, as well as their fractal dimension using box counting, and sandbox methods. It was found that the dielectric breakdown model produces monofractal clusters. As such, the dimension of the clusters can be represented by a single fractal dimension. In the DBM, the probability of a perimeter site connecting to the cluster is proportional to the strength of the local electric field raised to an exponent. If the exponent is a large positive number then perimeter sites which feel a stronger electric field are more likely to connect to the cluster. Increasing the exponent produces clusters which resemble lightning, with a fractal dimension lower than the dimension of the lattice. Similarly increasing the percentage of isolated nodes decreases the fractal dimension.

On the anharmonic, multiphonon, DebyeWaller contributions to the phononlimited resistivity of metals : applications to Na and KThe anharmonic, multiphonon (MP), and OebyeWaller factor (OW) contributions to the phonon limited resistivity (;0) of metals derived by Shukla and Muller (1979) by the doubletime temperature dependent Green function method have been numerically evaluated for Na and K in the high temperature limit. The anharmonic contributions arise from the cubic and quartic shift of phonons (CS, QS), and phonon width (W) and the interference term (1). The QS, MP and OW contributions to I' are also derived by the matrix element method and the results are in agreement with those of Shukla and Muller (1979). In the high temperature limit, the contributions to;O from each of the above mentioned terms are of the type BT2 For numerical calculations suitable expressions are derived for the anharmonic contributions to ~ in terms of the third and fourth rank tensors obtained by the Ewald procedure. The numerical calculation of the contributions to;O from the OW, MP term and the QS have been done exactly and from the CS, Wand I terms only approximately in the partial and total Einstein approximations (PEA, TEA), using a first principle approach (Shukla and Taylor (1976)). The results obtained indicate that there is a strong pairwise cancellation between the: OW and MP terms, the QS and CS and the Wand I terms. The sum total of these contributions to;O for Na and K amounts to 4 to 11% and 2 to 7%, respectively, in the PEA while in the TEA they amount to 3 to 7% and 1 to 4%, respectively, in the temperature range.

On the equation of state and atomic mean square displacement of crystalsWe have presented a Green's function method for the calculation of the atomic mean square displacement (MSD) for an anharmonic Hamil toni an . This method effectively sums a whole class of anharmonic contributions to MSD in the perturbation expansion in the high temperature limit. Using this formalism we have calculated the MSD for a nearest neighbour fcc Lennard Jones solid. The results show an improvement over the lowest order perturbation theory results, the difference with Monte Carlo calculations at temperatures close to melting is reduced from 11% to 3%. We also calculated the MSD for the Alkali metals Nat K/ Cs where a sixth neighbour interaction potential derived from the pseudopotential theory was employed in the calculations. The MSD by this method increases by 2.5% to 3.5% over the respective perturbation theory results. The MSD was calculated for Aluminum where different pseudopotential functions and a phenomenological Morse potential were used. The results show that the pseudopotentials provide better agreement with experimental data than the Morse potential. An excellent agreement with experiment over the whole temperature range is achieved with the Harrison modified pointion pseudopotential with HubbardSham screening function. We have calculated the thermodynamic properties of solid Kr by minimizing the total energy consisting of static and vibrational components, employing different schemes: The quasiharmonic theory (QH), ).2 and).4 perturbation theory, all terms up to 0 ().4) of the improved self consistent phonon theory (ISC), the ring diagrams up to o ().4) (RING), the iteration scheme (ITER) derived from the Greens's function method and a scheme consisting of ITER plus the remaining contributions of 0 ().4) which are not included in ITER which we call E(FULL). We have calculated the lattice constant, the volume expansion, the isothermal and adiabatic bulk modulus, the specific heat at constant volume and at constant pressure, and the Gruneisen parameter from two different potential functions: LennardJones and Aziz. The Aziz potential gives generally a better agreement with experimental data than the LJ potential for the QH, ).2, ).4 and E(FULL) schemes. When only a partial sum of the).4 diagrams is used in the calculations (e.g. RING and ISC) the LJ results are in better agreement with experiment. The iteration scheme brings a definitive improvement over the).2 PT for both potentials.

On the formulation and the calculation of the harmonic contributions to the DebyeWaller factor in metals (sodium)The algebraic expressions for the anharmonic contributions to the DebyeWaller factor up to 0(A ) and 0 L% ) £ where ^ is the scattering wavevector] have been derived in a form suitable for cubic metals with small ion cores where the interatomic potential extends to many neighbours. This has been achieved in terms of various wavevector dependent tensors, following the work of Shukla and Taylor (1974) on the cubic anharmonic Helmholtz free energy. The contribution to the various wavevector dependent tensors from the coulomb and the electronion terms in the interatomic metallic potential has been obtained by the Ewald procedure. All the restricted multiple whole B r i l l o u i n zone (B.Z.) sums are reduced to single whole B.Z. sums by using the plane wave representation of the delta function. These single whole B.Z. sums are further reduced to the •%?? portion of the B.Z. following Shukla and Wilk (1974) and Shukla and Taylor (1974). Numerical calculations have been performed for sodium where the BornMayer term in the interatomic potential has been neglected because i t is small £ Vosko (1964)3 • *n o^er to compare our calculated results with the experimental results of Dawton (1937), we have also calculated the r a t io of the intensities at different temperatures for the lowest five reflections (110), (200), (220), (310) and (400) . Our calculated quasiharmonic results agree reasonably well with the experimental results at temperatures (T) of the order of the Debye temperature ( 0 ). For T » © ^ 9 our calculated anharmonic results are found to be in good agreement with the experimental results.The anomalous terms in the DebyeWaller factor are found not to be negligible for certain reflections even for T ^ ©^ . At temperature T yy Op 9 where the temperature is of the order of the melting temperature (Xm) » "the anomalous terms are found to be important almost for all the f i ve reflections.

On the path integral formulation and the evaluation of quantum statistical averagesFour problems of physical interest have been solved in this thesis using the path integral formalism. Using the trigonometric expansion method of Burton and de Borde (1955), we found the kernel for two interacting one dimensional oscillators• The result is the same as one would obtain using a normal coordinate transformation, We next introduced the method of Papadopolous (1969), which is a systematic perturbation type method specifically geared to finding the partition function Z, or equivalently, the Helmholtz free energy F, of a system of interacting oscillators. We applied this method to the next three problems considered• First, by summing the perturbation expansion, we found F for a system of N interacting Einstein oscillators^ The result obtained is the same as the usual result obtained by Shukla and Muller (1972) • Next, we found F to 0(Xi)f where A is the usual Tan Hove ordering parameter* The results obtained are the same as those of Shukla and Oowley (1971), who have used a diagrammatic procedure, and did the necessary sums in Fourier space* We performed the work in temperature space• Finally, slightly modifying the method of Papadopolous, we found the finite temperature expressions for the Debyecaller factor in Bravais lattices, to 0(AZ) and u(/K/ j,where K is the scattering vector* The high temperature limit of the expressions obtained here, are in complete agreement with the classical results of Maradudin and Flinn (1963) .

Optical properties of organic superconductor K(BETS)2FeBr4 //c(BETS)2FeBr4 is the first antiferromagnetic organic superconductor with successive antiferromagnetic and superconducting transitions at Ta^=2.5K and Tc=l.lK respectively at ambient pressure. Polarized reflectance measurements were performed on three single crystalsamples of this material using a Briiker IFS66V/S Interferometer, and a Bolometer detector or an MCT detector, at seven temperatures between 4K and 300K, in both the farinfrared and midinfrared frequency range. After the reflectance results were obtained, the KramersKronig dispersion relation was apphed to determine the optical conductivity of /c(BETS)2FeBr4 at these seven temperatures. Additionally, the optical conductivity spectra were fitted with a Drude/Lorentz Oscillator model in order to study the evolution of the optical conductivity with temperature along the aaxis and caxis. The resistivities calculated from the Drude model parameters along the aaxis and caxis agreed reasonably with previous transport measurements.

Optical properties of organic superconductor K(BETS)2FeBr4 /K(BETS)2FeBr4 is a quasi2D charge transfer organic metal with interesting electronic and magnetic properties. It undergoes a transition to an antiferromagnetic (AF) state at ambient pressure at the Neel temperature (T^^) = 2.5 K, as well as to a superconducting (SC) state at 1.1 K [1]. The temperature dependence of the electrical resistivity shows a small decrease at T;v indicating the resistivity drops as a result of the onset of the ordering of Fe'*''" spins. A sharp drop in the resistivity at 1.1 K is due to its superconducting transition. The temperature dependence of the susceptibility indicates an antiferromagnetic spin structure with the easy axis parallel to the aaxis. The specific heat at zerofield shows a large peak at about 2.4 K, which corresponds to the antiferromagnetic transition temperature (Tat) and no anomaly is observed around the superconducting transition temperature (1.1 K) demonstrating that the magnetically ordered state is not destroyed by the appearance of another phase transition (the superconducting transition) in the 7relectron layers [1], [2]. This work presents an investigation of how the low frequency electromagnetic response is affected by the antiferromagnetic and superconducting states, as well as the onset of strong correlation. The location of the easy axis of three samples was determined and polarized thermal reflectance measurements of these «(BETS)2FeBr4 samples oriented with their vertical axis along the a and c axes were then carried out using a *He refrigerator cryostat and a MartinPuplett type polarizing interferometer at various temperatures (T = 0.5 K, 1.4 K. 1.9 K, 2.8 K) above and below the superconducting state and/or antiferromagnetic state. Comparison of the SC state to the normal state along the o and caxes indicates a rising thermal reflectance at low frequencies (below 10 cm"' ) which may be a manifestation of the superconducting energy gap. A dipHke feature is detected at low frequencies (below 15 cm"') in the thermal reflectance plots which probe the antiferromagnetic state along the two axes, and may be due to the opening of a gap in the excitation spectrum as a result of the antiferromagnetism. In another set of experiments, thermal reflectance measurements carried out along the a and caxes at higher temperatures (10 K80 K) show that the reflectivity decreases with increasing temperature to 60 K (the coherence temperature) above which it increases again. Comparison of the thermal reflectance plots along the a and caxes at higher temperatures reveals an anisotropy between these two axes. The HagenRubens thermal reflectance plots corresponding to an average over the acplane were calculated using experimental hterature resistivity values. Comparison of the HagenRubens plots with the experimental thermal reflectance along the a and caxes indicates that both exhibit the general trend of a decrease in thermal reflectance with increasing frequency, however the calculated HagenRubens thermal reflectance at different temperatures is much lower than the experimental curves.

Optical Properties of Sb2Te3, and Dilute Magnetic Semiconductors Sb1.97 VO.03 Te3 and Sb1.94 CrO.06 Te3This thesis reports on the optical properties of the dilute magnetic semiconductors, Sb1.97 V 0.03 Te3 and Sb1.94Cr0.06Te3, along with the parent compound Sb2Te3' These materials develop a ferromagnetic state at low temperature with Curie temperatures of 22 K and 16 K respectively. All three samples were oriented such that the electric field vector of the light was perpendicular to the caxis. The reflectance profile of these samples in the midinfrared (500 to 3000 cm1) shows a pronounced plasma edge which retracts with decreasing temperature. The farinfrared region of these samples exhibits a phonon at ~ 60 cm1 which softens as temperature decreases. KramersKronig analysis and a DrudeLorentz model were employed to determine the optical constants of the bulk samples. The real part of the optical conductivity is shown to consist of intraband contributions at frequencies below the energy gap (~0.26 eV) and interband contributions at frequencies above the energy gap. The temperature dependence of the scattering rate show that a mix of phonon and impurity scattering are present, while the signature of traditional spin disorder (magnetic) scattering was difficult to confirm.

Optical Study of (Nb0.5In0.5)0.02 Ti0.98O2 CrystalsThis work was a study of pure TiO2Rutile crystals, as well as Rutile crystals 2% codoped with Indium and Niobium (2NITO). There is much interest surrounding codoped TiO2recently, with several papers published on ’colossal permittivity’ in the lower frequency ranges (10^210^6Hz range). The aim of this work was to study the optical and Raman modes of pure and codoped crystals to determine the effects codoping has on these modes. Infrared reflectance along with Raman Spectroscopy were used for this purpose. In order to determine the dielectric function from the infrared data, the Factorized Model and KramersKronig analysis were used. Since TiO2has a tetragonal unit cell, infrared measurements of both the a and c axes of both doped and undoped crystals were done. Theaaxis is known to have 3 optical modes, whereas the caxis only has one. However an additional mode was seen in all spectra, believed to be caused by anharmonicity. In addition, the 136cm−1mode observed in polycrystalline conductivity spectra of 5 and 10NITO lines up directly with the A2u mode and the 793cm−1 mode also appears in single crystal TiO2, meaning these are not new modes. However the 447cm−1 and 654cm−1 modes do not appear in our data, and are likely a result of higher percentage codoping. The effect of codoping was observed to be an overall decrease in the reflectance of TiO2. We also observed sizable increases inγtofor all modes in 2NITO. In addition, the dielectric permittivity decreases below the first phonon mode; suggesting that the enhanced permittivity observed at lower frequencies is not caused by codoped changes to phonon modes. All expected Ramanactive modes were observed, however due to poor data resolution some of the peak positions appear to be slightly different than previously measured. Our Raman spectra showed new structures at around 300cm−1and 700cm−1 in the (100) surface spectra, it is possible these are combination lines.

Optimization of the valence energy variance of the CuH moleculeWe developed the concept of split't to deal with the large molecules (in terms of the number of electrons and nuclear charge Z). This naturally leads to partitioning the local energy into components due to each electron shell. The minimization of the variation of the valence shell local energy is used to optimize a simple two parameter CuH wave function. Molecular properties (spectroscopic constants and the dipole moment) are calculated for the optimized and nearly optimized wave functions using the Variational Quantum Monte Carlo method. Our best results are comparable to those from the single and double configuration interaction (SDCI) method.

Optimizing Computational Frameworks to Study the Influence of the Protein Environment on the Individual Site Energies of Chromophores in Photosystem II of PhotosynthesisPhotosynthesis is a process in which electromagnetic radiation is converted into chemical energy. Photosystems capture photons with chromophores and transfer their energy to reaction centers using chromophores as a medium. In the reaction center, the excitation energy is used to perform chemical reactions. Knowledge of chromophore site energies is crucial to the understanding of excitation energy transfer pathways in photosystems and the ability to compute the site energies in a fast and accurate manner is mandatory for investigating how protein dynamics effect the site energies and ultimately energy pathways with time. In this work we developed two software frameworks designed to optimize the calculations of chromophore site energies within a protein environment. The first is for performing quantum mechanical energy optimizations on molecules and the second is for computing site energies of chromophores in a fast and accurate manner using the polarizability embedding method. The two frameworks allow for the fast and accurate calculation of chromophore site energies within proteins, ultimately allowing for the effect of protein dynamics on energy pathways to be studied. We use these frameworks to compute the site energies of the eight chromophores in the reaction center of photosystem II (PSII) using a 1.9 Å resolution xray structure of photosystem II. We compare our results to conflicting experimental data obtained from both isolated intact PSII core preparations and the minimal reaction center preparation of PSII, and find our work more supportive of the former.

Order and membrane organization in chlorhexidinelipid mixtures /Formulations of a general bactericidal agent, chlorhexidine, mixed with a phospholipid at different concentrations are investigated using ^H NMR spectroscopy on a chaindeuterated lipid analog. Lipidchlorhexidine formulation is known to release the drug into an aqueous medium slowly, maintaining a comparable concentration of the drug for up to four times longer than a direct aqueous solution. The NMR data does not support the proposed liposomal entrapment of chlorhexidine in lipid compartments. Complex thermal history of the lipidchlorhexidine preparations is investigated in detail. In preparation for a counterpart measurement, using ^H NMR of deuterated chlorhexidine mixed with protonated lipid, the synthesis of a deuterated analog of chlorhexidine is performed.

Phonon dispersion curves and atomic mean square displacement for several fcc and bcc materialsThe atomic mean square displacement (MSD) and the phonon dispersion curves (PDC's) of a number of facecentred cubic (fcc) and bodycentred cubic (bcc) materials have been calclllated from the quasiharmonic (QH) theory, the lowest order (A2 ) perturbation theory (PT) and a recently proposed Green's function (GF) method by Shukla and Hiibschle. The latter method includes certain anharmonic effects to all orders of anharmonicity. In order to determine the effect of the range of the interatomic interaction upon the anharmonic contributions to the MSD we have carried out our calculations for a LennardJones (LJ) solid in the nearestneighbour (NN) and nextnearest neighbour (NNN) approximations. These results can be presented in dimensionless units but if the NN and NNN results are to be compared with each other they must be converted to that of a real solid. When this is done for Xe, the QH MSD for the NN and NNN approximations are found to differ from each other by about 2%. For the A2 and GF results this difference amounts to 8% and 7% respectively. For the NN case we have also compared our PT results, which have been calculated exactly, with PT results calculated using a frequencyshift approximation. We conclude that this frequencyshift approximation is a poor approximation. We have calculated the MSD of five alkali metals, five bcc transition metals and seven fcc transition metals. The model potentials we have used include the Morse, modified Morse, and Rydberg potentials. In general the results obtained from the Green's function method are in the best agreement with experiment. However, this improvement is mostly qualitative and the values of MSD calculated from the Green's function method are not in much better agreement with the experimental data than those calculated from the QH theory. We have calculated the phonon dispersion curves (PDC's) of Na and Cu, using the 4 parameter modified Morse potential. In the case of Na, our results for the PDC's are in poor agreement with experiment. In the case of eu, the agreement between the tlleory and experiment is much better and in addition the results for the PDC's calclliated from the GF method are in better agreement with experiment that those obtained from the QH theory.

Phonon spectra and temperature variation of thermodynamic properties of fcc metals via FinnisSinclair type many body potentials: SuttonChen and improved SuttonChen modelsVolume(density)independent pairpotentials cannot describe metallic cohesion adequately as the presence of the free electron gas renders the total energy strongly dependent on the electron density. The embedded atom method (EAM) addresses this issue by replacing part of the total energy with an explicitly densitydependent term called the embedding function. Finnis and Sinclair proposed a model where the embedding function is taken to be proportional to the square root of the electron density. Models of this type are known as FinnisSinclair many body potentials. In this work we study a particular parametrization of the FinnisSinclair type potential, called the "SuttonChen" model, and a later version, called the "Quantum SuttonChen" model, to study the phonon spectra and the temperature variation thermodynamic properties of fcc metals. Both models give poor results for thermal expansion, which can be traced to rapid softening of transverse phonon frequencies with increasing lattice parameter. We identify the power law decay of the electron density with distance assumed by the model as the main cause of this behaviour and show that an exponentially decaying form of charge density improves the results significantly. Results for SuttonChen and our improved version of SuttonChen models are compared for four fcc metals: Cu, Ag, Au and Pt. The calculated properties are the phonon spectra, thermal expansion coefficient, isobaric heat capacity, adiabatic and isothermal bulk moduli, atomic rootmeansquare displacement and Gr\"{u}neisen parameter. For the sake of comparison we have also considered two other models where the distancedependence of the charge density is an exponential multiplied by polynomials. None of these models exhibits the instability against thermal expansion (premature melting) as shown by the SuttonChen model. We also present results obtained via pure pair potential models, in order to identify advantages and disadvantages of methods used to obtain the parameters of these potentials.

Phonon spectra and thermal properties of some fcc metals using embeddedatom potentials /By employing the embeddedatom potentials of Mei et ai.[l], we have calculated the dynamical matrices and phonon dispersion curves for six fee metals (Cu,Ag,Au,Ni,Pd and Pt). We have also investigated, within the quasiharmonic approximation, some other thermal properties of these metals which depend on the phonon density of states, such as the temperature dependence of lattice constant, coefficient of linear thermal expansion, isothermal and adiabatic bulk moduli, heat capacities at constant volume and constant pressure, Griineisen parameter and Debye temperature. The computed results are compared with the experimental findings wherever possible. The comparison shows a generally good agreement between the theoretical values and experimental data for all properties except the discrepancies of phonon frequencies and Debye temperature for Pd, Pt and Au. Further, we modify the parameters of this model for Pd and Pt and obtain the phonon dispersion curves which is in good agreement with experimental data.

Phonon spectra and thermodynamic properties of rare gas solids based on empirical and semiempirical (ab initio) twobody potentials : a comparative study /We study the phonon dispersion, cohesive and thermal properties of raxe gas solids Ne, Ar, Kr, and Xe, using a variety of potentials obtained from different approaches; such as, fitting to crystal properties, purely ab initio calculations for molecules and dimers or ab initio calculations for solid crystalline phase, a combination of ab initio calculations and fitting to either gas phase data or sohd state properties. We explore whether potentials derived with a certain approaxih have any obvious benefit over the others in reproducing the solid state properties. In particular, we study phonon dispersion, isothermal ajid adiabatic bulk moduli, thermal expansion, and elastic (shear) constants as a function of temperatiue. Anharmonic effects on thermal expansion, specific heat, and bulk moduli have been studied using A^ perturbation theory in the high temperature limit using the neaxestneighbor central force (nncf) model as developed by Shukla and MacDonald [4]. In our study, we find that potentials based on fitting to the crystal properties have some advantage, particularly for Kr and Xe, in terms of reproducing the thermodynamic properties over an extended range of temperatiures, but agreement with the phonon frequencies with the measured values is not guaranteed. For the lighter element Ne, the LJ potential which is based on fitting to the gas phase data produces best results for the thermodynamic properties; however, the Eggenberger potential for Ne, where the potential is based on combining ab initio quantum chemical calculations and molecular dynamics simulations, produces results that have better agreement with the measured dispersion, and elastic (shear) values. For At, the Morsetype potential, which is based on M0llerPlesset perturbation theory to fourth order (MP4) ab initio calculations, yields the best results for the thermodynamic properties, elastic (shear) constants, and the phonon dispersion curves.

Planar Topological Defects in Unconventional SuperconductorsIn this work, we consider the properties of planar topological defects in unconventional superconductors. Specifically, we calculate microscopically the interaction energy of domain walls separating degenerate ground states in a chiral pwave fermionic superfluid. The interaction is mediated by the quasiparticles experiencing Andreev scattering at the domain walls. As a byproduct, we derive a useful general expression for the free energy of an arbitrary nonuniform texture of the order parameter in terms of the quasiparticle scattering matrix. The thesis is structured as follows. We begin with a historical review of the theories of superconductivity (Sec. 1.1), which led the way to the celebrated BardeenCooper Schrieffer (BCS) theory (Sec. 1.3). Then we proceed to the treatment of superconductors with socalled "unconventional pairing" in Sec. 1.4, and in Sec. 1.5 we introduce the specific case of chiral pwave superconductivity. After introducing in Sec. 2 the domain wall (DW) model that will be considered throughout the work, we derive the Bogoliubovde Gennes (BdG) equations in Sec. 3.1, which determine the quasiparticle excitation spectrum for a nonuniform superconductor. In this work, we use the semiclassical (Andreev) approximation, and solve the Andreev equations (which are a particular case of the BdG equations) in Sec. 4 to determine the quasiparticle spectrum for both the single and twoDW textures. The Andreev equations are derived in Sec. 3.2, and the formal properties of the Andreev scattering coefficients are discussed in the following subsection. In Sec. 5, we use the transfer matrix method to relate the interaction energy of the DWs to the scattering matrix of the Bogoliubov quasiparticles. This facilitates the derivation of an analytical expression for the interaction energy between the two DWs in Sec. 5.3. Finally, to illustrate the general applicability our method, we apply it in Sec. 6 to the interaction between phase solitons in a twoband swave superconductor.

Preparation of Single Crystal Molybdenum Bronzes and Polycrystalline Rubidium Molybdenum OxideLithium purple bronze (LiPB) Li0.9Mo6O17 was studied extensively in the 1980’s and has recently regained interest in the research community. The mechanism for the metalinsulator (M/I) transition at ∼25 K and superconductivity at ∼2 K is still unknown. The temperature gradient flux technique has been used to grow single crystal LiPB. A large superconducting LiPB single crystal was grown (di mensions: 4.7×3.0×1.2 mm3) and characterized and will be referred to throughout this work as ‘I33sc2’. For the nonsuperconducting single crystal oriented along the bcrystallographic axis, we found the maximum value of the energy gap to be 2 meV±0.1 meV at 9.16 K, which is different from the accepted literature value of ∼1 meV at 56 K [1,2]. Furthermore, heavier alkali metals (Na, K and Rb) were used in the temperature gradient flux technique. Small lilac brown single crystals were found in all three growths. In particular, the rubidium growth yielded polyhedral (denoted as ‘POLY’, having approximate dimensions: 0.5 × 0.3 × 0.3 mm3) and small bar shaped (denoted as ‘SBAR’, having approximate dimensions: 1×0.1×0.1 mm3) lilac brown single crystals which exhibited anomalous properties in the re sistivity and specific heat. The anomalous properties include a M/I transition at 250 K and a small phase transition in the resistivity at 94 K. These lilac brown single crystals were studied with powder Xray diffraction (XRD) and match very closely with molybdenum dioxide MoO2, however with slightly larger ‘a’ and ‘b’ lattice parameters. These lilac brown crystals are being further characterized with single crystal XRD. Polycrystalline rubidium molybdenum dioxide Rb0.03MoO2−δ was then grown to further understand what rubidium doping (if any) these lilac brown single crystals have. The resistivity of polycrystalline Rb0.03MoO2−δ has a M/I transition at 250 K, similar to the lilac brown single crystals, but lacks a phase transition at 94 K. Unlike the lilac brown single crystals, the slope of the M/I transition at 250 K is strongly affected by thermal cycling. Furthermore, poly crystalline Rb0.03MoO2−δ shows signs of a small superconducting volume fraction. This material is under further investigation.