Bulletin of the American Physical Society
2007 Joint Fall Meeting of the Texas Sections of the APS and AAPT; Zone 13 of SPS
Volume 52, Number 16
Thursday–Saturday, October 18–20, 2007; College Station, Texas
Session B6: P1: Poster Session I: Condensed Matter and Atomic, Molecular, and Optical |
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Chair: Andrea Burzo, Texas A&M University Room: MSC 292 |
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B6.00001: Optical Techniques for the Measurement of Nanoscale Hydrogel Crystal Particles Preston Kendall, Chenglin Chi, Wells Estabrook, Tong Cai, Zhibing Hu Two optical techniques can be used to measure the size of the hydrogel component particles at the nano scale: 1) a method using light absorbance and 2) a method using laser light diffraction. Both methods have advantages and disadvantages. The laser light diffraction method is effective for particles ranging from 400 nm to 1000 nm in diameter while the UV absorbance method is effective for particles ranging from 100 nm to 400 nm in diameter. [Preview Abstract] |
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B6.00002: Optical Characterization of the Ho$^{3+}$ Complex in HEMA Manuel Rodriguez III, Dhiraj Sardar, Kelly Nash, Raylon Yow, John Gruber The spectroscopic properties of the Ho$^{3+}$ complex embedded in 2-hydroxyethyl methacrylate (HEMA) are investigated. The intensities of the room temperature absorption spectra of the Ho$^{3+}$(4$f^{10})$ transitions in Ho(NO$_{3})_{3}$.5H$_{2}$O:HEMA have been analyzed using the Judd-Ofelt (J-O) model to obtain the phenomenological intensity parameters, $\Omega_{2}$, $\Omega_{4}$, and $\Omega_{6.}$ These parameters are used to calculate the spontaneous emission probabilities, radiative lifetimes, and branching ratios of the Ho$^{3+}$ transitions from the upper multiplet manifolds to the corresponding lower-lying multiplet manifolds of $^{2S+1}L_{J}$ Ho$^{3+}$(4$f^{10})$, which include $^{5}$G$_{4}+^{3}$K$_{7}^{(2)}$, $^{5}$G$_{5}$, $^{5}$G$_{6}+^{5}$F$_{1}$, $^{5}$F$_{2}+^{3}$K$_{8}^{(2)}$, $^{5}$F$_{3}$, $^{5}$F$_{4}+^{5}$S$_{2}$, and $^{5}$F$_{5.}$ The predicted room temperature fluorescence lifetime of $^{5}$I$_{7}$ to $^{5}$I$_{8}$ is about 0.5 ms, suggesting a reasonably strong interaction between the complex and the polymer. A comparative study of Ho$^{3+}$(4$f^{10})$ ions in different host materials suggests that Ho(NO$_{3})_{3}$.5H$_{2}$O:HEMA could be an excellent candidate for certain applications such as narrow band pass filters, especially in the visible-to-near infrared region of the spectrum. [Preview Abstract] |
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B6.00003: Effects of Sintering Temperature on Superconductivity in undoped and SiC-doped MgB$_{2}$/Ti Wires Cad Hoyt, Hui Fang, John Douglas, K. West, Gan Liang, Samaresh Guchhait, John Markert The effects of sintering temperature on the superconducting properties of both undoped and SiC-doped MgB$_{2}$ wires have been studied. The wires were fabricated by in situ powder-in-tube (PIT) method and characterized by x-ray diffraction, magnetization, scanning electron microscopy, and electrical resistivity measurements. Two groups of wire samples were prepared: the first group contains a pure MgB$_{2}$ core and the second contains MgB$_{2}$ core doped with 10 wt.{\%} of 20 nm SiC. Both groups of samples were sintered for 30 minutes at the following temperatures: 650 \r{ }C, 700 \r{ }C, 750 \r{ }C, 800\r{ }C, 850\r{ }C. It was found that the cores of these wires are almost in pure MgB$_{2}$ superconducting phase and the superconducting transition temperatures of the wires are about 36 K. For both groups of samples, the critical current density ($J_{c })$, measured at 5 K and 20 K in fields up to 7 Tesla, peaks up at sintering temperature 800\r{ }C. This result is in sharp contrast with recent results observed for Fe-sheathed wires for which the maximum $J_{c}$ was achieved at lower sintering temperatures. Detail discussion will be given to explain such dependence of $J_{c}$ on the sintering temperature. [Preview Abstract] |
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B6.00004: Spectroscopic and thermal studies of amino acid doped Potassium Dihydrogen Phosphate crystals Jayesh Govani, Mihir Joshi, Dipak Dave, Ketan Parikh, Felicia Manciu Potassium dihydrogen phosphate-based materials (KDP) are extensively used for non-linear optical applications. The samples for the current studies were prepared in 8 to 10 days by slow evaporation solution growth technique. Thermal gravimetric analysis of L-histidine amino acid doped KDP crystals demonstrate that the decomposition of the sample occurred at slightly lower temperatures with increasing doping amount. The powder X-ray diffraction patterns reveal a single phase nature with the unit cell parameters being unaltered by doping. Although the main bands observed in the infrared absorption spectra correspond to KDP crystals, the existence of vibrational lines at 1634 cm$^{-1}$, 1714 cm$^{-1}$, 2854 cm$^{-1}$, and 2923 cm$^{-1}$, which are attributed to the degenerate deformations of NH$_{3}$+ groups and of unprotonated monoclinic L-histidine ring, demonstrate that successful doping was achieved. This affirmation is supported with more evidences from FT-Raman measurements. [Preview Abstract] |
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B6.00005: Effects of Gasses on the Thermal Etching Properties of Graphene Nanoribbons Jason Jones, Phillip Ecton, Yudong Mo, Brian Gorman, David Diercks, Thomas Scharf, Jose Perez We investigate the rates of thermal etching for different thicknesses of graphene nanoribbons exposed to different gasses. Etching rates are determined by comparing the change in width of the ribbons to etching time. The thickness and width of the graphene nanoribbons are precisely measured using Atomic Fore Microscopy (AFM). We synthesize sheets of graphene by exfoliating Highly Ordered Pyrolytic Graphite (HOPG) onto a silicon substrate. Optical identification of the scattered graphene sheets is optimized by using a silicon substrate shielded with a 300nm thermal oxide layer, giving the substrate a deep blue color. Verification of monoatomic graphene is accomplished by Raman imaging. Graphene sheets are cut into ribbons using a Focused Ion Beam (FIB). Using FIB techniques, ribbons on the order of 50 to 100 nanometers are produced. The nanoribbons are placed in a furnace combined with a rough vacuum. The graphene nanoribbons are etched by exposure to oxygen gas at a pressure of 300 millitorr for 15 minutes at 650 degrees Celsius. We studied the effects of different gasses on the etching properties of the nanoribbons. [Preview Abstract] |
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B6.00006: Spectroscopic studies of particulate formation in fuel blends Felicia Manciu, Mahesh Subramanya, Jayesh Govani, Ahsan Choudhuri The Raman and infrared absorption spectroscopy were used to investigate the properties of carbon nanotubes (CNTs) flame-synthesized using CH$_{4}$-H$_{2}$ low calorific value gases. The development of large amounts of CNTs benefits from flame synthesis processes, where the fuel serves as both the heating and the reactant source. As a result of flame condition studies it was determined that the CNT growth region is at 20-30{\%} of the visible flame height and at a flow rate between 7.18E-07 m$^{3}$/s and 9.57E-07 m$^{3}$/s. Preliminary characterizations of the samples by Scanning Electron Microscopy demonstrate that the formation of nanostructure occurs only for $<$10{\%} H$_{2}$ concentration. The Raman analysis of the pristine samples shows the existence of distinctive multi-walled carbon nanotube (MWNT) D and G bands at 1321 cm$^{-1}$ and 1595 cm$^{-1}$, respectively. Besides the vibrational lines characteristic to MWNTs, infrared absorption measurements also reveal the presence of C-H bonds. [Preview Abstract] |
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B6.00007: Ion Beam Analysis of Thin Films on Silicon and Carbon Substrates Pelham Keahey, J'Nae Zwaschka, Lucas Phinney, Lee Mitchell, Khalid Hossian, Jerome Duggan Economics is the primary driving force behind the semiconductor industry's quest to make devices smaller and smaller. Such devices as transistors and integrated chips are produced by laying down very thin films of various materials, insulators and conductors, and masking them in such a way to produced the device. Ion Beam Analysis techniques, such as Rutherford Back Scattering (RBS) is commonly used to calculate the thickness of these layers and their integrity. To illustrate this type of analysis Aluminum, Copper and Gold were evaporated onto ultra pure Carbon and Silicon sheets (Figure 1). Using a 2.5MeV VandeGraff accelerator we use RBS with both a proton and alpha particle beam which impinged on the sample in an ultra high vacuum chamber (fig 2). From the data collected from RBS, we used two mathematical techniques and one simulation program to fit the experimental data. Mathematical methods include :1) Using known Rutherford cross section and experimental data 2) Comparing measured peaks to high precision standards. We used the simulation program (SIMNRA) to model the experimental results as shown in the following graphs. [Preview Abstract] |
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B6.00008: Growth and Characterization of Wide Band Gap Semiconductors (Zinc Oxide, Zinc Sulfide) Jeffrey Schwartz, Jose Perez, Yudong Mo Zinc Oxide and Zinc Sulfide nanostructures were grown on a variety of substrates using aqueous growth solutions. The chemical composition of the nanostructures was characterized using micro-Raman spectroscopy, energy-dispersive X-Ray spectroscopy, and X-Ray diffraction. A Scanning Electron Microscope reveals a well-aligned, uniform, layer of hexagonally shaped Zinc Oxide nanorods growing up perpendicular to the substrate surface while the Zinc Sulfide formed irregularly shaped spheres on the substrate. Depending on the growth conditions, the diameters of the ZnO nanorods ranged from a few hundred nanometers to about 1 $\mu $m. The field emission properties of the ZnO nanorods and the ZnS spheroids were studied, with turn-on voltages found to be around 36 v / $\mu $m, as well as the effects on ZnS after exposure to various gases which was found to increase the turn-on voltage in most cases. [Preview Abstract] |
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B6.00009: Thioindigo Interaction with Palygorskite and Sepiolite Alejandra Ramirez, Russell Chianelli, Sridhar Komarneni, Swati Kumar Pigments developed by the Mayan civilization are now known to be significantly `environmentally friendly' a technical skill developed circa 250-900 C.E! [1]. One such pigment called Maya Blue, has been the focus of numerous studies and is believed to be a mixture of palygorskite clay and indigo dye [2,3]. Several derivatives of this pigment have been now developed with intriguing properties. For instance, the dye, textit{thioindigo}, reacts with the \textit{palygorskite} clay to exhibit a broad range of colors from red to blue under UV-Vis excitation. The range of colors produced with \textit{sepiolite} clays is smaller. We present spectroscopic analyses of pigments derived from \textit{thioindigo:palygorskite} and \textit{thioindigo:sepiolite} mixtures. $^{27}$Al MAS-NMR spectra of \textit{sepiolite} mixtures clearly showed changes in the Al coordination upon reacting with \textit{thioindigo}. However, palygorskite-dye mixtures showed only slight changes in Al coordination. Future work will involve $^{27}$Al MAS-NMR analyses of \textit{thioindigo} and clays rich in tetrahedrally coordinated Al to confirm the coordination changes in Al in the presence of \textit{thioindigo}. [Preview Abstract] |
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B6.00010: Surface and optical analyses of a dye-mineral composite -- an XPS, FTIR and Raman study William Durrer, Felicia Manciu, Alejandra Ramirez, Russell Chianelli Maya Purple is a pigment produced by mixing the dye thioindigo with the clay mineral palygorskite. In this investigation, we address the questions of how the dye binds to the clay and how such binding might be affected by the organic-inorganic material ratio and of the heating time used in the preparation of the pigment. Synthetically prepared Maya Purple samples were examined using XPS, FTIR, and Raman spectroscopy. XPS measurements show that pigment preparation results in interactions between the dye and the mineral that give rise to several different binding states of the key elemental components oxygen, sulfur, and aluminum. These results are in good agreement with the Raman analysis, where the appearance and disappearance of bands in the 600 cm$^{-1}$, 1100 cm$^{-1}$, and 1600 cm$^{-1}$ regions demonstrate interaction affecting oxygen and sulfur. The data are further corroborated by vibrational line shifting in the FTIR data. [Preview Abstract] |
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B6.00011: Optical phonon modes of PbSe nanoparticles - a Raman and infrared study Francisco Carreto, Felicia Manciu, Yudhisthira Sahoo, Paras Prasad We here demonstrate the use of micro-probe Raman and far-infrared absorption spectroscopy in probing the existence of optical phonon modes of PbSe nanoparticles. The samples were prepared by colloidal chemistry and preliminary characterized by Transmission Electron Microscopy. The Raman results show evidence of the surface phonon (SP) mode. The frequency of this vibration is consistent with its prediction by a dielectric continuum model. While for different PbSe nanoparticle sizes the observed SP mode does not show any obvious change in its position, there is a clear shift by approximately 4 cm$^{-1}$ toward higher frequency in the appearance of the longitudinal optical mode in the Raman spectra from the 3 nm to the 7 nm PbSe nanoparticles. Far-infrared measurements demonstrate the presence of the transverse optical and of the coupled phonon modes. [Preview Abstract] |
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B6.00012: Anomalous Angular Nonstoichiometric Sputtering Yield of a Ga-In Eutectic Target Mariel Kerbacher, J.W. Deaton, L.R. Burns, N.T. Deoli, D.L. Weathers Sputtering is a thin film deposition technique in which an ion beam fired at a target ejects atoms from the top several layers of the target's surface allowing these atoms to deposit as a thin film on any nearby surface. We employed this technique to deposit the first layers of the Ga-In target onto an aluminum foil which we then analyzed using RBS to determine the angular distribution of sputtered material. The purpose of this experiment is to expand the base of scientific knowledge on sputtering and better understand the sputtering process in hopes of improving models of this process. The Ga-In eutectic alloy used in this experiment has a Gibbsian segregation, in which the first atomic monolayer of the surface is at least 94{\%} Indium, while the second layer is primarily Gallium, as reflected in the alloy's bulk concentration (16.5{\%} Indium). Therefore, the majority of Gallium deposited by sputtering originates from the second atomic monolayer or deeper in the sample. The eutectic alloy is a liquid at room temperature, which is ideal for sputtering processes. Liquid targets are self-healing; their composition does not change over time as atoms are ejected from their surface. Since we know that the majority of Gallium sputtered from the Ga-In target originates from below the first atomic monolayer, studying the angular distribution of Gallium isotopes reveals the behavior of atoms ejected from atomic layers beneath the first monolayer of a target during sputtering. [Preview Abstract] |
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B6.00013: Fabrication of Nanodopatterns Using Microphase Separation of Block Copolymer Andrew Bradshaw, Takuo Tanaka, Nobuyuki Takeyasu, Atsushi Taguchi, Satoshi Kawata Arranging and patterning on the nanoscale is of great importance to future efforts in data storage and nano-optical effects such as achieving negative permittivity and permeability at visible wavelengths. One way to achieve these nanoscale patterns is through the use of self-assembling block copolymer solutions. The diblock copolymer used, Polyethylene-block-Poly(ethylene glycol), was dissolved in a suitable solvent and then spin coated onto a substrate. During spin coating the diblock copolymer undergoes microphase separation to produce feature sizes on the scale of tens of nanometers. Further investigation into spin coating efficiency is researched through modification of properties such as solubility of the block copolymer, solvent volatility, and ambient humidity. [Preview Abstract] |
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B6.00014: Photon trajectories in multiple slit interference experiments with femtosecond pulses of light? Luis Grave de Peralta, Ayrton Bernussi A burst of pulses was observed at each output of the experimental arrangement, when a multiple slit was illuminated with a femtosecond pulse of light. Multiple times of fly became observably different and thus, each pulse in an output burst could be univocally associated with a particular slit. Nevertheless, interference between non-overlapping pulses was also observed. Previously, we have used a Fourier Optics approach to explain why interference was observed in conditions where which-path information was available [1]. We show in this work that the observed interference pattern can also be successfully described assuming that the energy of the light travels following well defined paths. Ref.: [1] ``Ultra fast response of arrayed waveguide gratings,'' L. Grave de Peralta, A.A. Bernussi and H. Temkin, IEEE Journal of Quantum Electronics, vol. 43, 473 (2007). [Preview Abstract] |
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B6.00015: Creating an Inexpensive Grid for Monte Carlo Calculations Sean Smith, Steve Alexander, Stephen Foster, Nathan Lindzey, Robert S. Potter, Walter M. Potter, Jon. T Rogers, Carl West, R.L. Coldwell, S. Datta We have developed software that converts an unused PC into a workstation that accepts jobs from a server and sends all results back to this server. Using a grid of up to 100 machines, a set of explicitly correlated wavefunctions optimized by Filippi and Umrigar and variational Monte Carlo we have plotted the electron density, the intracule density, the extracule density, the electron density difference, two forms of the kinetic energy density, the Laplacian of the electron density, the Laplacian of the intracule density and the Laplacian of the extracule density of the ground state of Li2, Be2, B2, C2, N2, O2 and F2 near their equilibrium distance. [Preview Abstract] |
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B6.00016: A Mathematical Model to Derive the Lorentz Factor, Zero Velocity, and Length Contractions (Finding a Privileged Reference Point) Richard Selvaggi, Charles Rogers This presentation uses Einsteinian concepts to derive the Lorentz factor, intersects observers A and B along a single axis to define a zero point, and uses the zero point to derive the Lorentz factor and understand length contractions. This zero point, as well as any other zero point, can be used by observer A to find A's velocity. The zero point mathematical model demonstrates that A finds that light only travels the hypotenuse distance and that, except at zero velocity, light does not travel the perpendicular distance. Finding a zero point defines a privileged reference point. [Preview Abstract] |
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B6.00017: Using Genetic Algorithms to Converge on Molecules with Specific Properties Stephen Foster, Nathan Lindzey, Jon Rogers, Carl West, Walt Potter, Sean Smith, Steven Alexander Although it can be a straightforward matter to determine the properties of a molecule from its structure, the inverse problem is much more difficult. We have chosen to generate molecules by using a genetic algorithm, a computer simulation that models biological evolution and natural selection. By creating a population of randomly generated molecules, we can apply a process of selection, mutation, and recombination to ensure that the best members of the population (i.e. those molecules that possess many of the qualities we are looking for) survive, while the worst members of the population ``die.'' The best members are then modified by random mutation and by ``mating'' with other molecules to produce ``offspring.'' After many hundreds (or thousands) of iterations, one hopes that the population will get better and better---that is, that the properties of the individuals in the population will more and more closely match the properties we want. [Preview Abstract] |
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B6.00018: Using Zero Velocity to Explain the Michelson-Morley and 2007 Rogers-Selvaggi-Chen Experiment Richard Selvaggi, Charles Rogers By accepting Lorentz's length contraction and zero velocity concepts, we can deduce that the speed of light produced in any reference frame is constant and that the produced light's direction is dependent on the velocity of the observer/laser. Since light production is the result of electron physical properties, the function describing the direction of emitted light by electrons to the velocity of the observer/laser is; \begin{center} sin \textit{$\theta $} = $\surd $(1-V$^{2}$/c$^{2})$ and 0 $<$ \textit{$\theta $} $\le $ 90\r{ }. \end{center} [Preview Abstract] |
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B6.00019: Spectroscopic properties of Ho$^{3+}$ in Ho$^{3+}$:Y$_{2}$O$_{3}$ Nanocrystals Douglas Dee, Kelly Nash, John Gruber, Dhiraj Sardar Spectroscopic properties are investigated for Ho$^{3+}$ in nanocrystalline Ho$^{3+}$:Y$_{2}$O$_{3}$. Room temperature absorption intensities of Ho$^{3+}$(4$f^{10})$ transitions in synthesized Ho$^{3+}$:Y$_{2}$O$_{3 }$nanocrystals have been analyzed using the Judd-Ofelt (J-O) approach in order to obtain the phenomenological intensity parameters. The J-O intensity parameters are used to calculate the spontaneous emission probabilities, radiative lifetimes, and branching ratios of the Ho$^{3+}_{ }$transitions from the upper multiplet manifolds to the corresponding lower-lying multiplet manifolds $^{2S+1}L_{J}$ of Ho$^{3+}$(4$f^{10})$. An 8K absorption spectra was also taken. From that spectra an in-depth crystal field splitting analysis was performed on selected manifolds. A comparison of the manifold splittings for Ho$^{3+}$:Y$_{2}$O$_{3}$ (nano) was made to that observed for Ho$^{3+}$ in large single crystals of Y$_{2}$O$_{3}$. Presently we are investigating the fluorescence properties of this nanocrystal. A comparative study of Ho$^{3+}$(4$f^{10})$ ions suggests that synthesized Ho$^{3+}$:Y$_{2}$O$_{3 }$nanocrystals could be an excellent alternative to single-crystal Ho$^{3+}$:Y$_{2}$O$_{3 }$for certain applications especially in the visible region. [Preview Abstract] |
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B6.00020: Absorption Intensities Analysis of Ho$^{3+}$:KPb$_{2}$Cl$_{5}$ Sreerenjini Chandrasekharan, Kelly L. Nash, John B. Gruber, Dhiraj K. Sardar Optical absorption and emission intensities were investigated for Ho$^{3+}$ in single crystal Ho$^{3+}$:KPb$_{2}$Cl$_{5}$. Room temperature absorption intensities of Ho$^{3+}$(4f$^{10})$ transitions in Ho$^{3+}$:KPb$_{2}$Cl$_{5}$ have been analyzed using the Judd-Ofelt (J-O) approach in order to obtain the phenomenological intensity parameters. The J-O intensity parameters are then used to calculate the spontaneous emission probabilities, radiative lifetimes, and branching ratios of the Ho$^{3+}$ transitions from the upper multiplet manifolds to the corresponding lower-lying multiplet manifolds $^{2S+1}$L$_{J }$of Ho$^{3+}$(4f$^{10})$. Presently we are measuring the room temperature fluorescence lifetime of this transition and it will be used to determine the quantum efficiency of Ho$^{3+}$:KPb$_{2}$Cl$_{5}$. From the fluorescence spectrum, the emission cross section of the important manifold $^{5}$I$_{7}\to ^{5}$I$_{8}$(2.0$\mu $m) transition will be determined. The 8K absorption spectrum was examined as well. Selected manifolds were analyzed in terms of the crystal field splitting using a charge-compensation model first developed for Er$^{3+}$ doped into KPb$_{2}$Cl$_{5.}$ The optical and spectroscopic characteristics of Ho$^{3+}$:KPb$_{2}$Cl$_{5}$ show that this material has a potential for 2.0$\mu $m laser system. [Preview Abstract] |
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B6.00021: Frequency Dependence of the Dielectric Response of Select Materials at Microwave Frequencies R. Caufield, J. Roberts, J. Dahiya, A. Anand, B. Johnson Select Materials were probed to determine their susceptibility to microwaves when they were subjected to microwaves over a range of frequencies. In the experiment, the effects sample loading on the magnetic and electrical interactions between a microwave field in a resonant cavity and select samples were monitored using standard perturbation techniques. This interaction is generally described by the equation: Z = f$_{1}$(O$_{e}$, E) + f$_{2}$(O$_{m}$, H) (1) Where f$_{1}$(O$_{e}$, E) is a function of the electric permeability O$_{e }$and the electric field E, while f$_{2}$(O$_{m}$, H) is a function of the magnetic permeability O$_{m }$and the magnetic field H. Changing the volume of the sample that is inserted into a resonant microwave cavity affects the microwave load in the resonant cavity, and thus produces frequency shifts in f$_{1 }$and f$_{2 }$, and changes in the Q factor of the cavity. Measurements were made for different materials as the microwaves were absorbed by the sample. The $\Delta $(1/Q) and $\Delta $f measurements describe the O$_{e}$ and O$_{m }$ interactions within the sample. The results were studied to find the fundamental electric and magnetic properties of the material loaded in the cavity. The results, as well as the behavior of electromagnetic fields allow us to understand the fundamental interaction processes within the sample of material acting as a load in the microwave cavity and to allow us to study frequency dependence on the sample load. [Preview Abstract] |
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B6.00022: A New Max-Min Variational, Semi-Definite Programming Based, Quantization Procedure. Carlos Handy A new variational quantization procedure is developed exploiting the moment problem based analysis underlying the Eigenvalue Moment Method (EMM) developed by Handy, Bessis, and co-workers [1-5]. The EMM procedure is the first to exploit Semidefinite Programming (SDP) analysis in solving quantum problems, and has played a pivotal role in defining new computational tools for tackling non-Hermitian problems such as those concerning PT-invariant (and symmetry breaking) systems [6] and Regge pole calculations for atomic-molecular scattering [7]. It offers a more rigorous (fool-proof) framework than other methods, including those based on a Hill determinant approach. By extension, these same properties are enjoyed by the new Max-Min variational procedure. We offer some illustrative examples which underscore important convexity properties of the underlying ``volcano-function'' [4,5]. [1] C. R. Handy and D. Bessis, Phys. Rev. Lett. 55, 931 (1985). [2] C. R. Handy, D. Bessis, T. D. Morley, Phys. Rev. A 37, 4557 (1988). [3] C. R. Handy, D. Bessis, G. Sigismondi, T. D. Morley, Phys. Rev. Lett. 60, 253 (1988). [4] C. R. Handy, K . Appiah, D. Bessis, Phys. Rev. A 50, 988 (1994). [5] C. R. Handy, Phys. Rev. A 52, 3468 (1995). [6] C. R. Handy, J. Phys. A 34, 5065 (2001). [7] C. R. Handy, C. J. Tymczak, A. Z. Msezane, Phys. Rev. A 66, 050701 (R) (2002). [Preview Abstract] |
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