Browsing by Author "Williams, Clayton D."

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    Application of classical non-linear Liouville dynamic approximations
    Harter, Terry Lee (Virginia Polytechnic Institute and State University, 1988)
    This dissertation examines the application of the Liouville operator to problems in classical mechanics. An approximation scheme or methodology is sought that would allow the calculation of the position and momentum of an object at a specified later time, given the initial values of the object's position and momentum at some specified earlier time. The approximation scheme utilizes matrix techniques to represent the Liouville operator. An approximation scheme using the Liouville operator is formulated and applied to several simple one-dimensional physical problems, whose solution is obtainable in terms of known analytic functions. The scheme is shown to be extendable relative to cross products and powers of the variables involved. The approximation scheme is applied to a more complicated one-dimensional problem, a quartic perturbed simple harmonic oscillator, whose solution is not capable of being expressed in terms of simple analytic functions. Data produced by the application of the approximation scheme to the perturbed quartic harmonic oscillator is analyzed statistically and graphically. The scheme is reapplied to the solution of the same problem with the incorporation of a drag term, and the results analyzed. The scheme is then applied to a simple physical pendulum having a functionalized potential in order to ascertain the limits of the approximation technique. The approximation scheme is next applied to a two-dimensional non-perturbed Kepler problem. The data produced is analyzed statistically and graphically. Conclusions are drawn and suggestions are made in order to continue the research in several of the areas presented.
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    An application of the Liouville resolvent method to the study of fermion-boson couplings
    Bressler, Barry Lee (Virginia Polytechnic Institute and State University, 1986)
    The Liouville resolvent method is an unconventional technique used for finding a Green function for a Hamiltonian. Implementation of the method entails the calculation of commutators of a second-quantized Hamiltonian operator with particular generalized stepping operators that are elements of a Hilbert space and that represent transitions between many-particle states. These commutators produce linear combinations of stepping operators, so the results can be arrayed as matrix elements of the Liouville operator L̂ in the Hilbert space of stepping operators. The resulting L̂ matrix is usually of infinite order, and in principle its eigenvalues and eigenvectors can be used to construct the Green function from the L̂ resolvent matrix. Approximations are usually necessary, at least in the form of truncation of the L̂ matrix, and if one produces a sequence of such matrices of increasing order and calculates the eigenvalues and eigenvectors of these matrices, a sequence of approximations for the L̂ resolvent matrix can be produced. This sequence is mathematically guaranteed to converge to the exact result for the L̂ resolvent matrix (except at its singularities). The accuracy of an approximation depends on the order of the matrix at which the sequence is truncated. Application of the method to a Hamiltonian representing interactions between fermions and bosons involves complications arising from the large number of terms generated by the commutation properties of boson operators. This dissertation describes the method and its use in the study of fermion-boson couplings. Approximations to second order in stepping operators are calculated for simplified Froehlich and Lee models. Limited thermodynamic results are obtained from the Lee model. Exact energy eigenvalues are obtained by operator algebra for simplified Froehlich, Lee and Dirac models. These exact solutions comprise the main contribution of this research and will prove to be valuable starting points for further research. Suggestions are made for further research.
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    Convergent methods for calculating thermodynamic Green functions
    Bowen, Samuel P.; Williams, Clayton D.; Mancini, Jay D. (American Physical Society, 1984-08)
    A convergent method of approximating thermodynamic Green functions is outlined briefly. The method constructs a sequence of approximants which converges independently of the strength of the Hamiltonian's coupling constants. Two new concepts associated with the approximants are introduced: the resolving power of the approximation, and conditional creation (annihilation) operators. These ideas are illustrated on an exactly soluble model and a numerical example. A convergent expression for the scattering rate in a field theory is also derived.
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    Cooperative Behavior in Driven Lattice Systems with Shifted Periodic Boundary Conditions
    Anderson, Mark Jule Jr. (Virginia Tech, 1998-04-17)
    We explore the nature of driven stochastic lattice systems with non-periodic boundary conditions. The systems consist of particle and holes which move by exchanges of nearest neighbor particle-hole pairs. These exchanges are controlled by the energetics associated with an internal Hamiltonian, an external drive and a stochastic coupling to a heat reservoir. The effect of the drive is to bias particle-hole exchanges along the field in such a way that a particle current can be established. Hard-core volume constraints limit the occupation of only one particle (hole) per lattice site. For certain regimes of the overall particle density and temperature, a system displays a homogeneous disordered phase. We investigate cooperative behavior in this phase by using two-point spatial correlation functions and structure factors. By varying the particle density and the temperature, the system orders into a phase separated state, consisting of particle-rich and particle-poor regions. The temperature and density for the co-existence state depend on the boundary conditions. By using Monte Carlo simulations, we establish co-existence curves for systems with shifted periodic boundary conditions.
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    The diagramatical solution of the two-impurity Kondo problem
    Zhou, Chen (Virginia Polytechnic Institute and State University, 1988)
    The problem of the two-impurity Kondo problem is studied via the perturbative diagrammatical method. The high-temperature magnetic susceptibility is calculated to fourth order in the coupling constant J for different regimes. The integral equations for the ground state energy are established and solved numerically. The two-stage Kondo effect and corresponding energy scales are found which agree with the scaling results.
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    Differential algebraic methods for obtaining approximate numerical solutions to the Hamilton-Jacobi equation
    Pusch, Gordon D. (Virginia Tech, 1990)
    I present two differential-algebraic (DA) methods for approximately solving the Hamilton- Jacobi (HJ) equation. I use the “automatic differentiation” property of DA to convert the nonlinear partial-differential HJ equation into a initial-value problem for a DA-valued first-order ordinary differential equation (ODE), the “HJ/DA equation”. The solution of either form of the HJ/DA equation is equivalent to a perturbative expansion of Hamilton’s principle function about some reference trajectory (RT) through the system. The HJ/DA method also extracts the equations of motion for the RT itself. Hamilton’s principle function generates the canonical transformation, or mapping, between the initial and final state of every trajectory through the system. Since the map is represented by a generating function, it must automatically be symplectic, even in the presence of round-off error. The DA-valued ODE produced by either form of HJ/DA is equivalent tc a hierarchically-ordered system of real-valued ODEs without “feedback” terms; therefore the hierarchy may be truncated at any (arbitrarily high) order without loss of self consistency. The HJ/DA equation may be numerically integrated using standard algorithms, if all mathematical operations are done in DA. I show that the norm of the DA-valued part of the solution is bounded by linear growth. The generating function may be used to track either particles or the moments of a particle distribution through the system. In the first method, all information about the perturbative dynamics is contained in the DA-valued generating function. I numerically integrate the HJ/DA equation, with the identity as the initial generating function. A difficulty with this approach is that not all canonical transformations can be represented by the class of generating functions connected to the identity; one finds that with the required initial conditions, the generating function becomes singular near caustics or foci. One may continue integrating through a caustic by using a Legendre transformation to obtain a new (but equivalent) generating function which is singular near the identity, but nonsingular near the caustic. However the Legendre transformation is a numerically costly procedure, so one would not want to do this often. This approach is therefore not practical for systems producing periodic motions, because one must perform a Legendre transformation four times per period. The second method avoids the caustic problem by representing only the nonlinear part of the dynamics by a generating function. The linearized dynamics is treated separately via matrix techniques. Since the nonlinear part of the dynamics may always be represented by a near-identity transformation, no problem occurs when passing through caustics. I successfully verify the HJ/DA method by applying it to three problems which can be solved in closed form. Finally, I demonstrate the method’s utility by using it to optimize the length of a lithium lens for minimum beam divergence via the moment-tracking technique.
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    Dynamic Nuclear Polarization in Samarium Doped Lanthanum Magnesium Nitrate
    Byvik, Charles E. (Virginia Tech, 1971-09-05)
    The dynamic nuclear polarization of hydrogen nuclei by the solid effect in single crystals of samarium doped lanthanum magnesium nitrate (Sm:LMN) has been studied theoretically and experimentally. The equations of evolution governing the dynamic nuclear polarization by the solid effect have been derived in detail using the spin temperature theory and the complete expression for the steady-state enhancement of the nuclear polarization has been calculated. For well-resolved solid effect transitions at microwave frequencies Ï ~ Ï e ± Ï n, the expression for the steady-state enhancement differs from the expression obtained by the rate equation approach by small terms which become zero at Ï ~ Ï e ± Ï n Experimental enhancements of the proton polarization were obtained for eight crystals at 9.2 GHz and liquid helium temperatures. The samarium concentration ranged from 0.1 percent to 1.1 percent as determined by X-ray fluorescence. A peak enhancement of 181 was measured for a 1.1 percent Sm:LMN crystal at 3.0" K. The maximum enhancements extrapolated with the theory using the experimental data for peak enhancement versus microwave power and correcting for leakage, agree with the ideal enhancement (24O in this experiment) within experimental error for three of the crystals. The calculated satellite separation was within 6 percent of the measured separation for each of the enhancement curves and the peak positive and negative enhancements were equal for all but two of the crystals. The nuclear spin"lattice relaxation time was measured for one of the crystals between l.6" K and 4.2" K. To account for nuclear spin"lattice relaxation, spin diffusion theory in the rapid airrusion limit was incorporated into the results of the spin temperature theory of the solid effect. The experimental results indicate that the spin temperature theory is a quantitatively correct approach for the description of dynamic nuclear polarization by the solid effect for well"resolved solid effect transitions.
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    (e,2e) spectroscopic investigations of the spectral momentum densities of thin carbon films
    Dennison, John Robert (Virginia Polytechnic Institute and State University, 1985)
    An (e,2e) electron scattering spectrometer has been constructed and used for the first time to investigate the spectral momentum density of the valence bands of a solid target. This technique provides fundamental information about the electronic structure of both crystalline and amorphous solids. The three fundamental quantities, the band structure, electron density of states, and electron momentum distribution can be simultaneously derived from the measured (e,2e) cross section. A review of single electron and (e,2e) scattering theory is given with an emphasis on scattering from solids. The effects of multiple scattering are discussed and a method of deconvoluting those effects from the measured (e,2e) cross section is developed. There is a detailed description of the spectrometer design and operation with particular attention given to the electron optics and voltage distribution. The algorithms and software for computer aided data acquisition and analysis are also outlined, as is error analysis. The techniques employed in the preparation and characterization of extremely thin film samples of a-C and single crystal graphite are described. An analysis of the data taken for a-C samples is given. The data are compared with the results of complementary experiments and theory for graphite, diamond, and a-C which are given in a review of the literature. The existence of a definite dispersion relation ε(q) in amorphous carbon is demonstrated. The a-C band structure appears to be more similar to that of graphite than to that of diamond, however it differs significantly from both in some respects. The measured spectral momentum density seems compatible with a model of a-C based on small, randomly-oriented islands of quasi-2D graphite-like continuous random network structures. However, no definitive interpretations can be made until higher resolution experiments are performed on both a-C and single crystal graphite.
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    Explicitly structured physics instruction
    Wright, David Shaw (Virginia Polytechnic Institute and State University, 1984)
    In an introductory physics course, problem solving skills are not traditionally taught. The instructor explains the physical theory and works example problems. Many students, however, are not able to develop the ability to solve problems implicitly. The program of Explicitly Structured Physics Instruction (ESPI) was developed to teach problem solving skills explicitly. It is designed to help students organize their work, increase their accuracy, eliminate initial panic or lack of direction in approaching a problem, increase confidence in problem solving, promote understanding instead of rote memory, and improve the students' ability to communicate with the instructor and other students. It provides not only an explicit strategy for problem solving, but also a structure for examining formulas called the formula fact sheet, and an opportunity for practice and feedback in a problem solving session which involves the use of out loud thinking. The program of ESPI was developed over five academic quarters of testing. A statistical analysis was performed on the data obtained, but the qualitative data obtained from student interviews and questionnaires, as well as the instructor reaction to the program, provided the main source of input in the development of the program and the measurement of its success. Reaction to the program in its final revised form was very positive. Over 90% said that they would use the strategy even if it were not required, and that the formula fact sheet had been very helpful. Over 75% said that the problem solving session was very helpful. Final grades of those who used the strategy were significantly higher than those who did not. Retention of students in the course was raised from 70% to 86%. The study indicates that a well integrated program built around the use of a problem solving strategy can help students focus on understanding physics and the problem sovling process.
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    Fermions in Yang-Mills gauge theories: invariance, covariance and topology
    Liang, Yigao (Virginia Polytechnic Institute and State University, 1987)
    I present a study on the invariance and covariance properties of the Dirac operator describing fermions in Yang-Mills fields. This includes the study of anomalies of the gauge currents. We are particularly interested in the geometric and topological features in the problem. The complicated topological structures and properties present in these theories are made clear by elementary calculations in several simple models. We show explicitly how non-trivial phase and sign ambiguities arise to give the so-called anomalies. The Atiyah-Singer index theorem is seen to be a very powerful tool to calculate the topological invariants that characterize the anomalies. The index theorem also gives topological invariants describing the failure of covariance of the fermion propagator.
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    Halphen's theorem and related results
    Culbertson, George Edward (Virginia Tech, 1970-03-05)
    Halphen's Theorem states that, "A necessary and sufficient condition for every dynamical trajectory in a positional field of force in E3 to be planar is that the field of force is either parallel or central." This result has been known for some time, however only the sufficiency part of the theorem is widely documented. A new analytic proof of the necessity part of Halphen's Theorem was developed. The details of this proof motivated the new concepts of a flat point in a field of force and a flat point on a dynamical trajectory in a positional field of force.
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    A High Order Correction of the Energy of a One Dimensional Model of an H2+ Molecule
    Humfeld, Keith Daniel (Virginia Tech, 1998-07-31)
    The ground state electron wavefunction of some molecules has a non-zero angular momentum about the internuclear axis. Molecular rotational momentum can couple with this angular momentum, splitting the energy degeneracy of the two directions of motion about the internuclear axis. Performing a Born-Oppenheimer approximation of such a system will break the relevant energy degeneracy at eighth order. This degeneracy breaking is known as L-doubling.
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    Impulse electrical breakdown of high-purity water
    Gehman, Victor H. (Virginia Tech, 1995-05-05)
    Experiments have been conducted on the electrical breakdown of high-purity water and water mixtures. The electrical regime of interest has been carefully defined and documented to consist of electrical impulses with approximately microsecond rise time and fall time greater than 65 microseconds, on approximately 81-square-centimeter-area planar electrodes with a dielectric gap of approximately one centimeter. The results of over 25,000 shots by a Marx generator have been distilled into database form in an Excel spreadsheet and analysis performed to try to find patterns or indirect evidence into the nature of the breakdown-initiation process. An extensive review of all the experiments, which had been conducted over eight years by the Naval Surface Warfare Center and which had been designed to find the largest water-breakdown fields, was conducted with the intention of delineating the physical factors that led to breakdown. A variety of theoretical models of breakdown initiation were compared to the data, until it became clear that many of the breakdowns were dominated by impurities of various sorts. An extensive study of old and new experiments led to a more detailed understanding of the phenomenology of impurity-dominated water breakdown (such as the process of "conditioning" the electrodes and hysteresis) and the proposal of a number of new experiments to further characterize the intrinsic role of electrode materials on determining high-electric-field dielectric breakdown in water.
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    Iterative image processing using a cavity with a phase-conjugate mirror: possibilities and limitations
    Lo, Kanwai Peter (Virginia Tech, 1991-09-05)
    An optical image feedback system utilizing a cavity with a phase-conjugate mirror (PCM) has been studied. A new theory, based on operators, is developed to describe the steady-state output of the cavity. The use of operators allows one to describe the various optical operations and transformations needed in the optical implementation of iterative algorithms. The characteristics of the cavity are discussed using an expansion of the cavity fields in the cavity eigenfunctions. Several image processing applications using a PCM cavity are proposed and are studied using computer simulations. These theoretical studies indicate that a PC11 cavity can be useful in many applications. Optical phase conjugation was realized using a single crystal of photorefractive BaTi0₃ in a degenerated four-wave mixing geometry. The reflectivity gain from the PCM was optimized experimentally by the geometrical parameters and by the beamintensity ratios. The ability of the PCM to remove phase distortion as predicted theoretically, was demonstrated experimentally. The output of a PCM cavity can be substantially influenced by self-oscillations of the cavity above threshold. This was experimentally studied by observing the time evolution of the input. To avoid the influence of self-oscillation, the cavity must be operated below threshold. It is found that the cavity decay time constant diverges at and about threshold. This can be used as an indicator to show whether the cavity has crossed the threshold or to measure how close to threshold the cavity operates. To verify that a PCM cavity can be used in iterative image processing, an experiment was set up to implement an image restoration scheme based on the Gerchberg algorithm. It is shown that an optical implementation of the Gerchberg algorithm is feasible for objects made of few pixels. The experiment confirmed .that image iteration in a PCm cavity is possible. The limitations of the cavity and the technical difficulties are discussed.
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    Liouville resolvent methods applied to highly correlated systems
    Holtz, Susan Lady (Virginia Polytechnic Institute and State University, 1986)
    In this dissertation we report on the application of the Liouville Operator Resolvent technique (LRM) to two hamiltonians used to model highly correlated systems: Falicov-Kimball and Anderson Lattice. We calculate specific heats, magnetic susceptibilities, thermal averages of physical operators, and energy bands. We demonstrate that the LRM is a viable method for investigating many body problems. For the Falicov-Kimball, an exact calculation of the atomic limit shows no sharp metal-insulator transition. A truncation approximation for the full hamiltonian has a smooth evolution from the atomic limit with the opening of a band for the conduction electrons. No phase transition was observed. A bose space calculation using the proper boson norm indicates that the conduction band induces a correlation between localized electrons on nearest-neighbor sites. It is not known if this effect is real or a by-product of the approximation. We applied the LRM to the Anderson Lattice and several of its limiting cases. In the limit of no hybridization, for both the symmetric and asymmetric (mixed-valence) parameter sets, we found that the thermodynamics could be described as competition between closely-lying energy levels. The effects that dominate are those that minimize the thermal average of the hamiltonian. A simple model is presented in which only hybridization between two localized orbitals is allowed. It shows that hybridization can give rise to mixed valence phenomena as the temperature approaches zero. For the full Anderson Lattice hybridization causes relatively small shifts in the occupation numbers of the localized and conduction electrons. However, these shifts can have dramatic effects on the physical properties as demonstrated by the magnetic susceptibilities. Band structures of the eigenenergies of the Liouville operator, for both parameter sets, reveal that low-lying excitations associated with some of the basis vector operators may split out from the fermi level and become significant at low temperatures. In addition, we report on progress toward extending the calculation to bose space using a commutator norm.
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    Monte Carlo computer simulation of the Lennard-Jones and Stockmayer fluid phase diagrams
    Gregory, Victor Paul (Virginia Tech, 1994-02-05)
    The isotherms of the Lennard-Jones fluid and the Stockmayer fluid are calculated by Monte Carlo computer simulation using the constant NpT ensemble. Empirical coefficients are determined for a truncated virial equation of state fitted to our data. Spinodal points are located for each temperature and fluid. For temperatures less than 0.90 of the critical temperature, we succeeded in temporarily isolating clusters during the gas to liquid transition for the LJ fluid. Density profiles are calculated for clusters at and above the spinodal pressures. The clusters above the spinodal pressure have liquid-like densities at their centers and are identified as critical condensation clusters. The clusters at the spinodal increase in size with temperature and have densities roughly half as dense as the equilibrium liquid at their centers. It is found that the results are essentially system size independent.
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    Monte Carlo simulation of aqueous dilute solutions of polyhydric alcohols
    Lilly, Arnys Clifton (Virginia Polytechnic Institute and State University, 1989)
    In order to investigate the details of hydrogen bonding and solution molecular conformation of complex alcohols in water, isobaric-isothermal Monte Carlo simulations were carried out on several systems. The solutes investigated were ethanol, ethylene glycol, 1,2-propylene glycol, 1,3-propylene glycol and glycerol. In addition, propane, which does not hydrogen bond but does form water hydrates, was simulated in aqueous solution. The complex alcohol-water systems are very nonideal in their behavior as a function of solute concentration down to very dilute solutions. The water model employed was TIP4P water¹ and the intermolecular potentials employed are of the Jorgensen type² in which the interactions between the molecules are represented by interaction sites usually located on nuclei. The interactions are represented by a sum of Coulomb and Lennard-Jones terms between all intermolecular pairs of sites. Intramolecular rotations in the solute are modeled by torsional potential energy functions taken from ethanol, 1-propanol and 2-propanol for C-O and C-C bond rotations. Quasi-component pair correlation functions were used to analyze the hydrogen bonding. Hydrogen bonds were classified as proton acceptor and proton donor bonds by analyzing the nearest neighbor pair correlation function between hydroxyl oxygen and hydrogen and between solvent-water hydrogen and oxygen. The results obtained for partial molar heats of solution are more negative than the experimental values by 3.0 to 14 kcal/mol. In solution, all solutes reached a contracted molecular geometry with the OH groups generally on one side of the molecule. There is a tendency for the solute OH groups to hydrogen bond with water, with more proton acceptor bonds than proton donor bonds. The water-solute binding energies correlate with experimental measurements of the water-binding properties of the solute. 1. Jorgensen, W.L. et al, J. Chem. Phys., 79, 926 (1983). 2. Jorgensen, W.L., J. Phys Chem., 87, 5304 (1983).
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    Non-equilibrium Phase Transitions and Steady States in Biased Diffusion of Two Species
    Korniss, György (Virginia Tech, 1997-04-21)
    We investigate the dynamics of a three-state stochastic lattice gas, consisting of holes and two oppositely "charged" species of particles, under the influence of an "electric" field, at zero total charge. Interacting only through an excluded volume constraint, particles can hop to nearest neighbor empty sites, but particle-particle exchange between oppositely charged particles is also allowed on a separate time scale. Controlled by this relative time scale, particle density and drive, the system orders into a charge-segregated state. Using a combination of Monte Carlo simulations and continuum field theory techniques, we study the order of these transitions and map out the steady state phase diagram of the system. On a single sheet of transitions, a line of multicritical points is found, separating the first order and continuous transitions. Furthermore, we study the steady-state structure factors in the disordered phase where homogeneous configurations are stable against small harmonic perturbations. The average structure factors show a discontinuity singularity at the origin which in real space predicts an intricate crossover between power laws of different kinds. We also seek for generic statistical properties of these quantities. The probability distributions of the structure factors are universal asymmetric exponential distributions. This research was supported in part by grants from the National Science Foundation through the Division of Materials Research.
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    Nonlinear optical effects in nematic liquid crystals
    Puang-ngern, Srisuda (Virginia Polytechnic Institute and State University, 1985)
    Theoretical studies of nonlinear optical effects in nematic liquid crystals including degenerate four-wave mixing are presented. The optically induced Freedericksz transition which is essential for these effects is also described. Experimental investigations are performed using a homeotropically aligned MBBA thin film. Good agreement is obtained between the theoretical predictions and the experiments. Some potential applications of phase conjugation obtained by the backward degenerate four wave-mixing process in the field of adaptive optics and image processing are demonstrated.
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    Optical characterization of processed gallium arsenide
    Siochi, Ramon Alfredo Carvalho (Virginia Tech, 1990)
    Raman scattering and ultraviolet-visible reflectivity have been used to characterize the structural and electronic changes that occur in GaAs during ion implantation and subsequent annealing. In this work, the damaged structure is modelled as an amorphous GaAs matrix embedded with GaAs microcrystals. The longitudinal-optic (LO) Raman-line characteristics were monitored to determine the amorphous volume fraction, the average microcrystal diameter and, for the annealed samples, the carrier concentration. An oscillator analysis of the reflectivity spectra, along with the effective medium approximation, was carried out to determine the linewidths of the interband peaks and the amorphous volume fraction in the damage layer. To determine damage depth profiles, spectra were taken as a function of the amount of material removed via chemical etch. A new method of interpreting reflectivity spectra was developed to deal with the etchant-induced roughness. This roughness reduced the reflectivity by a constant factor in the region between 4.5 and 5 eV. The ratio between reflectivities at 4.55 and 4.75 eV was monitored to determine qualitatively the amount of damage. The annealing studies show that structural recovery occurs at a lower temperature than that for which electrical activation occurs. The depth profile of a sample annealed at 400°C reveals that nucleation takes place not only at the boundary between the damaged and undamaged layers (i.e., "epitaxial regrowth") but also at the microcrystal/amorphous interfaces within the damage layer. The oscillator analysis of the dielectric properties was further developed, and a connection was established between the Strengths, positions, and linewidths of the interband oscillators and the shift in position of the LO Raman line. The results indicate that the static dielectric constant is independent of microcrystal size. A comparison between (211) and (100) oriented Si-implanted GaAs was done as well, showing greater near surface damage and a shallower total damage layer for the (211) samples. Finally, a method for characterizing damage, based on the observed shifts of the two-phonon ("2LO") Raman peak as the incident photon energy is varied around the E₁ interband energy (2.9 eV), has been developed. The results suggest that the total mass of the electron-hole pair involved in the scattering process increases even for large (>400 Å) microcrystals. The 525°C annealed sample had little damage, and was studied with this technique.